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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 ... |
| ax1w 13 | Weakening of ~ ax-1 . As ... |
| mp1i 14 | Inference detaching an ant... |
| a2i 15 | Inference distributing an ... |
| mpd 16 | A modus ponens deduction. ... |
| imim2i 17 | Inference adding common an... |
| syl 18 | An inference version of th... |
| 3syl 19 | Inference chaining two syl... |
| 4syl 20 | Inference chaining three s... |
| mpi 21 | A nested modus ponens infe... |
| mpisyl 22 | A syllogism combined with ... |
| id 23 | Principle of identity. Th... |
| idALT 24 | Alternate proof of ~ id . ... |
| idd 25 | Principle of identity ~ id... |
| a1d 26 | Deduction introducing an e... |
| 2a1d 27 | Deduction introducing two ... |
| a1i13 28 | Add two antecedents to a w... |
| 2a1 29 | A double form of ~ ax-1 . ... |
| a2d 30 | Deduction distributing an ... |
| sylcom 31 | Syllogism inference with c... |
| syl5com 32 | Syllogism inference with c... |
| com12 33 | Inference that swaps (comm... |
| syl11 34 | A syllogism inference. Co... |
| syl5 35 | A syllogism rule of infere... |
| syl6 36 | A syllogism rule of infere... |
| syl56 37 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 38 | Syllogism inference with c... |
| mpcom 39 | Modus ponens inference wit... |
| syli 40 | Syllogism inference with c... |
| syl2im 41 | Replace two antecedents. ... |
| syl2imc 42 | A commuted version of ~ sy... |
| pm2.27 43 | This theorem, sometimes ca... |
| mpdd 44 | A nested modus ponens dedu... |
| mpid 45 | A nested modus ponens dedu... |
| mpdi 46 | A nested modus ponens dedu... |
| mpii 47 | A doubly nested modus pone... |
| syld 48 | Syllogism deduction. Dedu... |
| syldc 49 | Syllogism deduction. Comm... |
| mp2d 50 | A double modus ponens dedu... |
| a1dd 51 | Double deduction introduci... |
| 2a1dd 52 | Double deduction introduci... |
| pm2.43i 53 | Inference absorbing redund... |
| pm2.43d 54 | Deduction absorbing redund... |
| pm2.43a 55 | Inference absorbing redund... |
| pm2.43b 56 | Inference absorbing redund... |
| pm2.43 57 | Absorption of redundant an... |
| imim2d 58 | Deduction adding nested an... |
| imim2 59 | A closed form of syllogism... |
| embantd 60 | Deduction embedding an ant... |
| 3syld 61 | Triple syllogism deduction... |
| sylsyld 62 | A double syllogism inferen... |
| imim12i 63 | Inference joining two impl... |
| imim1i 64 | Inference adding common co... |
| imim3i 65 | Inference adding three nes... |
| sylc 66 | A syllogism inference comb... |
| syl3c 67 | A syllogism inference comb... |
| syl6mpi 68 | A syllogism inference. (C... |
| mpsyl 69 | Modus ponens combined with... |
| mpsylsyld 70 | Modus ponens combined with... |
| syl6c 71 | Inference combining ~ syl6... |
| syl6ci 72 | A syllogism inference comb... |
| syldd 73 | Nested syllogism deduction... |
| syl5d 74 | A nested syllogism deducti... |
| syl7 75 | A syllogism rule of infere... |
| syl6d 76 | A nested syllogism deducti... |
| syl8 77 | A syllogism rule of infere... |
| syl9 78 | A nested syllogism inferen... |
| syl9r 79 | A nested syllogism inferen... |
| syl10 80 | A nested syllogism inferen... |
| a1ddd 81 | Triple deduction introduci... |
| imim12d 82 | Deduction combining antece... |
| imim1d 83 | Deduction adding nested co... |
| imim1 84 | A closed form of syllogism... |
| pm2.83 85 | Theorem *2.83 of [Whitehea... |
| peirceroll 86 | Over minimal implicational... |
| com23 87 | Commutation of antecedents... |
| com3r 88 | Commutation of antecedents... |
| com13 89 | Commutation of antecedents... |
| com3l 90 | Commutation of antecedents... |
| pm2.04 91 | Swap antecedents. Theorem... |
| com34 92 | Commutation of antecedents... |
| com4l 93 | Commutation of antecedents... |
| com4t 94 | Commutation of antecedents... |
| com4r 95 | Commutation of antecedents... |
| com24 96 | Commutation of antecedents... |
| com14 97 | Commutation of antecedents... |
| com45 98 | Commutation of antecedents... |
| com35 99 | Commutation of antecedents... |
| com25 100 | Commutation of antecedents... |
| com5l 101 | Commutation of antecedents... |
| com15 102 | Commutation of antecedents... |
| com52l 103 | Commutation of antecedents... |
| com52r 104 | Commutation of antecedents... |
| com5r 105 | Commutation of antecedents... |
| imim12 106 | Closed form of ~ imim12i a... |
| jarr 107 | Elimination of a nested an... |
| jarri 108 | Inference associated with ... |
| pm2.86d 109 | Deduction associated with ... |
| pm2.86 110 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 111 | Inference associated with ... |
| loolin 112 | The Linearity Axiom of the... |
| loowoz 113 | An alternate for the Linea... |
| con4 114 | Alias for ~ ax-3 to be use... |
| con4i 115 | Inference associated with ... |
| con4d 116 | Deduction associated with ... |
| mt4 117 | The rule of modus tollens.... |
| mt4d 118 | Modus tollens deduction. ... |
| mt4i 119 | Modus tollens inference. ... |
| pm2.21i 120 | A contradiction implies an... |
| pm2.24ii 121 | A contradiction implies an... |
| pm2.21d 122 | A contradiction implies an... |
| pm2.21ddALT 123 | Alternate proof of ~ pm2.2... |
| pm2.21 124 | From a wff and its negatio... |
| pm2.24 125 | Theorem *2.24 of [Whitehea... |
| jarl 126 | Elimination of a nested an... |
| jarli 127 | Inference associated with ... |
| pm2.18d 128 | Deduction form of the Clav... |
| pm2.18 129 | Clavius law, or "consequen... |
| pm2.18i 130 | Inference associated with ... |
| notnotr 131 | Double negation eliminatio... |
| notnotri 132 | Inference associated with ... |
| notnotriALT 133 | Alternate proof of ~ notno... |
| notnotrd 134 | Deduction associated with ... |
| con2d 135 | A contraposition deduction... |
| con2 136 | Contraposition. Theorem *... |
| mt2d 137 | Modus tollens deduction. ... |
| mt2i 138 | Modus tollens inference. ... |
| nsyl3 139 | A negated syllogism infere... |
| con2i 140 | A contraposition inference... |
| nsyl 141 | A negated syllogism infere... |
| nsyl2 142 | A negated syllogism infere... |
| notnot 143 | Double negation introducti... |
| notnoti 144 | Inference associated with ... |
| notnotd 145 | Deduction associated with ... |
| con1d 146 | A contraposition deduction... |
| con1 147 | Contraposition. Theorem *... |
| con1i 148 | A contraposition inference... |
| mt3d 149 | Modus tollens deduction. ... |
| mt3i 150 | Modus tollens inference. ... |
| pm2.24i 151 | Inference associated with ... |
| pm2.24d 152 | Deduction form of ~ pm2.24... |
| con3d 153 | A contraposition deduction... |
| con3 154 | Contraposition. Theorem *... |
| con3i 155 | A contraposition inference... |
| con3rr3 156 | Rotate through consequent ... |
| nsyld 157 | A negated syllogism deduct... |
| nsyli 158 | A negated syllogism infere... |
| nsyl4 159 | A negated syllogism infere... |
| nsyl5 160 | A negated syllogism infere... |
| pm3.2im 161 | Theorem *3.2 of [Whitehead... |
| jc 162 | Deduction joining the cons... |
| jcn 163 | Theorem joining the conseq... |
| jcnd 164 | Deduction joining the cons... |
| impi 165 | An importation inference. ... |
| expi 166 | An exportation inference. ... |
| simprim 167 | Simplification. Similar t... |
| simplim 168 | Simplification. Similar t... |
| pm2.5g 169 | General instance of Theore... |
| pm2.5 170 | Theorem *2.5 of [Whitehead... |
| conax1 171 | Contrapositive of ~ ax-1 .... |
| conax1k 172 | Weakening of ~ conax1 . G... |
| pm2.51 173 | Theorem *2.51 of [Whitehea... |
| pm2.52 174 | Theorem *2.52 of [Whitehea... |
| pm2.521g 175 | A general instance of Theo... |
| pm2.521g2 176 | A general instance of Theo... |
| pm2.521 177 | Theorem *2.521 of [Whitehe... |
| expt 178 | Exportation theorem ~ pm3.... |
| exptOLD 179 | Obsolete version of ~ expt... |
| impt 180 | Importation theorem ~ pm3.... |
| pm2.61d 181 | Deduction eliminating an a... |
| pm2.61d1 182 | Inference eliminating an a... |
| pm2.61d2 183 | Inference eliminating an a... |
| pm2.61i 184 | Inference eliminating an a... |
| pm2.61ii 185 | Inference eliminating two ... |
| pm2.61nii 186 | Inference eliminating two ... |
| pm2.61iii 187 | Inference eliminating thre... |
| ja 188 | Inference joining the ante... |
| jad 189 | Deduction form of ~ ja . ... |
| pm2.01 190 | Weak Clavius law. If a fo... |
| pm2.01i 191 | Inference associated with ... |
| pm2.01d 192 | Deduction based on reducti... |
| pm2.6 193 | Theorem *2.6 of [Whitehead... |
| pm2.61 194 | Theorem *2.61 of [Whitehea... |
| pm2.65 195 | Theorem *2.65 of [Whitehea... |
| pm2.65i 196 | Inference for proof by con... |
| pm2.65iOLD 197 | Obsolete version of ~ pm2.... |
| pm2.21dd 198 | A contradiction implies an... |
| pm2.65d 199 | Deduction for proof by con... |
| mto 200 | The rule of modus tollens.... |
| mtod 201 | Modus tollens deduction. ... |
| mtoi 202 | Modus tollens inference. ... |
| mt2 203 | A rule similar to modus to... |
| mt3 204 | A rule similar to modus to... |
| peirce 205 | Peirce's axiom. A non-int... |
| looinv 206 | The Inversion Axiom of the... |
| bijust0 207 | A self-implication (see ~ ... |
| bijust 208 | Theorem used to justify th... |
| impbi 211 | Property of the biconditio... |
| impbii 212 | Infer an equivalence from ... |
| impbidd 213 | Deduce an equivalence from... |
| impbid21d 214 | Deduce an equivalence from... |
| impbid 215 | Deduce an equivalence from... |
| dfbi1 216 | Relate the biconditional c... |
| dfbi1ALT 217 | Alternate proof of ~ dfbi1... |
| biimp 218 | Property of the biconditio... |
| biimpi 219 | Infer an implication from ... |
| sylbi 220 | A mixed syllogism inferenc... |
| sylib 221 | A mixed syllogism inferenc... |
| sylbb 222 | A mixed syllogism inferenc... |
| biimpr 223 | Property of the biconditio... |
| bicom1 224 | Commutative law for the bi... |
| bicom 225 | Commutative law for the bi... |
| bicomd 226 | Commute two sides of a bic... |
| bicomi 227 | Inference from commutative... |
| impbid1 228 | Infer an equivalence from ... |
| impbid2 229 | Infer an equivalence from ... |
| impcon4bid 230 | A variation on ~ impbid wi... |
| biimpri 231 | Infer a converse implicati... |
| biimpd 232 | Deduce an implication from... |
| mpbi 233 | An inference from a bicond... |
| mpbir 234 | An inference from a bicond... |
| mpbid 235 | A deduction from a bicondi... |
| mpbii 236 | An inference from a nested... |
| sylibr 237 | A mixed syllogism inferenc... |
| sylbir 238 | A mixed syllogism inferenc... |
| sylbbr 239 | A mixed syllogism inferenc... |
| sylbb1 240 | A mixed syllogism inferenc... |
| sylbb2 241 | A mixed syllogism inferenc... |
| sylibd 242 | A syllogism deduction. (C... |
| sylbid 243 | A syllogism deduction. (C... |
| mpbidi 244 | A deduction from a bicondi... |
| biimtrid 245 | A mixed syllogism inferenc... |
| biimtrrid 246 | A mixed syllogism inferenc... |
| imbitrid 247 | A mixed syllogism inferenc... |
| syl5ibcom 248 | A mixed syllogism inferenc... |
| imbitrrid 249 | A mixed syllogism inferenc... |
| syl5ibrcom 250 | A mixed syllogism inferenc... |
| biimprd 251 | Deduce a converse implicat... |
| biimpcd 252 | Deduce a commuted implicat... |
| biimprcd 253 | Deduce a converse commuted... |
| imbitrdi 254 | A mixed syllogism inferenc... |
| imbitrrdi 255 | A mixed syllogism inferenc... |
| biimtrdi 256 | A mixed syllogism inferenc... |
| biimtrrdi 257 | A mixed syllogism inferenc... |
| syl7bi 258 | A mixed syllogism inferenc... |
| syl8ib 259 | A syllogism rule of infere... |
| mpbird 260 | A deduction from a bicondi... |
| mpbiri 261 | An inference from a nested... |
| sylibrd 262 | A syllogism deduction. (C... |
| sylbird 263 | A syllogism deduction. (C... |
| biid 264 | Principle of identity for ... |
| biidd 265 | Principle of identity with... |
| pm5.1im 266 | Two propositions are equiv... |
| 2th 267 | Two truths are equivalent.... |
| 2thd 268 | Two truths are equivalent.... |
| monothetic 269 | Two self-implications (see... |
| ibi 270 | Inference that converts a ... |
| ibir 271 | Inference that converts a ... |
| ibd 272 | Deduction that converts a ... |
| pm5.74 273 | Distribution of implicatio... |
| pm5.74i 274 | Distribution of implicatio... |
| pm5.74ri 275 | Distribution of implicatio... |
| pm5.74d 276 | Distribution of implicatio... |
| pm5.74rd 277 | Distribution of implicatio... |
| bitri 278 | An inference from transiti... |
| bitr2i 279 | An inference from transiti... |
| bitr3i 280 | An inference from transiti... |
| bitr4i 281 | An inference from transiti... |
| bitrd 282 | Deduction form of ~ bitri ... |
| bitr2d 283 | Deduction form of ~ bitr2i... |
| bitr3d 284 | Deduction form of ~ bitr3i... |
| bitr4d 285 | Deduction form of ~ bitr4i... |
| bitrid 286 | A syllogism inference from... |
| bitr2id 287 | A syllogism inference from... |
| bitr3id 288 | A syllogism inference from... |
| bitr3di 289 | A syllogism inference from... |
| bitrdi 290 | A syllogism inference from... |
| bitr2di 291 | A syllogism inference from... |
| bitr4di 292 | A syllogism inference from... |
| bitr4id 293 | A syllogism inference from... |
| 3imtr3i 294 | A mixed syllogism inferenc... |
| 3imtr4i 295 | A mixed syllogism inferenc... |
| 3imtr3d 296 | More general version of ~ ... |
| 3imtr4d 297 | More general version of ~ ... |
| 3imtr3g 298 | More general version of ~ ... |
| 3imtr4g 299 | More general version of ~ ... |
| 3bitri 300 | A chained inference from t... |
| 3bitrri 301 | A chained inference from t... |
| 3bitr2i 302 | A chained inference from t... |
| 3bitr2ri 303 | A chained inference from t... |
| 3bitr3i 304 | A chained inference from t... |
| 3bitr3ri 305 | A chained inference from t... |
| 3bitr4i 306 | A chained inference from t... |
| 3bitr4ri 307 | A chained inference from t... |
| 3bitrd 308 | Deduction from transitivit... |
| 3bitrrd 309 | Deduction from transitivit... |
| 3bitr2d 310 | Deduction from transitivit... |
| 3bitr2rd 311 | Deduction from transitivit... |
| 3bitr3d 312 | Deduction from transitivit... |
| 3bitr3rd 313 | Deduction from transitivit... |
| 3bitr4d 314 | Deduction from transitivit... |
| 3bitr4rd 315 | Deduction from transitivit... |
| 3bitr3g 316 | More general version of ~ ... |
| 3bitr4g 317 | More general version of ~ ... |
| notnotb 318 | Double negation. Theorem ... |
| con34b 319 | A biconditional form of co... |
| con4bid 320 | A contraposition deduction... |
| notbid 321 | Deduction negating both si... |
| notbi 322 | Contraposition. Theorem *... |
| notbii 323 | Negate both sides of a log... |
| con4bii 324 | A contraposition inference... |
| mtbi 325 | An inference from a bicond... |
| mtbir 326 | An inference from a bicond... |
| mtbid 327 | A deduction from a bicondi... |
| mtbird 328 | A deduction from a bicondi... |
| mtbii 329 | An inference from a bicond... |
| mtbiri 330 | An inference from a bicond... |
| sylnib 331 | A mixed syllogism inferenc... |
| sylnibr 332 | A mixed syllogism inferenc... |
| sylnbi 333 | A mixed syllogism inferenc... |
| sylnbir 334 | A mixed syllogism inferenc... |
| xchnxbi 335 | Replacement of a subexpres... |
| xchnxbir 336 | Replacement of a subexpres... |
| xchbinx 337 | Replacement of a subexpres... |
| xchbinxr 338 | Replacement of a subexpres... |
| imbi2i 339 | Introduce an antecedent to... |
| bibi2i 340 | Inference adding a bicondi... |
| bibi1i 341 | Inference adding a bicondi... |
| bibi12i 342 | The equivalence of two equ... |
| imbi2d 343 | Deduction adding an antece... |
| imbi1d 344 | Deduction adding a consequ... |
| bibi2d 345 | Deduction adding a bicondi... |
| bibi1d 346 | Deduction adding a bicondi... |
| imbi12d 347 | Deduction joining two equi... |
| bibi12d 348 | Deduction joining two equi... |
| imbi12 349 | Closed form of ~ imbi12i .... |
| imbi1 350 | Theorem *4.84 of [Whitehea... |
| imbi2 351 | Theorem *4.85 of [Whitehea... |
| imbi1i 352 | Introduce a consequent to ... |
| imbi12i 353 | Join two logical equivalen... |
| bibi1 354 | Theorem *4.86 of [Whitehea... |
| bitr3 355 | Closed nested implication ... |
| con2bi 356 | Contraposition. Theorem *... |
| con2bid 357 | A contraposition deduction... |
| con1bid 358 | A contraposition deduction... |
| con1bii 359 | A contraposition inference... |
| con2bii 360 | A contraposition inference... |
| con1b 361 | Contraposition. Bidirecti... |
| con2b 362 | Contraposition. Bidirecti... |
| biimt 363 | A wff is equivalent to its... |
| pm5.5 364 | Theorem *5.5 of [Whitehead... |
| a1bi 365 | Inference introducing a th... |
| mt2bi 366 | A false consequent falsifi... |
| mtt 367 | Modus-tollens-like theorem... |
| imnot 368 | If a proposition is false,... |
| pm5.501 369 | Theorem *5.501 of [Whitehe... |
| ibib 370 | Implication in terms of im... |
| ibibr 371 | Implication in terms of im... |
| tbt 372 | A wff is equivalent to its... |
| nbn2 373 | The negation of a wff is e... |
| bibif 374 | Transfer negation via an e... |
| nbn 375 | The negation of a wff is e... |
| nbn3 376 | Transfer falsehood via equ... |
| pm5.21im 377 | Two propositions are equiv... |
| 2false 378 | Two falsehoods are equival... |
| 2falsed 379 | Two falsehoods are equival... |
| pm5.21ni 380 | Two propositions implying ... |
| pm5.21nii 381 | Eliminate an antecedent im... |
| pm5.21ndd 382 | Eliminate an antecedent im... |
| bija 383 | Combine antecedents into a... |
| pm5.18 384 | Theorem *5.18 of [Whitehea... |
| xor3 385 | Two ways to express "exclu... |
| nbbn 386 | Move negation outside of b... |
| nbbnOLD 387 | Obsolete version of ~ nbbn... |
| biass 388 | Associative law for the bi... |
| birot 389 | Rotation of the arguments ... |
| biluk 390 | Lukasiewicz's shortest axi... |
| pm5.19 391 | Theorem *5.19 of [Whitehea... |
| bi2.04 392 | Logical equivalence of com... |
| pm5.4 393 | Antecedent absorption impl... |
| imdi 394 | Distributive law for impli... |
| pm5.41 395 | Theorem *5.41 of [Whitehea... |
| imbibi 396 | The antecedent of one side... |
| imbibiOLD 397 | Obsolete version of ~ imbi... |
| pm4.8 398 | Theorem *4.8 of [Whitehead... |
| pm4.81 399 | A formula is equivalent to... |
| imim21b 400 | Simplify an implication be... |
| pm4.63 403 | Theorem *4.63 of [Whitehea... |
| pm4.67 404 | Theorem *4.67 of [Whitehea... |
| imnan 405 | Express an implication in ... |
| imnani 406 | Infer an implication from ... |
| iman 407 | Implication in terms of co... |
| pm3.24 408 | Law of noncontradiction. ... |
| annim 409 | Express a conjunction in t... |
| pm4.61 410 | Theorem *4.61 of [Whitehea... |
| pm4.65 411 | Theorem *4.65 of [Whitehea... |
| imp 412 | Importation inference. (C... |
| impcom 413 | Importation inference with... |
| con3dimp 414 | Variant of ~ con3d with im... |
| mpnanrd 415 | Eliminate the right side o... |
| impd 416 | Importation deduction. (C... |
| impcomd 417 | Importation deduction with... |
| ex 418 | Exportation inference. (T... |
| expcom 419 | Exportation inference with... |
| expdcom 420 | Commuted form of ~ expd . ... |
| expd 421 | Exportation deduction. (C... |
| expcomd 422 | Deduction form of ~ expcom... |
| imp31 423 | An importation inference. ... |
| imp32 424 | An importation inference. ... |
| exp31 425 | An exportation inference. ... |
| exp32 426 | An exportation inference. ... |
| imp4b 427 | An importation inference. ... |
| imp4a 428 | An importation inference. ... |
| imp4c 429 | An importation inference. ... |
| imp4d 430 | An importation inference. ... |
| imp41 431 | An importation inference. ... |
| imp42 432 | An importation inference. ... |
| imp43 433 | An importation inference. ... |
| imp44 434 | An importation inference. ... |
| imp45 435 | An importation inference. ... |
| exp4b 436 | An exportation inference. ... |
| exp4a 437 | An exportation inference. ... |
| exp4c 438 | An exportation inference. ... |
| exp4d 439 | An exportation inference. ... |
| exp41 440 | An exportation inference. ... |
| exp42 441 | An exportation inference. ... |
| exp43 442 | An exportation inference. ... |
| exp44 443 | An exportation inference. ... |
| exp45 444 | An exportation inference. ... |
| imp5d 445 | An importation inference. ... |
| imp5a 446 | An importation inference. ... |
| imp5g 447 | An importation inference. ... |
| imp55 448 | An importation inference. ... |
| imp511 449 | An importation inference. ... |
| exp5c 450 | An exportation inference. ... |
| exp5j 451 | An exportation inference. ... |
| exp5l 452 | An exportation inference. ... |
| exp53 453 | An exportation inference. ... |
| pm3.3 454 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 455 | Theorem *3.31 (Imp) of [Wh... |
| impexp 456 | Import-export theorem. Pa... |
| impancom 457 | Mixed importation/commutat... |
| expdimp 458 | A deduction version of exp... |
| expimpd 459 | Exportation followed by a ... |
| impr 460 | Import a wff into a right ... |
| impl 461 | Export a wff from a left c... |
| expr 462 | Export a wff from a right ... |
| expl 463 | Export a wff from a left c... |
| ancoms 464 | Inference commuting conjun... |
| pm3.22 465 | Theorem *3.22 of [Whitehea... |
| ancom 466 | Commutative law for conjun... |
| ancomd 467 | Commutation of conjuncts i... |
| biancomi 468 | Commuting conjunction in a... |
| biancomd 469 | Commuting conjunction in a... |
| ancomst 470 | Closed form of ~ ancoms . ... |
| ancomsd 471 | Deduction commuting conjun... |
| anasss 472 | Associative law for conjun... |
| anassrs 473 | Associative law for conjun... |
| anass 474 | Associative law for conjun... |
| pm3.2 475 | Join antecedents with conj... |
| pm3.2i 476 | Infer conjunction of premi... |
| pm3.21 477 | Join antecedents with conj... |
| pm3.43i 478 | Nested conjunction of ante... |
| pm3.43 479 | Theorem *3.43 (Comp) of [W... |
| dfbi2 480 | A theorem similar to the s... |
| dfbi 481 | Definition ~ df-bi rewritt... |
| biimpa 482 | Importation inference from... |
| biimpar 483 | Importation inference from... |
| biimpac 484 | Importation inference from... |
| biimparc 485 | Importation inference from... |
| adantr 486 | Inference adding a conjunc... |
| adantl 487 | Inference adding a conjunc... |
| simpl 488 | Elimination of a conjunct.... |
| simpli 489 | Inference eliminating a co... |
| simpr 490 | Elimination of a conjunct.... |
| simpri 491 | Inference eliminating a co... |
| intnan 492 | Introduction of conjunct i... |
| intnanr 493 | Introduction of conjunct i... |
| intnand 494 | Introduction of conjunct i... |
| intnanrd 495 | Introduction of conjunct i... |
| adantld 496 | Deduction adding a conjunc... |
| adantrd 497 | Deduction adding a conjunc... |
| pm3.41 498 | Theorem *3.41 of [Whitehea... |
| pm3.42 499 | Theorem *3.42 of [Whitehea... |
| simpld 500 | Deduction eliminating a co... |
| simprd 501 | Deduction eliminating a co... |
| simplbi 502 | Deduction eliminating a co... |
| simprbi 503 | Deduction eliminating a co... |
| simprbda 504 | Deduction eliminating a co... |
| simplbda 505 | Deduction eliminating a co... |
| simplbi2 506 | Deduction eliminating a co... |
| simplbi2comt 507 | Closed form of ~ simplbi2c... |
| simplbi2com 508 | A deduction eliminating a ... |
| birani 509 | Inference adding a conjunc... |
| bilani 510 | Inference adding a conjunc... |
| biranri 511 | Inference adding a conjunc... |
| bilanri 512 | Inference adding a conjunc... |
| simpl2im 513 | Implication from an elimin... |
| simplbiim 514 | Implication from an elimin... |
| impel 515 | An inference for implicati... |
| mpan9 516 | Modus ponens conjoining di... |
| sylan9 517 | Nested syllogism inference... |
| sylan9r 518 | Nested syllogism inference... |
| sylan9bb 519 | Nested syllogism inference... |
| sylan9bbr 520 | Nested syllogism inference... |
| jca 521 | Deduce conjunction of the ... |
| jcad 522 | Deduction conjoining the c... |
| jca2 523 | Inference conjoining the c... |
| jca31 524 | Join three consequents. (... |
| jca32 525 | Join three consequents. (... |
| jcai 526 | Deduction replacing implic... |
| jcab 527 | Distributive law for impli... |
| pm4.76 528 | Theorem *4.76 of [Whitehea... |
| jctil 529 | Inference conjoining a the... |
| jctir 530 | Inference conjoining a the... |
| jccir 531 | Inference conjoining a con... |
| jccil 532 | Inference conjoining a con... |
| jctl 533 | Inference conjoining a the... |
| jctr 534 | Inference conjoining a the... |
| jctild 535 | Deduction conjoining a the... |
| jctird 536 | Deduction conjoining a the... |
| iba 537 | Introduction of antecedent... |
| ibar 538 | Introduction of antecedent... |
| biantru 539 | A wff is equivalent to its... |
| biantrur 540 | A wff is equivalent to its... |
| biantrud 541 | A wff is equivalent to its... |
| biantrurd 542 | A wff is equivalent to its... |
| bianfi 543 | A wff conjoined with false... |
| bianfd 544 | A wff conjoined with false... |
| baib 545 | Move conjunction outside o... |
| baibr 546 | Move conjunction outside o... |
| rbaibr 547 | Move conjunction outside o... |
| rbaib 548 | Move conjunction outside o... |
| baibd 549 | Move conjunction outside o... |
| rbaibd 550 | Move conjunction outside o... |
| bianabs 551 | Absorb a hypothesis into t... |
| pm5.44 552 | Theorem *5.44 of [Whitehea... |
| pm5.42 553 | Theorem *5.42 of [Whitehea... |
| ancl 554 | Conjoin antecedent to left... |
| anclb 555 | Conjoin antecedent to left... |
| ancr 556 | Conjoin antecedent to righ... |
| ancrb 557 | Conjoin antecedent to righ... |
| ancli 558 | Deduction conjoining antec... |
| ancri 559 | Deduction conjoining antec... |
| ancld 560 | Deduction conjoining antec... |
| ancrd 561 | Deduction conjoining antec... |
| impac 562 | Importation with conjuncti... |
| anc2l 563 | Conjoin antecedent to left... |
| anc2r 564 | Conjoin antecedent to righ... |
| anc2li 565 | Deduction conjoining antec... |
| anc2ri 566 | Deduction conjoining antec... |
| pm4.71 567 | Implication in terms of bi... |
| pm4.71r 568 | Implication in terms of bi... |
| pm4.71i 569 | Inference converting an im... |
| pm4.71ri 570 | Inference converting an im... |
| pm4.71d 571 | Deduction converting an im... |
| pm4.71rd 572 | Deduction converting an im... |
| pm4.71da 573 | Deduction converting a bic... |
| pm4.24 574 | Theorem *4.24 of [Whitehea... |
| anidm 575 | Idempotent law for conjunc... |
| anidmdbi 576 | Conjunction idempotence wi... |
| anidms 577 | Inference from idempotent ... |
| imdistan 578 | Distribution of implicatio... |
| imdistani 579 | Distribution of implicatio... |
| imdistanri 580 | Distribution of implicatio... |
| imdistand 581 | Distribution of implicatio... |
| imdistanda 582 | Distribution of implicatio... |
| pm5.3 583 | Theorem *5.3 of [Whitehead... |
| pm5.32 584 | Distribution of implicatio... |
| pm5.32i 585 | Distribution of implicatio... |
| pm5.32ri 586 | Distribution of implicatio... |
| bianim 587 | Exchanging conjunction in ... |
| pm5.32d 588 | Distribution of implicatio... |
| pm5.32rd 589 | Distribution of implicatio... |
| pm5.32da 590 | Distribution of implicatio... |
| bian1d 591 | Adding a superfluous conju... |
| sylan 592 | A syllogism inference. (C... |
| sylanb 593 | A syllogism inference. (C... |
| sylanbr 594 | A syllogism inference. (C... |
| sylanbrc 595 | Syllogism inference. (Con... |
| syl2anc 596 | Syllogism inference combin... |
| syl2anc2 597 | Double syllogism inference... |
| sylancl 598 | Syllogism inference combin... |
| sylancr 599 | Syllogism inference combin... |
| sylancom 600 | Syllogism inference with c... |
| sylanblc 601 | Syllogism inference combin... |
| sylanblrc 602 | Syllogism inference combin... |
| syldan 603 | A syllogism deduction with... |
| sylbida 604 | A syllogism deduction. (C... |
| sylan2 605 | A syllogism inference. (C... |
| sylan2b 606 | A syllogism inference. (C... |
| sylan2br 607 | A syllogism inference. (C... |
| syl2an 608 | A double syllogism inferen... |
| syl2anr 609 | A double syllogism inferen... |
| syl2anb 610 | A double syllogism inferen... |
| syl2anbr 611 | A double syllogism inferen... |
| sylancb 612 | A syllogism inference comb... |
| sylancbr 613 | A syllogism inference comb... |
| syldanl 614 | A syllogism deduction with... |
| syland 615 | A syllogism deduction. (C... |
| sylani 616 | A syllogism inference. (C... |
| sylan2d 617 | A syllogism deduction. (C... |
| sylan2i 618 | A syllogism inference. (C... |
| syl2ani 619 | A syllogism inference. (C... |
| syl2and 620 | A syllogism deduction. (C... |
| anim12d 621 | Conjoin antecedents and co... |
| anim12d1 622 | Variant of ~ anim12d where... |
| anim1d 623 | Add a conjunct to right of... |
| anim2d 624 | Add a conjunct to left of ... |
| anim12i 625 | Conjoin antecedents and co... |
| anim12ci 626 | Variant of ~ anim12i with ... |
| anim1i 627 | Introduce conjunct to both... |
| anim1ci 628 | Introduce conjunct to both... |
| anim2i 629 | Introduce conjunct to both... |
| anim12ii 630 | Conjoin antecedents and co... |
| anim12dan 631 | Conjoin antecedents and co... |
| im2anan9 632 | Deduction joining nested i... |
| im2anan9r 633 | Deduction joining nested i... |
| pm3.45 634 | Theorem *3.45 (Fact) of [W... |
| anbi2i 635 | Introduce a left conjunct ... |
| anbi1i 636 | Introduce a right conjunct... |
| anbi2ci 637 | Variant of ~ anbi2i with c... |
| anbi1ci 638 | Variant of ~ anbi1i with c... |
| bianbi 639 | Exchanging conjunction in ... |
| anbi12i 640 | Conjoin both sides of two ... |
| anbi12ci 641 | Variant of ~ anbi12i with ... |
| anbi2d 642 | Deduction adding a left co... |
| anbi1d 643 | Deduction adding a right c... |
| anbi12d 644 | Deduction joining two equi... |
| anbi1 645 | Introduce a right conjunct... |
| anbi2 646 | Introduce a left conjunct ... |
| anbi1cd 647 | Introduce a proposition as... |
| an2anr 648 | Double commutation in conj... |
| pm4.38 649 | Theorem *4.38 of [Whitehea... |
| bi2anan9 650 | Deduction joining two equi... |
| bi2anan9r 651 | Deduction joining two equi... |
| bi2bian9 652 | Deduction joining two bico... |
| anbiim 653 | Adding biconditional when ... |
| anbiimOLD 654 | Obsolete version of ~ anbi... |
| bianass 655 | An inference to merge two ... |
| bianassc 656 | An inference to merge two ... |
| an21 657 | Swap two conjuncts. (Cont... |
| an12 658 | Swap two conjuncts. Note ... |
| an32 659 | A rearrangement of conjunc... |
| an13 660 | A rearrangement of conjunc... |
| an31 661 | A rearrangement of conjunc... |
| an12s 662 | Swap two conjuncts in ante... |
| ancom2s 663 | Inference commuting a nest... |
| an13s 664 | Swap two conjuncts in ante... |
| an32s 665 | Swap two conjuncts in ante... |
| ancom1s 666 | Inference commuting a nest... |
| an31s 667 | Swap two conjuncts in ante... |
| anass1rs 668 | Commutative-associative la... |
| an4 669 | Rearrangement of 4 conjunc... |
| an42 670 | Rearrangement of 4 conjunc... |
| an43 671 | Rearrangement of 4 conjunc... |
| an3 672 | A rearrangement of conjunc... |
| an4s 673 | Inference rearranging 4 co... |
| an42s 674 | Inference rearranging 4 co... |
| anabs1 675 | Absorption into embedded c... |
| anabs5 676 | Absorption into embedded c... |
| anabs7 677 | Absorption into embedded c... |
| anabsan 678 | Absorption of antecedent w... |
| anabss1 679 | Absorption of antecedent i... |
| anabss4 680 | Absorption of antecedent i... |
| anabss5 681 | Absorption of antecedent i... |
| anabsi5 682 | Absorption of antecedent i... |
| anabsi6 683 | Absorption of antecedent i... |
| anabsi7 684 | Absorption of antecedent i... |
| anabsi8 685 | Absorption of antecedent i... |
| anabss7 686 | Absorption of antecedent i... |
| anabsan2 687 | Absorption of antecedent w... |
| anabss3 688 | Absorption of antecedent i... |
| anandi 689 | Distribution of conjunctio... |
| anandir 690 | Distribution of conjunctio... |
| anandis 691 | Inference that undistribut... |
| anandirs 692 | Inference that undistribut... |
| sylanl1 693 | A syllogism inference. (C... |
| sylanl2 694 | A syllogism inference. (C... |
| sylanr1 695 | A syllogism inference. (C... |
| sylanr2 696 | A syllogism inference. (C... |
| syl6an 697 | A syllogism deduction comb... |
| syl2an2r 698 | ~ syl2anr with antecedents... |
| syl2an2 699 | ~ syl2an with antecedents ... |
| mpdan 700 | An inference based on modu... |
| mpancom 701 | An inference based on modu... |
| mpidan 702 | A deduction which "stacks"... |
| mpan 703 | An inference based on modu... |
| mpan2 704 | An inference based on modu... |
| mp2an 705 | An inference based on modu... |
| mp4an 706 | An inference based on modu... |
| mpan2d 707 | A deduction based on modus... |
| mpand 708 | A deduction based on modus... |
| mpani 709 | An inference based on modu... |
| mpan2i 710 | An inference based on modu... |
| mp2ani 711 | An inference based on modu... |
| mp2and 712 | A deduction based on modus... |
| mpanl1 713 | An inference based on modu... |
| mpanl2 714 | An inference based on modu... |
| mpanl12 715 | An inference based on modu... |
| mpanr1 716 | An inference based on modu... |
| mpanr2 717 | An inference based on modu... |
| mpanr12 718 | An inference based on modu... |
| mpanlr1 719 | An inference based on modu... |
| mpbirand 720 | Detach truth from conjunct... |
| mpbiran2d 721 | Detach truth from conjunct... |
| mpbiran 722 | Detach truth from conjunct... |
| mpbiran2 723 | Detach truth from conjunct... |
| mpbir2an 724 | Detach a conjunction of tr... |
| mpbi2and 725 | Detach a conjunction of tr... |
| mpbir2and 726 | Detach a conjunction of tr... |
| adantll 727 | Deduction adding a conjunc... |
| adantlr 728 | Deduction adding a conjunc... |
| adantrl 729 | Deduction adding a conjunc... |
| adantrr 730 | Deduction adding a conjunc... |
| adantlll 731 | Deduction adding a conjunc... |
| adantllr 732 | Deduction adding a conjunc... |
| adantlrl 733 | Deduction adding a conjunc... |
| adantlrr 734 | Deduction adding a conjunc... |
| adantrll 735 | Deduction adding a conjunc... |
| adantrlr 736 | Deduction adding a conjunc... |
| adantrrl 737 | Deduction adding a conjunc... |
| adantrrr 738 | Deduction adding a conjunc... |
| ad2antrr 739 | Deduction adding two conju... |
| ad2antlr 740 | Deduction adding two conju... |
| ad2antrl 741 | Deduction adding two conju... |
| ad2antll 742 | Deduction adding conjuncts... |
| ad3antrrr 743 | Deduction adding three con... |
| ad3antlr 744 | Deduction adding three con... |
| ad4antr 745 | Deduction adding 4 conjunc... |
| ad4antlr 746 | Deduction adding 4 conjunc... |
| ad5antr 747 | Deduction adding 5 conjunc... |
| ad5antlr 748 | Deduction adding 5 conjunc... |
| ad6antr 749 | Deduction adding 6 conjunc... |
| ad6antlr 750 | Deduction adding 6 conjunc... |
| ad7antr 751 | Deduction adding 7 conjunc... |
| ad7antlr 752 | Deduction adding 7 conjunc... |
| ad8antr 753 | Deduction adding 8 conjunc... |
| ad8antlr 754 | Deduction adding 8 conjunc... |
| ad9antr 755 | Deduction adding 9 conjunc... |
| ad9antlr 756 | Deduction adding 9 conjunc... |
| ad10antr 757 | Deduction adding 10 conjun... |
| ad10antlr 758 | Deduction adding 10 conjun... |
| ad2ant2l 759 | Deduction adding two conju... |
| ad2ant2r 760 | Deduction adding two conju... |
| ad2ant2lr 761 | Deduction adding two conju... |
| ad2ant2rl 762 | Deduction adding two conju... |
| adantl3r 763 | Deduction adding 1 conjunc... |
| ad4ant13 764 | Deduction adding conjuncts... |
| ad4ant14 765 | Deduction adding conjuncts... |
| ad4ant23 766 | Deduction adding conjuncts... |
| ad4ant24 767 | Deduction adding conjuncts... |
| adantl4r 768 | Deduction adding 1 conjunc... |
| ad5ant13 769 | Deduction adding conjuncts... |
| ad5ant14 770 | Deduction adding conjuncts... |
| ad5ant15 771 | Deduction adding conjuncts... |
| ad5ant23 772 | Deduction adding conjuncts... |
| ad5ant24 773 | Deduction adding conjuncts... |
| ad5ant25 774 | Deduction adding conjuncts... |
| adantl5r 775 | Deduction adding 1 conjunc... |
| adantl6r 776 | Deduction adding 1 conjunc... |
| pm3.33 777 | Theorem *3.33 (Syll) of [W... |
| pm3.34 778 | Theorem *3.34 (Syll) of [W... |
| simpll 779 | Simplification of a conjun... |
| simplld 780 | Deduction form of ~ simpll... |
| simplr 781 | Simplification of a conjun... |
| simplrd 782 | Deduction eliminating a do... |
| simprl 783 | Simplification of a conjun... |
| simprld 784 | Deduction eliminating a do... |
| simprr 785 | Simplification of a conjun... |
| simprrd 786 | Deduction form of ~ simprr... |
| simplll 787 | Simplification of a conjun... |
| simpllr 788 | Simplification of a conjun... |
| simplrl 789 | Simplification of a conjun... |
| simplrr 790 | Simplification of a conjun... |
| simprll 791 | Simplification of a conjun... |
| simprlr 792 | Simplification of a conjun... |
| simprrl 793 | Simplification of a conjun... |
| simprrr 794 | Simplification of a conjun... |
| simp-4l 795 | Simplification of a conjun... |
| simp-4r 796 | Simplification of a conjun... |
| simp-5l 797 | Simplification of a conjun... |
| simp-5r 798 | Simplification of a conjun... |
| simp-6l 799 | Simplification of a conjun... |
| simp-6r 800 | Simplification of a conjun... |
| simp-7l 801 | Simplification of a conjun... |
| simp-7r 802 | Simplification of a conjun... |
| simp-8l 803 | Simplification of a conjun... |
| simp-8r 804 | Simplification of a conjun... |
| simp-9l 805 | Simplification of a conjun... |
| simp-9r 806 | Simplification of a conjun... |
| simp-10l 807 | Simplification of a conjun... |
| simp-10r 808 | Simplification of a conjun... |
| simp-11l 809 | Simplification of a conjun... |
| simp-11r 810 | Simplification of a conjun... |
| pm2.01da 811 | Deduction based on reducti... |
| pm2.18da 812 | Deduction based on reducti... |
| impbida 813 | Deduce an equivalence from... |
| pm5.21nd 814 | Eliminate an antecedent im... |
| pm3.35 815 | Conjunctive detachment. T... |
| pm5.74da 816 | Distribution of implicatio... |
| bitr 817 | Theorem *4.22 of [Whitehea... |
| biantr 818 | A transitive law of equiva... |
| pm4.14 819 | Theorem *4.14 of [Whitehea... |
| pm3.37 820 | Theorem *3.37 (Transp) of ... |
| anim12 821 | Conjoin antecedents and co... |
| pm3.4 822 | Conjunction implies implic... |
| exbiri 823 | Inference form of ~ exbir ... |
| pm2.61ian 824 | Elimination of an antecede... |
| pm2.61dan 825 | Elimination of an antecede... |
| pm2.61ddan 826 | Elimination of two anteced... |
| pm2.61dda 827 | Elimination of two anteced... |
| mtand 828 | A modus tollens deduction.... |
| pm2.65da 829 | Deduction for proof by con... |
| condan 830 | Proof by contradiction. (... |
| biadan 831 | An implication is equivale... |
| biadani 832 | Inference associated with ... |
| biadaniALT 833 | Alternate proof of ~ biada... |
| biadanii 834 | Inference associated with ... |
| biadanid 835 | Deduction associated with ... |
| pm5.1 836 | Two propositions are equiv... |
| pm5.21 837 | Two propositions are equiv... |
| pm5.35 838 | Theorem *5.35 of [Whitehea... |
| abai 839 | Introduce one conjunct as ... |
| abab 840 | Introduce one conjunct as ... |
| pm4.45im 841 | Conjunction with implicati... |
| impimprbi 842 | An implication and its rev... |
| nan 843 | Theorem to move a conjunct... |
| pm5.31 844 | Theorem *5.31 of [Whitehea... |
| pm5.31r 845 | Variant of ~ pm5.31 . (Co... |
| pm4.15 846 | Theorem *4.15 of [Whitehea... |
| pm5.36 847 | Theorem *5.36 of [Whitehea... |
| annotanannot 848 | A conjunction with a negat... |
| pm5.33 849 | Theorem *5.33 of [Whitehea... |
| syl12anc 850 | Syllogism combined with co... |
| syl21anc 851 | Syllogism combined with co... |
| syl22anc 852 | Syllogism combined with co... |
| bibiad 853 | Eliminate an hypothesis ` ... |
| syl1111anc 854 | Four-hypothesis eliminatio... |
| syldbl2 855 | Stacked hypotheseis implie... |
| mpsyl4anc 856 | An elimination deduction. ... |
| pm4.87 857 | Theorem *4.87 of [Whitehea... |
| bimsc1 858 | Removal of conjunct from o... |
| a2and 859 | Deduction distributing a c... |
| animpimp2impd 860 | Deduction deriving nested ... |
| pm4.64 863 | Theorem *4.64 of [Whitehea... |
| pm4.66 864 | Theorem *4.66 of [Whitehea... |
| pm2.53 865 | Theorem *2.53 of [Whitehea... |
| pm2.54 866 | Theorem *2.54 of [Whitehea... |
| imor 867 | Implication in terms of di... |
| imori 868 | Infer disjunction from imp... |
| imorri 869 | Infer implication from dis... |
| pm4.62 870 | Theorem *4.62 of [Whitehea... |
| jaoi 871 | Inference disjoining the a... |
| jao1i 872 | Add a disjunct in the ante... |
| jaod 873 | Deduction disjoining the a... |
| mpjaod 874 | Eliminate a disjunction in... |
| ori 875 | Infer implication from dis... |
| orri 876 | Infer disjunction from imp... |
| orrd 877 | Deduce disjunction from im... |
| ord 878 | Deduce implication from di... |
| orci 879 | Deduction introducing a di... |
| olci 880 | Deduction introducing a di... |
| orc 881 | Introduction of a disjunct... |
| olc 882 | Introduction of a disjunct... |
| pm1.4 883 | Axiom *1.4 of [WhiteheadRu... |
| orcom 884 | Commutative law for disjun... |
| orcomd 885 | Commutation of disjuncts i... |
| orcoms 886 | Commutation of disjuncts i... |
| orcd 887 | Deduction introducing a di... |
| olcd 888 | Deduction introducing a di... |
| orcs 889 | Deduction eliminating disj... |
| olcs 890 | Deduction eliminating disj... |
| olcnd 891 | A lemma for Conjunctive No... |
| orcnd 892 | A lemma for Conjunctive No... |
| mtord 893 | A modus tollens deduction ... |
| pm3.2ni 894 | Infer negated disjunction ... |
| pm2.45 895 | Theorem *2.45 of [Whitehea... |
| pm2.46 896 | Theorem *2.46 of [Whitehea... |
| pm2.47 897 | Theorem *2.47 of [Whitehea... |
| pm2.48 898 | Theorem *2.48 of [Whitehea... |
| pm2.49 899 | Theorem *2.49 of [Whitehea... |
| norbi 900 | If neither of two proposit... |
| nbior 901 | If two propositions are no... |
| orel1 902 | Elimination of disjunction... |
| pm2.25 903 | Theorem *2.25 of [Whitehea... |
| orel2 904 | Elimination of disjunction... |
| pm2.67-2 905 | Slight generalization of T... |
| pm2.67 906 | Theorem *2.67 of [Whitehea... |
| curryax 907 | A non-intuitionistic posit... |
| exmid 908 | Law of excluded middle, al... |
| exmidd 909 | Law of excluded middle in ... |
| pm2.1 910 | Theorem *2.1 of [Whitehead... |
| pm2.13 911 | Theorem *2.13 of [Whitehea... |
| pm2.621 912 | Theorem *2.621 of [Whitehe... |
| pm2.62 913 | Theorem *2.62 of [Whitehea... |
| pm2.68 914 | Theorem *2.68 of [Whitehea... |
| dfor2 915 | Logical 'or' expressed in ... |
| pm2.07 916 | Theorem *2.07 of [Whitehea... |
| pm1.2 917 | Axiom *1.2 of [WhiteheadRu... |
| oridm 918 | Idempotent law for disjunc... |
| pm4.25 919 | Theorem *4.25 of [Whitehea... |
| pm2.4 920 | Theorem *2.4 of [Whitehead... |
| pm2.41 921 | Theorem *2.41 of [Whitehea... |
| orim12i 922 | Disjoin antecedents and co... |
| orim1i 923 | Introduce disjunct to both... |
| orim2i 924 | Introduce disjunct to both... |
| orim12dALT 925 | Alternate proof of ~ orim1... |
| orbi2i 926 | Inference adding a left di... |
| orbi1i 927 | Inference adding a right d... |
| orbi12i 928 | Infer the disjunction of t... |
| orbi2d 929 | Deduction adding a left di... |
| orbi1d 930 | Deduction adding a right d... |
| orbi1 931 | Theorem *4.37 of [Whitehea... |
| orbi12d 932 | Deduction joining two equi... |
| pm1.5 933 | Axiom *1.5 (Assoc) of [Whi... |
| or12 934 | Swap two disjuncts. (Cont... |
| orass 935 | Associative law for disjun... |
| pm2.31 936 | Theorem *2.31 of [Whitehea... |
| pm2.32 937 | Theorem *2.32 of [Whitehea... |
| pm2.3 938 | Theorem *2.3 of [Whitehead... |
| or32 939 | A rearrangement of disjunc... |
| or4 940 | Rearrangement of 4 disjunc... |
| or42 941 | Rearrangement of 4 disjunc... |
| orordi 942 | Distribution of disjunctio... |
| orordir 943 | Distribution of disjunctio... |
| orimdi 944 | Disjunction distributes ov... |
| pm2.76 945 | Theorem *2.76 of [Whitehea... |
| pm2.85 946 | Theorem *2.85 of [Whitehea... |
| pm2.75 947 | Theorem *2.75 of [Whitehea... |
| pm4.78 948 | Implication distributes ov... |
| biort 949 | A disjunction with a true ... |
| biorf 950 | A wff is equivalent to its... |
| biortn 951 | A wff is equivalent to its... |
| biorfi 952 | The dual of ~ biorf is not... |
| biorfri 953 | A wff is equivalent to its... |
| pm2.26 954 | Theorem *2.26 of [Whitehea... |
| pm2.63 955 | Theorem *2.63 of [Whitehea... |
| pm2.64 956 | Theorem *2.64 of [Whitehea... |
| pm2.42 957 | Theorem *2.42 of [Whitehea... |
| pm5.11g 958 | A general instance of Theo... |
| pm5.11 959 | Theorem *5.11 of [Whitehea... |
| pm5.12 960 | Theorem *5.12 of [Whitehea... |
| pm5.14 961 | Theorem *5.14 of [Whitehea... |
| pm5.13 962 | Theorem *5.13 of [Whitehea... |
| pm5.55 963 | Theorem *5.55 of [Whitehea... |
| pm4.72 964 | Implication in terms of bi... |
| imimorb 965 | Simplify an implication be... |
| oibabs 966 | Absorption of disjunction ... |
| orbidi 967 | Disjunction distributes ov... |
| pm5.7 968 | Disjunction distributes ov... |
| jaao 969 | Inference conjoining and d... |
| jaoa 970 | Inference disjoining and c... |
| jaoian 971 | Inference disjoining the a... |
| jaodan 972 | Deduction disjoining the a... |
| mpjaodan 973 | Eliminate a disjunction in... |
| pm3.44 974 | Theorem *3.44 of [Whitehea... |
| jao 975 | Disjunction of antecedents... |
| jaob 976 | Disjunction of antecedents... |
| pm4.77 977 | Theorem *4.77 of [Whitehea... |
| pm3.48 978 | Theorem *3.48 of [Whitehea... |
| orim12d 979 | Disjoin antecedents and co... |
| orim12da 980 | Deduce a disjunction from ... |
| orim1d 981 | Disjoin antecedents and co... |
| orim2d 982 | Disjoin antecedents and co... |
| orim2 983 | Axiom *1.6 (Sum) of [White... |
| pm2.38 984 | Theorem *2.38 of [Whitehea... |
| pm2.36 985 | Theorem *2.36 of [Whitehea... |
| pm2.37 986 | Theorem *2.37 of [Whitehea... |
| pm2.81 987 | Theorem *2.81 of [Whitehea... |
| pm2.8 988 | Theorem *2.8 of [Whitehead... |
| pm2.73 989 | Theorem *2.73 of [Whitehea... |
| pm2.74 990 | Theorem *2.74 of [Whitehea... |
| pm2.82 991 | Theorem *2.82 of [Whitehea... |
| pm4.39 992 | Theorem *4.39 of [Whitehea... |
| animorl 993 | Conjunction implies disjun... |
| animorr 994 | Conjunction implies disjun... |
| animorlr 995 | Conjunction implies disjun... |
| animorrl 996 | Conjunction implies disjun... |
| ianor 997 | Negated conjunction in ter... |
| anor 998 | Conjunction in terms of di... |
| ioran 999 | Negated disjunction in ter... |
| pm4.52 1000 | Theorem *4.52 of [Whitehea... |
| pm4.53 1001 | Theorem *4.53 of [Whitehea... |
| pm4.54 1002 | Theorem *4.54 of [Whitehea... |
| pm4.55 1003 | Theorem *4.55 of [Whitehea... |
| pm4.56 1004 | Theorem *4.56 of [Whitehea... |
| oran 1005 | Disjunction in terms of co... |
| pm4.57 1006 | Theorem *4.57 of [Whitehea... |
| pm3.1 1007 | Theorem *3.1 of [Whitehead... |
| pm3.11 1008 | Theorem *3.11 of [Whitehea... |
| pm3.12 1009 | Theorem *3.12 of [Whitehea... |
| pm3.13 1010 | Theorem *3.13 of [Whitehea... |
| pm3.14 1011 | Theorem *3.14 of [Whitehea... |
| pm4.44 1012 | Theorem *4.44 of [Whitehea... |
| pm4.45 1013 | Theorem *4.45 of [Whitehea... |
| orabs 1014 | Absorption of redundant in... |
| oranabs 1015 | Absorb a disjunct into a c... |
| pm5.61 1016 | Theorem *5.61 of [Whitehea... |
| pm5.6 1017 | Conjunction in antecedent ... |
| orcanai 1018 | Change disjunction in cons... |
| orsild 1019 | A lemma for not-or-not eli... |
| orsird 1020 | A lemma for not-or-not eli... |
| pm4.79 1021 | Theorem *4.79 of [Whitehea... |
| pm5.53 1022 | Theorem *5.53 of [Whitehea... |
| ordi 1023 | Distributive law for disju... |
| ordir 1024 | Distributive law for disju... |
| andi 1025 | Distributive law for conju... |
| andir 1026 | Distributive law for conju... |
| orddi 1027 | Double distributive law fo... |
| anddi 1028 | Double distributive law fo... |
| pm5.17 1029 | Theorem *5.17 of [Whitehea... |
| pm5.15 1030 | Theorem *5.15 of [Whitehea... |
| pm5.16 1031 | Theorem *5.16 of [Whitehea... |
| xor 1032 | Two ways to express exclus... |
| nbi2 1033 | Two ways to express "exclu... |
| xordi 1034 | Conjunction distributes ov... |
| pm5.54 1035 | Theorem *5.54 of [Whitehea... |
| pm5.62 1036 | Theorem *5.62 of [Whitehea... |
| pm5.63 1037 | Theorem *5.63 of [Whitehea... |
| niabn 1038 | Miscellaneous inference re... |
| ninba 1039 | Miscellaneous inference re... |
| pm4.43 1040 | Theorem *4.43 of [Whitehea... |
| pm4.82 1041 | Theorem *4.82 of [Whitehea... |
| pm4.83 1042 | Theorem *4.83 of [Whitehea... |
| pclem6 1043 | Negation inferred from emb... |
| bigolden 1044 | Dijkstra-Scholten's Golden... |
| pm5.71 1045 | Theorem *5.71 of [Whitehea... |
| pm5.75 1046 | Theorem *5.75 of [Whitehea... |
| ecase2d 1047 | Deduction for elimination ... |
| ecase3 1048 | Inference for elimination ... |
| ecase 1049 | Inference for elimination ... |
| ecase3d 1050 | Deduction for elimination ... |
| ecased 1051 | Deduction for elimination ... |
| ecase3ad 1052 | Deduction for elimination ... |
| ccase 1053 | Inference for combining ca... |
| ccased 1054 | Deduction for combining ca... |
| ccase2 1055 | Inference for combining ca... |
| 4cases 1056 | Inference eliminating two ... |
| 4casesdan 1057 | Deduction eliminating two ... |
| cases 1058 | Case disjunction according... |
| dedlem0a 1059 | Lemma for an alternate ver... |
| dedlem0b 1060 | Lemma for an alternate ver... |
| dedlema 1061 | Lemma for weak deduction t... |
| dedlemb 1062 | Lemma for weak deduction t... |
| cases2 1063 | Case disjunction according... |
| cases2ALT 1064 | Alternate proof of ~ cases... |
| dfbi3 1065 | An alternate definition of... |
| pm5.24 1066 | Theorem *5.24 of [Whitehea... |
| 4exmid 1067 | The disjunction of the fou... |
| consensus 1068 | The consensus theorem. Th... |
| pm4.42 1069 | Theorem *4.42 of [Whitehea... |
| prlem1 1070 | A specialized lemma for se... |
| prlem2 1071 | A specialized lemma for se... |
| oplem1 1072 | A specialized lemma for se... |
| dn1 1073 | A single axiom for Boolean... |
| bianir 1074 | A closed form of ~ mpbir ,... |
| jaoi2 1075 | Inference removing a negat... |
| jaoi3 1076 | Inference separating a dis... |
| ornld 1077 | Selecting one statement fr... |
| dfifp2 1080 | Alternate definition of th... |
| dfifp3 1081 | Alternate definition of th... |
| dfifp4 1082 | Alternate definition of th... |
| dfifp5 1083 | Alternate definition of th... |
| dfifp6 1084 | Alternate definition of th... |
| dfifp7 1085 | Alternate definition of th... |
| ifpdfbi 1086 | Define the biconditional a... |
| ifpdfbiOLD 1087 | Obsolete version of ~ ifpd... |
| anifp 1088 | The conditional operator i... |
| ifpor 1089 | The conditional operator i... |
| ifpn 1090 | Conditional operator for t... |
| ifptru 1091 | Value of the conditional o... |
| ifpfal 1092 | Value of the conditional o... |
| ifpid 1093 | Value of the conditional o... |
| casesifp 1094 | Version of ~ cases express... |
| ifpbi123d 1095 | Equivalence deduction for ... |
| ifpbi23d 1096 | Equivalence deduction for ... |
| ifpimpda 1097 | Separation of the values o... |
| 1fpid3 1098 | The value of the condition... |
| elimh 1099 | Hypothesis builder for the... |
| dedt 1100 | The weak deduction theorem... |
| con3ALT 1101 | Proof of ~ con3 from its a... |
| 3orass 1106 | Associative law for triple... |
| 3orel1 1107 | Partial elimination of a t... |
| 3orrot 1108 | Rotation law for triple di... |
| 3orcoma 1109 | Commutation law for triple... |
| 3orcomb 1110 | Commutation law for triple... |
| 3anass 1111 | Associative law for triple... |
| 3anan12 1112 | Convert triple conjunction... |
| 3anan32 1113 | Convert triple conjunction... |
| 3anan32OLD 1114 | Obsolete version of ~ 3ana... |
| 3ancoma 1115 | Commutation law for triple... |
| 3ancomb 1116 | Commutation law for triple... |
| 3anrot 1117 | Rotation law for triple co... |
| 3anrev 1118 | Reversal law for triple co... |
| anandi3 1119 | Distribution of triple con... |
| anandi3r 1120 | Distribution of triple con... |
| 3anidm 1121 | Idempotent law for conjunc... |
| 3an4anass 1122 | Associative law for four c... |
| 3ioran 1123 | Negated triple disjunction... |
| 3ianor 1124 | Negated triple conjunction... |
| 3anor 1125 | Triple conjunction express... |
| 3oran 1126 | Triple disjunction in term... |
| 3impa 1127 | Importation from double to... |
| 3imp 1128 | Importation inference. (C... |
| 3imp31 1129 | The importation inference ... |
| 3imp231 1130 | Importation inference. (C... |
| 3imp21 1131 | The importation inference ... |
| 3impb 1132 | Importation from double to... |
| bi23imp13 1133 | ~ 3imp with middle implica... |
| 3impib 1134 | Importation to triple conj... |
| 3impia 1135 | Importation to triple conj... |
| 3expa 1136 | Exportation from triple to... |
| 3exp 1137 | Exportation inference. (C... |
| 3expb 1138 | Exportation from triple to... |
| 3expia 1139 | Exportation from triple co... |
| 3expib 1140 | Exportation from triple co... |
| 3com12 1141 | Commutation in antecedent.... |
| 3com13 1142 | Commutation in antecedent.... |
| 3comr 1143 | Commutation in antecedent.... |
| 3com23 1144 | Commutation in antecedent.... |
| 3coml 1145 | Commutation in antecedent.... |
| 3jca 1146 | Join consequents with conj... |
| 3jcad 1147 | Deduction conjoining the c... |
| 3adant1 1148 | Deduction adding a conjunc... |
| 3adant2 1149 | Deduction adding a conjunc... |
| 3adant3 1150 | Deduction adding a conjunc... |
| 3ad2ant1 1151 | Deduction adding conjuncts... |
| 3ad2ant2 1152 | Deduction adding conjuncts... |
| 3ad2ant3 1153 | Deduction adding conjuncts... |
| simp1 1154 | Simplification of triple c... |
| simp2 1155 | Simplification of triple c... |
| simp3 1156 | Simplification of triple c... |
| simp1i 1157 | Infer a conjunct from a tr... |
| simp2i 1158 | Infer a conjunct from a tr... |
| simp3i 1159 | Infer a conjunct from a tr... |
| simp1d 1160 | Deduce a conjunct from a t... |
| simp2d 1161 | Deduce a conjunct from a t... |
| simp3d 1162 | Deduce a conjunct from a t... |
| simp1bi 1163 | Deduce a conjunct from a t... |
| simp2bi 1164 | Deduce a conjunct from a t... |
| simp3bi 1165 | Deduce a conjunct from a t... |
| 3simpa 1166 | Simplification of triple c... |
| 3simpb 1167 | Simplification of triple c... |
| 3simpc 1168 | Simplification of triple c... |
| 3anim123i 1169 | Join antecedents and conse... |
| 3anim1i 1170 | Add two conjuncts to antec... |
| 3anim2i 1171 | Add two conjuncts to antec... |
| 3anim3i 1172 | Add two conjuncts to antec... |
| 3anbi123i 1173 | Join 3 biconditionals with... |
| 3orbi123i 1174 | Join 3 biconditionals with... |
| 3anbi1i 1175 | Inference adding two conju... |
| 3anbi2i 1176 | Inference adding two conju... |
| 3anbi3i 1177 | Inference adding two conju... |
| syl3an 1178 | A triple syllogism inferen... |
| syl3anb 1179 | A triple syllogism inferen... |
| syl3anbr 1180 | A triple syllogism inferen... |
| syl3an1 1181 | A syllogism inference. (C... |
| syl3an2 1182 | A syllogism inference. (C... |
| syl3an3 1183 | A syllogism inference. (C... |
| syl3an132 1184 | ~ syl2an with antecedents ... |
| 3adantl1 1185 | Deduction adding a conjunc... |
| 3adantl2 1186 | Deduction adding a conjunc... |
| 3adantl3 1187 | Deduction adding a conjunc... |
| 3adantr1 1188 | Deduction adding a conjunc... |
| 3adantr2 1189 | Deduction adding a conjunc... |
| 3adantr3 1190 | Deduction adding a conjunc... |
| ad4ant123 1191 | Deduction adding conjuncts... |
| ad4ant124 1192 | Deduction adding conjuncts... |
| ad4ant134 1193 | Deduction adding conjuncts... |
| ad4ant234 1194 | Deduction adding conjuncts... |
| 3adant1l 1195 | Deduction adding a conjunc... |
| 3adant1r 1196 | Deduction adding a conjunc... |
| 3adant2l 1197 | Deduction adding a conjunc... |
| 3adant2r 1198 | Deduction adding a conjunc... |
| 3adant3l 1199 | Deduction adding a conjunc... |
| 3adant3r 1200 | Deduction adding a conjunc... |
| 3adant3r1 1201 | Deduction adding a conjunc... |
| 3adant3r2 1202 | Deduction adding a conjunc... |
| 3adant3r3 1203 | Deduction adding a conjunc... |
| 3ad2antl1 1204 | Deduction adding conjuncts... |
| 3ad2antl2 1205 | Deduction adding conjuncts... |
| 3ad2antl3 1206 | Deduction adding conjuncts... |
| 3ad2antr1 1207 | Deduction adding conjuncts... |
| 3ad2antr2 1208 | Deduction adding conjuncts... |
| 3ad2antr3 1209 | Deduction adding conjuncts... |
| simpl1 1210 | Simplification of conjunct... |
| simpl2 1211 | Simplification of conjunct... |
| simpl3 1212 | Simplification of conjunct... |
| simpr1 1213 | Simplification of conjunct... |
| simpr2 1214 | Simplification of conjunct... |
| simpr3 1215 | Simplification of conjunct... |
| simp1l 1216 | Simplification of triple c... |
| simp1r 1217 | Simplification of triple c... |
| simp2l 1218 | Simplification of triple c... |
| simp2r 1219 | Simplification of triple c... |
| simp3l 1220 | Simplification of triple c... |
| simp3r 1221 | Simplification of triple c... |
| simp11 1222 | Simplification of doubly t... |
| simp12 1223 | Simplification of doubly t... |
| simp13 1224 | Simplification of doubly t... |
| simp21 1225 | Simplification of doubly t... |
| simp22 1226 | Simplification of doubly t... |
| simp23 1227 | Simplification of doubly t... |
| simp31 1228 | Simplification of doubly t... |
| simp32 1229 | Simplification of doubly t... |
| simp33 1230 | Simplification of doubly t... |
| simpll1 1231 | Simplification of conjunct... |
| simpll2 1232 | Simplification of conjunct... |
| simpll3 1233 | Simplification of conjunct... |
| simplr1 1234 | Simplification of conjunct... |
| simplr2 1235 | Simplification of conjunct... |
| simplr3 1236 | Simplification of conjunct... |
| simprl1 1237 | Simplification of conjunct... |
| simprl2 1238 | Simplification of conjunct... |
| simprl3 1239 | Simplification of conjunct... |
| simprr1 1240 | Simplification of conjunct... |
| simprr2 1241 | Simplification of conjunct... |
| simprr3 1242 | Simplification of conjunct... |
| simpl1l 1243 | Simplification of conjunct... |
| simpl1r 1244 | Simplification of conjunct... |
| simpl2l 1245 | Simplification of conjunct... |
| simpl2r 1246 | Simplification of conjunct... |
| simpl3l 1247 | Simplification of conjunct... |
| simpl3r 1248 | Simplification of conjunct... |
| simpr1l 1249 | Simplification of conjunct... |
| simpr1r 1250 | Simplification of conjunct... |
| simpr2l 1251 | Simplification of conjunct... |
| simpr2r 1252 | Simplification of conjunct... |
| simpr3l 1253 | Simplification of conjunct... |
| simpr3r 1254 | Simplification of conjunct... |
| simp1ll 1255 | Simplification of conjunct... |
| simp1lr 1256 | Simplification of conjunct... |
| simp1rl 1257 | Simplification of conjunct... |
| simp1rr 1258 | Simplification of conjunct... |
| simp2ll 1259 | Simplification of conjunct... |
| simp2lr 1260 | Simplification of conjunct... |
| simp2rl 1261 | Simplification of conjunct... |
| simp2rr 1262 | Simplification of conjunct... |
| simp3ll 1263 | Simplification of conjunct... |
| simp3lr 1264 | Simplification of conjunct... |
| simp3rl 1265 | Simplification of conjunct... |
| simp3rr 1266 | Simplification of conjunct... |
| simpl11 1267 | Simplification of conjunct... |
| simpl12 1268 | Simplification of conjunct... |
| simpl13 1269 | Simplification of conjunct... |
| simpl21 1270 | Simplification of conjunct... |
| simpl22 1271 | Simplification of conjunct... |
| simpl23 1272 | Simplification of conjunct... |
| simpl31 1273 | Simplification of conjunct... |
| simpl32 1274 | Simplification of conjunct... |
| simpl33 1275 | Simplification of conjunct... |
| simpr11 1276 | Simplification of conjunct... |
| simpr12 1277 | Simplification of conjunct... |
| simpr13 1278 | Simplification of conjunct... |
| simpr21 1279 | Simplification of conjunct... |
| simpr22 1280 | Simplification of conjunct... |
| simpr23 1281 | Simplification of conjunct... |
| simpr31 1282 | Simplification of conjunct... |
| simpr32 1283 | Simplification of conjunct... |
| simpr33 1284 | Simplification of conjunct... |
| simp1l1 1285 | Simplification of conjunct... |
| simp1l2 1286 | Simplification of conjunct... |
| simp1l3 1287 | Simplification of conjunct... |
| simp1r1 1288 | Simplification of conjunct... |
| simp1r2 1289 | Simplification of conjunct... |
| simp1r3 1290 | Simplification of conjunct... |
| simp2l1 1291 | Simplification of conjunct... |
| simp2l2 1292 | Simplification of conjunct... |
| simp2l3 1293 | Simplification of conjunct... |
| simp2r1 1294 | Simplification of conjunct... |
| simp2r2 1295 | Simplification of conjunct... |
| simp2r3 1296 | Simplification of conjunct... |
| simp3l1 1297 | Simplification of conjunct... |
| simp3l2 1298 | Simplification of conjunct... |
| simp3l3 1299 | Simplification of conjunct... |
| simp3r1 1300 | Simplification of conjunct... |
| simp3r2 1301 | Simplification of conjunct... |
| simp3r3 1302 | Simplification of conjunct... |
| simp11l 1303 | Simplification of conjunct... |
| simp11r 1304 | Simplification of conjunct... |
| simp12l 1305 | Simplification of conjunct... |
| simp12r 1306 | Simplification of conjunct... |
| simp13l 1307 | Simplification of conjunct... |
| simp13r 1308 | Simplification of conjunct... |
| simp21l 1309 | Simplification of conjunct... |
| simp21r 1310 | Simplification of conjunct... |
| simp22l 1311 | Simplification of conjunct... |
| simp22r 1312 | Simplification of conjunct... |
| simp23l 1313 | Simplification of conjunct... |
| simp23r 1314 | Simplification of conjunct... |
| simp31l 1315 | Simplification of conjunct... |
| simp31r 1316 | Simplification of conjunct... |
| simp32l 1317 | Simplification of conjunct... |
| simp32r 1318 | Simplification of conjunct... |
| simp33l 1319 | Simplification of conjunct... |
| simp33r 1320 | Simplification of conjunct... |
| simp111 1321 | Simplification of conjunct... |
| simp112 1322 | Simplification of conjunct... |
| simp113 1323 | Simplification of conjunct... |
| simp121 1324 | Simplification of conjunct... |
| simp122 1325 | Simplification of conjunct... |
| simp123 1326 | Simplification of conjunct... |
| simp131 1327 | Simplification of conjunct... |
| simp132 1328 | Simplification of conjunct... |
| simp133 1329 | Simplification of conjunct... |
| simp211 1330 | Simplification of conjunct... |
| simp212 1331 | Simplification of conjunct... |
| simp213 1332 | Simplification of conjunct... |
| simp221 1333 | Simplification of conjunct... |
| simp222 1334 | Simplification of conjunct... |
| simp223 1335 | Simplification of conjunct... |
| simp231 1336 | Simplification of conjunct... |
| simp232 1337 | Simplification of conjunct... |
| simp233 1338 | Simplification of conjunct... |
| simp311 1339 | Simplification of conjunct... |
| simp312 1340 | Simplification of conjunct... |
| simp313 1341 | Simplification of conjunct... |
| simp321 1342 | Simplification of conjunct... |
| simp322 1343 | Simplification of conjunct... |
| simp323 1344 | Simplification of conjunct... |
| simp331 1345 | Simplification of conjunct... |
| simp332 1346 | Simplification of conjunct... |
| simp333 1347 | Simplification of conjunct... |
| 3anibar 1348 | Remove a hypothesis from t... |
| 3mix1 1349 | Introduction in triple dis... |
| 3mix2 1350 | Introduction in triple dis... |
| 3mix3 1351 | Introduction in triple dis... |
| 3mix1i 1352 | Introduction in triple dis... |
| 3mix2i 1353 | Introduction in triple dis... |
| 3mix3i 1354 | Introduction in triple dis... |
| 3mix1d 1355 | Deduction introducing trip... |
| 3mix2d 1356 | Deduction introducing trip... |
| 3mix3d 1357 | Deduction introducing trip... |
| 3pm3.2i 1358 | Infer conjunction of premi... |
| pm3.2an3 1359 | Version of ~ pm3.2 for a t... |
| mpbir3an 1360 | Detach a conjunction of tr... |
| mpbir3and 1361 | Detach a conjunction of tr... |
| syl3anbrc 1362 | Syllogism inference. (Con... |
| syl21anbrc 1363 | Syllogism inference. (Con... |
| 3imp3i2an 1364 | An elimination deduction. ... |
| ex3 1365 | Apply ~ ex to a hypothesis... |
| 3imp1 1366 | Importation to left triple... |
| 3impd 1367 | Importation deduction for ... |
| 3imp2 1368 | Importation to right tripl... |
| 3impdi 1369 | Importation inference (und... |
| 3impdir 1370 | Importation inference (und... |
| 3exp1 1371 | Exportation from left trip... |
| 3expd 1372 | Exportation deduction for ... |
| 3exp2 1373 | Exportation from right tri... |
| exp5o 1374 | A triple exportation infer... |
| exp516 1375 | A triple exportation infer... |
| exp520 1376 | A triple exportation infer... |
| 3impexp 1377 | Version of ~ impexp for a ... |
| 3an1rs 1378 | Swap conjuncts. (Contribu... |
| 13an22anass 1379 | Associative law for four c... |
| 3anasss 1380 | Associative law for conjun... |
| 3anassrs 1381 | Associative law for conjun... |
| 4anpull2 1382 | An equivalence of two four... |
| 4anpull2OLD 1383 | Obsolete version of ~ 4anp... |
| ad5ant245 1384 | Deduction adding conjuncts... |
| ad5ant234 1385 | Deduction adding conjuncts... |
| ad5ant235 1386 | Deduction adding conjuncts... |
| ad5ant123 1387 | Deduction adding conjuncts... |
| ad5ant124 1388 | Deduction adding conjuncts... |
| ad5ant124OLD 1389 | Obsolete version of ~ ad5a... |
| ad5ant125 1390 | Deduction adding conjuncts... |
| ad5ant125OLD 1391 | Obsolete version of ~ ad5a... |
| ad5ant134 1392 | Deduction adding conjuncts... |
| ad5ant134OLD 1393 | Obsolete version of ~ ad5a... |
| ad5ant135 1394 | Deduction adding conjuncts... |
| ad5ant135OLD 1395 | Obsolete version of ~ ad5a... |
| ad5ant145 1396 | Deduction adding conjuncts... |
| ad5ant2345 1397 | Deduction adding conjuncts... |
| syl3anc 1398 | Syllogism combined with co... |
| syl13anc 1399 | Syllogism combined with co... |
| syl31anc 1400 | Syllogism combined with co... |
| syl112anc 1401 | Syllogism combined with co... |
| syl121anc 1402 | Syllogism combined with co... |
| syl211anc 1403 | Syllogism combined with co... |
| syl23anc 1404 | Syllogism combined with co... |
| syl32anc 1405 | Syllogism combined with co... |
| syl122anc 1406 | Syllogism combined with co... |
| syl212anc 1407 | Syllogism combined with co... |
| syl221anc 1408 | Syllogism combined with co... |
| syl113anc 1409 | Syllogism combined with co... |
| syl131anc 1410 | Syllogism combined with co... |
| syl311anc 1411 | Syllogism combined with co... |
| syl33anc 1412 | Syllogism combined with co... |
| syl222anc 1413 | Syllogism combined with co... |
| syl123anc 1414 | Syllogism combined with co... |
| syl132anc 1415 | Syllogism combined with co... |
| syl213anc 1416 | Syllogism combined with co... |
| syl231anc 1417 | Syllogism combined with co... |
| syl312anc 1418 | Syllogism combined with co... |
| syl321anc 1419 | Syllogism combined with co... |
| syl133anc 1420 | Syllogism combined with co... |
| syl313anc 1421 | Syllogism combined with co... |
| syl331anc 1422 | Syllogism combined with co... |
| syl223anc 1423 | Syllogism combined with co... |
| syl232anc 1424 | Syllogism combined with co... |
| syl322anc 1425 | Syllogism combined with co... |
| syl233anc 1426 | Syllogism combined with co... |
| syl323anc 1427 | Syllogism combined with co... |
| syl332anc 1428 | Syllogism combined with co... |
| syl333anc 1429 | A syllogism inference comb... |
| syl3an1b 1430 | A syllogism inference. (C... |
| syl3an2b 1431 | A syllogism inference. (C... |
| syl3an3b 1432 | A syllogism inference. (C... |
| syl3an1br 1433 | A syllogism inference. (C... |
| syl3an2br 1434 | A syllogism inference. (C... |
| syl3an3br 1435 | A syllogism inference. (C... |
| syld3an3 1436 | A syllogism inference. (C... |
| syld3an1 1437 | A syllogism inference. (C... |
| syld3an2 1438 | A syllogism inference. (C... |
| syl3anl1 1439 | A syllogism inference. (C... |
| syl3anl2 1440 | A syllogism inference. (C... |
| syl3anl3 1441 | A syllogism inference. (C... |
| syl3anl 1442 | A triple syllogism inferen... |
| syl3anr1 1443 | A syllogism inference. (C... |
| syl3anr2 1444 | A syllogism inference. (C... |
| syl3anr3 1445 | A syllogism inference. (C... |
| 3anidm12 1446 | Inference from idempotent ... |
| 3anidm13 1447 | Inference from idempotent ... |
| 3anidm23 1448 | Inference from idempotent ... |
| syl2an3an 1449 | ~ syl3an with antecedents ... |
| syl2an23an 1450 | Deduction related to ~ syl... |
| 3ori 1451 | Infer implication from tri... |
| 3jao 1452 | Disjunction of three antec... |
| 3jaob 1453 | Disjunction of three antec... |
| 3jaoi 1454 | Disjunction of three antec... |
| 3jaoiOLD 1455 | Obsolete version of ~ 3jao... |
| 3jaod 1456 | Disjunction of three antec... |
| 3jaoian 1457 | Disjunction of three antec... |
| 3jaodan 1458 | Disjunction of three antec... |
| mpjao3dan 1459 | Eliminate a three-way disj... |
| 3jaao 1460 | Inference conjoining and d... |
| 3jaaoOLD 1461 | Obsolete version of ~ 3jaa... |
| syl3an9b 1462 | Nested syllogism inference... |
| 3orbi123d 1463 | Deduction joining 3 equiva... |
| 3anbi123d 1464 | Deduction joining 3 equiva... |
| 3anbi12d 1465 | Deduction conjoining and a... |
| 3anbi13d 1466 | Deduction conjoining and a... |
| 3anbi23d 1467 | Deduction conjoining and a... |
| 3anbi1d 1468 | Deduction adding conjuncts... |
| 3anbi2d 1469 | Deduction adding conjuncts... |
| 3anbi3d 1470 | Deduction adding conjuncts... |
| 3anim123d 1471 | Deduction joining 3 implic... |
| 3orim123d 1472 | Deduction joining 3 implic... |
| 3orim123da 1473 | Disjoin antecedents and co... |
| an6 1474 | Rearrangement of 6 conjunc... |
| 3an6 1475 | Analogue of ~ an4 for trip... |
| 3or6 1476 | Analogue of ~ or4 for trip... |
| mp3an1 1477 | An inference based on modu... |
| mp3an2 1478 | An inference based on modu... |
| mp3an3 1479 | An inference based on modu... |
| mp3an12 1480 | An inference based on modu... |
| mp3an13 1481 | An inference based on modu... |
| mp3an23 1482 | An inference based on modu... |
| mp3an1i 1483 | An inference based on modu... |
| mp3anl1 1484 | An inference based on modu... |
| mp3anl2 1485 | An inference based on modu... |
| mp3anl3 1486 | An inference based on modu... |
| mp3anr1 1487 | An inference based on modu... |
| mp3anr2 1488 | An inference based on modu... |
| mp3anr3 1489 | An inference based on modu... |
| mp3an 1490 | An inference based on modu... |
| mpd3an3 1491 | An inference based on modu... |
| mpd3an23 1492 | An inference based on modu... |
| mp3and 1493 | A deduction based on modus... |
| mp3an12i 1494 | ~ mp3an with antecedents i... |
| mp3an2i 1495 | ~ mp3an with antecedents i... |
| mp3an3an 1496 | ~ mp3an with antecedents i... |
| mp3an2ani 1497 | An elimination deduction. ... |
| biimp3a 1498 | Infer implication from a l... |
| biimp3ar 1499 | Infer implication from a l... |
| 3anandis 1500 | Inference that undistribut... |
| 3anandirs 1501 | Inference that undistribut... |
| ecase13d 1502 | Deduction for elimination ... |
| ecase23d 1503 | Deduction for elimination ... |
| ecase33d 1504 | Deduction for elimination ... |
| 3ecase 1505 | Inference for elimination ... |
| 3bior1fd 1506 | A disjunction is equivalen... |
| 3bior1fand 1507 | A disjunction is equivalen... |
| 3bior2fd 1508 | A wff is equivalent to its... |
| 3biant1d 1509 | A conjunction is equivalen... |
| intn3an1d 1510 | Introduction of a triple c... |
| intn3an2d 1511 | Introduction of a triple c... |
| intn3an3d 1512 | Introduction of a triple c... |
| an3andi 1513 | Distribution of conjunctio... |
| an33rean 1514 | Rearrange a 9-fold conjunc... |
| 3orel2 1515 | Partial elimination of a t... |
| 3orel2OLD 1516 | Obsolete version of ~ 3ore... |
| 3orel3 1517 | Partial elimination of a t... |
| 3orel13 1518 | Elimination of two disjunc... |
| 3pm3.2ni 1519 | Triple negated disjunction... |
| an42ds 1520 | Inference exchanging the l... |
| nanan 1523 | Conjunction in terms of al... |
| dfnan2 1524 | Alternative denial in term... |
| nanor 1525 | Alternative denial in term... |
| nancom 1526 | Alternative denial is comm... |
| nannan 1527 | Nested alternative denials... |
| nanim 1528 | Implication in terms of al... |
| nannot 1529 | Negation in terms of alter... |
| nanbi 1530 | Biconditional in terms of ... |
| nanbi1 1531 | Introduce a right anti-con... |
| nanbi2 1532 | Introduce a left anti-conj... |
| nanbi12 1533 | Join two logical equivalen... |
| nanbi1i 1534 | Introduce a right anti-con... |
| nanbi2i 1535 | Introduce a left anti-conj... |
| nanbi12i 1536 | Join two logical equivalen... |
| nanbi1d 1537 | Introduce a right anti-con... |
| nanbi2d 1538 | Introduce a left anti-conj... |
| nanbi12d 1539 | Join two logical equivalen... |
| nanass 1540 | A characterization of when... |
| xnor 1543 | Two ways to write XNOR (ex... |
| xorcom 1544 | The connector ` \/_ ` is c... |
| xorass 1545 | The connector ` \/_ ` is a... |
| excxor 1546 | This tautology shows that ... |
| xor2 1547 | Two ways to express "exclu... |
| xoror 1548 | Exclusive disjunction impl... |
| xornan 1549 | Exclusive disjunction impl... |
| xornan2 1550 | XOR implies NAND (written ... |
| xorneg2 1551 | The connector ` \/_ ` is n... |
| xorneg1 1552 | The connector ` \/_ ` is n... |
| xorneg 1553 | The connector ` \/_ ` is u... |
| xorbi12i 1554 | Equality property for excl... |
| xorbi12d 1555 | Equality property for excl... |
| anxordi 1556 | Conjunction distributes ov... |
| xorexmid 1557 | Exclusive-or variant of th... |
| norcom 1560 | The connector ` -\/ ` is c... |
| nornot 1561 | ` -. ` is expressible via ... |
| noran 1562 | ` /\ ` is expressible via ... |
| noror 1563 | ` \/ ` is expressible via ... |
| norasslem1 1564 | This lemma shows the equiv... |
| norasslem2 1565 | This lemma specializes ~ b... |
| norasslem3 1566 | This lemma specializes ~ b... |
| norass 1567 | A characterization of when... |
| trujust 1572 | Soundness justification th... |
| tru 1574 | The truth value ` T. ` is ... |
| dftru2 1575 | An alternate definition of... |
| trut 1576 | A proposition is equivalen... |
| mptru 1577 | Eliminate ` T. ` as an ant... |
| tbtru 1578 | A proposition is equivalen... |
| bitru 1579 | A theorem is equivalent to... |
| trud 1580 | Anything implies ` T. ` . ... |
| truan 1581 | True can be removed from a... |
| fal 1584 | The truth value ` F. ` is ... |
| nbfal 1585 | The negation of a proposit... |
| bifal 1586 | A contradiction is equival... |
| falim 1587 | The truth value ` F. ` imp... |
| falimd 1588 | The truth value ` F. ` imp... |
| dfnot 1589 | Given falsum ` F. ` , we c... |
| inegd 1590 | Negation introduction rule... |
| efald 1591 | Deduction based on reducti... |
| pm2.21fal 1592 | If a wff and its negation ... |
| truimtru 1593 | A ` -> ` identity. (Contr... |
| truimfal 1594 | A ` -> ` identity. (Contr... |
| falimtru 1595 | A ` -> ` identity. (Contr... |
| falimfal 1596 | A ` -> ` identity. (Contr... |
| nottru 1597 | A ` -. ` identity. (Contr... |
| notfal 1598 | A ` -. ` identity. (Contr... |
| trubitru 1599 | A ` <-> ` identity. (Cont... |
| falbitru 1600 | A ` <-> ` identity. (Cont... |
| trubifal 1601 | A ` <-> ` identity. (Cont... |
| falbifal 1602 | A ` <-> ` identity. (Cont... |
| truantru 1603 | A ` /\ ` identity. (Contr... |
| truanfal 1604 | A ` /\ ` identity. (Contr... |
| falantru 1605 | A ` /\ ` identity. (Contr... |
| falanfal 1606 | A ` /\ ` identity. (Contr... |
| truortru 1607 | A ` \/ ` identity. (Contr... |
| truorfal 1608 | A ` \/ ` identity. (Contr... |
| falortru 1609 | A ` \/ ` identity. (Contr... |
| falorfal 1610 | A ` \/ ` identity. (Contr... |
| trunantru 1611 | A ` -/\ ` identity. (Cont... |
| trunanfal 1612 | A ` -/\ ` identity. (Cont... |
| falnantru 1613 | A ` -/\ ` identity. (Cont... |
| falnanfal 1614 | A ` -/\ ` identity. (Cont... |
| truxortru 1615 | A ` \/_ ` identity. (Cont... |
| truxorfal 1616 | A ` \/_ ` identity. (Cont... |
| falxortru 1617 | A ` \/_ ` identity. (Cont... |
| falxorfal 1618 | A ` \/_ ` identity. (Cont... |
| trunortru 1619 | A ` -\/ ` identity. (Cont... |
| trunorfal 1620 | A ` -\/ ` identity. (Cont... |
| falnortru 1621 | A ` -\/ ` identity. (Cont... |
| falnorfal 1622 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1625 | Equality theorem for the a... |
| hadbi123i 1626 | Equality theorem for the a... |
| hadass 1627 | Associative law for the ad... |
| hadbi 1628 | The adder sum is the same ... |
| hadcoma 1629 | Commutative law for the ad... |
| hadcomb 1630 | Commutative law for the ad... |
| hadrot 1631 | Rotation law for the adder... |
| hadnot 1632 | The adder sum distributes ... |
| had1 1633 | If the first input is true... |
| had0 1634 | If the first input is fals... |
| had1OLD 1635 | Obsolete version of ~ had1... |
| had0OLD 1636 | Obsolete version of ~ had0... |
| hadifp 1637 | The value of the adder sum... |
| hadifpOLD 1638 | Obsolete version of ~ hadi... |
| cador 1641 | The adder carry in disjunc... |
| cadan 1642 | The adder carry in conjunc... |
| cadbi123d 1643 | Equality theorem for the a... |
| cadbi123i 1644 | Equality theorem for the a... |
| cadcoma 1645 | Commutative law for the ad... |
| cadcomb 1646 | Commutative law for the ad... |
| cadrot 1647 | Rotation law for the adder... |
| cadnot 1648 | The adder carry distribute... |
| cad11 1649 | If (at least) two inputs a... |
| cad1 1650 | If one input is true, then... |
| cad0 1651 | If one input is false, the... |
| cadifp 1652 | The value of the carry is,... |
| cadtru 1653 | The adder carry is true as... |
| minimp 1654 | A single axiom for minimal... |
| minimp-syllsimp 1655 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1656 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1657 | Derivation of a commuted f... |
| minimp-ax2 1658 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1659 | Derivation of ~ pm2.43 (al... |
| impsingle 1660 | The shortest single axiom ... |
| impsingle-step4 1661 | Derivation of impsingle-st... |
| impsingle-step8 1662 | Derivation of impsingle-st... |
| impsingle-ax1 1663 | Derivation of impsingle-ax... |
| impsingle-step15 1664 | Derivation of impsingle-st... |
| impsingle-step18 1665 | Derivation of impsingle-st... |
| impsingle-step19 1666 | Derivation of impsingle-st... |
| impsingle-step20 1667 | Derivation of impsingle-st... |
| impsingle-step21 1668 | Derivation of impsingle-st... |
| impsingle-step22 1669 | Derivation of impsingle-st... |
| impsingle-step25 1670 | Derivation of impsingle-st... |
| impsingle-imim1 1671 | Derivation of impsingle-im... |
| impsingle-peirce 1672 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1673 | Derivation of ~ ax-2 from ... |
| meredith 1674 | Carew Meredith's sole axio... |
| merlem1 1675 | Step 3 of Meredith's proof... |
| merlem2 1676 | Step 4 of Meredith's proof... |
| merlem3 1677 | Step 7 of Meredith's proof... |
| merlem4 1678 | Step 8 of Meredith's proof... |
| merlem5 1679 | Step 11 of Meredith's proo... |
| merlem6 1680 | Step 12 of Meredith's proo... |
| merlem7 1681 | Between steps 14 and 15 of... |
| merlem8 1682 | Step 15 of Meredith's proo... |
| merlem9 1683 | Step 18 of Meredith's proo... |
| merlem10 1684 | Step 19 of Meredith's proo... |
| merlem11 1685 | Step 20 of Meredith's proo... |
| merlem12 1686 | Step 28 of Meredith's proo... |
| merlem13 1687 | Step 35 of Meredith's proo... |
| luk-1 1688 | 1 of 3 axioms for proposit... |
| luk-2 1689 | 2 of 3 axioms for proposit... |
| luk-3 1690 | 3 of 3 axioms for proposit... |
| luklem1 1691 | Used to rederive standard ... |
| luklem2 1692 | Used to rederive standard ... |
| luklem3 1693 | Used to rederive standard ... |
| luklem4 1694 | Used to rederive standard ... |
| luklem5 1695 | Used to rederive standard ... |
| luklem6 1696 | Used to rederive standard ... |
| luklem7 1697 | Used to rederive standard ... |
| luklem8 1698 | Used to rederive standard ... |
| ax1 1699 | Standard propositional axi... |
| ax2 1700 | Standard propositional axi... |
| ax3 1701 | Standard propositional axi... |
| nic-dfim 1702 | This theorem "defines" imp... |
| nic-dfneg 1703 | This theorem "defines" neg... |
| nic-mp 1704 | Derive Nicod's rule of mod... |
| nic-mpALT 1705 | A direct proof of ~ nic-mp... |
| nic-ax 1706 | Nicod's axiom derived from... |
| nic-axALT 1707 | A direct proof of ~ nic-ax... |
| nic-imp 1708 | Inference for ~ nic-mp usi... |
| nic-idlem1 1709 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1710 | Lemma for ~ nic-id . Infe... |
| nic-id 1711 | Theorem ~ id expressed wit... |
| nic-swap 1712 | The connector ` -/\ ` is s... |
| nic-isw1 1713 | Inference version of ~ nic... |
| nic-isw2 1714 | Inference for swapping nes... |
| nic-iimp1 1715 | Inference version of ~ nic... |
| nic-iimp2 1716 | Inference version of ~ nic... |
| nic-idel 1717 | Inference to remove the tr... |
| nic-ich 1718 | Chained inference. (Contr... |
| nic-idbl 1719 | Double the terms. Since d... |
| nic-bijust 1720 | Biconditional justificatio... |
| nic-bi1 1721 | Inference to extract one s... |
| nic-bi2 1722 | Inference to extract the o... |
| nic-stdmp 1723 | Derive the standard modus ... |
| nic-luk1 1724 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1725 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1726 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1727 | This alternative axiom for... |
| lukshefth1 1728 | Lemma for ~ renicax . (Co... |
| lukshefth2 1729 | Lemma for ~ renicax . (Co... |
| renicax 1730 | A rederivation of ~ nic-ax... |
| tbw-bijust 1731 | Justification for ~ tbw-ne... |
| tbw-negdf 1732 | The definition of negation... |
| tbw-ax1 1733 | The first of four axioms i... |
| tbw-ax2 1734 | The second of four axioms ... |
| tbw-ax3 1735 | The third of four axioms i... |
| tbw-ax4 1736 | The fourth of four axioms ... |
| tbwsyl 1737 | Used to rederive the Lukas... |
| tbwlem1 1738 | Used to rederive the Lukas... |
| tbwlem2 1739 | Used to rederive the Lukas... |
| tbwlem3 1740 | Used to rederive the Lukas... |
| tbwlem4 1741 | Used to rederive the Lukas... |
| tbwlem5 1742 | Used to rederive the Lukas... |
| re1luk1 1743 | ~ luk-1 derived from the T... |
| re1luk2 1744 | ~ luk-2 derived from the T... |
| re1luk3 1745 | ~ luk-3 derived from the T... |
| merco1 1746 | A single axiom for proposi... |
| merco1lem1 1747 | Used to rederive the Tarsk... |
| retbwax4 1748 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1749 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1750 | Used to rederive the Tarsk... |
| merco1lem3 1751 | Used to rederive the Tarsk... |
| merco1lem4 1752 | Used to rederive the Tarsk... |
| merco1lem5 1753 | Used to rederive the Tarsk... |
| merco1lem6 1754 | Used to rederive the Tarsk... |
| merco1lem7 1755 | Used to rederive the Tarsk... |
| retbwax3 1756 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1757 | Used to rederive the Tarsk... |
| merco1lem9 1758 | Used to rederive the Tarsk... |
| merco1lem10 1759 | Used to rederive the Tarsk... |
| merco1lem11 1760 | Used to rederive the Tarsk... |
| merco1lem12 1761 | Used to rederive the Tarsk... |
| merco1lem13 1762 | Used to rederive the Tarsk... |
| merco1lem14 1763 | Used to rederive the Tarsk... |
| merco1lem15 1764 | Used to rederive the Tarsk... |
| merco1lem16 1765 | Used to rederive the Tarsk... |
| merco1lem17 1766 | Used to rederive the Tarsk... |
| merco1lem18 1767 | Used to rederive the Tarsk... |
| retbwax1 1768 | ~ tbw-ax1 rederived from ~... |
| merco2 1769 | A single axiom for proposi... |
| mercolem1 1770 | Used to rederive the Tarsk... |
| mercolem2 1771 | Used to rederive the Tarsk... |
| mercolem3 1772 | Used to rederive the Tarsk... |
| mercolem4 1773 | Used to rederive the Tarsk... |
| mercolem5 1774 | Used to rederive the Tarsk... |
| mercolem6 1775 | Used to rederive the Tarsk... |
| mercolem7 1776 | Used to rederive the Tarsk... |
| mercolem8 1777 | Used to rederive the Tarsk... |
| re1tbw1 1778 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1779 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1780 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1781 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1782 | Justification for ~ rb-imd... |
| rb-imdf 1783 | The definition of implicat... |
| anmp 1784 | Modus ponens for ` { \/ , ... |
| rb-ax1 1785 | The first of four axioms i... |
| rb-ax2 1786 | The second of four axioms ... |
| rb-ax3 1787 | The third of four axioms i... |
| rb-ax4 1788 | The fourth of four axioms ... |
| rbsyl 1789 | Used to rederive the Lukas... |
| rblem1 1790 | Used to rederive the Lukas... |
| rblem2 1791 | Used to rederive the Lukas... |
| rblem3 1792 | Used to rederive the Lukas... |
| rblem4 1793 | Used to rederive the Lukas... |
| rblem5 1794 | Used to rederive the Lukas... |
| rblem6 1795 | Used to rederive the Lukas... |
| rblem7 1796 | Used to rederive the Lukas... |
| re1axmp 1797 | ~ ax-mp derived from Russe... |
| re2luk1 1798 | ~ luk-1 derived from Russe... |
| re2luk2 1799 | ~ luk-2 derived from Russe... |
| re2luk3 1800 | ~ luk-3 derived from Russe... |
| mptnan 1801 | Modus ponendo tollens 1, o... |
| mptxor 1802 | Modus ponendo tollens 2, o... |
| mtpor 1803 | Modus tollendo ponens (inc... |
| mtpxor 1804 | Modus tollendo ponens (ori... |
| stoic1a 1805 | Stoic logic Thema 1 (part ... |
| stoic1b 1806 | Stoic logic Thema 1 (part ... |
| stoic2a 1807 | Stoic logic Thema 2 versio... |
| stoic2b 1808 | Stoic logic Thema 2 versio... |
| stoic3 1809 | Stoic logic Thema 3. Stat... |
| stoic4a 1810 | Stoic logic Thema 4 versio... |
| stoic4b 1811 | Stoic logic Thema 4 versio... |
| alnex 1814 | Universal quantification o... |
| eximal 1815 | An equivalence between an ... |
| nf2 1818 | Alternate definition of no... |
| nf3 1819 | Alternate definition of no... |
| nf4 1820 | Alternate definition of no... |
| nfi 1821 | Deduce that ` x ` is not f... |
| nfri 1822 | Consequence of the definit... |
| nfd 1823 | Deduce that ` x ` is not f... |
| nfrd 1824 | Consequence of the definit... |
| nftht 1825 | Closed form of ~ nfth . (... |
| nfntht 1826 | Closed form of ~ nfnth . ... |
| nfntht2 1827 | Closed form of ~ nfnth . ... |
| gen2 1829 | Generalization applied twi... |
| mpg 1830 | Modus ponens combined with... |
| mpgbi 1831 | Modus ponens on biconditio... |
| mpgbir 1832 | Modus ponens on biconditio... |
| nex 1833 | Generalization rule for ne... |
| nfth 1834 | No variable is (effectivel... |
| nfnth 1835 | No variable is (effectivel... |
| hbth 1836 | No variable is (effectivel... |
| nftru 1837 | The true constant has no f... |
| nffal 1838 | The false constant has no ... |
| sptruw 1839 | Version of ~ sp when ` ph ... |
| altru 1840 | For all sets, ` T. ` is tr... |
| alfal 1841 | For all sets, ` -. F. ` is... |
| alim 1843 | Restatement of Axiom ~ ax-... |
| alimi 1844 | Inference quantifying both... |
| 2alimi 1845 | Inference doubly quantifyi... |
| ala1 1846 | Add an antecedent in a uni... |
| al2im 1847 | Closed form of ~ al2imi . ... |
| al2imi 1848 | Inference quantifying ante... |
| alanimi 1849 | Variant of ~ al2imi with c... |
| alimdh 1850 | Deduction form of Theorem ... |
| albi 1851 | Theorem 19.15 of [Margaris... |
| albii 1852 | Inference adding universal... |
| 2albii 1853 | Inference adding two unive... |
| 3albii 1854 | Inference adding three uni... |
| sylgt 1855 | Closed form of ~ sylg . (... |
| sylg 1856 | A syllogism combined with ... |
| alrimih 1857 | Inference form of Theorem ... |
| hbxfrbi 1858 | A utility lemma to transfe... |
| alex 1859 | Universal quantifier in te... |
| exnal 1860 | Existential quantification... |
| 2nalexn 1861 | Part of theorem *11.5 in [... |
| 2exnaln 1862 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1863 | Theorem *11.25 in [Whitehe... |
| alimex 1864 | An equivalence between an ... |
| aleximi 1865 | A variant of ~ al2imi : in... |
| alexbii 1866 | Biconditional form of ~ al... |
| exim 1867 | Theorem 19.22 of [Margaris... |
| eximi 1868 | Inference adding existenti... |
| 2eximi 1869 | Inference adding two exist... |
| eximii 1870 | Inference associated with ... |
| exa1 1871 | Add an antecedent in an ex... |
| 19.38 1872 | Theorem 19.38 of [Margaris... |
| 19.38a 1873 | Under a nonfreeness hypoth... |
| 19.38b 1874 | Under a nonfreeness hypoth... |
| imnang 1875 | Quantified implication in ... |
| alinexa 1876 | A transformation of quanti... |
| exnalimn 1877 | Existential quantification... |
| alexn 1878 | A relationship between two... |
| 2exnexn 1879 | Theorem *11.51 in [Whitehe... |
| exbi 1880 | Theorem 19.18 of [Margaris... |
| exbii 1881 | Inference adding existenti... |
| 2exbii 1882 | Inference adding two exist... |
| 3exbii 1883 | Inference adding three exi... |
| nfbiit 1884 | Equivalence theorem for th... |
| nfbii 1885 | Equality theorem for the n... |
| nfxfr 1886 | A utility lemma to transfe... |
| nfxfrd 1887 | A utility lemma to transfe... |
| nfnbi 1888 | A variable is nonfree in a... |
| nfnt 1889 | If a variable is nonfree i... |
| nfn 1890 | Inference associated with ... |
| nfnd 1891 | Deduction associated with ... |
| exanali 1892 | A transformation of quanti... |
| 2exanali 1893 | Theorem *11.521 in [Whiteh... |
| exancom 1894 | Commutation of conjunction... |
| exan 1895 | Place a conjunct in the sc... |
| alrimdh 1896 | Deduction form of Theorem ... |
| eximdh 1897 | Deduction from Theorem 19.... |
| nexdh 1898 | Deduction for generalizati... |
| albidh 1899 | Formula-building rule for ... |
| exbidh 1900 | Formula-building rule for ... |
| exsimpl 1901 | Simplification of an exist... |
| exsimpr 1902 | Simplification of an exist... |
| 19.26 1903 | Theorem 19.26 of [Margaris... |
| 19.26-2 1904 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1905 | Theorem ~ 19.26 with tripl... |
| 19.29 1906 | Theorem 19.29 of [Margaris... |
| 19.29r 1907 | Variation of ~ 19.29 . (C... |
| 19.29r2 1908 | Variation of ~ 19.29r with... |
| 19.29x 1909 | Variation of ~ 19.29 with ... |
| 19.35 1910 | Theorem 19.35 of [Margaris... |
| 19.35i 1911 | Inference associated with ... |
| 19.35ri 1912 | Inference associated with ... |
| 19.25 1913 | Theorem 19.25 of [Margaris... |
| 19.30 1914 | Theorem 19.30 of [Margaris... |
| 19.43 1915 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1916 | Obsolete proof of ~ 19.43 ... |
| 19.33 1917 | Theorem 19.33 of [Margaris... |
| 19.33b 1918 | The antecedent provides a ... |
| 19.40 1919 | Theorem 19.40 of [Margaris... |
| 19.40-2 1920 | Theorem *11.42 in [Whitehe... |
| 19.40b 1921 | The antecedent provides a ... |
| albiim 1922 | Split a biconditional and ... |
| 2albiim 1923 | Split a biconditional and ... |
| exintrbi 1924 | Add/remove a conjunct in t... |
| exintr 1925 | Introduce a conjunct in th... |
| alsyl 1926 | Universally quantified and... |
| nfimd 1927 | If in a context ` x ` is n... |
| nfimt 1928 | Closed form of ~ nfim and ... |
| nfim 1929 | If ` x ` is not free in ` ... |
| nfand 1930 | If in a context ` x ` is n... |
| nf3and 1931 | Deduction form of bound-va... |
| nfan 1932 | If ` x ` is not free in ` ... |
| nfnan 1933 | If ` x ` is not free in ` ... |
| nf3an 1934 | If ` x ` is not free in ` ... |
| nfbid 1935 | If in a context ` x ` is n... |
| nfbi 1936 | If ` x ` is not free in ` ... |
| nfor 1937 | If ` x ` is not free in ` ... |
| nf3or 1938 | If ` x ` is not free in ` ... |
| empty 1939 | Two characterizations of t... |
| emptyex 1940 | On the empty domain, any e... |
| emptyal 1941 | On the empty domain, any u... |
| emptynf 1942 | On the empty domain, any v... |
| ax5d 1944 | Version of ~ ax-5 with ant... |
| ax5e 1945 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1946 | If a formula holds for som... |
| nfv 1947 | If ` x ` is not present in... |
| nfvd 1948 | ~ nfv with antecedent. Us... |
| alimdv 1949 | Deduction form of Theorem ... |
| eximdv 1950 | Deduction form of Theorem ... |
| 2alimdv 1951 | Deduction form of Theorem ... |
| 2eximdv 1952 | Deduction form of Theorem ... |
| albidv 1953 | Formula-building rule for ... |
| exbidv 1954 | Formula-building rule for ... |
| nfbidv 1955 | An equality theorem for no... |
| 2albidv 1956 | Formula-building rule for ... |
| 2exbidv 1957 | Formula-building rule for ... |
| 3exbidv 1958 | Formula-building rule for ... |
| 4exbidv 1959 | Formula-building rule for ... |
| alrimiv 1960 | Inference form of Theorem ... |
| alrimivv 1961 | Inference form of Theorem ... |
| alrimdv 1962 | Deduction form of Theorem ... |
| exlimiv 1963 | Inference form of Theorem ... |
| exlimiiv 1964 | Inference (Rule C) associa... |
| exlimivv 1965 | Inference form of Theorem ... |
| exlimdv 1966 | Deduction form of Theorem ... |
| exlimdvv 1967 | Deduction form of Theorem ... |
| exlimddv 1968 | Existential elimination ru... |
| nexdv 1969 | Deduction for generalizati... |
| 2ax5 1970 | Quantification of two vari... |
| stdpc5v 1971 | Version of ~ stdpc5 with a... |
| 19.21v 1972 | Version of ~ 19.21 with a ... |
| 19.32v 1973 | Version of ~ 19.32 with a ... |
| 19.31v 1974 | Version of ~ 19.31 with a ... |
| 19.23v 1975 | Version of ~ 19.23 with a ... |
| 19.23vv 1976 | Theorem ~ 19.23v extended ... |
| pm11.53v 1977 | Version of ~ pm11.53 with ... |
| 19.36imv 1978 | One direction of ~ 19.36v ... |
| 19.36iv 1979 | Inference associated with ... |
| 19.37imv 1980 | One direction of ~ 19.37v ... |
| 19.37iv 1981 | Inference associated with ... |
| 19.41v 1982 | Version of ~ 19.41 with a ... |
| 19.41vv 1983 | Version of ~ 19.41 with tw... |
| 19.41vvv 1984 | Version of ~ 19.41 with th... |
| 19.41vvvv 1985 | Version of ~ 19.41 with fo... |
| 19.42v 1986 | Version of ~ 19.42 with a ... |
| exdistr 1987 | Distribution of existentia... |
| exdistrv 1988 | Distribute a pair of exist... |
| 4exdistrv 1989 | Distribute two pairs of ex... |
| 19.42vv 1990 | Version of ~ 19.42 with tw... |
| exdistr2 1991 | Distribution of existentia... |
| 19.42vvv 1992 | Version of ~ 19.42 with th... |
| 3exdistr 1993 | Distribution of existentia... |
| 4exdistr 1994 | Distribution of existentia... |
| weq 1995 | Extend wff definition to i... |
| speimfw 1996 | Specialization, with addit... |
| speimfwALT 1997 | Alternate proof of ~ speim... |
| spimfw 1998 | Specialization, with addit... |
| ax12i 1999 | Inference that has ~ ax-12... |
| ax6v 2001 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 2002 | At least one individual ex... |
| spimw 2003 | Specialization. Lemma 8 o... |
| spimew 2004 | Existential introduction, ... |
| speiv 2005 | Inference from existential... |
| speivw 2006 | Version of ~ spei with a d... |
| exgen 2007 | Rule of existential genera... |
| extru 2008 | There exists a variable su... |
| 19.2 2009 | Theorem 19.2 of [Margaris]... |
| 19.2d 2010 | Deduction associated with ... |
| 19.8w 2011 | Weak version of ~ 19.8a an... |
| spnfw 2012 | Weak version of ~ sp . Us... |
| spfalw 2013 | Version of ~ sp when ` ph ... |
| spvw 2014 | Version of ~ sp when ` x `... |
| 19.3v 2015 | Version of ~ 19.3 with a d... |
| 19.8v 2016 | Version of ~ 19.8a with a ... |
| 19.9v 2017 | Version of ~ 19.9 with a d... |
| spimevw 2018 | Existential introduction, ... |
| spimvw 2019 | A weak form of specializat... |
| spsv 2020 | Generalization of antecede... |
| spvv 2021 | Specialization, using impl... |
| chvarvv 2022 | Implicit substitution of `... |
| 19.39 2023 | Theorem 19.39 of [Margaris... |
| 19.24 2024 | Theorem 19.24 of [Margaris... |
| 19.34 2025 | Theorem 19.34 of [Margaris... |
| 19.36v 2026 | Version of ~ 19.36 with a ... |
| 19.12vvv 2027 | Version of ~ 19.12vv with ... |
| 19.27v 2028 | Version of ~ 19.27 with a ... |
| 19.28v 2029 | Version of ~ 19.28 with a ... |
| 19.37v 2030 | Version of ~ 19.37 with a ... |
| 19.44v 2031 | Version of ~ 19.44 with a ... |
| 19.45v 2032 | Version of ~ 19.45 with a ... |
| equs4v 2033 | Version of ~ equs4 with a ... |
| alequexv 2034 | Version of ~ equs4v with i... |
| exsbim 2035 | One direction of the equiv... |
| equsv 2036 | If a formula does not cont... |
| equsalvw 2037 | Version of ~ equsalv with ... |
| equsexvw 2038 | Version of ~ equsexv with ... |
| cbvaliw 2039 | Change bound variable. Us... |
| cbvalivw 2040 | Change bound variable. Us... |
| ax7v 2042 | Weakened version of ~ ax-7... |
| ax7v1 2043 | First of two weakened vers... |
| ax7v2 2044 | Second of two weakened ver... |
| equid 2045 | Identity law for equality.... |
| nfequid 2046 | Bound-variable hypothesis ... |
| equcomiv 2047 | Weaker form of ~ equcomi w... |
| ax6evr 2048 | A commuted form of ~ ax6ev... |
| ax7 2049 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2050 | Commutative law for equali... |
| equcom 2051 | Commutative law for equali... |
| equcomd 2052 | Deduction form of ~ equcom... |
| equcoms 2053 | An inference commuting equ... |
| equtr 2054 | A transitive law for equal... |
| equtrr 2055 | A transitive law for equal... |
| equeuclr 2056 | Commuted version of ~ eque... |
| equeucl 2057 | Equality is a left-Euclide... |
| equequ1 2058 | An equivalence law for equ... |
| equequ2 2059 | An equivalence law for equ... |
| equtr2 2060 | Equality is a left-Euclide... |
| stdpc6 2061 | One of the two equality ax... |
| equvinv 2062 | A variable introduction la... |
| equvinva 2063 | A modified version of the ... |
| equvelv 2064 | A biconditional form of ~ ... |
| ax13b 2065 | An equivalence between two... |
| spfw 2066 | Weak version of ~ sp . Us... |
| spw 2067 | Weak version of the specia... |
| cbvalw 2068 | Change bound variable. Us... |
| cbvalvw 2069 | Change bound variable. Us... |
| cbvexvw 2070 | Change bound variable. Us... |
| cbvaldvaw 2071 | Rule used to change the bo... |
| cbvexdvaw 2072 | Rule used to change the bo... |
| cbval2vw 2073 | Rule used to change bound ... |
| cbvex2vw 2074 | Rule used to change bound ... |
| cbvex4vw 2075 | Rule used to change bound ... |
| alcomimw 2076 | Weak version of ~ ax-11 . ... |
| excomimw 2077 | Weak version of ~ excomim ... |
| alcomw 2078 | Weak version of ~ alcom an... |
| excomw 2079 | Weak version of ~ excom an... |
| hbn1fw 2080 | Weak version of ~ ax-10 fr... |
| hbn1w 2081 | Weak version of ~ hbn1 . ... |
| hba1w 2082 | Weak version of ~ hba1 . ... |
| hbe1w 2083 | Weak version of ~ hbe1 . ... |
| hbalw 2084 | Weak version of ~ hbal . ... |
| 19.8aw 2085 | If a formula is true, then... |
| exexw 2086 | Existential quantification... |
| spaev 2087 | A special instance of ~ sp... |
| cbvaev 2088 | Change bound variable in a... |
| aevlem0 2089 | Lemma for ~ aevlem . Inst... |
| aevlem 2090 | Lemma for ~ aev and ~ axc1... |
| aeveq 2091 | The antecedent ` A. x x = ... |
| aev 2092 | A "distinctor elimination"... |
| aev2 2093 | A version of ~ aev with tw... |
| hbaev 2094 | All variables are effectiv... |
| naev 2095 | If some set variables can ... |
| naev2 2096 | Generalization of ~ hbnaev... |
| hbnaev 2097 | Any variable is free in ` ... |
| sbjust 2098 | Justification theorem for ... |
| dfsb 2101 | Simplify definition ~ df-s... |
| sbtlem 2102 | In the case of ~ sbt , the... |
| sbt 2103 | A substitution into a theo... |
| sbtru 2104 | The result of substituting... |
| stdpc4 2105 | The specialization axiom o... |
| sbtALT 2106 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2107 | A double specialization us... |
| sbi1 2108 | Distribute substitution ov... |
| spsbim 2109 | Distribute substitution ov... |
| spsbbi 2110 | Biconditional property for... |
| sbimi 2111 | Distribute substitution ov... |
| sb2imi 2112 | Distribute substitution ov... |
| sbbii 2113 | Infer substitution into bo... |
| 2sbbii 2114 | Infer double substitution ... |
| sbimdv 2115 | Deduction substituting bot... |
| sbbidv 2116 | Deduction substituting bot... |
| sban 2117 | Conjunction inside and out... |
| sb3an 2118 | Threefold conjunction insi... |
| spsbe 2119 | Existential generalization... |
| sbequ 2120 | Equality property for subs... |
| sbequiOLD 2121 | Obsolete version of ~ sbeq... |
| sb6 2122 | Alternate definition of su... |
| 2sb6 2123 | Equivalence for double sub... |
| sb1v 2124 | One direction of ~ sb5 , p... |
| sbv 2125 | Substitution for a variabl... |
| sbcom4 2126 | Commutativity law for subs... |
| pm11.07 2127 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2128 | Substitution in an implica... |
| sbrimvwOLD 2129 | Obsolete version of ~ sbri... |
| sbbiiev 2130 | An equivalence of substitu... |
| sbievw 2131 | Conversion of implicit sub... |
| sbiedvw 2132 | Conversion of implicit sub... |
| 2sbievw 2133 | Conversion of double impli... |
| sbcom3vv 2134 | Substituting ` y ` for ` x... |
| sbievw2 2135 | ~ sbievw applied twice, av... |
| sbco2vv 2136 | A composition law for subs... |
| cbvsbv 2137 | Change the bound variable ... |
| sbco4lem 2138 | Lemma for ~ sbco4 . It re... |
| sbco4 2139 | Two ways of exchanging two... |
| equsb3 2140 | Substitution in an equalit... |
| equsb3r 2141 | Substitution applied to th... |
| equsb1v 2142 | Substitution applied to an... |
| nsb 2143 | Any substitution in an alw... |
| sbn1 2144 | One direction of ~ sbn , u... |
| wel 2146 | Extend wff definition to i... |
| ax8v 2148 | Weakened version of ~ ax-8... |
| ax8v1 2149 | First of two weakened vers... |
| ax8v2 2150 | Second of two weakened ver... |
| ax8 2151 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2152 | An identity law for the no... |
| elsb1 2153 | Substitution for the first... |
| cleljust 2154 | When the class variables i... |
| ax9v 2156 | Weakened version of ~ ax-9... |
| ax9v1 2157 | First of two weakened vers... |
| ax9v2 2158 | Second of two weakened ver... |
| ax9 2159 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2160 | An identity law for the no... |
| elequ2g 2161 | A form of ~ elequ2 with a ... |
| elsb2 2162 | Substitution for the secon... |
| elequ12 2163 | An identity law for the no... |
| ru0 2164 | The FOL statement used in ... |
| ax6dgen 2165 | Tarski's system uses the w... |
| ax10w 2166 | Weak version of ~ ax-10 fr... |
| ax11w 2167 | Weak version of ~ ax-11 fr... |
| ax11dgen 2168 | Degenerate instance of ~ a... |
| ax12wlem 2169 | Lemma for weak version of ... |
| ax12w 2170 | Weak version of ~ ax-12 fr... |
| ax12dgen 2171 | Degenerate instance of ~ a... |
| ax12wdemo 2172 | Example of an application ... |
| ax13w 2173 | Weak version (principal in... |
| ax13dgen1 2174 | Degenerate instance of ~ a... |
| ax13dgen2 2175 | Degenerate instance of ~ a... |
| ax13dgen3 2176 | Degenerate instance of ~ a... |
| ax13dgen4 2177 | Degenerate instance of ~ a... |
| hbn1 2179 | Alias for ~ ax-10 to be us... |
| hbe1 2180 | The setvar ` x ` is not fr... |
| hbe1a 2181 | Dual statement of ~ hbe1 .... |
| nf5-1 2182 | One direction of ~ nf5 can... |
| nf5i 2183 | Deduce that ` x ` is not f... |
| nf5dh 2184 | Deduce that ` x ` is not f... |
| nf5dv 2185 | Apply the definition of no... |
| nfnaew 2186 | All variables are effectiv... |
| nfe1 2187 | The setvar ` x ` is not fr... |
| nfa1 2188 | The setvar ` x ` is not fr... |
| nfna1 2189 | A convenience theorem part... |
| nfia1 2190 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2191 | The setvar ` x ` is not fr... |
| modal5 2192 | The analogue in our predic... |
| nfs1v 2193 | The setvar ` x ` is not fr... |
| alcoms 2195 | Swap quantifiers in an ant... |
| alcom 2196 | Theorem 19.5 of [Margaris]... |
| alrot3 2197 | Theorem *11.21 in [Whitehe... |
| alrot4 2198 | Rotate four universal quan... |
| excom 2199 | Theorem 19.11 of [Margaris... |
| excomim 2200 | One direction of Theorem 1... |
| excom13 2201 | Swap 1st and 3rd existenti... |
| exrot3 2202 | Rotate existential quantif... |
| exrot4 2203 | Rotate existential quantif... |
| hbal 2204 | If ` x ` is not free in ` ... |
| hbald 2205 | Deduction form of bound-va... |
| sbal 2206 | Move universal quantifier ... |
| sbalv 2207 | Quantify with new variable... |
| hbsbw 2208 | If ` z ` is not free in ` ... |
| sbcom2 2209 | Commutativity law for subs... |
| nfa2 2210 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2211 | Version of ~ nfex with the... |
| nfexa2 2212 | An inner universal quantif... |
| ax12v 2214 | This is essentially Axiom ... |
| ax12v2 2215 | It is possible to remove a... |
| ax12ev2 2216 | Version of ~ ax12v2 rewrit... |
| 19.8a 2217 | If a wff is true, it is tr... |
| 19.8ad 2218 | If a wff is true, it is tr... |
| sp 2219 | Specialization. A univers... |
| spi 2220 | Inference rule of universa... |
| sps 2221 | Generalization of antecede... |
| 2sp 2222 | A double specialization (s... |
| spsd 2223 | Deduction generalizing ant... |
| 19.2g 2224 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2225 | Inference form of ~ 19.21 ... |
| 19.21bbi 2226 | Inference removing two uni... |
| 19.23bi 2227 | Inference form of Theorem ... |
| nexr 2228 | Inference associated with ... |
| qexmid 2229 | Quantified excluded middle... |
| nf5r 2230 | Consequence of the definit... |
| nf5ri 2231 | Consequence of the definit... |
| nf5rd 2232 | Consequence of the definit... |
| spimedv 2233 | Deduction version of ~ spi... |
| spimefv 2234 | Version of ~ spime with a ... |
| nfim1 2235 | A closed form of ~ nfim . ... |
| nfan1 2236 | A closed form of ~ nfan . ... |
| 19.3t 2237 | Closed form of ~ 19.3 and ... |
| 19.3 2238 | A wff may be quantified wi... |
| 19.9d 2239 | A deduction version of one... |
| 19.9t 2240 | Closed form of ~ 19.9 and ... |
| 19.9 2241 | A wff may be existentially... |
| 19.21t 2242 | Closed form of Theorem 19.... |
| 19.21 2243 | Theorem 19.21 of [Margaris... |
| stdpc5 2244 | An axiom scheme of standar... |
| 19.21-2 2245 | Version of ~ 19.21 with tw... |
| 19.23t 2246 | Closed form of Theorem 19.... |
| 19.23 2247 | Theorem 19.23 of [Margaris... |
| alimd 2248 | Deduction form of Theorem ... |
| alrimi 2249 | Inference form of Theorem ... |
| alrimdd 2250 | Deduction form of Theorem ... |
| alrimd 2251 | Deduction form of Theorem ... |
| eximd 2252 | Deduction form of Theorem ... |
| exlimi 2253 | Inference associated with ... |
| exlimd 2254 | Deduction form of Theorem ... |
| exlimimdd 2255 | Existential elimination ru... |
| exlimdd 2256 | Existential elimination ru... |
| nexd 2257 | Deduction for generalizati... |
| albid 2258 | Formula-building rule for ... |
| exbid 2259 | Formula-building rule for ... |
| nfbidf 2260 | An equality theorem for ef... |
| 19.16 2261 | Theorem 19.16 of [Margaris... |
| 19.17 2262 | Theorem 19.17 of [Margaris... |
| 19.27 2263 | Theorem 19.27 of [Margaris... |
| 19.28 2264 | Theorem 19.28 of [Margaris... |
| 19.19 2265 | Theorem 19.19 of [Margaris... |
| 19.36 2266 | Theorem 19.36 of [Margaris... |
| 19.36i 2267 | Inference associated with ... |
| 19.37 2268 | Theorem 19.37 of [Margaris... |
| 19.32 2269 | Theorem 19.32 of [Margaris... |
| 19.31 2270 | Theorem 19.31 of [Margaris... |
| 19.41 2271 | Theorem 19.41 of [Margaris... |
| 19.42 2272 | Theorem 19.42 of [Margaris... |
| 19.44 2273 | Theorem 19.44 of [Margaris... |
| 19.45 2274 | Theorem 19.45 of [Margaris... |
| spimfv 2275 | Specialization, using impl... |
| chvarfv 2276 | Implicit substitution of `... |
| cbv3v2 2277 | Version of ~ cbv3 with two... |
| sbalex 2278 | Equivalence of two ways to... |
| sb4av 2279 | Version of ~ sb4a with a d... |
| sbimd 2280 | Deduction substituting bot... |
| sbbid 2281 | Deduction substituting bot... |
| 2sbbid 2282 | Deduction doubly substitut... |
| sbequ1 2283 | An equality theorem for su... |
| sbequ2 2284 | An equality theorem for su... |
| stdpc7 2285 | One of the two equality ax... |
| sbequ12 2286 | An equality theorem for su... |
| sbequ12r 2287 | An equality theorem for su... |
| sbelx 2288 | Elimination of substitutio... |
| sbequ12a 2289 | An equality theorem for su... |
| sbid 2290 | An identity theorem for su... |
| sbcov 2291 | A composition law for subs... |
| sb6a 2292 | Equivalence for substituti... |
| sbid2vw 2293 | Reverting substitution yie... |
| axc16g 2294 | Generalization of ~ axc16 ... |
| axc16 2295 | Proof of older axiom ~ ax-... |
| axc16gb 2296 | Biconditional strengthenin... |
| axc16nf 2297 | If ~ dtru is false, then t... |
| axc11v 2298 | Version of ~ axc11 with a ... |
| axc11rv 2299 | Version of ~ axc11r with a... |
| drsb2 2300 | Formula-building lemma for... |
| equsalv 2301 | An equivalence related to ... |
| equsexv 2302 | An equivalence related to ... |
| sbft 2303 | Substitution has no effect... |
| sbf 2304 | Substitution for a variabl... |
| sbf2 2305 | Substitution has no effect... |
| sbh 2306 | Substitution for a variabl... |
| hbs1 2307 | The setvar ` x ` is not fr... |
| nfs1f 2308 | If ` x ` is not free in ` ... |
| sb5 2309 | Alternate definition of su... |
| equs5av 2310 | A property related to subs... |
| 2sb5 2311 | Equivalence for double sub... |
| dfsb7 2312 | An alternate definition of... |
| sbn 2313 | Negation inside and outsid... |
| sbex 2314 | Move existential quantifie... |
| nf5 2315 | Alternate definition of ~ ... |
| nf6 2316 | An alternate definition of... |
| nf5d 2317 | Deduce that ` x ` is not f... |
| nf5di 2318 | Since the converse holds b... |
| 19.9h 2319 | A wff may be existentially... |
| 19.21h 2320 | Theorem 19.21 of [Margaris... |
| 19.23h 2321 | Theorem 19.23 of [Margaris... |
| exlimih 2322 | Inference associated with ... |
| exlimdh 2323 | Deduction form of Theorem ... |
| equsalhw 2324 | Version of ~ equsalh with ... |
| equsexhv 2325 | An equivalence related to ... |
| hba1 2326 | The setvar ` x ` is not fr... |
| hbnt 2327 | Closed theorem version of ... |
| hbn 2328 | If ` x ` is not free in ` ... |
| hbnd 2329 | Deduction form of bound-va... |
| hbim1 2330 | A closed form of ~ hbim . ... |
| hbimd 2331 | Deduction form of bound-va... |
| hbim 2332 | If ` x ` is not free in ` ... |
| hban 2333 | If ` x ` is not free in ` ... |
| hb3an 2334 | If ` x ` is not free in ` ... |
| sbi2 2335 | Introduction of implicatio... |
| sbim 2336 | Implication inside and out... |
| sbrim 2337 | Substitution in an implica... |
| sblim 2338 | Substitution in an implica... |
| sbor 2339 | Disjunction inside and out... |
| sbbi 2340 | Equivalence inside and out... |
| sblbis 2341 | Introduce left bicondition... |
| sbrbis 2342 | Introduce right biconditio... |
| sbrbif 2343 | Introduce right biconditio... |
| sbnf 2344 | Move nonfree predicate in ... |
| sbiev 2345 | Conversion of implicit sub... |
| sbiedw 2346 | Conversion of implicit sub... |
| axc7 2347 | Show that the original axi... |
| axc7e 2348 | Abbreviated version of ~ a... |
| modal-b 2349 | The analogue in our predic... |
| 19.9ht 2350 | A closed version of ~ 19.9... |
| axc4 2351 | Show that the original axi... |
| axc4i 2352 | Inference version of ~ axc... |
| nfal 2353 | If ` x ` is not free in ` ... |
| nfex 2354 | If ` x ` is not free in ` ... |
| hbex 2355 | If ` x ` is not free in ` ... |
| nfnf 2356 | If ` x ` is not free in ` ... |
| 19.12 2357 | Theorem 19.12 of [Margaris... |
| nfald 2358 | Deduction form of ~ nfal .... |
| nfexd 2359 | If ` x ` is not free in ` ... |
| nfsbv 2360 | If ` z ` is not free in ` ... |
| sbco2v 2361 | A composition law for subs... |
| aaan 2362 | Distribute universal quant... |
| eeor 2363 | Distribute existential qua... |
| cbv3v 2364 | Rule used to change bound ... |
| cbv1v 2365 | Rule used to change bound ... |
| cbv2w 2366 | Rule used to change bound ... |
| cbvaldw 2367 | Deduction used to change b... |
| cbvexdw 2368 | Deduction used to change b... |
| cbv3hv 2369 | Rule used to change bound ... |
| cbvalv1 2370 | Rule used to change bound ... |
| cbvexv1 2371 | Rule used to change bound ... |
| cbval2v 2372 | Rule used to change bound ... |
| cbvex2v 2373 | Rule used to change bound ... |
| dvelimhw 2374 | Proof of ~ dvelimh without... |
| pm11.53 2375 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2376 | Special case of ~ 19.12 wh... |
| eean 2377 | Distribute existential qua... |
| eeanv 2378 | Distribute a pair of exist... |
| eeeanv 2379 | Distribute three existenti... |
| ee4anv 2380 | Distribute two pairs of ex... |
| ee4anvOLD 2381 | Obsolete version of ~ ee4a... |
| sb8v 2382 | Substitution of variable i... |
| sb8f 2383 | Substitution of variable i... |
| sb8ef 2384 | Substitution of variable i... |
| 2sb8ef 2385 | An equivalent expression f... |
| sb6rfv 2386 | Reversed substitution. Ve... |
| sbnf2 2387 | Two ways of expressing " `... |
| exsb 2388 | An equivalent expression f... |
| 2exsb 2389 | An equivalent expression f... |
| sbbib 2390 | Reversal of substitution. ... |
| sbbibvv 2391 | Reversal of substitution. ... |
| cbvsbvf 2392 | Change the bound variable ... |
| cleljustALT 2393 | Alternate proof of ~ clelj... |
| cleljustALT2 2394 | Alternate proof of ~ clelj... |
| equs5aALT 2395 | Alternate proof of ~ equs5... |
| equs5eALT 2396 | Alternate proof of ~ equs5... |
| axc11r 2397 | Same as ~ axc11 but with r... |
| dral1v 2398 | Formula-building lemma for... |
| drex1v 2399 | Formula-building lemma for... |
| drnf1v 2400 | Formula-building lemma for... |
| ax13v 2402 | A weaker version of ~ ax-1... |
| ax13lem1 2403 | A version of ~ ax13v with ... |
| ax13 2404 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2405 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2406 | An equation between setvar... |
| dveeq2 2407 | Quantifier introduction wh... |
| nfeqf1 2408 | An equation between setvar... |
| dveeq1 2409 | Quantifier introduction wh... |
| nfeqf 2410 | A variable is effectively ... |
| axc9 2411 | Derive set.mm's original ~... |
| ax6e 2412 | At least one individual ex... |
| ax6 2413 | Theorem showing that ~ ax-... |
| axc10 2414 | Show that the original axi... |
| spimt 2415 | Closed theorem form of ~ s... |
| spim 2416 | Specialization, using impl... |
| spimed 2417 | Deduction version of ~ spi... |
| spime 2418 | Existential introduction, ... |
| spimv 2419 | A version of ~ spim with a... |
| spimvALT 2420 | Alternate proof of ~ spimv... |
| spimev 2421 | Distinct-variable version ... |
| spv 2422 | Specialization, using impl... |
| spei 2423 | Inference from existential... |
| chvar 2424 | Implicit substitution of `... |
| chvarv 2425 | Implicit substitution of `... |
| cbv3 2426 | Rule used to change bound ... |
| cbval 2427 | Rule used to change bound ... |
| cbvex 2428 | Rule used to change bound ... |
| cbvalv 2429 | Rule used to change bound ... |
| cbvexv 2430 | Rule used to change bound ... |
| cbv1 2431 | Rule used to change bound ... |
| cbv2 2432 | Rule used to change bound ... |
| cbv3h 2433 | Rule used to change bound ... |
| cbv1h 2434 | Rule used to change bound ... |
| cbv2h 2435 | Rule used to change bound ... |
| cbvald 2436 | Deduction used to change b... |
| cbvexd 2437 | Deduction used to change b... |
| cbvaldva 2438 | Rule used to change the bo... |
| cbvexdva 2439 | Rule used to change the bo... |
| cbval2 2440 | Rule used to change bound ... |
| cbvex2 2441 | Rule used to change bound ... |
| cbval2vv 2442 | Rule used to change bound ... |
| cbvex2vv 2443 | Rule used to change bound ... |
| cbvex4v 2444 | Rule used to change bound ... |
| equs4 2445 | Lemma used in proofs of im... |
| equsal 2446 | An equivalence related to ... |
| equsex 2447 | An equivalence related to ... |
| equsexALT 2448 | Alternate proof of ~ equse... |
| equsalh 2449 | An equivalence related to ... |
| equsexh 2450 | An equivalence related to ... |
| axc15 2451 | Derivation of set.mm's ori... |
| ax12 2452 | Rederivation of Axiom ~ ax... |
| ax12b 2453 | A bidirectional version of... |
| ax13ALT 2454 | Alternate proof of ~ ax13 ... |
| axc11n 2455 | Derive set.mm's original ~... |
| aecom 2456 | Commutation law for identi... |
| aecoms 2457 | A commutation rule for ide... |
| naecoms 2458 | A commutation rule for dis... |
| axc11 2459 | Show that ~ ax-c11 can be ... |
| hbae 2460 | All variables are effectiv... |
| hbnae 2461 | All variables are effectiv... |
| nfae 2462 | All variables are effectiv... |
| nfnae 2463 | All variables are effectiv... |
| hbnaes 2464 | Rule that applies ~ hbnae ... |
| axc16i 2465 | Inference with ~ axc16 as ... |
| axc16nfALT 2466 | Alternate proof of ~ axc16... |
| dral2 2467 | Formula-building lemma for... |
| dral1 2468 | Formula-building lemma for... |
| dral1ALT 2469 | Alternate proof of ~ dral1... |
| drex1 2470 | Formula-building lemma for... |
| drex2 2471 | Formula-building lemma for... |
| drnf1 2472 | Formula-building lemma for... |
| drnf2 2473 | Formula-building lemma for... |
| nfald2 2474 | Variation on ~ nfald which... |
| nfexd2 2475 | Variation on ~ nfexd which... |
| exdistrf 2476 | Distribution of existentia... |
| dvelimf 2477 | Version of ~ dvelimv witho... |
| dvelimdf 2478 | Deduction form of ~ dvelim... |
| dvelimh 2479 | Version of ~ dvelim withou... |
| dvelim 2480 | This theorem can be used t... |
| dvelimv 2481 | Similar to ~ dvelim with f... |
| dvelimnf 2482 | Version of ~ dvelim using ... |
| dveeq2ALT 2483 | Alternate proof of ~ dveeq... |
| equvini 2484 | A variable introduction la... |
| equvel 2485 | A variable elimination law... |
| equs5a 2486 | A property related to subs... |
| equs5e 2487 | A property related to subs... |
| equs45f 2488 | Two ways of expressing sub... |
| equs5 2489 | Lemma used in proofs of su... |
| dveel1 2490 | Quantifier introduction wh... |
| dveel2 2491 | Quantifier introduction wh... |
| axc14 2492 | Axiom ~ ax-c14 is redundan... |
| sb6x 2493 | Equivalence involving subs... |
| sbequ5 2494 | Substitution does not chan... |
| sbequ6 2495 | Substitution does not chan... |
| sb5rf 2496 | Reversed substitution. Us... |
| sb6rf 2497 | Reversed substitution. Fo... |
| ax12vALT 2498 | Alternate proof of ~ ax12v... |
| 2ax6elem 2499 | We can always find values ... |
| 2ax6e 2500 | We can always find values ... |
| 2sb5rf 2501 | Reversed double substituti... |
| 2sb6rf 2502 | Reversed double substituti... |
| sbel2x 2503 | Elimination of double subs... |
| sb4b 2504 | Simplified definition of s... |
| sb3b 2505 | Simplified definition of s... |
| sb3 2506 | One direction of a simplif... |
| sb1 2507 | One direction of a simplif... |
| sb2 2508 | One direction of a simplif... |
| sb4a 2509 | A version of one implicati... |
| dfsb1 2510 | Alternate definition of su... |
| hbsb2 2511 | Bound-variable hypothesis ... |
| nfsb2 2512 | Bound-variable hypothesis ... |
| hbsb2a 2513 | Special case of a bound-va... |
| sb4e 2514 | One direction of a simplif... |
| hbsb2e 2515 | Special case of a bound-va... |
| hbsb3 2516 | If ` y ` is not free in ` ... |
| nfs1 2517 | If ` y ` is not free in ` ... |
| axc16ALT 2518 | Alternate proof of ~ axc16... |
| axc16gALT 2519 | Alternate proof of ~ axc16... |
| equsb1 2520 | Substitution applied to an... |
| equsb2 2521 | Substitution applied to an... |
| dfsb2 2522 | An alternate definition of... |
| dfsb3 2523 | An alternate definition of... |
| drsb1 2524 | Formula-building lemma for... |
| sb2ae 2525 | In the case of two success... |
| sb6f 2526 | Equivalence for substituti... |
| sb5f 2527 | Equivalence for substituti... |
| nfsb4t 2528 | A variable not free in a p... |
| nfsb4 2529 | A variable not free in a p... |
| sbequ8 2530 | Elimination of equality fr... |
| sbie 2531 | Conversion of implicit sub... |
| sbied 2532 | Conversion of implicit sub... |
| sbiedv 2533 | Conversion of implicit sub... |
| 2sbiev 2534 | Conversion of double impli... |
| sbcom3 2535 | Substituting ` y ` for ` x... |
| sbco 2536 | A composition law for subs... |
| sbid2 2537 | An identity law for substi... |
| sbid2v 2538 | An identity law for substi... |
| sbidm 2539 | An idempotent law for subs... |
| sbco2 2540 | A composition law for subs... |
| sbco2d 2541 | A composition law for subs... |
| sbco3 2542 | A composition law for subs... |
| sbcom 2543 | A commutativity law for su... |
| sbtrt 2544 | Partially closed form of ~... |
| sbtr 2545 | A partial converse to ~ sb... |
| sb8 2546 | Substitution of variable i... |
| sb8e 2547 | Substitution of variable i... |
| sb9 2548 | Commutation of quantificat... |
| sb9i 2549 | Commutation of quantificat... |
| sbhb 2550 | Two ways of expressing " `... |
| nfsbd 2551 | Deduction version of ~ nfs... |
| nfsb 2552 | If ` z ` is not free in ` ... |
| hbsb 2553 | If ` z ` is not free in ` ... |
| sb7f 2554 | This version of ~ dfsb7 do... |
| sb7h 2555 | This version of ~ dfsb7 do... |
| sb10f 2556 | Hao Wang's identity axiom ... |
| sbal1 2557 | Check out ~ sbal for a ver... |
| sbal2 2558 | Move quantifier in and out... |
| 2sb8e 2559 | An equivalent expression f... |
| dfmoeu 2560 | An elementary proof of ~ m... |
| dfeumo 2561 | An elementary proof showin... |
| mojust 2563 | Soundness justification th... |
| dfmo 2565 | Simplify definition ~ df-m... |
| nexmo 2566 | Nonexistence implies uniqu... |
| exmo 2567 | Any proposition holds for ... |
| moabs 2568 | Absorption of existence co... |
| moim 2569 | The at-most-one quantifier... |
| moimi 2570 | The at-most-one quantifier... |
| moimdv 2571 | The at-most-one quantifier... |
| mobi 2572 | Equivalence theorem for th... |
| mobii 2573 | Formula-building rule for ... |
| mobidv 2574 | Formula-building rule for ... |
| mobid 2575 | Formula-building rule for ... |
| moa1 2576 | If an implication holds fo... |
| moan 2577 | "At most one" is still the... |
| moani 2578 | "At most one" is still tru... |
| moor 2579 | "At most one" is still the... |
| mooran1 2580 | "At most one" imports disj... |
| mooran2 2581 | "At most one" exports disj... |
| nfmo1 2582 | Bound-variable hypothesis ... |
| nfmod2 2583 | Bound-variable hypothesis ... |
| nfmodv 2584 | Bound-variable hypothesis ... |
| nfmov 2585 | Bound-variable hypothesis ... |
| nfmod 2586 | Bound-variable hypothesis ... |
| nfmo 2587 | Bound-variable hypothesis ... |
| mof 2588 | Version of ~ df-mo with di... |
| mo3 2589 | Alternate definition of th... |
| mo 2590 | Equivalent definitions of ... |
| mo4 2591 | At-most-one quantifier exp... |
| mo4f 2592 | At-most-one quantifier exp... |
| eu3v 2595 | An alternate way to expres... |
| eujust 2596 | Soundness justification th... |
| eujustALT 2597 | Alternate proof of ~ eujus... |
| eu6lem 2598 | Lemma of ~ eu6im . A diss... |
| eu6 2599 | Alternate definition of th... |
| eu6im 2600 | One direction of ~ eu6 nee... |
| euf 2601 | Version of ~ eu6 with disj... |
| euex 2602 | Existential uniqueness imp... |
| eumo 2603 | Existential uniqueness imp... |
| eumoi 2604 | Uniqueness inferred from e... |
| exmoeub 2605 | Existence implies that uni... |
| exmoeu 2606 | Existence is equivalent to... |
| moeuex 2607 | Uniqueness implies that ex... |
| moeu 2608 | Uniqueness is equivalent t... |
| eubi 2609 | Equivalence theorem for th... |
| eubii 2610 | Introduce unique existenti... |
| eubidv 2611 | Formula-building rule for ... |
| eubid 2612 | Formula-building rule for ... |
| nfeu1ALT 2613 | Alternate version of ~ nfe... |
| nfeu1 2614 | Bound-variable hypothesis ... |
| nfeud2 2615 | Bound-variable hypothesis ... |
| nfeudw 2616 | Bound-variable hypothesis ... |
| nfeud 2617 | Bound-variable hypothesis ... |
| nfeuw 2618 | Bound-variable hypothesis ... |
| nfeu 2619 | Bound-variable hypothesis ... |
| dfeu 2620 | Rederive ~ df-eu from the ... |
| dfmo2 2621 | Rederive ~ df-mo from the ... |
| euequ 2622 | There exists a unique set ... |
| sb8eulem 2623 | Lemma. Factor out the com... |
| sb8euv 2624 | Variable substitution in u... |
| sb8eu 2625 | Variable substitution in u... |
| sb8mo 2626 | Variable substitution for ... |
| cbvmovw 2627 | Change bound variable. Us... |
| cbvmow 2628 | Rule used to change bound ... |
| cbvmo 2629 | Rule used to change bound ... |
| cbveuvw 2630 | Change bound variable. Us... |
| cbveuw 2631 | Version of ~ cbveu with a ... |
| cbveu 2632 | Rule used to change bound ... |
| cbveuALT 2633 | Alternative proof of ~ cbv... |
| eu2 2634 | An alternate way of defini... |
| eu1 2635 | An alternate way to expres... |
| euor 2636 | Introduce a disjunct into ... |
| euorv 2637 | Introduce a disjunct into ... |
| euor2 2638 | Introduce or eliminate a d... |
| sbmo 2639 | Substitution into an at-mo... |
| eu4 2640 | Uniqueness using implicit ... |
| euimmo 2641 | Existential uniqueness imp... |
| euim 2642 | Add unique existential qua... |
| moanimlem 2643 | Factor out the common proo... |
| moanimv 2644 | Introduction of a conjunct... |
| moanim 2645 | Introduction of a conjunct... |
| euan 2646 | Introduction of a conjunct... |
| moanmo 2647 | Nested at-most-one quantif... |
| moaneu 2648 | Nested at-most-one and uni... |
| euanv 2649 | Introduction of a conjunct... |
| mopick 2650 | "At most one" picks a vari... |
| moexexlem 2651 | Factor out the proof skele... |
| 2moexv 2652 | Double quantification with... |
| moexexvw 2653 | "At most one" double quant... |
| 2moswapv 2654 | A condition allowing to sw... |
| 2euswapv 2655 | A condition allowing to sw... |
| 2euexv 2656 | Double quantification with... |
| 2exeuv 2657 | Double existential uniquen... |
| eupick 2658 | Existential uniqueness "pi... |
| eupicka 2659 | Version of ~ eupick with c... |
| eupickb 2660 | Existential uniqueness "pi... |
| eupickbi 2661 | Theorem *14.26 in [Whitehe... |
| mopick2 2662 | "At most one" can show the... |
| moexex 2663 | "At most one" double quant... |
| moexexv 2664 | "At most one" double quant... |
| 2moex 2665 | Double quantification with... |
| 2euex 2666 | Double quantification with... |
| 2eumo 2667 | Nested unique existential ... |
| 2eu2ex 2668 | Double existential uniquen... |
| 2moswap 2669 | A condition allowing to sw... |
| 2euswap 2670 | A condition allowing to sw... |
| 2exeu 2671 | Double existential uniquen... |
| 2mo2 2672 | Two ways of expressing "th... |
| 2mo 2673 | Two ways of expressing "th... |
| 2mos 2674 | Double "there exists at mo... |
| 2eu1 2675 | Double existential uniquen... |
| 2eu1v 2676 | Double existential uniquen... |
| 2eu2 2677 | Double existential uniquen... |
| 2eu3 2678 | Double existential uniquen... |
| 2eu4 2679 | This theorem provides us w... |
| 2eu5 2680 | An alternate definition of... |
| 2eu6 2681 | Two equivalent expressions... |
| 2eu7 2682 | Two equivalent expressions... |
| 2eu8 2683 | Two equivalent expressions... |
| euae 2684 | Two ways to express "exact... |
| exists1 2685 | Two ways to express "exact... |
| exists2 2686 | A condition implying that ... |
| barbara 2687 | "Barbara", one of the fund... |
| celarent 2688 | "Celarent", one of the syl... |
| darii 2689 | "Darii", one of the syllog... |
| dariiALT 2690 | Alternate proof of ~ darii... |
| ferio 2691 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2692 | Lemma for ~ barbari and th... |
| barbari 2693 | "Barbari", one of the syll... |
| barbariALT 2694 | Alternate proof of ~ barba... |
| celaront 2695 | "Celaront", one of the syl... |
| cesare 2696 | "Cesare", one of the syllo... |
| camestres 2697 | "Camestres", one of the sy... |
| festino 2698 | "Festino", one of the syll... |
| festinoALT 2699 | Alternate proof of ~ festi... |
| baroco 2700 | "Baroco", one of the syllo... |
| barocoALT 2701 | Alternate proof of ~ festi... |
| cesaro 2702 | "Cesaro", one of the syllo... |
| camestros 2703 | "Camestros", one of the sy... |
| datisi 2704 | "Datisi", one of the syllo... |
| disamis 2705 | "Disamis", one of the syll... |
| ferison 2706 | "Ferison", one of the syll... |
| bocardo 2707 | "Bocardo", one of the syll... |
| darapti 2708 | "Darapti", one of the syll... |
| daraptiALT 2709 | Alternate proof of ~ darap... |
| felapton 2710 | "Felapton", one of the syl... |
| calemes 2711 | "Calemes", one of the syll... |
| dimatis 2712 | "Dimatis", one of the syll... |
| fresison 2713 | "Fresison", one of the syl... |
| calemos 2714 | "Calemos", one of the syll... |
| fesapo 2715 | "Fesapo", one of the syllo... |
| bamalip 2716 | "Bamalip", one of the syll... |
| axia1 2717 | Left 'and' elimination (in... |
| axia2 2718 | Right 'and' elimination (i... |
| axia3 2719 | 'And' introduction (intuit... |
| axin1 2720 | 'Not' introduction (intuit... |
| axin2 2721 | 'Not' elimination (intuiti... |
| axio 2722 | Definition of 'or' (intuit... |
| axi4 2723 | Specialization (intuitioni... |
| axi5r 2724 | Converse of ~ axc4 (intuit... |
| axial 2725 | The setvar ` x ` is not fr... |
| axie1 2726 | The setvar ` x ` is not fr... |
| axie2 2727 | A key property of existent... |
| axi9 2728 | Axiom of existence (intuit... |
| axi10 2729 | Axiom of Quantifier Substi... |
| axi12 2730 | Axiom of Quantifier Introd... |
| axbnd 2731 | Axiom of Bundling (intuiti... |
| axexte 2733 | The axiom of extensionalit... |
| axextg 2734 | A generalization of the ax... |
| axextb 2735 | A bidirectional version of... |
| axextmo 2736 | There exists at most one s... |
| nulmo 2737 | There exists at most one e... |
| eleq1ab 2740 | Extension (in the sense of... |
| cleljustab 2741 | Extension of ~ cleljust fr... |
| abid 2742 | Simplification of class ab... |
| vexwt 2743 | A standard theorem of pred... |
| vexw 2744 | If ` ph ` is a theorem, th... |
| vextru 2745 | Every setvar is a member o... |
| nfsab1 2746 | Bound-variable hypothesis ... |
| hbab1 2747 | Bound-variable hypothesis ... |
| hbab 2748 | Bound-variable hypothesis ... |
| hbabg 2749 | Bound-variable hypothesis ... |
| nfsab 2750 | Bound-variable hypothesis ... |
| nfsabg 2751 | Bound-variable hypothesis ... |
| dfcleq 2753 | The defining characterizat... |
| cvjust 2754 | Every set is a class. Pro... |
| ax9ALT 2755 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2756 | A weaker version of ~ eleq... |
| eqriv 2757 | Infer equality of classes ... |
| eqrdv 2758 | Deduce equality of classes... |
| eqrdav 2759 | Deduce equality of classes... |
| eqid 2760 | Law of identity (reflexivi... |
| eqidd 2761 | Class identity law with an... |
| eqeq1d 2762 | Deduction from equality to... |
| eqeq1dALT 2763 | Alternate proof of ~ eqeq1... |
| eqeq1 2764 | Equality implies equivalen... |
| eqeq1i 2765 | Inference from equality to... |
| eqcomd 2766 | Deduction from commutative... |
| eqcom 2767 | Commutative law for class ... |
| eqcoms 2768 | Inference applying commuta... |
| eqcomi 2769 | Inference from commutative... |
| neqcomd 2770 | Commute an inequality. (C... |
| eqeq2d 2771 | Deduction from equality to... |
| eqeq2 2772 | Equality implies equivalen... |
| eqeq2i 2773 | Inference from equality to... |
| eqeqan12d 2774 | A useful inference for sub... |
| eqeqan12rd 2775 | A useful inference for sub... |
| eqeq12d 2776 | A useful inference for sub... |
| eqeq12 2777 | Equality relationship amon... |
| eqeq12i 2778 | A useful inference for sub... |
| eqeqan12dALT 2779 | Alternate proof of ~ eqeqa... |
| eqtr 2780 | Transitive law for class e... |
| eqtr2 2781 | A transitive law for class... |
| eqtr3 2782 | A transitive law for class... |
| eqtri 2783 | An equality transitivity i... |
| eqtr2i 2784 | An equality transitivity i... |
| eqtr3i 2785 | An equality transitivity i... |
| eqtr4i 2786 | An equality transitivity i... |
| 3eqtri 2787 | An inference from three ch... |
| 3eqtrri 2788 | An inference from three ch... |
| 3eqtr2i 2789 | An inference from three ch... |
| 3eqtr2ri 2790 | An inference from three ch... |
| 3eqtr3i 2791 | An inference from three ch... |
| 3eqtr3ri 2792 | An inference from three ch... |
| 3eqtr4i 2793 | An inference from three ch... |
| 3eqtr4ri 2794 | An inference from three ch... |
| eqtrd 2795 | An equality transitivity d... |
| eqtr2d 2796 | An equality transitivity d... |
| eqtr3d 2797 | An equality transitivity e... |
| eqtr4d 2798 | An equality transitivity e... |
| 3eqtrd 2799 | A deduction from three cha... |
| 3eqtrrd 2800 | A deduction from three cha... |
| 3eqtr2d 2801 | A deduction from three cha... |
| 3eqtr2rd 2802 | A deduction from three cha... |
| 3eqtr3d 2803 | A deduction from three cha... |
| 3eqtr3rd 2804 | A deduction from three cha... |
| 3eqtr4d 2805 | A deduction from three cha... |
| 3eqtr4rd 2806 | A deduction from three cha... |
| eqtrid 2807 | An equality transitivity d... |
| eqtr2id 2808 | An equality transitivity d... |
| eqtr3id 2809 | An equality transitivity d... |
| eqtr3di 2810 | An equality transitivity d... |
| eqtrdi 2811 | An equality transitivity d... |
| eqtr2di 2812 | An equality transitivity d... |
| eqtr4di 2813 | An equality transitivity d... |
| eqtr4id 2814 | An equality transitivity d... |
| sylan9eq 2815 | An equality transitivity d... |
| sylan9req 2816 | An equality transitivity d... |
| sylan9eqr 2817 | An equality transitivity d... |
| 3eqtr3g 2818 | A chained equality inferen... |
| 3eqtr3a 2819 | A chained equality inferen... |
| 3eqtr4g 2820 | A chained equality inferen... |
| 3eqtr4a 2821 | A chained equality inferen... |
| eq2tri 2822 | A compound transitive infe... |
| iseqsetvlem 2823 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2824 | Alternate proof of ~ iseqs... |
| abbi 2825 | Equivalent formulas yield ... |
| abbidv 2826 | Equivalent wff's yield equ... |
| abbii 2827 | Equivalent wff's yield equ... |
| abbid 2828 | Equivalent wff's yield equ... |
| abbib 2829 | Equal class abstractions r... |
| cbvabv 2830 | Rule used to change bound ... |
| cbvabw 2831 | Rule used to change bound ... |
| cbvab 2832 | Rule used to change bound ... |
| eqabbw 2833 | Version of ~ eqabb using i... |
| eqabcbw 2834 | Version of ~ eqabcb using ... |
| dfclel 2836 | Characterization of the el... |
| elex2 2837 | If a class contains anothe... |
| issettru 2838 | Weak version of ~ isset . ... |
| iseqsetv-clel 2839 | Alternate proof of ~ iseqs... |
| issetlem 2840 | Lemma for ~ elisset and ~ ... |
| elissetv 2841 | An element of a class exis... |
| elisset 2842 | An element of a class exis... |
| eleq1w 2843 | Weaker version of ~ eleq1 ... |
| eleq2w 2844 | Weaker version of ~ eleq2 ... |
| eleq1d 2845 | Deduction from equality to... |
| eleq2d 2846 | Deduction from equality to... |
| eleq2dALT 2847 | Alternate proof of ~ eleq2... |
| eleq1 2848 | Equality implies equivalen... |
| eleq2 2849 | Equality implies equivalen... |
| eleq12 2850 | Equality implies equivalen... |
| eleq1i 2851 | Inference from equality to... |
| eleq2i 2852 | Inference from equality to... |
| eleq12i 2853 | Inference from equality to... |
| eleq12d 2854 | Deduction from equality to... |
| eleq1a 2855 | A transitive-type law rela... |
| eqeltri 2856 | Substitution of equal clas... |
| eqeltrri 2857 | Substitution of equal clas... |
| eleqtri 2858 | Substitution of equal clas... |
| eleqtrri 2859 | Substitution of equal clas... |
| eqeltrd 2860 | Substitution of equal clas... |
| eqeltrrd 2861 | Deduction that substitutes... |
| eleqtrd 2862 | Deduction that substitutes... |
| eleqtrrd 2863 | Deduction that substitutes... |
| eqeltrid 2864 | A membership and equality ... |
| eqeltrrid 2865 | A membership and equality ... |
| eleqtrid 2866 | A membership and equality ... |
| eleqtrrid 2867 | A membership and equality ... |
| eqeltrdi 2868 | A membership and equality ... |
| eqeltrrdi 2869 | A membership and equality ... |
| eleqtrdi 2870 | A membership and equality ... |
| eleqtrrdi 2871 | A membership and equality ... |
| 3eltr3i 2872 | Substitution of equal clas... |
| 3eltr4i 2873 | Substitution of equal clas... |
| 3eltr3d 2874 | Substitution of equal clas... |
| 3eltr4d 2875 | Substitution of equal clas... |
| 3eltr3g 2876 | Substitution of equal clas... |
| 3eltr4g 2877 | Substitution of equal clas... |
| eleq2s 2878 | Substitution of equal clas... |
| eqneltri 2879 | If a class is not an eleme... |
| eqneltrd 2880 | If a class is not an eleme... |
| eqneltrrd 2881 | If a class is not an eleme... |
| neleqtrd 2882 | If a class is not an eleme... |
| neleqtrrd 2883 | If a class is not an eleme... |
| nelneq 2884 | A way of showing two class... |
| nelneq2 2885 | A way of showing two class... |
| eqsb1 2886 | Substitution for the left-... |
| clelsb1 2887 | Substitution for the first... |
| clelsb2 2888 | Substitution for the secon... |
| cleqh 2889 | Establish equality between... |
| hbxfreq 2890 | A utility lemma to transfe... |
| hblem 2891 | Change the free variable o... |
| hblemg 2892 | Change the free variable o... |
| eqabdv 2893 | Deduction from a wff to a ... |
| eqabcdv 2894 | Deduction from a wff to a ... |
| eqabi 2895 | Equality of a class variab... |
| abid1 2896 | Every class is equal to a ... |
| abid2 2897 | A simplification of class ... |
| eqab 2898 | One direction of ~ eqabb i... |
| eqabb 2899 | Equality of a class variab... |
| eqabcb 2900 | Equality of a class variab... |
| eqabrd 2901 | Equality of a class variab... |
| eqabri 2902 | Equality of a class variab... |
| eqabcri 2903 | Equality of a class variab... |
| clelab 2904 | Membership of a class vari... |
| clabel 2905 | Membership of a class abst... |
| sbab 2906 | The right-hand side of the... |
| nfcjust 2908 | Justification theorem for ... |
| nfci 2910 | Deduce that a class ` A ` ... |
| nfcii 2911 | Deduce that a class ` A ` ... |
| nfcr 2912 | Consequence of the not-fre... |
| nfcrALT 2913 | Alternate version of ~ nfc... |
| nfcri 2914 | Consequence of the not-fre... |
| nfcd 2915 | Deduce that a class ` A ` ... |
| nfcrd 2916 | Consequence of the not-fre... |
| nfcrii 2917 | Consequence of the not-fre... |
| nfceqdf 2918 | An equality theorem for ef... |
| nfceqi 2919 | Equality theorem for class... |
| nfcxfr 2920 | A utility lemma to transfe... |
| nfcxfrd 2921 | A utility lemma to transfe... |
| nfcv 2922 | If ` x ` is disjoint from ... |
| nfcvd 2923 | If ` x ` is disjoint from ... |
| nfab1 2924 | Bound-variable hypothesis ... |
| nfnfc1 2925 | The setvar ` x ` is bound ... |
| clelsb1fw 2926 | Substitution for the first... |
| clelsb1f 2927 | Substitution for the first... |
| nfab 2928 | Bound-variable hypothesis ... |
| nfabg 2929 | Bound-variable hypothesis ... |
| nfaba1 2930 | Bound-variable hypothesis ... |
| nfaba1g 2931 | Bound-variable hypothesis ... |
| nfeqd 2932 | Hypothesis builder for equ... |
| nfeld 2933 | Hypothesis builder for ele... |
| nfnfc 2934 | Hypothesis builder for ` F... |
| nfeq 2935 | Hypothesis builder for equ... |
| nfel 2936 | Hypothesis builder for ele... |
| nfeq1 2937 | Hypothesis builder for equ... |
| nfel1 2938 | Hypothesis builder for ele... |
| nfeq2 2939 | Hypothesis builder for equ... |
| nfel2 2940 | Hypothesis builder for ele... |
| drnfc1 2941 | Formula-building lemma for... |
| drnfc2 2942 | Formula-building lemma for... |
| nfabdw 2943 | Bound-variable hypothesis ... |
| nfabd 2944 | Bound-variable hypothesis ... |
| nfabd2 2945 | Bound-variable hypothesis ... |
| dvelimdc 2946 | Deduction form of ~ dvelim... |
| dvelimc 2947 | Version of ~ dvelim for cl... |
| nfcvf 2948 | If ` x ` and ` y ` are dis... |
| nfcvf2 2949 | If ` x ` and ` y ` are dis... |
| cleqf 2950 | Establish equality between... |
| eqabf 2951 | Equality of a class variab... |
| abid2f 2952 | A simplification of class ... |
| abid2fOLD 2953 | Obsolete version of ~ abid... |
| sbabel 2954 | Theorem to move a substitu... |
| neii 2957 | Inference associated with ... |
| neir 2958 | Inference associated with ... |
| nne 2959 | Negation of inequality. (... |
| neneqd 2960 | Deduction eliminating ineq... |
| neneq 2961 | From inequality to non-equ... |
| neqned 2962 | If it is not the case that... |
| neqne 2963 | From non-equality to inequ... |
| neirr 2964 | No class is unequal to its... |
| exmidne 2965 | Excluded middle with equal... |
| eqneqall 2966 | A contradiction concerning... |
| nonconne 2967 | Law of noncontradiction wi... |
| necon3ad 2968 | Contrapositive law deducti... |
| necon3bd 2969 | Contrapositive law deducti... |
| necon2ad 2970 | Contrapositive inference f... |
| necon2bd 2971 | Contrapositive inference f... |
| necon1ad 2972 | Contrapositive deduction f... |
| necon1bd 2973 | Contrapositive deduction f... |
| necon4ad 2974 | Contrapositive inference f... |
| necon4bd 2975 | Contrapositive inference f... |
| necon3d 2976 | Contrapositive law deducti... |
| necon1d 2977 | Contrapositive law deducti... |
| necon2d 2978 | Contrapositive inference f... |
| necon4d 2979 | Contrapositive inference f... |
| necon3ai 2980 | Contrapositive inference f... |
| necon3bi 2981 | Contrapositive inference f... |
| necon1ai 2982 | Contrapositive inference f... |
| necon1bi 2983 | Contrapositive inference f... |
| necon2ai 2984 | Contrapositive inference f... |
| necon2bi 2985 | Contrapositive inference f... |
| necon4ai 2986 | Contrapositive inference f... |
| necon3i 2987 | Contrapositive inference f... |
| necon1i 2988 | Contrapositive inference f... |
| necon2i 2989 | Contrapositive inference f... |
| necon4i 2990 | Contrapositive inference f... |
| necon3abid 2991 | Deduction from equality to... |
| necon3bbid 2992 | Deduction from equality to... |
| necon1abid 2993 | Contrapositive deduction f... |
| necon1bbid 2994 | Contrapositive inference f... |
| necon4abid 2995 | Contrapositive law deducti... |
| necon4bbid 2996 | Contrapositive law deducti... |
| necon2abid 2997 | Contrapositive deduction f... |
| necon2bbid 2998 | Contrapositive deduction f... |
| necon3bid 2999 | Deduction from equality to... |
| necon4bid 3000 | Contrapositive law deducti... |
| necon3abii 3001 | Deduction from equality to... |
| necon3bbii 3002 | Deduction from equality to... |
| necon1abii 3003 | Contrapositive inference f... |
| necon1bbii 3004 | Contrapositive inference f... |
| necon2abii 3005 | Contrapositive inference f... |
| necon2bbii 3006 | Contrapositive inference f... |
| necon3bii 3007 | Inference from equality to... |
| necom 3008 | Commutation of inequality.... |
| necomi 3009 | Inference from commutative... |
| necomd 3010 | Deduction from commutative... |
| nesym 3011 | Characterization of inequa... |
| nesymi 3012 | Inference associated with ... |
| nesymir 3013 | Inference associated with ... |
| neeq1d 3014 | Deduction for inequality. ... |
| neeq2d 3015 | Deduction for inequality. ... |
| neeq12d 3016 | Deduction for inequality. ... |
| neeq1 3017 | Equality theorem for inequ... |
| neeq2 3018 | Equality theorem for inequ... |
| neeq1i 3019 | Inference for inequality. ... |
| neeq2i 3020 | Inference for inequality. ... |
| neeq12i 3021 | Inference for inequality. ... |
| eqnetrd 3022 | Substitution of equal clas... |
| eqnetrrd 3023 | Substitution of equal clas... |
| neeqtrd 3024 | Substitution of equal clas... |
| eqnetri 3025 | Substitution of equal clas... |
| eqnetrri 3026 | Substitution of equal clas... |
| neeqtri 3027 | Substitution of equal clas... |
| neeqtrri 3028 | Substitution of equal clas... |
| neeqtrrd 3029 | Substitution of equal clas... |
| eqnetrrid 3030 | A chained equality inferen... |
| 3netr3d 3031 | Substitution of equality i... |
| 3netr4d 3032 | Substitution of equality i... |
| 3netr3g 3033 | Substitution of equality i... |
| 3netr4g 3034 | Substitution of equality i... |
| nebi 3035 | Contraposition law for ine... |
| pm13.18 3036 | Theorem *13.18 in [Whitehe... |
| pm13.181 3037 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3038 | Inference eliminating an i... |
| pm2.21ddne 3039 | A contradiction implies an... |
| pm2.61ne 3040 | Deduction eliminating an i... |
| pm2.61dne 3041 | Deduction eliminating an i... |
| pm2.61dane 3042 | Deduction eliminating an i... |
| pm2.61da2ne 3043 | Deduction eliminating two ... |
| pm2.61da3ne 3044 | Deduction eliminating thre... |
| pm2.61iine 3045 | Equality version of ~ pm2.... |
| mteqand 3046 | A modus tollens deduction ... |
| neor 3047 | Logical OR with an equalit... |
| neanior 3048 | A De Morgan's law for ineq... |
| ne3anior 3049 | A De Morgan's law for ineq... |
| neorian 3050 | A De Morgan's law for ineq... |
| nemtbir 3051 | An inference from an inequ... |
| nelne1 3052 | Two classes are different ... |
| nelne2 3053 | Two classes are different ... |
| elnelneqd 3054 | Two classes are not equal ... |
| elnelneq2d 3055 | Two classes are not equal ... |
| nelelne 3056 | Two classes are different ... |
| neneor 3057 | If two classes are differe... |
| nfne 3058 | Bound-variable hypothesis ... |
| nfned 3059 | Bound-variable hypothesis ... |
| nabbib 3060 | Not equivalent wff's corre... |
| neli 3063 | Inference associated with ... |
| nelir 3064 | Inference associated with ... |
| nelcon3d 3065 | Contrapositive law deducti... |
| neleq12d 3066 | Equality theorem for negat... |
| neleq1 3067 | Equality theorem for negat... |
| neleq2 3068 | Equality theorem for negat... |
| nfnel 3069 | Bound-variable hypothesis ... |
| nfneld 3070 | Bound-variable hypothesis ... |
| nnel 3071 | Negation of negated member... |
| elnelne1 3072 | Two classes are different ... |
| elnelne2 3073 | Two classes are different ... |
| pm2.24nel 3074 | A contradiction concerning... |
| pm2.61danel 3075 | Deduction eliminating an e... |
| rgen 3078 | Generalization rule for re... |
| ralel 3079 | All elements of a class ar... |
| rgenw 3080 | Generalization rule for re... |
| rgen2w 3081 | Generalization rule for re... |
| mprg 3082 | Modus ponens combined with... |
| mprgbir 3083 | Modus ponens on biconditio... |
| ralrid 3084 | Sufficient condition for t... |
| raln 3085 | Restricted universally qua... |
| ralnex 3088 | Relationship between restr... |
| dfrex2 3089 | Relationship between restr... |
| nrex 3090 | Inference adding restricte... |
| alral 3091 | Universal quantification i... |
| rexex 3092 | Restricted existence impli... |
| rextru 3093 | Two ways of expressing tha... |
| ralimi2 3094 | Inference quantifying both... |
| reximi2 3095 | Inference quantifying both... |
| ralimia 3096 | Inference quantifying both... |
| reximia 3097 | Inference quantifying both... |
| ralimiaa 3098 | Inference quantifying both... |
| ralimi 3099 | Inference quantifying both... |
| reximi 3100 | Inference quantifying both... |
| ral2imi 3101 | Inference quantifying ante... |
| ralim 3102 | Distribution of restricted... |
| rexim 3103 | Theorem 19.22 of [Margaris... |
| ralbii2 3104 | Inference adding different... |
| rexbii2 3105 | Inference adding different... |
| ralbiia 3106 | Inference adding restricte... |
| rexbiia 3107 | Inference adding restricte... |
| ralbii 3108 | Inference adding restricte... |
| rexbii 3109 | Inference adding restricte... |
| ralanid 3110 | Cancellation law for restr... |
| rexanid 3111 | Cancellation law for restr... |
| ralcom3 3112 | A commutation law for rest... |
| dfral2 3113 | Relationship between restr... |
| rexnal 3114 | Relationship between restr... |
| ralinexa 3115 | A transformation of restri... |
| rexanali 3116 | A transformation of restri... |
| ralbi 3117 | Distribute a restricted un... |
| rexbi 3118 | Distribute restricted quan... |
| ralrexbid 3119 | Formula-building rule for ... |
| r19.35 3120 | Restricted quantifier vers... |
| r19.26m 3121 | Version of ~ 19.26 and ~ r... |
| r19.26 3122 | Restricted quantifier vers... |
| r19.26-3 3123 | Version of ~ r19.26 with t... |
| ralbiim 3124 | Split a biconditional and ... |
| r19.29 3125 | Restricted quantifier vers... |
| r19.29r 3126 | Restricted quantifier vers... |
| r19.29imd 3127 | Theorem 19.29 of [Margaris... |
| r19.40 3128 | Restricted quantifier vers... |
| r19.30 3129 | Restricted quantifier vers... |
| r19.43 3130 | Restricted quantifier vers... |
| 3r19.43 3131 | Restricted quantifier vers... |
| 2ralimi 3132 | Inference quantifying both... |
| 3ralimi 3133 | Inference quantifying both... |
| 4ralimi 3134 | Inference quantifying both... |
| 5ralimi 3135 | Inference quantifying both... |
| 6ralimi 3136 | Inference quantifying both... |
| 2ralbii 3137 | Inference adding two restr... |
| 2rexbii 3138 | Inference adding two restr... |
| 3ralbii 3139 | Inference adding three res... |
| 4ralbii 3140 | Inference adding four rest... |
| 2ralbiim 3141 | Split a biconditional and ... |
| ralnex2 3142 | Relationship between two r... |
| ralnex3 3143 | Relationship between three... |
| rexnal2 3144 | Relationship between two r... |
| rexnal3 3145 | Relationship between three... |
| nrexralim 3146 | Negation of a complex pred... |
| r19.26-2 3147 | Restricted quantifier vers... |
| 2r19.29 3148 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3149 | Theorem 19.29 of [Margaris... |
| r2allem 3150 | Lemma factoring out common... |
| r2exlem 3151 | Lemma factoring out common... |
| hbralrimi 3152 | Inference from Theorem 19.... |
| ralrimiv 3153 | Inference from Theorem 19.... |
| ralrimiva 3154 | Inference from Theorem 19.... |
| rexlimiva 3155 | Inference from Theorem 19.... |
| rexlimiv 3156 | Inference from Theorem 19.... |
| nrexdv 3157 | Deduction adding restricte... |
| ralrimivw 3158 | Inference from Theorem 19.... |
| rexlimivw 3159 | Weaker version of ~ rexlim... |
| ralrimdv 3160 | Inference from Theorem 19.... |
| rexlimdv 3161 | Inference from Theorem 19.... |
| ralrimdva 3162 | Inference from Theorem 19.... |
| rexlimdva 3163 | Inference from Theorem 19.... |
| rexlimdvaa 3164 | Inference from Theorem 19.... |
| rexlimdva2 3165 | Inference from Theorem 19.... |
| r19.29an 3166 | A commonly used pattern in... |
| rexlimdv3a 3167 | Inference from Theorem 19.... |
| rexlimdvw 3168 | Inference from Theorem 19.... |
| rexlimddv 3169 | Restricted existential eli... |
| r19.29a 3170 | A commonly used pattern in... |
| ralimdv2 3171 | Inference quantifying both... |
| reximdv2 3172 | Deduction quantifying both... |
| reximdvai 3173 | Deduction quantifying both... |
| ralimdva 3174 | Deduction quantifying both... |
| reximdva 3175 | Deduction quantifying both... |
| ralimdv 3176 | Deduction quantifying both... |
| reximdv 3177 | Deduction from Theorem 19.... |
| reximddv 3178 | Deduction from Theorem 19.... |
| reximddv3 3179 | Deduction from Theorem 19.... |
| reximssdv 3180 | Derivation of a restricted... |
| ralbidv2 3181 | Formula-building rule for ... |
| rexbidv2 3182 | Formula-building rule for ... |
| ralbidva 3183 | Formula-building rule for ... |
| rexbidva 3184 | Formula-building rule for ... |
| ralbidv 3185 | Formula-building rule for ... |
| rexbidv 3186 | Formula-building rule for ... |
| r19.21v 3187 | Restricted quantifier vers... |
| r19.37v 3188 | Restricted quantifier vers... |
| r19.23v 3189 | Restricted quantifier vers... |
| r19.36v 3190 | Restricted quantifier vers... |
| r19.27v 3191 | Restricted quantitifer ver... |
| r19.41v 3192 | Restricted quantifier vers... |
| r19.28v 3193 | Restricted quantifier vers... |
| r19.42v 3194 | Restricted quantifier vers... |
| r19.32v 3195 | Restricted quantifier vers... |
| r19.45v 3196 | Restricted quantifier vers... |
| r19.44v 3197 | One direction of a restric... |
| r2al 3198 | Double restricted universa... |
| r2ex 3199 | Double restricted existent... |
| r3al 3200 | Triple restricted universa... |
| r3ex 3201 | Triple existential quantif... |
| rgen2 3202 | Generalization rule for re... |
| ralrimivv 3203 | Inference from Theorem 19.... |
| rexlimivv 3204 | Inference from Theorem 19.... |
| ralrimivva 3205 | Inference from Theorem 19.... |
| ralrimdvv 3206 | Inference from Theorem 19.... |
| rgen3 3207 | Generalization rule for re... |
| ralrimivvva 3208 | Inference from Theorem 19.... |
| ralimdvva 3209 | Deduction doubly quantifyi... |
| reximdvva 3210 | Deduction doubly quantifyi... |
| ralimdvv 3211 | Deduction doubly quantifyi... |
| ralimdvvOLD 3212 | Obsolete version of ~ rali... |
| ralimd4v 3213 | Deduction quadrupally quan... |
| ralimd4vOLD 3214 | Obsolete version of ~ rali... |
| ralimd6v 3215 | Deduction sextupally quant... |
| ralimd6vOLD 3216 | Obsolete version of ~ rali... |
| ralrimdvva 3217 | Inference from Theorem 19.... |
| rexlimdvv 3218 | Inference from Theorem 19.... |
| rexlimdvva 3219 | Inference from Theorem 19.... |
| rexlimdvvva 3220 | Inference from Theorem 19.... |
| reximddv2 3221 | Double deduction from Theo... |
| r19.29vva 3222 | A commonly used pattern ba... |
| 2rexbiia 3223 | Inference adding two restr... |
| 2ralbidva 3224 | Formula-building rule for ... |
| 2rexbidva 3225 | Formula-building rule for ... |
| 2ralbidv 3226 | Formula-building rule for ... |
| 2rexbidv 3227 | Formula-building rule for ... |
| rexralbidv 3228 | Formula-building rule for ... |
| 3ralbidv 3229 | Formula-building rule for ... |
| 4ralbidv 3230 | Formula-building rule for ... |
| 6ralbidv 3231 | Formula-building rule for ... |
| r19.41vv 3232 | Version of ~ r19.41v with ... |
| reeanlem 3233 | Lemma factoring out common... |
| reeanv 3234 | Rearrange restricted exist... |
| 3reeanv 3235 | Rearrange three restricted... |
| 2ralor 3236 | Distribute restricted univ... |
| risset 3237 | Two ways to say " ` A ` be... |
| nelb 3238 | A definition of ` -. A e. ... |
| rspw 3239 | Restricted specialization.... |
| cbvralvw 3240 | Change the bound variable ... |
| cbvrexvw 3241 | Change the bound variable ... |
| cbvraldva 3242 | Rule used to change the bo... |
| cbvrexdva 3243 | Rule used to change the bo... |
| cbvral2vw 3244 | Change bound variables of ... |
| cbvrex2vw 3245 | Change bound variables of ... |
| cbvral3vw 3246 | Change bound variables of ... |
| cbvral4vw 3247 | Change bound variables of ... |
| cbvral6vw 3248 | Change bound variables of ... |
| cbvral8vw 3249 | Change bound variables of ... |
| rsp 3250 | Restricted specialization.... |
| rspa 3251 | Restricted specialization.... |
| rspe 3252 | Restricted specialization.... |
| rspec 3253 | Specialization rule for re... |
| r19.21bi 3254 | Inference from Theorem 19.... |
| r19.21be 3255 | Inference from Theorem 19.... |
| r19.21t 3256 | Restricted quantifier vers... |
| r19.21 3257 | Restricted quantifier vers... |
| r19.23t 3258 | Closed theorem form of ~ r... |
| r19.23 3259 | Restricted quantifier vers... |
| ralrimi 3260 | Inference from Theorem 19.... |
| ralrimia 3261 | Inference from Theorem 19.... |
| rexlimi 3262 | Restricted quantifier vers... |
| ralimdaa 3263 | Deduction quantifying both... |
| reximdai 3264 | Deduction from Theorem 19.... |
| r19.37 3265 | Restricted quantifier vers... |
| r19.41 3266 | Restricted quantifier vers... |
| ralrimd 3267 | Inference from Theorem 19.... |
| rexlimd2 3268 | Version of ~ rexlimd with ... |
| rexlimd 3269 | Deduction form of ~ rexlim... |
| r19.29af2 3270 | A commonly used pattern ba... |
| r19.29af 3271 | A commonly used pattern ba... |
| reximd2a 3272 | Deduction quantifying both... |
| ralbida 3273 | Formula-building rule for ... |
| rexbida 3274 | Formula-building rule for ... |
| ralbid 3275 | Formula-building rule for ... |
| rexbid 3276 | Formula-building rule for ... |
| rexbidvALT 3277 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3278 | Alternate proof of ~ rexbi... |
| rsp2 3279 | Restricted specialization,... |
| rsp2e 3280 | Restricted specialization.... |
| rspec2 3281 | Specialization rule for re... |
| rspec3 3282 | Specialization rule for re... |
| r2alf 3283 | Double restricted universa... |
| r2exf 3284 | Double restricted existent... |
| 2ralbida 3285 | Formula-building rule for ... |
| nfra1 3286 | The setvar ` x ` is not fr... |
| nfre1 3287 | The setvar ` x ` is not fr... |
| ralcom4 3288 | Commutation of restricted ... |
| rexcom4 3289 | Commutation of restricted ... |
| ralcom 3290 | Commutation of restricted ... |
| rexcom 3291 | Commutation of restricted ... |
| rexcom4a 3292 | Specialized existential co... |
| ralrot3 3293 | Rotate three restricted un... |
| ralcom13 3294 | Swap first and third restr... |
| rexcom13 3295 | Swap first and third restr... |
| rexrot4 3296 | Rotate four restricted exi... |
| 2ex2rexrot 3297 | Rotate two existential qua... |
| nfra2w 3298 | Similar to Lemma 24 of [Mo... |
| hbra1 3299 | The setvar ` x ` is not fr... |
| ralcomf 3300 | Commutation of restricted ... |
| rexcomf 3301 | Commutation of restricted ... |
| cbvralfw 3302 | Rule used to change bound ... |
| cbvrexfw 3303 | Rule used to change bound ... |
| cbvralw 3304 | Rule used to change bound ... |
| cbvrexw 3305 | Rule used to change bound ... |
| hbral 3306 | Bound-variable hypothesis ... |
| nfraldw 3307 | Deduction version of ~ nfr... |
| nfrexdw 3308 | Deduction version of ~ nfr... |
| nfralw 3309 | Bound-variable hypothesis ... |
| nfrexw 3310 | Bound-variable hypothesis ... |
| r19.12 3311 | Restricted quantifier vers... |
| reean 3312 | Rearrange restricted exist... |
| cbvralsvw 3313 | Change bound variable by u... |
| cbvrexsvw 3314 | Change bound variable by u... |
| rexeq 3315 | Equality theorem for restr... |
| raleq 3316 | Equality theorem for restr... |
| raleqi 3317 | Equality inference for res... |
| rexeqi 3318 | Equality inference for res... |
| raleqdv 3319 | Equality deduction for res... |
| rexeqdv 3320 | Equality deduction for res... |
| raleqtrdv 3321 | Substitution of equal clas... |
| rexeqtrdv 3322 | Substitution of equal clas... |
| raleqtrrdv 3323 | Substitution of equal clas... |
| rexeqtrrdv 3324 | Substitution of equal clas... |
| raleqbidva 3325 | Equality deduction for res... |
| rexeqbidva 3326 | Equality deduction for res... |
| raleqbidvv 3327 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3328 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3329 | Equality deduction for res... |
| rexeqbi1dv 3330 | Equality deduction for res... |
| raleleq 3331 | All elements of a class ar... |
| raleqbii 3332 | Equality deduction for res... |
| rexeqbii 3333 | Equality deduction for res... |
| raleqbidv 3334 | Equality deduction for res... |
| rexeqbidv 3335 | Equality deduction for res... |
| cbvraldva2 3336 | Rule used to change the bo... |
| cbvrexdva2 3337 | Rule used to change the bo... |
| sbralie 3338 | Implicit to explicit subst... |
| sbralieALT 3339 | Alternative shorter proof ... |
| sbralieOLD 3340 | Obsolete version of ~ sbra... |
| raleqf 3341 | Equality theorem for restr... |
| rexeqf 3342 | Equality theorem for restr... |
| raleqbid 3343 | Equality deduction for res... |
| rexeqbid 3344 | Equality deduction for res... |
| cbvralf 3345 | Rule used to change bound ... |
| cbvrexf 3346 | Rule used to change bound ... |
| cbvral 3347 | Rule used to change bound ... |
| cbvrex 3348 | Rule used to change bound ... |
| cbvralv 3349 | Change the bound variable ... |
| cbvrexv 3350 | Change the bound variable ... |
| cbvralsv 3351 | Change bound variable by u... |
| cbvrexsv 3352 | Change bound variable by u... |
| cbvral2v 3353 | Change bound variables of ... |
| cbvrex2v 3354 | Change bound variables of ... |
| cbvral3v 3355 | Change bound variables of ... |
| rgen2a 3356 | Generalization rule for re... |
| nfrald 3357 | Deduction version of ~ nfr... |
| nfrexd 3358 | Deduction version of ~ nfr... |
| nfral 3359 | Bound-variable hypothesis ... |
| nfrex 3360 | Bound-variable hypothesis ... |
| nfra2 3361 | Similar to Lemma 24 of [Mo... |
| ralcom2 3362 | Commutation of restricted ... |
| reu5 3367 | Restricted uniqueness in t... |
| reurmo 3368 | Restricted existential uni... |
| reurex 3369 | Restricted unique existenc... |
| mormo 3370 | Unrestricted "at most one"... |
| rmobiia 3371 | Formula-building rule for ... |
| reubiia 3372 | Formula-building rule for ... |
| rmobii 3373 | Formula-building rule for ... |
| reubii 3374 | Formula-building rule for ... |
| rmoanid 3375 | Cancellation law for restr... |
| reuanid 3376 | Cancellation law for restr... |
| 2reu2rex 3377 | Double restricted existent... |
| rmobidva 3378 | Formula-building rule for ... |
| reubidva 3379 | Formula-building rule for ... |
| rmobidv 3380 | Formula-building rule for ... |
| reubidv 3381 | Formula-building rule for ... |
| reueubd 3382 | Restricted existential uni... |
| rmo5 3383 | Restricted "at most one" i... |
| nrexrmo 3384 | Nonexistence implies restr... |
| moel 3385 | "At most one" element in a... |
| cbvrmovw 3386 | Change the bound variable ... |
| cbvreuvw 3387 | Change the bound variable ... |
| rmobida 3388 | Formula-building rule for ... |
| reubida 3389 | Formula-building rule for ... |
| cbvrmow 3390 | Change the bound variable ... |
| cbvreuw 3391 | Change the bound variable ... |
| nfrmo1 3392 | The setvar ` x ` is not fr... |
| nfreu1 3393 | The setvar ` x ` is not fr... |
| nfrmow 3394 | Bound-variable hypothesis ... |
| nfreuw 3395 | Bound-variable hypothesis ... |
| rmoeq1 3396 | Equality theorem for restr... |
| reueq1 3397 | Equality theorem for restr... |
| rmoeqd 3398 | Equality deduction for res... |
| reueqd 3399 | Equality deduction for res... |
| reueqdv 3400 | Formula-building rule for ... |
| reueqbidv 3401 | Formula-building rule for ... |
| rmoeq1f 3402 | Equality theorem for restr... |
| reueq1f 3403 | Equality theorem for restr... |
| cbvreu 3404 | Change the bound variable ... |
| cbvrmo 3405 | Change the bound variable ... |
| cbvrmov 3406 | Change the bound variable ... |
| cbvreuv 3407 | Change the bound variable ... |
| nfrmod 3408 | Deduction version of ~ nfr... |
| nfreud 3409 | Deduction version of ~ nfr... |
| nfrmo 3410 | Bound-variable hypothesis ... |
| nfreu 3411 | Bound-variable hypothesis ... |
| rabbidva2 3414 | Equivalent wff's yield equ... |
| rabbia2 3415 | Equivalent wff's yield equ... |
| rabbiia 3416 | Equivalent formulas yield ... |
| rabbii 3417 | Equivalent wff's correspon... |
| rabbidva 3418 | Equivalent wff's yield equ... |
| rabbidv 3419 | Equivalent wff's yield equ... |
| rabbieq 3420 | Equivalent wff's correspon... |
| rabswap 3421 | Swap with a membership rel... |
| cbvrabv 3422 | Rule to change the bound v... |
| rabeqcda 3423 | When ` ps ` is always true... |
| rabeqc 3424 | A restricted class abstrac... |
| rabeqi 3425 | Equality theorem for restr... |
| rabeq 3426 | Equality theorem for restr... |
| rabeqdv 3427 | Equality of restricted cla... |
| rabeqbidva 3428 | Equality of restricted cla... |
| rabeqbidv 3429 | Equality of restricted cla... |
| rabrabi 3430 | Abstract builder restricte... |
| nfrab1 3431 | The abstraction variable i... |
| rabid 3432 | An "identity" law of concr... |
| rabidim1 3433 | Membership in a restricted... |
| reqabi 3434 | Inference from equality of... |
| rabrab 3435 | Abstract builder restricte... |
| rabbida4 3436 | Version of ~ rabbidva2 wit... |
| rabbida 3437 | Equivalent wff's yield equ... |
| rabbid 3438 | Version of ~ rabbidv with ... |
| rabeqd 3439 | Deduction form of ~ rabeq ... |
| rabeqbida 3440 | Version of ~ rabeqbidva wi... |
| rabbi 3441 | Equivalent wff's correspon... |
| rabid2f 3442 | An "identity" law for rest... |
| rabid2im 3443 | One direction of ~ rabid2 ... |
| rabid2 3444 | An "identity" law for rest... |
| rabeqf 3445 | Equality theorem for restr... |
| cbvrabw 3446 | Rule to change the bound v... |
| nfrabw 3447 | A variable not free in a w... |
| nfrab 3448 | A variable not free in a w... |
| cbvrab 3449 | Rule to change the bound v... |
| vjust 3451 | Justification theorem for ... |
| dfv2 3453 | Alternate definition of th... |
| vex 3454 | All setvar variables are s... |
| elv 3455 | If a proposition is implie... |
| elvd 3456 | If a proposition is implie... |
| el2v 3457 | If a proposition is implie... |
| el3v 3458 | If a proposition is implie... |
| el3v3 3459 | If a proposition is implie... |
| eqv 3460 | The universe contains ever... |
| eqvf 3461 | The universe contains ever... |
| abv 3462 | The class of sets verifyin... |
| abvALT 3463 | Alternate proof of ~ abv ,... |
| isset 3464 | Two ways to express that "... |
| cbvexeqsetf 3465 | The expression ` E. x x = ... |
| issetft 3466 | Closed theorem form of ~ i... |
| issetf 3467 | A version of ~ isset that ... |
| isseti 3468 | A way to say " ` A ` is a ... |
| issetri 3469 | A way to say " ` A ` is a ... |
| eqvisset 3470 | A class equal to a variabl... |
| elex 3471 | If a class is a member of ... |
| elexi 3472 | If a class is a member of ... |
| elexd 3473 | If a class is a member of ... |
| elex22 3474 | If two classes each contai... |
| prcnel 3475 | A proper class doesn't bel... |
| ralv 3476 | A universal quantifier res... |
| rexv 3477 | An existential quantifier ... |
| reuv 3478 | A unique existential quant... |
| rmov 3479 | An at-most-one quantifier ... |
| rabab 3480 | A class abstraction restri... |
| rexcom4b 3481 | Specialized existential co... |
| ceqsal1t 3482 | One direction of ~ ceqsalt... |
| ceqsalt 3483 | Closed theorem version of ... |
| ceqsralt 3484 | Restricted quantifier vers... |
| ceqsalg 3485 | A representation of explic... |
| ceqsalgALT 3486 | Alternate proof of ~ ceqsa... |
| ceqsal 3487 | A representation of explic... |
| ceqsalALT 3488 | A representation of explic... |
| ceqsalv 3489 | A representation of explic... |
| ceqsralv 3490 | Restricted quantifier vers... |
| gencl 3491 | Implicit substitution for ... |
| 2gencl 3492 | Implicit substitution for ... |
| 3gencl 3493 | Implicit substitution for ... |
| cgsexg 3494 | Implicit substitution infe... |
| cgsex2g 3495 | Implicit substitution infe... |
| cgsex4g 3496 | An implicit substitution i... |
| ceqsex 3497 | Elimination of an existent... |
| ceqsexv 3498 | Elimination of an existent... |
| ceqsexv2d 3499 | Elimination of an existent... |
| ceqsex2 3500 | Elimination of two existen... |
| ceqsex2v 3501 | Elimination of two existen... |
| ceqsex3v 3502 | Elimination of three exist... |
| ceqsex4v 3503 | Elimination of four existe... |
| ceqsex6v 3504 | Elimination of six existen... |
| ceqsex8v 3505 | Elimination of eight exist... |
| gencbvex 3506 | Change of bound variable u... |
| gencbvex2 3507 | Restatement of ~ gencbvex ... |
| gencbval 3508 | Change of bound variable u... |
| sbhypf 3509 | Introduce an explicit subs... |
| spcimgft 3510 | Closed theorem form of ~ s... |
| spcimgfi1 3511 | A closed version of ~ spci... |
| spcgft 3512 | A closed version of ~ spcg... |
| spcimgf 3513 | Rule of specialization, us... |
| spcimegf 3514 | Existential specialization... |
| vtoclgft 3515 | Closed theorem form of ~ v... |
| vtocleg 3516 | Implicit substitution of a... |
| vtoclg 3517 | Implicit substitution of a... |
| vtocle 3518 | Implicit substitution of a... |
| vtoclbg 3519 | Implicit substitution of a... |
| vtocl 3520 | Implicit substitution of a... |
| vtocldf 3521 | Implicit substitution of a... |
| vtocld 3522 | Implicit substitution of a... |
| vtocl2d 3523 | Implicit substitution of t... |
| vtoclef 3524 | Implicit substitution of a... |
| vtoclf 3525 | Implicit substitution of a... |
| vtocl2 3526 | Implicit substitution of c... |
| vtocl3 3527 | Implicit substitution of c... |
| vtoclb 3528 | Implicit substitution of a... |
| vtoclgf 3529 | Implicit substitution of a... |
| vtoclg1f 3530 | Version of ~ vtoclgf with ... |
| vtocl2gf 3531 | Implicit substitution of a... |
| vtocl3gf 3532 | Implicit substitution of a... |
| vtocl2g 3533 | Implicit substitution of 2... |
| vtocl3g 3534 | Implicit substitution of a... |
| vtoclgaf 3535 | Implicit substitution of a... |
| vtoclga 3536 | Implicit substitution of a... |
| vtocl2ga 3537 | Implicit substitution of 2... |
| vtocl2gaf 3538 | Implicit substitution of 2... |
| vtocl3gaf 3539 | Implicit substitution of 3... |
| vtocl3ga 3540 | Implicit substitution of 3... |
| vtocl4g 3541 | Implicit substitution of 4... |
| vtocl4ga 3542 | Implicit substitution of 4... |
| vtoclegft 3543 | Implicit substitution of a... |
| vtoclri 3544 | Implicit substitution of a... |
| spcgf 3545 | Rule of specialization, us... |
| spcegf 3546 | Existential specialization... |
| spcimdv 3547 | Restricted specialization,... |
| spcdv 3548 | Rule of specialization, us... |
| spcimedv 3549 | Restricted existential spe... |
| spcgv 3550 | Rule of specialization, us... |
| spcegv 3551 | Existential specialization... |
| spcedv 3552 | Existential specialization... |
| spc2egv 3553 | Existential specialization... |
| spc2gv 3554 | Specialization with two qu... |
| spc2ed 3555 | Existential specialization... |
| spc2d 3556 | Specialization with 2 quan... |
| spc3egv 3557 | Existential specialization... |
| spc3gv 3558 | Specialization with three ... |
| spcv 3559 | Rule of specialization, us... |
| spcev 3560 | Existential specialization... |
| spc2ev 3561 | Existential specialization... |
| rspct 3562 | A closed version of ~ rspc... |
| rspcdf 3563 | Restricted specialization,... |
| rspc 3564 | Restricted specialization,... |
| rspce 3565 | Restricted existential spe... |
| rspcimdv 3566 | Restricted specialization,... |
| rspcimedv 3567 | Restricted existential spe... |
| rspcdv 3568 | Restricted specialization,... |
| rspcedv 3569 | Restricted existential spe... |
| rspcebdv 3570 | Restricted existential spe... |
| rspcdv2 3571 | Restricted specialization,... |
| rspcv 3572 | Restricted specialization,... |
| rspccv 3573 | Restricted specialization,... |
| rspcva 3574 | Restricted specialization,... |
| rspccva 3575 | Restricted specialization,... |
| rspcev 3576 | Restricted existential spe... |
| rspcdva 3577 | Restricted specialization,... |
| rspcedvd 3578 | Restricted existential spe... |
| rspcedvdw 3579 | Version of ~ rspcedvd wher... |
| rspceb2dv 3580 | Restricted existential spe... |
| rspcime 3581 | Prove a restricted existen... |
| rspceaimv 3582 | Restricted existential spe... |
| rspcedeqvd 3583 | Restricted existential spe... |
| rspc2 3584 | Restricted specialization ... |
| rspc2gv 3585 | Restricted specialization ... |
| rspc2v 3586 | 2-variable restricted spec... |
| rspc2va 3587 | 2-variable restricted spec... |
| rspc2ev 3588 | 2-variable restricted exis... |
| 2rspcedvdw 3589 | Double application of ~ rs... |
| rspc2dv 3590 | 2-variable restricted spec... |
| rspc3v 3591 | 3-variable restricted spec... |
| rspc3ev 3592 | 3-variable restricted exis... |
| 3rspcedvdw 3593 | Triple application of ~ rs... |
| rspc3dv 3594 | 3-variable restricted spec... |
| rspc4v 3595 | 4-variable restricted spec... |
| rspc6v 3596 | 6-variable restricted spec... |
| rspc8v 3597 | 8-variable restricted spec... |
| rspceeqv 3598 | Restricted existential spe... |
| ralxpxfr2d 3599 | Transfer a universal quant... |
| rexraleqim 3600 | Statement following from e... |
| eqvincg 3601 | A variable introduction la... |
| eqvinc 3602 | A variable introduction la... |
| eqvincf 3603 | A variable introduction la... |
| alexeqg 3604 | Two ways to express substi... |
| ceqex 3605 | Equality implies equivalen... |
| ceqsexg 3606 | A representation of explic... |
| ceqsexgv 3607 | Elimination of an existent... |
| ceqsrexv 3608 | Elimination of a restricte... |
| ceqsrexbv 3609 | Elimination of a restricte... |
| ceqsralbv 3610 | Elimination of a restricte... |
| ceqsrex2v 3611 | Elimination of a restricte... |
| clel2g 3612 | Alternate definition of me... |
| clel2 3613 | Alternate definition of me... |
| clel3g 3614 | Alternate definition of me... |
| clel3 3615 | Alternate definition of me... |
| clel4g 3616 | Alternate definition of me... |
| clel4 3617 | Alternate definition of me... |
| clel5 3618 | Alternate definition of cl... |
| pm13.183 3619 | Compare theorem *13.183 in... |
| rr19.3v 3620 | Restricted quantifier vers... |
| rr19.28v 3621 | Restricted quantifier vers... |
| elab6g 3622 | Membership in a class abst... |
| elabd2 3623 | Membership in a class abst... |
| elabd3 3624 | Membership in a class abst... |
| elabgt 3625 | Membership in a class abst... |
| elabgtOLD 3626 | Obsolete version of ~ elab... |
| elabgf 3627 | Membership in a class abst... |
| elabf 3628 | Membership in a class abst... |
| elabg 3629 | Membership in a class abst... |
| elabgw 3630 | Membership in a class abst... |
| elab2gw 3631 | Membership in a class abst... |
| elab 3632 | Membership in a class abst... |
| elab2g 3633 | Membership in a class abst... |
| elabd 3634 | Explicit demonstration the... |
| elab2 3635 | Membership in a class abst... |
| elab4g 3636 | Membership in a class abst... |
| elab3gf 3637 | Membership in a class abst... |
| elab3g 3638 | Membership in a class abst... |
| elab3 3639 | Membership in a class abst... |
| elrabi 3640 | Implication for the member... |
| elrabf 3641 | Membership in a restricted... |
| rabtru 3642 | Abstract builder using the... |
| elrab3t 3643 | Membership in a restricted... |
| elrab 3644 | Membership in a restricted... |
| elrab3 3645 | Membership in a restricted... |
| elrabd 3646 | Membership in a restricted... |
| elrabrd 3647 | Deduction version of ~ elr... |
| elrab2 3648 | Membership in a restricted... |
| elrab2w 3649 | Membership in a restricted... |
| ralab 3650 | Universal quantification o... |
| ralrab 3651 | Universal quantification o... |
| rexab 3652 | Existential quantification... |
| rexrab 3653 | Existential quantification... |
| ralab2 3654 | Universal quantification o... |
| ralrab2 3655 | Universal quantification o... |
| rexab2 3656 | Existential quantification... |
| rexrab2 3657 | Existential quantification... |
| reurab 3658 | Restricted existential uni... |
| abidnf 3659 | Identity used to create cl... |
| dedhb 3660 | A deduction theorem for co... |
| class2seteq 3661 | Writing a set as a class a... |
| nelrdva 3662 | Deduce negative membership... |
| eqeu 3663 | A condition which implies ... |
| moeq 3664 | There exists at most one s... |
| eueq 3665 | A class is a set if and on... |
| eueqi 3666 | There exists a unique set ... |
| eueq2 3667 | Equality has existential u... |
| eueq3 3668 | Equality has existential u... |
| moeq3 3669 | "At most one" property of ... |
| mosub 3670 | "At most one" remains true... |
| mo2icl 3671 | Theorem for inferring "at ... |
| mob2 3672 | Consequence of "at most on... |
| moi2 3673 | Consequence of "at most on... |
| mob 3674 | Equality implied by "at mo... |
| moi 3675 | Equality implied by "at mo... |
| morex 3676 | Derive membership from uni... |
| euxfr2w 3677 | Transfer existential uniqu... |
| euxfrw 3678 | Transfer existential uniqu... |
| euxfr2 3679 | Transfer existential uniqu... |
| euxfr 3680 | Transfer existential uniqu... |
| euind 3681 | Existential uniqueness via... |
| reu2 3682 | A way to express restricte... |
| reu6 3683 | A way to express restricte... |
| reu3 3684 | A way to express restricte... |
| reu6i 3685 | A condition which implies ... |
| eqreu 3686 | A condition which implies ... |
| rmo4 3687 | Restricted "at most one" u... |
| reu4 3688 | Restricted uniqueness usin... |
| reu7 3689 | Restricted uniqueness usin... |
| reu8 3690 | Restricted uniqueness usin... |
| rmo3f 3691 | Restricted "at most one" u... |
| rmo4f 3692 | Restricted "at most one" u... |
| reu2eqd 3693 | Deduce equality from restr... |
| reueq 3694 | Equality has existential u... |
| rmoeq 3695 | Equality's restricted exis... |
| rmoan 3696 | Restricted "at most one" s... |
| rmoim 3697 | Restricted "at most one" i... |
| rmoimia 3698 | Restricted "at most one" i... |
| rmoimi 3699 | Restricted "at most one" i... |
| rmoimi2 3700 | Restricted "at most one" i... |
| 2reu5a 3701 | Double restricted existent... |
| reuimrmo 3702 | Restricted uniqueness impl... |
| 2reuswap 3703 | A condition allowing swap ... |
| 2reuswap2 3704 | A condition allowing swap ... |
| reuxfrd 3705 | Transfer existential uniqu... |
| reuxfr 3706 | Transfer existential uniqu... |
| reuxfr1d 3707 | Transfer existential uniqu... |
| reuxfr1ds 3708 | Transfer existential uniqu... |
| reuxfr1 3709 | Transfer existential uniqu... |
| reuind 3710 | Existential uniqueness via... |
| 2rmorex 3711 | Double restricted quantifi... |
| 2reu5lem1 3712 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3713 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3714 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3715 | Double restricted existent... |
| 2reurmo 3716 | Double restricted quantifi... |
| 2reurex 3717 | Double restricted quantifi... |
| 2rmoswap 3718 | A condition allowing to sw... |
| 2rexreu 3719 | Double restricted existent... |
| cdeqi 3722 | Deduce conditional equalit... |
| cdeqri 3723 | Property of conditional eq... |
| cdeqth 3724 | Deduce conditional equalit... |
| cdeqnot 3725 | Distribute conditional equ... |
| cdeqal 3726 | Distribute conditional equ... |
| cdeqab 3727 | Distribute conditional equ... |
| cdeqal1 3728 | Distribute conditional equ... |
| cdeqab1 3729 | Distribute conditional equ... |
| cdeqim 3730 | Distribute conditional equ... |
| cdeqcv 3731 | Conditional equality for s... |
| cdeqeq 3732 | Distribute conditional equ... |
| cdeqel 3733 | Distribute conditional equ... |
| nfcdeq 3734 | If we have a conditional e... |
| nfccdeq 3735 | Variation of ~ nfcdeq for ... |
| rru 3736 | Relative version of Russel... |
| ru 3737 | Russell's Paradox. Propos... |
| dfsbcq 3740 | Proper substitution of a c... |
| dfsbcq2 3741 | This theorem, which is sim... |
| sbsbc 3742 | Show that ~ df-sb and ~ df... |
| sbceq1d 3743 | Equality theorem for class... |
| sbceq1dd 3744 | Equality theorem for class... |
| sbceqbid 3745 | Equality theorem for class... |
| sbc8g 3746 | This is the closest we can... |
| sbc2or 3747 | The disjunction of two equ... |
| sbcex 3748 | By our definition of prope... |
| sbceq1a 3749 | Equality theorem for class... |
| sbceq2a 3750 | Equality theorem for class... |
| spsbc 3751 | Specialization: if a formu... |
| spsbcd 3752 | Specialization: if a formu... |
| sbcth 3753 | A substitution into a theo... |
| sbcthdv 3754 | Deduction version of ~ sbc... |
| sbcid 3755 | An identity theorem for su... |
| nfsbc1d 3756 | Deduction version of ~ nfs... |
| nfsbc1 3757 | Bound-variable hypothesis ... |
| nfsbc1v 3758 | Bound-variable hypothesis ... |
| nfsbcdw 3759 | Deduction version of ~ nfs... |
| nfsbcw 3760 | Bound-variable hypothesis ... |
| sbccow 3761 | A composition law for clas... |
| nfsbcd 3762 | Deduction version of ~ nfs... |
| nfsbc 3763 | Bound-variable hypothesis ... |
| sbcco 3764 | A composition law for clas... |
| sbcco2 3765 | A composition law for clas... |
| sbc5 3766 | An equivalence for class s... |
| sbc5ALT 3767 | Alternate proof of ~ sbc5 ... |
| sbc6g 3768 | An equivalence for class s... |
| sbc6 3769 | An equivalence for class s... |
| sbc7 3770 | An equivalence for class s... |
| cbvsbcw 3771 | Change bound variables in ... |
| cbvsbcvw 3772 | Change the bound variable ... |
| cbvsbc 3773 | Change bound variables in ... |
| cbvsbcv 3774 | Change the bound variable ... |
| sbciegft 3775 | Conversion of implicit sub... |
| sbciegf 3776 | Conversion of implicit sub... |
| sbcieg 3777 | Conversion of implicit sub... |
| sbcie2g 3778 | Conversion of implicit sub... |
| sbcie 3779 | Conversion of implicit sub... |
| sbciedf 3780 | Conversion of implicit sub... |
| sbcied 3781 | Conversion of implicit sub... |
| sbcied2 3782 | Conversion of implicit sub... |
| elrabsf 3783 | Membership in a restricted... |
| eqsbc1 3784 | Substitution for the left-... |
| sbcng 3785 | Move negation in and out o... |
| sbcimg 3786 | Distribution of class subs... |
| sbcan 3787 | Distribution of class subs... |
| sbcor 3788 | Distribution of class subs... |
| sbcbig 3789 | Distribution of class subs... |
| sbcn1 3790 | Move negation in and out o... |
| sbcim1 3791 | Distribution of class subs... |
| sbcbid 3792 | Formula-building deduction... |
| sbcbidv 3793 | Formula-building deduction... |
| sbcbii 3794 | Formula-building inference... |
| sbcbi1 3795 | Distribution of class subs... |
| sbcbi2 3796 | Substituting into equivale... |
| sbcal 3797 | Move universal quantifier ... |
| sbcex2 3798 | Move existential quantifie... |
| sbceqal 3799 | Class version of one impli... |
| sbeqalb 3800 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3801 | Substitution for the right... |
| sbc3an 3802 | Distribution of class subs... |
| sbcel1v 3803 | Class substitution into a ... |
| sbcel2gv 3804 | Class substitution into a ... |
| sbcel21v 3805 | Class substitution into a ... |
| sbcimdv 3806 | Substitution analogue of T... |
| sbctt 3807 | Substitution for a variabl... |
| sbcgf 3808 | Substitution for a variabl... |
| sbc19.21g 3809 | Substitution for a variabl... |
| sbcg 3810 | Substitution for a variabl... |
| sbcgfi 3811 | Substitution for a variabl... |
| sbc2iegf 3812 | Conversion of implicit sub... |
| sbc2ie 3813 | Conversion of implicit sub... |
| sbc2iedv 3814 | Conversion of implicit sub... |
| sbc3ie 3815 | Conversion of implicit sub... |
| sbccomlem 3816 | Lemma for ~ sbccom . (Con... |
| sbccom 3817 | Commutative law for double... |
| sbcralt 3818 | Interchange class substitu... |
| sbcrext 3819 | Interchange class substitu... |
| sbcralg 3820 | Interchange class substitu... |
| sbcrex 3821 | Interchange class substitu... |
| sbcreu 3822 | Interchange class substitu... |
| reu8nf 3823 | Restricted uniqueness usin... |
| sbcabel 3824 | Interchange class substitu... |
| rspsbc 3825 | Restricted quantifier vers... |
| rspsbca 3826 | Restricted quantifier vers... |
| rspesbca 3827 | Existence form of ~ rspsbc... |
| spesbc 3828 | Existence form of ~ spsbc ... |
| spesbcd 3829 | form of ~ spsbc . (Contri... |
| sbcth2 3830 | A substitution into a theo... |
| ra4v 3831 | Version of ~ ra4 with a di... |
| ra4 3832 | Restricted quantifier vers... |
| rmo2 3833 | Alternate definition of re... |
| rmo2i 3834 | Condition implying restric... |
| rmo3 3835 | Restricted "at most one" u... |
| rmob 3836 | Consequence of "at most on... |
| rmoi 3837 | Consequence of "at most on... |
| nrmod 3838 | Deduce the negation of a r... |
| rmob2 3839 | Consequence of "restricted... |
| rmoi2 3840 | Consequence of "restricted... |
| rmoanim 3841 | Introduction of a conjunct... |
| rmoanimALT 3842 | Alternate proof of ~ rmoan... |
| reuan 3843 | Introduction of a conjunct... |
| 2reu1 3844 | Double restricted existent... |
| 2reu2 3845 | Double restricted existent... |
| csb2 3848 | Alternate expression for t... |
| csbeq1 3849 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3850 | Equality deduction for pro... |
| csbeq2 3851 | Substituting into equivale... |
| csbeq2d 3852 | Formula-building deduction... |
| csbeq2dv 3853 | Formula-building deduction... |
| csbeq2i 3854 | Formula-building inference... |
| csbeq12dv 3855 | Formula-building inference... |
| cbvcsbw 3856 | Change bound variables in ... |
| cbvcsb 3857 | Change bound variables in ... |
| cbvcsbv 3858 | Change the bound variable ... |
| csbid 3859 | Analogue of ~ sbid for pro... |
| csbeq1a 3860 | Equality theorem for prope... |
| csbcow 3861 | Composition law for chaine... |
| csbco 3862 | Composition law for chaine... |
| csbtt 3863 | Substitution doesn't affec... |
| csbconstgf 3864 | Substitution doesn't affec... |
| csbconstg 3865 | Substitution doesn't affec... |
| csbgfi 3866 | Substitution for a variabl... |
| csbconstgi 3867 | The proper substitution of... |
| nfcsb1d 3868 | Bound-variable hypothesis ... |
| nfcsb1 3869 | Bound-variable hypothesis ... |
| nfcsb1v 3870 | Bound-variable hypothesis ... |
| nfcsbd 3871 | Deduction version of ~ nfc... |
| nfcsbw 3872 | Bound-variable hypothesis ... |
| nfcsb 3873 | Bound-variable hypothesis ... |
| csbhypf 3874 | Introduce an explicit subs... |
| csbiebt 3875 | Conversion of implicit sub... |
| csbiedf 3876 | Conversion of implicit sub... |
| csbieb 3877 | Bidirectional conversion b... |
| csbiebg 3878 | Bidirectional conversion b... |
| csbiegf 3879 | Conversion of implicit sub... |
| csbief 3880 | Conversion of implicit sub... |
| csbie 3881 | Conversion of implicit sub... |
| csbied 3882 | Conversion of implicit sub... |
| csbied2 3883 | Conversion of implicit sub... |
| csbie2t 3884 | Conversion of implicit sub... |
| csbie2 3885 | Conversion of implicit sub... |
| csbie2g 3886 | Conversion of implicit sub... |
| cbvrabcsfw 3887 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3888 | A more general version of ... |
| cbvrexcsf 3889 | A more general version of ... |
| cbvreucsf 3890 | A more general version of ... |
| cbvrabcsf 3891 | A more general version of ... |
| cbvralv2 3892 | Rule used to change the bo... |
| cbvrexv2 3893 | Rule used to change the bo... |
| rspc2vd 3894 | Deduction version of 2-var... |
| difjust 3900 | Soundness justification th... |
| unjust 3902 | Soundness justification th... |
| injust 3904 | Soundness justification th... |
| dfin5 3906 | Alternate definition for t... |
| dfdif2 3907 | Alternate definition of cl... |
| eldif 3908 | Expansion of membership in... |
| eldifd 3909 | If a class is in one class... |
| eldifad 3910 | If a class is in the diffe... |
| eldifbd 3911 | If a class is in the diffe... |
| elneeldif 3912 | The elements of a set diff... |
| velcomp 3913 | Characterization of setvar... |
| elin 3914 | Expansion of membership in... |
| dfss2 3916 | Alternate definition of th... |
| dfss 3917 | Variant of subclass defini... |
| dfss3 3919 | Alternate definition of su... |
| dfss6 3920 | Alternate definition of su... |
| dfssf 3921 | Equivalence for subclass r... |
| dfss3f 3922 | Equivalence for subclass r... |
| nfss 3923 | If ` x ` is not free in ` ... |
| ssel 3924 | Membership relationships f... |
| ssel2 3925 | Membership relationships f... |
| sseli 3926 | Membership implication fro... |
| sselii 3927 | Membership inference from ... |
| sselid 3928 | Membership inference from ... |
| sseld 3929 | Membership deduction from ... |
| sselda 3930 | Membership deduction from ... |
| sseldd 3931 | Membership inference from ... |
| ssneld 3932 | If a class is not in anoth... |
| ssneldd 3933 | If an element is not in a ... |
| ssriv 3934 | Inference based on subclas... |
| ssrd 3935 | Deduction based on subclas... |
| ssrdv 3936 | Deduction based on subclas... |
| sstr2 3937 | Transitivity of subclass r... |
| sstr 3938 | Transitivity of subclass r... |
| sstri 3939 | Subclass transitivity infe... |
| sstrd 3940 | Subclass transitivity dedu... |
| sstrid 3941 | Subclass transitivity dedu... |
| sstrdi 3942 | Subclass transitivity dedu... |
| sylan9ss 3943 | A subclass transitivity de... |
| sylan9ssr 3944 | A subclass transitivity de... |
| eqss 3945 | The subclass relationship ... |
| eqssi 3946 | Infer equality from two su... |
| eqssd 3947 | Equality deduction from tw... |
| sssseq 3948 | If a class is a subclass o... |
| eqrd 3949 | Deduce equality of classes... |
| eqri 3950 | Infer equality of classes ... |
| eqelssd 3951 | Equality deduction from su... |
| ssid 3952 | Any class is a subclass of... |
| ssidd 3953 | Weakening of ~ ssid . (Co... |
| ssv 3954 | Any class is a subclass of... |
| sseq1 3955 | Equality theorem for subcl... |
| sseq2 3956 | Equality theorem for the s... |
| sseq12 3957 | Equality theorem for the s... |
| sseq1i 3958 | An equality inference for ... |
| sseq2i 3959 | An equality inference for ... |
| sseq12i 3960 | An equality inference for ... |
| sseq1d 3961 | An equality deduction for ... |
| sseq2d 3962 | An equality deduction for ... |
| sseq12d 3963 | An equality deduction for ... |
| eqsstrd 3964 | Substitution of equality i... |
| eqsstrrd 3965 | Substitution of equality i... |
| sseqtrd 3966 | Substitution of equality i... |
| sseqtrrd 3967 | Substitution of equality i... |
| eqsstrid 3968 | A chained subclass and equ... |
| eqsstrrid 3969 | A chained subclass and equ... |
| sseqtrdi 3970 | A chained subclass and equ... |
| sseqtrrdi 3971 | A chained subclass and equ... |
| sseqtrid 3972 | Subclass transitivity dedu... |
| sseqtrrid 3973 | Subclass transitivity dedu... |
| eqsstrdi 3974 | A chained subclass and equ... |
| eqsstrrdi 3975 | A chained subclass and equ... |
| eqsstri 3976 | Substitution of equality i... |
| eqsstrri 3977 | Substitution of equality i... |
| sseqtri 3978 | Substitution of equality i... |
| sseqtrri 3979 | Substitution of equality i... |
| 3sstr3i 3980 | Substitution of equality i... |
| 3sstr4i 3981 | Substitution of equality i... |
| 3sstr3g 3982 | Substitution of equality i... |
| 3sstr4g 3983 | Substitution of equality i... |
| 3sstr3d 3984 | Substitution of equality i... |
| 3sstr4d 3985 | Substitution of equality i... |
| eqimssd 3986 | Equality implies inclusion... |
| eqimsscd 3987 | Equality implies inclusion... |
| eqimss 3988 | Equality implies inclusion... |
| eqimss2 3989 | Equality implies inclusion... |
| eqimssi 3990 | Infer subclass relationshi... |
| eqimss2i 3991 | Infer subclass relationshi... |
| nssne1 3992 | Two classes are different ... |
| nssne2 3993 | Two classes are different ... |
| nss 3994 | Negation of subclass relat... |
| nssrex 3995 | Negation of subclass relat... |
| nelss 3996 | Demonstrate by witnesses t... |
| ssrexf 3997 | Restricted existential qua... |
| ssrmof 3998 | "At most one" existential ... |
| ssralv 3999 | Quantification restricted ... |
| ssrexv 4000 | Existential quantification... |
| ss2ralv 4001 | Two quantifications restri... |
| ss2rexv 4002 | Two existential quantifica... |
| ralss 4003 | Restricted universal quant... |
| rexss 4004 | Restricted existential qua... |
| ralssOLD 4005 | Obsolete version of ~ rals... |
| rexssOLD 4006 | Obsolete version of ~ rexs... |
| ss2abim 4007 | Class abstractions in a su... |
| ss2ab 4008 | Class abstractions in a su... |
| abss 4009 | Class abstraction in a sub... |
| ssab 4010 | Subclass of a class abstra... |
| ssabral 4011 | The relation for a subclas... |
| ss2abdv 4012 | Deduction of abstraction s... |
| ss2abi 4013 | Inference of abstraction s... |
| abssdv 4014 | Deduction of abstraction s... |
| abssi 4015 | Inference of abstraction s... |
| ss2rab 4016 | Restricted abstraction cla... |
| rabss 4017 | Restricted class abstracti... |
| ssrab 4018 | Subclass of a restricted c... |
| ss2rabd 4019 | Subclass of a restricted c... |
| ssrabdv 4020 | Subclass of a restricted c... |
| rabssdv 4021 | Subclass of a restricted c... |
| ss2rabdv 4022 | Deduction of restricted ab... |
| ss2rabi 4023 | Inference of restricted ab... |
| rabss2 4024 | Subclass law for restricte... |
| rabss2OLD 4025 | Obsolete version of ~ rabs... |
| ssab2 4026 | Subclass relation for the ... |
| ssrab2 4027 | Subclass relation for a re... |
| rabss3d 4028 | Subclass law for restricte... |
| ssrab3 4029 | Subclass relation for a re... |
| rabssrabd 4030 | Subclass of a restricted c... |
| ssrabeq 4031 | If the restricting class o... |
| rabssab 4032 | A restricted class is a su... |
| eqrrabd 4033 | Deduce equality with a res... |
| uniiunlem 4034 | A subset relationship usef... |
| dfpss2 4035 | Alternate definition of pr... |
| dfpss3 4036 | Alternate definition of pr... |
| psseq1 4037 | Equality theorem for prope... |
| psseq2 4038 | Equality theorem for prope... |
| psseq1i 4039 | An equality inference for ... |
| psseq2i 4040 | An equality inference for ... |
| psseq12i 4041 | An equality inference for ... |
| psseq1d 4042 | An equality deduction for ... |
| psseq2d 4043 | An equality deduction for ... |
| psseq12d 4044 | An equality deduction for ... |
| pssss 4045 | A proper subclass is a sub... |
| pssne 4046 | Two classes in a proper su... |
| pssssd 4047 | Deduce subclass from prope... |
| pssned 4048 | Proper subclasses are uneq... |
| sspss 4049 | Subclass in terms of prope... |
| pssirr 4050 | Proper subclass is irrefle... |
| pssirrOLD 4051 | Obsolete version of ~ pssi... |
| pssn2lp 4052 | Proper subclass has no 2-c... |
| sspsstri 4053 | Two ways of stating tricho... |
| ssnpss 4054 | Partial trichotomy law for... |
| psstr 4055 | Transitive law for proper ... |
| sspsstr 4056 | Transitive law for subclas... |
| psssstr 4057 | Transitive law for subclas... |
| psstrd 4058 | Proper subclass inclusion ... |
| sspsstrd 4059 | Transitivity involving sub... |
| psssstrd 4060 | Transitivity involving sub... |
| npss 4061 | A class is not a proper su... |
| ssnelpss 4062 | A subclass missing a membe... |
| ssnelpssd 4063 | Subclass inclusion with on... |
| ssexnelpss 4064 | If there is an element of ... |
| dfdif3 4065 | Alternate definition of cl... |
| difeq1 4066 | Equality theorem for class... |
| difeq2 4067 | Equality theorem for class... |
| difeq12 4068 | Equality theorem for class... |
| difeq1i 4069 | Inference adding differenc... |
| difeq2i 4070 | Inference adding differenc... |
| difeq12i 4071 | Equality inference for cla... |
| difeq1d 4072 | Deduction adding differenc... |
| difeq2d 4073 | Deduction adding differenc... |
| difeq12d 4074 | Equality deduction for cla... |
| difeqri 4075 | Inference from membership ... |
| nfdif 4076 | Bound-variable hypothesis ... |
| eldifi 4077 | Implication of membership ... |
| eldifn 4078 | Implication of membership ... |
| elndif 4079 | A set does not belong to a... |
| neldif 4080 | Implication of membership ... |
| difdif 4081 | Double class difference. ... |
| difss 4082 | Subclass relationship for ... |
| difssd 4083 | A difference of two classe... |
| difss2 4084 | If a class is contained in... |
| difss2d 4085 | If a class is contained in... |
| ssdifss 4086 | Preservation of a subclass... |
| ddif 4087 | Double complement under un... |
| ssconb 4088 | Contraposition law for sub... |
| sscon 4089 | Contraposition law for sub... |
| ssdif 4090 | Difference law for subsets... |
| ssdifd 4091 | If ` A ` is contained in `... |
| sscond 4092 | If ` A ` is contained in `... |
| ssdifssd 4093 | If ` A ` is contained in `... |
| ssdif2d 4094 | If ` A ` is contained in `... |
| raldifb 4095 | Restricted universal quant... |
| rexdifi 4096 | Restricted existential qua... |
| complss 4097 | Complementation reverses i... |
| compleq 4098 | Two classes are equal if a... |
| elun 4099 | Expansion of membership in... |
| elunnel1 4100 | A member of a union that i... |
| elunnel2 4101 | A member of a union that i... |
| uneqri 4102 | Inference from membership ... |
| unidm 4103 | Idempotent law for union o... |
| uncom 4104 | Commutative law for union ... |
| equncom 4105 | If a class equals the unio... |
| equncomi 4106 | Inference form of ~ equnco... |
| uneq1 4107 | Equality theorem for the u... |
| uneq2 4108 | Equality theorem for the u... |
| uneq12 4109 | Equality theorem for the u... |
| uneq1i 4110 | Inference adding union to ... |
| uneq2i 4111 | Inference adding union to ... |
| uneq12i 4112 | Equality inference for the... |
| uneq1d 4113 | Deduction adding union to ... |
| uneq2d 4114 | Deduction adding union to ... |
| uneq12d 4115 | Equality deduction for the... |
| nfun 4116 | Bound-variable hypothesis ... |
| unass 4117 | Associative law for union ... |
| un12 4118 | A rearrangement of union. ... |
| un23 4119 | A rearrangement of union. ... |
| un4 4120 | A rearrangement of the uni... |
| unundi 4121 | Union distributes over its... |
| unundir 4122 | Union distributes over its... |
| ssun1 4123 | Subclass relationship for ... |
| ssun2 4124 | Subclass relationship for ... |
| ssun3 4125 | Subclass law for union of ... |
| ssun4 4126 | Subclass law for union of ... |
| elun1 4127 | Membership law for union o... |
| elun2 4128 | Membership law for union o... |
| elunant 4129 | A statement is true for ev... |
| unss1 4130 | Subclass law for union of ... |
| ssequn1 4131 | A relationship between sub... |
| unss2 4132 | Subclass law for union of ... |
| unss12 4133 | Subclass law for union of ... |
| ssequn2 4134 | A relationship between sub... |
| unss 4135 | The union of two subclasse... |
| unssi 4136 | An inference showing the u... |
| unssd 4137 | A deduction showing the un... |
| unssad 4138 | If ` ( A u. B ) ` is conta... |
| unssbd 4139 | If ` ( A u. B ) ` is conta... |
| ssun 4140 | A condition that implies i... |
| rexun 4141 | Restricted existential qua... |
| ralunb 4142 | Restricted quantification ... |
| ralun 4143 | Restricted quantification ... |
| elini 4144 | Membership in an intersect... |
| elind 4145 | Deduce membership in an in... |
| elinel1 4146 | Membership in an intersect... |
| elinel2 4147 | Membership in an intersect... |
| elin2 4148 | Membership in a class defi... |
| elin1d 4149 | Elementhood in the first s... |
| elin2d 4150 | Elementhood in the first s... |
| elin3 4151 | Membership in a class defi... |
| nel1nelin 4152 | Membership in an intersect... |
| nel2nelin 4153 | Membership in an intersect... |
| incom 4154 | Commutative law for inters... |
| ineqcom 4155 | Two ways of expressing tha... |
| ineqcomi 4156 | Two ways of expressing tha... |
| ineqri 4157 | Inference from membership ... |
| ineq1 4158 | Equality theorem for inter... |
| ineq2 4159 | Equality theorem for inter... |
| ineq12 4160 | Equality theorem for inter... |
| ineq1i 4161 | Equality inference for int... |
| ineq2i 4162 | Equality inference for int... |
| ineq12i 4163 | Equality inference for int... |
| ineq1d 4164 | Equality deduction for int... |
| ineq2d 4165 | Equality deduction for int... |
| ineq12d 4166 | Equality deduction for int... |
| ineqan12d 4167 | Equality deduction for int... |
| sseqin2 4168 | A relationship between sub... |
| nfin 4169 | Bound-variable hypothesis ... |
| rabbi2dva 4170 | Deduction from a wff to a ... |
| inidm 4171 | Idempotent law for interse... |
| inass 4172 | Associative law for inters... |
| in12 4173 | A rearrangement of interse... |
| in32 4174 | A rearrangement of interse... |
| in13 4175 | A rearrangement of interse... |
| in31 4176 | A rearrangement of interse... |
| inrot 4177 | Rotate the intersection of... |
| in4 4178 | Rearrangement of intersect... |
| inindi 4179 | Intersection distributes o... |
| inindir 4180 | Intersection distributes o... |
| inss1 4181 | The intersection of two cl... |
| inss2 4182 | The intersection of two cl... |
| ssin 4183 | Subclass of intersection. ... |
| ssini 4184 | An inference showing that ... |
| ssind 4185 | A deduction showing that a... |
| ssrin 4186 | Add right intersection to ... |
| sslin 4187 | Add left intersection to s... |
| ssrind 4188 | Add right intersection to ... |
| ss2in 4189 | Intersection of subclasses... |
| ssinss1 4190 | Intersection preserves sub... |
| ssinss1OLD 4191 | Obsolete version of ~ ssin... |
| ssinss1d 4192 | Intersection preserves sub... |
| inss 4193 | Inclusion of an intersecti... |
| ralin 4194 | Restricted universal quant... |
| rexin 4195 | Restricted existential qua... |
| dfss7 4196 | Alternate definition of su... |
| symdifcom 4199 | Symmetric difference is co... |
| symdifeq1 4200 | Equality theorem for symme... |
| symdifeq2 4201 | Equality theorem for symme... |
| nfsymdif 4202 | Hypothesis builder for sym... |
| elsymdif 4203 | Membership in a symmetric ... |
| dfsymdif4 4204 | Alternate definition of th... |
| elsymdifxor 4205 | Membership in a symmetric ... |
| dfsymdif2 4206 | Alternate definition of th... |
| symdifass 4207 | Symmetric difference is as... |
| difsssymdif 4208 | The symmetric difference c... |
| difsymssdifssd 4209 | If the symmetric differenc... |
| unabs 4210 | Absorption law for union. ... |
| inabs 4211 | Absorption law for interse... |
| nssinpss 4212 | Negation of subclass expre... |
| nsspssun 4213 | Negation of subclass expre... |
| dfss4 4214 | Subclass defined in terms ... |
| dfun2 4215 | An alternate definition of... |
| dfin2 4216 | An alternate definition of... |
| difin 4217 | Difference with intersecti... |
| ssdifim 4218 | Implication of a class dif... |
| ssdifsym 4219 | Symmetric class difference... |
| dfss5 4220 | Alternate definition of su... |
| dfun3 4221 | Union defined in terms of ... |
| dfin3 4222 | Intersection defined in te... |
| dfin4 4223 | Alternate definition of th... |
| invdif 4224 | Intersection with universa... |
| indif 4225 | Intersection with class di... |
| indif2 4226 | Bring an intersection in a... |
| indif1 4227 | Bring an intersection in a... |
| indifcom 4228 | Commutation law for inters... |
| indi 4229 | Distributive law for inter... |
| undi 4230 | Distributive law for union... |
| indir 4231 | Distributive law for inter... |
| undir 4232 | Distributive law for union... |
| unineq 4233 | Infer equality from equali... |
| uneqin 4234 | Equality of union and inte... |
| difundi 4235 | Distributive law for class... |
| difundir 4236 | Distributive law for class... |
| difindi 4237 | Distributive law for class... |
| difindir 4238 | Distributive law for class... |
| indifdi 4239 | Distribute intersection ov... |
| indifdir 4240 | Distribute intersection ov... |
| difdif2 4241 | Class difference by a clas... |
| undm 4242 | De Morgan's law for union.... |
| indm 4243 | De Morgan's law for inters... |
| difun1 4244 | A relationship involving d... |
| undif3 4245 | An equality involving clas... |
| difin2 4246 | Represent a class differen... |
| dif32 4247 | Swap second and third argu... |
| difabs 4248 | Absorption-like law for cl... |
| sscon34b 4249 | Relative complementation r... |
| rcompleq 4250 | Two subclasses are equal i... |
| dfsymdif3 4251 | Alternate definition of th... |
| unabw 4252 | Union of two class abstrac... |
| unab 4253 | Union of two class abstrac... |
| inab 4254 | Intersection of two class ... |
| difab 4255 | Difference of two class ab... |
| abanssl 4256 | A class abstraction with a... |
| abanssr 4257 | A class abstraction with a... |
| notabw 4258 | A class abstraction define... |
| notab 4259 | A class abstraction define... |
| unrab 4260 | Union of two restricted cl... |
| inrab 4261 | Intersection of two restri... |
| inrab2 4262 | Intersection with a restri... |
| difrab 4263 | Difference of two restrict... |
| dfrab3 4264 | Alternate definition of re... |
| dfrab2 4265 | Alternate definition of re... |
| rabdif 4266 | Move difference in and out... |
| notrab 4267 | Complementation of restric... |
| dfrab3ss 4268 | Restricted class abstracti... |
| rabun2 4269 | Abstraction restricted to ... |
| reuun2 4270 | Transfer uniqueness to a s... |
| reuss2 4271 | Transfer uniqueness to a s... |
| reuss 4272 | Transfer uniqueness to a s... |
| reuun1 4273 | Transfer uniqueness to a s... |
| reupick 4274 | Restricted uniqueness "pic... |
| reupick3 4275 | Restricted uniqueness "pic... |
| reupick2 4276 | Restricted uniqueness "pic... |
| euelss 4277 | Transfer uniqueness of an ... |
| dfnul4 4280 | Alternate definition of th... |
| dfnul2 4281 | Alternate definition of th... |
| dfnul3 4282 | Alternate definition of th... |
| noel 4283 | The empty set has no eleme... |
| nel02 4284 | The empty set has no eleme... |
| n0i 4285 | If a class has elements, t... |
| ne0i 4286 | If a class has elements, t... |
| ne0d 4287 | Deduction form of ~ ne0i .... |
| n0ii 4288 | If a class has elements, t... |
| ne0ii 4289 | If a class has elements, t... |
| vn0 4290 | The universal class is not... |
| vn0OLD 4291 | Obsolete version of ~ vn0 ... |
| vn0ALT 4292 | Alternate proof of ~ vn0 .... |
| eq0f 4293 | A class is equal to the em... |
| neq0f 4294 | A class is not empty if an... |
| n0f 4295 | A class is nonempty if and... |
| eq0 4296 | A class is equal to the em... |
| eq0ALT 4297 | Alternate proof of ~ eq0 .... |
| neq0 4298 | A class is not empty if an... |
| n0 4299 | A class is nonempty if and... |
| n0limd 4300 | Deduction rule for nonempt... |
| nel0 4301 | From the general negation ... |
| reximdva0 4302 | Restricted existence deduc... |
| rspn0 4303 | Specialization for restric... |
| n0rex 4304 | There is an element in a n... |
| ssn0rex 4305 | There is an element in a c... |
| n0moeu 4306 | A case of equivalence of "... |
| rex0 4307 | Vacuous restricted existen... |
| reu0 4308 | Vacuous restricted uniquen... |
| rmo0 4309 | Vacuous restricted at-most... |
| 0el 4310 | Membership of the empty se... |
| n0el 4311 | Negated membership of the ... |
| eqeuel 4312 | A condition which implies ... |
| ssdif0 4313 | Subclass expressed in term... |
| difn0 4314 | If the difference of two s... |
| pssdifn0 4315 | A proper subclass has a no... |
| pssdif 4316 | A proper subclass has a no... |
| ndisj 4317 | Express that an intersecti... |
| inn0f 4318 | A nonempty intersection. ... |
| inn0 4319 | A nonempty intersection. ... |
| difin0ss 4320 | Difference, intersection, ... |
| inssdif0 4321 | Intersection, subclass, an... |
| inssdif0OLD 4322 | Obsolete version of ~ inss... |
| inindif 4323 | The intersection and class... |
| difid 4324 | The difference between a c... |
| difidALT 4325 | Alternate proof of ~ difid... |
| dif0 4326 | The difference between a c... |
| ab0w 4327 | The class of sets verifyin... |
| ab0 4328 | The class of sets verifyin... |
| ab0ALT 4329 | Alternate proof of ~ ab0 ,... |
| dfnf5 4330 | Characterization of nonfre... |
| ab0orv 4331 | The class abstraction defi... |
| ab0orvALT 4332 | Alternate proof of ~ ab0or... |
| abn0 4333 | Nonempty class abstraction... |
| rab0 4334 | Any restricted class abstr... |
| rab0OLD 4335 | Obsolete version of ~ rab0... |
| rabeq0w 4336 | Condition for a restricted... |
| rabeq0 4337 | Condition for a restricted... |
| rabn0 4338 | Nonempty restricted class ... |
| rabxm 4339 | Law of excluded middle, in... |
| rabnc 4340 | Law of noncontradiction, i... |
| elneldisj 4341 | The set of elements ` s ` ... |
| elnelun 4342 | The union of the set of el... |
| un0 4343 | The union of a class with ... |
| in0 4344 | The intersection of a clas... |
| 0un 4345 | The union of the empty set... |
| 0in 4346 | The intersection of the em... |
| inv1 4347 | The intersection of a clas... |
| unv 4348 | The union of a class with ... |
| 0ss 4349 | The empty set is a subset ... |
| ss0b 4350 | Any subset of the empty se... |
| ss0 4351 | Any subset of the empty se... |
| sseq0b 4352 | The only subclass of the e... |
| sseq0 4353 | A subclass of an empty cla... |
| ssn0 4354 | A class with a nonempty su... |
| 0dif 4355 | The difference between the... |
| un00 4356 | Two classes are both empty... |
| vss 4357 | Only the universal class h... |
| vvin 4358 | Two classes are both the u... |
| 0pss 4359 | The empty set is a proper ... |
| npss0 4360 | No set is a proper subset ... |
| pssv 4361 | Any non-universal class is... |
| nvpss 4362 | No class strictly includes... |
| abf 4363 | A class abstraction determ... |
| 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... |
| disj 4402 | Two ways of saying that tw... |
| disjr 4403 | Two ways of saying that tw... |
| disj1 4404 | Two ways of saying that tw... |
| reldisj 4405 | Two ways of saying that tw... |
| disj3 4406 | Two ways of saying that tw... |
| disjne 4407 | Members of disjoint sets a... |
| disjeq0 4408 | Two disjoint sets are equa... |
| disjel 4409 | A set can't belong to both... |
| disj2 4410 | Two ways of saying that tw... |
| disj4 4411 | Two ways of saying that tw... |
| ssdisj 4412 | Intersection with a subcla... |
| disjpss 4413 | A class is a proper subset... |
| undisj1 4414 | The union of disjoint clas... |
| undisj2 4415 | The union of disjoint clas... |
| ssindif0 4416 | Subclass expressed in term... |
| inelcm 4417 | The intersection of classe... |
| minel 4418 | A minimum element of a cla... |
| undif4 4419 | Distribute union over diff... |
| disjssun 4420 | Subset relation for disjoi... |
| vdif0 4421 | Universal class equality i... |
| difrab0eq 4422 | If the difference between ... |
| pssnel 4423 | A proper subclass has a me... |
| disjdifg 4424 | A class does not intersect... |
| disjdif 4425 | A class and its relative c... |
| disjdifr 4426 | A class and its relative c... |
| difin0 4427 | The difference of a class ... |
| unvdif 4428 | The union of a class and i... |
| undif1 4429 | Absorption of difference b... |
| undif2 4430 | Absorption of difference b... |
| srcmpltd 4431 | If a statement is true for... |
| prsrcmpltd 4432 | If a statement is true for... |
| undifabs 4433 | Absorption of difference b... |
| inundif 4434 | The intersection and class... |
| disjdif2 4435 | The difference of a class ... |
| difun2 4436 | Absorption of union by dif... |
| undif 4437 | Union of complementary par... |
| undifr 4438 | Union of complementary par... |
| undif5 4439 | An equality involving clas... |
| ssdifin0 4440 | A subset of a difference d... |
| ssdifeq0 4441 | A class is a subclass of i... |
| ssundif 4442 | A condition equivalent to ... |
| difcom 4443 | Swap the arguments of a cl... |
| pssdifcom1 4444 | Two ways to express overla... |
| pssdifcom2 4445 | Two ways to express non-co... |
| difdifdir 4446 | Distributive law for class... |
| uneqdifeq 4447 | Two ways to say that ` A `... |
| raldifeq 4448 | Equality theorem for restr... |
| rzal 4449 | Vacuous quantification is ... |
| rzalALT 4450 | Alternate proof of ~ rzal ... |
| rexn0 4451 | Restricted existential qua... |
| ralf0 4452 | The quantification of a fa... |
| ral0 4453 | Vacuous universal quantifi... |
| r19.2z 4454 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4455 | A response to the notion t... |
| r19.3rz 4456 | Restricted quantification ... |
| r19.28z 4457 | Restricted quantifier vers... |
| r19.3rzv 4458 | Restricted quantification ... |
| r19.3rzvOLD 4459 | Obsolete version of ~ r19.... |
| r19.9rzv 4460 | Restricted quantification ... |
| r19.28zv 4461 | Restricted quantifier vers... |
| r19.37zv 4462 | Restricted quantifier vers... |
| r19.45zv 4463 | Restricted version of Theo... |
| r19.44zv 4464 | Restricted version of Theo... |
| r19.27z 4465 | Restricted quantifier vers... |
| r19.27zv 4466 | Restricted quantifier vers... |
| r19.36zv 4467 | Restricted quantifier vers... |
| ralnralall 4468 | A contradiction concerning... |
| falseral0 4469 | A false statement can only... |
| falseral0OLD 4470 | Obsolete version of ~ fals... |
| ralidmw 4471 | Idempotent law for restric... |
| ralidm 4472 | Idempotent law for restric... |
| raaan 4473 | Rearrange restricted quant... |
| raaanv 4474 | Rearrange restricted quant... |
| sbss 4475 | Set substitution into the ... |
| sbcssg 4476 | Distribute proper substitu... |
| raaan2 4477 | Rearrange restricted quant... |
| 2reu4lem 4478 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4479 | Definition of double restr... |
| csbdif 4480 | Distribution of class subs... |
| dfif2 4483 | An alternate definition of... |
| dfif6 4484 | An alternate definition of... |
| ifeq1 4485 | Equality theorem for condi... |
| ifeq2 4486 | Equality theorem for condi... |
| iftrue 4487 | Value of the conditional o... |
| iftruei 4488 | Inference associated with ... |
| iftrued 4489 | Value of the conditional o... |
| iffalse 4490 | Value of the conditional o... |
| iffalsei 4491 | Inference associated with ... |
| iffalsed 4492 | Value of the conditional o... |
| ifnefalse 4493 | When values are unequal, b... |
| iftrueb 4494 | When the branches are not ... |
| ifsb 4495 | Distribute a function over... |
| dfif3 4496 | Alternate definition of th... |
| dfif4 4497 | Alternate definition of th... |
| dfif5 4498 | Alternate definition of th... |
| ifssun 4499 | A conditional class is inc... |
| ifeq12 4500 | Equality theorem for condi... |
| ifeq1d 4501 | Equality deduction for con... |
| ifeq2d 4502 | Equality deduction for con... |
| ifeq12d 4503 | Equality deduction for con... |
| ifbi 4504 | Equivalence theorem for co... |
| ifbid 4505 | Equivalence deduction for ... |
| ifbieq1d 4506 | Equivalence/equality deduc... |
| ifbieq2i 4507 | Equivalence/equality infer... |
| ifbieq2d 4508 | Equivalence/equality deduc... |
| ifbieq12i 4509 | Equivalence deduction for ... |
| ifbieq12d 4510 | Equivalence deduction for ... |
| nfifd 4511 | Deduction form of ~ nfif .... |
| nfif 4512 | Bound-variable hypothesis ... |
| ifeq1da 4513 | Conditional equality. (Co... |
| ifeq2da 4514 | Conditional equality. (Co... |
| ifeq12da 4515 | Equivalence deduction for ... |
| ifbieq12d2 4516 | Equivalence deduction for ... |
| ifclda 4517 | Conditional closure. (Con... |
| ifeqda 4518 | Separation of the values o... |
| elimif 4519 | Elimination of a condition... |
| ifbothda 4520 | A wff ` th ` containing a ... |
| ifboth 4521 | A wff ` th ` containing a ... |
| ifid 4522 | Identical true and false a... |
| eqif 4523 | Expansion of an equality w... |
| ifval 4524 | Another expression of the ... |
| elif 4525 | Membership in a conditiona... |
| ifel 4526 | Membership of a conditiona... |
| ifcl 4527 | Membership (closure) of a ... |
| ifcld 4528 | Membership (closure) of a ... |
| ifcli 4529 | Inference associated with ... |
| ifexd 4530 | Existence of the condition... |
| ifexg 4531 | Existence of the condition... |
| ifex 4532 | Existence of the condition... |
| ifeqor 4533 | The possible values of a c... |
| ifnot 4534 | Negating the first argumen... |
| ifan 4535 | Rewrite a conjunction in a... |
| ifor 4536 | Rewrite a disjunction in a... |
| 2if2 4537 | Resolve two nested conditi... |
| ifcomnan 4538 | Commute the conditions in ... |
| csbif 4539 | Distribute proper substitu... |
| dedth 4540 | Weak deduction theorem tha... |
| dedth2h 4541 | Weak deduction theorem eli... |
| dedth3h 4542 | Weak deduction theorem eli... |
| dedth4h 4543 | Weak deduction theorem eli... |
| dedth2v 4544 | Weak deduction theorem for... |
| dedth3v 4545 | Weak deduction theorem for... |
| dedth4v 4546 | Weak deduction theorem for... |
| elimhyp 4547 | Eliminate a hypothesis con... |
| elimhyp2v 4548 | Eliminate a hypothesis con... |
| elimhyp3v 4549 | Eliminate a hypothesis con... |
| elimhyp4v 4550 | Eliminate a hypothesis con... |
| elimel 4551 | Eliminate a membership hyp... |
| elimdhyp 4552 | Version of ~ elimhyp where... |
| keephyp 4553 | Transform a hypothesis ` p... |
| keephyp2v 4554 | Keep a hypothesis containi... |
| keephyp3v 4555 | Keep a hypothesis containi... |
| pwjust 4557 | Soundness justification th... |
| elpwg 4559 | Membership in a power clas... |
| elpw 4560 | Membership in a power clas... |
| velpw 4561 | Setvar variable membership... |
| elpwd 4562 | Membership in a power clas... |
| elpwi 4563 | Subset relation implied by... |
| elpwb 4564 | Characterization of the el... |
| elpwid 4565 | An element of a power clas... |
| elelpwi 4566 | If ` A ` belongs to a part... |
| sspw 4567 | The powerclass preserves i... |
| sspwi 4568 | The powerclass preserves i... |
| sspwd 4569 | The powerclass preserves i... |
| pweq 4570 | Equality theorem for power... |
| pweqALT 4571 | Alternate proof of ~ pweq ... |
| pweqi 4572 | Equality inference for pow... |
| pweqd 4573 | Equality deduction for pow... |
| pwunss 4574 | The power class of the uni... |
| nfpw 4575 | Bound-variable hypothesis ... |
| pwidg 4576 | A set is an element of its... |
| pwidgOLD 4577 | Obsolete version of ~ pwid... |
| pwidb 4578 | A class is an element of i... |
| pwid 4579 | A set is a member of its p... |
| pwss 4580 | Subclass relationship for ... |
| pwundif 4581 | Break up the power class o... |
| snjust 4582 | Soundness justification th... |
| sneq 4593 | Equality theorem for singl... |
| sneqi 4594 | Equality inference for sin... |
| sneqd 4595 | Equality deduction for sin... |
| dfsn2 4596 | Alternate definition of si... |
| elsng 4597 | There is exactly one eleme... |
| elsn 4598 | There is exactly one eleme... |
| velsn 4599 | There is only one element ... |
| elsni 4600 | There is at most one eleme... |
| elsnd 4601 | There is at most one eleme... |
| rabsneq 4602 | Equality of class abstract... |
| absn 4603 | Condition for a class abst... |
| dfpr2 4604 | Alternate definition of a ... |
| dfsn2ALT 4605 | Alternate definition of si... |
| elprg 4606 | A member of a pair of clas... |
| elpri 4607 | If a class is an element o... |
| elpr 4608 | A member of a pair of clas... |
| elpr2g 4609 | A member of a pair of sets... |
| elpr2 4610 | A member of a pair of sets... |
| elprn1 4611 | A member of an unordered p... |
| elprn2 4612 | A member of an unordered p... |
| 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... |
| rabeqsnd 4629 | Conditions for a restricte... |
| ralsnsg 4630 | Substitution expressed in ... |
| rexsns 4631 | Restricted existential qua... |
| rexsngf 4632 | Restricted existential qua... |
| ralsngf 4633 | Restricted universal quant... |
| reusngf 4634 | Restricted existential uni... |
| ralsng 4635 | Substitution expressed in ... |
| rexsng 4636 | Restricted existential qua... |
| reusng 4637 | Restricted existential uni... |
| 2ralsng 4638 | Substitution expressed in ... |
| rexreusng 4639 | Restricted existential uni... |
| exsnrex 4640 | There is a set being the e... |
| ralsn 4641 | Convert a universal quanti... |
| rexsn 4642 | Convert an existential qua... |
| elunsn 4643 | Elementhood in a union wit... |
| 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... |
| el7g 4650 | Members of a set with seve... |
| dftp2 4651 | Alternate definition of un... |
| nfpr 4652 | Bound-variable hypothesis ... |
| ifpr 4653 | Membership of a conditiona... |
| ralprgf 4654 | Convert a restricted unive... |
| rexprgf 4655 | Convert a restricted exist... |
| ralprg 4656 | Convert a restricted unive... |
| rexprg 4657 | Convert a restricted exist... |
| raltpg 4658 | Convert a restricted unive... |
| rextpg 4659 | Convert a restricted exist... |
| ralpr 4660 | Convert a restricted unive... |
| rexpr 4661 | Convert a restricted exist... |
| reuprg0 4662 | Convert a restricted exist... |
| reuprg 4663 | Convert a restricted exist... |
| reurexprg 4664 | Convert a restricted exist... |
| raltp 4665 | Convert a universal quanti... |
| rextp 4666 | Convert an existential qua... |
| nfsn 4667 | Bound-variable hypothesis ... |
| csbsng 4668 | Distribute proper substitu... |
| csbprg 4669 | Distribute proper substitu... |
| elinsn 4670 | If the intersection of two... |
| disjsn 4671 | Intersection with the sing... |
| disjsn2 4672 | Two distinct singletons ar... |
| disjpr2 4673 | Two completely distinct un... |
| disjprsn 4674 | The disjoint intersection ... |
| disjtpsn 4675 | The disjoint intersection ... |
| disjtp2 4676 | Two completely distinct un... |
| snprc 4677 | The singleton of a proper ... |
| snnzb 4678 | A singleton is nonempty if... |
| rmosn 4679 | A restricted at-most-one q... |
| r19.12sn 4680 | Special case of ~ r19.12 w... |
| rabsn 4681 | Condition where a restrict... |
| rabsnifsb 4682 | A restricted class abstrac... |
| rabsnif 4683 | A restricted class abstrac... |
| rabrsn 4684 | A restricted class abstrac... |
| euabsn2 4685 | Another way to express exi... |
| euabsn 4686 | Another way to express exi... |
| reusn 4687 | A way to express restricte... |
| absneu 4688 | Restricted existential uni... |
| rabsneu 4689 | Restricted existential uni... |
| eusn 4690 | Two ways to express " ` A ... |
| rabsnt 4691 | Truth implied by equality ... |
| prcom 4692 | Commutative law for unorde... |
| preq1 4693 | Equality theorem for unord... |
| preq2 4694 | Equality theorem for unord... |
| preq12 4695 | Equality theorem for unord... |
| preq1i 4696 | Equality inference for uno... |
| preq2i 4697 | Equality inference for uno... |
| preq12i 4698 | Equality inference for uno... |
| preq1d 4699 | Equality deduction for uno... |
| preq2d 4700 | Equality deduction for uno... |
| preq12d 4701 | Equality deduction for uno... |
| tpeq1 4702 | Equality theorem for unord... |
| tpeq2 4703 | Equality theorem for unord... |
| tpeq3 4704 | Equality theorem for unord... |
| tpeq1d 4705 | Equality theorem for unord... |
| tpeq2d 4706 | Equality theorem for unord... |
| tpeq3d 4707 | Equality theorem for unord... |
| tpeq123d 4708 | Equality theorem for unord... |
| tprot 4709 | Rotation of the elements o... |
| tpcoma 4710 | Swap 1st and 2nd members o... |
| tpcomb 4711 | Swap 2nd and 3rd members o... |
| tpass 4712 | Split off the first elemen... |
| qdass 4713 | Two ways to write an unord... |
| qdassr 4714 | Two ways to write an unord... |
| tpidm12 4715 | Unordered triple ` { A , A... |
| tpidm13 4716 | Unordered triple ` { A , B... |
| tpidm23 4717 | Unordered triple ` { A , B... |
| tpidm 4718 | Unordered triple ` { A , A... |
| tppreq3 4719 | An unordered triple is an ... |
| prid1g 4720 | An unordered pair contains... |
| prid2g 4721 | An unordered pair contains... |
| prid1 4722 | An unordered pair contains... |
| prid2 4723 | An unordered pair contains... |
| ifpprsnss 4724 | An unordered pair is a sin... |
| prprc1 4725 | A proper class vanishes in... |
| prprc2 4726 | A proper class vanishes in... |
| prprc 4727 | An unordered pair containi... |
| tpid1 4728 | One of the three elements ... |
| tpid1g 4729 | Closed theorem form of ~ t... |
| tpid2 4730 | One of the three elements ... |
| tpid2g 4731 | Closed theorem form of ~ t... |
| tpid3g 4732 | Closed theorem form of ~ t... |
| tpid3 4733 | One of the three elements ... |
| snnzg 4734 | The singleton of a set is ... |
| snn0d 4735 | The singleton of a set is ... |
| snnz 4736 | The singleton of a set is ... |
| prnz 4737 | A pair containing a set is... |
| prnzg 4738 | A pair containing a set is... |
| tpnz 4739 | An unordered triple contai... |
| tpnzd 4740 | An unordered triple contai... |
| raltpd 4741 | Convert a universal quanti... |
| snssb 4742 | Characterization of the in... |
| snssg 4743 | The singleton formed on a ... |
| snss 4744 | The singleton of an elemen... |
| snssi 4745 | The singleton of an elemen... |
| snssd 4746 | The singleton of an elemen... |
| eldifsn 4747 | Membership in a set with a... |
| eldifsnbd 4748 | Membership in a set with a... |
| eldifsnd 4749 | Membership in a set with a... |
| ssdifsn 4750 | Subset of a set with an el... |
| elpwdifsn 4751 | A subset of a set is an el... |
| eldifsni 4752 | Membership in a set with a... |
| eldifsnneq 4753 | An element of a difference... |
| neldifsn 4754 | The class ` A ` is not in ... |
| neldifsnd 4755 | The class ` A ` is not in ... |
| rexdifsn 4756 | Restricted existential qua... |
| raldifsni 4757 | Rearrangement of a propert... |
| raldifsnb 4758 | Restricted universal quant... |
| eldifvsn 4759 | A set is an element of the... |
| difsn 4760 | An element not in a set ca... |
| difprsnss 4761 | Removal of a singleton fro... |
| difprsn1 4762 | Removal of a singleton fro... |
| difprsn2 4763 | Removal of a singleton fro... |
| diftpsn3 4764 | Removal of a singleton fro... |
| difpr 4765 | Removing two elements as p... |
| tpprceq3 4766 | An unordered triple is an ... |
| tppreqb 4767 | An unordered triple is an ... |
| difsnb 4768 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4769 | ` ( B \ { A } ) ` is a pro... |
| 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... |
| eqsnd 4790 | Deduce that a set is a sin... |
| issn 4791 | A sufficient condition for... |
| n0snor2el 4792 | A nonempty set is either a... |
| ssunpr 4793 | Possible values for a set ... |
| sspr 4794 | The subsets of a pair. (C... |
| sstp 4795 | The subsets of an unordere... |
| tpss 4796 | An unordered triple of ele... |
| tpssi 4797 | An unordered triple of ele... |
| sneqrg 4798 | Closed form of ~ sneqr . ... |
| sneqr 4799 | If the singletons of two s... |
| snsssn 4800 | If a singleton is a subset... |
| mosneq 4801 | There exists at most one s... |
| sneqbg 4802 | Two singletons of sets are... |
| snsspw 4803 | The singleton of a class i... |
| prsspw 4804 | An unordered pair belongs ... |
| preq1b 4805 | Biconditional equality lem... |
| preq2b 4806 | Biconditional equality lem... |
| preqr1 4807 | Reverse equality lemma for... |
| preqr2 4808 | Reverse equality lemma for... |
| preq12b 4809 | Equality relationship for ... |
| opthpr 4810 | An unordered pair has the ... |
| preqr1g 4811 | Reverse equality lemma for... |
| preq12bg 4812 | Closed form of ~ preq12b .... |
| prneimg 4813 | Two pairs are not equal if... |
| prneimg2 4814 | Two pairs are not equal if... |
| prnebg 4815 | A (proper) pair is not equ... |
| pr1eqbg 4816 | A (proper) pair is equal t... |
| pr1nebg 4817 | A (proper) pair is not equ... |
| preqsnd 4818 | Equivalence for a pair equ... |
| prnesn 4819 | A proper unordered pair is... |
| prneprprc 4820 | A proper unordered pair is... |
| preqsn 4821 | Equivalence for a pair equ... |
| preq12nebg 4822 | Equality relationship for ... |
| prel12g 4823 | Equality of two unordered ... |
| opthprneg 4824 | An unordered pair has the ... |
| elpreqprlem 4825 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4826 | Equality and membership ru... |
| elpreqprb 4827 | A set is an element of an ... |
| elpr2elpr 4828 | For an element ` A ` of an... |
| dfopif 4829 | Rewrite ~ df-op using ` if... |
| dfopg 4830 | Value of the ordered pair ... |
| dfop 4831 | Value of an ordered pair w... |
| opeq1 4832 | Equality theorem for order... |
| opeq2 4833 | Equality theorem for order... |
| opeq12 4834 | Equality theorem for order... |
| opeq1i 4835 | Equality inference for ord... |
| opeq2i 4836 | Equality inference for ord... |
| opeq12i 4837 | Equality inference for ord... |
| opeq1d 4838 | Equality deduction for ord... |
| opeq2d 4839 | Equality deduction for ord... |
| opeq12d 4840 | Equality deduction for ord... |
| oteq1 4841 | Equality theorem for order... |
| oteq2 4842 | Equality theorem for order... |
| oteq3 4843 | Equality theorem for order... |
| oteq1d 4844 | Equality deduction for ord... |
| oteq2d 4845 | Equality deduction for ord... |
| oteq3d 4846 | Equality deduction for ord... |
| oteq123d 4847 | Equality deduction for ord... |
| nfop 4848 | Bound-variable hypothesis ... |
| nfopd 4849 | Deduction version of bound... |
| csbopg 4850 | Distribution of class subs... |
| opidg 4851 | The ordered pair ` <. A , ... |
| opid 4852 | The ordered pair ` <. A , ... |
| ralunsn 4853 | Restricted quantification ... |
| 2ralunsn 4854 | Double restricted quantifi... |
| opprc 4855 | Expansion of an ordered pa... |
| opprc1 4856 | Expansion of an ordered pa... |
| opprc2 4857 | Expansion of an ordered pa... |
| oprcl 4858 | If an ordered pair has an ... |
| pwsn 4859 | The power set of a singlet... |
| pwpr 4860 | The power set of an unorde... |
| pwtp 4861 | The power set of an unorde... |
| pwpwpw0 4862 | Compute the power set of t... |
| pwv 4863 | The power class of the uni... |
| prproe 4864 | For an element of a proper... |
| 3elpr2eq 4865 | If there are three element... |
| dfuni2 4868 | Alternate definition of cl... |
| eluni 4869 | Membership in class union.... |
| eluni2 4870 | Membership in class union.... |
| elunii 4871 | Membership in class union.... |
| nfunid 4872 | Deduction version of ~ nfu... |
| nfuni 4873 | Bound-variable hypothesis ... |
| uniss 4874 | Subclass relationship for ... |
| unissi 4875 | Subclass relationship for ... |
| unissd 4876 | Subclass relationship for ... |
| unieq 4877 | Equality theorem for class... |
| 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... |
| unisng 4884 | A set equals the union of ... |
| unisn 4885 | A set equals the union of ... |
| unisnv 4886 | A set equals the union of ... |
| unisn3 4887 | Union of a singleton in th... |
| dfnfc2 4888 | An alternative statement o... |
| uniun 4889 | The class union of the uni... |
| uniin 4890 | The class union of the int... |
| uniinOLD 4891 | Obsolete version of ~ unii... |
| ssuni 4892 | Subclass relationship for ... |
| uni0b 4893 | The union of a set is empt... |
| uni0c 4894 | The union of a set is empt... |
| uni0 4895 | The union of the empty set... |
| uni0OLD 4896 | Obsolete version of ~ uni0... |
| 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... |
| uniss2 4901 | A subclass condition on th... |
| unidif 4902 | If the difference ` A \ B ... |
| ssunieq 4903 | Relationship implying unio... |
| unimax 4904 | Any member of a class is t... |
| pwuni 4905 | A class is a subclass of t... |
| dfint2 4908 | Alternate definition of cl... |
| inteq 4909 | Equality law for intersect... |
| inteqi 4910 | Equality inference for cla... |
| inteqd 4911 | Equality deduction for cla... |
| elint 4912 | Membership in class inters... |
| elint2 4913 | Membership in class inters... |
| elintg 4914 | Membership in class inters... |
| elinti 4915 | Membership in class inters... |
| nfint 4916 | Bound-variable hypothesis ... |
| elintabg 4917 | Two ways of saying a set i... |
| elintab 4918 | Membership in the intersec... |
| elintrab 4919 | Membership in the intersec... |
| elintrabg 4920 | Membership in the intersec... |
| int0 4921 | The intersection of the em... |
| intss1 4922 | An element of a class incl... |
| ssint 4923 | Subclass of a class inters... |
| ssintab 4924 | Subclass of the intersecti... |
| ssintub 4925 | Subclass of the least uppe... |
| ssmin 4926 | Subclass of the minimum va... |
| intmin 4927 | Any member of a class is t... |
| intss 4928 | Intersection of subclasses... |
| intssuni 4929 | The intersection of a none... |
| ssintrab 4930 | Subclass of the intersecti... |
| unissint 4931 | If the union of a class is... |
| intssuni2 4932 | Subclass relationship for ... |
| intminss 4933 | Under subset ordering, the... |
| intmin2 4934 | Any set is the smallest of... |
| intmin3 4935 | Under subset ordering, the... |
| intmin4 4936 | Elimination of a conjunct ... |
| intab 4937 | The intersection of a spec... |
| int0el 4938 | The intersection of a clas... |
| intun 4939 | The class intersection of ... |
| intprg 4940 | The intersection of a pair... |
| intpr 4941 | The intersection of a pair... |
| intsng 4942 | Intersection of a singleto... |
| intsn 4943 | The intersection of a sing... |
| uniintsn 4944 | Two ways to express " ` A ... |
| uniintab 4945 | The union and the intersec... |
| intunsn 4946 | Theorem joining a singleto... |
| rint0 4947 | Relative intersection of a... |
| elrint 4948 | Membership in a restricted... |
| elrint2 4949 | Membership in a restricted... |
| eliun 4954 | Membership in indexed unio... |
| eliin 4955 | Membership in indexed inte... |
| eliuni 4956 | Membership in an indexed u... |
| eliund 4957 | Membership in indexed unio... |
| iuncom 4958 | Commutation of indexed uni... |
| iuncom4 4959 | Commutation of union with ... |
| iunconst 4960 | Indexed union of a constan... |
| iinconst 4961 | Indexed intersection of a ... |
| iuneqconst 4962 | Indexed union of identical... |
| iuniin 4963 | Law combining indexed unio... |
| iinssiun 4964 | An indexed intersection is... |
| iunss1 4965 | Subclass theorem for index... |
| iinss1 4966 | Subclass theorem for index... |
| iuneq1 4967 | Equality theorem for index... |
| iineq1 4968 | Equality theorem for index... |
| ss2iun 4969 | Subclass theorem for index... |
| iuneq2 4970 | Equality theorem for index... |
| iineq2 4971 | Equality theorem for index... |
| iuneq2i 4972 | Equality inference for ind... |
| iineq2i 4973 | Equality inference for ind... |
| iineq2d 4974 | Equality deduction for ind... |
| iuneq2dv 4975 | Equality deduction for ind... |
| iineq2dv 4976 | Equality deduction for ind... |
| iuneq12df 4977 | Equality deduction for ind... |
| iuneq1d 4978 | Equality theorem for index... |
| iuneq12d 4979 | Equality deduction for ind... |
| iuneq2d 4980 | Equality deduction for ind... |
| nfiun 4981 | Bound-variable hypothesis ... |
| nfiin 4982 | Bound-variable hypothesis ... |
| nfiung 4983 | Bound-variable hypothesis ... |
| nfiing 4984 | Bound-variable hypothesis ... |
| nfiu1 4985 | Bound-variable hypothesis ... |
| nfii1 4986 | Bound-variable hypothesis ... |
| dfiun2g 4987 | Alternate definition of in... |
| dfiin2g 4988 | Alternate definition of in... |
| dfiun2 4989 | Alternate definition of in... |
| dfiin2 4990 | Alternate definition of in... |
| dfiunv2 4991 | Define double indexed unio... |
| cbviun 4992 | Rule used to change the bo... |
| cbviin 4993 | Change bound variables in ... |
| cbviung 4994 | Rule used to change the bo... |
| cbviing 4995 | Change bound variables in ... |
| cbviunv 4996 | Rule used to change the bo... |
| cbviinv 4997 | Change bound variables in ... |
| cbviunvg 4998 | Rule used to change the bo... |
| cbviinvg 4999 | Change bound variables in ... |
| iunssf 5000 | Subset theorem for an inde... |
| iunssfOLD 5001 | Obsolete version of ~ iuns... |
| iunss 5002 | Subset theorem for an inde... |
| iunssOLD 5003 | Obsolete version of ~ iuns... |
| ssiun 5004 | Subset implication for an ... |
| ssiun2 5005 | Identity law for subset of... |
| ssiun2s 5006 | Subset relationship for an... |
| iunss2 5007 | A subclass condition on th... |
| iunssd 5008 | Subset theorem for an inde... |
| iunab 5009 | The indexed union of a cla... |
| iunrab 5010 | The indexed union of a res... |
| iunxdif2 5011 | Indexed union with a class... |
| ssiinf 5012 | Subset theorem for an inde... |
| ssiin 5013 | Subset theorem for an inde... |
| iinss 5014 | Subset implication for an ... |
| iinss2 5015 | An indexed intersection is... |
| uniiun 5016 | Class union in terms of in... |
| intiin 5017 | Class intersection in term... |
| iunid 5018 | An indexed union of single... |
| iun0 5019 | An indexed union of the em... |
| 0iun 5020 | An empty indexed union is ... |
| 0iin 5021 | An empty indexed intersect... |
| viin 5022 | Indexed intersection with ... |
| iunsn 5023 | Indexed union of a singlet... |
| iunn0 5024 | There is a nonempty class ... |
| iinab 5025 | Indexed intersection of a ... |
| iinrab 5026 | Indexed intersection of a ... |
| iinrab2 5027 | Indexed intersection of a ... |
| iunin2 5028 | Indexed union of intersect... |
| iunin1 5029 | Indexed union of intersect... |
| iinun2 5030 | Indexed intersection of un... |
| iundif2 5031 | Indexed union of class dif... |
| uniin1 5032 | Union of intersection. Ge... |
| uniin2 5033 | Union of intersection. Ge... |
| iindif1 5034 | Indexed intersection of cl... |
| 2iunin 5035 | Rearrange indexed unions o... |
| iindif2 5036 | Indexed intersection of cl... |
| iinin2 5037 | Indexed intersection of in... |
| iinin1 5038 | Indexed intersection of in... |
| iinvdif 5039 | The indexed intersection o... |
| elriin 5040 | Elementhood in a relative ... |
| riin0 5041 | Relative intersection of a... |
| riinn0 5042 | Relative intersection of a... |
| riinrab 5043 | Relative intersection of a... |
| symdif0 5044 | Symmetric difference with ... |
| symdifv 5045 | The symmetric difference w... |
| symdifid 5046 | The symmetric difference o... |
| iinxsng 5047 | A singleton index picks ou... |
| iinxprg 5048 | Indexed intersection with ... |
| iunxsng 5049 | A singleton index picks ou... |
| iunxsn 5050 | A singleton index picks ou... |
| iunxsngf 5051 | A singleton index picks ou... |
| iunun 5052 | Separate a union in an ind... |
| iunxun 5053 | Separate a union in the in... |
| iunxdif3 5054 | An indexed union where som... |
| iunxprg 5055 | A pair index picks out two... |
| iunxiun 5056 | Separate an indexed union ... |
| iinuni 5057 | A relationship involving u... |
| iununi 5058 | A relationship involving u... |
| sspwuni 5059 | Subclass relationship for ... |
| pwssb 5060 | Two ways to express a coll... |
| elpwpw 5061 | Characterization of the el... |
| pwpwab 5062 | The double power class wri... |
| pwpwssunieq 5063 | The class of sets whose un... |
| elpwuni 5064 | Relationship for power cla... |
| iinpw 5065 | The power class of an inte... |
| iunpwss 5066 | Inclusion of an indexed un... |
| intss2 5067 | A nonempty intersection of... |
| rintn0 5068 | Relative intersection of a... |
| dfdisj2 5071 | Alternate definition for d... |
| disjss2 5072 | If each element of a colle... |
| disjeq2 5073 | Equality theorem for disjo... |
| disjeq2dv 5074 | Equality deduction for dis... |
| disjss1 5075 | A subset of a disjoint col... |
| disjeq1 5076 | Equality theorem for disjo... |
| disjeq1d 5077 | Equality theorem for disjo... |
| disjeq12d 5078 | Equality theorem for disjo... |
| cbvdisj 5079 | Change bound variables in ... |
| cbvdisjv 5080 | Change bound variables in ... |
| nfdisjw 5081 | Bound-variable hypothesis ... |
| nfdisj 5082 | Bound-variable hypothesis ... |
| nfdisj1 5083 | Bound-variable hypothesis ... |
| disjor 5084 | Two ways to say that a col... |
| disjors 5085 | Two ways to say that a col... |
| disji2 5086 | Property of a disjoint col... |
| disji 5087 | Property of a disjoint col... |
| invdisj 5088 | If there is a function ` C... |
| invdisjrab 5089 | The restricted class abstr... |
| disjiun 5090 | A disjoint collection yiel... |
| disjord 5091 | Conditions for a collectio... |
| disjiunb 5092 | Two ways to say that a col... |
| disjiund 5093 | Conditions for a collectio... |
| sndisj 5094 | Any collection of singleto... |
| 0disj 5095 | Any collection of empty se... |
| disjxsn 5096 | A singleton collection is ... |
| disjx0 5097 | An empty collection is dis... |
| disjprg 5098 | A pair collection is disjo... |
| disjxiun 5099 | An indexed union of a disj... |
| disjxun 5100 | The union of two disjoint ... |
| disjss3 5101 | Expand a disjoint collecti... |
| breq 5104 | Equality theorem for binar... |
| breq1 5105 | Equality theorem for a bin... |
| breq2 5106 | Equality theorem for a bin... |
| breq12 5107 | Equality theorem for a bin... |
| breqi 5108 | Equality inference for bin... |
| breq1i 5109 | Equality inference for a b... |
| breq2i 5110 | Equality inference for a b... |
| breq12i 5111 | Equality inference for a b... |
| breq1d 5112 | Equality deduction for a b... |
| breqd 5113 | Equality deduction for a b... |
| breq2d 5114 | Equality deduction for a b... |
| breq12d 5115 | Equality deduction for a b... |
| breq123d 5116 | Equality deduction for a b... |
| breqdi 5117 | Equality deduction for a b... |
| breqan12d 5118 | Equality deduction for a b... |
| breqan12rd 5119 | Equality deduction for a b... |
| breq1dd 5120 | Equality deduction for a b... |
| breq2dd 5121 | Equality deduction for a b... |
| eqnbrtrd 5122 | Substitution of equal clas... |
| nbrne1 5123 | Two classes are different ... |
| nbrne2 5124 | Two classes are different ... |
| eqbrtri 5125 | Substitution of equal clas... |
| eqbrtrd 5126 | Substitution of equal clas... |
| eqbrtrri 5127 | Substitution of equal clas... |
| eqbrtrrd 5128 | Substitution of equal clas... |
| breqtri 5129 | Substitution of equal clas... |
| breqtrd 5130 | Substitution of equal clas... |
| breqtrri 5131 | Substitution of equal clas... |
| breqtrrd 5132 | Substitution of equal clas... |
| 3brtr3i 5133 | Substitution of equality i... |
| 3brtr4i 5134 | Substitution of equality i... |
| 3brtr3d 5135 | Substitution of equality i... |
| 3brtr4d 5136 | Substitution of equality i... |
| 3brtr3g 5137 | Substitution of equality i... |
| 3brtr4g 5138 | Substitution of equality i... |
| eqbrtrid 5139 | A chained equality inferen... |
| eqbrtrrid 5140 | A chained equality inferen... |
| breqtrid 5141 | A chained equality inferen... |
| breqtrrid 5142 | A chained equality inferen... |
| eqbrtrdi 5143 | A chained equality inferen... |
| eqbrtrrdi 5144 | A chained equality inferen... |
| breqtrdi 5145 | A chained equality inferen... |
| breqtrrdi 5146 | A chained equality inferen... |
| ssbrd 5147 | Deduction from a subclass ... |
| ssbr 5148 | Implication from a subclas... |
| ssbri 5149 | Inference from a subclass ... |
| nfbrd 5150 | Deduction version of bound... |
| nfbr 5151 | Bound-variable hypothesis ... |
| brab1 5152 | Relationship between a bin... |
| br0 5153 | The empty binary relation ... |
| brne0 5154 | If two sets are in a binar... |
| brun 5155 | The union of two binary re... |
| brin 5156 | The intersection of two re... |
| brdif 5157 | The difference of two bina... |
| sbcbr123 5158 | Move substitution in and o... |
| sbcbr 5159 | Move substitution in and o... |
| sbcbr12g 5160 | Move substitution in and o... |
| sbcbr1g 5161 | Move substitution in and o... |
| sbcbr2g 5162 | Move substitution in and o... |
| brsymdif 5163 | Characterization of the sy... |
| brralrspcev 5164 | Restricted existential spe... |
| brimralrspcev 5165 | Restricted existential spe... |
| opabss 5168 | The collection of ordered ... |
| opabbid 5169 | Equivalent wff's yield equ... |
| opabbidv 5170 | Equivalent wff's yield equ... |
| opabbii 5171 | Equivalent wff's yield equ... |
| nfopabd 5172 | Bound-variable hypothesis ... |
| nfopab 5173 | Bound-variable hypothesis ... |
| nfopab1 5174 | The first abstraction vari... |
| nfopab2 5175 | The second abstraction var... |
| cbvopab 5176 | Rule used to change bound ... |
| cbvopabv 5177 | Rule used to change bound ... |
| cbvopab1 5178 | Change first bound variabl... |
| cbvopab1g 5179 | Change first bound variabl... |
| cbvopab2 5180 | Change second bound variab... |
| cbvopab1s 5181 | Change first bound variabl... |
| cbvopab1v 5182 | Rule used to change the fi... |
| cbvopab2v 5183 | Rule used to change the se... |
| unopab 5184 | Union of two ordered pair ... |
| mpteq12da 5187 | An equality inference for ... |
| mpteq12df 5188 | An equality inference for ... |
| mpteq12f 5189 | An equality theorem for th... |
| mpteq12dva 5190 | An equality inference for ... |
| mpteq12dv 5191 | An equality inference for ... |
| mpteq12 5192 | An equality theorem for th... |
| mpteq1 5193 | An equality theorem for th... |
| mpteq1d 5194 | An equality theorem for th... |
| mpteq1i 5195 | An equality theorem for th... |
| mpteq2da 5196 | Slightly more general equa... |
| mpteq2dva 5197 | Slightly more general equa... |
| mpteq2dv 5198 | An equality inference for ... |
| mpteq2ia 5199 | An equality inference for ... |
| mpteq2i 5200 | An equality inference for ... |
| mpteq12i 5201 | An equality inference for ... |
| nfmpt 5202 | Bound-variable hypothesis ... |
| nfmpt1 5203 | Bound-variable hypothesis ... |
| cbvmptf 5204 | Rule to change the bound v... |
| cbvmptfg 5205 | Rule to change the bound v... |
| cbvmpt 5206 | Rule to change the bound v... |
| cbvmptg 5207 | Rule to change the bound v... |
| cbvmptv 5208 | Rule to change the bound v... |
| cbvmptvg 5209 | Rule to change the bound v... |
| mptv 5210 | Function with universal do... |
| dftr2 5213 | An alternate way of defini... |
| dftr2c 5214 | Variant of ~ dftr2 with co... |
| dftr5 5215 | An alternate way of defini... |
| dftr3 5216 | An alternate way of defini... |
| dftr4 5217 | An alternate way of defini... |
| treq 5218 | Equality theorem for the t... |
| trel 5219 | In a transitive class, the... |
| trel3 5220 | In a transitive class, the... |
| trss 5221 | An element of a transitive... |
| trun 5222 | The union of transitive cl... |
| trin 5223 | The intersection of transi... |
| tr0 5224 | The empty set is transitiv... |
| trv 5225 | The universe is transitive... |
| triun 5226 | An indexed union of a clas... |
| truni 5227 | The union of a class of tr... |
| triin 5228 | An indexed intersection of... |
| trint 5229 | The intersection of a clas... |
| trintss 5230 | Any nonempty transitive cl... |
| axrep1 5232 | The version of the Axiom o... |
| axreplem 5233 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5234 | Axiom of Replacement expre... |
| axrep3 5235 | Axiom of Replacement sligh... |
| axrep4v 5236 | Version of ~ axrep4 with a... |
| axrep4 5237 | A more traditional version... |
| axrep5 5238 | Axiom of Replacement (simi... |
| axrep6 5239 | A condensed form of ~ ax-r... |
| replem 5240 | A lemma for variants of th... |
| zfrep6 5241 | A version of the Axiom of ... |
| axrep6g 5242 | ~ axrep6 in class notation... |
| zfrepclf 5243 | An inference based on the ... |
| zfrep3cl 5244 | An inference based on the ... |
| zfrep4 5245 | A version of Replacement u... |
| axsepgfromrep 5246 | A more general version ~ a... |
| axsep 5247 | Axiom scheme of separation... |
| axsepg 5249 | A more general version of ... |
| sepg 5250 | Version of the axiom of se... |
| sepgi 5251 | Inference associated with ... |
| zfausclOLD 5252 | Obsolete version of ~ sepg... |
| sepexlem 5253 | Lemma for ~ sepex . Use ~... |
| sepex 5254 | Convert implication to equ... |
| sepexi 5255 | Convert implication to equ... |
| ax6vsep 5256 | Derive ~ ax6v (a weakened ... |
| axnulALT 5257 | Alternate proof of ~ axnul... |
| axnul 5258 | The Null Set Axiom of ZF s... |
| 0ex 5260 | The Null Set Axiom of ZF s... |
| al0ssb 5261 | The empty set is the uniqu... |
| sseliALT 5262 | Alternate proof of ~ sseli... |
| csbexg 5263 | The existence of proper su... |
| csbex 5264 | The existence of proper su... |
| unisn2 5265 | A version of ~ unisn witho... |
| exnelv 5266 | For any set ` x ` , there ... |
| nalset 5267 | No set contains all sets. ... |
| nalsetOLD 5268 | Obsolete version of ~ nals... |
| vneqv 5269 | The universal class is not... |
| vnex 5270 | The universal class does n... |
| vnexOLD 5271 | Obsolete proof of ~ vnex a... |
| nvel 5272 | The universal class does n... |
| vprc 5273 | The universal class is not... |
| vprcOLD 5274 | Obsolete proof of ~ vprc ,... |
| nvelOLD 5275 | Obsolete proof of ~ nvel ,... |
| inex1 5276 | Separation Scheme (Aussond... |
| inex2 5277 | Separation Scheme (Aussond... |
| inex1g 5278 | Closed-form, generalized S... |
| inex2g 5279 | Sufficient condition for a... |
| ssexg 5280 | A subclass of a set is a s... |
| ssex 5281 | A subclass of a set is a s... |
| ssexOLD 5282 | Obsolete version of ~ ssex... |
| ssexi 5283 | A subclass of a set is a s... |
| ssexgOLD 5284 | Obsolete version of ~ ssex... |
| ssexd 5285 | A subclass of a set is a s... |
| abexd 5286 | Conditions for a class abs... |
| abex 5287 | Conditions for a class abs... |
| prcssprc 5288 | The superclass of a proper... |
| sselpwd 5289 | Membership in a power set.... |
| difexg 5290 | Existence of a difference.... |
| difexi 5291 | Existence of a difference,... |
| difexd 5292 | Existence of a difference.... |
| sepab 5293 | Separation Scheme (Aussond... |
| elpw2g 5294 | Membership in a power clas... |
| elpw2 5295 | Membership in a power clas... |
| elpwi2 5296 | Membership in a power clas... |
| rabelpw 5297 | A restricted class abstrac... |
| rabexg 5298 | Separation Scheme in terms... |
| rabex 5299 | Separation Scheme in terms... |
| rabexd 5300 | Separation Scheme in terms... |
| rabex2 5301 | Separation Scheme in terms... |
| rab2ex 5302 | A class abstraction based ... |
| elssabg 5303 | Membership in a class abst... |
| intex 5304 | The intersection of a none... |
| intnex 5305 | If a class intersection is... |
| intexab 5306 | The intersection of a none... |
| intexrab 5307 | The intersection of a none... |
| iinexg 5308 | The existence of a class i... |
| intabs 5309 | Absorption of a redundant ... |
| inuni 5310 | The intersection of a unio... |
| axpweq 5311 | Two equivalent ways to exp... |
| pwnss 5312 | The power set of a set is ... |
| pwne 5313 | No set equals its power se... |
| difelpw 5314 | A difference is an element... |
| class2set 5315 | The class of elements of `... |
| 0elpw 5316 | Every power class contains... |
| pwne0 5317 | A power class is never emp... |
| 0nep0 5318 | The empty set and its powe... |
| 0inp0 5319 | Something cannot be equal ... |
| unidif0 5320 | The removal of the empty s... |
| unidif0OLD 5321 | Obsolete version of ~ unid... |
| eqsnuniex 5322 | If a class is equal to the... |
| iin0 5323 | An indexed intersection of... |
| notsep 5324 | In the Separation Scheme ~... |
| intv 5325 | The intersection of the un... |
| zfpow 5327 | Axiom of Power Sets expres... |
| axpow2 5328 | A variant of the Axiom of ... |
| axpow3 5329 | A variant of the Axiom of ... |
| elALT2 5330 | Alternate proof of ~ el us... |
| dtruALT2 5331 | Alternate proof of ~ dtru ... |
| dtrucor 5332 | Corollary of ~ dtru . Thi... |
| dtrucor2 5333 | The theorem form of the de... |
| dvdemo1 5334 | Demonstration of a theorem... |
| dvdemo2 5335 | Demonstration of a theorem... |
| nfnid 5336 | A setvar variable is not f... |
| nfcvb 5337 | The "distinctor" expressio... |
| vpwex 5338 | Power set axiom: the power... |
| pwexg 5339 | Power set axiom expressed ... |
| pwexd 5340 | Deduction version of the p... |
| pwex 5341 | Power set axiom expressed ... |
| pwel 5342 | Quantitative version of ~ ... |
| abssexg 5343 | Existence of a class of su... |
| snexALT 5344 | Alternate proof of ~ snex ... |
| p0ex 5345 | The power set of the empty... |
| p0exALT 5346 | Alternate proof of ~ p0ex ... |
| pp0ex 5347 | The power set of the power... |
| ord3ex 5348 | The ordinal number 3 is a ... |
| dtruALT 5349 | Alternate proof of ~ dtru ... |
| axc16b 5350 | This theorem shows that Ax... |
| eunex 5351 | Existential uniqueness imp... |
| eusv1 5352 | Two ways to express single... |
| eusvnf 5353 | Even if ` x ` is free in `... |
| eusvnfb 5354 | Two ways to say that ` A (... |
| eusv2i 5355 | Two ways to express single... |
| eusv2nf 5356 | Two ways to express single... |
| eusv2 5357 | Two ways to express single... |
| reusv1 5358 | Two ways to express single... |
| reusv2lem1 5359 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5360 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5361 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5362 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5363 | Lemma for ~ reusv2 . (Con... |
| reusv2 5364 | Two ways to express single... |
| reusv3i 5365 | Two ways of expressing exi... |
| reusv3 5366 | Two ways to express single... |
| eusv4 5367 | Two ways to express single... |
| alxfr 5368 | Transfer universal quantif... |
| ralxfrd 5369 | Transfer universal quantif... |
| rexxfrd 5370 | Transfer existential quant... |
| ralxfr2d 5371 | Transfer universal quantif... |
| rexxfr2d 5372 | Transfer existential quant... |
| ralxfrd2 5373 | Transfer universal quantif... |
| rexxfrd2 5374 | Transfer existence from a ... |
| ralxfr 5375 | Transfer universal quantif... |
| ralxfrALT 5376 | Alternate proof of ~ ralxf... |
| rexxfr 5377 | Transfer existence from a ... |
| rabxfrd 5378 | Membership in a restricted... |
| rabxfr 5379 | Membership in a restricted... |
| reuhypd 5380 | A theorem useful for elimi... |
| reuhyp 5381 | A theorem useful for elimi... |
| zfpair 5382 | The Axiom of Pairing of Ze... |
| axprALT 5383 | Alternate proof of ~ axpr ... |
| axprlem1 5384 | Lemma for ~ axpr . There ... |
| axprlem2 5385 | Lemma for ~ axpr . There ... |
| axprlem3 5386 | Lemma for ~ axpr . Elimin... |
| axprlem4 5387 | Lemma for ~ axpr . If an ... |
| axpr 5388 | Unabbreviated version of t... |
| axprlem1OLD 5389 | Obsolete version of ~ axpr... |
| zfpair2 5391 | Derive the abbreviated ver... |
| vsnex 5392 | A singleton built on a set... |
| axprglem 5393 | Lemma for ~ axprg . (Cont... |
| axprg 5394 | Derive The Axiom of Pairin... |
| prex 5395 | The Axiom of Pairing using... |
| snex 5396 | A singleton is a set. The... |
| snexg 5397 | A singleton built on a set... |
| snexgALT 5398 | Alternate proof of ~ snexg... |
| snexOLD 5399 | Obsolete version of ~ snex... |
| prexOLD 5400 | Obsolete version of ~ prex... |
| exel 5401 | There exist two sets, one ... |
| exexneq 5402 | There exist two different ... |
| exneq 5403 | Given any set (the " ` y `... |
| dtru 5404 | Given any set (the " ` y `... |
| el 5405 | Any set is an element of s... |
| el.OLD 5406 | Obsolete version of ~ el a... |
| sels 5407 | If a class is a set, then ... |
| selsALT 5408 | Alternate proof of ~ sels ... |
| elALT 5409 | Alternate proof of ~ el , ... |
| snelpwg 5410 | A singleton of a set is a ... |
| snelpwi 5411 | If a set is a member of a ... |
| snelpw 5412 | A singleton of a set is a ... |
| prelpw 5413 | An unordered pair of two s... |
| prelpwi 5414 | If two sets are members of... |
| rext 5415 | A theorem similar to exten... |
| sspwb 5416 | The powerclass constructio... |
| unipw 5417 | A class equals the union o... |
| univ 5418 | The union of the universe ... |
| pwtr 5419 | A class is transitive iff ... |
| ssextss 5420 | An extensionality-like pri... |
| ssext 5421 | An extensionality-like pri... |
| nssss 5422 | Negation of subclass relat... |
| pweqb 5423 | Classes are equal if and o... |
| intidg 5424 | The intersection of all se... |
| moabex 5425 | "At most one" existence im... |
| moabexOLD 5426 | Obsolete version of ~ moab... |
| rmorabex 5427 | Restricted "at most one" e... |
| euabex 5428 | The abstraction of a wff w... |
| nnullss 5429 | A nonempty class (even if ... |
| exss 5430 | Restricted existence in a ... |
| opex 5431 | An ordered pair of classes... |
| opexOLD 5432 | Obsolete version of ~ opex... |
| otex 5433 | An ordered triple of class... |
| elopg 5434 | Characterization of the el... |
| elop 5435 | Characterization of the el... |
| opi1 5436 | One of the two elements in... |
| opi2 5437 | One of the two elements of... |
| opeluu 5438 | Each member of an ordered ... |
| op1stb 5439 | Extract the first member o... |
| brv 5440 | Two classes are always in ... |
| opnz 5441 | An ordered pair is nonempt... |
| opnzi 5442 | An ordered pair is nonempt... |
| opth1 5443 | Equality of the first memb... |
| opth 5444 | The ordered pair theorem. ... |
| opthg 5445 | Ordered pair theorem. ` C ... |
| opth1g 5446 | Equality of the first memb... |
| opthg2 5447 | Ordered pair theorem. (Co... |
| opth2 5448 | Ordered pair theorem. (Co... |
| opthneg 5449 | Two ordered pairs are not ... |
| opthne 5450 | Two ordered pairs are not ... |
| otth2 5451 | Ordered triple theorem, wi... |
| otth 5452 | Ordered triple theorem. (... |
| otthg 5453 | Ordered triple theorem, cl... |
| otthne 5454 | Contrapositive of the orde... |
| eqvinop 5455 | A variable introduction la... |
| sbcop1 5456 | The proper substitution of... |
| sbcop 5457 | The proper substitution of... |
| copsexgw 5458 | Version of ~ copsexg with ... |
| copsexgwOLD 5459 | Obsolete version of ~ cops... |
| copsexg 5460 | Substitution of class ` A ... |
| copsex2t 5461 | Closed theorem form of ~ c... |
| copsex2g 5462 | Implicit substitution infe... |
| copsex2dv 5463 | Implicit substitution dedu... |
| copsex4g 5464 | An implicit substitution i... |
| 0nelop 5465 | A property of ordered pair... |
| opwo0id 5466 | An ordered pair is equal t... |
| opeqex 5467 | Equivalence of existence i... |
| oteqex2 5468 | Equivalence of existence i... |
| oteqex 5469 | Equivalence of existence i... |
| opcom 5470 | An ordered pair commutes i... |
| moop2 5471 | "At most one" property of ... |
| opeqsng 5472 | Equivalence for an ordered... |
| opeqsn 5473 | Equivalence for an ordered... |
| opeqpr 5474 | Equivalence for an ordered... |
| snopeqop 5475 | Equivalence for an ordered... |
| propeqop 5476 | Equivalence for an ordered... |
| propssopi 5477 | If a pair of ordered pairs... |
| snopeqopsnid 5478 | Equivalence for an ordered... |
| mosubopt 5479 | "At most one" remains true... |
| mosubop 5480 | "At most one" remains true... |
| euop2 5481 | Transfer existential uniqu... |
| euotd 5482 | Prove existential uniquene... |
| opthwiener 5483 | Justification theorem for ... |
| uniop 5484 | The union of an ordered pa... |
| uniopel 5485 | Ordered pair membership is... |
| opthhausdorff 5486 | Justification theorem for ... |
| opthhausdorff0 5487 | Justification theorem for ... |
| otsndisj 5488 | The singletons consisting ... |
| otiunsndisj 5489 | The union of singletons co... |
| iunopeqop 5490 | Implication of an ordered ... |
| iunopeqopOLD 5491 | Obsolete version of ~ iuno... |
| brsnop 5492 | Binary relation for an ord... |
| brtp 5493 | A necessary and sufficient... |
| opabidw 5494 | The law of concretion. Sp... |
| opabid 5495 | The law of concretion. Sp... |
| elopabw 5496 | Membership in a class abst... |
| elopab 5497 | Membership in a class abst... |
| rexopabb 5498 | Restricted existential qua... |
| vopelopabsb 5499 | The law of concretion in t... |
| opelopabsb 5500 | The law of concretion in t... |
| brabsb 5501 | The law of concretion in t... |
| opelopabt 5502 | Closed theorem form of ~ o... |
| opelopabga 5503 | The law of concretion. Th... |
| brabga 5504 | The law of concretion for ... |
| opelopab2a 5505 | Ordered pair membership in... |
| opelopaba 5506 | The law of concretion. Th... |
| braba 5507 | The law of concretion for ... |
| brab2d 5508 | Expressing that two sets a... |
| opelopabg 5509 | The law of concretion. Th... |
| brabg 5510 | The law of concretion for ... |
| opelopabgf 5511 | The law of concretion. Th... |
| opelopab2 5512 | Ordered pair membership in... |
| opelopab 5513 | The law of concretion. Th... |
| brab 5514 | The law of concretion for ... |
| opelopabaf 5515 | The law of concretion. Th... |
| opelopabf 5516 | The law of concretion. Th... |
| ssopab2 5517 | Equivalence of ordered pai... |
| ssopab2bw 5518 | Equivalence of ordered pai... |
| eqopab2bw 5519 | Equivalence of ordered pai... |
| ssopab2b 5520 | Equivalence of ordered pai... |
| ssopab2i 5521 | Inference of ordered pair ... |
| ssopab2dv 5522 | Inference of ordered pair ... |
| eqopab2b 5523 | Equivalence of ordered pai... |
| opabn0 5524 | Nonempty ordered pair clas... |
| opab0 5525 | Empty ordered pair class a... |
| csbopab 5526 | Move substitution into a c... |
| csbopabw 5527 | Move substitution into a c... |
| csbmpt12 5528 | Move substitution into a m... |
| csbmpt2 5529 | Move substitution into the... |
| iunopab 5530 | Move indexed union inside ... |
| elopabr 5531 | Membership in an ordered-p... |
| elopabran 5532 | Membership in an ordered-p... |
| rbropapd 5533 | Properties of a pair in an... |
| rbropap 5534 | Properties of a pair in a ... |
| 2rbropap 5535 | Properties of a pair in a ... |
| 0nelopab 5536 | The empty set is never an ... |
| brabv 5537 | If two classes are in a re... |
| pwin 5538 | The power class of the int... |
| pwssun 5539 | The power class of the uni... |
| pwun 5540 | The power class of the uni... |
| dfid4 5543 | The identity function expr... |
| dfid2 5544 | Alternate definition of th... |
| dfid3 5545 | A stronger version of ~ df... |
| epelg 5548 | The membership relation an... |
| epeli 5549 | The membership relation an... |
| epel 5550 | The membership relation an... |
| 0sn0ep 5551 | An example for the members... |
| epn0 5552 | The membership relation is... |
| poss 5557 | Subset theorem for the par... |
| poeq1 5558 | Equality theorem for parti... |
| poeq2 5559 | Equality theorem for parti... |
| poeq12d 5560 | Equality deduction for par... |
| nfpo 5561 | Bound-variable hypothesis ... |
| nfso 5562 | Bound-variable hypothesis ... |
| pocl 5563 | Characteristic properties ... |
| ispod 5564 | Sufficient conditions for ... |
| swopolem 5565 | Perform the substitutions ... |
| swopo 5566 | A strict weak order is a p... |
| poirr 5567 | A partial order is irrefle... |
| potr 5568 | A partial order is a trans... |
| po2nr 5569 | A partial order has no 2-c... |
| po3nr 5570 | A partial order has no 3-c... |
| po2ne 5571 | Two sets related by a part... |
| po0 5572 | Any relation is a partial ... |
| pofun 5573 | The inverse image of a par... |
| sopo 5574 | A strict linear order is a... |
| soss 5575 | Subset theorem for the str... |
| soeq1 5576 | Equality theorem for the s... |
| soeq2 5577 | Equality theorem for the s... |
| soeq12d 5578 | Equality deduction for tot... |
| sonr 5579 | A strict order relation is... |
| sotr 5580 | A strict order relation is... |
| sotrd 5581 | Transitivity law for stric... |
| solin 5582 | A strict order relation is... |
| so2nr 5583 | A strict order relation ha... |
| so3nr 5584 | A strict order relation ha... |
| sotric 5585 | A strict order relation sa... |
| sotrieq 5586 | Trichotomy law for strict ... |
| sotrieq2 5587 | Trichotomy law for strict ... |
| soasym 5588 | Asymmetry law for strict o... |
| sotr2 5589 | A transitivity relation. ... |
| issod 5590 | An irreflexive, transitive... |
| issoi 5591 | An irreflexive, transitive... |
| isso2i 5592 | Deduce strict ordering fro... |
| so0 5593 | Any relation is a strict o... |
| somo 5594 | A totally ordered set has ... |
| sotrine 5595 | Trichotomy law for strict ... |
| sotr3 5596 | Transitivity law for stric... |
| dffr6 5603 | Alternate definition of ~ ... |
| frd 5604 | A nonempty subset of an ` ... |
| fri 5605 | A nonempty subset of an ` ... |
| seex 5606 | The ` R ` -preimage of an ... |
| exse 5607 | Any relation on a set is s... |
| dffr2 5608 | Alternate definition of we... |
| dffr2ALT 5609 | Alternate proof of ~ dffr2... |
| frc 5610 | Property of well-founded r... |
| frss 5611 | Subset theorem for the wel... |
| sess1 5612 | Subset theorem for the set... |
| sess2 5613 | Subset theorem for the set... |
| freq1 5614 | Equality theorem for the w... |
| freq2 5615 | Equality theorem for the w... |
| freq12d 5616 | Equality deduction for wel... |
| seeq1 5617 | Equality theorem for the s... |
| seeq2 5618 | Equality theorem for the s... |
| seeq12d 5619 | Equality deduction for the... |
| nffr 5620 | Bound-variable hypothesis ... |
| nfse 5621 | Bound-variable hypothesis ... |
| nfwe 5622 | Bound-variable hypothesis ... |
| frirr 5623 | A well-founded relation is... |
| fr2nr 5624 | A well-founded relation ha... |
| fr0 5625 | Any relation is well-found... |
| frminex 5626 | If an element of a well-fo... |
| efrirr 5627 | A well-founded class does ... |
| efrn2lp 5628 | A well-founded class conta... |
| epse 5629 | The membership relation is... |
| tz7.2 5630 | Similar to Theorem 7.2 of ... |
| dfepfr 5631 | An alternate way of saying... |
| epfrc 5632 | A subset of a well-founded... |
| wess 5633 | Subset theorem for the wel... |
| weeq1 5634 | Equality theorem for the w... |
| weeq2 5635 | Equality theorem for the w... |
| weeq12d 5636 | Equality deduction for wel... |
| wefr 5637 | A well-ordering is well-fo... |
| weso 5638 | A well-ordering is a stric... |
| wecmpep 5639 | The elements of a class we... |
| wetrep 5640 | On a class well-ordered by... |
| wefrc 5641 | A nonempty subclass of a c... |
| we0 5642 | Any relation is a well-ord... |
| wereu 5643 | A nonempty subset of an ` ... |
| wereu2 5644 | A nonempty subclass of an ... |
| xpeq1 5661 | Equality theorem for Carte... |
| xpss12 5662 | Subset theorem for Cartesi... |
| xpss 5663 | A Cartesian product is inc... |
| inxpssres 5664 | Intersection with a Cartes... |
| relxp 5665 | A Cartesian product is a r... |
| xpss1 5666 | Subset relation for Cartes... |
| xpss2 5667 | Subset relation for Cartes... |
| xpeq2 5668 | Equality theorem for Carte... |
| elxpi 5669 | Membership in a Cartesian ... |
| elxp 5670 | Membership in a Cartesian ... |
| elxp2 5671 | Membership in a Cartesian ... |
| xpeq12 5672 | Equality theorem for Carte... |
| xpeq1i 5673 | Equality inference for Car... |
| xpeq2i 5674 | Equality inference for Car... |
| xpeq12i 5675 | Equality inference for Car... |
| xpeq1d 5676 | Equality deduction for Car... |
| xpeq2d 5677 | Equality deduction for Car... |
| xpeq12d 5678 | Equality deduction for Car... |
| sqxpeqd 5679 | Equality deduction for a C... |
| nfxp 5680 | Bound-variable hypothesis ... |
| 0nelxp 5681 | The empty set is not a mem... |
| 0nelelxp 5682 | A member of a Cartesian pr... |
| opelxp 5683 | Ordered pair membership in... |
| opelxpi 5684 | Ordered pair membership in... |
| opelxpii 5685 | Ordered pair membership in... |
| opelxpd 5686 | Ordered pair membership in... |
| opelvv 5687 | Ordered pair membership in... |
| opelvvg 5688 | Ordered pair membership in... |
| opelxp1 5689 | The first member of an ord... |
| opelxp2 5690 | The second member of an or... |
| otelxp 5691 | Ordered triple membership ... |
| otelxp1 5692 | The first member of an ord... |
| otel3xp 5693 | An ordered triple is an el... |
| opabssxpd 5694 | An ordered-pair class abst... |
| rabxp 5695 | Class abstraction restrict... |
| brxp 5696 | Binary relation on a Carte... |
| pwvrel 5697 | A set is a binary relation... |
| pwvabrel 5698 | The powerclass of the cart... |
| brrelex12 5699 | Two classes related by a b... |
| brrelex1 5700 | If two classes are related... |
| brrelex2 5701 | If two classes are related... |
| brrelex12i 5702 | Two classes that are relat... |
| brrelex1i 5703 | The first argument of a bi... |
| brrelex2i 5704 | The second argument of a b... |
| nprrel12 5705 | Proper classes are not rel... |
| nprrel 5706 | No proper class is related... |
| 0nelrel0 5707 | A binary relation does not... |
| 0nelrel 5708 | A binary relation does not... |
| fconstmpt 5709 | Representation of a consta... |
| vtoclr 5710 | Variable to class conversi... |
| opthprc 5711 | Justification theorem for ... |
| brel 5712 | Two things in a binary rel... |
| elxp3 5713 | Membership in a Cartesian ... |
| opeliunxp 5714 | Membership in a union of C... |
| opeliun2xp 5715 | Membership of an ordered p... |
| xpundi 5716 | Distributive law for Carte... |
| xpundir 5717 | Distributive law for Carte... |
| xpiundi 5718 | Distributive law for Carte... |
| xpiundir 5719 | Distributive law for Carte... |
| iunxpconst 5720 | Membership in a union of C... |
| xpun 5721 | The Cartesian product of t... |
| elvv 5722 | Membership in universal cl... |
| elvvv 5723 | Membership in universal cl... |
| elvvuni 5724 | An ordered pair contains i... |
| brinxp2 5725 | Intersection of binary rel... |
| brinxp 5726 | Intersection of binary rel... |
| opelinxp 5727 | Ordered pair element in an... |
| poinxp 5728 | Intersection of partial or... |
| soinxp 5729 | Intersection of total orde... |
| frinxp 5730 | Intersection of well-found... |
| seinxp 5731 | Intersection of set-like r... |
| weinxp 5732 | Intersection of well-order... |
| posn 5733 | Partial ordering of a sing... |
| sosn 5734 | Strict ordering on a singl... |
| frsn 5735 | Founded relation on a sing... |
| wesn 5736 | Well-ordering of a singlet... |
| elopaelxp 5737 | Membership in an ordered-p... |
| bropaex12 5738 | Two classes related by an ... |
| opabssxp 5739 | An abstraction relation is... |
| brab2a 5740 | The law of concretion for ... |
| optocl 5741 | Implicit substitution of c... |
| optoclOLD 5742 | Obsolete version of ~ opto... |
| 2optocl 5743 | Implicit substitution of c... |
| 3optocl 5744 | Implicit substitution of c... |
| opbrop 5745 | Ordered pair membership in... |
| 0xp 5746 | The Cartesian product with... |
| xp0 5747 | The Cartesian product with... |
| csbxp 5748 | Distribute proper substitu... |
| releq 5749 | Equality theorem for the r... |
| releqi 5750 | Equality inference for the... |
| releqd 5751 | Equality deduction for the... |
| nfrel 5752 | Bound-variable hypothesis ... |
| sbcrel 5753 | Distribute proper substitu... |
| relss 5754 | Subclass theorem for relat... |
| ssrel 5755 | A subclass relationship de... |
| eqrel 5756 | Extensionality principle f... |
| ssrel2 5757 | A subclass relationship de... |
| ssrel3 5758 | Subclass relation in anoth... |
| relssi 5759 | Inference from subclass pr... |
| relssdv 5760 | Deduction from subclass pr... |
| eqrelriv 5761 | Inference from extensional... |
| eqrelriiv 5762 | Inference from extensional... |
| eqbrriv 5763 | Inference from extensional... |
| eqrelrdv 5764 | Deduce equality of relatio... |
| eqbrrdv 5765 | Deduction from extensional... |
| eqbrrdiv 5766 | Deduction from extensional... |
| eqrelrdv2 5767 | A version of ~ eqrelrdv . ... |
| ssrelrel 5768 | A subclass relationship de... |
| eqrelrel 5769 | Extensionality principle f... |
| elrel 5770 | A member of a relation is ... |
| elrelb 5771 | A member of a relation exp... |
| rel0 5772 | The empty set is a relatio... |
| nrelv 5773 | The universal class is not... |
| nrelvOLD 5774 | Obsolete version of ~ nrel... |
| relsng 5775 | A singleton is a relation ... |
| relsnb 5776 | An at-most-singleton is a ... |
| relsnopg 5777 | A singleton of an ordered ... |
| relsn 5778 | A singleton is a relation ... |
| relsnop 5779 | A singleton of an ordered ... |
| copsex2gb 5780 | Implicit substitution infe... |
| copsex2ga 5781 | Implicit substitution infe... |
| elopaba 5782 | Membership in an ordered-p... |
| xpsspw 5783 | A Cartesian product is inc... |
| unixpss 5784 | The double class union of ... |
| relun 5785 | The union of two relations... |
| relin1 5786 | The intersection with a re... |
| relin2 5787 | The intersection with a re... |
| relinxp 5788 | Intersection with a Cartes... |
| reldif 5789 | A difference cutting down ... |
| reliun 5790 | An indexed union is a rela... |
| reliin 5791 | An indexed intersection is... |
| reluni 5792 | The union of a class is a ... |
| relint 5793 | The intersection of a clas... |
| relopabiv 5794 | A class of ordered pairs i... |
| relopabv 5795 | A class of ordered pairs i... |
| relopabi 5796 | A class of ordered pairs i... |
| relopabiALT 5797 | Alternate proof of ~ relop... |
| relopab 5798 | A class of ordered pairs i... |
| mptrel 5799 | The maps-to notation alway... |
| reli 5800 | The identity relation is a... |
| rele 5801 | The membership relation is... |
| opabid2 5802 | A relation expressed as an... |
| inopab 5803 | Intersection of two ordere... |
| difopab 5804 | Difference of two ordered-... |
| inxp 5805 | Intersection of two Cartes... |
| xpindi 5806 | Distributive law for Carte... |
| xpindir 5807 | Distributive law for Carte... |
| xpiindi 5808 | Distributive law for Carte... |
| xpriindi 5809 | Distributive law for Carte... |
| eliunxp 5810 | Membership in a union of C... |
| opeliunxp2 5811 | Membership in a union of C... |
| raliunxp 5812 | Write a double restricted ... |
| rexiunxp 5813 | Write a double restricted ... |
| ralxp 5814 | Universal quantification r... |
| rexxp 5815 | Existential quantification... |
| el2xptp 5816 | A member of a nested Carte... |
| exopxfr 5817 | Transfer ordered-pair exis... |
| exopxfr2 5818 | Transfer ordered-pair exis... |
| djussxp 5819 | Disjoint union is a subset... |
| ralxpf 5820 | Version of ~ ralxp with bo... |
| rexxpf 5821 | Version of ~ rexxp with bo... |
| iunxpf 5822 | Indexed union on a Cartesi... |
| opabbi2dv 5823 | Deduce equality of a relat... |
| relop 5824 | A necessary and sufficient... |
| ideqg 5825 | For sets, the identity rel... |
| ideq 5826 | For sets, the identity rel... |
| ididg 5827 | A set is identical to itse... |
| issetid 5828 | Two ways of expressing set... |
| coss1 5829 | Subclass theorem for compo... |
| coss2 5830 | Subclass theorem for compo... |
| coeq1 5831 | Equality theorem for compo... |
| coeq2 5832 | Equality theorem for compo... |
| coeq1i 5833 | Equality inference for com... |
| coeq2i 5834 | Equality inference for com... |
| coeq1d 5835 | Equality deduction for com... |
| coeq2d 5836 | Equality deduction for com... |
| coeq12i 5837 | Equality inference for com... |
| coeq12d 5838 | Equality deduction for com... |
| nfco 5839 | Bound-variable hypothesis ... |
| brcog 5840 | Ordered pair membership in... |
| opelco2g 5841 | Ordered pair membership in... |
| brcogw 5842 | Ordered pair membership in... |
| eqbrrdva 5843 | Deduction from extensional... |
| brco 5844 | Binary relation on a compo... |
| opelco 5845 | Ordered pair membership in... |
| cnvss 5846 | Subset theorem for convers... |
| cnveq 5847 | Equality theorem for conve... |
| cnveqi 5848 | Equality inference for con... |
| cnveqd 5849 | Equality deduction for con... |
| elcnv 5850 | Membership in a converse r... |
| elcnv2 5851 | Membership in a converse r... |
| nfcnv 5852 | Bound-variable hypothesis ... |
| brcnvg 5853 | The converse of a binary r... |
| opelcnvg 5854 | Ordered-pair membership in... |
| opelcnv 5855 | Ordered-pair membership in... |
| brcnv 5856 | The converse of a binary r... |
| cnv0 5857 | The converse of the empty ... |
| cnv0OLD 5858 | Obsolete version of ~ cnv0... |
| cnvi 5859 | The converse of the identi... |
| csbcnv 5860 | Move class substitution in... |
| csbcnvOLD 5861 | Obsolete version of ~ csbc... |
| csbcnvgALTOLD 5862 | Obsolete version of ~ csbc... |
| cnvco 5863 | Distributive law of conver... |
| cnvuni 5864 | The converse of a class un... |
| dfdm3 5865 | Alternate definition of do... |
| dfrn2 5866 | Alternate definition of ra... |
| dfrn3 5867 | Alternate definition of ra... |
| elrn2g 5868 | Membership in a range. (C... |
| elrng 5869 | Membership in a range. (C... |
| elrn2 5870 | Membership in a range. (C... |
| elrn 5871 | Membership in a range. (C... |
| ssrelrn 5872 | If a relation is a subset ... |
| dfdm4 5873 | Alternate definition of do... |
| dfdmf 5874 | Definition of domain, usin... |
| csbdm 5875 | Distribute proper substitu... |
| eldmg 5876 | Domain membership. Theore... |
| eldm2g 5877 | Domain membership. Theore... |
| eldm 5878 | Membership in a domain. T... |
| eldm2 5879 | Membership in a domain. T... |
| dmss 5880 | Subset theorem for domain.... |
| dmeq 5881 | Equality theorem for domai... |
| dmeqi 5882 | Equality inference for dom... |
| dmeqd 5883 | Equality deduction for dom... |
| opeldmd 5884 | Membership of first of an ... |
| opeldm 5885 | Membership of first of an ... |
| breldm 5886 | Membership of first of a b... |
| breldmg 5887 | Membership of first of a b... |
| dmun 5888 | The domain of a union is t... |
| dmin 5889 | The domain of an intersect... |
| breldmd 5890 | Membership of first of a b... |
| dmiun 5891 | The domain of an indexed u... |
| dmuni 5892 | The domain of a union. Pa... |
| dmopab 5893 | The domain of a class of o... |
| dmopabelb 5894 | A set is an element of the... |
| dmopab2rex 5895 | The domain of an ordered p... |
| dmopabss 5896 | Upper bound for the domain... |
| dmopab3 5897 | The domain of a restricted... |
| dm0 5898 | The domain of the empty se... |
| dmi 5899 | The domain of the identity... |
| dmv 5900 | The domain of the universe... |
| dmep 5901 | The domain of the membersh... |
| dm0rn0 5902 | An empty domain is equival... |
| dm0rn0OLD 5903 | Obsolete version of ~ dm0r... |
| rn0 5904 | The range of the empty set... |
| rnep 5905 | The range of the membershi... |
| reldm0 5906 | A relation is empty iff it... |
| dmxp 5907 | The domain of a Cartesian ... |
| dmxpid 5908 | The domain of a Cartesian ... |
| dmxpin 5909 | The domain of the intersec... |
| xpid11 5910 | The Cartesian square is a ... |
| dmcnvcnv 5911 | The domain of the double c... |
| rncnvcnv 5912 | The range of the double co... |
| elreldm 5913 | The first member of an ord... |
| rneq 5914 | Equality theorem for range... |
| rneqi 5915 | Equality inference for ran... |
| rneqd 5916 | Equality deduction for ran... |
| rnss 5917 | Subset theorem for range. ... |
| rnssi 5918 | Subclass inference for ran... |
| brelrng 5919 | The second argument of a b... |
| brelrn 5920 | The second argument of a b... |
| opelrn 5921 | Membership of second membe... |
| releldm 5922 | The first argument of a bi... |
| relelrn 5923 | The second argument of a b... |
| releldmb 5924 | Membership in a domain. (... |
| relelrnb 5925 | Membership in a range. (C... |
| releldmi 5926 | The first argument of a bi... |
| relelrni 5927 | The second argument of a b... |
| dfrnf 5928 | Definition of range, using... |
| nfdm 5929 | Bound-variable hypothesis ... |
| nfrn 5930 | Bound-variable hypothesis ... |
| dmiin 5931 | Domain of an intersection.... |
| rnopab 5932 | The range of a class of or... |
| rnopabss 5933 | Upper bound for the range ... |
| rnopab3 5934 | The range of a restricted ... |
| rnmpt 5935 | The range of a function in... |
| elrnmpt 5936 | The range of a function in... |
| elrnmpt1s 5937 | Elementhood in an image se... |
| elrnmpt1 5938 | Elementhood in an image se... |
| elrnmptg 5939 | Membership in the range of... |
| elrnmpti 5940 | Membership in the range of... |
| elrnmptd 5941 | The range of a function in... |
| elrnmpt1d 5942 | Elementhood in an image se... |
| elrnmptdv 5943 | Elementhood in the range o... |
| elrnmpt2d 5944 | Elementhood in the range o... |
| nelrnmpt 5945 | Non-membership in the rang... |
| dfiun3g 5946 | Alternate definition of in... |
| dfiin3g 5947 | Alternate definition of in... |
| dfiun3 5948 | Alternate definition of in... |
| dfiin3 5949 | Alternate definition of in... |
| riinint 5950 | Express a relative indexed... |
| relrn0 5951 | A relation is empty iff it... |
| dmrnssfld 5952 | The domain and range of a ... |
| dmcoss 5953 | Domain of a composition. ... |
| dmcossOLD 5954 | Obsolete version of ~ dmco... |
| rncoss 5955 | Range of a composition. (... |
| dmcosseq 5956 | Domain of a composition. ... |
| dmcosseqOLD 5957 | Obsolete version of ~ dmco... |
| dmcoeq 5958 | Domain of a composition. ... |
| rncoeq 5959 | Range of a composition. (... |
| reseq1 5960 | Equality theorem for restr... |
| reseq2 5961 | Equality theorem for restr... |
| reseq1i 5962 | Equality inference for res... |
| reseq2i 5963 | Equality inference for res... |
| reseq12i 5964 | Equality inference for res... |
| reseq1d 5965 | Equality deduction for res... |
| reseq2d 5966 | Equality deduction for res... |
| reseq12d 5967 | Equality deduction for res... |
| nfres 5968 | Bound-variable hypothesis ... |
| csbres 5969 | Distribute proper substitu... |
| res0 5970 | A restriction to the empty... |
| dfres3 5971 | Alternate definition of re... |
| opelres 5972 | Ordered pair elementhood i... |
| brres 5973 | Binary relation on a restr... |
| opelresi 5974 | Ordered pair membership in... |
| brresi 5975 | Binary relation on a restr... |
| opres 5976 | Ordered pair membership in... |
| resieq 5977 | A restricted identity rela... |
| opelidres 5978 | ` <. A , A >. ` belongs to... |
| resres 5979 | The restriction of a restr... |
| resundi 5980 | Distributive law for restr... |
| resundir 5981 | Distributive law for restr... |
| resindi 5982 | Class restriction distribu... |
| resindir 5983 | Class restriction distribu... |
| inres 5984 | Move intersection into cla... |
| resdifcom 5985 | Commutative law for restri... |
| resiun1 5986 | Distribution of restrictio... |
| resiun2 5987 | Distribution of restrictio... |
| resss 5988 | A class includes its restr... |
| rescom 5989 | Commutative law for restri... |
| ssres 5990 | Subclass theorem for restr... |
| ssres2 5991 | Subclass theorem for restr... |
| relres 5992 | A restriction is a relatio... |
| resabs1 5993 | Absorption law for restric... |
| resabs1i 5994 | Absorption law for restric... |
| resabs1d 5995 | Absorption law for restric... |
| resabs2 5996 | Absorption law for restric... |
| residm 5997 | Idempotent law for restric... |
| dmresss 5998 | The domain of a restrictio... |
| dmres 5999 | The domain of a restrictio... |
| ssdmres 6000 | A domain restricted to a s... |
| dmresexg 6001 | The domain of a restrictio... |
| resima 6002 | A restriction to an image.... |
| resima2 6003 | Image under a restricted c... |
| rnresss 6004 | The range of a restriction... |
| xpssres 6005 | Restriction of a constant ... |
| elinxp 6006 | Membership in an intersect... |
| elres 6007 | Membership in a restrictio... |
| elsnres 6008 | Membership in restriction ... |
| relssres 6009 | Simplification law for res... |
| dmressnsn 6010 | The domain of a restrictio... |
| eldmressnsn 6011 | The element of the domain ... |
| eldmeldmressn 6012 | An element of the domain (... |
| resdm 6013 | A relation restricted to i... |
| resexg 6014 | The restriction of a set i... |
| resexd 6015 | The restriction of a set i... |
| resex 6016 | The restriction of a set i... |
| resindm 6017 | When restricting a class, ... |
| resindmOLD 6018 | Obsolete version of ~ resi... |
| resdmdfsn 6019 | Restricting a class to its... |
| resdmdfsnOLD 6020 | Obsolete version of ~ resd... |
| reldmun 6021 | Split a relation into two ... |
| reldisjunOLD 6022 | Obsolete version of ~ reld... |
| relresdm1 6023 | Restriction of a disjoint ... |
| resopab 6024 | Restriction of a class abs... |
| iss 6025 | A subclass of the identity... |
| resopab2 6026 | Restriction of a class abs... |
| resmpt 6027 | Restriction of the mapping... |
| resmpt3 6028 | Unconditional restriction ... |
| resmptf 6029 | Restriction of the mapping... |
| resmptd 6030 | Restriction of the mapping... |
| dfres2 6031 | Alternate definition of th... |
| mptss 6032 | Sufficient condition for i... |
| elimampt 6033 | Membership in the image of... |
| elidinxp 6034 | Characterization of the el... |
| elidinxpid 6035 | Characterization of the el... |
| elrid 6036 | Characterization of the el... |
| idinxpres 6037 | The intersection of the id... |
| idinxpresid 6038 | The intersection of the id... |
| idssxp 6039 | A diagonal set as a subset... |
| opabresid 6040 | The restricted identity re... |
| mptresid 6041 | The restricted identity re... |
| dmresi 6042 | The domain of a restricted... |
| restidsing 6043 | Restriction of the identit... |
| iresn0n0 6044 | The identity function rest... |
| imaeq1 6045 | Equality theorem for image... |
| imaeq2 6046 | Equality theorem for image... |
| imaeq1i 6047 | Equality theorem for image... |
| imaeq2i 6048 | Equality theorem for image... |
| imaeq1d 6049 | Equality theorem for image... |
| imaeq2d 6050 | Equality theorem for image... |
| imaeq12d 6051 | Equality theorem for image... |
| dfima2 6052 | Alternate definition of im... |
| dfima3 6053 | Alternate definition of im... |
| elimag 6054 | Membership in an image. T... |
| elima 6055 | Membership in an image. T... |
| elima2 6056 | Membership in an image. T... |
| elima3 6057 | Membership in an image. T... |
| nfima 6058 | Bound-variable hypothesis ... |
| nfimad 6059 | Deduction version of bound... |
| imadmrn 6060 | The image of the domain of... |
| imassrn 6061 | The image of a class is a ... |
| mptima 6062 | Image of a function in map... |
| mptimass 6063 | Image of a function in map... |
| imai 6064 | Image under the identity r... |
| rnresi 6065 | The range of the restricte... |
| resiima 6066 | The image of a restriction... |
| ima0 6067 | Image of the empty set. T... |
| 0ima 6068 | Image under the empty rela... |
| csbima12 6069 | Move class substitution in... |
| imadisj 6070 | A class whose image under ... |
| imadisjlnd 6071 | Deduction form of one nega... |
| cnvimass 6072 | A preimage under any class... |
| cnvimarndm 6073 | The preimage of the range ... |
| imasng 6074 | The image of a singleton. ... |
| relimasn 6075 | The image of a singleton. ... |
| elrelimasn 6076 | Elementhood in the image o... |
| elimasng1 6077 | Membership in an image of ... |
| elimasn1 6078 | Membership in an image of ... |
| elimasng 6079 | Membership in an image of ... |
| elimasn 6080 | Membership in an image of ... |
| elimasni 6081 | Membership in an image of ... |
| args 6082 | Two ways to express the cl... |
| elinisegg 6083 | Membership in the inverse ... |
| eliniseg 6084 | Membership in the inverse ... |
| epin 6085 | Any set is equal to its pr... |
| epini 6086 | Any set is equal to its pr... |
| iniseg 6087 | An idiom that signifies an... |
| inisegn0 6088 | Nonemptiness of an initial... |
| dffr3 6089 | Alternate definition of we... |
| dfse2 6090 | Alternate definition of se... |
| imass1 6091 | Subset theorem for image. ... |
| imass2 6092 | Subset theorem for image. ... |
| ndmima 6093 | The image of a singleton o... |
| relcnv 6094 | A converse is a relation. ... |
| relbrcnvg 6095 | When ` R ` is a relation, ... |
| eliniseg2 6096 | Eliminate the class existe... |
| relbrcnv 6097 | When ` R ` is a relation, ... |
| relco 6098 | A composition is a relatio... |
| cotrg 6099 | Two ways of saying that th... |
| cotr 6100 | Two ways of saying a relat... |
| idrefALT 6101 | Alternate proof of ~ idref... |
| cnvsym 6102 | Two ways of saying a relat... |
| intasym 6103 | Two ways of saying a relat... |
| asymref 6104 | Two ways of saying a relat... |
| asymref2 6105 | Two ways of saying a relat... |
| intirr 6106 | Two ways of saying a relat... |
| brcodir 6107 | Two ways of saying that tw... |
| codir 6108 | Two ways of saying a relat... |
| qfto 6109 | A quantifier-free way of e... |
| xpidtr 6110 | A Cartesian square is a tr... |
| trin2 6111 | The intersection of two tr... |
| poirr2 6112 | A partial order is irrefle... |
| trinxp 6113 | The relation induced by a ... |
| soirri 6114 | A strict order relation is... |
| sotri 6115 | A strict order relation is... |
| son2lpi 6116 | A strict order relation ha... |
| sotri2 6117 | A transitivity relation. ... |
| sotri3 6118 | A transitivity relation. ... |
| poleloe 6119 | Express "less than or equa... |
| poltletr 6120 | Transitive law for general... |
| somin1 6121 | Property of a minimum in a... |
| somincom 6122 | Commutativity of minimum i... |
| somin2 6123 | Property of a minimum in a... |
| soltmin 6124 | Being less than a minimum,... |
| cnvopab 6125 | The converse of a class ab... |
| mptcnv 6126 | The converse of a mapping ... |
| cnvun 6127 | The converse of a union is... |
| cnvdif 6128 | Distributive law for conve... |
| cnvin 6129 | Distributive law for conve... |
| rnun 6130 | Distributive law for range... |
| rnin 6131 | The range of an intersecti... |
| rninOLD 6132 | Obsolete version of ~ rnin... |
| rniun 6133 | The range of an indexed un... |
| rnuni 6134 | The range of a union. Par... |
| imaundi 6135 | Distributive law for image... |
| imaundir 6136 | The image of a union. (Co... |
| cnvimassrndm 6137 | The preimage of a superset... |
| dminss 6138 | An upper bound for interse... |
| imainss 6139 | An upper bound for interse... |
| inimass 6140 | The image of an intersecti... |
| inimasn 6141 | The intersection of the im... |
| cnvxp 6142 | The converse of a Cartesia... |
| cnvxpOLD 6143 | Obsolete version of ~ cnvx... |
| xp0OLD 6144 | Obsolete version of ~ xp0 ... |
| xpnz 6145 | The Cartesian product of n... |
| xpeq0 6146 | At least one member of an ... |
| xpdisj1 6147 | Cartesian products with di... |
| xpdisj2 6148 | Cartesian products with di... |
| xpsndisj 6149 | Cartesian products with tw... |
| difxp 6150 | Difference of Cartesian pr... |
| difxp1 6151 | Difference law for Cartesi... |
| difxp2 6152 | Difference law for Cartesi... |
| djudisj 6153 | Disjoint unions with disjo... |
| xpdifid 6154 | The set of distinct couple... |
| xpdifcnvepel 6155 | The set of couples in a Ca... |
| resdisj 6156 | A double restriction to di... |
| rnxp 6157 | The range of a Cartesian p... |
| dmxpss 6158 | The domain of a Cartesian ... |
| rnxpss 6159 | The range of a Cartesian p... |
| rnxpid 6160 | The range of a Cartesian s... |
| ssxpb 6161 | A Cartesian product subcla... |
| xp11 6162 | The Cartesian product of n... |
| xpcan 6163 | Cancellation law for Carte... |
| xpcan2 6164 | Cancellation law for Carte... |
| ssrnres 6165 | Two ways to express surjec... |
| rninxp 6166 | Two ways to express surjec... |
| dminxp 6167 | Two ways to express totali... |
| imainrect 6168 | Image by a restricted and ... |
| xpima 6169 | Direct image by a Cartesia... |
| xpima1 6170 | Direct image by a Cartesia... |
| xpima2 6171 | Direct image by a Cartesia... |
| xpimasn 6172 | Direct image of a singleto... |
| sossfld 6173 | The base set of a strict o... |
| sofld 6174 | The base set of a nonempty... |
| cnvcnv3 6175 | The set of all ordered pai... |
| dfrel2 6176 | Alternate definition of re... |
| dfrel4v 6177 | A relation can be expresse... |
| dfrel4 6178 | A relation can be expresse... |
| cnvcnv 6179 | The double converse of a c... |
| cnvcnv2 6180 | The double converse of a c... |
| cnvcnvss 6181 | The double converse of a c... |
| cnvcnvssOLD 6182 | Obsolete version of ~ cnvc... |
| cnvrescnv 6183 | Two ways to express the co... |
| cnveqb 6184 | Equality theorem for conve... |
| cnveq0 6185 | A relation empty iff its c... |
| dfrel3 6186 | Alternate definition of re... |
| elid 6187 | Characterization of the el... |
| dmresv 6188 | The domain of a universal ... |
| rnresv 6189 | The range of a universal r... |
| dfrn4 6190 | Range defined in terms of ... |
| imadifssran 6191 | Condition for the range of... |
| imadifssranOLD 6192 | Obsolete version of ~ imad... |
| csbrn 6193 | Distribute proper substitu... |
| rescnvcnv 6194 | The restriction of the dou... |
| cnvcnvres 6195 | The double converse of the... |
| imacnvcnv 6196 | The image of the double co... |
| dmsnn0 6197 | The domain of a singleton ... |
| rnsnn0 6198 | The range of a singleton i... |
| dmsn0 6199 | The domain of the singleto... |
| cnvsn0 6200 | The converse of the single... |
| dmsn0el 6201 | The domain of a singleton ... |
| relsn2 6202 | A singleton is a relation ... |
| dmsnopg 6203 | The domain of a singleton ... |
| dmsnopss 6204 | The domain of a singleton ... |
| dmpropg 6205 | The domain of an unordered... |
| dmsnop 6206 | The domain of a singleton ... |
| dmprop 6207 | The domain of an unordered... |
| dmtpop 6208 | The domain of an unordered... |
| cnvcnvsn 6209 | Double converse of a singl... |
| dmsnsnsn 6210 | The domain of the singleto... |
| rnsnopg 6211 | The range of a singleton o... |
| rnpropg 6212 | The range of a pair of ord... |
| cnvsng 6213 | Converse of a singleton of... |
| rnsnop 6214 | The range of a singleton o... |
| op1sta 6215 | Extract the first member o... |
| cnvsn 6216 | Converse of a singleton of... |
| op2ndb 6217 | Extract the second member ... |
| op2nda 6218 | Extract the second member ... |
| opswap 6219 | Swap the members of an ord... |
| cnvresima 6220 | An image under the convers... |
| resdm2 6221 | A class restricted to its ... |
| resdmres 6222 | Restriction to the domain ... |
| resresdm 6223 | A restriction by an arbitr... |
| imadmres 6224 | The image of the domain of... |
| resdmss 6225 | Subset relationship for th... |
| resdifdi 6226 | Distributive law for restr... |
| resdifdir 6227 | Distributive law for restr... |
| mptpreima 6228 | The preimage of a function... |
| mptiniseg 6229 | Converse singleton image o... |
| dmmpt 6230 | The domain of the mapping ... |
| dmmptss 6231 | The domain of a mapping is... |
| dmmptg 6232 | The domain of the mapping ... |
| rnmpt0f 6233 | The range of a function in... |
| rnmptn0 6234 | The range of a function in... |
| dfco2 6235 | Alternate definition of a ... |
| dfco2a 6236 | Generalization of ~ dfco2 ... |
| coundi 6237 | Class composition distribu... |
| coundir 6238 | Class composition distribu... |
| cores 6239 | Restricted first member of... |
| resco 6240 | Associative law for the re... |
| imaco 6241 | Image of the composition o... |
| rnco 6242 | The range of the compositi... |
| rncoOLD 6243 | Obsolete version of ~ rnco... |
| rnco2 6244 | The range of the compositi... |
| dmco 6245 | The domain of a compositio... |
| coeq0 6246 | A composition of two relat... |
| coiun 6247 | Composition with an indexe... |
| cocnvcnv1 6248 | A composition is not affec... |
| cocnvcnv2 6249 | A composition is not affec... |
| cores2 6250 | Absorption of a reverse (p... |
| co02 6251 | Composition with the empty... |
| co01 6252 | Composition with the empty... |
| coi1 6253 | Composition with the ident... |
| coi2 6254 | Composition with the ident... |
| coires1 6255 | Composition with a restric... |
| coass 6256 | Associative law for class ... |
| relcnvtrg 6257 | Subclass law for converse ... |
| relcnvtrgOLD 6258 | Obsolete form of ~ relcnvt... |
| relcnvtrOLD 6259 | Obsolete form of ~ relcnvt... |
| relssdmrn 6260 | A relation is included in ... |
| resssxp 6261 | If the ` R ` -image of a c... |
| cnvssrndm 6262 | The converse is a subset o... |
| cossxp 6263 | Composition as a subset of... |
| relrelss 6264 | Two ways to describe the s... |
| unielrel 6265 | The membership relation fo... |
| relfld 6266 | The double union of a rela... |
| relresfld 6267 | Restriction of a relation ... |
| relresfldOLD 6268 | Obsolete version of ~ relr... |
| relcoi2 6269 | Composition with the ident... |
| relcoi1 6270 | Composition with the ident... |
| unidmrn 6271 | The double union of the co... |
| relcnvfld 6272 | if ` R ` is a relation, it... |
| dfdm2 6273 | Alternate definition of do... |
| unixp 6274 | The double class union of ... |
| unixp0 6275 | A Cartesian product is emp... |
| unixpid 6276 | Field of a Cartesian squar... |
| ressn 6277 | Restriction of a class to ... |
| cnviin 6278 | The converse of an interse... |
| cnvpo 6279 | The converse of a partial ... |
| cnvso 6280 | The converse of a strict o... |
| xpco 6281 | Composition of two Cartesi... |
| xpcoid 6282 | Composition of two Cartesi... |
| elsnxp 6283 | Membership in a Cartesian ... |
| reu3op 6284 | There is a unique ordered ... |
| reuop 6285 | There is a unique ordered ... |
| opreu2reurex 6286 | There is a unique ordered ... |
| opreu2reu 6287 | If there is a unique order... |
| dfpo2 6288 | Quantifier-free definition... |
| csbcog 6289 | Distribute proper substitu... |
| snres0 6290 | Condition for restriction ... |
| imaindm 6291 | The image is unaffected by... |
| predeq123 6294 | Equality theorem for the p... |
| predeq1 6295 | Equality theorem for the p... |
| predeq2 6296 | Equality theorem for the p... |
| predeq3 6297 | Equality theorem for the p... |
| nfpred 6298 | Bound-variable hypothesis ... |
| csbpredg 6299 | Move class substitution in... |
| predpredss 6300 | If ` A ` is a subset of ` ... |
| predss 6301 | The predecessor class of `... |
| sspred 6302 | Another subset/predecessor... |
| dfpred2 6303 | An alternate definition of... |
| dfpred3 6304 | An alternate definition of... |
| dfpred3g 6305 | An alternate definition of... |
| elpredgg 6306 | Membership in a predecesso... |
| elpredg 6307 | Membership in a predecesso... |
| elpredimg 6308 | Membership in a predecesso... |
| elpredim 6309 | Membership in a predecesso... |
| elpred 6310 | Membership in a predecesso... |
| predexg 6311 | The predecessor class exis... |
| dffr4 6312 | Alternate definition of we... |
| predel 6313 | Membership in the predeces... |
| predtrss 6314 | If ` R ` is transitive ove... |
| predpo 6315 | Property of the predecesso... |
| predso 6316 | Property of the predecesso... |
| setlikespec 6317 | If ` R ` is set-like in ` ... |
| predidm 6318 | Idempotent law for the pre... |
| predin 6319 | Intersection law for prede... |
| predun 6320 | Union law for predecessor ... |
| preddif 6321 | Difference law for predece... |
| predep 6322 | The predecessor under the ... |
| trpred 6323 | The class of predecessors ... |
| preddowncl 6324 | A property of classes that... |
| predpoirr 6325 | Given a partial ordering, ... |
| predfrirr 6326 | Given a well-founded relat... |
| pred0 6327 | The predecessor class over... |
| dfse3 6328 | Alternate definition of se... |
| predrelss 6329 | Subset carries from relati... |
| predprc 6330 | The predecessor of a prope... |
| predres 6331 | Predecessor class is unaff... |
| frpomin 6332 | Every nonempty (possibly p... |
| frpomin2 6333 | Every nonempty (possibly p... |
| frpoind 6334 | The principle of well-foun... |
| frpoinsg 6335 | Well-Founded Induction Sch... |
| frpoins2fg 6336 | Well-Founded Induction sch... |
| frpoins2g 6337 | Well-Founded Induction sch... |
| frpoins3g 6338 | Well-Founded Induction sch... |
| tz6.26 6339 | All nonempty subclasses of... |
| tz6.26i 6340 | All nonempty subclasses of... |
| wfi 6341 | The Principle of Well-Orde... |
| wfii 6342 | The Principle of Well-Orde... |
| wfisg 6343 | Well-Ordered Induction Sch... |
| wfis 6344 | Well-Ordered Induction Sch... |
| wfis2fg 6345 | Well-Ordered Induction Sch... |
| wfis2f 6346 | Well-Ordered Induction sch... |
| wfis2g 6347 | Well-Ordered Induction Sch... |
| wfis2 6348 | Well-Ordered Induction sch... |
| wfis3 6349 | Well-Ordered Induction sch... |
| ordeq 6358 | Equality theorem for the o... |
| elong 6359 | An ordinal number is an or... |
| elon 6360 | An ordinal number is an or... |
| eloni 6361 | An ordinal number has the ... |
| elon2 6362 | An ordinal number is an or... |
| limeq 6363 | Equality theorem for the l... |
| ordwe 6364 | Membership well-orders eve... |
| ordtr 6365 | An ordinal class is transi... |
| ordfr 6366 | Membership is well-founded... |
| ordelss 6367 | An element of an ordinal c... |
| trssord 6368 | A transitive subclass of a... |
| ordirr 6369 | No ordinal class is a memb... |
| nordeq 6370 | A member of an ordinal cla... |
| ordn2lp 6371 | An ordinal class cannot be... |
| tz7.5 6372 | A nonempty subclass of an ... |
| ordelord 6373 | An element of an ordinal c... |
| tron 6374 | The class of all ordinal n... |
| ordelon 6375 | An element of an ordinal c... |
| onelon 6376 | An element of an ordinal n... |
| tz7.7 6377 | A transitive class belongs... |
| ordelssne 6378 | For ordinal classes, membe... |
| ordelpss 6379 | For ordinal classes, membe... |
| ordpss 6380 | ~ ordelpss with an anteced... |
| ordsseleq 6381 | For ordinal classes, inclu... |
| ordin 6382 | The intersection of two or... |
| onin 6383 | The intersection of two or... |
| ordtri3or 6384 | A trichotomy law for ordin... |
| ordtri1 6385 | A trichotomy law for ordin... |
| ontri1 6386 | A trichotomy law for ordin... |
| ordtri2 6387 | A trichotomy law for ordin... |
| ordtri3 6388 | A trichotomy law for ordin... |
| ordtri4 6389 | A trichotomy law for ordin... |
| orddisj 6390 | An ordinal class and its s... |
| onfr 6391 | The ordinal class is well-... |
| onelpss 6392 | Relationship between membe... |
| onsseleq 6393 | Relationship between subse... |
| onelss 6394 | An element of an ordinal n... |
| oneltri 6395 | The elementhood relation o... |
| ordtr1 6396 | Transitive law for ordinal... |
| ordtr2 6397 | Transitive law for ordinal... |
| ordtr3 6398 | Transitive law for ordinal... |
| ontr1 6399 | Transitive law for ordinal... |
| ontr2 6400 | Transitive law for ordinal... |
| onelssex 6401 | Ordinal less than is equiv... |
| ordunidif 6402 | The union of an ordinal st... |
| ordintdif 6403 | If ` B ` is smaller than `... |
| onintss 6404 | If a property is true for ... |
| oneqmini 6405 | A way to show that an ordi... |
| ord0 6406 | The empty set is an ordina... |
| 0elon 6407 | The empty set is an ordina... |
| ord0eln0 6408 | A nonempty ordinal contain... |
| on0eln0 6409 | An ordinal number contains... |
| dflim2 6410 | An alternate definition of... |
| inton 6411 | The intersection of the cl... |
| nlim0 6412 | The empty set is not a lim... |
| limord 6413 | A limit ordinal is ordinal... |
| limuni 6414 | A limit ordinal is its own... |
| limuni2 6415 | The union of a limit ordin... |
| 0ellim 6416 | A limit ordinal contains t... |
| limelon 6417 | A limit ordinal class that... |
| onn0 6418 | The class of all ordinal n... |
| suceqd 6419 | Deduction associated with ... |
| suceq 6420 | Equality of successors. (... |
| elsuci 6421 | Membership in a successor.... |
| elsucg 6422 | Membership in a successor.... |
| elsuc2g 6423 | Variant of membership in a... |
| elsuc 6424 | Membership in a successor.... |
| elsuc2 6425 | Membership in a successor.... |
| nfsuc 6426 | Bound-variable hypothesis ... |
| elelsuc 6427 | Membership in a successor.... |
| sucel 6428 | Membership of a successor ... |
| suc0 6429 | The successor of the empty... |
| sucprc 6430 | A proper class is its own ... |
| unisucs 6431 | The union of the successor... |
| unisucg 6432 | A transitive class is equa... |
| unisuc 6433 | A transitive class is equa... |
| sssucid 6434 | A class is included in its... |
| sucidg 6435 | Part of Proposition 7.23 o... |
| sucid 6436 | A set belongs to its succe... |
| nsuceq0 6437 | No successor is empty. (C... |
| eqelsuc 6438 | A set belongs to the succe... |
| iunsuc 6439 | Inductive definition for t... |
| suctr 6440 | The successor of a transit... |
| trsuc 6441 | A set whose successor belo... |
| trsucss 6442 | A member of the successor ... |
| ordsssuc 6443 | An ordinal is a subset of ... |
| onsssuc 6444 | A subset of an ordinal num... |
| ordsssuc2 6445 | An ordinal subset of an or... |
| onmindif 6446 | When its successor is subt... |
| ordnbtwn 6447 | There is no set between an... |
| onnbtwn 6448 | There is no set between an... |
| sucssel 6449 | A set whose successor is a... |
| orddif 6450 | Ordinal derived from its s... |
| orduniss 6451 | An ordinal class includes ... |
| ordtri2or 6452 | A trichotomy law for ordin... |
| ordtri2or2 6453 | A trichotomy law for ordin... |
| ordtri2or3 6454 | A consequence of total ord... |
| ordelinel 6455 | The intersection of two or... |
| ordssun 6456 | Property of a subclass of ... |
| ordequn 6457 | The maximum (i.e. union) o... |
| ordun 6458 | The maximum (i.e., union) ... |
| onunel 6459 | The union of two ordinals ... |
| ordunisssuc 6460 | A subclass relationship fo... |
| suc11 6461 | The successor operation be... |
| onun2 6462 | The union of two ordinals ... |
| ontr 6463 | An ordinal number is a tra... |
| onunisuc 6464 | An ordinal number is equal... |
| onordi 6465 | An ordinal number is an or... |
| onirri 6466 | An ordinal number is not a... |
| oneli 6467 | A member of an ordinal num... |
| onelssi 6468 | A member of an ordinal num... |
| onssneli 6469 | An ordering law for ordina... |
| onssnel2i 6470 | An ordering law for ordina... |
| onelini 6471 | An element of an ordinal n... |
| oneluni 6472 | An ordinal number equals i... |
| onunisuci 6473 | An ordinal number is equal... |
| onsseli 6474 | Subset is equivalent to me... |
| onun2i 6475 | The union of two ordinal n... |
| unizlim 6476 | An ordinal equal to its ow... |
| on0eqel 6477 | An ordinal number either e... |
| snsn0non 6478 | The singleton of the singl... |
| onxpdisj 6479 | Ordinal numbers and ordere... |
| onnev 6480 | The class of ordinal numbe... |
| iotajust 6482 | Soundness justification th... |
| dfiota2 6484 | Alternate definition for d... |
| nfiota1 6485 | Bound-variable hypothesis ... |
| nfiotadw 6486 | Deduction version of ~ nfi... |
| nfiotaw 6487 | Bound-variable hypothesis ... |
| nfiotad 6488 | Deduction version of ~ nfi... |
| nfiota 6489 | Bound-variable hypothesis ... |
| cbviotaw 6490 | Change bound variables in ... |
| cbviotavw 6491 | Change bound variables in ... |
| cbviota 6492 | Change bound variables in ... |
| cbviotav 6493 | Change bound variables in ... |
| sb8iota 6494 | Variable substitution in d... |
| iotaeq 6495 | Equality theorem for descr... |
| iotabi 6496 | Equivalence theorem for de... |
| uniabio 6497 | Part of Theorem 8.17 in [Q... |
| iotaval2 6498 | Version of ~ iotaval using... |
| iotauni2 6499 | Version of ~ iotauni using... |
| iotanul2 6500 | Version of ~ iotanul using... |
| iotaval 6501 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6502 | The ` iota ` class is a su... |
| iotaex 6503 | Theorem 8.23 in [Quine] p.... |
| iotauni 6504 | Equivalence between two di... |
| iotaint 6505 | Equivalence between two di... |
| iota1 6506 | Property of iota. (Contri... |
| iotanul 6507 | Theorem 8.22 in [Quine] p.... |
| iota4 6508 | Theorem *14.22 in [Whitehe... |
| iota4an 6509 | Theorem *14.23 in [Whitehe... |
| iota5 6510 | A method for computing iot... |
| iotabidv 6511 | Formula-building deduction... |
| iotabii 6512 | Formula-building deduction... |
| iotacl 6513 | Membership law for descrip... |
| iota2df 6514 | A condition that allows to... |
| iota2d 6515 | A condition that allows to... |
| iota2 6516 | The unique element such th... |
| iotan0 6517 | Representation of "the uni... |
| sniota 6518 | A class abstraction with a... |
| dfiota4 6519 | The ` iota ` operation usi... |
| csbiota 6520 | Class substitution within ... |
| dffun2 6537 | Alternate definition of a ... |
| dffun6 6538 | Alternate definition of a ... |
| dffun3 6539 | Alternate definition of fu... |
| dffun4 6540 | Alternate definition of a ... |
| dffun5 6541 | Alternate definition of fu... |
| dffun6f 6542 | Definition of function, us... |
| funmo 6543 | A function has at most one... |
| funrel 6544 | A function is a relation. ... |
| 0nelfun 6545 | A function does not contai... |
| funss 6546 | Subclass theorem for funct... |
| funeq 6547 | Equality theorem for funct... |
| funeqi 6548 | Equality inference for the... |
| funeqd 6549 | Equality deduction for the... |
| nffun 6550 | Bound-variable hypothesis ... |
| sbcfung 6551 | Distribute proper substitu... |
| sbcfungOLD 6552 | Obsolete version of ~ sbcf... |
| funeu 6553 | There is exactly one value... |
| funeu2 6554 | There is exactly one value... |
| dffun7 6555 | Alternate definition of a ... |
| dffun8 6556 | Alternate definition of a ... |
| dffun9 6557 | Alternate definition of a ... |
| funfn 6558 | A class is a function if a... |
| funfnd 6559 | A function is a function o... |
| funi 6560 | The identity relation is a... |
| nfunv 6561 | The universal class is not... |
| funopg 6562 | A Kuratowski ordered pair ... |
| funopab 6563 | A class of ordered pairs i... |
| funopabeq 6564 | A class of ordered pairs o... |
| funopab4 6565 | A class of ordered pairs o... |
| funmpt 6566 | A function in maps-to nota... |
| funmpt2 6567 | Functionality of a class g... |
| funco 6568 | The composition of two fun... |
| funresfunco 6569 | Composition of two functio... |
| funres 6570 | A restriction of a functio... |
| funresd 6571 | A restriction of a functio... |
| funssres 6572 | The restriction of a funct... |
| fun2ssres 6573 | Equality of restrictions o... |
| funun 6574 | The union of functions wit... |
| fununmo 6575 | If the union of classes is... |
| fununfun 6576 | If the union of classes is... |
| fundif 6577 | A function with removed el... |
| funcnvsn 6578 | The converse singleton of ... |
| funsng 6579 | A singleton of an ordered ... |
| fnsng 6580 | Functionality and domain o... |
| funsn 6581 | A singleton of an ordered ... |
| funprg 6582 | A set of two pairs is a fu... |
| funtpg 6583 | A set of three pairs is a ... |
| funpr 6584 | A function with a domain o... |
| funtp 6585 | A function with a domain o... |
| fnsn 6586 | Functionality and domain o... |
| fnprg 6587 | Function with a domain of ... |
| fntpg 6588 | Function with a domain of ... |
| fntp 6589 | A function with a domain o... |
| funcnvpr 6590 | The converse pair of order... |
| funcnvtp 6591 | The converse triple of ord... |
| funcnvqp 6592 | The converse quadruple of ... |
| fun0 6593 | The empty set is a functio... |
| funcnv0 6594 | The converse of the empty ... |
| funcnvcnv 6595 | The double converse of a f... |
| funcnv2 6596 | A simpler equivalence for ... |
| funcnv 6597 | The converse of a class is... |
| funcnv3 6598 | A condition showing a clas... |
| fun2cnv 6599 | The double converse of a c... |
| svrelfun 6600 | A single-valued relation i... |
| fncnv 6601 | Single-rootedness (see ~ f... |
| fun11 6602 | Two ways of stating that `... |
| fununi 6603 | The union of a chain (with... |
| funin 6604 | The intersection with a fu... |
| funres11 6605 | The restriction of a one-t... |
| funcnvres 6606 | The converse of a restrict... |
| cnvresid 6607 | Converse of a restricted i... |
| funcnvres2 6608 | The converse of a restrict... |
| funimacnv 6609 | The image of the preimage ... |
| funimass1 6610 | A kind of contraposition l... |
| funimass2 6611 | A kind of contraposition l... |
| imadif 6612 | The image of a difference ... |
| imain 6613 | The image of an intersecti... |
| funimaexg 6614 | Axiom of Replacement using... |
| funimaex 6615 | The image of a set under a... |
| isarep1 6616 | Part of a study of the Axi... |
| isarep2 6617 | Part of a study of the Axi... |
| fneq1 6618 | Equality theorem for funct... |
| fneq2 6619 | Equality theorem for funct... |
| fneq1d 6620 | Equality deduction for fun... |
| fneq2d 6621 | Equality deduction for fun... |
| fneq12d 6622 | Equality deduction for fun... |
| fneq12 6623 | Equality theorem for funct... |
| fneq1i 6624 | Equality inference for fun... |
| fneq2i 6625 | Equality inference for fun... |
| nffn 6626 | Bound-variable hypothesis ... |
| fnfun 6627 | A function with domain is ... |
| fnfund 6628 | A function with domain is ... |
| fnrel 6629 | A function with domain is ... |
| fndm 6630 | The domain of a function. ... |
| fndmi 6631 | The domain of a function. ... |
| fndmd 6632 | The domain of a function. ... |
| funfni 6633 | Inference to convert a fun... |
| fndmu 6634 | A function has a unique do... |
| fnbr 6635 | The first argument of bina... |
| fnop 6636 | The first argument of an o... |
| fneu 6637 | There is exactly one value... |
| fneu2 6638 | There is exactly one value... |
| fnunres1 6639 | Restriction of a disjoint ... |
| fnunres2 6640 | Restriction of a disjoint ... |
| fnun 6641 | The union of two functions... |
| fnund 6642 | The union of two functions... |
| fnunop 6643 | Extension of a function wi... |
| fncofn 6644 | Composition of a function ... |
| fnco 6645 | Composition of two functio... |
| fnresdm 6646 | A function does not change... |
| fnresdisj 6647 | A function restricted to a... |
| 2elresin 6648 | Membership in two function... |
| fnssresb 6649 | Restriction of a function ... |
| fnssres 6650 | Restriction of a function ... |
| fnssresd 6651 | Restriction of a function ... |
| fnresin1 6652 | Restriction of a function'... |
| fnresin2 6653 | Restriction of a function'... |
| fnres 6654 | An equivalence for functio... |
| idfn 6655 | The identity relation is a... |
| fnresi 6656 | The restricted identity re... |
| fnima 6657 | The image of a function's ... |
| fn0 6658 | A function with empty doma... |
| fnimadisj 6659 | A class that is disjoint w... |
| fnimaeq0 6660 | Images under a function ne... |
| dfmpt3 6661 | Alternate definition for t... |
| mptfnf 6662 | The maps-to notation defin... |
| fnmptf 6663 | The maps-to notation defin... |
| fnopabg 6664 | Functionality and domain o... |
| fnopab 6665 | Functionality and domain o... |
| mptfng 6666 | The maps-to notation defin... |
| fnmpt 6667 | The maps-to notation defin... |
| fnmptd 6668 | The maps-to notation defin... |
| mpt0 6669 | A mapping operation with e... |
| fnmpti 6670 | Functionality and domain o... |
| dmmpti 6671 | Domain of the mapping oper... |
| dmmptd 6672 | The domain of the mapping ... |
| mptun 6673 | Union of mappings which ar... |
| partfun 6674 | Rewrite a function defined... |
| feq1 6675 | Equality theorem for funct... |
| feq2 6676 | Equality theorem for funct... |
| feq3 6677 | Equality theorem for funct... |
| feq23 6678 | Equality theorem for funct... |
| feq1d 6679 | Equality deduction for fun... |
| feq1dd 6680 | Equality deduction for fun... |
| feq2d 6681 | Equality deduction for fun... |
| feq3d 6682 | Equality deduction for fun... |
| feq2dd 6683 | Equality deduction for fun... |
| feq3dd 6684 | Equality deduction for fun... |
| feq12d 6685 | Equality deduction for fun... |
| feq123d 6686 | Equality deduction for fun... |
| feq123 6687 | Equality theorem for funct... |
| feq1i 6688 | Equality inference for fun... |
| feq2i 6689 | Equality inference for fun... |
| feq12i 6690 | Equality inference for fun... |
| feq23i 6691 | Equality inference for fun... |
| feq23d 6692 | Equality deduction for fun... |
| nff 6693 | Bound-variable hypothesis ... |
| sbcfng 6694 | Distribute proper substitu... |
| sbcfg 6695 | Distribute proper substitu... |
| elimf 6696 | Eliminate a mapping hypoth... |
| ffn 6697 | A mapping is a function wi... |
| ffnd 6698 | A mapping is a function wi... |
| dffn2 6699 | Any function is a mapping ... |
| ffun 6700 | A mapping is a function. ... |
| ffunOLD 6701 | Obsolete version of ~ ffun... |
| ffund 6702 | A mapping is a function, d... |
| frel 6703 | A mapping is a relation. ... |
| freld 6704 | A mapping is a relation. ... |
| frn 6705 | The range of a mapping. (... |
| frnd 6706 | Deduction form of ~ frn . ... |
| fdm 6707 | The domain of a mapping. ... |
| fdmd 6708 | Deduction form of ~ fdm . ... |
| fdmi 6709 | Inference associated with ... |
| dffn3 6710 | A function maps to its ran... |
| ffrn 6711 | A function maps to its ran... |
| ffrnb 6712 | Characterization of a func... |
| ffrnbd 6713 | A function maps to its ran... |
| fss 6714 | Expanding the codomain of ... |
| fssd 6715 | Expanding the codomain of ... |
| fssdmd 6716 | Expressing that a class is... |
| fssdm 6717 | Expressing that a class is... |
| fimass 6718 | The image of a class under... |
| fimassd 6719 | The image of a class is a ... |
| fimacnv 6720 | The preimage of the codoma... |
| fcof 6721 | Composition of a function ... |
| fco 6722 | Composition of two functio... |
| fcod 6723 | Composition of two mapping... |
| fco2 6724 | Functionality of a composi... |
| fssxp 6725 | A mapping is a class of or... |
| funssxp 6726 | Two ways of specifying a p... |
| ffdm 6727 | A mapping is a partial fun... |
| ffdmd 6728 | The domain of a function. ... |
| fdmrn 6729 | A different way to write `... |
| funcofd 6730 | Composition of two functio... |
| opelf 6731 | The members of an ordered ... |
| fun 6732 | The union of two functions... |
| fun2 6733 | The union of two functions... |
| fun2d 6734 | The union of functions wit... |
| fnfco 6735 | Composition of two functio... |
| fssres 6736 | Restriction of a function ... |
| fssresd 6737 | Restriction of a function ... |
| fssres2 6738 | Restriction of a restricte... |
| fresin 6739 | An identity for the mappin... |
| resasplit 6740 | If two functions agree on ... |
| fresaun 6741 | The union of two functions... |
| fresaunres2 6742 | From the union of two func... |
| fresaunres1 6743 | From the union of two func... |
| fcoi1 6744 | Composition of a mapping a... |
| fcoi2 6745 | Composition of restricted ... |
| feu 6746 | There is exactly one value... |
| fcnvres 6747 | The converse of a restrict... |
| fimacnvdisj 6748 | The preimage of a class di... |
| fint 6749 | Function into an intersect... |
| fin 6750 | Mapping into an intersecti... |
| f0 6751 | The empty function. (Cont... |
| f00 6752 | A class is a function with... |
| f0bi 6753 | A function with empty doma... |
| f0dom0 6754 | A function is empty iff it... |
| f0rn0 6755 | If there is no element in ... |
| fconst 6756 | A Cartesian product with a... |
| fconstg 6757 | A Cartesian product with a... |
| fnconstg 6758 | A Cartesian product with a... |
| fconst6g 6759 | Constant function with loo... |
| fconst6 6760 | A constant function as a m... |
| f1eq1 6761 | Equality theorem for one-t... |
| f1eq2 6762 | Equality theorem for one-t... |
| f1eq3 6763 | Equality theorem for one-t... |
| nff1 6764 | Bound-variable hypothesis ... |
| dff12 6765 | Alternate definition of a ... |
| f1f 6766 | A one-to-one mapping is a ... |
| f1fn 6767 | A one-to-one mapping is a ... |
| f1fun 6768 | A one-to-one mapping is a ... |
| f1funOLD 6769 | Obsolete version of ~ f1fu... |
| f1rel 6770 | A one-to-one onto mapping ... |
| f1relOLD 6771 | Obsolete version of ~ f1re... |
| f1dm 6772 | The domain of a one-to-one... |
| f1ss 6773 | A function that is one-to-... |
| f1ssr 6774 | A function that is one-to-... |
| f1ssres 6775 | A function that is one-to-... |
| f1resf1 6776 | The restriction of an inje... |
| f1cnvcnv 6777 | Two ways to express that a... |
| f1cof1 6778 | Composition of two one-to-... |
| f1co 6779 | Composition of one-to-one ... |
| foeq1 6780 | Equality theorem for onto ... |
| foeq2 6781 | Equality theorem for onto ... |
| foeq3 6782 | Equality theorem for onto ... |
| nffo 6783 | Bound-variable hypothesis ... |
| fof 6784 | An onto mapping is a mappi... |
| fofun 6785 | An onto mapping is a funct... |
| fofn 6786 | An onto mapping is a funct... |
| forn 6787 | The codomain of an onto fu... |
| dffo2 6788 | Alternate definition of an... |
| foima 6789 | The image of the domain of... |
| dffn4 6790 | A function maps onto its r... |
| funforn 6791 | A function maps its domain... |
| fodmrnu 6792 | An onto function has uniqu... |
| fimadmfo 6793 | A function is a function o... |
| fores 6794 | Restriction of an onto fun... |
| fimadmfoALT 6795 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6796 | The preimage of the codoma... |
| focofo 6797 | Composition of onto functi... |
| foco 6798 | Composition of onto functi... |
| foconst 6799 | A nonzero constant functio... |
| f1oeq1 6800 | Equality theorem for one-t... |
| f1oeq2 6801 | Equality theorem for one-t... |
| f1oeq3 6802 | Equality theorem for one-t... |
| f1oeq23 6803 | Equality theorem for one-t... |
| f1eq123d 6804 | Equality deduction for one... |
| foeq123d 6805 | Equality deduction for ont... |
| f1oeq123d 6806 | Equality deduction for one... |
| f1oeq1d 6807 | Equality deduction for one... |
| f1oeq2d 6808 | Equality deduction for one... |
| f1oeq3d 6809 | Equality deduction for one... |
| nff1o 6810 | Bound-variable hypothesis ... |
| f1of1 6811 | A one-to-one onto mapping ... |
| f1of 6812 | A one-to-one onto mapping ... |
| f1ofn 6813 | A one-to-one onto mapping ... |
| f1ofun 6814 | A one-to-one onto mapping ... |
| f1orel 6815 | A one-to-one onto mapping ... |
| f1odm 6816 | The domain of a one-to-one... |
| f1odmOLD 6817 | Obsolete version of ~ f1od... |
| dff1o2 6818 | Alternate definition of on... |
| dff1o3 6819 | Alternate definition of on... |
| f1ofo 6820 | A one-to-one onto function... |
| dff1o4 6821 | Alternate definition of on... |
| dff1o5 6822 | Alternate definition of on... |
| f1orn 6823 | A one-to-one function maps... |
| f1f1orn 6824 | A one-to-one function maps... |
| f1ocnv 6825 | The converse of a one-to-o... |
| f1ocnvb 6826 | A relation is a one-to-one... |
| f1ores 6827 | The restriction of a one-t... |
| f1orescnv 6828 | The converse of a one-to-o... |
| f1imacnv 6829 | Preimage of an image. (Co... |
| foimacnv 6830 | A reverse version of ~ f1i... |
| foun 6831 | The union of two onto func... |
| f1oun 6832 | The union of two one-to-on... |
| f1un 6833 | The union of two one-to-on... |
| resdif 6834 | The restriction of a one-t... |
| resin 6835 | The restriction of a one-t... |
| f1oco 6836 | Composition of one-to-one ... |
| f1cnv 6837 | The converse of an injecti... |
| funcocnv2 6838 | Composition with the conve... |
| fococnv2 6839 | The composition of an onto... |
| f1ococnv2 6840 | The composition of a one-t... |
| f1cocnv2 6841 | Composition of an injectiv... |
| f1ococnv1 6842 | The composition of a one-t... |
| f1cocnv1 6843 | Composition of an injectiv... |
| funcoeqres 6844 | Express a constraint on a ... |
| f1ssf1 6845 | A subset of an injective f... |
| f10 6846 | The empty set maps one-to-... |
| f10d 6847 | The empty set maps one-to-... |
| f1o00 6848 | One-to-one onto mapping of... |
| fo00 6849 | Onto mapping of the empty ... |
| f1o0 6850 | One-to-one onto mapping of... |
| f1oi 6851 | A restriction of the ident... |
| f1oiOLD 6852 | Obsolete version of ~ f1oi... |
| f1ovi 6853 | The identity relation is a... |
| f1osn 6854 | A singleton of an ordered ... |
| f1osng 6855 | A singleton of an ordered ... |
| f1sng 6856 | A singleton of an ordered ... |
| fsnd 6857 | A singleton of an ordered ... |
| f1oprswap 6858 | A two-element swap is a bi... |
| f1oprg 6859 | An unordered pair of order... |
| tz6.12-2 6860 | Function value when ` F ` ... |
| tz6.12-2OLD 6861 | Obsolete version of ~ tz6.... |
| fveu 6862 | The value of a function at... |
| brprcneu 6863 | If ` A ` is a proper class... |
| brprcneuALT 6864 | Alternate proof of ~ brprc... |
| fvprc 6865 | A function's value at a pr... |
| fvprcALT 6866 | Alternate proof of ~ fvprc... |
| rnfvprc 6867 | The range of a function va... |
| fv2 6868 | Alternate definition of fu... |
| dffv3 6869 | A definition of function v... |
| dffv4 6870 | The previous definition of... |
| elfv 6871 | Membership in a function v... |
| fveq1 6872 | Equality theorem for funct... |
| fveq2 6873 | Equality theorem for funct... |
| fveq1i 6874 | Equality inference for fun... |
| fveq1d 6875 | Equality deduction for fun... |
| fveq2i 6876 | Equality inference for fun... |
| fveq2d 6877 | Equality deduction for fun... |
| 2fveq3 6878 | Equality theorem for neste... |
| fveq12i 6879 | Equality deduction for fun... |
| fveq12d 6880 | Equality deduction for fun... |
| fveqeq2d 6881 | Equality deduction for fun... |
| fveqeq2 6882 | Equality deduction for fun... |
| nffv 6883 | Bound-variable hypothesis ... |
| nffvmpt1 6884 | Bound-variable hypothesis ... |
| nffvd 6885 | Deduction version of bound... |
| fvex 6886 | The value of a class exist... |
| fvexi 6887 | The value of a class exist... |
| fvexd 6888 | The value of a class exist... |
| fvif 6889 | Move a conditional outside... |
| iffv 6890 | Move a conditional outside... |
| fv3 6891 | Alternate definition of th... |
| fvres 6892 | The value of a restricted ... |
| fvresd 6893 | The value of a restricted ... |
| funssfv 6894 | The value of a member of t... |
| tz6.12c 6895 | Corollary of Theorem 6.12(... |
| tz6.12-1 6896 | Function value. Theorem 6... |
| tz6.12 6897 | Function value. Theorem 6... |
| tz6.12f 6898 | Function value, using boun... |
| tz6.12i 6899 | Corollary of Theorem 6.12(... |
| fvbr0 6900 | Two possibilities for the ... |
| fvrn0 6901 | A function value is a memb... |
| fvn0fvelrn 6902 | If the value of a function... |
| elfvunirn 6903 | A function value is a subs... |
| fvssunirn 6904 | The result of a function v... |
| ndmfv 6905 | The value of a class outsi... |
| ndmfvrcl 6906 | Reverse closure law for fu... |
| elfvdm 6907 | If a function value has a ... |
| elfvex 6908 | If a function value has a ... |
| elfvexd 6909 | If a function value has a ... |
| eliman0 6910 | A nonempty function value ... |
| nfvres 6911 | The value of a non-member ... |
| nfunsn 6912 | If the restriction of a cl... |
| fvfundmfvn0 6913 | If the "value of a class" ... |
| 0fv 6914 | Function value of the empt... |
| fv2prc 6915 | A function value of a func... |
| elfv2ex 6916 | If a function value of a f... |
| fveqres 6917 | Equal values imply equal v... |
| csbfv12 6918 | Move class substitution in... |
| csbfv2g 6919 | Move class substitution in... |
| csbfv 6920 | Substitution for a functio... |
| funbrfv 6921 | The second argument of a b... |
| funopfv 6922 | The second element in an o... |
| fnbrfvb 6923 | Equivalence of function va... |
| fnopfvb 6924 | Equivalence of function va... |
| fvelima2 6925 | Function value in an image... |
| funbrfvb 6926 | Equivalence of function va... |
| funopfvb 6927 | Equivalence of function va... |
| fnbrfvb2 6928 | Version of ~ fnbrfvb for f... |
| fdmeu 6929 | There is exactly one codom... |
| funbrfv2b 6930 | Function value in terms of... |
| dffn5 6931 | Representation of a functi... |
| fnrnfv 6932 | The range of a function ex... |
| fvelrnb 6933 | A member of a function's r... |
| foelcdmi 6934 | A member of a surjective f... |
| dfimafn 6935 | Alternate definition of th... |
| dfimafn2 6936 | Alternate definition of th... |
| funimass4 6937 | Membership relation for th... |
| fvelima 6938 | Function value in an image... |
| funimassd 6939 | Sufficient condition for t... |
| fvelimad 6940 | Function value in an image... |
| feqmptd 6941 | Deduction form of ~ dffn5 ... |
| feqresmpt 6942 | Express a restricted funct... |
| feqmptdf 6943 | Deduction form of ~ dffn5f... |
| dffn5f 6944 | Representation of a functi... |
| fvelimab 6945 | Function value in an image... |
| fvelimabd 6946 | Deduction form of ~ fvelim... |
| fimarab 6947 | Expressing the image of a ... |
| unima 6948 | Image of a union. (Contri... |
| fvi 6949 | The value of the identity ... |
| fviss 6950 | The value of the identity ... |
| fniinfv 6951 | The indexed intersection o... |
| fnsnfv 6952 | Singleton of function valu... |
| opabiotafun 6953 | Define a function whose va... |
| opabiotadm 6954 | Define a function whose va... |
| opabiota 6955 | Define a function whose va... |
| fnimapr 6956 | The image of a pair under ... |
| fnimatpd 6957 | The image of an unordered ... |
| ssimaex 6958 | The existence of a subimag... |
| ssimaexg 6959 | The existence of a subimag... |
| funfv 6960 | A simplified expression fo... |
| funfv2 6961 | The value of a function. ... |
| funfv2f 6962 | The value of a function. ... |
| fvun 6963 | Value of the union of two ... |
| fvun1 6964 | The value of a union when ... |
| fvun2 6965 | The value of a union when ... |
| fvun1d 6966 | The value of a union when ... |
| fvun2d 6967 | The value of a union when ... |
| dffv2 6968 | Alternate definition of fu... |
| dmfco 6969 | Domains of a function comp... |
| fvco2 6970 | Value of a function compos... |
| fvco 6971 | Value of a function compos... |
| fvcod 6972 | Value of a function compos... |
| fvco3 6973 | Value of a function compos... |
| fvco3d 6974 | Value of a function compos... |
| fvco4i 6975 | Conditions for a compositi... |
| fvopab3g 6976 | Value of a function given ... |
| fvopab3ig 6977 | Value of a function given ... |
| brfvopabrbr 6978 | The binary relation of a f... |
| fvmptg 6979 | Value of a function given ... |
| fvmpti 6980 | Value of a function given ... |
| fvmpt 6981 | Value of a function given ... |
| fvmpt2f 6982 | Value of a function given ... |
| funcnvmpt 6983 | Condition for a function i... |
| fvtresfn 6984 | Functionality of a tuple-r... |
| fvmpts 6985 | Value of a function given ... |
| fvmpt3 6986 | Value of a function given ... |
| fvmpt3i 6987 | Value of a function given ... |
| fvmptdf 6988 | Deduction version of ~ fvm... |
| fvmptd 6989 | Deduction version of ~ fvm... |
| fvmptd2 6990 | Deduction version of ~ fvm... |
| mptrcl 6991 | Reverse closure for a mapp... |
| fvmpt2i 6992 | Value of a function given ... |
| fvmpt2 6993 | Value of a function given ... |
| fvmptss 6994 | If all the values of the m... |
| fvmpt2d 6995 | Deduction version of ~ fvm... |
| fvmptex 6996 | Express a function ` F ` w... |
| fvmptd3f 6997 | Alternate deduction versio... |
| fvmptd2f 6998 | Alternate deduction versio... |
| fvmptdv 6999 | Alternate deduction versio... |
| fvmptdv2 7000 | Alternate deduction versio... |
| mpteqb 7001 | Bidirectional equality the... |
| fvmptt 7002 | Closed theorem form of ~ f... |
| fvmptf 7003 | Value of a function given ... |
| fvmptnf 7004 | The value of a function gi... |
| fvmptd3 7005 | Deduction version of ~ fvm... |
| fvmptd4 7006 | Deduction version of ~ fvm... |
| fvmptn 7007 | This somewhat non-intuitiv... |
| fvmptss2 7008 | A mapping always evaluates... |
| elfvmptrab1w 7009 | Implications for the value... |
| elfvmptrab1 7010 | Implications for the value... |
| elfvmptrab 7011 | Implications for the value... |
| fvopab4ndm 7012 | Value of a function given ... |
| fvmptndm 7013 | Value of a function given ... |
| fvmptrabfv 7014 | Value of a function mappin... |
| fvopab5 7015 | The value of a function th... |
| fvopab6 7016 | Value of a function given ... |
| eqfnfv 7017 | Equality of functions is d... |
| eqfnfv2 7018 | Equality of functions is d... |
| eqfnfv3 7019 | Derive equality of functio... |
| eqfnfvd 7020 | Deduction for equality of ... |
| eqfnfv2f 7021 | Equality of functions is d... |
| fsneq 7022 | Equality condition for two... |
| eqfunfv 7023 | Equality of functions is d... |
| eqfnun 7024 | Two functions on ` A u. B ... |
| fvreseq0 7025 | Equality of restricted fun... |
| fvreseq1 7026 | Equality of a function res... |
| fvreseq 7027 | Equality of restricted fun... |
| fnmptfvd 7028 | A function with a given do... |
| fndmdif 7029 | Two ways to express the lo... |
| fndmdifcom 7030 | The difference set between... |
| fndmdifeq0 7031 | The difference set of two ... |
| fndmin 7032 | Two ways to express the lo... |
| fneqeql 7033 | Two functions are equal if... |
| fneqeql2 7034 | Two functions are equal if... |
| fnreseql 7035 | Two functions are equal on... |
| chfnrn 7036 | The range of a choice func... |
| funfvop 7037 | Ordered pair with function... |
| funfvbrb 7038 | Two ways to say that ` A `... |
| fvimacnvi 7039 | A member of a preimage is ... |
| fvimacnv 7040 | The argument of a function... |
| funimass3 7041 | A kind of contraposition l... |
| funimass5 7042 | A subclass of a preimage i... |
| funconstss 7043 | Two ways of specifying tha... |
| fvimacnvALT 7044 | Alternate proof of ~ fvima... |
| elpreima 7045 | Membership in the preimage... |
| elpreimad 7046 | Membership in the preimage... |
| fniniseg 7047 | Membership in the preimage... |
| fncnvima2 7048 | Inverse images under funct... |
| fniniseg2 7049 | Inverse point images under... |
| unpreima 7050 | Preimage of a union. (Con... |
| inpreima 7051 | Preimage of an intersectio... |
| difpreima 7052 | Preimage of a difference. ... |
| respreima 7053 | The preimage of a restrict... |
| cnvimainrn 7054 | The preimage of the inters... |
| sspreima 7055 | The preimage of a subset i... |
| iunpreima 7056 | Preimage of an indexed uni... |
| iinpreima 7057 | Preimage of an intersectio... |
| intpreima 7058 | Preimage of an intersectio... |
| fimacnvinrn 7059 | Taking the converse image ... |
| fimacnvinrn2 7060 | Taking the converse image ... |
| rescnvimafod 7061 | The restriction of a funct... |
| fvn0ssdmfun 7062 | If a class' function value... |
| fnopfv 7063 | Ordered pair with function... |
| fvelrn 7064 | A function's value belongs... |
| nelrnfvne 7065 | A function value cannot be... |
| fveqdmss 7066 | If the empty set is not co... |
| fveqressseq 7067 | If the empty set is not co... |
| fnfvelrn 7068 | A function's value belongs... |
| ffvelcdm 7069 | A function's value belongs... |
| fnfvelrnd 7070 | A function's value belongs... |
| ffvelcdmi 7071 | A function's value belongs... |
| ffvelcdmda 7072 | A function's value belongs... |
| ffvelcdmd 7073 | A function's value belongs... |
| feldmfvelcdm 7074 | A class is an element of t... |
| rexrn 7075 | Restricted existential qua... |
| ralrn 7076 | Restricted universal quant... |
| elrnrexdm 7077 | For any element in the ran... |
| elrnrexdmb 7078 | For any element in the ran... |
| eldmrexrn 7079 | For any element in the dom... |
| eldmrexrnb 7080 | For any element in the dom... |
| fvcofneq 7081 | The values of two function... |
| ralrnmptw 7082 | A restricted quantifier ov... |
| rexrnmptw 7083 | A restricted quantifier ov... |
| ralrnmpt 7084 | A restricted quantifier ov... |
| rexrnmpt 7085 | A restricted quantifier ov... |
| f0cli 7086 | Unconditional closure of a... |
| dff2 7087 | Alternate definition of a ... |
| dff3 7088 | Alternate definition of a ... |
| dff4 7089 | Alternate definition of a ... |
| dffo3 7090 | An onto mapping expressed ... |
| dffo4 7091 | Alternate definition of an... |
| dffo5 7092 | Alternate definition of an... |
| exfo 7093 | A relation equivalent to t... |
| dffo3f 7094 | An onto mapping expressed ... |
| foelrn 7095 | Property of a surjective f... |
| foelrnf 7096 | Property of a surjective f... |
| foco2 7097 | If a composition of two fu... |
| fmpt 7098 | Functionality of the mappi... |
| f1ompt 7099 | Express bijection for a ma... |
| fmpti 7100 | Functionality of the mappi... |
| fvmptelcdm 7101 | The value of a function at... |
| fmptd 7102 | Domain and codomain of the... |
| fmpttd 7103 | Version of ~ fmptd with in... |
| fmpt3d 7104 | Domain and codomain of the... |
| fmptdf 7105 | A version of ~ fmptd using... |
| fompt 7106 | Express being onto for a m... |
| ffnfv 7107 | A function maps to a class... |
| ffnfvf 7108 | A function maps to a class... |
| fnfvrnss 7109 | An upper bound for range d... |
| fcdmssb 7110 | A function is a function i... |
| rnmptss 7111 | The range of an operation ... |
| rnmptssd 7112 | The range of a function gi... |
| fmpt2d 7113 | Domain and codomain of the... |
| ffvresb 7114 | A necessary and sufficient... |
| fssrescdmd 7115 | Restriction of a function ... |
| f1oresrab 7116 | Build a bijection between ... |
| f1ossf1o 7117 | Restricting a bijection, w... |
| fmptco 7118 | Composition of two functio... |
| fmptcof 7119 | Version of ~ fmptco where ... |
| fmptcos 7120 | Composition of two functio... |
| cofmpt 7121 | Express composition of a m... |
| fcompt 7122 | Express composition of two... |
| fcoconst 7123 | Composition with a constan... |
| fsn 7124 | A function maps a singleto... |
| fsn2 7125 | A function that maps a sin... |
| fsng 7126 | A function maps a singleto... |
| fsn2g 7127 | A function that maps a sin... |
| xpsng 7128 | The Cartesian product of t... |
| xpsn 7129 | The Cartesian product of t... |
| xpprsng 7130 | The Cartesian product of a... |
| xpsnprg 7131 | The Cartesian product of a... |
| xpsntpg 7132 | The Cartesian product of a... |
| f1o2sn 7133 | A singleton consisting in ... |
| residpr 7134 | Restriction of the identit... |
| dfmpt 7135 | Alternate definition for t... |
| fnasrn 7136 | A function expressed as th... |
| idref 7137 | Two ways to state that a r... |
| funiun 7138 | A function is a union of s... |
| funopsn 7139 | If a function is an ordere... |
| funopsnOLD 7140 | Obsolete version of ~ funo... |
| funop 7141 | An ordered pair is a funct... |
| funopdmsn 7142 | The domain of a function w... |
| funsndifnop 7143 | A singleton of an ordered ... |
| funsneqopb 7144 | A singleton of an ordered ... |
| ressnop0 7145 | If ` A ` is not in ` C ` ,... |
| fpr 7146 | A function with a domain o... |
| fprg 7147 | A function with a domain o... |
| ftpg 7148 | A function with a domain o... |
| ftp 7149 | A function with a domain o... |
| fnressn 7150 | A function restricted to a... |
| funressn 7151 | A function restricted to a... |
| fressnfv 7152 | The value of a function re... |
| fvrnressn 7153 | If the value of a function... |
| fvressn 7154 | The value of a function re... |
| fvconst 7155 | The value of a constant fu... |
| fnsnr 7156 | If a class belongs to a fu... |
| fnsnbg 7157 | A function's domain is a s... |
| fnsnb 7158 | A function whose domain is... |
| fnsnbOLD 7159 | Obsolete version of ~ fnsn... |
| fmptsn 7160 | Express a singleton functi... |
| fmptsng 7161 | Express a singleton functi... |
| fmptsnd 7162 | Express a singleton functi... |
| fmptap 7163 | Append an additional value... |
| fmptapd 7164 | Append an additional value... |
| fmptpr 7165 | Express a pair function in... |
| fvresi 7166 | The value of a restricted ... |
| fninfp 7167 | Express the class of fixed... |
| fnelfp 7168 | Property of a fixed point ... |
| fndifnfp 7169 | Express the class of non-f... |
| fnelnfp 7170 | Property of a non-fixed po... |
| fnnfpeq0 7171 | A function is the identity... |
| fvunsn 7172 | Remove an ordered pair not... |
| fvsng 7173 | The value of a singleton o... |
| fvsn 7174 | The value of a singleton o... |
| fvsnun1 7175 | The value of a function wi... |
| fvsnun2 7176 | The value of a function wi... |
| fnsnsplit 7177 | Split a function into a si... |
| fsnunf 7178 | Adjoining a point to a fun... |
| fsnunf2 7179 | Adjoining a point to a pun... |
| fsnunfv 7180 | Recover the added point fr... |
| fsnunres 7181 | Recover the original funct... |
| funresdfunsn 7182 | Restricting a function to ... |
| fvpr1g 7183 | The value of a function wi... |
| fvpr2g 7184 | The value of a function wi... |
| fvpr1 7185 | The value of a function wi... |
| fvpr2 7186 | The value of a function wi... |
| fprb 7187 | A condition for functionho... |
| fvtp1 7188 | The first value of a funct... |
| fvtp2 7189 | The second value of a func... |
| fvtp3 7190 | The third value of a funct... |
| fvtp1g 7191 | The value of a function wi... |
| fvtp2g 7192 | The value of a function wi... |
| fvtp3g 7193 | The value of a function wi... |
| fvtp0 7194 | The undefined value of a f... |
| tpres 7195 | An unordered triple of ord... |
| fvconst2g 7196 | The value of a constant fu... |
| fconst2g 7197 | A constant function expres... |
| fvconst2 7198 | The value of a constant fu... |
| fconst2 7199 | A constant function expres... |
| fconst5 7200 | Two ways to express that a... |
| rnmptc 7201 | Range of a constant functi... |
| fnprb 7202 | A function whose domain ha... |
| fntpb 7203 | A function whose domain ha... |
| fnpr2g 7204 | A function whose domain ha... |
| fpr2g 7205 | A function that maps a pai... |
| fconstfv 7206 | A constant function expres... |
| fconst3 7207 | Two ways to express a cons... |
| fconst4 7208 | Two ways to express a cons... |
| resfunexg 7209 | The restriction of a funct... |
| resiexd 7210 | The restriction of the ide... |
| fnex 7211 | If the domain of a functio... |
| fnexd 7212 | If the domain of a functio... |
| funex 7213 | If the domain of a functio... |
| opabex 7214 | Existence of a function ex... |
| mptexg 7215 | If the domain of a functio... |
| mptexgf 7216 | If the domain of a functio... |
| mptex 7217 | If the domain of a functio... |
| mptexd 7218 | If the domain of a functio... |
| mptrabex 7219 | If the domain of a functio... |
| fex 7220 | If the domain of a mapping... |
| fexd 7221 | If the domain of a mapping... |
| mptfvmpt 7222 | A function in maps-to nota... |
| eufnfv 7223 | A function is uniquely det... |
| funfvima 7224 | A function's value in a pr... |
| funfvima2 7225 | A function's value in an i... |
| funfvima2d 7226 | A function's value in a pr... |
| fnfvima 7227 | The function value of an o... |
| fnfvimad 7228 | A function's value belongs... |
| resfvresima 7229 | The value of the function ... |
| funfvima3 7230 | A class including a functi... |
| ralima 7231 | Universal quantification u... |
| rexima 7232 | Existential quantification... |
| fvclss 7233 | Upper bound for the class ... |
| elabrex 7234 | Elementhood in an image se... |
| elabrexg 7235 | Elementhood in an image se... |
| abrexco 7236 | Composition of two image m... |
| imaiun 7237 | The image of an indexed un... |
| imauni 7238 | The image of a union is th... |
| fniunfv 7239 | The indexed union of a fun... |
| funiunfv 7240 | The indexed union of a fun... |
| funiunfvf 7241 | The indexed union of a fun... |
| eluniima 7242 | Membership in the union of... |
| elunirn 7243 | Membership in the union of... |
| elunirnALT 7244 | Alternate proof of ~ eluni... |
| fnunirn 7245 | Membership in a union of s... |
| dff13 7246 | A one-to-one function in t... |
| dff13f 7247 | A one-to-one function in t... |
| f1veqaeq 7248 | If the values of a one-to-... |
| f1cofveqaeq 7249 | If the values of a composi... |
| f1cofveqaeqALT 7250 | Alternate proof of ~ f1cof... |
| dff14i 7251 | A one-to-one function maps... |
| 2f1fvneq 7252 | If two one-to-one function... |
| f1mpt 7253 | Express injection for a ma... |
| f1fveq 7254 | Equality of function value... |
| f1elima 7255 | Membership in the image of... |
| f1imass 7256 | Taking images under a one-... |
| f1imaeq 7257 | Taking images under a one-... |
| f1imapss 7258 | Taking images under a one-... |
| fpropnf1 7259 | A function, given by an un... |
| f1dom3fv3dif 7260 | The function values for a ... |
| f1dom3el3dif 7261 | The codomain of a 1-1 func... |
| dff14a 7262 | A one-to-one function in t... |
| dff14b 7263 | A one-to-one function in t... |
| dff15 7264 | A one-to-one function in t... |
| f1resveqaeq 7265 | If a function restricted t... |
| f1resrcmplf1dlem 7266 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 7267 | If a function's restrictio... |
| f1ounsn 7268 | Extension of a bijection b... |
| f12dfv 7269 | A one-to-one function with... |
| f13dfv 7270 | A one-to-one function with... |
| dff1o6 7271 | A one-to-one onto function... |
| f1ocnvfv1 7272 | The converse value of the ... |
| f1ocnvfv2 7273 | The value of the converse ... |
| f1ocnvfv 7274 | Relationship between the v... |
| f1ocnvfvb 7275 | Relationship between the v... |
| nvof1o 7276 | An involution is a bijecti... |
| nvocnv 7277 | The converse of an involut... |
| f1cdmsn 7278 | If a one-to-one function w... |
| fsnex 7279 | Relate a function with a s... |
| f1prex 7280 | Relate a one-to-one functi... |
| f1ocnvdm 7281 | The value of the converse ... |
| f1ocnvfvrneq 7282 | If the values of a one-to-... |
| fcof1 7283 | An application is injectiv... |
| fcofo 7284 | An application is surjecti... |
| cbvfo 7285 | Change bound variable betw... |
| cbvexfo 7286 | Change bound variable betw... |
| cocan1 7287 | An injection is left-cance... |
| cocan2 7288 | A surjection is right-canc... |
| fcof1oinvd 7289 | Show that a function is th... |
| fcof1od 7290 | A function is bijective if... |
| 2fcoidinvd 7291 | Show that a function is th... |
| fcof1o 7292 | Show that two functions ar... |
| 2fvcoidd 7293 | Show that the composition ... |
| 2fvidf1od 7294 | A function is bijective if... |
| 2fvidinvd 7295 | Show that two functions ar... |
| foeqcnvco 7296 | Condition for function equ... |
| f1eqcocnv 7297 | Condition for function equ... |
| fveqf1o 7298 | Given a bijection ` F ` , ... |
| f1ocoima 7299 | The composition of two bij... |
| nf1const 7300 | A constant function from a... |
| nf1oconst 7301 | A constant function from a... |
| f1ofvswap 7302 | Swapping two values in a b... |
| fvf1pr 7303 | Values of a one-to-one fun... |
| fliftrel 7304 | ` F ` , a function lift, i... |
| fliftel 7305 | Elementhood in the relatio... |
| fliftel1 7306 | Elementhood in the relatio... |
| fliftcnv 7307 | Converse of the relation `... |
| fliftfun 7308 | The function ` F ` is the ... |
| fliftfund 7309 | The function ` F ` is the ... |
| fliftfuns 7310 | The function ` F ` is the ... |
| fliftf 7311 | The domain and range of th... |
| fliftval 7312 | The value of the function ... |
| isoeq1 7313 | Equality theorem for isomo... |
| isoeq2 7314 | Equality theorem for isomo... |
| isoeq3 7315 | Equality theorem for isomo... |
| isoeq4 7316 | Equality theorem for isomo... |
| isoeq5 7317 | Equality theorem for isomo... |
| nfiso 7318 | Bound-variable hypothesis ... |
| isof1o 7319 | An isomorphism is a one-to... |
| isof1oidb 7320 | A function is a bijection ... |
| isof1oopb 7321 | A function is a bijection ... |
| isorel 7322 | An isomorphism connects bi... |
| soisores 7323 | Express the condition of i... |
| soisoi 7324 | Infer isomorphism from one... |
| isoid 7325 | Identity law for isomorphi... |
| isocnv 7326 | Converse law for isomorphi... |
| isocnv2 7327 | Converse law for isomorphi... |
| isocnv3 7328 | Complementation law for is... |
| isores2 7329 | An isomorphism from one we... |
| isores1 7330 | An isomorphism from one we... |
| isores3 7331 | Induced isomorphism on a s... |
| isotr 7332 | Composition (transitive) l... |
| isomin 7333 | Isomorphisms preserve mini... |
| isoini 7334 | Isomorphisms preserve init... |
| isoini2 7335 | Isomorphisms are isomorphi... |
| isofrlem 7336 | Lemma for ~ isofr . (Cont... |
| isoselem 7337 | Lemma for ~ isose . (Cont... |
| isofr 7338 | An isomorphism preserves w... |
| isose 7339 | An isomorphism preserves s... |
| isofr2 7340 | A weak form of ~ isofr tha... |
| isopolem 7341 | Lemma for ~ isopo . (Cont... |
| isopo 7342 | An isomorphism preserves t... |
| isosolem 7343 | Lemma for ~ isoso . (Cont... |
| isoso 7344 | An isomorphism preserves t... |
| isowe 7345 | An isomorphism preserves t... |
| isowe2 7346 | A weak form of ~ isowe tha... |
| f1oiso 7347 | Any one-to-one onto functi... |
| f1oiso2 7348 | Any one-to-one onto functi... |
| f1owe 7349 | Well-ordering of isomorphi... |
| f1oweOLD 7350 | Obsolete version of f1owe ... |
| f1we 7351 | Pull back a well ordering ... |
| weniso 7352 | A set-like well-ordering h... |
| weisoeq 7353 | Thus, there is at most one... |
| weisoeq2 7354 | Thus, there is at most one... |
| knatar 7355 | The Knaster-Tarski theorem... |
| fvresval 7356 | The value of a restricted ... |
| funeldmb 7357 | If ` (/) ` is not part of ... |
| eqfunresadj 7358 | Law for adjoining an eleme... |
| eqfunressuc 7359 | Law for equality of restri... |
| fnssintima 7360 | Condition for subset of an... |
| fnimasnd 7361 | The image of a function by... |
| canth 7362 | No set ` A ` is equinumero... |
| ncanth 7363 | Cantor's theorem fails for... |
| riotaeqdv 7366 | Formula-building deduction... |
| riotabidv 7367 | Formula-building deduction... |
| riotaeqbidv 7368 | Equality deduction for res... |
| riotaex 7369 | Restricted iota is a set. ... |
| riotav 7370 | An iota restricted to the ... |
| riotauni 7371 | Restricted iota in terms o... |
| nfriota1 7372 | The abstraction variable i... |
| nfriotadw 7373 | Deduction version of ~ nfr... |
| cbvriotaw 7374 | Change bound variable in a... |
| cbvriotavw 7375 | Change bound variable in a... |
| nfriotad 7376 | Deduction version of ~ nfr... |
| nfriota 7377 | A variable not free in a w... |
| cbvriota 7378 | Change bound variable in a... |
| cbvriotav 7379 | Change bound variable in a... |
| csbriota 7380 | Interchange class substitu... |
| riotacl2 7381 | Membership law for "the un... |
| riotacl 7382 | Closure of restricted iota... |
| riotasbc 7383 | Substitution law for descr... |
| riotabidva 7384 | Equivalent wff's yield equ... |
| riotabiia 7385 | Equivalent wff's yield equ... |
| riota1 7386 | Property of restricted iot... |
| riota1a 7387 | Property of iota. (Contri... |
| riota2df 7388 | A deduction version of ~ r... |
| riota2f 7389 | This theorem shows a condi... |
| riota2 7390 | This theorem shows a condi... |
| riotaeqimp 7391 | If two restricted iota des... |
| riotaprop 7392 | Properties of a restricted... |
| riota5f 7393 | A method for computing res... |
| riota5 7394 | A method for computing res... |
| riotass2 7395 | Restriction of a unique el... |
| riotass 7396 | Restriction of a unique el... |
| moriotass 7397 | Restriction of a unique el... |
| snriota 7398 | A restricted class abstrac... |
| riotaxfrd 7399 | Change the variable ` x ` ... |
| eusvobj2 7400 | Specify the same property ... |
| eusvobj1 7401 | Specify the same object in... |
| f1ofveu 7402 | There is one domain elemen... |
| f1ocnvfv3 7403 | Value of the converse of a... |
| riotaund 7404 | Restricted iota equals the... |
| riotassuni 7405 | The restricted iota class ... |
| riotaclb 7406 | Bidirectional closure of r... |
| riotarab 7407 | Restricted iota of a restr... |
| oveq 7414 | Equality theorem for opera... |
| oveq1 7415 | Equality theorem for opera... |
| oveq2 7416 | Equality theorem for opera... |
| oveq12 7417 | Equality theorem for opera... |
| oveq1i 7418 | Equality inference for ope... |
| oveq2i 7419 | Equality inference for ope... |
| oveq12i 7420 | Equality inference for ope... |
| oveqi 7421 | Equality inference for ope... |
| oveq123i 7422 | Equality inference for ope... |
| oveq1d 7423 | Equality deduction for ope... |
| oveq2d 7424 | Equality deduction for ope... |
| oveqd 7425 | Equality deduction for ope... |
| oveq12d 7426 | Equality deduction for ope... |
| oveqan12d 7427 | Equality deduction for ope... |
| oveqan12rd 7428 | Equality deduction for ope... |
| oveq123d 7429 | Equality deduction for ope... |
| fvoveq1d 7430 | Equality deduction for nes... |
| fvoveq1 7431 | Equality theorem for neste... |
| ovanraleqv 7432 | Equality theorem for a con... |
| imbrov2fvoveq 7433 | Equality theorem for neste... |
| ovrspc2v 7434 | If an operation value is a... |
| oveqrspc2v 7435 | Restricted specialization ... |
| oveqdr 7436 | Equality of two operations... |
| nfovd 7437 | Deduction version of bound... |
| nfov 7438 | Bound-variable hypothesis ... |
| oprabidw 7439 | The law of concretion. Sp... |
| oprabid 7440 | The law of concretion. Sp... |
| ovex 7441 | The result of an operation... |
| ovexi 7442 | The result of an operation... |
| ovexd 7443 | The result of an operation... |
| ovssunirn 7444 | The result of an operation... |
| 0ov 7445 | Operation value of the emp... |
| ovprc 7446 | The value of an operation ... |
| ovprc1 7447 | The value of an operation ... |
| ovprc2 7448 | The value of an operation ... |
| ovrcl 7449 | Reverse closure for an ope... |
| elfvov1 7450 | Utility theorem: reverse c... |
| elfvov2 7451 | Utility theorem: reverse c... |
| csbov123 7452 | Move class substitution in... |
| csbov 7453 | Move class substitution in... |
| csbov12g 7454 | Move class substitution in... |
| csbov1g 7455 | Move class substitution in... |
| csbov2g 7456 | Move class substitution in... |
| rspceov 7457 | A frequently used special ... |
| elovimad 7458 | Elementhood of the image s... |
| fnbrovb 7459 | Value of a binary operatio... |
| fnotovb 7460 | Equivalence of operation v... |
| opabbrex 7461 | A collection of ordered pa... |
| opabresex2 7462 | Restrictions of a collecti... |
| fvmptopab 7463 | The function value of a ma... |
| f1opr 7464 | Condition for an operation... |
| brfvopab 7465 | The classes involved in a ... |
| dfoprab2 7466 | Class abstraction for oper... |
| reloprab 7467 | An operation class abstrac... |
| oprabv 7468 | If a pair and a class are ... |
| nfoprab1 7469 | The abstraction variables ... |
| nfoprab2 7470 | The abstraction variables ... |
| nfoprab3 7471 | The abstraction variables ... |
| nfoprab 7472 | Bound-variable hypothesis ... |
| oprabbid 7473 | Equivalent wff's yield equ... |
| oprabbidv 7474 | Equivalent wff's yield equ... |
| oprabbii 7475 | Equivalent wff's yield equ... |
| ssoprab2 7476 | Equivalence of ordered pai... |
| ssoprab2b 7477 | Equivalence of ordered pai... |
| eqoprab2bw 7478 | Equivalence of ordered pai... |
| eqoprab2b 7479 | Equivalence of ordered pai... |
| mpoeq123 7480 | An equality theorem for th... |
| mpoeq12 7481 | An equality theorem for th... |
| mpoeq123dva 7482 | An equality deduction for ... |
| mpoeq123dv 7483 | An equality deduction for ... |
| mpoeq123i 7484 | An equality inference for ... |
| mpoeq3dva 7485 | Slightly more general equa... |
| mpoeq3ia 7486 | An equality inference for ... |
| mpoeq3dv 7487 | An equality deduction for ... |
| nfmpo1 7488 | Bound-variable hypothesis ... |
| nfmpo2 7489 | Bound-variable hypothesis ... |
| nfmpo 7490 | Bound-variable hypothesis ... |
| 0mpo0 7491 | A mapping operation with e... |
| mpo0v 7492 | A mapping operation with e... |
| mpo0 7493 | A mapping operation with e... |
| oprab4 7494 | Two ways to state the doma... |
| cbvoprab1 7495 | Rule used to change first ... |
| cbvoprab2 7496 | Change the second bound va... |
| cbvoprab12 7497 | Rule used to change first ... |
| cbvoprab12v 7498 | Rule used to change first ... |
| cbvoprab3 7499 | Rule used to change the th... |
| cbvoprab3v 7500 | Rule used to change the th... |
| cbvmpox 7501 | Rule to change the bound v... |
| cbvmpo 7502 | Rule to change the bound v... |
| cbvmpov 7503 | Rule to change the bound v... |
| elimdelov 7504 | Eliminate a hypothesis whi... |
| brif1 7505 | Move a relation inside and... |
| ovif 7506 | Move a conditional outside... |
| ovif2 7507 | Move a conditional outside... |
| ovif12 7508 | Move a conditional outside... |
| ifov 7509 | Move a conditional outside... |
| ifmpt2v 7510 | Move a conditional inside ... |
| dmoprab 7511 | The domain of an operation... |
| dmoprabss 7512 | The domain of an operation... |
| rnoprab 7513 | The range of an operation ... |
| rnoprab2 7514 | The range of a restricted ... |
| reldmoprab 7515 | The domain of an operation... |
| oprabss 7516 | Structure of an operation ... |
| eloprabga 7517 | The law of concretion for ... |
| eloprabg 7518 | The law of concretion for ... |
| ssoprab2i 7519 | Inference of operation cla... |
| mpov 7520 | Operation with universal d... |
| mpomptx 7521 | Express a two-argument fun... |
| mpompt 7522 | Express a two-argument fun... |
| mpodifsnif 7523 | A mapping with two argumen... |
| mposnif 7524 | A mapping with two argumen... |
| fconstmpo 7525 | Representation of a consta... |
| resoprab 7526 | Restriction of an operatio... |
| resoprab2 7527 | Restriction of an operator... |
| resmpo 7528 | Restriction of the mapping... |
| funoprabg 7529 | "At most one" is a suffici... |
| funoprab 7530 | "At most one" is a suffici... |
| fnoprabg 7531 | Functionality and domain o... |
| mpofun 7532 | The maps-to notation for a... |
| fnoprab 7533 | Functionality and domain o... |
| ffnov 7534 | An operation maps to a cla... |
| fovcld 7535 | Closure law for an operati... |
| fovcl 7536 | Closure law for an operati... |
| eqfnov 7537 | Equality of two operations... |
| eqfnov2 7538 | Two operators with the sam... |
| fnov 7539 | Representation of a functi... |
| mpo2eqb 7540 | Bidirectional equality the... |
| rnmpo 7541 | The range of an operation ... |
| reldmmpo 7542 | The domain of an operation... |
| elrnmpog 7543 | Membership in the range of... |
| elrnmpo 7544 | Membership in the range of... |
| elimampo 7545 | Membership in the image of... |
| elrnmpores 7546 | Membership in the range of... |
| ralrnmpo 7547 | A restricted quantifier ov... |
| rexrnmpo 7548 | A restricted quantifier ov... |
| ovid 7549 | The value of an operation ... |
| ovidig 7550 | The value of an operation ... |
| ovidi 7551 | The value of an operation ... |
| ov 7552 | The value of an operation ... |
| ovigg 7553 | The value of an operation ... |
| ovig 7554 | The value of an operation ... |
| ovmpt4g 7555 | Value of a function given ... |
| ovmpos 7556 | Value of a function given ... |
| ov2gf 7557 | The value of an operation ... |
| ovmpodxf 7558 | Value of an operation give... |
| ovmpodx 7559 | Value of an operation give... |
| ovmpod 7560 | Value of an operation give... |
| ovmpox 7561 | The value of an operation ... |
| ovmpoga 7562 | Value of an operation give... |
| ovmpoa 7563 | Value of an operation give... |
| ovmpodf 7564 | Alternate deduction versio... |
| ovmpodv 7565 | Alternate deduction versio... |
| ovmpodv2 7566 | Alternate deduction versio... |
| ovmpog 7567 | Value of an operation give... |
| ovmpo 7568 | Value of an operation give... |
| ovmpot 7569 | The value of an operation ... |
| fvmpopr2d 7570 | Value of an operation give... |
| ov3 7571 | The value of an operation ... |
| ov6g 7572 | The value of an operation ... |
| ovg 7573 | The value of an operation ... |
| ovres 7574 | The value of a restricted ... |
| ovresd 7575 | Lemma for converting metri... |
| oprres 7576 | The restriction of an oper... |
| ovn0ssdmfun 7577 | If a class' operation valu... |
| oprssov 7578 | The value of a member of t... |
| fovcdm 7579 | An operation's value belon... |
| fovcdmda 7580 | An operation's value belon... |
| fovcdmd 7581 | An operation's value belon... |
| fnrnov 7582 | The range of an operation ... |
| foov 7583 | An onto mapping of an oper... |
| fnovrn 7584 | An operation's value belon... |
| ovelrn 7585 | A member of an operation's... |
| funimassov 7586 | Membership relation for th... |
| ovelimab 7587 | Operation value in an imag... |
| ovima0 7588 | An operation value is a me... |
| ovconst2 7589 | The value of a constant op... |
| oprssdm 7590 | Domain of closure of an op... |
| nssdmovg 7591 | The value of an operation ... |
| ndmovg 7592 | The value of an operation ... |
| ndmov 7593 | The value of an operation ... |
| ndmovcl 7594 | The closure of an operatio... |
| ndmovrcl 7595 | Reverse closure law, when ... |
| ndmovcom 7596 | Any operation is commutati... |
| ndmovass 7597 | Any operation is associati... |
| ndmovdistr 7598 | Any operation is distribut... |
| ndmovord 7599 | Elimination of redundant a... |
| ndmovordi 7600 | Elimination of redundant a... |
| caovclg 7601 | Convert an operation closu... |
| caovcld 7602 | Convert an operation closu... |
| caovcl 7603 | Convert an operation closu... |
| caovcomg 7604 | Convert an operation commu... |
| caovcomd 7605 | Convert an operation commu... |
| caovcom 7606 | Convert an operation commu... |
| caovassg 7607 | Convert an operation assoc... |
| caovassd 7608 | Convert an operation assoc... |
| caovass 7609 | Convert an operation assoc... |
| caovcang 7610 | Convert an operation cance... |
| caovcand 7611 | Convert an operation cance... |
| caovcanrd 7612 | Commute the arguments of a... |
| caovcan 7613 | Convert an operation cance... |
| caovordig 7614 | Convert an operation order... |
| caovordid 7615 | Convert an operation order... |
| caovordg 7616 | Convert an operation order... |
| caovordd 7617 | Convert an operation order... |
| caovord2d 7618 | Operation ordering law wit... |
| caovord3d 7619 | Ordering law. (Contribute... |
| caovord 7620 | Convert an operation order... |
| caovord2 7621 | Operation ordering law wit... |
| caovord3 7622 | Ordering law. (Contribute... |
| caovdig 7623 | Convert an operation distr... |
| caovdid 7624 | Convert an operation distr... |
| caovdir2d 7625 | Convert an operation distr... |
| caovdirg 7626 | Convert an operation rever... |
| caovdird 7627 | Convert an operation distr... |
| caovdi 7628 | Convert an operation distr... |
| caov32d 7629 | Rearrange arguments in a c... |
| caov12d 7630 | Rearrange arguments in a c... |
| caov31d 7631 | Rearrange arguments in a c... |
| caov13d 7632 | Rearrange arguments in a c... |
| caov4d 7633 | Rearrange arguments in a c... |
| caov411d 7634 | Rearrange arguments in a c... |
| caov42d 7635 | Rearrange arguments in a c... |
| caov32 7636 | Rearrange arguments in a c... |
| caov12 7637 | Rearrange arguments in a c... |
| caov31 7638 | Rearrange arguments in a c... |
| caov13 7639 | Rearrange arguments in a c... |
| caov4 7640 | Rearrange arguments in a c... |
| caov411 7641 | Rearrange arguments in a c... |
| caov42 7642 | Rearrange arguments in a c... |
| caovdir 7643 | Reverse distributive law. ... |
| caovdilem 7644 | Lemma used by real number ... |
| caovlem2 7645 | Lemma used in real number ... |
| caovmo 7646 | Uniqueness of inverse elem... |
| imaeqexov 7647 | Substitute an operation va... |
| imaeqalov 7648 | Substitute an operation va... |
| mpondm0 7649 | The value of an operation ... |
| elmpocl 7650 | If a two-parameter class i... |
| elmpocl1 7651 | If a two-parameter class i... |
| elmpocl2 7652 | If a two-parameter class i... |
| elovmpod 7653 | Utility lemma for two-para... |
| elovmpo 7654 | Utility lemma for two-para... |
| elovmporab 7655 | Implications for the value... |
| elovmporab1w 7656 | Implications for the value... |
| elovmporab1 7657 | Implications for the value... |
| 2mpo0 7658 | If the operation value of ... |
| relmptopab 7659 | Any function to sets of or... |
| f1ocnvd 7660 | Describe an implicit one-t... |
| f1od 7661 | Describe an implicit one-t... |
| f1ocnv2d 7662 | Describe an implicit one-t... |
| f1o2d 7663 | Describe an implicit one-t... |
| f1opw2 7664 | A one-to-one mapping induc... |
| f1opw 7665 | A one-to-one mapping induc... |
| elovmpt3imp 7666 | If the value of a function... |
| ovmpt3rab1 7667 | The value of an operation ... |
| ovmpt3rabdm 7668 | If the value of a function... |
| elovmpt3rab1 7669 | Implications for the value... |
| elovmpt3rab 7670 | Implications for the value... |
| mpt3mpt 7673 | Express a three-argument f... |
| ofeqd 7678 | Equality theorem for funct... |
| ofeq 7679 | Equality theorem for funct... |
| ofreq 7680 | Equality theorem for funct... |
| ofexg 7681 | A function operation restr... |
| nfof 7682 | Hypothesis builder for fun... |
| nfofr 7683 | Hypothesis builder for fun... |
| ofrfvalg 7684 | Value of a relation applie... |
| offval 7685 | Value of an operation appl... |
| ofrfval 7686 | Value of a relation applie... |
| ofval 7687 | Evaluate a function operat... |
| ofrval 7688 | Exhibit a function relatio... |
| offn 7689 | The function operation pro... |
| offun 7690 | The function operation pro... |
| offval2f 7691 | The function operation exp... |
| ofmresval 7692 | Value of a restriction of ... |
| fnfvof 7693 | Function value of a pointw... |
| off 7694 | The function operation pro... |
| ofres 7695 | Restrict the operands of a... |
| offval2 7696 | The function operation exp... |
| ofrfval2 7697 | The function relation acti... |
| offvalfv 7698 | The function operation exp... |
| ofmpteq 7699 | Value of a pointwise opera... |
| coof 7700 | The composition of a _homo... |
| ofco 7701 | The composition of a funct... |
| offveq 7702 | Convert an identity of the... |
| offveqb 7703 | Equivalent expressions for... |
| ofc1 7704 | Left operation by a consta... |
| ofc2 7705 | Right operation by a const... |
| ofc12 7706 | Function operation on two ... |
| caofref 7707 | Transfer a reflexive law t... |
| caofinvl 7708 | Transfer a left inverse la... |
| caofid0l 7709 | Transfer a left identity l... |
| caofid0r 7710 | Transfer a right identity ... |
| caofid1 7711 | Transfer a right absorptio... |
| caofid2 7712 | Transfer a right absorptio... |
| caofcom 7713 | Transfer a commutative law... |
| caofidlcan 7714 | Transfer a cancellation/id... |
| caofrss 7715 | Transfer a relation subset... |
| caofass 7716 | Transfer an associative la... |
| caoftrn 7717 | Transfer a transitivity la... |
| caofdi 7718 | Transfer a distributive la... |
| caofdir 7719 | Transfer a reverse distrib... |
| caonncan 7720 | Transfer ~ nncan -shaped l... |
| relrpss 7723 | The proper subset relation... |
| brrpssg 7724 | The proper subset relation... |
| brrpss 7725 | The proper subset relation... |
| porpss 7726 | Every class is partially o... |
| sorpss 7727 | Express strict ordering un... |
| sorpssi 7728 | Property of a chain of set... |
| sorpssun 7729 | A chain of sets is closed ... |
| sorpssin 7730 | A chain of sets is closed ... |
| sorpssuni 7731 | In a chain of sets, a maxi... |
| sorpssint 7732 | In a chain of sets, a mini... |
| sorpsscmpl 7733 | The componentwise compleme... |
| zfun 7735 | Axiom of Union expressed w... |
| axun2 7736 | A variant of the Axiom of ... |
| uniex2 7737 | The Axiom of Union using t... |
| uniex2OLD 7738 | Obsolete version of ~ unie... |
| vuniex 7739 | The union of a setvar is a... |
| uniexg 7740 | The ZF Axiom of Union in c... |
| uniex 7741 | The Axiom of Union in clas... |
| uniexd 7742 | Deduction version of the Z... |
| unexg 7743 | The union of two sets is a... |
| unex 7744 | The union of two sets is a... |
| tpex 7745 | An unordered triple of cla... |
| unexb 7746 | Existence of union is equi... |
| xpexg 7747 | The Cartesian product of t... |
| xpexd 7748 | The Cartesian product of t... |
| 3xpexg 7749 | The Cartesian product of t... |
| xpex 7750 | The Cartesian product of t... |
| unexd 7751 | The union of two sets is a... |
| sqxpexg 7752 | The Cartesian square of a ... |
| abnexg 7753 | Sufficient condition for a... |
| abnex 7754 | Sufficient condition for a... |
| snnex 7755 | The class of all singleton... |
| pwnex 7756 | The class of all power set... |
| difex2 7757 | If the subtrahend of a cla... |
| difsnexi 7758 | If the difference of a cla... |
| uniuni 7759 | Expression for double unio... |
| uniexr 7760 | Converse of the Axiom of U... |
| uniexb 7761 | The Axiom of Union and its... |
| pwexr 7762 | Converse of the Axiom of P... |
| pwexb 7763 | The Axiom of Power Sets an... |
| elpwpwel 7764 | A class belongs to a doubl... |
| eldifpw 7765 | Membership in a power clas... |
| elpwun 7766 | Membership in the power cl... |
| pwuncl 7767 | Power classes are closed u... |
| iunpw 7768 | An indexed union of a powe... |
| fr3nr 7769 | A well-founded relation ha... |
| epne3 7770 | A well-founded class conta... |
| dfwe2 7771 | Alternate definition of we... |
| epweon 7772 | The membership relation we... |
| epweonALT 7773 | Alternate proof of ~ epweo... |
| ordon 7774 | The class of all ordinal n... |
| onprc 7775 | No set contains all ordina... |
| ssorduni 7776 | The union of a class of or... |
| ssonuni 7777 | The union of a set of ordi... |
| ssonunii 7778 | The union of a set of ordi... |
| ordeleqon 7779 | A way to express the ordin... |
| ordsson 7780 | Any ordinal class is a sub... |
| dford5 7781 | A class is ordinal iff it ... |
| onss 7782 | An ordinal number is a sub... |
| predon 7783 | The predecessor of an ordi... |
| ssonprc 7784 | Two ways of saying a class... |
| onuni 7785 | The union of an ordinal nu... |
| orduni 7786 | The union of an ordinal cl... |
| onint 7787 | The intersection (infimum)... |
| onint0 7788 | The intersection of a clas... |
| onssmin 7789 | A nonempty class of ordina... |
| onminesb 7790 | If a property is true for ... |
| onminsb 7791 | If a property is true for ... |
| oninton 7792 | The intersection of a none... |
| onintrab 7793 | The intersection of a clas... |
| onintrab2 7794 | An existence condition equ... |
| onnmin 7795 | No member of a set of ordi... |
| onnminsb 7796 | An ordinal number smaller ... |
| oneqmin 7797 | A way to show that an ordi... |
| uniordint 7798 | The union of a set of ordi... |
| onminex 7799 | If a wff is true for an or... |
| sucon 7800 | The class of all ordinal n... |
| sucexb 7801 | A successor exists iff its... |
| sucexg 7802 | The successor of a set is ... |
| sucex 7803 | The successor of a set is ... |
| onmindif2 7804 | The minimum of a class of ... |
| ordsuci 7805 | The successor of an ordina... |
| sucexeloni 7806 | If the successor of an ord... |
| onsuc 7807 | The successor of an ordina... |
| ordsuc 7808 | A class is ordinal if and ... |
| ordpwsuc 7809 | The collection of ordinals... |
| onpwsuc 7810 | The collection of ordinal ... |
| onsucb 7811 | A class is an ordinal numb... |
| ordsucss 7812 | The successor of an elemen... |
| onpsssuc 7813 | An ordinal number is a pro... |
| ordelsuc 7814 | A set belongs to an ordina... |
| onsucmin 7815 | The successor of an ordina... |
| ordsucelsuc 7816 | Membership is inherited by... |
| ordsucsssuc 7817 | The subclass relationship ... |
| ordsucuniel 7818 | Given an element ` A ` of ... |
| ordsucun 7819 | The successor of the maxim... |
| ordunpr 7820 | The maximum of two ordinal... |
| ordunel 7821 | The maximum of two ordinal... |
| onsucuni 7822 | A class of ordinal numbers... |
| ordsucuni 7823 | An ordinal class is a subc... |
| orduniorsuc 7824 | An ordinal class is either... |
| unon 7825 | The class of all ordinal n... |
| ordunisuc 7826 | An ordinal class is equal ... |
| orduniss2 7827 | The union of the ordinal s... |
| onsucuni2 7828 | A successor ordinal is the... |
| 0elsuc 7829 | The successor of an ordina... |
| limon 7830 | The class of ordinal numbe... |
| onuniorsuc 7831 | An ordinal number is eithe... |
| onssi 7832 | An ordinal number is a sub... |
| onsuci 7833 | The successor of an ordina... |
| onuninsuci 7834 | An ordinal is equal to its... |
| onsucssi 7835 | A set belongs to an ordina... |
| nlimsucg 7836 | A successor is not a limit... |
| orduninsuc 7837 | An ordinal class is equal ... |
| ordunisuc2 7838 | An ordinal equal to its un... |
| ordzsl 7839 | An ordinal is zero, a succ... |
| onzsl 7840 | An ordinal number is zero,... |
| dflim3 7841 | An alternate definition of... |
| dflim4 7842 | An alternate definition of... |
| limsuc 7843 | The successor of a member ... |
| limsssuc 7844 | A class includes a limit o... |
| nlimon 7845 | Two ways to express the cl... |
| limuni3 7846 | The union of a nonempty cl... |
| tfi 7847 | The Principle of Transfini... |
| tfisg 7848 | A closed form of ~ tfis . ... |
| tfis 7849 | Transfinite Induction Sche... |
| tfis2f 7850 | Transfinite Induction Sche... |
| tfis2 7851 | Transfinite Induction Sche... |
| tfis3 7852 | Transfinite Induction Sche... |
| tfisi 7853 | A transfinite induction sc... |
| tfinds 7854 | Principle of Transfinite I... |
| tfindsg 7855 | Transfinite Induction (inf... |
| tfindsg2 7856 | Transfinite Induction (inf... |
| tfindes 7857 | Transfinite Induction with... |
| tfinds2 7858 | Transfinite Induction (inf... |
| tfinds3 7859 | Principle of Transfinite I... |
| dfom2 7862 | An alternate definition of... |
| elom 7863 | Membership in omega. The ... |
| omsson 7864 | Omega is a subset of ` On ... |
| limomss 7865 | The class of natural numbe... |
| nnon 7866 | A natural number is an ord... |
| nnoni 7867 | A natural number is an ord... |
| nnord 7868 | A natural number is ordina... |
| trom 7869 | The class of finite ordina... |
| ordom 7870 | The class of finite ordina... |
| elnn 7871 | A member of a natural numb... |
| omon 7872 | The class of natural numbe... |
| omelon2 7873 | Omega is an ordinal number... |
| nnlim 7874 | A natural number is not a ... |
| omssnlim 7875 | The class of natural numbe... |
| limom 7876 | Omega is a limit ordinal. ... |
| peano2b 7877 | A class belongs to omega i... |
| nnsuc 7878 | A nonzero natural number i... |
| omsucne 7879 | A natural number is not th... |
| ssnlim 7880 | An ordinal subclass of non... |
| omsinds 7881 | Strong (or "total") induct... |
| omun 7882 | The union of two finite or... |
| peano1 7883 | Zero is a natural number. ... |
| peano2 7884 | The successor of any natur... |
| peano3 7885 | The successor of any natur... |
| peano3OLD 7886 | Obsolete version of ~ pean... |
| peano4 7887 | Two natural numbers are eq... |
| peano5 7888 | The induction postulate: a... |
| nn0suc 7889 | A natural number is either... |
| find 7890 | The Principle of Finite In... |
| finds 7891 | Principle of Finite Induct... |
| findsg 7892 | Principle of Finite Induct... |
| finds2 7893 | Principle of Finite Induct... |
| finds1 7894 | Principle of Finite Induct... |
| findes 7895 | Finite induction with expl... |
| dmexg 7896 | The domain of a set is a s... |
| rnexg 7897 | The range of a set is a se... |
| dmexd 7898 | The domain of a set is a s... |
| fndmexd 7899 | If a function is a set, it... |
| dmfex 7900 | If a mapping is a set, its... |
| fndmexb 7901 | The domain of a function i... |
| fdmexb 7902 | The domain of a function i... |
| dmfexALT 7903 | Alternate proof of ~ dmfex... |
| dmex 7904 | The domain of a set is a s... |
| rnex 7905 | The range of a set is a se... |
| iprc 7906 | The identity function is a... |
| resiexg 7907 | The existence of a restric... |
| imaexg 7908 | The image of a set is a se... |
| imaex 7909 | The image of a set is a se... |
| rnexd 7910 | The range of a set is a se... |
| imaexd 7911 | The image of a set is a se... |
| exse2 7912 | Any set relation is set-li... |
| xpexr 7913 | If a Cartesian product is ... |
| xpexr2 7914 | If a nonempty Cartesian pr... |
| xpexcnv 7915 | A condition where the conv... |
| soex 7916 | If the relation in a stric... |
| elxp4 7917 | Membership in a Cartesian ... |
| elxp5 7918 | Membership in a Cartesian ... |
| cnvexg 7919 | The converse of a set is a... |
| cnvex 7920 | The converse of a set is a... |
| relcnvexb 7921 | A relation is a set iff it... |
| f1oexrnex 7922 | If the range of a 1-1 onto... |
| f1oexbi 7923 | There is a one-to-one onto... |
| coexg 7924 | The composition of two set... |
| coex 7925 | The composition of two set... |
| coexd 7926 | The composition of two set... |
| funcnvuni 7927 | The union of a chain (with... |
| fun11uni 7928 | The union of a chain (with... |
| resf1extb 7929 | Extension of an injection ... |
| resf1ext2b 7930 | Extension of an injection ... |
| fex2 7931 | A function with bounded do... |
| fabexd 7932 | Existence of a set of func... |
| fabexg 7933 | Existence of a set of func... |
| fabex 7934 | Existence of a set of func... |
| mapex 7935 | The class of all functions... |
| f1oabexg 7936 | The class of all 1-1-onto ... |
| fiunlem 7937 | Lemma for ~ fiun and ~ f1i... |
| fiun 7938 | The union of a chain (with... |
| f1iun 7939 | The union of a chain (with... |
| fviunfun 7940 | The function value of an i... |
| ffoss 7941 | Relationship between a map... |
| f11o 7942 | Relationship between one-t... |
| resfunexgALT 7943 | Alternate proof of ~ resfu... |
| cofunexg 7944 | Existence of a composition... |
| cofunex2g 7945 | Existence of a composition... |
| fnexALT 7946 | Alternate proof of ~ fnex ... |
| funexw 7947 | Weak version of ~ funex th... |
| mptexw 7948 | Weak version of ~ mptex th... |
| funrnex 7949 | If the domain of a functio... |
| zfrep6OLD 7950 | Obsolete proof of ~ zfrep6... |
| focdmex 7951 | If the domain of an onto f... |
| f1dmex 7952 | If the codomain of a one-t... |
| f1ovv 7953 | The codomain/range of a 1-... |
| fvclex 7954 | Existence of the class of ... |
| fvresex 7955 | Existence of the class of ... |
| abrexexg 7956 | Existence of a class abstr... |
| abrexex 7957 | Existence of a class abstr... |
| iunexg 7958 | The existence of an indexe... |
| abrexex2g 7959 | Existence of an existentia... |
| opabex3d 7960 | Existence of an ordered pa... |
| opabex3rd 7961 | Existence of an ordered pa... |
| opabex3 7962 | Existence of an ordered pa... |
| iunex 7963 | The existence of an indexe... |
| abrexex2 7964 | Existence of an existentia... |
| abexssex 7965 | Existence of a class abstr... |
| abexex 7966 | A condition where a class ... |
| f1oweALT 7967 | Alternate proof of one dir... |
| wemoiso 7968 | Thus, there is at most one... |
| wemoiso2 7969 | Thus, there is at most one... |
| oprabexd 7970 | Existence of an operator a... |
| oprabex 7971 | Existence of an operation ... |
| oprabex3 7972 | Existence of an operation ... |
| oprabrexex2 7973 | Existence of an existentia... |
| ab2rexex 7974 | Existence of a class abstr... |
| ab2rexex2 7975 | Existence of an existentia... |
| xpexgALT 7976 | Alternate proof of ~ xpexg... |
| offval3 7977 | General value of ` ( F oF ... |
| offres 7978 | Pointwise combination comm... |
| ofmres 7979 | Equivalent expressions for... |
| ofmresex 7980 | Existence of a restriction... |
| mptcnfimad 7981 | The converse of a mapping ... |
| 1stval 7986 | The value of the function ... |
| 2ndval 7987 | The value of the function ... |
| 1stnpr 7988 | Value of the first-member ... |
| 2ndnpr 7989 | Value of the second-member... |
| 1st0 7990 | The value of the first-mem... |
| 2nd0 7991 | The value of the second-me... |
| op1st 7992 | Extract the first member o... |
| op2nd 7993 | Extract the second member ... |
| op1std 7994 | Extract the first member o... |
| op2ndd 7995 | Extract the second member ... |
| op1stg 7996 | Extract the first member o... |
| op2ndg 7997 | Extract the second member ... |
| ot1stg 7998 | Extract the first member o... |
| ot2ndg 7999 | Extract the second member ... |
| ot3rdg 8000 | Extract the third member o... |
| 1stval2 8001 | Alternate value of the fun... |
| 2ndval2 8002 | Alternate value of the fun... |
| oteqimp 8003 | The components of an order... |
| fo1st 8004 | The ` 1st ` function maps ... |
| fo2nd 8005 | The ` 2nd ` function maps ... |
| br1steqg 8006 | Uniqueness condition for t... |
| br2ndeqg 8007 | Uniqueness condition for t... |
| f1stres 8008 | Mapping of a restriction o... |
| f2ndres 8009 | Mapping of a restriction o... |
| fo1stres 8010 | Onto mapping of a restrict... |
| fo2ndres 8011 | Onto mapping of a restrict... |
| 1st2val 8012 | Value of an alternate defi... |
| 2nd2val 8013 | Value of an alternate defi... |
| 1stcof 8014 | Composition of the first m... |
| 2ndcof 8015 | Composition of the second ... |
| xp1st 8016 | Location of the first elem... |
| xp2nd 8017 | Location of the second ele... |
| elxp6 8018 | Membership in a Cartesian ... |
| elxp7 8019 | Membership in a Cartesian ... |
| eqopi 8020 | Equality with an ordered p... |
| xp2 8021 | Representation of Cartesia... |
| unielxp 8022 | The membership relation fo... |
| 1st2nd2 8023 | Reconstruction of a member... |
| 1st2ndb 8024 | Reconstruction of an order... |
| xpopth 8025 | An ordered pair theorem fo... |
| eqop 8026 | Two ways to express equali... |
| eqop2 8027 | Two ways to express equali... |
| op1steq 8028 | Two ways of expressing tha... |
| opreuopreu 8029 | There is a unique ordered ... |
| el2xptp0 8030 | A member of a nested Carte... |
| el2xpss 8031 | Version of ~ elrel for tri... |
| 2nd1st 8032 | Swap the members of an ord... |
| 1st2nd 8033 | Reconstruction of a member... |
| 1stdm 8034 | The first ordered pair com... |
| 2ndrn 8035 | The second ordered pair co... |
| 1st2ndbr 8036 | Express an element of a re... |
| releldm2 8037 | Two ways of expressing mem... |
| reldm 8038 | An expression for the doma... |
| releldmdifi 8039 | One way of expressing memb... |
| funfv1st2nd 8040 | The function value for the... |
| funelss 8041 | If the first component of ... |
| funeldmdif 8042 | Two ways of expressing mem... |
| sbcopeq1a 8043 | Equality theorem for subst... |
| csbopeq1a 8044 | Equality theorem for subst... |
| sbcoteq1a 8045 | Equality theorem for subst... |
| dfopab2 8046 | A way to define an ordered... |
| dfoprab3s 8047 | A way to define an operati... |
| dfoprab3 8048 | Operation class abstractio... |
| dfoprab4 8049 | Operation class abstractio... |
| dfoprab4f 8050 | Operation class abstractio... |
| opabex2 8051 | Condition for an operation... |
| opabn1stprc 8052 | An ordered-pair class abst... |
| opiota 8053 | The property of a uniquely... |
| cnvoprab 8054 | The converse of a class ab... |
| dfxp3 8055 | Define the Cartesian produ... |
| elopabi 8056 | A consequence of membershi... |
| eloprabi 8057 | A consequence of membershi... |
| mpomptsx 8058 | Express a two-argument fun... |
| mpompts 8059 | Express a two-argument fun... |
| dmmpossx 8060 | The domain of a mapping is... |
| fmpox 8061 | Functionality, domain and ... |
| fmpo 8062 | Functionality, domain and ... |
| fnmpo 8063 | Functionality and domain o... |
| fnmpoi 8064 | Functionality and domain o... |
| dmmpo 8065 | Domain of a class given by... |
| fmpodg 8066 | Domain and codomain of the... |
| fmpod 8067 | Domain and codomain of the... |
| ovmpoelrn 8068 | An operation's value belon... |
| dmmpoga 8069 | Domain of an operation giv... |
| dmmpog 8070 | Domain of an operation giv... |
| mpoexxg 8071 | Existence of an operation ... |
| mpoexg 8072 | Existence of an operation ... |
| mpoexga 8073 | If the domain of an operat... |
| mpoexw 8074 | Weak version of ~ mpoex th... |
| mpoex 8075 | If the domain of an operat... |
| mpoexd 8076 | Existence of an operation ... |
| mptmpoopabbrd 8077 | The operation value of a f... |
| mptmpoopabovd 8078 | The operation value of a f... |
| el2mpocsbcl 8079 | If the operation value of ... |
| el2mpocl 8080 | If the operation value of ... |
| fnmpoovd 8081 | A function with a Cartesia... |
| offval22 8082 | The function operation exp... |
| brovpreldm 8083 | If a binary relation holds... |
| bropopvvv 8084 | If a binary relation holds... |
| bropfvvvvlem 8085 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8086 | If a binary relation holds... |
| ovmptss 8087 | If all the values of the m... |
| relmpoopab 8088 | Any function to sets of or... |
| fmpoco 8089 | Composition of two functio... |
| oprabco 8090 | Composition of a function ... |
| oprab2co 8091 | Composition of operator ab... |
| df1st2 8092 | An alternate possible defi... |
| df2nd2 8093 | An alternate possible defi... |
| 1stconst 8094 | The mapping of a restricti... |
| 2ndconst 8095 | The mapping of a restricti... |
| dfmpo 8096 | Alternate definition for t... |
| mposn 8097 | An operation (in maps-to n... |
| curry1 8098 | Composition with ` ``' ( 2... |
| curry1val 8099 | The value of a curried fun... |
| curry1f 8100 | Functionality of a curried... |
| curry2 8101 | Composition with ` ``' ( 1... |
| curry2f 8102 | Functionality of a curried... |
| curry2val 8103 | The value of a curried fun... |
| cnvf1olem 8104 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8105 | Describe a function that m... |
| fparlem1 8106 | Lemma for ~ fpar . (Contr... |
| fparlem2 8107 | Lemma for ~ fpar . (Contr... |
| fparlem3 8108 | Lemma for ~ fpar . (Contr... |
| fparlem4 8109 | Lemma for ~ fpar . (Contr... |
| fpar 8110 | Merge two functions in par... |
| fsplit 8111 | A function that can be use... |
| fsplitfpar 8112 | Merge two functions with a... |
| offsplitfpar 8113 | Express the function opera... |
| f2ndf 8114 | The ` 2nd ` (second compon... |
| fo2ndf 8115 | The ` 2nd ` (second compon... |
| f1o2ndf1 8116 | The ` 2nd ` (second compon... |
| opco1 8117 | Value of an operation prec... |
| opco2 8118 | Value of an operation prec... |
| opco1i 8119 | Inference form of ~ opco1 ... |
| mpof1o2d 8120 | Sufficient condition for a... |
| frxp 8121 | A lexicographical ordering... |
| xporderlem 8122 | Lemma for lexicographical ... |
| poxp 8123 | A lexicographical ordering... |
| soxp 8124 | A lexicographical ordering... |
| wexp 8125 | A lexicographical ordering... |
| fnwelem 8126 | Lemma for ~ fnwe . (Contr... |
| fnwe 8127 | A variant on lexicographic... |
| fnse 8128 | Condition for the well-ord... |
| fvproj 8129 | Value of a function on ord... |
| fimaproj 8130 | Image of a cartesian produ... |
| ralxpes 8131 | A version of ~ ralxp with ... |
| ralxp3f 8132 | Restricted for all over a ... |
| ralxp3 8133 | Restricted for all over a ... |
| ralxp3es 8134 | Restricted for-all over a ... |
| frpoins3xpg 8135 | Special case of founded pa... |
| frpoins3xp3g 8136 | Special case of founded pa... |
| xpord2lem 8137 | Lemma for Cartesian produc... |
| poxp2 8138 | Another way of partially o... |
| frxp2 8139 | Another way of giving a we... |
| xpord2pred 8140 | Calculate the predecessor ... |
| sexp2 8141 | Condition for the relation... |
| xpord2indlem 8142 | Induction over the Cartesi... |
| xpord2ind 8143 | Induction over the Cartesi... |
| xpord3lem 8144 | Lemma for triple ordering.... |
| poxp3 8145 | Triple Cartesian product p... |
| frxp3 8146 | Give well-foundedness over... |
| xpord3pred 8147 | Calculate the predecsessor... |
| sexp3 8148 | Show that the triple order... |
| xpord3inddlem 8149 | Induction over the triple ... |
| xpord3indd 8150 | Induction over the triple ... |
| xpord3ind 8151 | Induction over the triple ... |
| orderseqlem 8152 | Lemma for ~ poseq and ~ so... |
| poseq 8153 | A partial ordering of ordi... |
| soseq 8154 | A linear ordering of ordin... |
| suppval 8157 | The value of the operation... |
| supp0prc 8158 | The support of a class is ... |
| suppvalbr 8159 | The value of the operation... |
| supp0 8160 | The support of the empty s... |
| suppval1 8161 | The value of the operation... |
| suppvalfng 8162 | The value of the operation... |
| suppvalfn 8163 | The value of the operation... |
| elsuppfng 8164 | An element of the support ... |
| elsuppfn 8165 | An element of the support ... |
| fvdifsupp 8166 | Function value is zero out... |
| cnvimadfsn 8167 | The support of functions "... |
| suppimacnvss 8168 | The support of functions "... |
| suppimacnv 8169 | Support sets of functions ... |
| fsuppeq 8170 | Two ways of writing the su... |
| fsuppeqg 8171 | Version of ~ fsuppeq avoid... |
| suppssdm 8172 | The support of a function ... |
| suppsnop 8173 | The support of a singleton... |
| snopsuppss 8174 | The support of a singleton... |
| fvn0elsupp 8175 | If the function value for ... |
| fvn0elsuppb 8176 | The function value for a g... |
| rexsupp 8177 | Existential quantification... |
| ressuppss 8178 | The support of the restric... |
| suppun 8179 | The support of a class/fun... |
| ressuppssdif 8180 | The support of the restric... |
| mptsuppdifd 8181 | The support of a function ... |
| mptsuppd 8182 | The support of a function ... |
| extmptsuppeq 8183 | The support of an extended... |
| suppfnss 8184 | The support of a function ... |
| funsssuppss 8185 | The support of a function ... |
| fnsuppres 8186 | Two ways to express restri... |
| fnsuppeq0 8187 | The support of a function ... |
| fczsupp0 8188 | The support of a constant ... |
| suppss 8189 | Show that the support of a... |
| suppssr 8190 | A function is zero outside... |
| suppssrg 8191 | A function is zero outside... |
| suppssov1 8192 | Formula building theorem f... |
| suppssov2 8193 | Formula building theorem f... |
| suppssof1 8194 | Formula building theorem f... |
| suppss2 8195 | Show that the support of a... |
| suppsssn 8196 | Show that the support of a... |
| suppssfv 8197 | Formula building theorem f... |
| suppofssd 8198 | Condition for the support ... |
| suppofss1d 8199 | Condition for the support ... |
| suppofss2d 8200 | Condition for the support ... |
| suppco 8201 | The support of the composi... |
| suppcoss 8202 | The support of the composi... |
| supp0cosupp0 8203 | The support of the composi... |
| imacosupp 8204 | The image of the support o... |
| opeliunxp2f 8205 | Membership in a union of C... |
| mpoxeldm 8206 | If there is an element of ... |
| mpoxneldm 8207 | If the first argument of a... |
| mpoxopn0yelv 8208 | If there is an element of ... |
| mpoxopynvov0g 8209 | If the second argument of ... |
| mpoxopxnop0 8210 | If the first argument of a... |
| mpoxopx0ov0 8211 | If the first argument of a... |
| mpoxopxprcov0 8212 | If the components of the f... |
| mpoxopynvov0 8213 | If the second argument of ... |
| mpoxopoveq 8214 | Value of an operation give... |
| mpoxopovel 8215 | Element of the value of an... |
| mpoxopoveqd 8216 | Value of an operation give... |
| brovex 8217 | A binary relation of the v... |
| brovmpoex 8218 | A binary relation of the v... |
| sprmpod 8219 | The extension of a binary ... |
| tposss 8222 | Subset theorem for transpo... |
| tposeq 8223 | Equality theorem for trans... |
| tposeqd 8224 | Equality theorem for trans... |
| tposssxp 8225 | The transposition is a sub... |
| reltpos 8226 | The transposition is a rel... |
| brtpos2 8227 | Value of the transposition... |
| brtpos0 8228 | The behavior of ` tpos ` w... |
| reldmtpos 8229 | Necessary and sufficient c... |
| brtpos 8230 | The transposition swaps ar... |
| ottpos 8231 | The transposition swaps th... |
| relbrtpos 8232 | The transposition swaps ar... |
| dmtpos 8233 | The domain of ` tpos F ` w... |
| rntpos 8234 | The range of ` tpos F ` wh... |
| tposexg 8235 | The transposition of a set... |
| ovtpos 8236 | The transposition swaps th... |
| tposfun 8237 | The transposition of a fun... |
| dftpos2 8238 | Alternate definition of ` ... |
| dftpos3 8239 | Alternate definition of ` ... |
| dftpos4 8240 | Alternate definition of ` ... |
| tpostpos 8241 | Value of the double transp... |
| tpostpos2 8242 | Value of the double transp... |
| tposfn2 8243 | The domain of a transposit... |
| tposfo2 8244 | Condition for a surjective... |
| tposf2 8245 | The domain and codomain of... |
| tposf12 8246 | Condition for an injective... |
| tposf1o2 8247 | Condition of a bijective t... |
| tposfo 8248 | The domain and codomain/ra... |
| tposf 8249 | The domain and codomain of... |
| tposfn 8250 | Functionality of a transpo... |
| tpos0 8251 | Transposition of the empty... |
| tposco 8252 | Transposition of a composi... |
| tpossym 8253 | Two ways to say a function... |
| tposeqi 8254 | Equality theorem for trans... |
| tposex 8255 | A transposition is a set. ... |
| nftpos 8256 | Hypothesis builder for tra... |
| tposoprab 8257 | Transposition of a class o... |
| tposmpo 8258 | Transposition of a two-arg... |
| tposconst 8259 | The transposition of a con... |
| mpocurryd 8264 | The currying of an operati... |
| mpocurryvald 8265 | The value of a curried ope... |
| fvmpocurryd 8266 | The value of the value of ... |
| pwuninel2 8269 | Proof of ~ pwuninel under ... |
| pwuninel 8270 | The powerclass of the unio... |
| pwuninelOLD 8271 | Obsolete version of ~ pwun... |
| undefval 8272 | Value of the undefined val... |
| undefnel2 8273 | The undefined value genera... |
| undefnel 8274 | The undefined value genera... |
| undefne0 8275 | The undefined value genera... |
| frecseq123 8278 | Equality theorem for the w... |
| nffrecs 8279 | Bound-variable hypothesis ... |
| csbfrecsg 8280 | Move class substitution in... |
| fpr3g 8281 | Functions defined by well-... |
| frrlem1 8282 | Lemma for well-founded rec... |
| frrlem2 8283 | Lemma for well-founded rec... |
| frrlem3 8284 | Lemma for well-founded rec... |
| frrlem4 8285 | Lemma for well-founded rec... |
| frrlem5 8286 | Lemma for well-founded rec... |
| frrlem6 8287 | Lemma for well-founded rec... |
| frrlem7 8288 | Lemma for well-founded rec... |
| frrlem8 8289 | Lemma for well-founded rec... |
| frrlem9 8290 | Lemma for well-founded rec... |
| frrlem10 8291 | Lemma for well-founded rec... |
| frrlem11 8292 | Lemma for well-founded rec... |
| frrlem12 8293 | Lemma for well-founded rec... |
| frrlem13 8294 | Lemma for well-founded rec... |
| frrlem14 8295 | Lemma for well-founded rec... |
| fprlem1 8296 | Lemma for well-founded rec... |
| fprlem2 8297 | Lemma for well-founded rec... |
| fpr2a 8298 | Weak version of ~ fpr2 whi... |
| fpr1 8299 | Law of well-founded recurs... |
| fpr2 8300 | Law of well-founded recurs... |
| fpr3 8301 | Law of well-founded recurs... |
| frrrel 8302 | Show without using the axi... |
| frrdmss 8303 | Show without using the axi... |
| frrdmcl 8304 | Show without using the axi... |
| fprfung 8305 | A "function" defined by we... |
| fprresex 8306 | The restriction of a funct... |
| wrecseq123 8309 | General equality theorem f... |
| nfwrecs 8310 | Bound-variable hypothesis ... |
| wrecseq1 8311 | Equality theorem for the w... |
| wrecseq2 8312 | Equality theorem for the w... |
| wrecseq3 8313 | Equality theorem for the w... |
| csbwrecsg 8314 | Move class substitution in... |
| wfr3g 8315 | Functions defined by well-... |
| wfrrel 8316 | The well-ordered recursion... |
| wfrdmss 8317 | The domain of the well-ord... |
| wfrdmcl 8318 | The predecessor class of a... |
| wfrfun 8319 | The "function" generated b... |
| wfrresex 8320 | Show without using the axi... |
| wfr2a 8321 | A weak version of ~ wfr2 w... |
| wfr1 8322 | The Principle of Well-Orde... |
| wfr2 8323 | The Principle of Well-Orde... |
| wfr3 8324 | The principle of Well-Orde... |
| iunon 8325 | The indexed union of a set... |
| iinon 8326 | The nonempty indexed inter... |
| onfununi 8327 | A property of functions on... |
| onovuni 8328 | A variant of ~ onfununi fo... |
| onoviun 8329 | A variant of ~ onovuni wit... |
| onnseq 8330 | There are no length ` _om ... |
| dfsmo2 8333 | Alternate definition of a ... |
| issmo 8334 | Conditions for which ` A `... |
| issmo2 8335 | Alternate definition of a ... |
| smoeq 8336 | Equality theorem for stric... |
| smodm 8337 | The domain of a strictly m... |
| smores 8338 | A strictly monotone functi... |
| smores3 8339 | A strictly monotone functi... |
| smores2 8340 | A strictly monotone ordina... |
| smodm2 8341 | The domain of a strictly m... |
| smofvon2 8342 | The function values of a s... |
| iordsmo 8343 | The identity relation rest... |
| smo0 8344 | The empty set is a strictl... |
| smofvon 8345 | If ` B ` is a strictly mon... |
| smoel 8346 | If ` x ` is less than ` y ... |
| smoiun 8347 | The value of a strictly mo... |
| smoiso 8348 | If ` F ` is an isomorphism... |
| smoel2 8349 | A strictly monotone ordina... |
| smo11 8350 | A strictly monotone ordina... |
| smoord 8351 | A strictly monotone ordina... |
| smoword 8352 | A strictly monotone ordina... |
| smogt 8353 | A strictly monotone ordina... |
| smocdmdom 8354 | The codomain of a strictly... |
| smoiso2 8355 | The strictly monotone ordi... |
| dfrecs3 8358 | The old definition of tran... |
| recseq 8359 | Equality theorem for ` rec... |
| nfrecs 8360 | Bound-variable hypothesis ... |
| tfrlem1 8361 | A technical lemma for tran... |
| tfrlem3a 8362 | Lemma for transfinite recu... |
| tfrlem3 8363 | Lemma for transfinite recu... |
| tfrlem4 8364 | Lemma for transfinite recu... |
| tfrlem5 8365 | Lemma for transfinite recu... |
| recsfval 8366 | Lemma for transfinite recu... |
| tfrlem6 8367 | Lemma for transfinite recu... |
| tfrlem6OLD 8368 | Obsolete version of ~ tfrl... |
| tfrlem7 8369 | Lemma for transfinite recu... |
| tfrlem8 8370 | Lemma for transfinite recu... |
| tfrlem9 8371 | Lemma for transfinite recu... |
| tfrlem9a 8372 | Lemma for transfinite recu... |
| tfrlem10 8373 | Lemma for transfinite recu... |
| tfrlem11 8374 | Lemma for transfinite recu... |
| tfrlem12 8375 | Lemma for transfinite recu... |
| tfrlem13 8376 | Lemma for transfinite recu... |
| tfrlem14 8377 | Lemma for transfinite recu... |
| tfrlem15 8378 | Lemma for transfinite recu... |
| tfrlem16 8379 | Lemma for finite recursion... |
| tfr1a 8380 | A weak version of ~ tfr1 w... |
| tfr2a 8381 | A weak version of ~ tfr2 w... |
| tfr2b 8382 | Without assuming ~ ax-rep ... |
| tfr1 8383 | Principle of Transfinite R... |
| tfr2 8384 | Principle of Transfinite R... |
| tfr3 8385 | Principle of Transfinite R... |
| tfr1ALT 8386 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8387 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8388 | Alternate proof of ~ tfr3 ... |
| recsfnon 8389 | Strong transfinite recursi... |
| recsval 8390 | Strong transfinite recursi... |
| tz7.44lem1 8391 | The ordered pair abstracti... |
| tz7.44-1 8392 | The value of ` F ` at ` (/... |
| tz7.44-2 8393 | The value of ` F ` at a su... |
| tz7.44-3 8394 | The value of ` F ` at a li... |
| rdgeq1 8397 | Equality theorem for the r... |
| rdgeq2 8398 | Equality theorem for the r... |
| rdgeq12 8399 | Equality theorem for the r... |
| nfrdg 8400 | Bound-variable hypothesis ... |
| rdglem1 8401 | Lemma used with the recurs... |
| rdgfun 8402 | The recursive definition g... |
| rdgdmlim 8403 | The domain of the recursiv... |
| rdgfnon 8404 | The recursive definition g... |
| rdgvalg 8405 | Value of the recursive def... |
| rdgval 8406 | Value of the recursive def... |
| rdg0 8407 | The initial value of the r... |
| rdgseg 8408 | The initial segments of th... |
| rdgsucg 8409 | The value of the recursive... |
| rdgsuc 8410 | The value of the recursive... |
| rdglimg 8411 | The value of the recursive... |
| rdglim 8412 | The value of the recursive... |
| rdg0g 8413 | The initial value of the r... |
| rdgsucmptf 8414 | The value of the recursive... |
| rdgsucmptnf 8415 | The value of the recursive... |
| rdgsucmpt2 8416 | This version of ~ rdgsucmp... |
| rdgsucmpt 8417 | The value of the recursive... |
| rdglim2 8418 | The value of the recursive... |
| rdglim2a 8419 | The value of the recursive... |
| rdg0n 8420 | If ` A ` is a proper class... |
| frfnom 8421 | The function generated by ... |
| fr0g 8422 | The initial value resultin... |
| frsuc 8423 | The successor value result... |
| frsucmpt 8424 | The successor value result... |
| frsucmptn 8425 | The value of the finite re... |
| frsucmpt2 8426 | The successor value result... |
| onelfvnef1 8427 | A sufficient condition for... |
| tz7.48lem 8428 | A way of showing an ordina... |
| tz7.48lemOLD 8429 | Obsolete version of ~ tz7.... |
| tz7.48-2 8430 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8431 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8432 | Proposition 7.48(3) of [Ta... |
| tz7.49 8433 | Proposition 7.49 of [Takeu... |
| tz7.49c 8434 | Corollary of Proposition 7... |
| seqomlem0 8437 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8438 | Lemma for ` seqom ` . The... |
| seqomlem2 8439 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8440 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8441 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8442 | Equality theorem for ` seq... |
| fnseqom 8443 | An index-aware recursive d... |
| seqom0g 8444 | Value of an index-aware re... |
| seqomsuc 8445 | Value of an index-aware re... |
| omsucelsucb 8446 | Membership is inherited by... |
| df1o2 8461 | Expanded value of the ordi... |
| df2o3 8462 | Expanded value of the ordi... |
| df2o2 8463 | Expanded value of the ordi... |
| 1oex 8464 | Ordinal 1 is a set. (Cont... |
| 1oelpr 8465 | ` 1o ` is an element of ` ... |
| 2oex 8466 | ` 2o ` is a set. (Contrib... |
| 1on 8467 | Ordinal 1 is an ordinal nu... |
| 2on 8468 | Ordinal 2 is an ordinal nu... |
| 2on0 8469 | Ordinal two is not zero. ... |
| ord3 8470 | Ordinal 3 is an ordinal cl... |
| 3on 8471 | Ordinal 3 is an ordinal nu... |
| 4on 8472 | Ordinal 4 is an ordinal nu... |
| 1n0 8473 | Ordinal one is not equal t... |
| 1n0OLD 8474 | Obsolete version of ~ 1n0 ... |
| nlim1 8475 | 1 is not a limit ordinal. ... |
| nlim2 8476 | 2 is not a limit ordinal. ... |
| xp01disj 8477 | Cartesian products with th... |
| xp01disjl 8478 | Cartesian products with th... |
| ordgt0ge1 8479 | Two ways to express that a... |
| ordge1n0 8480 | An ordinal greater than or... |
| el1o 8481 | Membership in ordinal one.... |
| ord1eln01 8482 | An ordinal that is not 0 o... |
| ord2eln012 8483 | An ordinal that is not 0, ... |
| 1ellim 8484 | A limit ordinal contains 1... |
| 2ellim 8485 | A limit ordinal contains 2... |
| dif1o 8486 | Two ways to say that ` A `... |
| ondif1 8487 | Two ways to say that ` A `... |
| ondif2 8488 | Two ways to say that ` A `... |
| 2oconcl 8489 | Closure of the pair swappi... |
| 0lt1o 8490 | Ordinal zero is less than ... |
| dif20el 8491 | An ordinal greater than on... |
| 0we1 8492 | The empty set is a well-or... |
| brwitnlem 8493 | Lemma for relations which ... |
| fnoa 8494 | Functionality and domain o... |
| fnom 8495 | Functionality and domain o... |
| fnoe 8496 | Functionality and domain o... |
| oav 8497 | Value of ordinal addition.... |
| omv 8498 | Value of ordinal multiplic... |
| oe0lem 8499 | A helper lemma for ~ oe0 a... |
| oev 8500 | Value of ordinal exponenti... |
| oevn0 8501 | Value of ordinal exponenti... |
| oa0 8502 | Addition with zero. Propo... |
| om0 8503 | Ordinal multiplication wit... |
| oe0m 8504 | Value of zero raised to an... |
| om0x 8505 | Ordinal multiplication wit... |
| oe0m0 8506 | Ordinal exponentiation wit... |
| oe0m1 8507 | Ordinal exponentiation wit... |
| oe0 8508 | Ordinal exponentiation wit... |
| oev2 8509 | Alternate value of ordinal... |
| oasuc 8510 | Addition with successor. ... |
| oesuclem 8511 | Lemma for ~ oesuc . (Cont... |
| omsuc 8512 | Multiplication with succes... |
| oesuc 8513 | Ordinal exponentiation wit... |
| onasuc 8514 | Addition with successor. ... |
| onmsuc 8515 | Multiplication with succes... |
| onesuc 8516 | Exponentiation with a succ... |
| oa1suc 8517 | Addition with 1 is same as... |
| oalim 8518 | Ordinal addition with a li... |
| omlim 8519 | Ordinal multiplication wit... |
| oelim 8520 | Ordinal exponentiation wit... |
| oacl 8521 | Closure law for ordinal ad... |
| omcl 8522 | Closure law for ordinal mu... |
| oecl 8523 | Closure law for ordinal ex... |
| oa0r 8524 | Ordinal addition with zero... |
| om0r 8525 | Ordinal multiplication wit... |
| o1p1e2 8526 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8527 | 2 + 2 = 4 for ordinal numb... |
| om1 8528 | Ordinal multiplication wit... |
| om1r 8529 | Ordinal multiplication wit... |
| oe1 8530 | Ordinal exponentiation wit... |
| oe1m 8531 | Ordinal exponentiation wit... |
| oaordi 8532 | Ordering property of ordin... |
| oaord 8533 | Ordering property of ordin... |
| oacan 8534 | Left cancellation law for ... |
| oaword 8535 | Weak ordering property of ... |
| oawordri 8536 | Weak ordering property of ... |
| oaord1 8537 | An ordinal is less than it... |
| oaword1 8538 | An ordinal is less than or... |
| oaword2 8539 | An ordinal is less than or... |
| oawordeulem 8540 | Lemma for ~ oawordex . (C... |
| oawordeu 8541 | Existence theorem for weak... |
| oawordexr 8542 | Existence theorem for weak... |
| oawordex 8543 | Existence theorem for weak... |
| oaordex 8544 | Existence theorem for orde... |
| oa00 8545 | An ordinal sum is zero iff... |
| oalimcl 8546 | The ordinal sum with a lim... |
| oaass 8547 | Ordinal addition is associ... |
| oarec 8548 | Recursive definition of or... |
| oaf1o 8549 | Left addition by a constan... |
| oacomf1olem 8550 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8551 | Define a bijection from ` ... |
| omordi 8552 | Ordering property of ordin... |
| omord2 8553 | Ordering property of ordin... |
| omord 8554 | Ordering property of ordin... |
| omcan 8555 | Left cancellation law for ... |
| omword 8556 | Weak ordering property of ... |
| omwordi 8557 | Weak ordering property of ... |
| omwordri 8558 | Weak ordering property of ... |
| omword1 8559 | An ordinal is less than or... |
| omword2 8560 | An ordinal is less than or... |
| om00 8561 | The product of two ordinal... |
| om00el 8562 | The product of two nonzero... |
| omordlim 8563 | Ordering involving the pro... |
| omlimcl 8564 | The product of any nonzero... |
| odi 8565 | Distributive law for ordin... |
| omass 8566 | Multiplication of ordinal ... |
| oneo 8567 | If an ordinal number is ev... |
| omeulem1 8568 | Lemma for ~ omeu : existen... |
| omeulem2 8569 | Lemma for ~ omeu : uniquen... |
| omopth2 8570 | An ordered pair-like theor... |
| omeu 8571 | The division algorithm for... |
| om2 8572 | Two ways to double an ordi... |
| oen0 8573 | Ordinal exponentiation wit... |
| oeordi 8574 | Ordering law for ordinal e... |
| oeord 8575 | Ordering property of ordin... |
| oecan 8576 | Left cancellation law for ... |
| oeword 8577 | Weak ordering property of ... |
| oewordi 8578 | Weak ordering property of ... |
| oewordri 8579 | Weak ordering property of ... |
| oeworde 8580 | Ordinal exponentiation com... |
| oeordsuc 8581 | Ordering property of ordin... |
| oelim2 8582 | Ordinal exponentiation wit... |
| oeoalem 8583 | Lemma for ~ oeoa . (Contr... |
| oeoa 8584 | Sum of exponents law for o... |
| oeoelem 8585 | Lemma for ~ oeoe . (Contr... |
| oeoe 8586 | Product of exponents law f... |
| oelimcl 8587 | The ordinal exponential wi... |
| oeeulem 8588 | Lemma for ~ oeeu . (Contr... |
| oeeui 8589 | The division algorithm for... |
| oeeu 8590 | The division algorithm for... |
| nna0 8591 | Addition with zero. Theor... |
| nnm0 8592 | Multiplication with zero. ... |
| nnasuc 8593 | Addition with successor. ... |
| nnmsuc 8594 | Multiplication with succes... |
| nnesuc 8595 | Exponentiation with a succ... |
| nna0r 8596 | Addition to zero. Remark ... |
| nnm0r 8597 | Multiplication with zero. ... |
| nnacl 8598 | Closure of addition of nat... |
| nnmcl 8599 | Closure of multiplication ... |
| nnecl 8600 | Closure of exponentiation ... |
| nnacli 8601 | ` _om ` is closed under ad... |
| nnmcli 8602 | ` _om ` is closed under mu... |
| nnarcl 8603 | Reverse closure law for ad... |
| nnacom 8604 | Addition of natural number... |
| nnaordi 8605 | Ordering property of addit... |
| nnaord 8606 | Ordering property of addit... |
| nnaordr 8607 | Ordering property of addit... |
| nnawordi 8608 | Adding to both sides of an... |
| nnaass 8609 | Addition of natural number... |
| nndi 8610 | Distributive law for natur... |
| nnmass 8611 | Multiplication of natural ... |
| nnmsucr 8612 | Multiplication with succes... |
| nnmcom 8613 | Multiplication of natural ... |
| nnaword 8614 | Weak ordering property of ... |
| nnacan 8615 | Cancellation law for addit... |
| nnaword1 8616 | Weak ordering property of ... |
| nnaword2 8617 | Weak ordering property of ... |
| nnmordi 8618 | Ordering property of multi... |
| nnmord 8619 | Ordering property of multi... |
| nnmword 8620 | Weak ordering property of ... |
| nnmcan 8621 | Cancellation law for multi... |
| nnmwordi 8622 | Weak ordering property of ... |
| nnmwordri 8623 | Weak ordering property of ... |
| nnawordex 8624 | Equivalence for weak order... |
| nnaordex 8625 | Equivalence for ordering. ... |
| nnaordex2 8626 | Equivalence for ordering. ... |
| 1onn 8627 | The ordinal 1 is a natural... |
| 1onnALT 8628 | Shorter proof of ~ 1onn us... |
| 2onn 8629 | The ordinal 2 is a natural... |
| 2onnALT 8630 | Shorter proof of ~ 2onn us... |
| 3onn 8631 | The ordinal 3 is a natural... |
| 4onn 8632 | The ordinal 4 is a natural... |
| 1one2o 8633 | Ordinal one is not ordinal... |
| oaabslem 8634 | Lemma for ~ oaabs . (Cont... |
| oaabs 8635 | Ordinal addition absorbs a... |
| oaabs2 8636 | The absorption law ~ oaabs... |
| omabslem 8637 | Lemma for ~ omabs . (Cont... |
| omabs 8638 | Ordinal multiplication is ... |
| nnm1 8639 | Multiply an element of ` _... |
| nnm2 8640 | Multiply an element of ` _... |
| nn2m 8641 | Multiply an element of ` _... |
| nnneo 8642 | If a natural number is eve... |
| nneob 8643 | A natural number is even i... |
| omsmolem 8644 | Lemma for ~ omsmo . (Cont... |
| omsmo 8645 | A strictly monotonic ordin... |
| omopthlem1 8646 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8647 | Lemma for ~ omopthi . (Co... |
| omopthi 8648 | An ordered pair theorem fo... |
| omopth 8649 | An ordered pair theorem fo... |
| nnasmo 8650 | There is at most one left ... |
| eldifsucnn 8651 | Condition for membership i... |
| on2recsfn 8654 | Show that double recursion... |
| on2recsov 8655 | Calculate the value of the... |
| on2ind 8656 | Double induction over ordi... |
| on3ind 8657 | Triple induction over ordi... |
| coflton 8658 | Cofinality theorem for ord... |
| cofon1 8659 | Cofinality theorem for ord... |
| cofon2 8660 | Cofinality theorem for ord... |
| cofonr 8661 | Inverse cofinality law for... |
| naddfn 8662 | Natural addition is a func... |
| naddcllem 8663 | Lemma for ordinal addition... |
| naddcl 8664 | Closure law for natural ad... |
| naddov 8665 | The value of natural addit... |
| naddov2 8666 | Alternate expression for n... |
| naddcld 8667 | Closure law for natural ad... |
| naddov3 8668 | Alternate expression for n... |
| naddf 8669 | Function statement for nat... |
| naddcom 8670 | Natural addition is commut... |
| naddrid 8671 | Ordinal zero is the additi... |
| naddlid 8672 | Ordinal zero is the additi... |
| naddssim 8673 | Ordinal less-than-or-equal... |
| naddelim 8674 | Ordinal less-than is prese... |
| naddel1 8675 | Ordinal less-than is not a... |
| naddel2 8676 | Ordinal less-than is not a... |
| naddss1 8677 | Ordinal less-than-or-equal... |
| naddss2 8678 | Ordinal less-than-or-equal... |
| naddword1 8679 | Weak-ordering principle fo... |
| naddword2 8680 | Weak-ordering principle fo... |
| naddunif 8681 | Uniformity theorem for nat... |
| naddasslem1 8682 | Lemma for ~ naddass . Exp... |
| naddasslem2 8683 | Lemma for ~ naddass . Exp... |
| naddass 8684 | Natural ordinal addition i... |
| nadd32 8685 | Commutative/associative la... |
| nadd4 8686 | Rearragement of terms in a... |
| nadd42 8687 | Rearragement of terms in a... |
| naddel12 8688 | Natural addition to both s... |
| naddsuc2 8689 | Natural addition with succ... |
| naddoa 8690 | Natural addition of a natu... |
| omnaddcl 8691 | The naturals are closed un... |
| dfer2 8696 | Alternate definition of eq... |
| dfec2 8698 | Alternate definition of ` ... |
| ecexg 8699 | An equivalence class modul... |
| ecexr 8700 | A nonempty equivalence cla... |
| dfqs2 8702 | Alternate definition of qu... |
| ereq1 8703 | Equality theorem for equiv... |
| ereq2 8704 | Equality theorem for equiv... |
| errel 8705 | An equivalence relation is... |
| erdm 8706 | The domain of an equivalen... |
| ercl 8707 | Elementhood in the field o... |
| ersym 8708 | An equivalence relation is... |
| ercl2 8709 | Elementhood in the field o... |
| ersymb 8710 | An equivalence relation is... |
| ertr 8711 | An equivalence relation is... |
| ertrd 8712 | A transitivity relation fo... |
| ertr2d 8713 | A transitivity relation fo... |
| ertr3d 8714 | A transitivity relation fo... |
| ertr4d 8715 | A transitivity relation fo... |
| erref 8716 | An equivalence relation is... |
| ercnv 8717 | The converse of an equival... |
| errn 8718 | The range and domain of an... |
| erssxp 8719 | An equivalence relation is... |
| erex 8720 | An equivalence relation is... |
| erexb 8721 | An equivalence relation is... |
| iserd 8722 | A reflexive, symmetric, tr... |
| iseri 8723 | A reflexive, symmetric, tr... |
| iseriALT 8724 | Alternate proof of ~ iseri... |
| brinxper 8725 | Conditions for a reflexive... |
| brdifun 8726 | Evaluate the incomparabili... |
| swoer 8727 | Incomparability under a st... |
| swoord1 8728 | The incomparability equiva... |
| swoord2 8729 | The incomparability equiva... |
| swoso 8730 | If the incomparability rel... |
| eqerlem 8731 | Lemma for ~ eqer . (Contr... |
| eqer 8732 | Equivalence relation invol... |
| ider 8733 | The identity relation is a... |
| 0er 8734 | The empty set is an equiva... |
| eceq1 8735 | Equality theorem for equiv... |
| eceq1d 8736 | Equality theorem for equiv... |
| eceq2 8737 | Equality theorem for equiv... |
| eceq2i 8738 | Equality theorem for the `... |
| eceq2d 8739 | Equality theorem for the `... |
| elecg 8740 | Membership in an equivalen... |
| ecref 8741 | All elements are in their ... |
| elec 8742 | Membership in an equivalen... |
| relelec 8743 | Membership in an equivalen... |
| elecres 8744 | Elementhood in the restric... |
| elecreseq 8745 | The restricted coset of ` ... |
| elecex 8746 | Condition for a coset to b... |
| ecss 8747 | An equivalence class is a ... |
| ecdmn0 8748 | A representative of a none... |
| ereldm 8749 | Equality of equivalence cl... |
| erth 8750 | Basic property of equivale... |
| erth2 8751 | Basic property of equivale... |
| erthi 8752 | Basic property of equivale... |
| erdisj 8753 | Equivalence classes do not... |
| ecidsn 8754 | An equivalence class modul... |
| qseq1 8755 | Equality theorem for quoti... |
| qseq2 8756 | Equality theorem for quoti... |
| qseq2i 8757 | Equality theorem for quoti... |
| qseq1d 8758 | Equality theorem for quoti... |
| qseq2d 8759 | Equality theorem for quoti... |
| qseq12 8760 | Equality theorem for quoti... |
| 0qs 8761 | Quotient set with the empt... |
| elqsg 8762 | Closed form of ~ elqs . (... |
| elqs 8763 | Membership in a quotient s... |
| elqsi 8764 | Membership in a quotient s... |
| elqsecl 8765 | Membership in a quotient s... |
| ecelqs 8766 | Membership of an equivalen... |
| ecelqsw 8767 | Membership of an equivalen... |
| ecelqsi 8768 | Membership of an equivalen... |
| ecopqsi 8769 | "Closure" law for equivale... |
| qsexg 8770 | A quotient set exists. (C... |
| qsex 8771 | A quotient set exists. (C... |
| uniqs 8772 | The union of a quotient se... |
| uniqsw 8773 | The union of a quotient se... |
| qsss 8774 | A quotient set is a set of... |
| uniqs2 8775 | The union of a quotient se... |
| snecg 8776 | The singleton of a coset i... |
| snec 8777 | The singleton of an equiva... |
| ecqs 8778 | Equivalence class in terms... |
| ecid 8779 | A set is equal to its cose... |
| qsid 8780 | A set is equal to its quot... |
| ectocld 8781 | Implicit substitution of c... |
| ectocl 8782 | Implicit substitution of c... |
| elqsn0 8783 | A quotient set does not co... |
| ecelqsdm 8784 | Membership of an equivalen... |
| ecelqsdmb 8785 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8786 | ` R ` -coset in its domain... |
| xpider 8787 | A Cartesian square is an e... |
| iiner 8788 | The intersection of a none... |
| riiner 8789 | The relative intersection ... |
| erinxp 8790 | A restricted equivalence r... |
| ecinxp 8791 | Restrict the relation in a... |
| qsinxp 8792 | Restrict the equivalence r... |
| qsdisj 8793 | Members of a quotient set ... |
| qsdisj2 8794 | A quotient set is a disjoi... |
| qsel 8795 | If an element of a quotien... |
| uniinqs 8796 | Class union distributes ov... |
| qliftlem 8797 | Lemma for theorems about a... |
| qliftrel 8798 | ` F ` , a function lift, i... |
| qliftel 8799 | Elementhood in the relatio... |
| qliftel1 8800 | Elementhood in the relatio... |
| qliftfun 8801 | The function ` F ` is the ... |
| qliftfund 8802 | The function ` F ` is the ... |
| qliftfuns 8803 | The function ` F ` is the ... |
| qliftf 8804 | The domain and codomain of... |
| qliftval 8805 | The value of the function ... |
| ecoptocl 8806 | Implicit substitution of c... |
| 2ecoptocl 8807 | Implicit substitution of c... |
| 3ecoptocl 8808 | Implicit substitution of c... |
| brecop 8809 | Binary relation on a quoti... |
| brecop2 8810 | Binary relation on a quoti... |
| eroveu 8811 | Lemma for ~ erov and ~ ero... |
| erovlem 8812 | Lemma for ~ erov and ~ ero... |
| erov 8813 | The value of an operation ... |
| eroprf 8814 | Functionality of an operat... |
| erov2 8815 | The value of an operation ... |
| eroprf2 8816 | Functionality of an operat... |
| ecopoveq 8817 | This is the first of sever... |
| ecopovsym 8818 | Assuming the operation ` F... |
| ecopovtrn 8819 | Assuming that operation ` ... |
| ecopover 8820 | Assuming that operation ` ... |
| eceqoveq 8821 | Equality of equivalence re... |
| ecovcom 8822 | Lemma used to transfer a c... |
| ecovass 8823 | Lemma used to transfer an ... |
| ecovdi 8824 | Lemma used to transfer a d... |
| mapprc 8829 | When ` A ` is a proper cla... |
| pmex 8830 | The class of all partial f... |
| fnmap 8831 | Set exponentiation has a u... |
| fnpm 8832 | Partial function exponenti... |
| reldmmap 8833 | Set exponentiation is a we... |
| mapvalg 8834 | The value of set exponenti... |
| pmvalg 8835 | The value of the partial m... |
| mapval 8836 | The value of set exponenti... |
| elmapg 8837 | Membership relation for se... |
| elmapd 8838 | Deduction form of ~ elmapg... |
| elmapdd 8839 | Deduction associated with ... |
| mapdm0 8840 | The empty set is the only ... |
| elpmg 8841 | The predicate "is a partia... |
| elpm2g 8842 | The predicate "is a partia... |
| elpm2r 8843 | Sufficient condition for b... |
| elpmi 8844 | A partial function is a fu... |
| pmfun 8845 | A partial function is a fu... |
| elmapex 8846 | Eliminate antecedent for m... |
| elmapi 8847 | A mapping is a function, f... |
| elmaprd 8848 | Deduction associated with ... |
| elmaprdOLD 8849 | Obsolete version of ~ elma... |
| mapfset 8850 | If ` B ` is a set, the val... |
| mapssfset 8851 | The value of the set expon... |
| mapfoss 8852 | The value of the set expon... |
| fsetsspwxp 8853 | The class of all functions... |
| fset0 8854 | The set of functions from ... |
| fsetdmprc0 8855 | The set of functions with ... |
| fsetex 8856 | The set of functions betwe... |
| f1setex 8857 | The set of injections betw... |
| fosetex 8858 | The set of surjections bet... |
| f1osetex 8859 | The set of bijections betw... |
| fsetfcdm 8860 | The class of functions wit... |
| fsetfocdm 8861 | The class of functions wit... |
| fsetprcnex 8862 | The class of all functions... |
| fsetcdmex 8863 | The class of all functions... |
| fsetexb 8864 | The class of all functions... |
| elmapfn 8865 | A mapping is a function wi... |
| elmapfun 8866 | A mapping is always a func... |
| cureq 8867 | Equality theorem for curry... |
| curf 8868 | Functional property of cur... |
| uncf 8869 | Functional property of unc... |
| curfv 8870 | Value of currying. (Contr... |
| uncov 8871 | Value of uncurrying. (Con... |
| elmapssres 8872 | A restricted mapping is a ... |
| elmapssresd 8873 | A restricted mapping is a ... |
| fpmg 8874 | A total function is a part... |
| pmss12g 8875 | Subset relation for the se... |
| pmresg 8876 | Elementhood of a restricte... |
| elmap 8877 | Membership relation for se... |
| mapval2 8878 | Alternate expression for t... |
| elpm 8879 | The predicate "is a partia... |
| elpm2 8880 | The predicate "is a partia... |
| fpm 8881 | A total function is a part... |
| mapsspm 8882 | Set exponentiation is a su... |
| pmsspw 8883 | Partial maps are a subset ... |
| mapsspw 8884 | Set exponentiation is a su... |
| mapfvd 8885 | The value of a function th... |
| elmapresaun 8886 | ~ fresaun transposed to ma... |
| fvmptmap 8887 | Special case of ~ fvmpt fo... |
| map0e 8888 | Set exponentiation with an... |
| map0b 8889 | Set exponentiation with an... |
| map0g 8890 | Set exponentiation is empt... |
| 0map0sn0 8891 | The set of mappings of the... |
| mapsnd 8892 | The value of set exponenti... |
| map0 8893 | Set exponentiation is empt... |
| mapsn 8894 | The value of set exponenti... |
| mapss 8895 | Subset inheritance for set... |
| fdiagfn 8896 | Functionality of the diago... |
| fvdiagfn 8897 | Functionality of the diago... |
| mapsnconst 8898 | Every singleton map is a c... |
| mapsncnv 8899 | Expression for the inverse... |
| mapsnf1o2 8900 | Explicit bijection between... |
| mapsnf1o3 8901 | Explicit bijection in the ... |
| ralxpmap 8902 | Quantification over functi... |
| dfixp 8905 | Eliminate the expression `... |
| ixpsnval 8906 | The value of an infinite C... |
| elixp2 8907 | Membership in an infinite ... |
| fvixp 8908 | Projection of a factor of ... |
| ixpfn 8909 | A nuple is a function. (C... |
| elixp 8910 | Membership in an infinite ... |
| elixpconst 8911 | Membership in an infinite ... |
| ixpconstg 8912 | Infinite Cartesian product... |
| ixpconst 8913 | Infinite Cartesian product... |
| ixpeq1 8914 | Equality theorem for infin... |
| ixpeq1d 8915 | Equality theorem for infin... |
| ss2ixp 8916 | Subclass theorem for infin... |
| ixpeq2 8917 | Equality theorem for infin... |
| ixpeq2dva 8918 | Equality theorem for infin... |
| ixpeq2dv 8919 | Equality theorem for infin... |
| cbvixp 8920 | Change bound variable in a... |
| cbvixpv 8921 | Change bound variable in a... |
| nfixpw 8922 | Bound-variable hypothesis ... |
| nfixp 8923 | Bound-variable hypothesis ... |
| nfixp1 8924 | The index variable in an i... |
| ixpprc 8925 | A cartesian product of pro... |
| ixpf 8926 | A member of an infinite Ca... |
| uniixp 8927 | The union of an infinite C... |
| ixpexg 8928 | The existence of an infini... |
| ixpin 8929 | The intersection of two in... |
| ixpiin 8930 | The indexed intersection o... |
| ixpint 8931 | The intersection of a coll... |
| ixp0x 8932 | An infinite Cartesian prod... |
| ixpssmap2g 8933 | An infinite Cartesian prod... |
| ixpssmapg 8934 | An infinite Cartesian prod... |
| 0elixp 8935 | Membership of the empty se... |
| ixpn0 8936 | The infinite Cartesian pro... |
| ixp0 8937 | The infinite Cartesian pro... |
| ixpssmap 8938 | An infinite Cartesian prod... |
| resixp 8939 | Restriction of an element ... |
| undifixp 8940 | Union of two projections o... |
| mptelixpg 8941 | Condition for an explicit ... |
| resixpfo 8942 | Restriction of elements of... |
| elixpsn 8943 | Membership in a class of s... |
| ixpsnf1o 8944 | A bijection between a clas... |
| mapsnf1o 8945 | A bijection between a set ... |
| boxriin 8946 | A rectangular subset of a ... |
| boxcutc 8947 | The relative complement of... |
| relen 8956 | Equinumerosity is a relati... |
| reldom 8957 | Dominance is a relation. ... |
| relsdom 8958 | Strict dominance is a rela... |
| encv 8959 | If two classes are equinum... |
| breng 8960 | Equinumerosity relation. ... |
| bren 8961 | Equinumerosity relation. ... |
| brdom2g 8962 | Dominance relation. This ... |
| brdomg 8963 | Dominance relation. (Cont... |
| brdomi 8964 | Dominance relation. (Cont... |
| brdom 8965 | Dominance relation. (Cont... |
| domen 8966 | Dominance in terms of equi... |
| domeng 8967 | Dominance in terms of equi... |
| ctex 8968 | A countable set is a set. ... |
| f1oen4g 8969 | The domain and range of a ... |
| f1dom4g 8970 | The domain of a one-to-one... |
| f1oen3g 8971 | The domain and range of a ... |
| f1dom3g 8972 | The domain of a one-to-one... |
| f1oen2g 8973 | The domain and range of a ... |
| f1dom2g 8974 | The domain of a one-to-one... |
| f1oeng 8975 | The domain and range of a ... |
| f1domg 8976 | The domain of a one-to-one... |
| f1oen 8977 | The domain and range of a ... |
| f1dom 8978 | The domain of a one-to-one... |
| brsdom 8979 | Strict dominance relation,... |
| isfi 8980 | Express " ` A ` is finite"... |
| enssdom 8981 | Equinumerosity implies dom... |
| enssdomOLD 8982 | Obsolete version of ~ enss... |
| dfdom2 8983 | Alternate definition of do... |
| endom 8984 | Equinumerosity implies dom... |
| sdomdom 8985 | Strict dominance implies d... |
| sdomnen 8986 | Strict dominance implies n... |
| brdom2 8987 | Dominance in terms of stri... |
| bren2 8988 | Equinumerosity expressed i... |
| enrefg 8989 | Equinumerosity is reflexiv... |
| enref 8990 | Equinumerosity is reflexiv... |
| eqeng 8991 | Equality implies equinumer... |
| domrefg 8992 | Dominance is reflexive. (... |
| en2d 8993 | Equinumerosity inference f... |
| en3d 8994 | Equinumerosity inference f... |
| en2i 8995 | Equinumerosity inference f... |
| en3i 8996 | Equinumerosity inference f... |
| dom2lem 8997 | A mapping (first hypothesi... |
| dom2d 8998 | A mapping (first hypothesi... |
| dom3d 8999 | A mapping (first hypothesi... |
| dom2 9000 | A mapping (first hypothesi... |
| dom3 9001 | A mapping (first hypothesi... |
| idssen 9002 | Equality implies equinumer... |
| domssl 9003 | If ` A ` is a subset of ` ... |
| domssr 9004 | If ` C ` is a superset of ... |
| ssdomg 9005 | A set dominates its subset... |
| ener 9006 | Equinumerosity is an equiv... |
| ensymb 9007 | Symmetry of equinumerosity... |
| ensym 9008 | Symmetry of equinumerosity... |
| ensymi 9009 | Symmetry of equinumerosity... |
| ensymd 9010 | Symmetry of equinumerosity... |
| entr 9011 | Transitivity of equinumero... |
| domtr 9012 | Transitivity of dominance ... |
| entri 9013 | A chained equinumerosity i... |
| entr2i 9014 | A chained equinumerosity i... |
| entr3i 9015 | A chained equinumerosity i... |
| entr4i 9016 | A chained equinumerosity i... |
| endomtr 9017 | Transitivity of equinumero... |
| domentr 9018 | Transitivity of dominance ... |
| f1imaeng 9019 | If a function is one-to-on... |
| f1imaen2g 9020 | If a function is one-to-on... |
| f1imaen3g 9021 | If a set function is one-t... |
| f1imaen 9022 | If a function is one-to-on... |
| en0 9023 | The empty set is equinumer... |
| en0ALT 9024 | Shorter proof of ~ en0 , d... |
| en0r 9025 | The empty set is equinumer... |
| ensn1 9026 | A singleton is equinumerou... |
| ensn1g 9027 | A singleton is equinumerou... |
| enpr1g 9028 | ` { A , A } ` has only one... |
| en1 9029 | A set is equinumerous to o... |
| en1b 9030 | A set is equinumerous to o... |
| reuen1 9031 | Two ways to express "exact... |
| euen1 9032 | Two ways to express "exact... |
| euen1b 9033 | Two ways to express " ` A ... |
| funen1cnv 9034 | If a function is equinumer... |
| en1uniel 9035 | A singleton contains its s... |
| 2dom 9036 | A set that dominates ordin... |
| fundmen 9037 | A function is equinumerous... |
| fundmeng 9038 | A function is equinumerous... |
| cnven 9039 | A relational set is equinu... |
| cnvct 9040 | If a set is countable, so ... |
| fndmeng 9041 | A function is equinumerate... |
| mapsnend 9042 | Set exponentiation to a si... |
| mapsnen 9043 | Set exponentiation to a si... |
| snmapen 9044 | Set exponentiation: a sing... |
| snmapen1 9045 | Set exponentiation: a sing... |
| map1 9046 | Set exponentiation: ordina... |
| en2sn 9047 | Two singletons are equinum... |
| 0fi 9048 | The empty set is finite. ... |
| snfi 9049 | A singleton is finite. (C... |
| fiprc 9050 | The class of finite sets i... |
| unen 9051 | Equinumerosity of union of... |
| enrefnn 9052 | Equinumerosity is reflexiv... |
| en2prd 9053 | Two proper unordered pairs... |
| enpr2d 9054 | A pair with distinct eleme... |
| ssct 9055 | Any subset of a countable ... |
| difsnen 9056 | All decrements of a set ar... |
| domdifsn 9057 | Dominance over a set with ... |
| xpsnen 9058 | A set is equinumerous to i... |
| xpsneng 9059 | A set is equinumerous to i... |
| xp1en 9060 | One times a cardinal numbe... |
| endisj 9061 | Any two sets are equinumer... |
| undom 9062 | Dominance law for union. ... |
| xpcomf1o 9063 | The canonical bijection fr... |
| xpcomco 9064 | Composition with the bijec... |
| xpcomen 9065 | Commutative law for equinu... |
| xpcomeng 9066 | Commutative law for equinu... |
| xpsnen2g 9067 | A set is equinumerous to i... |
| xpassen 9068 | Associative law for equinu... |
| xpdom2 9069 | Dominance law for Cartesia... |
| xpdom2g 9070 | Dominance law for Cartesia... |
| xpdom1g 9071 | Dominance law for Cartesia... |
| xpdom3 9072 | A set is dominated by its ... |
| xpdom1 9073 | Dominance law for Cartesia... |
| domunsncan 9074 | A singleton cancellation l... |
| omxpenlem 9075 | Lemma for ~ omxpen . (Con... |
| omxpen 9076 | The cardinal and ordinal p... |
| omf1o 9077 | Construct an explicit bije... |
| pw2f1olem 9078 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9079 | The power set of a set is ... |
| pw2eng 9080 | The power set of a set is ... |
| pw2en 9081 | The power set of a set is ... |
| fopwdom 9082 | Covering implies injection... |
| enfixsn 9083 | Given two equipollent sets... |
| sbthlem1 9084 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9085 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9086 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9087 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9088 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9089 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9090 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9091 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9092 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9093 | Lemma for ~ sbth . (Contr... |
| sbth 9094 | Schroeder-Bernstein Theore... |
| sbthb 9095 | Schroeder-Bernstein Theore... |
| sbthcl 9096 | Schroeder-Bernstein Theore... |
| dfsdom2 9097 | Alternate definition of st... |
| brsdom2 9098 | Alternate definition of st... |
| sdomnsym 9099 | Strict dominance is asymme... |
| domnsym 9100 | Theorem 22(i) of [Suppes] ... |
| 0domg 9101 | Any set dominates the empt... |
| dom0 9102 | A set dominated by the emp... |
| 0sdomg 9103 | A set strictly dominates t... |
| 0dom 9104 | Any set dominates the empt... |
| 0sdom 9105 | A set strictly dominates t... |
| sdom0 9106 | The empty set does not str... |
| sdomdomtr 9107 | Transitivity of strict dom... |
| sdomentr 9108 | Transitivity of strict dom... |
| domsdomtr 9109 | Transitivity of dominance ... |
| ensdomtr 9110 | Transitivity of equinumero... |
| sdomirr 9111 | Strict dominance is irrefl... |
| sdomtr 9112 | Strict dominance is transi... |
| sdomn2lp 9113 | Strict dominance has no 2-... |
| enen1 9114 | Equality-like theorem for ... |
| enen2 9115 | Equality-like theorem for ... |
| domen1 9116 | Equality-like theorem for ... |
| domen2 9117 | Equality-like theorem for ... |
| sdomen1 9118 | Equality-like theorem for ... |
| sdomen2 9119 | Equality-like theorem for ... |
| domtriord 9120 | Dominance is trichotomous ... |
| sdomel 9121 | For ordinals, strict domin... |
| sdomdif 9122 | The difference of a set fr... |
| onsdominel 9123 | An ordinal with more eleme... |
| domunsn 9124 | Dominance over a set with ... |
| fodomr 9125 | There exists a mapping fro... |
| pwdom 9126 | Injection of sets implies ... |
| canth2 9127 | Cantor's Theorem. No set ... |
| canth2g 9128 | Cantor's theorem with the ... |
| 2pwuninel 9129 | The power set of the power... |
| 2pwne 9130 | No set equals the power se... |
| disjen 9131 | A stronger form of ~ pwuni... |
| disjenex 9132 | Existence version of ~ dis... |
| domss2 9133 | A corollary of ~ disjenex ... |
| domssex2 9134 | A corollary of ~ disjenex ... |
| domssex 9135 | Weakening of ~ domssex2 to... |
| xpf1o 9136 | Construct a bijection on a... |
| xpen 9137 | Equinumerosity law for Car... |
| mapen 9138 | Two set exponentiations ar... |
| mapdom1 9139 | Order-preserving property ... |
| mapxpen 9140 | Equinumerosity law for dou... |
| xpmapenlem 9141 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9142 | Equinumerosity law for set... |
| mapunen 9143 | Equinumerosity law for set... |
| map2xp 9144 | A cardinal power with expo... |
| mapdom2 9145 | Order-preserving property ... |
| mapdom3 9146 | Set exponentiation dominat... |
| pwen 9147 | If two sets are equinumero... |
| ssenen 9148 | Equinumerosity of equinume... |
| limenpsi 9149 | A limit ordinal is equinum... |
| limensuci 9150 | A limit ordinal is equinum... |
| limensuc 9151 | A limit ordinal is equinum... |
| infensuc 9152 | Any infinite ordinal is eq... |
| dif1enlem 9153 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9154 | If a set is equinumerous t... |
| dif1en 9155 | If a set ` A ` is equinume... |
| dif1ennn 9156 | If a set ` A ` is equinume... |
| findcard 9157 | Schema for induction on th... |
| findcard2 9158 | Schema for induction on th... |
| findcard2s 9159 | Variation of ~ findcard2 r... |
| findcard2d 9160 | Deduction version of ~ fin... |
| nnfi 9161 | Natural numbers are finite... |
| pssnn 9162 | A proper subset of a natur... |
| ssnnfi 9163 | A subset of a natural numb... |
| unfi 9164 | The union of two finite se... |
| unfid 9165 | The union of two finite se... |
| ssfi 9166 | A subset of a finite set i... |
| ssfiALT 9167 | Shorter proof of ~ ssfi us... |
| diffi 9168 | If ` A ` is finite, ` ( A ... |
| cnvfi 9169 | If a set is finite, its co... |
| pwssfi 9170 | Every element of the power... |
| fnfi 9171 | A version of ~ fnex for fi... |
| f1oenfi 9172 | If the domain of a one-to-... |
| f1oenfirn 9173 | If the range of a one-to-o... |
| f1domfi 9174 | If the codomain of a one-t... |
| f1domfi2 9175 | If the domain of a one-to-... |
| enreffi 9176 | Equinumerosity is reflexiv... |
| ensymfib 9177 | Symmetry of equinumerosity... |
| entrfil 9178 | Transitivity of equinumero... |
| enfii 9179 | A set equinumerous to a fi... |
| enfi 9180 | Equinumerous sets have the... |
| enfiALT 9181 | Shorter proof of ~ enfi us... |
| domfi 9182 | A set dominated by a finit... |
| entrfi 9183 | Transitivity of equinumero... |
| entrfir 9184 | Transitivity of equinumero... |
| domtrfil 9185 | Transitivity of dominance ... |
| domtrfi 9186 | Transitivity of dominance ... |
| domtrfir 9187 | Transitivity of dominance ... |
| f1imaenfi 9188 | If a function is one-to-on... |
| ssdomfi 9189 | A finite set dominates its... |
| ssdomfi2 9190 | A set dominates its finite... |
| sbthfilem 9191 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9192 | Schroeder-Bernstein Theore... |
| domnsymfi 9193 | If a set dominates a finit... |
| sdomdomtrfi 9194 | Transitivity of strict dom... |
| domsdomtrfi 9195 | Transitivity of dominance ... |
| sucdom2 9196 | Strict dominance of a set ... |
| phplem1 9197 | Lemma for Pigeonhole Princ... |
| phplem2 9198 | Lemma for Pigeonhole Princ... |
| nneneq 9199 | Two equinumerous natural n... |
| php 9200 | Pigeonhole Principle. A n... |
| php2 9201 | Corollary of Pigeonhole Pr... |
| php3 9202 | Corollary of Pigeonhole Pr... |
| php4 9203 | Corollary of the Pigeonhol... |
| php5 9204 | Corollary of the Pigeonhol... |
| phpeqd 9205 | Corollary of the Pigeonhol... |
| nndomog 9206 | Cardinal ordering agrees w... |
| onomeneq 9207 | An ordinal number equinume... |
| onfin 9208 | An ordinal number is finit... |
| ordfin 9209 | A generalization of ~ onfi... |
| onfin2 9210 | A set is a natural number ... |
| nndomo 9211 | Cardinal ordering agrees w... |
| nnsdomo 9212 | Cardinal ordering agrees w... |
| sucdom 9213 | Strict dominance of a set ... |
| snnen2o 9214 | A singleton ` { A } ` is n... |
| 0sdom1dom 9215 | Strict dominance over 0 is... |
| 0sdom1domALT 9216 | Alternate proof of ~ 0sdom... |
| 1sdom2 9217 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9218 | Alternate proof of ~ 1sdom... |
| sdom1 9219 | A set has less than one me... |
| modom 9220 | Two ways to express "at mo... |
| modom2 9221 | Two ways to express "at mo... |
| rex2dom 9222 | A set that has at least 2 ... |
| 1sdom2dom 9223 | Strict dominance over 1 is... |
| 1sdom 9224 | A set that strictly domina... |
| unxpdomlem1 9225 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9226 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9227 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9228 | Cartesian product dominate... |
| unxpdom2 9229 | Corollary of ~ unxpdom . ... |
| sucxpdom 9230 | Cartesian product dominate... |
| pssinf 9231 | A set equinumerous to a pr... |
| fisseneq 9232 | A finite set is equal to i... |
| ominf 9233 | The set of natural numbers... |
| isinf 9234 | Any set that is not finite... |
| fineqvlem 9235 | Lemma for ~ fineqv . (Con... |
| fineqv 9236 | If the Axiom of Infinity i... |
| xpfir 9237 | The components of a nonemp... |
| ssfid 9238 | A subset of a finite set i... |
| infi 9239 | The intersection of two se... |
| rabfi 9240 | A restricted class built f... |
| finresfin 9241 | The restriction of a finit... |
| f1finf1o 9242 | Any injection from one fin... |
| nfielex 9243 | If a class is not finite, ... |
| en1eqsn 9244 | A set with one element is ... |
| en1eqsnbi 9245 | A set containing an elemen... |
| dif1ennnALT 9246 | Alternate proof of ~ dif1e... |
| enp1ilem 9247 | Lemma for uses of ~ enp1i ... |
| enp1i 9248 | Proof induction for ~ en2 ... |
| en2 9249 | A set equinumerous to ordi... |
| en3 9250 | A set equinumerous to ordi... |
| en4 9251 | A set equinumerous to ordi... |
| findcard3 9252 | Schema for strong inductio... |
| ac6sfi 9253 | A version of ~ ac6s for fi... |
| frfi 9254 | A partial order is well-fo... |
| fimax2g 9255 | A finite set has a maximum... |
| fimaxg 9256 | A finite set has a maximum... |
| fisupg 9257 | Lemma showing existence an... |
| wofi 9258 | A total order on a finite ... |
| ordunifi 9259 | The maximum of a finite co... |
| fissorduni 9260 | The union (supremum) of a ... |
| nnunifi 9261 | The union (supremum) of a ... |
| unblem1 9262 | Lemma for ~ unbnn . After... |
| unblem2 9263 | Lemma for ~ unbnn . The v... |
| unblem3 9264 | Lemma for ~ unbnn . The v... |
| unblem4 9265 | Lemma for ~ unbnn . The f... |
| unbnn 9266 | Any unbounded subset of na... |
| unbnn2 9267 | Version of ~ unbnn that do... |
| isfinite2 9268 | Any set strictly dominated... |
| nnsdomg 9269 | Omega strictly dominates a... |
| isfiniteg 9270 | A set is finite iff it is ... |
| infsdomnn 9271 | An infinite set strictly d... |
| infn0 9272 | An infinite set is not emp... |
| infn0ALT 9273 | Shorter proof of ~ infn0 u... |
| fin2inf 9274 | This (useless) theorem, wh... |
| unfilem1 9275 | Lemma for proving that the... |
| unfilem2 9276 | Lemma for proving that the... |
| unfilem3 9277 | Lemma for proving that the... |
| unfir 9278 | If a union is finite, the ... |
| unfib 9279 | A union is finite if and o... |
| unfi2 9280 | The union of two finite se... |
| difinf 9281 | An infinite set ` A ` minu... |
| fodomfi 9282 | An onto function implies d... |
| fofi 9283 | If an onto function has a ... |
| f1fi 9284 | If a 1-to-1 function has a... |
| imafi 9285 | Images of finite sets are ... |
| pwfir 9286 | If the power set of a set ... |
| pwfilem 9287 | Lemma for ~ pwfi . (Contr... |
| pwfi 9288 | The power set of a finite ... |
| xpfi 9289 | The Cartesian product of t... |
| 3xpfi 9290 | The Cartesian product of t... |
| domunfican 9291 | A finite set union cancell... |
| infcntss 9292 | Every infinite set has a d... |
| prfi 9293 | An unordered pair is finit... |
| prfiALT 9294 | Shorter proof of ~ prfi us... |
| tpfi 9295 | An unordered triple is fin... |
| fiint 9296 | Equivalent ways of stating... |
| fodomfir 9297 | There exists a mapping fro... |
| fodomfib 9298 | Equivalence of an onto map... |
| fofinf1o 9299 | Any surjection from one fi... |
| rneqdmfinf1o 9300 | Any function from a finite... |
| fidomdm 9301 | Any finite set dominates i... |
| dmfi 9302 | The domain of a finite set... |
| fundmfibi 9303 | A function is finite if an... |
| resfnfinfin 9304 | The restriction of a funct... |
| residfi 9305 | A restricted identity func... |
| cnvfiALT 9306 | Shorter proof of ~ cnvfi u... |
| rnfi 9307 | The range of a finite set ... |
| f1dmvrnfibi 9308 | A one-to-one function whos... |
| f1vrnfibi 9309 | A one-to-one function whic... |
| iunfi 9310 | The finite union of finite... |
| unifi 9311 | The finite union of finite... |
| unifi2 9312 | The finite union of finite... |
| infssuni 9313 | If an infinite set ` A ` i... |
| unirnffid 9314 | The union of the range of ... |
| mapfi 9315 | Set exponentiation of fini... |
| ixpfi 9316 | A Cartesian product of fin... |
| ixpfi2 9317 | A Cartesian product of fin... |
| mptfi 9318 | A finite mapping set is fi... |
| abrexfi 9319 | An image set from a finite... |
| cnvimamptfin 9320 | A preimage of a mapping wi... |
| elfpw 9321 | Membership in a class of f... |
| unifpw 9322 | A set is the union of its ... |
| f1opwfi 9323 | A one-to-one mapping induc... |
| fissuni 9324 | A finite subset of a union... |
| fipreima 9325 | Given a finite subset ` A ... |
| finsschain 9326 | A finite subset of the uni... |
| indexfi 9327 | If for every element of a ... |
| imafi2 9328 | The image by a finite set ... |
| unifi3 9329 | If a union is finite, then... |
| tfsnfin2 9330 | A transfinite sequence is ... |
| relfsupp 9333 | The property of a function... |
| relprcnfsupp 9334 | A proper class is never fi... |
| isfsupp 9335 | The property of a class to... |
| isfsuppd 9336 | Deduction form of ~ isfsup... |
| funisfsupp 9337 | The property of a function... |
| fsuppimp 9338 | Implications of a class be... |
| fsuppimpd 9339 | A finitely supported funct... |
| fsuppfund 9340 | A finitely supported funct... |
| fisuppfi 9341 | A function on a finite set... |
| fidmfisupp 9342 | A function with a finite d... |
| finnzfsuppd 9343 | If a function is zero outs... |
| fdmfisuppfi 9344 | The support of a function ... |
| fdmfifsupp 9345 | A function with a finite d... |
| fsuppmptdm 9346 | A mapping with a finite do... |
| fndmfisuppfi 9347 | The support of a function ... |
| fndmfifsupp 9348 | A function with a finite d... |
| suppeqfsuppbi 9349 | If two functions have the ... |
| suppssfifsupp 9350 | If the support of a functi... |
| fsuppsssupp 9351 | If the support of a functi... |
| fsuppsssuppgd 9352 | If the support of a functi... |
| fsuppss 9353 | A subset of a finitely sup... |
| fsuppssov1 9354 | Formula building theorem f... |
| fsuppxpfi 9355 | The cartesian product of t... |
| fczfsuppd 9356 | A constant function with v... |
| fsuppun 9357 | The union of two finitely ... |
| fsuppunfi 9358 | The union of the support o... |
| fsuppunbi 9359 | If the union of two classe... |
| 0fsupp 9360 | The empty set is a finitel... |
| snopfsupp 9361 | A singleton containing an ... |
| funsnfsupp 9362 | Finite support for a funct... |
| fsuppres 9363 | The restriction of a finit... |
| fmptssfisupp 9364 | The restriction of a mappi... |
| ressuppfi 9365 | If the support of the rest... |
| resfsupp 9366 | If the restriction of a fu... |
| resfifsupp 9367 | The restriction of a funct... |
| ffsuppbi 9368 | Two ways of saying that a ... |
| fsuppmptif 9369 | A function mapping an argu... |
| sniffsupp 9370 | A function mapping all but... |
| fsuppcolem 9371 | Lemma for ~ fsuppco . For... |
| fsuppco 9372 | The composition of a 1-1 f... |
| fsuppco2 9373 | The composition of a funct... |
| fsuppcor 9374 | The composition of a funct... |
| mapfienlem1 9375 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9376 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9377 | Lemma 3 for ~ mapfien . (... |
| mapfien 9378 | A bijection of the base se... |
| mapfien2 9379 | Equinumerousity relation f... |
| fival 9382 | The set of all the finite ... |
| elfi 9383 | Specific properties of an ... |
| elfi2 9384 | The empty intersection nee... |
| elfir 9385 | Sufficient condition for a... |
| intrnfi 9386 | Sufficient condition for t... |
| iinfi 9387 | An indexed intersection of... |
| inelfi 9388 | The intersection of two se... |
| ssfii 9389 | Any element of a set ` A `... |
| fi0 9390 | The set of finite intersec... |
| fieq0 9391 | A set is empty iff the cla... |
| fiin 9392 | The elements of ` ( fi `` ... |
| dffi2 9393 | The set of finite intersec... |
| fiss 9394 | Subset relationship for fu... |
| inficl 9395 | A set which is closed unde... |
| fipwuni 9396 | The set of finite intersec... |
| fisn 9397 | A singleton is closed unde... |
| fiuni 9398 | The union of the finite in... |
| fipwss 9399 | If a set is a family of su... |
| elfiun 9400 | A finite intersection of e... |
| dffi3 9401 | The set of finite intersec... |
| fifo 9402 | Describe a surjection from... |
| marypha1lem 9403 | Core induction for Philip ... |
| marypha1 9404 | (Philip) Hall's marriage t... |
| marypha2lem1 9405 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9406 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9407 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9408 | Lemma for ~ marypha2 . Pr... |
| marypha2 9409 | Version of ~ marypha1 usin... |
| dfsup2 9414 | Quantifier-free definition... |
| supeq1 9415 | Equality theorem for supre... |
| supeq1d 9416 | Equality deduction for sup... |
| supeq1i 9417 | Equality inference for sup... |
| supeq2 9418 | Equality theorem for supre... |
| supeq3 9419 | Equality theorem for supre... |
| supeq123d 9420 | Equality deduction for sup... |
| nfsup 9421 | Hypothesis builder for sup... |
| supmo 9422 | Any class ` B ` has at mos... |
| supexd 9423 | A supremum is a set. (Con... |
| supeu 9424 | A supremum is unique. Sim... |
| supval2 9425 | Alternate expression for t... |
| eqsup 9426 | Sufficient condition for a... |
| eqsupd 9427 | Sufficient condition for a... |
| supcl 9428 | A supremum belongs to its ... |
| supub 9429 | A supremum is an upper bou... |
| suplub 9430 | A supremum is the least up... |
| suplub2 9431 | Bidirectional form of ~ su... |
| supnub 9432 | An upper bound is not less... |
| supssd 9433 | Inequality deduction for s... |
| supex 9434 | A supremum is a set. (Con... |
| sup00 9435 | The supremum under an empt... |
| sup0riota 9436 | The supremum of an empty s... |
| sup0 9437 | The supremum of an empty s... |
| supmax 9438 | The greatest element of a ... |
| fisup2g 9439 | A finite set satisfies the... |
| fisupcl 9440 | A nonempty finite set cont... |
| supgtoreq 9441 | The supremum of a finite s... |
| suppr 9442 | The supremum of a pair. (... |
| supsn 9443 | The supremum of a singleto... |
| supisolem 9444 | Lemma for ~ supiso . (Con... |
| supisoex 9445 | Lemma for ~ supiso . (Con... |
| supiso 9446 | Image of a supremum under ... |
| infeq1 9447 | Equality theorem for infim... |
| infeq1d 9448 | Equality deduction for inf... |
| infeq1i 9449 | Equality inference for inf... |
| infeq2 9450 | Equality theorem for infim... |
| infeq3 9451 | Equality theorem for infim... |
| infeq123d 9452 | Equality deduction for inf... |
| nfinf 9453 | Hypothesis builder for inf... |
| infexd 9454 | An infimum is a set. (Con... |
| eqinf 9455 | Sufficient condition for a... |
| eqinfd 9456 | Sufficient condition for a... |
| infval 9457 | Alternate expression for t... |
| infcllem 9458 | Lemma for ~ infcl , ~ infl... |
| infcl 9459 | An infimum belongs to its ... |
| inflb 9460 | An infimum is a lower boun... |
| infglb 9461 | An infimum is the greatest... |
| infglbb 9462 | Bidirectional form of ~ in... |
| infnlb 9463 | A lower bound is not great... |
| infssd 9464 | Inequality deduction for i... |
| infex 9465 | An infimum is a set. (Con... |
| infmin 9466 | The smallest element of a ... |
| infmo 9467 | Any class ` B ` has at mos... |
| infeu 9468 | An infimum is unique. (Co... |
| fimin2g 9469 | A finite set has a minimum... |
| fiming 9470 | A finite set has a minimum... |
| fiinfg 9471 | Lemma showing existence an... |
| fiinf2g 9472 | A finite set satisfies the... |
| fiinfcl 9473 | A nonempty finite set cont... |
| infltoreq 9474 | The infimum of a finite se... |
| infpr 9475 | The infimum of a pair. (C... |
| infsupprpr 9476 | The infimum of a proper pa... |
| infsn 9477 | The infimum of a singleton... |
| inf00 9478 | The infimum regarding an e... |
| infempty 9479 | The infimum of an empty se... |
| infiso 9480 | Image of an infimum under ... |
| dfoi 9483 | Rewrite ~ df-oi with abbre... |
| oieq1 9484 | Equality theorem for ordin... |
| oieq2 9485 | Equality theorem for ordin... |
| nfoi 9486 | Hypothesis builder for ord... |
| ordiso2 9487 | Generalize ~ ordiso to pro... |
| ordiso 9488 | Order-isomorphic ordinal n... |
| ordtypecbv 9489 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9490 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9491 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9492 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9493 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9494 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9495 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9496 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9497 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9498 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9499 | Lemma for ~ ordtype . Usi... |
| oi0 9500 | Definition of the ordinal ... |
| oicl 9501 | The order type of the well... |
| oif 9502 | The order isomorphism of t... |
| oiiso2 9503 | The order isomorphism of t... |
| ordtype 9504 | For any set-like well-orde... |
| oiiniseg 9505 | ` ran F ` is an initial se... |
| ordtype2 9506 | For any set-like well-orde... |
| oiexg 9507 | The order isomorphism on a... |
| oion 9508 | The order type of the well... |
| oiiso 9509 | The order isomorphism of t... |
| oien 9510 | The order type of a well-o... |
| oieu 9511 | Uniqueness of the unique o... |
| oismo 9512 | When ` A ` is a subclass o... |
| oiid 9513 | The order type of an ordin... |
| hartogslem1 9514 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9515 | Lemma for ~ hartogs . (Co... |
| hartogs 9516 | The class of ordinals domi... |
| wofib 9517 | The only sets which are we... |
| wemaplem1 9518 | Value of the lexicographic... |
| wemaplem2 9519 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9520 | Lemma for ~ wemapso . Tra... |
| wemappo 9521 | Construct lexicographic or... |
| wemapsolem 9522 | Lemma for ~ wemapso . (Co... |
| wemapso 9523 | Construct lexicographic or... |
| wemapso2lem 9524 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9525 | An alternative to having a... |
| card2on 9526 | The alternate definition o... |
| card2inf 9527 | The alternate definition o... |
| harf 9530 | Functionality of the Harto... |
| harcl 9531 | Values of the Hartogs func... |
| harval 9532 | Function value of the Hart... |
| elharval 9533 | The Hartogs number of a se... |
| harndom 9534 | The Hartogs number of a se... |
| harword 9535 | Weak ordering property of ... |
| relwdom 9538 | Weak dominance is a relati... |
| brwdom 9539 | Property of weak dominance... |
| brwdomi 9540 | Property of weak dominance... |
| brwdomn0 9541 | Weak dominance over nonemp... |
| 0wdom 9542 | Any set weakly dominates t... |
| fowdom 9543 | An onto function implies w... |
| wdomref 9544 | Reflexivity of weak domina... |
| brwdom2 9545 | Alternate characterization... |
| domwdom 9546 | Weak dominance is implied ... |
| wdomtr 9547 | Transitivity of weak domin... |
| wdomen1 9548 | Equality-like theorem for ... |
| wdomen2 9549 | Equality-like theorem for ... |
| wdompwdom 9550 | Weak dominance strengthens... |
| canthwdom 9551 | Cantor's Theorem, stated u... |
| wdom2d 9552 | Deduce weak dominance from... |
| wdomd 9553 | Deduce weak dominance from... |
| brwdom3 9554 | Condition for weak dominan... |
| brwdom3i 9555 | Weak dominance implies exi... |
| unwdomg 9556 | Weak dominance of a (disjo... |
| xpwdomg 9557 | Weak dominance of a Cartes... |
| wdomima2g 9558 | A set is weakly dominant o... |
| wdomimag 9559 | A set is weakly dominant o... |
| unxpwdom2 9560 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9561 | If a Cartesian product is ... |
| ixpiunwdom 9562 | Describe an onto function ... |
| harwdom 9563 | The value of the Hartogs f... |
| axreg2 9565 | Axiom of Regularity expres... |
| zfregcl 9566 | The Axiom of Regularity wi... |
| zfregclOLD 9567 | Obsolete version of ~ zfre... |
| zfreg 9568 | The Axiom of Regularity us... |
| elirrv 9569 | The membership relation is... |
| elirrvOLD 9570 | Obsolete version of ~ elir... |
| elirrvOLDOLD 9571 | Obsolete version of ~ elir... |
| elirr 9572 | No class is a member of it... |
| elneq 9573 | A class is not equal to an... |
| nelaneq 9574 | A class is not an element ... |
| nelaneqOLD 9575 | Obsolete version of ~ nela... |
| nelaneqOLDOLD 9576 | Obsolete version of ~ nela... |
| epinid0 9577 | The membership relation an... |
| sucprcreg 9578 | A class is equal to its su... |
| sucprcregOLD 9579 | Obsolete version of ~ sucp... |
| ruv 9580 | The Russell class is equal... |
| ruALT 9581 | Alternate proof of ~ ru , ... |
| disjcsn 9582 | A class is disjoint from i... |
| zfregfr 9583 | The membership relation is... |
| elirrvALT 9584 | Alternate proof of ~ elirr... |
| en2lp 9585 | No class has 2-cycle membe... |
| elnanel 9586 | Two classes are not elemen... |
| cnvepnep 9587 | The membership (epsilon) r... |
| epnsym 9588 | The membership (epsilon) r... |
| elnotel 9589 | A class cannot be an eleme... |
| elnel 9590 | A class cannot be an eleme... |
| en3lplem1 9591 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9592 | Lemma for ~ en3lp . (Cont... |
| en3lp 9593 | No class has 3-cycle membe... |
| preleqg 9594 | Equality of two unordered ... |
| preleq 9595 | Equality of two unordered ... |
| preleqALT 9596 | Alternate proof of ~ prele... |
| opthreg 9597 | Theorem for alternate repr... |
| suc11reg 9598 | The successor operation be... |
| dford2 9599 | Assuming ~ ax-reg , an ord... |
| inf0 9600 | Existence of ` _om ` impli... |
| inf1 9601 | Variation of Axiom of Infi... |
| inf2 9602 | Variation of Axiom of Infi... |
| inf3lema 9603 | Lemma for our Axiom of Inf... |
| inf3lemb 9604 | Lemma for our Axiom of Inf... |
| inf3lemc 9605 | Lemma for our Axiom of Inf... |
| inf3lemd 9606 | Lemma for our Axiom of Inf... |
| inf3lem1 9607 | Lemma for our Axiom of Inf... |
| inf3lem2 9608 | Lemma for our Axiom of Inf... |
| inf3lem3 9609 | Lemma for our Axiom of Inf... |
| inf3lem4 9610 | Lemma for our Axiom of Inf... |
| inf3lem5 9611 | Lemma for our Axiom of Inf... |
| inf3lem6 9612 | Lemma for our Axiom of Inf... |
| inf3lem7 9613 | Lemma for our Axiom of Inf... |
| inf3 9614 | Our Axiom of Infinity ~ ax... |
| infeq5i 9615 | Half of ~ infeq5 . (Contr... |
| infeq5 9616 | The statement "there exist... |
| zfinf 9618 | Axiom of Infinity expresse... |
| axinf2 9619 | A standard version of Axio... |
| zfinf2 9621 | A standard version of the ... |
| omex 9622 | The existence of omega (th... |
| axinf 9623 | The first version of the A... |
| inf5 9624 | The statement "there exist... |
| omelon 9625 | Omega is an ordinal number... |
| dfom3 9626 | The class of natural numbe... |
| elom3 9627 | A simplification of ~ elom... |
| dfom4 9628 | A simplification of ~ df-o... |
| dfom5 9629 | ` _om ` is the smallest li... |
| oancom 9630 | Ordinal addition is not co... |
| isfinite 9631 | A set is finite iff it is ... |
| fict 9632 | A finite set is countable ... |
| nnsdom 9633 | A natural number is strict... |
| omenps 9634 | Omega is equinumerous to a... |
| omensuc 9635 | The set of natural numbers... |
| infdifsn 9636 | Removing a singleton from ... |
| infdiffi 9637 | Removing a finite set from... |
| unbnn3 9638 | Any unbounded subset of na... |
| noinfep 9639 | Using the Axiom of Regular... |
| cantnffval 9642 | The value of the Cantor no... |
| cantnfdm 9643 | The domain of the Cantor n... |
| cantnfvalf 9644 | Lemma for ~ cantnf . The ... |
| cantnfs 9645 | Elementhood in the set of ... |
| cantnfcl 9646 | Basic properties of the or... |
| cantnfval 9647 | The value of the Cantor no... |
| cantnfval2 9648 | Alternate expression for t... |
| cantnfsuc 9649 | The value of the recursive... |
| cantnfle 9650 | A lower bound on the ` CNF... |
| cantnflt 9651 | An upper bound on the part... |
| cantnflt2 9652 | An upper bound on the ` CN... |
| cantnff 9653 | The ` CNF ` function is a ... |
| cantnf0 9654 | The value of the zero func... |
| cantnfrescl 9655 | A function is finitely sup... |
| cantnfres 9656 | The ` CNF ` function respe... |
| cantnfp1lem1 9657 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9658 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9659 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9660 | If ` F ` is created by add... |
| oemapso 9661 | The relation ` T ` is a st... |
| oemapval 9662 | Value of the relation ` T ... |
| oemapvali 9663 | If ` F < G ` , then there ... |
| cantnflem1a 9664 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9665 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9666 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9667 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9668 | Lemma for ~ cantnf . This... |
| cantnflem2 9669 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9670 | Lemma for ~ cantnf . Here... |
| cantnflem4 9671 | Lemma for ~ cantnf . Comp... |
| cantnf 9672 | The Cantor Normal Form the... |
| oemapwe 9673 | The lexicographic order on... |
| cantnffval2 9674 | An alternate definition of... |
| cantnff1o 9675 | Simplify the isomorphism o... |
| wemapwe 9676 | Construct lexicographic or... |
| oef1o 9677 | A bijection of the base se... |
| cnfcomlem 9678 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9679 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9680 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9681 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9682 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9683 | Any infinite ordinal ` B `... |
| cnfcom3clem 9684 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9685 | Wrap the construction of ~... |
| ttrcleq 9688 | Equality theorem for trans... |
| nfttrcld 9689 | Bound variable hypothesis ... |
| nfttrcl 9690 | Bound variable hypothesis ... |
| relttrcl 9691 | The transitive closure of ... |
| brttrcl 9692 | Characterization of elemen... |
| brttrcl2 9693 | Characterization of elemen... |
| ssttrcl 9694 | If ` R ` is a relation, th... |
| ttrcltr 9695 | The transitive closure of ... |
| ttrclresv 9696 | The transitive closure of ... |
| ttrclco 9697 | Composition law for the tr... |
| cottrcl 9698 | Composition law for the tr... |
| ttrclss 9699 | If ` R ` is a subclass of ... |
| dmttrcl 9700 | The domain of a transitive... |
| rnttrcl 9701 | The range of a transitive ... |
| ttrclexg 9702 | If ` R ` is a set, then so... |
| dfttrcl2 9703 | When ` R ` is a set and a ... |
| ttrclselem1 9704 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9705 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9706 | If ` R ` is set-like over ... |
| trcl 9707 | For any set ` A ` , show t... |
| tz9.1 9708 | Every set has a transitive... |
| tz9.1c 9709 | Alternate expression for t... |
| epfrs 9710 | The strong form of the Axi... |
| zfregs 9711 | The strong form of the Axi... |
| zfregs2 9712 | Alternate strong form of t... |
| tcvalg 9715 | Value of the transitive cl... |
| tcid 9716 | Defining property of the t... |
| tctr 9717 | Defining property of the t... |
| tcmin 9718 | Defining property of the t... |
| tc2 9719 | A variant of the definitio... |
| tcsni 9720 | The transitive closure of ... |
| tcss 9721 | The transitive closure fun... |
| tcel 9722 | The transitive closure fun... |
| tcidm 9723 | The transitive closure fun... |
| tc0 9724 | The transitive closure of ... |
| tc00 9725 | The transitive closure is ... |
| setind 9726 | Set (epsilon) induction. ... |
| setind2 9727 | Set (epsilon) induction, s... |
| setinds 9728 | Principle of set induction... |
| setinds2f 9729 | ` _E ` induction schema, u... |
| setinds2 9730 | ` _E ` induction schema, u... |
| frmin 9731 | Every (possibly proper) su... |
| frind 9732 | A subclass of a well-found... |
| frinsg 9733 | Well-Founded Induction Sch... |
| frins 9734 | Well-Founded Induction Sch... |
| frins2f 9735 | Well-Founded Induction sch... |
| frins2 9736 | Well-Founded Induction sch... |
| frins3 9737 | Well-Founded Induction sch... |
| frr3g 9738 | Functions defined by well-... |
| frrlem15 9739 | Lemma for general well-fou... |
| frrlem16 9740 | Lemma for general well-fou... |
| frr1 9741 | Law of general well-founde... |
| frr2 9742 | Law of general well-founde... |
| frr3 9743 | Law of general well-founde... |
| r1funlim 9748 | The cumulative hierarchy o... |
| r1fnon 9749 | The cumulative hierarchy o... |
| r10 9750 | Value of the cumulative hi... |
| r1sucg 9751 | Value of the cumulative hi... |
| r1suc 9752 | Value of the cumulative hi... |
| r1limg 9753 | Value of the cumulative hi... |
| r1lim 9754 | Value of the cumulative hi... |
| r1fin 9755 | The first ` _om ` levels o... |
| r1sdom 9756 | Each stage in the cumulati... |
| r111 9757 | The cumulative hierarchy i... |
| r1tr 9758 | The cumulative hierarchy o... |
| r1tr2 9759 | The union of a cumulative ... |
| r1ordg 9760 | Ordering relation for the ... |
| r1ord3g 9761 | Ordering relation for the ... |
| r1ord 9762 | Ordering relation for the ... |
| r1ord2 9763 | Ordering relation for the ... |
| r1ord3 9764 | Ordering relation for the ... |
| r1sssuc 9765 | The value of the cumulativ... |
| r1pwss 9766 | Each set of the cumulative... |
| r1sscl 9767 | Each set of the cumulative... |
| r1val1 9768 | The value of the cumulativ... |
| tz9.12lem1 9769 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9770 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9771 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9772 | A set is well-founded if a... |
| tz9.13 9773 | Every set is well-founded,... |
| tz9.13g 9774 | Every set is well-founded,... |
| rankwflemb 9775 | Two ways of saying a set i... |
| rankf 9776 | The domain and codomain of... |
| rankon 9777 | The rank of a set is an or... |
| r1elwf 9778 | Any member of the cumulati... |
| rankvalb 9779 | Value of the rank function... |
| rankr1ai 9780 | One direction of ~ rankr1a... |
| rankvaln 9781 | Value of the rank function... |
| rankidb 9782 | Identity law for the rank ... |
| rankdmr1 9783 | A rank is a member of the ... |
| rankr1ag 9784 | A version of ~ rankr1a tha... |
| rankr1bg 9785 | A relationship between ran... |
| r1rankidb 9786 | Any set is a subset of the... |
| r1elssi 9787 | The range of the ` R1 ` fu... |
| r1elss 9788 | The range of the ` R1 ` fu... |
| pwwf 9789 | A power set is well-founde... |
| sswf 9790 | A subset of a well-founded... |
| snwf 9791 | A singleton is well-founde... |
| unwf 9792 | A binary union is well-fou... |
| prwf 9793 | An unordered pair is well-... |
| opwf 9794 | An ordered pair is well-fo... |
| unir1 9795 | The cumulative hierarchy o... |
| jech9.3 9796 | Every set belongs to some ... |
| rankwflem 9797 | Every set is well-founded,... |
| rankval 9798 | Value of the rank function... |
| rankvalg 9799 | Value of the rank function... |
| rankval2 9800 | Value of an alternate defi... |
| uniwf 9801 | A union is well-founded if... |
| rankr1clem 9802 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9803 | A relationship between the... |
| rankidn 9804 | A relationship between the... |
| rankpwi 9805 | The rank of a power set. ... |
| rankelb 9806 | The membership relation is... |
| wfelirr 9807 | A well-founded set is not ... |
| rankval2b 9808 | Value of an alternate defi... |
| rankval3b 9809 | The value of the rank func... |
| ranksnb 9810 | The rank of a singleton. ... |
| rankonidlem 9811 | Lemma for ~ rankonid . (C... |
| rankonid 9812 | The rank of an ordinal num... |
| onwf 9813 | The ordinals are all well-... |
| r1wf 9814 | Each stage in the cumulati... |
| elwf 9815 | An element of a well-found... |
| onssr1 9816 | Initial segments of the or... |
| rankr1g 9817 | A relationship between the... |
| rankid 9818 | Identity law for the rank ... |
| rankr1 9819 | A relationship between the... |
| ssrankr1 9820 | A relationship between an ... |
| rankr1a 9821 | A relationship between ran... |
| r1val2 9822 | The value of the cumulativ... |
| r1val3 9823 | The value of the cumulativ... |
| rankel 9824 | The membership relation is... |
| rankelg 9825 | The membership relation is... |
| rankval3 9826 | The value of the rank func... |
| bndrank 9827 | Any class whose elements h... |
| unbndrank 9828 | The elements of a proper c... |
| rankpw 9829 | The rank of a power set. ... |
| rankpwg 9830 | The rank of a power set. ... |
| ranklim 9831 | The rank of a set belongs ... |
| r1pw 9832 | A stronger property of ` R... |
| r1pwALT 9833 | Alternate shorter proof of... |
| r1pwcl 9834 | The cumulative hierarchy o... |
| rankssb 9835 | The subset relation is inh... |
| rankss 9836 | The subset relation is inh... |
| rankunb 9837 | The rank of the union of t... |
| rankprb 9838 | The rank of an unordered p... |
| rankopb 9839 | The rank of an ordered pai... |
| rankuni2b 9840 | The value of the rank func... |
| ranksn 9841 | The rank of a singleton. ... |
| rankuni2 9842 | The rank of a union. Part... |
| rankun 9843 | The rank of the union of t... |
| rankpr 9844 | The rank of an unordered p... |
| rankop 9845 | The rank of an ordered pai... |
| rankung 9846 | The rank of the union of t... |
| ranksng 9847 | The rank of a singleton. ... |
| r1rankid 9848 | Any set is a subset of the... |
| rankeq0b 9849 | A set is empty iff its ran... |
| rankeq0 9850 | A set is empty iff its ran... |
| rankr1id 9851 | The rank of the hierarchy ... |
| rankuni 9852 | The rank of a union. Part... |
| rankval4b 9853 | The rank of a set is the s... |
| rankr1b 9854 | A relationship between ran... |
| ranksuc 9855 | The rank of a successor. ... |
| rankuniss 9856 | Upper bound of the rank of... |
| rankval4 9857 | The rank of a set is the s... |
| rankbnd 9858 | The rank of a set is bound... |
| rankbnd2 9859 | The rank of a set is bound... |
| rankc1 9860 | A relationship that can be... |
| rankc2 9861 | A relationship that can be... |
| rankelun 9862 | Rank membership is inherit... |
| rankelpr 9863 | Rank membership is inherit... |
| rankelop 9864 | Rank membership is inherit... |
| rankxpl 9865 | A lower bound on the rank ... |
| rankxpu 9866 | An upper bound on the rank... |
| rankfu 9867 | An upper bound on the rank... |
| rankmapu 9868 | An upper bound on the rank... |
| rankxplim 9869 | The rank of a Cartesian pr... |
| rankxplim2 9870 | If the rank of a Cartesian... |
| rankxplim3 9871 | The rank of a Cartesian pr... |
| rankxpsuc 9872 | The rank of a Cartesian pr... |
| tcwf 9873 | The transitive closure fun... |
| tcrank 9874 | This theorem expresses two... |
| rankfilimbi 9875 | If all elements of a finit... |
| r1filimi 9876 | If all elements of a finit... |
| elhf 9879 | Membership in the heredita... |
| hffi 9880 | Hereditarily finite sets a... |
| elhf2 9881 | Alternate form of membersh... |
| elhf2g 9882 | Hereditarily finiteness vi... |
| elhf4 9883 | A set is hereditarily fini... |
| elhf3 9884 | A set is hereditarily fini... |
| hfelhf 9885 | Any member of a hereditari... |
| hfsshf 9886 | Any subset of a hereditari... |
| hfelhfOLD 9887 | Obsolete version of ~ elhf... |
| 0hf 9888 | The empty set is a heredit... |
| hfun 9889 | The union of two hereditar... |
| hfunOLD 9890 | Obsolete version of ~ hfun... |
| hfsn 9891 | The singleton of a heredit... |
| hfsnOLD 9892 | Obsolete version of ~ hfsn... |
| hfadj 9893 | Adjoining one hereditarily... |
| elhf3OLD 9894 | Obsolete version of ~ elhf... |
| hfuni 9895 | The union of a hereditaril... |
| hfuniOLD 9896 | Obsolete version of ~ hfun... |
| hfpw 9897 | The power class of a hered... |
| hfpwOLD 9898 | Obsolete version of ~ hfpw... |
| scotteqd 9901 | Equality theorem for the S... |
| scotteq 9902 | Closed form of ~ scotteqd ... |
| nfscott 9903 | Bound-variable hypothesis ... |
| scottex 9904 | Scott's trick produces a s... |
| scottexOLD 9905 | Obsolete version of ~ scot... |
| scottss 9906 | Scott's trick produces a s... |
| scott0 9907 | Applying Scott's trick to ... |
| scott0b 9908 | Applying Scott's trick yie... |
| scott0OLD 9909 | Obsolete version of ~ scot... |
| scottabf 9910 | Value of the Scott operati... |
| scottab 9911 | Value of the Scott operati... |
| scottabes 9912 | Value of the Scott operati... |
| elscottab 9913 | An element of the output o... |
| scottexsOLD 9914 | Obsolete theorem as of 27-... |
| scott0bs 9915 | Theorem scheme version of ... |
| scott0bsOLD 9916 | Obsolete version of ~ scot... |
| scottex2OLD 9917 | Obsolete version of ~ scot... |
| scotteld 9918 | The Scott operation sends ... |
| scottelrankd 9919 | Property of a Scott's tric... |
| scottrankd 9920 | Rank of a nonempty Scott's... |
| cplem1 9921 | Lemma for the Collection P... |
| cplem1OLD 9922 | Obsolete version of ~ cple... |
| cplem2 9923 | Lemma for the Collection P... |
| cplem2OLD 9924 | Obsolete version of ~ cple... |
| cp 9925 | Collection Principle. Thi... |
| bnd 9926 | A very strong generalizati... |
| bnd2 9927 | A variant of the Boundedne... |
| kardex 9928 | The collection of all sets... |
| kardexOLD 9929 | Obsolete version of ~ kard... |
| karden 9930 | If we allow the Axiom of R... |
| kardenOLD 9931 | Obsolete version of ~ kard... |
| htalem 9932 | Lemma for defining an emul... |
| hta 9933 | A ZFC emulation of Hilbert... |
| htaOLD 9934 | Obsolete version of ~ hta ... |
| setrec1lem1 9937 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 9938 | Lemma for ~ setrec1 . If ... |
| bnd2d 9939 | Deduction form of ~ bnd2 .... |
| setrec1lem3 9940 | Lemma for ~ setrec1 . If ... |
| spcdvw 9941 | A version of ~ spcdv where... |
| setrec1lem4 9942 | Lemma for ~ setrec1 . If ... |
| setrec1 9943 | This is the first of two f... |
| setrec2fun 9944 | This is the second of two ... |
| setrec2lem1 9945 | Lemma for ~ setrec2 . The... |
| dffun3f 9946 | Alternate definition of fu... |
| setrec2lem2 9947 | Lemma for ~ setrec2 . The... |
| setrec2 9948 | This is the second of two ... |
| setrec2v 9949 | Version of ~ setrec2 with ... |
| djueq12 9956 | Equality theorem for disjo... |
| djueq1 9957 | Equality theorem for disjo... |
| djueq2 9958 | Equality theorem for disjo... |
| nfdju 9959 | Bound-variable hypothesis ... |
| djuex 9960 | The disjoint union of sets... |
| djuexb 9961 | The disjoint union of two ... |
| djulcl 9962 | Left closure of disjoint u... |
| djurcl 9963 | Right closure of disjoint ... |
| djulf1o 9964 | The left injection functio... |
| djurf1o 9965 | The right injection functi... |
| inlresf 9966 | The left injection restric... |
| inlresf1 9967 | The left injection restric... |
| inrresf 9968 | The right injection restri... |
| inrresf1 9969 | The right injection restri... |
| djuin 9970 | The images of any classes ... |
| djur 9971 | A member of a disjoint uni... |
| djuss 9972 | A disjoint union is a subc... |
| djuunxp 9973 | The union of a disjoint un... |
| djuexALT 9974 | Alternate proof of ~ djuex... |
| eldju1st 9975 | The first component of an ... |
| eldju2ndl 9976 | The second component of an... |
| eldju2ndr 9977 | The second component of an... |
| djuun 9978 | The disjoint union of two ... |
| 1stinl 9979 | The first component of the... |
| 2ndinl 9980 | The second component of th... |
| 1stinr 9981 | The first component of the... |
| 2ndinr 9982 | The second component of th... |
| updjudhf 9983 | The mapping of an element ... |
| updjudhcoinlf 9984 | The composition of the map... |
| updjudhcoinrg 9985 | The composition of the map... |
| updjud 9986 | Universal property of the ... |
| cardf2 9995 | The cardinality function i... |
| cardon 9996 | The cardinal number of a s... |
| isnum2 9997 | A way to express well-orde... |
| isnumi 9998 | A set equinumerous to an o... |
| ennum 9999 | Equinumerous sets are equi... |
| finnum 10000 | Every finite set is numera... |
| onenon 10001 | Every ordinal number is nu... |
| tskwe 10002 | A Tarski set is well-order... |
| xpnum 10003 | The cartesian product of n... |
| cardval3 10004 | An alternate definition of... |
| cardid2 10005 | Any numerable set is equin... |
| isnum3 10006 | A set is numerable iff it ... |
| oncardval 10007 | The value of the cardinal ... |
| oncardid 10008 | Any ordinal number is equi... |
| cardonle 10009 | The cardinal of an ordinal... |
| card0 10010 | The cardinality of the emp... |
| cardidm 10011 | The cardinality function i... |
| oncard 10012 | A set is a cardinal number... |
| ficardom 10013 | The cardinal number of a f... |
| ficardid 10014 | A finite set is equinumero... |
| cardnn 10015 | The cardinality of a natur... |
| cardnueq0 10016 | The empty set is the only ... |
| cardne 10017 | No member of a cardinal nu... |
| carden2a 10018 | If two sets have equal non... |
| carden2b 10019 | If two sets are equinumero... |
| card1 10020 | A set has cardinality one ... |
| cardsn 10021 | A singleton has cardinalit... |
| carddomi2 10022 | Two sets have the dominanc... |
| sdomsdomcardi 10023 | A set strictly dominates i... |
| cardlim 10024 | An infinite cardinal is a ... |
| cardsdomelir 10025 | A cardinal strictly domina... |
| cardsdomel 10026 | A cardinal strictly domina... |
| iscard 10027 | Two ways to express the pr... |
| iscard2 10028 | Two ways to express the pr... |
| carddom2 10029 | Two numerable sets have th... |
| harcard 10030 | The class of ordinal numbe... |
| cardprclem 10031 | Lemma for ~ cardprc . (Co... |
| cardprc 10032 | The class of all cardinal ... |
| carduni 10033 | The union of a set of card... |
| cardiun 10034 | The indexed union of a set... |
| cardennn 10035 | If ` A ` is equinumerous t... |
| cardsucinf 10036 | The cardinality of the suc... |
| cardsucnn 10037 | The cardinality of the suc... |
| cardom 10038 | The set of natural numbers... |
| carden2 10039 | Two numerable sets are equ... |
| cardsdom2 10040 | A numerable set is strictl... |
| domtri2 10041 | Trichotomy of dominance fo... |
| nnsdomel 10042 | Strict dominance and eleme... |
| cardval2 10043 | An alternate version of th... |
| isinffi 10044 | An infinite set contains s... |
| fidomtri 10045 | Trichotomy of dominance wi... |
| fidomtri2 10046 | Trichotomy of dominance wi... |
| harsdom 10047 | The Hartogs number of a we... |
| onsdom 10048 | Any well-orderable set is ... |
| harval2 10049 | An alternate expression fo... |
| harsucnn 10050 | The next cardinal after a ... |
| cardmin2 10051 | The smallest ordinal that ... |
| pm54.43lem 10052 | In Theorem *54.43 of [Whit... |
| pm54.43 10053 | Theorem *54.43 of [Whitehe... |
| enpr2 10054 | An unordered pair with dis... |
| pr2ne 10055 | If an unordered pair has t... |
| prdom2 10056 | An unordered pair has at m... |
| en2eqpr 10057 | Building a set with two el... |
| en2eleq 10058 | Express a set of pair card... |
| en2other2 10059 | Taking the other element t... |
| dif1card 10060 | The cardinality of a nonem... |
| leweon 10061 | Lexicographical order is a... |
| r0weon 10062 | A set-like well-ordering o... |
| infxpenlem 10063 | Lemma for ~ infxpen . (Co... |
| infxpen 10064 | Every infinite ordinal is ... |
| xpomen 10065 | The Cartesian product of o... |
| xpct 10066 | The cartesian product of t... |
| infxpidm2 10067 | Every infinite well-ordera... |
| infxpenc 10068 | A canonical version of ~ i... |
| infxpenc2lem1 10069 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10070 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10071 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10072 | Existence form of ~ infxpe... |
| iunmapdisj 10073 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10074 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10075 | Lemma for ~ fseqen . (Con... |
| fseqdom 10076 | One half of ~ fseqen . (C... |
| fseqen 10077 | A set that is equinumerous... |
| infpwfidom 10078 | The collection of finite s... |
| dfac8alem 10079 | Lemma for ~ dfac8a . If t... |
| dfac8a 10080 | Numeration theorem: every ... |
| dfac8b 10081 | The well-ordering theorem:... |
| dfac8clem 10082 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10083 | If the union of a set is w... |
| ac10ct 10084 | A proof of the well-orderi... |
| ween 10085 | A set is numerable iff it ... |
| ac5num 10086 | A version of ~ ac5b with t... |
| ondomen 10087 | If a set is dominated by a... |
| numdom 10088 | A set dominated by a numer... |
| ssnum 10089 | A subset of a numerable se... |
| onssnum 10090 | All subsets of the ordinal... |
| indcardi 10091 | Indirect strong induction ... |
| acnrcl 10092 | Reverse closure for the ch... |
| acneq 10093 | Equality theorem for the c... |
| isacn 10094 | The property of being a ch... |
| acni 10095 | The property of being a ch... |
| acni2 10096 | The property of being a ch... |
| acni3 10097 | The property of being a ch... |
| acnlem 10098 | Construct a mapping satisf... |
| numacn 10099 | A well-orderable set has c... |
| finacn 10100 | Every set has finite choic... |
| acndom 10101 | A set with long choice seq... |
| acnnum 10102 | A set ` X ` which has choi... |
| acnen 10103 | The class of choice sets o... |
| acndom2 10104 | A set smaller than one wit... |
| acnen2 10105 | The class of sets with cho... |
| fodomacn 10106 | A version of ~ fodom that ... |
| fodomnum 10107 | A version of ~ fodom that ... |
| fonum 10108 | A surjection maps numerabl... |
| numwdom 10109 | A surjection maps numerabl... |
| fodomfi2 10110 | Onto functions define domi... |
| wdomfil 10111 | Weak dominance agrees with... |
| infpwfien 10112 | Any infinite well-orderabl... |
| inffien 10113 | The set of finite intersec... |
| wdomnumr 10114 | Weak dominance agrees with... |
| alephfnon 10115 | The aleph function is a fu... |
| aleph0 10116 | The first infinite cardina... |
| alephlim 10117 | Value of the aleph functio... |
| alephsuc 10118 | Value of the aleph functio... |
| alephon 10119 | An aleph is an ordinal num... |
| alephcard 10120 | Every aleph is a cardinal ... |
| alephnbtwn 10121 | No cardinal can be sandwic... |
| alephnbtwn2 10122 | No set has equinumerosity ... |
| alephordilem1 10123 | Lemma for ~ alephordi . (... |
| alephordi 10124 | Strict ordering property o... |
| alephord 10125 | Ordering property of the a... |
| alephord2 10126 | Ordering property of the a... |
| alephord2i 10127 | Ordering property of the a... |
| alephord3 10128 | Ordering property of the a... |
| alephsucdom 10129 | A set dominated by an alep... |
| alephsuc2 10130 | An alternate representatio... |
| alephdom 10131 | Relationship between inclu... |
| alephgeom 10132 | Every aleph is greater tha... |
| alephislim 10133 | Every aleph is a limit ord... |
| aleph11 10134 | The aleph function is one-... |
| alephf1 10135 | The aleph function is a on... |
| alephsdom 10136 | If an ordinal is smaller t... |
| alephdom2 10137 | A dominated initial ordina... |
| alephle 10138 | The argument of the aleph ... |
| cardaleph 10139 | Given any transfinite card... |
| cardalephex 10140 | Every transfinite cardinal... |
| infenaleph 10141 | An infinite numerable set ... |
| isinfcard 10142 | Two ways to express the pr... |
| iscard3 10143 | Two ways to express the pr... |
| cardnum 10144 | Two ways to express the cl... |
| alephinit 10145 | An infinite initial ordina... |
| carduniima 10146 | The union of the image of ... |
| cardinfima 10147 | If a mapping to cardinals ... |
| alephiso 10148 | Aleph is an order isomorph... |
| alephprc 10149 | The class of all transfini... |
| alephsson 10150 | The class of transfinite c... |
| unialeph 10151 | The union of the class of ... |
| alephsmo 10152 | The aleph function is stri... |
| alephf1ALT 10153 | Alternate proof of ~ aleph... |
| alephfplem1 10154 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10155 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10156 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10157 | Lemma for ~ alephfp . (Co... |
| alephfp 10158 | The aleph function has a f... |
| alephfp2 10159 | The aleph function has at ... |
| alephval3 10160 | An alternate way to expres... |
| alephsucpw2 10161 | The power set of an aleph ... |
| mappwen 10162 | Power rule for cardinal ar... |
| finnisoeu 10163 | A finite totally ordered s... |
| iunfictbso 10164 | Countability of a countabl... |
| aceq1 10167 | Equivalence of two version... |
| aceq0 10168 | Equivalence of two version... |
| aceq2 10169 | Equivalence of two version... |
| aceq3lem 10170 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10171 | Equivalence of two version... |
| dfac4 10172 | Equivalence of two version... |
| dfac5lem1 10173 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10174 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10175 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10176 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10177 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10178 | Equivalence of two version... |
| dfac2a 10179 | Our Axiom of Choice (in th... |
| dfac2b 10180 | Axiom of Choice (first for... |
| dfac2 10181 | Axiom of Choice (first for... |
| dfac7 10182 | Equivalence of the Axiom o... |
| dfac0 10183 | Equivalence of two version... |
| dfac1 10184 | Equivalence of two version... |
| dfac8 10185 | A proof of the equivalency... |
| dfac9 10186 | Equivalence of the axiom o... |
| dfac10 10187 | Axiom of Choice equivalent... |
| dfac10c 10188 | Axiom of Choice equivalent... |
| dfac10b 10189 | Axiom of Choice equivalent... |
| acacni 10190 | A choice equivalent: every... |
| dfacacn 10191 | A choice equivalent: every... |
| dfac13 10192 | The axiom of choice holds ... |
| dfac12lem1 10193 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10194 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10195 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10196 | The axiom of choice holds ... |
| dfac12k 10197 | Equivalence of ~ dfac12 an... |
| dfac12a 10198 | The axiom of choice holds ... |
| dfac12 10199 | The axiom of choice holds ... |
| kmlem1 10200 | Lemma for 5-quantifier AC ... |
| kmlem2 10201 | Lemma for 5-quantifier AC ... |
| kmlem3 10202 | Lemma for 5-quantifier AC ... |
| kmlem4 10203 | Lemma for 5-quantifier AC ... |
| kmlem5 10204 | Lemma for 5-quantifier AC ... |
| kmlem6 10205 | Lemma for 5-quantifier AC ... |
| kmlem7 10206 | Lemma for 5-quantifier AC ... |
| kmlem8 10207 | Lemma for 5-quantifier AC ... |
| kmlem9 10208 | Lemma for 5-quantifier AC ... |
| kmlem10 10209 | Lemma for 5-quantifier AC ... |
| kmlem11 10210 | Lemma for 5-quantifier AC ... |
| kmlem12 10211 | Lemma for 5-quantifier AC ... |
| kmlem13 10212 | Lemma for 5-quantifier AC ... |
| kmlem14 10213 | Lemma for 5-quantifier AC ... |
| kmlem15 10214 | Lemma for 5-quantifier AC ... |
| kmlem16 10215 | Lemma for 5-quantifier AC ... |
| dfackm 10216 | Equivalence of the Axiom o... |
| undjudom 10217 | Cardinal addition dominate... |
| endjudisj 10218 | Equinumerosity of a disjoi... |
| djuen 10219 | Disjoint unions of equinum... |
| djuenun 10220 | Disjoint union is equinume... |
| dju1en 10221 | Cardinal addition with car... |
| dju1dif 10222 | Adding and subtracting one... |
| dju1p1e2 10223 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10224 | Alternate proof of ~ dju1p... |
| dju0en 10225 | Cardinal addition with car... |
| xp2dju 10226 | Two times a cardinal numbe... |
| djucomen 10227 | Commutative law for cardin... |
| djuassen 10228 | Associative law for cardin... |
| xpdjuen 10229 | Cardinal multiplication di... |
| mapdjuen 10230 | Sum of exponents law for c... |
| pwdjuen 10231 | Sum of exponents law for c... |
| djudom1 10232 | Ordering law for cardinal ... |
| djudom2 10233 | Ordering law for cardinal ... |
| djudoml 10234 | A set is dominated by its ... |
| djuxpdom 10235 | Cartesian product dominate... |
| djufi 10236 | The disjoint union of two ... |
| cdainflem 10237 | Any partition of omega int... |
| djuinf 10238 | A set is infinite iff the ... |
| infdju1 10239 | An infinite set is equinum... |
| pwdju1 10240 | The sum of a powerset with... |
| pwdjuidm 10241 | If the natural numbers inj... |
| djulepw 10242 | If ` A ` is idempotent und... |
| onadju 10243 | The cardinal and ordinal s... |
| cardadju 10244 | The cardinal sum is equinu... |
| djunum 10245 | The disjoint union of two ... |
| unnum 10246 | The union of two numerable... |
| nnadju 10247 | The cardinal and ordinal s... |
| nnadjuALT 10248 | Shorter proof of ~ nnadju ... |
| ficardadju 10249 | The disjoint union of fini... |
| ficardun 10250 | The cardinality of the uni... |
| ficardun2 10251 | The cardinality of the uni... |
| pwsdompw 10252 | Lemma for ~ domtriom . Th... |
| unctb 10253 | The union of two countable... |
| infdjuabs 10254 | Absorption law for additio... |
| infunabs 10255 | An infinite set is equinum... |
| infdju 10256 | The sum of two cardinal nu... |
| infdif 10257 | The cardinality of an infi... |
| infdif2 10258 | Cardinality ordering for a... |
| infxpdom 10259 | Dominance law for multipli... |
| infxpabs 10260 | Absorption law for multipl... |
| infunsdom1 10261 | The union of two sets that... |
| infunsdom 10262 | The union of two sets that... |
| infxp 10263 | Absorption law for multipl... |
| pwdjudom 10264 | A property of dominance ov... |
| infpss 10265 | Every infinite set has an ... |
| infmap2 10266 | An exponentiation law for ... |
| ackbij2lem1 10267 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10268 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10269 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10270 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10271 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10272 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10273 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10274 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10275 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10276 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10277 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10278 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10279 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10280 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10281 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10282 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10283 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10284 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10285 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10286 | The Ackermann bijection, p... |
| ackbij1b 10287 | The Ackermann bijection, p... |
| ackbij2lem2 10288 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10289 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10290 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10291 | The Ackermann bijection, p... |
| r1om 10292 | The set of hereditarily fi... |
| fictb 10293 | A set is countable iff its... |
| cflem 10294 | A lemma used to simplify c... |
| cfval 10295 | Value of the cofinality fu... |
| cff 10296 | Cofinality is a function o... |
| cfub 10297 | An upper bound on cofinali... |
| cflm 10298 | Value of the cofinality fu... |
| cf0 10299 | Value of the cofinality fu... |
| cardcf 10300 | Cofinality is a cardinal n... |
| cflecard 10301 | Cofinality is bounded by t... |
| cfle 10302 | Cofinality is bounded by i... |
| cfon 10303 | The cofinality of any set ... |
| cfonOLD 10304 | Obsolete version of ~ cfon... |
| cfeq0 10305 | Only the ordinal zero has ... |
| cfsuc 10306 | Value of the cofinality fu... |
| cff1 10307 | There is always a map from... |
| cfflb 10308 | If there is a cofinal map ... |
| cfval2 10309 | Another expression for the... |
| coflim 10310 | A simpler expression for t... |
| cflim3 10311 | Another expression for the... |
| cflim2 10312 | The cofinality function is... |
| cfom 10313 | Value of the cofinality fu... |
| cfss 10314 | There is a cofinal subset ... |
| cfslb 10315 | Any cofinal subset of ` A ... |
| cfslbn 10316 | Any subset of ` A ` smalle... |
| cfslb2n 10317 | Any small collection of sm... |
| cofsmo 10318 | Any cofinal map implies th... |
| cfsmolem 10319 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10320 | The map in ~ cff1 can be a... |
| cfcoflem 10321 | Lemma for ~ cfcof , showin... |
| coftr 10322 | If there is a cofinal map ... |
| cfcof 10323 | If there is a cofinal map ... |
| cfidm 10324 | The cofinality function is... |
| alephsing 10325 | The cofinality of a limit ... |
| sornom 10326 | The range of a single-step... |
| isfin1a 10341 | Definition of a Ia-finite ... |
| fin1ai 10342 | Property of a Ia-finite se... |
| isfin2 10343 | Definition of a II-finite ... |
| fin2i 10344 | Property of a II-finite se... |
| isfin3 10345 | Definition of a III-finite... |
| isfin4 10346 | Definition of a IV-finite ... |
| fin4i 10347 | Infer that a set is IV-inf... |
| isfin5 10348 | Definition of a V-finite s... |
| isfin6 10349 | Definition of a VI-finite ... |
| isfin7 10350 | Definition of a VII-finite... |
| sdom2en01 10351 | A set with less than two e... |
| infpssrlem1 10352 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10353 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10354 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10355 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10356 | Lemma for ~ infpssr . (Co... |
| infpssr 10357 | Dedekind infinity implies ... |
| fin4en1 10358 | Dedekind finite is a cardi... |
| ssfin4 10359 | Dedekind finite sets have ... |
| domfin4 10360 | A set dominated by a Dedek... |
| ominf4 10361 | ` _om ` is Dedekind infini... |
| infpssALT 10362 | Alternate proof of ~ infps... |
| isfin4-2 10363 | Alternate definition of IV... |
| isfin4p1 10364 | Alternate definition of IV... |
| fin23lem7 10365 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10366 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10367 | A II-finite set contains m... |
| isfin2-2 10368 | ` Fin2 ` expressed in term... |
| ssfin2 10369 | A subset of a II-finite se... |
| enfin2i 10370 | II-finiteness is a cardina... |
| fin23lem24 10371 | Lemma for ~ fin23 . In a ... |
| fincssdom 10372 | In a chain of finite sets,... |
| fin23lem25 10373 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10374 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10375 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10376 | Lemma for ~ fin23 but coul... |
| fin23lem27 10377 | The mapping constructed in... |
| isfin3ds 10378 | Property of a III-finite s... |
| ssfin3ds 10379 | A subset of a III-finite s... |
| fin23lem12 10380 | The beginning of the proof... |
| fin23lem13 10381 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10382 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10383 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10384 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10385 | Lemma for ~ fin23 . The f... |
| fin23lem20 10386 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10387 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10388 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10389 | Lemma for ~ fin23 . The r... |
| fin23lem29 10390 | Lemma for ~ fin23 . The r... |
| fin23lem30 10391 | Lemma for ~ fin23 . The r... |
| fin23lem31 10392 | Lemma for ~ fin23 . The r... |
| fin23lem32 10393 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10394 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10395 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10396 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10397 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10398 | Lemma for ~ fin23 . The c... |
| fin23lem39 10399 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10400 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10401 | Lemma for ~ fin23 . A set... |
| isf32lem1 10402 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10403 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10404 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10405 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10406 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10407 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10408 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10409 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10410 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10411 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10412 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10413 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10414 | One half of ~ isfin3-2 . ... |
| isf33lem 10415 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10416 | Weakly Dedekind-infinite s... |
| isfin3-3 10417 | Weakly Dedekind-infinite s... |
| fin33i 10418 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10419 | Lemma for ~ compsscnv . (... |
| compsscnv 10420 | Complementation on a power... |
| isf34lem1 10421 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10422 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10423 | Complementation is an anti... |
| isf34lem3 10424 | Lemma for ~ isfin3-4 . (C... |
| compss 10425 | Express image under of the... |
| isf34lem4 10426 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10427 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10428 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10429 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10430 | Inference from ~ isfin3-4 ... |
| isfin3-4 10431 | Weakly Dedekind-infinite s... |
| fin11a 10432 | Every I-finite set is Ia-f... |
| enfin1ai 10433 | Ia-finiteness is a cardina... |
| isfin1-2 10434 | A set is finite in the usu... |
| isfin1-3 10435 | A set is I-finite iff ever... |
| isfin1-4 10436 | A set is I-finite iff ever... |
| dffin1-5 10437 | Compact quantifier-free ve... |
| fin23 10438 | Every II-finite set (every... |
| fin34 10439 | Every III-finite set is IV... |
| isfin5-2 10440 | Alternate definition of V-... |
| fin45 10441 | Every IV-finite set is V-f... |
| fin56 10442 | Every V-finite set is VI-f... |
| fin17 10443 | Every I-finite set is VII-... |
| fin67 10444 | Every VI-finite set is VII... |
| isfin7-2 10445 | A set is VII-finite iff it... |
| fin71num 10446 | A well-orderable set is VI... |
| dffin7-2 10447 | Class form of ~ isfin7-2 .... |
| dfacfin7 10448 | Axiom of Choice equivalent... |
| fin1a2lem1 10449 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10450 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10451 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10452 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10453 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10454 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10455 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10456 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10457 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10458 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10459 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10460 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10461 | Lemma for ~ fin1a2 . (Con... |
| fin12 10462 | Weak theorem which skips I... |
| fin1a2s 10463 | An II-infinite set can hav... |
| fin1a2 10464 | Every Ia-finite set is II-... |
| itunifval 10465 | Function value of iterated... |
| itunifn 10466 | Functionality of the itera... |
| ituni0 10467 | A zero-fold iterated union... |
| itunisuc 10468 | Successor iterated union. ... |
| itunitc1 10469 | Each union iterate is a me... |
| itunitc 10470 | The union of all union ite... |
| ituniiun 10471 | Unwrap an iterated union f... |
| hsmexlem7 10472 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10473 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10474 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10475 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10476 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10477 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10478 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10479 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10480 | Lemma for ~ hsmex . (Cont... |
| hsmex 10481 | The collection of heredita... |
| hsmex2 10482 | The set of hereditary size... |
| hsmex3 10483 | The set of hereditary size... |
| axcc2lem 10485 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10486 | A possibly more useful ver... |
| axcc3 10487 | A possibly more useful ver... |
| axcc4 10488 | A version of ~ axcc3 that ... |
| acncc 10489 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10490 | Relax the constraint on ~ ... |
| domtriomlem 10491 | Lemma for ~ domtriom . (C... |
| domtriom 10492 | Trichotomy of equinumerosi... |
| fin41 10493 | Under countable choice, th... |
| dominf 10494 | A nonempty set that is a s... |
| dcomex 10496 | The Axiom of Dependent Cho... |
| axdc2lem 10497 | Lemma for ~ axdc2 . We co... |
| axdc2 10498 | An apparent strengthening ... |
| axdc3lem 10499 | The class ` S ` of finite ... |
| axdc3lem2 10500 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10501 | Simple substitution lemma ... |
| axdc3lem4 10502 | Lemma for ~ axdc3 . We ha... |
| axdc3 10503 | Dependent Choice. Axiom D... |
| axdc4lem 10504 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10505 | A more general version of ... |
| axcclem 10506 | Lemma for ~ axcc . (Contr... |
| axcc 10507 | Although CC can be proven ... |
| zfac 10509 | Axiom of Choice expressed ... |
| ac2 10510 | Axiom of Choice equivalent... |
| ac3 10511 | Axiom of Choice using abbr... |
| axac3 10513 | This theorem asserts that ... |
| ackm 10514 | A remarkable equivalent to... |
| axac2 10515 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10516 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10517 | Apply a choice equivalent.... |
| cardeqv 10518 | All sets are well-orderabl... |
| numth3 10519 | All sets are well-orderabl... |
| numth2 10520 | Numeration theorem: any se... |
| numth 10521 | Numeration theorem: every ... |
| ac7 10522 | An Axiom of Choice equival... |
| ac7g 10523 | An Axiom of Choice equival... |
| ac4 10524 | Equivalent of Axiom of Cho... |
| ac4c 10525 | Equivalent of Axiom of Cho... |
| ac5 10526 | An Axiom of Choice equival... |
| ac5b 10527 | Equivalent of Axiom of Cho... |
| ac6num 10528 | A version of ~ ac6 which t... |
| ac6 10529 | Equivalent of Axiom of Cho... |
| ac6c4 10530 | Equivalent of Axiom of Cho... |
| ac6c5 10531 | Equivalent of Axiom of Cho... |
| ac9 10532 | An Axiom of Choice equival... |
| ac6s 10533 | Equivalent of Axiom of Cho... |
| ac6n 10534 | Equivalent of Axiom of Cho... |
| ac6s2 10535 | Generalization of the Axio... |
| ac6s3 10536 | Generalization of the Axio... |
| ac6sg 10537 | ~ ac6s with sethood as ant... |
| ac6sf 10538 | Version of ~ ac6 with boun... |
| ac6s4 10539 | Generalization of the Axio... |
| ac6s5 10540 | Generalization of the Axio... |
| ac8 10541 | An Axiom of Choice equival... |
| ac9s 10542 | An Axiom of Choice equival... |
| numthcor 10543 | Any set is strictly domina... |
| weth 10544 | Well-ordering theorem: any... |
| zorn2lem1 10545 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10546 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10547 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10548 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10549 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10550 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10551 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10552 | Zorn's Lemma of [Monk1] p.... |
| zorng 10553 | Zorn's Lemma. If the unio... |
| zornn0g 10554 | Variant of Zorn's lemma ~ ... |
| zorn2 10555 | Zorn's Lemma of [Monk1] p.... |
| zorn 10556 | Zorn's Lemma. If the unio... |
| zornn0 10557 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10558 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10559 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10560 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10561 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10562 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10563 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10564 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10565 | The Teichmüller-Tukey... |
| ttukeyg 10566 | The Teichmüller-Tukey... |
| ttukey 10567 | The Teichmüller-Tukey... |
| axdclem 10568 | Lemma for ~ axdc . (Contr... |
| axdclem2 10569 | Lemma for ~ axdc . Using ... |
| axdc 10570 | This theorem derives ~ ax-... |
| fodomg 10571 | An onto function implies d... |
| fodom 10572 | An onto function implies d... |
| dmct 10573 | The domain of a countable ... |
| dmctOLD 10574 | Obsolete version of ~ dmct... |
| rnct 10575 | The range of a countable s... |
| fodomb 10576 | Equivalence of an onto map... |
| wdomac 10577 | When assuming AC, weak and... |
| brdom3 10578 | Equivalence to a dominance... |
| brdom5 10579 | An equivalence to a domina... |
| brdom4 10580 | An equivalence to a domina... |
| brdom7disj 10581 | An equivalence to a domina... |
| brdom6disj 10582 | An equivalence to a domina... |
| fin71ac 10583 | Once we allow AC, the "str... |
| imadomg 10584 | An image of a function und... |
| imadomnum 10585 | A version of ~ imadomg tha... |
| fimact 10586 | The image by a function of... |
| fimactOLD 10587 | Obsolete version of ~ fima... |
| fnrndomnum 10588 | A version of ~ fnrndomg th... |
| fnrndomg 10589 | The range of a function is... |
| fnrndomgOLD 10590 | Obsolete version of ~ fnrn... |
| fnct 10591 | If the domain of a functio... |
| fnctOLD 10592 | Obsolete version of ~ fnct... |
| mptct 10593 | A countable mapping set is... |
| iunfo 10594 | Existence of an onto funct... |
| iundom2g 10595 | An upper bound for the car... |
| iundomg 10596 | An upper bound for the car... |
| iundom 10597 | An upper bound for the car... |
| unidom 10598 | An upper bound for the car... |
| uniimadom 10599 | An upper bound for the car... |
| uniimadomf 10600 | An upper bound for the car... |
| cardval 10601 | The value of the cardinal ... |
| cardid 10602 | Any set is equinumerous to... |
| cardidg 10603 | Any set is equinumerous to... |
| cardidd 10604 | Any set is equinumerous to... |
| cardf 10605 | The cardinality function i... |
| carden 10606 | Two sets are equinumerous ... |
| cardeq0 10607 | Only the empty set has car... |
| unsnen 10608 | Equinumerosity of a set wi... |
| carddom 10609 | Two sets have the dominanc... |
| cardsdom 10610 | Two sets have the strict d... |
| domtri 10611 | Trichotomy law for dominan... |
| entric 10612 | Trichotomy of equinumerosi... |
| entri2 10613 | Trichotomy of dominance an... |
| entri3 10614 | Trichotomy of dominance. ... |
| sdomsdomcard 10615 | A set strictly dominates i... |
| canth3 10616 | Cantor's theorem in terms ... |
| infxpidm 10617 | Every infinite class is eq... |
| ondomon 10618 | The class of ordinals domi... |
| cardmin 10619 | The smallest ordinal that ... |
| ficard 10620 | A set is finite iff its ca... |
| infinfg 10621 | Equivalence between two in... |
| infinf 10622 | Equivalence between two in... |
| unirnfdomd 10623 | The union of the range of ... |
| konigthlem 10624 | Lemma for ~ konigth . (Co... |
| konigth 10625 | Konig's Theorem. If ` m (... |
| alephsucpw 10626 | The power set of an aleph ... |
| aleph1 10627 | The set exponentiation of ... |
| alephval2 10628 | An alternate way to expres... |
| dominfac 10629 | A nonempty set that is a s... |
| iunctb 10630 | The countable union of cou... |
| unictb 10631 | The countable union of cou... |
| infmap 10632 | An exponentiation law for ... |
| alephadd 10633 | The sum of two alephs is t... |
| alephmul 10634 | The product of two alephs ... |
| alephexp1 10635 | An exponentiation law for ... |
| alephsuc3 10636 | An alternate representatio... |
| alephexp2 10637 | An expression equinumerous... |
| alephreg 10638 | A successor aleph is regul... |
| pwcfsdom 10639 | A corollary of Konig's The... |
| cfpwsdom 10640 | A corollary of Konig's The... |
| alephom 10641 | From ~ canth2 , we know th... |
| smobeth 10642 | The beth function is stric... |
| nd1 10643 | A lemma for proving condit... |
| nd2 10644 | A lemma for proving condit... |
| nd3 10645 | A lemma for proving condit... |
| nd4 10646 | A lemma for proving condit... |
| axextnd 10647 | A version of the Axiom of ... |
| axrepndlem1 10648 | Lemma for the Axiom of Rep... |
| axrepndlem2 10649 | Lemma for the Axiom of Rep... |
| axrepnd 10650 | A version of the Axiom of ... |
| axunndlem1 10651 | Lemma for the Axiom of Uni... |
| axunnd 10652 | A version of the Axiom of ... |
| axpowndlem1 10653 | Lemma for the Axiom of Pow... |
| axpowndlem2 10654 | Lemma for the Axiom of Pow... |
| axpowndlem3 10655 | Lemma for the Axiom of Pow... |
| axpowndlem4 10656 | Lemma for the Axiom of Pow... |
| axpownd 10657 | A version of the Axiom of ... |
| axregndlem1 10658 | Lemma for the Axiom of Reg... |
| axregndlem2 10659 | Lemma for the Axiom of Reg... |
| axregnd 10660 | A version of the Axiom of ... |
| axinfndlem1 10661 | Lemma for the Axiom of Inf... |
| axinfnd 10662 | A version of the Axiom of ... |
| axacndlem1 10663 | Lemma for the Axiom of Cho... |
| axacndlem2 10664 | Lemma for the Axiom of Cho... |
| axacndlem3 10665 | Lemma for the Axiom of Cho... |
| axacndlem4 10666 | Lemma for the Axiom of Cho... |
| axacndlem5 10667 | Lemma for the Axiom of Cho... |
| axacnd 10668 | A version of the Axiom of ... |
| zfcndext 10669 | Axiom of Extensionality ~ ... |
| zfcndrep 10670 | Axiom of Replacement ~ ax-... |
| zfcndun 10671 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10672 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10673 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10674 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10675 | Axiom of Choice ~ ax-ac , ... |
| elgch 10678 | Elementhood in the collect... |
| fingch 10679 | A finite set is a GCH-set.... |
| gchi 10680 | The only GCH-sets which ha... |
| gchen1 10681 | If ` A <_ B < ~P A ` , and... |
| gchen2 10682 | If ` A < B <_ ~P A ` , and... |
| gchor 10683 | If ` A <_ B <_ ~P A ` , an... |
| engch 10684 | The property of being a GC... |
| gchdomtri 10685 | Under certain conditions, ... |
| fpwwe2cbv 10686 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10687 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10688 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10689 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10690 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10691 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10692 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10693 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10694 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10695 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10696 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10697 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10698 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10699 | Given any function ` F ` f... |
| fpwwecbv 10700 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10701 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10702 | Given any function ` F ` f... |
| canth4 10703 | An "effective" form of Can... |
| canthnumlem 10704 | Lemma for ~ canthnum . (C... |
| canthnum 10705 | The set of well-orderable ... |
| canthwelem 10706 | Lemma for ~ canthwe . (Co... |
| canthwe 10707 | The set of well-orders of ... |
| canthp1lem1 10708 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10709 | Lemma for ~ canthp1 . (Co... |
| canthp1 10710 | A slightly stronger form o... |
| finngch 10711 | The exclusion of finite se... |
| gchdju1 10712 | An infinite GCH-set is ide... |
| gchinf 10713 | An infinite GCH-set is Ded... |
| pwfseqlem1 10714 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10715 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10716 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10717 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10718 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10719 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10720 | The powerset of a Dedekind... |
| pwxpndom2 10721 | The powerset of a Dedekind... |
| pwxpndom 10722 | The powerset of a Dedekind... |
| pwdjundom 10723 | The powerset of a Dedekind... |
| gchdjuidm 10724 | An infinite GCH-set is ide... |
| gchxpidm 10725 | An infinite GCH-set is ide... |
| gchpwdom 10726 | A relationship between dom... |
| gchaleph 10727 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10728 | If ` ( aleph `` A ) ` and ... |
| hargch 10729 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10730 | If ` ( aleph `` suc A ) ` ... |
| gch2 10731 | It is sufficient to requir... |
| gch3 10732 | An equivalent formulation ... |
| gch-kn 10733 | The equivalence of two ver... |
| gchaclem 10734 | Lemma for ~ gchac (obsolet... |
| gchhar 10735 | A "local" form of ~ gchac ... |
| gchacg 10736 | A "local" form of ~ gchac ... |
| gchac 10737 | The Generalized Continuum ... |
| elwina 10742 | Conditions of weak inacces... |
| elina 10743 | Conditions of strong inacc... |
| winaon 10744 | A weakly inaccessible card... |
| inawinalem 10745 | Lemma for ~ inawina . (Co... |
| inawina 10746 | Every strongly inaccessibl... |
| omina 10747 | ` _om ` is a strongly inac... |
| winacard 10748 | A weakly inaccessible card... |
| winainflem 10749 | A weakly inaccessible card... |
| winainf 10750 | A weakly inaccessible card... |
| winalim 10751 | A weakly inaccessible card... |
| winalim2 10752 | A nontrivial weakly inacce... |
| winafp 10753 | A nontrivial weakly inacce... |
| winafpi 10754 | This theorem, which states... |
| gchina 10755 | Assuming the GCH, weakly a... |
| iswun 10760 | Properties of a weak unive... |
| wuntr 10761 | A weak universe is transit... |
| wununi 10762 | A weak universe is closed ... |
| wunpw 10763 | A weak universe is closed ... |
| wunelss 10764 | The elements of a weak uni... |
| wunpr 10765 | A weak universe is closed ... |
| wunun 10766 | A weak universe is closed ... |
| wuntp 10767 | A weak universe is closed ... |
| wunss 10768 | A weak universe is closed ... |
| wunin 10769 | A weak universe is closed ... |
| wundif 10770 | A weak universe is closed ... |
| wunint 10771 | A weak universe is closed ... |
| wunsn 10772 | A weak universe is closed ... |
| wunsuc 10773 | A weak universe is closed ... |
| wun0 10774 | A weak universe contains t... |
| wunr1om 10775 | A weak universe is infinit... |
| wunom 10776 | A weak universe contains a... |
| wunfi 10777 | A weak universe contains a... |
| wunop 10778 | A weak universe is closed ... |
| wunot 10779 | A weak universe is closed ... |
| wunxp 10780 | A weak universe is closed ... |
| wunpm 10781 | A weak universe is closed ... |
| wunmap 10782 | A weak universe is closed ... |
| wunf 10783 | A weak universe is closed ... |
| wundm 10784 | A weak universe is closed ... |
| wunrn 10785 | A weak universe is closed ... |
| wuncnv 10786 | A weak universe is closed ... |
| wunres 10787 | A weak universe is closed ... |
| wunfv 10788 | A weak universe is closed ... |
| wunco 10789 | A weak universe is closed ... |
| wuntpos 10790 | A weak universe is closed ... |
| intwun 10791 | The intersection of a coll... |
| r1limwun 10792 | Each limit stage in the cu... |
| r1wunlim 10793 | The weak universes in the ... |
| wunex2 10794 | Construct a weak universe ... |
| wunex 10795 | Construct a weak universe ... |
| uniwun 10796 | Every set is contained in ... |
| wunex3 10797 | Construct a weak universe ... |
| wuncval 10798 | Value of the weak universe... |
| wuncid 10799 | The weak universe closure ... |
| wunccl 10800 | The weak universe closure ... |
| wuncss 10801 | The weak universe closure ... |
| wuncidm 10802 | The weak universe closure ... |
| wuncval2 10803 | Our earlier expression for... |
| eltskg 10806 | Properties of a Tarski cla... |
| eltsk2g 10807 | Properties of a Tarski cla... |
| tskpwss 10808 | First axiom of a Tarski cl... |
| tskpw 10809 | Second axiom of a Tarski c... |
| tsken 10810 | Third axiom of a Tarski cl... |
| 0tsk 10811 | The empty set is a (transi... |
| tsksdom 10812 | An element of a Tarski cla... |
| tskssel 10813 | A part of a Tarski class s... |
| tskss 10814 | The subsets of an element ... |
| tskin 10815 | The intersection of two el... |
| tsksn 10816 | A singleton of an element ... |
| tsktrss 10817 | A transitive element of a ... |
| tsksuc 10818 | If an element of a Tarski ... |
| tsk0 10819 | A nonempty Tarski class co... |
| tsk1 10820 | One is an element of a non... |
| tsk2 10821 | Two is an element of a non... |
| 2domtsk 10822 | If a Tarski class is not e... |
| tskr1om 10823 | A nonempty Tarski class is... |
| tskr1om2 10824 | A nonempty Tarski class co... |
| tskinf 10825 | A nonempty Tarski class is... |
| tskpr 10826 | If ` A ` and ` B ` are mem... |
| tskop 10827 | If ` A ` and ` B ` are mem... |
| tskxpss 10828 | A Cartesian product of two... |
| tskwe2 10829 | A Tarski class is well-ord... |
| inttsk 10830 | The intersection of a coll... |
| inar1 10831 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10832 | Alternate proof of ~ r1om ... |
| rankcf 10833 | Any set must be at least a... |
| inatsk 10834 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10835 | The set of hereditarily fi... |
| tskord 10836 | A Tarski class contains al... |
| tskcard 10837 | An even more direct relati... |
| r1tskina 10838 | There is a direct relation... |
| tskuni 10839 | The union of an element of... |
| tskwun 10840 | A nonempty transitive Tars... |
| tskint 10841 | The intersection of an ele... |
| tskun 10842 | The union of two elements ... |
| tskxp 10843 | The Cartesian product of t... |
| tskmap 10844 | Set exponentiation is an e... |
| tskurn 10845 | A transitive Tarski class ... |
| elgrug 10848 | Properties of a Grothendie... |
| grutr 10849 | A Grothendieck universe is... |
| gruelss 10850 | A Grothendieck universe is... |
| grupw 10851 | A Grothendieck universe co... |
| gruss 10852 | Any subset of an element o... |
| grupr 10853 | A Grothendieck universe co... |
| gruurn 10854 | A Grothendieck universe co... |
| gruiun 10855 | If ` B ( x ) ` is a family... |
| gruuni 10856 | A Grothendieck universe co... |
| grurn 10857 | A Grothendieck universe co... |
| gruima 10858 | A Grothendieck universe co... |
| gruel 10859 | Any element of an element ... |
| grusn 10860 | A Grothendieck universe co... |
| gruop 10861 | A Grothendieck universe co... |
| gruun 10862 | A Grothendieck universe co... |
| gruxp 10863 | A Grothendieck universe co... |
| grumap 10864 | A Grothendieck universe co... |
| gruixp 10865 | A Grothendieck universe co... |
| gruiin 10866 | A Grothendieck universe co... |
| gruf 10867 | A Grothendieck universe co... |
| gruen 10868 | A Grothendieck universe co... |
| gruwun 10869 | A nonempty Grothendieck un... |
| intgru 10870 | The intersection of a fami... |
| ingru 10871 | The intersection of a univ... |
| wfgru 10872 | The wellfounded part of a ... |
| grudomon 10873 | Each ordinal that is compa... |
| gruina 10874 | If a Grothendieck universe... |
| grur1a 10875 | A characterization of Grot... |
| grur1 10876 | A characterization of Grot... |
| grutsk1 10877 | Grothendieck universes are... |
| grutsk 10878 | Grothendieck universes are... |
| axgroth5 10880 | The Tarski-Grothendieck ax... |
| axgroth2 10881 | Alternate version of the T... |
| grothpw 10882 | Derive the Axiom of Power ... |
| grothpwex 10883 | Derive the Axiom of Power ... |
| axgroth6 10884 | The Tarski-Grothendieck ax... |
| grothomex 10885 | The Tarski-Grothendieck Ax... |
| grothac 10886 | The Tarski-Grothendieck Ax... |
| axgroth3 10887 | Alternate version of the T... |
| axgroth4 10888 | Alternate version of the T... |
| grothprimlem 10889 | Lemma for ~ grothprim . E... |
| grothprim 10890 | The Tarski-Grothendieck Ax... |
| grothtsk 10891 | The Tarski-Grothendieck Ax... |
| inaprc 10892 | An equivalent to the Tarsk... |
| tskmval 10895 | Value of our tarski map. ... |
| tskmid 10896 | The set ` A ` is an elemen... |
| tskmcl 10897 | A Tarski class that contai... |
| sstskm 10898 | Being a part of ` ( tarski... |
| eltskm 10899 | Belonging to ` ( tarskiMap... |
| elni 10932 | Membership in the class of... |
| elni2 10933 | Membership in the class of... |
| pinn 10934 | A positive integer is a na... |
| pion 10935 | A positive integer is an o... |
| piord 10936 | A positive integer is ordi... |
| niex 10937 | The class of positive inte... |
| 0npi 10938 | The empty set is not a pos... |
| 1pi 10939 | Ordinal 'one' is a positiv... |
| addpiord 10940 | Positive integer addition ... |
| mulpiord 10941 | Positive integer multiplic... |
| mulidpi 10942 | 1 is an identity element f... |
| ltpiord 10943 | Positive integer 'less tha... |
| ltsopi 10944 | Positive integer 'less tha... |
| ltrelpi 10945 | Positive integer 'less tha... |
| dmaddpi 10946 | Domain of addition on posi... |
| dmmulpi 10947 | Domain of multiplication o... |
| addclpi 10948 | Closure of addition of pos... |
| mulclpi 10949 | Closure of multiplication ... |
| addcompi 10950 | Addition of positive integ... |
| addasspi 10951 | Addition of positive integ... |
| mulcompi 10952 | Multiplication of positive... |
| mulasspi 10953 | Multiplication of positive... |
| distrpi 10954 | Multiplication of positive... |
| addcanpi 10955 | Addition cancellation law ... |
| mulcanpi 10956 | Multiplication cancellatio... |
| addnidpi 10957 | There is no identity eleme... |
| ltexpi 10958 | Ordering on positive integ... |
| ltapi 10959 | Ordering property of addit... |
| ltmpi 10960 | Ordering property of multi... |
| 1lt2pi 10961 | One is less than two (one ... |
| nlt1pi 10962 | No positive integer is les... |
| indpi 10963 | Principle of Finite Induct... |
| enqbreq 10975 | Equivalence relation for p... |
| enqbreq2 10976 | Equivalence relation for p... |
| enqer 10977 | The equivalence relation f... |
| enqex 10978 | The equivalence relation f... |
| nqex 10979 | The class of positive frac... |
| 0nnq 10980 | The empty set is not a pos... |
| elpqn 10981 | Each positive fraction is ... |
| ltrelnq 10982 | Positive fraction 'less th... |
| pinq 10983 | The representatives of pos... |
| 1nq 10984 | The positive fraction 'one... |
| nqereu 10985 | There is a unique element ... |
| nqerf 10986 | Corollary of ~ nqereu : th... |
| nqercl 10987 | Corollary of ~ nqereu : cl... |
| nqerrel 10988 | Any member of ` ( N. X. N.... |
| nqerid 10989 | Corollary of ~ nqereu : th... |
| enqeq 10990 | Corollary of ~ nqereu : if... |
| nqereq 10991 | The function ` /Q ` acts a... |
| addpipq2 10992 | Addition of positive fract... |
| addpipq 10993 | Addition of positive fract... |
| addpqnq 10994 | Addition of positive fract... |
| mulpipq2 10995 | Multiplication of positive... |
| mulpipq 10996 | Multiplication of positive... |
| mulpqnq 10997 | Multiplication of positive... |
| ordpipq 10998 | Ordering of positive fract... |
| ordpinq 10999 | Ordering of positive fract... |
| addpqf 11000 | Closure of addition on pos... |
| addclnq 11001 | Closure of addition on pos... |
| mulpqf 11002 | Closure of multiplication ... |
| mulclnq 11003 | Closure of multiplication ... |
| addnqf 11004 | Domain of addition on posi... |
| mulnqf 11005 | Domain of multiplication o... |
| addcompq 11006 | Addition of positive fract... |
| addcomnq 11007 | Addition of positive fract... |
| mulcompq 11008 | Multiplication of positive... |
| mulcomnq 11009 | Multiplication of positive... |
| adderpqlem 11010 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 11011 | Lemma for ~ mulerpq . (Co... |
| adderpq 11012 | Addition is compatible wit... |
| mulerpq 11013 | Multiplication is compatib... |
| addassnq 11014 | Addition of positive fract... |
| mulassnq 11015 | Multiplication of positive... |
| mulcanenq 11016 | Lemma for distributive law... |
| distrnq 11017 | Multiplication of positive... |
| 1nqenq 11018 | The equivalence class of r... |
| mulidnq 11019 | Multiplication identity el... |
| recmulnq 11020 | Relationship between recip... |
| recidnq 11021 | A positive fraction times ... |
| recclnq 11022 | Closure law for positive f... |
| recrecnq 11023 | Reciprocal of reciprocal o... |
| dmrecnq 11024 | Domain of reciprocal on po... |
| ltsonq 11025 | 'Less than' is a strict or... |
| lterpq 11026 | Compatibility of ordering ... |
| ltanq 11027 | Ordering property of addit... |
| ltmnq 11028 | Ordering property of multi... |
| 1lt2nq 11029 | One is less than two (one ... |
| ltaddnq 11030 | The sum of two fractions i... |
| ltexnq 11031 | Ordering on positive fract... |
| halfnq 11032 | One-half of any positive f... |
| nsmallnq 11033 | The is no smallest positiv... |
| ltbtwnnq 11034 | There exists a number betw... |
| ltrnq 11035 | Ordering property of recip... |
| archnq 11036 | For any fraction, there is... |
| npex 11042 | The class of positive real... |
| elnp 11043 | Membership in positive rea... |
| elnpi 11044 | Membership in positive rea... |
| prn0 11045 | A positive real is not emp... |
| prpssnq 11046 | A positive real is a subse... |
| elprnq 11047 | A positive real is a set o... |
| 0npr 11048 | The empty set is not a pos... |
| prcdnq 11049 | A positive real is closed ... |
| prub 11050 | A positive fraction not in... |
| prnmax 11051 | A positive real has no lar... |
| npomex 11052 | A simplifying observation,... |
| prnmadd 11053 | A positive real has no lar... |
| ltrelpr 11054 | Positive real 'less than' ... |
| genpv 11055 | Value of general operation... |
| genpelv 11056 | Membership in value of gen... |
| genpprecl 11057 | Pre-closure law for genera... |
| genpdm 11058 | Domain of general operatio... |
| genpn0 11059 | The result of an operation... |
| genpss 11060 | The result of an operation... |
| genpnnp 11061 | The result of an operation... |
| genpcd 11062 | Downward closure of an ope... |
| genpnmax 11063 | An operation on positive r... |
| genpcl 11064 | Closure of an operation on... |
| genpass 11065 | Associativity of an operat... |
| plpv 11066 | Value of addition on posit... |
| mpv 11067 | Value of multiplication on... |
| dmplp 11068 | Domain of addition on posi... |
| dmmp 11069 | Domain of multiplication o... |
| nqpr 11070 | The canonical embedding of... |
| 1pr 11071 | The positive real number '... |
| addclprlem1 11072 | Lemma to prove downward cl... |
| addclprlem2 11073 | Lemma to prove downward cl... |
| addclpr 11074 | Closure of addition on pos... |
| mulclprlem 11075 | Lemma to prove downward cl... |
| mulclpr 11076 | Closure of multiplication ... |
| addcompr 11077 | Addition of positive reals... |
| addasspr 11078 | Addition of positive reals... |
| mulcompr 11079 | Multiplication of positive... |
| mulasspr 11080 | Multiplication of positive... |
| distrlem1pr 11081 | Lemma for distributive law... |
| distrlem4pr 11082 | Lemma for distributive law... |
| distrlem5pr 11083 | Lemma for distributive law... |
| distrpr 11084 | Multiplication of positive... |
| 1idpr 11085 | 1 is an identity element f... |
| ltprord 11086 | Positive real 'less than' ... |
| psslinpr 11087 | Proper subset is a linear ... |
| ltsopr 11088 | Positive real 'less than' ... |
| prlem934 11089 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11090 | The sum of two positive re... |
| ltaddpr2 11091 | The sum of two positive re... |
| ltexprlem1 11092 | Lemma for Proposition 9-3.... |
| ltexprlem2 11093 | Lemma for Proposition 9-3.... |
| ltexprlem3 11094 | Lemma for Proposition 9-3.... |
| ltexprlem4 11095 | Lemma for Proposition 9-3.... |
| ltexprlem5 11096 | Lemma for Proposition 9-3.... |
| ltexprlem6 11097 | Lemma for Proposition 9-3.... |
| ltexprlem7 11098 | Lemma for Proposition 9-3.... |
| ltexpri 11099 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11100 | Lemma for Proposition 9-3.... |
| ltapr 11101 | Ordering property of addit... |
| addcanpr 11102 | Addition cancellation law ... |
| prlem936 11103 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11104 | Lemma for Proposition 9-3.... |
| reclem3pr 11105 | Lemma for Proposition 9-3.... |
| reclem4pr 11106 | Lemma for Proposition 9-3.... |
| recexpr 11107 | The reciprocal of a positi... |
| suplem1pr 11108 | The union of a nonempty, b... |
| suplem2pr 11109 | The union of a set of posi... |
| supexpr 11110 | The union of a nonempty, b... |
| enrer 11119 | The equivalence relation f... |
| nrex1 11120 | The class of signed reals ... |
| enrbreq 11121 | Equivalence relation for s... |
| enreceq 11122 | Equivalence class equality... |
| enrex 11123 | The equivalence relation f... |
| ltrelsr 11124 | Signed real 'less than' is... |
| addcmpblnr 11125 | Lemma showing compatibilit... |
| mulcmpblnrlem 11126 | Lemma used in lemma showin... |
| mulcmpblnr 11127 | Lemma showing compatibilit... |
| prsrlem1 11128 | Decomposing signed reals i... |
| addsrmo 11129 | There is at most one resul... |
| mulsrmo 11130 | There is at most one resul... |
| addsrpr 11131 | Addition of signed reals i... |
| mulsrpr 11132 | Multiplication of signed r... |
| ltsrpr 11133 | Ordering of signed reals i... |
| gt0srpr 11134 | Greater than zero in terms... |
| 0nsr 11135 | The empty set is not a sig... |
| 0r 11136 | The constant ` 0R ` is a s... |
| 1sr 11137 | The constant ` 1R ` is a s... |
| m1r 11138 | The constant ` -1R ` is a ... |
| addclsr 11139 | Closure of addition on sig... |
| mulclsr 11140 | Closure of multiplication ... |
| dmaddsr 11141 | Domain of addition on sign... |
| dmmulsr 11142 | Domain of multiplication o... |
| addcomsr 11143 | Addition of signed reals i... |
| addasssr 11144 | Addition of signed reals i... |
| mulcomsr 11145 | Multiplication of signed r... |
| mulasssr 11146 | Multiplication of signed r... |
| distrsr 11147 | Multiplication of signed r... |
| m1p1sr 11148 | Minus one plus one is zero... |
| m1m1sr 11149 | Minus one times minus one ... |
| ltsosr 11150 | Signed real 'less than' is... |
| 0lt1sr 11151 | 0 is less than 1 for signe... |
| 1ne0sr 11152 | 1 and 0 are distinct for s... |
| 0idsr 11153 | The signed real number 0 i... |
| 1idsr 11154 | 1 is an identity element f... |
| 00sr 11155 | A signed real times 0 is 0... |
| ltasr 11156 | Ordering property of addit... |
| pn0sr 11157 | A signed real plus its neg... |
| negexsr 11158 | Existence of negative sign... |
| recexsrlem 11159 | The reciprocal of a positi... |
| addgt0sr 11160 | The sum of two positive si... |
| mulgt0sr 11161 | The product of two positiv... |
| sqgt0sr 11162 | The square of a nonzero si... |
| recexsr 11163 | The reciprocal of a nonzer... |
| mappsrpr 11164 | Mapping from positive sign... |
| ltpsrpr 11165 | Mapping of order from posi... |
| map2psrpr 11166 | Equivalence for positive s... |
| supsrlem 11167 | Lemma for supremum theorem... |
| supsr 11168 | A nonempty, bounded set of... |
| opelcn 11185 | Ordered pair membership in... |
| opelreal 11186 | Ordered pair membership in... |
| elreal 11187 | Membership in class of rea... |
| elreal2 11188 | Ordered pair membership in... |
| 0ncn 11189 | The empty set is not a com... |
| ltrelre 11190 | 'Less than' is a relation ... |
| addcnsr 11191 | Addition of complex number... |
| mulcnsr 11192 | Multiplication of complex ... |
| eqresr 11193 | Equality of real numbers i... |
| addresr 11194 | Addition of real numbers i... |
| mulresr 11195 | Multiplication of real num... |
| ltresr 11196 | Ordering of real subset of... |
| ltresr2 11197 | Ordering of real subset of... |
| dfcnqs 11198 | Technical trick to permit ... |
| addcnsrec 11199 | Technical trick to permit ... |
| mulcnsrec 11200 | Technical trick to permit ... |
| axaddf 11201 | Addition is an operation o... |
| axmulf 11202 | Multiplication is an opera... |
| axcnex 11203 | The complex numbers form a... |
| axresscn 11204 | The real numbers are a sub... |
| ax1cn 11205 | 1 is a complex number. Ax... |
| axicn 11206 | ` _i ` is a complex number... |
| axaddcl 11207 | Closure law for addition o... |
| axaddrcl 11208 | Closure law for addition i... |
| axmulcl 11209 | Closure law for multiplica... |
| axmulrcl 11210 | Closure law for multiplica... |
| axmulcom 11211 | Multiplication of complex ... |
| axaddass 11212 | Addition of complex number... |
| axmulass 11213 | Multiplication of complex ... |
| axdistr 11214 | Distributive law for compl... |
| axi2m1 11215 | i-squared equals -1 (expre... |
| ax1ne0 11216 | 1 and 0 are distinct. Axi... |
| ax1rid 11217 | ` 1 ` is an identity eleme... |
| axrnegex 11218 | Existence of negative of r... |
| axrrecex 11219 | Existence of reciprocal of... |
| axcnre 11220 | A complex number can be ex... |
| axpre-lttri 11221 | Ordering on reals satisfie... |
| axpre-lttrn 11222 | Ordering on reals is trans... |
| axpre-ltadd 11223 | Ordering property of addit... |
| axpre-mulgt0 11224 | The product of two positiv... |
| axpre-sup 11225 | A nonempty, bounded-above ... |
| wuncn 11226 | A weak universe containing... |
| cnex 11252 | Alias for ~ ax-cnex . See... |
| addcl 11253 | Alias for ~ ax-addcl , for... |
| readdcl 11254 | Alias for ~ ax-addrcl , fo... |
| mulcl 11255 | Alias for ~ ax-mulcl , for... |
| remulcl 11256 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11257 | Alias for ~ ax-mulcom , fo... |
| addass 11258 | Alias for ~ ax-addass , fo... |
| mulass 11259 | Alias for ~ ax-mulass , fo... |
| adddi 11260 | Alias for ~ ax-distr , for... |
| recn 11261 | A real number is a complex... |
| reex 11262 | The real numbers form a se... |
| reelprrecn 11263 | Reals are a subset of the ... |
| cnelprrecn 11264 | Complex numbers are a subs... |
| mpoaddf 11265 | Addition is an operation o... |
| mpomulf 11266 | Multiplication is an opera... |
| elimne0 11267 | Hypothesis for weak deduct... |
| adddir 11268 | Distributive law for compl... |
| 0cn 11269 | Zero is a complex number. ... |
| 0cnd 11270 | Zero is a complex number, ... |
| c0ex 11271 | Zero is a set. (Contribut... |
| 0elpr01 11272 | 0 is an element of ` { 0 ,... |
| 1cnd 11273 | One is a complex number, d... |
| 1ex 11274 | One is a set. (Contribute... |
| 1elpr01 11275 | 1 is an element of ` { 0 ,... |
| cnre 11276 | Alias for ~ ax-cnre , for ... |
| mulrid 11277 | The number 1 is an identit... |
| mullid 11278 | Identity law for multiplic... |
| 1re 11279 | The number 1 is real. Thi... |
| 1red 11280 | The number 1 is real, dedu... |
| 0re 11281 | The number 0 is real. Rem... |
| 0red 11282 | The number 0 is real, dedu... |
| pr01ssre 11283 | The pair ` { 0 , 1 } ` is ... |
| mulridi 11284 | Identity law for multiplic... |
| mullidi 11285 | Identity law for multiplic... |
| addcli 11286 | Closure law for addition. ... |
| mulcli 11287 | Closure law for multiplica... |
| mulcomi 11288 | Commutative law for multip... |
| mulcomli 11289 | Commutative law for multip... |
| addassi 11290 | Associative law for additi... |
| mulassi 11291 | Associative law for multip... |
| adddii 11292 | Distributive law (left-dis... |
| adddiri 11293 | Distributive law (right-di... |
| recni 11294 | A real number is a complex... |
| readdcli 11295 | Closure law for addition o... |
| remulcli 11296 | Closure law for multiplica... |
| mulridd 11297 | Identity law for multiplic... |
| mullidd 11298 | Identity law for multiplic... |
| addcld 11299 | Closure law for addition. ... |
| mulcld 11300 | Closure law for multiplica... |
| mulcomd 11301 | Commutative law for multip... |
| addassd 11302 | Associative law for additi... |
| mulassd 11303 | Associative law for multip... |
| adddid 11304 | Distributive law (left-dis... |
| adddird 11305 | Distributive law (right-di... |
| adddirp1d 11306 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11307 | Join AB+CB into (A+C) on L... |
| recnd 11308 | Deduction from real number... |
| readdcld 11309 | Closure law for addition o... |
| remulcld 11310 | Closure law for multiplica... |
| pnfnre 11321 | Plus infinity is not a rea... |
| pnfnre2 11322 | Plus infinity is not a rea... |
| mnfnre 11323 | Minus infinity is not a re... |
| ressxr 11324 | The standard reals are a s... |
| rexpssxrxp 11325 | The Cartesian product of s... |
| rexr 11326 | A standard real is an exte... |
| 0xr 11327 | Zero is an extended real. ... |
| renepnf 11328 | No (finite) real equals pl... |
| renemnf 11329 | No real equals minus infin... |
| rexrd 11330 | A standard real is an exte... |
| renepnfd 11331 | No (finite) real equals pl... |
| renemnfd 11332 | No real equals minus infin... |
| pnfex 11333 | Plus infinity exists. (Co... |
| pnfxr 11334 | Plus infinity belongs to t... |
| pnfnemnf 11335 | Plus and minus infinity ar... |
| mnfnepnf 11336 | Minus and plus infinity ar... |
| mnfxr 11337 | Minus infinity belongs to ... |
| rexri 11338 | A standard real is an exte... |
| 1xr 11339 | ` 1 ` is an extended real ... |
| renfdisj 11340 | The reals and the infiniti... |
| ltrelxr 11341 | "Less than" is a relation ... |
| ltrel 11342 | "Less than" is a relation.... |
| lerelxr 11343 | "Less than or equal to" is... |
| lerel 11344 | "Less than or equal to" is... |
| xrlenlt 11345 | "Less than or equal to" ex... |
| xrlenltd 11346 | "Less than or equal to" ex... |
| xrltnle 11347 | "Less than" expressed in t... |
| xrltnled 11348 | 'Less than' in terms of 'l... |
| xrnltled 11349 | "Not less than" implies "l... |
| ssxr 11350 | The three (non-exclusive) ... |
| ltxrlt 11351 | The standard less-than ` <... |
| axlttri 11352 | Ordering on reals satisfie... |
| axlttrn 11353 | Ordering on reals is trans... |
| axltadd 11354 | Ordering property of addit... |
| axmulgt0 11355 | The product of two positiv... |
| axsup 11356 | A nonempty, bounded-above ... |
| lttr 11357 | Alias for ~ axlttrn , for ... |
| mulgt0 11358 | The product of two positiv... |
| lenlt 11359 | 'Less than or equal to' ex... |
| ltnle 11360 | 'Less than' expressed in t... |
| ltso 11361 | 'Less than' is a strict or... |
| gtso 11362 | 'Greater than' is a strict... |
| lttri2 11363 | Consequence of trichotomy.... |
| lttri3 11364 | Trichotomy law for 'less t... |
| lttri4 11365 | Trichotomy law for 'less t... |
| letri3 11366 | Trichotomy law. (Contribu... |
| leloe 11367 | 'Less than or equal to' ex... |
| eqlelt 11368 | Equality in terms of 'less... |
| ltle 11369 | 'Less than' implies 'less ... |
| leltne 11370 | 'Less than or equal to' im... |
| lelttr 11371 | Transitive law. (Contribu... |
| leltletr 11372 | Transitive law, weaker for... |
| ltletr 11373 | Transitive law. (Contribu... |
| ltleletr 11374 | Transitive law, weaker for... |
| letr 11375 | Transitive law. (Contribu... |
| ltnr 11376 | 'Less than' is irreflexive... |
| leid 11377 | 'Less than or equal to' is... |
| ltne 11378 | 'Less than' implies not eq... |
| ltnsym 11379 | 'Less than' is not symmetr... |
| ltnsym2 11380 | 'Less than' is antisymmetr... |
| letric 11381 | Trichotomy law. (Contribu... |
| ltlen 11382 | 'Less than' expressed in t... |
| eqle 11383 | Equality implies 'less tha... |
| eqled 11384 | Equality implies 'less tha... |
| ltadd2 11385 | Addition to both sides of ... |
| ne0gt0 11386 | A nonzero nonnegative numb... |
| lecasei 11387 | Ordering elimination by ca... |
| lelttric 11388 | Trichotomy law. (Contribu... |
| ltlecasei 11389 | Ordering elimination by ca... |
| ltnri 11390 | 'Less than' is irreflexive... |
| eqlei 11391 | Equality implies 'less tha... |
| eqlei2 11392 | Equality implies 'less tha... |
| gtneii 11393 | 'Less than' implies not eq... |
| ltneii 11394 | 'Greater than' implies not... |
| lttri2i 11395 | Consequence of trichotomy.... |
| lttri3i 11396 | Consequence of trichotomy.... |
| letri3i 11397 | Consequence of trichotomy.... |
| leloei 11398 | 'Less than or equal to' in... |
| ltleni 11399 | 'Less than' expressed in t... |
| ltnsymi 11400 | 'Less than' is not symmetr... |
| lenlti 11401 | 'Less than or equal to' in... |
| ltnlei 11402 | 'Less than' in terms of 'l... |
| ltlei 11403 | 'Less than' implies 'less ... |
| ltleii 11404 | 'Less than' implies 'less ... |
| ltnei 11405 | 'Less than' implies not eq... |
| letrii 11406 | Trichotomy law for 'less t... |
| lttri 11407 | 'Less than' is transitive.... |
| lelttri 11408 | 'Less than or equal to', '... |
| ltletri 11409 | 'Less than', 'less than or... |
| letri 11410 | 'Less than or equal to' is... |
| le2tri3i 11411 | Extended trichotomy law fo... |
| ltadd2i 11412 | Addition to both sides of ... |
| mulgt0i 11413 | The product of two positiv... |
| mulgt0ii 11414 | The product of two positiv... |
| ltnrd 11415 | 'Less than' is irreflexive... |
| gtned 11416 | 'Less than' implies not eq... |
| ltned 11417 | 'Greater than' implies not... |
| ne0gt0d 11418 | A nonzero nonnegative numb... |
| lttrid 11419 | Ordering on reals satisfie... |
| lttri2d 11420 | Consequence of trichotomy.... |
| lttri3d 11421 | Consequence of trichotomy.... |
| lttri4d 11422 | Trichotomy law for 'less t... |
| letri3d 11423 | Consequence of trichotomy.... |
| leloed 11424 | 'Less than or equal to' in... |
| eqleltd 11425 | Equality in terms of 'less... |
| ltlend 11426 | 'Less than' expressed in t... |
| lenltd 11427 | 'Less than or equal to' in... |
| ltnled 11428 | 'Less than' in terms of 'l... |
| ltled 11429 | 'Less than' implies 'less ... |
| ltnsymd 11430 | 'Less than' implies 'less ... |
| nltled 11431 | 'Not less than ' implies '... |
| lensymd 11432 | 'Less than or equal to' im... |
| letrid 11433 | Trichotomy law for 'less t... |
| leltned 11434 | 'Less than or equal to' im... |
| leneltd 11435 | 'Less than or equal to' an... |
| mulgt0d 11436 | The product of two positiv... |
| ltadd2d 11437 | Addition to both sides of ... |
| letrd 11438 | Transitive law deduction f... |
| lelttrd 11439 | Transitive law deduction f... |
| ltadd2dd 11440 | Addition to both sides of ... |
| ltletrd 11441 | Transitive law deduction f... |
| lttrd 11442 | Transitive law deduction f... |
| lelttrdi 11443 | If a number is less than a... |
| dedekind 11444 | The Dedekind cut theorem. ... |
| dedekindle 11445 | The Dedekind cut theorem, ... |
| mul12 11446 | Commutative/associative la... |
| mul32 11447 | Commutative/associative la... |
| mul31 11448 | Commutative/associative la... |
| mul4 11449 | Rearrangement of 4 factors... |
| mul4r 11450 | Rearrangement of 4 factors... |
| muladd11 11451 | A simple product of sums e... |
| 1p1times 11452 | Two times a number. (Cont... |
| peano2cn 11453 | A theorem for complex numb... |
| peano2re 11454 | A theorem for reals analog... |
| readdcan 11455 | Cancellation law for addit... |
| 00id 11456 | ` 0 ` is its own additive ... |
| mul02lem1 11457 | Lemma for ~ mul02 . If an... |
| mul02lem2 11458 | Lemma for ~ mul02 . Zero ... |
| mul02 11459 | Multiplication by ` 0 ` . ... |
| mul01 11460 | Multiplication by ` 0 ` . ... |
| addrid 11461 | ` 0 ` is an additive ident... |
| cnegex 11462 | Existence of the negative ... |
| cnegex2 11463 | Existence of a left invers... |
| addlid 11464 | ` 0 ` is a left identity f... |
| addcan 11465 | Cancellation law for addit... |
| addcan2 11466 | Cancellation law for addit... |
| addcom 11467 | Addition is commutative. ... |
| addridi 11468 | ` 0 ` is an additive ident... |
| addlidi 11469 | ` 0 ` is a left identity f... |
| mul02i 11470 | Multiplication by 0. Theo... |
| mul01i 11471 | Multiplication by ` 0 ` . ... |
| addcomi 11472 | Addition is commutative. ... |
| addcomli 11473 | Addition is commutative. ... |
| addcani 11474 | Cancellation law for addit... |
| addcan2i 11475 | Cancellation law for addit... |
| mul12i 11476 | Commutative/associative la... |
| mul32i 11477 | Commutative/associative la... |
| mul4i 11478 | Rearrangement of 4 factors... |
| mul02d 11479 | Multiplication by 0. Theo... |
| mul01d 11480 | Multiplication by ` 0 ` . ... |
| addridd 11481 | ` 0 ` is an additive ident... |
| addlidd 11482 | ` 0 ` is a left identity f... |
| addcomd 11483 | Addition is commutative. ... |
| addcand 11484 | Cancellation law for addit... |
| addcan2d 11485 | Cancellation law for addit... |
| addcanad 11486 | Cancelling a term on the l... |
| addcan2ad 11487 | Cancelling a term on the r... |
| addneintrd 11488 | Introducing a term on the ... |
| addneintr2d 11489 | Introducing a term on the ... |
| mul12d 11490 | Commutative/associative la... |
| mul32d 11491 | Commutative/associative la... |
| mul31d 11492 | Commutative/associative la... |
| mul4d 11493 | Rearrangement of 4 factors... |
| muladd11r 11494 | A simple product of sums e... |
| comraddd 11495 | Commute RHS addition, in d... |
| comraddi 11496 | Commute RHS addition. See... |
| ltaddneg 11497 | Adding a negative number t... |
| ltaddnegr 11498 | Adding a negative number t... |
| add12 11499 | Commutative/associative la... |
| add32 11500 | Commutative/associative la... |
| add32r 11501 | Commutative/associative la... |
| add4 11502 | Rearrangement of 4 terms i... |
| add42 11503 | Rearrangement of 4 terms i... |
| add12i 11504 | Commutative/associative la... |
| add32i 11505 | Commutative/associative la... |
| add4i 11506 | Rearrangement of 4 terms i... |
| add42i 11507 | Rearrangement of 4 terms i... |
| add12d 11508 | Commutative/associative la... |
| add32d 11509 | Commutative/associative la... |
| add4d 11510 | Rearrangement of 4 terms i... |
| add42d 11511 | Rearrangement of 4 terms i... |
| 0cnALT 11516 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11517 | Alternate proof of ~ 0cnAL... |
| negeu 11518 | Existential uniqueness of ... |
| subval 11519 | Value of subtraction, whic... |
| negeq 11520 | Equality theorem for negat... |
| negeqi 11521 | Equality inference for neg... |
| negeqd 11522 | Equality deduction for neg... |
| nfnegd 11523 | Deduction version of ~ nfn... |
| nfneg 11524 | Bound-variable hypothesis ... |
| csbnegg 11525 | Move class substitution in... |
| negex 11526 | A negative is a set. (Con... |
| subcl 11527 | Closure law for subtractio... |
| negcl 11528 | Closure law for negative. ... |
| negicn 11529 | ` -u _i ` is a complex num... |
| subf 11530 | Subtraction is an operatio... |
| subadd 11531 | Relationship between subtr... |
| subadd2 11532 | Relationship between subtr... |
| subsub23 11533 | Swap subtrahend and result... |
| pncan 11534 | Cancellation law for subtr... |
| pncan2 11535 | Cancellation law for subtr... |
| pncan3 11536 | Subtraction and addition o... |
| npcan 11537 | Cancellation law for subtr... |
| addsubass 11538 | Associative-type law for a... |
| addsub 11539 | Law for addition and subtr... |
| subadd23 11540 | Commutative/associative la... |
| addsub12 11541 | Commutative/associative la... |
| 2addsub 11542 | Law for subtraction and ad... |
| addsubeq4 11543 | Relation between sums and ... |
| pncan3oi 11544 | Subtraction and addition o... |
| mvrraddi 11545 | Move the right term in a s... |
| mvrladdi 11546 | Move the left term in a su... |
| mvlladdi 11547 | Move the left term in a su... |
| subid 11548 | Subtraction of a number fr... |
| subid1 11549 | Identity law for subtracti... |
| npncan 11550 | Cancellation law for subtr... |
| nppcan 11551 | Cancellation law for subtr... |
| nnpcan 11552 | Cancellation law for subtr... |
| nppcan3 11553 | Cancellation law for subtr... |
| subcan2 11554 | Cancellation law for subtr... |
| subeq0 11555 | If the difference between ... |
| npncan2 11556 | Cancellation law for subtr... |
| subsub2 11557 | Law for double subtraction... |
| nncan 11558 | Cancellation law for subtr... |
| subsub 11559 | Law for double subtraction... |
| nppcan2 11560 | Cancellation law for subtr... |
| subsub3 11561 | Law for double subtraction... |
| subsub4 11562 | Law for double subtraction... |
| sub32 11563 | Swap the second and third ... |
| nnncan 11564 | Cancellation law for subtr... |
| nnncan1 11565 | Cancellation law for subtr... |
| nnncan2 11566 | Cancellation law for subtr... |
| npncan3 11567 | Cancellation law for subtr... |
| pnpcan 11568 | Cancellation law for mixed... |
| pnpcan2 11569 | Cancellation law for mixed... |
| pnncan 11570 | Cancellation law for mixed... |
| ppncan 11571 | Cancellation law for mixed... |
| addsub4 11572 | Rearrangement of 4 terms i... |
| subadd4 11573 | Rearrangement of 4 terms i... |
| sub4 11574 | Rearrangement of 4 terms i... |
| neg0 11575 | Minus 0 equals 0. (Contri... |
| negid 11576 | Addition of a number and i... |
| negsub 11577 | Relationship between subtr... |
| subneg 11578 | Relationship between subtr... |
| negneg 11579 | A number is equal to the n... |
| neg11 11580 | Negative is one-to-one. (... |
| negcon1 11581 | Negative contraposition la... |
| negcon2 11582 | Negative contraposition la... |
| negeq0 11583 | A number is zero iff its n... |
| subcan 11584 | Cancellation law for subtr... |
| negsubdi 11585 | Distribution of negative o... |
| negdi 11586 | Distribution of negative o... |
| negdi2 11587 | Distribution of negative o... |
| negsubdi2 11588 | Distribution of negative o... |
| neg2sub 11589 | Relationship between subtr... |
| renegcli 11590 | Closure law for negative o... |
| resubcli 11591 | Closure law for subtractio... |
| renegcl 11592 | Closure law for negative o... |
| resubcl 11593 | Closure law for subtractio... |
| negreb 11594 | The negative of a real is ... |
| peano2cnm 11595 | "Reverse" second Peano pos... |
| peano2rem 11596 | "Reverse" second Peano pos... |
| negcli 11597 | Closure law for negative. ... |
| negidi 11598 | Addition of a number and i... |
| negnegi 11599 | A number is equal to the n... |
| subidi 11600 | Subtraction of a number fr... |
| subid1i 11601 | Identity law for subtracti... |
| negne0bi 11602 | A number is nonzero iff it... |
| negrebi 11603 | The negative of a real is ... |
| negne0i 11604 | The negative of a nonzero ... |
| subcli 11605 | Closure law for subtractio... |
| pncan3i 11606 | Subtraction and addition o... |
| negsubi 11607 | Relationship between subtr... |
| subnegi 11608 | Relationship between subtr... |
| subeq0i 11609 | If the difference between ... |
| neg11i 11610 | Negative is one-to-one. (... |
| negcon1i 11611 | Negative contraposition la... |
| negcon2i 11612 | Negative contraposition la... |
| negdii 11613 | Distribution of negative o... |
| negsubdii 11614 | Distribution of negative o... |
| negsubdi2i 11615 | Distribution of negative o... |
| subaddi 11616 | Relationship between subtr... |
| subadd2i 11617 | Relationship between subtr... |
| subaddrii 11618 | Relationship between subtr... |
| subsub23i 11619 | Swap subtrahend and result... |
| addsubassi 11620 | Associative-type law for s... |
| addsubi 11621 | Law for subtraction and ad... |
| subcani 11622 | Cancellation law for subtr... |
| subcan2i 11623 | Cancellation law for subtr... |
| pnncani 11624 | Cancellation law for mixed... |
| addsub4i 11625 | Rearrangement of 4 terms i... |
| 0reALT 11626 | Alternate proof of ~ 0re .... |
| negcld 11627 | Closure law for negative. ... |
| subidd 11628 | Subtraction of a number fr... |
| subid1d 11629 | Identity law for subtracti... |
| negidd 11630 | Addition of a number and i... |
| negnegd 11631 | A number is equal to the n... |
| negeq0d 11632 | A number is zero iff its n... |
| negne0bd 11633 | A number is nonzero iff it... |
| negcon1d 11634 | Contraposition law for una... |
| negcon1ad 11635 | Contraposition law for una... |
| neg11ad 11636 | The negatives of two compl... |
| negned 11637 | If two complex numbers are... |
| negne0d 11638 | The negative of a nonzero ... |
| negrebd 11639 | The negative of a real is ... |
| subcld 11640 | Closure law for subtractio... |
| pncand 11641 | Cancellation law for subtr... |
| pncan2d 11642 | Cancellation law for subtr... |
| pncan3d 11643 | Subtraction and addition o... |
| npcand 11644 | Cancellation law for subtr... |
| nncand 11645 | Cancellation law for subtr... |
| negsubd 11646 | Relationship between subtr... |
| subnegd 11647 | Relationship between subtr... |
| subeq0ad 11648 | The difference of two comp... |
| subeq0d 11649 | If the difference between ... |
| subne0d 11650 | Two unequal numbers have n... |
| subne0ad 11651 | If the difference of two c... |
| neg11d 11652 | If the difference between ... |
| negdid 11653 | Distribution of negative o... |
| negdi2d 11654 | Distribution of negative o... |
| negsubdid 11655 | Distribution of negative o... |
| negsubdi2d 11656 | Distribution of negative o... |
| neg2subd 11657 | Relationship between subtr... |
| subaddd 11658 | Relationship between subtr... |
| subadd2d 11659 | Relationship between subtr... |
| addsubassd 11660 | Associative-type law for s... |
| addsubd 11661 | Law for subtraction and ad... |
| subadd23d 11662 | Commutative/associative la... |
| addsub12d 11663 | Commutative/associative la... |
| npncand 11664 | Cancellation law for subtr... |
| nppcand 11665 | Cancellation law for subtr... |
| nppcan2d 11666 | Cancellation law for subtr... |
| nppcan3d 11667 | Cancellation law for subtr... |
| subsubd 11668 | Law for double subtraction... |
| subsub2d 11669 | Law for double subtraction... |
| subsub3d 11670 | Law for double subtraction... |
| subsub4d 11671 | Law for double subtraction... |
| sub32d 11672 | Swap the second and third ... |
| nnncand 11673 | Cancellation law for subtr... |
| nnncan1d 11674 | Cancellation law for subtr... |
| nnncan2d 11675 | Cancellation law for subtr... |
| npncan3d 11676 | Cancellation law for subtr... |
| pnpcand 11677 | Cancellation law for mixed... |
| pnpcan2d 11678 | Cancellation law for mixed... |
| pnncand 11679 | Cancellation law for mixed... |
| ppncand 11680 | Cancellation law for mixed... |
| subcand 11681 | Cancellation law for subtr... |
| subcan2d 11682 | Cancellation law for subtr... |
| subcanad 11683 | Cancellation law for subtr... |
| subneintrd 11684 | Introducing subtraction on... |
| subcan2ad 11685 | Cancellation law for subtr... |
| subneintr2d 11686 | Introducing subtraction on... |
| addsub4d 11687 | Rearrangement of 4 terms i... |
| subadd4d 11688 | Rearrangement of 4 terms i... |
| sub4d 11689 | Rearrangement of 4 terms i... |
| 2addsubd 11690 | Law for subtraction and ad... |
| addsubeq4d 11691 | Relation between sums and ... |
| subsubadd23 11692 | Swap the second and the th... |
| addsubsub23 11693 | Swap the second and the th... |
| subeqxfrd 11694 | Transfer two terms of a su... |
| mvlraddd 11695 | Move the right term in a s... |
| mvlladdd 11696 | Move the left term in a su... |
| mvrraddd 11697 | Move the right term in a s... |
| mvrladdd 11698 | Move the left term in a su... |
| assraddsubd 11699 | Associate RHS addition-sub... |
| subaddeqd 11700 | Transfer two terms of a su... |
| addlsub 11701 | Left-subtraction: Subtrac... |
| addrsub 11702 | Right-subtraction: Subtra... |
| subexsub 11703 | A subtraction law: Exchan... |
| addid0 11704 | If adding a number to a an... |
| addn0nid 11705 | Adding a nonzero number to... |
| pnpncand 11706 | Addition/subtraction cance... |
| subeqrev 11707 | Reverse the order of subtr... |
| addeq0 11708 | Two complex numbers add up... |
| pncan1 11709 | Cancellation law for addit... |
| npcan1 11710 | Cancellation law for subtr... |
| subeq0bd 11711 | If two complex numbers are... |
| renegcld 11712 | Closure law for negative o... |
| resubcld 11713 | Closure law for subtractio... |
| negn0 11714 | The image under negation o... |
| negf1o 11715 | Negation is an isomorphism... |
| kcnktkm1cn 11716 | k times k minus 1 is a com... |
| muladd 11717 | Product of two sums. (Con... |
| subdi 11718 | Distribution of multiplica... |
| subdir 11719 | Distribution of multiplica... |
| ine0 11720 | The imaginary unit ` _i ` ... |
| mulneg1 11721 | Product with negative is n... |
| mulneg2 11722 | The product with a negativ... |
| mulneg12 11723 | Swap the negative sign in ... |
| mul2neg 11724 | Product of two negatives. ... |
| submul2 11725 | Convert a subtraction to a... |
| mulm1 11726 | Product with minus one is ... |
| addneg1mul 11727 | Addition with product with... |
| mulsub 11728 | Product of two differences... |
| mulsub2 11729 | Swap the order of subtract... |
| mulm1i 11730 | Product with minus one is ... |
| mulneg1i 11731 | Product with negative is n... |
| mulneg2i 11732 | Product with negative is n... |
| mul2negi 11733 | Product of two negatives. ... |
| subdii 11734 | Distribution of multiplica... |
| subdiri 11735 | Distribution of multiplica... |
| muladdi 11736 | Product of two sums. (Con... |
| mulm1d 11737 | Product with minus one is ... |
| mulneg1d 11738 | Product with negative is n... |
| mulneg2d 11739 | Product with negative is n... |
| mul2negd 11740 | Product of two negatives. ... |
| subdid 11741 | Distribution of multiplica... |
| subdird 11742 | Distribution of multiplica... |
| muladdd 11743 | Product of two sums. (Con... |
| mulsubd 11744 | Product of two differences... |
| muls1d 11745 | Multiplication by one minu... |
| mulsubfacd 11746 | Multiplication followed by... |
| addmulsub 11747 | The product of a sum and a... |
| subaddmulsub 11748 | The difference with a prod... |
| mulsubaddmulsub 11749 | A special difference of a ... |
| gt0ne0 11750 | Positive implies nonzero. ... |
| lt0ne0 11751 | A number which is less tha... |
| ltadd1 11752 | Addition to both sides of ... |
| leadd1 11753 | Addition to both sides of ... |
| leadd2 11754 | Addition to both sides of ... |
| ltsubadd 11755 | 'Less than' relationship b... |
| ltsubadd2 11756 | 'Less than' relationship b... |
| lesubadd 11757 | 'Less than or equal to' re... |
| lesubadd2 11758 | 'Less than or equal to' re... |
| ltaddsub 11759 | 'Less than' relationship b... |
| ltaddsub2 11760 | 'Less than' relationship b... |
| leaddsub 11761 | 'Less than or equal to' re... |
| leaddsub2 11762 | 'Less than or equal to' re... |
| suble 11763 | Swap subtrahends in an ine... |
| lesub 11764 | Swap subtrahends in an ine... |
| ltsub23 11765 | 'Less than' relationship b... |
| ltsub13 11766 | 'Less than' relationship b... |
| le2add 11767 | Adding both sides of two '... |
| ltleadd 11768 | Adding both sides of two o... |
| leltadd 11769 | Adding both sides of two o... |
| lt2add 11770 | Adding both sides of two '... |
| addgt0 11771 | The sum of 2 positive numb... |
| addgegt0 11772 | The sum of nonnegative and... |
| addgtge0 11773 | The sum of nonnegative and... |
| addge0 11774 | The sum of 2 nonnegative n... |
| ltaddpos 11775 | Adding a positive number t... |
| ltaddpos2 11776 | Adding a positive number t... |
| ltsubpos 11777 | Subtracting a positive num... |
| posdif 11778 | Comparison of two numbers ... |
| lesub1 11779 | Subtraction from both side... |
| lesub2 11780 | Subtraction of both sides ... |
| ltsub1 11781 | Subtraction from both side... |
| ltsub2 11782 | Subtraction of both sides ... |
| lt2sub 11783 | Subtracting both sides of ... |
| le2sub 11784 | Subtracting both sides of ... |
| ltneg 11785 | Negative of both sides of ... |
| ltnegcon1 11786 | Contraposition of negative... |
| ltnegcon2 11787 | Contraposition of negative... |
| leneg 11788 | Negative of both sides of ... |
| lenegcon1 11789 | Contraposition of negative... |
| lenegcon2 11790 | Contraposition of negative... |
| lt0neg1 11791 | Comparison of a number and... |
| lt0neg2 11792 | Comparison of a number and... |
| le0neg1 11793 | Comparison of a number and... |
| le0neg2 11794 | Comparison of a number and... |
| addge01 11795 | A number is less than or e... |
| addge02 11796 | A number is less than or e... |
| add20 11797 | Two nonnegative numbers ar... |
| subge0 11798 | Nonnegative subtraction. ... |
| suble0 11799 | Nonpositive subtraction. ... |
| leaddle0 11800 | The sum of a real number a... |
| subge02 11801 | Nonnegative subtraction. ... |
| lesub0 11802 | Lemma to show a nonnegativ... |
| mulge0 11803 | The product of two nonnega... |
| mullt0 11804 | The product of two negativ... |
| msqgt0 11805 | A nonzero square is positi... |
| msqge0 11806 | A square is nonnegative. ... |
| 0lt1 11807 | 0 is less than 1. Theorem... |
| 0le1 11808 | 0 is less than or equal to... |
| relin01 11809 | An interval law for less t... |
| ltordlem 11810 | Lemma for ~ ltord1 . (Con... |
| ltord1 11811 | Infer an ordering relation... |
| leord1 11812 | Infer an ordering relation... |
| eqord1 11813 | A strictly increasing real... |
| ltord2 11814 | Infer an ordering relation... |
| leord2 11815 | Infer an ordering relation... |
| eqord2 11816 | A strictly decreasing real... |
| wloglei 11817 | Form of ~ wlogle where bot... |
| wlogle 11818 | If the predicate ` ch ( x ... |
| leidi 11819 | 'Less than or equal to' is... |
| gt0ne0i 11820 | Positive means nonzero (us... |
| gt0ne0ii 11821 | Positive implies nonzero. ... |
| msqgt0i 11822 | A nonzero square is positi... |
| msqge0i 11823 | A square is nonnegative. ... |
| addgt0i 11824 | Addition of 2 positive num... |
| addge0i 11825 | Addition of 2 nonnegative ... |
| addgegt0i 11826 | Addition of nonnegative an... |
| addgt0ii 11827 | Addition of 2 positive num... |
| add20i 11828 | Two nonnegative numbers ar... |
| ltnegi 11829 | Negative of both sides of ... |
| lenegi 11830 | Negative of both sides of ... |
| ltnegcon2i 11831 | Contraposition of negative... |
| mulge0i 11832 | The product of two nonnega... |
| lesub0i 11833 | Lemma to show a nonnegativ... |
| ltaddposi 11834 | Adding a positive number t... |
| posdifi 11835 | Comparison of two numbers ... |
| ltnegcon1i 11836 | Contraposition of negative... |
| lenegcon1i 11837 | Contraposition of negative... |
| subge0i 11838 | Nonnegative subtraction. ... |
| ltadd1i 11839 | Addition to both sides of ... |
| leadd1i 11840 | Addition to both sides of ... |
| leadd2i 11841 | Addition to both sides of ... |
| ltsubaddi 11842 | 'Less than' relationship b... |
| lesubaddi 11843 | 'Less than or equal to' re... |
| ltsubadd2i 11844 | 'Less than' relationship b... |
| lesubadd2i 11845 | 'Less than or equal to' re... |
| ltaddsubi 11846 | 'Less than' relationship b... |
| lt2addi 11847 | Adding both side of two in... |
| le2addi 11848 | Adding both side of two in... |
| gt0ne0d 11849 | Positive implies nonzero. ... |
| lt0ne0d 11850 | Something less than zero i... |
| leidd 11851 | 'Less than or equal to' is... |
| msqgt0d 11852 | A nonzero square is positi... |
| msqge0d 11853 | A square is nonnegative. ... |
| lt0neg1d 11854 | Comparison of a number and... |
| lt0neg2d 11855 | Comparison of a number and... |
| le0neg1d 11856 | Comparison of a number and... |
| le0neg2d 11857 | Comparison of a number and... |
| addgegt0d 11858 | Addition of nonnegative an... |
| addgtge0d 11859 | Addition of positive and n... |
| addgt0d 11860 | Addition of 2 positive num... |
| addge0d 11861 | Addition of 2 nonnegative ... |
| mulge0d 11862 | The product of two nonnega... |
| ltnegd 11863 | Negative of both sides of ... |
| lenegd 11864 | Negative of both sides of ... |
| ltnegcon1d 11865 | Contraposition of negative... |
| ltnegcon2d 11866 | Contraposition of negative... |
| lenegcon1d 11867 | Contraposition of negative... |
| lenegcon2d 11868 | Contraposition of negative... |
| ltaddposd 11869 | Adding a positive number t... |
| ltaddpos2d 11870 | Adding a positive number t... |
| ltsubposd 11871 | Subtracting a positive num... |
| posdifd 11872 | Comparison of two numbers ... |
| addge01d 11873 | A number is less than or e... |
| addge02d 11874 | A number is less than or e... |
| subge0d 11875 | Nonnegative subtraction. ... |
| suble0d 11876 | Nonpositive subtraction. ... |
| subge02d 11877 | Nonnegative subtraction. ... |
| ltadd1d 11878 | Addition to both sides of ... |
| leadd1d 11879 | Addition to both sides of ... |
| leadd2d 11880 | Addition to both sides of ... |
| ltsubaddd 11881 | 'Less than' relationship b... |
| lesubaddd 11882 | 'Less than or equal to' re... |
| ltsubadd2d 11883 | 'Less than' relationship b... |
| lesubadd2d 11884 | 'Less than or equal to' re... |
| ltaddsubd 11885 | 'Less than' relationship b... |
| ltaddsub2d 11886 | 'Less than' relationship b... |
| leaddsub2d 11887 | 'Less than or equal to' re... |
| subled 11888 | Swap subtrahends in an ine... |
| lesubd 11889 | Swap subtrahends in an ine... |
| ltsub23d 11890 | 'Less than' relationship b... |
| ltsub13d 11891 | 'Less than' relationship b... |
| lesub1d 11892 | Subtraction from both side... |
| lesub2d 11893 | Subtraction of both sides ... |
| ltsub1d 11894 | Subtraction from both side... |
| ltsub2d 11895 | Subtraction of both sides ... |
| ltadd1dd 11896 | Addition to both sides of ... |
| ltsub1dd 11897 | Subtraction from both side... |
| ltsub2dd 11898 | Subtraction of both sides ... |
| leadd1dd 11899 | Addition to both sides of ... |
| leadd2dd 11900 | Addition to both sides of ... |
| lesub1dd 11901 | Subtraction from both side... |
| lesub2dd 11902 | Subtraction of both sides ... |
| lesub3d 11903 | The result of subtracting ... |
| le2addd 11904 | Adding both side of two in... |
| le2subd 11905 | Subtracting both sides of ... |
| ltleaddd 11906 | Adding both sides of two o... |
| leltaddd 11907 | Adding both sides of two o... |
| lt2addd 11908 | Adding both side of two in... |
| lt2subd 11909 | Subtracting both sides of ... |
| possumd 11910 | Condition for a positive s... |
| sublt0d 11911 | When a subtraction gives a... |
| ltaddsublt 11912 | Addition and subtraction o... |
| 1le1 11913 | One is less than or equal ... |
| ixi 11914 | ` _i ` times itself is min... |
| recextlem1 11915 | Lemma for ~ recex . (Cont... |
| recextlem2 11916 | Lemma for ~ recex . (Cont... |
| recex 11917 | Existence of reciprocal of... |
| mulcand 11918 | Cancellation law for multi... |
| mulcan2d 11919 | Cancellation law for multi... |
| mulcanad 11920 | Cancellation of a nonzero ... |
| mulcan2ad 11921 | Cancellation of a nonzero ... |
| mulcan 11922 | Cancellation law for multi... |
| mulcan2 11923 | Cancellation law for multi... |
| mulcani 11924 | Cancellation law for multi... |
| mul0or 11925 | If a product is zero, one ... |
| mulne0b 11926 | The product of two nonzero... |
| mulne0 11927 | The product of two nonzero... |
| mulne0i 11928 | The product of two nonzero... |
| muleqadd 11929 | Property of numbers whose ... |
| receu 11930 | Existential uniqueness of ... |
| mulnzcnf 11931 | Multiplication maps nonzer... |
| mul0ori 11932 | If a product is zero, one ... |
| mul0ord 11933 | If a product is zero, one ... |
| msq0i 11934 | A number is zero iff its s... |
| msq0d 11935 | A number is zero iff its s... |
| mulne0bd 11936 | The product of two nonzero... |
| mulne0d 11937 | The product of two nonzero... |
| mulcan1g 11938 | A generalized form of the ... |
| mulcan2g 11939 | A generalized form of the ... |
| mulne0bad 11940 | A factor of a nonzero comp... |
| mulne0bbd 11941 | A factor of a nonzero comp... |
| 1div0 11944 | You can't divide by zero, ... |
| divval 11945 | Value of division: if ` A ... |
| divmul 11946 | Relationship between divis... |
| divmul2 11947 | Relationship between divis... |
| divmul3 11948 | Relationship between divis... |
| divcl 11949 | Closure law for division. ... |
| reccl 11950 | Closure law for reciprocal... |
| divcan2 11951 | A cancellation law for div... |
| divcan1 11952 | A cancellation law for div... |
| diveq0 11953 | A ratio is zero iff the nu... |
| divne0b 11954 | The ratio of nonzero numbe... |
| divne0 11955 | The ratio of nonzero numbe... |
| recne0 11956 | The reciprocal of a nonzer... |
| recid 11957 | Multiplication of a number... |
| recid2 11958 | Multiplication of a number... |
| divrec 11959 | Relationship between divis... |
| divrec2 11960 | Relationship between divis... |
| divass 11961 | An associative law for div... |
| div23 11962 | A commutative/associative ... |
| div32 11963 | A commutative/associative ... |
| div13 11964 | A commutative/associative ... |
| div12 11965 | A commutative/associative ... |
| divmulass 11966 | An associative law for div... |
| divmulasscom 11967 | An associative/commutative... |
| divdir 11968 | Distribution of division o... |
| divcan3 11969 | A cancellation law for div... |
| divcan4 11970 | A cancellation law for div... |
| div11 11971 | One-to-one relationship fo... |
| diveq1 11972 | Equality in terms of unit ... |
| divid 11973 | A number divided by itself... |
| div0 11974 | Division into zero is zero... |
| div1 11975 | A number divided by 1 is i... |
| 1div1e1 11976 | 1 divided by 1 is 1. (Con... |
| divneg 11977 | Move negative sign inside ... |
| muldivdir 11978 | Distribution of division o... |
| divsubdir 11979 | Distribution of division o... |
| muldivdid 11980 | Distribution of division o... |
| subdivcomb1 11981 | Bring a term in a subtract... |
| subdivcomb2 11982 | Bring a term in a subtract... |
| recrec 11983 | A number is equal to the r... |
| rec11 11984 | Reciprocal is one-to-one. ... |
| rec11r 11985 | Mutual reciprocals. (Cont... |
| divmuldiv 11986 | Multiplication of two rati... |
| divdivdiv 11987 | Division of two ratios. T... |
| divcan5 11988 | Cancellation of common fac... |
| divmul13 11989 | Swap the denominators in t... |
| divmul24 11990 | Swap the numerators in the... |
| divmuleq 11991 | Cross-multiply in an equal... |
| recdiv 11992 | The reciprocal of a ratio.... |
| divcan6 11993 | Cancellation of inverted f... |
| divdiv32 11994 | Swap denominators in a div... |
| divcan7 11995 | Cancel equal divisors in a... |
| dmdcan 11996 | Cancellation law for divis... |
| divdiv1 11997 | Division into a fraction. ... |
| divdiv2 11998 | Division by a fraction. (... |
| recdiv2 11999 | Division into a reciprocal... |
| ddcan 12000 | Cancellation in a double d... |
| divadddiv 12001 | Addition of two ratios. T... |
| divsubdiv 12002 | Subtraction of two ratios.... |
| conjmul 12003 | Two numbers whose reciproc... |
| rereccl 12004 | Closure law for reciprocal... |
| redivcl 12005 | Closure law for division o... |
| eqneg 12006 | A number equal to its nega... |
| eqnegd 12007 | A complex number equals it... |
| eqnegad 12008 | If a complex number equals... |
| div2neg 12009 | Quotient of two negatives.... |
| divneg2 12010 | Move negative sign inside ... |
| recclzi 12011 | Closure law for reciprocal... |
| recne0zi 12012 | The reciprocal of a nonzer... |
| recidzi 12013 | Multiplication of a number... |
| div1i 12014 | A number divided by 1 is i... |
| eqnegi 12015 | A number equal to its nega... |
| reccli 12016 | Closure law for reciprocal... |
| recidi 12017 | Multiplication of a number... |
| recreci 12018 | A number is equal to the r... |
| dividi 12019 | A number divided by itself... |
| div0i 12020 | Division into zero is zero... |
| divclzi 12021 | Closure law for division. ... |
| divcan1zi 12022 | A cancellation law for div... |
| divcan2zi 12023 | A cancellation law for div... |
| divreczi 12024 | Relationship between divis... |
| divcan3zi 12025 | A cancellation law for div... |
| divcan4zi 12026 | A cancellation law for div... |
| rec11i 12027 | Reciprocal is one-to-one. ... |
| divcli 12028 | Closure law for division. ... |
| divcan2i 12029 | A cancellation law for div... |
| divcan1i 12030 | A cancellation law for div... |
| divreci 12031 | Relationship between divis... |
| divcan3i 12032 | A cancellation law for div... |
| divcan4i 12033 | A cancellation law for div... |
| divne0i 12034 | The ratio of nonzero numbe... |
| rec11ii 12035 | Reciprocal is one-to-one. ... |
| divasszi 12036 | An associative law for div... |
| divmulzi 12037 | Relationship between divis... |
| divdirzi 12038 | Distribution of division o... |
| divdiv23zi 12039 | Swap denominators in a div... |
| divmuli 12040 | Relationship between divis... |
| divdiv32i 12041 | Swap denominators in a div... |
| divassi 12042 | An associative law for div... |
| divdiri 12043 | Distribution of division o... |
| div23i 12044 | A commutative/associative ... |
| div11i 12045 | One-to-one relationship fo... |
| divmuldivi 12046 | Multiplication of two rati... |
| divmul13i 12047 | Swap denominators of two r... |
| divadddivi 12048 | Addition of two ratios. T... |
| divdivdivi 12049 | Division of two ratios. T... |
| rerecclzi 12050 | Closure law for reciprocal... |
| rereccli 12051 | Closure law for reciprocal... |
| redivclzi 12052 | Closure law for division o... |
| redivcli 12053 | Closure law for division o... |
| div1d 12054 | A number divided by 1 is i... |
| reccld 12055 | Closure law for reciprocal... |
| recne0d 12056 | The reciprocal of a nonzer... |
| recidd 12057 | Multiplication of a number... |
| recid2d 12058 | Multiplication of a number... |
| recrecd 12059 | A number is equal to the r... |
| dividd 12060 | A number divided by itself... |
| div0d 12061 | Division into zero is zero... |
| divcld 12062 | Closure law for division. ... |
| divcan1d 12063 | A cancellation law for div... |
| divcan2d 12064 | A cancellation law for div... |
| divrecd 12065 | Relationship between divis... |
| divrec2d 12066 | Relationship between divis... |
| divcan3d 12067 | A cancellation law for div... |
| divcan4d 12068 | A cancellation law for div... |
| diveq0d 12069 | A ratio is zero iff the nu... |
| diveq1d 12070 | Equality in terms of unit ... |
| diveq1ad 12071 | The quotient of two comple... |
| diveq0ad 12072 | A fraction of complex numb... |
| divne1d 12073 | If two complex numbers are... |
| divne0bd 12074 | A ratio is zero iff the nu... |
| divnegd 12075 | Move negative sign inside ... |
| divneg2d 12076 | Move negative sign inside ... |
| div2negd 12077 | Quotient of two negatives.... |
| divne0d 12078 | The ratio of nonzero numbe... |
| recdivd 12079 | The reciprocal of a ratio.... |
| recdiv2d 12080 | Division into a reciprocal... |
| divcan6d 12081 | Cancellation of inverted f... |
| ddcand 12082 | Cancellation in a double d... |
| rec11d 12083 | Reciprocal is one-to-one. ... |
| divmuld 12084 | Relationship between divis... |
| div32d 12085 | A commutative/associative ... |
| div13d 12086 | A commutative/associative ... |
| divdiv32d 12087 | Swap denominators in a div... |
| divcan5d 12088 | Cancellation of common fac... |
| divcan5rd 12089 | Cancellation of common fac... |
| divcan7d 12090 | Cancel equal divisors in a... |
| dmdcand 12091 | Cancellation law for divis... |
| dmdcan2d 12092 | Cancellation law for divis... |
| divdiv1d 12093 | Division into a fraction. ... |
| divdiv2d 12094 | Division by a fraction. (... |
| divmul2d 12095 | Relationship between divis... |
| divmul3d 12096 | Relationship between divis... |
| divassd 12097 | An associative law for div... |
| div12d 12098 | A commutative/associative ... |
| div23d 12099 | A commutative/associative ... |
| divdird 12100 | Distribution of division o... |
| divsubdird 12101 | Distribution of division o... |
| div11d 12102 | One-to-one relationship fo... |
| divmuldivd 12103 | Multiplication of two rati... |
| divmul13d 12104 | Swap denominators of two r... |
| divmul24d 12105 | Swap the numerators in the... |
| divadddivd 12106 | Addition of two ratios. T... |
| divsubdivd 12107 | Subtraction of two ratios.... |
| divmuleqd 12108 | Cross-multiply in an equal... |
| divdivdivd 12109 | Division of two ratios. T... |
| diveq1bd 12110 | If two complex numbers are... |
| div2sub 12111 | Swap the order of subtract... |
| div2subd 12112 | Swap subtrahend and minuen... |
| rereccld 12113 | Closure law for reciprocal... |
| redivcld 12114 | Closure law for division o... |
| subrecd 12115 | Subtraction of reciprocals... |
| subrec 12116 | Subtraction of reciprocals... |
| subreci 12117 | Subtraction of reciprocals... |
| mvllmuld 12118 | Move the left term in a pr... |
| mvllmuli 12119 | Move the left term in a pr... |
| ldiv 12120 | Left-division. (Contribut... |
| rdiv 12121 | Right-division. (Contribu... |
| mdiv 12122 | A division law. (Contribu... |
| lineq 12123 | Solution of a (scalar) lin... |
| elimgt0 12124 | Hypothesis for weak deduct... |
| elimge0 12125 | Hypothesis for weak deduct... |
| ltp1 12126 | A number is less than itse... |
| lep1 12127 | A number is less than or e... |
| ltm1 12128 | A number minus 1 is less t... |
| lem1 12129 | A number minus 1 is less t... |
| letrp1 12130 | A transitive property of '... |
| p1le 12131 | A transitive property of p... |
| recgt0 12132 | The reciprocal of a positi... |
| prodgt0 12133 | Infer that a multiplicand ... |
| prodgt02 12134 | Infer that a multiplier is... |
| ltmul1a 12135 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12136 | Multiplication of both sid... |
| ltmul2 12137 | Multiplication of both sid... |
| lemul1 12138 | Multiplication of both sid... |
| lemul2 12139 | Multiplication of both sid... |
| lemul1a 12140 | Multiplication of both sid... |
| lemul2a 12141 | Multiplication of both sid... |
| ltmul12a 12142 | Comparison of product of t... |
| lemul12b 12143 | Comparison of product of t... |
| lemul12a 12144 | Comparison of product of t... |
| ltmulgt11 12145 | Multiplication by a number... |
| ltmulgt12 12146 | Multiplication by a number... |
| mulgt1 12147 | The product of two numbers... |
| lemulge11 12148 | Multiplication by a number... |
| lemulge12 12149 | Multiplication by a number... |
| ltdiv1 12150 | Division of both sides of ... |
| lediv1 12151 | Division of both sides of ... |
| gt0div 12152 | Division of a positive num... |
| ge0div 12153 | Division of a nonnegative ... |
| divgt0 12154 | The ratio of two positive ... |
| divge0 12155 | The ratio of nonnegative a... |
| mulge0b 12156 | A condition for multiplica... |
| mulle0b 12157 | A condition for multiplica... |
| mulsuble0b 12158 | A condition for multiplica... |
| ltmuldiv 12159 | 'Less than' relationship b... |
| ltmuldiv2 12160 | 'Less than' relationship b... |
| ltdivmul 12161 | 'Less than' relationship b... |
| ledivmul 12162 | 'Less than or equal to' re... |
| ltdivmul2 12163 | 'Less than' relationship b... |
| lt2mul2div 12164 | 'Less than' relationship b... |
| ledivmul2 12165 | 'Less than or equal to' re... |
| lemuldiv 12166 | 'Less than or equal' relat... |
| lemuldiv2 12167 | 'Less than or equal' relat... |
| ltrec 12168 | The reciprocal of both sid... |
| lerec 12169 | The reciprocal of both sid... |
| lt2msq1 12170 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12171 | Two nonnegative numbers co... |
| ltdiv2 12172 | Division of a positive num... |
| ltrec1 12173 | Reciprocal swap in a 'less... |
| lerec2 12174 | Reciprocal swap in a 'less... |
| ledivdiv 12175 | Invert ratios of positive ... |
| lediv2 12176 | Division of a positive num... |
| ltdiv23 12177 | Swap denominator with othe... |
| lediv23 12178 | Swap denominator with othe... |
| lediv12a 12179 | Comparison of ratio of two... |
| lediv2a 12180 | Division of both sides of ... |
| reclt1 12181 | The reciprocal of a positi... |
| recgt1 12182 | The reciprocal of a positi... |
| recgt1i 12183 | The reciprocal of a number... |
| recp1lt1 12184 | Construct a number less th... |
| recreclt 12185 | Given a positive number ` ... |
| le2msq 12186 | The square function on non... |
| msq11 12187 | The square of a nonnegativ... |
| ledivp1 12188 | "Less than or equal to" an... |
| squeeze0 12189 | If a nonnegative number is... |
| ltp1i 12190 | A number is less than itse... |
| recgt0i 12191 | The reciprocal of a positi... |
| recgt0ii 12192 | The reciprocal of a positi... |
| prodgt0i 12193 | Infer that a multiplicand ... |
| divgt0i 12194 | The ratio of two positive ... |
| divge0i 12195 | The ratio of nonnegative a... |
| ltreci 12196 | The reciprocal of both sid... |
| lereci 12197 | The reciprocal of both sid... |
| lt2msqi 12198 | The square function on non... |
| le2msqi 12199 | The square function on non... |
| msq11i 12200 | The square of a nonnegativ... |
| divgt0i2i 12201 | The ratio of two positive ... |
| ltrecii 12202 | The reciprocal of both sid... |
| divgt0ii 12203 | The ratio of two positive ... |
| ltmul1i 12204 | Multiplication of both sid... |
| ltdiv1i 12205 | Division of both sides of ... |
| ltmuldivi 12206 | 'Less than' relationship b... |
| ltmul2i 12207 | Multiplication of both sid... |
| lemul1i 12208 | Multiplication of both sid... |
| lemul2i 12209 | Multiplication of both sid... |
| ltdiv23i 12210 | Swap denominator with othe... |
| ledivp1i 12211 | "Less than or equal to" an... |
| ltdivp1i 12212 | Less-than and division rel... |
| ltdiv23ii 12213 | Swap denominator with othe... |
| ltmul1ii 12214 | Multiplication of both sid... |
| ltdiv1ii 12215 | Division of both sides of ... |
| ltp1d 12216 | A number is less than itse... |
| lep1d 12217 | A number is less than or e... |
| ltm1d 12218 | A number minus 1 is less t... |
| lem1d 12219 | A number minus 1 is less t... |
| recgt0d 12220 | The reciprocal of a positi... |
| divgt0d 12221 | The ratio of two positive ... |
| mulgt1d 12222 | The product of two numbers... |
| lemulge11d 12223 | Multiplication by a number... |
| lemulge12d 12224 | Multiplication by a number... |
| lemul1ad 12225 | Multiplication of both sid... |
| lemul2ad 12226 | Multiplication of both sid... |
| ltmul12ad 12227 | Comparison of product of t... |
| lemul12ad 12228 | Comparison of product of t... |
| lemul12bd 12229 | Comparison of product of t... |
| fimaxre 12230 | A finite set of real numbe... |
| fimaxre2 12231 | A nonempty finite set of r... |
| fimaxre3 12232 | A nonempty finite set of r... |
| fiminre 12233 | A nonempty finite set of r... |
| fiminre2 12234 | A nonempty finite set of r... |
| negfi 12235 | The negation of a finite s... |
| lbreu 12236 | If a set of reals contains... |
| lbcl 12237 | If a set of reals contains... |
| lble 12238 | If a set of reals contains... |
| lbinf 12239 | If a set of reals contains... |
| lbinfcl 12240 | If a set of reals contains... |
| lbinfle 12241 | If a set of reals contains... |
| sup2 12242 | A nonempty, bounded-above ... |
| sup3 12243 | A version of the completen... |
| infm3lem 12244 | Lemma for ~ infm3 . (Cont... |
| infm3 12245 | The completeness axiom for... |
| suprcl 12246 | Closure of supremum of a n... |
| suprub 12247 | A member of a nonempty bou... |
| suprubd 12248 | Natural deduction form of ... |
| suprcld 12249 | Natural deduction form of ... |
| suprlub 12250 | The supremum of a nonempty... |
| suprnub 12251 | An upper bound is not less... |
| suprleub 12252 | The supremum of a nonempty... |
| supaddc 12253 | The supremum function dist... |
| supadd 12254 | The supremum function dist... |
| supmul1 12255 | The supremum function dist... |
| supmullem1 12256 | Lemma for ~ supmul . (Con... |
| supmullem2 12257 | Lemma for ~ supmul . (Con... |
| supmul 12258 | The supremum function dist... |
| sup3ii 12259 | A version of the completen... |
| suprclii 12260 | Closure of supremum of a n... |
| suprubii 12261 | A member of a nonempty bou... |
| suprlubii 12262 | The supremum of a nonempty... |
| suprnubii 12263 | An upper bound is not less... |
| suprleubii 12264 | The supremum of a nonempty... |
| riotaneg 12265 | The negative of the unique... |
| negiso 12266 | Negation is an order anti-... |
| dfinfre 12267 | The infimum of a set of re... |
| infrecl 12268 | Closure of infimum of a no... |
| infrenegsup 12269 | The infimum of a set of re... |
| infregelb 12270 | Any lower bound of a nonem... |
| infrelb 12271 | If a nonempty set of real ... |
| infrefilb 12272 | The infimum of a finite se... |
| supfirege 12273 | The supremum of a finite s... |
| neg1cn 12274 | -1 is a complex number. (... |
| neg1rr 12275 | -1 is a real number. (Con... |
| neg1ne0 12276 | -1 is nonzero. (Contribut... |
| neg1lt0 12277 | -1 is less than 0. (Contr... |
| negneg1e1 12278 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12279 | The imaginary unit ` _i ` ... |
| rimul 12280 | A real number times the im... |
| cru 12281 | The representation of comp... |
| crne0 12282 | The real representation of... |
| creur 12283 | The real part of a complex... |
| creui 12284 | The imaginary part of a co... |
| cju 12285 | The complex conjugate of a... |
| ofsubeq0 12286 | Function analogue of ~ sub... |
| ofnegsub 12287 | Function analogue of ~ neg... |
| ofsubge0 12288 | Function analogue of ~ sub... |
| indv 12291 | Value of the indicator fun... |
| indval 12292 | Value of the indicator fun... |
| indval0 12293 | The indicator function gen... |
| indval2 12294 | Alternate value of the ind... |
| indf 12295 | An indicator function as a... |
| indfval 12296 | Value of the indicator fun... |
| fvindre 12297 | The range of the indicator... |
| ind1 12298 | Value of the indicator fun... |
| ind0 12299 | Value of the indicator fun... |
| ind1a 12300 | Value of the indicator fun... |
| indconst0 12301 | Indicator of the empty set... |
| indconst1 12302 | Indicator of the whole set... |
| indpi1 12303 | Preimage of the singleton ... |
| nnexALT 12306 | Alternate proof of ~ nnex ... |
| peano5nni 12307 | Peano's inductive postulat... |
| nnssre 12308 | The positive integers are ... |
| nnsscn 12309 | The positive integers are ... |
| nnex 12310 | The set of positive intege... |
| nnre 12311 | A positive integer is a re... |
| nncn 12312 | A positive integer is a co... |
| nnrei 12313 | A positive integer is a re... |
| nncni 12314 | A positive integer is a co... |
| 1nn 12315 | Peano postulate: 1 is a po... |
| peano2nn 12316 | Peano postulate: a success... |
| dfnn2 12317 | Alternate definition of th... |
| dfnn3 12318 | Alternate definition of th... |
| nnred 12319 | A positive integer is a re... |
| nncnd 12320 | A positive integer is a co... |
| peano2nnd 12321 | Peano postulate: a success... |
| nnind 12322 | Principle of Mathematical ... |
| nnindALT 12323 | Principle of Mathematical ... |
| nnindd 12324 | Principle of Mathematical ... |
| nn1m1nn 12325 | Every positive integer is ... |
| nn1suc 12326 | If a statement holds for 1... |
| nnaddcl 12327 | Closure of addition of pos... |
| nnmulcl 12328 | Closure of multiplication ... |
| nnmulcli 12329 | Closure of multiplication ... |
| nnadd1com 12330 | Addition with 1 is commuta... |
| nnaddcom 12331 | Addition is commutative fo... |
| nnaddcomli 12332 | Version of ~ addcomli for ... |
| nnmtmip 12333 | "Minus times minus is plus... |
| nn2ge 12334 | There exists a positive in... |
| nnge1 12335 | A positive integer is one ... |
| nngt1ne1 12336 | A positive integer is grea... |
| nnle1eq1 12337 | A positive integer is less... |
| nngt0 12338 | A positive integer is posi... |
| nnnlt1 12339 | A positive integer is not ... |
| nnnle0 12340 | A positive integer is not ... |
| nnne0 12341 | A positive integer is nonz... |
| nnneneg 12342 | No positive integer is equ... |
| 0nnn 12343 | Zero is not a positive int... |
| 0nnnALT 12344 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12345 | Alternate version of ~ nnn... |
| nngt0i 12346 | A positive integer is posi... |
| nnne0i 12347 | A positive integer is nonz... |
| nndivre 12348 | The quotient of a real and... |
| nnrecre 12349 | The reciprocal of a positi... |
| nnrecgt0 12350 | The reciprocal of a positi... |
| nnsub 12351 | Subtraction of positive in... |
| nnsubi 12352 | Subtraction of positive in... |
| nndiv 12353 | Two ways to express " ` A ... |
| nndivtr 12354 | Transitive property of div... |
| nnge1d 12355 | A positive integer is one ... |
| nngt0d 12356 | A positive integer is posi... |
| nnne0d 12357 | A positive integer is nonz... |
| nnrecred 12358 | The reciprocal of a positi... |
| nnaddcld 12359 | Closure of addition of pos... |
| nnmulcld 12360 | Closure of multiplication ... |
| nndivred 12361 | A positive integer is one ... |
| 1t1e1ALT 12362 | Alternate proof of ~ 1t1e1... |
| nnadddir 12363 | Right-distributivity for n... |
| nnmul1com 12364 | Multiplication with 1 is c... |
| nnmulcom 12365 | Multiplication is commutat... |
| 1eltp012 12382 | 1 is an element of ` { 0 ,... |
| 0ne1 12383 | Zero is different from one... |
| 1m1e0 12384 | One minus one equals zero.... |
| 2nn 12385 | 2 is a positive integer. ... |
| 2re 12386 | The number 2 is real. (Co... |
| 2cn 12387 | The number 2 is a complex ... |
| 2cnALT 12388 | Alternate proof of ~ 2cn .... |
| 2ex 12389 | The number 2 is a set. (C... |
| 2cnd 12390 | The number 2 is a complex ... |
| 3nn 12391 | 3 is a positive integer. ... |
| 3re 12392 | The number 3 is real. (Co... |
| 3cn 12393 | The number 3 is a complex ... |
| 3ex 12394 | The number 3 is a set. (C... |
| 4nn 12395 | 4 is a positive integer. ... |
| 4re 12396 | The number 4 is real. (Co... |
| 4cn 12397 | The number 4 is a complex ... |
| 5nn 12398 | 5 is a positive integer. ... |
| 5re 12399 | The number 5 is real. (Co... |
| 5cn 12400 | The number 5 is a complex ... |
| 6nn 12401 | 6 is a positive integer. ... |
| 6re 12402 | The number 6 is real. (Co... |
| 6cn 12403 | The number 6 is a complex ... |
| 7nn 12404 | 7 is a positive integer. ... |
| 7re 12405 | The number 7 is real. (Co... |
| 7cn 12406 | The number 7 is a complex ... |
| 8nn 12407 | 8 is a positive integer. ... |
| 8re 12408 | The number 8 is real. (Co... |
| 8cn 12409 | The number 8 is a complex ... |
| 9nn 12410 | 9 is a positive integer. ... |
| 9re 12411 | The number 9 is real. (Co... |
| 9cn 12412 | The number 9 is a complex ... |
| 0le0 12413 | Zero is nonnegative. (Con... |
| 0le2 12414 | The number 0 is less than ... |
| 0le2OLD 12415 | Obsolete version of ~ 0le2... |
| 2pos 12416 | The number 2 is positive. ... |
| 2posOLD 12417 | Obsolete version of ~ 2pos... |
| 2ne0 12418 | The number 2 is nonzero. ... |
| 2thalfe1 12419 | 2 times one half equals 1.... |
| 3pos 12420 | The number 3 is positive. ... |
| 3ne0 12421 | The number 3 is nonzero. ... |
| 4pos 12422 | The number 4 is positive. ... |
| 4ne0 12423 | The number 4 is nonzero. ... |
| 5pos 12424 | The number 5 is positive. ... |
| 6pos 12425 | The number 6 is positive. ... |
| 7pos 12426 | The number 7 is positive. ... |
| 8pos 12427 | The number 8 is positive. ... |
| 9pos 12428 | The number 9 is positive. ... |
| 1pneg1e0 12429 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12430 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12431 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12432 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12433 | Function value at ` N + 1 ... |
| 1p0e1 12434 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12435 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12436 | 2 - 1 = 1. The result is ... |
| 2m1e1OLD 12437 | Obsolete version of ~ 2m1e... |
| 1e2m1 12438 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12439 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12440 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12441 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12442 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12443 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12444 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12445 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12446 | Two plus two equals four. ... |
| 2times 12447 | Two times a number. (Cont... |
| times2 12448 | A number times 2. (Contri... |
| 2timesi 12449 | Two times a number. (Cont... |
| times2i 12450 | A number times 2. (Contri... |
| 2txmxeqx 12451 | Two times a complex number... |
| 2div2e1 12452 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12453 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12454 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12455 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12456 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12457 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12458 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12459 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12460 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12461 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12462 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12463 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12464 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12465 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12466 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12467 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12468 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12469 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12470 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12471 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12472 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12473 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12474 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12475 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12476 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12477 | 3 times 2 equals 6. (Cont... |
| 2t3e6 12478 | 2 times 3 equals 6. (Cont... |
| 3t3e9 12479 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12480 | 4 times 2 equals 8. (Cont... |
| 2t4e8 12481 | 2 times 4 equals 8. (Cont... |
| 2t0e0 12482 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12483 | One half of four is two. ... |
| 1lt2 12484 | 1 is less than 2. (Contri... |
| 2lt3 12485 | 2 is less than 3. (Contri... |
| 2le3 12486 | 2 is less than or equal to... |
| 1lt3 12487 | 1 is less than 3. (Contri... |
| 3lt4 12488 | 3 is less than 4. (Contri... |
| 2lt4 12489 | 2 is less than 4. (Contri... |
| 1lt4 12490 | 1 is less than 4. (Contri... |
| 4lt5 12491 | 4 is less than 5. (Contri... |
| 3lt5 12492 | 3 is less than 5. (Contri... |
| 2lt5 12493 | 2 is less than 5. (Contri... |
| 1lt5 12494 | 1 is less than 5. (Contri... |
| 5lt6 12495 | 5 is less than 6. (Contri... |
| 4lt6 12496 | 4 is less than 6. (Contri... |
| 3lt6 12497 | 3 is less than 6. (Contri... |
| 2lt6 12498 | 2 is less than 6. (Contri... |
| 1lt6 12499 | 1 is less than 6. (Contri... |
| 6lt7 12500 | 6 is less than 7. (Contri... |
| 5lt7 12501 | 5 is less than 7. (Contri... |
| 4lt7 12502 | 4 is less than 7. (Contri... |
| 3lt7 12503 | 3 is less than 7. (Contri... |
| 2lt7 12504 | 2 is less than 7. (Contri... |
| 1lt7 12505 | 1 is less than 7. (Contri... |
| 7lt8 12506 | 7 is less than 8. (Contri... |
| 6lt8 12507 | 6 is less than 8. (Contri... |
| 5lt8 12508 | 5 is less than 8. (Contri... |
| 4lt8 12509 | 4 is less than 8. (Contri... |
| 3lt8 12510 | 3 is less than 8. (Contri... |
| 2lt8 12511 | 2 is less than 8. (Contri... |
| 1lt8 12512 | 1 is less than 8. (Contri... |
| 8lt9 12513 | 8 is less than 9. (Contri... |
| 7lt9 12514 | 7 is less than 9. (Contri... |
| 6lt9 12515 | 6 is less than 9. (Contri... |
| 5lt9 12516 | 5 is less than 9. (Contri... |
| 4lt9 12517 | 4 is less than 9. (Contri... |
| 3lt9 12518 | 3 is less than 9. (Contri... |
| 2lt9 12519 | 2 is less than 9. (Contri... |
| 1lt9 12520 | 1 is less than 9. (Contri... |
| 0ne2 12521 | 0 is not equal to 2. (Con... |
| 1ne2 12522 | 1 is not equal to 2. (Con... |
| 1le2 12523 | 1 is less than or equal to... |
| 2cnne0 12524 | 2 is a nonzero complex num... |
| 2rene0 12525 | 2 is a nonzero real number... |
| 1le3 12526 | 1 is less than or equal to... |
| neg1mulneg1e1 12527 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12528 | One-half is real. (Contri... |
| halfcn 12529 | One-half is a complex numb... |
| halfgt0 12530 | One-half is greater than z... |
| halfge0 12531 | One-half is not negative. ... |
| halflt1 12532 | One-half is less than one.... |
| 2halves 12533 | Two halves make a whole. ... |
| 1mhlfehlf 12534 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12535 | An eighth of four thirds i... |
| halfthird 12536 | Half minus a third. (Cont... |
| halfpm6th 12537 | One half plus or minus one... |
| it0e0 12538 | i times 0 equals 0. (Cont... |
| 2mulicn 12539 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12540 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12541 | Closure of half of a numbe... |
| rehalfcl 12542 | Real closure of half. (Co... |
| half0 12543 | Half of a number is zero i... |
| halfpos2 12544 | A number is positive iff i... |
| halfpos 12545 | A positive number is great... |
| halfnneg2 12546 | A number is nonnegative if... |
| halfaddsubcl 12547 | Closure of half-sum and ha... |
| halfaddsub 12548 | Sum and difference of half... |
| subhalfhalf 12549 | Subtracting the half of a ... |
| lt2halves 12550 | A sum is less than the who... |
| addltmul 12551 | Sum is less than product f... |
| nominpos 12552 | There is no smallest posit... |
| avglt1 12553 | Ordering property for aver... |
| avglt2 12554 | Ordering property for aver... |
| avgle1 12555 | Ordering property for aver... |
| avgle2 12556 | Ordering property for aver... |
| avgle 12557 | The average of two numbers... |
| 2timesd 12558 | Two times a number. (Cont... |
| times2d 12559 | A number times 2. (Contri... |
| halfcld 12560 | Closure of half of a numbe... |
| 2halvesd 12561 | Two halves make a whole. ... |
| rehalfcld 12562 | Real closure of half. (Co... |
| lt2halvesd 12563 | A sum is less than the who... |
| rehalfcli 12564 | Half a real number is real... |
| lt2addmuld 12565 | If two real numbers are le... |
| add1p1 12566 | Adding two times 1 to a nu... |
| sub1m1 12567 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12568 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12569 | A complex number increased... |
| div4p1lem1div2 12570 | An integer greater than 5,... |
| nnunb 12571 | The set of positive intege... |
| arch 12572 | Archimedean property of re... |
| nnrecl 12573 | There exists a positive in... |
| bndndx 12574 | A bounded real sequence ` ... |
| elnn0 12577 | Nonnegative integers expre... |
| nnssnn0 12578 | Positive naturals are a su... |
| nn0ssre 12579 | Nonnegative integers are a... |
| nn0sscn 12580 | Nonnegative integers are a... |
| nn0ex 12581 | The set of nonnegative int... |
| nnnn0 12582 | A positive integer is a no... |
| nnnn0i 12583 | A positive integer is a no... |
| nn0re 12584 | A nonnegative integer is a... |
| nn0cn 12585 | A nonnegative integer is a... |
| nn0rei 12586 | A nonnegative integer is a... |
| nn0cni 12587 | A nonnegative integer is a... |
| dfn2 12588 | The set of positive intege... |
| elnnne0 12589 | The positive integer prope... |
| 0nn0 12590 | 0 is a nonnegative integer... |
| 1nn0 12591 | 1 is a nonnegative integer... |
| 2nn0 12592 | 2 is a nonnegative integer... |
| 3nn0 12593 | 3 is a nonnegative integer... |
| 4nn0 12594 | 4 is a nonnegative integer... |
| 5nn0 12595 | 5 is a nonnegative integer... |
| 6nn0 12596 | 6 is a nonnegative integer... |
| 7nn0 12597 | 7 is a nonnegative integer... |
| 8nn0 12598 | 8 is a nonnegative integer... |
| 9nn0 12599 | 9 is a nonnegative integer... |
| nn0ge0 12600 | A nonnegative integer is g... |
| nn0nlt0 12601 | A nonnegative integer is n... |
| nn0ge0i 12602 | Nonnegative integers are n... |
| nn0le0eq0 12603 | A nonnegative integer is l... |
| nn0p1gt0 12604 | A nonnegative integer incr... |
| nnnn0addcl 12605 | A positive integer plus a ... |
| nn0nnaddcl 12606 | A nonnegative integer plus... |
| 0mnnnnn0 12607 | The result of subtracting ... |
| un0addcl 12608 | If ` S ` is closed under a... |
| un0mulcl 12609 | If ` S ` is closed under m... |
| nn0addcl 12610 | Closure of addition of non... |
| nn0mulcl 12611 | Closure of multiplication ... |
| nn0addcli 12612 | Closure of addition of non... |
| nn0mulcli 12613 | Closure of multiplication ... |
| nn0p1nn 12614 | A nonnegative integer plus... |
| peano2nn0 12615 | Second Peano postulate for... |
| nnm1nn0 12616 | A positive integer minus 1... |
| elnn0nn 12617 | The nonnegative integer pr... |
| elnnnn0 12618 | The positive integer prope... |
| elnnnn0b 12619 | The positive integer prope... |
| elnnnn0c 12620 | The positive integer prope... |
| nn0addge1 12621 | A number is less than or e... |
| nn0addge2 12622 | A number is less than or e... |
| nn0addge1i 12623 | A number is less than or e... |
| nn0addge2i 12624 | A number is less than or e... |
| nn0sub 12625 | Subtraction of nonnegative... |
| ltsubnn0 12626 | Subtracting a nonnegative ... |
| nn0negleid 12627 | A nonnegative integer is g... |
| difgtsumgt 12628 | If the difference of a rea... |
| nn0le2x 12629 | A nonnegative integer is l... |
| nn0le2xi 12630 | A nonnegative integer is l... |
| nn0lele2xi 12631 | 'Less than or equal to' im... |
| fcdmnn0supp 12632 | Two ways to write the supp... |
| fcdmnn0fsupp 12633 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12634 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12635 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12636 | A positive integer is a no... |
| nn0red 12637 | A nonnegative integer is a... |
| nn0cnd 12638 | A nonnegative integer is a... |
| nn0ge0d 12639 | A nonnegative integer is g... |
| nn0addcld 12640 | Closure of addition of non... |
| nn0mulcld 12641 | Closure of multiplication ... |
| nn0readdcl 12642 | Closure law for addition o... |
| nn0n0n1ge2 12643 | A nonnegative integer whic... |
| nn0n0n1ge2b 12644 | A nonnegative integer is n... |
| nn0ge2m1nn 12645 | If a nonnegative integer i... |
| nn0ge2m1nn0 12646 | If a nonnegative integer i... |
| nn0nndivcl 12647 | Closure law for dividing o... |
| elxnn0 12650 | An extended nonnegative in... |
| nn0ssxnn0 12651 | The standard nonnegative i... |
| nn0xnn0 12652 | A standard nonnegative int... |
| xnn0xr 12653 | An extended nonnegative in... |
| 0xnn0 12654 | Zero is an extended nonneg... |
| pnf0xnn0 12655 | Positive infinity is an ex... |
| nn0nepnf 12656 | No standard nonnegative in... |
| nn0xnn0d 12657 | A standard nonnegative int... |
| nn0nepnfd 12658 | No standard nonnegative in... |
| xnn0nemnf 12659 | No extended nonnegative in... |
| xnn0xrnemnf 12660 | The extended nonnegative i... |
| xnn0nnn0pnf 12661 | An extended nonnegative in... |
| elz 12664 | Membership in the set of i... |
| nnnegz 12665 | The negative of a positive... |
| zre 12666 | An integer is a real. (Co... |
| zcn 12667 | An integer is a complex nu... |
| zrei 12668 | An integer is a real numbe... |
| zssre 12669 | The integers are a subset ... |
| zsscn 12670 | The integers are a subset ... |
| zex 12671 | The set of integers exists... |
| elnnz 12672 | Positive integer property ... |
| 0z 12673 | Zero is an integer. (Cont... |
| 0zd 12674 | Zero is an integer, deduct... |
| elnn0z 12675 | Nonnegative integer proper... |
| elznn0nn 12676 | Integer property expressed... |
| elznn0 12677 | Integer property expressed... |
| elznn 12678 | Integer property expressed... |
| zle0orge1 12679 | There is no integer in the... |
| elz2 12680 | Membership in the set of i... |
| dfz2 12681 | Alternative definition of ... |
| zexALT 12682 | Alternate proof of ~ zex .... |
| nnz 12683 | A positive integer is an i... |
| nnssz 12684 | Positive integers are a su... |
| nn0ssz 12685 | Nonnegative integers are a... |
| nn0z 12686 | A nonnegative integer is a... |
| nn0zd 12687 | A nonnegative integer is a... |
| nnzd 12688 | A positive integer is an i... |
| nnzi 12689 | A positive integer is an i... |
| nn0zi 12690 | A nonnegative integer is a... |
| elnnz1 12691 | Positive integer property ... |
| znnnlt1 12692 | An integer is not a positi... |
| nnzrab 12693 | Positive integers expresse... |
| nn0zrab 12694 | Nonnegative integers expre... |
| 1z 12695 | One is an integer. (Contr... |
| 1zzd 12696 | One is an integer, deducti... |
| 2z 12697 | 2 is an integer. (Contrib... |
| 3z 12698 | 3 is an integer. (Contrib... |
| 4z 12699 | 4 is an integer. (Contrib... |
| znegcl 12700 | Closure law for negative i... |
| neg1z 12701 | -1 is an integer. (Contri... |
| znegclb 12702 | A complex number is an int... |
| nn0negz 12703 | The negative of a nonnegat... |
| nn0negzi 12704 | The negative of a nonnegat... |
| zaddcl 12705 | Closure of addition of int... |
| peano2z 12706 | Second Peano postulate gen... |
| zsubcl 12707 | Closure of subtraction of ... |
| peano2zm 12708 | "Reverse" second Peano pos... |
| zletr 12709 | Transitive law of ordering... |
| zrevaddcl 12710 | Reverse closure law for ad... |
| znnsub 12711 | The positive difference of... |
| znn0sub 12712 | The nonnegative difference... |
| nzadd 12713 | The sum of a real number n... |
| zmulcl 12714 | Closure of multiplication ... |
| zltp1le 12715 | Integer ordering relation.... |
| zleltp1 12716 | Integer ordering relation.... |
| zlem1lt 12717 | Integer ordering relation.... |
| zltlem1 12718 | Integer ordering relation.... |
| zltlem1d 12719 | Integer ordering relation,... |
| zltp1led 12720 | Integer ordering relation,... |
| zgt0ge1 12721 | An integer greater than ` ... |
| 0nn0m1nnn0 12722 | A number is zero if and on... |
| nnleltp1 12723 | Positive integer ordering ... |
| nnltp1le 12724 | Positive integer ordering ... |
| nnaddm1cl 12725 | Closure of addition of pos... |
| nn0ltp1le 12726 | Nonnegative integer orderi... |
| nn0leltp1 12727 | Nonnegative integer orderi... |
| nn0ltlem1 12728 | Nonnegative integer orderi... |
| nn0sub2 12729 | Subtraction of nonnegative... |
| nn0lt10b 12730 | A nonnegative integer less... |
| nn0lt2 12731 | A nonnegative integer less... |
| nn0le2is012 12732 | A nonnegative integer whic... |
| nn0lem1lt 12733 | Nonnegative integer orderi... |
| nnlem1lt 12734 | Positive integer ordering ... |
| nnltlem1 12735 | Positive integer ordering ... |
| nnm1ge0 12736 | A positive integer decreas... |
| nn0ge0div 12737 | Division of a nonnegative ... |
| zdiv 12738 | Two ways to express " ` M ... |
| zdivadd 12739 | Property of divisibility: ... |
| zdivmul 12740 | Property of divisibility: ... |
| zextle 12741 | An extensionality-like pro... |
| zextlt 12742 | An extensionality-like pro... |
| recnz 12743 | The reciprocal of a number... |
| btwnnz 12744 | A number between an intege... |
| gtndiv 12745 | A larger number does not d... |
| halfnz 12746 | One-half is not an integer... |
| 3halfnz 12747 | Three halves is not an int... |
| suprzcl 12748 | The supremum of a bounded-... |
| prime 12749 | Two ways to express " ` A ... |
| msqznn 12750 | The square of a nonzero in... |
| zneo 12751 | No even integer equals an ... |
| nneo 12752 | A positive integer is even... |
| nneoi 12753 | A positive integer is even... |
| zeo 12754 | An integer is even or odd.... |
| zeo2 12755 | An integer is even or odd ... |
| peano2uz2 12756 | Second Peano postulate for... |
| peano5uzi 12757 | Peano's inductive postulat... |
| peano5uzti 12758 | Peano's inductive postulat... |
| dfuzi 12759 | An expression for the uppe... |
| uzind 12760 | Induction on the upper int... |
| uzind2 12761 | Induction on the upper int... |
| uzind3 12762 | Induction on the upper int... |
| nn0ind 12763 | Principle of Mathematical ... |
| nn0indALT 12764 | Principle of Mathematical ... |
| nn0indd 12765 | Principle of Mathematical ... |
| fzind 12766 | Induction on the integers ... |
| fnn0ind 12767 | Induction on the integers ... |
| nn0ind-raph 12768 | Principle of Mathematical ... |
| zindd 12769 | Principle of Mathematical ... |
| fzindd 12770 | Induction on the integers ... |
| btwnz 12771 | Any real number can be san... |
| zred 12772 | An integer is a real numbe... |
| zcnd 12773 | An integer is a complex nu... |
| znegcld 12774 | Closure law for negative i... |
| peano2zd 12775 | Deduction from second Pean... |
| zaddcld 12776 | Closure of addition of int... |
| zsubcld 12777 | Closure of subtraction of ... |
| zmulcld 12778 | Closure of multiplication ... |
| znnn0nn 12779 | The negative of a negative... |
| zadd2cl 12780 | Increasing an integer by 2... |
| zriotaneg 12781 | The negative of the unique... |
| suprfinzcl 12782 | The supremum of a nonempty... |
| 9p1e10 12785 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12786 | Version of the definition ... |
| decex 12787 | A decimal number is a set.... |
| deceq1 12788 | Equality theorem for the d... |
| deceq2 12789 | Equality theorem for the d... |
| deceq1i 12790 | Equality theorem for the d... |
| deceq2i 12791 | Equality theorem for the d... |
| deceq12i 12792 | Equality theorem for the d... |
| numnncl 12793 | Closure for a numeral (wit... |
| num0u 12794 | Add a zero in the units pl... |
| num0h 12795 | Add a zero in the higher p... |
| numcl 12796 | Closure for a decimal inte... |
| numsuc 12797 | The successor of a decimal... |
| deccl 12798 | Closure for a numeral. (C... |
| 11nn0 12799 | 11 is a nonnegative intege... |
| 12nn0 12800 | 12 is a nonnegative intege... |
| 16nn0 12801 | 16 is a nonnegative intege... |
| 25nn0 12802 | 25 is a nonnegative intege... |
| 10nn 12803 | 10 is a positive integer. ... |
| 10pos 12804 | The number 10 is positive.... |
| 10nn0 12805 | 10 is a nonnegative intege... |
| 10re 12806 | The number 10 is real. (C... |
| decnncl 12807 | Closure for a numeral. (C... |
| 11nn 12808 | 11 is a positive integer. ... |
| dec0u 12809 | Add a zero in the units pl... |
| dec0h 12810 | Add a zero in the higher p... |
| numnncl2 12811 | Closure for a decimal inte... |
| decnncl2 12812 | Closure for a decimal inte... |
| numlt 12813 | Comparing two decimal inte... |
| numltc 12814 | Comparing two decimal inte... |
| le9lt10 12815 | A "decimal digit" (i.e. a ... |
| declt 12816 | Comparing two decimal inte... |
| decltc 12817 | Comparing two decimal inte... |
| declth 12818 | Comparing two decimal inte... |
| decsuc 12819 | The successor of a decimal... |
| 3declth 12820 | Comparing two decimal inte... |
| 3decltc 12821 | Comparing two decimal inte... |
| decle 12822 | Comparing two decimal inte... |
| decleh 12823 | Comparing two decimal inte... |
| declei 12824 | Comparing a digit to a dec... |
| numlti 12825 | Comparing a digit to a dec... |
| declti 12826 | Comparing a digit to a dec... |
| decltdi 12827 | Comparing a digit to a dec... |
| numsucc 12828 | The successor of a decimal... |
| decsucc 12829 | The successor of a decimal... |
| 1e0p1 12830 | The successor of zero. (C... |
| dec10p 12831 | Ten plus an integer. (Con... |
| numma 12832 | Perform a multiply-add of ... |
| nummac 12833 | Perform a multiply-add of ... |
| numma2c 12834 | Perform a multiply-add of ... |
| numadd 12835 | Add two decimal integers `... |
| numaddc 12836 | Add two decimal integers `... |
| nummul1c 12837 | The product of a decimal i... |
| nummul2c 12838 | The product of a decimal i... |
| decma 12839 | Perform a multiply-add of ... |
| decmac 12840 | Perform a multiply-add of ... |
| decma2c 12841 | Perform a multiply-add of ... |
| decadd 12842 | Add two numerals ` M ` and... |
| decaddc 12843 | Add two numerals ` M ` and... |
| decaddc2 12844 | Add two numerals ` M ` and... |
| decrmanc 12845 | Perform a multiply-add of ... |
| decrmac 12846 | Perform a multiply-add of ... |
| decaddm10 12847 | The sum of two multiples o... |
| decaddi 12848 | Add two numerals ` M ` and... |
| decaddci 12849 | Add two numerals ` M ` and... |
| decaddci2 12850 | Add two numerals ` M ` and... |
| decsubi 12851 | Difference between a numer... |
| decmul1 12852 | The product of a numeral w... |
| decmul1c 12853 | The product of a numeral w... |
| decmul2c 12854 | The product of a numeral w... |
| decmulnc 12855 | The product of a numeral w... |
| 11multnc 12856 | The product of 11 (as nume... |
| decmul10add 12857 | A multiplication of a numb... |
| 6p5lem 12858 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12859 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12860 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12861 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12862 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12863 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12864 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12865 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12866 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12867 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12868 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12869 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12870 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12871 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12872 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12873 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12874 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12875 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12876 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12877 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12878 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12879 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12880 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12881 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12882 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12883 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12884 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12885 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12886 | 4 times 3 equals 12. (Con... |
| 4t4e16 12887 | 4 times 4 equals 16. (Con... |
| 5t2e10 12888 | 5 times 2 equals 10. (Con... |
| 5t3e15 12889 | 5 times 3 equals 15. (Con... |
| 5t4e20 12890 | 5 times 4 equals 20. (Con... |
| 5t5e25 12891 | 5 times 5 equals 25. (Con... |
| 6t2e12 12892 | 6 times 2 equals 12. (Con... |
| 6t3e18 12893 | 6 times 3 equals 18. (Con... |
| 6t4e24 12894 | 6 times 4 equals 24. (Con... |
| 6t5e30 12895 | 6 times 5 equals 30. (Con... |
| 6t6e36 12896 | 6 times 6 equals 36. (Con... |
| 7t2e14 12897 | 7 times 2 equals 14. (Con... |
| 7t3e21 12898 | 7 times 3 equals 21. (Con... |
| 7t4e28 12899 | 7 times 4 equals 28. (Con... |
| 7t5e35 12900 | 7 times 5 equals 35. (Con... |
| 7t6e42 12901 | 7 times 6 equals 42. (Con... |
| 7t7e49 12902 | 7 times 7 equals 49. (Con... |
| 8t2e16 12903 | 8 times 2 equals 16. (Con... |
| 8t3e24 12904 | 8 times 3 equals 24. (Con... |
| 8t4e32 12905 | 8 times 4 equals 32. (Con... |
| 8t5e40 12906 | 8 times 5 equals 40. (Con... |
| 8t6e48 12907 | 8 times 6 equals 48. (Con... |
| 8t7e56 12908 | 8 times 7 equals 56. (Con... |
| 8t8e64 12909 | 8 times 8 equals 64. (Con... |
| 9t2e18 12910 | 9 times 2 equals 18. (Con... |
| 9t3e27 12911 | 9 times 3 equals 27. (Con... |
| 9t4e36 12912 | 9 times 4 equals 36. (Con... |
| 9t5e45 12913 | 9 times 5 equals 45. (Con... |
| 9t6e54 12914 | 9 times 6 equals 54. (Con... |
| 9t7e63 12915 | 9 times 7 equals 63. (Con... |
| 9t8e72 12916 | 9 times 8 equals 72. (Con... |
| 9t9e81 12917 | 9 times 9 equals 81. (Con... |
| 9t11e99 12918 | 9 times 11 equals 99. (Co... |
| 9t11e99OLD 12919 | Obsolete version of ~ 9t11... |
| 9lt10 12920 | 9 is less than 10. (Contr... |
| 8lt10 12921 | 8 is less than 10. (Contr... |
| 7lt10 12922 | 7 is less than 10. (Contr... |
| 6lt10 12923 | 6 is less than 10. (Contr... |
| 5lt10 12924 | 5 is less than 10. (Contr... |
| 4lt10 12925 | 4 is less than 10. (Contr... |
| 3lt10 12926 | 3 is less than 10. (Contr... |
| 2lt10 12927 | 2 is less than 10. (Contr... |
| 1lt10 12928 | 1 is less than 10. (Contr... |
| 1lt10OLD 12929 | Obsolete version of ~ 1lt1... |
| decbin0 12930 | Decompose base 4 into base... |
| decbin2 12931 | Decompose base 4 into base... |
| decbin3 12932 | Decompose base 4 into base... |
| 5recm6rec 12933 | One fifth minus one sixth.... |
| uzval 12936 | The value of the upper int... |
| uzf 12937 | The domain and codomain of... |
| eluz1 12938 | Membership in the upper se... |
| eluzel2 12939 | Implication of membership ... |
| eluz2 12940 | Membership in an upper set... |
| eluzmn 12941 | Membership in an earlier u... |
| eluz1i 12942 | Membership in an upper set... |
| eluzuzle 12943 | An integer in an upper set... |
| eluzelz 12944 | A member of an upper set o... |
| eluzelre 12945 | A member of an upper set o... |
| eluzelcn 12946 | A member of an upper set o... |
| eluzle 12947 | Implication of membership ... |
| eluz 12948 | Membership in an upper set... |
| uzid 12949 | Membership of the least me... |
| uzidd 12950 | Membership of the least me... |
| uzn0 12951 | The upper integers are all... |
| uztrn 12952 | Transitive law for sets of... |
| uztrn2 12953 | Transitive law for sets of... |
| uzneg 12954 | Contraposition law for upp... |
| uzssz 12955 | An upper set of integers i... |
| uzssre 12956 | An upper set of integers i... |
| uzss 12957 | Subset relationship for tw... |
| uztric 12958 | Totality of the ordering r... |
| uz11 12959 | The upper integers functio... |
| eluzp1m1 12960 | Membership in the next upp... |
| eluzp1l 12961 | Strict ordering implied by... |
| eluzp1p1 12962 | Membership in the next upp... |
| eluzadd 12963 | Membership in a later uppe... |
| eluzsub 12964 | Membership in an earlier u... |
| eluzaddi 12965 | Membership in a later uppe... |
| eluzsubi 12966 | Membership in an earlier u... |
| subeluzsub 12967 | Membership of a difference... |
| uzm1 12968 | Choices for an element of ... |
| uznn0sub 12969 | The nonnegative difference... |
| uzin 12970 | Intersection of two upper ... |
| uzp1 12971 | Choices for an element of ... |
| nn0uz 12972 | Nonnegative integers expre... |
| nnuz 12973 | Positive integers expresse... |
| elnnuz 12974 | A positive integer express... |
| elnn0uz 12975 | A nonnegative integer expr... |
| 1eluzge0 12976 | 1 is an integer greater th... |
| 2eluzge0 12977 | 2 is an integer greater th... |
| 2eluzge1 12978 | 2 is an integer greater th... |
| 5eluz3 12979 | 5 is an integer greater th... |
| uzuzle23 12980 | An integer greater than or... |
| uzuzle24 12981 | An integer greater than or... |
| uzuzle34 12982 | An integer greater than or... |
| uzuzle35 12983 | An integer greater than or... |
| eluz2nn 12984 | An integer greater than or... |
| eluz3nn 12985 | An integer greater than or... |
| eluz4nn 12986 | An integer greater than or... |
| eluz5nn 12987 | An integer greater than or... |
| eluzge2nn0 12988 | If an integer is greater t... |
| eluz2n0 12989 | An integer greater than or... |
| uz3m2nn 12990 | An integer greater than or... |
| uznnssnn 12991 | The upper integers startin... |
| raluz 12992 | Restricted universal quant... |
| raluz2 12993 | Restricted universal quant... |
| rexuz 12994 | Restricted existential qua... |
| rexuz2 12995 | Restricted existential qua... |
| 2rexuz 12996 | Double existential quantif... |
| peano2uz 12997 | Second Peano postulate for... |
| peano2uzs 12998 | Second Peano postulate for... |
| peano2uzr 12999 | Reversed second Peano axio... |
| uzaddcl 13000 | Addition closure law for a... |
| nn0pzuz 13001 | The sum of a nonnegative i... |
| uzind4 13002 | Induction on the upper set... |
| uzind4ALT 13003 | Induction on the upper set... |
| uzind4s 13004 | Induction on the upper set... |
| uzind4s2 13005 | Induction on the upper set... |
| uzind4i 13006 | Induction on the upper int... |
| uzwo 13007 | Well-ordering principle: a... |
| uzwo2 13008 | Well-ordering principle: a... |
| nnwo 13009 | Well-ordering principle: a... |
| nnwof 13010 | Well-ordering principle: a... |
| nnwos 13011 | Well-ordering principle: a... |
| indstr 13012 | Strong Mathematical Induct... |
| eluznn0 13013 | Membership in a nonnegativ... |
| eluznn 13014 | Membership in a positive u... |
| eluz2b1 13015 | Two ways to say "an intege... |
| eluz2gt1 13016 | An integer greater than or... |
| eluz2b2 13017 | Two ways to say "an intege... |
| eluz2b3 13018 | Two ways to say "an intege... |
| uz2m1nn 13019 | One less than an integer g... |
| 1nuz2 13020 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 13021 | A positive integer is eith... |
| uz2mulcl 13022 | Closure of multiplication ... |
| indstr2 13023 | Strong Mathematical Induct... |
| uzinfi 13024 | Extract the lower bound of... |
| nninf 13025 | The infimum of the set of ... |
| nn0inf 13026 | The infimum of the set of ... |
| infssuzle 13027 | The infimum of a subset of... |
| infssuzcl 13028 | The infimum of a subset of... |
| ublbneg 13029 | The image under negation o... |
| eqreznegel 13030 | Two ways to express the im... |
| supminf 13031 | The supremum of a bounded-... |
| lbzbi 13032 | If a set of reals is bound... |
| zsupss 13033 | Any nonempty bounded subse... |
| suprzcl2 13034 | The supremum of a bounded-... |
| suprzub 13035 | The supremum of a bounded-... |
| uzsupss 13036 | Any bounded subset of an u... |
| nn01to3 13037 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 13038 | Alternate proof of ~ nn0ge... |
| uzwo3 13039 | Well-ordering principle: a... |
| zmin 13040 | There is a unique smallest... |
| zmax 13041 | There is a unique largest ... |
| zbtwnre 13042 | There is a unique integer ... |
| rebtwnz 13043 | There is a unique greatest... |
| elq 13046 | Membership in the set of r... |
| qmulz 13047 | If ` A ` is rational, then... |
| znq 13048 | The ratio of an integer an... |
| qre 13049 | A rational number is a rea... |
| zq 13050 | An integer is a rational n... |
| qred 13051 | A rational number is a rea... |
| zssq 13052 | The integers are a subset ... |
| nn0ssq 13053 | The nonnegative integers a... |
| nnssq 13054 | The positive integers are ... |
| qssre 13055 | The rationals are a subset... |
| qsscn 13056 | The rationals are a subset... |
| qex 13057 | The set of rational number... |
| nnq 13058 | A positive integer is rati... |
| qcn 13059 | A rational number is a com... |
| qexALT 13060 | Alternate proof of ~ qex .... |
| 1q 13061 | The number 1 is rational. ... |
| qaddcl 13062 | Closure of addition of rat... |
| qnegcl 13063 | Closure law for the negati... |
| qmulcl 13064 | Closure of multiplication ... |
| qsubcl 13065 | Closure of subtraction of ... |
| qreccl 13066 | Closure of reciprocal of r... |
| qdivcl 13067 | Closure of division of rat... |
| qrevaddcl 13068 | Reverse closure law for ad... |
| nnrecq 13069 | The reciprocal of a positi... |
| irradd 13070 | The sum of an irrational n... |
| irrmul 13071 | The product of an irration... |
| elpq 13072 | A positive rational is the... |
| elpqb 13073 | A class is a positive rati... |
| rpnnen1lem2 13074 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 13075 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 13076 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 13077 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 13078 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 13079 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 13080 | One half of ~ rpnnen , whe... |
| reexALT 13081 | Alternate proof of ~ reex ... |
| cnref1o 13082 | There is a natural one-to-... |
| cnexALT 13083 | The set of complex numbers... |
| xrex 13084 | The set of extended reals ... |
| mpoaddex 13085 | The addition operation is ... |
| addex 13086 | The addition operation is ... |
| mpomulex 13087 | The multiplication operati... |
| mulex 13088 | The multiplication operati... |
| elrp 13091 | Membership in the set of p... |
| elrpii 13092 | Membership in the set of p... |
| 1rp 13093 | 1 is a positive real. (Co... |
| 2rp 13094 | 2 is a positive real. (Co... |
| 3rp 13095 | 3 is a positive real. (Co... |
| 5rp 13096 | 5 is a positive real. (Co... |
| rpssre 13097 | The positive reals are a s... |
| rpre 13098 | A positive real is a real.... |
| rpxr 13099 | A positive real is an exte... |
| rpcn 13100 | A positive real is a compl... |
| nnrp 13101 | A positive integer is a po... |
| rpgt0 13102 | A positive real is greater... |
| rpge0 13103 | A positive real is greater... |
| rpregt0 13104 | A positive real is a posit... |
| rprege0 13105 | A positive real is a nonne... |
| rpne0 13106 | A positive real is nonzero... |
| rprene0 13107 | A positive real is a nonze... |
| rpcnne0 13108 | A positive real is a nonze... |
| neglt 13109 | The negative of a positive... |
| rpcndif0 13110 | A positive real number is ... |
| ralrp 13111 | Quantification over positi... |
| rexrp 13112 | Quantification over positi... |
| rpaddcl 13113 | Closure law for addition o... |
| rpmulcl 13114 | Closure law for multiplica... |
| rpmtmip 13115 | "Minus times minus is plus... |
| rpdivcl 13116 | Closure law for division o... |
| rpreccl 13117 | Closure law for reciprocat... |
| rphalfcl 13118 | Closure law for half of a ... |
| rpgecl 13119 | A number greater than or e... |
| rphalflt 13120 | Half of a positive real is... |
| rerpdivcl 13121 | Closure law for division o... |
| ge0p1rp 13122 | A nonnegative number plus ... |
| rpneg 13123 | Either a nonzero real or i... |
| negelrp 13124 | Elementhood of a negation ... |
| negelrpd 13125 | The negation of a negative... |
| 0nrp 13126 | Zero is not a positive rea... |
| ltsubrp 13127 | Subtracting a positive rea... |
| ltaddrp 13128 | Adding a positive number t... |
| difrp 13129 | Two ways to say one number... |
| elrpd 13130 | Membership in the set of p... |
| nnrpd 13131 | A positive integer is a po... |
| zgt1rpn0n1 13132 | An integer greater than 1 ... |
| rpred 13133 | A positive real is a real.... |
| rpxrd 13134 | A positive real is an exte... |
| rpcnd 13135 | A positive real is a compl... |
| rpgt0d 13136 | A positive real is greater... |
| rpge0d 13137 | A positive real is greater... |
| rpne0d 13138 | A positive real is nonzero... |
| rpregt0d 13139 | A positive real is real an... |
| rprege0d 13140 | A positive real is real an... |
| rprene0d 13141 | A positive real is a nonze... |
| rpcnne0d 13142 | A positive real is a nonze... |
| rpreccld 13143 | Closure law for reciprocat... |
| rprecred 13144 | Closure law for reciprocat... |
| rphalfcld 13145 | Closure law for half of a ... |
| reclt1d 13146 | The reciprocal of a positi... |
| recgt1d 13147 | The reciprocal of a positi... |
| rpaddcld 13148 | Closure law for addition o... |
| rpmulcld 13149 | Closure law for multiplica... |
| rpdivcld 13150 | Closure law for division o... |
| ltrecd 13151 | The reciprocal of both sid... |
| lerecd 13152 | The reciprocal of both sid... |
| ltrec1d 13153 | Reciprocal swap in a 'less... |
| lerec2d 13154 | Reciprocal swap in a 'less... |
| lediv2ad 13155 | Division of both sides of ... |
| ltdiv2d 13156 | Division of a positive num... |
| lediv2d 13157 | Division of a positive num... |
| ledivdivd 13158 | Invert ratios of positive ... |
| divge1 13159 | The ratio of a number over... |
| divlt1lt 13160 | A real number divided by a... |
| divle1le 13161 | A real number divided by a... |
| ledivge1le 13162 | If a number is less than o... |
| ge0p1rpd 13163 | A nonnegative number plus ... |
| rerpdivcld 13164 | Closure law for division o... |
| ltsubrpd 13165 | Subtracting a positive rea... |
| ltaddrpd 13166 | Adding a positive number t... |
| ltaddrp2d 13167 | Adding a positive number t... |
| ltmulgt11d 13168 | Multiplication by a number... |
| ltmulgt12d 13169 | Multiplication by a number... |
| gt0divd 13170 | Division of a positive num... |
| ge0divd 13171 | Division of a nonnegative ... |
| rpgecld 13172 | A number greater than or e... |
| divge0d 13173 | The ratio of nonnegative a... |
| ltmul1d 13174 | The ratio of nonnegative a... |
| ltmul2d 13175 | Multiplication of both sid... |
| lemul1d 13176 | Multiplication of both sid... |
| lemul2d 13177 | Multiplication of both sid... |
| ltdiv1d 13178 | Division of both sides of ... |
| lediv1d 13179 | Division of both sides of ... |
| ltmuldivd 13180 | 'Less than' relationship b... |
| ltmuldiv2d 13181 | 'Less than' relationship b... |
| lemuldivd 13182 | 'Less than or equal to' re... |
| lemuldiv2d 13183 | 'Less than or equal to' re... |
| ltdivmuld 13184 | 'Less than' relationship b... |
| ltdivmul2d 13185 | 'Less than' relationship b... |
| ledivmuld 13186 | 'Less than or equal to' re... |
| ledivmul2d 13187 | 'Less than or equal to' re... |
| ltmul1dd 13188 | The ratio of nonnegative a... |
| ltmul2dd 13189 | Multiplication of both sid... |
| ltdiv1dd 13190 | Division of both sides of ... |
| lediv1dd 13191 | Division of both sides of ... |
| lediv12ad 13192 | Comparison of ratio of two... |
| mul2lt0rlt0 13193 | If the result of a multipl... |
| mul2lt0rgt0 13194 | If the result of a multipl... |
| mul2lt0llt0 13195 | If the result of a multipl... |
| mul2lt0lgt0 13196 | If the result of a multipl... |
| mul2lt0bi 13197 | If the result of a multipl... |
| prodge0rd 13198 | Infer that a multiplicand ... |
| prodge0ld 13199 | Infer that a multiplier is... |
| ltdiv23d 13200 | Swap denominator with othe... |
| lediv23d 13201 | Swap denominator with othe... |
| lt2mul2divd 13202 | The ratio of nonnegative a... |
| nnledivrp 13203 | Division of a positive int... |
| nn0ledivnn 13204 | Division of a nonnegative ... |
| addlelt 13205 | If the sum of a real numbe... |
| ge2halflem1 13206 | Half of an integer greater... |
| ltxr 13213 | The 'less than' binary rel... |
| elxr 13214 | Membership in the set of e... |
| xrnemnf 13215 | An extended real other tha... |
| xrnepnf 13216 | An extended real other tha... |
| xrltnr 13217 | The extended real 'less th... |
| ltpnf 13218 | Any (finite) real is less ... |
| ltpnfd 13219 | Any (finite) real is less ... |
| 0ltpnf 13220 | Zero is less than plus inf... |
| mnflt 13221 | Minus infinity is less tha... |
| mnfltd 13222 | Minus infinity is less tha... |
| mnflt0 13223 | Minus infinity is less tha... |
| mnfltpnf 13224 | Minus infinity is less tha... |
| mnfltxr 13225 | Minus infinity is less tha... |
| pnfnlt 13226 | No extended real is greate... |
| nltmnf 13227 | No extended real is less t... |
| pnfge 13228 | Plus infinity is an upper ... |
| pnfged 13229 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13230 | An extended nonnegative in... |
| 0lepnf 13231 | 0 less than or equal to po... |
| xnn0ge0 13232 | An extended nonnegative in... |
| mnfle 13233 | Minus infinity is less tha... |
| mnfled 13234 | Minus infinity is less tha... |
| xrltnsym 13235 | Ordering on the extended r... |
| xrltnsym2 13236 | 'Less than' is antisymmetr... |
| xrlttri 13237 | Ordering on the extended r... |
| xrlttr 13238 | Ordering on the extended r... |
| xrltso 13239 | 'Less than' is a strict or... |
| xrlttri2 13240 | Trichotomy law for 'less t... |
| xrlttri3 13241 | Trichotomy law for 'less t... |
| xrleloe 13242 | 'Less than or equal' expre... |
| xrleltne 13243 | 'Less than or equal to' im... |
| xrltlen 13244 | 'Less than' expressed in t... |
| dfle2 13245 | Alternative definition of ... |
| dflt2 13246 | Alternative definition of ... |
| xrltle 13247 | 'Less than' implies 'less ... |
| xrltled 13248 | 'Less than' implies 'less ... |
| xrleid 13249 | 'Less than or equal to' is... |
| xrleidd 13250 | 'Less than or equal to' is... |
| xrletri 13251 | Trichotomy law for extende... |
| xrletri3 13252 | Trichotomy law for extende... |
| xrletrid 13253 | Trichotomy law for extende... |
| xrlelttr 13254 | Transitive law for orderin... |
| xrltletr 13255 | Transitive law for orderin... |
| xrletr 13256 | Transitive law for orderin... |
| xrlttrd 13257 | Transitive law for orderin... |
| xrlelttrd 13258 | Transitive law for orderin... |
| xrltletrd 13259 | Transitive law for orderin... |
| xrletrd 13260 | Transitive law for orderin... |
| xrltne 13261 | 'Less than' implies not eq... |
| xrgtned 13262 | 'Greater than' implies not... |
| nltpnft 13263 | An extended real is not le... |
| xgepnf 13264 | An extended real which is ... |
| ngtmnft 13265 | An extended real is not gr... |
| xlemnf 13266 | An extended real which is ... |
| xrrebnd 13267 | An extended real is real i... |
| xrre 13268 | A way of proving that an e... |
| xrre2 13269 | An extended real between t... |
| xrre3 13270 | A way of proving that an e... |
| ge0gtmnf 13271 | A nonnegative extended rea... |
| ge0nemnf 13272 | A nonnegative extended rea... |
| xrrege0 13273 | A nonnegative extended rea... |
| xrmax1 13274 | An extended real is less t... |
| xrmax2 13275 | An extended real is less t... |
| xrmin1 13276 | The minimum of two extende... |
| xrmin2 13277 | The minimum of two extende... |
| xrmaxeq 13278 | The maximum of two extende... |
| xrmineq 13279 | The minimum of two extende... |
| xrmaxlt 13280 | Two ways of saying the max... |
| xrltmin 13281 | Two ways of saying an exte... |
| xrmaxle 13282 | Two ways of saying the max... |
| xrlemin 13283 | Two ways of saying a numbe... |
| max1 13284 | A number is less than or e... |
| max1ALT 13285 | A number is less than or e... |
| max2 13286 | A number is less than or e... |
| 2resupmax 13287 | The supremum of two real n... |
| min1 13288 | The minimum of two numbers... |
| min2 13289 | The minimum of two numbers... |
| maxle 13290 | Two ways of saying the max... |
| lemin 13291 | Two ways of saying a numbe... |
| maxlt 13292 | Two ways of saying the max... |
| ltmin 13293 | Two ways of saying a numbe... |
| lemaxle 13294 | A real number which is les... |
| max0sub 13295 | Decompose a real number in... |
| ifle 13296 | An if statement transforms... |
| z2ge 13297 | There exists an integer gr... |
| qbtwnre 13298 | The rational numbers are d... |
| qbtwnxr 13299 | The rational numbers are d... |
| qsqueeze 13300 | If a nonnegative real is l... |
| qextltlem 13301 | Lemma for ~ qextlt and qex... |
| qextlt 13302 | An extensionality-like pro... |
| qextle 13303 | An extensionality-like pro... |
| xralrple 13304 | Show that ` A ` is less th... |
| alrple 13305 | Show that ` A ` is less th... |
| xnegeq 13306 | Equality of two extended n... |
| xnegex 13307 | A negative extended real e... |
| xnegpnf 13308 | Minus ` +oo ` . Remark of... |
| xnegmnf 13309 | Minus ` -oo ` . Remark of... |
| rexneg 13310 | Minus a real number. Rema... |
| xneg0 13311 | The negative of zero. (Co... |
| xnegcl 13312 | Closure of extended real n... |
| xnegneg 13313 | Extended real version of ~... |
| xneg11 13314 | Extended real version of ~... |
| xltnegi 13315 | Forward direction of ~ xlt... |
| xltneg 13316 | Extended real version of ~... |
| xleneg 13317 | Extended real version of ~... |
| xlt0neg1 13318 | Extended real version of ~... |
| xlt0neg2 13319 | Extended real version of ~... |
| xle0neg1 13320 | Extended real version of ~... |
| xle0neg2 13321 | Extended real version of ~... |
| xaddval 13322 | Value of the extended real... |
| xaddf 13323 | The extended real addition... |
| xmulval 13324 | Value of the extended real... |
| xaddpnf1 13325 | Addition of positive infin... |
| xaddpnf2 13326 | Addition of positive infin... |
| xaddmnf1 13327 | Addition of negative infin... |
| xaddmnf2 13328 | Addition of negative infin... |
| pnfaddmnf 13329 | Addition of positive and n... |
| mnfaddpnf 13330 | Addition of negative and p... |
| rexadd 13331 | The extended real addition... |
| rexsub 13332 | Extended real subtraction ... |
| rexaddd 13333 | The extended real addition... |
| xnn0xaddcl 13334 | The extended nonnegative i... |
| xaddnemnf 13335 | Closure of extended real a... |
| xaddnepnf 13336 | Closure of extended real a... |
| xnegid 13337 | Extended real version of ~... |
| xaddcl 13338 | The extended real addition... |
| xaddcom 13339 | The extended real addition... |
| xaddrid 13340 | Extended real version of ~... |
| xaddlid 13341 | Extended real version of ~... |
| xaddridd 13342 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13343 | Extended nonnegative integ... |
| xnn0lenn0nn0 13344 | An extended nonnegative in... |
| xnn0le2is012 13345 | An extended nonnegative in... |
| xnn0xadd0 13346 | The sum of two extended no... |
| xnegdi 13347 | Extended real version of ~... |
| xaddass 13348 | Associativity of extended ... |
| xaddass2 13349 | Associativity of extended ... |
| xpncan 13350 | Extended real version of ~... |
| xnpcan 13351 | Extended real version of ~... |
| xleadd1a 13352 | Extended real version of ~... |
| xleadd2a 13353 | Commuted form of ~ xleadd1... |
| xleadd1 13354 | Weakened version of ~ xlea... |
| xltadd1 13355 | Extended real version of ~... |
| xltadd2 13356 | Extended real version of ~... |
| xaddge0 13357 | The sum of nonnegative ext... |
| xle2add 13358 | Extended real version of ~... |
| xlt2add 13359 | Extended real version of ~... |
| xsubge0 13360 | Extended real version of ~... |
| xposdif 13361 | Extended real version of ~... |
| xlesubadd 13362 | Under certain conditions, ... |
| xmullem 13363 | Lemma for ~ rexmul . (Con... |
| xmullem2 13364 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13365 | Extended real multiplicati... |
| xmul01 13366 | Extended real version of ~... |
| xmul02 13367 | Extended real version of ~... |
| xmulneg1 13368 | Extended real version of ~... |
| xmulneg2 13369 | Extended real version of ~... |
| rexmul 13370 | The extended real multipli... |
| xmulf 13371 | The extended real multipli... |
| xmulcl 13372 | Closure of extended real m... |
| xmulpnf1 13373 | Multiplication by plus inf... |
| xmulpnf2 13374 | Multiplication by plus inf... |
| xmulmnf1 13375 | Multiplication by minus in... |
| xmulmnf2 13376 | Multiplication by minus in... |
| xmulpnf1n 13377 | Multiplication by plus inf... |
| xmulrid 13378 | Extended real version of ~... |
| xmullid 13379 | Extended real version of ~... |
| xmulm1 13380 | Extended real version of ~... |
| xmulasslem2 13381 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13382 | Extended real version of ~... |
| xmulge0 13383 | Extended real version of ~... |
| xmulasslem 13384 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13385 | Lemma for ~ xmulass . (Co... |
| xmulass 13386 | Associativity of the exten... |
| xlemul1a 13387 | Extended real version of ~... |
| xlemul2a 13388 | Extended real version of ~... |
| xlemul1 13389 | Extended real version of ~... |
| xlemul2 13390 | Extended real version of ~... |
| xltmul1 13391 | Extended real version of ~... |
| xltmul2 13392 | Extended real version of ~... |
| xadddilem 13393 | Lemma for ~ xadddi . (Con... |
| xadddi 13394 | Distributive property for ... |
| xadddir 13395 | Commuted version of ~ xadd... |
| xadddi2 13396 | The assumption that the mu... |
| xadddi2r 13397 | Commuted version of ~ xadd... |
| x2times 13398 | Extended real version of ~... |
| xnegcld 13399 | Closure of extended real n... |
| xaddcld 13400 | The extended real addition... |
| xmulcld 13401 | Closure of extended real m... |
| xadd4d 13402 | Rearrangement of 4 terms i... |
| xnn0add4d 13403 | Rearrangement of 4 terms i... |
| xrsupexmnf 13404 | Adding minus infinity to a... |
| xrinfmexpnf 13405 | Adding plus infinity to a ... |
| xrsupsslem 13406 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13407 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13408 | Any subset of extended rea... |
| xrinfmss 13409 | Any subset of extended rea... |
| xrinfmss2 13410 | Any subset of extended rea... |
| xrub 13411 | By quantifying only over r... |
| supxr 13412 | The supremum of a set of e... |
| supxr2 13413 | The supremum of a set of e... |
| supxrcl 13414 | The supremum of an arbitra... |
| supxrun 13415 | The supremum of the union ... |
| supxrmnf 13416 | Adding minus infinity to a... |
| supxrpnf 13417 | The supremum of a set of e... |
| supxrunb1 13418 | The supremum of an unbound... |
| supxrunb2 13419 | The supremum of an unbound... |
| supxrbnd1 13420 | The supremum of a bounded-... |
| supxrbnd2 13421 | The supremum of a bounded-... |
| xrsup0 13422 | The supremum of an empty s... |
| supxrub 13423 | A member of a set of exten... |
| supxrlub 13424 | The supremum of a set of e... |
| supxrleub 13425 | The supremum of a set of e... |
| supxrre 13426 | The real and extended real... |
| supxrbnd 13427 | The supremum of a bounded-... |
| supxrgtmnf 13428 | The supremum of a nonempty... |
| supxrre1 13429 | The supremum of a nonempty... |
| supxrre2 13430 | The supremum of a nonempty... |
| supxrss 13431 | Smaller sets of extended r... |
| xrsupssd 13432 | Inequality deduction for s... |
| infxrcl 13433 | The infimum of an arbitrar... |
| infxrlb 13434 | A member of a set of exten... |
| infxrgelb 13435 | The infimum of a set of ex... |
| infxrre 13436 | The real and extended real... |
| infxrmnf 13437 | The infinimum of a set of ... |
| xrinf0 13438 | The infimum of the empty s... |
| infxrss 13439 | Larger sets of extended re... |
| reltre 13440 | For all real numbers there... |
| rpltrp 13441 | For all positive real numb... |
| reltxrnmnf 13442 | For all extended real numb... |
| infmremnf 13443 | The infimum of the reals i... |
| infmrp1 13444 | The infimum of the positiv... |
| ixxval 13453 | Value of the interval func... |
| elixx1 13454 | Membership in an interval ... |
| ixxf 13455 | The set of intervals of ex... |
| ixxex 13456 | The set of intervals of ex... |
| ixxssxr 13457 | The set of intervals of ex... |
| elixx3g 13458 | Membership in a set of ope... |
| ixxssixx 13459 | An interval is a subset of... |
| ixxdisj 13460 | Split an interval into dis... |
| ixxun 13461 | Split an interval into two... |
| ixxin 13462 | Intersection of two interv... |
| ixxss1 13463 | Subset relationship for in... |
| ixxss2 13464 | Subset relationship for in... |
| ixxss12 13465 | Subset relationship for in... |
| ixxub 13466 | Extract the upper bound of... |
| ixxlb 13467 | Extract the lower bound of... |
| iooex 13468 | The set of open intervals ... |
| iooval 13469 | Value of the open interval... |
| ioo0 13470 | An empty open interval of ... |
| ioon0 13471 | An open interval of extend... |
| ndmioo 13472 | The open interval function... |
| iooid 13473 | An open interval with iden... |
| elioo3g 13474 | Membership in a set of ope... |
| elioore 13475 | A member of an open interv... |
| lbioo 13476 | An open interval does not ... |
| ubioo 13477 | An open interval does not ... |
| iooval2 13478 | Value of the open interval... |
| iooin 13479 | Intersection of two open i... |
| iooss1 13480 | Subset relationship for op... |
| iooss2 13481 | Subset relationship for op... |
| iocval 13482 | Value of the open-below, c... |
| icoval 13483 | Value of the closed-below,... |
| iccval 13484 | Value of the closed interv... |
| elioo1 13485 | Membership in an open inte... |
| elioo2 13486 | Membership in an open inte... |
| elioc1 13487 | Membership in an open-belo... |
| elico1 13488 | Membership in a closed-bel... |
| elicc1 13489 | Membership in a closed int... |
| iccid 13490 | A closed interval with ide... |
| ico0 13491 | An empty open interval of ... |
| ioc0 13492 | An empty open interval of ... |
| icc0 13493 | An empty closed interval o... |
| dfrp2 13494 | Alternate definition of th... |
| elicod 13495 | Membership in a left-close... |
| icogelb 13496 | An element of a left-close... |
| icogelbd 13497 | An element of a left-close... |
| elicore 13498 | A member of a left-closed ... |
| ubioc1 13499 | The upper bound belongs to... |
| lbico1 13500 | The lower bound belongs to... |
| iccleub 13501 | An element of a closed int... |
| iccgelb 13502 | An element of a closed int... |
| elioo5 13503 | Membership in an open inte... |
| eliooxr 13504 | A nonempty open interval s... |
| eliooord 13505 | Ordering implied by a memb... |
| elioo4g 13506 | Membership in an open inte... |
| ioossre 13507 | An open interval is a set ... |
| ioosscn 13508 | An open interval is a set ... |
| elioc2 13509 | Membership in an open-belo... |
| elico2 13510 | Membership in a closed-bel... |
| elicc2 13511 | Membership in a closed rea... |
| elicc2i 13512 | Inference for membership i... |
| elicc4 13513 | Membership in a closed rea... |
| iccss 13514 | Condition for a closed int... |
| iccssioo 13515 | Condition for a closed int... |
| icossico 13516 | Condition for a closed-bel... |
| iccss2 13517 | Condition for a closed int... |
| iccssico 13518 | Condition for a closed int... |
| iccssioo2 13519 | Condition for a closed int... |
| iccssico2 13520 | Condition for a closed int... |
| icossico2d 13521 | Condition for a closed-bel... |
| ioomax 13522 | The open interval from min... |
| iccmax 13523 | The closed interval from m... |
| ioopos 13524 | The set of positive reals ... |
| ioorp 13525 | The set of positive reals ... |
| iooshf 13526 | Shift the arguments of the... |
| iocssre 13527 | A closed-above interval wi... |
| icossre 13528 | A closed-below interval wi... |
| iccssre 13529 | A closed real interval is ... |
| iccssxr 13530 | A closed interval is a set... |
| iocssxr 13531 | An open-below, closed-abov... |
| icossxr 13532 | A closed-below, open-above... |
| ioossicc 13533 | An open interval is a subs... |
| iccssred 13534 | A closed real interval is ... |
| eliccxr 13535 | A member of a closed inter... |
| icossicc 13536 | A closed-below, open-above... |
| iocssicc 13537 | A closed-above, open-below... |
| ioossico 13538 | An open interval is a subs... |
| iocssioo 13539 | Condition for a closed int... |
| icossioo 13540 | Condition for a closed int... |
| ioossioo 13541 | Condition for an open inte... |
| iccsupr 13542 | A nonempty subset of a clo... |
| elioopnf 13543 | Membership in an unbounded... |
| elioomnf 13544 | Membership in an unbounded... |
| elicopnf 13545 | Membership in a closed unb... |
| repos 13546 | Two ways of saying that a ... |
| ioof 13547 | The set of open intervals ... |
| iccf 13548 | The set of closed interval... |
| unirnioo 13549 | The union of the range of ... |
| dfioo2 13550 | Alternate definition of th... |
| ioorebas 13551 | Open intervals are element... |
| xrge0neqmnf 13552 | A nonnegative extended rea... |
| xrge0nre 13553 | An extended real which is ... |
| elrege0 13554 | The predicate "is a nonneg... |
| nn0rp0 13555 | A nonnegative integer is a... |
| rge0ssre 13556 | Nonnegative real numbers a... |
| elxrge0 13557 | Elementhood in the set of ... |
| 0e0icopnf 13558 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13559 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13560 | The nonnegative reals are ... |
| ge0mulcl 13561 | The nonnegative reals are ... |
| ge0xaddcl 13562 | The nonnegative reals are ... |
| ge0xmulcl 13563 | The nonnegative extended r... |
| lbicc2 13564 | The lower bound of a close... |
| ubicc2 13565 | The upper bound of a close... |
| elicc01 13566 | Membership in the closed r... |
| elunitrn 13567 | The closed unit interval i... |
| elunitcn 13568 | The closed unit interval i... |
| 0elunit 13569 | Zero is an element of the ... |
| 1elunit 13570 | One is an element of the c... |
| iooneg 13571 | Membership in a negated op... |
| iccneg 13572 | Membership in a negated cl... |
| icoshft 13573 | A shifted real is a member... |
| icoshftf1o 13574 | Shifting a closed-below, o... |
| icoun 13575 | The union of two adjacent ... |
| icodisj 13576 | Adjacent left-closed right... |
| ioounsn 13577 | The union of an open inter... |
| snunioo 13578 | The closure of one end of ... |
| snunico 13579 | The closure of the open en... |
| snunioc 13580 | The closure of the open en... |
| prunioo 13581 | The closure of an open rea... |
| ioodisj 13582 | If the upper bound of one ... |
| ioojoin 13583 | Join two open intervals to... |
| difreicc 13584 | The class difference of ` ... |
| iccsplit 13585 | Split a closed interval in... |
| iccshftr 13586 | Membership in a shifted in... |
| iccshftri 13587 | Membership in a shifted in... |
| iccshftl 13588 | Membership in a shifted in... |
| iccshftli 13589 | Membership in a shifted in... |
| iccdil 13590 | Membership in a dilated in... |
| iccdili 13591 | Membership in a dilated in... |
| icccntr 13592 | Membership in a contracted... |
| icccntri 13593 | Membership in a contracted... |
| divelunit 13594 | A condition for a ratio to... |
| lincmb01cmp 13595 | A linear combination of tw... |
| iccf1o 13596 | Describe a bijection from ... |
| iccen 13597 | Any nontrivial closed inte... |
| xov1plusxeqvd 13598 | A complex number ` X ` is ... |
| unitssre 13599 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13600 | The closed unit interval i... |
| supicc 13601 | Supremum of a bounded set ... |
| supiccub 13602 | The supremum of a bounded ... |
| supicclub 13603 | The supremum of a bounded ... |
| supicclub2 13604 | The supremum of a bounded ... |
| zltaddlt1le 13605 | The sum of an integer and ... |
| xnn0xrge0 13606 | An extended nonnegative in... |
| nnge2recico01 13607 | The reciprocal of an integ... |
| fzval 13610 | The value of a finite set ... |
| fzval2 13611 | An alternative way of expr... |
| fzf 13612 | Establish the domain and c... |
| elfz1 13613 | Membership in a finite set... |
| elfz 13614 | Membership in a finite set... |
| elfz2 13615 | Membership in a finite set... |
| elfzd 13616 | Membership in a finite set... |
| elfz5 13617 | Membership in a finite set... |
| elfz4 13618 | Membership in a finite set... |
| elfzuzb 13619 | Membership in a finite set... |
| eluzfz 13620 | Membership in a finite set... |
| elfzuz 13621 | A member of a finite set o... |
| elfzuz3 13622 | Membership in a finite set... |
| elfzel2 13623 | Membership in a finite set... |
| elfzel1 13624 | Membership in a finite set... |
| elfzelz 13625 | A member of a finite set o... |
| elfzelzd 13626 | A member of a finite set o... |
| fzssz 13627 | A finite sequence of integ... |
| elfzle1 13628 | A member of a finite set o... |
| elfzle2 13629 | A member of a finite set o... |
| elfzuz2 13630 | Implication of membership ... |
| elfzle3 13631 | Membership in a finite set... |
| eluzfz1 13632 | Membership in a finite set... |
| eluzfz2 13633 | Membership in a finite set... |
| eluzfz2b 13634 | Membership in a finite set... |
| elfz3 13635 | Membership in a finite set... |
| elfz1eq 13636 | Membership in a finite set... |
| elfzubelfz 13637 | If there is a member in a ... |
| peano2fzr 13638 | A Peano-postulate-like the... |
| fzn0 13639 | Properties of a finite int... |
| fz0 13640 | A finite set of sequential... |
| fzn 13641 | A finite set of sequential... |
| fzen 13642 | A shifted finite set of se... |
| fz1n 13643 | A 1-based finite set of se... |
| 0nelfz1 13644 | 0 is not an element of a f... |
| 0fz1 13645 | Two ways to say a finite 1... |
| fz10 13646 | There are no integers betw... |
| fz00m1 13647 | There are no integers betw... |
| uzsubsubfz 13648 | Membership of an integer g... |
| uzsubsubfz1 13649 | Membership of an integer g... |
| ige3m2fz 13650 | Membership of an integer g... |
| fzsplit2 13651 | Split a finite interval of... |
| fzsplit 13652 | Split a finite interval of... |
| fzdisj 13653 | Condition for two finite i... |
| fz01en 13654 | 0-based and 1-based finite... |
| elfznn 13655 | A member of a finite set o... |
| elfz1end 13656 | A nonempty finite range of... |
| fz1ssnn 13657 | A finite set of positive i... |
| fznn0sub 13658 | Subtraction closure for a ... |
| fzmmmeqm 13659 | Subtracting the difference... |
| fzaddel 13660 | Membership of a sum in a f... |
| fzadd2 13661 | Membership of a sum in a f... |
| fzsubel 13662 | Membership of a difference... |
| fzopth 13663 | A finite set of sequential... |
| fzass4 13664 | Two ways to express a nond... |
| fzss1 13665 | Subset relationship for fi... |
| fzss2 13666 | Subset relationship for fi... |
| fzssuz 13667 | A finite set of sequential... |
| fzsn 13668 | A finite interval of integ... |
| fzssp1 13669 | Subset relationship for fi... |
| fzssnn 13670 | Finite sets of sequential ... |
| ssfzunsnext 13671 | A subset of a finite seque... |
| ssfzunsn 13672 | A subset of a finite seque... |
| fzsuc 13673 | Join a successor to the en... |
| fzpred 13674 | Join a predecessor to the ... |
| fzpreddisj 13675 | A finite set of sequential... |
| elfzp1 13676 | Append an element to a fin... |
| fzp1ss 13677 | Subset relationship for fi... |
| fzelp1 13678 | Membership in a set of seq... |
| fzp1elp1 13679 | Add one to an element of a... |
| fznatpl1 13680 | Shift membership in a fini... |
| fzpr 13681 | A finite interval of integ... |
| fztp 13682 | A finite interval of integ... |
| fz12pr 13683 | An integer range between 1... |
| fzsuc2 13684 | Join a successor to the en... |
| fzp1disj 13685 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13686 | Remove a successor from th... |
| fzprval 13687 | Two ways of defining the f... |
| fztpval 13688 | Two ways of defining the f... |
| fzrev 13689 | Reversal of start and end ... |
| fzrev2 13690 | Reversal of start and end ... |
| fzrev2i 13691 | Reversal of start and end ... |
| fzrev3 13692 | The "complement" of a memb... |
| fzrev3i 13693 | The "complement" of a memb... |
| fznn 13694 | Finite set of sequential i... |
| elfz1b 13695 | Membership in a 1-based fi... |
| elfz1uz 13696 | Membership in a 1-based fi... |
| elfzm11 13697 | Membership in a finite set... |
| uzsplit 13698 | Express an upper integer s... |
| uzdisj 13699 | The first ` N ` elements o... |
| fseq1p1m1 13700 | Add/remove an item to/from... |
| fseq1m1p1 13701 | Add/remove an item to/from... |
| fz1sbc 13702 | Quantification over a one-... |
| elfzp1b 13703 | An integer is a member of ... |
| elfzm1b 13704 | An integer is a member of ... |
| elfzp12 13705 | Options for membership in ... |
| fzne1 13706 | Elementhood in a finite se... |
| fzdif1 13707 | Split the first element of... |
| fz0dif1 13708 | Split the first element of... |
| fzm1 13709 | Choices for an element of ... |
| fzneuz 13710 | No finite set of sequentia... |
| fznuz 13711 | Disjointness of the upper ... |
| uznfz 13712 | Disjointness of the upper ... |
| fzp1nel 13713 | One plus the upper bound o... |
| fzrevral 13714 | Reversal of scanning order... |
| fzrevral2 13715 | Reversal of scanning order... |
| fzrevral3 13716 | Reversal of scanning order... |
| fzshftral 13717 | Shift the scanning order i... |
| ige2m1fz1 13718 | Membership of an integer g... |
| ige2m1fz 13719 | Membership in a 0-based fi... |
| elfz2nn0 13720 | Membership in a finite set... |
| fznn0 13721 | Characterization of a fini... |
| elfznn0 13722 | A member of a finite set o... |
| elfz3nn0 13723 | The upper bound of a nonem... |
| fz0ssnn0 13724 | Finite sets of sequential ... |
| fz1ssfz0 13725 | Subset relationship for fi... |
| 0elfz 13726 | 0 is an element of a finit... |
| nn0fz0 13727 | A nonnegative integer is a... |
| elfz0add 13728 | An element of a finite set... |
| fz0sn 13729 | An integer range from 0 to... |
| fz0tp 13730 | An integer range from 0 to... |
| fz0to3un2pr 13731 | An integer range from 0 to... |
| fz0to4untppr 13732 | An integer range from 0 to... |
| fz0to5un2tp 13733 | An integer range from 0 to... |
| elfz0ubfz0 13734 | An element of a finite set... |
| elfz0fzfz0 13735 | A member of a finite set o... |
| fz0fzelfz0 13736 | If a member of a finite se... |
| fznn0sub2 13737 | Subtraction closure for a ... |
| uzsubfz0 13738 | Membership of an integer g... |
| fz0fzdiffz0 13739 | The difference of an integ... |
| elfzmlbm 13740 | Subtracting the lower boun... |
| elfzmlbp 13741 | Subtracting the lower boun... |
| fzctr 13742 | Lemma for theorems about t... |
| difelfzle 13743 | The difference of two inte... |
| difelfznle 13744 | The difference of two inte... |
| nn0split 13745 | Express the set of nonnega... |
| nn0disj 13746 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13747 | A finite set of sequential... |
| fvffz0 13748 | The function value of a fu... |
| 1fv 13749 | A function on a singleton.... |
| 4fvwrd4 13750 | The first four function va... |
| 2ffzeq 13751 | Two functions over 0-based... |
| preduz 13752 | The value of the predecess... |
| prednn 13753 | The value of the predecess... |
| prednn0 13754 | The value of the predecess... |
| predfz 13755 | Calculate the predecessor ... |
| fzof 13758 | Functionality of the half-... |
| elfzoel1 13759 | Reverse closure for half-o... |
| elfzoel2 13760 | Reverse closure for half-o... |
| elfzoelz 13761 | Reverse closure for half-o... |
| fzoval 13762 | Value of the half-open int... |
| elfzo 13763 | Membership in a half-open ... |
| elfzo2 13764 | Membership in a half-open ... |
| elfzod 13765 | Membership in a half-open ... |
| elfzouz 13766 | Membership in a half-open ... |
| nelfzo 13767 | An integer not being a mem... |
| fzolb 13768 | The left endpoint of a hal... |
| fzolb2 13769 | The left endpoint of a hal... |
| elfzole1 13770 | A member in a half-open in... |
| elfzolt2 13771 | A member in a half-open in... |
| elfzolt3 13772 | Membership in a half-open ... |
| elfzolt2b 13773 | A member in a half-open in... |
| elfzolt3b 13774 | Membership in a half-open ... |
| elfzop1le2 13775 | A member in a half-open in... |
| fzonel 13776 | A half-open range does not... |
| elfzouz2 13777 | The upper bound of a half-... |
| elfzofz 13778 | A half-open range is conta... |
| elfzo3 13779 | Express membership in a ha... |
| fzon0 13780 | A half-open integer interv... |
| fzossfz 13781 | A half-open range is conta... |
| fzossz 13782 | A half-open integer interv... |
| fzon 13783 | A half-open set of sequent... |
| fzo0n 13784 | A half-open range of nonne... |
| fzonlt0 13785 | A half-open integer range ... |
| fzo0 13786 | Half-open sets with equal ... |
| fzonnsub 13787 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13788 | If ` M < N ` then ` N - M ... |
| fzoss1 13789 | Subset relationship for ha... |
| fzoss2 13790 | Subset relationship for ha... |
| fzossrbm1 13791 | Subset of a half-open rang... |
| fzo0ss1 13792 | Subset relationship for ha... |
| fzossnn0 13793 | A half-open integer range ... |
| fzospliti 13794 | One direction of splitting... |
| fzosplit 13795 | Split a half-open integer ... |
| fzodisj 13796 | Abutting half-open integer... |
| fzouzsplit 13797 | Split an upper integer set... |
| fzouzdisj 13798 | A half-open integer range ... |
| fzoun 13799 | A half-open integer range ... |
| fzodisjsn 13800 | A half-open integer range ... |
| prinfzo0 13801 | The intersection of a half... |
| lbfzo0 13802 | An integer is strictly gre... |
| elfzo0 13803 | Membership in a half-open ... |
| elfzo0z 13804 | Membership in a half-open ... |
| nn0p1elfzo 13805 | A nonnegative integer incr... |
| elfzo0le 13806 | A member in a half-open ra... |
| elfzolem1 13807 | A member in a half-open in... |
| elfzo0subge1 13808 | The difference of the uppe... |
| elfzo0suble 13809 | The difference of the uppe... |
| elfzonn0 13810 | A member of a half-open ra... |
| fzonmapblen 13811 | The result of subtracting ... |
| fzofzim 13812 | If a nonnegative integer i... |
| fz1fzo0m1 13813 | Translation of one between... |
| fzossnn 13814 | Half-open integer ranges s... |
| elfzo1 13815 | Membership in a half-open ... |
| fzo1lb 13816 | 1 is the left endpoint of ... |
| 1elfzo1 13817 | 1 is in a half-open range ... |
| fzo1fzo0n0 13818 | An integer between 1 and a... |
| fzo0n0 13819 | A half-open integer range ... |
| fzoaddel 13820 | Translate membership in a ... |
| fzo0addel 13821 | Translate membership in a ... |
| fzo0addelr 13822 | Translate membership in a ... |
| fzoaddel2 13823 | Translate membership in a ... |
| elfzoextl 13824 | Membership of an integer i... |
| elfzoext 13825 | Membership of an integer i... |
| elincfzoext 13826 | Membership of an increased... |
| fzosubel 13827 | Translate membership in a ... |
| fzosubel2 13828 | Membership in a translated... |
| fzosubel3 13829 | Membership in a translated... |
| eluzgtdifelfzo 13830 | Membership of the differen... |
| ige2m2fzo 13831 | Membership of an integer g... |
| fzocatel 13832 | Translate membership in a ... |
| ubmelfzo 13833 | If an integer in a 1-based... |
| elfzodifsumelfzo 13834 | If an integer is in a half... |
| elfzom1elp1fzo 13835 | Membership of an integer i... |
| elfzom1elfzo 13836 | Membership in a half-open ... |
| fzval3 13837 | Expressing a closed intege... |
| fz0add1fz1 13838 | Translate membership in a ... |
| fzosn 13839 | Expressing a singleton as ... |
| elfzomin 13840 | Membership of an integer i... |
| zpnn0elfzo 13841 | Membership of an integer i... |
| zpnn0elfzo1 13842 | Membership of an integer i... |
| fzosplitsnm1 13843 | Removing a singleton from ... |
| elfzonlteqm1 13844 | If an element of a half-op... |
| fzonn0p1 13845 | A nonnegative integer is a... |
| fzossfzop1 13846 | A half-open range of nonne... |
| fzonn0p1p1 13847 | If a nonnegative integer i... |
| elfzom1p1elfzo 13848 | Increasing an element of a... |
| fzo0ssnn0 13849 | Half-open integer ranges s... |
| fzo01 13850 | Expressing the singleton o... |
| fzo12sn 13851 | A 1-based half-open intege... |
| fzo13pr 13852 | A 1-based half-open intege... |
| fzo0to2pr 13853 | A half-open integer range ... |
| fz01pr 13854 | An integer range between 0... |
| fzo0to3tp 13855 | A half-open integer range ... |
| fzo0to42pr 13856 | A half-open integer range ... |
| fzo1to4tp 13857 | A half-open integer range ... |
| fzo0sn0fzo1 13858 | A half-open range of nonne... |
| elfzo0l 13859 | A member of a half-open ra... |
| fzoend 13860 | The endpoint of a half-ope... |
| fzo0end 13861 | The endpoint of a zero-bas... |
| ssfzo12 13862 | Subset relationship for ha... |
| ssfzoulel 13863 | If a half-open integer ran... |
| ssfzo12bi 13864 | Subset relationship for ha... |
| fzoopth 13865 | A half-open integer range ... |
| ubmelm1fzo 13866 | The result of subtracting ... |
| fzofzp1 13867 | If a point is in a half-op... |
| fzofzp1b 13868 | If a point is in a half-op... |
| elfzom1b 13869 | An integer is a member of ... |
| elfzom1elp1fzo1 13870 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13871 | Membership of a positive i... |
| elfzonelfzo 13872 | If an element of a half-op... |
| elfzodif0 13873 | If an integer ` M ` is in ... |
| fzonfzoufzol 13874 | If an element of a half-op... |
| elfzomelpfzo 13875 | An integer increased by an... |
| elfznelfzo 13876 | A value in a finite set of... |
| elfznelfzob 13877 | A value in a finite set of... |
| peano2fzor 13878 | A Peano-postulate-like the... |
| fzosplitsn 13879 | Extending a half-open rang... |
| fzosplitpr 13880 | Extending a half-open inte... |
| fzosplitprm1 13881 | Extending a half-open inte... |
| fzosplitsni 13882 | Membership in a half-open ... |
| fzisfzounsn 13883 | A finite interval of integ... |
| elfzr 13884 | A member of a finite inter... |
| elfzlmr 13885 | A member of a finite inter... |
| elfz0lmr 13886 | A member of a finite inter... |
| fzone1 13887 | Elementhood in a half-open... |
| fzom1ne1 13888 | Elementhood in a half-open... |
| fzostep1 13889 | Two possibilities for a nu... |
| fzoshftral 13890 | Shift the scanning order i... |
| fzind2 13891 | Induction on the integers ... |
| fvinim0ffz 13892 | The function values for th... |
| injresinjlem 13893 | Lemma for ~ injresinj . (... |
| injresinj 13894 | A function whose restricti... |
| f1resfz0f1d 13895 | If a function with a seque... |
| subfzo0 13896 | The difference between two... |
| fvf1tp 13897 | Values of a one-to-one fun... |
| flval 13902 | Value of the floor (greate... |
| flcl 13903 | The floor (greatest intege... |
| reflcl 13904 | The floor (greatest intege... |
| fllelt 13905 | A basic property of the fl... |
| flcld 13906 | The floor (greatest intege... |
| flle 13907 | A basic property of the fl... |
| flltp1 13908 | A basic property of the fl... |
| fllep1 13909 | A basic property of the fl... |
| fraclt1 13910 | The fractional part of a r... |
| fracle1 13911 | The fractional part of a r... |
| fracge0 13912 | The fractional part of a r... |
| flge 13913 | The floor function value i... |
| fllt 13914 | The floor function value i... |
| flflp1 13915 | Move floor function betwee... |
| flid 13916 | An integer is its own floo... |
| flidm 13917 | The floor function is idem... |
| flidz 13918 | A real number equals its f... |
| flltnz 13919 | The floor of a non-integer... |
| flwordi 13920 | Ordering relation for the ... |
| flword2 13921 | Ordering relation for the ... |
| flval2 13922 | An alternate way to define... |
| flval3 13923 | An alternate way to define... |
| flbi 13924 | A condition equivalent to ... |
| flbi2 13925 | A condition equivalent to ... |
| adddivflid 13926 | The floor of a sum of an i... |
| ico01fl0 13927 | The floor of a real number... |
| flge0nn0 13928 | The floor of a number grea... |
| flge1nn 13929 | The floor of a number grea... |
| fldivnn0 13930 | The floor function of a di... |
| refldivcl 13931 | The floor function of a di... |
| divfl0 13932 | The floor of a fraction is... |
| fladdz 13933 | An integer can be moved in... |
| flzadd 13934 | An integer can be moved in... |
| flmulnn0 13935 | Move a nonnegative integer... |
| btwnzge0 13936 | A real bounded between an ... |
| 2tnp1ge0ge0 13937 | Two times an integer plus ... |
| flhalf 13938 | Ordering relation for the ... |
| fldivle 13939 | The floor function of a di... |
| fldivnn0le 13940 | The floor function of a di... |
| flltdivnn0lt 13941 | The floor function of a di... |
| ltdifltdiv 13942 | If the dividend of a divis... |
| fldiv4p1lem1div2 13943 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13944 | The floor of an integer gr... |
| fldiv4lem1div2 13945 | The floor of a positive in... |
| ceilval 13946 | The value of the ceiling f... |
| dfceil2 13947 | Alternative definition of ... |
| ceilval2 13948 | The value of the ceiling f... |
| ceicl 13949 | The ceiling function retur... |
| ceilcl 13950 | Closure of the ceiling fun... |
| ceilcld 13951 | Closure of the ceiling fun... |
| ceige 13952 | The ceiling of a real numb... |
| ceilge 13953 | The ceiling of a real numb... |
| ceilged 13954 | The ceiling of a real numb... |
| ceim1l 13955 | One less than the ceiling ... |
| ceilm1lt 13956 | One less than the ceiling ... |
| ceile 13957 | The ceiling of a real numb... |
| ceille 13958 | The ceiling of a real numb... |
| ceilid 13959 | An integer is its own ceil... |
| ceilidz 13960 | A real number equals its c... |
| flleceil 13961 | The floor of a real number... |
| fleqceilz 13962 | A real number is an intege... |
| quoremz 13963 | Quotient and remainder of ... |
| quoremnn0 13964 | Quotient and remainder of ... |
| quoremnn0ALT 13965 | Alternate proof of ~ quore... |
| intfrac2 13966 | Decompose a real into inte... |
| intfracq 13967 | Decompose a rational numbe... |
| fldiv 13968 | Cancellation of the embedd... |
| fldiv2 13969 | Cancellation of an embedde... |
| fznnfl 13970 | Finite set of sequential i... |
| uzsup 13971 | An upper set of integers i... |
| ioopnfsup 13972 | An upper set of reals is u... |
| icopnfsup 13973 | An upper set of reals is u... |
| rpsup 13974 | The positive reals are unb... |
| resup 13975 | The real numbers are unbou... |
| xrsup 13976 | The extended real numbers ... |
| modval 13979 | The value of the modulo op... |
| modvalr 13980 | The value of the modulo op... |
| modcl 13981 | Closure law for the modulo... |
| flpmodeq 13982 | Partition of a division in... |
| modcld 13983 | Closure law for the modulo... |
| mod0 13984 | ` A mod B ` is zero iff ` ... |
| mulmod0 13985 | The product of an integer ... |
| negmod0 13986 | ` A ` is divisible by ` B ... |
| modge0 13987 | The modulo operation is no... |
| modlt 13988 | The modulo operation is le... |
| modelico 13989 | Modular reduction produces... |
| moddiffl 13990 | Value of the modulo operat... |
| moddifz 13991 | The modulo operation diffe... |
| modfrac 13992 | The fractional part of a n... |
| flmod 13993 | The floor function express... |
| intfrac 13994 | Break a number into its in... |
| zmod10 13995 | An integer modulo 1 is 0. ... |
| zmod1congr 13996 | Two arbitrary integers are... |
| modmulnn 13997 | Move a positive integer in... |
| modvalp1 13998 | The value of the modulo op... |
| zmodcl 13999 | Closure law for the modulo... |
| zmodcld 14000 | Closure law for the modulo... |
| zmodfz 14001 | An integer mod ` B ` lies ... |
| zmodfzo 14002 | An integer mod ` B ` lies ... |
| zmodfzp1 14003 | An integer mod ` B ` lies ... |
| modid 14004 | Identity law for modulo. ... |
| modid0 14005 | A positive real number mod... |
| modid2 14006 | Identity law for modulo. ... |
| zmodid2 14007 | Identity law for modulo re... |
| zmodidfzo 14008 | Identity law for modulo re... |
| zmodidfzoimp 14009 | Identity law for modulo re... |
| 0mod 14010 | Special case: 0 modulo a p... |
| 1mod 14011 | Special case: 1 modulo a r... |
| modabs 14012 | Absorption law for modulo.... |
| modabs2 14013 | Absorption law for modulo.... |
| modcyc 14014 | The modulo operation is pe... |
| modcyc2 14015 | The modulo operation is pe... |
| modadd1 14016 | Addition property of the m... |
| modaddb 14017 | Addition property of the m... |
| modaddid 14018 | The sums of two nonnegativ... |
| modaddabs 14019 | Absorption law for modulo.... |
| modaddmod 14020 | The sum of a real number m... |
| muladdmodid 14021 | The sum of a positive real... |
| mulp1mod1 14022 | The product of an integer ... |
| muladdmod 14023 | A real number is the sum o... |
| modmuladd 14024 | Decomposition of an intege... |
| modmuladdim 14025 | Implication of a decomposi... |
| modmuladdnn0 14026 | Implication of a decomposi... |
| negmod 14027 | The negation of a number m... |
| m1modnnsub1 14028 | Minus one modulo a positiv... |
| m1modge3gt1 14029 | Minus one modulo an intege... |
| addmodid 14030 | The sum of a positive inte... |
| addmodidr 14031 | The sum of a positive inte... |
| modadd2mod 14032 | The sum of a real number m... |
| modm1p1mod0 14033 | If a real number modulo a ... |
| modltm1p1mod 14034 | If a real number modulo a ... |
| modmul1 14035 | Multiplication property of... |
| modmul12d 14036 | Multiplication property of... |
| modnegd 14037 | Negation property of the m... |
| modadd12d 14038 | Additive property of the m... |
| modsub12d 14039 | Subtraction property of th... |
| modsubmod 14040 | The difference of a real n... |
| modsubmodmod 14041 | The difference of a real n... |
| 2txmodxeq0 14042 | Two times a positive real ... |
| 2submod 14043 | If a real number is betwee... |
| modifeq2int 14044 | If a nonnegative integer i... |
| modaddmodup 14045 | The sum of an integer modu... |
| modaddmodlo 14046 | The sum of an integer modu... |
| modmulmod 14047 | The product of a real numb... |
| modmulmodr 14048 | The product of an integer ... |
| modaddmulmod 14049 | The sum of a real number a... |
| moddi 14050 | Distribute multiplication ... |
| modsubdir 14051 | Distribute the modulo oper... |
| modeqmodmin 14052 | A real number equals the d... |
| modirr 14053 | A number modulo an irratio... |
| modfzo0difsn 14054 | For a number within a half... |
| modsumfzodifsn 14055 | The sum of a number within... |
| modlteq 14056 | Two nonnegative integers l... |
| addmodlteq 14057 | Two nonnegative integers l... |
| om2uz0i 14058 | The mapping ` G ` is a one... |
| om2uzsuci 14059 | The value of ` G ` (see ~ ... |
| om2uzuzi 14060 | The value ` G ` (see ~ om2... |
| om2uzlti 14061 | Less-than relation for ` G... |
| om2uzlt2i 14062 | The mapping ` G ` (see ~ o... |
| om2uzrani 14063 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 14064 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 14065 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 14066 | An alternative definition ... |
| om2uzrdg 14067 | A helper lemma for the val... |
| uzrdglem 14068 | A helper lemma for the val... |
| uzrdgfni 14069 | The recursive definition g... |
| uzrdg0i 14070 | Initial value of a recursi... |
| uzrdgsuci 14071 | Successor value of a recur... |
| ltweuz 14072 | ` < ` is a well-founded re... |
| ltwenn 14073 | Less than well-orders the ... |
| ltwefz 14074 | Less than well-orders a se... |
| uzenom 14075 | An upper integer set is de... |
| uzinf 14076 | An upper integer set is in... |
| nnnfi 14077 | The set of positive intege... |
| uzrdgxfr 14078 | Transfer the value of the ... |
| fzennn 14079 | The cardinality of a finit... |
| fzen2 14080 | The cardinality of a finit... |
| cardfz 14081 | The cardinality of a finit... |
| hashgf1o 14082 | ` G ` maps ` _om ` one-to-... |
| fzfi 14083 | A finite interval of integ... |
| fzfid 14084 | Commonly used special case... |
| fzofi 14085 | Half-open integer sets are... |
| fsequb 14086 | The values of a finite rea... |
| fsequb2 14087 | The values of a finite rea... |
| fseqsupcl 14088 | The values of a finite rea... |
| fseqsupubi 14089 | The values of a finite rea... |
| nn0ennn 14090 | The nonnegative integers a... |
| nnenom 14091 | The set of positive intege... |
| nnct 14092 | ` NN ` is countable. (Con... |
| uzindi 14093 | Indirect strong induction ... |
| axdc4uzlem 14094 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 14095 | A version of ~ axdc4 that ... |
| ssnn0fi 14096 | A subset of the nonnegativ... |
| rabssnn0fi 14097 | A subset of the nonnegativ... |
| uzsinds 14098 | Strong (or "total") induct... |
| nnsinds 14099 | Strong (or "total") induct... |
| nn0sinds 14100 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 14101 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 14102 | If all functions of a fini... |
| fsuppmapnn0fiubex 14103 | If all functions of a fini... |
| fsuppmapnn0fiub0 14104 | If all functions of a fini... |
| suppssfz 14105 | Condition for a function o... |
| fsuppmapnn0ub 14106 | If a function over the non... |
| fsuppmapnn0fz 14107 | If a function over the non... |
| mptnn0fsupp 14108 | A mapping from the nonnega... |
| mptnn0fsuppd 14109 | A mapping from the nonnega... |
| mptnn0fsuppr 14110 | A finitely supported mappi... |
| f13idfv 14111 | A one-to-one function with... |
| seqex 14114 | Existence of the sequence ... |
| seqeq1 14115 | Equality theorem for the s... |
| seqeq2 14116 | Equality theorem for the s... |
| seqeq3 14117 | Equality theorem for the s... |
| seqeq1d 14118 | Equality deduction for the... |
| seqeq2d 14119 | Equality deduction for the... |
| seqeq3d 14120 | Equality deduction for the... |
| seqeq123d 14121 | Equality deduction for the... |
| nfseq 14122 | Hypothesis builder for the... |
| seqval 14123 | Value of the sequence buil... |
| seqfn 14124 | The sequence builder funct... |
| seq1 14125 | Value of the sequence buil... |
| seq1i 14126 | Value of the sequence buil... |
| seqp1 14127 | Value of the sequence buil... |
| seqexw 14128 | Weak version of ~ seqex th... |
| seqp1d 14129 | Value of the sequence buil... |
| seqm1 14130 | Value of the sequence buil... |
| seqcl2 14131 | Closure properties of the ... |
| seqf2 14132 | Range of the recursive seq... |
| seqcl 14133 | Closure properties of the ... |
| seqf 14134 | Range of the recursive seq... |
| seqfveq2 14135 | Equality of sequences. (C... |
| seqfeq2 14136 | Equality of sequences. (C... |
| seqfveq 14137 | Equality of sequences. (C... |
| seqfeq 14138 | Equality of sequences. (C... |
| seqshft2 14139 | Shifting the index set of ... |
| seqres 14140 | Restricting its characteri... |
| serf 14141 | An infinite series of comp... |
| serfre 14142 | An infinite series of real... |
| monoord 14143 | Ordering relation for a mo... |
| monoord2 14144 | Ordering relation for a mo... |
| sermono 14145 | The partial sums in an inf... |
| seqsplit 14146 | Split a sequence into two ... |
| seq1p 14147 | Removing the first term fr... |
| seqcaopr3 14148 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14149 | The sum of two infinite se... |
| seqcaopr 14150 | The sum of two infinite se... |
| seqf1olem2a 14151 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14152 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14153 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14154 | Rearrange a sum via an arb... |
| seradd 14155 | The sum of two infinite se... |
| sersub 14156 | The difference of two infi... |
| seqid3 14157 | A sequence that consists e... |
| seqid 14158 | Discarding the first few t... |
| seqid2 14159 | The last few partial sums ... |
| seqhomo 14160 | Apply a homomorphism to a ... |
| seqz 14161 | If the operation ` .+ ` ha... |
| seqfeq4 14162 | Equality of series under d... |
| seqfeq3 14163 | Equality of series under d... |
| seqdistr 14164 | The distributive property ... |
| ser0 14165 | The value of the partial s... |
| ser0f 14166 | A zero-valued infinite ser... |
| serge0 14167 | A finite sum of nonnegativ... |
| serle 14168 | Comparison of partial sums... |
| ser1const 14169 | Value of the partial serie... |
| seqof 14170 | Distribute function operat... |
| seqof2 14171 | Distribute function operat... |
| expval 14174 | Value of exponentiation to... |
| expnnval 14175 | Value of exponentiation to... |
| exp0 14176 | Value of a complex number ... |
| 0exp0e1 14177 | The zeroth power of zero e... |
| exp1 14178 | Value of a complex number ... |
| expp1 14179 | Value of a complex number ... |
| expneg 14180 | Value of a complex number ... |
| expneg2 14181 | Value of a complex number ... |
| expn1 14182 | A complex number raised to... |
| expcllem 14183 | Lemma for proving nonnegat... |
| expcl2lem 14184 | Lemma for proving integer ... |
| nnexpcl 14185 | Closure of exponentiation ... |
| nn0expcl 14186 | Closure of exponentiation ... |
| zexpcl 14187 | Closure of exponentiation ... |
| qexpcl 14188 | Closure of exponentiation ... |
| reexpcl 14189 | Closure of exponentiation ... |
| expcl 14190 | Closure law for nonnegativ... |
| rpexpcl 14191 | Closure law for integer ex... |
| qexpclz 14192 | Closure of integer exponen... |
| reexpclz 14193 | Closure of integer exponen... |
| expclzlem 14194 | Lemma for ~ expclz . (Con... |
| expclz 14195 | Closure law for integer ex... |
| m1expcl2 14196 | Closure of integer exponen... |
| m1expcl 14197 | Closure of exponentiation ... |
| zexpcld 14198 | Closure of exponentiation ... |
| nn0expcli 14199 | Closure of exponentiation ... |
| nn0sqcl 14200 | The square of a nonnegativ... |
| expm1t 14201 | Exponentiation in terms of... |
| 1exp 14202 | Value of 1 raised to an in... |
| expeq0 14203 | A positive integer power i... |
| expne0 14204 | A positive integer power i... |
| expne0i 14205 | An integer power is nonzer... |
| expgt0 14206 | A positive real raised to ... |
| expnegz 14207 | Value of a nonzero complex... |
| 0exp 14208 | Value of zero raised to a ... |
| expge0 14209 | A nonnegative real raised ... |
| expge1 14210 | A real greater than or equ... |
| expgt1 14211 | A real greater than 1 rais... |
| mulexp 14212 | Nonnegative integer expone... |
| mulexpz 14213 | Integer exponentiation of ... |
| exprec 14214 | Integer exponentiation of ... |
| expadd 14215 | Sum of exponents law for n... |
| expaddzlem 14216 | Lemma for ~ expaddz . (Co... |
| expaddz 14217 | Sum of exponents law for i... |
| expmul 14218 | Product of exponents law f... |
| expmulz 14219 | Product of exponents law f... |
| m1expeven 14220 | Exponentiation of negative... |
| expsub 14221 | Exponent subtraction law f... |
| expp1z 14222 | Value of a nonzero complex... |
| expm1 14223 | Value of a nonzero complex... |
| expdiv 14224 | Nonnegative integer expone... |
| sqval 14225 | Value of the square of a c... |
| sqneg 14226 | The square of the negative... |
| sqnegd 14227 | The square of the negative... |
| sqsubswap 14228 | Swap the order of subtract... |
| sqcl 14229 | Closure of square. (Contr... |
| sqmul 14230 | Distribution of squaring o... |
| sqeq0 14231 | A complex number is zero i... |
| sqdiv 14232 | Distribution of squaring o... |
| sqdivid 14233 | The square of a nonzero co... |
| sqne0 14234 | A complex number is nonzer... |
| resqcl 14235 | Closure of squaring in rea... |
| resqcld 14236 | Closure of squaring in rea... |
| sqgt0 14237 | The square of a nonzero re... |
| sqn0rp 14238 | The square of a nonzero re... |
| nnsqcl 14239 | The positive naturals are ... |
| zsqcl 14240 | Integers are closed under ... |
| qsqcl 14241 | The square of a rational i... |
| sq11 14242 | The square function is one... |
| nn0sq11 14243 | The square function is one... |
| lt2sq 14244 | The square function is inc... |
| le2sq 14245 | The square function is non... |
| le2sq2 14246 | The square function is non... |
| sqge0 14247 | The square of a real is no... |
| sqge0d 14248 | The square of a real is no... |
| zsqcl2 14249 | The square of an integer i... |
| 0expd 14250 | Value of zero raised to a ... |
| exp0d 14251 | Value of a complex number ... |
| exp1d 14252 | Value of a complex number ... |
| expeq0d 14253 | If a positive integer powe... |
| sqvald 14254 | Value of square. Inferenc... |
| sqcld 14255 | Closure of square. (Contr... |
| sqeq0d 14256 | A number is zero iff its s... |
| expcld 14257 | Closure law for nonnegativ... |
| expp1d 14258 | Value of a complex number ... |
| expaddd 14259 | Sum of exponents law for n... |
| expmuld 14260 | Product of exponents law f... |
| sqrecd 14261 | Square of reciprocal is re... |
| expclzd 14262 | Closure law for integer ex... |
| expne0d 14263 | A nonnegative integer powe... |
| expnegd 14264 | Value of a nonzero complex... |
| exprecd 14265 | An integer power of a reci... |
| expp1zd 14266 | Value of a nonzero complex... |
| expm1d 14267 | Value of a nonzero complex... |
| expsubd 14268 | Exponent subtraction law f... |
| sqmuld 14269 | Distribution of squaring o... |
| sqdivd 14270 | Distribution of squaring o... |
| expdivd 14271 | Nonnegative integer expone... |
| mulexpd 14272 | Nonnegative integer expone... |
| znsqcld 14273 | The square of a nonzero in... |
| reexpcld 14274 | Closure of exponentiation ... |
| expge0d 14275 | A nonnegative real raised ... |
| expge1d 14276 | A real greater than or equ... |
| ltexp2a 14277 | Exponent ordering relation... |
| expmordi 14278 | Base ordering relationship... |
| rpexpmord 14279 | Base ordering relationship... |
| expcan 14280 | Cancellation law for integ... |
| ltexp2 14281 | Strict ordering law for ex... |
| leexp2 14282 | Ordering law for exponenti... |
| leexp2a 14283 | Weak ordering relationship... |
| ltexp2r 14284 | The integer powers of a fi... |
| leexp2r 14285 | Weak ordering relationship... |
| leexp1a 14286 | Weak base ordering relatio... |
| leexp1ad 14287 | Weak base ordering relatio... |
| exple1 14288 | A real between 0 and 1 inc... |
| expubnd 14289 | An upper bound on ` A ^ N ... |
| sumsqeq0 14290 | The sum of two squres of r... |
| sqvali 14291 | Value of square. Inferenc... |
| sqcli 14292 | Closure of square. (Contr... |
| sqeq0i 14293 | A complex number is zero i... |
| sqrecii 14294 | The square of a reciprocal... |
| sqmuli 14295 | Distribution of squaring o... |
| sqdivi 14296 | Distribution of squaring o... |
| resqcli 14297 | Closure of square in reals... |
| sqgt0i 14298 | The square of a nonzero re... |
| sqge0i 14299 | The square of a real is no... |
| lt2sqi 14300 | The square function on non... |
| le2sqi 14301 | The square function on non... |
| sq11i 14302 | The square function is one... |
| sq0 14303 | The square of 0 is 0. (Co... |
| sq0i 14304 | If a number is zero, then ... |
| sq0id 14305 | If a number is zero, then ... |
| sq1 14306 | The square of 1 is 1. (Co... |
| neg1sqe1 14307 | The square of ` -u 1 ` is ... |
| sq2 14308 | The square of 2 is 4. (Co... |
| sq3 14309 | The square of 3 is 9. (Co... |
| sq4e2t8 14310 | The square of 4 is 2 times... |
| cu2 14311 | The cube of 2 is 8. (Cont... |
| irec 14312 | The reciprocal of ` _i ` .... |
| i2 14313 | ` _i ` squared. (Contribu... |
| i3 14314 | ` _i ` cubed. (Contribute... |
| i4 14315 | ` _i ` to the fourth power... |
| nnlesq 14316 | A positive integer is less... |
| zzlesq 14317 | An integer is less than or... |
| iexpcyc 14318 | Taking ` _i ` to the ` K `... |
| expnass 14319 | A counterexample showing t... |
| sqlecan 14320 | Cancel one factor of a squ... |
| subsq 14321 | Factor the difference of t... |
| subsq2 14322 | Express the difference of ... |
| binom2i 14323 | The square of a binomial. ... |
| subsqi 14324 | Factor the difference of t... |
| sqeqori 14325 | The squares of two complex... |
| subsq0i 14326 | The two solutions to the d... |
| sqeqor 14327 | The squares of two complex... |
| binom2 14328 | The square of a binomial. ... |
| binom2d 14329 | Deduction form of ~ binom2... |
| binom21 14330 | Special case of ~ binom2 w... |
| binom2sub 14331 | Expand the square of a sub... |
| binom2sub1 14332 | Special case of ~ binom2su... |
| binom2subi 14333 | Expand the square of a sub... |
| mulbinom2 14334 | The square of a binomial w... |
| binom3 14335 | The cube of a binomial. (... |
| sq01 14336 | If a complex number equals... |
| zesq 14337 | An integer is even iff its... |
| nnesq 14338 | A positive integer is even... |
| crreczi 14339 | Reciprocal of a complex nu... |
| bernneq 14340 | Bernoulli's inequality, du... |
| bernneq2 14341 | Variation of Bernoulli's i... |
| bernneq3 14342 | A corollary of ~ bernneq .... |
| expnbnd 14343 | Exponentiation with a base... |
| expnlbnd 14344 | The reciprocal of exponent... |
| expnlbnd2 14345 | The reciprocal of exponent... |
| expmulnbnd 14346 | Exponentiation with a base... |
| digit2 14347 | Two ways to express the ` ... |
| digit1 14348 | Two ways to express the ` ... |
| modexp 14349 | Exponentiation property of... |
| discr1 14350 | A nonnegative quadratic fo... |
| discr 14351 | If a quadratic polynomial ... |
| expnngt1 14352 | If an integer power with a... |
| expnngt1b 14353 | An integer power with an i... |
| sqoddm1div8 14354 | A squared odd number minus... |
| nnsqcld 14355 | The naturals are closed un... |
| nnexpcld 14356 | Closure of exponentiation ... |
| nn0expcld 14357 | Closure of exponentiation ... |
| rpexpcld 14358 | Closure law for exponentia... |
| ltexp2rd 14359 | The power of a positive nu... |
| reexpclzd 14360 | Closure of exponentiation ... |
| sqgt0d 14361 | The square of a nonzero re... |
| ltexp2d 14362 | Ordering relationship for ... |
| leexp2d 14363 | Ordering law for exponenti... |
| expcand 14364 | Ordering relationship for ... |
| leexp2ad 14365 | Ordering relationship for ... |
| leexp2rd 14366 | Ordering relationship for ... |
| lt2sqd 14367 | The square function on non... |
| le2sqd 14368 | The square function on non... |
| sq11d 14369 | The square function is one... |
| ltexp1d 14370 | Elevating to a positive po... |
| ltexp1dd 14371 | Raising both sides of 'les... |
| exp11nnd 14372 | The function elevating non... |
| mulsubdivbinom2 14373 | The square of a binomial w... |
| muldivbinom2 14374 | The square of a binomial w... |
| sq10 14375 | The square of 10 is 100. ... |
| sq10e99m1 14376 | The square of 10 is 99 plu... |
| 3dec 14377 | A "decimal constructor" wh... |
| nn0le2msqi 14378 | The square function on non... |
| nn0opthlem1 14379 | A rather pretty lemma for ... |
| nn0opthlem2 14380 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14381 | An ordered pair theorem fo... |
| nn0opth2i 14382 | An ordered pair theorem fo... |
| nn0opth2 14383 | An ordered pair theorem fo... |
| facnn 14386 | Value of the factorial fun... |
| fac0 14387 | The factorial of 0. (Cont... |
| fac1 14388 | The factorial of 1. (Cont... |
| facp1 14389 | The factorial of a success... |
| fac2 14390 | The factorial of 2. (Cont... |
| fac3 14391 | The factorial of 3. (Cont... |
| fac4 14392 | The factorial of 4. (Cont... |
| facnn2 14393 | Value of the factorial fun... |
| faccl 14394 | Closure of the factorial f... |
| faccld 14395 | Closure of the factorial f... |
| facmapnn 14396 | The factorial function res... |
| facne0 14397 | The factorial function is ... |
| facdiv 14398 | A positive integer divides... |
| facndiv 14399 | No positive integer (great... |
| facwordi 14400 | Ordering property of facto... |
| faclbnd 14401 | A lower bound for the fact... |
| faclbnd2 14402 | A lower bound for the fact... |
| faclbnd3 14403 | A lower bound for the fact... |
| faclbnd4lem1 14404 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14405 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14406 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14407 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14408 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14409 | The factorial function gro... |
| faclbnd6 14410 | Geometric lower bound for ... |
| facubnd 14411 | An upper bound for the fac... |
| facavg 14412 | The product of two factori... |
| bcval 14415 | Value of the binomial coef... |
| bcval2 14416 | Value of the binomial coef... |
| bcval3 14417 | Value of the binomial coef... |
| bcval4 14418 | Value of the binomial coef... |
| bcrpcl 14419 | Closure of the binomial co... |
| bccmpl 14420 | "Complementing" its second... |
| bcn0 14421 | ` N ` choose 0 is 1. Rema... |
| bc0k 14422 | The binomial coefficient "... |
| bcnn 14423 | ` N ` choose ` N ` is 1. ... |
| bcn1 14424 | Binomial coefficient: ` N ... |
| bcnp1n 14425 | Binomial coefficient: ` N ... |
| bcm1k 14426 | The proportion of one bino... |
| bcp1n 14427 | The proportion of one bino... |
| bcp1nk 14428 | The proportion of one bino... |
| bcval5 14429 | Write out the top and bott... |
| bcn2 14430 | Binomial coefficient: ` N ... |
| bcp1m1 14431 | Compute the binomial coeff... |
| bcpasc 14432 | Pascal's rule for the bino... |
| bccl 14433 | A binomial coefficient, in... |
| bccl2 14434 | A binomial coefficient, in... |
| bcn2m1 14435 | Compute the binomial coeff... |
| bcn2p1 14436 | Compute the binomial coeff... |
| permnn 14437 | The number of permutations... |
| bcnm1 14438 | The binomial coefficient o... |
| 4bc3eq4 14439 | The value of four choose t... |
| 4bc2eq6 14440 | The value of four choose t... |
| hashkf 14443 | The finite part of the siz... |
| hashgval 14444 | The value of the ` # ` fun... |
| hashginv 14445 | The converse of ` G ` maps... |
| hashinf 14446 | The value of the ` # ` fun... |
| hashbnd 14447 | If ` A ` has size bounded ... |
| hashfxnn0 14448 | The size function is a fun... |
| hashf 14449 | The size function maps all... |
| hashxnn0 14450 | The value of the hash func... |
| hashresfn 14451 | Restriction of the domain ... |
| dmhashres 14452 | Restriction of the domain ... |
| hashnn0pnf 14453 | The value of the hash func... |
| hashnnn0genn0 14454 | If the size of a set is no... |
| hashnemnf 14455 | The size of a set is never... |
| hashv01gt1 14456 | The size of a set is eithe... |
| hashfz1 14457 | The set ` ( 1 ... N ) ` ha... |
| hashen 14458 | Two finite sets have the s... |
| hasheni 14459 | Equinumerous sets have the... |
| hasheqf1o 14460 | The size of two finite set... |
| fiinfnf1o 14461 | There is no bijection betw... |
| hasheqf1oi 14462 | The size of two sets is eq... |
| hashf1rn 14463 | The size of a finite set w... |
| hasheqf1od 14464 | The size of two sets is eq... |
| fz1eqb 14465 | Two possibly-empty 1-based... |
| hashcard 14466 | The size function of the c... |
| hashcl 14467 | Closure of the ` # ` funct... |
| hashxrcl 14468 | Extended real closure of t... |
| hashclb 14469 | Reverse closure of the ` #... |
| nfile 14470 | The size of any infinite s... |
| hashvnfin 14471 | A set of finite size is a ... |
| hashnfinnn0 14472 | The size of an infinite se... |
| isfinite4 14473 | A finite set is equinumero... |
| hasheq0 14474 | Two ways of saying a set i... |
| hashneq0 14475 | Two ways of saying a set i... |
| hashgt0n0 14476 | If the size of a set is gr... |
| hashnncl 14477 | Positive natural closure o... |
| hash0 14478 | The empty set has size zer... |
| hashelne0d 14479 | A set with an element has ... |
| hashsng 14480 | The size of a singleton. ... |
| hashen1 14481 | A set has size 1 if and on... |
| hash1elsn 14482 | A set of size 1 with a kno... |
| hashrabrsn 14483 | The size of a restricted c... |
| hashrabsn01 14484 | The size of a restricted c... |
| hashrabsn1 14485 | If the size of a restricte... |
| hashfn 14486 | A function is equinumerous... |
| fseq1hash 14487 | The value of the size func... |
| hashgadd 14488 | ` G ` maps ordinal additio... |
| hashgval2 14489 | A short expression for the... |
| hashdom 14490 | Dominance relation for the... |
| hashdomi 14491 | Non-strict order relation ... |
| hashsdom 14492 | Strict dominance relation ... |
| hashun 14493 | The size of the union of d... |
| hashun2 14494 | The size of the union of f... |
| hashun3 14495 | The size of the union of f... |
| hashinfxadd 14496 | The extended real addition... |
| hashunx 14497 | The size of the union of d... |
| hashge0 14498 | The cardinality of a set i... |
| hashgt0 14499 | The cardinality of a nonem... |
| hashge1 14500 | The cardinality of a nonem... |
| 1elfz0hash 14501 | 1 is an element of the fin... |
| hashnn0n0nn 14502 | If a nonnegative integer i... |
| hashunsng 14503 | The size of the union of a... |
| hashunsngx 14504 | The size of the union of a... |
| hashunsnggt 14505 | The size of a set is great... |
| hashprg 14506 | The size of an unordered p... |
| elprchashprn2 14507 | If one element of an unord... |
| hashprb 14508 | The size of an unordered p... |
| hashprdifel 14509 | The elements of an unorder... |
| prhash2ex 14510 | There is (at least) one se... |
| hashle00 14511 | If the size of a set is le... |
| hashgt0elex 14512 | If the size of a set is gr... |
| hashgt0elexb 14513 | The size of a set is great... |
| hashp1i 14514 | Size of a finite ordinal. ... |
| hash1 14515 | Size of a finite ordinal. ... |
| hash2 14516 | Size of a finite ordinal. ... |
| hash3 14517 | Size of a finite ordinal. ... |
| hash4 14518 | Size of a finite ordinal. ... |
| pr0hash2ex 14519 | There is (at least) one se... |
| hashss 14520 | The size of a subset is le... |
| hashpss 14521 | The size of a proper subse... |
| prsshashgt1 14522 | The size of a superset of ... |
| hashin 14523 | The size of the intersecti... |
| hashssdif 14524 | The size of the difference... |
| hashdif 14525 | The size of the difference... |
| hashdifsn 14526 | The size of the difference... |
| hashdifpr 14527 | The size of the difference... |
| hashsn01 14528 | The size of a singleton is... |
| hashsnle1 14529 | The size of a singleton is... |
| hashsnlei 14530 | Get an upper bound on a co... |
| hash1snb 14531 | The size of a set is 1 if ... |
| euhash1 14532 | The size of a set is 1 in ... |
| hash1n0 14533 | If the size of a set is 1 ... |
| hashgt12el 14534 | In a set with more than on... |
| hashgt12el2 14535 | In a set with more than on... |
| hashgt23el 14536 | A set with more than two e... |
| hashunlei 14537 | Get an upper bound on a co... |
| hashsslei 14538 | Get an upper bound on a co... |
| hashfz 14539 | Value of the numeric cardi... |
| fzsdom2 14540 | Condition for finite range... |
| hashfzo 14541 | Cardinality of a half-open... |
| hashfzo0 14542 | Cardinality of a half-open... |
| hashfzp1 14543 | Value of the numeric cardi... |
| hashfz0 14544 | Value of the numeric cardi... |
| hashxplem 14545 | Lemma for ~ hashxp . (Con... |
| hashxp 14546 | The size of the Cartesian ... |
| hashmap 14547 | The size of the set expone... |
| hashpw 14548 | The size of the power set ... |
| hashfun 14549 | A finite set is a function... |
| hashres 14550 | The number of elements of ... |
| hashreshashfun 14551 | The number of elements of ... |
| hashimarn 14552 | The size of the image of a... |
| hashimarni 14553 | If the size of the image o... |
| hashfundm 14554 | The size of a set function... |
| hashf1dmrn 14555 | The size of the domain of ... |
| hashf1dmcdm 14556 | The size of the domain of ... |
| resunimafz0 14557 | TODO-AV: Revise using ` F... |
| fnfz0hash 14558 | The size of a function on ... |
| ffz0hash 14559 | The size of a function on ... |
| fnfz0hashnn0 14560 | The size of a function on ... |
| ffzo0hash 14561 | The size of a function on ... |
| fnfzo0hash 14562 | The size of a function on ... |
| fnfzo0hashnn0 14563 | The value of the size func... |
| hashbclem 14564 | Lemma for ~ hashbc : induc... |
| hashbc 14565 | The binomial coefficient c... |
| hashfacen 14566 | The number of bijections b... |
| hashf1lem1 14567 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14568 | Lemma for ~ hashf1 . (Con... |
| hashf1 14569 | The permutation number ` |... |
| hashfac 14570 | A factorial counts the num... |
| leiso 14571 | Two ways to write a strict... |
| leisorel 14572 | Version of ~ isorel for st... |
| fz1isolem 14573 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14574 | Any finite ordered set has... |
| ishashinf 14575 | Any set that is not finite... |
| seqcoll 14576 | The function ` F ` contain... |
| seqcoll2 14577 | The function ` F ` contain... |
| phphashd 14578 | Corollary of the Pigeonhol... |
| phphashrd 14579 | Corollary of the Pigeonhol... |
| hashprlei 14580 | An unordered pair has at m... |
| hash2pr 14581 | A set of size two is an un... |
| hash2prde 14582 | A set of size two is an un... |
| hash2exprb 14583 | A set of size two is an un... |
| hash2prb 14584 | A set of size two is a pro... |
| prprrab 14585 | The set of proper pairs of... |
| nehash2 14586 | The cardinality of a set w... |
| hash2prd 14587 | A set of size two is an un... |
| hash2pwpr 14588 | If the size of a subset of... |
| hashle2pr 14589 | A nonempty set of size les... |
| hashle2prv 14590 | A nonempty subset of a pow... |
| pr2pwpr 14591 | The set of subsets of a pa... |
| hashge2el2dif 14592 | A set with size at least 2... |
| hashge2el2difr 14593 | A set with at least 2 diff... |
| hashge2el2difb 14594 | A set has size at least 2 ... |
| hashdmpropge2 14595 | The size of the domain of ... |
| hashtplei 14596 | An unordered triple has at... |
| hashtpg 14597 | The size of an unordered t... |
| hash7g 14598 | The size of an unordered s... |
| hashge3el3dif 14599 | A set with size at least 3... |
| elss2prb 14600 | An element of the set of s... |
| hash2sspr 14601 | A subset of size two is an... |
| exprelprel 14602 | If there is an element of ... |
| hash3tr 14603 | A set of size three is an ... |
| hash1to3 14604 | If the size of a set is be... |
| hash3tpde 14605 | A set of size three is an ... |
| hash3tpexb 14606 | A set of size three is an ... |
| hash3tpb 14607 | A set of size three is a p... |
| tpf1ofv0 14608 | The value of a one-to-one ... |
| tpf1ofv1 14609 | The value of a one-to-one ... |
| tpf1ofv2 14610 | The value of a one-to-one ... |
| tpf 14611 | A function into a (proper)... |
| tpfo 14612 | A function onto a (proper)... |
| tpf1o 14613 | A bijection onto a (proper... |
| fundmge2nop0 14614 | A function with a domain c... |
| fundmge2nop 14615 | A function with a domain c... |
| fun2dmnop0 14616 | A function with a domain c... |
| fun2dmnop 14617 | A function with a domain c... |
| hashdifsnp1 14618 | If the size of a set is a ... |
| fi1uzind 14619 | Properties of an ordered p... |
| brfi1uzind 14620 | Properties of a binary rel... |
| brfi1ind 14621 | Properties of a binary rel... |
| brfi1indALT 14622 | Alternate proof of ~ brfi1... |
| opfi1uzind 14623 | Properties of an ordered p... |
| opfi1ind 14624 | Properties of an ordered p... |
| iswrd 14627 | Property of being a word o... |
| wrdval 14628 | Value of the set of words ... |
| iswrdi 14629 | A zero-based sequence is a... |
| wrdf 14630 | A word is a zero-based seq... |
| wrdfd 14631 | A word is a zero-based seq... |
| iswrdb 14632 | A word over an alphabet is... |
| wrddm 14633 | The indices of a word (i.e... |
| sswrd 14634 | The set of words respects ... |
| snopiswrd 14635 | A singleton of an ordered ... |
| wrdexg 14636 | The set of words over a se... |
| wrdexb 14637 | The set of words over a se... |
| wrdexi 14638 | The set of words over a se... |
| wrdsymbcl 14639 | A symbol within a word ove... |
| wrdfn 14640 | A word is a function with ... |
| wrdv 14641 | A word over an alphabet is... |
| wrdlndm 14642 | The length of a word is no... |
| iswrdsymb 14643 | An arbitrary word is a wor... |
| wrdfin 14644 | A word is a finite set. (... |
| lencl 14645 | The length of a word is a ... |
| lennncl 14646 | The length of a nonempty w... |
| wrdffz 14647 | A word is a function from ... |
| wrdeq 14648 | Equality theorem for the s... |
| wrdeqi 14649 | Equality theorem for the s... |
| iswrddm0 14650 | A function with empty doma... |
| wrd0 14651 | The empty set is a word (t... |
| 0wrd0 14652 | The empty word is the only... |
| ffz0iswrd 14653 | A sequence with zero-based... |
| wrdsymb 14654 | A word is a word over the ... |
| nfwrd 14655 | Hypothesis builder for ` W... |
| csbwrdg 14656 | Class substitution for the... |
| wrdnval 14657 | Words of a fixed length ar... |
| wrdmap 14658 | Words as a mapping. (Cont... |
| hashwrdn 14659 | If there is only a finite ... |
| wrdnfi 14660 | If there is only a finite ... |
| wrdsymb0 14661 | A symbol at a position "ou... |
| wrdlenge1n0 14662 | A word with length at leas... |
| len0nnbi 14663 | The length of a word is a ... |
| wrdlenge2n0 14664 | A word with length at leas... |
| wrdsymb1 14665 | The first symbol of a none... |
| wrdlen1 14666 | A word of length 1 starts ... |
| fstwrdne 14667 | The first symbol of a none... |
| fstwrdne0 14668 | The first symbol of a none... |
| eqwrd 14669 | Two words are equal iff th... |
| elovmpowrd 14670 | Implications for the value... |
| elovmptnn0wrd 14671 | Implications for the value... |
| wrdred1 14672 | A word truncated by a symb... |
| wrdred1hash 14673 | The length of a word trunc... |
| lsw 14676 | Extract the last symbol of... |
| lsw0 14677 | The last symbol of an empt... |
| lsw0g 14678 | The last symbol of an empt... |
| lsw1 14679 | The last symbol of a word ... |
| lswcl 14680 | Closure of the last symbol... |
| lswlgt0cl 14681 | The last symbol of a nonem... |
| ccatfn 14684 | The concatenation operator... |
| ccatfval 14685 | Value of the concatenation... |
| ccatcl 14686 | The concatenation of two w... |
| ccatlen 14687 | The length of a concatenat... |
| ccat0 14688 | The concatenation of two w... |
| ccatval1 14689 | Value of a symbol in the l... |
| ccatval2 14690 | Value of a symbol in the r... |
| ccatval3 14691 | Value of a symbol in the r... |
| elfzelfzccat 14692 | An element of a finite set... |
| ccatvalfn 14693 | The concatenation of two w... |
| ccatdmss 14694 | The domain of a concatenat... |
| ccatsymb 14695 | The symbol at a given posi... |
| ccatfv0 14696 | The first symbol of a conc... |
| ccatval1lsw 14697 | The last symbol of the lef... |
| ccatval21sw 14698 | The first symbol of the ri... |
| ccatlid 14699 | Concatenation of a word by... |
| ccatrid 14700 | Concatenation of a word by... |
| ccatass 14701 | Associative law for concat... |
| ccatrn 14702 | The range of a concatenate... |
| ccatf1 14703 | Conditions for a concatena... |
| ccatidid 14704 | Concatenation of the empty... |
| lswccatn0lsw 14705 | The last symbol of a word ... |
| lswccat0lsw 14706 | The last symbol of a word ... |
| ccatalpha 14707 | A concatenation of two arb... |
| ccatrcl1 14708 | Reverse closure of a conca... |
| ids1 14711 | Identity function protecti... |
| s1val 14712 | Value of a singleton word.... |
| s1rn 14713 | The range of a singleton w... |
| s1eq 14714 | Equality theorem for a sin... |
| s1eqd 14715 | Equality theorem for a sin... |
| s1cl 14716 | A singleton word is a word... |
| s1cld 14717 | A singleton word is a word... |
| s1prc 14718 | Value of a singleton word ... |
| s1cli 14719 | A singleton word is a word... |
| s1len 14720 | Length of a singleton word... |
| s1nz 14721 | A singleton word is not th... |
| s1dm 14722 | The domain of a singleton ... |
| s1f1 14723 | Conditions for a length 1 ... |
| s1dmALT 14724 | Alternate version of ~ s1d... |
| s1fv 14725 | Sole symbol of a singleton... |
| lsws1 14726 | The last symbol of a singl... |
| eqs1 14727 | A word of length 1 is a si... |
| wrdl1exs1 14728 | A word of length 1 is a si... |
| wrdl1s1 14729 | A word of length 1 is a si... |
| s111 14730 | The singleton word functio... |
| ccatws1cl 14731 | The concatenation of a wor... |
| ccatws1clv 14732 | The concatenation of a wor... |
| ccat2s1cl 14733 | The concatenation of two s... |
| ccats1alpha 14734 | A concatenation of a word ... |
| ccatws1len 14735 | The length of the concaten... |
| ccatws1lenp1b 14736 | The length of a word is ` ... |
| wrdlenccats1lenm1 14737 | The length of a word is th... |
| ccat2s1len 14738 | The length of the concaten... |
| ccatw2s1cl 14739 | The concatenation of a wor... |
| ccatw2s1len 14740 | The length of the concaten... |
| ccats1val1 14741 | Value of a symbol in the l... |
| ccats1val2 14742 | Value of the symbol concat... |
| ccat1st1st 14743 | The first symbol of a word... |
| ccat2s1p1 14744 | Extract the first of two c... |
| ccat2s1p2 14745 | Extract the second of two ... |
| ccatw2s1ass 14746 | Associative law for a conc... |
| ccatws1n0 14747 | The concatenation of a wor... |
| ccatws1ls 14748 | The last symbol of the con... |
| lswccats1 14749 | The last symbol of a word ... |
| lswccats1fst 14750 | The last symbol of a nonem... |
| ccatw2s1p1 14751 | Extract the symbol of the ... |
| ccatw2s1p2 14752 | Extract the second of two ... |
| ccat2s1fvw 14753 | Extract a symbol of a word... |
| ccat2s1fst 14754 | The first symbol of the co... |
| swrdnznd 14757 | The value of a subword ope... |
| swrdval 14758 | Value of a subword. (Cont... |
| swrd00 14759 | A zero length substring. ... |
| swrdcl 14760 | Closure of the subword ext... |
| swrdval2 14761 | Value of the subword extra... |
| swrdlen 14762 | Length of an extracted sub... |
| swrdfv 14763 | A symbol in an extracted s... |
| swrdfv0 14764 | The first symbol in an ext... |
| swrdf 14765 | A subword of a word is a f... |
| swrdf1 14766 | Condition for a subword to... |
| swrdvalfn 14767 | Value of the subword extra... |
| swrdrn 14768 | The range of a subword of ... |
| swrdrn3 14769 | Express the range of a sub... |
| swrdlend 14770 | The value of the subword e... |
| swrdnd 14771 | The value of the subword e... |
| swrdnd2 14772 | Value of the subword extra... |
| swrdnnn0nd 14773 | The value of a subword ope... |
| swrdnd0 14774 | The value of a subword ope... |
| swrd0 14775 | A subword of an empty set ... |
| swrdrlen 14776 | Length of a right-anchored... |
| swrdlen2 14777 | Length of an extracted sub... |
| swrdfv2 14778 | A symbol in an extracted s... |
| swrdwrdsymb 14779 | A subword is a word over t... |
| swrdsb0eq 14780 | Two subwords with the same... |
| swrdsbslen 14781 | Two subwords with the same... |
| swrdspsleq 14782 | Two words have a common su... |
| swrds1 14783 | Extract a single symbol fr... |
| swrdlsw 14784 | Extract the last single sy... |
| ccatswrd 14785 | Joining two adjacent subwo... |
| swrdccat2 14786 | Recover the right half of ... |
| pfxnndmnd 14789 | The value of a prefix oper... |
| pfxval 14790 | Value of a prefix operatio... |
| pfx00 14791 | The zero length prefix is ... |
| pfx0 14792 | A prefix of an empty set i... |
| pfxval0 14793 | Value of a prefix operatio... |
| pfxcl 14794 | Closure of the prefix extr... |
| pfxmpt 14795 | Value of the prefix extrac... |
| pfxres 14796 | Value of the prefix extrac... |
| pfxf 14797 | A prefix of a word is a fu... |
| pfxfn 14798 | Value of the prefix extrac... |
| pfxfv 14799 | A symbol in a prefix of a ... |
| pfxlen 14800 | Length of a prefix. (Cont... |
| pfxid 14801 | A word is a prefix of itse... |
| pfxrn 14802 | The range of a prefix of a... |
| pfxn0 14803 | A prefix consisting of at ... |
| pfxnd 14804 | The value of a prefix oper... |
| pfxnd0 14805 | The value of a prefix oper... |
| pfxwrdsymb 14806 | A prefix of a word is a wo... |
| addlenpfx 14807 | The sum of the lengths of ... |
| pfxfv0 14808 | The first symbol of a pref... |
| pfxtrcfv 14809 | A symbol in a word truncat... |
| pfxtrcfv0 14810 | The first symbol in a word... |
| pfxfvlsw 14811 | The last symbol in a nonem... |
| pfxeq 14812 | The prefixes of two words ... |
| pfxtrcfvl 14813 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14814 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14815 | Two (nonempty) words are e... |
| disjwrdpfx 14816 | Sets of words are disjoint... |
| ccatpfx 14817 | Concatenating a prefix wit... |
| pfxccat1 14818 | Recover the left half of a... |
| pfx1 14819 | The prefix of length one o... |
| swrdswrdlem 14820 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14821 | A subword of a subword is ... |
| pfxswrd 14822 | A prefix of a subword is a... |
| swrdpfx 14823 | A subword of a prefix is a... |
| pfxpfx 14824 | A prefix of a prefix is a ... |
| pfxpfxid 14825 | A prefix of a prefix with ... |
| pfxcctswrd 14826 | The concatenation of the p... |
| lenpfxcctswrd 14827 | The length of the concaten... |
| lenrevpfxcctswrd 14828 | The length of the concaten... |
| pfxlswccat 14829 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14830 | The last symbol of a word ... |
| ccats1pfxeqrex 14831 | There exists a symbol such... |
| ccatopth 14832 | An ~ opth -like theorem fo... |
| ccatopth2 14833 | An ~ opth -like theorem fo... |
| ccatlcan 14834 | Concatenation of words is ... |
| ccatrcan 14835 | Concatenation of words is ... |
| wrdeqs1cat 14836 | Decompose a nonempty word ... |
| cats1un 14837 | Express a word with an ext... |
| wrdind 14838 | Perform induction over the... |
| wrd2ind 14839 | Perform induction over the... |
| swrdccatfn 14840 | The subword of a concatena... |
| swrdccatin1 14841 | The subword of a concatena... |
| pfxccatin12lem4 14842 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14843 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14844 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14845 | The subword of a concatena... |
| pfxccatin12lem2c 14846 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14847 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14848 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14849 | The subword of a concatena... |
| pfxccat3 14850 | The subword of a concatena... |
| swrdccat 14851 | The subword of a concatena... |
| pfxccatpfx1 14852 | A prefix of a concatenatio... |
| pfxccatpfx2 14853 | A prefix of a concatenatio... |
| pfxccat3a 14854 | A prefix of a concatenatio... |
| swrdccat3blem 14855 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14856 | A suffix of a concatenatio... |
| pfxccatid 14857 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14858 | A word is a prefix of a wo... |
| swrdccatin1d 14859 | The subword of a concatena... |
| swrdccatin2d 14860 | The subword of a concatena... |
| pfxccatin12d 14861 | The subword of a concatena... |
| reuccatpfxs1lem 14862 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14863 | There is a unique word hav... |
| reuccatpfxs1v 14864 | There is a unique word hav... |
| splval 14867 | Value of the substring rep... |
| splcl 14868 | Closure of the substring r... |
| splid 14869 | Splicing a subword for the... |
| spllen 14870 | The length of a splice. (... |
| splfv1 14871 | Symbols to the left of a s... |
| splfv2a 14872 | Symbols within the replace... |
| splval2 14873 | Value of a splice, assumin... |
| revval 14876 | Value of the word reversin... |
| revcl 14877 | The reverse of a word is a... |
| revlen 14878 | The reverse of a word has ... |
| revfv 14879 | Reverse of a word at a poi... |
| rev0 14880 | The empty word is its own ... |
| revs1 14881 | Singleton words are their ... |
| revccat 14882 | Antiautomorphic property o... |
| revrev 14883 | Reversal is an involution ... |
| revpfxsfxrev 14884 | The reverse of a prefix of... |
| swrdrevpfx 14885 | A subword expressed in ter... |
| reps 14888 | Construct a function mappi... |
| repsundef 14889 | A function mapping a half-... |
| repsconst 14890 | Construct a function mappi... |
| repsf 14891 | The constructed function m... |
| repswsymb 14892 | The symbols of a "repeated... |
| repsw 14893 | A function mapping a half-... |
| repswlen 14894 | The length of a "repeated ... |
| repsw0 14895 | The "repeated symbol word"... |
| repsdf2 14896 | Alternative definition of ... |
| repswsymball 14897 | All the symbols of a "repe... |
| repswsymballbi 14898 | A word is a "repeated symb... |
| repswfsts 14899 | The first symbol of a none... |
| repswlsw 14900 | The last symbol of a nonem... |
| repsw1 14901 | The "repeated symbol word"... |
| repswswrd 14902 | A subword of a "repeated s... |
| repswpfx 14903 | A prefix of a repeated sym... |
| repswccat 14904 | The concatenation of two "... |
| repswrevw 14905 | The reverse of a "repeated... |
| cshfn 14908 | Perform a cyclical shift f... |
| cshword 14909 | Perform a cyclical shift f... |
| cshnz 14910 | A cyclical shift is the em... |
| 0csh0 14911 | Cyclically shifting an emp... |
| cshw0 14912 | A word cyclically shifted ... |
| cshwmodn 14913 | Cyclically shifting a word... |
| cshwsublen 14914 | Cyclically shifting a word... |
| cshwn 14915 | A word cyclically shifted ... |
| cshwcl 14916 | A cyclically shifted word ... |
| cshwlen 14917 | The length of a cyclically... |
| cshwf 14918 | A cyclically shifted word ... |
| cshwfn 14919 | A cyclically shifted word ... |
| cshwrn 14920 | The range of a cyclically ... |
| cshwidxmod 14921 | The symbol at a given inde... |
| cshwidxmodr 14922 | The symbol at a given inde... |
| cshwidx0mod 14923 | The symbol at index 0 of a... |
| cshwidx0 14924 | The symbol at index 0 of a... |
| cshwidxm1 14925 | The symbol at index ((n-N)... |
| cshwidxm 14926 | The symbol at index (n-N) ... |
| cshwidxn 14927 | The symbol at index (n-1) ... |
| cshf1 14928 | Cyclically shifting a word... |
| cshinj 14929 | If a word is injectiv (reg... |
| repswcshw 14930 | A cyclically shifted "repe... |
| 2cshw 14931 | Cyclically shifting a word... |
| 2cshwid 14932 | Cyclically shifting a word... |
| lswcshw 14933 | The last symbol of a word ... |
| 2cshwcom 14934 | Cyclically shifting a word... |
| cshwleneq 14935 | If the results of cyclical... |
| 3cshw 14936 | Cyclically shifting a word... |
| cshweqdif2 14937 | If cyclically shifting two... |
| cshweqdifid 14938 | If cyclically shifting a w... |
| cshweqrep 14939 | If cyclically shifting a w... |
| cshw1 14940 | If cyclically shifting a w... |
| cshw1repsw 14941 | If cyclically shifting a w... |
| cshwsexa 14942 | The class of (different!) ... |
| 2cshwcshw 14943 | If a word is a cyclically ... |
| scshwfzeqfzo 14944 | For a nonempty word the se... |
| cshwcshid 14945 | A cyclically shifted word ... |
| cshwcsh2id 14946 | A cyclically shifted word ... |
| cshimadifsn 14947 | The image of a cyclically ... |
| cshimadifsn0 14948 | The image of a cyclically ... |
| wrdco 14949 | Mapping a word by a functi... |
| lenco 14950 | Length of a mapped word is... |
| s1co 14951 | Mapping of a singleton wor... |
| revco 14952 | Mapping of words (i.e., a ... |
| ccatco 14953 | Mapping of words commutes ... |
| cshco 14954 | Mapping of words commutes ... |
| swrdco 14955 | Mapping of words commutes ... |
| pfxco 14956 | Mapping of words commutes ... |
| lswco 14957 | Mapping of (nonempty) word... |
| repsco 14958 | Mapping of words commutes ... |
| cats1cld 14973 | Closure of concatenation w... |
| cats1co 14974 | Closure of concatenation w... |
| cats1cli 14975 | Closure of concatenation w... |
| cats1fvn 14976 | The last symbol of a conca... |
| cats1fv 14977 | A symbol other than the la... |
| cats1len 14978 | The length of concatenatio... |
| cats1cat 14979 | Closure of concatenation w... |
| cats2cat 14980 | Closure of concatenation o... |
| s2eqd 14981 | Equality theorem for a dou... |
| s3eqd 14982 | Equality theorem for a len... |
| s4eqd 14983 | Equality theorem for a len... |
| s5eqd 14984 | Equality theorem for a len... |
| s6eqd 14985 | Equality theorem for a len... |
| s7eqd 14986 | Equality theorem for a len... |
| s8eqd 14987 | Equality theorem for a len... |
| s3eq2 14988 | Equality theorem for a len... |
| s2cld 14989 | A doubleton word is a word... |
| s3cld 14990 | A length 3 string is a wor... |
| s4cld 14991 | A length 4 string is a wor... |
| s5cld 14992 | A length 5 string is a wor... |
| s6cld 14993 | A length 6 string is a wor... |
| s7cld 14994 | A length 7 string is a wor... |
| s8cld 14995 | A length 8 string is a wor... |
| s2cl 14996 | A doubleton word is a word... |
| s3cl 14997 | A length 3 string is a wor... |
| s2cli 14998 | A doubleton word is a word... |
| s3cli 14999 | A length 3 string is a wor... |
| s4cli 15000 | A length 4 string is a wor... |
| s5cli 15001 | A length 5 string is a wor... |
| s6cli 15002 | A length 6 string is a wor... |
| s7cli 15003 | A length 7 string is a wor... |
| s8cli 15004 | A length 8 string is a wor... |
| s2fv0 15005 | Extract the first symbol f... |
| s2fv1 15006 | Extract the second symbol ... |
| s2len 15007 | The length of a doubleton ... |
| s2dm 15008 | The domain of a doubleton ... |
| s3fv0 15009 | Extract the first symbol f... |
| s3fv1 15010 | Extract the second symbol ... |
| s3fv2 15011 | Extract the third symbol f... |
| s3len 15012 | The length of a length 3 s... |
| s4fv0 15013 | Extract the first symbol f... |
| s4fv1 15014 | Extract the second symbol ... |
| s4fv2 15015 | Extract the third symbol f... |
| s4fv3 15016 | Extract the fourth symbol ... |
| s4len 15017 | The length of a length 4 s... |
| s5len 15018 | The length of a length 5 s... |
| s6len 15019 | The length of a length 6 s... |
| s7len 15020 | The length of a length 7 s... |
| s8len 15021 | The length of a length 8 s... |
| lsws2 15022 | The last symbol of a doubl... |
| lsws3 15023 | The last symbol of a 3 let... |
| lsws4 15024 | The last symbol of a 4 let... |
| s2prop 15025 | A length 2 word is an unor... |
| s2dmALT 15026 | Alternate version of ~ s2d... |
| s3tpop 15027 | A length 3 word is an unor... |
| s4prop 15028 | A length 4 word is a union... |
| s3fn 15029 | A length 3 word is a funct... |
| funcnvs1 15030 | The converse of a singleto... |
| funcnvs2 15031 | The converse of a length 2... |
| funcnvs3 15032 | The converse of a length 3... |
| funcnvs4 15033 | The converse of a length 4... |
| s2f1o 15034 | A length 2 word with mutua... |
| f1oun2prg 15035 | A union of unordered pairs... |
| s4f1o 15036 | A length 4 word with mutua... |
| s4dom 15037 | The domain of a length 4 w... |
| s2co 15038 | Mapping a doubleton word b... |
| s3co 15039 | Mapping a length 3 string ... |
| s0s1 15040 | Concatenation of fixed len... |
| s1s2 15041 | Concatenation of fixed len... |
| s1s3 15042 | Concatenation of fixed len... |
| s1s4 15043 | Concatenation of fixed len... |
| s1s5 15044 | Concatenation of fixed len... |
| s1s6 15045 | Concatenation of fixed len... |
| s1s7 15046 | Concatenation of fixed len... |
| s2s2 15047 | Concatenation of fixed len... |
| s4s2 15048 | Concatenation of fixed len... |
| s4s3 15049 | Concatenation of fixed len... |
| s4s4 15050 | Concatenation of fixed len... |
| s3s4 15051 | Concatenation of fixed len... |
| s2s5 15052 | Concatenation of fixed len... |
| s5s2 15053 | Concatenation of fixed len... |
| s2eq2s1eq 15054 | Two length 2 words are equ... |
| s2eq2seq 15055 | Two length 2 words are equ... |
| s3eqs2s1eq 15056 | Two length 3 words are equ... |
| s3eq3seq 15057 | Two length 3 words are equ... |
| swrds2 15058 | Extract two adjacent symbo... |
| swrds2m 15059 | Extract two adjacent symbo... |
| wrdlen2i 15060 | Implications of a word of ... |
| wrd2pr2op 15061 | A word of length two repre... |
| wrdlen2 15062 | A word of length two. (Co... |
| wrdlen2s2 15063 | A word of length two as do... |
| wrdl2exs2 15064 | A word of length two is a ... |
| pfx2 15065 | A prefix of length two. (... |
| wrd3tpop 15066 | A word of length three rep... |
| wrdlen3s3 15067 | A word of length three as ... |
| s3rex 15068 | Membership in a family of ... |
| s3rexrd 15069 | Converse of ~ s3rex , dedu... |
| repsw2 15070 | The "repeated symbol word"... |
| repsw3 15071 | The "repeated symbol word"... |
| swrd2lsw 15072 | Extract the last two symbo... |
| 2swrd2eqwrdeq 15073 | Two words of length at lea... |
| ccatw2s1ccatws2 15074 | The concatenation of a wor... |
| ccat2s1fvwALT 15075 | Alternate proof of ~ ccat2... |
| wwlktovf 15076 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 15077 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 15078 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 15079 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 15080 | There is a bijection betwe... |
| eqwrds3 15081 | A word is equal with a len... |
| wrdl3s3 15082 | A word of length 3 is a le... |
| s2rn 15083 | Range of a length 2 string... |
| s3rn 15084 | Range of a length 3 string... |
| s7rn 15085 | Range of a length 7 string... |
| s7f1o 15086 | A length 7 word with mutua... |
| s3sndisj 15087 | The singletons consisting ... |
| s3iunsndisj 15088 | The union of singletons co... |
| ofccat 15089 | Letterwise operations on w... |
| ofs1 15090 | Letterwise operations on a... |
| ofs2 15091 | Letterwise operations on a... |
| coss12d 15092 | Subset deduction for compo... |
| trrelssd 15093 | The composition of subclas... |
| xpcogend 15094 | The most interesting case ... |
| xpcoidgend 15095 | If two classes are not dis... |
| cotr2g 15096 | Two ways of saying that th... |
| cotr2 15097 | Two ways of saying a relat... |
| cotr3 15098 | Two ways of saying a relat... |
| coemptyd 15099 | Deduction about compositio... |
| xptrrel 15100 | The cross product is alway... |
| 0trrel 15101 | The empty class is a trans... |
| cleq1lem 15102 | Equality implies bijection... |
| cleq1 15103 | Equality of relations impl... |
| clsslem 15104 | The closure of a subclass ... |
| trcleq1 15109 | Equality of relations impl... |
| trclsslem 15110 | The transitive closure (as... |
| trcleq2lem 15111 | Equality implies bijection... |
| cvbtrcl 15112 | Change of bound variable i... |
| trcleq12lem 15113 | Equality implies bijection... |
| trclexlem 15114 | Existence of relation impl... |
| trclublem 15115 | If a relation exists then ... |
| trclubi 15116 | The Cartesian product of t... |
| trclubgi 15117 | The union with the Cartesi... |
| trclub 15118 | The Cartesian product of t... |
| trclubg 15119 | The union with the Cartesi... |
| trclfv 15120 | The transitive closure of ... |
| brintclab 15121 | Two ways to express a bina... |
| brtrclfv 15122 | Two ways of expressing the... |
| brcnvtrclfv 15123 | Two ways of expressing the... |
| brtrclfvcnv 15124 | Two ways of expressing the... |
| brcnvtrclfvcnv 15125 | Two ways of expressing the... |
| trclfvss 15126 | The transitive closure (as... |
| trclfvub 15127 | The transitive closure of ... |
| trclfvlb 15128 | The transitive closure of ... |
| trclfvcotr 15129 | The transitive closure of ... |
| trclfvlb2 15130 | The transitive closure of ... |
| trclfvlb3 15131 | The transitive closure of ... |
| cotrtrclfv 15132 | The transitive closure of ... |
| trclidm 15133 | The transitive closure of ... |
| trclun 15134 | Transitive closure of a un... |
| trclfvg 15135 | The value of the transitiv... |
| trclfvcotrg 15136 | The value of the transitiv... |
| reltrclfv 15137 | The transitive closure of ... |
| dmtrclfv 15138 | The domain of the transiti... |
| reldmrelexp 15141 | The domain of the repeated... |
| relexp0g 15142 | A relation composed zero t... |
| relexp0 15143 | A relation composed zero t... |
| relexp0d 15144 | A relation composed zero t... |
| relexpsucnnr 15145 | A reduction for relation e... |
| relexp1g 15146 | A relation composed once i... |
| dfid5 15147 | Identity relation is equal... |
| dfid6 15148 | Identity relation expresse... |
| relexp1d 15149 | A relation composed once i... |
| relexpsucnnl 15150 | A reduction for relation e... |
| relexpsucl 15151 | A reduction for relation e... |
| relexpsucr 15152 | A reduction for relation e... |
| relexpsucrd 15153 | A reduction for relation e... |
| relexpsucld 15154 | A reduction for relation e... |
| relexpcnv 15155 | Commutation of converse an... |
| relexpcnvd 15156 | Commutation of converse an... |
| relexp0rel 15157 | The exponentiation of a cl... |
| relexprelg 15158 | The exponentiation of a cl... |
| relexprel 15159 | The exponentiation of a re... |
| relexpreld 15160 | The exponentiation of a re... |
| relexpnndm 15161 | The domain of an exponenti... |
| relexpdmg 15162 | The domain of an exponenti... |
| relexpdm 15163 | The domain of an exponenti... |
| relexpdmd 15164 | The domain of an exponenti... |
| relexpnnrn 15165 | The range of an exponentia... |
| relexprng 15166 | The range of an exponentia... |
| relexprn 15167 | The range of an exponentia... |
| relexprnd 15168 | The range of an exponentia... |
| relexpfld 15169 | The field of an exponentia... |
| relexpfldd 15170 | The field of an exponentia... |
| relexpaddnn 15171 | Relation composition becom... |
| relexpuzrel 15172 | The exponentiation of a cl... |
| relexpaddg 15173 | Relation composition becom... |
| relexpaddd 15174 | Relation composition becom... |
| rtrclreclem1 15177 | The reflexive, transitive ... |
| dfrtrclrec2 15178 | If two elements are connec... |
| rtrclreclem2 15179 | The reflexive, transitive ... |
| rtrclreclem3 15180 | The reflexive, transitive ... |
| rtrclreclem4 15181 | The reflexive, transitive ... |
| dfrtrcl2 15182 | The two definitions ` t* `... |
| relexpindlem 15183 | Principle of transitive in... |
| relexpind 15184 | Principle of transitive in... |
| rtrclind 15185 | Principle of transitive in... |
| shftlem 15188 | Two ways to write a shifte... |
| shftuz 15189 | A shift of the upper integ... |
| shftfval 15190 | The value of the sequence ... |
| shftdm 15191 | Domain of a relation shift... |
| shftfib 15192 | Value of a fiber of the re... |
| shftfn 15193 | Functionality and domain o... |
| shftval 15194 | Value of a sequence shifte... |
| shftval2 15195 | Value of a sequence shifte... |
| shftval3 15196 | Value of a sequence shifte... |
| shftval4 15197 | Value of a sequence shifte... |
| shftval5 15198 | Value of a shifted sequenc... |
| shftf 15199 | Functionality of a shifted... |
| 2shfti 15200 | Composite shift operations... |
| shftidt2 15201 | Identity law for the shift... |
| shftidt 15202 | Identity law for the shift... |
| shftcan1 15203 | Cancellation law for the s... |
| shftcan2 15204 | Cancellation law for the s... |
| seqshft 15205 | Shifting the index set of ... |
| sgnval 15208 | Value of the signum functi... |
| sgn0 15209 | The signum of 0 is 0. (Co... |
| sgnp 15210 | The signum of a positive e... |
| sgnrrp 15211 | The signum of a positive r... |
| sgn1 15212 | The signum of 1 is 1. (Co... |
| sgnpnf 15213 | The signum of ` +oo ` is 1... |
| sgnn 15214 | The signum of a negative e... |
| sgnmnf 15215 | The signum of ` -oo ` is -... |
| sgndm 15216 | The domain of the signum f... |
| sgncl 15217 | Closure of the signum: Th... |
| sgnrn 15218 | The range of the signum fu... |
| sgnfo 15219 | The signum function as ont... |
| sgnneg 15220 | Negation of the signum. (... |
| sgn3da 15221 | A conditional containing a... |
| sgnclre 15222 | Closure of the signum for ... |
| sgn0bi 15223 | Zero signum. (Contributed... |
| sgnnbi 15224 | Negative signum. (Contrib... |
| sgnpbi 15225 | Positive signum. (Contrib... |
| sgnsub 15226 | Signum of a difference wit... |
| sgnmul 15227 | Signum of a product. (Con... |
| sgnmulrp2 15228 | Multiplication by a positi... |
| sgnmulsgn 15229 | If two real numbers are of... |
| cjval 15236 | The value of the conjugate... |
| cjth 15237 | The defining property of t... |
| cjf 15238 | Domain and codomain of the... |
| cjcl 15239 | The conjugate of a complex... |
| reval 15240 | The value of the real part... |
| imval 15241 | The value of the imaginary... |
| imre 15242 | The imaginary part of a co... |
| reim 15243 | The real part of a complex... |
| recl 15244 | The real part of a complex... |
| imcl 15245 | The imaginary part of a co... |
| ref 15246 | Domain and codomain of the... |
| imf 15247 | Domain and codomain of the... |
| crre 15248 | The real part of a complex... |
| crim 15249 | The real part of a complex... |
| replim 15250 | Reconstruct a complex numb... |
| remim 15251 | Value of the conjugate of ... |
| reim0 15252 | The imaginary part of a re... |
| reim0b 15253 | A number is real iff its i... |
| rereb 15254 | A number is real iff it eq... |
| mulre 15255 | A product with a nonzero r... |
| rere 15256 | A real number equals its r... |
| cjreb 15257 | A number is real iff it eq... |
| recj 15258 | Real part of a complex con... |
| reneg 15259 | Real part of negative. (C... |
| readd 15260 | Real part distributes over... |
| resub 15261 | Real part distributes over... |
| remullem 15262 | Lemma for ~ remul , ~ immu... |
| remul 15263 | Real part of a product. (... |
| remul2 15264 | Real part of a product. (... |
| rediv 15265 | Real part of a division. ... |
| imcj 15266 | Imaginary part of a comple... |
| imneg 15267 | The imaginary part of a ne... |
| imadd 15268 | Imaginary part distributes... |
| imsub 15269 | Imaginary part distributes... |
| immul 15270 | Imaginary part of a produc... |
| immul2 15271 | Imaginary part of a produc... |
| imdiv 15272 | Imaginary part of a divisi... |
| cjre 15273 | A real number equals its c... |
| cjcj 15274 | The conjugate of the conju... |
| cjadd 15275 | Complex conjugate distribu... |
| cjmul 15276 | Complex conjugate distribu... |
| ipcnval 15277 | Standard inner product on ... |
| cjmulrcl 15278 | A complex number times its... |
| cjmulval 15279 | A complex number times its... |
| cjmulge0 15280 | A complex number times its... |
| cjneg 15281 | Complex conjugate of negat... |
| addcj 15282 | A number plus its conjugat... |
| cjsub 15283 | Complex conjugate distribu... |
| cjexp 15284 | Complex conjugate of posit... |
| imval2 15285 | The imaginary part of a nu... |
| re0 15286 | The real part of zero. (C... |
| im0 15287 | The imaginary part of zero... |
| re1 15288 | The real part of one. (Co... |
| im1 15289 | The imaginary part of one.... |
| rei 15290 | The real part of ` _i ` . ... |
| imi 15291 | The imaginary part of ` _i... |
| cj0 15292 | The conjugate of zero. (C... |
| cji 15293 | The complex conjugate of t... |
| cjreim 15294 | The conjugate of a represe... |
| cjreim2 15295 | The conjugate of the repre... |
| cj11 15296 | Complex conjugate is a one... |
| cjne0 15297 | A number is nonzero iff it... |
| cjdiv 15298 | Complex conjugate distribu... |
| cnrecnv 15299 | The inverse to the canonic... |
| sqeqd 15300 | A deduction for showing tw... |
| recli 15301 | The real part of a complex... |
| imcli 15302 | The imaginary part of a co... |
| cjcli 15303 | Closure law for complex co... |
| replimi 15304 | Construct a complex number... |
| cjcji 15305 | The conjugate of the conju... |
| reim0bi 15306 | A number is real iff its i... |
| rerebi 15307 | A real number equals its r... |
| cjrebi 15308 | A number is real iff it eq... |
| recji 15309 | Real part of a complex con... |
| imcji 15310 | Imaginary part of a comple... |
| cjmulrcli 15311 | A complex number times its... |
| cjmulvali 15312 | A complex number times its... |
| cjmulge0i 15313 | A complex number times its... |
| renegi 15314 | Real part of negative. (C... |
| imnegi 15315 | Imaginary part of negative... |
| cjnegi 15316 | Complex conjugate of negat... |
| addcji 15317 | A number plus its conjugat... |
| readdi 15318 | Real part distributes over... |
| imaddi 15319 | Imaginary part distributes... |
| remuli 15320 | Real part of a product. (... |
| immuli 15321 | Imaginary part of a produc... |
| cjaddi 15322 | Complex conjugate distribu... |
| cjmuli 15323 | Complex conjugate distribu... |
| ipcni 15324 | Standard inner product on ... |
| cjdivi 15325 | Complex conjugate distribu... |
| crrei 15326 | The real part of a complex... |
| crimi 15327 | The imaginary part of a co... |
| recld 15328 | The real part of a complex... |
| imcld 15329 | The imaginary part of a co... |
| cjcld 15330 | Closure law for complex co... |
| replimd 15331 | Construct a complex number... |
| remimd 15332 | Value of the conjugate of ... |
| cjcjd 15333 | The conjugate of the conju... |
| reim0bd 15334 | A number is real iff its i... |
| rerebd 15335 | A real number equals its r... |
| cjrebd 15336 | A number is real iff it eq... |
| cjne0d 15337 | A number is nonzero iff it... |
| recjd 15338 | Real part of a complex con... |
| imcjd 15339 | Imaginary part of a comple... |
| cjmulrcld 15340 | A complex number times its... |
| cjmulvald 15341 | A complex number times its... |
| cjmulge0d 15342 | A complex number times its... |
| renegd 15343 | Real part of negative. (C... |
| imnegd 15344 | Imaginary part of negative... |
| cjnegd 15345 | Complex conjugate of negat... |
| addcjd 15346 | A number plus its conjugat... |
| cjexpd 15347 | Complex conjugate of posit... |
| readdd 15348 | Real part distributes over... |
| imaddd 15349 | Imaginary part distributes... |
| resubd 15350 | Real part distributes over... |
| imsubd 15351 | Imaginary part distributes... |
| remuld 15352 | Real part of a product. (... |
| immuld 15353 | Imaginary part of a produc... |
| cjaddd 15354 | Complex conjugate distribu... |
| cjmuld 15355 | Complex conjugate distribu... |
| ipcnd 15356 | Standard inner product on ... |
| cjdivd 15357 | Complex conjugate distribu... |
| rered 15358 | A real number equals its r... |
| reim0d 15359 | The imaginary part of a re... |
| cjred 15360 | A real number equals its c... |
| remul2d 15361 | Real part of a product. (... |
| immul2d 15362 | Imaginary part of a produc... |
| redivd 15363 | Real part of a division. ... |
| imdivd 15364 | Imaginary part of a divisi... |
| crred 15365 | The real part of a complex... |
| crimd 15366 | The imaginary part of a co... |
| sqrtval 15371 | Value of square root funct... |
| absval 15372 | The absolute value (modulu... |
| rennim 15373 | A real number does not lie... |
| cnpart 15374 | The specification of restr... |
| sqrt0 15375 | The square root of zero is... |
| 01sqrexlem1 15376 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15377 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15378 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15379 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15380 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15381 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15382 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15383 | Existence of a square root... |
| resqrex 15384 | Existence of a square root... |
| sqrmo 15385 | Uniqueness for the square ... |
| resqreu 15386 | Existence and uniqueness f... |
| resqrtcl 15387 | Closure of the square root... |
| resqrtthlem 15388 | Lemma for ~ resqrtth . (C... |
| resqrtth 15389 | Square root theorem over t... |
| remsqsqrt 15390 | Square of square root. (C... |
| sqrtge0 15391 | The square root function i... |
| sqrtgt0 15392 | The square root function i... |
| sqrtmul 15393 | Square root distributes ov... |
| sqrtle 15394 | Square root is monotonic. ... |
| sqrtlt 15395 | Square root is strictly mo... |
| sqrt11 15396 | The square root function i... |
| sqrt00 15397 | A square root is zero iff ... |
| rpsqrtcl 15398 | The square root of a posit... |
| sqrtdiv 15399 | Square root distributes ov... |
| sqrtneglem 15400 | The square root of a negat... |
| sqrtneg 15401 | The square root of a negat... |
| sqrtsq2 15402 | Relationship between squar... |
| sqrtsq 15403 | Square root of square. (C... |
| sqrtmsq 15404 | Square root of square. (C... |
| sqrt1 15405 | The square root of 1 is 1.... |
| sqrt4 15406 | The square root of 4 is 2.... |
| sqrt9 15407 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15408 | The square root of 2 is bo... |
| sqrtm1 15409 | The imaginary unit is the ... |
| nn0sqeq1 15410 | A natural number with squa... |
| absneg 15411 | Absolute value of the nega... |
| abscl 15412 | Real closure of absolute v... |
| abscj 15413 | The absolute value of a nu... |
| absvalsq 15414 | Square of value of absolut... |
| absvalsq2 15415 | Square of value of absolut... |
| sqabsadd 15416 | Square of absolute value o... |
| sqabssub 15417 | Square of absolute value o... |
| absval2 15418 | Value of absolute value fu... |
| abs0 15419 | The absolute value of 0. ... |
| absi 15420 | The absolute value of the ... |
| absge0 15421 | Absolute value is nonnegat... |
| absrpcl 15422 | The absolute value of a no... |
| abs00 15423 | The absolute value of a nu... |
| abs00ad 15424 | A complex number is zero i... |
| abs00bd 15425 | If a complex number is zer... |
| absreimsq 15426 | Square of the absolute val... |
| absreim 15427 | Absolute value of a number... |
| absmul 15428 | Absolute value distributes... |
| absdiv 15429 | Absolute value distributes... |
| absid 15430 | A nonnegative number is it... |
| abs1 15431 | The absolute value of one ... |
| absnid 15432 | For a negative number, its... |
| leabs 15433 | A real number is less than... |
| absor 15434 | The absolute value of a re... |
| absre 15435 | Absolute value of a real n... |
| absresq 15436 | Square of the absolute val... |
| absmod0 15437 | ` A ` is divisible by ` B ... |
| absexp 15438 | Absolute value of positive... |
| absexpz 15439 | Absolute value of integer ... |
| abssq 15440 | Square can be moved in and... |
| sqabs 15441 | The squares of two reals a... |
| absrele 15442 | The absolute value of a co... |
| absimle 15443 | The absolute value of a co... |
| max0add 15444 | The sum of the positive an... |
| absz 15445 | A real number is an intege... |
| nn0abscl 15446 | The absolute value of an i... |
| zabscl 15447 | The absolute value of an i... |
| zabs0b 15448 | An integer has an absolute... |
| abslt 15449 | Absolute value and 'less t... |
| absle 15450 | Absolute value and 'less t... |
| abssubne0 15451 | If the absolute value of a... |
| absdiflt 15452 | The absolute value of a di... |
| absdifle 15453 | The absolute value of a di... |
| elicc4abs 15454 | Membership in a symmetric ... |
| lenegsq 15455 | Comparison to a nonnegativ... |
| releabs 15456 | The real part of a number ... |
| recval 15457 | Reciprocal expressed with ... |
| absidm 15458 | The absolute value functio... |
| absgt0 15459 | The absolute value of a no... |
| nnabscl 15460 | The absolute value of a no... |
| abssub 15461 | Swapping order of subtract... |
| abssubge0 15462 | Absolute value of a nonneg... |
| abssuble0 15463 | Absolute value of a nonpos... |
| absmax 15464 | The maximum of two numbers... |
| abstri 15465 | Triangle inequality for ab... |
| abs3dif 15466 | Absolute value of differen... |
| abs2dif 15467 | Difference of absolute val... |
| abs2dif2 15468 | Difference of absolute val... |
| abs2difabs 15469 | Absolute value of differen... |
| abs1m 15470 | For any complex number, th... |
| recan 15471 | Cancellation law involving... |
| absf 15472 | Mapping domain and codomai... |
| abs3lem 15473 | Lemma involving absolute v... |
| abslem2 15474 | Lemma involving absolute v... |
| rddif 15475 | The difference between a r... |
| absrdbnd 15476 | Bound on the absolute valu... |
| fzomaxdiflem 15477 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15478 | A bound on the separation ... |
| uzin2 15479 | The upper integers are clo... |
| rexanuz 15480 | Combine two different uppe... |
| rexanre 15481 | Combine two different uppe... |
| rexfiuz 15482 | Combine finitely many diff... |
| rexuz3 15483 | Restrict the base of the u... |
| rexanuz2 15484 | Combine two different uppe... |
| r19.29uz 15485 | A version of ~ 19.29 for u... |
| r19.2uz 15486 | A version of ~ r19.2z for ... |
| rexuzre 15487 | Convert an upper real quan... |
| rexico 15488 | Restrict the base of an up... |
| cau3lem 15489 | Lemma for ~ cau3 . (Contr... |
| cau3 15490 | Convert between three-quan... |
| cau4 15491 | Change the base of a Cauch... |
| caubnd2 15492 | A Cauchy sequence of compl... |
| caubnd 15493 | A Cauchy sequence of compl... |
| sqreulem 15494 | Lemma for ~ sqreu : write ... |
| sqreu 15495 | Existence and uniqueness f... |
| sqrtcl 15496 | Closure of the square root... |
| sqrtthlem 15497 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15498 | Mapping domain and codomai... |
| sqrtth 15499 | Square root theorem over t... |
| sqrtrege0 15500 | The square root function m... |
| eqsqrtor 15501 | Solve an equation containi... |
| eqsqrtd 15502 | A deduction for showing th... |
| eqsqrt2d 15503 | A deduction for showing th... |
| amgm2 15504 | Arithmetic-geometric mean ... |
| sqrtthi 15505 | Square root theorem. Theo... |
| sqrtcli 15506 | The square root of a nonne... |
| sqrtgt0i 15507 | The square root of a posit... |
| sqrtmsqi 15508 | Square root of square. (C... |
| sqrtsqi 15509 | Square root of square. (C... |
| sqsqrti 15510 | Square of square root. (C... |
| sqrtge0i 15511 | The square root of a nonne... |
| absidi 15512 | A nonnegative number is it... |
| absnidi 15513 | A negative number is the n... |
| leabsi 15514 | A real number is less than... |
| absori 15515 | The absolute value of a re... |
| absrei 15516 | Absolute value of a real n... |
| sqrtpclii 15517 | The square root of a posit... |
| sqrtgt0ii 15518 | The square root of a posit... |
| sqrt11i 15519 | The square root function i... |
| sqrtmuli 15520 | Square root distributes ov... |
| sqrtmulii 15521 | Square root distributes ov... |
| sqrtmsq2i 15522 | Relationship between squar... |
| sqrtlei 15523 | Square root is monotonic. ... |
| sqrtlti 15524 | Square root is strictly mo... |
| abslti 15525 | Absolute value and 'less t... |
| abslei 15526 | Absolute value and 'less t... |
| cnsqrt00 15527 | A square root of a complex... |
| absvalsqi 15528 | Square of value of absolut... |
| absvalsq2i 15529 | Square of value of absolut... |
| abscli 15530 | Real closure of absolute v... |
| absge0i 15531 | Absolute value is nonnegat... |
| absval2i 15532 | Value of absolute value fu... |
| abs00i 15533 | The absolute value of a nu... |
| absgt0i 15534 | The absolute value of a no... |
| absnegi 15535 | Absolute value of negative... |
| abscji 15536 | The absolute value of a nu... |
| releabsi 15537 | The real part of a number ... |
| abssubi 15538 | Swapping order of subtract... |
| absmuli 15539 | Absolute value distributes... |
| sqabsaddi 15540 | Square of absolute value o... |
| sqabssubi 15541 | Square of absolute value o... |
| absdivzi 15542 | Absolute value distributes... |
| abstrii 15543 | Triangle inequality for ab... |
| abs3difi 15544 | Absolute value of differen... |
| abs3lemi 15545 | Lemma involving absolute v... |
| rpsqrtcld 15546 | The square root of a posit... |
| sqrtgt0d 15547 | The square root of a posit... |
| absnidd 15548 | A negative number is the n... |
| leabsd 15549 | A real number is less than... |
| absord 15550 | The absolute value of a re... |
| absred 15551 | Absolute value of a real n... |
| resqrtcld 15552 | The square root of a nonne... |
| sqrtmsqd 15553 | Square root of square. (C... |
| sqrtsqd 15554 | Square root of square. (C... |
| sqrtge0d 15555 | The square root of a nonne... |
| sqrtnegd 15556 | The square root of a negat... |
| absidd 15557 | A nonnegative number is it... |
| sqrtdivd 15558 | Square root distributes ov... |
| sqrtmuld 15559 | Square root distributes ov... |
| sqrtsq2d 15560 | Relationship between squar... |
| sqrtled 15561 | Square root is monotonic. ... |
| sqrtltd 15562 | Square root is strictly mo... |
| sqr11d 15563 | The square root function i... |
| nn0absid 15564 | A nonnegative integer is i... |
| nn0absidi 15565 | A nonnegative integer is i... |
| absltd 15566 | Absolute value and 'less t... |
| absled 15567 | Absolute value and 'less t... |
| abssubge0d 15568 | Absolute value of a nonneg... |
| abssuble0d 15569 | Absolute value of a nonpos... |
| absdifltd 15570 | The absolute value of a di... |
| absdifled 15571 | The absolute value of a di... |
| icodiamlt 15572 | Two elements in a half-ope... |
| abscld 15573 | Real closure of absolute v... |
| sqrtcld 15574 | Closure of the square root... |
| sqrtrege0d 15575 | The real part of the squar... |
| sqsqrtd 15576 | Square root theorem. Theo... |
| msqsqrtd 15577 | Square root theorem. Theo... |
| sqr00d 15578 | A square root is zero iff ... |
| absvalsqd 15579 | Square of value of absolut... |
| absvalsq2d 15580 | Square of value of absolut... |
| absge0d 15581 | Absolute value is nonnegat... |
| absval2d 15582 | Value of absolute value fu... |
| abs00d 15583 | The absolute value of a nu... |
| absne0d 15584 | The absolute value of a nu... |
| absrpcld 15585 | The absolute value of a no... |
| absnegd 15586 | Absolute value of negative... |
| abscjd 15587 | The absolute value of a nu... |
| releabsd 15588 | The real part of a number ... |
| absexpd 15589 | Absolute value of positive... |
| abssubd 15590 | Swapping order of subtract... |
| absmuld 15591 | Absolute value distributes... |
| absdivd 15592 | Absolute value distributes... |
| abstrid 15593 | Triangle inequality for ab... |
| abs2difd 15594 | Difference of absolute val... |
| abs2dif2d 15595 | Difference of absolute val... |
| abs2difabsd 15596 | Absolute value of differen... |
| abs3difd 15597 | Absolute value of differen... |
| abs3lemd 15598 | Lemma involving absolute v... |
| reusq0 15599 | A complex number is the sq... |
| bhmafibid1cn 15600 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15601 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15602 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15603 | The Brahmagupta-Fibonacci ... |
| limsupgord 15606 | Ordering property of the s... |
| limsupcl 15607 | Closure of the superior li... |
| limsupval 15608 | The superior limit of an i... |
| limsupgf 15609 | Closure of the superior li... |
| limsupgval 15610 | Value of the superior limi... |
| limsupgle 15611 | The defining property of t... |
| limsuple 15612 | The defining property of t... |
| limsuplt 15613 | The defining property of t... |
| limsupval2 15614 | The superior limit, relati... |
| limsupgre 15615 | If a sequence of real numb... |
| limsupbnd1 15616 | If a sequence is eventuall... |
| limsupbnd2 15617 | If a sequence is eventuall... |
| climrel 15626 | The limit relation is a re... |
| rlimrel 15627 | The limit relation is a re... |
| clim 15628 | Express the predicate: Th... |
| rlim 15629 | Express the predicate: Th... |
| rlim2 15630 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15631 | Use strictly less-than in ... |
| rlim3 15632 | Restrict the range of the ... |
| climcl 15633 | Closure of the limit of a ... |
| rlimpm 15634 | Closure of a function with... |
| rlimf 15635 | Closure of a function with... |
| rlimss 15636 | Domain closure of a functi... |
| rlimcl 15637 | Closure of the limit of a ... |
| clim2 15638 | Express the predicate: Th... |
| clim2c 15639 | Express the predicate ` F ... |
| clim0 15640 | Express the predicate ` F ... |
| clim0c 15641 | Express the predicate ` F ... |
| rlim0 15642 | Express the predicate ` B ... |
| rlim0lt 15643 | Use strictly less-than in ... |
| climi 15644 | Convergence of a sequence ... |
| climi2 15645 | Convergence of a sequence ... |
| climi0 15646 | Convergence of a sequence ... |
| rlimi 15647 | Convergence at infinity of... |
| rlimi2 15648 | Convergence at infinity of... |
| ello1 15649 | Elementhood in the set of ... |
| ello12 15650 | Elementhood in the set of ... |
| ello12r 15651 | Sufficient condition for e... |
| lo1f 15652 | An eventually upper bounde... |
| lo1dm 15653 | An eventually upper bounde... |
| lo1bdd 15654 | The defining property of a... |
| ello1mpt 15655 | Elementhood in the set of ... |
| ello1mpt2 15656 | Elementhood in the set of ... |
| ello1d 15657 | Sufficient condition for e... |
| lo1bdd2 15658 | If an eventually bounded f... |
| lo1bddrp 15659 | Refine ~ o1bdd2 to give a ... |
| elo1 15660 | Elementhood in the set of ... |
| elo12 15661 | Elementhood in the set of ... |
| elo12r 15662 | Sufficient condition for e... |
| o1f 15663 | An eventually bounded func... |
| o1dm 15664 | An eventually bounded func... |
| o1bdd 15665 | The defining property of a... |
| lo1o1 15666 | A function is eventually b... |
| lo1o12 15667 | A function is eventually b... |
| elo1mpt 15668 | Elementhood in the set of ... |
| elo1mpt2 15669 | Elementhood in the set of ... |
| elo1d 15670 | Sufficient condition for e... |
| o1lo1 15671 | A real function is eventua... |
| o1lo12 15672 | A lower bounded real funct... |
| o1lo1d 15673 | A real eventually bounded ... |
| icco1 15674 | Derive eventual boundednes... |
| o1bdd2 15675 | If an eventually bounded f... |
| o1bddrp 15676 | Refine ~ o1bdd2 to give a ... |
| climconst 15677 | An (eventually) constant s... |
| rlimconst 15678 | A constant sequence conver... |
| rlimclim1 15679 | Forward direction of ~ rli... |
| rlimclim 15680 | A sequence on an upper int... |
| climrlim2 15681 | Produce a real limit from ... |
| climconst2 15682 | A constant sequence conver... |
| climz 15683 | The zero sequence converge... |
| rlimuni 15684 | A real function whose doma... |
| rlimdm 15685 | Two ways to express that a... |
| climuni 15686 | An infinite sequence of co... |
| fclim 15687 | The limit relation is func... |
| climdm 15688 | Two ways to express that a... |
| climeu 15689 | An infinite sequence of co... |
| climreu 15690 | An infinite sequence of co... |
| climmo 15691 | An infinite sequence of co... |
| rlimres 15692 | The restriction of a funct... |
| lo1res 15693 | The restriction of an even... |
| o1res 15694 | The restriction of an even... |
| rlimres2 15695 | The restriction of a funct... |
| lo1res2 15696 | The restriction of a funct... |
| o1res2 15697 | The restriction of a funct... |
| lo1resb 15698 | The restriction of a funct... |
| rlimresb 15699 | The restriction of a funct... |
| o1resb 15700 | The restriction of a funct... |
| climeq 15701 | Two functions that are eve... |
| lo1eq 15702 | Two functions that are eve... |
| rlimeq 15703 | Two functions that are eve... |
| o1eq 15704 | Two functions that are eve... |
| climmpt 15705 | Exhibit a function ` G ` w... |
| 2clim 15706 | If two sequences converge ... |
| climmpt2 15707 | Relate an integer limit on... |
| climshftlem 15708 | A shifted function converg... |
| climres 15709 | A function restricted to u... |
| climshft 15710 | A shifted function converg... |
| serclim0 15711 | The zero series converges ... |
| rlimcld2 15712 | If ` D ` is a closed set i... |
| rlimrege0 15713 | The limit of a sequence of... |
| rlimrecl 15714 | The limit of a real sequen... |
| rlimge0 15715 | The limit of a sequence of... |
| climshft2 15716 | A shifted function converg... |
| climrecl 15717 | The limit of a convergent ... |
| climge0 15718 | A nonnegative sequence con... |
| climabs0 15719 | Convergence to zero of the... |
| o1co 15720 | Sufficient condition for t... |
| o1compt 15721 | Sufficient condition for t... |
| rlimcn1 15722 | Image of a limit under a c... |
| rlimcn1b 15723 | Image of a limit under a c... |
| rlimcn3 15724 | Image of a limit under a c... |
| rlimcn2 15725 | Image of a limit under a c... |
| climcn1 15726 | Image of a limit under a c... |
| climcn2 15727 | Image of a limit under a c... |
| addcn2 15728 | Complex number addition is... |
| subcn2 15729 | Complex number subtraction... |
| mulcn2 15730 | Complex number multiplicat... |
| reccn2 15731 | The reciprocal function is... |
| cn1lem 15732 | A sufficient condition for... |
| abscn2 15733 | The absolute value functio... |
| cjcn2 15734 | The complex conjugate func... |
| recn2 15735 | The real part function is ... |
| imcn2 15736 | The imaginary part functio... |
| climcn1lem 15737 | The limit of a continuous ... |
| climabs 15738 | Limit of the absolute valu... |
| climcj 15739 | Limit of the complex conju... |
| climre 15740 | Limit of the real part of ... |
| climim 15741 | Limit of the imaginary par... |
| rlimmptrcl 15742 | Reverse closure for a real... |
| rlimabs 15743 | Limit of the absolute valu... |
| rlimcj 15744 | Limit of the complex conju... |
| rlimre 15745 | Limit of the real part of ... |
| rlimim 15746 | Limit of the imaginary par... |
| o1of2 15747 | Show that a binary operati... |
| o1add 15748 | The sum of two eventually ... |
| o1mul 15749 | The product of two eventua... |
| o1sub 15750 | The difference of two even... |
| rlimo1 15751 | Any function with a finite... |
| rlimdmo1 15752 | A convergent function is e... |
| o1rlimmul 15753 | The product of an eventual... |
| o1const 15754 | A constant function is eve... |
| lo1const 15755 | A constant function is eve... |
| lo1mptrcl 15756 | Reverse closure for an eve... |
| o1mptrcl 15757 | Reverse closure for an eve... |
| o1add2 15758 | The sum of two eventually ... |
| o1mul2 15759 | The product of two eventua... |
| o1sub2 15760 | The product of two eventua... |
| lo1add 15761 | The sum of two eventually ... |
| lo1mul 15762 | The product of an eventual... |
| lo1mul2 15763 | The product of an eventual... |
| o1dif 15764 | If the difference of two f... |
| lo1sub 15765 | The difference of an event... |
| climadd 15766 | Limit of the sum of two co... |
| climmul 15767 | Limit of the product of tw... |
| climsub 15768 | Limit of the difference of... |
| climaddc1 15769 | Limit of a constant ` C ` ... |
| climaddc2 15770 | Limit of a constant ` C ` ... |
| climmulc2 15771 | Limit of a sequence multip... |
| climsubc1 15772 | Limit of a constant ` C ` ... |
| climsubc2 15773 | Limit of a constant ` C ` ... |
| climle 15774 | Comparison of the limits o... |
| climsqz 15775 | Convergence of a sequence ... |
| climsqz2 15776 | Convergence of a sequence ... |
| rlimadd 15777 | Limit of the sum of two co... |
| rlimsub 15778 | Limit of the difference of... |
| rlimmul 15779 | Limit of the product of tw... |
| rlimdiv 15780 | Limit of the quotient of t... |
| rlimneg 15781 | Limit of the negative of a... |
| rlimle 15782 | Comparison of the limits o... |
| rlimsqzlem 15783 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15784 | Convergence of a sequence ... |
| rlimsqz2 15785 | Convergence of a sequence ... |
| lo1le 15786 | Transfer eventual upper bo... |
| o1le 15787 | Transfer eventual boundedn... |
| rlimno1 15788 | A function whose inverse c... |
| clim2ser 15789 | The limit of an infinite s... |
| clim2ser2 15790 | The limit of an infinite s... |
| iserex 15791 | An infinite series converg... |
| isermulc2 15792 | Multiplication of an infin... |
| climlec2 15793 | Comparison of a constant t... |
| iserle 15794 | Comparison of the limits o... |
| iserge0 15795 | The limit of an infinite s... |
| climub 15796 | The limit of a monotonic s... |
| climserle 15797 | The partial sums of a conv... |
| isershft 15798 | Index shift of the limit o... |
| isercolllem1 15799 | Lemma for ~ isercoll . (C... |
| isercolllem2 15800 | Lemma for ~ isercoll . (C... |
| isercolllem3 15801 | Lemma for ~ isercoll . (C... |
| isercoll 15802 | Rearrange an infinite seri... |
| isercoll2 15803 | Generalize ~ isercoll so t... |
| climsup 15804 | A bounded monotonic sequen... |
| climcau 15805 | A converging sequence of c... |
| climbdd 15806 | A converging sequence of c... |
| caucvgrlem 15807 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15808 | A Cauchy sequence of real ... |
| caucvgrlem2 15809 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15810 | A Cauchy sequence of compl... |
| caurcvg 15811 | A Cauchy sequence of real ... |
| caurcvg2 15812 | A Cauchy sequence of real ... |
| caucvg 15813 | A Cauchy sequence of compl... |
| caucvgb 15814 | A function is convergent i... |
| serf0 15815 | If an infinite series conv... |
| iseraltlem1 15816 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15817 | Lemma for ~ iseralt . The... |
| iseraltlem3 15818 | Lemma for ~ iseralt . Fro... |
| iseralt 15819 | The alternating series tes... |
| sumex 15822 | A sum is a set. (Contribu... |
| sumeq1 15823 | Equality theorem for a sum... |
| nfsum1 15824 | Bound-variable hypothesis ... |
| nfsum 15825 | Bound-variable hypothesis ... |
| sumeq2w 15826 | Equality theorem for sum, ... |
| sumeq2ii 15827 | Equality theorem for sum, ... |
| sumeq2 15828 | Equality theorem for sum. ... |
| cbvsum 15829 | Change bound variable in a... |
| cbvsumv 15830 | Change bound variable in a... |
| sumeq1i 15831 | Equality inference for sum... |
| sumeq2i 15832 | Equality inference for sum... |
| sumeq12i 15833 | Equality inference for sum... |
| sumeq1d 15834 | Equality deduction for sum... |
| sumeq2d 15835 | Equality deduction for sum... |
| sumeq2dv 15836 | Equality deduction for sum... |
| sumeq2sdv 15837 | Equality deduction for sum... |
| 2sumeq2dv 15838 | Equality deduction for dou... |
| sumeq12dv 15839 | Equality deduction for sum... |
| sumeq12rdv 15840 | Equality deduction for sum... |
| sum2id 15841 | The second class argument ... |
| sumfc 15842 | A lemma to facilitate conv... |
| fz1f1o 15843 | A lemma for working with f... |
| sumrblem 15844 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15845 | The sequence of partial su... |
| sumrb 15846 | Rebase the starting point ... |
| summolem3 15847 | Lemma for ~ summo . (Cont... |
| summolem2a 15848 | Lemma for ~ summo . (Cont... |
| summolem2 15849 | Lemma for ~ summo . (Cont... |
| summo 15850 | A sum has at most one limi... |
| zsum 15851 | Series sum with index set ... |
| isum 15852 | Series sum with an upper i... |
| fsum 15853 | The value of a sum over a ... |
| sum0 15854 | Any sum over the empty set... |
| sumz 15855 | Any sum of zero over a sum... |
| fsumf1o 15856 | Re-index a finite sum usin... |
| sumss 15857 | Change the index set to a ... |
| fsumss 15858 | Change the index set to a ... |
| sumss2 15859 | Change the index set of a ... |
| fsumcvg2 15860 | The sequence of partial su... |
| fsumsers 15861 | Special case of series sum... |
| fsumcvg3 15862 | A finite sum is convergent... |
| fsumser 15863 | A finite sum expressed in ... |
| fsumcl2lem 15864 | - Lemma for finite sum clo... |
| fsumcllem 15865 | - Lemma for finite sum clo... |
| fsumcl 15866 | Closure of a finite sum of... |
| fsumrecl 15867 | Closure of a finite sum of... |
| fsumzcl 15868 | Closure of a finite sum of... |
| fsumnn0cl 15869 | Closure of a finite sum of... |
| fsumrpcl 15870 | Closure of a finite sum of... |
| fsumclf 15871 | Closure of a finite sum of... |
| fsumzcl2 15872 | A finite sum with integer ... |
| fsumadd 15873 | The sum of two finite sums... |
| fsumsplit 15874 | Split a sum into two parts... |
| fsumsplitf 15875 | Split a sum into two parts... |
| sumsnf 15876 | A sum of a singleton is th... |
| fsumsplitsn 15877 | Separate out a term in a f... |
| fsumsplit1 15878 | Separate out a term in a f... |
| sumsn 15879 | A sum of a singleton is th... |
| fsum1 15880 | The finite sum of ` A ( k ... |
| sumpr 15881 | A sum over a pair is the s... |
| sumtp 15882 | A sum over a triple is the... |
| sumsns 15883 | A sum of a singleton is th... |
| fsumm1 15884 | Separate out the last term... |
| fzosump1 15885 | Separate out the last term... |
| fsum1p 15886 | Separate out the first ter... |
| fsummsnunz 15887 | A finite sum all of whose ... |
| fsumsplitsnun 15888 | Separate out a term in a f... |
| fsump1 15889 | The addition of the next t... |
| isumclim 15890 | An infinite sum equals the... |
| isumclim2 15891 | A converging series conver... |
| isumclim3 15892 | The sequence of partial fi... |
| sumnul 15893 | The sum of a non-convergen... |
| isumcl 15894 | The sum of a converging in... |
| isummulc2 15895 | An infinite sum multiplied... |
| isummulc1 15896 | An infinite sum multiplied... |
| isumdivc 15897 | An infinite sum divided by... |
| isumrecl 15898 | The sum of a converging in... |
| isumge0 15899 | An infinite sum of nonnega... |
| isumadd 15900 | Addition of infinite sums.... |
| sumsplit 15901 | Split a sum into two parts... |
| fsump1i 15902 | Optimized version of ~ fsu... |
| fsum2dlem 15903 | Lemma for ~ fsum2d - induc... |
| fsum2d 15904 | Write a double sum as a su... |
| fsumxp 15905 | Combine two sums into a si... |
| fsumcnv 15906 | Transform a region of summ... |
| fsumcom2 15907 | Interchange order of summa... |
| fsumcom 15908 | Interchange order of summa... |
| fsum0diaglem 15909 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15910 | Two ways to express "the s... |
| mptfzshft 15911 | 1-1 onto function in maps-... |
| fsumrev 15912 | Reversal of a finite sum. ... |
| fsumshft 15913 | Index shift of a finite su... |
| fsumshftm 15914 | Negative index shift of a ... |
| fsumrev2 15915 | Reversal of a finite sum. ... |
| fsum0diag2 15916 | Two ways to express "the s... |
| fsummulc2 15917 | A finite sum multiplied by... |
| fsummulc1 15918 | A finite sum multiplied by... |
| fsumdivc 15919 | A finite sum divided by a ... |
| fsumneg 15920 | Negation of a finite sum. ... |
| fsumsub 15921 | Split a finite sum over a ... |
| fsum2mul 15922 | Separate the nested sum of... |
| fsumconst 15923 | The sum of constant terms ... |
| fsumconst1 15924 | The sum of 1 over a finite... |
| fsumdifsnconst 15925 | The sum of constant terms ... |
| modfsummodslem1 15926 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15927 | Induction step for ~ modfs... |
| modfsummod 15928 | A finite sum modulo a posi... |
| fsumge0 15929 | If all of the terms of a f... |
| fsumless 15930 | A shorter sum of nonnegati... |
| fsumge1 15931 | A sum of nonnegative numbe... |
| fsum00 15932 | A sum of nonnegative numbe... |
| fsumle 15933 | If all of the terms of fin... |
| fsumlt 15934 | If every term in one finit... |
| fsumabs 15935 | Generalized triangle inequ... |
| telfsumo 15936 | Sum of a telescoping serie... |
| telfsumo2 15937 | Sum of a telescoping serie... |
| telfsum 15938 | Sum of a telescoping serie... |
| telfsum2 15939 | Sum of a telescoping serie... |
| fsumparts 15940 | Summation by parts. (Cont... |
| fsumrelem 15941 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15942 | The real part of a sum. (... |
| fsumim 15943 | The imaginary part of a su... |
| fsumcj 15944 | The complex conjugate of a... |
| fsumrlim 15945 | Limit of a finite sum of c... |
| fsumo1 15946 | The finite sum of eventual... |
| o1fsum 15947 | If ` A ( k ) ` is O(1), th... |
| seqabs 15948 | Generalized triangle inequ... |
| iserabs 15949 | Generalized triangle inequ... |
| cvgcmp 15950 | A comparison test for conv... |
| cvgcmpub 15951 | An upper bound for the lim... |
| cvgcmpce 15952 | A comparison test for conv... |
| abscvgcvg 15953 | An absolutely convergent s... |
| climfsum 15954 | Limit of a finite sum of c... |
| fsumiun 15955 | Sum over a disjoint indexe... |
| hashiun 15956 | The cardinality of a disjo... |
| hash2iun 15957 | The cardinality of a neste... |
| hash2iun1dif1 15958 | The cardinality of a neste... |
| hashrabrex 15959 | The number of elements in ... |
| hashuni 15960 | The cardinality of a disjo... |
| qshash 15961 | The cardinality of a set w... |
| indsum 15962 | Finite sum of a product wi... |
| indsumhash 15963 | The finite sum of the indi... |
| ackbijnn 15964 | Translate the Ackermann bi... |
| binomlem 15965 | Lemma for ~ binom (binomia... |
| binom 15966 | The binomial theorem: ` ( ... |
| binom1p 15967 | Special case of the binomi... |
| binom11 15968 | Special case of the binomi... |
| binom1dif 15969 | A summation for the differ... |
| bcxmaslem1 15970 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15971 | Parallel summation (Christ... |
| incexclem 15972 | Lemma for ~ incexc . (Con... |
| incexc 15973 | The inclusion/exclusion pr... |
| incexc2 15974 | The inclusion/exclusion pr... |
| isumshft 15975 | Index shift of an infinite... |
| isumsplit 15976 | Split off the first ` N ` ... |
| isum1p 15977 | The infinite sum of a conv... |
| isumnn0nn 15978 | Sum from 0 to infinity in ... |
| isumrpcl 15979 | The infinite sum of positi... |
| isumle 15980 | Comparison of two infinite... |
| isumless 15981 | A finite sum of nonnegativ... |
| isumsup2 15982 | An infinite sum of nonnega... |
| isumsup 15983 | An infinite sum of nonnega... |
| isumltss 15984 | A partial sum of a series ... |
| climcndslem1 15985 | Lemma for ~ climcnds : bou... |
| climcndslem2 15986 | Lemma for ~ climcnds : bou... |
| climcnds 15987 | The Cauchy condensation te... |
| divrcnv 15988 | The sequence of reciprocal... |
| divcnv 15989 | The sequence of reciprocal... |
| flo1 15990 | The floor function satisfi... |
| divcnvshft 15991 | Limit of a ratio function.... |
| supcvg 15992 | Extract a sequence ` f ` i... |
| infcvgaux1i 15993 | Auxiliary theorem for appl... |
| infcvgaux2i 15994 | Auxiliary theorem for appl... |
| harmonic 15995 | The harmonic series ` H ` ... |
| arisum 15996 | Arithmetic series sum of t... |
| arisum2 15997 | Arithmetic series sum of t... |
| trireciplem 15998 | Lemma for ~ trirecip . Sh... |
| trirecip 15999 | The sum of the reciprocals... |
| expcnv 16000 | A sequence of powers of a ... |
| explecnv 16001 | A sequence of terms conver... |
| geoserg 16002 | The value of the finite ge... |
| geoser 16003 | The value of the finite ge... |
| pwdif 16004 | The difference of two numb... |
| pwm1geoser 16005 | The n-th power of a number... |
| geolim 16006 | The partial sums in the in... |
| geolim2 16007 | The partial sums in the ge... |
| georeclim 16008 | The limit of a geometric s... |
| geo2sum 16009 | The value of the finite ge... |
| geo2sum2 16010 | The value of the finite ge... |
| geo2lim 16011 | The value of the infinite ... |
| geomulcvg 16012 | The geometric series conve... |
| geoisum 16013 | The infinite sum of ` 1 + ... |
| geoisumr 16014 | The infinite sum of recipr... |
| geoisum1 16015 | The infinite sum of ` A ^ ... |
| geoisum1c 16016 | The infinite sum of ` A x.... |
| 0.999... 16017 | The recurring decimal 0.99... |
| geoihalfsum 16018 | Prove that the infinite ge... |
| cvgrat 16019 | Ratio test for convergence... |
| mertenslem1 16020 | Lemma for ~ mertens . (Co... |
| mertenslem2 16021 | Lemma for ~ mertens . (Co... |
| mertens 16022 | Mertens' theorem. If ` A ... |
| prodf 16023 | An infinite product of com... |
| clim2prod 16024 | The limit of an infinite p... |
| clim2div 16025 | The limit of an infinite p... |
| prodfmul 16026 | The product of two infinit... |
| prodf1 16027 | The value of the partial p... |
| prodf1f 16028 | A one-valued infinite prod... |
| prodfclim1 16029 | The constant one product c... |
| prodfn0 16030 | No term of a nonzero infin... |
| prodfrec 16031 | The reciprocal of an infin... |
| prodfdiv 16032 | The quotient of two infini... |
| ntrivcvg 16033 | A non-trivially converging... |
| ntrivcvgn0 16034 | A product that converges t... |
| ntrivcvgfvn0 16035 | Any value of a product seq... |
| ntrivcvgtail 16036 | A tail of a non-trivially ... |
| ntrivcvgmullem 16037 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 16038 | The product of two non-tri... |
| prodex 16041 | A product is a set. (Cont... |
| prodeq1f 16042 | Equality theorem for a pro... |
| prodeq1 16043 | Equality theorem for a pro... |
| nfcprod1 16044 | Bound-variable hypothesis ... |
| nfcprod 16045 | Bound-variable hypothesis ... |
| prodeq2w 16046 | Equality theorem for produ... |
| prodeq2ii 16047 | Equality theorem for produ... |
| prodeq2 16048 | Equality theorem for produ... |
| cbvprod 16049 | Change bound variable in a... |
| cbvprodv 16050 | Change bound variable in a... |
| cbvprodi 16051 | Change bound variable in a... |
| prodeq1i 16052 | Equality inference for pro... |
| prodeq2i 16053 | Equality inference for pro... |
| prodeq12i 16054 | Equality inference for pro... |
| prodeq1d 16055 | Equality deduction for pro... |
| prodeq2d 16056 | Equality deduction for pro... |
| prodeq2dv 16057 | Equality deduction for pro... |
| prodeq2sdv 16058 | Equality deduction for pro... |
| 2cprodeq2dv 16059 | Equality deduction for dou... |
| prodeq12dv 16060 | Equality deduction for pro... |
| prodeq12rdv 16061 | Equality deduction for pro... |
| prod2id 16062 | The second class argument ... |
| prodrblem 16063 | Lemma for ~ prodrb . (Con... |
| fprodcvg 16064 | The sequence of partial pr... |
| prodrblem2 16065 | Lemma for ~ prodrb . (Con... |
| prodrb 16066 | Rebase the starting point ... |
| prodmolem3 16067 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 16068 | Lemma for ~ prodmo . (Con... |
| prodmolem2 16069 | Lemma for ~ prodmo . (Con... |
| prodmo 16070 | A product has at most one ... |
| zprod 16071 | Series product with index ... |
| iprod 16072 | Series product with an upp... |
| zprodn0 16073 | Nonzero series product wit... |
| iprodn0 16074 | Nonzero series product wit... |
| fprod 16075 | The value of a product ove... |
| fprodntriv 16076 | A non-triviality lemma for... |
| prod0 16077 | A product over the empty s... |
| prod1 16078 | Any product of one over a ... |
| prodfc 16079 | A lemma to facilitate conv... |
| fprodf1o 16080 | Re-index a finite product ... |
| prodss 16081 | Change the index set to a ... |
| fprodss 16082 | Change the index set to a ... |
| fprodser 16083 | A finite product expressed... |
| fprodcl2lem 16084 | Finite product closure lem... |
| fprodcllem 16085 | Finite product closure lem... |
| fprodcl 16086 | Closure of a finite produc... |
| fprodrecl 16087 | Closure of a finite produc... |
| fprodzcl 16088 | Closure of a finite produc... |
| fprodnncl 16089 | Closure of a finite produc... |
| fprodrpcl 16090 | Closure of a finite produc... |
| fprodnn0cl 16091 | Closure of a finite produc... |
| fprodcllemf 16092 | Finite product closure lem... |
| fprodreclf 16093 | Closure of a finite produc... |
| fprodmul 16094 | The product of two finite ... |
| fproddiv 16095 | The quotient of two finite... |
| prodsn 16096 | A product of a singleton i... |
| fprod1 16097 | A finite product of only o... |
| prodsnf 16098 | A product of a singleton i... |
| climprod1 16099 | The limit of a product ove... |
| fprodsplit 16100 | Split a finite product int... |
| fprodm1 16101 | Separate out the last term... |
| fprod1p 16102 | Separate out the first ter... |
| fprodp1 16103 | Multiply in the last term ... |
| fprodm1s 16104 | Separate out the last term... |
| fprodp1s 16105 | Multiply in the last term ... |
| prodsns 16106 | A product of the singleton... |
| fprodfac 16107 | Factorial using product no... |
| fprodabs 16108 | The absolute value of a fi... |
| fprodeq0 16109 | Any finite product contain... |
| fprodshft 16110 | Shift the index of a finit... |
| fprodrev 16111 | Reversal of a finite produ... |
| fprodconst 16112 | The product of constant te... |
| fprodn0 16113 | A finite product of nonzer... |
| fprod2dlem 16114 | Lemma for ~ fprod2d - indu... |
| fprod2d 16115 | Write a double product as ... |
| fprodxp 16116 | Combine two products into ... |
| fprodcnv 16117 | Transform a product region... |
| fprodcom2 16118 | Interchange order of multi... |
| fprodcom 16119 | Interchange product order.... |
| fprod0diag 16120 | Two ways to express "the p... |
| fproddivf 16121 | The quotient of two finite... |
| fprodsplitf 16122 | Split a finite product int... |
| fprodsplitsn 16123 | Separate out a term in a f... |
| fprodsplit1f 16124 | Separate out a term in a f... |
| fprodn0f 16125 | A finite product of nonzer... |
| fprodclf 16126 | Closure of a finite produc... |
| fprodge0 16127 | If all the terms of a fini... |
| fprodeq0g 16128 | Any finite product contain... |
| fprodge1 16129 | If all of the terms of a f... |
| fprodle 16130 | If all the terms of two fi... |
| fprodmodd 16131 | If all factors of two fini... |
| iprodclim 16132 | An infinite product equals... |
| iprodclim2 16133 | A converging product conve... |
| iprodclim3 16134 | The sequence of partial fi... |
| iprodcl 16135 | The product of a non-trivi... |
| iprodrecl 16136 | The product of a non-trivi... |
| iprodmul 16137 | Multiplication of infinite... |
| risefacval 16142 | The value of the rising fa... |
| fallfacval 16143 | The value of the falling f... |
| risefacval2 16144 | One-based value of rising ... |
| fallfacval2 16145 | One-based value of falling... |
| fallfacval3 16146 | A product representation o... |
| risefaccllem 16147 | Lemma for rising factorial... |
| fallfaccllem 16148 | Lemma for falling factoria... |
| risefaccl 16149 | Closure law for rising fac... |
| fallfaccl 16150 | Closure law for falling fa... |
| rerisefaccl 16151 | Closure law for rising fac... |
| refallfaccl 16152 | Closure law for falling fa... |
| nnrisefaccl 16153 | Closure law for rising fac... |
| zrisefaccl 16154 | Closure law for rising fac... |
| zfallfaccl 16155 | Closure law for falling fa... |
| nn0risefaccl 16156 | Closure law for rising fac... |
| rprisefaccl 16157 | Closure law for rising fac... |
| risefallfac 16158 | A relationship between ris... |
| fallrisefac 16159 | A relationship between fal... |
| risefac0 16160 | The value of the rising fa... |
| fallfac0 16161 | The value of the falling f... |
| risefacp1 16162 | The value of the rising fa... |
| fallfacp1 16163 | The value of the falling f... |
| risefacp1d 16164 | The value of the rising fa... |
| fallfacp1d 16165 | The value of the falling f... |
| risefac1 16166 | The value of rising factor... |
| fallfac1 16167 | The value of falling facto... |
| risefacfac 16168 | Relate rising factorial to... |
| fallfacfwd 16169 | The forward difference of ... |
| 0fallfac 16170 | The value of the zero fall... |
| 0risefac 16171 | The value of the zero risi... |
| binomfallfaclem1 16172 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16173 | Lemma for ~ binomfallfac .... |
| binomfallfac 16174 | A version of the binomial ... |
| binomrisefac 16175 | A version of the binomial ... |
| fallfacval4 16176 | Represent the falling fact... |
| bcfallfac 16177 | Binomial coefficient in te... |
| fallfacfac 16178 | Relate falling factorial t... |
| bpolylem 16181 | Lemma for ~ bpolyval . (C... |
| bpolyval 16182 | The value of the Bernoulli... |
| bpoly0 16183 | The value of the Bernoulli... |
| bpoly1 16184 | The value of the Bernoulli... |
| bpolycl 16185 | Closure law for Bernoulli ... |
| bpolysum 16186 | A sum for Bernoulli polyno... |
| bpolydiflem 16187 | Lemma for ~ bpolydif . (C... |
| bpolydif 16188 | Calculate the difference b... |
| fsumkthpow 16189 | A closed-form expression f... |
| bpoly2 16190 | The Bernoulli polynomials ... |
| bpoly3 16191 | The Bernoulli polynomials ... |
| bpoly4 16192 | The Bernoulli polynomials ... |
| fsumcube 16193 | Express the sum of cubes i... |
| eftcl 16206 | Closure of a term in the s... |
| reeftcl 16207 | The terms of the series ex... |
| eftabs 16208 | The absolute value of a te... |
| eftval 16209 | The value of a term in the... |
| efcllem 16210 | Lemma for ~ efcl . The se... |
| ef0lem 16211 | The series defining the ex... |
| efval 16212 | Value of the exponential f... |
| esum 16213 | Value of Euler's constant ... |
| eff 16214 | Domain and codomain of the... |
| efcl 16215 | Closure law for the expone... |
| efcld 16216 | Closure law for the expone... |
| efval2 16217 | Value of the exponential f... |
| efcvg 16218 | The series that defines th... |
| efcvgfsum 16219 | Exponential function conve... |
| reefcl 16220 | The exponential function i... |
| reefcld 16221 | The exponential function i... |
| ere 16222 | Euler's constant ` _e ` = ... |
| ege2le3 16223 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16224 | Value of the exponential f... |
| efcj 16225 | The exponential of a compl... |
| efaddlem 16226 | Lemma for ~ efadd (exponen... |
| efadd 16227 | Sum of exponents law for e... |
| fprodefsum 16228 | Move the exponential funct... |
| efcan 16229 | Cancellation law for expon... |
| efne0d 16230 | The exponential of a compl... |
| efne0 16231 | The exponential of a compl... |
| efne0OLD 16232 | Obsolete version of ~ efne... |
| efneg 16233 | The exponential of the opp... |
| eff2 16234 | The exponential function m... |
| efsub 16235 | Difference of exponents la... |
| efexp 16236 | The exponential of an inte... |
| efzval 16237 | Value of the exponential f... |
| efgt0 16238 | The exponential of a real ... |
| rpefcl 16239 | The exponential of a real ... |
| rpefcld 16240 | The exponential of a real ... |
| eftlcvg 16241 | The tail series of the exp... |
| eftlcl 16242 | Closure of the sum of an i... |
| reeftlcl 16243 | Closure of the sum of an i... |
| eftlub 16244 | An upper bound on the abso... |
| efsep 16245 | Separate out the next term... |
| effsumlt 16246 | The partial sums of the se... |
| eft0val 16247 | The value of the first ter... |
| ef4p 16248 | Separate out the first fou... |
| efgt1p2 16249 | The exponential of a posit... |
| efgt1p 16250 | The exponential of a posit... |
| efgt1 16251 | The exponential of a posit... |
| eflt 16252 | The exponential function o... |
| efle 16253 | The exponential function o... |
| reef11 16254 | The exponential function o... |
| reeff1 16255 | The exponential function m... |
| eflegeo 16256 | The exponential function o... |
| sinval 16257 | Value of the sine function... |
| cosval 16258 | Value of the cosine functi... |
| sinf 16259 | Domain and codomain of the... |
| cosf 16260 | Domain and codomain of the... |
| sincl 16261 | Closure of the sine functi... |
| coscl 16262 | Closure of the cosine func... |
| tanval 16263 | Value of the tangent funct... |
| tancl 16264 | The closure of the tangent... |
| sincld 16265 | Closure of the sine functi... |
| coscld 16266 | Closure of the cosine func... |
| tancld 16267 | Closure of the tangent fun... |
| tanval2 16268 | Express the tangent functi... |
| tanval3 16269 | Express the tangent functi... |
| resinval 16270 | The sine of a real number ... |
| recosval 16271 | The cosine of a real numbe... |
| efi4p 16272 | Separate out the first fou... |
| resin4p 16273 | Separate out the first fou... |
| recos4p 16274 | Separate out the first fou... |
| resincl 16275 | The sine of a real number ... |
| recoscl 16276 | The cosine of a real numbe... |
| retancl 16277 | The closure of the tangent... |
| resincld 16278 | Closure of the sine functi... |
| recoscld 16279 | Closure of the cosine func... |
| retancld 16280 | Closure of the tangent fun... |
| sinneg 16281 | The sine of a negative is ... |
| cosneg 16282 | The cosines of a number an... |
| tanneg 16283 | The tangent of a negative ... |
| sin0 16284 | Value of the sine function... |
| cos0 16285 | Value of the cosine functi... |
| tan0 16286 | The value of the tangent f... |
| efival 16287 | The exponential function i... |
| efmival 16288 | The exponential function i... |
| sinhval 16289 | Value of the hyperbolic si... |
| coshval 16290 | Value of the hyperbolic co... |
| resinhcl 16291 | The hyperbolic sine of a r... |
| rpcoshcl 16292 | The hyperbolic cosine of a... |
| recoshcl 16293 | The hyperbolic cosine of a... |
| retanhcl 16294 | The hyperbolic tangent of ... |
| tanhlt1 16295 | The hyperbolic tangent of ... |
| tanhbnd 16296 | The hyperbolic tangent of ... |
| efeul 16297 | Eulerian representation of... |
| efieq 16298 | The exponentials of two im... |
| sinadd 16299 | Addition formula for sine.... |
| cosadd 16300 | Addition formula for cosin... |
| tanaddlem 16301 | A useful intermediate step... |
| tanadd 16302 | Addition formula for tange... |
| sinsub 16303 | Sine of difference. (Cont... |
| cossub 16304 | Cosine of difference. (Co... |
| addsin 16305 | Sum of sines. (Contribute... |
| subsin 16306 | Difference of sines. (Con... |
| sinmul 16307 | Product of sines can be re... |
| cosmul 16308 | Product of cosines can be ... |
| addcos 16309 | Sum of cosines. (Contribu... |
| subcos 16310 | Difference of cosines. (C... |
| sincossq 16311 | Sine squared plus cosine s... |
| sin2t 16312 | Double-angle formula for s... |
| cos2t 16313 | Double-angle formula for c... |
| cos2tsin 16314 | Double-angle formula for c... |
| sinbnd 16315 | The sine of a real number ... |
| cosbnd 16316 | The cosine of a real numbe... |
| sinbnd2 16317 | The sine of a real number ... |
| cosbnd2 16318 | The cosine of a real numbe... |
| ef01bndlem 16319 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16320 | Bounds on the sine of a po... |
| cos01bnd 16321 | Bounds on the cosine of a ... |
| cos1bnd 16322 | Bounds on the cosine of 1.... |
| cos2bnd 16323 | Bounds on the cosine of 2.... |
| sinltx 16324 | The sine of a positive rea... |
| sin01gt0 16325 | The sine of a positive rea... |
| cos01gt0 16326 | The cosine of a positive r... |
| sin02gt0 16327 | The sine of a positive rea... |
| sincos1sgn 16328 | The signs of the sine and ... |
| sincos2sgn 16329 | The signs of the sine and ... |
| sin4lt0 16330 | The sine of 4 is negative.... |
| absefi 16331 | The absolute value of the ... |
| absef 16332 | The absolute value of the ... |
| absefib 16333 | A complex number is real i... |
| efieq1re 16334 | A number whose imaginary e... |
| demoivre 16335 | De Moivre's Formula. Proo... |
| demoivreALT 16336 | Alternate proof of ~ demoi... |
| eirrlem 16339 | Lemma for ~ eirr . (Contr... |
| eirr 16340 | ` _e ` is irrational. (Co... |
| egt2lt3 16341 | Euler's constant ` _e ` = ... |
| epos 16342 | Euler's constant ` _e ` is... |
| epr 16343 | Euler's constant ` _e ` is... |
| ene0 16344 | ` _e ` is not 0. (Contrib... |
| ene1 16345 | ` _e ` is not 1. (Contrib... |
| xpnnen 16346 | The Cartesian product of t... |
| znnen 16347 | The set of integers and th... |
| qnnen 16348 | The rational numbers are c... |
| rpnnen2lem1 16349 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16350 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16351 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16352 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16353 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16354 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16355 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16356 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16357 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16358 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16359 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16360 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16361 | The other half of ~ rpnnen... |
| rpnnen 16362 | The cardinality of the con... |
| rexpen 16363 | The real numbers are equin... |
| cpnnen 16364 | The complex numbers are eq... |
| rucALT 16365 | Alternate proof of ~ ruc .... |
| ruclem1 16366 | Lemma for ~ ruc (the reals... |
| ruclem2 16367 | Lemma for ~ ruc . Orderin... |
| ruclem3 16368 | Lemma for ~ ruc . The con... |
| ruclem4 16369 | Lemma for ~ ruc . Initial... |
| ruclem6 16370 | Lemma for ~ ruc . Domain ... |
| ruclem7 16371 | Lemma for ~ ruc . Success... |
| ruclem8 16372 | Lemma for ~ ruc . The int... |
| ruclem9 16373 | Lemma for ~ ruc . The fir... |
| ruclem10 16374 | Lemma for ~ ruc . Every f... |
| ruclem11 16375 | Lemma for ~ ruc . Closure... |
| ruclem12 16376 | Lemma for ~ ruc . The sup... |
| ruclem13 16377 | Lemma for ~ ruc . There i... |
| ruc 16378 | The set of positive intege... |
| resdomq 16379 | The set of rationals is st... |
| aleph1re 16380 | There are at least aleph-o... |
| aleph1irr 16381 | There are at least aleph-o... |
| cnso 16382 | The complex numbers can be... |
| sqrt2irrlem 16383 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16384 | The square root of 2 is ir... |
| sqrt2re 16385 | The square root of 2 exist... |
| sqrt2irr0 16386 | The square root of 2 is an... |
| nthruc 16387 | The sequence ` NN ` , ` ZZ... |
| nthruz 16388 | The sequence ` NN ` , ` NN... |
| divides 16391 | Define the divides relatio... |
| dvdsval2 16392 | One nonzero integer divide... |
| dvdsval3 16393 | One nonzero integer divide... |
| dvdszrcl 16394 | Reverse closure for the di... |
| dvdsmod0 16395 | If a positive integer divi... |
| p1modz1 16396 | If a number greater than 1... |
| dvdsmodexp 16397 | If a positive integer divi... |
| nndivdvds 16398 | Strong form of ~ dvdsval2 ... |
| nndivides 16399 | Definition of the divides ... |
| moddvds 16400 | Two ways to say ` A == B `... |
| modm1div 16401 | An integer greater than on... |
| addmulmodb 16402 | An integer plus a product ... |
| dvds0lem 16403 | A lemma to assist theorems... |
| dvds1lem 16404 | A lemma to assist theorems... |
| dvds2lem 16405 | A lemma to assist theorems... |
| iddvds 16406 | An integer divides itself.... |
| 1dvds 16407 | 1 divides any integer. Th... |
| dvds0 16408 | Any integer divides 0. Th... |
| negdvdsb 16409 | An integer divides another... |
| dvdsnegb 16410 | An integer divides another... |
| absdvdsb 16411 | An integer divides another... |
| dvdsabsb 16412 | An integer divides another... |
| 0dvds 16413 | Only 0 is divisible by 0. ... |
| dvdsmul1 16414 | An integer divides a multi... |
| dvdsmul2 16415 | An integer divides a multi... |
| iddvdsexp 16416 | An integer divides a posit... |
| muldvds1 16417 | If a product divides an in... |
| muldvds2 16418 | If a product divides an in... |
| dvdscmul 16419 | Multiplication by a consta... |
| dvdsmulc 16420 | Multiplication by a consta... |
| dvdscmulr 16421 | Cancellation law for the d... |
| dvdsmulcr 16422 | Cancellation law for the d... |
| summodnegmod 16423 | The sum of two integers mo... |
| difmod0 16424 | The difference of two inte... |
| modmulconst 16425 | Constant multiplication in... |
| dvds2ln 16426 | If an integer divides each... |
| dvds2add 16427 | If an integer divides each... |
| dvds2sub 16428 | If an integer divides each... |
| dvds2addd 16429 | Deduction form of ~ dvds2a... |
| dvds2subd 16430 | Deduction form of ~ dvds2s... |
| dvdstr 16431 | The divides relation is tr... |
| dvdstrd 16432 | The divides relation is tr... |
| dvdsmultr1 16433 | If an integer divides anot... |
| dvdsmultr1d 16434 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16435 | If an integer divides anot... |
| dvdsmultr2d 16436 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16437 | If an integer divides eith... |
| dvdssub2 16438 | If an integer divides a di... |
| dvdsadd 16439 | An integer divides another... |
| dvdsaddr 16440 | An integer divides another... |
| dvdssub 16441 | An integer divides another... |
| dvdssubr 16442 | An integer divides another... |
| dvdsadd2b 16443 | Adding a multiple of the b... |
| dvdsaddre2b 16444 | Adding a multiple of the b... |
| fsumdvds 16445 | If every term in a sum is ... |
| dvdslelem 16446 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16447 | The divisors of a positive... |
| dvdsleabs 16448 | The divisors of a nonzero ... |
| dvdsleabs2 16449 | Transfer divisibility to a... |
| dvdsabseq 16450 | If two integers divide eac... |
| dvdseq 16451 | If two nonnegative integer... |
| divconjdvds 16452 | If a nonzero integer ` M `... |
| dvdsdivcl 16453 | The complement of a diviso... |
| dvdsflip 16454 | An involution of the divis... |
| dvdsssfz1 16455 | The set of divisors of a n... |
| dvds1 16456 | The only nonnegative integ... |
| alzdvds 16457 | Only 0 is divisible by all... |
| dvdsext 16458 | Poset extensionality for d... |
| fzm1ndvds 16459 | No number between ` 1 ` an... |
| fzo0dvdseq 16460 | Zero is the only one of th... |
| fzocongeq 16461 | Two different elements of ... |
| addmodlteqALT 16462 | Two nonnegative integers l... |
| dvdsfac 16463 | A positive integer divides... |
| dvdsexp2im 16464 | If an integer divides anot... |
| dvdsexp 16465 | A power divides a power wi... |
| dvdsmod 16466 | Any number ` K ` whose mod... |
| mulmoddvds 16467 | If an integer is divisible... |
| 3dvds 16468 | A rule for divisibility by... |
| 3dvdsdec 16469 | A decimal number is divisi... |
| 3dvds2dec 16470 | A decimal number is divisi... |
| fprodfvdvdsd 16471 | A finite product of intege... |
| fproddvdsd 16472 | A finite product of intege... |
| evenelz 16473 | An even number is an integ... |
| zeo3 16474 | An integer is even or odd.... |
| zeo4 16475 | An integer is even or odd ... |
| zeneo 16476 | No even integer equals an ... |
| odd2np1lem 16477 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16478 | An integer is odd iff it i... |
| even2n 16479 | An integer is even iff it ... |
| oddm1even 16480 | An integer is odd iff its ... |
| oddp1even 16481 | An integer is odd iff its ... |
| oexpneg 16482 | The exponential of the neg... |
| mod2eq0even 16483 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16484 | An integer is 1 modulo 2 i... |
| oddnn02np1 16485 | A nonnegative integer is o... |
| oddge22np1 16486 | An integer greater than on... |
| evennn02n 16487 | A nonnegative integer is e... |
| evennn2n 16488 | A positive integer is even... |
| 2tp1odd 16489 | A number which is twice an... |
| mulsucdiv2z 16490 | An integer multiplied with... |
| sqoddm1div8z 16491 | A squared odd number minus... |
| 2teven 16492 | A number which is twice an... |
| zeo5 16493 | An integer is either even ... |
| evend2 16494 | An integer is even iff its... |
| oddp1d2 16495 | An integer is odd iff its ... |
| zob 16496 | Alternate characterization... |
| oddm1d2 16497 | An integer is odd iff its ... |
| ltoddhalfle 16498 | An integer is less than ha... |
| halfleoddlt 16499 | An integer is greater than... |
| opoe 16500 | The sum of two odds is eve... |
| omoe 16501 | The difference of two odds... |
| opeo 16502 | The sum of an odd and an e... |
| omeo 16503 | The difference of an odd a... |
| z0even 16504 | 2 divides 0. That means 0... |
| n2dvds1 16505 | 2 does not divide 1. That... |
| n2dvdsm1 16506 | 2 does not divide -1. Tha... |
| z2even 16507 | 2 divides 2. That means 2... |
| n2dvds3 16508 | 2 does not divide 3. That... |
| z4even 16509 | 2 divides 4. That means 4... |
| 4dvdseven 16510 | An integer which is divisi... |
| m1expe 16511 | Exponentiation of -1 by an... |
| m1expo 16512 | Exponentiation of -1 by an... |
| m1exp1 16513 | Exponentiation of negative... |
| nn0enne 16514 | A positive integer is an e... |
| nn0ehalf 16515 | The half of an even nonneg... |
| nnehalf 16516 | The half of an even positi... |
| nn0onn 16517 | An odd nonnegative integer... |
| nn0o1gt2 16518 | An odd nonnegative integer... |
| nno 16519 | An alternate characterizat... |
| nn0o 16520 | An alternate characterizat... |
| nn0ob 16521 | Alternate characterization... |
| nn0oddm1d2 16522 | A positive integer is odd ... |
| nnoddm1d2 16523 | A positive integer is odd ... |
| sumeven 16524 | If every term in a sum is ... |
| sumodd 16525 | If every term in a sum is ... |
| evensumodd 16526 | If every term in a sum wit... |
| oddsumodd 16527 | If every term in a sum wit... |
| pwp1fsum 16528 | The n-th power of a number... |
| oddpwp1fsum 16529 | An odd power of a number i... |
| divalglem0 16530 | Lemma for ~ divalg . (Con... |
| divalglem1 16531 | Lemma for ~ divalg . (Con... |
| divalglem2 16532 | Lemma for ~ divalg . (Con... |
| divalglem4 16533 | Lemma for ~ divalg . (Con... |
| divalglem5 16534 | Lemma for ~ divalg . (Con... |
| divalglem6 16535 | Lemma for ~ divalg . (Con... |
| divalglem7 16536 | Lemma for ~ divalg . (Con... |
| divalglem8 16537 | Lemma for ~ divalg . (Con... |
| divalglem9 16538 | Lemma for ~ divalg . (Con... |
| divalglem10 16539 | Lemma for ~ divalg . (Con... |
| divalg 16540 | The division algorithm (th... |
| divalgb 16541 | Express the division algor... |
| divalg2 16542 | The division algorithm (th... |
| divalgmod 16543 | The result of the ` mod ` ... |
| divalgmodcl 16544 | The result of the ` mod ` ... |
| modremain 16545 | The result of the modulo o... |
| ndvdssub 16546 | Corollary of the division ... |
| ndvdsadd 16547 | Corollary of the division ... |
| ndvdsp1 16548 | Special case of ~ ndvdsadd... |
| ndvdsi 16549 | A quick test for non-divis... |
| 5ndvds3 16550 | 5 does not divide 3. (Con... |
| 5ndvds6 16551 | 5 does not divide 6. (Con... |
| flodddiv4 16552 | The floor of an odd intege... |
| fldivndvdslt 16553 | The floor of an integer di... |
| flodddiv4lt 16554 | The floor of an odd number... |
| flodddiv4t2lthalf 16555 | The floor of an odd number... |
| bitsfval 16560 | Expand the definition of t... |
| bitsval 16561 | Expand the definition of t... |
| bitsval2 16562 | Expand the definition of t... |
| bitsss 16563 | The set of bits of an inte... |
| bitsf 16564 | The ` bits ` function is a... |
| bits0 16565 | Value of the zeroth bit. ... |
| bits0e 16566 | The zeroth bit of an even ... |
| bits0o 16567 | The zeroth bit of an odd n... |
| bitsp1 16568 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16569 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16570 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16571 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16572 | The bits of a number are a... |
| bitsmod 16573 | Truncating the bit sequenc... |
| bitsfi 16574 | Every number is associated... |
| bitscmp 16575 | The bit complement of ` N ... |
| 0bits 16576 | The bits of zero. (Contri... |
| m1bits 16577 | The bits of negative one. ... |
| bitsinv1lem 16578 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16579 | There is an explicit inver... |
| bitsinv2 16580 | There is an explicit inver... |
| bitsf1ocnv 16581 | The ` bits ` function rest... |
| bitsf1o 16582 | The ` bits ` function rest... |
| bitsf1 16583 | The ` bits ` function is a... |
| 2ebits 16584 | The bits of a power of two... |
| bitsinv 16585 | The inverse of the ` bits ... |
| bitsinvp1 16586 | Recursive definition of th... |
| sadadd2lem2 16587 | The core of the proof of ~... |
| sadfval 16589 | Define the addition of two... |
| sadcf 16590 | The carry sequence is a se... |
| sadc0 16591 | The initial element of the... |
| sadcp1 16592 | The carry sequence (which ... |
| sadval 16593 | The full adder sequence is... |
| sadcaddlem 16594 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16595 | Non-recursive definition o... |
| sadadd2lem 16596 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16597 | Sum of initial segments of... |
| sadadd3 16598 | Sum of initial segments of... |
| sadcl 16599 | The sum of two sequences i... |
| sadcom 16600 | The adder sequence functio... |
| saddisjlem 16601 | Lemma for ~ sadadd . (Con... |
| saddisj 16602 | The sum of disjoint sequen... |
| sadaddlem 16603 | Lemma for ~ sadadd . (Con... |
| sadadd 16604 | For sequences that corresp... |
| sadid1 16605 | The adder sequence functio... |
| sadid2 16606 | The adder sequence functio... |
| sadasslem 16607 | Lemma for ~ sadass . (Con... |
| sadass 16608 | Sequence addition is assoc... |
| sadeq 16609 | Any element of a sequence ... |
| bitsres 16610 | Restrict the bits of a num... |
| bitsuz 16611 | The bits of a number are a... |
| bitsshft 16612 | Shifting a bit sequence to... |
| smufval 16614 | The multiplication of two ... |
| smupf 16615 | The sequence of partial su... |
| smup0 16616 | The initial element of the... |
| smupp1 16617 | The initial element of the... |
| smuval 16618 | Define the addition of two... |
| smuval2 16619 | The partial sum sequence s... |
| smupvallem 16620 | If ` A ` only has elements... |
| smucl 16621 | The product of two sequenc... |
| smu01lem 16622 | Lemma for ~ smu01 and ~ sm... |
| smu01 16623 | Multiplication of a sequen... |
| smu02 16624 | Multiplication of a sequen... |
| smupval 16625 | Rewrite the elements of th... |
| smup1 16626 | Rewrite ~ smupp1 using onl... |
| smueqlem 16627 | Any element of a sequence ... |
| smueq 16628 | Any element of a sequence ... |
| smumullem 16629 | Lemma for ~ smumul . (Con... |
| smumul 16630 | For sequences that corresp... |
| gcdval 16633 | The value of the ` gcd ` o... |
| gcd0val 16634 | The value, by convention, ... |
| gcdn0val 16635 | The value of the ` gcd ` o... |
| gcdcllem1 16636 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16637 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16638 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16639 | Closure of the ` gcd ` ope... |
| gcddvds 16640 | The gcd of two integers di... |
| dvdslegcd 16641 | An integer which divides b... |
| nndvdslegcd 16642 | A positive integer which d... |
| gcdcl 16643 | Closure of the ` gcd ` ope... |
| gcdnncl 16644 | Closure of the ` gcd ` ope... |
| gcdcld 16645 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16646 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16647 | An integer is the product ... |
| divgcdz 16648 | An integer divided by the ... |
| gcdf 16649 | Domain and codomain of the... |
| gcdcom 16650 | The ` gcd ` operator is co... |
| gcdcomd 16651 | The ` gcd ` operator is co... |
| divgcdnn 16652 | A positive integer divided... |
| divgcdnnr 16653 | A positive integer divided... |
| gcdeq0 16654 | The gcd of two integers is... |
| gcdn0gt0 16655 | The gcd of two integers is... |
| gcd0id 16656 | The gcd of 0 and an intege... |
| gcdid0 16657 | The gcd of an integer and ... |
| nn0gcdid0 16658 | The gcd of a nonnegative i... |
| gcdneg 16659 | Negating one operand of th... |
| neggcd 16660 | Negating one operand of th... |
| gcdaddmlem 16661 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16662 | Adding a multiple of one o... |
| gcdadd 16663 | The GCD of two numbers is ... |
| gcdid 16664 | The gcd of a number and it... |
| gcd1 16665 | The gcd of a number with 1... |
| gcdabs1 16666 | ` gcd ` of the absolute va... |
| gcdabs2 16667 | ` gcd ` of the absolute va... |
| gcdabs 16668 | The gcd of two integers is... |
| modgcd 16669 | The gcd remains unchanged ... |
| 1gcd 16670 | The GCD of one and an inte... |
| gcdmultipled 16671 | The greatest common diviso... |
| gcdmultiplez 16672 | The GCD of a multiple of a... |
| gcdmultiple 16673 | The GCD of a multiple of a... |
| dvdsgcdidd 16674 | The greatest common diviso... |
| 6gcd4e2 16675 | The greatest common diviso... |
| bezoutlem1 16676 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16677 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16678 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16679 | Lemma for ~ bezout . (Con... |
| bezout 16680 | Bézout's identity: ... |
| dvdsgcd 16681 | An integer which divides e... |
| dvdsgcdb 16682 | Biconditional form of ~ dv... |
| dfgcd2 16683 | Alternate definition of th... |
| gcdass 16684 | Associative law for ` gcd ... |
| mulgcd 16685 | Distribute multiplication ... |
| absmulgcd 16686 | Distribute absolute value ... |
| mulgcdr 16687 | Reverse distribution law f... |
| gcddiv 16688 | Division law for GCD. (Con... |
| gcdzeq 16689 | A positive integer ` A ` i... |
| gcdeq 16690 | ` A ` is equal to its gcd ... |
| dvdssqim 16691 | Unidirectional form of ~ d... |
| dvdsexpim 16692 | If two numbers are divisib... |
| dvdsmulgcd 16693 | A divisibility equivalent ... |
| rpmulgcd 16694 | If ` K ` and ` M ` are rel... |
| rplpwr 16695 | If ` A ` and ` B ` are rel... |
| rprpwr 16696 | If ` A ` and ` B ` are rel... |
| rppwr 16697 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16698 | If ` A ` and ` B ` are rel... |
| sqgcd 16699 | Square distributes over gc... |
| expgcd 16700 | Exponentiation distributes... |
| nn0expgcd 16701 | Exponentiation distributes... |
| zexpgcd 16702 | Exponentiation distributes... |
| dvdssqlem 16703 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16704 | Two numbers are divisible ... |
| bezoutr 16705 | Partial converse to ~ bezo... |
| bezoutr1 16706 | Converse of ~ bezout for w... |
| nn0seqcvgd 16707 | A strictly-decreasing nonn... |
| seq1st 16708 | A sequence whose iteration... |
| algr0 16709 | The value of the algorithm... |
| algrf 16710 | An algorithm is a step fun... |
| algrp1 16711 | The value of the algorithm... |
| alginv 16712 | If ` I ` is an invariant o... |
| algcvg 16713 | One way to prove that an a... |
| algcvgblem 16714 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16715 | Two ways of expressing tha... |
| algcvga 16716 | The countdown function ` C... |
| algfx 16717 | If ` F ` reaches a fixed p... |
| eucalgval2 16718 | The value of the step func... |
| eucalgval 16719 | Euclid's Algorithm ~ eucal... |
| eucalgf 16720 | Domain and codomain of the... |
| eucalginv 16721 | The invariant of the step ... |
| eucalglt 16722 | The second member of the s... |
| eucalgcvga 16723 | Once Euclid's Algorithm ha... |
| eucalg 16724 | Euclid's Algorithm compute... |
| lcmval 16729 | Value of the ` lcm ` opera... |
| lcmcom 16730 | The ` lcm ` operator is co... |
| lcm0val 16731 | The value, by convention, ... |
| lcmn0val 16732 | The value of the ` lcm ` o... |
| lcmcllem 16733 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16734 | Closure of the ` lcm ` ope... |
| dvdslcm 16735 | The lcm of two integers is... |
| lcmledvds 16736 | A positive integer which b... |
| lcmeq0 16737 | The lcm of two integers is... |
| lcmcl 16738 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16739 | The greatest common diviso... |
| lcmneg 16740 | Negating one operand of th... |
| neglcm 16741 | Negating one operand of th... |
| lcmabs 16742 | The lcm of two integers is... |
| lcmgcdlem 16743 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16744 | The product of two numbers... |
| lcmdvds 16745 | The lcm of two integers di... |
| lcmid 16746 | The lcm of an integer and ... |
| lcm1 16747 | The lcm of an integer and ... |
| lcmgcdnn 16748 | The product of two positiv... |
| lcmgcdeq 16749 | Two integers' absolute val... |
| lcmdvdsb 16750 | Biconditional form of ~ lc... |
| lcmass 16751 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16752 | The least common multiple ... |
| 6lcm4e12 16753 | The least common multiple ... |
| absproddvds 16754 | The absolute value of the ... |
| absprodnn 16755 | The absolute value of the ... |
| fissn0dvds 16756 | For each finite subset of ... |
| fissn0dvdsn0 16757 | For each finite subset of ... |
| lcmfval 16758 | Value of the ` _lcm ` func... |
| lcmf0val 16759 | The value, by convention, ... |
| lcmfn0val 16760 | The value of the ` _lcm ` ... |
| lcmfnnval 16761 | The value of the ` _lcm ` ... |
| lcmfcllem 16762 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16763 | Closure of the ` _lcm ` fu... |
| lcmfpr 16764 | The value of the ` _lcm ` ... |
| lcmfcl 16765 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16766 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16767 | The least common multiple ... |
| dvdslcmf 16768 | The least common multiple ... |
| lcmfledvds 16769 | A positive integer which i... |
| lcmf 16770 | Characterization of the le... |
| lcmf0 16771 | The least common multiple ... |
| lcmfsn 16772 | The least common multiple ... |
| lcmftp 16773 | The least common multiple ... |
| lcmfunsnlem1 16774 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16775 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16776 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16777 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16778 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16779 | The least common multiple ... |
| lcmfdvdsb 16780 | Biconditional form of ~ lc... |
| lcmfunsn 16781 | The ` _lcm ` function for ... |
| lcmfun 16782 | The ` _lcm ` function for ... |
| lcmfass 16783 | Associative law for the ` ... |
| lcmf2a3a4e12 16784 | The least common multiple ... |
| lcmflefac 16785 | The least common multiple ... |
| coprmgcdb 16786 | Two positive integers are ... |
| ncoprmgcdne1b 16787 | Two positive integers are ... |
| ncoprmgcdgt1b 16788 | Two positive integers are ... |
| coprmdvds1 16789 | If two positive integers a... |
| coprmdvds 16790 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16791 | If an integer is divisible... |
| mulgcddvds 16792 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16793 | If ` M ` is relatively pri... |
| qredeq 16794 | Two equal reduced fraction... |
| qredeu 16795 | Every rational number has ... |
| rpmul 16796 | If ` K ` is relatively pri... |
| rpdvds 16797 | If ` K ` is relatively pri... |
| coprmprod 16798 | The product of the element... |
| coprmproddvdslem 16799 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16800 | If a positive integer is d... |
| congr 16801 | Definition of congruence b... |
| divgcdcoprm0 16802 | Integers divided by gcd ar... |
| divgcdcoprmex 16803 | Integers divided by gcd ar... |
| cncongr1 16804 | One direction of the bicon... |
| cncongr2 16805 | The other direction of the... |
| cncongr 16806 | Cancellability of Congruen... |
| cncongrcoprm 16807 | Corollary 1 of Cancellabil... |
| isprm 16810 | The predicate "is a prime ... |
| prmnn 16811 | A prime number is a positi... |
| prmz 16812 | A prime number is an integ... |
| prmssnn 16813 | The prime numbers are a su... |
| prmex 16814 | The set of prime numbers e... |
| 0nprm 16815 | 0 is not a prime number. ... |
| 1nprm 16816 | 1 is not a prime number. ... |
| 1idssfct 16817 | The positive divisors of a... |
| isprm2lem 16818 | Lemma for ~ isprm2 . (Con... |
| isprm2 16819 | The predicate "is a prime ... |
| isprm3 16820 | The predicate "is a prime ... |
| isprm4 16821 | The predicate "is a prime ... |
| prmind2 16822 | A variation on ~ prmind as... |
| prmind 16823 | Perform induction over the... |
| dvdsprime 16824 | If ` M ` divides a prime, ... |
| nprm 16825 | A product of two integers ... |
| nprmi 16826 | An inference for composite... |
| dvdsnprmd 16827 | If a number is divisible b... |
| prm2orodd 16828 | A prime number is either 2... |
| 2prm 16829 | 2 is a prime number. (Con... |
| 2mulprm 16830 | A multiple of two is prime... |
| 3prm 16831 | 3 is a prime number. (Con... |
| 4nprm 16832 | 4 is not a prime number. ... |
| prmuz2 16833 | A prime number is an integ... |
| prmssuz2 16834 | The primes are integers gr... |
| prmgt1 16835 | A prime number is an integ... |
| prmm2nn0 16836 | Subtracting 2 from a prime... |
| oddprmgt2 16837 | An odd prime is greater th... |
| oddprmge3 16838 | An odd prime is greater th... |
| ge2nprmge4 16839 | A composite integer greate... |
| sqnprm 16840 | A square is never prime. ... |
| dvdsprm 16841 | An integer greater than or... |
| exprmfct 16842 | Every integer greater than... |
| prmdvdsfz 16843 | Each integer greater than ... |
| nprmdvds1 16844 | No prime number divides 1.... |
| isprm5 16845 | One need only check prime ... |
| isprm7 16846 | One need only check prime ... |
| maxprmfct 16847 | The set of prime factors o... |
| divgcdodd 16848 | Either ` A / ( A gcd B ) `... |
| coprm 16849 | A prime number either divi... |
| prmrp 16850 | Unequal prime numbers are ... |
| euclemma 16851 | Euclid's lemma. A prime n... |
| isprm6 16852 | A number is prime iff it s... |
| prmdvdsexp 16853 | A prime divides a positive... |
| prmdvdsexpb 16854 | A prime divides a positive... |
| prmdvdsexpr 16855 | If a prime divides a nonne... |
| prmdvdssq 16856 | Condition for a prime divi... |
| prmexpb 16857 | Two positive prime powers ... |
| prmfac1 16858 | The factorial of a number ... |
| dvdszzq 16859 | Divisibility for an intege... |
| rpexp 16860 | If two numbers ` A ` and `... |
| rpexp1i 16861 | Relative primality passes ... |
| rpexp12i 16862 | Relative primality passes ... |
| prmndvdsfaclt 16863 | A prime number does not di... |
| prmdvdsbc 16864 | Condition for a prime numb... |
| prmdvdsncoprmbd 16865 | Two positive integers are ... |
| ncoprmlnprm 16866 | If two positive integers a... |
| cncongrprm 16867 | Corollary 2 of Cancellabil... |
| isevengcd2 16868 | The predicate "is an even ... |
| isoddgcd1 16869 | The predicate "is an odd n... |
| 3lcm2e6 16870 | The least common multiple ... |
| qnumval 16875 | Value of the canonical num... |
| qdenval 16876 | Value of the canonical den... |
| qnumdencl 16877 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16878 | The canonical numerator of... |
| qdencl 16879 | The canonical denominator ... |
| fnum 16880 | Canonical numerator define... |
| fden 16881 | Canonical denominator defi... |
| qnumdenbi 16882 | Two numbers are the canoni... |
| qnumdencoprm 16883 | The canonical representati... |
| qeqnumdivden 16884 | Recover a rational number ... |
| qmuldeneqnum 16885 | Multiplying a rational by ... |
| divnumden 16886 | Calculate the reduced form... |
| divdenle 16887 | Reducing a quotient never ... |
| qnumgt0 16888 | A rational is positive iff... |
| qgt0numnn 16889 | A rational is positive iff... |
| nn0gcdsq 16890 | Squaring commutes with GCD... |
| zgcdsq 16891 | ~ nn0gcdsq extended to int... |
| numdensq 16892 | Squaring a rational square... |
| numsq 16893 | Square commutes with canon... |
| densq 16894 | Square commutes with canon... |
| qden1elz 16895 | A rational is an integer i... |
| zsqrtelqelz 16896 | If an integer has a ration... |
| nonsq 16897 | Any integer strictly betwe... |
| numdenexp 16898 | Elevating a rational numbe... |
| numexp 16899 | Elevating to a nonnegative... |
| denexp 16900 | Elevating to a nonnegative... |
| phival 16905 | Value of the Euler ` phi `... |
| phicl2 16906 | Bounds and closure for the... |
| phicl 16907 | Closure for the value of t... |
| phibndlem 16908 | Lemma for ~ phibnd . (Con... |
| phibnd 16909 | A slightly tighter bound o... |
| phicld 16910 | Closure for the value of t... |
| phi1 16911 | Value of the Euler ` phi `... |
| dfphi2 16912 | Alternate definition of th... |
| hashdvds 16913 | The number of numbers in a... |
| phiprmpw 16914 | Value of the Euler ` phi `... |
| phiprm 16915 | Value of the Euler ` phi `... |
| crth 16916 | The Chinese Remainder Theo... |
| phimullem 16917 | Lemma for ~ phimul . (Con... |
| phimul 16918 | The Euler ` phi ` function... |
| eulerthlem1 16919 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16920 | Lemma for ~ eulerth . (Co... |
| eulerth 16921 | Euler's theorem, a general... |
| fermltl 16922 | Fermat's little theorem. ... |
| prmdiv 16923 | Show an explicit expressio... |
| prmdiveq 16924 | The modular inverse of ` A... |
| prmdivdiv 16925 | The (modular) inverse of t... |
| hashgcdlem 16926 | A correspondence between e... |
| dvdsfi 16927 | A natural number has finit... |
| hashgcdeq 16928 | Number of initial positive... |
| phisum 16929 | The divisor sum identity o... |
| odzval 16930 | Value of the order functio... |
| odzcllem 16931 | - Lemma for ~ odzcl , show... |
| odzcl 16932 | The order of a group eleme... |
| odzid 16933 | Any element raised to the ... |
| odzdvds 16934 | The only powers of ` A ` t... |
| odzphi 16935 | The order of any group ele... |
| modprm1div 16936 | A prime number divides an ... |
| m1dvdsndvds 16937 | If an integer minus 1 is d... |
| modprminv 16938 | Show an explicit expressio... |
| modprminveq 16939 | The modular inverse of ` A... |
| vfermltl 16940 | Variant of Fermat's little... |
| vfermltlALT 16941 | Alternate proof of ~ vferm... |
| powm2modprm 16942 | If an integer minus 1 is d... |
| reumodprminv 16943 | For any prime number and f... |
| modprm0 16944 | For two positive integers ... |
| nnnn0modprm0 16945 | For a positive integer and... |
| modprmn0modprm0 16946 | For an integer not being 0... |
| coprimeprodsq 16947 | If three numbers are copri... |
| coprimeprodsq2 16948 | If three numbers are copri... |
| oddprm 16949 | A prime not equal to ` 2 `... |
| nnoddn2prm 16950 | A prime not equal to ` 2 `... |
| oddn2prm 16951 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16952 | A number is a prime number... |
| prm23lt5 16953 | A prime less than 5 is eit... |
| prm23ge5 16954 | A prime is either 2 or 3 o... |
| pythagtriplem1 16955 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16956 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16957 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16958 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16959 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16960 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16961 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16962 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16963 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16964 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16965 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16966 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16967 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16968 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16969 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16970 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16971 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16972 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16973 | Parameterize the Pythagore... |
| iserodd 16974 | Collect the odd terms in a... |
| pclem 16977 | - Lemma for the prime powe... |
| pcprecl 16978 | Closure of the prime power... |
| pcprendvds 16979 | Non-divisibility property ... |
| pcprendvds2 16980 | Non-divisibility property ... |
| pcpre1 16981 | Value of the prime power p... |
| pcpremul 16982 | Multiplicative property of... |
| pcval 16983 | The value of the prime pow... |
| pceulem 16984 | Lemma for ~ pceu . (Contr... |
| pceu 16985 | Uniqueness for the prime p... |
| pczpre 16986 | Connect the prime count pr... |
| pczcl 16987 | Closure of the prime power... |
| pccl 16988 | Closure of the prime power... |
| pccld 16989 | Closure of the prime power... |
| pcmul 16990 | Multiplication property of... |
| pcdiv 16991 | Division property of the p... |
| pcqmul 16992 | Multiplication property of... |
| pc0 16993 | The value of the prime pow... |
| pc1 16994 | Value of the prime count f... |
| pcqcl 16995 | Closure of the general pri... |
| pcqdiv 16996 | Division property of the p... |
| pcrec 16997 | Prime power of a reciproca... |
| pcexp 16998 | Prime power of an exponent... |
| pcxnn0cl 16999 | Extended nonnegative integ... |
| pcxcl 17000 | Extended real closure of t... |
| pcge0 17001 | The prime count of an inte... |
| pczdvds 17002 | Defining property of the p... |
| pcdvds 17003 | Defining property of the p... |
| pczndvds 17004 | Defining property of the p... |
| pcndvds 17005 | Defining property of the p... |
| pczndvds2 17006 | The remainder after dividi... |
| pcndvds2 17007 | The remainder after dividi... |
| pcdvdsb 17008 | ` P ^ A ` divides ` N ` if... |
| pcelnn 17009 | There are a positive numbe... |
| pceq0 17010 | There are zero powers of a... |
| pcidlem 17011 | The prime count of a prime... |
| pcid 17012 | The prime count of a prime... |
| pcneg 17013 | The prime count of a negat... |
| pcabs 17014 | The prime count of an abso... |
| pcdvdstr 17015 | The prime count increases ... |
| pcgcd1 17016 | The prime count of a GCD i... |
| pcgcd 17017 | The prime count of a GCD i... |
| pc2dvds 17018 | A characterization of divi... |
| pc11 17019 | The prime count function, ... |
| pcz 17020 | The prime count function c... |
| pcprmpw2 17021 | Self-referential expressio... |
| pcprmpw 17022 | Self-referential expressio... |
| dvdsprmpweq 17023 | If a positive integer divi... |
| dvdsprmpweqnn 17024 | If an integer greater than... |
| dvdsprmpweqle 17025 | If a positive integer divi... |
| difsqpwdvds 17026 | If the difference of two s... |
| pcaddlem 17027 | Lemma for ~ pcadd . The o... |
| pcadd 17028 | An inequality for the prim... |
| pcadd2 17029 | The inequality of ~ pcadd ... |
| pcmptcl 17030 | Closure for the prime powe... |
| pcmpt 17031 | Construct a function with ... |
| pcmpt2 17032 | Dividing two prime count m... |
| pcmptdvds 17033 | The partial products of th... |
| pcprod 17034 | The product of the primes ... |
| sumhash 17035 | The sum of 1 over a set is... |
| fldivp1 17036 | The difference between the... |
| pcfaclem 17037 | Lemma for ~ pcfac . (Cont... |
| pcfac 17038 | Calculate the prime count ... |
| pcbc 17039 | Calculate the prime count ... |
| qexpz 17040 | If a power of a rational n... |
| expnprm 17041 | A second or higher power o... |
| oddprmdvds 17042 | Every positive integer whi... |
| prmpwdvds 17043 | A relation involving divis... |
| pockthlem 17044 | Lemma for ~ pockthg . (Co... |
| pockthg 17045 | The generalized Pocklingto... |
| pockthi 17046 | Pocklington's theorem, whi... |
| unbenlem 17047 | Lemma for ~ unben . (Cont... |
| unben 17048 | An unbounded set of positi... |
| infpnlem1 17049 | Lemma for ~ infpn . The s... |
| infpnlem2 17050 | Lemma for ~ infpn . For a... |
| infpn 17051 | There exist infinitely man... |
| infpn2 17052 | There exist infinitely man... |
| prmunb 17053 | The primes are unbounded. ... |
| prminf 17054 | There are an infinite numb... |
| prmreclem1 17055 | Lemma for ~ prmrec . Prop... |
| prmreclem2 17056 | Lemma for ~ prmrec . Ther... |
| prmreclem3 17057 | Lemma for ~ prmrec . The ... |
| prmreclem4 17058 | Lemma for ~ prmrec . Show... |
| prmreclem5 17059 | Lemma for ~ prmrec . Here... |
| prmreclem6 17060 | Lemma for ~ prmrec . If t... |
| prmrec 17061 | The sum of the reciprocals... |
| 1arithlem1 17062 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 17063 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 17064 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 17065 | Lemma for ~ 1arith . (Con... |
| 1arith 17066 | Fundamental theorem of ari... |
| 1arith2 17067 | Fundamental theorem of ari... |
| elgz 17070 | Elementhood in the gaussia... |
| gzcn 17071 | A gaussian integer is a co... |
| zgz 17072 | An integer is a gaussian i... |
| igz 17073 | ` _i ` is a gaussian integ... |
| gznegcl 17074 | The gaussian integers are ... |
| gzcjcl 17075 | The gaussian integers are ... |
| gzaddcl 17076 | The gaussian integers are ... |
| gzmulcl 17077 | The gaussian integers are ... |
| gzreim 17078 | Construct a gaussian integ... |
| gzsubcl 17079 | The gaussian integers are ... |
| gzabssqcl 17080 | The squared norm of a gaus... |
| 4sqlem5 17081 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 17082 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 17083 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 17084 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 17085 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 17086 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 17087 | Lemma for ~ 4sq . The set... |
| 4sqlem2 17088 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 17089 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 17090 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 17091 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 17092 | Lemma for ~ mul4sq : algeb... |
| mul4sq 17093 | Euler's four-square identi... |
| 4sqlem11 17094 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 17095 | Lemma for ~ 4sq . For any... |
| 4sqlem13 17096 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 17097 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 17098 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 17099 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 17100 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 17101 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 17102 | Lemma for ~ 4sq . The pro... |
| 4sq 17103 | Lagrange's four-square the... |
| vdwapfval 17110 | Define the arithmetic prog... |
| vdwapf 17111 | The arithmetic progression... |
| vdwapval 17112 | Value of the arithmetic pr... |
| vdwapun 17113 | Remove the first element o... |
| vdwapid1 17114 | The first element of an ar... |
| vdwap0 17115 | Value of a length-1 arithm... |
| vdwap1 17116 | Value of a length-1 arithm... |
| vdwmc 17117 | The predicate " The ` <. R... |
| vdwmc2 17118 | Expand out the definition ... |
| vdwpc 17119 | The predicate " The colori... |
| vdwlem1 17120 | Lemma for ~ vdw . (Contri... |
| vdwlem2 17121 | Lemma for ~ vdw . (Contri... |
| vdwlem3 17122 | Lemma for ~ vdw . (Contri... |
| vdwlem4 17123 | Lemma for ~ vdw . (Contri... |
| vdwlem5 17124 | Lemma for ~ vdw . (Contri... |
| vdwlem6 17125 | Lemma for ~ vdw . (Contri... |
| vdwlem7 17126 | Lemma for ~ vdw . (Contri... |
| vdwlem8 17127 | Lemma for ~ vdw . (Contri... |
| vdwlem9 17128 | Lemma for ~ vdw . (Contri... |
| vdwlem10 17129 | Lemma for ~ vdw . Set up ... |
| vdwlem11 17130 | Lemma for ~ vdw . (Contri... |
| vdwlem12 17131 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 17132 | Lemma for ~ vdw . Main in... |
| vdw 17133 | Van der Waerden's theorem.... |
| vdwnnlem1 17134 | Corollary of ~ vdw , and l... |
| vdwnnlem2 17135 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 17136 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 17137 | Van der Waerden's theorem,... |
| ramtlecl 17139 | The set ` T ` of numbers w... |
| hashbcval 17141 | Value of the "binomial set... |
| hashbccl 17142 | The binomial set is a fini... |
| hashbcss 17143 | Subset relation for the bi... |
| hashbc0 17144 | The set of subsets of size... |
| hashbc2 17145 | The size of the binomial s... |
| 0hashbc 17146 | There are no subsets of th... |
| ramval 17147 | The value of the Ramsey nu... |
| ramcl2lem 17148 | Lemma for extended real cl... |
| ramtcl 17149 | The Ramsey number has the ... |
| ramtcl2 17150 | The Ramsey number is an in... |
| ramtub 17151 | The Ramsey number is a low... |
| ramub 17152 | The Ramsey number is a low... |
| ramub2 17153 | It is sufficient to check ... |
| rami 17154 | The defining property of a... |
| ramcl2 17155 | The Ramsey number is eithe... |
| ramxrcl 17156 | The Ramsey number is an ex... |
| ramubcl 17157 | If the Ramsey number is up... |
| ramlb 17158 | Establish a lower bound on... |
| 0ram 17159 | The Ramsey number when ` M... |
| 0ram2 17160 | The Ramsey number when ` M... |
| ram0 17161 | The Ramsey number when ` R... |
| 0ramcl 17162 | Lemma for ~ ramcl : Exist... |
| ramz2 17163 | The Ramsey number when ` F... |
| ramz 17164 | The Ramsey number when ` F... |
| ramub1lem1 17165 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 17166 | Lemma for ~ ramub1 . (Con... |
| ramub1 17167 | Inductive step for Ramsey'... |
| ramcl 17168 | Ramsey's theorem: the Rams... |
| ramsey 17169 | Ramsey's theorem with the ... |
| prmoval 17172 | Value of the primorial fun... |
| prmocl 17173 | Closure of the primorial f... |
| prmone0 17174 | The primorial function is ... |
| prmo0 17175 | The primorial of 0. (Cont... |
| prmo1 17176 | The primorial of 1. (Cont... |
| prmop1 17177 | The primorial of a success... |
| prmonn2 17178 | Value of the primorial fun... |
| prmo2 17179 | The primorial of 2. (Cont... |
| prmo3 17180 | The primorial of 3. (Cont... |
| prmdvdsprmo 17181 | The primorial of a number ... |
| prmdvdsprmop 17182 | The primorial of a number ... |
| fvprmselelfz 17183 | The value of the prime sel... |
| fvprmselgcd1 17184 | The greatest common diviso... |
| prmolefac 17185 | The primorial of a positiv... |
| prmodvdslcmf 17186 | The primorial of a nonnega... |
| prmolelcmf 17187 | The primorial of a positiv... |
| prmgaplem1 17188 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17189 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17190 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17191 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17192 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17193 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17194 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17195 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17196 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17197 | Lemma for ~ prmgap . (Con... |
| prmgap 17198 | The prime gap theorem: for... |
| prmgaplcm 17199 | Alternate proof of ~ prmga... |
| prmgapprmolem 17200 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17201 | Alternate proof of ~ prmga... |
| dec2dvds 17202 | Divisibility by two is obv... |
| dec5dvds 17203 | Divisibility by five is ob... |
| dec5dvds2 17204 | Divisibility by five is ob... |
| dec5nprm 17205 | A decimal number greater t... |
| dec2nprm 17206 | A decimal number greater t... |
| modxai 17207 | Add exponents in a power m... |
| mod2xi 17208 | Double exponents in a powe... |
| modxp1i 17209 | Add one to an exponent in ... |
| mod2xnegi 17210 | Version of ~ mod2xi where ... |
| modsubi 17211 | Subtract from within a mod... |
| gcdi 17212 | Calculate a GCD via Euclid... |
| gcdmodi 17213 | Calculate a GCD via Euclid... |
| numexp0 17214 | Calculate an integer power... |
| numexp1 17215 | Calculate an integer power... |
| numexpp1 17216 | Calculate an integer power... |
| numexp2x 17217 | Double an integer power. ... |
| decsplit0b 17218 | Split a decimal number int... |
| decsplit0 17219 | Split a decimal number int... |
| decsplit1 17220 | Split a decimal number int... |
| decsplit 17221 | Split a decimal number int... |
| karatsuba 17222 | The Karatsuba multiplicati... |
| 2exp4 17223 | Two to the fourth power is... |
| 2exp5 17224 | Two to the fifth power is ... |
| 2exp6 17225 | Two to the sixth power is ... |
| 2exp7 17226 | Two to the seventh power i... |
| 2exp8 17227 | Two to the eighth power is... |
| 2exp11 17228 | Two to the eleventh power ... |
| 2exp16 17229 | Two to the sixteenth power... |
| 3exp3 17230 | Three to the third power i... |
| 2expltfac 17231 | The factorial grows faster... |
| cshwsidrepsw 17232 | If cyclically shifting a w... |
| cshwsidrepswmod0 17233 | If cyclically shifting a w... |
| cshwshashlem1 17234 | If cyclically shifting a w... |
| cshwshashlem2 17235 | If cyclically shifting a w... |
| cshwshashlem3 17236 | If cyclically shifting a w... |
| cshwsdisj 17237 | The singletons resulting b... |
| cshwsiun 17238 | The set of (different!) wo... |
| cshwsex 17239 | The class of (different!) ... |
| cshws0 17240 | The size of the set of (di... |
| cshwrepswhash1 17241 | The size of the set of (di... |
| cshwshashnsame 17242 | If a word (not consisting ... |
| cshwshash 17243 | If a word has a length bei... |
| prmlem0 17244 | Lemma for ~ prmlem1a and ~... |
| prmlem1a 17245 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1 17246 | A quick proof skeleton to ... |
| 5prm 17247 | 5 is a prime number. (Con... |
| 6nprm 17248 | 6 is not a prime number. ... |
| 7prm 17249 | 7 is a prime number. (Con... |
| 8nprm 17250 | 8 is not a prime number. ... |
| 9nprm 17251 | 9 is not a prime number. ... |
| 10nprm 17252 | 10 is not a prime number. ... |
| 10nprmOLD 17253 | Obsolete version of ~ 10np... |
| 11prm 17254 | 11 is a prime number. (Co... |
| 13prm 17255 | 13 is a prime number. (Co... |
| 17prm 17256 | 17 is a prime number. (Co... |
| 19prm 17257 | 19 is a prime number. (Co... |
| 23prm 17258 | 23 is a prime number. (Co... |
| prmlem2 17259 | Our last proving session g... |
| 37prm 17260 | 37 is a prime number. (Co... |
| 43prm 17261 | 43 is a prime number. (Co... |
| 83prm 17262 | 83 is a prime number. (Co... |
| 139prm 17263 | 139 is a prime number. (C... |
| 163prm 17264 | 163 is a prime number. (C... |
| 317prm 17265 | 317 is a prime number. (C... |
| 631prm 17266 | 631 is a prime number. (C... |
| prmo4 17267 | The primorial of 4. (Cont... |
| prmo5 17268 | The primorial of 5. (Cont... |
| prmo6 17269 | The primorial of 6. (Cont... |
| 1259lem1 17270 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17271 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17272 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17273 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17274 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17275 | 1259 is a prime number. (... |
| 2503lem1 17276 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17277 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17278 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17279 | 2503 is a prime number. (... |
| 4001lem1 17280 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17281 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17282 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17283 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17284 | 4001 is a prime number. (... |
| brstruct 17287 | The structure relation is ... |
| isstruct2 17288 | The property of being a st... |
| structex 17289 | A structure is a set. (Co... |
| structn0fun 17290 | A structure without the em... |
| isstruct 17291 | The property of being a st... |
| structcnvcnv 17292 | Two ways to express the re... |
| structfung 17293 | The converse of the conver... |
| structfun 17294 | Convert between two kinds ... |
| structfn 17295 | Convert between two kinds ... |
| strleun 17296 | Combine two structures int... |
| strle1 17297 | Make a structure from a si... |
| strle2 17298 | Make a structure from a pa... |
| strle3 17299 | Make a structure from a tr... |
| sbcie2s 17300 | A special version of class... |
| sbcie3s 17301 | A special version of class... |
| reldmsets 17304 | The structure override ope... |
| setsvalg 17305 | Value of the structure rep... |
| setsval 17306 | Value of the structure rep... |
| fvsetsid 17307 | The value of the structure... |
| fsets 17308 | The structure replacement ... |
| setsdm 17309 | The domain of a structure ... |
| setsfun 17310 | A structure with replaceme... |
| setsfun0 17311 | A structure with replaceme... |
| setsn0fun 17312 | The value of the structure... |
| setsstruct2 17313 | An extensible structure wi... |
| setsexstruct2 17314 | An extensible structure wi... |
| setsstruct 17315 | An extensible structure wi... |
| wunsets 17316 | Closure of structure repla... |
| setsres 17317 | The structure replacement ... |
| setsabs 17318 | Replacing the same compone... |
| setscom 17319 | Different components can b... |
| sloteq 17322 | Equality theorem for the `... |
| slotfn 17323 | A slot is a function on se... |
| strfvnd 17324 | Deduction version of ~ str... |
| strfvn 17325 | Value of a structure compo... |
| strfvss 17326 | A structure component extr... |
| wunstr 17327 | Closure of a structure ind... |
| str0 17328 | All components of the empt... |
| strfvi 17329 | Structure slot extractors ... |
| fveqprc 17330 | Lemma for showing the equa... |
| oveqprc 17331 | Lemma for showing the equa... |
| wunndx 17334 | Closure of the index extra... |
| ndxarg 17335 | Get the numeric argument f... |
| ndxid 17336 | A structure component extr... |
| strndxid 17337 | The value of a structure c... |
| setsidvald 17338 | Value of the structure rep... |
| strfvd 17339 | Deduction version of ~ str... |
| strfv2d 17340 | Deduction version of ~ str... |
| strfv2 17341 | A variation on ~ strfv to ... |
| strfv 17342 | Extract a structure compon... |
| strfv3 17343 | Variant on ~ strfv for lar... |
| strssd 17344 | Deduction version of ~ str... |
| strss 17345 | Propagate component extrac... |
| setsid 17346 | Value of the structure rep... |
| setsnid 17347 | Value of the structure rep... |
| baseval 17350 | Value of the base set extr... |
| baseid 17351 | Utility theorem: index-ind... |
| basfn 17352 | The base set extractor is ... |
| base0 17353 | The base set of the empty ... |
| elbasfv 17354 | Utility theorem: reverse c... |
| elbasov 17355 | Utility theorem: reverse c... |
| strov2rcl 17356 | Partial reverse closure fo... |
| basendx 17357 | Index value of the base se... |
| basendxnn 17358 | The index value of the bas... |
| basndxelwund 17359 | The index of the base set ... |
| basprssdmsets 17360 | The pair of the base index... |
| opelstrbas 17361 | The base set of a structur... |
| 1strstr 17362 | A constructed one-slot str... |
| 1strbas 17363 | The base set of a construc... |
| 1strwunbndx 17364 | A constructed one-slot str... |
| 1strwun 17365 | A constructed one-slot str... |
| 2strstr 17366 | A constructed two-slot str... |
| 2strbas 17367 | The base set of a construc... |
| 2strop 17368 | The other slot of a constr... |
| reldmress 17371 | The structure restriction ... |
| ressval 17372 | Value of structure restric... |
| ressid2 17373 | General behavior of trivia... |
| ressval2 17374 | Value of nontrivial struct... |
| ressbas 17375 | Base set of a structure re... |
| ressbasssg 17376 | The base set of a restrict... |
| ressbas2 17377 | Base set of a structure re... |
| ressbasss 17378 | The base set of a restrict... |
| ressbasssOLD 17379 | Obsolete version of ~ ress... |
| ressbasss2 17380 | The base set of a restrict... |
| resseqnbas 17381 | The components of an exten... |
| ress0 17382 | All restrictions of the em... |
| ressid 17383 | Behavior of trivial restri... |
| ressinbas 17384 | Restriction only cares abo... |
| ressval3d 17385 | Value of structure restric... |
| ressress 17386 | Restriction composition la... |
| ressabs 17387 | Restriction absorption law... |
| wunress 17388 | Closure of structure restr... |
| plusgndx 17415 | Index value of the ~ df-pl... |
| plusgid 17416 | Utility theorem: index-ind... |
| plusgndxnn 17417 | The index of the slot for ... |
| basendxltplusgndx 17418 | The index of the slot for ... |
| basendxnplusgndx 17419 | The slot for the base set ... |
| grpstr 17420 | A constructed group is a s... |
| grpbase 17421 | The base set of a construc... |
| grpplusg 17422 | The operation of a constru... |
| ressplusg 17423 | ` +g ` is unaffected by re... |
| grpbasex 17424 | The base of an explicitly ... |
| grpplusgx 17425 | The operation of an explic... |
| mulrndx 17426 | Index value of the ~ df-mu... |
| mulridx 17427 | Utility theorem: index-ind... |
| basendxnmulrndx 17428 | The slot for the base set ... |
| plusgndxnmulrndx 17429 | The slot for the group (ad... |
| rngstr 17430 | A constructed ring is a st... |
| rngbase 17431 | The base set of a construc... |
| rngplusg 17432 | The additive operation of ... |
| rngmulr 17433 | The multiplicative operati... |
| starvndx 17434 | Index value of the ~ df-st... |
| starvid 17435 | Utility theorem: index-ind... |
| starvndxnbasendx 17436 | The slot for the involutio... |
| starvndxnplusgndx 17437 | The slot for the involutio... |
| starvndxnmulrndx 17438 | The slot for the involutio... |
| ressmulr 17439 | ` .r ` is unaffected by re... |
| ressstarv 17440 | ` *r ` is unaffected by re... |
| srngstr 17441 | A constructed star ring is... |
| srngbase 17442 | The base set of a construc... |
| srngplusg 17443 | The addition operation of ... |
| srngmulr 17444 | The multiplication operati... |
| srnginvl 17445 | The involution function of... |
| scandx 17446 | Index value of the ~ df-sc... |
| scaid 17447 | Utility theorem: index-ind... |
| scandxnbasendx 17448 | The slot for the scalar is... |
| scandxnplusgndx 17449 | The slot for the scalar fi... |
| scandxnmulrndx 17450 | The slot for the scalar fi... |
| vscandx 17451 | Index value of the ~ df-vs... |
| vscaid 17452 | Utility theorem: index-ind... |
| vscandxnbasendx 17453 | The slot for the scalar pr... |
| vscandxnplusgndx 17454 | The slot for the scalar pr... |
| vscandxnmulrndx 17455 | The slot for the scalar pr... |
| vscandxnscandx 17456 | The slot for the scalar pr... |
| lmodstr 17457 | A constructed left module ... |
| lmodbase 17458 | The base set of a construc... |
| lmodplusg 17459 | The additive operation of ... |
| lmodsca 17460 | The set of scalars of a co... |
| lmodvsca 17461 | The scalar product operati... |
| ipndx 17462 | Index value of the ~ df-ip... |
| ipid 17463 | Utility theorem: index-ind... |
| ipndxnbasendx 17464 | The slot for the inner pro... |
| ipndxnplusgndx 17465 | The slot for the inner pro... |
| ipndxnmulrndx 17466 | The slot for the inner pro... |
| slotsdifipndx 17467 | The slot for the scalar is... |
| ipsstr 17468 | Lemma to shorten proofs of... |
| ipsbase 17469 | The base set of a construc... |
| ipsaddg 17470 | The additive operation of ... |
| ipsmulr 17471 | The multiplicative operati... |
| ipssca 17472 | The set of scalars of a co... |
| ipsvsca 17473 | The scalar product operati... |
| ipsip 17474 | The multiplicative operati... |
| resssca 17475 | ` Scalar ` is unaffected b... |
| ressvsca 17476 | ` .s ` is unaffected by re... |
| ressip 17477 | The inner product is unaff... |
| phlstr 17478 | A constructed pre-Hilbert ... |
| phlbase 17479 | The base set of a construc... |
| phlplusg 17480 | The additive operation of ... |
| phlsca 17481 | The ring of scalars of a c... |
| phlvsca 17482 | The scalar product operati... |
| phlip 17483 | The inner product (Hermiti... |
| tsetndx 17484 | Index value of the ~ df-ts... |
| tsetid 17485 | Utility theorem: index-ind... |
| tsetndxnn 17486 | The index of the slot for ... |
| basendxlttsetndx 17487 | The index of the slot for ... |
| tsetndxnbasendx 17488 | The slot for the topology ... |
| tsetndxnplusgndx 17489 | The slot for the topology ... |
| tsetndxnmulrndx 17490 | The slot for the topology ... |
| tsetndxnstarvndx 17491 | The slot for the topology ... |
| slotstnscsi 17492 | The slots ` Scalar ` , ` .... |
| topgrpstr 17493 | A constructed topological ... |
| topgrpbas 17494 | The base set of a construc... |
| topgrpplusg 17495 | The additive operation of ... |
| topgrptset 17496 | The topology of a construc... |
| resstset 17497 | ` TopSet ` is unaffected b... |
| plendx 17498 | Index value of the ~ df-pl... |
| pleid 17499 | Utility theorem: self-refe... |
| plendxnn 17500 | The index value of the ord... |
| basendxltplendx 17501 | The index value of the ` B... |
| plendxnbasendx 17502 | The slot for the order is ... |
| plendxnplusgndx 17503 | The slot for the "less tha... |
| plendxnmulrndx 17504 | The slot for the "less tha... |
| plendxnscandx 17505 | The slot for the "less tha... |
| plendxnvscandx 17506 | The slot for the "less tha... |
| slotsdifplendx 17507 | The index of the slot for ... |
| otpsstr 17508 | Functionality of a topolog... |
| otpsbas 17509 | The base set of a topologi... |
| otpstset 17510 | The open sets of a topolog... |
| otpsle 17511 | The order of a topological... |
| ressle 17512 | ` le ` is unaffected by re... |
| ocndx 17513 | Index value of the ~ df-oc... |
| ocid 17514 | Utility theorem: index-ind... |
| basendxnocndx 17515 | The slot for the orthocomp... |
| plendxnocndx 17516 | The slot for the orthocomp... |
| dsndx 17517 | Index value of the ~ df-ds... |
| dsid 17518 | Utility theorem: index-ind... |
| dsndxnn 17519 | The index of the slot for ... |
| basendxltdsndx 17520 | The index of the slot for ... |
| dsndxnbasendx 17521 | The slot for the distance ... |
| dsndxnplusgndx 17522 | The slot for the distance ... |
| dsndxnmulrndx 17523 | The slot for the distance ... |
| slotsdnscsi 17524 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17525 | The slot for the distance ... |
| slotsdifdsndx 17526 | The index of the slot for ... |
| unifndx 17527 | Index value of the ~ df-un... |
| unifid 17528 | Utility theorem: index-ind... |
| unifndxnn 17529 | The index of the slot for ... |
| basendxltunifndx 17530 | The index of the slot for ... |
| unifndxnbasendx 17531 | The slot for the uniform s... |
| unifndxntsetndx 17532 | The slot for the uniform s... |
| slotsdifunifndx 17533 | The index of the slot for ... |
| ressunif 17534 | ` UnifSet ` is unaffected ... |
| odrngstr 17535 | Functionality of an ordere... |
| odrngbas 17536 | The base set of an ordered... |
| odrngplusg 17537 | The addition operation of ... |
| odrngmulr 17538 | The multiplication operati... |
| odrngtset 17539 | The open sets of an ordere... |
| odrngle 17540 | The order of an ordered me... |
| odrngds 17541 | The metric of an ordered m... |
| ressds 17542 | ` dist ` is unaffected by ... |
| homndx 17543 | Index value of the ~ df-ho... |
| homid 17544 | Utility theorem: index-ind... |
| ccondx 17545 | Index value of the ~ df-cc... |
| ccoid 17546 | Utility theorem: index-ind... |
| slotsbhcdif 17547 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17548 | The index of the slot for ... |
| slotsdifocndx 17549 | The index of the slot for ... |
| resshom 17550 | ` Hom ` is unaffected by r... |
| ressco 17551 | ` comp ` is unaffected by ... |
| restfn 17556 | The subspace topology oper... |
| topnfn 17557 | The topology extractor fun... |
| restval 17558 | The subspace topology indu... |
| elrest 17559 | The predicate "is an open ... |
| elrestr 17560 | Sufficient condition for b... |
| 0rest 17561 | Value of the structure res... |
| restid2 17562 | The subspace topology over... |
| restsspw 17563 | The subspace topology is a... |
| firest 17564 | The finite intersections o... |
| restid 17565 | The subspace topology of t... |
| topnval 17566 | Value of the topology extr... |
| topnid 17567 | Value of the topology extr... |
| topnpropd 17568 | The topology extractor fun... |
| reldmprds 17580 | The structure product is a... |
| prdsbasex 17582 | Lemma for structure produc... |
| imasvalstr 17583 | An image structure value i... |
| prdsvalstr 17584 | Structure product value is... |
| prdsbaslem 17585 | Lemma for ~ prdsbas and si... |
| prdsvallem 17586 | Lemma for ~ prdsval . (Co... |
| prdsval 17587 | Value of the structure pro... |
| prdssca 17588 | Scalar ring of a structure... |
| prdsbas 17589 | Base set of a structure pr... |
| prdsplusg 17590 | Addition in a structure pr... |
| prdsmulr 17591 | Multiplication in a struct... |
| prdsvsca 17592 | Scalar multiplication in a... |
| prdsip 17593 | Inner product in a structu... |
| prdsle 17594 | Structure product weak ord... |
| prdsless 17595 | Closure of the order relat... |
| prdsds 17596 | Structure product distance... |
| prdsdsfn 17597 | Structure product distance... |
| prdstset 17598 | Structure product topology... |
| prdshom 17599 | Structure product hom-sets... |
| prdsco 17600 | Structure product composit... |
| prdsbas2 17601 | The base set of a structur... |
| prdsbasmpt 17602 | A constructed tuple is a p... |
| prdsbasfn 17603 | Points in the structure pr... |
| prdsbasprj 17604 | Each point in a structure ... |
| prdsplusgval 17605 | Value of a componentwise s... |
| prdsplusgfval 17606 | Value of a structure produ... |
| prdsmulrval 17607 | Value of a componentwise r... |
| prdsmulrfval 17608 | Value of a structure produ... |
| prdsleval 17609 | Value of the product order... |
| prdsdsval 17610 | Value of the metric in a s... |
| prdsvscaval 17611 | Scalar multiplication in a... |
| prdsvscafval 17612 | Scalar multiplication of a... |
| prdsbas3 17613 | The base set of an indexed... |
| prdsbasmpt2 17614 | A constructed tuple is a p... |
| prdsbascl 17615 | An element of the base has... |
| prdsdsval2 17616 | Value of the metric in a s... |
| prdsdsval3 17617 | Value of the metric in a s... |
| pwsval 17618 | Value of a structure power... |
| pwsbas 17619 | Base set of a structure po... |
| pwselbasb 17620 | Membership in the base set... |
| pwselbas 17621 | An element of a structure ... |
| pwselbasr 17622 | The reverse direction of ~... |
| pwsplusgval 17623 | Value of addition in a str... |
| pwsmulrval 17624 | Value of multiplication in... |
| pwsle 17625 | Ordering in a structure po... |
| pwsleval 17626 | Ordering in a structure po... |
| pwsvscafval 17627 | Scalar multiplication in a... |
| pwsvscaval 17628 | Scalar multiplication of a... |
| pwssca 17629 | The ring of scalars of a s... |
| pwsdiagel 17630 | Membership of diagonal ele... |
| pwssnf1o 17631 | Triviality of singleton po... |
| imasval 17644 | Value of an image structur... |
| imasbas 17645 | The base set of an image s... |
| imasds 17646 | The distance function of a... |
| imasdsfn 17647 | The distance function is a... |
| imasdsval 17648 | The distance function of a... |
| imasdsval2 17649 | The distance function of a... |
| imasplusg 17650 | The group operation in an ... |
| imasmulr 17651 | The ring multiplication in... |
| imassca 17652 | The scalar field of an ima... |
| imasvsca 17653 | The scalar multiplication ... |
| imasip 17654 | The inner product of an im... |
| imastset 17655 | The topology of an image s... |
| imasle 17656 | The ordering of an image s... |
| f1ocpbllem 17657 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17658 | An injection is compatible... |
| f1ovscpbl 17659 | An injection is compatible... |
| f1olecpbl 17660 | An injection is compatible... |
| imasaddfnlem 17661 | The image structure operat... |
| imasaddvallem 17662 | The operation of an image ... |
| imasaddflem 17663 | The image set operations a... |
| imasaddfn 17664 | The image structure's grou... |
| imasaddval 17665 | The value of an image stru... |
| imasaddf 17666 | The image structure's grou... |
| imasmulfn 17667 | The image structure's ring... |
| imasmulval 17668 | The value of an image stru... |
| imasmulf 17669 | The image structure's ring... |
| imasvscafn 17670 | The image structure's scal... |
| imasvscaval 17671 | The value of an image stru... |
| imasvscaf 17672 | The image structure's scal... |
| imasless 17673 | The order relation defined... |
| imasleval 17674 | The value of the image str... |
| qusval 17675 | Value of a quotient struct... |
| quslem 17676 | The function in ~ qusval i... |
| qusin 17677 | Restrict the equivalence r... |
| qusbas 17678 | Base set of a quotient str... |
| quss 17679 | The scalar field of a quot... |
| divsfval 17680 | Value of the function in ~... |
| ercpbllem 17681 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17682 | Translate the function com... |
| erlecpbl 17683 | Translate the relation com... |
| qusaddvallem 17684 | Value of an operation defi... |
| qusaddflem 17685 | The operation of a quotien... |
| qusaddval 17686 | The addition in a quotient... |
| qusaddf 17687 | The addition in a quotient... |
| qusmulval 17688 | The multiplication in a qu... |
| qusmulf 17689 | The multiplication in a qu... |
| fnpr2o 17690 | Function with a domain of ... |
| fnpr2ob 17691 | Biconditional version of ~... |
| fvpr0o 17692 | The value of a function wi... |
| fvpr1o 17693 | The value of a function wi... |
| fvprif 17694 | The value of the pair func... |
| xpsfrnel 17695 | Elementhood in the target ... |
| xpsfeq 17696 | A function on ` 2o ` is de... |
| xpsfrnel2 17697 | Elementhood in the target ... |
| xpscf 17698 | Equivalent condition for t... |
| xpsfval 17699 | The value of the function ... |
| xpsff1o 17700 | The function appearing in ... |
| xpsfrn 17701 | A short expression for the... |
| xpsff1o2 17702 | The function appearing in ... |
| xpsval 17703 | Value of the binary struct... |
| xpsrnbas 17704 | The indexed structure prod... |
| xpsbas 17705 | The base set of the binary... |
| xpsaddlem 17706 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17707 | Value of the addition oper... |
| xpsmul 17708 | Value of the multiplicatio... |
| xpssca 17709 | Value of the scalar field ... |
| xpsvsca 17710 | Value of the scalar multip... |
| xpsless 17711 | Closure of the ordering in... |
| xpsle 17712 | Value of the ordering in a... |
| ismre 17721 | Property of being a Moore ... |
| fnmre 17722 | The Moore collection gener... |
| mresspw 17723 | A Moore collection is a su... |
| mress 17724 | A Moore-closed subset is a... |
| mre1cl 17725 | In any Moore collection th... |
| mreintcl 17726 | A nonempty collection of c... |
| mreiincl 17727 | A nonempty indexed interse... |
| mrerintcl 17728 | The relative intersection ... |
| mreriincl 17729 | The relative intersection ... |
| mreincl 17730 | Two closed sets have a clo... |
| mreuni 17731 | Since the entire base set ... |
| mreunirn 17732 | Two ways to express the no... |
| ismred 17733 | Properties that determine ... |
| ismred2 17734 | Properties that determine ... |
| mremre 17735 | The Moore collections of s... |
| submre 17736 | The subcollection of a clo... |
| xrsle 17737 | The ordering of the extend... |
| xrge0le 17738 | The "less than or equal to... |
| xrsbas 17739 | The base set of the extend... |
| xrge0base 17740 | The base of the extended n... |
| mrcflem 17741 | The domain and codomain of... |
| fnmrc 17742 | Moore-closure is a well-be... |
| mrcfval 17743 | Value of the function expr... |
| mrcf 17744 | The Moore closure is a fun... |
| mrcval 17745 | Evaluation of the Moore cl... |
| mrccl 17746 | The Moore closure of a set... |
| mrcsncl 17747 | The Moore closure of a sin... |
| mrcid 17748 | The closure of a closed se... |
| mrcssv 17749 | The closure of a set is a ... |
| mrcidb 17750 | A set is closed iff it is ... |
| mrcss 17751 | Closure preserves subset o... |
| mrcssid 17752 | The closure of a set is a ... |
| mrcidb2 17753 | A set is closed iff it con... |
| mrcidm 17754 | The closure operation is i... |
| mrcsscl 17755 | The closure is the minimal... |
| mrcuni 17756 | Idempotence of closure und... |
| mrcun 17757 | Idempotence of closure und... |
| mrcssvd 17758 | The Moore closure of a set... |
| mrcssd 17759 | Moore closure preserves su... |
| mrcssidd 17760 | A set is contained in its ... |
| mrcidmd 17761 | Moore closure is idempoten... |
| mressmrcd 17762 | In a Moore system, if a se... |
| submrc 17763 | In a closure system which ... |
| mrieqvlemd 17764 | In a Moore system, if ` Y ... |
| mrisval 17765 | Value of the set of indepe... |
| ismri 17766 | Criterion for a set to be ... |
| ismri2 17767 | Criterion for a subset of ... |
| ismri2d 17768 | Criterion for a subset of ... |
| ismri2dd 17769 | Definition of independence... |
| mriss 17770 | An independent set of a Mo... |
| mrissd 17771 | An independent set of a Mo... |
| ismri2dad 17772 | Consequence of a set in a ... |
| mrieqvd 17773 | In a Moore system, a set i... |
| mrieqv2d 17774 | In a Moore system, a set i... |
| mrissmrcd 17775 | In a Moore system, if an i... |
| mrissmrid 17776 | In a Moore system, subsets... |
| mreexd 17777 | In a Moore system, the clo... |
| mreexmrid 17778 | In a Moore system whose cl... |
| mreexexlemd 17779 | This lemma is used to gene... |
| mreexexlem2d 17780 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17781 | Base case of the induction... |
| mreexexlem4d 17782 | Induction step of the indu... |
| mreexexd 17783 | Exchange-type theorem. In... |
| mreexdomd 17784 | In a Moore system whose cl... |
| mreexfidimd 17785 | In a Moore system whose cl... |
| isacs 17786 | A set is an algebraic clos... |
| acsmre 17787 | Algebraic closure systems ... |
| isacs2 17788 | In the definition of an al... |
| acsfiel 17789 | A set is closed in an alge... |
| acsfiel2 17790 | A set is closed in an alge... |
| acsmred 17791 | An algebraic closure syste... |
| isacs1i 17792 | A closure system determine... |
| mreacs 17793 | Algebraicity is a composab... |
| acsfn 17794 | Algebraicity of a conditio... |
| acsfn0 17795 | Algebraicity of a point cl... |
| acsfn1 17796 | Algebraicity of a one-argu... |
| acsfn1c 17797 | Algebraicity of a one-argu... |
| acsfn2 17798 | Algebraicity of a two-argu... |
| iscat 17807 | The predicate "is a catego... |
| iscatd 17808 | Properties that determine ... |
| catidex 17809 | Each object in a category ... |
| catideu 17810 | Each object in a category ... |
| cidfval 17811 | Each object in a category ... |
| cidval 17812 | Each object in a category ... |
| cidffn 17813 | The identity arrow constru... |
| cidfn 17814 | The identity arrow operato... |
| catidd 17815 | Deduce the identity arrow ... |
| iscatd2 17816 | Version of ~ iscatd with a... |
| catidcl 17817 | Each object in a category ... |
| catlid 17818 | Left identity property of ... |
| catrid 17819 | Right identity property of... |
| catcocl 17820 | Closure of a composition a... |
| catass 17821 | Associativity of compositi... |
| catcone0 17822 | Composition of non-empty h... |
| 0catg 17823 | Any structure with an empt... |
| 0cat 17824 | The empty set is a categor... |
| homffval 17825 | Value of the functionalize... |
| fnhomeqhomf 17826 | If the Hom-set operation i... |
| homfval 17827 | Value of the functionalize... |
| homffn 17828 | The functionalized Hom-set... |
| homfeq 17829 | Condition for two categori... |
| homfeqd 17830 | If two structures have the... |
| homfeqbas 17831 | Deduce equality of base se... |
| homfeqval 17832 | Value of the functionalize... |
| comfffval 17833 | Value of the functionalize... |
| comffval 17834 | Value of the functionalize... |
| comfval 17835 | Value of the functionalize... |
| comfffval2 17836 | Value of the functionalize... |
| comffval2 17837 | Value of the functionalize... |
| comfval2 17838 | Value of the functionalize... |
| comfffn 17839 | The functionalized composi... |
| comffn 17840 | The functionalized composi... |
| comfeq 17841 | Condition for two categori... |
| comfeqd 17842 | Condition for two categori... |
| comfeqval 17843 | Equality of two compositio... |
| catpropd 17844 | Two structures with the sa... |
| cidpropd 17845 | Two structures with the sa... |
| oppcval 17848 | Value of the opposite cate... |
| oppchomfval 17849 | Hom-sets of the opposite c... |
| oppchom 17850 | Hom-sets of the opposite c... |
| oppccofval 17851 | Composition in the opposit... |
| oppcco 17852 | Composition in the opposit... |
| oppcbas 17853 | Base set of an opposite ca... |
| oppccatid 17854 | Lemma for ~ oppccat . (Co... |
| oppchomf 17855 | Hom-sets of the opposite c... |
| oppcid 17856 | Identity function of an op... |
| oppccat 17857 | An opposite category is a ... |
| 2oppcbas 17858 | The double opposite catego... |
| 2oppchomf 17859 | The double opposite catego... |
| 2oppccomf 17860 | The double opposite catego... |
| oppchomfpropd 17861 | If two categories have the... |
| oppccomfpropd 17862 | If two categories have the... |
| oppccatf 17863 | ` oppCat ` restricted to `... |
| monfval 17868 | Definition of a monomorphi... |
| ismon 17869 | Definition of a monomorphi... |
| ismon2 17870 | Write out the monomorphism... |
| monhom 17871 | A monomorphism is a morphi... |
| moni 17872 | Property of a monomorphism... |
| monpropd 17873 | If two categories have the... |
| oppcmon 17874 | A monomorphism in the oppo... |
| oppcepi 17875 | An epimorphism in the oppo... |
| isepi 17876 | Definition of an epimorphi... |
| isepi2 17877 | Write out the epimorphism ... |
| epihom 17878 | An epimorphism is a morphi... |
| epii 17879 | Property of an epimorphism... |
| sectffval 17886 | Value of the section opera... |
| sectfval 17887 | Value of the section relat... |
| sectss 17888 | The section relation is a ... |
| issect 17889 | The property " ` F ` is a ... |
| issect2 17890 | Property of being a sectio... |
| sectcan 17891 | If ` G ` is a section of `... |
| sectco 17892 | Composition of two section... |
| isofval 17893 | Function value of the func... |
| invffval 17894 | Value of the inverse relat... |
| invfval 17895 | Value of the inverse relat... |
| isinv 17896 | Value of the inverse relat... |
| invss 17897 | The inverse relation is a ... |
| invsym 17898 | The inverse relation is sy... |
| invsym2 17899 | The inverse relation is sy... |
| invfun 17900 | The inverse relation is a ... |
| isoval 17901 | The isomorphisms are the d... |
| inviso1 17902 | If ` G ` is an inverse to ... |
| inviso2 17903 | If ` G ` is an inverse to ... |
| invf 17904 | The inverse relation is a ... |
| invf1o 17905 | The inverse relation is a ... |
| invinv 17906 | The inverse of the inverse... |
| invco 17907 | The composition of two iso... |
| dfiso2 17908 | Alternate definition of an... |
| dfiso3 17909 | Alternate definition of an... |
| inveq 17910 | If there are two inverses ... |
| isofn 17911 | The function value of the ... |
| isohom 17912 | An isomorphism is a homomo... |
| isoco 17913 | The composition of two iso... |
| oppcsect 17914 | A section in the opposite ... |
| oppcsect2 17915 | A section in the opposite ... |
| oppcinv 17916 | An inverse in the opposite... |
| oppciso 17917 | An isomorphism in the oppo... |
| sectmon 17918 | If ` F ` is a section of `... |
| monsect 17919 | If ` F ` is a monomorphism... |
| sectepi 17920 | If ` F ` is a section of `... |
| episect 17921 | If ` F ` is an epimorphism... |
| sectid 17922 | The identity is a section ... |
| invid 17923 | The inverse of the identit... |
| idiso 17924 | The identity is an isomorp... |
| idinv 17925 | The inverse of the identit... |
| invisoinvl 17926 | The inverse of an isomorph... |
| invisoinvr 17927 | The inverse of an isomorph... |
| invcoisoid 17928 | The inverse of an isomorph... |
| isocoinvid 17929 | The inverse of an isomorph... |
| rcaninv 17930 | Right cancellation of an i... |
| cicfval 17933 | The set of isomorphic obje... |
| brcic 17934 | The relation "is isomorphi... |
| cic 17935 | Objects ` X ` and ` Y ` in... |
| brcici 17936 | Prove that two objects are... |
| cicref 17937 | Isomorphism is reflexive. ... |
| ciclcl 17938 | Isomorphism implies the le... |
| cicrcl 17939 | Isomorphism implies the ri... |
| cicsym 17940 | Isomorphism is symmetric. ... |
| cictr 17941 | Isomorphism is transitive.... |
| cicer 17942 | Isomorphism is an equivale... |
| sscrel 17949 | The subcategory subset rel... |
| brssc 17950 | The subcategory subset rel... |
| sscpwex 17951 | An analogue of ~ pwex for ... |
| subcrcl 17952 | Reverse closure for the su... |
| sscfn1 17953 | The subcategory subset rel... |
| sscfn2 17954 | The subcategory subset rel... |
| ssclem 17955 | Lemma for ~ ssc1 and simil... |
| isssc 17956 | Value of the subcategory s... |
| ssc1 17957 | Infer subset relation on o... |
| ssc2 17958 | Infer subset relation on m... |
| sscres 17959 | Any function restricted to... |
| sscid 17960 | The subcategory subset rel... |
| ssctr 17961 | The subcategory subset rel... |
| ssceq 17962 | The subcategory subset rel... |
| rescval 17963 | Value of the category rest... |
| rescval2 17964 | Value of the category rest... |
| rescbas 17965 | Base set of the category r... |
| reschom 17966 | Hom-sets of the category r... |
| reschomf 17967 | Hom-sets of the category r... |
| rescco 17968 | Composition in the categor... |
| rescabs 17969 | Restriction absorption law... |
| rescabs2 17970 | Restriction absorption law... |
| issubc 17971 | Elementhood in the set of ... |
| issubc2 17972 | Elementhood in the set of ... |
| 0ssc 17973 | For any category ` C ` , t... |
| 0subcat 17974 | For any category ` C ` , t... |
| catsubcat 17975 | For any category ` C ` , `... |
| subcssc 17976 | An element in the set of s... |
| subcfn 17977 | An element in the set of s... |
| subcss1 17978 | The objects of a subcatego... |
| subcss2 17979 | The morphisms of a subcate... |
| subcidcl 17980 | The identity of the origin... |
| subccocl 17981 | A subcategory is closed un... |
| subccatid 17982 | A subcategory is a categor... |
| subcid 17983 | The identity in a subcateg... |
| subccat 17984 | A subcategory is a categor... |
| issubc3 17985 | Alternate definition of a ... |
| fullsubc 17986 | The full subcategory gener... |
| fullresc 17987 | The category formed by str... |
| resscat 17988 | A category restricted to a... |
| subsubc 17989 | A subcategory of a subcate... |
| relfunc 17998 | The set of functors is a r... |
| funcrcl 17999 | Reverse closure for a func... |
| isfunc 18000 | Value of the set of functo... |
| isfuncd 18001 | Deduce that an operation i... |
| funcf1 18002 | The object part of a funct... |
| funcixp 18003 | The morphism part of a fun... |
| funcf2 18004 | The morphism part of a fun... |
| funcfn2 18005 | The morphism part of a fun... |
| funcid 18006 | A functor maps each identi... |
| funcco 18007 | A functor maps composition... |
| funcsect 18008 | The image of a section und... |
| funcinv 18009 | The image of an inverse un... |
| funciso 18010 | The image of an isomorphis... |
| funcoppc 18011 | A functor on categories yi... |
| idfuval 18012 | Value of the identity func... |
| idfu2nd 18013 | Value of the morphism part... |
| idfu2 18014 | Value of the morphism part... |
| idfu1st 18015 | Value of the object part o... |
| idfu1 18016 | Value of the object part o... |
| idfucl 18017 | The identity functor is a ... |
| cofuval 18018 | Value of the composition o... |
| cofu1st 18019 | Value of the object part o... |
| cofu1 18020 | Value of the object part o... |
| cofu2nd 18021 | Value of the morphism part... |
| cofu2 18022 | Value of the morphism part... |
| cofuval2 18023 | Value of the composition o... |
| cofucl 18024 | The composition of two fun... |
| cofuass 18025 | Functor composition is ass... |
| cofulid 18026 | The identity functor is a ... |
| cofurid 18027 | The identity functor is a ... |
| resfval 18028 | Value of the functor restr... |
| resfval2 18029 | Value of the functor restr... |
| resf1st 18030 | Value of the functor restr... |
| resf2nd 18031 | Value of the functor restr... |
| funcres 18032 | A functor restricted to a ... |
| funcres2b 18033 | Condition for a functor to... |
| funcres2 18034 | A functor into a restricte... |
| idfusubc0 18035 | The identity functor for a... |
| idfusubc 18036 | The identity functor for a... |
| wunfunc 18037 | A weak universe is closed ... |
| funcpropd 18038 | If two categories have the... |
| funcres2c 18039 | Condition for a functor to... |
| fullfunc 18044 | A full functor is a functo... |
| fthfunc 18045 | A faithful functor is a fu... |
| relfull 18046 | The set of full functors i... |
| relfth 18047 | The set of faithful functo... |
| isfull 18048 | Value of the set of full f... |
| isfull2 18049 | Equivalent condition for a... |
| fullfo 18050 | The morphism map of a full... |
| fulli 18051 | The morphism map of a full... |
| isfth 18052 | Value of the set of faithf... |
| isfth2 18053 | Equivalent condition for a... |
| isffth2 18054 | A fully faithful functor i... |
| fthf1 18055 | The morphism map of a fait... |
| fthi 18056 | The morphism map of a fait... |
| ffthf1o 18057 | The morphism map of a full... |
| fullpropd 18058 | If two categories have the... |
| fthpropd 18059 | If two categories have the... |
| fulloppc 18060 | The opposite functor of a ... |
| fthoppc 18061 | The opposite functor of a ... |
| ffthoppc 18062 | The opposite functor of a ... |
| fthsect 18063 | A faithful functor reflect... |
| fthinv 18064 | A faithful functor reflect... |
| fthmon 18065 | A faithful functor reflect... |
| fthepi 18066 | A faithful functor reflect... |
| ffthiso 18067 | A fully faithful functor r... |
| fthres2b 18068 | Condition for a faithful f... |
| fthres2c 18069 | Condition for a faithful f... |
| fthres2 18070 | A faithful functor into a ... |
| idffth 18071 | The identity functor is a ... |
| cofull 18072 | The composition of two ful... |
| cofth 18073 | The composition of two fai... |
| coffth 18074 | The composition of two ful... |
| rescfth 18075 | The inclusion functor from... |
| ressffth 18076 | The inclusion functor from... |
| fullres2c 18077 | Condition for a full funct... |
| ffthres2c 18078 | Condition for a fully fait... |
| inclfusubc 18079 | The "inclusion functor" fr... |
| fnfuc 18084 | The ` FuncCat ` operation ... |
| natfval 18085 | Value of the function givi... |
| isnat 18086 | Property of being a natura... |
| isnat2 18087 | Property of being a natura... |
| natffn 18088 | The natural transformation... |
| natrcl 18089 | Reverse closure for a natu... |
| nat1st2nd 18090 | Rewrite the natural transf... |
| natixp 18091 | A natural transformation i... |
| natcl 18092 | A component of a natural t... |
| natfn 18093 | A natural transformation i... |
| nati 18094 | Naturality property of a n... |
| wunnat 18095 | A weak universe is closed ... |
| catstr 18096 | A category structure is a ... |
| fucval 18097 | Value of the functor categ... |
| fuccofval 18098 | Value of the functor categ... |
| fucbas 18099 | The objects of the functor... |
| fuchom 18100 | The morphisms in the funct... |
| fucco 18101 | Value of the composition o... |
| fuccoval 18102 | Value of the functor categ... |
| fuccocl 18103 | The composition of two nat... |
| fucidcl 18104 | The identity natural trans... |
| fuclid 18105 | Left identity of natural t... |
| fucrid 18106 | Right identity of natural ... |
| fucass 18107 | Associativity of natural t... |
| fuccatid 18108 | The functor category is a ... |
| fuccat 18109 | The functor category is a ... |
| fucid 18110 | The identity morphism in t... |
| fucsect 18111 | Two natural transformation... |
| fucinv 18112 | Two natural transformation... |
| invfuc 18113 | If ` V ( x ) ` is an inver... |
| fuciso 18114 | A natural transformation i... |
| natpropd 18115 | If two categories have the... |
| fucpropd 18116 | If two categories have the... |
| initofn 18123 | ` InitO ` is a function on... |
| termofn 18124 | ` TermO ` is a function on... |
| zeroofn 18125 | ` ZeroO ` is a function on... |
| initorcl 18126 | Reverse closure for an ini... |
| termorcl 18127 | Reverse closure for a term... |
| zeroorcl 18128 | Reverse closure for a zero... |
| initoval 18129 | The value of the initial o... |
| termoval 18130 | The value of the terminal ... |
| zerooval 18131 | The value of the zero obje... |
| isinito 18132 | The predicate "is an initi... |
| istermo 18133 | The predicate "is a termin... |
| iszeroo 18134 | The predicate "is a zero o... |
| isinitoi 18135 | Implication of a class bei... |
| istermoi 18136 | Implication of a class bei... |
| initoid 18137 | For an initial object, the... |
| termoid 18138 | For a terminal object, the... |
| dfinito2 18139 | An initial object is a ter... |
| dftermo2 18140 | A terminal object is an in... |
| dfinito3 18141 | An alternate definition of... |
| dftermo3 18142 | An alternate definition of... |
| initoo 18143 | An initial object is an ob... |
| termoo 18144 | A terminal object is an ob... |
| iszeroi 18145 | Implication of a class bei... |
| 2initoinv 18146 | Morphisms between two init... |
| initoeu1 18147 | Initial objects are essent... |
| initoeu1w 18148 | Initial objects are essent... |
| initoeu2lem0 18149 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18150 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18151 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18152 | Initial objects are essent... |
| 2termoinv 18153 | Morphisms between two term... |
| termoeu1 18154 | Terminal objects are essen... |
| termoeu1w 18155 | Terminal objects are essen... |
| homarcl 18164 | Reverse closure for an arr... |
| homafval 18165 | Value of the disjointified... |
| homaf 18166 | Functionality of the disjo... |
| homaval 18167 | Value of the disjointified... |
| elhoma 18168 | Value of the disjointified... |
| elhomai 18169 | Produce an arrow from a mo... |
| elhomai2 18170 | Produce an arrow from a mo... |
| homarcl2 18171 | Reverse closure for the do... |
| homarel 18172 | An arrow is an ordered pai... |
| homa1 18173 | The first component of an ... |
| homahom2 18174 | The second component of an... |
| homahom 18175 | The second component of an... |
| homadm 18176 | The domain of an arrow wit... |
| homacd 18177 | The codomain of an arrow w... |
| homadmcd 18178 | Decompose an arrow into do... |
| arwval 18179 | The set of arrows is the u... |
| arwrcl 18180 | The first component of an ... |
| arwhoma 18181 | An arrow is contained in t... |
| homarw 18182 | A hom-set is a subset of t... |
| arwdm 18183 | The domain of an arrow is ... |
| arwcd 18184 | The codomain of an arrow i... |
| dmaf 18185 | The domain function is a f... |
| cdaf 18186 | The codomain function is a... |
| arwhom 18187 | The second component of an... |
| arwdmcd 18188 | Decompose an arrow into do... |
| idafval 18193 | Value of the identity arro... |
| idaval 18194 | Value of the identity arro... |
| ida2 18195 | Morphism part of the ident... |
| idahom 18196 | Domain and codomain of the... |
| idadm 18197 | Domain of the identity arr... |
| idacd 18198 | Codomain of the identity a... |
| idaf 18199 | The identity arrow functio... |
| coafval 18200 | The value of the compositi... |
| eldmcoa 18201 | A pair ` <. G , F >. ` is ... |
| dmcoass 18202 | The domain of composition ... |
| homdmcoa 18203 | If ` F : X --> Y ` and ` G... |
| coaval 18204 | Value of composition for c... |
| coa2 18205 | The morphism part of arrow... |
| coahom 18206 | The composition of two com... |
| coapm 18207 | Composition of arrows is a... |
| arwlid 18208 | Left identity of a categor... |
| arwrid 18209 | Right identity of a catego... |
| arwass 18210 | Associativity of compositi... |
| setcval 18213 | Value of the category of s... |
| setcbas 18214 | Set of objects of the cate... |
| setchomfval 18215 | Set of arrows of the categ... |
| setchom 18216 | Set of arrows of the categ... |
| elsetchom 18217 | A morphism of sets is a fu... |
| setccofval 18218 | Composition in the categor... |
| setcco 18219 | Composition in the categor... |
| setccatid 18220 | Lemma for ~ setccat . (Co... |
| setccat 18221 | The category of sets is a ... |
| setcid 18222 | The identity arrow in the ... |
| setcmon 18223 | A monomorphism of sets is ... |
| setcepi 18224 | An epimorphism of sets is ... |
| setcsect 18225 | A section in the category ... |
| setcinv 18226 | An inverse in the category... |
| setciso 18227 | An isomorphism in the cate... |
| resssetc 18228 | The restriction of the cat... |
| funcsetcres2 18229 | A functor into a smaller c... |
| setc2obas 18230 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18231 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18232 | The category of sets in a ... |
| cat1 18233 | The definition of category... |
| catcval 18236 | Value of the category of c... |
| catcbas 18237 | Set of objects of the cate... |
| catchomfval 18238 | Set of arrows of the categ... |
| catchom 18239 | Set of arrows of the categ... |
| catccofval 18240 | Composition in the categor... |
| catcco 18241 | Composition in the categor... |
| catccatid 18242 | Lemma for ~ catccat . (Co... |
| catcid 18243 | The identity arrow in the ... |
| catccat 18244 | The category of categories... |
| resscatc 18245 | The restriction of the cat... |
| catcisolem 18246 | Lemma for ~ catciso . (Co... |
| catciso 18247 | A functor is an isomorphis... |
| catcbascl 18248 | An element of the base set... |
| catcslotelcl 18249 | A slot entry of an element... |
| catcbaselcl 18250 | The base set of an element... |
| catchomcl 18251 | The Hom-set of an element ... |
| catcccocl 18252 | The composition operation ... |
| catcoppccl 18253 | The category of categories... |
| catcfuccl 18254 | The category of categories... |
| fncnvimaeqv 18255 | The inverse images of the ... |
| bascnvimaeqv 18256 | The inverse image of the u... |
| estrcval 18259 | Value of the category of e... |
| estrcbas 18260 | Set of objects of the cate... |
| estrchomfval 18261 | Set of morphisms ("arrows"... |
| estrchom 18262 | The morphisms between exte... |
| elestrchom 18263 | A morphism between extensi... |
| estrccofval 18264 | Composition in the categor... |
| estrcco 18265 | Composition in the categor... |
| estrcbasbas 18266 | An element of the base set... |
| estrccatid 18267 | Lemma for ~ estrccat . (C... |
| estrccat 18268 | The category of extensible... |
| estrcid 18269 | The identity arrow in the ... |
| estrchomfn 18270 | The Hom-set operation in t... |
| estrchomfeqhom 18271 | The functionalized Hom-set... |
| estrreslem1 18272 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18273 | Lemma 2 for ~ estrres . (... |
| estrres 18274 | Any restriction of a categ... |
| funcestrcsetclem1 18275 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18276 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18277 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18278 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18279 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18280 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18281 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18282 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18283 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18284 | The "natural forgetful fun... |
| fthestrcsetc 18285 | The "natural forgetful fun... |
| fullestrcsetc 18286 | The "natural forgetful fun... |
| equivestrcsetc 18287 | The "natural forgetful fun... |
| setc1strwun 18288 | A constructed one-slot str... |
| funcsetcestrclem1 18289 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18290 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18291 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18292 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18293 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18294 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18295 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18296 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18297 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18298 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18299 | The "embedding functor" fr... |
| fthsetcestrc 18300 | The "embedding functor" fr... |
| fullsetcestrc 18301 | The "embedding functor" fr... |
| embedsetcestrc 18302 | The "embedding functor" fr... |
| fnxpc 18311 | The binary product of cate... |
| xpcval 18312 | Value of the binary produc... |
| xpcbas 18313 | Set of objects of the bina... |
| xpchomfval 18314 | Set of morphisms of the bi... |
| xpchom 18315 | Set of morphisms of the bi... |
| relxpchom 18316 | A hom-set in the binary pr... |
| xpccofval 18317 | Value of composition in th... |
| xpcco 18318 | Value of composition in th... |
| xpcco1st 18319 | Value of composition in th... |
| xpcco2nd 18320 | Value of composition in th... |
| xpchom2 18321 | Value of the set of morphi... |
| xpcco2 18322 | Value of composition in th... |
| xpccatid 18323 | The product of two categor... |
| xpcid 18324 | The identity morphism in t... |
| xpccat 18325 | The product of two categor... |
| 1stfval 18326 | Value of the first project... |
| 1stf1 18327 | Value of the first project... |
| 1stf2 18328 | Value of the first project... |
| 2ndfval 18329 | Value of the first project... |
| 2ndf1 18330 | Value of the first project... |
| 2ndf2 18331 | Value of the first project... |
| 1stfcl 18332 | The first projection funct... |
| 2ndfcl 18333 | The second projection func... |
| prfval 18334 | Value of the pairing funct... |
| prf1 18335 | Value of the pairing funct... |
| prf2fval 18336 | Value of the pairing funct... |
| prf2 18337 | Value of the pairing funct... |
| prfcl 18338 | The pairing of functors ` ... |
| prf1st 18339 | Cancellation of pairing wi... |
| prf2nd 18340 | Cancellation of pairing wi... |
| 1st2ndprf 18341 | Break a functor into a pro... |
| catcxpccl 18342 | The category of categories... |
| xpcpropd 18343 | If two categories have the... |
| evlfval 18352 | Value of the evaluation fu... |
| evlf2 18353 | Value of the evaluation fu... |
| evlf2val 18354 | Value of the evaluation na... |
| evlf1 18355 | Value of the evaluation fu... |
| evlfcllem 18356 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18357 | The evaluation functor is ... |
| curfval 18358 | Value of the curry functor... |
| curf1fval 18359 | Value of the object part o... |
| curf1 18360 | Value of the object part o... |
| curf11 18361 | Value of the double evalua... |
| curf12 18362 | The partially evaluated cu... |
| curf1cl 18363 | The partially evaluated cu... |
| curf2 18364 | Value of the curry functor... |
| curf2val 18365 | Value of a component of th... |
| curf2cl 18366 | The curry functor at a mor... |
| curfcl 18367 | The curry functor of a fun... |
| curfpropd 18368 | If two categories have the... |
| uncfval 18369 | Value of the uncurry funct... |
| uncfcl 18370 | The uncurry operation take... |
| uncf1 18371 | Value of the uncurry funct... |
| uncf2 18372 | Value of the uncurry funct... |
| curfuncf 18373 | Cancellation of curry with... |
| uncfcurf 18374 | Cancellation of uncurry wi... |
| diagval 18375 | Define the diagonal functo... |
| diagcl 18376 | The diagonal functor is a ... |
| diag1cl 18377 | The constant functor of ` ... |
| diag11 18378 | Value of the constant func... |
| diag12 18379 | Value of the constant func... |
| diag2 18380 | Value of the diagonal func... |
| diag2cl 18381 | The diagonal functor at a ... |
| curf2ndf 18382 | As shown in ~ diagval , th... |
| hofval 18387 | Value of the Hom functor, ... |
| hof1fval 18388 | The object part of the Hom... |
| hof1 18389 | The object part of the Hom... |
| hof2fval 18390 | The morphism part of the H... |
| hof2val 18391 | The morphism part of the H... |
| hof2 18392 | The morphism part of the H... |
| hofcllem 18393 | Lemma for ~ hofcl . (Cont... |
| hofcl 18394 | Closure of the Hom functor... |
| oppchofcl 18395 | Closure of the opposite Ho... |
| yonval 18396 | Value of the Yoneda embedd... |
| yoncl 18397 | The Yoneda embedding is a ... |
| yon1cl 18398 | The Yoneda embedding at an... |
| yon11 18399 | Value of the Yoneda embedd... |
| yon12 18400 | Value of the Yoneda embedd... |
| yon2 18401 | Value of the Yoneda embedd... |
| hofpropd 18402 | If two categories have the... |
| yonpropd 18403 | If two categories have the... |
| oppcyon 18404 | Value of the opposite Yone... |
| oyoncl 18405 | The opposite Yoneda embedd... |
| oyon1cl 18406 | The opposite Yoneda embedd... |
| yonedalem1 18407 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18408 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18409 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18410 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18411 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18412 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18413 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18414 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18415 | Lemma for ~ yoneda . (Con... |
| yonedainv 18416 | The Yoneda Lemma with expl... |
| yonffthlem 18417 | Lemma for ~ yonffth . (Co... |
| yoneda 18418 | The Yoneda Lemma. There i... |
| yonffth 18419 | The Yoneda Lemma. The Yon... |
| yoniso 18420 | If the codomain is recover... |
| oduval 18423 | Value of an order dual str... |
| oduleval 18424 | Value of the less-equal re... |
| oduleg 18425 | Truth of the less-equal re... |
| odubas 18426 | Base set of an order dual ... |
| isprs 18431 | Property of being a preord... |
| prslem 18432 | Lemma for ~ prsref and ~ p... |
| prsref 18433 | "Less than or equal to" is... |
| prstr 18434 | "Less than or equal to" is... |
| oduprs 18435 | Being a proset is a self-d... |
| isdrs 18436 | Property of being a direct... |
| drsdir 18437 | Direction of a directed se... |
| drsprs 18438 | A directed set is a proset... |
| drsbn0 18439 | The base of a directed set... |
| drsdirfi 18440 | Any _finite_ number of ele... |
| isdrs2 18441 | Directed sets may be defin... |
| ispos 18449 | The predicate "is a poset"... |
| ispos2 18450 | A poset is an antisymmetri... |
| posprs 18451 | A poset is a proset. (Con... |
| posi 18452 | Lemma for poset properties... |
| posref 18453 | A poset ordering is reflex... |
| posasymb 18454 | A poset ordering is asymme... |
| postr 18455 | A poset ordering is transi... |
| 0pos 18456 | Technical lemma to simplif... |
| isposd 18457 | Properties that determine ... |
| isposi 18458 | Properties that determine ... |
| isposix 18459 | Properties that determine ... |
| pospropd 18460 | Posethood is determined on... |
| odupos 18461 | Being a poset is a self-du... |
| oduposb 18462 | Being a poset is a self-du... |
| pltfval 18464 | Value of the less-than rel... |
| pltval 18465 | Less-than relation. ( ~ d... |
| pltle 18466 | "Less than" implies "less ... |
| pltne 18467 | The "less than" relation i... |
| pltirr 18468 | The "less than" relation i... |
| pleval2i 18469 | One direction of ~ pleval2... |
| pleval2 18470 | "Less than or equal to" in... |
| pltnle 18471 | "Less than" implies not co... |
| pltval3 18472 | Alternate expression for t... |
| pltnlt 18473 | The less-than relation imp... |
| pltn2lp 18474 | The less-than relation has... |
| plttr 18475 | The less-than relation is ... |
| pltletr 18476 | Transitive law for chained... |
| plelttr 18477 | Transitive law for chained... |
| pospo 18478 | Write a poset structure in... |
| lubfval 18483 | Value of the least upper b... |
| lubdm 18484 | Domain of the least upper ... |
| lubfun 18485 | The LUB is a function. (C... |
| lubeldm 18486 | Member of the domain of th... |
| lubelss 18487 | A member of the domain of ... |
| lubeu 18488 | Unique existence proper of... |
| lubval 18489 | Value of the least upper b... |
| lubcl 18490 | The least upper bound func... |
| lubprop 18491 | Properties of greatest low... |
| luble 18492 | The greatest lower bound i... |
| lublecllem 18493 | Lemma for ~ lublecl and ~ ... |
| lublecl 18494 | The set of all elements le... |
| lubid 18495 | The LUB of elements less t... |
| glbfval 18496 | Value of the greatest lowe... |
| glbdm 18497 | Domain of the greatest low... |
| glbfun 18498 | The GLB is a function. (C... |
| glbeldm 18499 | Member of the domain of th... |
| glbelss 18500 | A member of the domain of ... |
| glbeu 18501 | Unique existence proper of... |
| glbval 18502 | Value of the greatest lowe... |
| glbcl 18503 | The least upper bound func... |
| glbprop 18504 | Properties of greatest low... |
| glble 18505 | The greatest lower bound i... |
| joinfval 18506 | Value of join function for... |
| joinfval2 18507 | Value of join function for... |
| joindm 18508 | Domain of join function fo... |
| joindef 18509 | Two ways to say that a joi... |
| joinval 18510 | Join value. Since both si... |
| joincl 18511 | Closure of join of element... |
| joindmss 18512 | Subset property of domain ... |
| joinval2lem 18513 | Lemma for ~ joinval2 and ~... |
| joinval2 18514 | Value of join for a poset ... |
| joineu 18515 | Uniqueness of join of elem... |
| joinlem 18516 | Lemma for join properties.... |
| lejoin1 18517 | A join's first argument is... |
| lejoin2 18518 | A join's second argument i... |
| joinle 18519 | A join is less than or equ... |
| meetfval 18520 | Value of meet function for... |
| meetfval2 18521 | Value of meet function for... |
| meetdm 18522 | Domain of meet function fo... |
| meetdef 18523 | Two ways to say that a mee... |
| meetval 18524 | Meet value. Since both si... |
| meetcl 18525 | Closure of meet of element... |
| meetdmss 18526 | Subset property of domain ... |
| meetval2lem 18527 | Lemma for ~ meetval2 and ~... |
| meetval2 18528 | Value of meet for a poset ... |
| meeteu 18529 | Uniqueness of meet of elem... |
| meetlem 18530 | Lemma for meet properties.... |
| lemeet1 18531 | A meet's first argument is... |
| lemeet2 18532 | A meet's second argument i... |
| meetle 18533 | A meet is less than or equ... |
| joincomALT 18534 | The join of a poset is com... |
| joincom 18535 | The join of a poset is com... |
| meetcomALT 18536 | The meet of a poset is com... |
| meetcom 18537 | The meet of a poset is com... |
| join0 18538 | Lemma for ~ odumeet . (Co... |
| meet0 18539 | Lemma for ~ odujoin . (Co... |
| odulub 18540 | Least upper bounds in a du... |
| odujoin 18541 | Joins in a dual order are ... |
| oduglb 18542 | Greatest lower bounds in a... |
| odumeet 18543 | Meets in a dual order are ... |
| poslubmo 18544 | Least upper bounds in a po... |
| posglbmo 18545 | Greatest lower bounds in a... |
| poslubd 18546 | Properties which determine... |
| poslubdg 18547 | Properties which determine... |
| posglbdg 18548 | Properties which determine... |
| istos 18551 | The predicate "is a toset"... |
| tosso 18552 | Write the totally ordered ... |
| tospos 18553 | A Toset is a Poset. (Cont... |
| tleile 18554 | In a Toset, any two elemen... |
| tltnle 18555 | In a Toset, "less than" is... |
| p0val 18560 | Value of poset zero. (Con... |
| p1val 18561 | Value of poset zero. (Con... |
| p0le 18562 | Any element is less than o... |
| ple1 18563 | Any element is less than o... |
| resspos 18564 | The restriction of a Poset... |
| resstos 18565 | The restriction of a Toset... |
| islat 18568 | The predicate "is a lattic... |
| odulatb 18569 | Being a lattice is self-du... |
| odulat 18570 | Being a lattice is self-du... |
| latcl2 18571 | The join and meet of any t... |
| latlem 18572 | Lemma for lattice properti... |
| latpos 18573 | A lattice is a poset. (Co... |
| latjcl 18574 | Closure of join operation ... |
| latmcl 18575 | Closure of meet operation ... |
| latref 18576 | A lattice ordering is refl... |
| latasymb 18577 | A lattice ordering is asym... |
| latasym 18578 | A lattice ordering is asym... |
| lattr 18579 | A lattice ordering is tran... |
| latasymd 18580 | Deduce equality from latti... |
| lattrd 18581 | A lattice ordering is tran... |
| latjcom 18582 | The join of a lattice comm... |
| latlej1 18583 | A join's first argument is... |
| latlej2 18584 | A join's second argument i... |
| latjle12 18585 | A join is less than or equ... |
| latleeqj1 18586 | "Less than or equal to" in... |
| latleeqj2 18587 | "Less than or equal to" in... |
| latjlej1 18588 | Add join to both sides of ... |
| latjlej2 18589 | Add join to both sides of ... |
| latjlej12 18590 | Add join to both sides of ... |
| latnlej 18591 | An idiom to express that a... |
| latnlej1l 18592 | An idiom to express that a... |
| latnlej1r 18593 | An idiom to express that a... |
| latnlej2 18594 | An idiom to express that a... |
| latnlej2l 18595 | An idiom to express that a... |
| latnlej2r 18596 | An idiom to express that a... |
| latjidm 18597 | Lattice join is idempotent... |
| latmcom 18598 | The join of a lattice comm... |
| latmle1 18599 | A meet is less than or equ... |
| latmle2 18600 | A meet is less than or equ... |
| latlem12 18601 | An element is less than or... |
| latleeqm1 18602 | "Less than or equal to" in... |
| latleeqm2 18603 | "Less than or equal to" in... |
| latmlem1 18604 | Add meet to both sides of ... |
| latmlem2 18605 | Add meet to both sides of ... |
| latmlem12 18606 | Add join to both sides of ... |
| latnlemlt 18607 | Negation of "less than or ... |
| latnle 18608 | Equivalent expressions for... |
| latmidm 18609 | Lattice meet is idempotent... |
| latabs1 18610 | Lattice absorption law. F... |
| latabs2 18611 | Lattice absorption law. F... |
| latledi 18612 | An ortholattice is distrib... |
| latmlej11 18613 | Ordering of a meet and joi... |
| latmlej12 18614 | Ordering of a meet and joi... |
| latmlej21 18615 | Ordering of a meet and joi... |
| latmlej22 18616 | Ordering of a meet and joi... |
| lubsn 18617 | The least upper bound of a... |
| latjass 18618 | Lattice join is associativ... |
| latj12 18619 | Swap 1st and 2nd members o... |
| latj32 18620 | Swap 2nd and 3rd members o... |
| latj13 18621 | Swap 1st and 3rd members o... |
| latj31 18622 | Swap 2nd and 3rd members o... |
| latjrot 18623 | Rotate lattice join of 3 c... |
| latj4 18624 | Rearrangement of lattice j... |
| latj4rot 18625 | Rotate lattice join of 4 c... |
| latjjdi 18626 | Lattice join distributes o... |
| latjjdir 18627 | Lattice join distributes o... |
| mod1ile 18628 | The weak direction of the ... |
| mod2ile 18629 | The weak direction of the ... |
| latmass 18630 | Lattice meet is associativ... |
| latdisdlem 18631 | Lemma for ~ latdisd . (Co... |
| latdisd 18632 | In a lattice, joins distri... |
| isclat 18635 | The predicate "is a comple... |
| clatpos 18636 | A complete lattice is a po... |
| clatlem 18637 | Lemma for properties of a ... |
| clatlubcl 18638 | Any subset of the base set... |
| clatlubcl2 18639 | Any subset of the base set... |
| clatglbcl 18640 | Any subset of the base set... |
| clatglbcl2 18641 | Any subset of the base set... |
| oduclatb 18642 | Being a complete lattice i... |
| clatl 18643 | A complete lattice is a la... |
| isglbd 18644 | Properties that determine ... |
| lublem 18645 | Lemma for the least upper ... |
| lubub 18646 | The LUB of a complete latt... |
| lubl 18647 | The LUB of a complete latt... |
| lubss 18648 | Subset law for least upper... |
| lubel 18649 | An element of a set is les... |
| lubun 18650 | The LUB of a union. (Cont... |
| clatglb 18651 | Properties of greatest low... |
| clatglble 18652 | The greatest lower bound i... |
| clatleglb 18653 | Two ways of expressing "le... |
| clatglbss 18654 | Subset law for greatest lo... |
| isdlat 18657 | Property of being a distri... |
| dlatmjdi 18658 | In a distributive lattice,... |
| dlatl 18659 | A distributive lattice is ... |
| odudlatb 18660 | The dual of a distributive... |
| dlatjmdi 18661 | In a distributive lattice,... |
| ipostr 18664 | The structure of ~ df-ipo ... |
| ipoval 18665 | Value of the inclusion pos... |
| ipobas 18666 | Base set of the inclusion ... |
| ipolerval 18667 | Relation of the inclusion ... |
| ipotset 18668 | Topology of the inclusion ... |
| ipole 18669 | Weak order condition of th... |
| ipolt 18670 | Strict order condition of ... |
| ipopos 18671 | The inclusion poset on a f... |
| isipodrs 18672 | Condition for a family of ... |
| ipodrscl 18673 | Direction by inclusion as ... |
| ipodrsfi 18674 | Finite upper bound propert... |
| fpwipodrs 18675 | The finite subsets of any ... |
| ipodrsima 18676 | The monotone image of a di... |
| isacs3lem 18677 | An algebraic closure syste... |
| acsdrsel 18678 | An algebraic closure syste... |
| isacs4lem 18679 | In a closure system in whi... |
| isacs5lem 18680 | If closure commutes with d... |
| acsdrscl 18681 | In an algebraic closure sy... |
| acsficl 18682 | A closure in an algebraic ... |
| isacs5 18683 | A closure system is algebr... |
| isacs4 18684 | A closure system is algebr... |
| isacs3 18685 | A closure system is algebr... |
| acsficld 18686 | In an algebraic closure sy... |
| acsficl2d 18687 | In an algebraic closure sy... |
| acsfiindd 18688 | In an algebraic closure sy... |
| acsmapd 18689 | In an algebraic closure sy... |
| acsmap2d 18690 | In an algebraic closure sy... |
| acsinfd 18691 | In an algebraic closure sy... |
| acsdomd 18692 | In an algebraic closure sy... |
| acsinfdimd 18693 | In an algebraic closure sy... |
| acsexdimd 18694 | In an algebraic closure sy... |
| mrelatglb 18695 | Greatest lower bounds in a... |
| mrelatglb0 18696 | The empty intersection in ... |
| mrelatlub 18697 | Least upper bounds in a Mo... |
| mreclatBAD 18698 | A Moore space is a complet... |
| isps 18703 | The predicate "is a poset"... |
| psrel 18704 | A poset is a relation. (C... |
| psref2 18705 | A poset is antisymmetric a... |
| pstr2 18706 | A poset is transitive. (C... |
| pslem 18707 | Lemma for ~ psref and othe... |
| psdmrn 18708 | The domain and range of a ... |
| psref 18709 | A poset is reflexive. (Co... |
| psrn 18710 | The range of a poset equal... |
| psasym 18711 | A poset is antisymmetric. ... |
| pstr 18712 | A poset is transitive. (C... |
| cnvps 18713 | The converse of a poset is... |
| cnvpsb 18714 | The converse of a poset is... |
| psss 18715 | Any subset of a partially ... |
| psssdm2 18716 | Field of a subposet. (Con... |
| psssdm 18717 | Field of a subposet. (Con... |
| istsr 18718 | The predicate is a toset. ... |
| istsr2 18719 | The predicate is a toset. ... |
| tsrlin 18720 | A toset is a linear order.... |
| tsrlemax 18721 | Two ways of saying a numbe... |
| tsrps 18722 | A toset is a poset. (Cont... |
| cnvtsr 18723 | The converse of a toset is... |
| tsrss 18724 | Any subset of a totally or... |
| ledm 18725 | The domain of ` <_ ` is ` ... |
| lern 18726 | The range of ` <_ ` is ` R... |
| lefld 18727 | The field of the 'less or ... |
| letsr 18728 | The "less than or equal to... |
| isdir 18733 | A condition for a relation... |
| reldir 18734 | A direction is a relation.... |
| dirdm 18735 | A direction's domain is eq... |
| dirref 18736 | A direction is reflexive. ... |
| dirtr 18737 | A direction is transitive.... |
| dirge 18738 | For any two elements of a ... |
| tsrdir 18739 | A totally ordered set is a... |
| ischn 18742 | Property of being a chain.... |
| chnwrd 18743 | A chain is an ordered sequ... |
| chnltm1 18744 | Basic property of a chain.... |
| pfxchn 18745 | A prefix of a chain is sti... |
| nfchnd 18746 | Bound-variable hypothesis ... |
| chneq1 18747 | Equality theorem for chain... |
| chneq2 18748 | Equality theorem for chain... |
| chneq12 18749 | Equality theorem for chain... |
| chnrss 18750 | Chains under a relation ar... |
| chndss 18751 | Chains with an alphabet ar... |
| chnrdss 18752 | Subset theorem for chains.... |
| chnexg 18753 | Chains with a set given fo... |
| nulchn 18754 | Empty set is an increasing... |
| s1chn 18755 | A singleton word is always... |
| chnind 18756 | Induction over a chain. S... |
| chnub 18757 | In a chain, the last eleme... |
| chnlt 18758 | Compare any two elements i... |
| chnso 18759 | A chain induces a total or... |
| chnccats1 18760 | Extend a chain with a sing... |
| chnccat 18761 | Concatenate two chains. (... |
| chnrev 18762 | Reverse of a chain is chai... |
| chnflenfi 18763 | There is a finite number o... |
| chnf 18764 | A chain is a zero-based fi... |
| chnpof1 18765 | A chain under relation whi... |
| chnpoadomd 18766 | A chain under relation whi... |
| chnpolleha 18767 | A chain under relation whi... |
| chnpolfz 18768 | Provided that chain's rela... |
| chnfi 18769 | There is a finite number o... |
| chninf 18770 | There is an infinite numbe... |
| chnfibg 18771 | Given a partial order, the... |
| ex-chn1 18772 | Example: a doubleton of tw... |
| ex-chn2 18773 | Example: sequence <" ZZ NN... |
| ismgm 18778 | The predicate "is a magma"... |
| ismgmn0 18779 | The predicate "is a magma"... |
| mgmcl 18780 | Closure of the operation o... |
| isnmgm 18781 | A condition for a structur... |
| mgmsscl 18782 | If the base set of a magma... |
| plusffval 18783 | The group addition operati... |
| plusfval 18784 | The group addition operati... |
| plusfeq 18785 | If the addition operation ... |
| plusffn 18786 | The group addition operati... |
| mgmplusf 18787 | The group addition functio... |
| mgmn0plusgf 18788 | The restriction of the gro... |
| mgmn0plusgplusf 18789 | The group addition functio... |
| mgmpropd 18790 | If two structures have the... |
| ismgmd 18791 | Deduce a magma from its pr... |
| issstrmgm 18792 | Characterize a substructur... |
| intopsn 18793 | The internal operation for... |
| mgmb1mgm1 18794 | The only magma with a base... |
| mgm0 18795 | Any set with an empty base... |
| mgm0b 18796 | The structure with an empt... |
| mgm1 18797 | The structure with one ele... |
| opifismgm 18798 | A structure with a group a... |
| mgmidmo 18799 | A two-sided identity eleme... |
| mgmideud 18800 | Uniqueness of the left and... |
| grpidval 18801 | The value of the group ide... |
| idvalriota 18802 | The unique value of the gr... |
| grpidpropd 18803 | If two structures have the... |
| fn0g 18804 | The group identity extract... |
| 0g0 18805 | The identity element funct... |
| ismgmid 18806 | Conditions for a class to ... |
| mgmidcl 18807 | The identity element of a ... |
| mgmlrid 18808 | The identity element of a ... |
| 0gisid 18809 | In a structure with an ide... |
| ismgmid2 18810 | Show that a given element ... |
| lidrideqd 18811 | If there is a left and rig... |
| lidrididd 18812 | If there is a left and rig... |
| grpidd 18813 | Deduce the identity elemen... |
| mgmidsssn0 18814 | Property of the set of ide... |
| grpinvalem 18815 | Lemma for ~ grpinva . (Co... |
| grpinva 18816 | Deduce right inverse from ... |
| grprida 18817 | Deduce right identity from... |
| mgmidpfod 18818 | The operation of a magma w... |
| mgmidprnd 18819 | Range of an operation with... |
| mgmfod 18820 | The operation of a magma w... |
| idressidex0 18821 | The restriction of a struc... |
| idressidex 18822 | The restriction of a struc... |
| idressid 18823 | The restriction of a struc... |
| imasmgm2 18824 | The image structure of a u... |
| qusmgm 18825 | Prove that a quotient stru... |
| gsumvalx 18826 | Expand out the substitutio... |
| gsumval 18827 | Expand out the substitutio... |
| gsumpropd 18828 | The group sum depends only... |
| gsumpropd2lem 18829 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18830 | A stronger version of ~ gs... |
| gsummgmpropd 18831 | A stronger version of ~ gs... |
| gsumress 18832 | The group sum in a substru... |
| gsumval1 18833 | Value of the group sum ope... |
| gsum0 18834 | Value of the empty group s... |
| gsumval2a 18835 | Value of the group sum ope... |
| gsumval2 18836 | Value of the group sum ope... |
| gsumsplit1r 18837 | Splitting off the rightmos... |
| gsumprval 18838 | Value of the group sum ope... |
| gsumpr12val 18839 | Value of the group sum ope... |
| mgmhmrcl 18844 | Reverse closure of a magma... |
| submgmrcl 18845 | Reverse closure for submag... |
| ismgmhm 18846 | Property of a magma homomo... |
| mgmhmf 18847 | A magma homomorphism is a ... |
| mgmhmpropd 18848 | Magma homomorphism depends... |
| mgmhmlin 18849 | A magma homomorphism prese... |
| mgmhmf1o 18850 | A magma homomorphism is bi... |
| idmgmhm 18851 | The identity homomorphism ... |
| issubmgm 18852 | Expand definition of a sub... |
| issubmgm2 18853 | Submagmas are subsets that... |
| rabsubmgmd 18854 | Deduction for proving that... |
| submgmss 18855 | Submagmas are subsets of t... |
| submgmid 18856 | Every magma is trivially a... |
| submgmcl 18857 | Submagmas are closed under... |
| submgmmgm 18858 | Submagmas are themselves m... |
| submgmbas 18859 | The base set of a submagma... |
| subsubmgm 18860 | A submagma of a submagma i... |
| resmgmhm 18861 | Restriction of a magma hom... |
| resmgmhm2 18862 | One direction of ~ resmgmh... |
| resmgmhm2b 18863 | Restriction of the codomai... |
| mgmhmco 18864 | The composition of magma h... |
| mgmhmima 18865 | The homomorphic image of a... |
| mgmhmeql 18866 | The equalizer of two magma... |
| submgmacs 18867 | Submagmas are an algebraic... |
| issgrp 18870 | The predicate "is a semigr... |
| issgrpv 18871 | The predicate "is a semigr... |
| issgrpn0 18872 | The predicate "is a semigr... |
| isnsgrp 18873 | A condition for a structur... |
| sgrpmgm 18874 | A semigroup is a magma. (... |
| sgrpass 18875 | A semigroup operation is a... |
| sgrpcl 18876 | Closure of the operation o... |
| sgrp0 18877 | Any set with an empty base... |
| sgrp0b 18878 | The structure with an empt... |
| sgrp1 18879 | The structure with one ele... |
| issgrpd 18880 | Deduce a semigroup from it... |
| sgrppropd 18881 | If two structures are sets... |
| prdsplusgsgrpcl 18882 | Structure product pointwis... |
| prdssgrpd 18883 | The product of a family of... |
| ismnddef 18886 | The predicate "is a monoid... |
| ismnd 18887 | The predicate "is a monoid... |
| isnmnd 18888 | A condition for a structur... |
| sgrpidmnd 18889 | A semigroup with an identi... |
| mndsgrp 18890 | A monoid is a semigroup. ... |
| mndmgm 18891 | A monoid is a magma. (Con... |
| mndcl 18892 | Closure of the operation o... |
| mndass 18893 | A monoid operation is asso... |
| mndid 18894 | A monoid has a two-sided i... |
| mndideu 18895 | The two-sided identity ele... |
| mndideuOLD 18896 | Obsolete version of ~ mndi... |
| mnd32g 18897 | Commutative/associative la... |
| mnd12g 18898 | Commutative/associative la... |
| mnd4g 18899 | Commutative/associative la... |
| mndidcl 18900 | The identity element of a ... |
| mndbn0 18901 | The base set of a monoid i... |
| hashfinmndnn 18902 | A finite monoid has positi... |
| mndplusf 18903 | The group addition operati... |
| mndlrid 18904 | A monoid's identity elemen... |
| mndlid 18905 | The identity element of a ... |
| mndrid 18906 | The identity element of a ... |
| ismndd 18907 | Deduce a monoid from its p... |
| mndpfo 18908 | The addition operation of ... |
| mndfo 18909 | The addition operation of ... |
| mndpfoOLD 18910 | Obsolete version of ~ mndp... |
| mndfoOLD 18911 | Obsolete version of ~ mndf... |
| mndpropd 18912 | If two structures have the... |
| mndprop 18913 | If two structures have the... |
| issubmnd 18914 | Characterize a submonoid b... |
| ress0g 18915 | ` 0g ` is unaffected by re... |
| ress0gOLD 18916 | Obsolete version of ~ ress... |
| submnd0 18917 | The zero of a submonoid is... |
| submnd0OLD 18918 | Obsolete version of ~ subm... |
| mndinvmod 18919 | Uniqueness of an inverse e... |
| mndpsuppss 18920 | The support of a mapping o... |
| mndpsuppfi 18921 | The support of a mapping o... |
| mndpfsupp 18922 | A mapping of a scalar mult... |
| prdsplusgcl 18923 | Structure product pointwis... |
| prdsidlem 18924 | Characterization of identi... |
| prdsmndd 18925 | The product of a family of... |
| prds0g 18926 | The identity in a product ... |
| pwsmnd 18927 | The structure power of a m... |
| pws0g 18928 | The identity in a structur... |
| imasmnd2 18929 | The image structure of a m... |
| imasmnd 18930 | The image structure of a m... |
| imasmndf1 18931 | The image of a monoid unde... |
| xpsmnd 18932 | The binary product of mono... |
| xpsmnd0 18933 | The identity element of a ... |
| mnd1 18934 | The (smallest) structure r... |
| mnd1id 18935 | The singleton element of a... |
| qusmnd 18936 | Prove that a quotient stru... |
| ismhm 18941 | Property of a monoid homom... |
| ismhmd 18942 | Deduction version of ~ ism... |
| mhmrcl1 18943 | Reverse closure of a monoi... |
| mhmrcl2 18944 | Reverse closure of a monoi... |
| mhmf 18945 | A monoid homomorphism is a... |
| ismhm0 18946 | Property of a monoid homom... |
| mhmismgmhm 18947 | Each monoid homomorphism i... |
| mhmpropd 18948 | Monoid homomorphism depend... |
| mhmlin 18949 | A monoid homomorphism comm... |
| mhm0 18950 | A monoid homomorphism pres... |
| idmhm 18951 | The identity homomorphism ... |
| mhmf1o 18952 | A monoid homomorphism is b... |
| mndvcl 18953 | Tuple-wise additive closur... |
| mndvass 18954 | Tuple-wise associativity i... |
| mndvlid 18955 | Tuple-wise left identity i... |
| mndvrid 18956 | Tuple-wise right identity ... |
| mhmvlin 18957 | Tuple extension of monoid ... |
| submrcl 18958 | Reverse closure for submon... |
| issubm 18959 | Expand definition of a sub... |
| issubm2 18960 | Submonoids are subsets tha... |
| issubmndb 18961 | The submonoid predicate. ... |
| issubmd 18962 | Deduction for proving a su... |
| mndissubm 18963 | If the base set of a monoi... |
| resmndismnd 18964 | If the base set of a monoi... |
| submss 18965 | Submonoids are subsets of ... |
| submid 18966 | Every monoid is trivially ... |
| subm0cl 18967 | Submonoids contain zero. ... |
| submcl 18968 | Submonoids are closed unde... |
| submcld 18969 | Submonoids are closed unde... |
| submmnd 18970 | Submonoids are themselves ... |
| submbas 18971 | The base set of a submonoi... |
| subm0 18972 | Submonoids have the same i... |
| subsubm 18973 | A submonoid of a submonoid... |
| 0subm 18974 | The zero submonoid of an a... |
| insubm 18975 | The intersection of two su... |
| 0mhm 18976 | The constant zero linear f... |
| resmhm 18977 | Restriction of a monoid ho... |
| resmhm2 18978 | One direction of ~ resmhm2... |
| resmhm2b 18979 | Restriction of the codomai... |
| mhmco 18980 | The composition of monoid ... |
| mhmimalem 18981 | Lemma for ~ mhmima and sim... |
| mhmima 18982 | The homomorphic image of a... |
| mhmeql 18983 | The equalizer of two monoi... |
| submacs 18984 | Submonoids are an algebrai... |
| mndind 18985 | Induction in a monoid. In... |
| prdspjmhm 18986 | A projection from a produc... |
| pwspjmhm 18987 | A projection from a struct... |
| pwsdiagmhm 18988 | Diagonal monoid homomorphi... |
| pwsco1mhm 18989 | Right composition with a f... |
| pwsco2mhm 18990 | Left composition with a mo... |
| gsumvallem2 18991 | Lemma for properties of th... |
| gsumsubm 18992 | Evaluate a group sum in a ... |
| gsumz 18993 | Value of a group sum over ... |
| gsumwsubmcl 18994 | Closure of the composite i... |
| gsumws1 18995 | A singleton composite reco... |
| gsumwcl 18996 | Closure of the composite o... |
| gsumsgrpccat 18997 | Homomorphic property of no... |
| gsumccat 18998 | Homomorphic property of co... |
| gsumws2 18999 | Valuation of a pair in a m... |
| gsumccatsn 19000 | Homomorphic property of co... |
| gsumspl 19001 | The primary purpose of the... |
| gsumwmhm 19002 | Behavior of homomorphisms ... |
| gsumwspan 19003 | The submonoid generated by... |
| frmdval 19008 | Value of the free monoid c... |
| frmdbas 19009 | The base set of a free mon... |
| frmdelbas 19010 | An element of the base set... |
| frmdplusg 19011 | The monoid operation of a ... |
| frmdadd 19012 | Value of the monoid operat... |
| vrmdfval 19013 | The canonical injection fr... |
| vrmdval 19014 | The value of the generatin... |
| vrmdf 19015 | The mapping from the index... |
| frmdmnd 19016 | A free monoid is a monoid.... |
| frmd0 19017 | The identity of the free m... |
| frmdsssubm 19018 | The set of words taking va... |
| frmdgsum 19019 | Any word in a free monoid ... |
| frmdss2 19020 | A subset of generators is ... |
| frmdup1 19021 | Any assignment of the gene... |
| frmdup2 19022 | The evaluation map has the... |
| frmdup3lem 19023 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 19024 | Universal property of the ... |
| efmnd 19027 | The monoid of endofunction... |
| efmndbas 19028 | The base set of the monoid... |
| efmndbasabf 19029 | The base set of the monoid... |
| elefmndbas 19030 | Two ways of saying a funct... |
| elefmndbas2 19031 | Two ways of saying a funct... |
| efmndbasf 19032 | Elements in the monoid of ... |
| efmndhash 19033 | The monoid of endofunction... |
| efmndbasfi 19034 | The monoid of endofunction... |
| efmndfv 19035 | The function value of an e... |
| efmndtset 19036 | The topology of the monoid... |
| efmndplusg 19037 | The group operation of a m... |
| efmndov 19038 | The value of the group ope... |
| efmndcl 19039 | The group operation of the... |
| efmndtopn 19040 | The topology of the monoid... |
| symggrplem 19041 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 19042 | The monoid of endofunction... |
| efmndsgrp 19043 | The monoid of endofunction... |
| ielefmnd 19044 | The identity function rest... |
| efmndid 19045 | The identity function rest... |
| efmndmnd 19046 | The monoid of endofunction... |
| efmnd0nmnd 19047 | Even the monoid of endofun... |
| efmndbas0 19048 | The base set of the monoid... |
| efmnd1hash 19049 | The monoid of endofunction... |
| efmnd1bas 19050 | The monoid of endofunction... |
| efmnd2hash 19051 | The monoid of endofunction... |
| submefmnd 19052 | If the base set of a monoi... |
| sursubmefmnd 19053 | The set of surjective endo... |
| injsubmefmnd 19054 | The set of injective endof... |
| idressubmefmnd 19055 | The singleton containing o... |
| idresefmnd 19056 | The structure with the sin... |
| smndex1ibas 19057 | The modulo function ` I ` ... |
| smndex1iidm 19058 | The modulo function ` I ` ... |
| smndex1gbas 19059 | The constant functions ` (... |
| smndex1gbasOLD 19060 | Obsolete version of ~ smnd... |
| smndex1gid 19061 | The composition of a const... |
| smndex1gidOLD 19062 | Obsolete version of ~ smnd... |
| smndex1igid 19063 | The composition of the mod... |
| smndex1igidOLD 19064 | Obsolete version of ~ smnd... |
| smndex1basss 19065 | The modulo function ` I ` ... |
| smndex1bas 19066 | The base set of the monoid... |
| smndex1mgm 19067 | The monoid of endofunction... |
| smndex1sgrp 19068 | The monoid of endofunction... |
| smndex1mndlem 19069 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 19070 | The monoid of endofunction... |
| smndex1id 19071 | The modulo function ` I ` ... |
| smndex1n0mnd 19072 | The identity of the monoid... |
| nsmndex1 19073 | The base set ` B ` of the ... |
| smndex2dbas 19074 | The doubling function ` D ... |
| smndex2dnrinv 19075 | The doubling function ` D ... |
| smndex2hbas 19076 | The halving functions ` H ... |
| smndex2dlinvh 19077 | The halving functions ` H ... |
| mgm2nsgrplem1 19078 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 19079 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 19080 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 19081 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 19082 | A small magma (with two el... |
| sgrp2nmndlem1 19083 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 19084 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 19085 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 19086 | A small semigroup (with tw... |
| sgrp2rid2ex 19087 | A small semigroup (with tw... |
| sgrp2nmndlem4 19088 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 19089 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 19090 | A small semigroup (with tw... |
| mgmnsgrpex 19091 | There is a magma which is ... |
| sgrpnmndex 19092 | There is a semigroup which... |
| sgrpssmgm 19093 | The class of all semigroup... |
| mndsssgrp 19094 | The class of all monoids i... |
| degenmgmopdm 19095 | The domain of the operatio... |
| degenmgmbas 19096 | The base set of a degenera... |
| degenmgmnfn 19097 | The operation of a degener... |
| degenmgm 19098 | A degenerate magma: althou... |
| degenmgm2opdm 19099 | The domain of the operatio... |
| degenmgm2nfun 19100 | The operation of a second ... |
| degenmgm2 19101 | A degenerate magma: althou... |
| pwmndgplus 19102 | The operation of the monoi... |
| pwmndid 19103 | The identity of the monoid... |
| pwmnd 19104 | The power set of a class `... |
| isgrp 19111 | The predicate "is a group"... |
| grpmnd 19112 | A group is a monoid. (Con... |
| grpcl 19113 | Closure of the operation o... |
| grpass 19114 | A group operation is assoc... |
| grpinvex 19115 | Every member of a group ha... |
| grpideu 19116 | The two-sided identity ele... |
| grpassd 19117 | A group operation is assoc... |
| grpmndd 19118 | A group is a monoid. (Con... |
| grpcld 19119 | Closure of the operation o... |
| grpplusf 19120 | The group addition operati... |
| grpplusfo 19121 | The group addition operati... |
| resgrpplusfrn 19122 | The underlying set of a gr... |
| grppropd 19123 | If two structures have the... |
| grpprop 19124 | If two structures have the... |
| grppropstr 19125 | Generalize a specific 2-el... |
| grpss 19126 | Show that a structure exte... |
| isgrpd2e 19127 | Deduce a group from its pr... |
| isgrpd2 19128 | Deduce a group from its pr... |
| isgrpde 19129 | Deduce a group from its pr... |
| isgrpd 19130 | Deduce a group from its pr... |
| isgrpi 19131 | Properties that determine ... |
| grpsgrp 19132 | A group is a semigroup. (... |
| grpmgmd 19133 | A group is a magma, deduct... |
| dfgrp2 19134 | Alternate definition of a ... |
| dfgrp2e 19135 | Alternate definition of a ... |
| isgrpix 19136 | Properties that determine ... |
| grpidcl 19137 | The identity element of a ... |
| grpbn0 19138 | The base set of a group is... |
| grplid 19139 | The identity element of a ... |
| grprid 19140 | The identity element of a ... |
| grplidd 19141 | The identity element of a ... |
| grpridd 19142 | The identity element of a ... |
| grpn0 19143 | A group is not empty. (Co... |
| hashfingrpnn 19144 | A finite group has positiv... |
| grprcan 19145 | Right cancellation law for... |
| grpinveu 19146 | The left inverse element o... |
| grpid 19147 | Two ways of saying that an... |
| isgrpid2 19148 | Properties showing that an... |
| grpidd2 19149 | Deduce the identity elemen... |
| grpinvfval 19150 | The inverse function of a ... |
| grpinvfvalALT 19151 | Shorter proof of ~ grpinvf... |
| grpinvval 19152 | The inverse of a group ele... |
| grpinvfn 19153 | Functionality of the group... |
| grpinvfvi 19154 | The group inverse function... |
| grpsubfval 19155 | Group subtraction (divisio... |
| grpsubfvalALT 19156 | Shorter proof of ~ grpsubf... |
| grpsubval 19157 | Group subtraction (divisio... |
| grpinvf 19158 | The group inversion operat... |
| grpinvcl 19159 | A group element's inverse ... |
| grpinvcld 19160 | A group element's inverse ... |
| grplinv 19161 | The left inverse of a grou... |
| grprinv 19162 | The right inverse of a gro... |
| grpinvid1 19163 | The inverse of a group ele... |
| grpinvid2 19164 | The inverse of a group ele... |
| isgrpinv 19165 | Properties showing that a ... |
| grplinvd 19166 | The left inverse of a grou... |
| grprinvd 19167 | The right inverse of a gro... |
| grplrinv 19168 | In a group, every member h... |
| grpidinv2 19169 | A group's properties using... |
| grpidinv 19170 | A group has a left and rig... |
| grpinvid 19171 | The inverse of the identit... |
| grplcan 19172 | Left cancellation law for ... |
| grpasscan1 19173 | An associative cancellatio... |
| grpasscan2 19174 | An associative cancellatio... |
| grpidrcan 19175 | If right adding an element... |
| grpidlcan 19176 | If left adding an element ... |
| grpinvinv 19177 | Double inverse law for gro... |
| grpinvcnv 19178 | The group inverse is its o... |
| grpinv11 19179 | The group inverse is one-t... |
| grpinvf1o 19180 | The group inverse is a one... |
| grpinvnz 19181 | The inverse of a nonzero g... |
| grpinvnzcl 19182 | The inverse of a nonzero g... |
| grpsubinv 19183 | Subtraction of an inverse.... |
| grplmulf1o 19184 | Left multiplication by a g... |
| grpraddf1o 19185 | Right addition by a group ... |
| grpinvpropd 19186 | If two structures have the... |
| grpidssd 19187 | If the base set of a group... |
| grpinvssd 19188 | If the base set of a group... |
| grpinvadd 19189 | The inverse of the group o... |
| grpsubf 19190 | Functionality of group sub... |
| grpsubcl 19191 | Closure of group subtracti... |
| grpsubrcan 19192 | Right cancellation law for... |
| grpinvsub 19193 | Inverse of a group subtrac... |
| grpinvval2 19194 | A ~ df-neg -like equation ... |
| grpsubid 19195 | Subtraction of a group ele... |
| grpsubid1 19196 | Subtraction of the identit... |
| grpsubeq0 19197 | If the difference between ... |
| grpsubadd0sub 19198 | Subtraction expressed as a... |
| grpsubadd 19199 | Relationship between group... |
| grpsubsub 19200 | Double group subtraction. ... |
| grpaddsubass 19201 | Associative-type law for g... |
| grppncan 19202 | Cancellation law for subtr... |
| grpnpcan 19203 | Cancellation law for subtr... |
| grpsubsub4 19204 | Double group subtraction (... |
| grppnpcan2 19205 | Cancellation law for mixed... |
| grpnpncan 19206 | Cancellation law for group... |
| grpnpncan0 19207 | Cancellation law for group... |
| grpnnncan2 19208 | Cancellation law for group... |
| dfgrp3lem 19209 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19210 | Alternate definition of a ... |
| dfgrp3e 19211 | Alternate definition of a ... |
| grplactfval 19212 | The left group action of e... |
| grplactval 19213 | The value of the left grou... |
| grplactcnv 19214 | The left group action of e... |
| grplactf1o 19215 | The left group action of e... |
| grpsubpropd 19216 | Weak property deduction fo... |
| grpsubpropd2 19217 | Strong property deduction ... |
| grp1 19218 | The (smallest) structure r... |
| grp1inv 19219 | The inverse function of th... |
| prdsinvlem 19220 | Characterization of invers... |
| prdsgrpd 19221 | The product of a family of... |
| prdsinvgd 19222 | Negation in a product of g... |
| pwsgrp 19223 | A structure power of a gro... |
| pwsinvg 19224 | Negation in a structure po... |
| pwssub 19225 | Subtraction in a structure... |
| imasgrp2 19226 | The image structure of a g... |
| imasgrp 19227 | The image structure of a g... |
| imasgrpf1 19228 | The image of a group under... |
| qusgrp2 19229 | Prove that a quotient stru... |
| xpsgrp 19230 | The binary product of grou... |
| xpsinv 19231 | Value of the negation oper... |
| xpsgrpsub 19232 | Value of the subtraction o... |
| mhmlem 19233 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19234 | A surjective monoid morphi... |
| mhmmnd 19235 | The image of a monoid ` G ... |
| mhmfmhm 19236 | The function fulfilling th... |
| ghmgrp 19237 | The image of a group ` G `... |
| mulgfval 19240 | Group multiple (exponentia... |
| mulgfvalALT 19241 | Shorter proof of ~ mulgfva... |
| mulgval 19242 | Value of the group multipl... |
| mulgfn 19243 | Functionality of the group... |
| mulgfvi 19244 | The group multiple operati... |
| mulg0 19245 | Group multiple (exponentia... |
| mulgnn 19246 | Group multiple (exponentia... |
| ressmulgnn 19247 | Values for the group multi... |
| ressmulgnn0 19248 | Values for the group multi... |
| ressmulgnnd 19249 | Values for the group multi... |
| mulgnngsum 19250 | Group multiple (exponentia... |
| mulgnn0gsum 19251 | Group multiple (exponentia... |
| mulg1 19252 | Group multiple (exponentia... |
| mulgnnp1 19253 | Group multiple (exponentia... |
| mulg2 19254 | Group multiple (exponentia... |
| mulgnegnn 19255 | Group multiple (exponentia... |
| mulgnn0p1 19256 | Group multiple (exponentia... |
| mulgnnsubcl 19257 | Closure of the group multi... |
| mulgnn0subcl 19258 | Closure of the group multi... |
| mulgsubcl 19259 | Closure of the group multi... |
| mulgnncl 19260 | Closure of the group multi... |
| mulgnn0cl 19261 | Closure of the group multi... |
| mulgcl 19262 | Closure of the group multi... |
| mulgneg 19263 | Group multiple (exponentia... |
| mulgnegneg 19264 | The inverse of a negative ... |
| mulgm1 19265 | Group multiple (exponentia... |
| mulgnn0cld 19266 | Closure of the group multi... |
| mulgcld 19267 | Deduction associated with ... |
| mulgaddcomlem 19268 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19269 | The group multiple operato... |
| mulginvcom 19270 | The group multiple operato... |
| mulginvinv 19271 | The group multiple operato... |
| mulgnn0z 19272 | A group multiple of the id... |
| mulgz 19273 | A group multiple of the id... |
| mulgnndir 19274 | Sum of group multiples, fo... |
| mulgnn0dir 19275 | Sum of group multiples, ge... |
| mulgdirlem 19276 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19277 | Sum of group multiples, ge... |
| mulgp1 19278 | Group multiple (exponentia... |
| mulgneg2 19279 | Group multiple (exponentia... |
| mulgnnass 19280 | Product of group multiples... |
| mulgnn0ass 19281 | Product of group multiples... |
| mulgass 19282 | Product of group multiples... |
| mulgassr 19283 | Reversed product of group ... |
| mulgmodid 19284 | Casting out multiples of t... |
| mulgsubdir 19285 | Distribution of group mult... |
| mhmmulg 19286 | A homomorphism of monoids ... |
| mulgpropd 19287 | Two structures with the sa... |
| submmulgcl 19288 | Closure of the group multi... |
| submmulg 19289 | A group multiple is the sa... |
| pwsmulg 19290 | Value of a group multiple ... |
| issubg 19297 | The subgroup predicate. (... |
| subgss 19298 | A subgroup is a subset. (... |
| subgid 19299 | A group is a subgroup of i... |
| subggrp 19300 | A subgroup is a group. (C... |
| subgbas 19301 | The base of the restricted... |
| subgrcl 19302 | Reverse closure for the su... |
| subg0 19303 | A subgroup of a group must... |
| subginv 19304 | The inverse of an element ... |
| subg0cl 19305 | The group identity is an e... |
| subginvcl 19306 | The inverse of an element ... |
| subgcl 19307 | A subgroup is closed under... |
| subgcld 19308 | A subgroup is closed under... |
| subgsubcl 19309 | A subgroup is closed under... |
| subgsub 19310 | The subtraction of element... |
| subgmulgcl 19311 | Closure of the group multi... |
| subgmulg 19312 | A group multiple is the sa... |
| issubg2 19313 | Characterize the subgroups... |
| issubgrpd2 19314 | Prove a subgroup by closur... |
| issubgrpd 19315 | Prove a subgroup by closur... |
| issubg3 19316 | A subgroup is a symmetric ... |
| issubg4 19317 | A subgroup is a nonempty s... |
| grpissubg 19318 | If the base set of a group... |
| resgrpisgrp 19319 | If the base set of a group... |
| subgsubm 19320 | A subgroup is a submonoid.... |
| subsubg 19321 | A subgroup of a subgroup i... |
| subgint 19322 | The intersection of a none... |
| 0subg 19323 | The zero subgroup of an ar... |
| trivsubgd 19324 | The only subgroup of a tri... |
| trivsubgsnd 19325 | The only subgroup of a tri... |
| isnsg 19326 | Property of being a normal... |
| isnsg2 19327 | Weaken the condition of ~ ... |
| nsgbi 19328 | Defining property of a nor... |
| nsgsubg 19329 | A normal subgroup is a sub... |
| nsgconj 19330 | The conjugation of an elem... |
| isnsg3 19331 | A subgroup is normal iff t... |
| subgacs 19332 | Subgroups are an algebraic... |
| nsgacs 19333 | Normal subgroups form an a... |
| elnmz 19334 | Elementhood in the normali... |
| nmzbi 19335 | Defining property of the n... |
| nmzsubg 19336 | The normalizer N_G(S) of a... |
| ssnmz 19337 | A subgroup is a subset of ... |
| isnsg4 19338 | A subgroup is normal iff i... |
| nmznsg 19339 | Any subgroup is a normal s... |
| 0nsg 19340 | The zero subgroup is norma... |
| nsgid 19341 | The whole group is a norma... |
| 0idnsgd 19342 | The whole group and the ze... |
| trivnsgd 19343 | The only normal subgroup o... |
| triv1nsgd 19344 | A trivial group has exactl... |
| 1nsgtrivd 19345 | A group with exactly one n... |
| releqg 19346 | The left coset equivalence... |
| ecxpid 19347 | The equivalence class of a... |
| qsxpid 19348 | The quotient set of a cart... |
| eqgfval 19349 | Value of the subgroup left... |
| eqgval 19350 | Value of the subgroup left... |
| eqger 19351 | The subgroup coset equival... |
| eqglact 19352 | A left coset can be expres... |
| eqgid 19353 | The left coset containing ... |
| eqgen 19354 | Each coset is equipotent t... |
| eqgcpbl 19355 | The subgroup coset equival... |
| qusxpid 19356 | The Group quotient equival... |
| qustriv 19357 | The quotient of a group ` ... |
| qustrivr 19358 | Converse of ~ qustriv . (... |
| eqg0el 19359 | Equivalence class of a quo... |
| quselbas 19360 | Membership in the base set... |
| quseccl0 19361 | Closure of the quotient ma... |
| qusgrp 19362 | If ` Y ` is a normal subgr... |
| quseccl 19363 | Closure of the quotient ma... |
| qusadd 19364 | Value of the group operati... |
| qus0 19365 | Value of the group identit... |
| qusinv 19366 | Value of the group inverse... |
| qussub 19367 | Value of the group subtrac... |
| ecqusaddd 19368 | Addition of equivalence cl... |
| ecqusaddcl 19369 | Closure of the addition in... |
| lagsubg2 19370 | Lagrange's theorem for fin... |
| lagsubg 19371 | Lagrange's theorem for Gro... |
| eqg0subg 19372 | The coset equivalence rela... |
| eqg0subgecsn 19373 | The equivalence classes mo... |
| qus0subgbas 19374 | The base set of a quotient... |
| qus0subgadd 19375 | The addition in a quotient... |
| cycsubmel 19376 | Characterization of an ele... |
| cycsubmcl 19377 | The set of nonnegative int... |
| cycsubm 19378 | The set of nonnegative int... |
| cyccom 19379 | Condition for an operation... |
| cycsubmcom 19380 | The operation of a monoid ... |
| cycsubggend 19381 | The cyclic subgroup genera... |
| cycsubgcl 19382 | The set of integer powers ... |
| cycsubgss 19383 | The cyclic subgroup genera... |
| cycsubg 19384 | The cyclic group generated... |
| cycsubgcld 19385 | The cyclic subgroup genera... |
| cycsubg2 19386 | The subgroup generated by ... |
| cycsubg2cl 19387 | Any multiple of an element... |
| reldmghm 19390 | Lemma for group homomorphi... |
| isghm 19391 | Property of being a homomo... |
| isghm3 19392 | Property of a group homomo... |
| ghmgrp1 19393 | A group homomorphism is on... |
| ghmgrp2 19394 | A group homomorphism is on... |
| ghmf 19395 | A group homomorphism is a ... |
| ghmlin 19396 | A homomorphism of groups i... |
| ghmid 19397 | A homomorphism of groups p... |
| ghminv 19398 | A homomorphism of groups p... |
| ghmsub 19399 | Linearity of subtraction t... |
| isghmd 19400 | Deduction for a group homo... |
| ghmmhm 19401 | A group homomorphism is a ... |
| ghmmhmb 19402 | Group homomorphisms and mo... |
| ghmmulg 19403 | A group homomorphism prese... |
| ghmrn 19404 | The range of a homomorphis... |
| 0ghm 19405 | The constant zero linear f... |
| idghm 19406 | The identity homomorphism ... |
| resghm 19407 | Restriction of a homomorph... |
| resghm2 19408 | One direction of ~ resghm2... |
| resghm2b 19409 | Restriction of the codomai... |
| ghmghmrn 19410 | A group homomorphism from ... |
| ghmco 19411 | The composition of group h... |
| ghmima 19412 | The image of a subgroup un... |
| ghmpreima 19413 | The inverse image of a sub... |
| ghmeql 19414 | The equalizer of two group... |
| ghmnsgima 19415 | The image of a normal subg... |
| ghmnsgpreima 19416 | The inverse image of a nor... |
| ghmker 19417 | The kernel of a homomorphi... |
| ghmeqker 19418 | Two source points map to t... |
| pwsdiagghm 19419 | Diagonal homomorphism into... |
| f1ghm0to0 19420 | If a group homomorphism ` ... |
| ghmf1 19421 | Two ways of saying a group... |
| kerf1ghm 19422 | A group homomorphism ` F `... |
| ghmf1o 19423 | A bijective group homomorp... |
| conjghm 19424 | Conjugation is an automorp... |
| conjsubg 19425 | A conjugated subgroup is a... |
| conjsubgen 19426 | A conjugated subgroup is e... |
| conjnmz 19427 | A subgroup is unchanged un... |
| conjnmzb 19428 | Alternative condition for ... |
| conjnsg 19429 | A normal subgroup is uncha... |
| qusghm 19430 | If ` Y ` is a normal subgr... |
| ghmpropd 19431 | Group homomorphism depends... |
| gimfn 19436 | The group isomorphism func... |
| isgim 19437 | An isomorphism of groups i... |
| gimf1o 19438 | An isomorphism of groups i... |
| gimghm 19439 | An isomorphism of groups i... |
| isgim2 19440 | A group isomorphism is a h... |
| subggim 19441 | Behavior of subgroups unde... |
| gimcnv 19442 | The converse of a group is... |
| gimco 19443 | The composition of group i... |
| gim0to0 19444 | A group isomorphism maps t... |
| brgic 19445 | The relation "is isomorphi... |
| brgici 19446 | Prove isomorphic by an exp... |
| gicref 19447 | Isomorphism is reflexive. ... |
| giclcl 19448 | Isomorphism implies the le... |
| gicrcl 19449 | Isomorphism implies the ri... |
| gicsym 19450 | Isomorphism is symmetric. ... |
| gictr 19451 | Isomorphism is transitive.... |
| gicer 19452 | Isomorphism is an equivale... |
| gicen 19453 | Isomorphic groups have equ... |
| gicsubgen 19454 | A less trivial example of ... |
| ghmqusnsglem1 19455 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19456 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19457 | The mapping ` H ` induced ... |
| ghmquskerlem1 19458 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19459 | In the case of theorem ~ g... |
| ghmquskerlem2 19460 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19461 | The mapping ` H ` induced ... |
| ghmqusker 19462 | A surjective group homomor... |
| gicqusker 19463 | The image ` H ` of a group... |
| isga 19466 | The predicate "is a (left)... |
| gagrp 19467 | The left argument of a gro... |
| gaset 19468 | The right argument of a gr... |
| gagrpid 19469 | The identity of the group ... |
| gaf 19470 | The mapping of the group a... |
| gafo 19471 | A group action is onto its... |
| gaass 19472 | An "associative" property ... |
| ga0 19473 | The action of a group on t... |
| gaid 19474 | The trivial action of a gr... |
| subgga 19475 | A subgroup acts on its par... |
| gass 19476 | A subset of a group action... |
| gasubg 19477 | The restriction of a group... |
| gaid2 19478 | A group operation is a lef... |
| galcan 19479 | The action of a particular... |
| gacan 19480 | Group inverses cancel in a... |
| gapm 19481 | The action of a particular... |
| gaorb 19482 | The orbit equivalence rela... |
| gaorber 19483 | The orbit equivalence rela... |
| gastacl 19484 | The stabilizer subgroup in... |
| gastacos 19485 | Write the coset relation f... |
| orbstafun 19486 | Existence and uniqueness f... |
| orbstaval 19487 | Value of the function at a... |
| orbsta 19488 | The Orbit-Stabilizer theor... |
| orbsta2 19489 | Relation between the size ... |
| cntrval 19494 | Substitute definition of t... |
| cntzfval 19495 | First level substitution f... |
| cntzval 19496 | Definition substitution fo... |
| elcntz 19497 | Elementhood in the central... |
| cntzel 19498 | Membership in a centralize... |
| cntzsnval 19499 | Special substitution for t... |
| elcntzsn 19500 | Value of the centralizer o... |
| sscntz 19501 | A centralizer expression f... |
| cntzrcl 19502 | Reverse closure for elemen... |
| cntzssv 19503 | The centralizer is uncondi... |
| cntzi 19504 | Membership in a centralize... |
| elcntr 19505 | Elementhood in the center ... |
| cntrss 19506 | The center is a subset of ... |
| cntri 19507 | Defining property of the c... |
| resscntz 19508 | Centralizer in a substruct... |
| cntzsgrpcl 19509 | Centralizers are closed un... |
| cntz2ss 19510 | Centralizers reverse the s... |
| cntzrec 19511 | Reciprocity relationship f... |
| cntziinsn 19512 | Express any centralizer as... |
| cntzsubm 19513 | Centralizers in a monoid a... |
| cntzsubg 19514 | Centralizers in a group ar... |
| cntzidss 19515 | If the elements of ` S ` c... |
| cntzmhm 19516 | Centralizers in a monoid a... |
| cntzmhm2 19517 | Centralizers in a monoid a... |
| cntrsubgnsg 19518 | A central subgroup is norm... |
| cntrnsg 19519 | The center of a group is a... |
| oppgval 19522 | Value of the opposite grou... |
| oppgplusfval 19523 | Value of the addition oper... |
| oppgplus 19524 | Value of the addition oper... |
| setsplusg 19525 | The other components of an... |
| oppgbas 19526 | Base set of an opposite gr... |
| oppgtset 19527 | Topology of an opposite gr... |
| oppgtopn 19528 | Topology of an opposite gr... |
| oppgmnd 19529 | The opposite of a monoid i... |
| oppgmndb 19530 | Bidirectional form of ~ op... |
| oppgid 19531 | Zero in a monoid is a symm... |
| oppggrp 19532 | The opposite of a group is... |
| oppggrpb 19533 | Bidirectional form of ~ op... |
| oppginv 19534 | Inverses in a group are a ... |
| invoppggim 19535 | The inverse is an antiauto... |
| oppggic 19536 | Every group is (naturally)... |
| oppgsubm 19537 | Being a submonoid is a sym... |
| oppgsubg 19538 | Being a subgroup is a symm... |
| oppgcntz 19539 | A centralizer in a group i... |
| oppgcntr 19540 | The center of a group is t... |
| gsumwrev 19541 | A sum in an opposite monoi... |
| oppgle 19542 | less-than relation of an o... |
| oppglt 19543 | less-than relation of an o... |
| symgval 19546 | The value of the symmetric... |
| symgbas 19547 | The base set of the symmet... |
| elsymgbas2 19548 | Two ways of saying a funct... |
| elsymgbas 19549 | Two ways of saying a funct... |
| symgbasf1o 19550 | Elements in the symmetric ... |
| symgbasf 19551 | A permutation (element of ... |
| symgbasmap 19552 | A permutation (element of ... |
| symghash 19553 | The symmetric group on ` n... |
| symgbasfi 19554 | The symmetric group on a f... |
| symgfv 19555 | The function value of a pe... |
| symgfvne 19556 | The function values of a p... |
| symgressbas 19557 | The symmetric group on ` A... |
| symgplusg 19558 | The group operation of a s... |
| symgov 19559 | The value of the group ope... |
| symgcl 19560 | The group operation of the... |
| idresperm 19561 | The identity function rest... |
| symgmov1 19562 | For a permutation of a set... |
| symgmov2 19563 | For a permutation of a set... |
| symgbas0 19564 | The base set of the symmet... |
| symg1hash 19565 | The symmetric group on a s... |
| symg1bas 19566 | The symmetric group on a s... |
| symg2hash 19567 | The symmetric group on a (... |
| symg2bas 19568 | The symmetric group on a p... |
| 0symgefmndeq 19569 | The symmetric group on the... |
| snsymgefmndeq 19570 | The symmetric group on a s... |
| symgpssefmnd 19571 | For a set ` A ` with more ... |
| symgvalstruct 19572 | The value of the symmetric... |
| symgsubmefmnd 19573 | The symmetric group on a s... |
| symgtset 19574 | The topology of the symmet... |
| symggrp 19575 | The symmetric group on a s... |
| symgid 19576 | The group identity element... |
| symginv 19577 | The group inverse in the s... |
| symgsubmefmndALT 19578 | The symmetric group on a s... |
| galactghm 19579 | The currying of a group ac... |
| lactghmga 19580 | The converse of ~ galactgh... |
| symgtopn 19581 | The topology of the symmet... |
| symgga 19582 | The symmetric group induce... |
| pgrpsubgsymgbi 19583 | Every permutation group is... |
| pgrpsubgsymg 19584 | Every permutation group is... |
| idressubgsymg 19585 | The singleton containing o... |
| idrespermg 19586 | The structure with the sin... |
| cayleylem1 19587 | Lemma for ~ cayley . (Con... |
| cayleylem2 19588 | Lemma for ~ cayley . (Con... |
| cayley 19589 | Cayley's Theorem (construc... |
| cayleyth 19590 | Cayley's Theorem (existenc... |
| symgfix2 19591 | If a permutation does not ... |
| symgextf 19592 | The extension of a permuta... |
| symgextfv 19593 | The function value of the ... |
| symgextfve 19594 | The function value of the ... |
| symgextf1lem 19595 | Lemma for ~ symgextf1 . (... |
| symgextf1 19596 | The extension of a permuta... |
| symgextfo 19597 | The extension of a permuta... |
| symgextf1o 19598 | The extension of a permuta... |
| symgextsymg 19599 | The extension of a permuta... |
| symgextres 19600 | The restriction of the ext... |
| gsumccatsymgsn 19601 | Homomorphic property of co... |
| gsmsymgrfixlem1 19602 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19603 | The composition of permuta... |
| fvcosymgeq 19604 | The values of two composit... |
| gsmsymgreqlem1 19605 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19606 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19607 | Two combination of permuta... |
| symgfixelq 19608 | A permutation of a set fix... |
| symgfixels 19609 | The restriction of a permu... |
| symgfixelsi 19610 | The restriction of a permu... |
| symgfixf 19611 | The mapping of a permutati... |
| symgfixf1 19612 | The mapping of a permutati... |
| symgfixfolem1 19613 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19614 | The mapping of a permutati... |
| symgfixf1o 19615 | The mapping of a permutati... |
| f1omvdmvd 19618 | A permutation of any class... |
| f1omvdcnv 19619 | A permutation and its inve... |
| mvdco 19620 | Composing two permutations... |
| f1omvdconj 19621 | Conjugation of a permutati... |
| f1otrspeq 19622 | A transposition is charact... |
| f1omvdco2 19623 | If exactly one of two perm... |
| f1omvdco3 19624 | If a point is moved by exa... |
| pmtrfval 19625 | The function generating tr... |
| pmtrval 19626 | A generated transposition,... |
| pmtrfv 19627 | General value of mapping a... |
| pmtrprfv 19628 | In a transposition of two ... |
| pmtrprfv3 19629 | In a transposition of two ... |
| pmtrf 19630 | Functionality of a transpo... |
| pmtrmvd 19631 | A transposition moves prec... |
| pmtrrn 19632 | Transposing two points giv... |
| pmtrfrn 19633 | A transposition (as a kind... |
| pmtrffv 19634 | Mapping of a point under a... |
| pmtrrn2 19635 | For any transposition ther... |
| pmtrfinv 19636 | A transposition function i... |
| pmtrfmvdn0 19637 | A transposition moves at l... |
| pmtrff1o 19638 | A transposition function i... |
| pmtrfcnv 19639 | A transposition function i... |
| pmtrfb 19640 | An intrinsic characterizat... |
| pmtrfconj 19641 | Any conjugate of a transpo... |
| symgsssg 19642 | The symmetric group has su... |
| symgfisg 19643 | The symmetric group has a ... |
| symgtrf 19644 | Transpositions are element... |
| symggen 19645 | The span of the transposit... |
| symggen2 19646 | A finite permutation group... |
| symgtrinv 19647 | To invert a permutation re... |
| pmtr3ncomlem1 19648 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19649 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19650 | Transpositions over sets w... |
| pmtrdifellem1 19651 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19652 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19653 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19654 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19655 | A transposition of element... |
| pmtrdifwrdellem1 19656 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19657 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19658 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19659 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19660 | A sequence of transpositio... |
| pmtrdifwrdel2 19661 | A sequence of transpositio... |
| pmtrprfval 19662 | The transpositions on a pa... |
| pmtrprfvalrn 19663 | The range of the transposi... |
| psgnunilem1 19668 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19669 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19670 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19671 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19672 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19673 | Addition and subtraction o... |
| psgnuni 19674 | If the same permutation ca... |
| psgnfval 19675 | Function definition of the... |
| psgnfn 19676 | Functionality and domain o... |
| psgndmsubg 19677 | The finitary permutations ... |
| psgneldm 19678 | Property of being a finita... |
| psgneldm2 19679 | The finitary permutations ... |
| psgneldm2i 19680 | A sequence of transpositio... |
| psgneu 19681 | A finitary permutation has... |
| psgnval 19682 | Value of the permutation s... |
| psgnvali 19683 | A finitary permutation has... |
| psgnvalii 19684 | Any representation of a pe... |
| psgnpmtr 19685 | All transpositions are odd... |
| psgn0fv0 19686 | The permutation sign funct... |
| sygbasnfpfi 19687 | The class of non-fixed poi... |
| psgnfvalfi 19688 | Function definition of the... |
| psgnvalfi 19689 | Value of the permutation s... |
| psgnran 19690 | The range of the permutati... |
| gsmtrcl 19691 | The group sum of transposi... |
| psgnfitr 19692 | A permutation of a finite ... |
| psgnfieu 19693 | A permutation of a finite ... |
| pmtrsn 19694 | The value of the transposi... |
| psgnsn 19695 | The permutation sign funct... |
| psgnprfval 19696 | The permutation sign funct... |
| psgnprfval1 19697 | The permutation sign of th... |
| psgnprfval2 19698 | The permutation sign of th... |
| odfval 19707 | Value of the order functio... |
| odfvalALT 19708 | Shorter proof of ~ odfval ... |
| odval 19709 | Second substitution for th... |
| odlem1 19710 | The group element order is... |
| odcl 19711 | The order of a group eleme... |
| odf 19712 | Functionality of the group... |
| odid 19713 | Any element to the power o... |
| odlem2 19714 | Any positive annihilator o... |
| odmodnn0 19715 | Reduce the argument of a g... |
| mndodconglem 19716 | Lemma for ~ mndodcong . (... |
| mndodcong 19717 | If two multipliers are con... |
| mndodcongi 19718 | If two multipliers are con... |
| oddvdsnn0 19719 | The only multiples of ` A ... |
| odnncl 19720 | If a nonzero multiple of a... |
| odmod 19721 | Reduce the argument of a g... |
| oddvds 19722 | The only multiples of ` A ... |
| oddvdsi 19723 | Any group element is annih... |
| odcong 19724 | If two multipliers are con... |
| odeq 19725 | The ~ oddvds property uniq... |
| odval2 19726 | A non-conditional definiti... |
| odcld 19727 | The order of a group eleme... |
| odm1inv 19728 | The (order-1)th multiple o... |
| odmulgid 19729 | A relationship between the... |
| odmulg2 19730 | The order of a multiple di... |
| odmulg 19731 | Relationship between the o... |
| odmulgeq 19732 | A multiple of a point of f... |
| odbezout 19733 | If ` N ` is coprime to the... |
| od1 19734 | The order of the group ide... |
| odeq1 19735 | The group identity is the ... |
| odinv 19736 | The order of the inverse o... |
| odf1 19737 | The multiples of an elemen... |
| odinf 19738 | The multiples of an elemen... |
| dfod2 19739 | An alternative definition ... |
| odcl2 19740 | The order of an element of... |
| oddvds2 19741 | The order of an element of... |
| finodsubmsubg 19742 | A submonoid whose elements... |
| 0subgALT 19743 | A shorter proof of ~ 0subg... |
| submod 19744 | The order of an element is... |
| subgod 19745 | The order of an element is... |
| odsubdvds 19746 | The order of an element of... |
| odf1o1 19747 | An element with zero order... |
| odf1o2 19748 | An element with nonzero or... |
| odhash 19749 | An element of zero order g... |
| odhash2 19750 | If an element has nonzero ... |
| odhash3 19751 | An element which generates... |
| odngen 19752 | A cyclic subgroup of size ... |
| gexval 19753 | Value of the exponent of a... |
| gexlem1 19754 | The group element order is... |
| gexcl 19755 | The exponent of a group is... |
| gexid 19756 | Any element to the power o... |
| gexlem2 19757 | Any positive annihilator o... |
| gexdvdsi 19758 | Any group element is annih... |
| gexdvds 19759 | The only ` N ` that annihi... |
| gexdvds2 19760 | An integer divides the gro... |
| gexod 19761 | Any group element is annih... |
| gexcl3 19762 | If the order of every grou... |
| gexnnod 19763 | Every group element has fi... |
| gexcl2 19764 | The exponent of a finite g... |
| gexdvds3 19765 | The exponent of a finite g... |
| gex1 19766 | A group or monoid has expo... |
| ispgp 19767 | A group is a ` P ` -group ... |
| pgpprm 19768 | Reverse closure for the fi... |
| pgpgrp 19769 | Reverse closure for the se... |
| pgpfi1 19770 | A finite group with order ... |
| pgp0 19771 | The identity subgroup is a... |
| subgpgp 19772 | A subgroup of a p-group is... |
| sylow1lem1 19773 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19774 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19775 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19776 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19777 | Lemma for ~ sylow1 . Usin... |
| sylow1 19778 | Sylow's first theorem. If... |
| odcau 19779 | Cauchy's theorem for the o... |
| pgpfi 19780 | The converse to ~ pgpfi1 .... |
| pgpfi2 19781 | Alternate version of ~ pgp... |
| pgphash 19782 | The order of a p-group. (... |
| isslw 19783 | The property of being a Sy... |
| slwprm 19784 | Reverse closure for the fi... |
| slwsubg 19785 | A Sylow ` P ` -subgroup is... |
| slwispgp 19786 | Defining property of a Syl... |
| slwpss 19787 | A proper superset of a Syl... |
| slwpgp 19788 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19789 | Every ` P ` -subgroup is c... |
| slwn0 19790 | Every finite group contain... |
| subgslw 19791 | A Sylow subgroup that is c... |
| sylow2alem1 19792 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19793 | Lemma for ~ sylow2a . All... |
| sylow2a 19794 | A named lemma of Sylow's s... |
| sylow2blem1 19795 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19796 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19797 | Sylow's second theorem. P... |
| sylow2b 19798 | Sylow's second theorem. A... |
| slwhash 19799 | A sylow subgroup has cardi... |
| fislw 19800 | The sylow subgroups of a f... |
| sylow2 19801 | Sylow's second theorem. S... |
| sylow3lem1 19802 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19803 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19804 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19805 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19806 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19807 | Lemma for ~ sylow3 , secon... |
| sylow3 19808 | Sylow's third theorem. Th... |
| lsmfval 19813 | The subgroup sum function ... |
| lsmvalx 19814 | Subspace sum value (for a ... |
| lsmelvalx 19815 | Subspace sum membership (f... |
| lsmelvalix 19816 | Subspace sum membership (f... |
| oppglsm 19817 | The subspace sum operation... |
| lsmssv 19818 | Subgroup sum is a subset o... |
| lsmless1x 19819 | Subset implies subgroup su... |
| lsmless2x 19820 | Subset implies subgroup su... |
| lsmub1x 19821 | Subgroup sum is an upper b... |
| lsmub2x 19822 | Subgroup sum is an upper b... |
| lsmval 19823 | Subgroup sum value (for a ... |
| lsmelval 19824 | Subgroup sum membership (f... |
| lsmelvali 19825 | Subgroup sum membership (f... |
| lsmelvalm 19826 | Subgroup sum membership an... |
| lsmelvalmi 19827 | Membership of vector subtr... |
| lsmsubm 19828 | The sum of two commuting s... |
| lsmsubg 19829 | The sum of two commuting s... |
| lsmcom2 19830 | Subgroup sum commutes. (C... |
| smndlsmidm 19831 | The direct product is idem... |
| lsmub1 19832 | Subgroup sum is an upper b... |
| lsmub2 19833 | Subgroup sum is an upper b... |
| lsmunss 19834 | Union of subgroups is a su... |
| lsmless1 19835 | Subset implies subgroup su... |
| lsmless2 19836 | Subset implies subgroup su... |
| lsmless12 19837 | Subset implies subgroup su... |
| lsmidm 19838 | Subgroup sum is idempotent... |
| lsmlub 19839 | The least upper bound prop... |
| lsmss1 19840 | Subgroup sum with a subset... |
| lsmss1b 19841 | Subgroup sum with a subset... |
| lsmss2 19842 | Subgroup sum with a subset... |
| lsmss2b 19843 | Subgroup sum with a subset... |
| lsmass 19844 | Subgroup sum is associativ... |
| mndlsmidm 19845 | Subgroup sum is idempotent... |
| lsm01 19846 | Subgroup sum with the zero... |
| lsm02 19847 | Subgroup sum with the zero... |
| subglsm 19848 | The subgroup sum evaluated... |
| lssnle 19849 | Equivalent expressions for... |
| lsmmod 19850 | The modular law holds for ... |
| lsmmod2 19851 | Modular law dual for subgr... |
| lsmpropd 19852 | If two structures have the... |
| cntzrecd 19853 | Commute the "subgroups com... |
| lsmcntz 19854 | The "subgroups commute" pr... |
| lsmcntzr 19855 | The "subgroups commute" pr... |
| lsmdisj 19856 | Disjointness from a subgro... |
| lsmdisj2 19857 | Association of the disjoin... |
| lsmdisj3 19858 | Association of the disjoin... |
| lsmdisjr 19859 | Disjointness from a subgro... |
| lsmdisj2r 19860 | Association of the disjoin... |
| lsmdisj3r 19861 | Association of the disjoin... |
| lsmdisj2a 19862 | Association of the disjoin... |
| lsmdisj2b 19863 | Association of the disjoin... |
| lsmdisj3a 19864 | Association of the disjoin... |
| lsmdisj3b 19865 | Association of the disjoin... |
| subgdisj1 19866 | Vectors belonging to disjo... |
| subgdisj2 19867 | Vectors belonging to disjo... |
| subgdisjb 19868 | Vectors belonging to disjo... |
| pj1fval 19869 | The left projection functi... |
| pj1val 19870 | The left projection functi... |
| pj1eu 19871 | Uniqueness of a left proje... |
| pj1f 19872 | The left projection functi... |
| pj2f 19873 | The right projection funct... |
| pj1id 19874 | Any element of a direct su... |
| pj1eq 19875 | Any element of a direct su... |
| pj1lid 19876 | The left projection functi... |
| pj1rid 19877 | The left projection functi... |
| pj1ghm 19878 | The left projection functi... |
| pj1ghm2 19879 | The left projection functi... |
| lsmhash 19880 | The order of the direct pr... |
| efgmval 19887 | Value of the formal invers... |
| efgmf 19888 | The formal inverse operati... |
| efgmnvl 19889 | The inversion function on ... |
| efgrcl 19890 | Lemma for ~ efgval . (Con... |
| efglem 19891 | Lemma for ~ efgval . (Con... |
| efgval 19892 | Value of the free group co... |
| efger 19893 | Value of the free group co... |
| efgi 19894 | Value of the free group co... |
| efgi0 19895 | Value of the free group co... |
| efgi1 19896 | Value of the free group co... |
| efgtf 19897 | Value of the free group co... |
| efgtval 19898 | Value of the extension fun... |
| efgval2 19899 | Value of the free group co... |
| efgi2 19900 | Value of the free group co... |
| efgtlen 19901 | Value of the free group co... |
| efginvrel2 19902 | The inverse of the reverse... |
| efginvrel1 19903 | The inverse of the reverse... |
| efgsf 19904 | Value of the auxiliary fun... |
| efgsdm 19905 | Elementhood in the domain ... |
| efgsval 19906 | Value of the auxiliary fun... |
| efgsdmi 19907 | Property of the last link ... |
| efgsval2 19908 | Value of the auxiliary fun... |
| efgsrel 19909 | The start and end of any e... |
| efgs1 19910 | A singleton of an irreduci... |
| efgs1b 19911 | Every extension sequence e... |
| efgsp1 19912 | If ` F ` is an extension s... |
| efgsres 19913 | An initial segment of an e... |
| efgsfo 19914 | For any word, there is a s... |
| efgredlema 19915 | The reduced word that form... |
| efgredlemf 19916 | Lemma for ~ efgredleme . ... |
| efgredlemg 19917 | Lemma for ~ efgred . (Con... |
| efgredleme 19918 | Lemma for ~ efgred . (Con... |
| efgredlemd 19919 | The reduced word that form... |
| efgredlemc 19920 | The reduced word that form... |
| efgredlemb 19921 | The reduced word that form... |
| efgredlem 19922 | The reduced word that form... |
| efgred 19923 | The reduced word that form... |
| efgrelexlema 19924 | If two words ` A , B ` are... |
| efgrelexlemb 19925 | If two words ` A , B ` are... |
| efgrelex 19926 | If two words ` A , B ` are... |
| efgredeu 19927 | There is a unique reduced ... |
| efgred2 19928 | Two extension sequences ha... |
| efgcpbllema 19929 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19930 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19931 | Two extension sequences ha... |
| efgcpbl2 19932 | Two extension sequences ha... |
| frgpval 19933 | Value of the free group co... |
| frgpcpbl 19934 | Compatibility of the group... |
| frgp0 19935 | The free group is a group.... |
| frgpeccl 19936 | Closure of the quotient ma... |
| frgpgrp 19937 | The free group is a group.... |
| frgpadd 19938 | Addition in the free group... |
| frgpinv 19939 | The inverse of an element ... |
| frgpmhm 19940 | The "natural map" from wor... |
| vrgpfval 19941 | The canonical injection fr... |
| vrgpval 19942 | The value of the generatin... |
| vrgpf 19943 | The mapping from the index... |
| vrgpinv 19944 | The inverse of a generatin... |
| frgpuptf 19945 | Any assignment of the gene... |
| frgpuptinv 19946 | Any assignment of the gene... |
| frgpuplem 19947 | Any assignment of the gene... |
| frgpupf 19948 | Any assignment of the gene... |
| frgpupval 19949 | Any assignment of the gene... |
| frgpup1 19950 | Any assignment of the gene... |
| frgpup2 19951 | The evaluation map has the... |
| frgpup3lem 19952 | The evaluation map has the... |
| frgpup3 19953 | Universal property of the ... |
| 0frgp 19954 | The free group on zero gen... |
| isabl 19959 | The predicate "is an Abeli... |
| ablgrp 19960 | An Abelian group is a grou... |
| ablgrpd 19961 | An Abelian group is a grou... |
| ablcmn 19962 | An Abelian group is a comm... |
| ablcmnd 19963 | An Abelian group is a comm... |
| iscmn 19964 | The predicate "is a commut... |
| isabl2 19965 | The predicate "is an Abeli... |
| cmnpropd 19966 | If two structures have the... |
| ablpropd 19967 | If two structures have the... |
| ablprop 19968 | If two structures have the... |
| iscmnd 19969 | Properties that determine ... |
| isabld 19970 | Properties that determine ... |
| isabli 19971 | Properties that determine ... |
| cmnmnd 19972 | A commutative monoid is a ... |
| cmncom 19973 | A commutative monoid is co... |
| ablcom 19974 | An Abelian group operation... |
| cmn32 19975 | Commutative/associative la... |
| cmn4 19976 | Commutative/associative la... |
| cmn12 19977 | Commutative/associative la... |
| abl32 19978 | Commutative/associative la... |
| cmnmndd 19979 | A commutative monoid is a ... |
| cmnbascntr 19980 | The base set of a commutat... |
| rinvmod 19981 | Uniqueness of a right inve... |
| ablinvadd 19982 | The inverse of an Abelian ... |
| ablsub2inv 19983 | Abelian group subtraction ... |
| ablsubadd 19984 | Relationship between Abeli... |
| ablsub4 19985 | Commutative/associative su... |
| abladdsub4 19986 | Abelian group addition/sub... |
| abladdsub 19987 | Associative-type law for g... |
| ablsubadd23 19988 | Commutative/associative la... |
| ablsubaddsub 19989 | Double subtraction and add... |
| ablpncan2 19990 | Cancellation law for subtr... |
| ablpncan3 19991 | A cancellation law for Abe... |
| ablsubsub 19992 | Law for double subtraction... |
| ablsubsub4 19993 | Law for double subtraction... |
| ablpnpcan 19994 | Cancellation law for mixed... |
| ablnncan 19995 | Cancellation law for group... |
| ablsub32 19996 | Swap the second and third ... |
| ablnnncan 19997 | Cancellation law for group... |
| ablnnncan1 19998 | Cancellation law for group... |
| ablsubsub23 19999 | Swap subtrahend and result... |
| mulgnn0di 20000 | Group multiple of a sum, f... |
| mulgdi 20001 | Group multiple of a sum. ... |
| mulgmhm 20002 | The map from ` x ` to ` n ... |
| mulgghm 20003 | The map from ` x ` to ` n ... |
| mulgsubdi 20004 | Group multiple of a differ... |
| ghmfghm 20005 | The function fulfilling th... |
| ghmcmn 20006 | The image of a commutative... |
| ghmabl 20007 | The image of an abelian gr... |
| invghm 20008 | The inversion map is a gro... |
| eqgabl 20009 | Value of the subgroup cose... |
| qusecsub 20010 | Two subgroup cosets are eq... |
| subgabl 20011 | A subgroup of an abelian g... |
| subcmn 20012 | A submonoid of a commutati... |
| submcmn 20013 | A submonoid of a commutati... |
| submcmn2 20014 | A submonoid is commutative... |
| cntzcmn 20015 | The centralizer of any sub... |
| cntzcmnss 20016 | Any subset in a commutativ... |
| cntrcmnd 20017 | The center of a monoid is ... |
| cntrabl 20018 | The center of a group is a... |
| cntzspan 20019 | If the generators commute,... |
| cntzcmnf 20020 | Discharge the centralizer ... |
| ghmplusg 20021 | The pointwise sum of two l... |
| ablnsg 20022 | Every subgroup of an abeli... |
| odadd1 20023 | The order of a product in ... |
| odadd2 20024 | The order of a product in ... |
| odadd 20025 | The order of a product is ... |
| gex2abl 20026 | A group with exponent 2 (o... |
| gexexlem 20027 | Lemma for ~ gexex . (Cont... |
| gexex 20028 | In an abelian group with f... |
| torsubg 20029 | The set of all elements of... |
| oddvdssubg 20030 | The set of all elements wh... |
| lsmcomx 20031 | Subgroup sum commutes (ext... |
| ablcntzd 20032 | All subgroups in an abelia... |
| lsmcom 20033 | Subgroup sum commutes. (C... |
| lsmsubg2 20034 | The sum of two subgroups i... |
| lsm4 20035 | Commutative/associative la... |
| prdscmnd 20036 | The product of a family of... |
| prdsabld 20037 | The product of a family of... |
| pwscmn 20038 | The structure power on a c... |
| pwsabl 20039 | The structure power on an ... |
| qusabl 20040 | If ` Y ` is a subgroup of ... |
| abl1 20041 | The (smallest) structure r... |
| abln0 20042 | Abelian groups (and theref... |
| cnaddablx 20043 | The complex numbers are an... |
| cnaddabl 20044 | The complex numbers are an... |
| cnaddid 20045 | The group identity element... |
| cnaddinv 20046 | Value of the group inverse... |
| zaddablx 20047 | The integers are an Abelia... |
| frgpnabllem1 20048 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 20049 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 20050 | The free group on two or m... |
| imasabl 20051 | The image structure of an ... |
| iscyg 20054 | Definition of a cyclic gro... |
| iscyggen 20055 | The property of being a cy... |
| iscyggen2 20056 | The property of being a cy... |
| iscyg2 20057 | A cyclic group is a group ... |
| cyggeninv 20058 | The inverse of a cyclic ge... |
| cyggenod 20059 | An element is the generato... |
| cyggenod2 20060 | In an infinite cyclic grou... |
| iscyg3 20061 | Definition of a cyclic gro... |
| iscygd 20062 | Definition of a cyclic gro... |
| iscygodd 20063 | Show that a group with an ... |
| cycsubmcmn 20064 | The set of nonnegative int... |
| cyggrp 20065 | A cyclic group is a group.... |
| cygabl 20066 | A cyclic group is abelian.... |
| cygctb 20067 | A cyclic group is countabl... |
| 0cyg 20068 | The trivial group is cycli... |
| prmcyg 20069 | A group with prime order i... |
| lt6abl 20070 | A group with fewer than ` ... |
| ghmcyg 20071 | The image of a cyclic grou... |
| cyggex2 20072 | The exponent of a cyclic g... |
| cyggex 20073 | The exponent of a finite c... |
| cyggexb 20074 | A finite abelian group is ... |
| giccyg 20075 | Cyclicity is a group prope... |
| cycsubgcyg 20076 | The cyclic subgroup genera... |
| cycsubgcyg2 20077 | The cyclic subgroup genera... |
| gsumval3a 20078 | Value of the group sum ope... |
| gsumval3eu 20079 | The group sum as defined i... |
| gsumval3lem1 20080 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 20081 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 20082 | Value of the group sum ope... |
| gsumcllem 20083 | Lemma for ~ gsumcl and rel... |
| gsumzres 20084 | Extend a finite group sum ... |
| gsumzcl2 20085 | Closure of a finite group ... |
| gsumzcl 20086 | Closure of a finite group ... |
| gsumzf1o 20087 | Re-index a finite group su... |
| gsumres 20088 | Extend a finite group sum ... |
| gsumcl2 20089 | Closure of a finite group ... |
| gsumcl 20090 | Closure of a finite group ... |
| gsumf1o 20091 | Re-index a finite group su... |
| gsumreidx 20092 | Re-index a finite group su... |
| gsumzsubmcl 20093 | Closure of a group sum in ... |
| gsumsubmcl 20094 | Closure of a group sum in ... |
| gsumsubgcl 20095 | Closure of a group sum in ... |
| gsumzaddlem 20096 | The sum of two group sums.... |
| gsumzadd 20097 | The sum of two group sums.... |
| gsumadd 20098 | The sum of two group sums.... |
| gsummptfsadd 20099 | The sum of two group sums ... |
| gsummptfidmadd 20100 | The sum of two group sums ... |
| gsummptfidmadd2 20101 | The sum of two group sums ... |
| gsumzsplit 20102 | Split a group sum into two... |
| gsumsplit 20103 | Split a group sum into two... |
| gsumsplit2 20104 | Split a group sum into two... |
| gsummptfidmsplit 20105 | Split a group sum expresse... |
| gsummptfidmsplitres 20106 | Split a group sum expresse... |
| gsummptfzsplit 20107 | Split a group sum expresse... |
| gsummptfzsplitl 20108 | Split a group sum expresse... |
| gsumconst 20109 | Sum of a constant series. ... |
| gsumconstf 20110 | Sum of a constant series. ... |
| gsummptshft 20111 | Index shift of a finite gr... |
| gsumzmhm 20112 | Apply a group homomorphism... |
| gsummhm 20113 | Apply a group homomorphism... |
| gsummhm2 20114 | Apply a group homomorphism... |
| gsummptmhm 20115 | Apply a group homomorphism... |
| gsummulglem 20116 | Lemma for ~ gsummulg and ~... |
| gsummulg 20117 | Nonnegative multiple of a ... |
| gsummulgz 20118 | Integer multiple of a grou... |
| gsumzoppg 20119 | The opposite of a group su... |
| gsumzinv 20120 | Inverse of a group sum. (... |
| gsuminv 20121 | Inverse of a group sum. (... |
| gsummptfidminv 20122 | Inverse of a group sum exp... |
| gsumsub 20123 | The difference of two grou... |
| gsummptfssub 20124 | The difference of two grou... |
| gsummptfidmsub 20125 | The difference of two grou... |
| gsumsnfd 20126 | Group sum of a singleton, ... |
| gsumsnd 20127 | Group sum of a singleton, ... |
| gsumsnf 20128 | Group sum of a singleton, ... |
| gsumsn 20129 | Group sum of a singleton. ... |
| gsumpr 20130 | Group sum of a pair. (Con... |
| gsumzunsnd 20131 | Append an element to a fin... |
| gsumunsnfd 20132 | Append an element to a fin... |
| gsumunsnd 20133 | Append an element to a fin... |
| gsumunsnf 20134 | Append an element to a fin... |
| gsumunsn 20135 | Append an element to a fin... |
| gsumdifsnd 20136 | Extract a summand from a f... |
| gsumpt 20137 | Sum of a family that is no... |
| gsummptf1o 20138 | Re-index a finite group su... |
| gsummptun 20139 | Group sum of a disjoint un... |
| gsummpt1n0 20140 | If only one summand in a f... |
| gsummptif1n0 20141 | If only one summand in a f... |
| gsummptcl 20142 | Closure of a finite group ... |
| gsummptfif1o 20143 | Re-index a finite group su... |
| gsummptfzcl 20144 | Closure of a finite group ... |
| gsum2dlem1 20145 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 20146 | Lemma for ~ gsum2d . (Con... |
| gsum2d 20147 | Write a sum over a two-dim... |
| gsum2d2lem 20148 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 20149 | Write a group sum over a t... |
| gsumcom2 20150 | Two-dimensional commutatio... |
| gsumxp 20151 | Write a group sum over a c... |
| gsumcom 20152 | Commute the arguments of a... |
| gsumcom3 20153 | A commutative law for fini... |
| gsumcom3fi 20154 | A commutative law for fini... |
| gsumxp2 20155 | Write a group sum over a c... |
| prdsgsum 20156 | Finite commutative sums in... |
| pwsgsum 20157 | Finite commutative sums in... |
| fsfnn0gsumfsffz 20158 | Replacing a finitely suppo... |
| nn0gsumfz 20159 | Replacing a finitely suppo... |
| nn0gsumfz0 20160 | Replacing a finitely suppo... |
| gsummptnn0fz 20161 | A final group sum over a f... |
| gsummptnn0fzfv 20162 | A final group sum over a f... |
| telgsumfzslem 20163 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 20164 | Telescoping group sum rang... |
| telgsumfz 20165 | Telescoping group sum rang... |
| telgsumfz0s 20166 | Telescoping finite group s... |
| telgsumfz0 20167 | Telescoping finite group s... |
| telgsums 20168 | Telescoping finitely suppo... |
| telgsum 20169 | Telescoping finitely suppo... |
| reldmdprd 20174 | The domain of the internal... |
| dmdprd 20175 | The domain of definition o... |
| dmdprdd 20176 | Show that a given family i... |
| dprddomprc 20177 | A family of subgroups inde... |
| dprddomcld 20178 | If a family of subgroups i... |
| dprdval0prc 20179 | The internal direct produc... |
| dprdval 20180 | The value of the internal ... |
| eldprd 20181 | A class ` A ` is an intern... |
| dprdgrp 20182 | Reverse closure for the in... |
| dprdf 20183 | The function ` S ` is a fa... |
| dprdf2 20184 | The function ` S ` is a fa... |
| dprdcntz 20185 | The function ` S ` is a fa... |
| dprddisj 20186 | The function ` S ` is a fa... |
| dprdw 20187 | The property of being a fi... |
| dprdwd 20188 | A mapping being a finitely... |
| dprdff 20189 | A finitely supported funct... |
| dprdfcl 20190 | A finitely supported funct... |
| dprdffsupp 20191 | A finitely supported funct... |
| dprdfcntz 20192 | A function on the elements... |
| dprdssv 20193 | The internal direct produc... |
| dprdfid 20194 | A function mapping all but... |
| eldprdi 20195 | The domain of definition o... |
| dprdfinv 20196 | Take the inverse of a grou... |
| dprdfadd 20197 | Take the sum of group sums... |
| dprdfsub 20198 | Take the difference of gro... |
| dprdfeq0 20199 | The zero function is the o... |
| dprdf11 20200 | Two group sums over a dire... |
| dprdsubg 20201 | The internal direct produc... |
| dprdub 20202 | Each factor is a subset of... |
| dprdlub 20203 | The direct product is smal... |
| dprdspan 20204 | The direct product is the ... |
| dprdres 20205 | Restriction of a direct pr... |
| dprdss 20206 | Create a direct product by... |
| dprdz 20207 | A family consisting entire... |
| dprd0 20208 | The empty family is an int... |
| dprdf1o 20209 | Rearrange the index set of... |
| dprdf1 20210 | Rearrange the index set of... |
| subgdmdprd 20211 | A direct product in a subg... |
| subgdprd 20212 | A direct product in a subg... |
| dprdsn 20213 | A singleton family is an i... |
| dmdprdsplitlem 20214 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20215 | The function ` S ` is a fa... |
| dprddisj2 20216 | The function ` S ` is a fa... |
| dprd2dlem2 20217 | The direct product of a co... |
| dprd2dlem1 20218 | The direct product of a co... |
| dprd2da 20219 | The direct product of a co... |
| dprd2db 20220 | The direct product of a co... |
| dprd2d2 20221 | The direct product of a co... |
| dmdprdsplit2lem 20222 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20223 | The direct product splits ... |
| dmdprdsplit 20224 | The direct product splits ... |
| dprdsplit 20225 | The direct product is the ... |
| dmdprdpr 20226 | A singleton family is an i... |
| dprdpr 20227 | A singleton family is an i... |
| dpjlem 20228 | Lemma for theorems about d... |
| dpjcntz 20229 | The two subgroups that app... |
| dpjdisj 20230 | The two subgroups that app... |
| dpjlsm 20231 | The two subgroups that app... |
| dpjfval 20232 | Value of the direct produc... |
| dpjval 20233 | Value of the direct produc... |
| dpjf 20234 | The ` X ` -th index projec... |
| dpjidcl 20235 | The key property of projec... |
| dpjeq 20236 | Decompose a group sum into... |
| dpjid 20237 | The key property of projec... |
| dpjlid 20238 | The ` X ` -th index projec... |
| dpjrid 20239 | The ` Y ` -th index projec... |
| dpjghm 20240 | The direct product is the ... |
| dpjghm2 20241 | The direct product is the ... |
| ablfacrplem 20242 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20243 | A finite abelian group who... |
| ablfacrp2 20244 | The factors ` K , L ` of ~... |
| ablfac1lem 20245 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20246 | The factors of ~ ablfac1b ... |
| ablfac1b 20247 | Any abelian group is the d... |
| ablfac1c 20248 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20249 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20250 | The factorization of ~ abl... |
| pgpfac1lem1 20251 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20252 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20253 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20254 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20255 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20256 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20257 | Factorization of a finite ... |
| pgpfaclem1 20258 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20259 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20260 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20261 | Full factorization of a fi... |
| ablfaclem1 20262 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20263 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20264 | Lemma for ~ ablfac . (Con... |
| ablfac 20265 | The Fundamental Theorem of... |
| ablfac2 20266 | Choose generators for each... |
| issimpg 20269 | The predicate "is a simple... |
| issimpgd 20270 | Deduce a simple group from... |
| simpggrp 20271 | A simple group is a group.... |
| simpggrpd 20272 | A simple group is a group.... |
| simpg2nsg 20273 | A simple group has two nor... |
| trivnsimpgd 20274 | Trivial groups are not sim... |
| simpgntrivd 20275 | Simple groups are nontrivi... |
| simpgnideld 20276 | A simple group contains a ... |
| simpgnsgd 20277 | The only normal subgroups ... |
| simpgnsgeqd 20278 | A normal subgroup of a sim... |
| 2nsgsimpgd 20279 | If any normal subgroup of ... |
| simpgnsgbid 20280 | A nontrivial group is simp... |
| ablsimpnosubgd 20281 | A subgroup of an abelian s... |
| ablsimpg1gend 20282 | An abelian simple group is... |
| ablsimpgcygd 20283 | An abelian simple group is... |
| ablsimpgfindlem1 20284 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20285 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20286 | Relationship between the o... |
| ablsimpgfind 20287 | An abelian simple group is... |
| fincygsubgd 20288 | The subgroup referenced in... |
| fincygsubgodd 20289 | Calculate the order of a s... |
| fincygsubgodexd 20290 | A finite cyclic group has ... |
| prmgrpsimpgd 20291 | A group of prime order is ... |
| ablsimpgprmd 20292 | An abelian simple group ha... |
| ablsimpgd 20293 | An abelian group is simple... |
| isomnd 20298 | A (left) ordered monoid is... |
| isogrp 20299 | A (left-)ordered group is ... |
| ogrpgrp 20300 | A left-ordered group is a ... |
| omndmnd 20301 | A left-ordered monoid is a... |
| omndtos 20302 | A left-ordered monoid is a... |
| omndadd 20303 | In an ordered monoid, the ... |
| omndaddr 20304 | In a right ordered monoid,... |
| omndadd2d 20305 | In a commutative left orde... |
| omndadd2rd 20306 | In a left- and right- orde... |
| submomnd 20307 | A submonoid of an ordered ... |
| omndmul2 20308 | In an ordered monoid, the ... |
| omndmul3 20309 | In an ordered monoid, the ... |
| omndmul 20310 | In a commutative ordered m... |
| ogrpinv0le 20311 | In an ordered group, the o... |
| ogrpsub 20312 | In an ordered group, the o... |
| ogrpaddlt 20313 | In an ordered group, stric... |
| ogrpaddltbi 20314 | In a right ordered group, ... |
| ogrpaddltrd 20315 | In a right ordered group, ... |
| ogrpaddltrbid 20316 | In a right ordered group, ... |
| ogrpsublt 20317 | In an ordered group, stric... |
| ogrpinv0lt 20318 | In an ordered group, the o... |
| ogrpinvlt 20319 | In an ordered group, the o... |
| gsumle 20320 | A finite sum in an ordered... |
| fnmgp 20323 | The multiplicative group o... |
| mgpval 20324 | Value of the multiplicatio... |
| mgpplusg 20325 | Value of the group operati... |
| mgpbas 20326 | Base set of the multiplica... |
| mgpsca 20327 | The multiplication monoid ... |
| mgptset 20328 | Topology component of the ... |
| mgptopn 20329 | Topology of the multiplica... |
| mgpds 20330 | Distance function of the m... |
| mgpress 20331 | Subgroup commutes with the... |
| prdsmgp 20332 | The multiplicative monoid ... |
| elmgplsm 20333 | Membership in a product of... |
| elmgplsmd 20334 | Membership in a product of... |
| isrng 20337 | The predicate "is a non-un... |
| rngabl 20338 | A non-unital ring is an (a... |
| rngmgp 20339 | A non-unital ring is a sem... |
| rngmgpf 20340 | Restricted functionality o... |
| rnggrp 20341 | A non-unital ring is a (ad... |
| rngass 20342 | Associative law for the mu... |
| rngdi 20343 | Distributive law for the m... |
| rngdir 20344 | Distributive law for the m... |
| rngacl 20345 | Closure of the addition op... |
| rng0cl 20346 | The zero element of a non-... |
| rngcl 20347 | Closure of the multiplicat... |
| rnglz 20348 | The zero of a non-unital r... |
| rngrz 20349 | The zero of a non-unital r... |
| rngmneg1 20350 | Negation of a product in a... |
| rngmneg2 20351 | Negation of a product in a... |
| rngm2neg 20352 | Double negation of a produ... |
| rngansg 20353 | Every additive subgroup of... |
| rngsubdi 20354 | Ring multiplication distri... |
| rngsubdir 20355 | Ring multiplication distri... |
| isrngd 20356 | Properties that determine ... |
| rngpropd 20357 | If two structures have the... |
| prdsmulrngcl 20358 | Closure of the multiplicat... |
| prdsrngd 20359 | A product of non-unital ri... |
| imasrng 20360 | The image structure of a n... |
| imasrngf1 20361 | The image of a non-unital ... |
| xpsrngd 20362 | A product of two non-unita... |
| qusrng 20363 | The quotient structure of ... |
| rng1zrlem 20364 | Lemma for ~ rng1zr and ~ s... |
| rng1zr 20365 | The only ring with a base ... |
| rngen1zr 20366 | The only ring with one ele... |
| rngen1zr0 20367 | The only ring with one ele... |
| ringidval 20370 | The value of the unity ele... |
| dfur2 20371 | The multiplicative identit... |
| ringurd 20372 | Deduce the unity element o... |
| issrg 20375 | The predicate "is a semiri... |
| srgcmn 20376 | A semiring is a commutativ... |
| srgmnd 20377 | A semiring is a monoid. (... |
| srgmgp 20378 | A semiring is a monoid und... |
| srgdilem 20379 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20380 | Closure of the multiplicat... |
| srgass 20381 | Associative law for the mu... |
| srgideu 20382 | The unity element of a sem... |
| srgfcl 20383 | Functionality of the multi... |
| srgdi 20384 | Distributive law for the m... |
| srgdir 20385 | Distributive law for the m... |
| srgidcl 20386 | The unity element of a sem... |
| srg0cl 20387 | The zero element of a semi... |
| srgidmlem 20388 | Lemma for ~ srglidm and ~ ... |
| srglidm 20389 | The unity element of a sem... |
| srgridm 20390 | The unity element of a sem... |
| issrgid 20391 | Properties showing that an... |
| srgacl 20392 | Closure of the addition op... |
| srgcom 20393 | Commutativity of the addit... |
| srgrz 20394 | The zero of a semiring is ... |
| srglz 20395 | The zero of a semiring is ... |
| srgisid 20396 | In a semiring, the only le... |
| o2timesd 20397 | An element of a ring-like ... |
| rglcom4d 20398 | Restricted commutativity o... |
| srgo2times 20399 | A semiring element plus it... |
| srgcom4lem 20400 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20401 | Restricted commutativity o... |
| srg1zr 20402 | The only semiring with a b... |
| srgen1zr0 20403 | The only semiring with one... |
| srgmulgass 20404 | An associative property be... |
| srgpcomp 20405 | If two elements of a semir... |
| srgpcompp 20406 | If two elements of a semir... |
| srgpcomppsc 20407 | If two elements of a semir... |
| srglmhm 20408 | Left-multiplication in a s... |
| srgrmhm 20409 | Right-multiplication in a ... |
| srgsummulcr 20410 | A finite semiring sum mult... |
| sgsummulcl 20411 | A finite semiring sum mult... |
| srg1expzeq1 20412 | The exponentiation (by a n... |
| srgbinomlem1 20413 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20414 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20415 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20416 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20417 | Lemma for ~ srgbinom . In... |
| srgbinom 20418 | The binomial theorem for c... |
| csrgbinom 20419 | The binomial theorem for c... |
| isring 20424 | The predicate "is a (unita... |
| ringgrp 20425 | A ring is a group. (Contr... |
| ringmgp 20426 | A ring is a monoid under m... |
| iscrng 20427 | A commutative ring is a ri... |
| crngmgp 20428 | A commutative ring's multi... |
| ringbn0 20429 | The base set of a ring is ... |
| ringgrpd 20430 | A ring is a group. (Contr... |
| ringmnd 20431 | A ring is a monoid under a... |
| ringmgm 20432 | A ring is a magma. (Contr... |
| crngring 20433 | A commutative ring is a ri... |
| crngringd 20434 | A commutative ring is a ri... |
| crnggrpd 20435 | A commutative ring is a gr... |
| mgpf 20436 | Restricted functionality o... |
| ringdilem 20437 | Properties of a unital rin... |
| ringcl 20438 | Closure of the multiplicat... |
| crngcom 20439 | A commutative ring's multi... |
| iscrng2 20440 | A commutative ring is a ri... |
| ringass 20441 | Associative law for multip... |
| ringideu 20442 | The unity element of a rin... |
| crngcomd 20443 | Multiplication is commutat... |
| crngbascntr 20444 | The base set of a commutat... |
| ringcld 20445 | Closure of the multiplicat... |
| ringassd 20446 | Associative law for multip... |
| crng12d 20447 | Commutative/associative la... |
| crng32d 20448 | Commutative/associative la... |
| crng4 20449 | Commutative/associative la... |
| ringdi 20450 | Distributive law for the m... |
| ringdir 20451 | Distributive law for the m... |
| ringdid 20452 | Distributive law for the m... |
| ringdird 20453 | Distributive law for the m... |
| ringdi22 20454 | Expand the product of two ... |
| ringidcl 20455 | The unity element of a rin... |
| ringidcld 20456 | The unity element of a rin... |
| ring0cl 20457 | The zero element of a ring... |
| ringidmlem 20458 | Lemma for ~ ringlidm and ~... |
| ringlidm 20459 | The unity element of a rin... |
| ringridm 20460 | The unity element of a rin... |
| isringid 20461 | Properties showing that an... |
| ringlidmd 20462 | The unity element of a rin... |
| ringridmd 20463 | The unity element of a rin... |
| ringid 20464 | The multiplication operati... |
| ringo2times 20465 | A ring element plus itself... |
| ringadd2 20466 | A ring element plus itself... |
| ringidss 20467 | A subset of the multiplica... |
| ringacl 20468 | Closure of the addition op... |
| ringcomlem 20469 | Lemma for ~ ringcom . Thi... |
| ringcom 20470 | Commutativity of the addit... |
| ringabl 20471 | A ring is an Abelian group... |
| ringcmn 20472 | A ring is a commutative mo... |
| ringabld 20473 | A ring is an Abelian group... |
| ringcmnd 20474 | A ring is a commutative mo... |
| ringrng 20475 | A unital ring is a non-uni... |
| ringssrng 20476 | The unital rings are non-u... |
| isringrng 20477 | The predicate "is a unital... |
| dfring2 20478 | The predicate "is a unital... |
| dfring3 20479 | The predicate "is a (unita... |
| ringpropd 20480 | If two structures have the... |
| crngpropd 20481 | If two structures have the... |
| ringprop 20482 | If two structures have the... |
| isringd 20483 | Properties that determine ... |
| iscrngd 20484 | Properties that determine ... |
| ringlz 20485 | The zero of a unital ring ... |
| ringrz 20486 | The zero of a unital ring ... |
| ringlzd 20487 | The zero of a unital ring ... |
| ringrzd 20488 | The zero of a unital ring ... |
| ringsrg 20489 | Any ring is also a semirin... |
| ring1eq0 20490 | If one and zero are equal,... |
| ring1ne0 20491 | If a ring has at least two... |
| ringinvnz1ne0 20492 | In a unital ring, a left i... |
| ringinvnzdiv 20493 | In a unital ring, a left i... |
| ringnegl 20494 | Negation in a ring is the ... |
| ringnegr 20495 | Negation in a ring is the ... |
| ringmneg1 20496 | Negation of a product in a... |
| ringmneg2 20497 | Negation of a product in a... |
| ringm2neg 20498 | Double negation of a produ... |
| ringsubdi 20499 | Ring multiplication distri... |
| ringsubdir 20500 | Ring multiplication distri... |
| mulgass2 20501 | An associative property be... |
| ring1 20502 | The (smallest) structure r... |
| ringn0 20503 | Rings exist. (Contributed... |
| ringlghm 20504 | Left-multiplication in a r... |
| ringrghm 20505 | Right-multiplication in a ... |
| gsummulc1 20506 | A finite ring sum multipli... |
| gsummulc2 20507 | A finite ring sum multipli... |
| gsummgp0 20508 | If one factor in a finite ... |
| gsumdixp 20509 | Distribute a binary produc... |
| prdsmulrcl 20510 | A structure product of rin... |
| prdsringd 20511 | A product of rings is a ri... |
| prdscrngd 20512 | A product of commutative r... |
| prds1 20513 | Value of the ring unity in... |
| pwsring 20514 | A structure power of a rin... |
| pws1 20515 | Value of the ring unity in... |
| pwscrng 20516 | A structure power of a com... |
| pwsmgp 20517 | The multiplicative group o... |
| pwspjmhmmgpd 20518 | The projection given by ~ ... |
| pwsexpg 20519 | Value of a group exponenti... |
| pwsgprod 20520 | Finite products in a power... |
| imasring 20521 | The image structure of a r... |
| imasringf1 20522 | The image of a ring under ... |
| xpsringd 20523 | A product of two rings is ... |
| xpsring1d 20524 | The multiplicative identit... |
| qusring2 20525 | The quotient structure of ... |
| crngbinom 20526 | The binomial theorem for c... |
| opprval 20529 | Value of the opposite ring... |
| opprmulfval 20530 | Value of the multiplicatio... |
| opprmul 20531 | Value of the multiplicatio... |
| crngoppr 20532 | In a commutative ring, the... |
| opprlem 20533 | Lemma for ~ opprbas and ~ ... |
| opprbas 20534 | Base set of an opposite ri... |
| oppradd 20535 | Addition operation of an o... |
| opprrng 20536 | An opposite non-unital rin... |
| opprrngb 20537 | A class is a non-unital ri... |
| opprring 20538 | An opposite ring is a ring... |
| opprringb 20539 | Bidirectional form of ~ op... |
| oppr0 20540 | Additive identity of an op... |
| oppr1 20541 | Multiplicative identity of... |
| opprneg 20542 | The negative function in a... |
| opprsubg 20543 | Being a subgroup is a symm... |
| mulgass3 20544 | An associative property be... |
| reldvdsr 20551 | The divides relation is a ... |
| dvdsrval 20552 | Value of the divides relat... |
| dvdsr 20553 | Value of the divides relat... |
| dvdsr2 20554 | Value of the divides relat... |
| dvdsrmul 20555 | A left-multiple of ` X ` i... |
| dvdsrcl 20556 | Closure of a dividing elem... |
| dvdsrcl2 20557 | Closure of a dividing elem... |
| dvdsrid 20558 | An element in a (unital) r... |
| dvdsrtr 20559 | Divisibility is transitive... |
| dvdsrmul1 20560 | The divisibility relation ... |
| dvdsrneg 20561 | An element divides its neg... |
| dvdsr01 20562 | In a ring, zero is divisib... |
| dvdsr02 20563 | Only zero is divisible by ... |
| isunit 20564 | Property of being a unit o... |
| 1unit 20565 | The multiplicative identit... |
| unitcl 20566 | A unit is an element of th... |
| unitss 20567 | The set of units is contai... |
| opprunit 20568 | Being a unit is a symmetri... |
| crngunit 20569 | Property of being a unit i... |
| dvdsunit 20570 | A divisor of a unit is a u... |
| unitmulcl 20571 | The product of units is a ... |
| unitmulclb 20572 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20573 | The base set of the group ... |
| unitgrp 20574 | The group of units is a gr... |
| unitabl 20575 | The group of units of a co... |
| unitgrpid 20576 | The identity of the group ... |
| unitsubm 20577 | The group of units is a su... |
| invrfval 20580 | Multiplicative inverse fun... |
| unitinvcl 20581 | The inverse of a unit exis... |
| unitinvinv 20582 | The inverse of the inverse... |
| ringinvcl 20583 | The inverse of a unit is a... |
| unitlinv 20584 | A unit times its inverse i... |
| unitrinv 20585 | A unit times its inverse i... |
| 1rinv 20586 | The inverse of the ring un... |
| 0unit 20587 | The additive identity is a... |
| unitnegcl 20588 | The negative of a unit is ... |
| ringunitnzdiv 20589 | In a unitary ring, a unit ... |
| ring1nzdiv 20590 | In a unitary ring, the rin... |
| dvrfval 20593 | Division operation in a ri... |
| dvrval 20594 | Division operation in a ri... |
| dvrcl 20595 | Closure of division operat... |
| unitdvcl 20596 | The units are closed under... |
| dvrid 20597 | A ring element divided by ... |
| dvr1 20598 | A ring element divided by ... |
| dvrass 20599 | An associative law for div... |
| dvrcan1 20600 | A cancellation law for div... |
| dvrcan3 20601 | A cancellation law for div... |
| dvreq1 20602 | Equality in terms of ratio... |
| dvrdir 20603 | Distributive law for the d... |
| rdivmuldivd 20604 | Multiplication of two rati... |
| ringinvdv 20605 | Write the inverse function... |
| rngidpropd 20606 | The ring unity depends onl... |
| dvdsrpropd 20607 | The divisibility relation ... |
| unitpropd 20608 | The set of units depends o... |
| invrpropd 20609 | The ring inverse function ... |
| isirred 20610 | An irreducible element of ... |
| isnirred 20611 | The property of being a no... |
| isirred2 20612 | Expand out the class diffe... |
| opprirred 20613 | Irreducibility is symmetri... |
| irredn0 20614 | The additive identity is n... |
| irredcl 20615 | An irreducible element is ... |
| irrednu 20616 | An irreducible element is ... |
| irredn1 20617 | The multiplicative identit... |
| irredrmul 20618 | The product of an irreduci... |
| irredlmul 20619 | The product of a unit and ... |
| irredmul 20620 | If product of two elements... |
| irredneg 20621 | The negative of an irreduc... |
| irrednegb 20622 | An element is irreducible ... |
| rnghmrcl 20629 | Reverse closure of a non-u... |
| rnghmfn 20630 | The mapping of two non-uni... |
| rnghmval 20631 | The set of the non-unital ... |
| isrnghm 20632 | A function is a non-unital... |
| isrnghmmul 20633 | A function is a non-unital... |
| rnghmmgmhm 20634 | A non-unital ring homomorp... |
| rnghmval2 20635 | The non-unital ring homomo... |
| isrngim 20636 | An isomorphism of non-unit... |
| rngimrcl 20637 | Reverse closure for an iso... |
| rnghmghm 20638 | A non-unital ring homomorp... |
| rnghmf 20639 | A ring homomorphism is a f... |
| rnghmmul 20640 | A homomorphism of non-unit... |
| isrnghm2d 20641 | Demonstration of non-unita... |
| isrnghmd 20642 | Demonstration of non-unita... |
| rnghmf1o 20643 | A non-unital ring homomorp... |
| isrngim2 20644 | An isomorphism of non-unit... |
| rngimf1o 20645 | An isomorphism of non-unit... |
| rngimrnghm 20646 | An isomorphism of non-unit... |
| rngimcnv 20647 | The converse of an isomorp... |
| rnghmco 20648 | The composition of non-uni... |
| idrnghm 20649 | The identity homomorphism ... |
| c0mgm 20650 | The constant mapping to ze... |
| c0mhm 20651 | The constant mapping to ze... |
| c0ghm 20652 | The constant mapping to ze... |
| c0snmgmhm 20653 | The constant mapping to ze... |
| c0snmhm 20654 | The constant mapping to ze... |
| c0snghm 20655 | The constant mapping to ze... |
| rngisomfv1 20656 | If there is a non-unital r... |
| rngisom1 20657 | If there is a non-unital r... |
| rngisomring 20658 | If there is a non-unital r... |
| rngisomring1 20659 | If there is a non-unital r... |
| dfrhm2 20665 | The property of a ring hom... |
| rhmval0 20666 | The set of ring homomorphi... |
| isrhm0 20667 | The predicate "is a ring h... |
| rhmrcl1 20668 | Reverse closure of a ring ... |
| rhmrcl2 20669 | Reverse closure of a ring ... |
| isrhm 20670 | A function is a ring homom... |
| rhmmhm 20671 | A ring homomorphism is a h... |
| rhmisrnghm 20672 | Each unital ring homomorph... |
| rimrcl 20673 | Reverse closure for an iso... |
| isrim0 20674 | A ring isomorphism is a ho... |
| rhmghm 20675 | A ring homomorphism is an ... |
| rhmf 20676 | A ring homomorphism is a f... |
| rhmcl 20677 | Closure law for a ring hom... |
| rimcnv 20678 | The converse of a ring iso... |
| rhmadd 20679 | Ring homomorphisms preserv... |
| rhmsub 20680 | Ring homomorphisms preserv... |
| rhmmul 20681 | A homomorphism of rings pr... |
| isrhm2d 20682 | Demonstration of ring homo... |
| isrhmd 20683 | Demonstration of ring homo... |
| rhm0 20684 | A ring homomorphism preser... |
| rhm1 20685 | Ring homomorphisms are req... |
| idrhm 20686 | The identity homomorphism ... |
| crngrhmfo 20687 | The image of a surjective ... |
| rhmf1o 20688 | A ring homomorphism is bij... |
| isrim 20689 | An isomorphism of rings is... |
| rimf1o 20690 | An isomorphism of rings is... |
| rimval 20691 | The set of ring isomorphis... |
| rimrhm 20692 | A ring isomorphism is a ho... |
| rimrcl1 20693 | Reverse closure of a ring ... |
| rimrcl2 20694 | Reverse closure of a ring ... |
| rimgim 20695 | An isomorphism of rings is... |
| rimisrngim 20696 | Each unital ring isomorphi... |
| rhmfn 20697 | The mapping of two rings t... |
| rimfn 20698 | The mapping of two rings t... |
| rhmval 20699 | The ring homomorphisms bet... |
| rhmco 20700 | The composition of ring ho... |
| rhmkerinj 20701 | A ring homomorphism is inj... |
| pwsco1rhm 20702 | Right composition with a f... |
| pwsco2rhm 20703 | Left composition with a ri... |
| ricrel 20705 | The domain of the ring iso... |
| brric 20706 | The relation "is isomorphi... |
| brrici 20707 | Prove isomorphic by an exp... |
| rimco 20708 | The composition of ring is... |
| ricref 20709 | Ring isomorphism is reflex... |
| riclcl 20710 | Ring isomorphism implies t... |
| ricrcl 20711 | Ring isomorphism implies t... |
| ricsym 20712 | Ring isomorphism is symmet... |
| rictr 20713 | Ring isomorphism is transi... |
| isbrric2 20714 | The relation "is isomorphi... |
| brric2 20715 | The ring isomorphism relat... |
| ricgic 20716 | If two rings are (ring) is... |
| ricer 20717 | Ring isomorphism is an equ... |
| dfric2 20718 | Alternate definition of th... |
| rhmdvdsr 20719 | A ring homomorphism preser... |
| rhmopp 20720 | A ring homomorphism is als... |
| elrhmunit 20721 | Ring homomorphisms preserv... |
| rhmunitinv 20722 | Ring homomorphisms preserv... |
| isnzr 20725 | Property of a nonzero ring... |
| nzrnz 20726 | One and zero are different... |
| nzrring 20727 | A nonzero ring is a ring. ... |
| nzrringOLD 20728 | Obsolete version of ~ nzrr... |
| isnzr2 20729 | Equivalent characterizatio... |
| drnglidl1ne0 20730 | In a nonzero ring, the zer... |
| isnzr2hash 20731 | Equivalent characterizatio... |
| nzrpropd 20732 | If two structures have the... |
| opprnzrb 20733 | The opposite of a nonzero ... |
| opprnzr 20734 | The opposite of a nonzero ... |
| ringelnzr 20735 | A ring is nonzero if it ha... |
| nzrunit 20736 | A unit is nonzero in any n... |
| 0ringnnzr 20737 | A ring is a zero ring iff ... |
| 0ring 20738 | If a ring has only one ele... |
| 0ringdif 20739 | A zero ring is a ring whic... |
| 0ringbas 20740 | The base set of a zero rin... |
| 0ring01eq 20741 | In a ring with only one el... |
| 01eq0ring 20742 | If the zero and the identi... |
| 01eq0ringOLD 20743 | Obsolete version of ~ 01eq... |
| 0ring01eqbi2 20744 | In a ring, ` 0 = 1 ` iff t... |
| 0ring01eqbi 20745 | In a unital ring the zero ... |
| 0ring1eq0 20746 | In a zero ring, a ring whi... |
| c0rhm 20747 | The constant mapping to ze... |
| c0rnghm 20748 | The constant mapping to ze... |
| zrrnghm 20749 | The constant mapping to ze... |
| nrhmzr 20750 | There is no ring homomorph... |
| islring 20753 | The predicate "is a local ... |
| lringnzr 20754 | A local ring is a nonzero ... |
| lringring 20755 | A local ring is a ring. (... |
| lringnz 20756 | A local ring is a nonzero ... |
| lringuplu 20757 | If the sum of two elements... |
| issubrng 20760 | The subring of non-unital ... |
| subrngss 20761 | A subring is a subset. (C... |
| subrngid 20762 | Every non-unital ring is a... |
| subrngrng 20763 | A subring is a non-unital ... |
| subrngrcl 20764 | Reverse closure for a subr... |
| subrngsubg 20765 | A subring is a subgroup. ... |
| subrngringnsg 20766 | A subring is a normal subg... |
| subrngbas 20767 | Base set of a subring stru... |
| subrng0 20768 | A subring always has the s... |
| subrngacl 20769 | A subring is closed under ... |
| subrngmcl 20770 | A subring is closed under ... |
| issubrng2 20771 | Characterize the subrings ... |
| opprsubrng 20772 | Being a subring is a symme... |
| subrngint 20773 | The intersection of a none... |
| subrngin 20774 | The intersection of two su... |
| subrngmre 20775 | The subrings of a non-unit... |
| subsubrng 20776 | A subring of a subring is ... |
| subsubrng2 20777 | The set of subrings of a s... |
| rhmimasubrnglem 20778 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20779 | The homomorphic image of a... |
| cntzsubrng 20780 | Centralizers in a non-unit... |
| subrngpropd 20781 | If two structures have the... |
| issubrg 20784 | The subring predicate. (C... |
| subrgss 20785 | A subring is a subset. (C... |
| subrgid 20786 | Every ring is a subring of... |
| subrgring 20787 | A subring is a ring. (Con... |
| subrgcrng 20788 | A subring of a commutative... |
| subrgrcl 20789 | Reverse closure for a subr... |
| subrgsubg 20790 | A subring is a subgroup. ... |
| subrgsubrng 20791 | A subring of a unital ring... |
| subrg0 20792 | A subring always has the s... |
| subrg1cl 20793 | A subring contains the mul... |
| subrgbas 20794 | Base set of a subring stru... |
| subrg1 20795 | A subring always has the s... |
| subrgacl 20796 | A subring is closed under ... |
| subrgmcl 20797 | A subring is closed under ... |
| subrgsubm 20798 | A subring is a submonoid o... |
| subrgdvds 20799 | If an element divides anot... |
| subrguss 20800 | A unit of a subring is a u... |
| subrginv 20801 | A subring always has the s... |
| subrgdv 20802 | A subring always has the s... |
| subrgunit 20803 | An element of a ring is a ... |
| subrgugrp 20804 | The units of a subring for... |
| issubrg2 20805 | Characterize the subrings ... |
| opprsubrg 20806 | Being a subring is a symme... |
| subrgnzr 20807 | A subring of a nonzero rin... |
| subrgint 20808 | The intersection of a none... |
| subrgin 20809 | The intersection of two su... |
| subrgmre 20810 | The subrings of a ring are... |
| subsubrg 20811 | A subring of a subring is ... |
| subsubrg2 20812 | The set of subrings of a s... |
| issubrg3 20813 | A subring is an additive s... |
| resrhm 20814 | Restriction of a ring homo... |
| resrhm2b 20815 | Restriction of the codomai... |
| rhmeql 20816 | The equalizer of two ring ... |
| rhmima 20817 | The homomorphic image of a... |
| rnrhmsubrg 20818 | The range of a ring homomo... |
| cntzsubr 20819 | Centralizers in a ring are... |
| pwsdiagrhm 20820 | Diagonal homomorphism into... |
| subrgpropd 20821 | If two structures have the... |
| rhmpropd 20822 | Ring homomorphism depends ... |
| rgspnval 20825 | Value of the ring-span of ... |
| rgspncl 20826 | The ring-span of a set is ... |
| rgspnssid 20827 | The ring-span of a set con... |
| rgspnmin 20828 | The ring-span is contained... |
| rngcval 20831 | Value of the category of n... |
| rnghmresfn 20832 | The class of non-unital ri... |
| rnghmresel 20833 | An element of the non-unit... |
| rngcbas 20834 | Set of objects of the cate... |
| rngchomfval 20835 | Set of arrows of the categ... |
| rngchom 20836 | Set of arrows of the categ... |
| elrngchom 20837 | A morphism of non-unital r... |
| rngchomfeqhom 20838 | The functionalized Hom-set... |
| rngccofval 20839 | Composition in the categor... |
| rngcco 20840 | Composition in the categor... |
| dfrngc2 20841 | Alternate definition of th... |
| rnghmsscmap2 20842 | The non-unital ring homomo... |
| rnghmsscmap 20843 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20844 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20845 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20846 | The non-unital ring homomo... |
| rngccat 20847 | The category of non-unital... |
| rngcid 20848 | The identity arrow in the ... |
| rngcsect 20849 | A section in the category ... |
| rngcinv 20850 | An inverse in the category... |
| rngciso 20851 | An isomorphism in the cate... |
| rngcifuestrc 20852 | The "inclusion functor" fr... |
| funcrngcsetc 20853 | The "natural forgetful fun... |
| funcrngcsetcALT 20854 | Alternate proof of ~ funcr... |
| zrinitorngc 20855 | The zero ring is an initia... |
| zrtermorngc 20856 | The zero ring is a termina... |
| zrzeroorngc 20857 | The zero ring is a zero ob... |
| ringcval 20860 | Value of the category of u... |
| rhmresfn 20861 | The class of unital ring h... |
| rhmresel 20862 | An element of the unital r... |
| ringcbas 20863 | Set of objects of the cate... |
| ringchomfval 20864 | Set of arrows of the categ... |
| ringchom 20865 | Set of arrows of the categ... |
| elringchom 20866 | A morphism of unital rings... |
| ringchomfeqhom 20867 | The functionalized Hom-set... |
| ringccofval 20868 | Composition in the categor... |
| ringcco 20869 | Composition in the categor... |
| dfringc2 20870 | Alternate definition of th... |
| rhmsscmap2 20871 | The unital ring homomorphi... |
| rhmsscmap 20872 | The unital ring homomorphi... |
| rhmsubcsetclem1 20873 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20874 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20875 | The unital ring homomorphi... |
| ringccat 20876 | The category of unital rin... |
| ringcid 20877 | The identity arrow in the ... |
| rhmsscrnghm 20878 | The unital ring homomorphi... |
| rhmsubcrngclem1 20879 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20880 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20881 | The unital ring homomorphi... |
| rngcresringcat 20882 | The restriction of the cat... |
| ringcsect 20883 | A section in the category ... |
| ringcinv 20884 | An inverse in the category... |
| ringciso 20885 | An isomorphism in the cate... |
| ringcbasbas 20886 | An element of the base set... |
| funcringcsetc 20887 | The "natural forgetful fun... |
| zrtermoringc 20888 | The zero ring is a termina... |
| zrninitoringc 20889 | The zero ring is not an in... |
| srhmsubclem1 20890 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20891 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20892 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20893 | According to ~ df-subc , t... |
| sringcat 20894 | The restriction of the cat... |
| crhmsubc 20895 | According to ~ df-subc , t... |
| cringcat 20896 | The restriction of the cat... |
| rngcrescrhm 20897 | The category of non-unital... |
| rhmsubclem1 20898 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20899 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20900 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20901 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20902 | According to ~ df-subc , t... |
| rhmsubccat 20903 | The restriction of the cat... |
| rrgval 20910 | Value of the set or left-r... |
| isrrg 20911 | Membership in the set of l... |
| rrgeq0i 20912 | Property of a left-regular... |
| rrgeq0 20913 | Left-multiplication by a l... |
| rrgsupp 20914 | Left multiplication by a l... |
| rrgss 20915 | Left-regular elements are ... |
| unitrrg 20916 | Units are regular elements... |
| rrgnz 20917 | In a nonzero ring, the zer... |
| isdomn 20918 | Expand definition of a dom... |
| domnnzr 20919 | A domain is a nonzero ring... |
| domnring 20920 | A domain is a ring. (Cont... |
| domneq0 20921 | In a domain, a product is ... |
| domnmuln0 20922 | In a domain, a product of ... |
| isdomn5 20923 | The equivalence between th... |
| isdomn2 20924 | A ring is a domain iff all... |
| domnrrg 20925 | In a domain, a nonzero ele... |
| isdomn6 20926 | A ring is a domain iff the... |
| isdomn3 20927 | Nonzero elements form a mu... |
| isdomn4 20928 | A ring is a domain iff it ... |
| opprdomnb 20929 | A class is a domain if and... |
| opprdomn 20930 | The opposite of a domain i... |
| isdomn4r 20931 | A ring is a domain iff it ... |
| domnlcanb 20932 | Left-cancellation law for ... |
| domnlcan 20933 | Left-cancellation law for ... |
| domnrcanb 20934 | Right-cancellation law for... |
| domnrcan 20935 | Right-cancellation law for... |
| domneq0r 20936 | Right multiplication by a ... |
| isidom 20937 | An integral domain is a co... |
| idomdomd 20938 | An integral domain is a do... |
| idomcringd 20939 | An integral domain is a co... |
| idomringd 20940 | An integral domain is a ri... |
| isdrng 20945 | The predicate "is a divisi... |
| drngunit 20946 | Elementhood in the set of ... |
| drngui 20947 | The set of units of a divi... |
| drngring 20948 | A division ring is a ring.... |
| drngringd 20949 | A division ring is a ring.... |
| drnggrpd 20950 | A division ring is a group... |
| drnggrp 20951 | A division ring is a group... |
| ringinveu 20952 | If a ring unit element ` X... |
| isdrng4 20953 | A division ring is a ring ... |
| isfld 20954 | A field is a commutative d... |
| flddrngd 20955 | A field is a division ring... |
| fldcrngd 20956 | A field is a commutative r... |
| drngprops 20957 | Properties of a division r... |
| isdrng2 20958 | A division ring can equiva... |
| drngprop 20959 | If two structures have the... |
| drngmgp 20960 | A division ring contains a... |
| drngid 20961 | A division ring's unity is... |
| drngunz 20962 | A division ring's unity is... |
| drngnzr 20963 | A division ring is a nonze... |
| drngdomn 20964 | A division ring is a domai... |
| isdrng3lem0 20965 | Lemma for ~ isdrng3 : The... |
| isdrng3lem1 20966 | Lemma for ~ isdrng3 (for t... |
| isdrng3lem2 20967 | Lemma for ~ isdrng3 (for t... |
| isdrng3 20968 | A division ring is a ring ... |
| isdrng5 20969 | A division ring is a ring ... |
| drngmcl 20970 | The product of two nonzero... |
| drngid2 20971 | Properties showing that an... |
| drnginvrcl 20972 | Closure of the multiplicat... |
| drnginvrn0 20973 | The multiplicative inverse... |
| drnginvrcld 20974 | Closure of the multiplicat... |
| drnginvrl 20975 | Property of the multiplica... |
| drnginvrr 20976 | Property of the multiplica... |
| drnginvrld 20977 | Property of the multiplica... |
| drnginvrrd 20978 | Property of the multiplica... |
| drngmul0or 20979 | A product is zero iff one ... |
| drngmulne0 20980 | A product is nonzero iff b... |
| drngmuleq0 20981 | An element is zero iff its... |
| opprdrng 20982 | The opposite of a division... |
| isdrngd 20983 | Properties that characteri... |
| isdrngrd 20984 | Properties that characteri... |
| isdrngdOLD 20985 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20986 | Obsolete version of ~ isdr... |
| zrdrng 20987 | A zero ring is not a divis... |
| drngpropd 20988 | If two structures have the... |
| fldpropd 20989 | If two structures have the... |
| fldidom 20990 | A field is an integral dom... |
| fidomndrnglem 20991 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20992 | A finite domain is a divis... |
| fiidomfld 20993 | A finite integral domain i... |
| rng1nnzr 20994 | The (smallest) structure r... |
| ring1zr 20995 | The only unital ring with ... |
| ringen1zr0 20996 | The only unital ring with ... |
| rng1nfld 20997 | The zero ring is not a fie... |
| issubdrg 20998 | Characterize the subfields... |
| drhmsubc 20999 | According to ~ df-subc , t... |
| drngcat 21000 | The restriction of the cat... |
| fldcat 21001 | The restriction of the cat... |
| fldc 21002 | The restriction of the cat... |
| fldhmsubc 21003 | According to ~ df-subc , t... |
| issdrg 21006 | Property of a division sub... |
| sdrgrcl 21007 | Reverse closure for a sub-... |
| sdrgdrng 21008 | A sub-division-ring is a d... |
| sdrgsubrg 21009 | A sub-division-ring is a s... |
| sdrgid 21010 | Every division ring is a d... |
| sdrgss 21011 | A division subring is a su... |
| sdrgbas 21012 | Base set of a sub-division... |
| issdrg2 21013 | Property of a division sub... |
| sdrgunit 21014 | A unit of a sub-division-r... |
| imadrhmcl 21015 | The image of a (nontrivial... |
| fldsdrgfld 21016 | A sub-division-ring of a f... |
| acsfn1p 21017 | Construction of a closure ... |
| subrgacs 21018 | Closure property of subrin... |
| sdrgacs 21019 | Closure property of divisi... |
| cntzsdrg 21020 | Centralizers in division r... |
| subdrgint 21021 | The intersection of a none... |
| sdrgint 21022 | The intersection of a none... |
| primefld 21023 | The smallest sub division ... |
| primefld0cl 21024 | The prime field contains t... |
| primefld1cl 21025 | The prime field contains t... |
| abvfval 21028 | Value of the set of absolu... |
| isabv 21029 | Elementhood in the set of ... |
| isabvd 21030 | Properties that determine ... |
| abvrcl 21031 | Reverse closure for the ab... |
| abvfge0 21032 | An absolute value is a fun... |
| abvf 21033 | An absolute value is a fun... |
| abvcl 21034 | An absolute value is a fun... |
| abvge0 21035 | The absolute value of a nu... |
| abveq0 21036 | The value of an absolute v... |
| abvne0 21037 | The absolute value of a no... |
| abvgt0 21038 | The absolute value of a no... |
| abvmul 21039 | An absolute value distribu... |
| abvtri 21040 | An absolute value satisfie... |
| abv0 21041 | The absolute value of zero... |
| abv1z 21042 | The absolute value of one ... |
| abv1 21043 | The absolute value of one ... |
| abvneg 21044 | The absolute value of a ne... |
| abvsubtri 21045 | An absolute value satisfie... |
| abvrec 21046 | The absolute value distrib... |
| abvdiv 21047 | The absolute value distrib... |
| abvdom 21048 | Any ring with an absolute ... |
| abvres 21049 | The restriction of an abso... |
| abvtrivd 21050 | The trivial absolute value... |
| abvtrivg 21051 | The trivial absolute value... |
| abvtriv 21052 | The trivial absolute value... |
| abvpropd 21053 | If two structures have the... |
| abvn0b 21054 | Another characterization o... |
| staffval 21059 | The functionalization of t... |
| stafval 21060 | The functionalization of t... |
| staffn 21061 | The functionalization is e... |
| issrng 21062 | The predicate "is a star r... |
| srngrhm 21063 | The involution function in... |
| srngring 21064 | A star ring is a ring. (C... |
| srngcnv 21065 | The involution function in... |
| srngf1o 21066 | The involution function in... |
| srngcl 21067 | The involution function in... |
| srngnvl 21068 | The involution function in... |
| srngadd 21069 | The involution function in... |
| srngmul 21070 | The involution function in... |
| srng1 21071 | The conjugate of the ring ... |
| srng0 21072 | The conjugate of the ring ... |
| issrngd 21073 | Properties that determine ... |
| idsrngd 21074 | A commutative ring is a st... |
| isorng 21079 | An ordered ring is a ring ... |
| orngring 21080 | An ordered ring is a ring.... |
| orngogrp 21081 | An ordered ring is an orde... |
| isofld 21082 | An ordered field is a fiel... |
| orngmul 21083 | In an ordered ring, the or... |
| orngsqr 21084 | In an ordered ring, all sq... |
| ornglmulle 21085 | In an ordered ring, multip... |
| orngrmulle 21086 | In an ordered ring, multip... |
| ornglmullt 21087 | In an ordered ring, multip... |
| orngrmullt 21088 | In an ordered ring, multip... |
| orngmullt 21089 | In an ordered ring, the st... |
| ofldfld 21090 | An ordered field is a fiel... |
| ofldtos 21091 | An ordered field is a tota... |
| orng0le1 21092 | In an ordered ring, the ri... |
| ofldlt1 21093 | In an ordered field, the r... |
| suborng 21094 | Every subring of an ordere... |
| subofld 21095 | Every subfield of an order... |
| islmod 21100 | The predicate "is a left m... |
| lmodlema 21101 | Lemma for properties of a ... |
| islmodd 21102 | Properties that determine ... |
| lmodgrp 21103 | A left module is a group. ... |
| lmodring 21104 | The scalar component of a ... |
| lmodfgrp 21105 | The scalar component of a ... |
| lmodgrpd 21106 | A left module is a group. ... |
| lmodbn0 21107 | The base set of a left mod... |
| lmodacl 21108 | Closure of ring addition f... |
| lmodmcl 21109 | Closure of ring multiplica... |
| lmodsn0 21110 | The set of scalars in a le... |
| lmodvacl 21111 | Closure of vector addition... |
| lmodass 21112 | Left module vector sum is ... |
| lmodlcan 21113 | Left cancellation law for ... |
| lmodvscl 21114 | Closure of scalar product ... |
| lmodvscld 21115 | Closure of scalar product ... |
| scaffval 21116 | The scalar multiplication ... |
| scafval 21117 | The scalar multiplication ... |
| scafeq 21118 | If the scalar multiplicati... |
| scaffn 21119 | The scalar multiplication ... |
| lmodscaf 21120 | The scalar multiplication ... |
| lmodvsdi 21121 | Distributive law for scala... |
| lmodvsdir 21122 | Distributive law for scala... |
| lmodvsass 21123 | Associative law for scalar... |
| lmod0cl 21124 | The ring zero in a left mo... |
| lmod1cl 21125 | The ring unity in a left m... |
| lmodvs1 21126 | Scalar product with the ri... |
| lmod0vcl 21127 | The zero vector is a vecto... |
| lmod0vlid 21128 | Left identity law for the ... |
| lmod0vrid 21129 | Right identity law for the... |
| lmod0vid 21130 | Identity equivalent to the... |
| lmod0vs 21131 | Zero times a vector is the... |
| lmodvs0 21132 | Anything times the zero ve... |
| lmodvsmmulgdi 21133 | Distributive law for a gro... |
| lmodfopnelem1 21134 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 21135 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 21136 | The (functionalized) opera... |
| lcomf 21137 | A linear-combination sum i... |
| lcomfsupp 21138 | A linear-combination sum i... |
| lmodvnegcl 21139 | Closure of vector negative... |
| lmodvnegid 21140 | Addition of a vector with ... |
| lmodvneg1 21141 | Minus 1 times a vector is ... |
| lmodvsneg 21142 | Multiplication of a vector... |
| lmodvsubcl 21143 | Closure of vector subtract... |
| lmodcom 21144 | Left module vector sum is ... |
| lmodabl 21145 | A left module is an abelia... |
| lmodcmn 21146 | A left module is a commuta... |
| lmodnegadd 21147 | Distribute negation throug... |
| lmod4 21148 | Commutative/associative la... |
| lmodvsubadd 21149 | Relationship between vecto... |
| lmodvaddsub4 21150 | Vector addition/subtractio... |
| lmodvpncan 21151 | Addition/subtraction cance... |
| lmodvnpcan 21152 | Cancellation law for vecto... |
| lmodvsubval2 21153 | Value of vector subtractio... |
| lmodsubvs 21154 | Subtraction of a scalar pr... |
| lmodsubdi 21155 | Scalar multiplication dist... |
| lmodsubdir 21156 | Scalar multiplication dist... |
| lmodsubeq0 21157 | If the difference between ... |
| lmodsubid 21158 | Subtraction of a vector fr... |
| lmodvsghm 21159 | Scalar multiplication of t... |
| lmodprop2d 21160 | If two structures have the... |
| lmodpropd 21161 | If two structures have the... |
| gsumvsmul 21162 | Pull a scalar multiplicati... |
| mptscmfsupp0 21163 | A mapping to a scalar prod... |
| mptscmfsuppd 21164 | A function mapping to a sc... |
| rmodislmodlem 21165 | Lemma for ~ rmodislmod . ... |
| rmodislmod 21166 | The right module ` R ` ind... |
| lssset 21169 | The set of all (not necess... |
| islss 21170 | The predicate "is a subspa... |
| islssd 21171 | Properties that determine ... |
| lssss 21172 | A subspace is a set of vec... |
| lssel 21173 | A subspace member is a vec... |
| lss1 21174 | The set of vectors in a le... |
| lssuni 21175 | The union of all subspaces... |
| lssn0 21176 | A subspace is not empty. ... |
| 00lss 21177 | The empty structure has no... |
| lsscl 21178 | Closure property of a subs... |
| lssvacl 21179 | Closure of vector addition... |
| lssvsubcl 21180 | Closure of vector subtract... |
| lssvancl1 21181 | Non-closure: if one vector... |
| lssvancl2 21182 | Non-closure: if one vector... |
| lss0cl 21183 | The zero vector belongs to... |
| lsssn0 21184 | The singleton of the zero ... |
| lss0ss 21185 | The zero subspace is inclu... |
| lssle0 21186 | No subspace is smaller tha... |
| lssne0 21187 | A nonzero subspace has a n... |
| lssvneln0 21188 | A vector ` X ` which doesn... |
| lssneln0 21189 | A vector ` X ` which doesn... |
| lssssr 21190 | Conclude subspace ordering... |
| lssvscl 21191 | Closure of scalar product ... |
| lssvnegcl 21192 | Closure of negative vector... |
| lsssubg 21193 | All subspaces are subgroup... |
| lsssssubg 21194 | All subspaces are subgroup... |
| islss3 21195 | A linear subspace of a mod... |
| lsslmod 21196 | A submodule is a module. ... |
| lsslss 21197 | The subspaces of a subspac... |
| islss4 21198 | A linear subspace is a sub... |
| lss1d 21199 | One-dimensional subspace (... |
| lssintcl 21200 | The intersection of a none... |
| lssincl 21201 | The intersection of two su... |
| lssmre 21202 | The subspaces of a module ... |
| lssacs 21203 | Submodules are an algebrai... |
| prdsvscacl 21204 | Pointwise scalar multiplic... |
| prdslmodd 21205 | The product of a family of... |
| pwslmod 21206 | A structure power of a lef... |
| lspfval 21209 | The span function for a le... |
| lspf 21210 | The span function on a lef... |
| lspval 21211 | The span of a set of vecto... |
| lspcl 21212 | The span of a set of vecto... |
| lspsncl 21213 | The span of a singleton is... |
| lspprcl 21214 | The span of a pair is a su... |
| lsptpcl 21215 | The span of an unordered t... |
| lspsnsubg 21216 | The span of a singleton is... |
| 00lsp 21217 | ~ fvco4i lemma for linear ... |
| lspid 21218 | The span of a subspace is ... |
| lspssv 21219 | A span is a set of vectors... |
| lspss 21220 | Span preserves subset orde... |
| lspssid 21221 | A set of vectors is a subs... |
| lspidm 21222 | The span of a set of vecto... |
| lspun 21223 | The span of union is the s... |
| lspssp 21224 | If a set of vectors is a s... |
| mrclsp 21225 | Moore closure generalizes ... |
| lspsnss 21226 | The span of the singleton ... |
| ellspsn3 21227 | A member of the span of th... |
| lspprss 21228 | The span of a pair of vect... |
| lspsnid 21229 | A vector belongs to the sp... |
| ellspsn6 21230 | Relationship between a vec... |
| ellspsn5b 21231 | Relationship between a vec... |
| ellspsn5 21232 | Relationship between a vec... |
| lspprid1 21233 | A member of a pair of vect... |
| lspprid2 21234 | A member of a pair of vect... |
| lspprvacl 21235 | The sum of two vectors bel... |
| lssats2 21236 | A way to express atomistic... |
| ellspsni 21237 | A scalar product with a ve... |
| lspsn 21238 | Span of the singleton of a... |
| ellspsn 21239 | Member of span of the sing... |
| lspsnvsi 21240 | Span of a scalar product o... |
| lspsnss2 21241 | Comparable spans of single... |
| lspsnneg 21242 | Negation does not change t... |
| lspsnsub 21243 | Swapping subtraction order... |
| lspsn0 21244 | Span of the singleton of t... |
| lsp0 21245 | Span of the empty set. (C... |
| lspuni0 21246 | Union of the span of the e... |
| lspun0 21247 | The span of a union with t... |
| lspsneq0 21248 | Span of the singleton is t... |
| lspsneq0b 21249 | Equal singleton spans impl... |
| lmodindp1 21250 | Two independent (non-colin... |
| lsslsp 21251 | Spans in submodules corres... |
| lss0v 21252 | The zero vector in a submo... |
| lsspropd 21253 | If two structures have the... |
| lsppropd 21254 | If two structures have the... |
| reldmlmhm 21261 | Lemma for module homomorph... |
| lmimfn 21262 | Lemma for module isomorphi... |
| islmhm 21263 | Property of being a homomo... |
| islmhm3 21264 | Property of a module homom... |
| lmhmlem 21265 | Non-quantified consequence... |
| lmhmsca 21266 | A homomorphism of left mod... |
| lmghm 21267 | A homomorphism of left mod... |
| lmhmlmod2 21268 | A homomorphism of left mod... |
| lmhmlmod1 21269 | A homomorphism of left mod... |
| lmhmf 21270 | A homomorphism of left mod... |
| lmhmlin 21271 | A homomorphism of left mod... |
| lmodvsinv 21272 | Multiplication of a vector... |
| lmodvsinv2 21273 | Multiplying a negated vect... |
| islmhm2 21274 | A one-equation proof of li... |
| islmhmd 21275 | Deduction for a module hom... |
| 0lmhm 21276 | The constant zero linear f... |
| idlmhm 21277 | The identity function on a... |
| invlmhm 21278 | The negative function on a... |
| lmhmco 21279 | The composition of two mod... |
| lmhmplusg 21280 | The pointwise sum of two l... |
| lmhmvsca 21281 | The pointwise scalar produ... |
| lmhmf1o 21282 | A bijective module homomor... |
| lmhmima 21283 | The image of a subspace un... |
| lmhmpreima 21284 | The inverse image of a sub... |
| lmhmlsp 21285 | Homomorphisms preserve spa... |
| lmhmrnlss 21286 | The range of a homomorphis... |
| lmhmkerlss 21287 | The kernel of a homomorphi... |
| reslmhm 21288 | Restriction of a homomorph... |
| reslmhm2 21289 | Expansion of the codomain ... |
| reslmhm2b 21290 | Expansion of the codomain ... |
| lmhmeql 21291 | The equalizer of two modul... |
| lspextmo 21292 | A linear function is compl... |
| pwsdiaglmhm 21293 | Diagonal homomorphism into... |
| pwssplit0 21294 | Splitting for structure po... |
| pwssplit1 21295 | Splitting for structure po... |
| pwssplit2 21296 | Splitting for structure po... |
| pwssplit3 21297 | Splitting for structure po... |
| islmim 21298 | An isomorphism of left mod... |
| lmimf1o 21299 | An isomorphism of left mod... |
| lmimlmhm 21300 | An isomorphism of modules ... |
| lmimgim 21301 | An isomorphism of modules ... |
| islmim2 21302 | An isomorphism of left mod... |
| lmimcnv 21303 | The converse of a bijectiv... |
| brlmic 21304 | The relation "is isomorphi... |
| brlmici 21305 | Prove isomorphic by an exp... |
| lmiclcl 21306 | Isomorphism implies the le... |
| lmicrcl 21307 | Isomorphism implies the ri... |
| lmicsym 21308 | Module isomorphism is symm... |
| lmhmpropd 21309 | Module homomorphism depend... |
| islbs 21312 | The predicate " ` B ` is a... |
| lbsss 21313 | A basis is a set of vector... |
| lbsel 21314 | An element of a basis is a... |
| lbssp 21315 | The span of a basis is the... |
| lbsind 21316 | A basis is linearly indepe... |
| lbsind2 21317 | A basis is linearly indepe... |
| lbspss 21318 | No proper subset of a basi... |
| lsmcl 21319 | The sum of two subspaces i... |
| lsmspsn 21320 | Member of subspace sum of ... |
| lsmelval2 21321 | Subspace sum membership in... |
| lsmsp 21322 | Subspace sum in terms of s... |
| lsmsp2 21323 | Subspace sum of spans of s... |
| lsmssspx 21324 | Subspace sum (in its exten... |
| lsmpr 21325 | The span of a pair of vect... |
| lsppreli 21326 | A vector expressed as a su... |
| lsmelpr 21327 | Two ways to say that a vec... |
| lsppr0 21328 | The span of a vector paire... |
| lsppr 21329 | Span of a pair of vectors.... |
| lspprel 21330 | Member of the span of a pa... |
| lspprabs 21331 | Absorption of vector sum i... |
| lspvadd 21332 | The span of a vector sum i... |
| lspsntri 21333 | Triangle-type inequality f... |
| lspsntrim 21334 | Triangle-type inequality f... |
| lbspropd 21335 | If two structures have the... |
| pj1lmhm 21336 | The left projection functi... |
| pj1lmhm2 21337 | The left projection functi... |
| islvec 21340 | The predicate "is a left v... |
| lvecdrng 21341 | The set of scalars of a le... |
| lveclmod 21342 | A left vector space is a l... |
| lveclmodd 21343 | A vector space is a left m... |
| lvecgrpd 21344 | A vector space is a group.... |
| lsslvec 21345 | A vector subspace is a vec... |
| lmhmlvec 21346 | The property for modules t... |
| lvecvs0or 21347 | If a scalar product is zer... |
| lvecvsn0 21348 | A scalar product is nonzer... |
| lssvs0or 21349 | If a scalar product belong... |
| lvecvscan 21350 | Cancellation law for scala... |
| lvecvscan2 21351 | Cancellation law for scala... |
| lvecinv 21352 | Invert coefficient of scal... |
| lspsnvs 21353 | A nonzero scalar product d... |
| lspsneleq 21354 | Membership relation that i... |
| lspsncmp 21355 | Comparable spans of nonzer... |
| lspsnne1 21356 | Two ways to express that v... |
| lspsnne2 21357 | Two ways to express that v... |
| lspsnnecom 21358 | Swap two vectors with diff... |
| lspabs2 21359 | Absorption law for span of... |
| lspabs3 21360 | Absorption law for span of... |
| lspsneq 21361 | Equal spans of singletons ... |
| lspsneu 21362 | Nonzero vectors with equal... |
| ellspsn4 21363 | A member of the span of th... |
| lspdisj 21364 | The span of a vector not i... |
| lspdisjb 21365 | A nonzero vector is not in... |
| lspdisj2 21366 | Unequal spans are disjoint... |
| lspfixed 21367 | Show membership in the spa... |
| lspexch 21368 | Exchange property for span... |
| lspexchn1 21369 | Exchange property for span... |
| lspexchn2 21370 | Exchange property for span... |
| lspindpi 21371 | Partial independence prope... |
| lspindp1 21372 | Alternate way to say 3 vec... |
| lspindp2l 21373 | Alternate way to say 3 vec... |
| lspindp2 21374 | Alternate way to say 3 vec... |
| lspindp3 21375 | Independence of 2 vectors ... |
| lspindp4 21376 | (Partial) independence of ... |
| lvecindp 21377 | Compute the ` X ` coeffici... |
| lvecindp2 21378 | Sums of independent vector... |
| lspsnsubn0 21379 | Unequal singleton spans im... |
| lsmcv 21380 | Subspace sum has the cover... |
| lspsolvlem 21381 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21382 | If ` X ` is in the span of... |
| lssacsex 21383 | In a vector space, subspac... |
| lspsnat 21384 | There is no subspace stric... |
| lspsncv0 21385 | The span of a singleton co... |
| lsppratlem1 21386 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21387 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21388 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21389 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21390 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21391 | Lemma for ~ lspprat . Neg... |
| lspprat 21392 | A proper subspace of the s... |
| islbs2 21393 | An equivalent formulation ... |
| islbs3 21394 | An equivalent formulation ... |
| lbsacsbs 21395 | Being a basis in a vector ... |
| lvecdim 21396 | The dimension theorem for ... |
| lbsextlem1 21397 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21398 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21399 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21400 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21401 | For any linearly independe... |
| lbsext 21402 | For any linearly independe... |
| lbsexg 21403 | Every vector space has a b... |
| lbsex 21404 | Every vector space has a b... |
| lvecprop2d 21405 | If two structures have the... |
| lvecpropd 21406 | If two structures have the... |
| sraval 21411 | Lemma for ~ srabase throug... |
| sralem 21412 | Lemma for ~ srabase and si... |
| srabase 21413 | Base set of a subring alge... |
| sraaddg 21414 | Additive operation of a su... |
| sramulr 21415 | Multiplicative operation o... |
| srasca 21416 | The set of scalars of a su... |
| sravsca 21417 | The scalar product operati... |
| sraip 21418 | The inner product operatio... |
| sratset 21419 | Topology component of a su... |
| sratopn 21420 | Topology component of a su... |
| srads 21421 | Distance function of a sub... |
| sraring 21422 | Condition for a subring al... |
| sralmod 21423 | The subring algebra is a l... |
| sralmod0 21424 | The subring module inherit... |
| issubrgd 21425 | Prove a subring by closure... |
| rlmfn 21426 | ` ringLMod ` is a function... |
| rlmval 21427 | Value of the ring module. ... |
| rlmval2 21428 | Value of the ring module e... |
| rlmbas 21429 | Base set of the ring modul... |
| rlmplusg 21430 | Vector addition in the rin... |
| rlm0 21431 | Zero vector in the ring mo... |
| rlmsub 21432 | Subtraction in the ring mo... |
| rlmmulr 21433 | Ring multiplication in the... |
| rlmsca 21434 | Scalars in the ring module... |
| rlmsca2 21435 | Scalars in the ring module... |
| rlmvsca 21436 | Scalar multiplication in t... |
| rlmtopn 21437 | Topology component of the ... |
| rlmds 21438 | Metric component of the ri... |
| rlmlmod 21439 | The ring module is a modul... |
| rlmlvec 21440 | The ring module over a div... |
| rlmlsm 21441 | Subgroup sum of the ring m... |
| rlmvneg 21442 | Vector negation in the rin... |
| rlmscaf 21443 | Functionalized scalar mult... |
| ixpsnbasval 21444 | The value of an infinite C... |
| lidlval 21449 | Value of the set of ring i... |
| rspval 21450 | Value of the ring span fun... |
| lidlss 21451 | An ideal is a subset of th... |
| lidlbasel 21452 | An element of an ideal is ... |
| lidlssbas 21453 | The base set of the restri... |
| lidlbas 21454 | A (left) ideal of a ring i... |
| islidl 21455 | Predicate of being a (left... |
| rnglidlmcl 21456 | A (left) ideal containing ... |
| rngridlmcl 21457 | A right ideal (which is a ... |
| dflidl2rng 21458 | Alternate (the usual textb... |
| isridlrng 21459 | A right ideal is a left id... |
| lidl0cl 21460 | An ideal contains 0. (Con... |
| lidlacl 21461 | An ideal is closed under a... |
| lidlnegcl 21462 | An ideal contains negative... |
| lidlsubg 21463 | An ideal is a subgroup of ... |
| lidlsubcl 21464 | An ideal is closed under s... |
| lidlmcl 21465 | An ideal is closed under l... |
| lidl1el 21466 | An ideal contains 1 iff it... |
| lidlmcld 21467 | An ideal is closed under l... |
| dflidl2 21468 | Alternate (the usual textb... |
| lidl0ALT 21469 | Alternate proof for ~ lidl... |
| rnglidl0 21470 | Every non-unital ring cont... |
| lidl0 21471 | Every ring contains a zero... |
| lidl1ALT 21472 | Alternate proof for ~ lidl... |
| rnglidl1 21473 | The base set of every non-... |
| lidl1 21474 | Every ring contains a unit... |
| 0ringidl 21475 | The zero ideal is the only... |
| lidlunin0 21476 | The union of a nonempty su... |
| unichnlidl 21477 | The union of a nonempty ch... |
| lidlacs 21478 | The ideal system is an alg... |
| rspcl 21479 | The span of a set of ring ... |
| rspssid 21480 | The span of a set of ring ... |
| rsp1 21481 | The span of the identity e... |
| rsp0 21482 | The span of the zero eleme... |
| rspssp 21483 | The ideal span of a set of... |
| rspvalint 21484 | The ideal generated by a s... |
| rspprop 21485 | Properties of a class to b... |
| elrspsn 21486 | Membership in a principal ... |
| rspsn0 21487 | A principal ideal (an idea... |
| rspsnid 21488 | A principal ideal contains... |
| pidlnz 21489 | A principal ideal generate... |
| mrcrsp 21490 | Moore closure generalizes ... |
| lidlnz 21491 | A nonzero ideal contains a... |
| drngnidl 21492 | A division ring has only t... |
| lidlrsppropd 21493 | The left ideals and ring s... |
| rnglidlmmgm 21494 | The multiplicative group o... |
| rnglidlmsgrp 21495 | The multiplicative group o... |
| rnglidlrng 21496 | A (left) ideal of a non-un... |
| lidlnsg 21497 | An ideal is a normal subgr... |
| lsmidllsp 21498 | The sum of two ideals is t... |
| lsmidl 21499 | The sum of two ideals is a... |
| drngidl 21500 | A nonzero ring is a divisi... |
| isfieldidl 21501 | Determine if a ring is a f... |
| isfieldidl2 21502 | Determine if a ring is a f... |
| 2idlval 21505 | Definition of a two-sided ... |
| isridl 21506 | A right ideal is a left id... |
| 2idlelb 21507 | Membership in a two-sided ... |
| 2idllidld 21508 | A two-sided ideal is a lef... |
| 2idlridld 21509 | A two-sided ideal is a rig... |
| 2idl1el 21510 | A two-sided ideal contains... |
| df2idl2rng 21511 | Alternate (the usual textb... |
| df2idl2 21512 | Alternate (the usual textb... |
| ridl0 21513 | Every ring contains a zero... |
| ridl1 21514 | Every ring contains a unit... |
| 2idl0 21515 | Every ring contains a zero... |
| 2idl1 21516 | Every ring contains a unit... |
| 2idlss 21517 | A two-sided ideal is a sub... |
| 2idlbas 21518 | The base set of a two-side... |
| 2idlelbas 21519 | The base set of a two-side... |
| rng2idlsubrng 21520 | A two-sided ideal of a non... |
| rng2idlnsg 21521 | A two-sided ideal of a non... |
| rng2idl0 21522 | The zero (additive identit... |
| rng2idlsubgsubrng 21523 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21524 | A two-sided ideal of a non... |
| rng2idlsubg0 21525 | The zero (additive identit... |
| 2idlcpblrng 21526 | The coset equivalence rela... |
| 2idlcpbl 21527 | The coset equivalence rela... |
| qus2idrng 21528 | The quotient of a non-unit... |
| qus1 21529 | The multiplicative identit... |
| qusring 21530 | If ` S ` is a two-sided id... |
| qusrhm 21531 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21532 | The preimage of an ideal b... |
| kerlidl 21533 | The kernel of a ring homom... |
| ker2idl 21534 | The kernel of a ring homom... |
| qusmul2idl 21535 | Value of the ring operatio... |
| crngridl 21536 | In a commutative ring, the... |
| crng2idl 21537 | In a commutative ring, a t... |
| df2idl2crng 21538 | The predicate "is an ideal... |
| qusmulrng 21539 | Value of the multiplicatio... |
| quscrng 21540 | The quotient of a commutat... |
| qusmulcrng 21541 | Value of the ring operatio... |
| rhmqusnsg 21542 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21543 | The ring unity of a two-si... |
| rngqiprngghmlem1 21544 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21545 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21546 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21547 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21548 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21549 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21550 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21551 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21552 | The base set of the produc... |
| rngqiprng 21553 | The product of the quotien... |
| rngqiprngimf 21554 | ` F ` is a function from (... |
| rngqiprngimfv 21555 | The value of the function ... |
| rngqiprngghm 21556 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21557 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21558 | ` F ` is a function from (... |
| rngqiprnglin 21559 | ` F ` is linear with respe... |
| rngqiprngho 21560 | ` F ` is a homomorphism of... |
| rngqiprngim 21561 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21562 | The unity of a two-sided i... |
| rngringbdlem1 21563 | In a unital ring, the quot... |
| rngringbdlem2 21564 | A non-unital ring is unita... |
| rngringbd 21565 | A non-unital ring is unita... |
| ring2idlqus 21566 | For every unital ring ther... |
| ring2idlqusb 21567 | A non-unital ring is unita... |
| rngqiprngfulem1 21568 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21569 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21570 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21571 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21572 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21573 | The ring unity of the prod... |
| rngqiprngfu 21574 | The function value of ` F ... |
| rngqiprngu 21575 | If a non-unital ring has a... |
| ring2idlqus1 21576 | If a non-unital ring has a... |
| prmidlval 21579 | The class of prime ideals ... |
| isprmidl 21580 | The predicate "is a prime ... |
| prmidlnr 21581 | A prime ideal is a proper ... |
| prmidl 21582 | The main property of a pri... |
| prmidl2 21583 | A condition that shows an ... |
| idlmulssprm 21584 | Let ` P ` be a prime ideal... |
| pridln1 21585 | A proper ideal cannot cont... |
| prmidlidl 21586 | A prime ideal is an ideal.... |
| prmidlssidl 21587 | Prime ideals as a subset o... |
| cringm4 21588 | Commutative/associative la... |
| isprmidlc 21589 | The predicate "is prime id... |
| prmidlc 21590 | Property of a prime ideal ... |
| prmidlc2 21591 | Property of a prime ideal ... |
| cmprmidlmcl 21592 | The complement of a prime ... |
| prmidlprop 21593 | Property of prime ideals. ... |
| 0ringprmidl 21594 | The trivial ring does not ... |
| prmidl0 21595 | The zero ideal of a commut... |
| rhmpreimaprmidl 21596 | The preimage of a prime id... |
| qsidomlem1 21597 | If the quotient ring of a ... |
| qsidomlem2 21598 | A quotient by a prime idea... |
| qsidom 21599 | An ideal ` I ` in the comm... |
| qsnzr 21600 | A quotient of a nonzero ri... |
| ssdifidllem 21601 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 21602 | Let ` R ` be a ring, and l... |
| ssdifidlprm 21603 | If the set ` S ` of ~ ssdi... |
| prmidlsubm 21604 | The complement of a prime ... |
| lpival 21609 | Value of the set of princi... |
| islpidl 21610 | Property of being a princi... |
| lpi0 21611 | The zero ideal is always p... |
| lpi1 21612 | The unit ideal is always p... |
| islpir 21613 | Principal ideal rings are ... |
| lpiss 21614 | Principal ideals are a sub... |
| islpir2 21615 | Principal ideal rings are ... |
| lpirring 21616 | Principal ideal rings are ... |
| drnglpir 21617 | Division rings are princip... |
| rspsn 21618 | Membership in principal id... |
| lidldvgen 21619 | An element generates an id... |
| lpigen 21620 | An ideal is principal iff ... |
| cnfldstr 21641 | The field of complex numbe... |
| cnfldex 21642 | The field of complex numbe... |
| cnfldbas 21643 | The base set of the field ... |
| mpocnfldadd 21644 | The addition operation of ... |
| cnfldadd 21645 | The addition operation of ... |
| mpocnfldmul 21646 | The multiplication operati... |
| cnfldmul 21647 | The multiplication operati... |
| cnfldcj 21648 | The conjugation operation ... |
| cnfldtset 21649 | The topology component of ... |
| cnfldle 21650 | The ordering of the field ... |
| cnfldds 21651 | The metric of the field of... |
| cnfldunif 21652 | The uniform structure comp... |
| cnfldfun 21653 | The field of complex numbe... |
| cnfldfunALT 21654 | The field of complex numbe... |
| xrsstr 21655 | The extended real structur... |
| xrsex 21656 | The extended real structur... |
| xrsadd 21657 | The addition operation of ... |
| xrsmul 21658 | The multiplication operati... |
| xrstset 21659 | The topology component of ... |
| cncrng 21660 | The complex numbers form a... |
| cnring 21661 | The complex numbers form a... |
| xrsmcmn 21662 | The "multiplicative group"... |
| cnfld0 21663 | Zero is the zero element o... |
| cnfld1 21664 | One is the unity element o... |
| cnfldneg 21665 | The additive inverse in th... |
| cnfldplusf 21666 | The functionalized additio... |
| cnfldsub 21667 | The subtraction operator i... |
| cndrng 21668 | The complex numbers form a... |
| cnflddiv 21669 | The division operation in ... |
| cnfldinv 21670 | The multiplicative inverse... |
| cnfldmulg 21671 | The group multiple functio... |
| cnfldexp 21672 | The exponentiation operato... |
| cnsrng 21673 | The complex numbers form a... |
| xrsmgm 21674 | The "additive group" of th... |
| xrsnsgrp 21675 | The "additive group" of th... |
| xrsmgmdifsgrp 21676 | The "additive group" of th... |
| xrsds 21677 | The metric of the extended... |
| xrsdsval 21678 | The metric of the extended... |
| xrsdsreval 21679 | The metric of the extended... |
| xrsdsreclblem 21680 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21681 | The metric of the extended... |
| cnsubmlem 21682 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21683 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21684 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 21685 | Lemma for ~ resubdrg and f... |
| qsubdrg 21686 | The rational numbers form ... |
| zsubrg 21687 | The integers form a subrin... |
| gzsubrg 21688 | The gaussian integers form... |
| nn0subm 21689 | The nonnegative integers f... |
| rege0subm 21690 | The nonnegative reals form... |
| absabv 21691 | The regular absolute value... |
| zsssubrg 21692 | The integers are a subset ... |
| qsssubdrg 21693 | The rational numbers are a... |
| cnsubrg 21694 | There are no subrings of t... |
| cnmgpabl 21695 | The unit group of the comp... |
| cnmgpid 21696 | The group identity element... |
| cnmsubglem 21697 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21698 | The positive reals form a ... |
| gzrngunitlem 21699 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21700 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21701 | Relate a group sum on ` CC... |
| regsumfsum 21702 | Relate a group sum on ` ( ... |
| expmhm 21703 | Exponentiation is a monoid... |
| nn0srg 21704 | The nonnegative integers f... |
| rge0srg 21705 | The nonnegative real numbe... |
| xrge0plusg 21706 | The additive law of the ex... |
| xrs1mnd 21707 | The extended real numbers,... |
| xrs10 21708 | The zero of the extended r... |
| xrs1cmn 21709 | The extended real numbers ... |
| xrge0subm 21710 | The nonnegative extended r... |
| xrge0cmn 21711 | The nonnegative extended r... |
| xrge0omnd 21712 | The nonnegative extended r... |
| zringcrng 21715 | The ring of integers is a ... |
| zringring 21716 | The ring of integers is a ... |
| zringrng 21717 | The ring of integers is a ... |
| zringabl 21718 | The ring of integers is an... |
| zringgrp 21719 | The ring of integers is an... |
| zringbas 21720 | The integers are the base ... |
| zringplusg 21721 | The addition operation of ... |
| zringsub 21722 | The subtraction of element... |
| zringmulg 21723 | The multiplication (group ... |
| zringmulr 21724 | The multiplication operati... |
| zring0 21725 | The zero element of the ri... |
| zring1 21726 | The unity element of the r... |
| zringnzr 21727 | The ring of integers is a ... |
| dvdsrzring 21728 | Ring divisibility in the r... |
| zringlpirlem1 21729 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21730 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21731 | Lemma for ~ zringlpir . A... |
| zringinvg 21732 | The additive inverse of an... |
| zringunit 21733 | The units of ` ZZ ` are th... |
| zringlpir 21734 | The integers are a princip... |
| zringndrg 21735 | The integers are not a div... |
| zringcyg 21736 | The integers are a cyclic ... |
| zringsubgval 21737 | Subtraction in the ring of... |
| zringmpg 21738 | The multiplicative group o... |
| prmirredlem 21739 | A positive integer is irre... |
| dfprm2 21740 | The positive irreducible e... |
| prmirred 21741 | The irreducible elements o... |
| expghm 21742 | Exponentiation is a group ... |
| mulgghm2 21743 | The powers of a group elem... |
| mulgrhm 21744 | The powers of the element ... |
| mulgrhm2 21745 | The powers of the element ... |
| irinitoringc 21746 | The ring of integers is an... |
| nzerooringczr 21747 | There is no zero object in... |
| pzriprnglem1 21748 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21749 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21750 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21751 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21752 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21753 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21754 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21755 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21756 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21757 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21758 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21759 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21760 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21761 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21762 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21763 | The ring unity of the ring... |
| pzriprng 21764 | The non-unital ring ` ( ZZ... |
| pzriprng1 21765 | The ring unity of the ring... |
| zrhval 21774 | Define the unique homomorp... |
| zrhval2 21775 | Alternate value of the ` Z... |
| zrhmulg 21776 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21777 | The ` ZRHom ` homomorphism... |
| zrhrhm 21778 | The ` ZRHom ` homomorphism... |
| zrh1 21779 | Interpretation of 1 in a r... |
| zrh0 21780 | Interpretation of 0 in a r... |
| zrhpropd 21781 | The ` ZZ ` ring homomorphi... |
| zlmval 21782 | Augment an abelian group w... |
| zlmlem 21783 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21784 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21785 | Group operation of a ` ZZ ... |
| zlmmulr 21786 | Ring operation of a ` ZZ `... |
| zlmsca 21787 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21788 | Scalar multiplication oper... |
| zlmlmod 21789 | The ` ZZ ` -module operati... |
| chrval 21790 | Definition substitution of... |
| chrcl 21791 | Closure of the characteris... |
| chrid 21792 | The canonical ` ZZ ` ring ... |
| chrdvds 21793 | The ` ZZ ` ring homomorphi... |
| chrcong 21794 | If two integers are congru... |
| dvdschrmulg 21795 | In a ring, any multiple of... |
| fermltlchr 21796 | A generalization of Fermat... |
| chrnzr 21797 | Nonzero rings are precisel... |
| chrrhm 21798 | The characteristic restric... |
| domnchr 21799 | The characteristic of a do... |
| znlidl 21800 | The set ` n ZZ ` is an ide... |
| zncrng2 21801 | Making a commutative ring ... |
| znval 21802 | The value of the ` Z/nZ ` ... |
| znle 21803 | The value of the ` Z/nZ ` ... |
| znval2 21804 | Self-referential expressio... |
| znbaslem 21805 | Lemma for ~ znbas . (Cont... |
| znbas2 21806 | The base set of ` Z/nZ ` i... |
| znadd 21807 | The additive structure of ... |
| znmul 21808 | The multiplicative structu... |
| znzrh 21809 | The ` ZZ ` ring homomorphi... |
| znbas 21810 | The base set of ` Z/nZ ` s... |
| zncrng 21811 | ` Z/nZ ` is a commutative ... |
| znzrh2 21812 | The ` ZZ ` ring homomorphi... |
| znzrhval 21813 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21814 | The ` ZZ ` ring homomorphi... |
| zncyg 21815 | The group ` ZZ / n ZZ ` is... |
| zndvds 21816 | Express equality of equiva... |
| zndvds0 21817 | Special case of ~ zndvds w... |
| znf1o 21818 | The function ` F ` enumera... |
| zzngim 21819 | The ` ZZ ` ring homomorphi... |
| znle2 21820 | The ordering of the ` Z/nZ... |
| znleval 21821 | The ordering of the ` Z/nZ... |
| znleval2 21822 | The ordering of the ` Z/nZ... |
| zntoslem 21823 | Lemma for ~ zntos . (Cont... |
| zntos 21824 | The ` Z/nZ ` structure is ... |
| znhash 21825 | The ` Z/nZ ` structure has... |
| znfi 21826 | The ` Z/nZ ` structure is ... |
| znfld 21827 | The ` Z/nZ ` structure is ... |
| znidomb 21828 | The ` Z/nZ ` structure is ... |
| znchr 21829 | Cyclic rings are defined b... |
| znunit 21830 | The units of ` Z/nZ ` are ... |
| znunithash 21831 | The size of the unit group... |
| znrrg 21832 | The regular elements of ` ... |
| cygznlem1 21833 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21834 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21835 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21836 | A cyclic group with ` n ` ... |
| cygzn 21837 | A cyclic group with ` n ` ... |
| cygth 21838 | The "fundamental theorem o... |
| cyggic 21839 | Cyclic groups are isomorph... |
| frgpcyg 21840 | A free group is cyclic iff... |
| freshmansdream 21841 | For a prime number ` P ` ,... |
| frobrhm 21842 | In a commutative ring with... |
| ofldchr 21843 | The characteristic of an o... |
| cnmsgnsubg 21844 | The signs form a multiplic... |
| cnmsgnbas 21845 | The base set of the sign s... |
| cnmsgngrp 21846 | The group of signs under m... |
| psgnghm 21847 | The sign is a homomorphism... |
| psgnghm2 21848 | The sign is a homomorphism... |
| psgninv 21849 | The sign of a permutation ... |
| psgnco 21850 | Multiplicativity of the pe... |
| zrhpsgnmhm 21851 | Embedding of permutation s... |
| zrhpsgninv 21852 | The embedded sign of a per... |
| evpmss 21853 | Even permutations are perm... |
| psgnevpmb 21854 | A class is an even permuta... |
| psgnodpm 21855 | A permutation which is odd... |
| psgnevpm 21856 | A permutation which is eve... |
| psgnodpmr 21857 | If a permutation has sign ... |
| zrhpsgnevpm 21858 | The sign of an even permut... |
| zrhpsgnodpm 21859 | The sign of an odd permuta... |
| cofipsgn 21860 | Composition of any class `... |
| zrhpsgnelbas 21861 | Embedding of permutation s... |
| zrhcopsgnelbas 21862 | Embedding of permutation s... |
| evpmodpmf1o 21863 | The function for performin... |
| pmtrodpm 21864 | A transposition is an odd ... |
| psgnfix1 21865 | A permutation of a finite ... |
| psgnfix2 21866 | A permutation of a finite ... |
| psgndiflemB 21867 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21868 | Lemma 2 for ~ psgndif . (... |
| psgndif 21869 | Embedding of permutation s... |
| copsgndif 21870 | Embedding of permutation s... |
| rebase 21873 | The base of the field of r... |
| remulg 21874 | The multiplication (group ... |
| resubdrg 21875 | The real numbers form a di... |
| resubgval 21876 | Subtraction in the field o... |
| replusg 21877 | The addition operation of ... |
| remulr 21878 | The multiplication operati... |
| re0g 21879 | The zero element of the fi... |
| re1r 21880 | The unity element of the f... |
| rele2 21881 | The ordering relation of t... |
| relt 21882 | The ordering relation of t... |
| reds 21883 | The distance of the field ... |
| redvr 21884 | The division operation of ... |
| retos 21885 | The real numbers are a tot... |
| refld 21886 | The real numbers form a fi... |
| refldcj 21887 | The conjugation operation ... |
| resrng 21888 | The real numbers form a st... |
| regsumsupp 21889 | The group sum over the rea... |
| rzgrp 21890 | The quotient group ` RR / ... |
| isphl 21895 | The predicate "is a genera... |
| phllvec 21896 | A pre-Hilbert space is a l... |
| phllmod 21897 | A pre-Hilbert space is a l... |
| phlsrng 21898 | The scalar ring of a pre-H... |
| phllmhm 21899 | The inner product of a pre... |
| ipcl 21900 | Closure of the inner produ... |
| ipcj 21901 | Conjugate of an inner prod... |
| iporthcom 21902 | Orthogonality (meaning inn... |
| ip0l 21903 | Inner product with a zero ... |
| ip0r 21904 | Inner product with a zero ... |
| ipeq0 21905 | The inner product of a vec... |
| ipdir 21906 | Distributive law for inner... |
| ipdi 21907 | Distributive law for inner... |
| ip2di 21908 | Distributive law for inner... |
| ipsubdir 21909 | Distributive law for inner... |
| ipsubdi 21910 | Distributive law for inner... |
| ip2subdi 21911 | Distributive law for inner... |
| ipass 21912 | Associative law for inner ... |
| ipassr 21913 | "Associative" law for seco... |
| ipassr2 21914 | "Associative" law for inne... |
| ipffval 21915 | The inner product operatio... |
| ipfval 21916 | The inner product operatio... |
| ipfeq 21917 | If the inner product opera... |
| ipffn 21918 | The inner product operatio... |
| phlipf 21919 | The inner product operatio... |
| ip2eq 21920 | Two vectors are equal iff ... |
| isphld 21921 | Properties that determine ... |
| phlpropd 21922 | If two structures have the... |
| ssipeq 21923 | The inner product on a sub... |
| phssipval 21924 | The inner product on a sub... |
| phssip 21925 | The inner product (as a fu... |
| phlssphl 21926 | A subspace of an inner pro... |
| ocvfval 21933 | The orthocomplement operat... |
| ocvval 21934 | Value of the orthocompleme... |
| elocv 21935 | Elementhood in the orthoco... |
| ocvi 21936 | Property of a member of th... |
| ocvss 21937 | The orthocomplement of a s... |
| ocvocv 21938 | A set is contained in its ... |
| ocvlss 21939 | The orthocomplement of a s... |
| ocv2ss 21940 | Orthocomplements reverse s... |
| ocvin 21941 | An orthocomplement has tri... |
| ocvsscon 21942 | Two ways to say that ` S `... |
| ocvlsp 21943 | The orthocomplement of a l... |
| ocv0 21944 | The orthocomplement of the... |
| ocvz 21945 | The orthocomplement of the... |
| ocv1 21946 | The orthocomplement of the... |
| unocv 21947 | The orthocomplement of a u... |
| iunocv 21948 | The orthocomplement of an ... |
| cssval 21949 | The set of closed subspace... |
| iscss 21950 | The predicate "is a closed... |
| cssi 21951 | Property of a closed subsp... |
| cssss 21952 | A closed subspace is a sub... |
| iscss2 21953 | It is sufficient to prove ... |
| ocvcss 21954 | The orthocomplement of any... |
| cssincl 21955 | The zero subspace is a clo... |
| css0 21956 | The zero subspace is a clo... |
| css1 21957 | The whole space is a close... |
| csslss 21958 | A closed subspace of a pre... |
| lsmcss 21959 | A subset of a pre-Hilbert ... |
| cssmre 21960 | The closed subspaces of a ... |
| mrccss 21961 | The Moore closure correspo... |
| thlval 21962 | Value of the Hilbert latti... |
| thlbas 21963 | Base set of the Hilbert la... |
| thlle 21964 | Ordering on the Hilbert la... |
| thlleval 21965 | Ordering on the Hilbert la... |
| thloc 21966 | Orthocomplement on the Hil... |
| pjfval 21973 | The value of the projectio... |
| pjdm 21974 | A subspace is in the domai... |
| pjpm 21975 | The projection map is a pa... |
| pjfval2 21976 | Value of the projection ma... |
| pjval 21977 | Value of the projection ma... |
| pjdm2 21978 | A subspace is in the domai... |
| pjff 21979 | A projection is a linear o... |
| pjf 21980 | A projection is a function... |
| pjf2 21981 | A projection is a function... |
| pjfo 21982 | A projection is a surjecti... |
| pjcss 21983 | A projection subspace is a... |
| ocvpj 21984 | The orthocomplement of a p... |
| ishil 21985 | The predicate "is a Hilber... |
| ishil2 21986 | The predicate "is a Hilber... |
| isobs 21987 | The predicate "is an ortho... |
| obsip 21988 | The inner product of two e... |
| obsipid 21989 | A basis element has length... |
| obsrcl 21990 | Reverse closure for an ort... |
| obsss 21991 | An orthonormal basis is a ... |
| obsne0 21992 | A basis element is nonzero... |
| obsocv 21993 | An orthonormal basis has t... |
| obs2ocv 21994 | The double orthocomplement... |
| obselocv 21995 | A basis element is in the ... |
| obs2ss 21996 | A basis has no proper subs... |
| obslbs 21997 | An orthogonal basis is a l... |
| reldmdsmm 22000 | The direct sum is a well-b... |
| dsmmval 22001 | Value of the module direct... |
| dsmmbase 22002 | Base set of the module dir... |
| dsmmval2 22003 | Self-referential definitio... |
| dsmmbas2 22004 | Base set of the direct sum... |
| dsmmfi 22005 | For finite products, the d... |
| dsmmelbas 22006 | Membership in the finitely... |
| dsmm0cl 22007 | The all-zero vector is con... |
| dsmmacl 22008 | The finite hull is closed ... |
| prdsinvgd2 22009 | Negation of a single coord... |
| dsmmsubg 22010 | The finite hull of a produ... |
| dsmmlss 22011 | The finite hull of a produ... |
| dsmmlmod 22012 | The direct sum of a family... |
| frlmval 22015 | Value of the "free module"... |
| frlmlmod 22016 | The free module is a modul... |
| frlmpws 22017 | The free module as a restr... |
| frlmlss 22018 | The base set of the free m... |
| frlmpwsfi 22019 | The finite free module is ... |
| frlmsca 22020 | The ring of scalars of a f... |
| frlm0 22021 | Zero in a free module (rin... |
| frlmbas 22022 | Base set of the free modul... |
| frlmelbas 22023 | Membership in the base set... |
| frlmrcl 22024 | If a free module is inhabi... |
| frlmbasfsupp 22025 | Elements of the free modul... |
| frlmbasmap 22026 | Elements of the free modul... |
| frlmbasf 22027 | Elements of the free modul... |
| frlmlvec 22028 | The free module over a div... |
| frlmfibas 22029 | The base set of the finite... |
| elfrlmbasn0 22030 | If the dimension of a free... |
| frlmplusgval 22031 | Addition in a free module.... |
| frlmsubgval 22032 | Subtraction in a free modu... |
| frlmvscafval 22033 | Scalar multiplication in a... |
| frlmvplusgvalc 22034 | Coordinates of a sum with ... |
| frlmvscaval 22035 | Coordinates of a scalar mu... |
| frlmplusgvalb 22036 | Addition in a free module ... |
| frlmvscavalb 22037 | Scalar multiplication in a... |
| frlmvplusgscavalb 22038 | Addition combined with sca... |
| frlmgsum 22039 | Finite commutative sums in... |
| frlmsplit2 22040 | Restriction is homomorphic... |
| frlmsslss 22041 | A subset of a free module ... |
| frlmsslss2 22042 | A subset of a free module ... |
| frlmbas3 22043 | An element of the base set... |
| mpofrlmd 22044 | Elements of the free modul... |
| frlmip 22045 | The inner product of a fre... |
| frlmipval 22046 | The inner product of a fre... |
| frlmphllem 22047 | Lemma for ~ frlmphl . (Co... |
| frlmphl 22048 | Conditions for a free modu... |
| uvcfval 22051 | Value of the unit-vector g... |
| uvcval 22052 | Value of a single unit vec... |
| uvcvval 22053 | Value of a unit vector coo... |
| uvcvvcl 22054 | A coordinate of a unit vec... |
| uvcvvcl2 22055 | A unit vector coordinate i... |
| uvcvv1 22056 | The unit vector is one at ... |
| uvcvv0 22057 | The unit vector is zero at... |
| uvcff 22058 | Domain and codomain of the... |
| uvcf1 22059 | In a nonzero ring, each un... |
| uvcresum 22060 | Any element of a free modu... |
| frlmssuvc1 22061 | A scalar multiple of a uni... |
| frlmssuvc2 22062 | A nonzero scalar multiple ... |
| frlmsslsp 22063 | A subset of a free module ... |
| frlmlbs 22064 | The unit vectors comprise ... |
| frlmup1 22065 | Any assignment of unit vec... |
| frlmup2 22066 | The evaluation map has the... |
| frlmup3 22067 | The range of such an evalu... |
| frlmup4 22068 | Universal property of the ... |
| ellspd 22069 | The elements of the span o... |
| elfilspd 22070 | Simplified version of ~ el... |
| rellindf 22075 | The independent-family pre... |
| islinds 22076 | Property of an independent... |
| linds1 22077 | An independent set of vect... |
| linds2 22078 | An independent set of vect... |
| islindf 22079 | Property of an independent... |
| islinds2 22080 | Expanded property of an in... |
| islindf2 22081 | Property of an independent... |
| lindff 22082 | Functional property of a l... |
| lindfind 22083 | A linearly independent fam... |
| lindsind 22084 | A linearly independent set... |
| lindfind2 22085 | In a linearly independent ... |
| lindsind2 22086 | In a linearly independent ... |
| lindff1 22087 | A linearly independent fam... |
| lindfrn 22088 | The range of an independen... |
| f1lindf 22089 | Rearranging and deleting e... |
| lindfres 22090 | Any restriction of an inde... |
| lindsss 22091 | Any subset of an independe... |
| f1linds 22092 | A family constructed from ... |
| islindf3 22093 | In a nonzero ring, indepen... |
| lindfmm 22094 | Linear independence of a f... |
| lindsmm 22095 | Linear independence of a s... |
| lindsmm2 22096 | The monomorphic image of a... |
| lsslindf 22097 | Linear independence is unc... |
| lsslinds 22098 | Linear independence is unc... |
| islbs4 22099 | A basis is an independent ... |
| lbslinds 22100 | A basis is independent. (... |
| islinds3 22101 | A subset is linearly indep... |
| islinds4 22102 | A set is independent in a ... |
| lmimlbs 22103 | The isomorphic image of a ... |
| lmiclbs 22104 | Having a basis is an isomo... |
| islindf4 22105 | A family is independent if... |
| islindf5 22106 | A family is independent if... |
| indlcim 22107 | An independent, spanning f... |
| lbslcic 22108 | A module with a basis is i... |
| lmisfree 22109 | A module has a basis iff i... |
| lvecisfrlm 22110 | Every vector space is isom... |
| lmimco 22111 | The composition of two iso... |
| lmictra 22112 | Module isomorphism is tran... |
| uvcf1o 22113 | In a nonzero ring, the map... |
| uvcendim 22114 | In a nonzero ring, the num... |
| frlmisfrlm 22115 | A free module is isomorphi... |
| frlmiscvec 22116 | Every free module is isomo... |
| lindsdom 22117 | A linearly independent set... |
| lindsenlbs 22118 | A maximal linearly indepen... |
| isassa 22125 | The properties of an assoc... |
| assalem 22126 | The properties of an assoc... |
| assaass 22127 | Left-associative property ... |
| assaassr 22128 | Right-associative property... |
| assalmod 22129 | An associative algebra is ... |
| assaring 22130 | An associative algebra is ... |
| assasca 22131 | The scalars of an associat... |
| assa2ass 22132 | Left- and right-associativ... |
| assa2ass2 22133 | Left- and right-associativ... |
| isassad 22134 | Sufficient condition for b... |
| issubassa3 22135 | A subring that is also a s... |
| issubassa 22136 | The subalgebras of an asso... |
| sraassab 22137 | A subring algebra is an as... |
| sraassa 22138 | The subring algebra over a... |
| rlmassa 22139 | The ring module over a com... |
| assapropd 22140 | If two structures have the... |
| aspval 22141 | Value of the algebraic clo... |
| asplss 22142 | The algebraic span of a se... |
| aspid 22143 | The algebraic span of a su... |
| aspsubrg 22144 | The algebraic span of a se... |
| aspss 22145 | Span preserves subset orde... |
| aspssid 22146 | A set of vectors is a subs... |
| asclfval 22147 | Function value of the alge... |
| asclval 22148 | Value of a mapped algebra ... |
| asclfn 22149 | Unconditional functionalit... |
| asclf 22150 | The algebra scalar lifting... |
| asclghm 22151 | The algebra scalar lifting... |
| asclelbas 22152 | Lifted scalars are in the ... |
| ascl0 22153 | The scalar 0 embedded into... |
| ascl1 22154 | The scalar 1 embedded into... |
| asclmul1 22155 | Left multiplication by a l... |
| asclmul2 22156 | Right multiplication by a ... |
| ascldimul 22157 | The algebra scalar lifting... |
| asclinvg 22158 | The group inverse (negatio... |
| asclrhm 22159 | The algebra scalar lifting... |
| rnascl 22160 | The set of lifted scalars ... |
| issubassa2 22161 | A subring of a unital alge... |
| rnasclsubrg 22162 | The scalar multiples of th... |
| rnasclmulcl 22163 | (Vector) multiplication is... |
| rnasclassa 22164 | The scalar multiples of th... |
| ressascl 22165 | The lifting of scalars is ... |
| asclpropd 22166 | If two structures have the... |
| aspval2 22167 | The algebraic closure is t... |
| assamulgscmlem1 22168 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 22169 | Lemma for ~ assamulgscm (i... |
| assamulgscm 22170 | Exponentiation of a scalar... |
| asclmulg 22171 | Apply group multiplication... |
| zlmassa 22172 | The ` ZZ ` -module operati... |
| reldmpsr 22183 | The multivariate power ser... |
| psrval 22184 | Value of the multivariate ... |
| psrvalstr 22185 | The multivariate power ser... |
| psrbag 22186 | Elementhood in the set of ... |
| psrbagf 22187 | A finite bag is a function... |
| psrbagfsupp 22188 | Finite bags have finite su... |
| snifpsrbag 22189 | A bag containing one eleme... |
| fczpsrbag 22190 | The constant function equa... |
| psrbaglesupp 22191 | The support of a dominated... |
| psrbaglecl 22192 | The set of finite bags is ... |
| psrbagaddcl 22193 | The sum of two finite bags... |
| psrbagcon 22194 | The analogue of the statem... |
| psrbaglefi 22195 | There are finitely many ba... |
| psrbagconcl 22196 | The complement of a bag is... |
| psrbagleadd1 22197 | The analogue of " ` X <_ F... |
| psrbagconf1o 22198 | Bag complementation is a b... |
| psrbagres 22199 | Restrict a bag of variable... |
| gsumbagdiaglem 22200 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 22201 | Two-dimensional commutatio... |
| psrass1lem 22202 | A group sum commutation us... |
| psrbas 22203 | The base set of the multiv... |
| psrelbas 22204 | An element of the set of p... |
| psrelbasfun 22205 | An element of the set of p... |
| psrplusg 22206 | The addition operation of ... |
| psradd 22207 | The addition operation of ... |
| psraddcl 22208 | Closure of the power serie... |
| rhmpsrlem1 22209 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 22210 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 22211 | The multiplication operati... |
| psrmulfval 22212 | The multiplication operati... |
| psrmulval 22213 | The multiplication operati... |
| psrmulcllem 22214 | Closure of the power serie... |
| psrmulcl 22215 | Closure of the power serie... |
| psrsca 22216 | The scalar field of the mu... |
| psrvscafval 22217 | The scalar multiplication ... |
| psrvsca 22218 | The scalar multiplication ... |
| psrvscaval 22219 | The scalar multiplication ... |
| psrvscacl 22220 | Closure of the power serie... |
| psr0cl 22221 | The zero element of the ri... |
| psr0lid 22222 | The zero element of the ri... |
| psrnegcl 22223 | The negative function in t... |
| psrlinv 22224 | The negative function in t... |
| psrgrp 22225 | The ring of power series i... |
| psr0 22226 | The zero element of the ri... |
| psrneg 22227 | The negative function of t... |
| psrlmod 22228 | The ring of power series i... |
| psr1cl 22229 | The identity element of th... |
| psrlidm 22230 | The identity element of th... |
| psrridm 22231 | The identity element of th... |
| psrass1 22232 | Associative identity for t... |
| psrdi 22233 | Distributive law for the r... |
| psrdir 22234 | Distributive law for the r... |
| psrass23l 22235 | Associative identity for t... |
| psrcom 22236 | Commutative law for the ri... |
| psrass23 22237 | Associative identities for... |
| psrring 22238 | The ring of power series i... |
| psr1 22239 | The identity element of th... |
| psrcrng 22240 | The ring of power series i... |
| psrassa 22241 | The ring of power series i... |
| resspsrbas 22242 | A restricted power series ... |
| resspsradd 22243 | A restricted power series ... |
| resspsrmul 22244 | A restricted power series ... |
| resspsrvsca 22245 | A restricted power series ... |
| subrgpsr 22246 | A subring of the base ring... |
| psrascl 22247 | Value of the scalar inject... |
| psrasclcl 22248 | A scalar is lifted into a ... |
| mvrfval 22249 | Value of the generating el... |
| mvrval 22250 | Value of the generating el... |
| mvrval2 22251 | Value of the generating el... |
| mvrid 22252 | The ` X i ` -th coefficien... |
| mvrf 22253 | The power series variable ... |
| mvrf1 22254 | The power series variable ... |
| mvrcl2 22255 | A power series variable is... |
| reldmmpl 22256 | The multivariate polynomia... |
| mplval 22257 | Value of the set of multiv... |
| mplbas 22258 | Base set of the set of mul... |
| mplelbas 22259 | Property of being a polyno... |
| mvrcl 22260 | A power series variable is... |
| mvrf2 22261 | The power series/polynomia... |
| mplrcl 22262 | Reverse closure for the po... |
| mplelsfi 22263 | A polynomial treated as a ... |
| mplval2 22264 | Self-referential expressio... |
| mplbasss 22265 | The set of polynomials is ... |
| mplelf 22266 | A polynomial is defined as... |
| mplsubglem 22267 | If ` A ` is an ideal of se... |
| mpllsslem 22268 | If ` A ` is an ideal of su... |
| mplsubglem2 22269 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 22270 | The set of polynomials is ... |
| mpllss 22271 | The set of polynomials is ... |
| mplsubrglem 22272 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 22273 | The set of polynomials is ... |
| mpl0 22274 | The zero polynomial. (Con... |
| mplplusg 22275 | Value of addition in a pol... |
| mplmulr 22276 | Value of multiplication in... |
| mpladd 22277 | The addition operation on ... |
| mplneg 22278 | The negative function on m... |
| mplmul 22279 | The multiplication operati... |
| mpl1 22280 | The identity element of th... |
| mplsca 22281 | The scalar field of a mult... |
| mplvsca2 22282 | The scalar multiplication ... |
| mplvsca 22283 | The scalar multiplication ... |
| mplvscaval 22284 | The scalar multiplication ... |
| mplgrp 22285 | The polynomial ring is a g... |
| mpllmod 22286 | The polynomial ring is a l... |
| mplring 22287 | The polynomial ring is a r... |
| mpllvec 22288 | The polynomial ring is a v... |
| mplcrng 22289 | The polynomial ring is a c... |
| mplassa 22290 | The polynomial ring is an ... |
| mplringd 22291 | The polynomial ring is a r... |
| mplcrngd 22292 | The polynomial ring is a c... |
| mpllmodd 22293 | The polynomial ring is a l... |
| mplascl0 22294 | The zero scalar as a polyn... |
| mplascl1 22295 | The one scalar as a polyno... |
| ressmplbas2 22296 | The base set of a restrict... |
| ressmplbas 22297 | A restricted polynomial al... |
| ressmpladd 22298 | A restricted polynomial al... |
| ressmplmul 22299 | A restricted polynomial al... |
| ressmplvsca 22300 | A restricted power series ... |
| subrgmpl 22301 | A subring of the base ring... |
| mplsubrgcl 22302 | An element of a polynomial... |
| subrgmvr 22303 | The variables in a subring... |
| subrgmvrf 22304 | The variables in a polynom... |
| mplmon 22305 | A monomial is a polynomial... |
| mplmonmul 22306 | The product of two monomia... |
| mplcoe1 22307 | Decompose a polynomial int... |
| mplcoe3 22308 | Decompose a monomial in on... |
| mplcoe5lem 22309 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22310 | Decompose a monomial into ... |
| mplcoe2 22311 | Decompose a monomial into ... |
| mplbas2 22312 | An alternative expression ... |
| ltbval 22313 | Value of the well-order on... |
| ltbwe 22314 | The finite bag order is a ... |
| reldmopsr 22315 | Lemma for ordered power se... |
| opsrval 22316 | The value of the "ordered ... |
| opsrle 22317 | An alternative expression ... |
| opsrval2 22318 | Self-referential expressio... |
| opsrbaslem 22319 | Get a component of the ord... |
| opsrbas 22320 | The base set of the ordere... |
| opsrplusg 22321 | The addition operation of ... |
| opsrmulr 22322 | The multiplication operati... |
| opsrvsca 22323 | The scalar product operati... |
| opsrsca 22324 | The scalar ring of the ord... |
| opsrtoslem1 22325 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22326 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22327 | The ordered power series s... |
| opsrso 22328 | The ordered power series s... |
| opsrcrng 22329 | The ring of ordered power ... |
| opsrassa 22330 | The ring of ordered power ... |
| mplmon2 22331 | Express a scaled monomial.... |
| psrbag0 22332 | The empty bag is a bag. (... |
| psrbagsn 22333 | A singleton bag is a bag. ... |
| mplascl 22334 | Value of the scalar inject... |
| mplasclf 22335 | The scalar injection is a ... |
| subrgascl 22336 | The scalar injection funct... |
| subrgasclcl 22337 | The scalars in a polynomia... |
| mplmon2cl 22338 | A scaled monomial is a pol... |
| mplmon2mul 22339 | Product of scaled monomial... |
| mplind 22340 | Prove a property of polyno... |
| mplcoe4 22341 | Decompose a polynomial int... |
| evlslem4 22346 | The support of a tensor pr... |
| psrbagev1 22347 | A bag of multipliers provi... |
| psrbagev2 22348 | Closure of a sum using a b... |
| evlslem2 22349 | A linear function on the p... |
| evlslem3 22350 | Lemma for ~ evlseu . Poly... |
| evlslem6 22351 | Lemma for ~ evlseu . Fini... |
| evlslem1 22352 | Lemma for ~ evlseu , give ... |
| evlseu 22353 | For a given interpretation... |
| reldmevls 22354 | Well-behaved binary operat... |
| mpfrcl 22355 | Reverse closure for the se... |
| evlsval 22356 | Value of the polynomial ev... |
| evlsval2 22357 | Characterizing properties ... |
| evlsrhm 22358 | Polynomial evaluation is a... |
| evlsval3 22359 | Give a formula for the pol... |
| evlsvval 22360 | Give a formula for the eva... |
| evlsvvvallem 22361 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22362 | Lemma for theorems using ~... |
| evlsvvval 22363 | Give a formula for the eva... |
| evlssca 22364 | Polynomial evaluation maps... |
| evlsvar 22365 | Polynomial evaluation maps... |
| evlsgsumadd 22366 | Polynomial evaluation maps... |
| evlsgsummul 22367 | Polynomial evaluation maps... |
| evlspw 22368 | Polynomial evaluation for ... |
| evlsvarpw 22369 | Polynomial evaluation for ... |
| evlval 22370 | Value of the simple/same r... |
| evlrhm 22371 | The simple evaluation map ... |
| evlcl 22372 | A polynomial over the ring... |
| evladdval 22373 | Polynomial evaluation buil... |
| evlmulval 22374 | Polynomial evaluation buil... |
| evlsscasrng 22375 | The evaluation of a scalar... |
| evlsca 22376 | Simple polynomial evaluati... |
| evlsvarsrng 22377 | The evaluation of the vari... |
| evlvar 22378 | Simple polynomial evaluati... |
| mpfconst 22379 | Constants are multivariate... |
| mpfproj 22380 | Projections are multivaria... |
| mpfsubrg 22381 | Polynomial functions are a... |
| mpff 22382 | Polynomial functions are f... |
| mpfaddcl 22383 | The sum of multivariate po... |
| mpfmulcl 22384 | The product of multivariat... |
| mpfind 22385 | Prove a property of polyno... |
| selvffval 22388 | Value of the "variable sel... |
| selvfval 22389 | Value of the "variable sel... |
| selvval 22390 | Value of the "variable sel... |
| mhmcompl 22391 | The composition of a monoi... |
| mplmapghm 22392 | The function ` H ` mapping... |
| mhmcoaddmpl 22393 | Show that the ring homomor... |
| rhmcomulmpl 22394 | Show that the ring homomor... |
| evlscl 22395 | A polynomial over the ring... |
| evlsscaval 22396 | Polynomial evaluation buil... |
| evlsvarval 22397 | Polynomial evaluation buil... |
| evlsexpval 22398 | Polynomial evaluation buil... |
| evlsaddval 22399 | Polynomial evaluation buil... |
| evlsmulval 22400 | Polynomial evaluation buil... |
| evlsmaprhm 22401 | The function ` F ` mapping... |
| evlsevl 22402 | Evaluation in a subring is... |
| evlvvval 22403 | Give a formula for the eva... |
| selvcllem1 22404 | ` T ` is an associative al... |
| selvcllem2 22405 | ` D ` is a ring homomorphi... |
| selvcllem3 22406 | The third argument passed ... |
| selvcllemh 22407 | Apply the third argument (... |
| selvcllem4 22408 | The fourth argument passed... |
| selvcllem5 22409 | The fifth argument passed ... |
| selvcl 22410 | Closure of the "variable s... |
| selvval2 22411 | Value of the "variable sel... |
| selvvvval 22412 | Recover the original polyn... |
| selvadd 22413 | The "variable selection" f... |
| selvmul 22414 | The "variable selection" f... |
| reldmmhp 22419 | The domain of the homogene... |
| mhpfval 22420 | Value of the "homogeneous ... |
| mhpval 22421 | Value of the "homogeneous ... |
| ismhp 22422 | Property of being a homoge... |
| ismhp2 22423 | Deduce a homogeneous polyn... |
| ismhp3 22424 | A polynomial is homogeneou... |
| mhprcl 22425 | Reverse closure for homoge... |
| mhpmpl 22426 | A homogeneous polynomial i... |
| mhpdeg 22427 | All nonzero terms of a hom... |
| mhp0cl 22428 | The zero polynomial is hom... |
| mhpsclcl 22429 | A scalar (or constant) pol... |
| mhpvarcl 22430 | A power series variable is... |
| mhpmulcl 22431 | A product of homogeneous p... |
| mhppwdeg 22432 | Degree of a homogeneous po... |
| mhpaddcl 22433 | Homogeneous polynomials ar... |
| mhpinvcl 22434 | Homogeneous polynomials ar... |
| mhpsubg 22435 | Homogeneous polynomials fo... |
| mhpvscacl 22436 | Homogeneous polynomials ar... |
| mhplss 22437 | Homogeneous polynomials fo... |
| psdffval 22439 | Value of the power series ... |
| psdfval 22440 | Give a map between power s... |
| psdval 22441 | Evaluate the partial deriv... |
| psdcoef 22442 | Coefficient of a term of t... |
| psdcl 22443 | The derivative of a power ... |
| psdmplcl 22444 | The derivative of a polyno... |
| psdadd 22445 | The derivative of a sum is... |
| psdvsca 22446 | The derivative of a scaled... |
| psdmullem 22447 | Lemma for ~ psdmul . Tran... |
| psdmul 22448 | Product rule for power ser... |
| psd1 22449 | The derivative of one is z... |
| psdascl 22450 | The derivative of a consta... |
| psdmvr 22451 | The partial derivative of ... |
| psdpw 22452 | Power rule for partial der... |
| psr1baslem 22464 | The set of finite bags on ... |
| psr1val 22465 | Value of the ring of univa... |
| psr1crng 22466 | The ring of univariate pow... |
| psr1assa 22467 | The ring of univariate pow... |
| psr1tos 22468 | The ordered power series s... |
| psr1bas2 22469 | The base set of the ring o... |
| psr1bas 22470 | The base set of the ring o... |
| vr1val 22471 | The value of the generator... |
| vr1cl2 22472 | The variable ` X ` is a me... |
| ply1val 22473 | The value of the set of un... |
| ply1bas 22474 | The value of the base set ... |
| ply1lss 22475 | Univariate polynomials for... |
| ply1subrg 22476 | Univariate polynomials for... |
| ply1crng 22477 | The ring of univariate pol... |
| ply1assa 22478 | The ring of univariate pol... |
| psr1bascl 22479 | A univariate power series ... |
| psr1basf 22480 | Univariate power series ba... |
| ply1basf 22481 | Univariate polynomial base... |
| ply1bascl 22482 | A univariate polynomial is... |
| ply1bascl2 22483 | A univariate polynomial is... |
| coe1fval 22484 | Value of the univariate po... |
| coe1fv 22485 | Value of an evaluated coef... |
| fvcoe1 22486 | Value of a multivariate co... |
| coe1fval3 22487 | Univariate power series co... |
| coe1f2 22488 | Functionality of univariat... |
| coe1fval2 22489 | Univariate polynomial coef... |
| coe1f 22490 | Functionality of univariat... |
| coe1fvalcl 22491 | A coefficient of a univari... |
| coe1sfi 22492 | Finite support of univaria... |
| coe1fsupp 22493 | The coefficient vector of ... |
| mptcoe1fsupp 22494 | A mapping involving coeffi... |
| coe1ae0 22495 | The coefficient vector of ... |
| vr1cl 22496 | The generator of a univari... |
| opsr0 22497 | Zero in the ordered power ... |
| opsr1 22498 | One in the ordered power s... |
| psr1plusg 22499 | Value of addition in a uni... |
| psr1vsca 22500 | Value of scalar multiplica... |
| psr1mulr 22501 | Value of multiplication in... |
| ply1plusg 22502 | Value of addition in a uni... |
| ply1vsca 22503 | Value of scalar multiplica... |
| ply1mulr 22504 | Value of multiplication in... |
| ply1ass23l 22505 | Associative identity with ... |
| ressply1bas2 22506 | The base set of a restrict... |
| ressply1bas 22507 | A restricted polynomial al... |
| ressply1add 22508 | A restricted polynomial al... |
| ressply1mul 22509 | A restricted polynomial al... |
| ressply1vsca 22510 | A restricted power series ... |
| subrgply1 22511 | A subring of the base ring... |
| gsumply1subr 22512 | Evaluate a group sum in a ... |
| psrbaspropd 22513 | Property deduction for pow... |
| psrplusgpropd 22514 | Property deduction for pow... |
| mplbaspropd 22515 | Property deduction for pol... |
| psropprmul 22516 | Reversing multiplication i... |
| ply1opprmul 22517 | Reversing multiplication i... |
| 00ply1bas 22518 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22519 | Protection compatibility o... |
| ply1plusgfvi 22520 | Protection compatibility o... |
| ply1baspropd 22521 | Property deduction for uni... |
| ply1plusgpropd 22522 | Property deduction for uni... |
| opsrring 22523 | Ordered power series form ... |
| opsrlmod 22524 | Ordered power series form ... |
| psr1ring 22525 | Univariate power series fo... |
| ply1ring 22526 | Univariate polynomials for... |
| psr1lmod 22527 | Univariate power series fo... |
| psr1sca 22528 | Scalars of a univariate po... |
| psr1sca2 22529 | Scalars of a univariate po... |
| ply1lmod 22530 | Univariate polynomials for... |
| ply1sca 22531 | Scalars of a univariate po... |
| ply1sca2 22532 | Scalars of a univariate po... |
| ply1ascl0 22533 | The zero scalar as a polyn... |
| ply1ascl1 22534 | The multiplicative identit... |
| ply1mpl0 22535 | The univariate polynomial ... |
| ply10s0 22536 | Zero times a univariate po... |
| ply1mpl1 22537 | The univariate polynomial ... |
| ply1ascl 22538 | The univariate polynomial ... |
| subrg1ascl 22539 | The scalar injection funct... |
| subrg1asclcl 22540 | The scalars in a polynomia... |
| subrgvr1 22541 | The variables in a subring... |
| subrgvr1cl 22542 | The variables in a polynom... |
| coe1z 22543 | The coefficient vector of ... |
| coe1add 22544 | The coefficient vector of ... |
| coe1addfv 22545 | A particular coefficient o... |
| coe1subfv 22546 | A particular coefficient o... |
| coe1mul2lem1 22547 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22548 | An equivalence for ~ coe1m... |
| coe1mul2 22549 | The coefficient vector of ... |
| coe1mul 22550 | The coefficient vector of ... |
| ply1moncl 22551 | Closure of the expression ... |
| ply1tmcl 22552 | Closure of the expression ... |
| coe1tm 22553 | Coefficient vector of a po... |
| coe1tmfv1 22554 | Nonzero coefficient of a p... |
| coe1tmfv2 22555 | Zero coefficient of a poly... |
| coe1tmmul2 22556 | Coefficient vector of a po... |
| coe1tmmul 22557 | Coefficient vector of a po... |
| coe1tmmul2fv 22558 | Function value of a right-... |
| coe1pwmul 22559 | Coefficient vector of a po... |
| coe1pwmulfv 22560 | Function value of a right-... |
| ply1scltm 22561 | A scalar is a term with ze... |
| coe1sclmul 22562 | Coefficient vector of a po... |
| coe1sclmulfv 22563 | A single coefficient of a ... |
| coe1sclmul2 22564 | Coefficient vector of a po... |
| ply1sclf 22565 | A scalar polynomial is a p... |
| ply1sclcl 22566 | The value of the algebra s... |
| coe1scl 22567 | Coefficient vector of a sc... |
| ply1sclid 22568 | Recover the base scalar fr... |
| ply1sclf1 22569 | The polynomial scalar func... |
| ply1scl0 22570 | The zero scalar is zero. ... |
| ply1scln0 22571 | Nonzero scalars create non... |
| ply1scl1 22572 | The one scalar is the unit... |
| coe1id 22573 | Coefficient vector of the ... |
| ply1idvr1 22574 | The identity of a polynomi... |
| cply1mul 22575 | The product of two constan... |
| ply1coefsupp 22576 | The decomposition of a uni... |
| ply1coe 22577 | Decompose a univariate pol... |
| eqcoe1ply1eq 22578 | Two polynomials over the s... |
| ply1coe1eq 22579 | Two polynomials over the s... |
| cply1coe0 22580 | All but the first coeffici... |
| cply1coe0bi 22581 | A polynomial is constant (... |
| coe1fzgsumdlem 22582 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22583 | Value of an evaluated coef... |
| ply1scleq 22584 | Equality of a constant pol... |
| ply1chr 22585 | The characteristic of a po... |
| gsumsmonply1 22586 | A finite group sum of scal... |
| gsummoncoe1 22587 | A coefficient of the polyn... |
| gsumply1eq 22588 | Two univariate polynomials... |
| lply1binom 22589 | The binomial theorem for l... |
| lply1binomsc 22590 | The binomial theorem for l... |
| ply1fermltlchr 22591 | Fermat's little theorem fo... |
| reldmevls1 22596 | Well-behaved binary operat... |
| ply1frcl 22597 | Reverse closure for the se... |
| evls1fval 22598 | Value of the univariate po... |
| evls1val 22599 | Value of the univariate po... |
| evls1rhmlem 22600 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22601 | Polynomial evaluation is a... |
| evls1sca 22602 | Univariate polynomial eval... |
| evls1gsumadd 22603 | Univariate polynomial eval... |
| evls1gsummul 22604 | Univariate polynomial eval... |
| evls1pw 22605 | Univariate polynomial eval... |
| evls1varpw 22606 | Univariate polynomial eval... |
| evl1fval 22607 | Value of the simple/same r... |
| evl1val 22608 | Value of the simple/same r... |
| evl1fval1lem 22609 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22610 | Value of the simple/same r... |
| evl1rhm 22611 | Polynomial evaluation is a... |
| fveval1fvcl 22612 | The function value of the ... |
| evl1sca 22613 | Polynomial evaluation maps... |
| evl1scad 22614 | Polynomial evaluation buil... |
| evl1var 22615 | Polynomial evaluation maps... |
| evl1vard 22616 | Polynomial evaluation buil... |
| evls1var 22617 | Univariate polynomial eval... |
| evls1scasrng 22618 | The evaluation of a scalar... |
| evls1varsrng 22619 | The evaluation of the vari... |
| evl1addd 22620 | Polynomial evaluation buil... |
| evl1subd 22621 | Polynomial evaluation buil... |
| evl1muld 22622 | Polynomial evaluation buil... |
| evl1vsd 22623 | Polynomial evaluation buil... |
| evl1expd 22624 | Polynomial evaluation buil... |
| pf1const 22625 | Constants are polynomial f... |
| pf1id 22626 | The identity is a polynomi... |
| pf1subrg 22627 | Polynomial functions are a... |
| pf1rcl 22628 | Reverse closure for the se... |
| pf1f 22629 | Polynomial functions are f... |
| mpfpf1 22630 | Convert a multivariate pol... |
| pf1mpf 22631 | Convert a univariate polyn... |
| pf1addcl 22632 | The sum of multivariate po... |
| pf1mulcl 22633 | The product of multivariat... |
| pf1ind 22634 | Prove a property of polyno... |
| evl1gsumdlem 22635 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22636 | Polynomial evaluation buil... |
| evl1gsumadd 22637 | Univariate polynomial eval... |
| evl1gsumaddval 22638 | Value of a univariate poly... |
| evl1gsummul 22639 | Univariate polynomial eval... |
| evl1varpw 22640 | Univariate polynomial eval... |
| evl1varpwval 22641 | Value of a univariate poly... |
| evl1scvarpw 22642 | Univariate polynomial eval... |
| evl1scvarpwval 22643 | Value of a univariate poly... |
| evl1gsummon 22644 | Value of a univariate poly... |
| evls1scafv 22645 | Value of the univariate po... |
| evls1expd 22646 | Univariate polynomial eval... |
| evls1varpwval 22647 | Univariate polynomial eval... |
| evls1fpws 22648 | Evaluation of a univariate... |
| ressply1evl 22649 | Evaluation of a univariate... |
| evls1addd 22650 | Univariate polynomial eval... |
| evls1muld 22651 | Univariate polynomial eval... |
| evls1vsca 22652 | Univariate polynomial eval... |
| asclply1subcl 22653 | Closure of the algebra sca... |
| evls1fvcl 22654 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22655 | The function ` F ` mapping... |
| evls1maplmhm 22656 | The function ` F ` mapping... |
| evls1maprnss 22657 | The function ` F ` mapping... |
| evl1maprhm 22658 | The function ` F ` mapping... |
| rhmmpl 22659 | Provide a ring homomorphis... |
| ply1vscl 22660 | Closure of scalar multipli... |
| mhmcoply1 22661 | The composition of a monoi... |
| rhmply1 22662 | Provide a ring homomorphis... |
| rhmply1vr1 22663 | A ring homomorphism betwee... |
| rhmply1vsca 22664 | Apply a ring homomorphism ... |
| rhmply1mon 22665 | Apply a ring homomorphism ... |
| mamufval 22668 | Functional value of the ma... |
| mamuval 22669 | Multiplication of two matr... |
| mamufv 22670 | A cell in the multiplicati... |
| mamudm 22671 | The domain of the matrix m... |
| mamufacex 22672 | Every solution of the equa... |
| mamures 22673 | Rows in a matrix product a... |
| grpvlinv 22674 | Tuple-wise left inverse in... |
| grpvrinv 22675 | Tuple-wise right inverse i... |
| ringvcl 22676 | Tuple-wise multiplication ... |
| mamucl 22677 | Operation closure of matri... |
| mamuass 22678 | Matrix multiplication is a... |
| mamudi 22679 | Matrix multiplication dist... |
| mamudir 22680 | Matrix multiplication dist... |
| mamuvs1 22681 | Matrix multiplication dist... |
| mamuvs2 22682 | Matrix multiplication dist... |
| matbas0pc 22685 | There is no matrix with a ... |
| matbas0 22686 | There is no matrix for a n... |
| matval 22687 | Value of the matrix algebr... |
| matrcl 22688 | Reverse closure for the ma... |
| matbas 22689 | The matrix ring has the sa... |
| matplusg 22690 | The matrix ring has the sa... |
| matsca 22691 | The matrix ring has the sa... |
| matvsca 22692 | The matrix ring has the sa... |
| mat0 22693 | The matrix ring has the sa... |
| matinvg 22694 | The matrix ring has the sa... |
| mat0op 22695 | Value of a zero matrix as ... |
| matsca2 22696 | The scalars of the matrix ... |
| matbas2 22697 | The base set of the matrix... |
| matbas2i 22698 | A matrix is a function. (... |
| matbas2d 22699 | The base set of the matrix... |
| eqmat 22700 | Two square matrices of the... |
| matecl 22701 | Each entry (according to W... |
| matecld 22702 | Each entry (according to W... |
| matplusg2 22703 | Addition in the matrix rin... |
| matvsca2 22704 | Scalar multiplication in t... |
| matlmod 22705 | The matrix ring is a linea... |
| matgrp 22706 | The matrix ring is a group... |
| matvscl 22707 | Closure of the scalar mult... |
| matsubg 22708 | The matrix ring has the sa... |
| matplusgcell 22709 | Addition in the matrix rin... |
| matsubgcell 22710 | Subtraction in the matrix ... |
| matinvgcell 22711 | Additive inversion in the ... |
| matvscacell 22712 | Scalar multiplication in t... |
| matgsum 22713 | Finite commutative sums in... |
| matmulr 22714 | Multiplication in the matr... |
| mamumat1cl 22715 | The identity matrix (as op... |
| mat1comp 22716 | The components of the iden... |
| mamulid 22717 | The identity matrix (as op... |
| mamurid 22718 | The identity matrix (as op... |
| matring 22719 | Existence of the matrix ri... |
| matassa 22720 | Existence of the matrix al... |
| matmulcell 22721 | Multiplication in the matr... |
| mpomatmul 22722 | Multiplication of two N x ... |
| mat1 22723 | Value of an identity matri... |
| mat1ov 22724 | Entries of an identity mat... |
| mat1bas 22725 | The identity matrix is a m... |
| matsc 22726 | The identity matrix multip... |
| ofco2 22727 | Distribution law for the f... |
| oftpos 22728 | The transposition of the v... |
| mattposcl 22729 | The transpose of a square ... |
| mattpostpos 22730 | The transpose of the trans... |
| mattposvs 22731 | The transposition of a mat... |
| mattpos1 22732 | The transposition of the i... |
| tposmap 22733 | The transposition of an I ... |
| mamutpos 22734 | Behavior of transposes in ... |
| mattposm 22735 | Multiplying two transposed... |
| matgsumcl 22736 | Closure of a group sum ove... |
| madetsumid 22737 | The identity summand in th... |
| matepmcl 22738 | Each entry of a matrix wit... |
| matepm2cl 22739 | Each entry of a matrix wit... |
| madetsmelbas 22740 | A summand of the determina... |
| madetsmelbas2 22741 | A summand of the determina... |
| mat0dimbas0 22742 | The empty set is the one a... |
| mat0dim0 22743 | The zero of the algebra of... |
| mat0dimid 22744 | The identity of the algebr... |
| mat0dimscm 22745 | The scalar multiplication ... |
| mat0dimcrng 22746 | The algebra of matrices wi... |
| mat1dimelbas 22747 | A matrix with dimension 1 ... |
| mat1dimbas 22748 | A matrix with dimension 1 ... |
| mat1dim0 22749 | The zero of the algebra of... |
| mat1dimid 22750 | The identity of the algebr... |
| mat1dimscm 22751 | The scalar multiplication ... |
| mat1dimmul 22752 | The ring multiplication in... |
| mat1dimcrng 22753 | The algebra of matrices wi... |
| mat1f1o 22754 | There is a 1-1 function fr... |
| mat1rhmval 22755 | The value of the ring homo... |
| mat1rhmelval 22756 | The value of the ring homo... |
| mat1rhmcl 22757 | The value of the ring homo... |
| mat1f 22758 | There is a function from a... |
| mat1ghm 22759 | There is a group homomorph... |
| mat1mhm 22760 | There is a monoid homomorp... |
| mat1rhm 22761 | There is a ring homomorphi... |
| mat1rngiso 22762 | There is a ring isomorphis... |
| mat1ric 22763 | A ring is isomorphic to th... |
| dmatval 22768 | The set of ` N ` x ` N ` d... |
| dmatel 22769 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22770 | An ` N ` x ` N ` diagonal ... |
| dmatid 22771 | The identity matrix is a d... |
| dmatelnd 22772 | An extradiagonal entry of ... |
| dmatmul 22773 | The product of two diagona... |
| dmatsubcl 22774 | The difference of two diag... |
| dmatsgrp 22775 | The set of diagonal matric... |
| dmatmulcl 22776 | The product of two diagona... |
| dmatsrng 22777 | The set of diagonal matric... |
| dmatcrng 22778 | The subring of diagonal ma... |
| dmatscmcl 22779 | The multiplication of a di... |
| scmatval 22780 | The set of ` N ` x ` N ` s... |
| scmatel 22781 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22782 | A scalar matrix can be exp... |
| scmatscmide 22783 | An entry of a scalar matri... |
| scmatscmiddistr 22784 | Distributive law for scala... |
| scmatmat 22785 | An ` N ` x ` N ` scalar ma... |
| scmate 22786 | An entry of an ` N ` x ` N... |
| scmatmats 22787 | The set of an ` N ` x ` N ... |
| scmateALT 22788 | Alternate proof of ~ scmat... |
| scmatscm 22789 | The multiplication of a ma... |
| scmatid 22790 | The identity matrix is a s... |
| scmatdmat 22791 | A scalar matrix is a diago... |
| scmataddcl 22792 | The sum of two scalar matr... |
| scmatsubcl 22793 | The difference of two scal... |
| scmatmulcl 22794 | The product of two scalar ... |
| scmatsgrp 22795 | The set of scalar matrices... |
| scmatsrng 22796 | The set of scalar matrices... |
| scmatcrng 22797 | The subring of scalar matr... |
| scmatsgrp1 22798 | The set of scalar matrices... |
| scmatsrng1 22799 | The set of scalar matrices... |
| smatvscl 22800 | Closure of the scalar mult... |
| scmatlss 22801 | The set of scalar matrices... |
| scmatstrbas 22802 | The set of scalar matrices... |
| scmatrhmval 22803 | The value of the ring homo... |
| scmatrhmcl 22804 | The value of the ring homo... |
| scmatf 22805 | There is a function from a... |
| scmatfo 22806 | There is a function from a... |
| scmatf1 22807 | There is a 1-1 function fr... |
| scmatf1o 22808 | There is a bijection betwe... |
| scmatghm 22809 | There is a group homomorph... |
| scmatmhm 22810 | There is a monoid homomorp... |
| scmatrhm 22811 | There is a ring homomorphi... |
| scmatrngiso 22812 | There is a ring isomorphis... |
| scmatric 22813 | A ring is isomorphic to ev... |
| mat0scmat 22814 | The empty matrix over a ri... |
| mat1scmat 22815 | A 1-dimensional matrix ove... |
| mvmulfval 22818 | Functional value of the ma... |
| mvmulval 22819 | Multiplication of a vector... |
| mvmulfv 22820 | A cell/element in the vect... |
| mavmulval 22821 | Multiplication of a vector... |
| mavmulfv 22822 | A cell/element in the vect... |
| mavmulcl 22823 | Multiplication of an NxN m... |
| 1mavmul 22824 | Multiplication of the iden... |
| mavmulass 22825 | Associativity of the multi... |
| mavmuldm 22826 | The domain of the matrix v... |
| mavmulsolcl 22827 | Every solution of the equa... |
| mavmul0 22828 | Multiplication of a 0-dime... |
| mavmul0g 22829 | The result of the 0-dimens... |
| mvmumamul1 22830 | The multiplication of an M... |
| mavmumamul1 22831 | The multiplication of an N... |
| marrepfval 22836 | First substitution for the... |
| marrepval0 22837 | Second substitution for th... |
| marrepval 22838 | Third substitution for the... |
| marrepeval 22839 | An entry of a matrix with ... |
| marrepcl 22840 | Closure of the row replace... |
| marepvfval 22841 | First substitution for the... |
| marepvval0 22842 | Second substitution for th... |
| marepvval 22843 | Third substitution for the... |
| marepveval 22844 | An entry of a matrix with ... |
| marepvcl 22845 | Closure of the column repl... |
| ma1repvcl 22846 | Closure of the column repl... |
| ma1repveval 22847 | An entry of an identity ma... |
| mulmarep1el 22848 | Element by element multipl... |
| mulmarep1gsum1 22849 | The sum of element by elem... |
| mulmarep1gsum2 22850 | The sum of element by elem... |
| 1marepvmarrepid 22851 | Replacing the ith row by 0... |
| submabas 22854 | Any subset of the index se... |
| submafval 22855 | First substitution for a s... |
| submaval0 22856 | Second substitution for a ... |
| submaval 22857 | Third substitution for a s... |
| submaeval 22858 | An entry of a submatrix of... |
| 1marepvsma1 22859 | The submatrix of the ident... |
| mdetfval 22862 | First substitution for the... |
| mdetleib 22863 | Full substitution of our d... |
| mdetleib2 22864 | Leibniz' formula can also ... |
| nfimdetndef 22865 | The determinant is not def... |
| mdetfval1 22866 | First substitution of an a... |
| mdetleib1 22867 | Full substitution of an al... |
| mdet0pr 22868 | The determinant function f... |
| mdet0f1o 22869 | The determinant function f... |
| mdet0fv0 22870 | The determinant of the emp... |
| mdetf 22871 | Functionality of the deter... |
| mdetcl 22872 | The determinant evaluates ... |
| m1detdiag 22873 | The determinant of a 1-dim... |
| mdetdiaglem 22874 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22875 | The determinant of a diago... |
| mdetdiagid 22876 | The determinant of a diago... |
| mdet1 22877 | The determinant of the ide... |
| mdetrlin 22878 | The determinant function i... |
| mdetrsca 22879 | The determinant function i... |
| mdetrsca2 22880 | The determinant function i... |
| mdetr0 22881 | The determinant of a matri... |
| mdet0 22882 | The determinant of the zer... |
| mdetrlin2 22883 | The determinant function i... |
| mdetralt 22884 | The determinant function i... |
| mdetralt2 22885 | The determinant function i... |
| mdetero 22886 | The determinant function i... |
| mdettpos 22887 | Determinant is invariant u... |
| mdetunilem1 22888 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22889 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22890 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22891 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22892 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22893 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22894 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22895 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22896 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22897 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22898 | According to the definitio... |
| mdetmul 22899 | Multiplicativity of the de... |
| m2detleiblem1 22900 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22901 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22902 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22903 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22904 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22905 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22906 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22907 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22912 | Lemma for ~ maducoeval2 . ... |
| madufval 22913 | First substitution for the... |
| maduval 22914 | Second substitution for th... |
| maducoeval 22915 | An entry of the adjunct (c... |
| maducoeval2 22916 | An entry of the adjunct (c... |
| maduf 22917 | Creating the adjunct of ma... |
| madutpos 22918 | The adjuct of a transposed... |
| madugsum 22919 | The determinant of a matri... |
| madurid 22920 | Multiplying a matrix with ... |
| madulid 22921 | Multiplying the adjunct of... |
| minmar1fval 22922 | First substitution for the... |
| minmar1val0 22923 | Second substitution for th... |
| minmar1val 22924 | Third substitution for the... |
| minmar1eval 22925 | An entry of a matrix for a... |
| minmar1marrep 22926 | The minor matrix is a spec... |
| minmar1cl 22927 | Closure of the row replace... |
| maducoevalmin1 22928 | The coefficients of an adj... |
| symgmatr01lem 22929 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22930 | Applying a permutation tha... |
| gsummatr01lem1 22931 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22932 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22933 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22934 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22935 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22936 | Replacing a row of a squar... |
| smadiadetlem0 22937 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22938 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22939 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22940 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22941 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22942 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22943 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22944 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22945 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22946 | The determinant of a subma... |
| smadiadetglem1 22947 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22948 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22949 | The determinant of a squar... |
| smadiadetg0 22950 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22951 | The determinant of a squar... |
| invrvald 22952 | If a matrix multiplied wit... |
| matinv 22953 | The inverse of a matrix is... |
| matunit 22954 | A matrix is a unit in the ... |
| matunitlindflem1 22955 | One direction of ~ matunit... |
| matunitlindflem2 22956 | One direction of ~ matunit... |
| matunitlindf 22957 | A matrix over a field is i... |
| slesolvec 22958 | Every solution of a system... |
| slesolinv 22959 | The solution of a system o... |
| slesolinvbi 22960 | The solution of a system o... |
| slesolex 22961 | Every system of linear equ... |
| cramerimplem1 22962 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22963 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22964 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22965 | One direction of Cramer's ... |
| cramerlem1 22966 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22967 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22968 | Lemma 3 for ~ cramer . (C... |
| cramer0 22969 | Special case of Cramer's r... |
| cramer 22970 | Cramer's rule. According ... |
| pmatring 22971 | The set of polynomial matr... |
| pmatlmod 22972 | The set of polynomial matr... |
| pmatassa 22973 | The set of polynomial matr... |
| pmat0op 22974 | The zero polynomial matrix... |
| pmat1op 22975 | The identity polynomial ma... |
| pmat1ovd 22976 | Entries of the identity po... |
| pmat0opsc 22977 | The zero polynomial matrix... |
| pmat1opsc 22978 | The identity polynomial ma... |
| pmat1ovscd 22979 | Entries of the identity po... |
| pmatcoe1fsupp 22980 | For a polynomial matrix th... |
| 1pmatscmul 22981 | The scalar product of the ... |
| cpmat 22988 | Value of the constructor o... |
| cpmatpmat 22989 | A constant polynomial matr... |
| cpmatel 22990 | Property of a constant pol... |
| cpmatelimp 22991 | Implication of a set being... |
| cpmatel2 22992 | Another property of a cons... |
| cpmatelimp2 22993 | Another implication of a s... |
| 1elcpmat 22994 | The identity of the ring o... |
| cpmatacl 22995 | The set of all constant po... |
| cpmatinvcl 22996 | The set of all constant po... |
| cpmatmcllem 22997 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22998 | The set of all constant po... |
| cpmatsubgpmat 22999 | The set of all constant po... |
| cpmatsrgpmat 23000 | The set of all constant po... |
| 0elcpmat 23001 | The zero of the ring of al... |
| mat2pmatfval 23002 | Value of the matrix transf... |
| mat2pmatval 23003 | The result of a matrix tra... |
| mat2pmatvalel 23004 | A (matrix) element of the ... |
| mat2pmatbas 23005 | The result of a matrix tra... |
| mat2pmatbas0 23006 | The result of a matrix tra... |
| mat2pmatf 23007 | The matrix transformation ... |
| mat2pmatf1 23008 | The matrix transformation ... |
| mat2pmatghm 23009 | The transformation of matr... |
| mat2pmatmul 23010 | The transformation of matr... |
| mat2pmat1 23011 | The transformation of the ... |
| mat2pmatmhm 23012 | The transformation of matr... |
| mat2pmatrhm 23013 | The transformation of matr... |
| mat2pmatlin 23014 | The transformation of matr... |
| 0mat2pmat 23015 | The transformed zero matri... |
| idmatidpmat 23016 | The transformed identity m... |
| d0mat2pmat 23017 | The transformed empty set ... |
| d1mat2pmat 23018 | The transformation of a ma... |
| mat2pmatscmxcl 23019 | A transformed matrix multi... |
| m2cpm 23020 | The result of a matrix tra... |
| m2cpmf 23021 | The matrix transformation ... |
| m2cpmf1 23022 | The matrix transformation ... |
| m2cpmghm 23023 | The transformation of matr... |
| m2cpmmhm 23024 | The transformation of matr... |
| m2cpmrhm 23025 | The transformation of matr... |
| m2pmfzmap 23026 | The transformed values of ... |
| m2pmfzgsumcl 23027 | Closure of the sum of scal... |
| cpm2mfval 23028 | Value of the inverse matri... |
| cpm2mval 23029 | The result of an inverse m... |
| cpm2mvalel 23030 | A (matrix) element of the ... |
| cpm2mf 23031 | The inverse matrix transfo... |
| m2cpminvid 23032 | The inverse transformation... |
| m2cpminvid2lem 23033 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 23034 | The transformation applied... |
| m2cpmfo 23035 | The matrix transformation ... |
| m2cpmf1o 23036 | The matrix transformation ... |
| m2cpmrngiso 23037 | The transformation of matr... |
| matcpmric 23038 | The ring of matrices over ... |
| m2cpminv 23039 | The inverse matrix transfo... |
| m2cpminv0 23040 | The inverse matrix transfo... |
| decpmatval0 23043 | The matrix consisting of t... |
| decpmatval 23044 | The matrix consisting of t... |
| decpmate 23045 | An entry of the matrix con... |
| decpmatcl 23046 | Closure of the decompositi... |
| decpmataa0 23047 | The matrix consisting of t... |
| decpmatfsupp 23048 | The mapping to the matrice... |
| decpmatid 23049 | The matrix consisting of t... |
| decpmatmullem 23050 | Lemma for ~ decpmatmul . ... |
| decpmatmul 23051 | The matrix consisting of t... |
| decpmatmulsumfsupp 23052 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 23053 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 23054 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 23055 | Write a polynomial matrix ... |
| pmatcollpw2lem 23056 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 23057 | Write a polynomial matrix ... |
| monmatcollpw 23058 | The matrix consisting of t... |
| pmatcollpwlem 23059 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 23060 | Write a polynomial matrix ... |
| pmatcollpwfi 23061 | Write a polynomial matrix ... |
| pmatcollpw3lem 23062 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 23063 | Write a polynomial matrix ... |
| pmatcollpw3fi 23064 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 23065 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 23066 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 23067 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 23068 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 23069 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 23070 | Write a scalar matrix over... |
| pm2mpf1lem 23073 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 23074 | Value of the transformatio... |
| pm2mpfval 23075 | A polynomial matrix transf... |
| pm2mpcl 23076 | The transformation of poly... |
| pm2mpf 23077 | The transformation of poly... |
| pm2mpf1 23078 | The transformation of poly... |
| pm2mpcoe1 23079 | A coefficient of the polyn... |
| idpm2idmp 23080 | The transformation of the ... |
| mptcoe1matfsupp 23081 | The mapping extracting the... |
| mply1topmatcllem 23082 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 23083 | A polynomial over matrices... |
| mply1topmatcl 23084 | A polynomial over matrices... |
| mp2pm2mplem1 23085 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 23086 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 23087 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 23088 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 23089 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 23090 | A polynomial over matrices... |
| pm2mpghmlem2 23091 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 23092 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 23093 | The transformation of poly... |
| pm2mpf1o 23094 | The transformation of poly... |
| pm2mpghm 23095 | The transformation of poly... |
| pm2mpgrpiso 23096 | The transformation of poly... |
| pm2mpmhmlem1 23097 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 23098 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 23099 | The transformation of poly... |
| pm2mprhm 23100 | The transformation of poly... |
| pm2mprngiso 23101 | The transformation of poly... |
| pmmpric 23102 | The ring of polynomial mat... |
| monmat2matmon 23103 | The transformation of a po... |
| pm2mp 23104 | The transformation of a su... |
| chmatcl 23107 | Closure of the characteris... |
| chmatval 23108 | The entries of the charact... |
| chpmatfval 23109 | Value of the characteristi... |
| chpmatval 23110 | The characteristic polynom... |
| chpmatply1 23111 | The characteristic polynom... |
| chpmatval2 23112 | The characteristic polynom... |
| chpmat0d 23113 | The characteristic polynom... |
| chpmat1dlem 23114 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 23115 | The characteristic polynom... |
| chpdmatlem0 23116 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 23117 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 23118 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 23119 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 23120 | The characteristic polynom... |
| chpscmat 23121 | The characteristic polynom... |
| chpscmat0 23122 | The characteristic polynom... |
| chpscmatgsumbin 23123 | The characteristic polynom... |
| chpscmatgsummon 23124 | The characteristic polynom... |
| chp0mat 23125 | The characteristic polynom... |
| chpidmat 23126 | The characteristic polynom... |
| chmaidscmat 23127 | The characteristic polynom... |
| fvmptnn04if 23128 | The function values of a m... |
| fvmptnn04ifa 23129 | The function value of a ma... |
| fvmptnn04ifb 23130 | The function value of a ma... |
| fvmptnn04ifc 23131 | The function value of a ma... |
| fvmptnn04ifd 23132 | The function value of a ma... |
| chfacfisf 23133 | The "characteristic factor... |
| chfacfisfcpmat 23134 | The "characteristic factor... |
| chfacffsupp 23135 | The "characteristic factor... |
| chfacfscmulcl 23136 | Closure of a scaled value ... |
| chfacfscmul0 23137 | A scaled value of the "cha... |
| chfacfscmulfsupp 23138 | A mapping of scaled values... |
| chfacfscmulgsum 23139 | Breaking up a sum of value... |
| chfacfpmmulcl 23140 | Closure of the value of th... |
| chfacfpmmul0 23141 | The value of the "characte... |
| chfacfpmmulfsupp 23142 | A mapping of values of the... |
| chfacfpmmulgsum 23143 | Breaking up a sum of value... |
| chfacfpmmulgsum2 23144 | Breaking up a sum of value... |
| cayhamlem1 23145 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 23146 | The right-hand fundamental... |
| cpmidgsum 23147 | Representation of the iden... |
| cpmidgsumm2pm 23148 | Representation of the iden... |
| cpmidpmatlem1 23149 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 23150 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 23151 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 23152 | Representation of the iden... |
| cpmadugsumlemB 23153 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 23154 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 23155 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 23156 | The product of the charact... |
| cpmadugsum 23157 | The product of the charact... |
| cpmidgsum2 23158 | Representation of the iden... |
| cpmidg2sum 23159 | Equality of two sums repre... |
| cpmadumatpolylem1 23160 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 23161 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 23162 | The product of the charact... |
| cayhamlem2 23163 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 23164 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 23165 | The coefficients of the ch... |
| cayhamlem3 23166 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 23167 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 23168 | The Cayley-Hamilton theore... |
| cayleyhamilton 23169 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 23170 | Alternate proof of ~ cayle... |
| cayleyhamilton1 23171 | The Cayley-Hamilton theore... |
| istopg 23174 | Express the predicate " ` ... |
| istop2g 23175 | Express the predicate " ` ... |
| uniopn 23176 | The union of a subset of a... |
| iunopn 23177 | The indexed union of a sub... |
| inopn 23178 | The intersection of two op... |
| fitop 23179 | A topology is closed under... |
| fiinopn 23180 | The intersection of a none... |
| iinopn 23181 | The intersection of a none... |
| unopn 23182 | The union of two open sets... |
| 0opn 23183 | The empty set is an open s... |
| 0ntop 23184 | The empty set is not a top... |
| topopn 23185 | The underlying set of a to... |
| eltopss 23186 | A member of a topology is ... |
| riinopn 23187 | A finite indexed relative ... |
| rintopn 23188 | A finite relative intersec... |
| istopon 23191 | Property of being a topolo... |
| topontop 23192 | A topology on a given base... |
| toponuni 23193 | The base set of a topology... |
| topontopi 23194 | A topology on a given base... |
| toponunii 23195 | The base set of a topology... |
| toptopon 23196 | Alternative definition of ... |
| toptopon2 23197 | A topology is the same thi... |
| topontopon 23198 | A topology on a set is a t... |
| funtopon 23199 | The class ` TopOn ` is a f... |
| toponrestid 23200 | Given a topology on a set,... |
| toponsspwpw 23201 | The set of topologies on a... |
| dmtopon 23202 | The domain of ` TopOn ` is... |
| fntopon 23203 | The class ` TopOn ` is a f... |
| toprntopon 23204 | A topology is the same thi... |
| toponmax 23205 | The base set of a topology... |
| toponss 23206 | A member of a topology is ... |
| toponcom 23207 | If ` K ` is a topology on ... |
| toponcomb 23208 | Biconditional form of ~ to... |
| topgele 23209 | The topologies over the sa... |
| topsn 23210 | The only topology on a sin... |
| istps 23213 | Express the predicate "is ... |
| istps2 23214 | Express the predicate "is ... |
| tpsuni 23215 | The base set of a topologi... |
| tpstop 23216 | The topology extractor on ... |
| tpspropd 23217 | A topological space depend... |
| tpsprop2d 23218 | A topological space depend... |
| topontopn 23219 | Express the predicate "is ... |
| tsettps 23220 | If the topology component ... |
| istpsi 23221 | Properties that determine ... |
| eltpsg 23222 | Properties that determine ... |
| eltpsi 23223 | Properties that determine ... |
| isbasisg 23226 | Express the predicate "the... |
| isbasis2g 23227 | Express the predicate "the... |
| isbasis3g 23228 | Express the predicate "the... |
| basis1 23229 | Property of a basis. (Con... |
| basis2 23230 | Property of a basis. (Con... |
| fiinbas 23231 | If a set is closed under f... |
| basdif0 23232 | A basis is not affected by... |
| baspartn 23233 | A disjoint system of sets ... |
| tgval 23234 | The topology generated by ... |
| tgval2 23235 | Definition of a topology g... |
| eltg 23236 | Membership in a topology g... |
| eltg2 23237 | Membership in a topology g... |
| eltg2b 23238 | Membership in a topology g... |
| eltg4i 23239 | An open set in a topology ... |
| eltg3i 23240 | The union of a set of basi... |
| eltg3 23241 | Membership in a topology g... |
| tgval3 23242 | Alternate expression for t... |
| tg1 23243 | Property of a member of a ... |
| tg2 23244 | Property of a member of a ... |
| bastg 23245 | A member of a basis is a s... |
| unitg 23246 | The topology generated by ... |
| tgss 23247 | Subset relation for genera... |
| tgcl 23248 | Show that a basis generate... |
| tgclb 23249 | The property ~ tgcl can be... |
| tgtopon 23250 | A basis generates a topolo... |
| topbas 23251 | A topology is its own basi... |
| tgtop 23252 | A topology is its own basi... |
| eltop 23253 | Membership in a topology, ... |
| eltop2 23254 | Membership in a topology. ... |
| eltop3 23255 | Membership in a topology. ... |
| fibas 23256 | A collection of finite int... |
| tgdom 23257 | A space has no more open s... |
| tgiun 23258 | The indexed union of a set... |
| tgidm 23259 | The topology generator fun... |
| bastop 23260 | Two ways to express that a... |
| tgtop11 23261 | The topology generation fu... |
| 0top 23262 | The singleton of the empty... |
| en1top 23263 | ` { (/) } ` is the only to... |
| en2top 23264 | If a topology has two elem... |
| tgss3 23265 | A criterion for determinin... |
| tgss2 23266 | A criterion for determinin... |
| basgen 23267 | Given a topology ` J ` , s... |
| basgen2 23268 | Given a topology ` J ` , s... |
| 2basgen 23269 | Conditions that determine ... |
| tgfiss 23270 | If a subbase is included i... |
| tgdif0 23271 | A generated topology is no... |
| bastop1 23272 | A subset of a topology is ... |
| bastop2 23273 | A version of ~ bastop1 tha... |
| distop 23274 | The discrete topology on a... |
| topnex 23275 | The class of all topologie... |
| distopon 23276 | The discrete topology on a... |
| sn0topon 23277 | The singleton of the empty... |
| sn0top 23278 | The singleton of the empty... |
| indislem 23279 | A lemma to eliminate some ... |
| indistopon 23280 | The indiscrete topology on... |
| indistop 23281 | The indiscrete topology on... |
| indisuni 23282 | The base set of the indisc... |
| fctop 23283 | The finite complement topo... |
| fctop2 23284 | The finite complement topo... |
| cctop 23285 | The countable complement t... |
| ppttop 23286 | The particular point topol... |
| pptbas 23287 | The particular point topol... |
| epttop 23288 | The excluded point topolog... |
| indistpsx 23289 | The indiscrete topology on... |
| indistps 23290 | The indiscrete topology on... |
| indistps2 23291 | The indiscrete topology on... |
| indistpsALT 23292 | The indiscrete topology on... |
| indistps2ALT 23293 | The indiscrete topology on... |
| distps 23294 | The discrete topology on a... |
| fncld 23301 | The closed-set generator i... |
| cldval 23302 | The set of closed sets of ... |
| ntrfval 23303 | The interior function on t... |
| clsfval 23304 | The closure function on th... |
| cldrcl 23305 | Reverse closure of the clo... |
| iscld 23306 | The predicate "the class `... |
| iscld2 23307 | A subset of the underlying... |
| cldss 23308 | A closed set is a subset o... |
| cldss2 23309 | The set of closed sets is ... |
| cldopn 23310 | The complement of a closed... |
| isopn2 23311 | A subset of the underlying... |
| opncld 23312 | The complement of an open ... |
| difopn 23313 | The difference of a closed... |
| topcld 23314 | The underlying set of a to... |
| ntrval 23315 | The interior of a subset o... |
| clsval 23316 | The closure of a subset of... |
| 0cld 23317 | The empty set is closed. ... |
| iincld 23318 | The indexed intersection o... |
| intcld 23319 | The intersection of a set ... |
| uncld 23320 | The union of two closed se... |
| cldcls 23321 | A closed subset equals its... |
| incld 23322 | The intersection of two cl... |
| riincld 23323 | An indexed relative inters... |
| iuncld 23324 | A finite indexed union of ... |
| unicld 23325 | A finite union of closed s... |
| clscld 23326 | The closure of a subset of... |
| clsf 23327 | The closure function is a ... |
| ntropn 23328 | The interior of a subset o... |
| clsval2 23329 | Express closure in terms o... |
| ntrval2 23330 | Interior expressed in term... |
| ntrdif 23331 | An interior of a complemen... |
| clsdif 23332 | A closure of a complement ... |
| clsss 23333 | Subset relationship for cl... |
| ntrss 23334 | Subset relationship for in... |
| sscls 23335 | A subset of a topology's u... |
| ntrss2 23336 | A subset includes its inte... |
| ssntr 23337 | An open subset of a set is... |
| clsss3 23338 | The closure of a subset of... |
| ntrss3 23339 | The interior of a subset o... |
| ntrin 23340 | A pairwise intersection of... |
| cmclsopn 23341 | The complement of a closur... |
| cmntrcld 23342 | The complement of an inter... |
| iscld3 23343 | A subset is closed iff it ... |
| iscld4 23344 | A subset is closed iff it ... |
| isopn3 23345 | A subset is open iff it eq... |
| clsidm 23346 | The closure operation is i... |
| ntridm 23347 | The interior operation is ... |
| clstop 23348 | The closure of a topology'... |
| ntrtop 23349 | The interior of a topology... |
| 0ntr 23350 | A subset with an empty int... |
| clsss2 23351 | If a subset is included in... |
| elcls 23352 | Membership in a closure. ... |
| elcls2 23353 | Membership in a closure. ... |
| clsndisj 23354 | Any open set containing a ... |
| ntrcls0 23355 | A subset whose closure has... |
| ntreq0 23356 | Two ways to say that a sub... |
| cldmre 23357 | The closed sets of a topol... |
| mrccls 23358 | Moore closure generalizes ... |
| cls0 23359 | The closure of the empty s... |
| ntr0 23360 | The interior of the empty ... |
| isopn3i 23361 | An open subset equals its ... |
| elcls3 23362 | Membership in a closure in... |
| opncldf1 23363 | A bijection useful for con... |
| opncldf2 23364 | The values of the open-clo... |
| opncldf3 23365 | The values of the converse... |
| isclo 23366 | A set ` A ` is clopen iff ... |
| isclo2 23367 | A set ` A ` is clopen iff ... |
| discld 23368 | The open sets of a discret... |
| sn0cld 23369 | The closed sets of the top... |
| indiscld 23370 | The closed sets of an indi... |
| mretopd 23371 | A Moore collection which i... |
| toponmre 23372 | The topologies over a give... |
| cldmreon 23373 | The closed sets of a topol... |
| iscldtop 23374 | A family is the closed set... |
| mreclatdemoBAD 23375 | The closed subspaces of a ... |
| neifval 23378 | Value of the neighborhood ... |
| neif 23379 | The neighborhood function ... |
| neiss2 23380 | A set with a neighborhood ... |
| neival 23381 | Value of the set of neighb... |
| isnei 23382 | The predicate "the class `... |
| neiint 23383 | An intuitive definition of... |
| isneip 23384 | The predicate "the class `... |
| neii1 23385 | A neighborhood is included... |
| neisspw 23386 | The neighborhoods of any s... |
| neii2 23387 | Property of a neighborhood... |
| neiss 23388 | Any neighborhood of a set ... |
| ssnei 23389 | A set is included in any o... |
| elnei 23390 | A point belongs to any of ... |
| 0nnei 23391 | The empty set is not a nei... |
| neips 23392 | A neighborhood of a set is... |
| opnneissb 23393 | An open set is a neighborh... |
| opnssneib 23394 | Any superset of an open se... |
| ssnei2 23395 | Any subset ` M ` of ` X ` ... |
| neindisj 23396 | Any neighborhood of an ele... |
| opnneiss 23397 | An open set is a neighborh... |
| opnneip 23398 | An open set is a neighborh... |
| opnnei 23399 | A set is open iff it is a ... |
| tpnei 23400 | The underlying set of a to... |
| neiuni 23401 | The union of the neighborh... |
| neindisj2 23402 | A point ` P ` belongs to t... |
| topssnei 23403 | A finer topology has more ... |
| innei 23404 | The intersection of two ne... |
| opnneiid 23405 | Only an open set is a neig... |
| neissex 23406 | For any neighborhood ` N `... |
| 0nei 23407 | The empty set is a neighbo... |
| neipeltop 23408 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23409 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23410 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23411 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23412 | If, to each element ` P ` ... |
| lpfval 23417 | The limit point function o... |
| lpval 23418 | The set of limit points of... |
| islp 23419 | The predicate "the class `... |
| lpsscls 23420 | The limit points of a subs... |
| lpss 23421 | The limit points of a subs... |
| lpdifsn 23422 | ` P ` is a limit point of ... |
| lpss3 23423 | Subset relationship for li... |
| islp2 23424 | The predicate " ` P ` is a... |
| islp3 23425 | The predicate " ` P ` is a... |
| maxlp 23426 | A point is a limit point o... |
| clslp 23427 | The closure of a subset of... |
| islpi 23428 | A point belonging to a set... |
| cldlp 23429 | A subset of a topological ... |
| isperf 23430 | Definition of a perfect sp... |
| isperf2 23431 | Definition of a perfect sp... |
| isperf3 23432 | A perfect space is a topol... |
| perflp 23433 | The limit points of a perf... |
| perfi 23434 | Property of a perfect spac... |
| perftop 23435 | A perfect space is a topol... |
| restrcl 23436 | Reverse closure for the su... |
| restbas 23437 | A subspace topology basis ... |
| tgrest 23438 | A subspace can be generate... |
| resttop 23439 | A subspace topology is a t... |
| resttopon 23440 | A subspace topology is a t... |
| restuni 23441 | The underlying set of a su... |
| stoig 23442 | The topological space buil... |
| restco 23443 | Composition of subspaces. ... |
| restabs 23444 | Equivalence of being a sub... |
| restin 23445 | When the subspace region i... |
| restuni2 23446 | The underlying set of a su... |
| resttopon2 23447 | The underlying set of a su... |
| rest0 23448 | The subspace topology indu... |
| restsn 23449 | The only subspace topology... |
| restsn2 23450 | The subspace topology indu... |
| restcld 23451 | A closed set of a subspace... |
| restcldi 23452 | A closed set is closed in ... |
| restcldr 23453 | A set which is closed in t... |
| restopnb 23454 | If ` B ` is an open subset... |
| ssrest 23455 | If ` K ` is a finer topolo... |
| restopn2 23456 | If ` A ` is open, then ` B... |
| restdis 23457 | A subspace of a discrete t... |
| restfpw 23458 | The restriction of the set... |
| neitr 23459 | The neighborhood of a trac... |
| restcls 23460 | A closure in a subspace to... |
| restntr 23461 | An interior in a subspace ... |
| restlp 23462 | The limit points of a subs... |
| restperf 23463 | Perfection of a subspace. ... |
| perfopn 23464 | An open subset of a perfec... |
| resstopn 23465 | The topology of a restrict... |
| resstps 23466 | A restricted topological s... |
| ordtbaslem 23467 | Lemma for ~ ordtbas . In ... |
| ordtval 23468 | Value of the order topolog... |
| ordtuni 23469 | Value of the order topolog... |
| ordtbas2 23470 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23471 | In a total order, the fini... |
| ordttopon 23472 | Value of the order topolog... |
| ordtopn1 23473 | An upward ray ` ( P , +oo ... |
| ordtopn2 23474 | A downward ray ` ( -oo , P... |
| ordtopn3 23475 | An open interval ` ( A , B... |
| ordtcld1 23476 | A downward ray ` ( -oo , P... |
| ordtcld2 23477 | An upward ray ` [ P , +oo ... |
| ordtcld3 23478 | A closed interval ` [ A , ... |
| ordttop 23479 | The order topology is a to... |
| ordtcnv 23480 | The order dual generates t... |
| ordtrest 23481 | The subspace topology of a... |
| ordtrest2lem 23482 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23483 | An interval-closed set ` A... |
| letopon 23484 | The topology of the extend... |
| letop 23485 | The topology of the extend... |
| letopuni 23486 | The topology of the extend... |
| xrstopn 23487 | The topology component of ... |
| xrstps 23488 | The extended real number s... |
| leordtvallem1 23489 | Lemma for ~ leordtval . (... |
| leordtvallem2 23490 | Lemma for ~ leordtval . (... |
| leordtval2 23491 | The topology of the extend... |
| leordtval 23492 | The topology of the extend... |
| iccordt 23493 | A closed interval is close... |
| iocpnfordt 23494 | An unbounded above open in... |
| icomnfordt 23495 | An unbounded above open in... |
| iooordt 23496 | An open interval is open i... |
| reordt 23497 | The real numbers are an op... |
| lecldbas 23498 | The set of closed interval... |
| pnfnei 23499 | A neighborhood of ` +oo ` ... |
| mnfnei 23500 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23501 | The restriction of the les... |
| ordtresticc 23502 | The restriction of the les... |
| lmrel 23509 | The topological space conv... |
| lmrcl 23510 | Reverse closure for the co... |
| lmfval 23511 | The relation "sequence ` f... |
| cnfval 23512 | The set of all continuous ... |
| cnpfval 23513 | The function mapping the p... |
| iscn 23514 | The predicate "the class `... |
| cnpval 23515 | The set of all functions f... |
| iscnp 23516 | The predicate "the class `... |
| iscn2 23517 | The predicate "the class `... |
| iscnp2 23518 | The predicate "the class `... |
| cntop1 23519 | Reverse closure for a cont... |
| cntop2 23520 | Reverse closure for a cont... |
| cnptop1 23521 | Reverse closure for a func... |
| cnptop2 23522 | Reverse closure for a func... |
| iscnp3 23523 | The predicate "the class `... |
| cnprcl 23524 | Reverse closure for a func... |
| cnf 23525 | A continuous function is a... |
| cnpf 23526 | A continuous function at p... |
| cnpcl 23527 | The value of a continuous ... |
| cnf2 23528 | A continuous function is a... |
| cnpf2 23529 | A continuous function at p... |
| cnprcl2 23530 | Reverse closure for a func... |
| tgcn 23531 | The continuity predicate w... |
| tgcnp 23532 | The "continuous at a point... |
| subbascn 23533 | The continuity predicate w... |
| ssidcn 23534 | The identity function is a... |
| cnpimaex 23535 | Property of a function con... |
| idcn 23536 | A restricted identity func... |
| lmbr 23537 | Express the binary relatio... |
| lmbr2 23538 | Express the binary relatio... |
| lmbrf 23539 | Express the binary relatio... |
| lmconst 23540 | A constant sequence conver... |
| lmcvg 23541 | Convergence property of a ... |
| iscnp4 23542 | The predicate "the class `... |
| cnpnei 23543 | A condition for continuity... |
| cnima 23544 | An open subset of the codo... |
| cnco 23545 | The composition of two con... |
| cnpco 23546 | The composition of a funct... |
| cnclima 23547 | A closed subset of the cod... |
| iscncl 23548 | A characterization of a co... |
| cncls2i 23549 | Property of the preimage o... |
| cnntri 23550 | Property of the preimage o... |
| cnclsi 23551 | Property of the image of a... |
| cncls2 23552 | Continuity in terms of clo... |
| cncls 23553 | Continuity in terms of clo... |
| cnntr 23554 | Continuity in terms of int... |
| cnss1 23555 | If the topology ` K ` is f... |
| cnss2 23556 | If the topology ` K ` is f... |
| cncnpi 23557 | A continuous function is c... |
| cnsscnp 23558 | The set of continuous func... |
| cncnp 23559 | A continuous function is c... |
| cncnp2 23560 | A continuous function is c... |
| cnnei 23561 | Continuity in terms of nei... |
| cnconst2 23562 | A constant function is con... |
| cnconst 23563 | A constant function is con... |
| cnrest 23564 | Continuity of a restrictio... |
| cnrest2 23565 | Equivalence of continuity ... |
| cnrest2r 23566 | Equivalence of continuity ... |
| cnpresti 23567 | One direction of ~ cnprest... |
| cnprest 23568 | Equivalence of continuity ... |
| cnprest2 23569 | Equivalence of point-conti... |
| cndis 23570 | Every function is continuo... |
| cnindis 23571 | Every function is continuo... |
| cnpdis 23572 | If ` A ` is an isolated po... |
| paste 23573 | Pasting lemma. If ` A ` a... |
| lmfpm 23574 | If ` F ` converges, then `... |
| lmfss 23575 | Inclusion of a function ha... |
| lmcl 23576 | Closure of a limit. (Cont... |
| lmss 23577 | Limit on a subspace. (Con... |
| sslm 23578 | A finer topology has fewer... |
| lmres 23579 | A function converges iff i... |
| lmff 23580 | If ` F ` converges, there ... |
| lmcls 23581 | Any convergent sequence of... |
| lmcld 23582 | Any convergent sequence of... |
| lmcnp 23583 | The image of a convergent ... |
| lmcn 23584 | The image of a convergent ... |
| ist0 23599 | The predicate "is a T_0 sp... |
| ist1 23600 | The predicate "is a T_1 sp... |
| ishaus 23601 | The predicate "is a Hausdo... |
| iscnrm 23602 | The property of being comp... |
| t0sep 23603 | Any two topologically indi... |
| t0dist 23604 | Any two distinct points in... |
| t1sncld 23605 | In a T_1 space, singletons... |
| t1ficld 23606 | In a T_1 space, finite set... |
| hausnei 23607 | Neighborhood property of a... |
| t0top 23608 | A T_0 space is a topologic... |
| t1top 23609 | A T_1 space is a topologic... |
| haustop 23610 | A Hausdorff space is a top... |
| isreg 23611 | The predicate "is a regula... |
| regtop 23612 | A regular space is a topol... |
| regsep 23613 | In a regular space, every ... |
| isnrm 23614 | The predicate "is a normal... |
| nrmtop 23615 | A normal space is a topolo... |
| cnrmtop 23616 | A completely normal space ... |
| iscnrm2 23617 | The property of being comp... |
| ispnrm 23618 | The property of being perf... |
| pnrmnrm 23619 | A perfectly normal space i... |
| pnrmtop 23620 | A perfectly normal space i... |
| pnrmcld 23621 | A closed set in a perfectl... |
| pnrmopn 23622 | An open set in a perfectly... |
| ist0-2 23623 | The predicate "is a T_0 sp... |
| ist0-3 23624 | The predicate "is a T_0 sp... |
| cnt0 23625 | The preimage of a T_0 topo... |
| ist1-2 23626 | An alternate characterizat... |
| t1t0 23627 | A T_1 space is a T_0 space... |
| ist1-3 23628 | A space is T_1 iff every p... |
| cnt1 23629 | The preimage of a T_1 topo... |
| ishaus2 23630 | Express the predicate " ` ... |
| haust1 23631 | A Hausdorff space is a T_1... |
| hausnei2 23632 | The Hausdorff condition st... |
| cnhaus 23633 | The preimage of a Hausdorf... |
| nrmsep3 23634 | In a normal space, given a... |
| nrmsep2 23635 | In a normal space, any two... |
| nrmsep 23636 | In a normal space, disjoin... |
| isnrm2 23637 | An alternate characterizat... |
| isnrm3 23638 | A topological space is nor... |
| cnrmi 23639 | A subspace of a completely... |
| cnrmnrm 23640 | A completely normal space ... |
| restcnrm 23641 | A subspace of a completely... |
| resthauslem 23642 | Lemma for ~ resthaus and s... |
| lpcls 23643 | The limit points of the cl... |
| perfcls 23644 | A subset of a perfect spac... |
| restt0 23645 | A subspace of a T_0 topolo... |
| restt1 23646 | A subspace of a T_1 topolo... |
| resthaus 23647 | A subspace of a Hausdorff ... |
| t1sep2 23648 | Any two points in a T_1 sp... |
| t1sep 23649 | Any two distinct points in... |
| sncld 23650 | A singleton is closed in a... |
| sshauslem 23651 | Lemma for ~ sshaus and sim... |
| sst0 23652 | A topology finer than a T_... |
| sst1 23653 | A topology finer than a T_... |
| sshaus 23654 | A topology finer than a Ha... |
| regsep2 23655 | In a regular space, a clos... |
| isreg2 23656 | A topological space is reg... |
| dnsconst 23657 | If a continuous mapping to... |
| ordtt1 23658 | The order topology is T_1 ... |
| lmmo 23659 | A sequence in a Hausdorff ... |
| lmfun 23660 | The convergence relation i... |
| dishaus 23661 | A discrete topology is Hau... |
| ordthauslem 23662 | Lemma for ~ ordthaus . (C... |
| ordthaus 23663 | The order topology of a to... |
| xrhaus 23664 | The topology of the extend... |
| iscmp 23667 | The predicate "is a compac... |
| cmpcov 23668 | An open cover of a compact... |
| cmpcov2 23669 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23670 | Combine ~ cmpcov with ~ ac... |
| cncmp 23671 | Compactness is respected b... |
| fincmp 23672 | A finite topology is compa... |
| 0cmp 23673 | The singleton of the empty... |
| cmptop 23674 | A compact topology is a to... |
| rncmp 23675 | The image of a compact set... |
| imacmp 23676 | The image of a compact set... |
| discmp 23677 | A discrete topology is com... |
| cmpsublem 23678 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23679 | Two equivalent ways of des... |
| tgcmp 23680 | A topology generated by a ... |
| cmpcld 23681 | A closed subset of a compa... |
| uncmp 23682 | The union of two compact s... |
| fiuncmp 23683 | A finite union of compact ... |
| sscmp 23684 | A subset of a compact topo... |
| hauscmplem 23685 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23686 | A compact subspace of a T2... |
| cmpfi 23687 | If a topology is compact a... |
| cmpfii 23688 | In a compact topology, a s... |
| bwth 23689 | The glorious Bolzano-Weier... |
| isconn 23692 | The predicate ` J ` is a c... |
| isconn2 23693 | The predicate ` J ` is a c... |
| connclo 23694 | The only nonempty clopen s... |
| conndisj 23695 | If a topology is connected... |
| conntop 23696 | A connected topology is a ... |
| indisconn 23697 | The indiscrete topology (o... |
| dfconn2 23698 | An alternate definition of... |
| connsuba 23699 | Connectedness for a subspa... |
| connsub 23700 | Two equivalent ways of say... |
| cnconn 23701 | Connectedness is respected... |
| nconnsubb 23702 | Disconnectedness for a sub... |
| connsubclo 23703 | If a clopen set meets a co... |
| connima 23704 | The image of a connected s... |
| conncn 23705 | A continuous function from... |
| iunconnlem 23706 | Lemma for ~ iunconn . (Co... |
| iunconn 23707 | The indexed union of conne... |
| unconn 23708 | The union of two connected... |
| clsconn 23709 | The closure of a connected... |
| conncompid 23710 | The connected component co... |
| conncompconn 23711 | The connected component co... |
| conncompss 23712 | The connected component co... |
| conncompcld 23713 | The connected component co... |
| conncompclo 23714 | The connected component co... |
| t1connperf 23715 | A connected T_1 space is p... |
| is1stc 23720 | The predicate "is a first-... |
| is1stc2 23721 | An equivalent way of sayin... |
| 1stctop 23722 | A first-countable topology... |
| 1stcclb 23723 | A property of points in a ... |
| 1stcfb 23724 | For any point ` A ` in a f... |
| is2ndc 23725 | The property of being seco... |
| 2ndctop 23726 | A second-countable topolog... |
| 2ndci 23727 | A countable basis generate... |
| 2ndcsb 23728 | Having a countable subbase... |
| 2ndcredom 23729 | A second-countable space h... |
| 2ndc1stc 23730 | A second-countable space i... |
| 1stcrestlem 23731 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23732 | A subspace of a first-coun... |
| 2ndcrest 23733 | A subspace of a second-cou... |
| abrexct 23734 | An image set of a countabl... |
| 2ndcctbss 23735 | If a topology is second-co... |
| 2ndcdisj 23736 | Any disjoint family of ope... |
| 2ndcdisj2 23737 | Any disjoint collection of... |
| 2ndcomap 23738 | A surjective continuous op... |
| 2ndcsep 23739 | A second-countable topolog... |
| dis2ndc 23740 | A discrete space is second... |
| 1stcelcls 23741 | A point belongs to the clo... |
| 1stccnp 23742 | A mapping is continuous at... |
| 1stccn 23743 | A mapping ` X --> Y ` , wh... |
| islly 23748 | The property of being a lo... |
| isnlly 23749 | The property of being an n... |
| llyeq 23750 | Equality theorem for the `... |
| nllyeq 23751 | Equality theorem for the `... |
| llytop 23752 | A locally ` A ` space is a... |
| nllytop 23753 | A locally ` A ` space is a... |
| llyi 23754 | The property of a locally ... |
| nllyi 23755 | The property of an n-local... |
| nlly2i 23756 | Eliminate the neighborhood... |
| llynlly 23757 | A locally ` A ` space is n... |
| llyssnlly 23758 | A locally ` A ` space is n... |
| llyss 23759 | The "locally" predicate re... |
| nllyss 23760 | The "n-locally" predicate ... |
| subislly 23761 | The property of a subspace... |
| restnlly 23762 | If the property ` A ` pass... |
| restlly 23763 | If the property ` A ` pass... |
| islly2 23764 | An alternative expression ... |
| llyrest 23765 | An open subspace of a loca... |
| nllyrest 23766 | An open subspace of an n-l... |
| loclly 23767 | If ` A ` is a local proper... |
| llyidm 23768 | Idempotence of the "locall... |
| nllyidm 23769 | Idempotence of the "n-loca... |
| toplly 23770 | A topology is locally a to... |
| topnlly 23771 | A topology is n-locally a ... |
| hauslly 23772 | A Hausdorff space is local... |
| hausnlly 23773 | A Hausdorff space is n-loc... |
| hausllycmp 23774 | A compact Hausdorff space ... |
| cldllycmp 23775 | A closed subspace of a loc... |
| lly1stc 23776 | First-countability is a lo... |
| dislly 23777 | The discrete space ` ~P X ... |
| disllycmp 23778 | A discrete space is locall... |
| dis1stc 23779 | A discrete space is first-... |
| hausmapdom 23780 | If ` X ` is a first-counta... |
| hauspwdom 23781 | Simplify the cardinal ` A ... |
| refrel 23788 | Refinement is a relation. ... |
| isref 23789 | The property of being a re... |
| refbas 23790 | A refinement covers the sa... |
| refssex 23791 | Every set in a refinement ... |
| ssref 23792 | A subcover is a refinement... |
| refref 23793 | Reflexivity of refinement.... |
| reftr 23794 | Refinement is transitive. ... |
| refun0 23795 | Adding the empty set prese... |
| isptfin 23796 | The statement "is a point-... |
| islocfin 23797 | The statement "is a locall... |
| finptfin 23798 | A finite cover is a point-... |
| ptfinfin 23799 | A point covered by a point... |
| finlocfin 23800 | A finite cover of a topolo... |
| locfintop 23801 | A locally finite cover cov... |
| locfinbas 23802 | A locally finite cover mus... |
| locfinnei 23803 | A point covered by a local... |
| lfinpfin 23804 | A locally finite cover is ... |
| lfinun 23805 | Adding a finite set preser... |
| locfincmp 23806 | For a compact space, the l... |
| unisngl 23807 | Taking the union of the se... |
| dissnref 23808 | The set of singletons is a... |
| dissnlocfin 23809 | The set of singletons is l... |
| locfindis 23810 | The locally finite covers ... |
| locfincf 23811 | A locally finite cover in ... |
| comppfsc 23812 | A space where every open c... |
| kgenval 23815 | Value of the compact gener... |
| elkgen 23816 | Value of the compact gener... |
| kgeni 23817 | Property of the open sets ... |
| kgentopon 23818 | The compact generator gene... |
| kgenuni 23819 | The base set of the compac... |
| kgenftop 23820 | The compact generator gene... |
| kgenf 23821 | The compact generator is a... |
| kgentop 23822 | A compactly generated spac... |
| kgenss 23823 | The compact generator gene... |
| kgenhaus 23824 | The compact generator gene... |
| kgencmp 23825 | The compact generator topo... |
| kgencmp2 23826 | The compact generator topo... |
| kgenidm 23827 | The compact generator is i... |
| iskgen2 23828 | A space is compactly gener... |
| iskgen3 23829 | Derive the usual definitio... |
| llycmpkgen2 23830 | A locally compact space is... |
| cmpkgen 23831 | A compact space is compact... |
| llycmpkgen 23832 | A locally compact space is... |
| 1stckgenlem 23833 | The one-point compactifica... |
| 1stckgen 23834 | A first-countable space is... |
| kgen2ss 23835 | The compact generator pres... |
| kgencn 23836 | A function from a compactl... |
| kgencn2 23837 | A function ` F : J --> K `... |
| kgencn3 23838 | The set of continuous func... |
| kgen2cn 23839 | A continuous function is a... |
| txval 23844 | Value of the binary topolo... |
| txuni2 23845 | The underlying set of the ... |
| txbasex 23846 | The basis for the product ... |
| txbas 23847 | The set of Cartesian produ... |
| eltx 23848 | A set in a product is open... |
| txtop 23849 | The product of two topolog... |
| ptval 23850 | The value of the product t... |
| ptpjpre1 23851 | The preimage of a projecti... |
| elpt 23852 | Elementhood in the bases o... |
| elptr 23853 | A basic open set in the pr... |
| elptr2 23854 | A basic open set in the pr... |
| ptbasid 23855 | The base set of the produc... |
| ptuni2 23856 | The base set for the produ... |
| ptbasin 23857 | The basis for a product to... |
| ptbasin2 23858 | The basis for a product to... |
| ptbas 23859 | The basis for a product to... |
| ptpjpre2 23860 | The basis for a product to... |
| ptbasfi 23861 | The basis for the product ... |
| pttop 23862 | The product topology is a ... |
| ptopn 23863 | A basic open set in the pr... |
| ptopn2 23864 | A sub-basic open set in th... |
| xkotf 23865 | Functionality of function ... |
| xkobval 23866 | Alternative expression for... |
| xkoval 23867 | Value of the compact-open ... |
| xkotop 23868 | The compact-open topology ... |
| xkoopn 23869 | A basic open set of the co... |
| txtopi 23870 | The product of two topolog... |
| txtopon 23871 | The underlying set of the ... |
| txuni 23872 | The underlying set of the ... |
| txunii 23873 | The underlying set of the ... |
| ptuni 23874 | The base set for the produ... |
| ptunimpt 23875 | Base set of a product topo... |
| pttopon 23876 | The base set for the produ... |
| pttoponconst 23877 | The base set for a product... |
| ptuniconst 23878 | The base set for a product... |
| xkouni 23879 | The base set of the compac... |
| xkotopon 23880 | The base set of the compac... |
| ptval2 23881 | The value of the product t... |
| txopn 23882 | The product of two open se... |
| txcld 23883 | The product of two closed ... |
| txcls 23884 | Closure of a rectangle in ... |
| txss12 23885 | Subset property of the top... |
| txbasval 23886 | It is sufficient to consid... |
| neitx 23887 | The Cartesian product of t... |
| txcnpi 23888 | Continuity of a two-argume... |
| tx1cn 23889 | Continuity of the first pr... |
| tx2cn 23890 | Continuity of the second p... |
| ptpjcn 23891 | Continuity of a projection... |
| ptpjopn 23892 | The projection map is an o... |
| ptcld 23893 | A closed box in the produc... |
| ptcldmpt 23894 | A closed box in the produc... |
| ptclsg 23895 | The closure of a box in th... |
| ptcls 23896 | The closure of a box in th... |
| dfac14lem 23897 | Lemma for ~ dfac14 . By e... |
| dfac14 23898 | Theorem ~ ptcls is an equi... |
| xkoccn 23899 | The "constant function" fu... |
| txcnp 23900 | If two functions are conti... |
| ptcnplem 23901 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23902 | If every projection of a f... |
| upxp 23903 | Universal property of the ... |
| txcnmpt 23904 | A map into the product of ... |
| uptx 23905 | Universal property of the ... |
| txcn 23906 | A map into the product of ... |
| ptcn 23907 | If every projection of a f... |
| prdstopn 23908 | Topology of a structure pr... |
| prdstps 23909 | A structure product of top... |
| pwstps 23910 | A structure power of a top... |
| txrest 23911 | The subspace of a topologi... |
| txdis 23912 | The topological product of... |
| txindislem 23913 | Lemma for ~ txindis . (Co... |
| txindis 23914 | The topological product of... |
| txdis1cn 23915 | A function is jointly cont... |
| txlly 23916 | If the property ` A ` is p... |
| txnlly 23917 | If the property ` A ` is p... |
| pthaus 23918 | The product of a collectio... |
| ptrescn 23919 | Restriction is a continuou... |
| txtube 23920 | The "tube lemma". If ` X ... |
| txcmplem1 23921 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23922 | Lemma for ~ txcmp . (Cont... |
| txcmp 23923 | The topological product of... |
| txcmpb 23924 | The topological product of... |
| hausdiag 23925 | A topology is Hausdorff if... |
| hauseqlcld 23926 | In a Hausdorff topology, t... |
| txhaus 23927 | The topological product of... |
| txlm 23928 | Two sequences converge iff... |
| lmcn2 23929 | The image of a convergent ... |
| tx1stc 23930 | The topological product of... |
| tx2ndc 23931 | The topological product of... |
| txkgen 23932 | The topological product of... |
| xkohaus 23933 | If the codomain space is H... |
| xkoptsub 23934 | The compact-open topology ... |
| xkopt 23935 | The compact-open topology ... |
| xkopjcn 23936 | Continuity of a projection... |
| xkoco1cn 23937 | If ` F ` is a continuous f... |
| xkoco2cn 23938 | If ` F ` is a continuous f... |
| xkococnlem 23939 | Continuity of the composit... |
| xkococn 23940 | Continuity of the composit... |
| cnmptid 23941 | The identity function is c... |
| cnmptc 23942 | A constant function is con... |
| cnmpt11 23943 | The composition of continu... |
| cnmpt11f 23944 | The composition of continu... |
| cnmpt1t 23945 | The composition of continu... |
| cnmpt12f 23946 | The composition of continu... |
| cnmpt12 23947 | The composition of continu... |
| cnmpt1st 23948 | The projection onto the fi... |
| cnmpt2nd 23949 | The projection onto the se... |
| cnmpt2c 23950 | A constant function is con... |
| cnmpt21 23951 | The composition of continu... |
| cnmpt21f 23952 | The composition of continu... |
| cnmpt2t 23953 | The composition of continu... |
| cnmpt22 23954 | The composition of continu... |
| cnmpt22f 23955 | The composition of continu... |
| cnmpt1res 23956 | The restriction of a conti... |
| cnmpt2res 23957 | The restriction of a conti... |
| cnmptcom 23958 | The argument converse of a... |
| cnmptkc 23959 | The curried first projecti... |
| cnmptkp 23960 | The evaluation of the inne... |
| cnmptk1 23961 | The composition of a curri... |
| cnmpt1k 23962 | The composition of a one-a... |
| cnmptkk 23963 | The composition of two cur... |
| xkofvcn 23964 | Joint continuity of the fu... |
| cnmptk1p 23965 | The evaluation of a currie... |
| cnmptk2 23966 | The uncurrying of a currie... |
| xkoinjcn 23967 | Continuity of "injection",... |
| cnmpt2k 23968 | The currying of a two-argu... |
| txconn 23969 | The topological product of... |
| imasnopn 23970 | If a relation graph is ope... |
| imasncld 23971 | If a relation graph is clo... |
| imasncls 23972 | If a relation graph is clo... |
| qtopval 23975 | Value of the quotient topo... |
| qtopval2 23976 | Value of the quotient topo... |
| elqtop 23977 | Value of the quotient topo... |
| qtopres 23978 | The quotient topology is u... |
| qtoptop2 23979 | The quotient topology is a... |
| qtoptop 23980 | The quotient topology is a... |
| elqtop2 23981 | Value of the quotient topo... |
| qtopuni 23982 | The base set of the quotie... |
| elqtop3 23983 | Value of the quotient topo... |
| qtoptopon 23984 | The base set of the quotie... |
| qtopid 23985 | A quotient map is a contin... |
| idqtop 23986 | The quotient topology indu... |
| qtopcmplem 23987 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23988 | A quotient of a compact sp... |
| qtopconn 23989 | A quotient of a connected ... |
| qtopkgen 23990 | A quotient of a compactly ... |
| basqtop 23991 | An injection maps bases to... |
| tgqtop 23992 | An injection maps generate... |
| qtopcld 23993 | The property of being a cl... |
| qtopcn 23994 | Universal property of a qu... |
| qtopss 23995 | A surjective continuous fu... |
| qtopeu 23996 | Universal property of the ... |
| qtoprest 23997 | If ` A ` is a saturated op... |
| qtopomap 23998 | If ` F ` is a surjective c... |
| qtopcmap 23999 | If ` F ` is a surjective c... |
| imastopn 24000 | The topology of an image s... |
| imastps 24001 | The image of a topological... |
| qustps 24002 | A quotient structure is a ... |
| kqfval 24003 | Value of the function appe... |
| kqfeq 24004 | Two points in the Kolmogor... |
| kqffn 24005 | The topological indistingu... |
| kqval 24006 | Value of the quotient topo... |
| kqtopon 24007 | The Kolmogorov quotient is... |
| kqid 24008 | The topological indistingu... |
| ist0-4 24009 | The topological indistingu... |
| kqfvima 24010 | When the image set is open... |
| kqsat 24011 | Any open set is saturated ... |
| kqdisj 24012 | A version of ~ imain for t... |
| kqcldsat 24013 | Any closed set is saturate... |
| kqopn 24014 | The topological indistingu... |
| kqcld 24015 | The topological indistingu... |
| kqt0lem 24016 | Lemma for ~ kqt0 . (Contr... |
| isr0 24017 | The property " ` J ` is an... |
| r0cld 24018 | The analogue of the T_1 ax... |
| regr1lem 24019 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 24020 | A Kolmogorov quotient of a... |
| kqreglem1 24021 | A Kolmogorov quotient of a... |
| kqreglem2 24022 | If the Kolmogorov quotient... |
| kqnrmlem1 24023 | A Kolmogorov quotient of a... |
| kqnrmlem2 24024 | If the Kolmogorov quotient... |
| kqtop 24025 | The Kolmogorov quotient is... |
| kqt0 24026 | The Kolmogorov quotient is... |
| kqf 24027 | The Kolmogorov quotient is... |
| r0sep 24028 | The separation property of... |
| nrmr0reg 24029 | A normal R_0 space is also... |
| regr1 24030 | A regular space is R_1, wh... |
| kqreg 24031 | The Kolmogorov quotient of... |
| kqnrm 24032 | The Kolmogorov quotient of... |
| hmeofn 24037 | The set of homeomorphisms ... |
| hmeofval 24038 | The set of all the homeomo... |
| ishmeo 24039 | The predicate F is a homeo... |
| hmeocn 24040 | A homeomorphism is continu... |
| hmeocnvcn 24041 | The converse of a homeomor... |
| hmeocnv 24042 | The converse of a homeomor... |
| hmeof1o2 24043 | A homeomorphism is a 1-1-o... |
| hmeof1o 24044 | A homeomorphism is a 1-1-o... |
| hmeoima 24045 | The image of an open set b... |
| hmeoopn 24046 | Homeomorphisms preserve op... |
| hmeocld 24047 | Homeomorphisms preserve cl... |
| hmeocls 24048 | Homeomorphisms preserve cl... |
| hmeontr 24049 | Homeomorphisms preserve in... |
| hmeoimaf1o 24050 | The function mapping open ... |
| hmeores 24051 | The restriction of a homeo... |
| hmeoco 24052 | The composite of two homeo... |
| idhmeo 24053 | The identity function is a... |
| hmeocnvb 24054 | The converse of a homeomor... |
| hmeoqtop 24055 | A homeomorphism is a quoti... |
| hmph 24056 | Express the predicate ` J ... |
| hmphi 24057 | If there is a homeomorphis... |
| hmphtop 24058 | Reverse closure for the ho... |
| hmphtop1 24059 | The relation "being homeom... |
| hmphtop2 24060 | The relation "being homeom... |
| hmphref 24061 | "Is homeomorphic to" is re... |
| hmphsym 24062 | "Is homeomorphic to" is sy... |
| hmphtr 24063 | "Is homeomorphic to" is tr... |
| hmpher 24064 | "Is homeomorphic to" is an... |
| hmphen 24065 | Homeomorphisms preserve th... |
| hmphsymb 24066 | "Is homeomorphic to" is sy... |
| haushmphlem 24067 | Lemma for ~ haushmph and s... |
| cmphmph 24068 | Compactness is a topologic... |
| connhmph 24069 | Connectedness is a topolog... |
| t0hmph 24070 | T_0 is a topological prope... |
| t1hmph 24071 | T_1 is a topological prope... |
| haushmph 24072 | Hausdorff-ness is a topolo... |
| reghmph 24073 | Regularity is a topologica... |
| nrmhmph 24074 | Normality is a topological... |
| hmph0 24075 | A topology homeomorphic to... |
| hmphdis 24076 | Homeomorphisms preserve to... |
| hmphindis 24077 | Homeomorphisms preserve to... |
| indishmph 24078 | Equinumerous sets equipped... |
| hmphen2 24079 | Homeomorphisms preserve th... |
| cmphaushmeo 24080 | A continuous bijection fro... |
| ordthmeolem 24081 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 24082 | An order isomorphism is a ... |
| txhmeo 24083 | Lift a pair of homeomorphi... |
| txswaphmeolem 24084 | Show inverse for the "swap... |
| txswaphmeo 24085 | There is a homeomorphism f... |
| pt1hmeo 24086 | The canonical homeomorphis... |
| ptuncnv 24087 | Exhibit the converse funct... |
| ptunhmeo 24088 | Define a homeomorphism fro... |
| xpstopnlem1 24089 | The function ` F ` used in... |
| xpstps 24090 | A binary product of topolo... |
| xpstopnlem2 24091 | Lemma for ~ xpstopn . (Co... |
| xpstopn 24092 | The topology on a binary p... |
| ptcmpfi 24093 | A topological product of f... |
| xkocnv 24094 | The inverse of the "curryi... |
| xkohmeo 24095 | The Exponential Law for to... |
| qtopf1 24096 | If a quotient map is injec... |
| qtophmeo 24097 | If two functions on a base... |
| t0kq 24098 | A topological space is T_0... |
| kqhmph 24099 | A topological space is T_0... |
| ist1-5lem 24100 | Lemma for ~ ist1-5 and sim... |
| t1r0 24101 | A T_1 space is R_0. That ... |
| ist1-5 24102 | A topological space is T_1... |
| ishaus3 24103 | A topological space is Hau... |
| nrmreg 24104 | A normal T_1 space is regu... |
| reghaus 24105 | A regular T_0 space is Hau... |
| nrmhaus 24106 | A T_1 normal space is Haus... |
| elmptrab 24107 | Membership in a one-parame... |
| elmptrab2 24108 | Membership in a one-parame... |
| isfbas 24109 | The predicate " ` F ` is a... |
| fbasne0 24110 | There are no empty filter ... |
| 0nelfb 24111 | No filter base contains th... |
| fbsspw 24112 | A filter base on a set is ... |
| fbelss 24113 | An element of the filter b... |
| fbdmn0 24114 | The domain of a filter bas... |
| isfbas2 24115 | The predicate " ` F ` is a... |
| fbasssin 24116 | A filter base contains sub... |
| fbssfi 24117 | A filter base contains sub... |
| fbssint 24118 | A filter base contains sub... |
| fbncp 24119 | A filter base does not con... |
| fbun 24120 | A necessary and sufficient... |
| fbfinnfr 24121 | No filter base containing ... |
| opnfbas 24122 | The collection of open sup... |
| trfbas2 24123 | Conditions for the trace o... |
| trfbas 24124 | Conditions for the trace o... |
| isfil 24127 | The predicate "is a filter... |
| filfbas 24128 | A filter is a filter base.... |
| 0nelfil 24129 | The empty set doesn't belo... |
| fileln0 24130 | An element of a filter is ... |
| filsspw 24131 | A filter is a subset of th... |
| filelss 24132 | An element of a filter is ... |
| filss 24133 | A filter is closed under t... |
| filin 24134 | A filter is closed under t... |
| filtop 24135 | The underlying set belongs... |
| isfil2 24136 | Derive the standard axioms... |
| isfildlem 24137 | Lemma for ~ isfild . (Con... |
| isfild 24138 | Sufficient condition for a... |
| filfi 24139 | A filter is closed under t... |
| filinn0 24140 | The intersection of two el... |
| filintn0 24141 | A filter has the finite in... |
| filn0 24142 | The empty set is not a fil... |
| infil 24143 | The intersection of two fi... |
| snfil 24144 | A singleton is a filter. ... |
| fbasweak 24145 | A filter base on any set i... |
| snfbas 24146 | Condition for a singleton ... |
| fsubbas 24147 | A condition for a set to g... |
| fbasfip 24148 | A filter base has the fini... |
| fbunfip 24149 | A helpful lemma for showin... |
| fgval 24150 | The filter generating clas... |
| elfg 24151 | A condition for elements o... |
| ssfg 24152 | A filter base is a subset ... |
| fgss 24153 | A bigger base generates a ... |
| fgss2 24154 | A condition for a filter t... |
| fgfil 24155 | A filter generates itself.... |
| elfilss 24156 | An element belongs to a fi... |
| filfinnfr 24157 | No filter containing a fin... |
| fgcl 24158 | A generated filter is a fi... |
| fgabs 24159 | Absorption law for filter ... |
| neifil 24160 | The neighborhoods of a non... |
| filunibas 24161 | Recover the base set from ... |
| filunirn 24162 | Two ways to express a filt... |
| filconn 24163 | A filter gives rise to a c... |
| fbasrn 24164 | Given a filter on a domain... |
| filuni 24165 | The union of a nonempty se... |
| trfil1 24166 | Conditions for the trace o... |
| trfil2 24167 | Conditions for the trace o... |
| trfil3 24168 | Conditions for the trace o... |
| trfilss 24169 | If ` A ` is a member of th... |
| fgtr 24170 | If ` A ` is a member of th... |
| trfg 24171 | The trace operation and th... |
| trnei 24172 | The trace, over a set ` A ... |
| cfinfil 24173 | Relative complements of th... |
| csdfil 24174 | The set of all elements wh... |
| supfil 24175 | The supersets of a nonempt... |
| zfbas 24176 | The set of upper sets of i... |
| uzrest 24177 | The restriction of the set... |
| uzfbas 24178 | The set of upper sets of i... |
| isufil 24183 | The property of being an u... |
| ufilfil 24184 | An ultrafilter is a filter... |
| ufilss 24185 | For any subset of the base... |
| ufilb 24186 | The complement is in an ul... |
| ufilmax 24187 | Any filter finer than an u... |
| isufil2 24188 | The maximal property of an... |
| ufprim 24189 | An ultrafilter is a prime ... |
| trufil 24190 | Conditions for the trace o... |
| filssufilg 24191 | A filter is contained in s... |
| filssufil 24192 | A filter is contained in s... |
| isufl 24193 | Define the (strong) ultraf... |
| ufli 24194 | Property of a set that sat... |
| numufl 24195 | Consequence of ~ filssufil... |
| fiufl 24196 | A finite set satisfies the... |
| acufl 24197 | The axiom of choice implie... |
| ssufl 24198 | If ` Y ` is a subset of ` ... |
| ufileu 24199 | If the ultrafilter contain... |
| filufint 24200 | A filter is equal to the i... |
| uffix 24201 | Lemma for ~ fixufil and ~ ... |
| fixufil 24202 | The condition describing a... |
| uffixfr 24203 | An ultrafilter is either f... |
| uffix2 24204 | A classification of fixed ... |
| uffixsn 24205 | The singleton of the gener... |
| ufildom1 24206 | An ultrafilter is generate... |
| uffinfix 24207 | An ultrafilter containing ... |
| cfinufil 24208 | An ultrafilter is free iff... |
| ufinffr 24209 | An infinite subset is cont... |
| ufilen 24210 | Any infinite set has an ul... |
| ufildr 24211 | An ultrafilter gives rise ... |
| fin1aufil 24212 | There are no definable fre... |
| fmval 24223 | Introduce a function that ... |
| fmfil 24224 | A mapping filter is a filt... |
| fmf 24225 | Pushing-forward via a func... |
| fmss 24226 | A finer filter produces a ... |
| elfm 24227 | An element of a mapping fi... |
| elfm2 24228 | An element of a mapping fi... |
| fmfg 24229 | The image filter of a filt... |
| elfm3 24230 | An alternate formulation o... |
| imaelfm 24231 | An image of a filter eleme... |
| rnelfmlem 24232 | Lemma for ~ rnelfm . (Con... |
| rnelfm 24233 | A condition for a filter t... |
| fmfnfmlem1 24234 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 24235 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 24236 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 24237 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 24238 | A filter finer than an ima... |
| fmufil 24239 | An image filter of an ultr... |
| fmid 24240 | The filter map applied to ... |
| fmco 24241 | Composition of image filte... |
| ufldom 24242 | The ultrafilter lemma prop... |
| flimval 24243 | The set of limit points of... |
| elflim2 24244 | The predicate "is a limit ... |
| flimtop 24245 | Reverse closure for the li... |
| flimneiss 24246 | A filter contains the neig... |
| flimnei 24247 | A filter contains all of t... |
| flimelbas 24248 | A limit point of a filter ... |
| flimfil 24249 | Reverse closure for the li... |
| flimtopon 24250 | Reverse closure for the li... |
| elflim 24251 | The predicate "is a limit ... |
| flimss2 24252 | A limit point of a filter ... |
| flimss1 24253 | A limit point of a filter ... |
| neiflim 24254 | A point is a limit point o... |
| flimopn 24255 | The condition for being a ... |
| fbflim 24256 | A condition for a filter t... |
| fbflim2 24257 | A condition for a filter b... |
| flimclsi 24258 | The convergent points of a... |
| hausflimlem 24259 | If ` A ` and ` B ` are bot... |
| hausflimi 24260 | One direction of ~ hausfli... |
| hausflim 24261 | A condition for a topology... |
| flimcf 24262 | Fineness is properly chara... |
| flimrest 24263 | The set of limit points in... |
| flimclslem 24264 | Lemma for ~ flimcls . (Co... |
| flimcls 24265 | Closure in terms of filter... |
| flimsncls 24266 | If ` A ` is a limit point ... |
| hauspwpwf1 24267 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 24268 | If ` X ` is a Hausdorff sp... |
| flffval 24269 | Given a topology and a fil... |
| flfval 24270 | Given a function from a fi... |
| flfnei 24271 | The property of being a li... |
| flfneii 24272 | A neighborhood of a limit ... |
| isflf 24273 | The property of being a li... |
| flfelbas 24274 | A limit point of a functio... |
| flffbas 24275 | Limit points of a function... |
| flftg 24276 | Limit points of a function... |
| hausflf 24277 | If a function has its valu... |
| hausflf2 24278 | If a convergent function h... |
| cnpflfi 24279 | Forward direction of ~ cnp... |
| cnpflf2 24280 | ` F ` is continuous at poi... |
| cnpflf 24281 | Continuity of a function a... |
| cnflf 24282 | A function is continuous i... |
| cnflf2 24283 | A function is continuous i... |
| flfcnp 24284 | A continuous function pres... |
| lmflf 24285 | The topological limit rela... |
| txflf 24286 | Two sequences converge in ... |
| flfcnp2 24287 | The image of a convergent ... |
| fclsval 24288 | The set of all cluster poi... |
| isfcls 24289 | A cluster point of a filte... |
| fclsfil 24290 | Reverse closure for the cl... |
| fclstop 24291 | Reverse closure for the cl... |
| fclstopon 24292 | Reverse closure for the cl... |
| isfcls2 24293 | A cluster point of a filte... |
| fclsopn 24294 | Write the cluster point co... |
| fclsopni 24295 | An open neighborhood of a ... |
| fclselbas 24296 | A cluster point is in the ... |
| fclsneii 24297 | A neighborhood of a cluste... |
| fclssscls 24298 | The set of cluster points ... |
| fclsnei 24299 | Cluster points in terms of... |
| supnfcls 24300 | The filter of supersets of... |
| fclsbas 24301 | Cluster points in terms of... |
| fclsss1 24302 | A finer topology has fewer... |
| fclsss2 24303 | A finer filter has fewer c... |
| fclsrest 24304 | The set of cluster points ... |
| fclscf 24305 | Characterization of finene... |
| flimfcls 24306 | A limit point is a cluster... |
| fclsfnflim 24307 | A filter clusters at a poi... |
| flimfnfcls 24308 | A filter converges to a po... |
| fclscmpi 24309 | Forward direction of ~ fcl... |
| fclscmp 24310 | A space is compact iff eve... |
| uffclsflim 24311 | The cluster points of an u... |
| ufilcmp 24312 | A space is compact iff eve... |
| fcfval 24313 | The set of cluster points ... |
| isfcf 24314 | The property of being a cl... |
| fcfnei 24315 | The property of being a cl... |
| fcfelbas 24316 | A cluster point of a funct... |
| fcfneii 24317 | A neighborhood of a cluste... |
| flfssfcf 24318 | A limit point of a functio... |
| uffcfflf 24319 | If the domain filter is an... |
| cnpfcfi 24320 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24321 | A function ` F ` is contin... |
| cnfcf 24322 | Continuity of a function i... |
| flfcntr 24323 | A continuous function's va... |
| alexsublem 24324 | Lemma for ~ alexsub . (Co... |
| alexsub 24325 | The Alexander Subbase Theo... |
| alexsubb 24326 | Biconditional form of the ... |
| alexsubALTlem1 24327 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24328 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24329 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24330 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24331 | The Alexander Subbase Theo... |
| ptcmplem1 24332 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24333 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24334 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24335 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24336 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24337 | Tychonoff's theorem: The ... |
| ptcmp 24338 | Tychonoff's theorem: The ... |
| cnextval 24341 | The function applying cont... |
| cnextfval 24342 | The continuous extension o... |
| cnextrel 24343 | In the general case, a con... |
| cnextfun 24344 | If the target space is Hau... |
| cnextfvval 24345 | The value of the continuou... |
| cnextf 24346 | Extension by continuity. ... |
| cnextcn 24347 | Extension by continuity. ... |
| cnextfres1 24348 | ` F ` and its extension by... |
| cnextfres 24349 | ` F ` and its extension by... |
| istmd 24354 | The predicate "is a topolo... |
| tmdmnd 24355 | A topological monoid is a ... |
| tmdtps 24356 | A topological monoid is a ... |
| istgp 24357 | The predicate "is a topolo... |
| tgpgrp 24358 | A topological group is a g... |
| tgptmd 24359 | A topological group is a t... |
| tgptps 24360 | A topological group is a t... |
| tmdtopon 24361 | The topology of a topologi... |
| tgptopon 24362 | The topology of a topologi... |
| tmdcn 24363 | In a topological monoid, t... |
| tgpcn 24364 | In a topological group, th... |
| tgpinv 24365 | In a topological group, th... |
| grpinvhmeo 24366 | The inverse function in a ... |
| cnmpt1plusg 24367 | Continuity of the group su... |
| cnmpt2plusg 24368 | Continuity of the group su... |
| tmdcn2 24369 | Write out the definition o... |
| tgpsubcn 24370 | In a topological group, th... |
| istgp2 24371 | A group with a topology is... |
| tmdmulg 24372 | In a topological monoid, t... |
| tgpmulg 24373 | In a topological group, th... |
| tgpmulg2 24374 | In a topological monoid, t... |
| tmdgsum 24375 | In a topological monoid, t... |
| tmdgsum2 24376 | For any neighborhood ` U `... |
| oppgtmd 24377 | The opposite of a topologi... |
| oppgtgp 24378 | The opposite of a topologi... |
| distgp 24379 | Any group equipped with th... |
| indistgp 24380 | Any group equipped with th... |
| efmndtmd 24381 | The monoid of endofunction... |
| tmdlactcn 24382 | The left group action of e... |
| tgplacthmeo 24383 | The left group action of e... |
| submtmd 24384 | A submonoid of a topologic... |
| subgtgp 24385 | A subgroup of a topologica... |
| symgtgp 24386 | The symmetric group is a t... |
| subgntr 24387 | A subgroup of a topologica... |
| opnsubg 24388 | An open subgroup of a topo... |
| clssubg 24389 | The closure of a subgroup ... |
| clsnsg 24390 | The closure of a normal su... |
| cldsubg 24391 | A subgroup of finite index... |
| tgpconncompeqg 24392 | The connected component co... |
| tgpconncomp 24393 | The identity component, th... |
| tgpconncompss 24394 | The identity component is ... |
| ghmcnp 24395 | A group homomorphism on to... |
| snclseqg 24396 | The coset of the closure o... |
| tgphaus 24397 | A topological group is Hau... |
| tgpt1 24398 | Hausdorff and T1 are equiv... |
| tgpt0 24399 | Hausdorff and T0 are equiv... |
| qustgpopn 24400 | A quotient map in a topolo... |
| qustgplem 24401 | Lemma for ~ qustgp . (Con... |
| qustgp 24402 | The quotient of a topologi... |
| qustgphaus 24403 | The quotient of a topologi... |
| prdstmdd 24404 | The product of a family of... |
| prdstgpd 24405 | The product of a family of... |
| tsmsfbas 24408 | The collection of all sets... |
| tsmslem1 24409 | The finite partial sums of... |
| tsmsval2 24410 | Definition of the topologi... |
| tsmsval 24411 | Definition of the topologi... |
| tsmspropd 24412 | The group sum depends only... |
| eltsms 24413 | The property of being a su... |
| tsmsi 24414 | The property of being a su... |
| tsmscl 24415 | A sum in a topological gro... |
| haustsms 24416 | In a Hausdorff topological... |
| haustsms2 24417 | In a Hausdorff topological... |
| tsmscls 24418 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24419 | The convergent points of a... |
| tsmsid 24420 | If a sum is finite, the us... |
| haustsmsid 24421 | In a Hausdorff topological... |
| tsms0 24422 | The sum of zero is zero. ... |
| tsmssubm 24423 | Evaluate an infinite group... |
| tsmsres 24424 | Extend an infinite group s... |
| tsmsf1o 24425 | Re-index an infinite group... |
| tsmsmhm 24426 | Apply a continuous group h... |
| tsmsadd 24427 | The sum of two infinite gr... |
| tsmsinv 24428 | Inverse of an infinite gro... |
| tsmssub 24429 | The difference of two infi... |
| tgptsmscls 24430 | A sum in a topological gro... |
| tgptsmscld 24431 | The set of limit points to... |
| tsmssplit 24432 | Split a topological group ... |
| tsmsxplem1 24433 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24434 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24435 | Write a sum over a two-dim... |
| istrg 24444 | Express the predicate " ` ... |
| trgtmd 24445 | The multiplicative monoid ... |
| istdrg 24446 | Express the predicate " ` ... |
| tdrgunit 24447 | The unit group of a topolo... |
| trgtgp 24448 | A topological ring is a to... |
| trgtmd2 24449 | A topological ring is a to... |
| trgtps 24450 | A topological ring is a to... |
| trgring 24451 | A topological ring is a ri... |
| trggrp 24452 | A topological ring is a gr... |
| tdrgtrg 24453 | A topological division rin... |
| tdrgdrng 24454 | A topological division rin... |
| tdrgring 24455 | A topological division rin... |
| tdrgtmd 24456 | A topological division rin... |
| tdrgtps 24457 | A topological division rin... |
| istdrg2 24458 | A topological-ring divisio... |
| mulrcn 24459 | The functionalization of t... |
| invrcn2 24460 | The multiplicative inverse... |
| invrcn 24461 | The multiplicative inverse... |
| cnmpt1mulr 24462 | Continuity of ring multipl... |
| cnmpt2mulr 24463 | Continuity of ring multipl... |
| dvrcn 24464 | The division function is c... |
| istlm 24465 | The predicate " ` W ` is a... |
| vscacn 24466 | The scalar multiplication ... |
| tlmtmd 24467 | A topological module is a ... |
| tlmtps 24468 | A topological module is a ... |
| tlmlmod 24469 | A topological module is a ... |
| tlmtrg 24470 | The scalar ring of a topol... |
| tlmscatps 24471 | The scalar ring of a topol... |
| istvc 24472 | A topological vector space... |
| tvctdrg 24473 | The scalar field of a topo... |
| cnmpt1vsca 24474 | Continuity of scalar multi... |
| cnmpt2vsca 24475 | Continuity of scalar multi... |
| tlmtgp 24476 | A topological vector space... |
| tvctlm 24477 | A topological vector space... |
| tvclmod 24478 | A topological vector space... |
| tvclvec 24479 | A topological vector space... |
| ustfn 24482 | The defined uniform struct... |
| ustval 24483 | The class of all uniform s... |
| isust 24484 | The predicate " ` U ` is a... |
| ustssxp 24485 | Entourages are subsets of ... |
| ustssel 24486 | A uniform structure is upw... |
| ustbasel 24487 | The full set is always an ... |
| ustincl 24488 | A uniform structure is clo... |
| ustdiag 24489 | The diagonal set is includ... |
| ustinvel 24490 | If ` V ` is an entourage, ... |
| ustexhalf 24491 | For each entourage ` V ` t... |
| ustrel 24492 | The elements of uniform st... |
| ustfilxp 24493 | A uniform structure on a n... |
| ustne0 24494 | A uniform structure cannot... |
| ustssco 24495 | In an uniform structure, a... |
| ustexsym 24496 | In an uniform structure, f... |
| ustex2sym 24497 | In an uniform structure, f... |
| ustex3sym 24498 | In an uniform structure, f... |
| ustref 24499 | Any element of the base se... |
| ust0 24500 | The unique uniform structu... |
| ustn0 24501 | The empty set is not an un... |
| ustund 24502 | If two intersecting sets `... |
| ustelimasn 24503 | Any point ` A ` is near en... |
| ustneism 24504 | For a point ` A ` in ` X `... |
| ustbas2 24505 | Second direction for ~ ust... |
| ustuni 24506 | The set union of a uniform... |
| ustbas 24507 | Recover the base of an uni... |
| ustimasn 24508 | Lemma for ~ ustuqtop . (C... |
| trust 24509 | The trace of a uniform str... |
| utopval 24512 | The topology induced by a ... |
| elutop 24513 | Open sets in the topology ... |
| utoptop 24514 | The topology induced by a ... |
| utopbas 24515 | The base of the topology i... |
| utoptopon 24516 | Topology induced by a unif... |
| restutop 24517 | Restriction of a topology ... |
| restutopopn 24518 | The restriction of the top... |
| ustuqtoplem 24519 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24520 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24521 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24522 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24523 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24524 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24525 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24526 | For a given uniform struct... |
| utopsnneiplem 24527 | The neighborhoods of a poi... |
| utopsnneip 24528 | The neighborhoods of a poi... |
| utopsnnei 24529 | Images of singletons by en... |
| utop2nei 24530 | For any symmetrical entour... |
| utop3cls 24531 | Relation between a topolog... |
| utopreg 24532 | All Hausdorff uniform spac... |
| ussval 24539 | The uniform structure on u... |
| ussid 24540 | In case the base of the ` ... |
| isusp 24541 | The predicate ` W ` is a u... |
| ressuss 24542 | Value of the uniform struc... |
| ressust 24543 | The uniform structure of a... |
| ressusp 24544 | The restriction of a unifo... |
| tusval 24545 | The value of the uniform s... |
| tuslem 24546 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24547 | The base set of a construc... |
| tusunif 24548 | The uniform structure of a... |
| tususs 24549 | The uniform structure of a... |
| tustopn 24550 | The topology induced by a ... |
| tususp 24551 | A constructed uniform spac... |
| tustps 24552 | A constructed uniform spac... |
| uspreg 24553 | If a uniform space is Haus... |
| ucnval 24556 | The set of all uniformly c... |
| isucn 24557 | The predicate " ` F ` is a... |
| isucn2 24558 | The predicate " ` F ` is a... |
| ucnimalem 24559 | Reformulate the ` G ` func... |
| ucnima 24560 | An equivalent statement of... |
| ucnprima 24561 | The preimage by a uniforml... |
| iducn 24562 | The identity is uniformly ... |
| cstucnd 24563 | A constant function is uni... |
| ucncn 24564 | Uniform continuity implies... |
| iscfilu 24567 | The predicate " ` F ` is a... |
| cfilufbas 24568 | A Cauchy filter base is a ... |
| cfiluexsm 24569 | For a Cauchy filter base a... |
| fmucndlem 24570 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24571 | The image of a Cauchy filt... |
| cfilufg 24572 | The filter generated by a ... |
| trcfilu 24573 | Condition for the trace of... |
| cfiluweak 24574 | A Cauchy filter base is al... |
| neipcfilu 24575 | In an uniform space, a nei... |
| iscusp 24578 | The predicate " ` W ` is a... |
| cuspusp 24579 | A complete uniform space i... |
| cuspcvg 24580 | In a complete uniform spac... |
| iscusp2 24581 | The predicate " ` W ` is a... |
| cnextucn 24582 | Extension by continuity. ... |
| ucnextcn 24583 | Extension by continuity. ... |
| ispsmet 24584 | Express the predicate " ` ... |
| psmetdmdm 24585 | Recover the base set from ... |
| psmetf 24586 | The distance function of a... |
| psmetcl 24587 | Closure of the distance fu... |
| psmet0 24588 | The distance function of a... |
| psmettri2 24589 | Triangle inequality for th... |
| psmetsym 24590 | The distance function of a... |
| psmettri 24591 | Triangle inequality for th... |
| psmetge0 24592 | The distance function of a... |
| psmetxrge0 24593 | The distance function of a... |
| psmetres2 24594 | Restriction of a pseudomet... |
| psmetlecl 24595 | Real closure of an extende... |
| distspace 24596 | A set ` X ` together with ... |
| ismet 24603 | Express the predicate " ` ... |
| isxmet 24604 | Express the predicate " ` ... |
| ismeti 24605 | Properties that determine ... |
| isxmetd 24606 | Properties that determine ... |
| isxmet2d 24607 | It is safe to only require... |
| metflem 24608 | Lemma for ~ metf and other... |
| xmetf 24609 | Mapping of the distance fu... |
| metf 24610 | Mapping of the distance fu... |
| xmetcl 24611 | Closure of the distance fu... |
| metcl 24612 | Closure of the distance fu... |
| ismet2 24613 | An extended metric is a me... |
| metxmet 24614 | A metric is an extended me... |
| xmetdmdm 24615 | Recover the base set from ... |
| metdmdm 24616 | Recover the base set from ... |
| xmetunirn 24617 | Two ways to express an ext... |
| xmeteq0 24618 | The value of an extended m... |
| meteq0 24619 | The value of a metric is z... |
| xmettri2 24620 | Triangle inequality for th... |
| mettri2 24621 | Triangle inequality for th... |
| xmet0 24622 | The distance function of a... |
| met0 24623 | The distance function of a... |
| xmetge0 24624 | The distance function of a... |
| metge0 24625 | The distance function of a... |
| xmetlecl 24626 | Real closure of an extende... |
| xmetsym 24627 | The distance function of a... |
| xmetpsmet 24628 | An extended metric is a ps... |
| xmettpos 24629 | The distance function of a... |
| metsym 24630 | The distance function of a... |
| xmettri 24631 | Triangle inequality for th... |
| mettri 24632 | Triangle inequality for th... |
| xmettri3 24633 | Triangle inequality for th... |
| mettri3 24634 | Triangle inequality for th... |
| xmetrtri 24635 | One half of the reverse tr... |
| xmetrtri2 24636 | The reverse triangle inequ... |
| metrtri 24637 | Reverse triangle inequalit... |
| xmetgt0 24638 | The distance function of a... |
| metgt0 24639 | The distance function of a... |
| metn0 24640 | A metric space is nonempty... |
| xmetres2 24641 | Restriction of an extended... |
| metreslem 24642 | Lemma for ~ metres . (Con... |
| metres2 24643 | Lemma for ~ metres . (Con... |
| xmetres 24644 | A restriction of an extend... |
| metres 24645 | A restriction of a metric ... |
| 0met 24646 | The empty metric. (Contri... |
| prdsdsf 24647 | The product metric is a fu... |
| prdsxmetlem 24648 | The product metric is an e... |
| prdsxmet 24649 | The product metric is an e... |
| prdsmet 24650 | The product metric is a me... |
| ressprdsds 24651 | Restriction of a product m... |
| resspwsds 24652 | Restriction of a power met... |
| imasdsf1olem 24653 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24654 | The distance function is t... |
| imasf1oxmet 24655 | The image of an extended m... |
| imasf1omet 24656 | The image of a metric is a... |
| xpsdsfn 24657 | Closure of the metric in a... |
| xpsdsfn2 24658 | Closure of the metric in a... |
| xpsxmetlem 24659 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24660 | A product metric of extend... |
| xpsdsval 24661 | Value of the metric in a b... |
| xpsmet 24662 | The direct product of two ... |
| blfvalps 24663 | The value of the ball func... |
| blfval 24664 | The value of the ball func... |
| blvalps 24665 | The ball around a point ` ... |
| blval 24666 | The ball around a point ` ... |
| elblps 24667 | Membership in a ball. (Co... |
| elbl 24668 | Membership in a ball. (Co... |
| elbl2ps 24669 | Membership in a ball. (Co... |
| elbl2 24670 | Membership in a ball. (Co... |
| elbl3ps 24671 | Membership in a ball, with... |
| elbl3 24672 | Membership in a ball, with... |
| blcomps 24673 | Commute the arguments to t... |
| blcom 24674 | Commute the arguments to t... |
| xblpnfps 24675 | The infinity ball in an ex... |
| xblpnf 24676 | The infinity ball in an ex... |
| blpnf 24677 | The infinity ball in a sta... |
| bldisj 24678 | Two balls are disjoint if ... |
| blgt0 24679 | A nonempty ball implies th... |
| bl2in 24680 | Two balls are disjoint if ... |
| xblss2ps 24681 | One ball is contained in a... |
| xblss2 24682 | One ball is contained in a... |
| blss2ps 24683 | One ball is contained in a... |
| blss2 24684 | One ball is contained in a... |
| blhalf 24685 | A ball of radius ` R / 2 `... |
| blfps 24686 | Mapping of a ball. (Contr... |
| blf 24687 | Mapping of a ball. (Contr... |
| blrnps 24688 | Membership in the range of... |
| blrn 24689 | Membership in the range of... |
| xblcntrps 24690 | A ball contains its center... |
| xblcntr 24691 | A ball contains its center... |
| blcntrps 24692 | A ball contains its center... |
| blcntr 24693 | A ball contains its center... |
| xbln0 24694 | A ball is nonempty iff the... |
| bln0 24695 | A ball is not empty. (Con... |
| blelrnps 24696 | A ball belongs to the set ... |
| blelrn 24697 | A ball belongs to the set ... |
| blssm 24698 | A ball is a subset of the ... |
| unirnblps 24699 | The union of the set of ba... |
| unirnbl 24700 | The union of the set of ba... |
| blin 24701 | The intersection of two ba... |
| ssblps 24702 | The size of a ball increas... |
| ssbl 24703 | The size of a ball increas... |
| blssps 24704 | Any point ` P ` in a ball ... |
| blss 24705 | Any point ` P ` in a ball ... |
| blssexps 24706 | Two ways to express the ex... |
| blssex 24707 | Two ways to express the ex... |
| ssblex 24708 | A nested ball exists whose... |
| blin2 24709 | Given any two balls and a ... |
| blbas 24710 | The balls of a metric spac... |
| blres 24711 | A ball in a restricted met... |
| xmeterval 24712 | Value of the "finitely sep... |
| xmeter 24713 | The "finitely separated" r... |
| xmetec 24714 | The equivalence classes un... |
| blssec 24715 | A ball centered at ` P ` i... |
| blpnfctr 24716 | The infinity ball in an ex... |
| xmetresbl 24717 | An extended metric restric... |
| mopnval 24718 | An open set is a subset of... |
| mopntopon 24719 | The set of open sets of a ... |
| mopntop 24720 | The set of open sets of a ... |
| mopnuni 24721 | The union of all open sets... |
| elmopn 24722 | The defining property of a... |
| mopnfss 24723 | The family of open sets of... |
| mopnm 24724 | The base set of a metric s... |
| elmopn2 24725 | A defining property of an ... |
| mopnss 24726 | An open set of a metric sp... |
| isxms 24727 | Express the predicate " ` ... |
| isxms2 24728 | Express the predicate " ` ... |
| isms 24729 | Express the predicate " ` ... |
| isms2 24730 | Express the predicate " ` ... |
| xmstopn 24731 | The topology component of ... |
| mstopn 24732 | The topology component of ... |
| xmstps 24733 | An extended metric space i... |
| msxms 24734 | A metric space is an exten... |
| mstps 24735 | A metric space is a topolo... |
| xmsxmet 24736 | The distance function, sui... |
| msmet 24737 | The distance function, sui... |
| msf 24738 | The distance function of a... |
| xmsxmet2 24739 | The distance function, sui... |
| msmet2 24740 | The distance function, sui... |
| mscl 24741 | Closure of the distance fu... |
| xmscl 24742 | Closure of the distance fu... |
| xmsge0 24743 | The distance function in a... |
| xmseq0 24744 | The distance between two p... |
| xmssym 24745 | The distance function in a... |
| xmstri2 24746 | Triangle inequality for th... |
| mstri2 24747 | Triangle inequality for th... |
| xmstri 24748 | Triangle inequality for th... |
| mstri 24749 | Triangle inequality for th... |
| xmstri3 24750 | Triangle inequality for th... |
| mstri3 24751 | Triangle inequality for th... |
| msrtri 24752 | Reverse triangle inequalit... |
| xmspropd 24753 | Property deduction for an ... |
| mspropd 24754 | Property deduction for a m... |
| setsmsbas 24755 | The base set of a construc... |
| setsmsds 24756 | The distance function of a... |
| setsmstset 24757 | The topology of a construc... |
| setsmstopn 24758 | The topology of a construc... |
| setsxms 24759 | The constructed metric spa... |
| setsms 24760 | The constructed metric spa... |
| tmsval 24761 | For any metric there is an... |
| tmslem 24762 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24763 | The base set of a construc... |
| tmsds 24764 | The metric of a constructe... |
| tmstopn 24765 | The topology of a construc... |
| tmsxms 24766 | The constructed metric spa... |
| tmsms 24767 | The constructed metric spa... |
| imasf1obl 24768 | The image of a metric spac... |
| imasf1oxms 24769 | The image of a metric spac... |
| imasf1oms 24770 | The image of a metric spac... |
| prdsbl 24771 | A ball in the product metr... |
| mopni 24772 | An open set of a metric sp... |
| mopni2 24773 | An open set of a metric sp... |
| mopni3 24774 | An open set of a metric sp... |
| blssopn 24775 | The balls of a metric spac... |
| unimopn 24776 | The union of a collection ... |
| mopnin 24777 | The intersection of two op... |
| mopn0 24778 | The empty set is an open s... |
| rnblopn 24779 | A ball of a metric space i... |
| blopn 24780 | A ball of a metric space i... |
| neibl 24781 | The neighborhoods around a... |
| blnei 24782 | A ball around a point is a... |
| lpbl 24783 | Every ball around a limit ... |
| blsscls2 24784 | A smaller closed ball is c... |
| blcld 24785 | A "closed ball" in a metri... |
| blcls 24786 | The closure of an open bal... |
| blsscls 24787 | If two concentric balls ha... |
| metss 24788 | Two ways of saying that me... |
| metequiv 24789 | Two ways of saying that tw... |
| metequiv2 24790 | If there is a sequence of ... |
| metss2lem 24791 | Lemma for ~ metss2 . (Con... |
| metss2 24792 | If the metric ` D ` is "st... |
| comet 24793 | The composition of an exte... |
| stdbdmetval 24794 | Value of the standard boun... |
| stdbdxmet 24795 | The standard bounded metri... |
| stdbdmet 24796 | The standard bounded metri... |
| stdbdbl 24797 | The standard bounded metri... |
| stdbdmopn 24798 | The standard bounded metri... |
| mopnex 24799 | The topology generated by ... |
| methaus 24800 | The topology generated by ... |
| met1stc 24801 | The topology generated by ... |
| met2ndci 24802 | A separable metric space (... |
| met2ndc 24803 | A metric space is second-c... |
| metrest 24804 | Two alternate formulations... |
| ressxms 24805 | The restriction of a metri... |
| ressms 24806 | The restriction of a metri... |
| prdsmslem1 24807 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24808 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24809 | Lemma for ~ prdsxms . The... |
| prdsxms 24810 | The indexed product struct... |
| prdsms 24811 | The indexed product struct... |
| pwsxms 24812 | A power of an extended met... |
| pwsms 24813 | A power of a metric space ... |
| xpsxms 24814 | A binary product of metric... |
| xpsms 24815 | A binary product of metric... |
| tmsxps 24816 | Express the product of two... |
| tmsxpsmopn 24817 | Express the product of two... |
| tmsxpsval 24818 | Value of the product of tw... |
| tmsxpsval2 24819 | Value of the product of tw... |
| metcnp3 24820 | Two ways to express that `... |
| metcnp 24821 | Two ways to say a mapping ... |
| metcnp2 24822 | Two ways to say a mapping ... |
| metcn 24823 | Two ways to say a mapping ... |
| metcnpi 24824 | Epsilon-delta property of ... |
| metcnpi2 24825 | Epsilon-delta property of ... |
| metcnpi3 24826 | Epsilon-delta property of ... |
| txmetcnp 24827 | Continuity of a binary ope... |
| txmetcn 24828 | Continuity of a binary ope... |
| metuval 24829 | Value of the uniform struc... |
| metustel 24830 | Define a filter base ` F `... |
| metustss 24831 | Range of the elements of t... |
| metustrel 24832 | Elements of the filter bas... |
| metustto 24833 | Any two elements of the fi... |
| metustid 24834 | The identity diagonal is i... |
| metustsym 24835 | Elements of the filter bas... |
| metustexhalf 24836 | For any element ` A ` of t... |
| metustfbas 24837 | The filter base generated ... |
| metust 24838 | The uniform structure gene... |
| cfilucfil 24839 | Given a metric ` D ` and a... |
| metuust 24840 | The uniform structure gene... |
| cfilucfil2 24841 | Given a metric ` D ` and a... |
| blval2 24842 | The ball around a point ` ... |
| elbl4 24843 | Membership in a ball, alte... |
| metuel 24844 | Elementhood in the uniform... |
| metuel2 24845 | Elementhood in the uniform... |
| metustbl 24846 | The "section" image of an ... |
| psmetutop 24847 | The topology induced by a ... |
| xmetutop 24848 | The topology induced by a ... |
| xmsusp 24849 | If the uniform set of a me... |
| restmetu 24850 | The uniform structure gene... |
| metucn 24851 | Uniform continuity in metr... |
| dscmet 24852 | The discrete metric on any... |
| dscopn 24853 | The discrete metric genera... |
| nrmmetd 24854 | Show that a group norm gen... |
| abvmet 24855 | An absolute value ` F ` ge... |
| nmfval 24868 | The value of the norm func... |
| nmval 24869 | The value of the norm as t... |
| nmfval0 24870 | The value of the norm func... |
| nmfval2 24871 | The value of the norm func... |
| nmval2 24872 | The value of the norm on a... |
| nmf2 24873 | The norm on a metric group... |
| nmpropd 24874 | Weak property deduction fo... |
| nmpropd2 24875 | Strong property deduction ... |
| isngp 24876 | The property of being a no... |
| isngp2 24877 | The property of being a no... |
| isngp3 24878 | The property of being a no... |
| ngpgrp 24879 | A normed group is a group.... |
| ngpms 24880 | A normed group is a metric... |
| ngpxms 24881 | A normed group is an exten... |
| ngptps 24882 | A normed group is a topolo... |
| ngpmet 24883 | The (induced) metric of a ... |
| ngpds 24884 | Value of the distance func... |
| ngpdsr 24885 | Value of the distance func... |
| ngpds2 24886 | Write the distance between... |
| ngpds2r 24887 | Write the distance between... |
| ngpds3 24888 | Write the distance between... |
| ngpds3r 24889 | Write the distance between... |
| ngprcan 24890 | Cancel right addition insi... |
| ngplcan 24891 | Cancel left addition insid... |
| isngp4 24892 | Express the property of be... |
| ngpinvds 24893 | Two elements are the same ... |
| ngpsubcan 24894 | Cancel right subtraction i... |
| nmf 24895 | The norm on a normed group... |
| nmcl 24896 | The norm of a normed group... |
| nmge0 24897 | The norm of a normed group... |
| nmeq0 24898 | The identity is the only e... |
| nmne0 24899 | The norm of a nonzero elem... |
| nmrpcl 24900 | The norm of a nonzero elem... |
| nminv 24901 | The norm of a negated elem... |
| nmmtri 24902 | The triangle inequality fo... |
| nmsub 24903 | The norm of the difference... |
| nmrtri 24904 | Reverse triangle inequalit... |
| nm2dif 24905 | Inequality for the differe... |
| nmtri 24906 | The triangle inequality fo... |
| nmtri2 24907 | Triangle inequality for th... |
| ngpi 24908 | The properties of a normed... |
| nm0 24909 | Norm of the identity eleme... |
| nmgt0 24910 | The norm of a nonzero elem... |
| sgrim 24911 | The induced metric on a su... |
| sgrimval 24912 | The induced metric on a su... |
| subgnm 24913 | The norm in a subgroup. (... |
| subgnm2 24914 | A substructure assigns the... |
| subgngp 24915 | A normed group restricted ... |
| ngptgp 24916 | A normed abelian group is ... |
| ngppropd 24917 | Property deduction for a n... |
| reldmtng 24918 | The function ` toNrmGrp ` ... |
| tngval 24919 | Value of the function whic... |
| tnglem 24920 | Lemma for ~ tngbas and sim... |
| tngbas 24921 | The base set of a structur... |
| tngplusg 24922 | The group addition of a st... |
| tng0 24923 | The group identity of a st... |
| tngmulr 24924 | The ring multiplication of... |
| tngsca 24925 | The scalar ring of a struc... |
| tngvsca 24926 | The scalar multiplication ... |
| tngip 24927 | The inner product operatio... |
| tngds 24928 | The metric function of a s... |
| tngtset 24929 | The topology generated by ... |
| tngtopn 24930 | The topology generated by ... |
| tngnm 24931 | The topology generated by ... |
| tngngp2 24932 | A norm turns a group into ... |
| tngngpd 24933 | Derive the axioms for a no... |
| tngngp 24934 | Derive the axioms for a no... |
| tnggrpr 24935 | If a structure equipped wi... |
| tngngp3 24936 | Alternate definition of a ... |
| nrmtngdist 24937 | The augmentation of a norm... |
| nrmtngnrm 24938 | The augmentation of a norm... |
| tngngpim 24939 | The induced metric of a no... |
| isnrg 24940 | A normed ring is a ring wi... |
| nrgabv 24941 | The norm of a normed ring ... |
| nrgngp 24942 | A normed ring is a normed ... |
| nrgring 24943 | A normed ring is a ring. ... |
| nmmul 24944 | The norm of a product in a... |
| nrgdsdi 24945 | Distribute a distance calc... |
| nrgdsdir 24946 | Distribute a distance calc... |
| nm1 24947 | The norm of one in a nonze... |
| unitnmn0 24948 | The norm of a unit is nonz... |
| nminvr 24949 | The norm of an inverse in ... |
| nmdvr 24950 | The norm of a division in ... |
| nrgdomn 24951 | A nonzero normed ring is a... |
| nrgtgp 24952 | A normed ring is a topolog... |
| subrgnrg 24953 | A normed ring restricted t... |
| tngnrg 24954 | Given any absolute value o... |
| isnlm 24955 | A normed (left) module is ... |
| nmvs 24956 | Defining property of a nor... |
| nlmngp 24957 | A normed module is a norme... |
| nlmlmod 24958 | A normed module is a left ... |
| nlmnrg 24959 | The scalar component of a ... |
| nlmngp2 24960 | The scalar component of a ... |
| nlmdsdi 24961 | Distribute a distance calc... |
| nlmdsdir 24962 | Distribute a distance calc... |
| nlmmul0or 24963 | If a scalar product is zer... |
| sranlm 24964 | The subring algebra over a... |
| nlmvscnlem2 24965 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24966 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24967 | The scalar multiplication ... |
| rlmnlm 24968 | The ring module over a nor... |
| rlmnm 24969 | The norm function in the r... |
| nrgtrg 24970 | A normed ring is a topolog... |
| nrginvrcnlem 24971 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24972 | The ring inverse function ... |
| nrgtdrg 24973 | A normed division ring is ... |
| nlmtlm 24974 | A normed module is a topol... |
| isnvc 24975 | A normed vector space is j... |
| nvcnlm 24976 | A normed vector space is a... |
| nvclvec 24977 | A normed vector space is a... |
| nvclmod 24978 | A normed vector space is a... |
| isnvc2 24979 | A normed vector space is j... |
| nvctvc 24980 | A normed vector space is a... |
| lssnlm 24981 | A subspace of a normed mod... |
| lssnvc 24982 | A subspace of a normed vec... |
| rlmnvc 24983 | The ring module over a nor... |
| ngpocelbl 24984 | Membership of an off-cente... |
| nmoffn 24991 | The function producing ope... |
| reldmnghm 24992 | Lemma for normed group hom... |
| reldmnmhm 24993 | Lemma for module homomorph... |
| nmofval 24994 | Value of the operator norm... |
| nmoval 24995 | Value of the operator norm... |
| nmogelb 24996 | Property of the operator n... |
| nmolb 24997 | Any upper bound on the val... |
| nmolb2d 24998 | Any upper bound on the val... |
| nmof 24999 | The operator norm is a fun... |
| nmocl 25000 | The operator norm of an op... |
| nmoge0 25001 | The operator norm of an op... |
| nghmfval 25002 | A normed group homomorphis... |
| isnghm 25003 | A normed group homomorphis... |
| isnghm2 25004 | A normed group homomorphis... |
| isnghm3 25005 | A normed group homomorphis... |
| bddnghm 25006 | A bounded group homomorphi... |
| nghmcl 25007 | A normed group homomorphis... |
| nmoi 25008 | The operator norm achieves... |
| nmoix 25009 | The operator norm is a bou... |
| nmoi2 25010 | The operator norm is a bou... |
| nmoleub 25011 | The operator norm, defined... |
| nghmrcl1 25012 | Reverse closure for a norm... |
| nghmrcl2 25013 | Reverse closure for a norm... |
| nghmghm 25014 | A normed group homomorphis... |
| nmo0 25015 | The operator norm of the z... |
| nmoeq0 25016 | The operator norm is zero ... |
| nmoco 25017 | An upper bound on the oper... |
| nghmco 25018 | The composition of normed ... |
| nmotri 25019 | Triangle inequality for th... |
| nghmplusg 25020 | The sum of two bounded lin... |
| 0nghm 25021 | The zero operator is a nor... |
| nmoid 25022 | The operator norm of the i... |
| idnghm 25023 | The identity operator is a... |
| nmods 25024 | Upper bound for the distan... |
| nghmcn 25025 | A normed group homomorphis... |
| isnmhm 25026 | A normed module homomorphi... |
| nmhmrcl1 25027 | Reverse closure for a norm... |
| nmhmrcl2 25028 | Reverse closure for a norm... |
| nmhmlmhm 25029 | A normed module homomorphi... |
| nmhmnghm 25030 | A normed module homomorphi... |
| nmhmghm 25031 | A normed module homomorphi... |
| isnmhm2 25032 | A normed module homomorphi... |
| nmhmcl 25033 | A normed module homomorphi... |
| idnmhm 25034 | The identity operator is a... |
| 0nmhm 25035 | The zero operator is a bou... |
| nmhmco 25036 | The composition of bounded... |
| nmhmplusg 25037 | The sum of two bounded lin... |
| qtopbaslem 25038 | The set of open intervals ... |
| qtopbas 25039 | The set of open intervals ... |
| retopbas 25040 | A basis for the standard t... |
| retop 25041 | The standard topology on t... |
| uniretop 25042 | The underlying set of the ... |
| retopon 25043 | The standard topology on t... |
| retps 25044 | The standard topological s... |
| iooretop 25045 | Open intervals are open se... |
| icccld 25046 | Closed intervals are close... |
| icopnfcld 25047 | Right-unbounded closed int... |
| iocmnfcld 25048 | Left-unbounded closed inte... |
| qdensere 25049 | ` QQ ` is dense in the sta... |
| cnmetdval 25050 | Value of the distance func... |
| cnmet 25051 | The absolute value metric ... |
| cnxmet 25052 | The absolute value metric ... |
| cnbl0 25053 | Two ways to write the open... |
| cnblcld 25054 | Two ways to write the clos... |
| cnfldms 25055 | The complex number field i... |
| cnfldxms 25056 | The complex number field i... |
| cnfldtps 25057 | The complex number field i... |
| cnfldnm 25058 | The norm of the field of c... |
| cnngp 25059 | The complex numbers form a... |
| cnnrg 25060 | The complex numbers form a... |
| cnfldtopn 25061 | The topology of the comple... |
| cnfldtopon 25062 | The topology of the comple... |
| cnfldtop 25063 | The topology of the comple... |
| cnfldhaus 25064 | The topology of the comple... |
| unicntop 25065 | The underlying set of the ... |
| cnopn 25066 | The set of complex numbers... |
| cnn0opn 25067 | The set of nonzero complex... |
| zringnrg 25068 | The ring of integers is a ... |
| remetdval 25069 | Value of the distance func... |
| remet 25070 | The absolute value metric ... |
| rexmet 25071 | The absolute value metric ... |
| bl2ioo 25072 | A ball in terms of an open... |
| ioo2bl 25073 | An open interval of reals ... |
| ioo2blex 25074 | An open interval of reals ... |
| blssioo 25075 | The balls of the standard ... |
| tgioo 25076 | The topology generated by ... |
| qdensere2 25077 | ` QQ ` is dense in ` RR ` ... |
| blcvx 25078 | An open ball in the comple... |
| rehaus 25079 | The standard topology on t... |
| tgqioo 25080 | The topology generated by ... |
| re2ndc 25081 | The standard topology on t... |
| resubmet 25082 | The subspace topology indu... |
| tgioo2 25083 | The standard topology on t... |
| rerest 25084 | The subspace topology indu... |
| tgioo4 25085 | The standard topology on t... |
| tgioo3 25086 | The standard topology on t... |
| xrtgioo 25087 | The topology on the extend... |
| xrrest 25088 | The subspace topology indu... |
| xrrest2 25089 | The subspace topology indu... |
| xrsxmet 25090 | The metric on the extended... |
| xrsdsre 25091 | The metric on the extended... |
| xrsblre 25092 | Any ball of the metric of ... |
| xrsmopn 25093 | The metric on the extended... |
| zcld 25094 | The integers are a closed ... |
| recld2 25095 | The real numbers are a clo... |
| zcld2 25096 | The integers are a closed ... |
| zdis 25097 | The integers are a discret... |
| sszcld 25098 | Every subset of the intege... |
| reperflem 25099 | A subset of the real numbe... |
| reperf 25100 | The real numbers are a per... |
| cnperf 25101 | The complex numbers are a ... |
| iccntr 25102 | The interior of a closed i... |
| icccmplem1 25103 | Lemma for ~ icccmp . (Con... |
| icccmplem2 25104 | Lemma for ~ icccmp . (Con... |
| icccmplem3 25105 | Lemma for ~ icccmp . (Con... |
| icccmp 25106 | A closed interval in ` RR ... |
| reconnlem1 25107 | Lemma for ~ reconn . Conn... |
| reconnlem2 25108 | Lemma for ~ reconn . (Con... |
| reconn 25109 | A subset of the reals is c... |
| retopconn 25110 | Corollary of ~ reconn . T... |
| iccconn 25111 | A closed interval is conne... |
| opnreen 25112 | Every nonempty open set is... |
| rectbntr0 25113 | A countable subset of the ... |
| xrge0gsumle 25114 | A finite sum in the nonneg... |
| xrge0tsms 25115 | Any finite or infinite sum... |
| xrge0tsms2 25116 | Any finite or infinite sum... |
| metdcnlem 25117 | The metric function of a m... |
| xmetdcn2 25118 | The metric function of an ... |
| xmetdcn 25119 | The metric function of an ... |
| metdcn2 25120 | The metric function of a m... |
| metdcn 25121 | The metric function of a m... |
| msdcn 25122 | The metric function of a m... |
| cnmpt1ds 25123 | Continuity of the metric f... |
| cnmpt2ds 25124 | Continuity of the metric f... |
| nmcn 25125 | The norm of a normed group... |
| ngnmcncn 25126 | The norm of a normed group... |
| abscn 25127 | The absolute value functio... |
| metdsval 25128 | Value of the "distance to ... |
| metdsf 25129 | The distance from a point ... |
| metdsge 25130 | The distance from the poin... |
| metds0 25131 | If a point is in a set, it... |
| metdstri 25132 | A generalization of the tr... |
| metdsle 25133 | The distance from a point ... |
| metdsre 25134 | The distance from a point ... |
| metdseq0 25135 | The distance from a point ... |
| metdscnlem 25136 | Lemma for ~ metdscn . (Co... |
| metdscn 25137 | The function ` F ` which g... |
| metdscn2 25138 | The function ` F ` which g... |
| metnrmlem1a 25139 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 25140 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 25141 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 25142 | Lemma for ~ metnrm . (Con... |
| metnrm 25143 | A metric space is normal. ... |
| metreg 25144 | A metric space is regular.... |
| addcnlem 25145 | Lemma for ~ addcn , ~ subc... |
| addcn 25146 | Complex number addition is... |
| subcn 25147 | Complex number subtraction... |
| mulcn 25148 | Complex number multiplicat... |
| mpomulcn 25149 | Complex number multiplicat... |
| divcn 25150 | Complex number division is... |
| cnfldtgp 25151 | The complex numbers form a... |
| fsumcn 25152 | A finite sum of functions ... |
| fsum2cn 25153 | Version of ~ fsumcn for tw... |
| expcn 25154 | The power function on comp... |
| divccn 25155 | Division by a nonzero cons... |
| sqcn 25156 | The square function on com... |
| iitopon 25161 | The unit interval is a top... |
| iitop 25162 | The unit interval is a top... |
| iiuni 25163 | The base set of the unit i... |
| dfii2 25164 | Alternate definition of th... |
| dfii3 25165 | Alternate definition of th... |
| dfii4 25166 | Alternate definition of th... |
| dfii5 25167 | The unit interval expresse... |
| iicmp 25168 | The unit interval is compa... |
| iiconn 25169 | The unit interval is conne... |
| cncfval 25170 | The value of the continuou... |
| elcncf 25171 | Membership in the set of c... |
| elcncf2 25172 | Version of ~ elcncf with a... |
| cncfrss 25173 | Reverse closure of the con... |
| cncfrss2 25174 | Reverse closure of the con... |
| cncff 25175 | A continuous complex funct... |
| cncfi 25176 | Defining property of a con... |
| elcncf1di 25177 | Membership in the set of c... |
| elcncf1ii 25178 | Membership in the set of c... |
| rescncf 25179 | A continuous complex funct... |
| cncfcdm 25180 | Change the codomain of a c... |
| cncfss 25181 | The set of continuous func... |
| climcncf 25182 | Image of a limit under a c... |
| abscncf 25183 | Absolute value is continuo... |
| recncf 25184 | Real part is continuous. ... |
| imcncf 25185 | Imaginary part is continuo... |
| cjcncf 25186 | Complex conjugate is conti... |
| mulc1cncf 25187 | Multiplication by a consta... |
| divccncf 25188 | Division by a constant is ... |
| cncfco 25189 | The composition of two con... |
| cncfcompt2 25190 | Composition of continuous ... |
| cncfmet 25191 | Relate complex function co... |
| cncfcn 25192 | Relate complex function co... |
| cncfcn1 25193 | Relate complex function co... |
| cncfmptc 25194 | A constant function is a c... |
| cncfmptid 25195 | The identity function is a... |
| cncfmpt1f 25196 | Composition of continuous ... |
| cncfmpt2f 25197 | Composition of continuous ... |
| cncfmpt2ss 25198 | Composition of continuous ... |
| addccncf 25199 | Adding a constant is a con... |
| idcncf 25200 | The identity function is a... |
| sub1cncf 25201 | Subtracting a constant is ... |
| sub2cncf 25202 | Subtraction from a constan... |
| cdivcncf 25203 | Division with a constant n... |
| negcncf 25204 | The negative function is c... |
| negfcncf 25205 | The negative of a continuo... |
| abscncfALT 25206 | Absolute value is continuo... |
| cncfcnvcn 25207 | Rewrite ~ cmphaushmeo for ... |
| expcncf 25208 | The power function on comp... |
| cnmptre 25209 | Lemma for ~ iirevcn and re... |
| cnmpopc 25210 | Piecewise definition of a ... |
| iirev 25211 | Reverse the unit interval.... |
| iirevcn 25212 | The reversion function is ... |
| iihalf1 25213 | Map the first half of ` II... |
| iihalf1cn 25214 | The first half function is... |
| iihalf2 25215 | Map the second half of ` I... |
| iihalf2cn 25216 | The second half function i... |
| elii1 25217 | Divide the unit interval i... |
| elii2 25218 | Divide the unit interval i... |
| iimulcl 25219 | The unit interval is close... |
| iimulcn 25220 | Multiplication is a contin... |
| icoopnst 25221 | A half-open interval start... |
| iocopnst 25222 | A half-open interval endin... |
| icchmeo 25223 | The natural bijection from... |
| icopnfcnv 25224 | Define a bijection from ` ... |
| icopnfhmeo 25225 | The defined bijection from... |
| iccpnfcnv 25226 | Define a bijection from ` ... |
| iccpnfhmeo 25227 | The defined bijection from... |
| xrhmeo 25228 | The bijection from ` [ -u ... |
| xrhmph 25229 | The extended reals are hom... |
| xrcmp 25230 | The topology of the extend... |
| xrconn 25231 | The topology of the extend... |
| icccvx 25232 | A linear combination of tw... |
| oprpiece1res1 25233 | Restriction to the first p... |
| oprpiece1res2 25234 | Restriction to the second ... |
| cnrehmeo 25235 | The canonical bijection fr... |
| cnheiborlem 25236 | Lemma for ~ cnheibor . (C... |
| cnheibor 25237 | Heine-Borel theorem for co... |
| cnllycmp 25238 | The topology on the comple... |
| rellycmp 25239 | The topology on the reals ... |
| bndth 25240 | The Boundedness Theorem. ... |
| evth 25241 | The Extreme Value Theorem.... |
| evth2 25242 | The Extreme Value Theorem,... |
| lebnumlem1 25243 | Lemma for ~ lebnum . The ... |
| lebnumlem2 25244 | Lemma for ~ lebnum . As a... |
| lebnumlem3 25245 | Lemma for ~ lebnum . By t... |
| lebnum 25246 | The Lebesgue number lemma,... |
| xlebnum 25247 | Generalize ~ lebnum to ext... |
| lebnumii 25248 | Specialize the Lebesgue nu... |
| ishtpy 25254 | Membership in the class of... |
| htpycn 25255 | A homotopy is a continuous... |
| htpyi 25256 | A homotopy evaluated at it... |
| ishtpyd 25257 | Deduction for membership i... |
| htpycom 25258 | Given a homotopy from ` F ... |
| htpyid 25259 | A homotopy from a function... |
| htpyco1 25260 | Compose a homotopy with a ... |
| htpyco2 25261 | Compose a homotopy with a ... |
| htpycc 25262 | Concatenate two homotopies... |
| isphtpy 25263 | Membership in the class of... |
| phtpyhtpy 25264 | A path homotopy is a homot... |
| phtpycn 25265 | A path homotopy is a conti... |
| phtpyi 25266 | Membership in the class of... |
| phtpy01 25267 | Two path-homotopic paths h... |
| isphtpyd 25268 | Deduction for membership i... |
| isphtpy2d 25269 | Deduction for membership i... |
| phtpycom 25270 | Given a homotopy from ` F ... |
| phtpyid 25271 | A homotopy from a path to ... |
| phtpyco2 25272 | Compose a path homotopy wi... |
| phtpycc 25273 | Concatenate two path homot... |
| phtpcrel 25275 | The path homotopy relation... |
| isphtpc 25276 | The relation "is path homo... |
| phtpcer 25277 | Path homotopy is an equiva... |
| phtpc01 25278 | Path homotopic paths have ... |
| reparphti 25279 | Lemma for ~ reparpht . (C... |
| reparpht 25280 | Reparametrization lemma. ... |
| phtpcco2 25281 | Compose a path homotopy wi... |
| pcofval 25292 | The value of the path conc... |
| pcoval 25293 | The concatenation of two p... |
| pcovalg 25294 | Evaluate the concatenation... |
| pcoval1 25295 | Evaluate the concatenation... |
| pco0 25296 | The starting point of a pa... |
| pco1 25297 | The ending point of a path... |
| pcoval2 25298 | Evaluate the concatenation... |
| pcocn 25299 | The concatenation of two p... |
| copco 25300 | The composition of a conca... |
| pcohtpylem 25301 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25302 | Homotopy invariance of pat... |
| pcoptcl 25303 | A constant function is a p... |
| pcopt 25304 | Concatenation with a point... |
| pcopt2 25305 | Concatenation with a point... |
| pcoass 25306 | Order of concatenation doe... |
| pcorevcl 25307 | Closure for a reversed pat... |
| pcorevlem 25308 | Lemma for ~ pcorev . Prov... |
| pcorev 25309 | Concatenation with the rev... |
| pcorev2 25310 | Concatenation with the rev... |
| pcophtb 25311 | The path homotopy equivale... |
| om1val 25312 | The definition of the loop... |
| om1bas 25313 | The base set of the loop s... |
| om1elbas 25314 | Elementhood in the base se... |
| om1addcl 25315 | Closure of the group opera... |
| om1plusg 25316 | The group operation (which... |
| om1tset 25317 | The topology of the loop s... |
| om1opn 25318 | The topology of the loop s... |
| pi1val 25319 | The definition of the fund... |
| pi1bas 25320 | The base set of the fundam... |
| pi1blem 25321 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25322 | Another way to write the l... |
| pi1bas2 25323 | The base set of the fundam... |
| pi1eluni 25324 | Elementhood in the base se... |
| pi1bas3 25325 | The base set of the fundam... |
| pi1cpbl 25326 | The group operation, loop ... |
| elpi1 25327 | The elements of the fundam... |
| elpi1i 25328 | The elements of the fundam... |
| pi1addf 25329 | The group operation of ` p... |
| pi1addval 25330 | The concatenation of two p... |
| pi1grplem 25331 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25332 | The fundamental group is a... |
| pi1id 25333 | The identity element of th... |
| pi1inv 25334 | An inverse in the fundamen... |
| pi1xfrf 25335 | Functionality of the loop ... |
| pi1xfrval 25336 | The value of the loop tran... |
| pi1xfr 25337 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25338 | Given a path ` F ` between... |
| pi1xfrcnv 25339 | Given a path ` F ` between... |
| pi1xfrgim 25340 | The mapping ` G ` between ... |
| pi1cof 25341 | Functionality of the loop ... |
| pi1coval 25342 | The value of the loop tran... |
| pi1coghm 25343 | The mapping ` G ` between ... |
| isclm 25346 | A subcomplex module is a l... |
| clmsca 25347 | The ring of scalars ` F ` ... |
| clmsubrg 25348 | The base set of the ring o... |
| clmlmod 25349 | A subcomplex module is a l... |
| clmgrp 25350 | A subcomplex module is an ... |
| clmabl 25351 | A subcomplex module is an ... |
| clmring 25352 | The scalar ring of a subco... |
| clmfgrp 25353 | The scalar ring of a subco... |
| clm0 25354 | The zero of the scalar rin... |
| clm1 25355 | The identity of the scalar... |
| clmadd 25356 | The addition of the scalar... |
| clmmul 25357 | The multiplication of the ... |
| clmcj 25358 | The conjugation of the sca... |
| isclmi 25359 | Reverse direction of ~ isc... |
| clmzss 25360 | The scalar ring of a subco... |
| clmsscn 25361 | The scalar ring of a subco... |
| clmsub 25362 | Subtraction in the scalar ... |
| clmneg 25363 | Negation in the scalar rin... |
| clmneg1 25364 | Minus one is in the scalar... |
| clmabs 25365 | Norm in the scalar ring of... |
| clmacl 25366 | Closure of ring addition f... |
| clmmcl 25367 | Closure of ring multiplica... |
| clmsubcl 25368 | Closure of ring subtractio... |
| lmhmclm 25369 | The domain of a linear ope... |
| clmvscl 25370 | Closure of scalar product ... |
| clmvsass 25371 | Associative law for scalar... |
| clmvscom 25372 | Commutative law for the sc... |
| clmvsdir 25373 | Distributive law for scala... |
| clmvsdi 25374 | Distributive law for scala... |
| clmvs1 25375 | Scalar product with ring u... |
| clmvs2 25376 | A vector plus itself is tw... |
| clm0vs 25377 | Zero times a vector is the... |
| clmopfne 25378 | The (functionalized) opera... |
| isclmp 25379 | The predicate "is a subcom... |
| isclmi0 25380 | Properties that determine ... |
| clmvneg1 25381 | Minus 1 times a vector is ... |
| clmvsneg 25382 | Multiplication of a vector... |
| clmmulg 25383 | The group multiple functio... |
| clmsubdir 25384 | Scalar multiplication dist... |
| clmpm1dir 25385 | Subtractive distributive l... |
| clmnegneg 25386 | Double negative of a vecto... |
| clmnegsubdi2 25387 | Distribution of negative o... |
| clmsub4 25388 | Rearrangement of 4 terms i... |
| clmvsrinv 25389 | A vector minus itself. (C... |
| clmvslinv 25390 | Minus a vector plus itself... |
| clmvsubval 25391 | Value of vector subtractio... |
| clmvsubval2 25392 | Value of vector subtractio... |
| clmvz 25393 | Two ways to express the ne... |
| zlmclm 25394 | The ` ZZ ` -module operati... |
| clmzlmvsca 25395 | The scalar product of a su... |
| nmoleub2lem 25396 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25397 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25398 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25399 | The operator norm is the s... |
| nmoleub2b 25400 | The operator norm is the s... |
| nmoleub3 25401 | The operator norm is the s... |
| nmhmcn 25402 | A linear operator over a n... |
| cmodscexp 25403 | The powers of ` _i ` belon... |
| cmodscmulexp 25404 | The scalar product of a ve... |
| cvslvec 25407 | A subcomplex vector space ... |
| cvsclm 25408 | A subcomplex vector space ... |
| iscvs 25409 | A subcomplex vector space ... |
| iscvsp 25410 | The predicate "is a subcom... |
| iscvsi 25411 | Properties that determine ... |
| cvsi 25412 | The properties of a subcom... |
| cvsunit 25413 | Unit group of the scalar r... |
| cvsdiv 25414 | Division of the scalar rin... |
| cvsdivcl 25415 | The scalar field of a subc... |
| cvsmuleqdivd 25416 | An equality involving rati... |
| cvsdiveqd 25417 | An equality involving rati... |
| cnlmodlem1 25418 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25419 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25420 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25421 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25422 | The set of complex numbers... |
| cnstrcvs 25423 | The set of complex numbers... |
| cnrbas 25424 | The set of complex numbers... |
| cnrlmod 25425 | The complex left module of... |
| cnrlvec 25426 | The complex left module of... |
| cncvs 25427 | The complex left module of... |
| recvs 25428 | The field of the real numb... |
| qcvs 25429 | The field of rational numb... |
| zclmncvs 25430 | The ring of integers as le... |
| isncvsngp 25431 | A normed subcomplex vector... |
| isncvsngpd 25432 | Properties that determine ... |
| ncvsi 25433 | The properties of a normed... |
| ncvsprp 25434 | Proportionality property o... |
| ncvsge0 25435 | The norm of a scalar produ... |
| ncvsm1 25436 | The norm of the opposite o... |
| ncvsdif 25437 | The norm of the difference... |
| ncvspi 25438 | The norm of a vector plus ... |
| ncvs1 25439 | From any nonzero vector of... |
| cnrnvc 25440 | The module of complex numb... |
| cnncvs 25441 | The module of complex numb... |
| cnnm 25442 | The norm of the normed sub... |
| ncvspds 25443 | Value of the distance func... |
| cnindmet 25444 | The metric induced on the ... |
| cnncvsaddassdemo 25445 | Derive the associative law... |
| cnncvsmulassdemo 25446 | Derive the associative law... |
| cnncvsabsnegdemo 25447 | Derive the absolute value ... |
| iscph 25452 | A subcomplex pre-Hilbert s... |
| cphphl 25453 | A subcomplex pre-Hilbert s... |
| cphnlm 25454 | A subcomplex pre-Hilbert s... |
| cphngp 25455 | A subcomplex pre-Hilbert s... |
| cphlmod 25456 | A subcomplex pre-Hilbert s... |
| cphlvec 25457 | A subcomplex pre-Hilbert s... |
| cphnvc 25458 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25459 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25460 | Lemma for ~ cphreccl . (C... |
| cphsca 25461 | A subcomplex pre-Hilbert s... |
| cphsubrg 25462 | The scalar field of a subc... |
| cphreccl 25463 | The scalar field of a subc... |
| cphdivcl 25464 | The scalar field of a subc... |
| cphcjcl 25465 | The scalar field of a subc... |
| cphsqrtcl 25466 | The scalar field of a subc... |
| cphabscl 25467 | The scalar field of a subc... |
| cphsqrtcl2 25468 | The scalar field of a subc... |
| cphsqrtcl3 25469 | If the scalar field of a s... |
| cphqss 25470 | The scalar field of a subc... |
| cphclm 25471 | A subcomplex pre-Hilbert s... |
| cphnmvs 25472 | Norm of a scalar product. ... |
| cphipcl 25473 | An inner product is a memb... |
| cphnmfval 25474 | The value of the norm in a... |
| cphnm 25475 | The square of the norm is ... |
| nmsq 25476 | The square of the norm is ... |
| cphnmf 25477 | The norm of a vector is a ... |
| cphnmcl 25478 | The norm of a vector is a ... |
| reipcl 25479 | An inner product of an ele... |
| ipge0 25480 | The inner product in a sub... |
| cphipcj 25481 | Conjugate of an inner prod... |
| cphipipcj 25482 | An inner product times its... |
| cphorthcom 25483 | Orthogonality (meaning inn... |
| cphip0l 25484 | Inner product with a zero ... |
| cphip0r 25485 | Inner product with a zero ... |
| cphipeq0 25486 | The inner product of a vec... |
| cphdir 25487 | Distributive law for inner... |
| cphdi 25488 | Distributive law for inner... |
| cph2di 25489 | Distributive law for inner... |
| cphsubdir 25490 | Distributive law for inner... |
| cphsubdi 25491 | Distributive law for inner... |
| cph2subdi 25492 | Distributive law for inner... |
| cphass 25493 | Associative law for inner ... |
| cphassr 25494 | "Associative" law for seco... |
| cph2ass 25495 | Move scalar multiplication... |
| cphassi 25496 | Associative law for the fi... |
| cphassir 25497 | "Associative" law for the ... |
| cphpyth 25498 | The pythagorean theorem fo... |
| tcphex 25499 | Lemma for ~ tcphbas and si... |
| tcphval 25500 | Define a function to augme... |
| tcphbas 25501 | The base set of a subcompl... |
| tchplusg 25502 | The addition operation of ... |
| tcphsub 25503 | The subtraction operation ... |
| tcphmulr 25504 | The ring operation of a su... |
| tcphsca 25505 | The scalar field of a subc... |
| tcphvsca 25506 | The scalar multiplication ... |
| tcphip 25507 | The inner product of a sub... |
| tcphtopn 25508 | The topology of a subcompl... |
| tcphphl 25509 | Augmentation of a subcompl... |
| tchnmfval 25510 | The norm of a subcomplex p... |
| tcphnmval 25511 | The norm of a subcomplex p... |
| cphtcphnm 25512 | The norm of a norm-augment... |
| tcphds 25513 | The distance of a pre-Hilb... |
| phclm 25514 | A pre-Hilbert space whose ... |
| tcphcphlem3 25515 | Lemma for ~ tcphcph : real... |
| ipcau2 25516 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25517 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25518 | Lemma for ~ tcphcph : homo... |
| tcphcph 25519 | The standard definition of... |
| ipcau 25520 | The Cauchy-Schwarz inequal... |
| nmparlem 25521 | Lemma for ~ nmpar . (Cont... |
| nmpar 25522 | A subcomplex pre-Hilbert s... |
| cphipval2 25523 | Value of the inner product... |
| 4cphipval2 25524 | Four times the inner produ... |
| cphipval 25525 | Value of the inner product... |
| ipcnlem2 25526 | The inner product operatio... |
| ipcnlem1 25527 | The inner product operatio... |
| ipcn 25528 | The inner product operatio... |
| cnmpt1ip 25529 | Continuity of inner produc... |
| cnmpt2ip 25530 | Continuity of inner produc... |
| csscld 25531 | A "closed subspace" in a s... |
| clsocv 25532 | The orthogonal complement ... |
| cphsscph 25533 | A subspace of a subcomplex... |
| lmmbr 25540 | Express the binary relatio... |
| lmmbr2 25541 | Express the binary relatio... |
| lmmbr3 25542 | Express the binary relatio... |
| lmmcvg 25543 | Convergence property of a ... |
| lmmbrf 25544 | Express the binary relatio... |
| lmnn 25545 | A condition that implies c... |
| cfilfval 25546 | The set of Cauchy filters ... |
| iscfil 25547 | The property of being a Ca... |
| iscfil2 25548 | The property of being a Ca... |
| cfilfil 25549 | A Cauchy filter is a filte... |
| cfili 25550 | Property of a Cauchy filte... |
| cfil3i 25551 | A Cauchy filter contains b... |
| cfilss 25552 | A filter finer than a Cauc... |
| fgcfil 25553 | The Cauchy filter conditio... |
| fmcfil 25554 | The Cauchy filter conditio... |
| iscfil3 25555 | A filter is Cauchy iff it ... |
| cfilfcls 25556 | Similar to ultrafilters ( ... |
| caufval 25557 | The set of Cauchy sequence... |
| iscau 25558 | Express the property " ` F... |
| iscau2 25559 | Express the property " ` F... |
| iscau3 25560 | Express the Cauchy sequenc... |
| iscau4 25561 | Express the property " ` F... |
| iscauf 25562 | Express the property " ` F... |
| caun0 25563 | A metric with a Cauchy seq... |
| caufpm 25564 | Inclusion of a Cauchy sequ... |
| caucfil 25565 | A Cauchy sequence predicat... |
| iscmet 25566 | The property " ` D ` is a ... |
| cmetcvg 25567 | The convergence of a Cauch... |
| cmetmet 25568 | A complete metric space is... |
| cmetmeti 25569 | A complete metric space is... |
| cmetcaulem 25570 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25571 | The convergence of a Cauch... |
| iscmet3lem3 25572 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25573 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25574 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25575 | The property " ` D ` is a ... |
| iscmet2 25576 | A metric ` D ` is complete... |
| cfilresi 25577 | A Cauchy filter on a metri... |
| cfilres 25578 | Cauchy filter on a metric ... |
| caussi 25579 | Cauchy sequence on a metri... |
| causs 25580 | Cauchy sequence on a metri... |
| equivcfil 25581 | If the metric ` D ` is "st... |
| equivcau 25582 | If the metric ` D ` is "st... |
| lmle 25583 | If the distance from each ... |
| nglmle 25584 | If the norm of each member... |
| lmclim 25585 | Relate a limit on the metr... |
| lmclimf 25586 | Relate a limit on the metr... |
| metelcls 25587 | A point belongs to the clo... |
| metcld 25588 | A subset of a metric space... |
| metcld2 25589 | A subset of a metric space... |
| caubl 25590 | Sufficient condition to en... |
| caublcls 25591 | The convergent point of a ... |
| metcnp4 25592 | Two ways to say a mapping ... |
| metcn4 25593 | Two ways to say a mapping ... |
| iscmet3i 25594 | Properties that determine ... |
| lmcau 25595 | Every convergent sequence ... |
| flimcfil 25596 | Every convergent filter in... |
| metsscmetcld 25597 | A complete subspace of a m... |
| cmetss 25598 | A subspace of a complete m... |
| equivcmet 25599 | If two metrics are strongl... |
| relcmpcmet 25600 | If ` D ` is a metric space... |
| cmpcmet 25601 | A compact metric space is ... |
| cfilucfil3 25602 | Given a metric ` D ` and a... |
| cfilucfil4 25603 | Given a metric ` D ` and a... |
| cncmet 25604 | The set of complex numbers... |
| recmet 25605 | The real numbers are a com... |
| bcthlem1 25606 | Lemma for ~ bcth . Substi... |
| bcthlem2 25607 | Lemma for ~ bcth . The ba... |
| bcthlem3 25608 | Lemma for ~ bcth . The li... |
| bcthlem4 25609 | Lemma for ~ bcth . Given ... |
| bcthlem5 25610 | Lemma for ~ bcth . The pr... |
| bcth 25611 | Baire's Category Theorem. ... |
| bcth2 25612 | Baire's Category Theorem, ... |
| bcth3 25613 | Baire's Category Theorem, ... |
| isbn 25620 | A Banach space is a normed... |
| bnsca 25621 | The scalar field of a Bana... |
| bnnvc 25622 | A Banach space is a normed... |
| bnnlm 25623 | A Banach space is a normed... |
| bnngp 25624 | A Banach space is a normed... |
| bnlmod 25625 | A Banach space is a left m... |
| bncms 25626 | A Banach space is a comple... |
| iscms 25627 | A complete metric space is... |
| cmscmet 25628 | The induced metric on a co... |
| bncmet 25629 | The induced metric on Bana... |
| cmsms 25630 | A complete metric space is... |
| cmspropd 25631 | Property deduction for a c... |
| cmssmscld 25632 | The restriction of a metri... |
| cmsss 25633 | The restriction of a compl... |
| lssbn 25634 | A subspace of a Banach spa... |
| cmetcusp1 25635 | If the uniform set of a co... |
| cmetcusp 25636 | The uniform space generate... |
| cncms 25637 | The field of complex numbe... |
| cnflduss 25638 | The uniform structure of t... |
| cnfldcusp 25639 | The field of complex numbe... |
| resscdrg 25640 | The real numbers are a sub... |
| cncdrg 25641 | The only complete subfield... |
| srabn 25642 | The subring algebra over a... |
| rlmbn 25643 | The ring module over a com... |
| ishl 25644 | The predicate "is a subcom... |
| hlbn 25645 | Every subcomplex Hilbert s... |
| hlcph 25646 | Every subcomplex Hilbert s... |
| hlphl 25647 | Every subcomplex Hilbert s... |
| hlcms 25648 | Every subcomplex Hilbert s... |
| hlprlem 25649 | Lemma for ~ hlpr . (Contr... |
| hlress 25650 | The scalar field of a subc... |
| hlpr 25651 | The scalar field of a subc... |
| ishl2 25652 | A Hilbert space is a compl... |
| cphssphl 25653 | A Banach subspace of a sub... |
| cmslssbn 25654 | A complete linear subspace... |
| cmscsscms 25655 | A closed subspace of a com... |
| bncssbn 25656 | A closed subspace of a Ban... |
| cssbn 25657 | A complete subspace of a n... |
| csschl 25658 | A complete subspace of a c... |
| cmslsschl 25659 | A complete linear subspace... |
| chlcsschl 25660 | A closed subspace of a sub... |
| retopn 25661 | The topology of the real n... |
| recms 25662 | The real numbers form a co... |
| reust 25663 | The Uniform structure of t... |
| recusp 25664 | The real numbers form a co... |
| rrxval 25669 | Value of the generalized E... |
| rrxbase 25670 | The base of the generalize... |
| rrxprds 25671 | Expand the definition of t... |
| rrxip 25672 | The inner product of the g... |
| rrxnm 25673 | The norm of the generalize... |
| rrxcph 25674 | Generalized Euclidean real... |
| rrxds 25675 | The distance over generali... |
| rrxvsca 25676 | The scalar product over ge... |
| rrxplusgvscavalb 25677 | The result of the addition... |
| rrxsca 25678 | The field of real numbers ... |
| rrx0 25679 | The zero ("origin") in a g... |
| rrx0el 25680 | The zero ("origin") in a g... |
| csbren 25681 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25682 | Triangle inequality in R^n... |
| rrxf 25683 | Euclidean vectors as funct... |
| rrxfsupp 25684 | Euclidean vectors are of f... |
| rrxsuppss 25685 | Support of Euclidean vecto... |
| rrxmvallem 25686 | Support of the function us... |
| rrxmval 25687 | The value of the Euclidean... |
| rrxmfval 25688 | The value of the Euclidean... |
| rrxmetlem 25689 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25690 | Euclidean space is a metri... |
| rrxdstprj1 25691 | The distance between two p... |
| rrxbasefi 25692 | The base of the generalize... |
| rrxdsfi 25693 | The distance over generali... |
| rrxmetfi 25694 | Euclidean space is a metri... |
| rrxdsfival 25695 | The value of the Euclidean... |
| ehlval 25696 | Value of the Euclidean spa... |
| ehlbase 25697 | The base of the Euclidean ... |
| ehl0base 25698 | The base of the Euclidean ... |
| ehl0 25699 | The Euclidean space of dim... |
| ehleudis 25700 | The Euclidean distance fun... |
| ehleudisval 25701 | The value of the Euclidean... |
| ehl1eudis 25702 | The Euclidean distance fun... |
| ehl1eudisval 25703 | The value of the Euclidean... |
| ehl2eudis 25704 | The Euclidean distance fun... |
| ehl2eudisval 25705 | The value of the Euclidean... |
| minveclem1 25706 | Lemma for ~ minvec . The ... |
| minveclem4c 25707 | Lemma for ~ minvec . The ... |
| minveclem2 25708 | Lemma for ~ minvec . Any ... |
| minveclem3a 25709 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25710 | Lemma for ~ minvec . The ... |
| minveclem3 25711 | Lemma for ~ minvec . The ... |
| minveclem4a 25712 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25713 | Lemma for ~ minvec . The ... |
| minveclem4 25714 | Lemma for ~ minvec . The ... |
| minveclem5 25715 | Lemma for ~ minvec . Disc... |
| minveclem6 25716 | Lemma for ~ minvec . Any ... |
| minveclem7 25717 | Lemma for ~ minvec . Sinc... |
| minvec 25718 | Minimizing vector theorem,... |
| pjthlem1 25719 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25720 | Lemma for ~ pjth . (Contr... |
| pjth 25721 | Projection Theorem: Any H... |
| pjth2 25722 | Projection Theorem with ab... |
| cldcss 25723 | Corollary of the Projectio... |
| cldcss2 25724 | Corollary of the Projectio... |
| hlhil 25725 | Corollary of the Projectio... |
| addcncf 25726 | The addition of two contin... |
| subcncf 25727 | The subtraction of two con... |
| mulcncf 25728 | The multiplication of two ... |
| divcncf 25729 | The quotient of two contin... |
| pmltpclem1 25730 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25731 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25732 | Any function on the reals ... |
| ivthlem1 25733 | Lemma for ~ ivth . The se... |
| ivthlem2 25734 | Lemma for ~ ivth . Show t... |
| ivthlem3 25735 | Lemma for ~ ivth , the int... |
| ivth 25736 | The intermediate value the... |
| ivth2 25737 | The intermediate value the... |
| ivthle 25738 | The intermediate value the... |
| ivthle2 25739 | The intermediate value the... |
| ivthicc 25740 | The interval between any t... |
| evthicc 25741 | Specialization of the Extr... |
| evthicc2 25742 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25743 | A continuous function on a... |
| ovolfcl 25748 | Closure for the interval e... |
| ovolfioo 25749 | Unpack the interval coveri... |
| ovolficc 25750 | Unpack the interval coveri... |
| ovolficcss 25751 | Any (closed) interval cove... |
| ovolfsval 25752 | The value of the interval ... |
| ovolfsf 25753 | Closure for the interval l... |
| ovolsf 25754 | Closure for the partial su... |
| ovolval 25755 | The value of the outer mea... |
| elovolmlem 25756 | Lemma for ~ elovolm and re... |
| elovolm 25757 | Elementhood in the set ` M... |
| elovolmr 25758 | Sufficient condition for e... |
| ovolmge0 25759 | The set ` M ` is composed ... |
| ovolcl 25760 | The volume of a set is an ... |
| ovollb 25761 | The outer volume is a lowe... |
| ovolgelb 25762 | The outer volume is the gr... |
| ovolge0 25763 | The volume of a set is alw... |
| ovolf 25764 | The domain and codomain of... |
| ovollecl 25765 | If an outer volume is boun... |
| ovolsslem 25766 | Lemma for ~ ovolss . (Con... |
| ovolss 25767 | The volume of a set is mon... |
| ovolsscl 25768 | If a set is contained in a... |
| ovolssnul 25769 | A subset of a nullset is n... |
| ovollb2lem 25770 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25771 | It is often more convenien... |
| ovolctb 25772 | The volume of a denumerabl... |
| ovolq 25773 | The rational numbers have ... |
| ovolctb2 25774 | The volume of a countable ... |
| ovol0 25775 | The empty set has 0 outer ... |
| ovolfi 25776 | A finite set has 0 outer L... |
| ovolsn 25777 | A singleton has 0 outer Le... |
| ovolunlem1a 25778 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25779 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25780 | Lemma for ~ ovolun . (Con... |
| ovolun 25781 | The Lebesgue outer measure... |
| ovolunnul 25782 | Adding a nullset does not ... |
| ovolfiniun 25783 | The Lebesgue outer measure... |
| ovoliunlem1 25784 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25785 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25786 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25787 | The Lebesgue outer measure... |
| ovoliun2 25788 | The Lebesgue outer measure... |
| ovoliunnul 25789 | A countable union of nulls... |
| shft2rab 25790 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25791 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25792 | Lemma for ~ ovolshft . (C... |
| ovolshft 25793 | The Lebesgue outer measure... |
| sca2rab 25794 | If ` B ` is a scale of ` A... |
| ovolscalem1 25795 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25796 | Lemma for ~ ovolshft . (C... |
| ovolsca 25797 | The Lebesgue outer measure... |
| ovolicc1 25798 | The measure of a closed in... |
| ovolicc2lem1 25799 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25800 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25801 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25802 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25803 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25804 | The measure of a closed in... |
| ovolicc 25805 | The measure of a closed in... |
| ovolicopnf 25806 | The measure of a right-unb... |
| ovolre 25807 | The measure of the real nu... |
| ismbl 25808 | The predicate " ` A ` is L... |
| ismbl2 25809 | From ~ ovolun , it suffice... |
| volres 25810 | A self-referencing abbrevi... |
| volf 25811 | The domain and codomain of... |
| mblvol 25812 | The volume of a measurable... |
| mblss 25813 | A measurable set is a subs... |
| mblsplit 25814 | The defining property of m... |
| volss 25815 | The Lebesgue measure is mo... |
| cmmbl 25816 | The complement of a measur... |
| nulmbl 25817 | A nullset is measurable. ... |
| nulmbl2 25818 | A set of outer measure zer... |
| unmbl 25819 | A union of measurable sets... |
| shftmbl 25820 | A shift of a measurable se... |
| 0mbl 25821 | The empty set is measurabl... |
| rembl 25822 | The set of all real number... |
| unidmvol 25823 | The union of the Lebesgue ... |
| inmbl 25824 | An intersection of measura... |
| difmbl 25825 | A difference of measurable... |
| finiunmbl 25826 | A finite union of measurab... |
| volun 25827 | The Lebesgue measure funct... |
| volinun 25828 | Addition of non-disjoint s... |
| volfiniun 25829 | The volume of a disjoint f... |
| iundisj 25830 | Rewrite a countable union ... |
| iundisj2 25831 | A disjoint union is disjoi... |
| voliunlem1 25832 | Lemma for ~ voliun . (Con... |
| voliunlem2 25833 | Lemma for ~ voliun . (Con... |
| voliunlem3 25834 | Lemma for ~ voliun . (Con... |
| iunmbl 25835 | The measurable sets are cl... |
| voliun 25836 | The Lebesgue measure funct... |
| volsuplem 25837 | Lemma for ~ volsup . (Con... |
| volsup 25838 | The volume of the limit of... |
| iunmbl2 25839 | The measurable sets are cl... |
| ioombl1lem1 25840 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25841 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25842 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25843 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25844 | An open right-unbounded in... |
| icombl1 25845 | A closed unbounded-above i... |
| icombl 25846 | A closed-below, open-above... |
| ioombl 25847 | An open real interval is m... |
| iccmbl 25848 | A closed real interval is ... |
| iccvolcl 25849 | A closed real interval has... |
| ovolioo 25850 | The measure of an open int... |
| volioo 25851 | The measure of an open int... |
| ioovolcl 25852 | An open real interval has ... |
| ovolfs2 25853 | Alternative expression for... |
| ioorcl2 25854 | An open interval with fini... |
| ioorf 25855 | Define a function from ope... |
| ioorval 25856 | Define a function from ope... |
| ioorinv2 25857 | The function ` F ` is an "... |
| ioorinv 25858 | The function ` F ` is an "... |
| ioorcl 25859 | The function ` F ` does no... |
| uniiccdif 25860 | A union of closed interval... |
| uniioovol 25861 | A disjoint union of open i... |
| uniiccvol 25862 | An almost-disjoint union o... |
| uniioombllem1 25863 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25864 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25865 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25866 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25867 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25868 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25869 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25870 | Lemma for ~ uniioombl . (... |
| uniioombl 25871 | A disjoint union of open i... |
| uniiccmbl 25872 | An almost-disjoint union o... |
| dyadf 25873 | The function ` F ` returns... |
| dyadval 25874 | Value of the dyadic ration... |
| dyadovol 25875 | Volume of a dyadic rationa... |
| dyadss 25876 | Two closed dyadic rational... |
| dyaddisjlem 25877 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25878 | Two closed dyadic rational... |
| dyadmaxlem 25879 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25880 | Any nonempty set of dyadic... |
| dyadmbllem 25881 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25882 | Any union of dyadic ration... |
| opnmbllem 25883 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25884 | All open sets are measurab... |
| opnmblALT 25885 | All open sets are measurab... |
| subopnmbl 25886 | Sets which are open in a m... |
| volsup2 25887 | The volume of ` A ` is the... |
| volcn 25888 | The function formed by res... |
| volivth 25889 | The Intermediate Value The... |
| vitalilem1 25890 | Lemma for ~ vitali . (Con... |
| vitalilem2 25891 | Lemma for ~ vitali . (Con... |
| vitalilem3 25892 | Lemma for ~ vitali . (Con... |
| vitalilem4 25893 | Lemma for ~ vitali . (Con... |
| vitalilem5 25894 | Lemma for ~ vitali . (Con... |
| vitali 25895 | If the reals can be well-o... |
| ismbf1 25906 | The predicate " ` F ` is a... |
| mbff 25907 | A measurable function is a... |
| mbfdm 25908 | The domain of a measurable... |
| mbfconstlem 25909 | Lemma for ~ mbfconst and r... |
| ismbf 25910 | The predicate " ` F ` is a... |
| ismbfcn 25911 | A complex function is meas... |
| mbfima 25912 | Definitional property of a... |
| mbfimaicc 25913 | The preimage of any closed... |
| mbfimasn 25914 | The preimage of a point un... |
| mbfconst 25915 | A constant function is mea... |
| mbf0 25916 | The empty function is meas... |
| mbfid 25917 | The identity function is m... |
| mbfmptcl 25918 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25919 | The domain of a measurable... |
| ismbfcn2 25920 | A complex function is meas... |
| ismbfd 25921 | Deduction to prove measura... |
| ismbf2d 25922 | Deduction to prove measura... |
| mbfeqalem1 25923 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25924 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25925 | If two functions are equal... |
| mbfres 25926 | The restriction of a measu... |
| mbfres2 25927 | Measurability of a piecewi... |
| mbfss 25928 | Change the domain of a mea... |
| mbfmulc2lem 25929 | Multiplication by a consta... |
| mbfmulc2re 25930 | Multiplication by a consta... |
| mbfmax 25931 | The maximum of two functio... |
| mbfneg 25932 | The negative of a measurab... |
| mbfpos 25933 | The positive part of a mea... |
| mbfposr 25934 | Converse to ~ mbfpos . (C... |
| mbfposb 25935 | A function is measurable i... |
| ismbf3d 25936 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25937 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25938 | The preimage of any open s... |
| mbfimaopn2 25939 | The preimage of any set op... |
| cncombf 25940 | The composition of a conti... |
| cnmbf 25941 | A continuous function is m... |
| mbfaddlem 25942 | The sum of two measurable ... |
| mbfadd 25943 | The sum of two measurable ... |
| mbfsub 25944 | The difference of two meas... |
| mbfmulc2 25945 | A complex constant times a... |
| mbfsup 25946 | The supremum of a sequence... |
| mbfinf 25947 | The infimum of a sequence ... |
| mbflimsup 25948 | The limit supremum of a se... |
| mbflimlem 25949 | The pointwise limit of a s... |
| mbflim 25950 | The pointwise limit of a s... |
| 0pval 25953 | The zero function evaluate... |
| 0plef 25954 | Two ways to say that the f... |
| 0pledm 25955 | Adjust the domain of the l... |
| isi1f 25956 | The predicate " ` F ` is a... |
| i1fmbf 25957 | Simple functions are measu... |
| i1ff 25958 | A simple function is a fun... |
| i1frn 25959 | A simple function has fini... |
| i1fima 25960 | Any preimage of a simple f... |
| i1fima2 25961 | Any preimage of a simple f... |
| i1fima2sn 25962 | Preimage of a singleton. ... |
| i1fd 25963 | A simplified set of assump... |
| i1f0rn 25964 | Any simple function takes ... |
| itg1val 25965 | The value of the integral ... |
| itg1val2 25966 | The value of the integral ... |
| itg1cl 25967 | Closure of the integral on... |
| itg1ge0 25968 | Closure of the integral on... |
| i1f0 25969 | The zero function is simpl... |
| itg10 25970 | The zero function has zero... |
| i1f1lem 25971 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25972 | Base case simple functions... |
| itg11 25973 | The integral of an indicat... |
| itg1addlem1 25974 | Decompose a preimage, whic... |
| i1faddlem 25975 | Decompose the preimage of ... |
| i1fmullem 25976 | Decompose the preimage of ... |
| i1fadd 25977 | The sum of two simple func... |
| i1fmul 25978 | The pointwise product of t... |
| itg1addlem2 25979 | Lemma for ~ itg1add . The... |
| itg1addlem3 25980 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25981 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25982 | Lemma for ~ itg1add . (Co... |
| itg1add 25983 | The integral of a sum of s... |
| i1fmulclem 25984 | Decompose the preimage of ... |
| i1fmulc 25985 | A nonnegative constant tim... |
| itg1mulc 25986 | The integral of a constant... |
| i1fres 25987 | The "restriction" of a sim... |
| i1fpos 25988 | The positive part of a sim... |
| i1fposd 25989 | Deduction form of ~ i1fpos... |
| i1fsub 25990 | The difference of two simp... |
| itg1sub 25991 | The integral of a differen... |
| itg10a 25992 | The integral of a simple f... |
| itg1ge0a 25993 | The integral of an almost ... |
| itg1lea 25994 | Approximate version of ~ i... |
| itg1le 25995 | If one simple function dom... |
| itg1climres 25996 | Restricting the simple fun... |
| mbfi1fseqlem1 25997 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25998 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25999 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 26000 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 26001 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 26002 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 26003 | A characterization of meas... |
| mbfi1flimlem 26004 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 26005 | Any real measurable functi... |
| mbfmullem2 26006 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 26007 | Lemma for ~ mbfmul . (Con... |
| mbfmul 26008 | The product of two measura... |
| itg2lcl 26009 | The set of lower sums is a... |
| itg2val 26010 | Value of the integral on n... |
| itg2l 26011 | Elementhood in the set ` L... |
| itg2lr 26012 | Sufficient condition for e... |
| xrge0f 26013 | A real function is a nonne... |
| itg2cl 26014 | The integral of a nonnegat... |
| itg2ub 26015 | The integral of a nonnegat... |
| itg2leub 26016 | Any upper bound on the int... |
| itg2ge0 26017 | The integral of a nonnegat... |
| itg2itg1 26018 | The integral of a nonnegat... |
| itg20 26019 | The integral of the zero f... |
| itg2lecl 26020 | If an ` S.2 ` integral is ... |
| itg2le 26021 | If one function dominates ... |
| itg2const 26022 | Integral of a constant fun... |
| itg2const2 26023 | When the base set of a con... |
| itg2seq 26024 | Definitional property of t... |
| itg2uba 26025 | Approximate version of ~ i... |
| itg2lea 26026 | Approximate version of ~ i... |
| itg2eqa 26027 | Approximate equality of in... |
| itg2mulclem 26028 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 26029 | The integral of a nonnegat... |
| itg2splitlem 26030 | Lemma for ~ itg2split . (... |
| itg2split 26031 | The ` S.2 ` integral split... |
| itg2monolem1 26032 | Lemma for ~ itg2mono . We... |
| itg2monolem2 26033 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 26034 | Lemma for ~ itg2mono . (C... |
| itg2mono 26035 | The Monotone Convergence T... |
| itg2i1fseqle 26036 | Subject to the conditions ... |
| itg2i1fseq 26037 | Subject to the conditions ... |
| itg2i1fseq2 26038 | In an extension to the res... |
| itg2i1fseq3 26039 | Special case of ~ itg2i1fs... |
| itg2addlem 26040 | Lemma for ~ itg2add . (Co... |
| itg2add 26041 | The ` S.2 ` integral is li... |
| itg2gt0 26042 | If the function ` F ` is s... |
| itg2cnlem1 26043 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 26044 | Lemma for ~ itgcn . (Cont... |
| itg2cn 26045 | A sort of absolute continu... |
| ibllem 26046 | Conditioned equality theor... |
| isibl 26047 | The predicate " ` F ` is i... |
| isibl2 26048 | The predicate " ` F ` is i... |
| iblmbf 26049 | An integrable function is ... |
| iblitg 26050 | If a function is integrabl... |
| dfitg 26051 | Evaluate the class substit... |
| itgex 26052 | An integral is a set. (Co... |
| itgeq1f 26053 | Equality theorem for an in... |
| itgeq1 26054 | Equality theorem for an in... |
| nfitg1 26055 | Bound-variable hypothesis ... |
| nfitg 26056 | Bound-variable hypothesis ... |
| cbvitg 26057 | Change bound variable in a... |
| cbvitgv 26058 | Change bound variable in a... |
| itgeq2 26059 | Equality theorem for an in... |
| itgresr 26060 | The domain of an integral ... |
| itg0 26061 | The integral of anything o... |
| itgz 26062 | The integral of zero on an... |
| itgeq2dv 26063 | Equality theorem for an in... |
| itgmpt 26064 | Change bound variable in a... |
| itgcl 26065 | The integral of an integra... |
| itgvallem 26066 | Substitution lemma. (Cont... |
| itgvallem3 26067 | Lemma for ~ itgposval and ... |
| ibl0 26068 | The zero function is integ... |
| iblcnlem1 26069 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 26070 | Expand out the universal q... |
| itgcnlem 26071 | Expand out the sum in ~ df... |
| iblrelem 26072 | Integrability of a real fu... |
| iblposlem 26073 | Lemma for ~ iblpos . (Con... |
| iblpos 26074 | Integrability of a nonnega... |
| iblre 26075 | Integrability of a real fu... |
| itgrevallem1 26076 | Lemma for ~ itgposval and ... |
| itgposval 26077 | The integral of a nonnegat... |
| itgreval 26078 | Decompose the integral of ... |
| itgrecl 26079 | Real closure of an integra... |
| iblcn 26080 | Integrability of a complex... |
| itgcnval 26081 | Decompose the integral of ... |
| itgre 26082 | Real part of an integral. ... |
| itgim 26083 | Imaginary part of an integ... |
| iblneg 26084 | The negative of an integra... |
| itgneg 26085 | Negation of an integral. ... |
| iblss 26086 | A subset of an integrable ... |
| iblss2 26087 | Change the domain of an in... |
| itgitg2 26088 | Transfer an integral using... |
| i1fibl 26089 | A simple function is integ... |
| itgitg1 26090 | Transfer an integral using... |
| itgle 26091 | Monotonicity of an integra... |
| itgge0 26092 | The integral of a positive... |
| itgss 26093 | Expand the set of an integ... |
| itgss2 26094 | Expand the set of an integ... |
| itgeqa 26095 | Approximate equality of in... |
| itgss3 26096 | Expand the set of an integ... |
| itgioo 26097 | Equality of integrals on o... |
| itgless 26098 | Expand the integral of a n... |
| iblconst 26099 | A constant function is int... |
| itgconst 26100 | Integral of a constant fun... |
| ibladdlem 26101 | Lemma for ~ ibladd . (Con... |
| ibladd 26102 | Add two integrals over the... |
| iblsub 26103 | Subtract two integrals ove... |
| itgaddlem1 26104 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 26105 | Lemma for ~ itgadd . (Con... |
| itgadd 26106 | Add two integrals over the... |
| itgsub 26107 | Subtract two integrals ove... |
| itgfsum 26108 | Take a finite sum of integ... |
| iblabslem 26109 | Lemma for ~ iblabs . (Con... |
| iblabs 26110 | The absolute value of an i... |
| iblabsr 26111 | A measurable function is i... |
| iblmulc2 26112 | Multiply an integral by a ... |
| itgmulc2lem1 26113 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 26114 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 26115 | Multiply an integral by a ... |
| itgabs 26116 | The triangle inequality fo... |
| itgsplit 26117 | The ` S. ` integral splits... |
| itgspliticc 26118 | The ` S. ` integral splits... |
| itgsplitioo 26119 | The ` S. ` integral splits... |
| bddmulibl 26120 | A bounded function times a... |
| bddibl 26121 | A bounded function is inte... |
| cniccibl 26122 | A continuous function on a... |
| bddiblnc 26123 | Choice-free proof of ~ bdd... |
| cnicciblnc 26124 | Choice-free proof of ~ cni... |
| itggt0 26125 | The integral of a strictly... |
| itgcn 26126 | Transfer ~ itg2cn to the f... |
| ditgeq1 26129 | Equality theorem for the d... |
| ditgeq2 26130 | Equality theorem for the d... |
| ditgeq3 26131 | Equality theorem for the d... |
| ditgeq3dv 26132 | Equality theorem for the d... |
| ditgex 26133 | A directed integral is a s... |
| ditg0 26134 | Value of the directed inte... |
| cbvditg 26135 | Change bound variable in a... |
| cbvditgv 26136 | Change bound variable in a... |
| ditgpos 26137 | Value of the directed inte... |
| ditgneg 26138 | Value of the directed inte... |
| ditgcl 26139 | Closure of a directed inte... |
| ditgswap 26140 | Reverse a directed integra... |
| ditgsplitlem 26141 | Lemma for ~ ditgsplit . (... |
| ditgsplit 26142 | This theorem is the raison... |
| reldv 26151 | The derivative function is... |
| limcvallem 26152 | Lemma for ~ ellimc . (Con... |
| limcfval 26153 | Value and set bounds on th... |
| ellimc 26154 | Value of the limit predica... |
| limcrcl 26155 | Reverse closure for the li... |
| limccl 26156 | Closure of the limit opera... |
| limcdif 26157 | It suffices to consider fu... |
| ellimc2 26158 | Write the definition of a ... |
| limcnlp 26159 | If ` B ` is not a limit po... |
| ellimc3 26160 | Write the epsilon-delta de... |
| limcflflem 26161 | Lemma for ~ limcflf . (Co... |
| limcflf 26162 | The limit operator can be ... |
| limcmo 26163 | If ` B ` is a limit point ... |
| limcmpt 26164 | Express the limit operator... |
| limcmpt2 26165 | Express the limit operator... |
| limcresi 26166 | Any limit of ` F ` is also... |
| limcres 26167 | If ` B ` is an interior po... |
| cnplimc 26168 | A function is continuous a... |
| cnlimc 26169 | ` F ` is a continuous func... |
| cnlimci 26170 | If ` F ` is a continuous f... |
| cnmptlimc 26171 | If ` F ` is a continuous f... |
| limccnp 26172 | If the limit of ` F ` at `... |
| limccnp2 26173 | The image of a convergent ... |
| limcco 26174 | Composition of two limits.... |
| limciun 26175 | A point is a limit of ` F ... |
| limcun 26176 | A point is a limit of ` F ... |
| dvlem 26177 | Closure for a difference q... |
| dvfval 26178 | Value and set bounds on th... |
| eldv 26179 | The differentiable predica... |
| dvcl 26180 | The derivative function ta... |
| dvbssntr 26181 | The set of differentiable ... |
| dvbss 26182 | The set of differentiable ... |
| dvbsss 26183 | The set of differentiable ... |
| perfdvf 26184 | The derivative is a functi... |
| recnprss 26185 | Both ` RR ` and ` CC ` are... |
| recnperf 26186 | Both ` RR ` and ` CC ` are... |
| dvfg 26187 | Explicitly write out the f... |
| dvf 26188 | The derivative is a functi... |
| dvfcn 26189 | The derivative is a functi... |
| dvreslem 26190 | Lemma for ~ dvres . (Cont... |
| dvres2lem 26191 | Lemma for ~ dvres2 . (Con... |
| dvres 26192 | Restriction of a derivativ... |
| dvres2 26193 | Restriction of the base se... |
| dvres3 26194 | Restriction of a complex d... |
| dvres3a 26195 | Restriction of a complex d... |
| dvidlem 26196 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 26197 | Derivative of a function r... |
| dvconst 26198 | Derivative of a constant f... |
| dvid 26199 | Derivative of the identity... |
| dvcnp 26200 | The difference quotient is... |
| dvcnp2 26201 | A function is continuous a... |
| dvcn 26202 | A differentiable function ... |
| dvnfval 26203 | Value of the iterated deri... |
| dvnff 26204 | The iterated derivative is... |
| dvn0 26205 | Zero times iterated deriva... |
| dvnp1 26206 | Successor iterated derivat... |
| dvn1 26207 | One times iterated derivat... |
| dvnf 26208 | The N-times derivative is ... |
| dvnbss 26209 | The set of N-times differe... |
| dvnadd 26210 | The ` N ` -th derivative o... |
| dvn2bss 26211 | An N-times differentiable ... |
| dvnres 26212 | Multiple derivative versio... |
| cpnfval 26213 | Condition for n-times cont... |
| fncpn 26214 | The ` C^n ` object is a fu... |
| elcpn 26215 | Condition for n-times cont... |
| cpnord 26216 | ` C^n ` conditions are ord... |
| cpncn 26217 | A ` C^n ` function is cont... |
| cpnres 26218 | The restriction of a ` C^n... |
| dvaddbr 26219 | The sum rule for derivativ... |
| dvmulbr 26220 | The product rule for deriv... |
| dvadd 26221 | The sum rule for derivativ... |
| dvmul 26222 | The product rule for deriv... |
| dvaddf 26223 | The sum rule for everywher... |
| dvmulf 26224 | The product rule for every... |
| dvcmul 26225 | The product rule when one ... |
| dvcmulf 26226 | The product rule when one ... |
| dvcobr 26227 | The chain rule for derivat... |
| dvco 26228 | The chain rule for derivat... |
| dvcof 26229 | The chain rule for everywh... |
| dvcjbr 26230 | The derivative of the conj... |
| dvcj 26231 | The derivative of the conj... |
| dvfre 26232 | The derivative of a real f... |
| dvnfre 26233 | The ` N ` -th derivative o... |
| dvexp 26234 | Derivative of a power func... |
| dvexp2 26235 | Derivative of an exponenti... |
| dvrec 26236 | Derivative of the reciproc... |
| dvmptres3 26237 | Function-builder for deriv... |
| dvmptid 26238 | Function-builder for deriv... |
| dvmptc 26239 | Function-builder for deriv... |
| dvmptcl 26240 | Closure lemma for ~ dvmptc... |
| dvmptadd 26241 | Function-builder for deriv... |
| dvmptmul 26242 | Function-builder for deriv... |
| dvmptres2 26243 | Function-builder for deriv... |
| dvmptres 26244 | Function-builder for deriv... |
| dvmptcmul 26245 | Function-builder for deriv... |
| dvmptdivc 26246 | Function-builder for deriv... |
| dvmptneg 26247 | Function-builder for deriv... |
| dvmptsub 26248 | Function-builder for deriv... |
| dvmptcj 26249 | Function-builder for deriv... |
| dvmptre 26250 | Function-builder for deriv... |
| dvmptim 26251 | Function-builder for deriv... |
| dvmptntr 26252 | Function-builder for deriv... |
| dvmptco 26253 | Function-builder for deriv... |
| dvrecg 26254 | Derivative of the reciproc... |
| dvmptdiv 26255 | Function-builder for deriv... |
| dvmptfsum 26256 | Function-builder for deriv... |
| dvcnvlem 26257 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 26258 | A weak version of ~ dvcnvr... |
| dvexp3 26259 | Derivative of an exponenti... |
| dveflem 26260 | Derivative of the exponent... |
| dvef 26261 | Derivative of the exponent... |
| dvsincos 26262 | Derivative of the sine and... |
| dvsin 26263 | Derivative of the sine fun... |
| dvcos 26264 | Derivative of the cosine f... |
| dvferm1lem 26265 | Lemma for ~ dvferm . (Con... |
| dvferm1 26266 | One-sided version of ~ dvf... |
| dvferm2lem 26267 | Lemma for ~ dvferm . (Con... |
| dvferm2 26268 | One-sided version of ~ dvf... |
| dvferm 26269 | Fermat's theorem on statio... |
| rollelem 26270 | Lemma for ~ rolle . (Cont... |
| rolle 26271 | Rolle's theorem. If ` F `... |
| cmvth 26272 | Cauchy's Mean Value Theore... |
| mvth 26273 | The Mean Value Theorem. I... |
| dvlip 26274 | A function with derivative... |
| dvlipcn 26275 | A complex function with de... |
| dvlip2 26276 | Combine the results of ~ d... |
| c1liplem1 26277 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 26278 | C^1 functions are Lipschit... |
| c1lip2 26279 | C^1 functions are Lipschit... |
| c1lip3 26280 | C^1 functions are Lipschit... |
| dveq0 26281 | If a continuous function h... |
| dv11cn 26282 | Two functions defined on a... |
| dvgt0lem1 26283 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 26284 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 26285 | A function on a closed int... |
| dvlt0 26286 | A function on a closed int... |
| dvge0 26287 | A function on a closed int... |
| dvle 26288 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 26289 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 26290 | Lemma for ~ dvivth . (Con... |
| dvivth 26291 | Darboux' theorem, or the i... |
| dvne0 26292 | A function on a closed int... |
| dvne0f1 26293 | A function on a closed int... |
| lhop1lem 26294 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26295 | L'Hôpital's Rule for... |
| lhop2 26296 | L'Hôpital's Rule for... |
| lhop 26297 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26298 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26299 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26300 | The derivative rule for in... |
| dvcvx 26301 | A real function with stric... |
| dvfsumle 26302 | Compare a finite sum to an... |
| dvfsumge 26303 | Compare a finite sum to an... |
| dvfsumabs 26304 | Compare a finite sum to an... |
| dvmptrecl 26305 | Real closure of a derivati... |
| dvfsumrlimf 26306 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26307 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26308 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 26309 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26310 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26311 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26312 | Compare a finite sum to an... |
| dvfsumrlim2 26313 | Compare a finite sum to an... |
| dvfsumrlim3 26314 | Conjoin the statements of ... |
| dvfsum2 26315 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26316 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26317 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26318 | The Fundamental Theorem of... |
| ftc1lem3 26319 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26320 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26321 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26322 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26323 | The Fundamental Theorem of... |
| ftc1cn 26324 | Strengthen the assumptions... |
| ftc2 26325 | The Fundamental Theorem of... |
| ftc2ditglem 26326 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26327 | Directed integral analogue... |
| itgparts 26328 | Integration by parts. If ... |
| itgsubstlem 26329 | Lemma for ~ itgsubst . (C... |
| itgsubst 26330 | Integration by ` u ` -subs... |
| itgpowd 26331 | The integral of a monomial... |
| reldmmdeg 26336 | Multivariate degree is a b... |
| tdeglem1 26337 | Functionality of the total... |
| tdeglem3 26338 | Additivity of the total de... |
| tdeglem4 26339 | There is only one multi-in... |
| tdeglem2 26340 | Simplification of total de... |
| mdegfval 26341 | Value of the multivariate ... |
| mdegval 26342 | Value of the multivariate ... |
| mdegleb 26343 | Property of being of limit... |
| mdeglt 26344 | If there is an upper limit... |
| mdegldg 26345 | A nonzero polynomial has s... |
| mdegxrcl 26346 | Closure of polynomial degr... |
| mdegxrf 26347 | Functionality of polynomia... |
| mdegcl 26348 | Sharp closure for multivar... |
| mdeg0 26349 | Degree of the zero polynom... |
| mdegnn0cl 26350 | Degree of a nonzero polyno... |
| degltlem1 26351 | Theorem on arithmetic of e... |
| degltp1le 26352 | Theorem on arithmetic of e... |
| mdegaddle 26353 | The degree of a sum is at ... |
| mdegvscale 26354 | The degree of a scalar mul... |
| mdegvsca 26355 | The degree of a scalar mul... |
| mdegle0 26356 | A polynomial has nonpositi... |
| mdegmullem 26357 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26358 | The multivariate degree of... |
| deg1fval 26359 | Relate univariate polynomi... |
| deg1xrf 26360 | Functionality of univariat... |
| deg1xrcl 26361 | Closure of univariate poly... |
| deg1cl 26362 | Sharp closure of univariat... |
| mdegpropd 26363 | Property deduction for pol... |
| deg1fvi 26364 | Univariate polynomial degr... |
| deg1propd 26365 | Property deduction for pol... |
| deg1z 26366 | Degree of the zero univari... |
| deg1nn0cl 26367 | Degree of a nonzero univar... |
| deg1n0ima 26368 | Degree image of a set of p... |
| deg1nn0clb 26369 | A polynomial is nonzero if... |
| deg1lt0 26370 | A polynomial is zero iff i... |
| deg1ldg 26371 | A nonzero univariate polyn... |
| deg1ldgn 26372 | An index at which a polyno... |
| deg1ldgdomn 26373 | A nonzero univariate polyn... |
| deg1leb 26374 | Property of being of limit... |
| deg1val 26375 | Value of the univariate de... |
| deg1lt 26376 | If the degree of a univari... |
| deg1ge 26377 | Conversely, a nonzero coef... |
| coe1mul3 26378 | The coefficient vector of ... |
| coe1mul4 26379 | Value of the "leading" coe... |
| deg1addle 26380 | The degree of a sum is at ... |
| deg1addle2 26381 | If both factors have degre... |
| deg1add 26382 | Exact degree of a sum of t... |
| deg1vscale 26383 | The degree of a scalar tim... |
| deg1vsca 26384 | The degree of a scalar tim... |
| deg1invg 26385 | The degree of the negated ... |
| deg1suble 26386 | The degree of a difference... |
| deg1sub 26387 | Exact degree of a differen... |
| deg1mulle2 26388 | Produce a bound on the pro... |
| deg1sublt 26389 | Subtraction of two polynom... |
| deg1le0 26390 | A polynomial has nonpositi... |
| deg1sclle 26391 | A scalar polynomial has no... |
| deg1scl 26392 | A nonzero scalar polynomia... |
| deg1mul2 26393 | Degree of multiplication o... |
| deg1mul 26394 | Degree of multiplication o... |
| deg1mul3 26395 | Degree of multiplication o... |
| deg1mul3le 26396 | Degree of multiplication o... |
| deg1tmle 26397 | Limiting degree of a polyn... |
| deg1tm 26398 | Exact degree of a polynomi... |
| deg1pwle 26399 | Limiting degree of a varia... |
| deg1pw 26400 | Exact degree of a variable... |
| ply1nz 26401 | Univariate polynomials ove... |
| ply1nzb 26402 | Univariate polynomials are... |
| ply1domn 26403 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26404 | The ring of univariate pol... |
| ply1divmo 26415 | Uniqueness of a quotient i... |
| ply1divex 26416 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26417 | The division algorithm for... |
| ply1divalg2 26418 | Reverse the order of multi... |
| uc1pval 26419 | Value of the set of unitic... |
| isuc1p 26420 | Being a unitic polynomial.... |
| mon1pval 26421 | Value of the set of monic ... |
| ismon1p 26422 | Being a monic polynomial. ... |
| uc1pcl 26423 | Unitic polynomials are pol... |
| mon1pcl 26424 | Monic polynomials are poly... |
| uc1pn0 26425 | Unitic polynomials are not... |
| mon1pn0 26426 | Monic polynomials are not ... |
| uc1pdeg 26427 | Unitic polynomials have no... |
| uc1pldg 26428 | Unitic polynomials have un... |
| mon1pldg 26429 | Unitic polynomials have on... |
| mon1puc1p 26430 | Monic polynomials are unit... |
| uc1pmon1p 26431 | Make a unitic polynomial m... |
| deg1submon1p 26432 | The difference of two moni... |
| mon1pid 26433 | Monicity and degree of the... |
| q1pval 26434 | Value of the univariate po... |
| q1peqb 26435 | Characterizing property of... |
| q1pcl 26436 | Closure of the quotient by... |
| r1pval 26437 | Value of the polynomial re... |
| r1pcl 26438 | Closure of remainder follo... |
| r1pdeglt 26439 | The remainder has a degree... |
| r1pid 26440 | Express the original polyn... |
| r1pid2 26441 | Identity law for polynomia... |
| dvdsq1p 26442 | Divisibility in a polynomi... |
| dvdsr1p 26443 | Divisibility in a polynomi... |
| ply1remlem 26444 | A term of the form ` x - N... |
| ply1rem 26445 | The polynomial remainder t... |
| facth1 26446 | The factor theorem and its... |
| fta1glem1 26447 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26448 | Lemma for ~ fta1g . (Cont... |
| fta1g 26449 | The one-sided fundamental ... |
| fta1blem 26450 | Lemma for ~ fta1b . (Cont... |
| fta1b 26451 | The assumption that ` R ` ... |
| idomrootle 26452 | No element of an integral ... |
| drnguc1p 26453 | Over a division ring, all ... |
| ig1peu 26454 | There is a unique monic po... |
| ig1pval 26455 | Substitutions for the poly... |
| ig1pval2 26456 | Generator of the zero idea... |
| ig1pval3 26457 | Characterizing properties ... |
| ig1pcl 26458 | The monic generator of an ... |
| ig1pdvds 26459 | The monic generator of an ... |
| ig1prsp 26460 | Any ideal of polynomials o... |
| ply1lpir 26461 | The ring of polynomials ov... |
| ply1pid 26462 | The polynomials over a fie... |
| plyco0 26471 | Two ways to say that a fun... |
| plyval 26472 | Value of the polynomial se... |
| plybss 26473 | Reverse closure of the par... |
| elply 26474 | Definition of a polynomial... |
| elply2 26475 | The coefficient function c... |
| plyun0 26476 | The set of polynomials is ... |
| plyf 26477 | A polynomial is a function... |
| plyss 26478 | The polynomial set functio... |
| plyssc 26479 | Every polynomial ring is c... |
| elplyr 26480 | Sufficient condition for e... |
| elplyd 26481 | Sufficient condition for e... |
| ply1termlem 26482 | Lemma for ~ ply1term . (C... |
| ply1term 26483 | A one-term polynomial. (C... |
| plypow 26484 | A power is a polynomial. ... |
| plyconst 26485 | A constant function is a p... |
| ne0p 26486 | A test to show that a poly... |
| ply0 26487 | The zero function is a pol... |
| plyid 26488 | The identity function is a... |
| idpfv 26489 | Value of the identity poly... |
| plyeq0lem 26490 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26491 | If a polynomial is zero at... |
| plypf1 26492 | Write the set of complex p... |
| plyaddlem1 26493 | Derive the coefficient fun... |
| plymullem1 26494 | Derive the coefficient fun... |
| plyaddlem 26495 | Lemma for ~ plyadd . (Con... |
| plymullem 26496 | Lemma for ~ plymul . (Con... |
| plyadd 26497 | The sum of two polynomials... |
| plymul 26498 | The product of two polynom... |
| plysub 26499 | The difference of two poly... |
| plyaddcl 26500 | The sum of two polynomials... |
| plymulcl 26501 | The product of two polynom... |
| plysubcl 26502 | The difference of two poly... |
| coeval 26503 | Value of the coefficient f... |
| coeeulem 26504 | Lemma for ~ coeeu . (Cont... |
| coeeu 26505 | Uniqueness of the coeffici... |
| coelem 26506 | Lemma for properties of th... |
| coeeq 26507 | If ` A ` satisfies the pro... |
| dgrval 26508 | Value of the degree functi... |
| dgrlem 26509 | Lemma for ~ dgrcl and simi... |
| coef 26510 | The domain and codomain of... |
| coef2 26511 | The domain and codomain of... |
| coef3 26512 | The domain and codomain of... |
| dgrcl 26513 | The degree of any polynomi... |
| dgrub 26514 | If the ` M ` -th coefficie... |
| dgrub2 26515 | All the coefficients above... |
| dgrlb 26516 | If all the coefficients ab... |
| coeidlem 26517 | Lemma for ~ coeid . (Cont... |
| coeid 26518 | Reconstruct a polynomial a... |
| coeid2 26519 | Reconstruct a polynomial a... |
| coeid3 26520 | Reconstruct a polynomial a... |
| plyco 26521 | The composition of two pol... |
| coeeq2 26522 | Compute the coefficient fu... |
| dgrle 26523 | Given an explicit expressi... |
| dgreq 26524 | If the highest term in a p... |
| 0dgr 26525 | A constant function has de... |
| 0dgrb 26526 | A function has degree zero... |
| dgrnznn 26527 | A nonzero polynomial with ... |
| coefv0 26528 | The result of evaluating a... |
| coeaddlem 26529 | Lemma for ~ coeadd and ~ d... |
| coemullem 26530 | Lemma for ~ coemul and ~ d... |
| coeadd 26531 | The coefficient function o... |
| coemul 26532 | A coefficient of a product... |
| coe11 26533 | The coefficient function i... |
| coemulhi 26534 | The leading coefficient of... |
| coemulc 26535 | The coefficient function i... |
| coe0 26536 | The coefficients of the ze... |
| coesub 26537 | The coefficient function o... |
| coe1termlem 26538 | The coefficient function o... |
| coe1term 26539 | The coefficient function o... |
| dgr1term 26540 | The degree of a monomial. ... |
| plycn 26541 | A polynomial is a continuo... |
| dgr0 26542 | The degree of the zero pol... |
| coeidp 26543 | The coefficients of the id... |
| dgrid 26544 | The degree of the identity... |
| dgreq0 26545 | The leading coefficient of... |
| dgrlt 26546 | Two ways to say that the d... |
| dgradd 26547 | The degree of a sum of pol... |
| dgradd2 26548 | The degree of a sum of pol... |
| dgrmul2 26549 | The degree of a product of... |
| dgrmul 26550 | The degree of a product of... |
| dgrmulc 26551 | Scalar multiplication by a... |
| dgrsub 26552 | The degree of a difference... |
| dgrcolem1 26553 | The degree of a compositio... |
| dgrcolem2 26554 | Lemma for ~ dgrco . (Cont... |
| dgrco 26555 | The degree of a compositio... |
| plycjlem 26556 | Lemma for ~ plycj and ~ co... |
| plycj 26557 | The double conjugation of ... |
| coecj 26558 | Double conjugation of a po... |
| plycjOLD 26559 | Obsolete version of ~ plyc... |
| coecjOLD 26560 | Obsolete version of ~ coec... |
| plyrecj 26561 | A polynomial with real coe... |
| plymul0or 26562 | Polynomial multiplication ... |
| ofmulrt 26563 | The set of roots of a prod... |
| plymul02 26564 | Product of a polynomial wi... |
| plyn0mulidp 26565 | Coefficients of a non-zero... |
| plymulidp 26566 | Coefficients of a polynomi... |
| plyreres 26567 | Real-coefficient polynomia... |
| dvply1 26568 | Derivative of a polynomial... |
| dvply2g 26569 | The derivative of a polyno... |
| dvply2 26570 | The derivative of a polyno... |
| dvnply2 26571 | Polynomials have polynomia... |
| dvnply 26572 | Polynomials have polynomia... |
| plycpn 26573 | Polynomials are smooth. (... |
| quotval 26576 | Value of the quotient func... |
| plydivlem1 26577 | Lemma for ~ plydivalg . (... |
| plydivlem2 26578 | Lemma for ~ plydivalg . (... |
| plydivlem3 26579 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26580 | Lemma for ~ plydivex . In... |
| plydivex 26581 | Lemma for ~ plydivalg . (... |
| plydiveu 26582 | Lemma for ~ plydivalg . (... |
| plydivalg 26583 | The division algorithm on ... |
| quotlem 26584 | Lemma for properties of th... |
| quotcl 26585 | The quotient of two polyno... |
| quotcl2 26586 | Closure of the quotient fu... |
| quotdgr 26587 | Remainder property of the ... |
| plyremlem 26588 | Closure of a linear factor... |
| plyrem 26589 | The polynomial remainder t... |
| facth 26590 | The factor theorem. If a ... |
| fta1lem 26591 | Lemma for ~ fta1 . (Contr... |
| fta1 26592 | The easy direction of the ... |
| rnplynfin 26593 | The range of a nonconstant... |
| plyconz 26594 | The composition of a nonze... |
| quotcan 26595 | Exact division with a mult... |
| vieta1lem1 26596 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26597 | Lemma for ~ vieta1 : induc... |
| vieta1 26598 | The first-order Vieta's fo... |
| plyexmo 26599 | An infinite set of values ... |
| elaa 26602 | Elementhood in the set of ... |
| aacn 26603 | An algebraic number is a c... |
| aasscn 26604 | The algebraic numbers are ... |
| elqaalem1 26605 | Lemma for ~ elqaa . The f... |
| elqaalem2 26606 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26607 | Lemma for ~ elqaa . (Cont... |
| elqaa 26608 | The set of numbers generat... |
| preimaaa 26609 | An element of the preimage... |
| qaa 26610 | Every rational number is a... |
| qssaa 26611 | The rational numbers are c... |
| 0aa 26612 | Zero is algebraic. (Contr... |
| 1aa 26613 | One is algebraic. (Contri... |
| iaa 26614 | The imaginary unit is alge... |
| iaaOLD 26615 | Obsolete version of ~ iaa ... |
| aareccl 26616 | The reciprocal of an algeb... |
| aacjcl 26617 | The conjugate of an algebr... |
| aannenlem1 26618 | Lemma for ~ aannen . (Con... |
| aannenlem2 26619 | Lemma for ~ aannen . (Con... |
| aannenlem3 26620 | The algebraic numbers are ... |
| aannen 26621 | The algebraic numbers are ... |
| aalioulem1 26622 | Lemma for ~ aaliou . An i... |
| aalioulem2 26623 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26624 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26625 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26626 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26627 | Lemma for ~ aaliou . (Con... |
| aaliou 26628 | Liouville's theorem on dio... |
| geolim3 26629 | Geometric series convergen... |
| aaliou2 26630 | Liouville's approximation ... |
| aaliou2b 26631 | Liouville's approximation ... |
| aaliou3lem1 26632 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26633 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26634 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26635 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26636 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26637 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26638 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26639 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26640 | Example of a "Liouville nu... |
| aaliou3 26641 | Example of a "Liouville nu... |
| aaliou3r 26642 | The sum presented above is... |
| taylfvallem1 26647 | Lemma for ~ taylfval . (C... |
| taylfvallem 26648 | Lemma for ~ taylfval . (C... |
| taylfval 26649 | Define the Taylor polynomi... |
| eltayl 26650 | Value of the Taylor series... |
| taylf 26651 | The Taylor series defines ... |
| tayl0 26652 | The Taylor series is alway... |
| taylplem1 26653 | Lemma for ~ taylpfval and ... |
| taylplem2 26654 | Lemma for ~ taylpfval and ... |
| taylpfval 26655 | Define the Taylor polynomi... |
| taylpf 26656 | The Taylor polynomial is a... |
| taylpval 26657 | Value of the Taylor polyno... |
| taylply2 26658 | The Taylor polynomial is a... |
| taylply 26659 | The Taylor polynomial is a... |
| dvtaylp 26660 | The derivative of the Tayl... |
| dvntaylp 26661 | The ` M ` -th derivative o... |
| dvntaylp0 26662 | The first ` N ` derivative... |
| taylthlem1 26663 | Lemma for ~ taylth . This... |
| taylthlem2 26664 | Lemma for ~ taylth . (Con... |
| taylth 26665 | Taylor's theorem. The Tay... |
| ulmrel 26668 | The uniform limit relation... |
| ulmscl 26669 | Closure of the base set in... |
| ulmval 26670 | Express the predicate: Th... |
| ulmcl 26671 | Closure of a uniform limit... |
| ulmf 26672 | Closure of a uniform limit... |
| ulmpm 26673 | Closure of a uniform limit... |
| ulmf2 26674 | Closure of a uniform limit... |
| ulm2 26675 | Simplify ~ ulmval when ` F... |
| ulmi 26676 | The uniform limit property... |
| ulmclm 26677 | A uniform limit of functio... |
| ulmres 26678 | A sequence of functions co... |
| ulmshftlem 26679 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26680 | A sequence of functions co... |
| ulm0 26681 | Every function converges u... |
| ulmuni 26682 | A sequence of functions un... |
| ulmdm 26683 | Two ways to express that a... |
| ulmcaulem 26684 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26685 | A sequence of functions co... |
| ulmcau2 26686 | A sequence of functions co... |
| ulmss 26687 | A uniform limit of functio... |
| ulmbdd 26688 | A uniform limit of bounded... |
| ulmcn 26689 | A uniform limit of continu... |
| ulmdvlem1 26690 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26691 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26692 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26693 | If ` F ` is a sequence of ... |
| mtest 26694 | The Weierstrass M-test. I... |
| mtestbdd 26695 | Given the hypotheses of th... |
| mbfulm 26696 | A uniform limit of measura... |
| iblulm 26697 | A uniform limit of integra... |
| itgulm 26698 | A uniform limit of integra... |
| itgulm2 26699 | A uniform limit of integra... |
| pserval 26700 | Value of the function ` G ... |
| pserval2 26701 | Value of the function ` G ... |
| psergf 26702 | The sequence of terms in t... |
| radcnvlem1 26703 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26704 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26705 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26706 | Zero is always a convergen... |
| radcnvcl 26707 | The radius of convergence ... |
| radcnvlt1 26708 | If ` X ` is within the ope... |
| radcnvlt2 26709 | If ` X ` is within the ope... |
| radcnvle 26710 | If ` X ` is a convergent p... |
| dvradcnv 26711 | The radius of convergence ... |
| pserulm 26712 | If ` S ` is a region conta... |
| psercn2 26713 | Since by ~ pserulm the ser... |
| psercnlem2 26714 | Lemma for ~ psercn . (Con... |
| psercnlem1 26715 | Lemma for ~ psercn . (Con... |
| psercn 26716 | An infinite series converg... |
| pserdvlem1 26717 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26718 | Lemma for ~ pserdv . (Con... |
| pserdv 26719 | The derivative of a power ... |
| pserdv2 26720 | The derivative of a power ... |
| abelthlem1 26721 | Lemma for ~ abelth . (Con... |
| abelthlem2 26722 | Lemma for ~ abelth . The ... |
| abelthlem3 26723 | Lemma for ~ abelth . (Con... |
| abelthlem4 26724 | Lemma for ~ abelth . (Con... |
| abelthlem5 26725 | Lemma for ~ abelth . (Con... |
| abelthlem6 26726 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26727 | Lemma for ~ abelth . (Con... |
| abelthlem7 26728 | Lemma for ~ abelth . (Con... |
| abelthlem8 26729 | Lemma for ~ abelth . (Con... |
| abelthlem9 26730 | Lemma for ~ abelth . By a... |
| abelth 26731 | Abel's theorem. If the po... |
| abelth2 26732 | Abel's theorem, restricted... |
| efcn 26733 | The exponential function i... |
| sincn 26734 | Sine is continuous. (Cont... |
| coscn 26735 | Cosine is continuous. (Co... |
| reeff1olem 26736 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26737 | The real exponential funct... |
| reefiso 26738 | The exponential function o... |
| efcvx 26739 | The exponential function o... |
| reefgim 26740 | The exponential function i... |
| pilem1 26741 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26742 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26743 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26744 | ` _pi ` is between 2 and 4... |
| sinpi 26745 | The sine of ` _pi ` is 0. ... |
| pire 26746 | ` _pi ` is a real number. ... |
| 2pire 26747 | ` ( 2 x. _pi ) ` is a real... |
| picn 26748 | ` _pi ` is a complex numbe... |
| 2picn 26749 | ` ( 2 x. _pi ) ` is a comp... |
| pipos 26750 | ` _pi ` is positive. (Con... |
| pige0 26751 | ` _pi ` is nonnegative. (... |
| pine0 26752 | ` _pi ` is nonzero. (Cont... |
| pirp 26753 | ` _pi ` is a positive real... |
| negpicn 26754 | ` -u _pi ` is a complex nu... |
| sinhalfpilem 26755 | Lemma for ~ sinhalfpi and ... |
| halfpire 26756 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26757 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26758 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26759 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26760 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26761 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26762 | The cosine of ` -u _pi / 2... |
| efhalfpi 26763 | The exponential of ` _i _p... |
| cospi 26764 | The cosine of ` _pi ` is `... |
| efipi 26765 | The exponential of ` _i x.... |
| eulerid 26766 | Euler's identity. (Contri... |
| sin2pi 26767 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26768 | The cosine of ` 2 _pi ` is... |
| ef2pi 26769 | The exponential of ` 2 _pi... |
| ef2kpi 26770 | If ` K ` is an integer, th... |
| efper 26771 | The exponential function i... |
| sinperlem 26772 | Lemma for ~ sinper and ~ c... |
| sinper 26773 | The sine function is perio... |
| cosper 26774 | The cosine function is per... |
| sin2kpi 26775 | If ` K ` is an integer, th... |
| cos2kpi 26776 | If ` K ` is an integer, th... |
| sin2pim 26777 | Sine of a number subtracte... |
| cos2pim 26778 | Cosine of a number subtrac... |
| sinmpi 26779 | Sine of a number less ` _p... |
| cosmpi 26780 | Cosine of a number less ` ... |
| sinppi 26781 | Sine of a number plus ` _p... |
| cosppi 26782 | Cosine of a number plus ` ... |
| efimpi 26783 | The exponential function a... |
| sinhalfpip 26784 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26785 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26786 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26787 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26788 | Ptolemy's Theorem. This t... |
| sincosq1lem 26789 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26790 | The signs of the sine and ... |
| sincosq2sgn 26791 | The signs of the sine and ... |
| sincosq3sgn 26792 | The signs of the sine and ... |
| sincosq4sgn 26793 | The signs of the sine and ... |
| coseq00topi 26794 | Location of the zeroes of ... |
| coseq0negpitopi 26795 | Location of the zeroes of ... |
| tanrpcl 26796 | Positive real closure of t... |
| tangtx 26797 | The tangent function is gr... |
| tanabsge 26798 | The tangent function is gr... |
| sinq12gt0 26799 | The sine of a number stric... |
| sinq12ge0 26800 | The sine of a number betwe... |
| sinq34lt0t 26801 | The sine of a number stric... |
| cosq14gt0 26802 | The cosine of a number str... |
| cosq14ge0 26803 | The cosine of a number bet... |
| sincosq1eq 26804 | Complementarity of the sin... |
| sincos4thpi 26805 | The sine and cosine of ` _... |
| tan4thpi 26806 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26807 | The sine and cosine of ` _... |
| sincos3rdpi 26808 | The sine and cosine of ` _... |
| pigt3 26809 | ` _pi ` is greater than 3.... |
| pige3 26810 | ` _pi ` is greater than or... |
| pige3ALT 26811 | Alternate proof of ~ pige3... |
| abssinper 26812 | The absolute value of sine... |
| sinkpi 26813 | The sine of an integer mul... |
| coskpi 26814 | The absolute value of the ... |
| sineq0 26815 | A complex number whose sin... |
| coseq1 26816 | A complex number whose cos... |
| cos02pilt1 26817 | Cosine is less than one be... |
| cosq34lt1 26818 | Cosine is less than one in... |
| efeq1 26819 | A complex number whose exp... |
| cosne0 26820 | The cosine function has no... |
| cosordlem 26821 | Lemma for ~ cosord . (Con... |
| cosord 26822 | Cosine is decreasing over ... |
| cos0pilt1 26823 | Cosine is between minus on... |
| cos11 26824 | Cosine is one-to-one over ... |
| sinord 26825 | Sine is increasing over th... |
| recosf1o 26826 | The cosine function is a b... |
| resinf1o 26827 | The sine function is a bij... |
| tanord1 26828 | The tangent function is st... |
| tanord 26829 | The tangent function is st... |
| tanregt0 26830 | The real part of the tange... |
| negpitopissre 26831 | The interval ` ( -u _pi (,... |
| efgh 26832 | The exponential function o... |
| efif1olem1 26833 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26834 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26835 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26836 | The exponential function o... |
| efif1o 26837 | The exponential function o... |
| efifo 26838 | The exponential function o... |
| eff1olem 26839 | The exponential function m... |
| eff1o 26840 | The exponential function m... |
| efabl 26841 | The image of a subgroup of... |
| efsubm 26842 | The image of a subgroup of... |
| circgrp 26843 | The circle group ` T ` is ... |
| circsubm 26844 | The circle group ` T ` is ... |
| logrn 26849 | The range of the natural l... |
| ellogrn 26850 | Write out the property ` A... |
| dflog2 26851 | The natural logarithm func... |
| relogrn 26852 | The range of the natural l... |
| logrncn 26853 | The range of the natural l... |
| eff1o2 26854 | The exponential function r... |
| logf1o 26855 | The natural logarithm func... |
| dfrelog 26856 | The natural logarithm func... |
| relogf1o 26857 | The natural logarithm func... |
| logrncl 26858 | Closure of the natural log... |
| logcl 26859 | Closure of the natural log... |
| logimcl 26860 | Closure of the imaginary p... |
| logcld 26861 | The logarithm of a nonzero... |
| logimcld 26862 | The imaginary part of the ... |
| logimclad 26863 | The imaginary part of the ... |
| abslogimle 26864 | The imaginary part of the ... |
| logrnaddcl 26865 | The range of the natural l... |
| relogcl 26866 | Closure of the natural log... |
| eflog 26867 | Relationship between the n... |
| logeq0im1 26868 | If the logarithm of a numb... |
| logccne0 26869 | The logarithm isn't 0 if i... |
| logne0 26870 | Logarithm of a non-1 posit... |
| reeflog 26871 | Relationship between the n... |
| logef 26872 | Relationship between the n... |
| relogef 26873 | Relationship between the n... |
| logeftb 26874 | Relationship between the n... |
| relogeftb 26875 | Relationship between the n... |
| log1 26876 | The natural logarithm of `... |
| loge 26877 | The natural logarithm of `... |
| logi 26878 | The natural logarithm of `... |
| logneg 26879 | The natural logarithm of a... |
| logm1 26880 | The natural logarithm of n... |
| lognegb 26881 | If a number has imaginary ... |
| relogoprlem 26882 | Lemma for ~ relogmul and ~... |
| relogmul 26883 | The natural logarithm of t... |
| relogdiv 26884 | The natural logarithm of t... |
| explog 26885 | Exponentiation of a nonzer... |
| reexplog 26886 | Exponentiation of a positi... |
| relogexp 26887 | The natural logarithm of p... |
| relog 26888 | Real part of a logarithm. ... |
| relogiso 26889 | The natural logarithm func... |
| reloggim 26890 | The natural logarithm is a... |
| logltb 26891 | The natural logarithm func... |
| logfac 26892 | The logarithm of a factori... |
| eflogeq 26893 | Solve an equation involvin... |
| logleb 26894 | Natural logarithm preserve... |
| rplogcl 26895 | Closure of the logarithm f... |
| logge0 26896 | The logarithm of a number ... |
| logcj 26897 | The natural logarithm dist... |
| efiarg 26898 | The exponential of the "ar... |
| cosargd 26899 | The cosine of the argument... |
| cosarg0d 26900 | The cosine of the argument... |
| argregt0 26901 | Closure of the argument of... |
| argrege0 26902 | Closure of the argument of... |
| argimgt0 26903 | Closure of the argument of... |
| argimlt0 26904 | Closure of the argument of... |
| logimul 26905 | Multiplying a number by ` ... |
| logneg2 26906 | The logarithm of the negat... |
| logmul2 26907 | Generalization of ~ relogm... |
| logdiv2 26908 | Generalization of ~ relogd... |
| abslogle 26909 | Bound on the magnitude of ... |
| tanarg 26910 | The basic relation between... |
| logdivlti 26911 | The ` log x / x ` function... |
| logdivlt 26912 | The ` log x / x ` function... |
| logdivle 26913 | The ` log x / x ` function... |
| relogcld 26914 | Closure of the natural log... |
| reeflogd 26915 | Relationship between the n... |
| relogmuld 26916 | The natural logarithm of t... |
| relogdivd 26917 | The natural logarithm of t... |
| logled 26918 | Natural logarithm preserve... |
| relogefd 26919 | Relationship between the n... |
| rplogcld 26920 | Closure of the logarithm f... |
| logge0d 26921 | The logarithm of a number ... |
| logge0b 26922 | The logarithm of a number ... |
| loggt0b 26923 | The logarithm of a number ... |
| logle1b 26924 | The logarithm of a number ... |
| loglt1b 26925 | The logarithm of a number ... |
| divlogrlim 26926 | The inverse logarithm func... |
| logno1 26927 | The logarithm function is ... |
| dvrelog 26928 | The derivative of the real... |
| relogcn 26929 | The real logarithm functio... |
| ellogdm 26930 | Elementhood in the "contin... |
| logdmn0 26931 | A number in the continuous... |
| logdmnrp 26932 | A number in the continuous... |
| logdmss 26933 | The continuity domain of `... |
| logcnlem2 26934 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26935 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26936 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26937 | Lemma for ~ logcn . (Cont... |
| logcn 26938 | The logarithm function is ... |
| dvloglem 26939 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26940 | The "continuous domain" of... |
| logf1o2 26941 | The logarithm maps its con... |
| dvlog 26942 | The derivative of the comp... |
| dvlog2lem 26943 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26944 | The derivative of the comp... |
| advlog 26945 | The antiderivative of the ... |
| advlogexp 26946 | The antiderivative of a po... |
| efopnlem1 26947 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26948 | Lemma for ~ efopn . (Cont... |
| efopn 26949 | The exponential map is an ... |
| logtayllem 26950 | Lemma for ~ logtayl . (Co... |
| logtayl 26951 | The Taylor series for ` -u... |
| logtaylsum 26952 | The Taylor series for ` -u... |
| logtayl2 26953 | Power series expression fo... |
| logccv 26954 | The natural logarithm func... |
| cxpval 26955 | Value of the complex power... |
| cxpef 26956 | Value of the complex power... |
| 0cxp 26957 | Value of the complex power... |
| cxpexpz 26958 | Relate the complex power f... |
| cxpexp 26959 | Relate the complex power f... |
| logcxp 26960 | Logarithm of a complex pow... |
| cxp0 26961 | Value of the complex power... |
| cxp1 26962 | Value of the complex power... |
| 1cxp 26963 | Value of the complex power... |
| ecxp 26964 | Write the exponential func... |
| cxpcl 26965 | Closure of the complex pow... |
| recxpcl 26966 | Real closure of the comple... |
| rpcxpcl 26967 | Positive real closure of t... |
| cxpne0 26968 | Complex exponentiation is ... |
| cxpeq0 26969 | Complex exponentiation is ... |
| cxpadd 26970 | Sum of exponents law for c... |
| cxpp1 26971 | Value of a nonzero complex... |
| cxpneg 26972 | Value of a complex number ... |
| cxpsub 26973 | Exponent subtraction law f... |
| cxpge0 26974 | Nonnegative exponentiation... |
| mulcxplem 26975 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26976 | Complex exponentiation of ... |
| cxprec 26977 | Complex exponentiation of ... |
| divcxp 26978 | Complex exponentiation of ... |
| cxpmul 26979 | Product of exponents law f... |
| cxpmul2 26980 | Product of exponents law f... |
| cxproot 26981 | The complex power function... |
| cxpmul2z 26982 | Generalize ~ cxpmul2 to ne... |
| abscxp 26983 | Absolute value of a power,... |
| abscxp2 26984 | Absolute value of a power,... |
| cxplt 26985 | Ordering property for comp... |
| cxple 26986 | Ordering property for comp... |
| cxplea 26987 | Ordering property for comp... |
| cxple2 26988 | Ordering property for comp... |
| cxplt2 26989 | Ordering property for comp... |
| cxple2a 26990 | Ordering property for comp... |
| cxplt3 26991 | Ordering property for comp... |
| cxple3 26992 | Ordering property for comp... |
| cxpsqrtlem 26993 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26994 | The complex exponential fu... |
| logsqrt 26995 | Logarithm of a square root... |
| cxp0d 26996 | Value of the complex power... |
| cxp1d 26997 | Value of the complex power... |
| 1cxpd 26998 | Value of the complex power... |
| cxpcld 26999 | Closure of the complex pow... |
| cxpmul2d 27000 | Product of exponents law f... |
| 0cxpd 27001 | Value of the complex power... |
| cxpexpzd 27002 | Relate the complex power f... |
| cxpefd 27003 | Value of the complex power... |
| cxpne0d 27004 | Complex exponentiation is ... |
| cxpp1d 27005 | Value of a nonzero complex... |
| cxpnegd 27006 | Value of a complex number ... |
| cxpmul2zd 27007 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 27008 | Sum of exponents law for c... |
| cxpsubd 27009 | Exponent subtraction law f... |
| cxpltd 27010 | Ordering property for comp... |
| cxpled 27011 | Ordering property for comp... |
| cxplead 27012 | Ordering property for comp... |
| divcxpd 27013 | Complex exponentiation of ... |
| recxpcld 27014 | Positive real closure of t... |
| cxpge0d 27015 | Nonnegative exponentiation... |
| cxple2ad 27016 | Ordering property for comp... |
| cxplt2d 27017 | Ordering property for comp... |
| cxple2d 27018 | Ordering property for comp... |
| mulcxpd 27019 | Complex exponentiation of ... |
| recxpf1lem 27020 | Complex exponentiation on ... |
| cxpsqrtth 27021 | Square root theorem over t... |
| 2irrexpq 27022 | There exist irrational num... |
| cxprecd 27023 | Complex exponentiation of ... |
| rpcxpcld 27024 | Positive real closure of t... |
| logcxpd 27025 | Logarithm of a complex pow... |
| cxplt3d 27026 | Ordering property for comp... |
| cxple3d 27027 | Ordering property for comp... |
| cxpmuld 27028 | Product of exponents law f... |
| cxpgt0d 27029 | A positive real raised to ... |
| cxpcom 27030 | Commutative law for real e... |
| dvcxp1 27031 | The derivative of a comple... |
| dvcxp2 27032 | The derivative of a comple... |
| dvsqrt 27033 | The derivative of the real... |
| dvcncxp1 27034 | Derivative of complex powe... |
| dvcnsqrt 27035 | Derivative of square root ... |
| cxpcn 27036 | Domain of continuity of th... |
| cxpcn2 27037 | Continuity of the complex ... |
| cxpcn3lem 27038 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 27039 | Extend continuity of the c... |
| resqrtcn 27040 | Continuity of the real squ... |
| sqrtcn 27041 | Continuity of the square r... |
| cxpaddlelem 27042 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 27043 | Ordering property for comp... |
| abscxpbnd 27044 | Bound on the absolute valu... |
| root1id 27045 | Property of an ` N ` -th r... |
| root1eq1 27046 | The only powers of an ` N ... |
| root1cj 27047 | Within the ` N ` -th roots... |
| cxpeq 27048 | Solve an equation involvin... |
| zrtelqelz 27049 | If the ` N ` -th root of a... |
| zrtdvds 27050 | A positive integer root di... |
| rtprmirr 27051 | The root of a prime number... |
| loglesqrt 27052 | An upper bound on the loga... |
| logreclem 27053 | Symmetry of the natural lo... |
| logrec 27054 | Logarithm of a reciprocal ... |
| logbval 27057 | Define the value of the ` ... |
| logbcl 27058 | General logarithm closure.... |
| logbid1 27059 | General logarithm is 1 whe... |
| logb1 27060 | The logarithm of ` 1 ` to ... |
| elogb 27061 | The general logarithm of a... |
| logbchbase 27062 | Change of base for logarit... |
| relogbval 27063 | Value of the general logar... |
| relogbcl 27064 | Closure of the general log... |
| relogbzcl 27065 | Closure of the general log... |
| relogbreexp 27066 | Power law for the general ... |
| relogbzexp 27067 | Power law for the general ... |
| relogbmul 27068 | The logarithm of the produ... |
| relogbmulexp 27069 | The logarithm of the produ... |
| relogbdiv 27070 | The logarithm of the quoti... |
| relogbexp 27071 | Identity law for general l... |
| nnlogbexp 27072 | Identity law for general l... |
| logbrec 27073 | Logarithm of a reciprocal ... |
| logbleb 27074 | The general logarithm func... |
| logblt 27075 | The general logarithm func... |
| relogbcxp 27076 | Identity law for the gener... |
| cxplogb 27077 | Identity law for the gener... |
| relogbcxpb 27078 | The logarithm is the inver... |
| logbmpt 27079 | The general logarithm to a... |
| logbf 27080 | The general logarithm to a... |
| logbfval 27081 | The general logarithm of a... |
| relogbf 27082 | The general logarithm to a... |
| logblog 27083 | The general logarithm to t... |
| logbgt0b 27084 | The logarithm of a positiv... |
| logbgcd1irr 27085 | The logarithm of an intege... |
| 2logb9irr 27086 | Example for ~ logbgcd1irr ... |
| logbprmirr 27087 | The logarithm of a prime t... |
| 2logb3irr 27088 | Example for ~ logbprmirr .... |
| 2logb9irrALT 27089 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 27090 | The square root of two to ... |
| 2irrexpqALT 27091 | Alternate proof of ~ 2irre... |
| angval 27092 | Define the angle function,... |
| angcan 27093 | Cancel a constant multipli... |
| angneg 27094 | Cancel a negative sign in ... |
| angvald 27095 | The (signed) angle between... |
| angcld 27096 | The (signed) angle between... |
| angrteqvd 27097 | Two vectors are at a right... |
| cosangneg2d 27098 | The cosine of the angle be... |
| angrtmuld 27099 | Perpendicularity of two ve... |
| ang180lem1 27100 | Lemma for ~ ang180 . Show... |
| ang180lem2 27101 | Lemma for ~ ang180 . Show... |
| ang180lem3 27102 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 27103 | Lemma for ~ ang180 . Redu... |
| ang180lem5 27104 | Lemma for ~ ang180 : Redu... |
| ang180 27105 | The sum of angles ` m A B ... |
| lawcoslem1 27106 | Lemma for ~ lawcos . Here... |
| lawcos 27107 | Law of cosines (also known... |
| pythag 27108 | Pythagorean theorem. Give... |
| isosctrlem1 27109 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 27110 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 27111 | Lemma for ~ isosctr . Cor... |
| isosctr 27112 | Isosceles triangle theorem... |
| ssscongptld 27113 | If two triangles have equa... |
| affineequiv 27114 | Equivalence between two wa... |
| affineequiv2 27115 | Equivalence between two wa... |
| affineequiv3 27116 | Equivalence between two wa... |
| affineequiv4 27117 | Equivalence between two wa... |
| affineequivne 27118 | Equivalence between two wa... |
| angpieqvdlem 27119 | Equivalence used in the pr... |
| angpieqvdlem2 27120 | Equivalence used in ~ angp... |
| angpined 27121 | If the angle at ABC is ` _... |
| angpieqvd 27122 | The angle ABC is ` _pi ` i... |
| chordthmlem 27123 | If ` M ` is the midpoint o... |
| chordthmlem2 27124 | If M is the midpoint of AB... |
| chordthmlem3 27125 | If M is the midpoint of AB... |
| chordthmlem4 27126 | If P is on the segment AB ... |
| chordthmlem5 27127 | If P is on the segment AB ... |
| chordthm 27128 | The intersecting chords th... |
| heron 27129 | Heron's formula gives the ... |
| quad2 27130 | The quadratic equation, wi... |
| quad 27131 | The quadratic equation. (... |
| 1cubrlem 27132 | The cube roots of unity. ... |
| 1cubr 27133 | The cube roots of unity. ... |
| dcubic1lem 27134 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 27135 | Reverse direction of ~ dcu... |
| dcubic1 27136 | Forward direction of ~ dcu... |
| dcubic 27137 | Solutions to the depressed... |
| mcubic 27138 | Solutions to a monic cubic... |
| cubic2 27139 | The solution to the genera... |
| cubic 27140 | The cubic equation, which ... |
| binom4 27141 | Work out a quartic binomia... |
| dquartlem1 27142 | Lemma for ~ dquart . (Con... |
| dquartlem2 27143 | Lemma for ~ dquart . (Con... |
| dquart 27144 | Solve a depressed quartic ... |
| quart1cl 27145 | Closure lemmas for ~ quart... |
| quart1lem 27146 | Lemma for ~ quart1 . (Con... |
| quart1 27147 | Depress a quartic equation... |
| quartlem1 27148 | Lemma for ~ quart . (Cont... |
| quartlem2 27149 | Closure lemmas for ~ quart... |
| quartlem3 27150 | Closure lemmas for ~ quart... |
| quartlem4 27151 | Closure lemmas for ~ quart... |
| quart 27152 | The quartic equation, writ... |
| asinlem 27159 | The argument to the logari... |
| asinlem2 27160 | The argument to the logari... |
| asinlem3a 27161 | Lemma for ~ asinlem3 . (C... |
| asinlem3 27162 | The argument to the logari... |
| asinf 27163 | Domain and codomain of the... |
| asincl 27164 | Closure for the arcsin fun... |
| acosf 27165 | Domain and codoamin of the... |
| acoscl 27166 | Closure for the arccos fun... |
| atandm 27167 | Since the property is a li... |
| atandm2 27168 | This form of ~ atandm is a... |
| atandm3 27169 | A compact form of ~ atandm... |
| atandm4 27170 | A compact form of ~ atandm... |
| atanf 27171 | Domain and codoamin of the... |
| atancl 27172 | Closure for the arctan fun... |
| asinval 27173 | Value of the arcsin functi... |
| acosval 27174 | Value of the arccos functi... |
| atanval 27175 | Value of the arctan functi... |
| atanre 27176 | A real number is in the do... |
| asinneg 27177 | The arcsine function is od... |
| acosneg 27178 | The negative symmetry rela... |
| efiasin 27179 | The exponential of the arc... |
| sinasin 27180 | The arcsine function is an... |
| cosacos 27181 | The arccosine function is ... |
| asinsinlem 27182 | Lemma for ~ asinsin . (Co... |
| asinsin 27183 | The arcsine function compo... |
| acoscos 27184 | The arccosine function is ... |
| asin1 27185 | The arcsine of ` 1 ` is ` ... |
| acos1 27186 | The arccosine of ` 1 ` is ... |
| reasinsin 27187 | The arcsine function compo... |
| asinsinb 27188 | Relationship between sine ... |
| acoscosb 27189 | Relationship between cosin... |
| asinbnd 27190 | The arcsine function has r... |
| acosbnd 27191 | The arccosine function has... |
| asinrebnd 27192 | Bounds on the arcsine func... |
| asinrecl 27193 | The arcsine function is re... |
| acosrecl 27194 | The arccosine function is ... |
| cosasin 27195 | The cosine of the arcsine ... |
| sinacos 27196 | The sine of the arccosine ... |
| atandmneg 27197 | The domain of the arctange... |
| atanneg 27198 | The arctangent function is... |
| atan0 27199 | The arctangent of zero is ... |
| atandmcj 27200 | The arctangent function di... |
| atancj 27201 | The arctangent function di... |
| atanrecl 27202 | The arctangent function is... |
| efiatan 27203 | Value of the exponential o... |
| atanlogaddlem 27204 | Lemma for ~ atanlogadd . ... |
| atanlogadd 27205 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 27206 | Lemma for ~ atanlogsub . ... |
| atanlogsub 27207 | A variation on ~ atanlogad... |
| efiatan2 27208 | Value of the exponential o... |
| 2efiatan 27209 | Value of the exponential o... |
| tanatan 27210 | The arctangent function is... |
| atandmtan 27211 | The tangent function has r... |
| cosatan 27212 | The cosine of an arctangen... |
| cosatanne0 27213 | The arctangent function ha... |
| atantan 27214 | The arctangent function is... |
| atantanb 27215 | Relationship between tange... |
| atanbndlem 27216 | Lemma for ~ atanbnd . (Co... |
| atanbnd 27217 | The arctangent function is... |
| atanord 27218 | The arctangent function is... |
| atan1 27219 | The arctangent of ` 1 ` is... |
| bndatandm 27220 | A point in the open unit d... |
| atans 27221 | The "domain of continuity"... |
| atans2 27222 | It suffices to show that `... |
| atansopn 27223 | The domain of continuity o... |
| atansssdm 27224 | The domain of continuity o... |
| ressatans 27225 | The real number line is a ... |
| dvatan 27226 | The derivative of the arct... |
| atancn 27227 | The arctangent is a contin... |
| atantayl 27228 | The Taylor series for ` ar... |
| atantayl2 27229 | The Taylor series for ` ar... |
| atantayl3 27230 | The Taylor series for ` ar... |
| leibpilem1 27231 | Lemma for ~ leibpi . (Con... |
| leibpilem2 27232 | The Leibniz formula for ` ... |
| leibpi 27233 | The Leibniz formula for ` ... |
| leibpisum 27234 | The Leibniz formula for ` ... |
| log2cnv 27235 | Using the Taylor series fo... |
| log2tlbnd 27236 | Bound the error term in th... |
| log2ublem1 27237 | Lemma for ~ log2ub . The ... |
| log2ublem2 27238 | Lemma for ~ log2ub . (Con... |
| log2ublem3 27239 | Lemma for ~ log2ub . In d... |
| log2ub 27240 | ` log 2 ` is less than ` 2... |
| log2le1 27241 | ` log 2 ` is less than ` 1... |
| birthdaylem1 27242 | Lemma for ~ birthday . (C... |
| birthdaylem2 27243 | For general ` N ` and ` K ... |
| birthdaylem3 27244 | For general ` N ` and ` K ... |
| birthday 27245 | The Birthday Problem. The... |
| dmarea 27248 | The domain of the area fun... |
| areambl 27249 | The fibers of a measurable... |
| areass 27250 | A measurable region is a s... |
| dfarea 27251 | Rewrite ~ df-area self-ref... |
| areaf 27252 | Area measurement is a func... |
| areacl 27253 | The area of a measurable r... |
| areage0 27254 | The area of a measurable r... |
| areaval 27255 | The area of a measurable r... |
| rlimcnp 27256 | Relate a limit of a real-v... |
| rlimcnp2 27257 | Relate a limit of a real-v... |
| rlimcnp3 27258 | Relate a limit of a real-v... |
| xrlimcnp 27259 | Relate a limit of a real-v... |
| efrlim 27260 | The limit of the sequence ... |
| dfef2 27261 | The limit of the sequence ... |
| cxplim 27262 | A power to a negative expo... |
| sqrtlim 27263 | The inverse square root fu... |
| rlimcxp 27264 | Any power to a positive ex... |
| o1cxp 27265 | An eventually bounded func... |
| cxp2limlem 27266 | A linear factor grows slow... |
| cxp2lim 27267 | Any power grows slower tha... |
| cxploglim 27268 | The logarithm grows slower... |
| cxploglim2 27269 | Every power of the logarit... |
| divsqrtsumlem 27270 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 27271 | The function ` F ` used in... |
| divsqrsum 27272 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 27273 | A bound on the distance of... |
| divsqrtsumo1 27274 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 27275 | Closure of a 0-1 linear co... |
| scvxcvx 27276 | A strictly convex function... |
| jensenlem1 27277 | Lemma for ~ jensen . (Con... |
| jensenlem2 27278 | Lemma for ~ jensen . (Con... |
| jensen 27279 | Jensen's inequality, a fin... |
| amgmlem 27280 | Lemma for ~ amgm . (Contr... |
| amgm 27281 | Inequality of arithmetic a... |
| logdifbnd 27284 | Bound on the difference of... |
| logdiflbnd 27285 | Lower bound on the differe... |
| emcllem1 27286 | Lemma for ~ emcl . The se... |
| emcllem2 27287 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 27288 | Lemma for ~ emcl . The fu... |
| emcllem4 27289 | Lemma for ~ emcl . The di... |
| emcllem5 27290 | Lemma for ~ emcl . The pa... |
| emcllem6 27291 | Lemma for ~ emcl . By the... |
| emcllem7 27292 | Lemma for ~ emcl and ~ har... |
| emcl 27293 | Closure and bounds for the... |
| harmonicbnd 27294 | A bound on the harmonic se... |
| harmonicbnd2 27295 | A bound on the harmonic se... |
| emre 27296 | The Euler-Mascheroni const... |
| emgt0 27297 | The Euler-Mascheroni const... |
| harmonicbnd3 27298 | A bound on the harmonic se... |
| harmoniclbnd 27299 | A bound on the harmonic se... |
| harmonicubnd 27300 | A bound on the harmonic se... |
| harmonicbnd4 27301 | The asymptotic behavior of... |
| fsumharmonic 27302 | Bound a finite sum based o... |
| zetacvg 27305 | The zeta series is converg... |
| eldmgm 27312 | Elementhood in the set of ... |
| dmgmaddn0 27313 | If ` A ` is not a nonposit... |
| dmlogdmgm 27314 | If ` A ` is in the continu... |
| rpdmgm 27315 | A positive real number is ... |
| dmgmn0 27316 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27317 | If ` A ` is not a nonposit... |
| dmgmdivn0 27318 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27319 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27320 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27321 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27322 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27323 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27324 | The series ` G ` is unifor... |
| lgamgulm 27325 | The series ` G ` is unifor... |
| lgamgulm2 27326 | Rewrite the limit of the s... |
| lgambdd 27327 | The log-Gamma function is ... |
| lgamucov 27328 | The ` U ` regions used in ... |
| lgamucov2 27329 | The ` U ` regions used in ... |
| lgamcvglem 27330 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27331 | The log-Gamma function is ... |
| lgamf 27332 | The log-Gamma function is ... |
| gamf 27333 | The Gamma function is a co... |
| gamcl 27334 | The exponential of the log... |
| eflgam 27335 | The exponential of the log... |
| gamne0 27336 | The Gamma function is neve... |
| igamval 27337 | Value of the inverse Gamma... |
| igamz 27338 | Value of the inverse Gamma... |
| igamgam 27339 | Value of the inverse Gamma... |
| igamlgam 27340 | Value of the inverse Gamma... |
| igamf 27341 | Closure of the inverse Gam... |
| igamcl 27342 | Closure of the inverse Gam... |
| gamigam 27343 | The Gamma function is the ... |
| lgamcvg 27344 | The series ` G ` converges... |
| lgamcvg2 27345 | The series ` G ` converges... |
| gamcvg 27346 | The pointwise exponential ... |
| lgamp1 27347 | The functional equation of... |
| gamp1 27348 | The functional equation of... |
| gamcvg2lem 27349 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27350 | An infinite product expres... |
| regamcl 27351 | The Gamma function is real... |
| relgamcl 27352 | The log-Gamma function is ... |
| rpgamcl 27353 | The log-Gamma function is ... |
| lgam1 27354 | The log-Gamma function at ... |
| gam1 27355 | The log-Gamma function at ... |
| facgam 27356 | The Gamma function general... |
| gamfac 27357 | The Gamma function general... |
| wilthlem1 27358 | The only elements that are... |
| wilthlem2 27359 | Lemma for ~ wilth : induct... |
| wilthlem3 27360 | Lemma for ~ wilth . Here ... |
| wilth 27361 | Wilson's theorem. A numbe... |
| wilthimp 27362 | The forward implication of... |
| ftalem1 27363 | Lemma for ~ fta : "growth... |
| ftalem2 27364 | Lemma for ~ fta . There e... |
| ftalem3 27365 | Lemma for ~ fta . There e... |
| ftalem4 27366 | Lemma for ~ fta : Closure... |
| ftalem5 27367 | Lemma for ~ fta : Main pr... |
| ftalem6 27368 | Lemma for ~ fta : Dischar... |
| ftalem7 27369 | Lemma for ~ fta . Shift t... |
| fta 27370 | The Fundamental Theorem of... |
| basellem1 27371 | Lemma for ~ basel . Closu... |
| basellem2 27372 | Lemma for ~ basel . Show ... |
| basellem3 27373 | Lemma for ~ basel . Using... |
| basellem4 27374 | Lemma for ~ basel . By ~ ... |
| basellem5 27375 | Lemma for ~ basel . Using... |
| basellem6 27376 | Lemma for ~ basel . The f... |
| basellem7 27377 | Lemma for ~ basel . The f... |
| basellem8 27378 | Lemma for ~ basel . The f... |
| basellem9 27379 | Lemma for ~ basel . Since... |
| basel 27380 | The sum of the inverse squ... |
| efnnfsumcl 27393 | Finite sum closure in the ... |
| ppisval 27394 | The set of primes less tha... |
| ppisval2 27395 | The set of primes less tha... |
| ppifi 27396 | The set of primes less tha... |
| prmdvdsfi 27397 | The set of prime divisors ... |
| chtf 27398 | Domain and codoamin of the... |
| chtcl 27399 | Real closure of the Chebys... |
| chtval 27400 | Value of the Chebyshev fun... |
| efchtcl 27401 | The Chebyshev function is ... |
| chtge0 27402 | The Chebyshev function is ... |
| vmaval 27403 | Value of the von Mangoldt ... |
| isppw 27404 | Two ways to say that ` A `... |
| isppw2 27405 | Two ways to say that ` A `... |
| vmappw 27406 | Value of the von Mangoldt ... |
| vmaprm 27407 | Value of the von Mangoldt ... |
| vmacl 27408 | Closure for the von Mangol... |
| vmaf 27409 | Functionality of the von M... |
| efvmacl 27410 | The von Mangoldt is closed... |
| vmage0 27411 | The von Mangoldt function ... |
| chpval 27412 | Value of the second Chebys... |
| chpf 27413 | Functionality of the secon... |
| chpcl 27414 | Closure for the second Che... |
| efchpcl 27415 | The second Chebyshev funct... |
| chpge0 27416 | The second Chebyshev funct... |
| ppival 27417 | Value of the prime-countin... |
| ppival2 27418 | Value of the prime-countin... |
| ppival2g 27419 | Value of the prime-countin... |
| ppif 27420 | Domain and codomain of the... |
| ppicl 27421 | Real closure of the prime-... |
| muval 27422 | The value of the Möbi... |
| muval1 27423 | The value of the Möbi... |
| muval2 27424 | The value of the Möbi... |
| isnsqf 27425 | Two ways to say that a num... |
| issqf 27426 | Two ways to say that a num... |
| sqfpc 27427 | The prime count of a squar... |
| dvdssqf 27428 | A divisor of a squarefree ... |
| sqf11 27429 | A squarefree number is com... |
| muf 27430 | The Möbius function i... |
| mucl 27431 | Closure of the Möbius... |
| sgmval 27432 | The value of the divisor f... |
| sgmval2 27433 | The value of the divisor f... |
| 0sgm 27434 | The value of the sum-of-di... |
| sgmf 27435 | The divisor function is a ... |
| sgmcl 27436 | Closure of the divisor fun... |
| sgmnncl 27437 | Closure of the divisor fun... |
| mule1 27438 | The Möbius function t... |
| chtfl 27439 | The Chebyshev function doe... |
| chpfl 27440 | The second Chebyshev funct... |
| ppiprm 27441 | The prime-counting functio... |
| ppinprm 27442 | The prime-counting functio... |
| chtprm 27443 | The Chebyshev function at ... |
| chtnprm 27444 | The Chebyshev function at ... |
| chpp1 27445 | The second Chebyshev funct... |
| chtwordi 27446 | The Chebyshev function is ... |
| chpwordi 27447 | The second Chebyshev funct... |
| chtdif 27448 | The difference of the Cheb... |
| efchtdvds 27449 | The exponentiated Chebyshe... |
| ppifl 27450 | The prime-counting functio... |
| ppip1le 27451 | The prime-counting functio... |
| ppiwordi 27452 | The prime-counting functio... |
| ppidif 27453 | The difference of the prim... |
| ppi1 27454 | The prime-counting functio... |
| cht1 27455 | The Chebyshev function at ... |
| vma1 27456 | The von Mangoldt function ... |
| chp1 27457 | The second Chebyshev funct... |
| ppi1i 27458 | Inference form of ~ ppiprm... |
| ppi2i 27459 | Inference form of ~ ppinpr... |
| ppi2 27460 | The prime-counting functio... |
| ppi3 27461 | The prime-counting functio... |
| cht2 27462 | The Chebyshev function at ... |
| cht3 27463 | The Chebyshev function at ... |
| ppinncl 27464 | Closure of the prime-count... |
| chtrpcl 27465 | Closure of the Chebyshev f... |
| ppieq0 27466 | The prime-counting functio... |
| ppiltx 27467 | The prime-counting functio... |
| prmorcht 27468 | Relate the primorial (prod... |
| mumullem1 27469 | Lemma for ~ mumul . A mul... |
| mumullem2 27470 | Lemma for ~ mumul . The p... |
| mumul 27471 | The Möbius function i... |
| sqff1o 27472 | There is a bijection from ... |
| fsumdvdsdiaglem 27473 | A "diagonal commutation" o... |
| fsumdvdsdiag 27474 | A "diagonal commutation" o... |
| fsumdvdscom 27475 | A double commutation of di... |
| dvdsppwf1o 27476 | A bijection between the di... |
| dvdsflf1o 27477 | A bijection from the numbe... |
| dvdsflsumcom 27478 | A sum commutation from ` s... |
| fsumfldivdiaglem 27479 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27480 | The right-hand side of ~ d... |
| musum 27481 | The sum of the Möbius... |
| musumsum 27482 | Evaluate a collapsing sum ... |
| muinv 27483 | The Möbius inversion ... |
| mpodvdsmulf1o 27484 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27485 | Product of two divisor sum... |
| dvdsmulf1o 27486 | If ` M ` and ` N ` are two... |
| sgmppw 27487 | The value of the divisor f... |
| 0sgmppw 27488 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27489 | The sum of divisors for a ... |
| 1sgm2ppw 27490 | The sum of the divisors of... |
| sgmmul 27491 | The divisor function for f... |
| ppiublem1 27492 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27493 | A prime greater than ` 3 `... |
| ppiub 27494 | An upper bound on the prim... |
| vmalelog 27495 | The von Mangoldt function ... |
| chtlepsi 27496 | The first Chebyshev functi... |
| chprpcl 27497 | Closure of the second Cheb... |
| chpeq0 27498 | The second Chebyshev funct... |
| chteq0 27499 | The first Chebyshev functi... |
| chtleppi 27500 | Upper bound on the ` theta... |
| chtublem 27501 | Lemma for ~ chtub . (Cont... |
| chtub 27502 | An upper bound on the Cheb... |
| fsumvma 27503 | Rewrite a sum over the von... |
| fsumvma2 27504 | Apply ~ fsumvma for the co... |
| pclogsum 27505 | The logarithmic analogue o... |
| vmasum 27506 | The sum of the von Mangold... |
| logfac2 27507 | Another expression for the... |
| chpval2 27508 | Express the second Chebysh... |
| chpchtsum 27509 | The second Chebyshev funct... |
| chpub 27510 | An upper bound on the seco... |
| logfacubnd 27511 | A simple upper bound on th... |
| logfaclbnd 27512 | A lower bound on the logar... |
| logfacbnd3 27513 | Show the stronger statemen... |
| logfacrlim 27514 | Combine the estimates ~ lo... |
| logexprlim 27515 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27516 | Write out ~ logfacrlim as ... |
| mersenne 27517 | A Mersenne prime is a prim... |
| perfect1 27518 | Euclid's contribution to t... |
| perfectlem1 27519 | Lemma for ~ perfect . (Co... |
| perfectlem2 27520 | Lemma for ~ perfect . (Co... |
| perfect 27521 | The Euclid-Euler theorem, ... |
| dchrval 27524 | Value of the group of Diri... |
| dchrbas 27525 | Base set of the group of D... |
| dchrelbas 27526 | A Dirichlet character is a... |
| dchrelbas2 27527 | A Dirichlet character is a... |
| dchrelbas3 27528 | A Dirichlet character is a... |
| dchrelbasd 27529 | A Dirichlet character is a... |
| dchrrcl 27530 | Reverse closure for a Diri... |
| dchrmhm 27531 | A Dirichlet character is a... |
| dchrf 27532 | A Dirichlet character is a... |
| dchrelbas4 27533 | A Dirichlet character is a... |
| dchrzrh1 27534 | Value of a Dirichlet chara... |
| dchrzrhcl 27535 | A Dirichlet character take... |
| dchrzrhmul 27536 | A Dirichlet character is c... |
| dchrplusg 27537 | Group operation on the gro... |
| dchrmul 27538 | Group operation on the gro... |
| dchrmulcl 27539 | Closure of the group opera... |
| dchrn0 27540 | A Dirichlet character is n... |
| dchr1cl 27541 | Closure of the principal D... |
| dchrmullid 27542 | Left identity for the prin... |
| dchrinvcl 27543 | Closure of the group inver... |
| dchrabl 27544 | The set of Dirichlet chara... |
| dchrfi 27545 | The group of Dirichlet cha... |
| dchrghm 27546 | A Dirichlet character rest... |
| dchr1 27547 | Value of the principal Dir... |
| dchreq 27548 | A Dirichlet character is d... |
| dchrresb 27549 | A Dirichlet character is d... |
| dchrabs 27550 | A Dirichlet character take... |
| dchrinv 27551 | The inverse of a Dirichlet... |
| dchrabs2 27552 | A Dirichlet character take... |
| dchr1re 27553 | The principal Dirichlet ch... |
| dchrptlem1 27554 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27555 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27556 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27557 | For any element other than... |
| dchrsum2 27558 | An orthogonality relation ... |
| dchrsum 27559 | An orthogonality relation ... |
| sumdchr2 27560 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27561 | There are exactly ` phi ( ... |
| sumdchr 27562 | An orthogonality relation ... |
| dchr2sum 27563 | An orthogonality relation ... |
| sum2dchr 27564 | An orthogonality relation ... |
| bcctr 27565 | Value of the central binom... |
| pcbcctr 27566 | Prime count of a central b... |
| bcmono 27567 | The binomial coefficient i... |
| bcmax 27568 | The binomial coefficient t... |
| bcp1ctr 27569 | Ratio of two central binom... |
| bclbnd 27570 | A bound on the binomial co... |
| efexple 27571 | Convert a bound on a power... |
| bpos1lem 27572 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27573 | Bertrand's postulate, chec... |
| bposlem1 27574 | An upper bound on the prim... |
| bposlem2 27575 | There are no odd primes in... |
| bposlem3 27576 | Lemma for ~ bpos . Since ... |
| bposlem4 27577 | Lemma for ~ bpos . (Contr... |
| bposlem5 27578 | Lemma for ~ bpos . Bound ... |
| bposlem6 27579 | Lemma for ~ bpos . By usi... |
| bposlem7 27580 | Lemma for ~ bpos . The fu... |
| bposlem8 27581 | Lemma for ~ bpos . Show t... |
| bposlem9 27582 | Lemma for ~ bpos . Derive... |
| bpos 27583 | Bertrand's postulate: ther... |
| zabsle1 27586 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27587 | When ` a ` is coprime to t... |
| lgslem2 27588 | The set ` Z ` of all integ... |
| lgslem3 27589 | The set ` Z ` of all integ... |
| lgslem4 27590 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27591 | Value of the Legendre symb... |
| lgsfval 27592 | Value of the function ` F ... |
| lgsfcl2 27593 | The function ` F ` is clos... |
| lgscllem 27594 | The Legendre symbol is an ... |
| lgsfcl 27595 | Closure of the function ` ... |
| lgsfle1 27596 | The function ` F ` has mag... |
| lgsval2lem 27597 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27598 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27599 | The Legendre symbol is an ... |
| lgs0 27600 | The Legendre symbol when t... |
| lgscl 27601 | The Legendre symbol is an ... |
| lgsle1 27602 | The Legendre symbol has ab... |
| lgsval2 27603 | The Legendre symbol at a p... |
| lgs2 27604 | The Legendre symbol at ` 2... |
| lgsval3 27605 | The Legendre symbol at an ... |
| lgsvalmod 27606 | The Legendre symbol is equ... |
| lgsval4 27607 | Restate ~ lgsval for nonze... |
| lgsfcl3 27608 | Closure of the function ` ... |
| lgsval4a 27609 | Same as ~ lgsval4 for posi... |
| lgscl1 27610 | The value of the Legendre ... |
| lgsneg 27611 | The Legendre symbol is eit... |
| lgsneg1 27612 | The Legendre symbol for no... |
| lgsmod 27613 | The Legendre (Jacobi) symb... |
| lgsdilem 27614 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27615 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27616 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27617 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27618 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27619 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27620 | The Legendre symbol is com... |
| lgsdirprm 27621 | The Legendre symbol is com... |
| lgsdir 27622 | The Legendre symbol is com... |
| lgsdilem2 27623 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27624 | The Legendre symbol is com... |
| lgsne0 27625 | The Legendre symbol is non... |
| lgsabs1 27626 | The Legendre symbol is non... |
| lgssq 27627 | The Legendre symbol at a s... |
| lgssq2 27628 | The Legendre symbol at a s... |
| lgsprme0 27629 | The Legendre symbol at any... |
| 1lgs 27630 | The Legendre symbol at ` 1... |
| lgs1 27631 | The Legendre symbol at ` 1... |
| lgsmodeq 27632 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27633 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27634 | Variation on ~ lgsdir vali... |
| lgsdinn0 27635 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27636 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27637 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27638 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27639 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27640 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27641 | The Legendre symbol for od... |
| lgsqrmod 27642 | If the Legendre symbol of ... |
| lgsqrmodndvds 27643 | If the Legendre symbol of ... |
| lgsdchrval 27644 | The Legendre symbol functi... |
| lgsdchr 27645 | The Legendre symbol functi... |
| gausslemma2dlem0a 27646 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27647 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27648 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27649 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27650 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27651 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27652 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27653 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27654 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27655 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27656 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27657 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27658 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27659 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27660 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27661 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27662 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27663 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27664 | Gauss' Lemma (see also the... |
| lgseisenlem1 27665 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27666 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27667 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27668 | Lemma for ~ lgseisen . (C... |
| lgseisen 27669 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27670 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27671 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27672 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27673 | The Law of Quadratic Recip... |
| lgsquad2lem1 27674 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27675 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27676 | Extend ~ lgsquad to coprim... |
| lgsquad3 27677 | Extend ~ lgsquad2 to integ... |
| m1lgs 27678 | The first supplement to th... |
| 2lgslem1a1 27679 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27680 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27681 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27682 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27683 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27684 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27685 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27686 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27687 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27688 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27689 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27690 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27691 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27692 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27693 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27694 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27695 | The Legendre symbol for ` ... |
| 2lgslem4 27696 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27697 | The second supplement to t... |
| 2lgsoddprmlem1 27698 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27699 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27700 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27701 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27702 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27703 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27704 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27705 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27706 | The second supplement to t... |
| 2sqlem1 27707 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27708 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27709 | Fibonacci's identity (actu... |
| 2sqlem3 27710 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27711 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27712 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27713 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27714 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27715 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27716 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27717 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27718 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27719 | Lemma for ~ 2sq . (Contri... |
| 2sq 27720 | All primes of the form ` 4... |
| 2sqblem 27721 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27722 | The converse to ~ 2sq . (... |
| 2sq2 27723 | ` 2 ` is the sum of square... |
| 2sqn0 27724 | If the sum of two squares ... |
| 2sqcoprm 27725 | If the sum of two squares ... |
| 2sqmod 27726 | Given two decompositions o... |
| 2sqmo 27727 | There exists at most one d... |
| 2sqnn0 27728 | All primes of the form ` 4... |
| 2sqnn 27729 | All primes of the form ` 4... |
| addsq2reu 27730 | For each complex number ` ... |
| addsqn2reu 27731 | For each complex number ` ... |
| addsqrexnreu 27732 | For each complex number, t... |
| addsqnreup 27733 | There is no unique decompo... |
| addsq2nreurex 27734 | For each complex number ` ... |
| addsqn2reurex2 27735 | For each complex number ` ... |
| 2sqreulem1 27736 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27737 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27738 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27739 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27740 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27741 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27742 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27743 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27744 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27745 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27746 | There exists a unique deco... |
| 2sqreunn 27747 | There exists a unique deco... |
| 2sqreult 27748 | There exists a unique deco... |
| 2sqreultb 27749 | There exists a unique deco... |
| 2sqreunnlt 27750 | There exists a unique deco... |
| 2sqreunnltb 27751 | There exists a unique deco... |
| 2sqreuop 27752 | There exists a unique deco... |
| 2sqreuopnn 27753 | There exists a unique deco... |
| 2sqreuoplt 27754 | There exists a unique deco... |
| 2sqreuopltb 27755 | There exists a unique deco... |
| 2sqreuopnnlt 27756 | There exists a unique deco... |
| 2sqreuopnnltb 27757 | There exists a unique deco... |
| 2sqreuopb 27758 | There exists a unique deco... |
| chebbnd1lem1 27759 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27760 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27761 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27762 | The Chebyshev bound: The ... |
| chtppilimlem1 27763 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27764 | Lemma for ~ chtppilim . (... |
| chtppilim 27765 | The ` theta ` function is ... |
| chto1ub 27766 | The ` theta ` function is ... |
| chebbnd2 27767 | The Chebyshev bound, part ... |
| chto1lb 27768 | The ` theta ` function is ... |
| chpchtlim 27769 | The ` psi ` and ` theta ` ... |
| chpo1ub 27770 | The ` psi ` function is up... |
| chpo1ubb 27771 | The ` psi ` function is up... |
| vmadivsum 27772 | The sum of the von Mangold... |
| vmadivsumb 27773 | Give a total bound on the ... |
| rplogsumlem1 27774 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27775 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27776 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27777 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27778 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27779 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27780 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27781 | Lemma for ~ dchrisum . Le... |
| dchrisum 27782 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27783 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27784 | The sum of the Möbius... |
| dchrvmasumlem1 27785 | An alternative expression ... |
| dchrvmasum2lem 27786 | Give an expression for ` l... |
| dchrvmasum2if 27787 | Combine the results of ~ d... |
| dchrvmasumlem2 27788 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27789 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27790 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27791 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27792 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27793 | An asymptotic approximatio... |
| dchrvmaeq0 27794 | The set ` W ` is the colle... |
| dchrisum0fval 27795 | Value of the function ` F ... |
| dchrisum0fmul 27796 | The function ` F ` , the d... |
| dchrisum0ff 27797 | The function ` F ` is a re... |
| dchrisum0flblem1 27798 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27799 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27800 | The divisor sum of a real ... |
| dchrisum0fno1 27801 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27802 | A partial result along the... |
| dchrisum0re 27803 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27804 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27805 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27806 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27807 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27808 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27809 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27810 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27811 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27812 | The sum of the Möbius... |
| dchrvmasumlem 27813 | The sum of the Möbius... |
| dchrmusum 27814 | The sum of the Möbius... |
| dchrvmasum 27815 | The sum of the von Mangold... |
| rpvmasum 27816 | The sum of the von Mangold... |
| rplogsum 27817 | The sum of ` log p / p ` o... |
| dirith2 27818 | Dirichlet's theorem: there... |
| dirith 27819 | Dirichlet's theorem: there... |
| mudivsum 27820 | Asymptotic formula for ` s... |
| mulogsumlem 27821 | Lemma for ~ mulogsum . (C... |
| mulogsum 27822 | Asymptotic formula for ... |
| logdivsum 27823 | Asymptotic analysis of ... |
| mulog2sumlem1 27824 | Asymptotic formula for ... |
| mulog2sumlem2 27825 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27826 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27827 | Asymptotic formula for ... |
| vmalogdivsum2 27828 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27829 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27830 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27831 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27832 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27833 | The Möbius inverse of... |
| log2sumbnd 27834 | Bound on the difference be... |
| selberglem1 27835 | Lemma for ~ selberg . Est... |
| selberglem2 27836 | Lemma for ~ selberg . (Co... |
| selberglem3 27837 | Lemma for ~ selberg . Est... |
| selberg 27838 | Selberg's symmetry formula... |
| selbergb 27839 | Convert eventual boundedne... |
| selberg2lem 27840 | Lemma for ~ selberg2 . Eq... |
| selberg2 27841 | Selberg's symmetry formula... |
| selberg2b 27842 | Convert eventual boundedne... |
| chpdifbndlem1 27843 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27844 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27845 | A bound on the difference ... |
| logdivbnd 27846 | A bound on a sum of logs, ... |
| selberg3lem1 27847 | Introduce a log weighting ... |
| selberg3lem2 27848 | Lemma for ~ selberg3 . Eq... |
| selberg3 27849 | Introduce a log weighting ... |
| selberg4lem1 27850 | Lemma for ~ selberg4 . Eq... |
| selberg4 27851 | The Selberg symmetry formu... |
| pntrval 27852 | Define the residual of the... |
| pntrf 27853 | Functionality of the resid... |
| pntrmax 27854 | There is a bound on the re... |
| pntrsumo1 27855 | A bound on a sum over ` R ... |
| pntrsumbnd 27856 | A bound on a sum over ` R ... |
| pntrsumbnd2 27857 | A bound on a sum over ` R ... |
| selbergr 27858 | Selberg's symmetry formula... |
| selberg3r 27859 | Selberg's symmetry formula... |
| selberg4r 27860 | Selberg's symmetry formula... |
| selberg34r 27861 | The sum of ~ selberg3r and... |
| pntsval 27862 | Define the "Selberg functi... |
| pntsf 27863 | Functionality of the Selbe... |
| selbergs 27864 | Selberg's symmetry formula... |
| selbergsb 27865 | Selberg's symmetry formula... |
| pntsval2 27866 | The Selberg function can b... |
| pntrlog2bndlem1 27867 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27868 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27869 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27870 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27871 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27872 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27873 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27874 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27875 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27876 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27877 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27878 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27879 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27880 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27881 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27882 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27883 | Lemma for ~ pnt . Establi... |
| pntlemd 27884 | Lemma for ~ pnt . Closure... |
| pntlemc 27885 | Lemma for ~ pnt . Closure... |
| pntlema 27886 | Lemma for ~ pnt . Closure... |
| pntlemb 27887 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27888 | Lemma for ~ pnt . Closure... |
| pntlemh 27889 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27890 | Lemma for ~ pnt . The "na... |
| pntlemq 27891 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27892 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27893 | Lemma for ~ pnt . The ind... |
| pntlemi 27894 | Lemma for ~ pnt . Elimina... |
| pntlemf 27895 | Lemma for ~ pnt . Add up ... |
| pntlemk 27896 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27897 | Lemma for ~ pnt . Combine... |
| pntleme 27898 | Lemma for ~ pnt . Package... |
| pntlem3 27899 | Lemma for ~ pnt . Equatio... |
| pntlemp 27900 | Lemma for ~ pnt . Wrappin... |
| pntleml 27901 | Lemma for ~ pnt . Equatio... |
| pnt3 27902 | The Prime Number Theorem, ... |
| pnt2 27903 | The Prime Number Theorem, ... |
| pnt 27904 | The Prime Number Theorem: ... |
| abvcxp 27905 | Raising an absolute value ... |
| padicfval 27906 | Value of the p-adic absolu... |
| padicval 27907 | Value of the p-adic absolu... |
| ostth2lem1 27908 | Lemma for ~ ostth2 , altho... |
| qrngbas 27909 | The base set of the field ... |
| qdrng 27910 | The rationals form a divis... |
| qrng0 27911 | The zero element of the fi... |
| qrng1 27912 | The unity element of the f... |
| qrngneg 27913 | The additive inverse in th... |
| qrngdiv 27914 | The division operation in ... |
| qabvle 27915 | By using induction on ` N ... |
| qabvexp 27916 | Induct the product rule ~ ... |
| ostthlem1 27917 | Lemma for ~ ostth . If tw... |
| ostthlem2 27918 | Lemma for ~ ostth . Refin... |
| qabsabv 27919 | The regular absolute value... |
| padicabv 27920 | The p-adic absolute value ... |
| padicabvf 27921 | The p-adic absolute value ... |
| padicabvcxp 27922 | All positive powers of the... |
| ostth1 27923 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27924 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27925 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27926 | Lemma for ~ ostth2 . (Con... |
| ostth2 27927 | - Lemma for ~ ostth : regu... |
| ostth3 27928 | - Lemma for ~ ostth : p-ad... |
| ostth 27929 | Ostrowski's theorem, which... |
| elno 27936 | Membership in the surreals... |
| ltsval 27937 | The value of the surreal l... |
| bdayval 27938 | The value of the birthday ... |
| nofun 27939 | A surreal is a function. ... |
| nodmon 27940 | The domain of a surreal is... |
| norn 27941 | The range of a surreal is ... |
| nofnbday 27942 | A surreal is a function ov... |
| nodmord 27943 | The domain of a surreal ha... |
| elno2 27944 | An alternative condition f... |
| elno3 27945 | Another condition for memb... |
| ltsval2 27946 | Alternate expression for s... |
| nofv 27947 | The function value of a su... |
| nosgnn0 27948 | ` (/) ` is not a surreal s... |
| nosgnn0i 27949 | If ` X ` is a surreal sign... |
| noreson 27950 | The restriction of a surre... |
| ltsintdifex 27951 |
If ` A |
| ltsres 27952 | If the restrictions of two... |
| noxp1o 27953 | The Cartesian product of a... |
| noseponlem 27954 | Lemma for ~ nosepon . Con... |
| nosepon 27955 | Given two unequal surreals... |
| noextend 27956 | Extending a surreal by one... |
| noextendseq 27957 | Extend a surreal by a sequ... |
| noextenddif 27958 | Calculate the place where ... |
| noextendlt 27959 | Extending a surreal with a... |
| noextendgt 27960 | Extending a surreal with a... |
| nolesgn2o 27961 | Given ` A ` less-than or e... |
| nolesgn2ores 27962 | Given ` A ` less-than or e... |
| nogesgn1o 27963 | Given ` A ` greater than o... |
| nogesgn1ores 27964 | Given ` A ` greater than o... |
| ltssolem1 27965 | Lemma for ~ ltsso . The "... |
| ltsso 27966 | Less-than totally orders t... |
| bdayfo 27967 | The birthday function maps... |
| fvnobday 27968 | The value of a surreal at ... |
| nosepnelem 27969 | Lemma for ~ nosepne . (Co... |
| nosepne 27970 | The value of two non-equal... |
| nosep1o 27971 | If the value of a surreal ... |
| nosep2o 27972 | If the value of a surreal ... |
| nosepdmlem 27973 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27974 | The first place two surrea... |
| nosepeq 27975 | The values of two surreals... |
| nosepssdm 27976 | Given two non-equal surrea... |
| nodenselem4 27977 | Lemma for ~ nodense . Sho... |
| nodenselem5 27978 | Lemma for ~ nodense . If ... |
| nodenselem6 27979 | The restriction of a surre... |
| nodenselem7 27980 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27981 | Lemma for ~ nodense . Giv... |
| nodense 27982 | Given two distinct surreal... |
| bdayimaon 27983 | Lemma for full-eta propert... |
| nolt02olem 27984 | Lemma for ~ nolt02o . If ... |
| nolt02o 27985 | Given ` A ` less-than ` B ... |
| nogt01o 27986 | Given ` A ` greater than `... |
| noresle 27987 | Restriction law for surrea... |
| nomaxmo 27988 | A class of surreals has at... |
| nominmo 27989 | A class of surreals has at... |
| nosupprefixmo 27990 | In any class of surreals, ... |
| noinfprefixmo 27991 | In any class of surreals, ... |
| nosupcbv 27992 | Lemma to change bound vari... |
| nosupno 27993 | The next several theorems ... |
| nosupdm 27994 | The domain of the surreal ... |
| nosupbday 27995 | Birthday bounding law for ... |
| nosupfv 27996 | The value of surreal supre... |
| nosupres 27997 | A restriction law for surr... |
| nosupbnd1lem1 27998 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27999 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 28000 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 28001 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 28002 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 28003 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 28004 | Bounding law from below fo... |
| nosupbnd2lem1 28005 | Bounding law from above wh... |
| nosupbnd2 28006 | Bounding law from above fo... |
| noinfcbv 28007 | Change bound variables for... |
| noinfno 28008 | The next several theorems ... |
| noinfdm 28009 | Next, we calculate the dom... |
| noinfbday 28010 | Birthday bounding law for ... |
| noinffv 28011 | The value of surreal infim... |
| noinfres 28012 | The restriction of surreal... |
| noinfbnd1lem1 28013 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 28014 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 28015 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 28016 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 28017 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 28018 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 28019 | Bounding law from above fo... |
| noinfbnd2lem1 28020 | Bounding law from below wh... |
| noinfbnd2 28021 | Bounding law from below fo... |
| nosupinfsep 28022 | Given two sets of surreals... |
| noetasuplem1 28023 | Lemma for ~ noeta . Estab... |
| noetasuplem2 28024 | Lemma for ~ noeta . The r... |
| noetasuplem3 28025 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 28026 | Lemma for ~ noeta . When ... |
| noetainflem1 28027 | Lemma for ~ noeta . Estab... |
| noetainflem2 28028 | Lemma for ~ noeta . The r... |
| noetainflem3 28029 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 28030 | Lemma for ~ noeta . If ` ... |
| noetalem1 28031 | Lemma for ~ noeta . Eithe... |
| noetalem2 28032 | Lemma for ~ noeta . The f... |
| noeta 28033 | The full-eta axiom for the... |
| ltsirr 28036 | Surreal less-than is irref... |
| ltstr 28037 | Surreal less-than is trans... |
| ltsasym 28038 | Surreal less-than is asymm... |
| ltslin 28039 | Surreal less-than obeys tr... |
| ltstrieq2 28040 | Trichotomy law for surreal... |
| ltstrine 28041 | Trichotomy law for surreal... |
| lenlts 28042 | Surreal less-than or equal... |
| ltnles 28043 | Surreal less-than in terms... |
| lesloe 28044 | Surreal less-than or equal... |
| lestri3 28045 | Trichotomy law for surreal... |
| lesnltd 28046 | Surreal less-than or equal... |
| ltsnled 28047 | Surreal less-than in terms... |
| lesloed 28048 | Surreal less-than or equal... |
| lestri3d 28049 | Trichotomy law for surreal... |
| ltlestr 28050 | Surreal transitive law. (... |
| leltstr 28051 | Surreal transitive law. (... |
| lestr 28052 | Surreal transitive law. (... |
| ltstrd 28053 | Surreal less-than is trans... |
| ltlestrd 28054 | Surreal less-than is trans... |
| leltstrd 28055 | Surreal less-than is trans... |
| lestrd 28056 | Surreal less-than or equal... |
| lesid 28057 | Surreal less-than or equal... |
| lestric 28058 | Surreal trichotomy law. (... |
| maxs1 28059 | A surreal is less than or ... |
| maxs2 28060 | A surreal is less than or ... |
| mins1 28061 | The minimum of two surreal... |
| mins2 28062 | The minimum of two surreal... |
| ltlesd 28063 | Surreal less-than implies ... |
| ltsne 28064 | Surreal less-than implies ... |
| ltlesnd 28065 | Surreal less-than in terms... |
| bdayfun 28066 | The birthday function is a... |
| bdayfn 28067 | The birthday function is a... |
| bdaydm 28068 | The birthday function's do... |
| bdaydmOLD 28069 | Obsolete version of ~ bday... |
| bdayrn 28070 | The birthday function's ra... |
| bdayon 28071 | The value of the birthday ... |
| nobdaymin 28072 | Any non-empty class of sur... |
| nocvxminlem 28073 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 28074 | Given a nonempty convex cl... |
| noprc 28075 | The surreal numbers are a ... |
| noeta2 28080 | A version of ~ noeta with ... |
| brslts 28081 | Binary relation form of th... |
| sltsex1 28082 | The first argument of surr... |
| sltsex2 28083 | The second argument of sur... |
| sltsss1 28084 | The first argument of surr... |
| sltsss2 28085 | The second argument of sur... |
| sltssep 28086 | The separation property of... |
| sltsd 28087 | Deduce surreal set less-th... |
| sltssnb 28088 | Surreal set less-than of t... |
| sltssn 28089 | Surreal set less-than of t... |
| sltssepc 28090 | Two elements of separated ... |
| sltssepcd 28091 | Two elements of separated ... |
| ssslts1 28092 | Relation between surreal s... |
| ssslts2 28093 | Relation between surreal s... |
| nulslts 28094 | The empty set is less-than... |
| nulsgts 28095 | The empty set is greater t... |
| nulsltsd 28096 | The empty set is less-than... |
| nulsgtsd 28097 | The empty set is greater t... |
| conway 28098 | Conway's Simplicity Theore... |
| cutsval 28099 | The value of the surreal c... |
| cutcuts 28100 | Cut properties of the surr... |
| cutscl 28101 | Closure law for surreal cu... |
| cutscld 28102 | Closure law for surreal cu... |
| cutbday 28103 | The birthday of the surrea... |
| eqcuts 28104 | Condition for equality to ... |
| eqcuts2 28105 | Condition for equality to ... |
| sltstr 28106 | Transitive law for surreal... |
| sltsun1 28107 | Union law for surreal set ... |
| sltsun2 28108 | Union law for surreal set ... |
| cutsun12 28109 | Union law for surreal cuts... |
| dmcuts 28110 | The domain of the surreal ... |
| cutsf 28111 | Functionality statement fo... |
| etaslts 28112 | A restatement of ~ noeta u... |
| etaslts2 28113 | A version of ~ etaslts wit... |
| cutbdaybnd 28114 | An upper bound on the birt... |
| cutbdaybnd2 28115 | An upper bound on the birt... |
| cutbdaybnd2lim 28116 | An upper bound on the birt... |
| cutbdaylt 28117 | If a surreal lies in a gap... |
| lesrec 28118 | A comparison law for surre... |
| lesrecd 28119 | A comparison law for surre... |
| ltsrec 28120 | A comparison law for surre... |
| ltsrecd 28121 | A comparison law for surre... |
| sltsdisj 28122 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 28123 | A variant of the simplicit... |
| 0no 28128 | Surreal zero is a surreal.... |
| 1no 28129 | Surreal one is a surreal. ... |
| bday0 28130 | Calculate the birthday of ... |
| 0lt1s 28131 | Surreal zero is less than ... |
| bday0b 28132 | The only surreal with birt... |
| bday1 28133 | The birthday of surreal on... |
| cuteq0 28134 | Condition for a surreal cu... |
| cutneg 28135 | The simplest number greate... |
| cuteq1 28136 | Condition for a surreal cu... |
| gt0ne0s 28137 | A positive surreal is not ... |
| gt0ne0sd 28138 | A positive surreal is not ... |
| 1ne0s 28139 | Surreal zero does not equa... |
| rightge0 28140 | A surreal is non-negative ... |
| madeval 28151 | The value of the made by f... |
| madeval2 28152 | Alternative characterizati... |
| oldval 28153 | The value of the old optio... |
| newval 28154 | The value of the new optio... |
| madef 28155 | The made function is a fun... |
| oldf 28156 | The older function is a fu... |
| newf 28157 | The new function is a func... |
| old0 28158 | No surreal is older than `... |
| madessno 28159 | Made sets are surreals. (... |
| oldssno 28160 | Old sets are surreals. (C... |
| newssno 28161 | New sets are surreals. (C... |
| madeno 28162 | An element of a made set i... |
| oldno 28163 | An element of an old set i... |
| newno 28164 | An element of a new set is... |
| madenod 28165 | An element of a made set i... |
| oldnod 28166 | An element of an old set i... |
| newnod 28167 | An element of a new set is... |
| leftval 28168 | The value of the left opti... |
| rightval 28169 | The value of the right opt... |
| elleft 28170 | Membership in the left set... |
| elright 28171 | Membership in the right se... |
| leftlt 28172 | A member of a surreal's le... |
| rightgt 28173 | A member of a surreal's ri... |
| leftf 28174 | The functionality of the l... |
| rightf 28175 | The functionality of the r... |
| elmade 28176 | Membership in the made fun... |
| elmade2 28177 | Membership in the made fun... |
| elold 28178 | Membership in an old set. ... |
| sltsleft 28179 | A surreal is greater than ... |
| sltsright 28180 | A surreal is less than its... |
| lltr 28181 | The left options of a surr... |
| made0 28182 | The only surreal made on d... |
| new0 28183 | The only surreal new on da... |
| old1 28184 | The only surreal older tha... |
| madess 28185 | If ` A ` is less than or e... |
| oldssmade 28186 | The older-than set is a su... |
| oldmade 28187 | An element of an old set i... |
| oldmaded 28188 | An element of an old set i... |
| oldss 28189 | If ` A ` is less than or e... |
| leftssold 28190 | The left options are a sub... |
| rightssold 28191 | The right options are a su... |
| leftssno 28192 | The left set of a surreal ... |
| rightssno 28193 | The right set of a surreal... |
| leftold 28194 | An element of a left set i... |
| rightold 28195 | An element of a right set ... |
| leftno 28196 | An element of a left set i... |
| rightno 28197 | An element of a right set ... |
| leftoldd 28198 | An element of a left set i... |
| leftnod 28199 | An element of a left set i... |
| rightoldd 28200 | An element of a right set ... |
| rightnod 28201 | An element of a right set ... |
| madecut 28202 | Given a section that is a ... |
| madeun 28203 | The made set is the union ... |
| madeoldsuc 28204 | The made set is the old se... |
| oldsuc 28205 | The value of the old set a... |
| oldlim 28206 | The value of the old set a... |
| madebdayim 28207 | If a surreal is a member o... |
| oldbdayim 28208 | If ` X ` is in the old set... |
| oldirr 28209 | No surreal is a member of ... |
| leftirr 28210 | No surreal is a member of ... |
| rightirr 28211 | No surreal is a member of ... |
| left0s 28212 | The left set of ` 0s ` is ... |
| right0s 28213 | The right set of ` 0s ` is... |
| left1s 28214 | The left set of ` 1s ` is ... |
| right1s 28215 | The right set of ` 1s ` is... |
| lrold 28216 | The union of the left and ... |
| madebdaylemold 28217 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 28218 | Lemma for ~ madebday . If... |
| madebday 28219 | A surreal is part of the s... |
| oldbday 28220 | A surreal is part of the s... |
| newbday 28221 | A surreal is an element of... |
| newbdayim 28222 | One direction of the bicon... |
| lrcut 28223 | A surreal is equal to the ... |
| cutsfo 28224 | The surreal cut function i... |
| ltsn0 28225 | If ` X ` is less than ` Y ... |
| lruneq 28226 | If two surreals share a bi... |
| ltslpss 28227 | If two surreals share a bi... |
| leslss 28228 | If two surreals ` A ` and ... |
| 0elold 28229 | Zero is in the old set of ... |
| 0elleft 28230 | Zero is in the left set of... |
| 0elright 28231 | Zero is in the right set o... |
| madefi 28232 | The made set of an ordinal... |
| oldfi 28233 | The old set of an ordinal ... |
| bdayiun 28234 | The birthday of a surreal ... |
| bdayle 28235 | A condition for bounding a... |
| sltsbday 28236 | Birthday comparison rule f... |
| cofslts 28237 | If every element of ` A ` ... |
| coinitslts 28238 | If ` B ` is coinitial with... |
| cofcut1 28239 | If ` C ` is cofinal with `... |
| cofcut1d 28240 | If ` C ` is cofinal with `... |
| cofcut2 28241 | If ` A ` and ` C ` are mut... |
| cofcut2d 28242 | If ` A ` and ` C ` are mut... |
| cofcutr 28243 | If ` X ` is the cut of ` A... |
| cofcutr1d 28244 | If ` X ` is the cut of ` A... |
| cofcutr2d 28245 | If ` X ` is the cut of ` A... |
| cofcutrtime 28246 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 28247 | If ` X ` is a timely cut o... |
| cofcutrtime2d 28248 | If ` X ` is a timely cut o... |
| cofss 28249 | Cofinality for a subset. ... |
| coiniss 28250 | Coinitiality for a subset.... |
| cutlt 28251 | Eliminating all elements b... |
| cutpos 28252 | Reduce the elements of a c... |
| cutmax 28253 | If ` A ` has a maximum, th... |
| cutmin 28254 | If ` B ` has a minimum, th... |
| cutminmax 28255 | If the left set of ` X ` h... |
| lrrecval 28258 | The next step in the devel... |
| lrrecval2 28259 | Next, we establish an alte... |
| lrrecpo 28260 | Now, we establish that ` R... |
| lrrecse 28261 | Next, we show that ` R ` i... |
| lrrecfr 28262 | Now we show that ` R ` is ... |
| lrrecpred 28263 | Finally, we calculate the ... |
| noinds 28264 | Induction principle for a ... |
| norecfn 28265 | Surreal recursion over one... |
| norecov 28266 | Calculate the value of the... |
| noxpordpo 28269 | To get through most of the... |
| noxpordfr 28270 | Next we establish the foun... |
| noxpordse 28271 | Next we establish the set-... |
| noxpordpred 28272 | Next we calculate the pred... |
| no2indlesm 28273 | Double induction on surrea... |
| no2inds 28274 | Double induction on surrea... |
| norec2fn 28275 | The double-recursion opera... |
| norec2ov 28276 | The value of the double-re... |
| no3inds 28277 | Triple induction over surr... |
| addsfn 28280 | Surreal addition is a func... |
| addsval 28281 | The value of surreal addit... |
| addsval2 28282 | The value of surreal addit... |
| addsrid 28283 | Surreal addition to zero i... |
| addsridd 28284 | Surreal addition to zero i... |
| addscom 28285 | Surreal addition is commut... |
| addscomd 28286 | Surreal addition is commut... |
| addslid 28287 | Surreal addition to zero i... |
| addsproplem1 28288 | Lemma for surreal addition... |
| addsproplem2 28289 | Lemma for surreal addition... |
| addsproplem3 28290 | Lemma for surreal addition... |
| addsproplem4 28291 | Lemma for surreal addition... |
| addsproplem5 28292 | Lemma for surreal addition... |
| addsproplem6 28293 | Lemma for surreal addition... |
| addsproplem7 28294 | Lemma for surreal addition... |
| addsprop 28295 | Inductively show that surr... |
| addcutslem 28296 | Lemma for ~ addcuts . Sho... |
| addcuts 28297 | Demonstrate the cut proper... |
| addcuts2 28298 | Show that the cut involved... |
| addscld 28299 | Surreal numbers are closed... |
| addscl 28300 | Surreal numbers are closed... |
| addsf 28301 | Function statement for sur... |
| addsfo 28302 | Surreal addition is onto. ... |
| peano2no 28303 | A theorem for surreals tha... |
| ltadds1im 28304 | Surreal less-than is prese... |
| ltadds2im 28305 | Surreal less-than is prese... |
| leadds1im 28306 | Surreal less-than or equal... |
| leadds2im 28307 | Surreal less-than or equal... |
| leadds1 28308 | Addition to both sides of ... |
| leadds2 28309 | Addition to both sides of ... |
| ltadds2 28310 | Addition to both sides of ... |
| ltadds1 28311 | Addition to both sides of ... |
| addscan2 28312 | Cancellation law for surre... |
| addscan1 28313 | Cancellation law for surre... |
| leadds1d 28314 | Addition to both sides of ... |
| leadds2d 28315 | Addition to both sides of ... |
| ltadds2d 28316 | Addition to both sides of ... |
| ltadds1d 28317 | Addition to both sides of ... |
| addscan2d 28318 | Cancellation law for surre... |
| addscan1d 28319 | Cancellation law for surre... |
| addsuniflem 28320 | Lemma for ~ addsunif . St... |
| addsunif 28321 | Uniformity theorem for sur... |
| addsasslem1 28322 | Lemma for addition associa... |
| addsasslem2 28323 | Lemma for addition associa... |
| addsass 28324 | Surreal addition is associ... |
| addsassd 28325 | Surreal addition is associ... |
| adds32d 28326 | Commutative/associative la... |
| adds12d 28327 | Commutative/associative la... |
| adds4d 28328 | Rearrangement of four term... |
| adds42d 28329 | Rearrangement of four term... |
| ltaddspos1d 28330 | Addition of a positive num... |
| ltaddspos2d 28331 | Addition of a positive num... |
| lt2addsd 28332 | Adding both sides of two s... |
| addsgt0d 28333 | The sum of two positive su... |
| ltsp1d 28334 | A surreal is less than its... |
| addsge01d 28335 | A surreal is less-than or ... |
| addbdaylem 28336 | Lemma for ~ addbday . (Co... |
| addbday 28337 | The birthday of the sum of... |
| negsfn 28342 | Surreal negation is a func... |
| subsfn 28343 | Surreal subtraction is a f... |
| negsval 28344 | The value of the surreal n... |
| neg0s 28345 | Negative surreal zero is s... |
| neg1s 28346 | An expression for negative... |
| negsproplem1 28347 | Lemma for surreal negation... |
| negsproplem2 28348 | Lemma for surreal negation... |
| negsproplem3 28349 | Lemma for surreal negation... |
| negsproplem4 28350 | Lemma for surreal negation... |
| negsproplem5 28351 | Lemma for surreal negation... |
| negsproplem6 28352 | Lemma for surreal negation... |
| negsproplem7 28353 | Lemma for surreal negation... |
| negsprop 28354 | Show closure and ordering ... |
| negscl 28355 | The surreals are closed un... |
| negscld 28356 | The surreals are closed un... |
| ltnegsim 28357 | The forward direction of t... |
| negcut 28358 | The cut properties of surr... |
| negcut2 28359 | The cut that defines surre... |
| negsid 28360 | Surreal addition of a numb... |
| negsidd 28361 | Surreal addition of a numb... |
| negsex 28362 | Every surreal has a negati... |
| negnegs 28363 | A surreal is equal to the ... |
| ltnegs 28364 | Negative of both sides of ... |
| lenegs 28365 | Negative of both sides of ... |
| ltnegsd 28366 | Negative of both sides of ... |
| lenegsd 28367 | Negative of both sides of ... |
| negs11 28368 | Surreal negation is one-to... |
| negsdi 28369 | Distribution of surreal ne... |
| lt0negs2d 28370 | Comparison of a surreal an... |
| negsf 28371 | Function statement for sur... |
| negsfo 28372 | Function statement for sur... |
| negsf1o 28373 | Surreal negation is a bije... |
| negsunif 28374 | Uniformity property for su... |
| negbdaylem 28375 | Lemma for ~ negbday . Bou... |
| negbday 28376 | Negation of a surreal numb... |
| negleft 28377 | The left set of the negati... |
| negright 28378 | The right set of the negat... |
| subsval 28379 | The value of surreal subtr... |
| subsvald 28380 | The value of surreal subtr... |
| subscl 28381 | Closure law for surreal su... |
| subscld 28382 | Closure law for surreal su... |
| subsf 28383 | Function statement for sur... |
| subsfo 28384 | Surreal subtraction is an ... |
| negsval2 28385 | Surreal negation in terms ... |
| negsval2d 28386 | Surreal negation in terms ... |
| subsid1 28387 | Identity law for subtracti... |
| subsid 28388 | Subtraction of a surreal f... |
| subadds 28389 | Relationship between addit... |
| subaddsd 28390 | Relationship between addit... |
| pncans 28391 | Cancellation law for surre... |
| pncan3s 28392 | Subtraction and addition o... |
| pncan2s 28393 | Cancellation law for surre... |
| npcans 28394 | Cancellation law for surre... |
| ltsubs1 28395 | Subtraction from both side... |
| ltsubs2 28396 | Subtraction from both side... |
| ltsubs1d 28397 | Subtraction from both side... |
| ltsubs2d 28398 | Subtraction from both side... |
| negsubsdi2d 28399 | Distribution of negative o... |
| addsubsassd 28400 | Associative-type law for s... |
| addsubsd 28401 | Law for surreal addition a... |
| ltsubsubsbd 28402 | Equivalence for the surrea... |
| ltsubsubs2bd 28403 | Equivalence for the surrea... |
| ltsubsubs3bd 28404 | Equivalence for the surrea... |
| lesubsubsbd 28405 | Equivalence for the surrea... |
| lesubsubs2bd 28406 | Equivalence for the surrea... |
| lesubsubs3bd 28407 | Equivalence for the surrea... |
| ltsubaddsd 28408 | Surreal less-than relation... |
| ltsubadds2d 28409 | Surreal less-than relation... |
| ltaddsubsd 28410 | Surreal less-than relation... |
| ltaddsubs2d 28411 | Surreal less-than relation... |
| lesubaddsd 28412 | Surreal less-than or equal... |
| subsubs4d 28413 | Law for double surreal sub... |
| subsubs2d 28414 | Law for double surreal sub... |
| lesubsd 28415 | Swap subtrahends in a surr... |
| nncansd 28416 | Cancellation law for surre... |
| posdifsd 28417 | Comparison of two surreals... |
| ltsubsposd 28418 | Subtraction of a positive ... |
| subsge0d 28419 | Non-negative subtraction. ... |
| addsubs4d 28420 | Rearrangement of four term... |
| ltsm1d 28421 | A surreal is greater than ... |
| subscan1d 28422 | Cancellation law for surre... |
| subscan2d 28423 | Cancellation law for surre... |
| subseq0d 28424 | The difference between two... |
| mulsfn 28427 | Surreal multiplication is ... |
| mulsval 28428 | The value of surreal multi... |
| mulsval2lem 28429 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28430 | The value of surreal multi... |
| muls01 28431 | Surreal multiplication by ... |
| mulsrid 28432 | Surreal one is a right ide... |
| mulsridd 28433 | Surreal one is a right ide... |
| mulsproplemcbv 28434 | Lemma for surreal multipli... |
| mulsproplem1 28435 | Lemma for surreal multipli... |
| mulsproplem2 28436 | Lemma for surreal multipli... |
| mulsproplem3 28437 | Lemma for surreal multipli... |
| mulsproplem4 28438 | Lemma for surreal multipli... |
| mulsproplem5 28439 | Lemma for surreal multipli... |
| mulsproplem6 28440 | Lemma for surreal multipli... |
| mulsproplem7 28441 | Lemma for surreal multipli... |
| mulsproplem8 28442 | Lemma for surreal multipli... |
| mulsproplem9 28443 | Lemma for surreal multipli... |
| mulsproplem10 28444 | Lemma for surreal multipli... |
| mulsproplem11 28445 | Lemma for surreal multipli... |
| mulsproplem12 28446 | Lemma for surreal multipli... |
| mulsproplem13 28447 | Lemma for surreal multipli... |
| mulsproplem14 28448 | Lemma for surreal multipli... |
| mulsprop 28449 | Surreals are closed under ... |
| mulcutlem 28450 | Lemma for ~ mulcut . Stat... |
| mulcut 28451 | Show the cut properties of... |
| mulcut2 28452 | Show that the cut involved... |
| mulscl 28453 | The surreals are closed un... |
| mulscld 28454 | The surreals are closed un... |
| ltmuls 28455 | An ordering relationship f... |
| ltmulsd 28456 | An ordering relationship f... |
| lemulsd 28457 | An ordering relationship f... |
| mulscom 28458 | Surreal multiplication is ... |
| mulscomd 28459 | Surreal multiplication is ... |
| muls02 28460 | Surreal multiplication by ... |
| mulslid 28461 | Surreal one is a left iden... |
| mulslidd 28462 | Surreal one is a left iden... |
| mulsgt0 28463 | The product of two positiv... |
| mulsgt0d 28464 | The product of two positiv... |
| mulsge0d 28465 | The product of two non-neg... |
| sltmuls1 28466 | One surreal set less-than ... |
| sltmuls2 28467 | One surreal set less-than ... |
| mulsuniflem 28468 | Lemma for ~ mulsunif . St... |
| mulsunif 28469 | Surreal multiplication has... |
| addsdilem1 28470 | Lemma for surreal distribu... |
| addsdilem2 28471 | Lemma for surreal distribu... |
| addsdilem3 28472 | Lemma for ~ addsdi . Show... |
| addsdilem4 28473 | Lemma for ~ addsdi . Show... |
| addsdi 28474 | Distributive law for surre... |
| addsdid 28475 | Distributive law for surre... |
| addsdird 28476 | Distributive law for surre... |
| subsdid 28477 | Distribution of surreal mu... |
| subsdird 28478 | Distribution of surreal mu... |
| mulnegs1d 28479 | Product with negative is n... |
| mulnegs2d 28480 | Product with negative is n... |
| mul2negsd 28481 | Surreal product of two neg... |
| mulsasslem1 28482 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28483 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28484 | Lemma for ~ mulsass . Dem... |
| mulsass 28485 | Associative law for surrea... |
| mulsassd 28486 | Associative law for surrea... |
| muls4d 28487 | Rearrangement of four surr... |
| mulsunif2lem 28488 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28489 | Alternate expression for s... |
| ltmuls2 28490 | Multiplication of both sid... |
| ltmuls2d 28491 | Multiplication of both sid... |
| ltmuls1d 28492 | Multiplication of both sid... |
| lemuls2d 28493 | Multiplication of both sid... |
| lemuls1d 28494 | Multiplication of both sid... |
| ltmulnegs1d 28495 | Multiplication of both sid... |
| ltmulnegs2d 28496 | Multiplication of both sid... |
| mulscan2dlem 28497 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28498 | Cancellation of surreal mu... |
| mulscan1d 28499 | Cancellation of surreal mu... |
| muls12d 28500 | Commutative/associative la... |
| lemuls1ad 28501 | Multiplication of both sid... |
| ltmuls12ad 28502 | Comparison of the product ... |
| divsmo 28503 | Uniqueness of surreal inve... |
| muls0ord 28504 | If a surreal product is ze... |
| mulsne0bd 28505 | The product of two nonzero... |
| divsval 28508 | The value of surreal divis... |
| norecdiv 28509 | If a surreal has a recipro... |
| noreceuw 28510 | If a surreal has a recipro... |
| recsne0 28511 | If a surreal has a recipro... |
| divmulsw 28512 | Relationship between surre... |
| divmulswd 28513 | Relationship between surre... |
| divsclw 28514 | Weak division closure law.... |
| divsclwd 28515 | Weak division closure law.... |
| divscan2wd 28516 | A weak cancellation law fo... |
| divscan1wd 28517 | A weak cancellation law fo... |
| ltdivmulswd 28518 | Surreal less-than relation... |
| ltdivmuls2wd 28519 | Surreal less-than relation... |
| ltmuldivswd 28520 | Surreal less-than relation... |
| ltmuldivs2wd 28521 | Surreal less-than relation... |
| divsasswd 28522 | An associative law for sur... |
| divs1 28523 | A surreal divided by one i... |
| divs1d 28524 | A surreal divided by one i... |
| precsexlemcbv 28525 | Lemma for surreal reciproc... |
| precsexlem1 28526 | Lemma for surreal reciproc... |
| precsexlem2 28527 | Lemma for surreal reciproc... |
| precsexlem3 28528 | Lemma for surreal reciproc... |
| precsexlem4 28529 | Lemma for surreal reciproc... |
| precsexlem5 28530 | Lemma for surreal reciproc... |
| precsexlem6 28531 | Lemma for surreal reciproc... |
| precsexlem7 28532 | Lemma for surreal reciproc... |
| precsexlem8 28533 | Lemma for surreal reciproc... |
| precsexlem9 28534 | Lemma for surreal reciproc... |
| precsexlem10 28535 | Lemma for surreal reciproc... |
| precsexlem11 28536 | Lemma for surreal reciproc... |
| precsex 28537 | Every positive surreal has... |
| recsex 28538 | A nonzero surreal has a re... |
| recsexd 28539 | A nonzero surreal has a re... |
| divmuls 28540 | Relationship between surre... |
| divmulsd 28541 | Relationship between surre... |
| divscl 28542 | Surreal division closure l... |
| divscld 28543 | Surreal division closure l... |
| divscan2d 28544 | A cancellation law for sur... |
| divscan1d 28545 | A cancellation law for sur... |
| ltdivmulsd 28546 | Surreal less-than relation... |
| ltdivmuls2d 28547 | Surreal less-than relation... |
| ltmuldivsd 28548 | Surreal less-than relation... |
| ltmuldivs2d 28549 | Surreal less-than relation... |
| divsassd 28550 | An associative law for sur... |
| divmuldivsd 28551 | Multiplication of two surr... |
| divdivs1d 28552 | Surreal division into a fr... |
| divsrecd 28553 | Relationship between surre... |
| divsdird 28554 | Distribution of surreal di... |
| divscan3d 28555 | A cancellation law for sur... |
| abssval 28558 | The value of surreal absol... |
| absscl 28559 | Closure law for surreal ab... |
| abssid 28560 | The absolute value of a no... |
| abs0s 28561 | The absolute value of surr... |
| abssnid 28562 | For a negative surreal, it... |
| absmuls 28563 | Surreal absolute value dis... |
| abssge0 28564 | The absolute value of a su... |
| abssor 28565 | The absolute value of a su... |
| absnegs 28566 | Surreal absolute value of ... |
| leabss 28567 | A surreal is less than or ... |
| abslts 28568 | Surreal absolute value and... |
| abssubs 28569 | Swapping order of surreal ... |
| elons 28572 | Membership in the class of... |
| onssno 28573 | The surreal ordinals are a... |
| onno 28574 | A surreal ordinal is a sur... |
| 0ons 28575 | Surreal zero is a surreal ... |
| 1ons 28576 | Surreal one is a surreal o... |
| elons2 28577 | A surreal is ordinal iff i... |
| elons2d 28578 | The cut of any set of surr... |
| onleft 28579 | The left set of a surreal ... |
| ltonold 28580 | The class of ordinals less... |
| ltonsex 28581 | The class of ordinals less... |
| oncutleft 28582 | A surreal ordinal is equal... |
| oncutlt 28583 | A surreal ordinal is the s... |
| bday11on 28584 | The birthday function is o... |
| onnolt 28585 | If a surreal ordinal is le... |
| onlts 28586 | Less-than is the same as b... |
| onles 28587 | Less-than or equal is the ... |
| onltsd 28588 | Less-than is the same as b... |
| onlesd 28589 | Less-than or equal is the ... |
| oniso 28590 | The birthday function rest... |
| onswe 28591 | Surreal less-than well-ord... |
| onsse 28592 | Surreal less-than is set-l... |
| onsis 28593 | Transfinite induction sche... |
| ons2ind 28594 | Double induction schema fo... |
| bdayons 28595 | The birthday of a surreal ... |
| onaddscl 28596 | The surreal ordinals are c... |
| onmulscl 28597 | The surreal ordinals are c... |
| addonbday 28598 | The birthday of the sum of... |
| peano2ons 28599 | The successor of a surreal... |
| onsbnd 28600 | The surreals of a given bi... |
| onsbnd2 28601 | The surreals of a given bi... |
| seqsex 28604 | Existence of the surreal s... |
| seqseq123d 28605 | Equality deduction for the... |
| nfseqs 28606 | Hypothesis builder for the... |
| seqsval 28607 | The value of the surreal s... |
| noseqex 28608 | The next several theorems ... |
| noseq0 28609 | The surreal ` A ` is a mem... |
| noseqp1 28610 | One plus an element of ` Z... |
| noseqind 28611 | Peano's inductive postulat... |
| noseqinds 28612 | Induction schema for surre... |
| noseqssno 28613 | A surreal sequence is a su... |
| noseqno 28614 | An element of a surreal se... |
| om2noseq0 28615 | The mapping ` G ` is a one... |
| om2noseqsuc 28616 | The value of ` G ` at a su... |
| om2noseqfo 28617 | Function statement for ` G... |
| om2noseqlt 28618 | Surreal less-than relation... |
| om2noseqlt2 28619 | The mapping ` G ` preserve... |
| om2noseqf1o 28620 | ` G ` is a bijection. (Co... |
| om2noseqiso 28621 | ` G ` is an isomorphism fr... |
| om2noseqoi 28622 | An alternative definition ... |
| om2noseqrdg 28623 | A helper lemma for the val... |
| noseqrdglem 28624 | A helper lemma for the val... |
| noseqrdgfn 28625 | The recursive definition g... |
| noseqrdg0 28626 | Initial value of a recursi... |
| noseqrdgsuc 28627 | Successor value of a recur... |
| seqsfn 28628 | The surreal sequence build... |
| seqs1 28629 | The value of the surreal s... |
| seqsp1 28630 | The value of the surreal s... |
| n0sexg 28635 | The set of all non-negativ... |
| n0sex 28636 | The set of all non-negativ... |
| nnsex 28637 | The set of all positive su... |
| peano5n0s 28638 | Peano's inductive postulat... |
| n0ssno 28639 | The non-negative surreal i... |
| nnssn0s 28640 | The positive surreal integ... |
| nnssno 28641 | The positive surreal integ... |
| n0no 28642 | A non-negative surreal int... |
| nnno 28643 | A positive surreal integer... |
| n0nod 28644 | A non-negative surreal int... |
| nnnod 28645 | A positive surreal integer... |
| nnn0s 28646 | A positive surreal integer... |
| nnn0sd 28647 | A positive surreal integer... |
| 0n0s 28648 | Peano postulate: ` 0s ` is... |
| peano2n0s 28649 | Peano postulate: the succe... |
| peano2n0sd 28650 | Peano postulate: the succe... |
| dfn0s2 28651 | Alternate definition of th... |
| n0sind 28652 | Principle of Mathematical ... |
| n0cut 28653 | A cut form for non-negativ... |
| n0cut2 28654 | A cut form for the success... |
| n0on 28655 | A surreal natural is a sur... |
| nnne0s 28656 | A surreal positive integer... |
| n0sge0 28657 | A non-negative integer is ... |
| nnsgt0 28658 | A positive integer is grea... |
| elnns 28659 | Membership in the positive... |
| elnns2 28660 | A positive surreal integer... |
| n0s0suc 28661 | A non-negative surreal int... |
| nnsge1 28662 | A positive surreal integer... |
| n0addscl 28663 | The non-negative surreal i... |
| n0mulscl 28664 | The non-negative surreal i... |
| nnaddscl 28665 | The positive surreal integ... |
| nnmulscl 28666 | The positive surreal integ... |
| 1n0s 28667 | Surreal one is a non-negat... |
| 1nns 28668 | Surreal one is a positive ... |
| peano2nns 28669 | Peano postulate for positi... |
| nnsrecgt0d 28670 | The reciprocal of a positi... |
| n0bday 28671 | A non-negative surreal int... |
| n0ssoldg 28672 | The non-negative surreal i... |
| n0ssold 28673 | The non-negative surreal i... |
| n0fincut 28674 | The simplest number greate... |
| onsfi 28675 | A surreal ordinal with a f... |
| eln0s2 28676 | A non-negative surreal int... |
| onltn0s 28677 | A surreal ordinal that is ... |
| n0cutlt 28678 | A non-negative surreal int... |
| seqn0sfn 28679 | The surreal sequence build... |
| eln0s 28680 | A non-negative surreal int... |
| n0s0m1 28681 | Every non-negative surreal... |
| n0subs 28682 | Subtraction of non-negativ... |
| n0subs2 28683 | Subtraction of non-negativ... |
| n0ltsp1le 28684 | Non-negative surreal order... |
| n0lesltp1 28685 | Non-negative surreal order... |
| n0lesm1lt 28686 | Non-negative surreal order... |
| n0lts1e0 28687 | A non-negative surreal int... |
| bdayn0p1 28688 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28689 | The birthday function rest... |
| n0p1nns 28690 | One plus a non-negative su... |
| dfnns2 28691 | Alternate definition of th... |
| nnsind 28692 | Principle of Mathematical ... |
| nn1m1nns 28693 | Every positive surreal int... |
| nnm1n0s 28694 | A positive surreal integer... |
| eucliddivs 28695 | Euclid's division lemma fo... |
| oldfib 28696 | The old set of an ordinal ... |
| zsex 28699 | The surreal integers form ... |
| zssno 28700 | The surreal integers are a... |
| zno 28701 | A surreal integer is a sur... |
| znod 28702 | A surreal integer is a sur... |
| elzs 28703 | Membership in the set of s... |
| nnzsubs 28704 | The difference of two surr... |
| nnzs 28705 | A positive surreal integer... |
| nnzsd 28706 | A positive surreal integer... |
| 0zs 28707 | Zero is a surreal integer.... |
| n0zs 28708 | A non-negative surreal int... |
| n0zsd 28709 | A non-negative surreal int... |
| 1zs 28710 | One is a surreal integer. ... |
| znegscl 28711 | The surreal integers are c... |
| znegscld 28712 | The surreal integers are c... |
| zaddscl 28713 | The surreal integers are c... |
| zaddscld 28714 | The surreal integers are c... |
| zsubscld 28715 | The surreal integers are c... |
| zmulscld 28716 | The surreal integers are c... |
| elzn0s 28717 | A surreal integer is a sur... |
| elzs2 28718 | A surreal integer is eithe... |
| eln0zs 28719 | Non-negative surreal integ... |
| elnnzs 28720 | Positive surreal integer p... |
| elznns 28721 | Surreal integer property e... |
| zn0subs 28722 | The non-negative differenc... |
| peano5uzs 28723 | Peano's inductive postulat... |
| uzsind 28724 | Induction on the upper sur... |
| zsbday 28725 | A surreal integer has a fi... |
| zcuts 28726 | A cut expression for surre... |
| zcuts0 28727 | Either the left or right s... |
| zsoring 28728 | The surreal integers form ... |
| 1p1e2s 28735 | One plus one is two. Surr... |
| no2times 28736 | Version of ~ 2times for su... |
| 2nns 28737 | Surreal two is a surreal n... |
| 2no 28738 | Surreal two is a surreal n... |
| 2ne0s 28739 | Surreal two is nonzero. (... |
| n0seo 28740 | A non-negative surreal int... |
| zseo 28741 | A surreal integer is eithe... |
| twocut 28742 | Two times the cut of zero ... |
| nohalf 28743 | An explicit expression for... |
| expsval 28744 | The value of surreal expon... |
| expnnsval 28745 | Value of surreal exponenti... |
| exps0 28746 | Surreal exponentiation to ... |
| exps1 28747 | Surreal exponentiation to ... |
| expsp1 28748 | Value of a surreal number ... |
| expscllem 28749 | Lemma for proving non-nega... |
| expscl 28750 | Closure law for surreal ex... |
| n0expscl 28751 | Closure law for non-negati... |
| nnexpscl 28752 | Closure law for positive s... |
| zexpscl 28753 | Closure law for surreal in... |
| expadds 28754 | Sum of exponents law for s... |
| expsne0 28755 | A non-negative surreal int... |
| expsgt0 28756 | A non-negative surreal int... |
| pw2recs 28757 | Any power of two has a mul... |
| pw2divscld 28758 | Division closure for power... |
| pw2divmulsd 28759 | Relationship between surre... |
| pw2divscan3d 28760 | Cancellation law for surre... |
| pw2divscan2d 28761 | A cancellation law for sur... |
| pw2divsassd 28762 | An associative law for div... |
| pw2divscan4d 28763 | Cancellation law for divis... |
| pw2gt0divsd 28764 | Division of a positive sur... |
| pw2ge0divsd 28765 | Divison of a non-negative ... |
| pw2divsrecd 28766 | Relationship between surre... |
| pw2divsdird 28767 | Distribution of surreal di... |
| pw2divsnegd 28768 | Move negative sign inside ... |
| pw2ltdivmulsd 28769 | Surreal less-than relation... |
| pw2ltmuldivs2d 28770 | Surreal less-than relation... |
| pw2ltsdiv1d 28771 | Surreal less-than relation... |
| avglts1d 28772 | Ordering property for aver... |
| avglts2d 28773 | Ordering property for aver... |
| pw2divs0d 28774 | Division into zero is zero... |
| pw2divsidd 28775 | Identity law for division ... |
| pw2ltdivmuls2d 28776 | Surreal less-than relation... |
| halfcut 28777 | Relate the cut of twice of... |
| addhalfcut 28778 | The cut of a surreal non-n... |
| pw2cut 28779 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28780 | Simplify ~ pw2cut in the c... |
| pw2cut2 28781 | Cut expression for powers ... |
| bdaypw2n0bndlem 28782 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28783 | Upper bound for the birthd... |
| bdaypw2bnd 28784 | Birthday bounding rule for... |
| bdayfinbndcbv 28785 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28786 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28787 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28788 | Given a non-negative integ... |
| z12bdaylem1 28789 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28790 | Lemma for ~ z12bday . Sho... |
| elz12s 28791 | Membership in the dyadic f... |
| elz12si 28792 | Inference form of membersh... |
| z12sex 28793 | The class of dyadic fracti... |
| zz12s 28794 | A surreal integer is a dya... |
| z12no 28795 | A dyadic is a surreal. (C... |
| z12addscl 28796 | The dyadics are closed und... |
| z12negscl 28797 | The dyadics are closed und... |
| z12subscl 28798 | The dyadics are closed und... |
| z12shalf 28799 | Half of a dyadic is a dyad... |
| z12negsclb 28800 | A surreal is a dyadic frac... |
| z12zsodd 28801 | A dyadic fraction is eithe... |
| z12sge0 28802 | An expression for non-nega... |
| z12bdaylem 28803 | Lemma for ~ z12bday . Han... |
| z12bday 28804 | A dyadic fraction has a fi... |
| bdayfinlem 28805 | Lemma for ~ bdayfin . Han... |
| bdayfin 28806 | A surreal has a finite bir... |
| dfz12s2 28807 | The set of dyadic fraction... |
| elreno 28810 | Membership in the set of s... |
| reno 28811 | A surreal real is a surrea... |
| renod 28812 | A surreal real is a surrea... |
| recut 28813 | The cut involved in defini... |
| elreno2 28814 | Alternate characterization... |
| 0reno 28815 | Surreal zero is a surreal ... |
| 1reno 28816 | Surreal one is a surreal r... |
| renegscl 28817 | The surreal reals are clos... |
| readdscl 28818 | The surreal reals are clos... |
| remulscllem1 28819 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28820 | Lemma for ~ remulscl . Bo... |
| remulscl 28821 | The surreal reals are clos... |
| itvndx 28832 | Index value of the Interva... |
| lngndx 28833 | Index value of the "line" ... |
| itvid 28834 | Utility theorem: index-ind... |
| lngid 28835 | Utility theorem: index-ind... |
| slotsinbpsd 28836 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28837 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28838 | The slot for the line is n... |
| trkgstr 28839 | Functionality of a Tarski ... |
| trkgbas 28840 | The base set of a Tarski g... |
| trkgdist 28841 | The measure of a distance ... |
| trkgitv 28842 | The congruence relation in... |
| istrkgc 28849 | Property of being a Tarski... |
| istrkgb 28850 | Property of being a Tarski... |
| istrkgcb 28851 | Property of being a Tarski... |
| istrkge 28852 | Property of fulfilling Euc... |
| istrkgl 28853 | Building lines from the se... |
| istrkgld 28854 | Property of fulfilling the... |
| istrkg2ld 28855 | Property of fulfilling the... |
| istrkg3ld 28856 | Property of fulfilling the... |
| axtgcgrrflx 28857 | Axiom of reflexivity of co... |
| axtgcgrid 28858 | Axiom of identity of congr... |
| axtgsegcon 28859 | Axiom of segment construct... |
| axtg5seg 28860 | Five segments axiom, Axiom... |
| axtgbtwnid 28861 | Identity of Betweenness. ... |
| axtgpasch 28862 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28863 | Axiom of Continuity. Axio... |
| axtgcont 28864 | Axiom of Continuity. Axio... |
| axtglowdim2 28865 | Lower dimension axiom for ... |
| axtgupdim2 28866 | Upper dimension axiom for ... |
| axtgeucl 28867 | Euclid's Axiom. Axiom A10... |
| tgjustf 28868 | Given any function ` F ` ,... |
| tgjustr 28869 | Given any equivalence rela... |
| tgjustc1 28870 | A justification for using ... |
| tgjustc2 28871 | A justification for using ... |
| tgcgrcomimp 28872 | Congruence commutes on the... |
| tgcgrcomr 28873 | Congruence commutes on the... |
| tgcgrcoml 28874 | Congruence commutes on the... |
| tgcgrcomlr 28875 | Congruence commutes on bot... |
| tgcgreqb 28876 | Congruence and equality. ... |
| tgcgreq 28877 | Congruence and equality. ... |
| tgcgrneq 28878 | Congruence and equality. ... |
| tgcgrtriv 28879 | Degenerate segments are co... |
| tgcgrextend 28880 | Link congruence over a pai... |
| tgsegconeq 28881 | Two points that satisfy th... |
| tgsegconeu 28882 | The point constructed in ~... |
| tgbtwntriv2 28883 | Betweenness always holds f... |
| tgbtwncom 28884 | Betweenness commutes. The... |
| tgbtwncomb 28885 | Betweenness commutes, bico... |
| tgbtwnne 28886 | Betweenness and inequality... |
| tgbtwntriv1 28887 | Betweenness always holds f... |
| tgbtwnswapid 28888 | If you can swap the first ... |
| tgbtwnintr 28889 | Inner transitivity law for... |
| tgbtwnexch3 28890 | Exchange the first endpoin... |
| tgbtwnouttr2 28891 | Outer transitivity law for... |
| tgbtwnexch2 28892 | Exchange the outer point o... |
| tgbtwnouttr 28893 | Outer transitivity law for... |
| tgbtwnexch 28894 | Outer transitivity law for... |
| tgtrisegint 28895 | A line segment between two... |
| tglowdim1 28896 | Lower dimension axiom for ... |
| tglowdim1i 28897 | Lower dimension axiom for ... |
| tgldimor 28898 | Excluded-middle like state... |
| tgldim0eq 28899 | In dimension zero, any two... |
| tgldim0itv 28900 | In dimension zero, any two... |
| tgldim0cgr 28901 | In dimension zero, any two... |
| tgbtwndiff 28902 | There is always a ` c ` di... |
| tgdim01 28903 | In geometries of dimension... |
| tgifscgr 28904 | Inner five segment congrue... |
| tgcgrsub 28905 | Removing identical parts f... |
| iscgrg 28908 | The congruence property fo... |
| iscgrgd 28909 | The property for two seque... |
| iscgrglt 28910 | The property for two seque... |
| trgcgrg 28911 | The property for two trian... |
| trgcgr 28912 | Triangle congruence. (Con... |
| ercgrg 28913 | The shape congruence relat... |
| tgcgrxfr 28914 | A line segment can be divi... |
| cgr3id 28915 | Reflexivity law for three-... |
| cgr3simp1 28916 | Deduce segment congruence ... |
| cgr3simp2 28917 | Deduce segment congruence ... |
| cgr3simp3 28918 | Deduce segment congruence ... |
| cgr3swap12 28919 | Permutation law for three-... |
| cgr3swap23 28920 | Permutation law for three-... |
| cgr3swap13 28921 | Permutation law for three-... |
| cgr3rotr 28922 | Permutation law for three-... |
| cgr3rotl 28923 | Permutation law for three-... |
| trgcgrcom 28924 | Commutative law for three-... |
| cgr3tr 28925 | Transitivity law for three... |
| tgbtwnxfr 28926 | A condition for extending ... |
| tgcgr4 28927 | Two quadrilaterals to be c... |
| isismt 28930 | Property of being an isome... |
| ismot 28931 | Property of being an isome... |
| motcgr 28932 | Property of a motion: dist... |
| idmot 28933 | The identity is a motion. ... |
| motf1o 28934 | Motions are bijections. (... |
| motcl 28935 | Closure of motions. (Cont... |
| motco 28936 | The composition of two mot... |
| cnvmot 28937 | The converse of a motion i... |
| motplusg 28938 | The operation for motions ... |
| motgrp 28939 | The motions of a geometry ... |
| motcgrg 28940 | Property of a motion: dist... |
| motcgr3 28941 | Property of a motion: dist... |
| tglng 28942 | Lines of a Tarski Geometry... |
| tglnfn 28943 | Lines as functions. (Cont... |
| tglnunirn 28944 | Lines are sets of points. ... |
| tglnpt 28945 | Lines are sets of points. ... |
| tglngne 28946 | It takes two different poi... |
| tglngval 28947 | The line going through poi... |
| tglnssp 28948 | Lines are subset of the ge... |
| tgellng 28949 | Property of lying on the l... |
| tgcolg 28950 | We choose the notation ` (... |
| btwncolg1 28951 | Betweenness implies coline... |
| btwncolg2 28952 | Betweenness implies coline... |
| btwncolg3 28953 | Betweenness implies coline... |
| colcom 28954 | Swapping the points defini... |
| colrot1 28955 | Rotating the points defini... |
| colrot2 28956 | Rotating the points defini... |
| ncolcom 28957 | Swapping non-colinear poin... |
| ncolrot1 28958 | Rotating non-colinear poin... |
| ncolrot2 28959 | Rotating non-colinear poin... |
| tgdim01ln 28960 | In geometries of dimension... |
| ncoltgdim2 28961 | If there are three non-col... |
| lnxfr 28962 | Transfer law for colineari... |
| lnext 28963 | Extend a line with a missi... |
| tgfscgr 28964 | Congruence law for the gen... |
| lncgr 28965 | Congruence rule for lines.... |
| lnid 28966 | Identity law for points on... |
| tgidinside 28967 | Law for finding a point in... |
| tgbtwnconn1lem1 28968 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28969 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28970 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28971 | Connectivity law for betwe... |
| tgbtwnconn2 28972 | Another connectivity law f... |
| tgbtwnconn3 28973 | Inner connectivity law for... |
| tgbtwnconnln3 28974 | Derive colinearity from be... |
| tgbtwnconn22 28975 | Double connectivity law fo... |
| tgbtwnconnln1 28976 | Derive colinearity from be... |
| tgbtwnconnln2 28977 | Derive colinearity from be... |
| legval 28980 | Value of the less-than rel... |
| legov 28981 | Value of the less-than rel... |
| legov2 28982 | An equivalent definition o... |
| legid 28983 | Reflexivity of the less-th... |
| btwnleg 28984 | Betweenness implies less-t... |
| legtrd 28985 | Transitivity of the less-t... |
| legtri3 28986 | Equality from the less-tha... |
| legtrid 28987 | Trichotomy law for the les... |
| leg0 28988 | Degenerated (zero-length) ... |
| legeq 28989 | Deduce equality from "less... |
| legbtwn 28990 | Deduce betweenness from "l... |
| tgcgrsub2 28991 | Removing identical parts f... |
| ltgseg 28992 | The set ` E ` denotes the ... |
| ltgov 28993 | Strict "shorter than" geom... |
| legov3 28994 | An equivalent definition o... |
| legso 28995 | The "shorter than" relatio... |
| ishlg2 28998 | Alternate version of ~ ish... |
| hlgrcl1 28999 | Reverse closure for rays. ... |
| hlgrcl2 29000 | Reverse closure for rays. ... |
| ishlg 29001 | Rays : Definition 6.1 of ... |
| hlcomb 29002 | The half-line relation is ... |
| hlcomd 29003 | The half-line relation is ... |
| hlne1 29004 | The half-line relation imp... |
| hlne2 29005 | The half-line relation imp... |
| hlln 29006 | The half-line relation imp... |
| hleqnid 29007 | The endpoint does not belo... |
| hlid 29008 | The half-line relation is ... |
| hltr 29009 | The half-line relation is ... |
| hlbtwn 29010 | Betweenness is a sufficien... |
| btwnhl1 29011 | Deduce half-line from betw... |
| btwnhl2 29012 | Deduce half-line from betw... |
| btwnhl 29013 | Swap betweenness for a hal... |
| lnhl 29014 | Either a point ` C ` on th... |
| hlcgrex 29015 | Construct a point on a hal... |
| hlcgreulem 29016 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 29017 | The point constructed in ~... |
| hlcgreq 29018 | A constructed point on a h... |
| tghlsub 29019 | Removing identical parts f... |
| btwnlng1 29020 | Betweenness implies coline... |
| btwnlng2 29021 | Betweenness implies coline... |
| btwnlng3 29022 | Betweenness implies coline... |
| lncom 29023 | Swapping the points defini... |
| lnrot1 29024 | Rotating the points defini... |
| lnrot2 29025 | Rotating the points defini... |
| ncolne1 29026 | Non-colinear points are di... |
| ncolne2 29027 | Non-colinear points are di... |
| tgisline 29028 | The property of being a pr... |
| tglnne 29029 | It takes two different poi... |
| tglndim0 29030 | There are no lines in dime... |
| tgelrnln 29031 | The property of being a pr... |
| tglineeltr 29032 | Transitivity law for lines... |
| tglineelsb2 29033 | If ` S ` lies on PQ , then... |
| tglinerflx1 29034 | Reflexivity law for line m... |
| tglinerflx2 29035 | Reflexivity law for line m... |
| tglinecom 29036 | Commutativity law for line... |
| tglinethru 29037 | If ` A ` is a line contain... |
| tghilberti1 29038 | There is a line through an... |
| tghilberti2 29039 | There is at most one line ... |
| tglinethrueu 29040 | There is a unique line goi... |
| tglinesseq 29041 | If a line is a subset of a... |
| tglnne0 29042 | A line ` A ` has at least ... |
| tglineintmo 29043 | Two distinct lines interse... |
| tglineineq 29044 | Two distinct lines interse... |
| tglineinsn 29045 | If two distinct lines inte... |
| tglineneq 29046 | Given three non-colinear p... |
| tglineinteq 29047 | Two distinct lines interse... |
| ncolncol 29048 | Deduce non-colinearity fro... |
| coltr 29049 | A transitivity law for col... |
| coltr3 29050 | A transitivity law for col... |
| colline 29051 | Three points are colinear ... |
| tglowdim2l 29052 | Reformulation of the lower... |
| tglowdim2ln 29053 | There is always one point ... |
| tglnpt2 29054 | Find a second point on a l... |
| tglnpt3 29055 | Find a third point on a li... |
| tglnpt4 29056 | Find a second point on a l... |
| mirreu3 29059 | Existential uniqueness of ... |
| mirval 29060 | Value of the point inversi... |
| mirfv 29061 | Value of the point inversi... |
| mircgr 29062 | Property of the image by t... |
| mirbtwn 29063 | Property of the image by t... |
| ismir 29064 | Property of the image by t... |
| mirf 29065 | Point inversion as functio... |
| mircl 29066 | Closure of the point inver... |
| mirmir 29067 | The point inversion functi... |
| mircom 29068 | Variation on ~ mirmir . (... |
| mirreu 29069 | Any point has a unique ant... |
| mireq 29070 | Equality deduction for poi... |
| mirinv 29071 | The only invariant point o... |
| mirne 29072 | Mirror of non-center point... |
| mircinv 29073 | The center point is invari... |
| mirf1o 29074 | The point inversion functi... |
| miriso 29075 | The point inversion functi... |
| mirbtwni 29076 | Point inversion preserves ... |
| mirbtwnb 29077 | Point inversion preserves ... |
| mircgrs 29078 | Point inversion preserves ... |
| mirmir2 29079 | Point inversion of a point... |
| mirmot 29080 | Point investion is a motio... |
| mirln 29081 | If two points are on the s... |
| mirln2 29082 | If a point and its mirror ... |
| mirconn 29083 | Point inversion of connect... |
| mirhl 29084 | If two points ` X ` and ` ... |
| mirbtwnhl 29085 | If the center of the point... |
| mirhl2 29086 | Deduce half-line relation ... |
| mircgrextend 29087 | Link congruence over a pai... |
| mirtrcgr 29088 | Point inversion of one poi... |
| mirauto 29089 | Point inversion preserves ... |
| miduniq 29090 | Uniqueness of the middle p... |
| miduniq1 29091 | Uniqueness of the middle p... |
| miduniq2 29092 | If two point inversions co... |
| colmid 29093 | Colinearity and equidistan... |
| symquadlem 29094 | Lemma of the symmetrical q... |
| krippenlem 29095 | Lemma for ~ krippen . We ... |
| krippen 29096 | Krippenlemma (German for c... |
| midexlem 29097 | Lemma for the existence of... |
| symquadprlnglem 29098 | Lemma for ~ symquadprlngle... |
| mirleqb 29099 | Equality theorem for point... |
| mirlni 29100 | The mirror of a point ` X ... |
| israg 29105 | Property for 3 points A, B... |
| ragcom 29106 | Commutative rule for right... |
| ragcol 29107 | The right angle property i... |
| ragmir 29108 | Right angle property is pr... |
| mirrag 29109 | Right angle is conserved b... |
| ragtrivb 29110 | Trivial right angle. Theo... |
| ragflat2 29111 | Deduce equality from two r... |
| ragflat 29112 | Deduce equality from two r... |
| ragtriva 29113 | Trivial right angle. Theo... |
| ragflat3 29114 | Right angle and colinearit... |
| ragcgr 29115 | Right angle and colinearit... |
| motrag 29116 | Right angles are preserved... |
| ragncol 29117 | Right angle implies non-co... |
| perpln1 29118 | Derive a line from perpend... |
| perpln2 29119 | Derive a line from perpend... |
| isperp 29120 | Property for 2 lines A, B ... |
| perpcom 29121 | The "perpendicular" relati... |
| perpneq 29122 | Two perpendicular lines ar... |
| isperp2 29123 | Property for 2 lines A, B,... |
| isperp2d 29124 | One direction of ~ isperp2... |
| ragperp 29125 | Deduce that two lines are ... |
| footexALT 29126 | Alternative version of ~ f... |
| footexlem1 29127 | Lemma for ~ footex . (Con... |
| footexlem2 29128 | Lemma for ~ footex . (Con... |
| footex 29129 | From a point ` C ` outside... |
| foot 29130 | From a point ` C ` outside... |
| footne 29131 | Uniqueness of the foot poi... |
| footeq 29132 | Uniqueness of the foot poi... |
| perpin 29133 | If two lines ` A ` and ` B... |
| hlperpnel 29134 | A point on a half-line whi... |
| perprag 29135 | Deduce a right angle from ... |
| perpdragALT 29136 | Deduce a right angle from ... |
| perpdrag 29137 | Deduce a right angle from ... |
| colperp 29138 | Deduce a perpendicularity ... |
| colperpexlem1 29139 | Lemma for ~ colperp . Fir... |
| colperpexlem2 29140 | Lemma for ~ colperpex . S... |
| colperpexlem3 29141 | Lemma for ~ colperpex . C... |
| colperpex 29142 | In dimension 2 and above, ... |
| mideulem2 29143 | Lemma for ~ opphllem , whi... |
| opphllem 29144 | Lemma 8.24 of [Schwabhause... |
| mideulem 29145 | Lemma for ~ mideu . We ca... |
| midex 29146 | Existence of the midpoint,... |
| mideu 29147 | Existence and uniqueness o... |
| islnopp 29148 | The property for two point... |
| islnoppd 29149 | Deduce that ` A ` and ` B ... |
| oppne1 29150 | Points lying on opposite s... |
| oppne2 29151 | Points lying on opposite s... |
| oppne3 29152 | Points lying on opposite s... |
| oppcom 29153 | Commutativity rule for "op... |
| opptgdim2 29154 | If two points opposite to ... |
| oppnid 29155 | The "opposite to a line" r... |
| opphllem1 29156 | Lemma for ~ opphl . (Cont... |
| opphllem2 29157 | Lemma for ~ opphl . Lemma... |
| opphllem3 29158 | Lemma for ~ opphl : We as... |
| opphllem4 29159 | Lemma for ~ opphl . (Cont... |
| opphllem5 29160 | Second part of Lemma 9.4 o... |
| opphllem6 29161 | First part of Lemma 9.4 of... |
| oppperpex 29162 | Restating ~ colperpex usin... |
| opphl 29163 | If two points ` A ` and ` ... |
| lnoppinn0 29164 | The segment between two po... |
| oppmir 29165 | The mirror point with rega... |
| outpasch 29166 | Axiom of Pasch, outer form... |
| hlpasch 29167 | An application of the axio... |
| ishpg 29170 | Value of the half-plane re... |
| hpgbr 29171 | Half-planes : property for... |
| hpgne1 29172 | Points on the open half pl... |
| hpgne2 29173 | Points on the open half pl... |
| lnopp2hpgb 29174 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 29175 | If two points lie on the o... |
| hpgerlem 29176 | Lemma for the proof that t... |
| hpgid 29177 | The half-plane relation is... |
| hpgcom 29178 | The half-plane relation is... |
| hpgtr 29179 | The half-plane relation is... |
| colopp 29180 | Opposite sides of a line f... |
| colhp 29181 | Half-plane relation for co... |
| hphl 29182 | If two points are on the s... |
| hlopp 29183 | If two points ` X ` and ` ... |
| tgplnfn 29186 | The plane generating funct... |
| tgelrnpln 29187 | The property of being a pl... |
| plngval 29188 | The plane defined by a lin... |
| isplng 29189 | The property of being a pl... |
| plngrnssp 29190 | Planes are sets of points.... |
| elplng 29191 | Elementhood in the plane d... |
| plngssp 29192 | Planes are sets of points.... |
| elplngid 29193 | The point ` R ` is itself ... |
| elplnglnid 29194 | The line ` A ` itself is a... |
| lnincplng 29195 | If two lines ` A ` and ` B... |
| plngcplem 29196 | Lemma for ~ plngcp . (Con... |
| plngcp 29197 | The plane defined by a lin... |
| plngrotlem1 29198 | Lemma for ~ plngrot . (Co... |
| plngrotlem2 29199 | Lemma for ~ plngrot . (Co... |
| plngrotlem3 29200 | Lemma for ~ plngrot . (Co... |
| plngrot 29201 | The plane defined by a lin... |
| lnssplnglem 29202 | Lemma for ~ lnssplng . (C... |
| lnssplng 29203 | A line defined by two poin... |
| lnssplng1 29204 | A line defined by two poin... |
| plngmiropp 29205 | Given a line ` A ` and a p... |
| mirplncl 29206 | The mirror of a point with... |
| hpgssplng 29207 | Any point ` X ` on a half ... |
| plng3p 29208 | If ` H ` is a plane contai... |
| nhpmirhp 29209 | If a point ` Z ` is on the... |
| midf 29214 | Midpoint as a function. (... |
| midcl 29215 | Closure of the midpoint. ... |
| ismidb 29216 | Property of the midpoint. ... |
| midbtwn 29217 | Betweenness of midpoint. ... |
| midcgr 29218 | Congruence of midpoint. (... |
| midid 29219 | Midpoint of a null segment... |
| midcom 29220 | Commutativity rule for the... |
| mirmid 29221 | Point inversion preserves ... |
| lmieu 29222 | Uniqueness of the line mir... |
| lmif 29223 | Line mirror as a function.... |
| lmicl 29224 | Closure of the line mirror... |
| islmib 29225 | Property of the line mirro... |
| lmicom 29226 | The line mirroring functio... |
| lmilmi 29227 | Line mirroring is an invol... |
| lmireu 29228 | Any point has a unique ant... |
| lmieq 29229 | Equality deduction for lin... |
| lmiinv 29230 | The invariants of the line... |
| lmicinv 29231 | The mirroring line is an i... |
| lmimid 29232 | If we have a right angle, ... |
| lmif1o 29233 | The line mirroring functio... |
| lmiisolem 29234 | Lemma for ~ lmiiso . (Con... |
| lmiiso 29235 | The line mirroring functio... |
| lmimot 29236 | Line mirroring is a motion... |
| symquadmid 29237 | In a symmetrical quadrilat... |
| hypcgrlem1 29238 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 29239 | Lemma for ~ hypcgr , case ... |
| hypcgr 29240 | If the catheti of two righ... |
| lmiopp 29241 | Line mirroring produces po... |
| lnperpex 29242 | Existence of a perpendicul... |
| lnperpexs 29243 | Existence of a perpendicul... |
| trgcopy 29244 | Triangle construction: a c... |
| trgcopyeulem 29245 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 29246 | Triangle construction: a c... |
| iscgra 29249 | Property for two angles AB... |
| iscgra1 29250 | A special version of ~ isc... |
| iscgrad 29251 | Sufficient conditions for ... |
| cgrane1 29252 | Angles imply inequality. ... |
| cgrane2 29253 | Angles imply inequality. ... |
| cgrane3 29254 | Angles imply inequality. ... |
| cgrane4 29255 | Angles imply inequality. ... |
| cgrahl1 29256 | Angle congruence is indepe... |
| cgrahl2 29257 | Angle congruence is indepe... |
| cgracgr 29258 | First direction of proposi... |
| cgraid 29259 | Angle congruence is reflex... |
| cgraswap 29260 | Swap rays in a congruence ... |
| cgrcgra 29261 | Triangle congruence implie... |
| cgracom 29262 | Angle congruence commutes.... |
| cgratr 29263 | Angle congruence is transi... |
| zerocgra 29264 | Zero angles are congruent.... |
| flatcgra 29265 | Flat angles are congruent.... |
| cgraswaplr 29266 | Swap both side of angle co... |
| cgrabtwn 29267 | Angle congruence preserves... |
| cgrahl 29268 | Angle congruence preserves... |
| cgracol 29269 | Angle congruence preserves... |
| cgrancol 29270 | Angle congruence preserves... |
| dfcgra2 29271 | This is the full statement... |
| sacgr 29272 | Supplementary angles of co... |
| oacgr 29273 | Vertical angle theorem. V... |
| acopy 29274 | Angle construction. Theor... |
| acopyeu 29275 | Angle construction. Theor... |
| ragcgra 29276 | Right angles are congruent... |
| cgrarag 29277 | Any angle ` <" A B C "> ` ... |
| ragsupplcgra 29278 | An angle ` <" X Y Z "> ` i... |
| ragraghl 29279 | Drawing two right angles a... |
| perpeqlem 29280 | Lemma for ~ perpeq . (Con... |
| perpeq 29281 | Uniqueness of the perpendi... |
| tgaaddcpbllem1 29282 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbllem2 29283 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbllem3 29284 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbl 29285 | The angular addition is co... |
| tgaaddcpbl2 29286 | The angular addition is co... |
| isinag 29290 | Property for point ` X ` t... |
| isinagd 29291 | Sufficient conditions for ... |
| inagflat 29292 | Any point lies in a flat a... |
| inagswap 29293 | Swap the order of the half... |
| inagne1 29294 | Deduce inequality from the... |
| inagne2 29295 | Deduce inequality from the... |
| inagne3 29296 | Deduce inequality from the... |
| inaghl 29297 | The "point lie in angle" r... |
| isleag 29299 | Geometrical "less than" pr... |
| isleagd 29300 | Sufficient condition for "... |
| leagne1 29301 | Deduce inequality from the... |
| leagne2 29302 | Deduce inequality from the... |
| leagne3 29303 | Deduce inequality from the... |
| leagne4 29304 | Deduce inequality from the... |
| cgrg3col4 29305 | Lemma 11.28 of [Schwabhaus... |
| elcgrabasi 29308 | Helper theorem for the mem... |
| elcgrabasrd 29309 | Helper theorem for the mem... |
| cgraer 29310 | The angle congruence relat... |
| cgrabasimass 29311 | The angle congruence relat... |
| angmgmaddeu1 29312 | There exists a unique poin... |
| angmgmaddeu2 29313 | Existence of a unique poin... |
| angmgmaddeu3 29314 | Existence of a unique poin... |
| angmgmaddeu4 29315 | There exists a unique poin... |
| angmgmaddeu5 29316 | There exists a unique poin... |
| angmgmaddeu6 29317 | There exists a unique poin... |
| angmgmaddeu7 29318 | There exists a unique poin... |
| angmgmaddov1lem 29319 | Lemma for ~ angmgmaddov1 .... |
| angmgmaddov2lem 29320 | Lemma for ~ angmgmaddov2 .... |
| angmgmaddov1 29321 | Value of the addition oper... |
| angmgmaddov2 29322 | Value of the addition oper... |
| angmgmaddcpbl 29323 | Addition of angles is comp... |
| angmgmaddcl 29324 | Closure of the addition of... |
| angmgmaddlid 29325 | The left identity element ... |
| angmgmaddrid 29326 | The right identity element... |
| angmgmval 29327 | Explicit the value of the ... |
| angmgmlem 29328 | Lemma for ~ angmgm . (Con... |
| angmgm0g 29329 | The identity element of th... |
| angmgm 29330 | The angle addition magma i... |
| angmgmbas 29331 | The base set of the angle ... |
| tgsas1 29332 | First congruence theorem: ... |
| tgsas 29333 | First congruence theorem: ... |
| tgsas2 29334 | First congruence theorem: ... |
| tgsas3 29335 | First congruence theorem: ... |
| tgasa1 29336 | Second congruence theorem:... |
| tgasa 29337 | Second congruence theorem:... |
| tgsss1 29338 | Third congruence theorem: ... |
| tgsss2 29339 | Third congruence theorem: ... |
| tgsss3 29340 | Third congruence theorem: ... |
| dfcgrg2 29341 | Congruence for two triangl... |
| isoas 29342 | Congruence theorem for iso... |
| iseqlg 29345 | Property of a triangle bei... |
| iseqlgd 29346 | Condition for a triangle t... |
| brprlng 29349 | Property of two lines ` A ... |
| prlngd 29350 | Deduce parallelism between... |
| prlngref 29351 | Parallelism is reflexive. ... |
| prlngsym 29352 | Parallelism is symmetric. ... |
| prlngrcl1 29353 | Reverse closure for parall... |
| prlngrcl2 29354 | Reverse closure for parall... |
| prlngin0 29355 | Two parallel lines do not ... |
| prlngpln 29356 | Two parallel lines are on ... |
| prlnghpg 29357 | If two lines ` A ` and ` B... |
| dfprlng2 29358 | Alternate definition of (s... |
| dfprlng3 29359 | Alternate definition of (s... |
| prlngpln3 29360 | Two parallel lines are on ... |
| perpprlng 29361 | If two lines ` A ` and ` B... |
| prlngex 29362 | There exists at least one ... |
| prlngmolem1 29363 | Lemma for ~ prlngmo : Con... |
| prlngmolem2 29364 | Lemma for ~ prlngmo . (Co... |
| prlngmo 29365 | Playfair's axiom. Given a... |
| prlngeu 29366 | Given a line ` A ` and a p... |
| prlngmo2 29367 | Playfair's axiom, without ... |
| prlngeq 29368 | Playfair's axiom, written ... |
| prlngpln4 29369 | Building a parallel line c... |
| prlngplngtr 29370 | Transitivity of parallelis... |
| prlnginn0 29371 | A line ` C ` intersecting ... |
| prlngmid2 29372 | If the midpoints of two se... |
| symquadprlng 29373 | Symmetrical quadrilaterals... |
| prlngsymquadlem 29374 | Lemma for ~ prlngsymquad .... |
| prlngsymquad 29375 | All parallelograms are sym... |
| prlngsymquadopp 29376 | In parallelograms, opposin... |
| quadcgrprlng 29377 | Nontrivial quadrilaterals ... |
| tgaltai 29378 | Parallelism implies altern... |
| f1otrgds 29379 | Convenient lemma for ~ f1o... |
| f1otrgitv 29380 | Convenient lemma for ~ f1o... |
| f1otrg 29381 | A bijection between bases ... |
| f1otrge 29382 | A bijection between bases ... |
| ttgval 29385 | Define a function to augme... |
| ttglem 29386 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 29387 | The base set of a subcompl... |
| ttgplusg 29388 | The addition operation of ... |
| ttgsub 29389 | The subtraction operation ... |
| ttgvsca 29390 | The scalar product of a su... |
| ttgds 29391 | The metric of a subcomplex... |
| ttgitvval 29392 | Betweenness for a subcompl... |
| ttgelitv 29393 | Betweenness for a subcompl... |
| ttgbtwnid 29394 | Any subcomplex module equi... |
| ttgcontlem1 29395 | Lemma for % ttgcont . (Co... |
| xmstrkgc 29396 | Any metric space fulfills ... |
| cchhllem 29397 | Lemma for chlbas and chlvs... |
| elee 29404 | Membership in a Euclidean ... |
| mptelee 29405 | A condition for a mapping ... |
| mpteleeOLD 29406 | Obsolete version of ~ mpte... |
| eleenn 29407 | If ` A ` is in ` ( EE `` N... |
| eleei 29408 | The forward direction of ~... |
| eedimeq 29409 | A point belongs to at most... |
| brbtwn 29410 | The binary relation form o... |
| brcgr 29411 | The binary relation form o... |
| fveere 29412 | The function value of a po... |
| fveecn 29413 | The function value of a po... |
| eqeefv 29414 | Two points are equal iff t... |
| eqeelen 29415 | Two points are equal iff t... |
| brbtwn2 29416 | Alternate characterization... |
| colinearalglem1 29417 | Lemma for ~ colinearalg . ... |
| colinearalglem2 29418 | Lemma for ~ colinearalg . ... |
| colinearalglem3 29419 | Lemma for ~ colinearalg . ... |
| colinearalglem4 29420 | Lemma for ~ colinearalg . ... |
| colinearalg 29421 | An algebraic characterizat... |
| eleesub 29422 | Membership of a subtractio... |
| eleesubd 29423 | Membership of a subtractio... |
| axdimuniq 29424 | The unique dimension axiom... |
| axcgrrflx 29425 | ` A ` is as far from ` B `... |
| axcgrtr 29426 | Congruence is transitive. ... |
| axcgrid 29427 | If there is no distance be... |
| axsegconlem1 29428 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 29429 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 29430 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 29431 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 29432 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 29433 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 29434 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 29435 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 29436 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 29437 | Lemma for ~ axsegcon . Sh... |
| axsegcon 29438 | Any segment ` A B ` can be... |
| ax5seglem1 29439 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 29440 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 29441 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 29442 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 29443 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 29444 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 29445 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 29446 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 29447 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 29448 | Lemma for ~ ax5seg . Take... |
| ax5seg 29449 | The five segment axiom. T... |
| axbtwnid 29450 | Points are indivisible. T... |
| axpaschlem 29451 | Lemma for ~ axpasch . Set... |
| axpasch 29452 | The inner Pasch axiom. Ta... |
| axlowdimlem1 29453 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29454 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29455 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29456 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29457 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29458 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29459 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29460 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29461 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29462 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29463 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29464 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29465 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29466 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29467 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29468 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29469 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29470 | The lower dimension axiom ... |
| axlowdim2 29471 | The lower two-dimensional ... |
| axlowdim 29472 | The general lower dimensio... |
| axeuclidlem 29473 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29474 | Euclid's axiom. Take an a... |
| axcontlem1 29475 | Lemma for ~ axcont . Chan... |
| axcontlem2 29476 | Lemma for ~ axcont . The ... |
| axcontlem3 29477 | Lemma for ~ axcont . Give... |
| axcontlem4 29478 | Lemma for ~ axcont . Give... |
| axcontlem5 29479 | Lemma for ~ axcont . Comp... |
| axcontlem6 29480 | Lemma for ~ axcont . Stat... |
| axcontlem7 29481 | Lemma for ~ axcont . Give... |
| axcontlem8 29482 | Lemma for ~ axcont . A po... |
| axcontlem9 29483 | Lemma for ~ axcont . Give... |
| axcontlem10 29484 | Lemma for ~ axcont . Give... |
| axcontlem11 29485 | Lemma for ~ axcont . Elim... |
| axcontlem12 29486 | Lemma for ~ axcont . Elim... |
| axcont 29487 | The axiom of continuity. ... |
| eengv 29490 | The value of the Euclidean... |
| eengstr 29491 | The Euclidean geometry as ... |
| eengbas 29492 | The Base of the Euclidean ... |
| ebtwntg 29493 | The betweenness relation u... |
| ecgrtg 29494 | The congruence relation us... |
| elntg 29495 | The line definition in the... |
| elntg2 29496 | The line definition in the... |
| eengtrkg 29497 | The geometry structure for... |
| eengtrkge 29498 | The geometry structure for... |
| edgfid 29501 | Utility theorem: index-ind... |
| edgfndx 29502 | Index value of the ~ df-ed... |
| edgfndxnn 29503 | The index value of the edg... |
| edgfndxid 29504 | The value of the edge func... |
| basendxltedgfndx 29505 | The index value of the ` B... |
| basendxnedgfndx 29506 | The slots ` Base ` and ` .... |
| vtxval 29511 | The set of vertices of a g... |
| iedgval 29512 | The set of indexed edges o... |
| 1vgrex 29513 | A graph with at least one ... |
| opvtxval 29514 | The set of vertices of a g... |
| opvtxfv 29515 | The set of vertices of a g... |
| opvtxov 29516 | The set of vertices of a g... |
| opiedgval 29517 | The set of indexed edges o... |
| opiedgfv 29518 | The set of indexed edges o... |
| opiedgov 29519 | The set of indexed edges o... |
| opvtxfvi 29520 | The set of vertices of a g... |
| opiedgfvi 29521 | The set of indexed edges o... |
| funvtxdmge2val 29522 | The set of vertices of an ... |
| funiedgdmge2val 29523 | The set of indexed edges o... |
| funvtxdm2val 29524 | The set of vertices of an ... |
| funiedgdm2val 29525 | The set of indexed edges o... |
| funvtxval0 29526 | The set of vertices of an ... |
| basvtxval 29527 | The set of vertices of a g... |
| edgfiedgval 29528 | The set of indexed edges o... |
| funvtxval 29529 | The set of vertices of a g... |
| funiedgval 29530 | The set of indexed edges o... |
| structvtxvallem 29531 | Lemma for ~ structvtxval a... |
| structvtxval 29532 | The set of vertices of an ... |
| structiedg0val 29533 | The set of indexed edges o... |
| structgrssvtxlem 29534 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29535 | The set of vertices of a g... |
| structgrssiedg 29536 | The set of indexed edges o... |
| struct2grstr 29537 | A graph represented as an ... |
| struct2grvtx 29538 | The set of vertices of a g... |
| struct2griedg 29539 | The set of indexed edges o... |
| graop 29540 | Any representation of a gr... |
| grastruct 29541 | Any representation of a gr... |
| gropd 29542 | If any representation of a... |
| grstructd 29543 | If any representation of a... |
| gropeld 29544 | If any representation of a... |
| grstructeld 29545 | If any representation of a... |
| setsvtx 29546 | The vertices of a structur... |
| setsiedg 29547 | The (indexed) edges of a s... |
| snstrvtxval 29548 | The set of vertices of a g... |
| snstriedgval 29549 | The set of indexed edges o... |
| vtxval0 29550 | Degenerated case 1 for ver... |
| iedgval0 29551 | Degenerated case 1 for edg... |
| vtxvalsnop 29552 | Degenerated case 2 for ver... |
| iedgvalsnop 29553 | Degenerated case 2 for edg... |
| vtxval3sn 29554 | Degenerated case 3 for ver... |
| iedgval3sn 29555 | Degenerated case 3 for edg... |
| vtxvalprc 29556 | Degenerated case 4 for ver... |
| iedgvalprc 29557 | Degenerated case 4 for edg... |
| edgval 29560 | The edges of a graph. (Co... |
| iedgedg 29561 | An indexed edge is an edge... |
| edgopval 29562 | The edges of a graph repre... |
| edgov 29563 | The edges of a graph repre... |
| edgstruct 29564 | The edges of a graph repre... |
| edgiedgb 29565 | A set is an edge iff it is... |
| edg0iedg0 29566 | There is no edge in a grap... |
| isuhgr 29571 | The predicate "is an undir... |
| isushgr 29572 | The predicate "is an undir... |
| uhgrf 29573 | The edge function of an un... |
| ushgrf 29574 | The edge function of an un... |
| uhgrss 29575 | An edge is a subset of ver... |
| uhgreq12g 29576 | If two sets have the same ... |
| uhgrfun 29577 | The edge function of an un... |
| uhgrn0 29578 | An edge is a nonempty subs... |
| lpvtx 29579 | The endpoints of a loop (w... |
| ushgruhgr 29580 | An undirected simple hyper... |
| isuhgrop 29581 | The property of being an u... |
| uhgr0e 29582 | The empty graph, with vert... |
| uhgr0vb 29583 | The null graph, with no ve... |
| uhgr0 29584 | The null graph represented... |
| uhgrun 29585 | The union ` U ` of two (un... |
| uhgrunop 29586 | The union of two (undirect... |
| ushgrun 29587 | The union ` U ` of two (un... |
| ushgrunop 29588 | The union of two (undirect... |
| uhgrstrrepe 29589 | Replacing (or adding) the ... |
| incistruhgr 29590 | An _incidence structure_ `... |
| isupgr 29595 | The property of being an u... |
| wrdupgr 29596 | The property of being an u... |
| upgrf 29597 | The edge function of an un... |
| upgrfn 29598 | The edge function of an un... |
| upgrss 29599 | An edge is a subset of ver... |
| upgrn0 29600 | An edge is a nonempty subs... |
| upgrle 29601 | An edge of an undirected p... |
| upgrfi 29602 | An edge is a finite subset... |
| upgrex 29603 | An edge is an unordered pa... |
| upgrbi 29604 | Show that an unordered pai... |
| upgrop 29605 | A pseudograph represented ... |
| isumgr 29606 | The property of being an u... |
| isumgrs 29607 | The simplified property of... |
| wrdumgr 29608 | The property of being an u... |
| umgrf 29609 | The edge function of an un... |
| umgrfn 29610 | The edge function of an un... |
| umgredg2 29611 | An edge of a multigraph ha... |
| umgrbi 29612 | Show that an unordered pai... |
| upgruhgr 29613 | An undirected pseudograph ... |
| umgrupgr 29614 | An undirected multigraph i... |
| umgruhgr 29615 | An undirected multigraph i... |
| upgrle2 29616 | An edge of an undirected p... |
| umgrnloopv 29617 | In a multigraph, there is ... |
| umgredgprv 29618 | In a multigraph, an edge i... |
| umgrnloop 29619 | In a multigraph, there is ... |
| umgrnloop0 29620 | A multigraph has no loops.... |
| umgr0e 29621 | The empty graph, with vert... |
| upgr0e 29622 | The empty graph, with vert... |
| upgr1elem 29623 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29624 | A pseudograph with one edg... |
| upgr0eop 29625 | The empty graph, with vert... |
| upgr1eop 29626 | A pseudograph with one edg... |
| upgr0eopALT 29627 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29628 | Alternate proof of ~ upgr1... |
| upgrun 29629 | The union ` U ` of two pse... |
| upgrunop 29630 | The union of two pseudogra... |
| umgrun 29631 | The union ` U ` of two mul... |
| umgrunop 29632 | The union of two multigrap... |
| umgrislfupgrlem 29633 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29634 | A multigraph is a loop-fre... |
| lfgredgge2 29635 | An edge of a loop-free gra... |
| lfgrnloop 29636 | A loop-free graph has no l... |
| uhgredgiedgb 29637 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29638 | A hypergraph has no edges ... |
| uhgredgn0 29639 | An edge of a hypergraph is... |
| edguhgr 29640 | An edge of a hypergraph is... |
| uhgredgrnv 29641 | An edge of a hypergraph co... |
| uhgredgss 29642 | The set of edges of a hype... |
| upgredgss 29643 | The set of edges of a pseu... |
| umgredgss 29644 | The set of edges of a mult... |
| edgupgr 29645 | Properties of an edge of a... |
| edgumgr 29646 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29647 | In a hypergraph, a vertex ... |
| upgredg 29648 | For each edge in a pseudog... |
| umgredg 29649 | For each edge in a multigr... |
| upgrpredgv 29650 | An edge of a pseudograph a... |
| umgrpredgv 29651 | An edge of a multigraph al... |
| upgredg2vtx 29652 | For a vertex incident to a... |
| upgredgpr 29653 | If a proper pair (of verti... |
| edglnl 29654 | The edges incident with a ... |
| numedglnl 29655 | The number of edges incide... |
| umgredgne 29656 | An edge of a multigraph al... |
| umgrnloop2 29657 | A multigraph has no loops.... |
| umgredgnlp 29658 | An edge of a multigraph is... |
| lfuhgr 29659 | A hypergraph is loop-free ... |
| lfuhgr2 29660 | A hypergraph is loop-free ... |
| lfuhgr3 29661 | A hypergraph is loop-free ... |
| isuspgr 29666 | The property of being a si... |
| isusgr 29667 | The property of being a si... |
| uspgrf 29668 | The edge function of a sim... |
| usgrf 29669 | The edge function of a sim... |
| isusgrs 29670 | The property of being a si... |
| usgrfs 29671 | The edge function of a sim... |
| usgrfun 29672 | The edge function of a sim... |
| usgredgss 29673 | The set of edges of a simp... |
| edgusgr 29674 | An edge of a simple graph ... |
| isuspgrop 29675 | The property of being an u... |
| isusgrop 29676 | The property of being an u... |
| usgrop 29677 | A simple graph represented... |
| isausgr 29678 | The property of an ordered... |
| ausgrusgrb 29679 | The equivalence of the def... |
| usgrausgri 29680 | A simple graph represented... |
| ausgrumgri 29681 | If an alternatively define... |
| ausgrusgri 29682 | The equivalence of the def... |
| usgrausgrb 29683 | The equivalence of the def... |
| usgredgop 29684 | An edge of a simple graph ... |
| usgrf1o 29685 | The edge function of a sim... |
| usgrf1 29686 | The edge function of a sim... |
| uspgrf1oedg 29687 | The edge function of a sim... |
| usgrss 29688 | An edge is a subset of ver... |
| uspgredgiedg 29689 | In a simple pseudograph, f... |
| uspgriedgedg 29690 | In a simple pseudograph, f... |
| uspgrushgr 29691 | A simple pseudograph is an... |
| uspgrupgr 29692 | A simple pseudograph is an... |
| uspgrupgrushgr 29693 | A graph is a simple pseudo... |
| usgruspgr 29694 | A simple graph is a simple... |
| usgrumgr 29695 | A simple graph is an undir... |
| usgrumgruspgr 29696 | A graph is a simple graph ... |
| usgruspgrb 29697 | A class is a simple graph ... |
| uspgruhgr 29698 | An undirected simple pseud... |
| usgrupgr 29699 | A simple graph is an undir... |
| usgruhgr 29700 | A simple graph is an undir... |
| usgrislfuspgr 29701 | A simple graph is a loop-f... |
| uspgrun 29702 | The union ` U ` of two sim... |
| uspgrunop 29703 | The union of two simple ps... |
| usgrun 29704 | The union ` U ` of two sim... |
| usgrunop 29705 | The union of two simple gr... |
| usgredg2 29706 | The value of the "edge fun... |
| usgredg2ALT 29707 | Alternate proof of ~ usgre... |
| usgredgprv 29708 | In a simple graph, an edge... |
| usgredgprvALT 29709 | Alternate proof of ~ usgre... |
| usgredgppr 29710 | An edge of a simple graph ... |
| usgrpredgv 29711 | An edge of a simple graph ... |
| edgssv2 29712 | An edge of a simple graph ... |
| usgredg 29713 | For each edge in a simple ... |
| usgrnloopv 29714 | In a simple graph, there i... |
| usgrnloopvALT 29715 | Alternate proof of ~ usgrn... |
| usgrnloop 29716 | In a simple graph, there i... |
| usgrnloopALT 29717 | Alternate proof of ~ usgrn... |
| usgrnloop0 29718 | A simple graph has no loop... |
| usgrnloop0ALT 29719 | Alternate proof of ~ usgrn... |
| usgredgne 29720 | An edge of a simple graph ... |
| usgrf1oedg 29721 | The edge function of a sim... |
| uhgr2edg 29722 | If a vertex is adjacent to... |
| umgr2edg 29723 | If a vertex is adjacent to... |
| usgr2edg 29724 | If a vertex is adjacent to... |
| umgr2edg1 29725 | If a vertex is adjacent to... |
| usgr2edg1 29726 | If a vertex is adjacent to... |
| umgrvad2edg 29727 | If a vertex is adjacent to... |
| umgr2edgneu 29728 | If a vertex is adjacent to... |
| usgrsizedg 29729 | In a simple graph, the siz... |
| usgredg3 29730 | The value of the "edge fun... |
| usgredg4 29731 | For a vertex incident to a... |
| usgredgreu 29732 | For a vertex incident to a... |
| usgredg2vtx 29733 | For a vertex incident to a... |
| uspgredg2vtxeu 29734 | For a vertex incident to a... |
| usgredg2vtxeu 29735 | For a vertex incident to a... |
| usgredg2vtxeuALT 29736 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29737 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29738 | In a simple pseudograph, t... |
| usgredg2vlem1 29739 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29740 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29741 | In a simple graph, the map... |
| usgriedgleord 29742 | Alternate version of ~ usg... |
| ushgredgedg 29743 | In a simple hypergraph the... |
| usgredgedg 29744 | In a simple graph there is... |
| ushgredgedgloop 29745 | In a simple hypergraph the... |
| uspgredgleord 29746 | In a simple pseudograph th... |
| usgredgleord 29747 | In a simple graph the numb... |
| usgredgleordALT 29748 | Alternate proof for ~ usgr... |
| usgrstrrepe 29749 | Replacing (or adding) the ... |
| usgr0e 29750 | The empty graph, with vert... |
| usgr0vb 29751 | The null graph, with no ve... |
| uhgr0v0e 29752 | The null graph, with no ve... |
| uhgr0vsize0 29753 | The size of a hypergraph w... |
| uhgr0edgfi 29754 | A graph of order 0 (i.e. w... |
| usgr0v 29755 | The null graph, with no ve... |
| uhgr0vusgr 29756 | The null graph, with no ve... |
| usgr0 29757 | The null graph represented... |
| uspgr1e 29758 | A simple pseudograph with ... |
| usgr1e 29759 | A simple graph with one ed... |
| usgr0eop 29760 | The empty graph, with vert... |
| uspgr1eop 29761 | A simple pseudograph with ... |
| uspgr1ewop 29762 | A simple pseudograph with ... |
| uspgr1v1eop 29763 | A simple pseudograph with ... |
| usgr1eop 29764 | A simple graph with (at le... |
| uspgr2v1e2w 29765 | A simple pseudograph with ... |
| usgr2v1e2w 29766 | A simple graph with two ve... |
| edg0usgr 29767 | A class without edges is a... |
| lfuhgr1v0e 29768 | A loop-free hypergraph wit... |
| usgr1vr 29769 | A simple graph with one ve... |
| usgr1v 29770 | A class with one (or no) v... |
| usgr1v0edg 29771 | A class with one (or no) v... |
| usgrexmpldifpr 29772 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29773 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29774 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29775 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29776 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29777 | ` G ` is a simple graph of... |
| griedg0prc 29778 | The class of empty graphs ... |
| griedg0ssusgr 29779 | The class of all simple gr... |
| usgrprc 29780 | The class of simple graphs... |
| relsubgr 29783 | The class of the subgraph ... |
| subgrv 29784 | If a class is a subgraph o... |
| issubgr 29785 | The property of a set to b... |
| issubgr2 29786 | The property of a set to b... |
| subgrprop 29787 | The properties of a subgra... |
| subgrprop2 29788 | The properties of a subgra... |
| uhgrissubgr 29789 | The property of a hypergra... |
| subgrprop3 29790 | The properties of a subgra... |
| egrsubgr 29791 | An empty graph consisting ... |
| 0grsubgr 29792 | The null graph (represente... |
| 0uhgrsubgr 29793 | The null graph (as hypergr... |
| uhgrsubgrself 29794 | A hypergraph is a subgraph... |
| subgrfun 29795 | The edge function of a sub... |
| subgruhgrfun 29796 | The edge function of a sub... |
| subgreldmiedg 29797 | An element of the domain o... |
| subgruhgredgd 29798 | An edge of a subgraph of a... |
| subumgredg2 29799 | An edge of a subgraph of a... |
| subuhgr 29800 | A subgraph of a hypergraph... |
| subupgr 29801 | A subgraph of a pseudograp... |
| subumgr 29802 | A subgraph of a multigraph... |
| subusgr 29803 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29804 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29805 | A spanning subgraph ` S ` ... |
| uhgrspan 29806 | A spanning subgraph ` S ` ... |
| upgrspan 29807 | A spanning subgraph ` S ` ... |
| umgrspan 29808 | A spanning subgraph ` S ` ... |
| usgrspan 29809 | A spanning subgraph ` S ` ... |
| uhgrspanop 29810 | A spanning subgraph of a h... |
| upgrspanop 29811 | A spanning subgraph of a p... |
| umgrspanop 29812 | A spanning subgraph of a m... |
| usgrspanop 29813 | A spanning subgraph of a s... |
| uhgrspan1lem1 29814 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29815 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29816 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29817 | The induced subgraph ` S `... |
| upgrreslem 29818 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29819 | Lemma for ~ umgrres and ~ ... |
| upgrres 29820 | A subgraph obtained by rem... |
| umgrres 29821 | A subgraph obtained by rem... |
| usgrres 29822 | A subgraph obtained by rem... |
| upgrres1lem1 29823 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29824 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29825 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29826 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29827 | A pseudograph obtained by ... |
| umgrres1 29828 | A multigraph obtained by r... |
| usgrres1 29829 | Restricting a simple graph... |
| isfusgr 29832 | The property of being a fi... |
| fusgrvtxfi 29833 | A finite simple graph has ... |
| isfusgrf1 29834 | The property of being a fi... |
| isfusgrcl 29835 | The property of being a fi... |
| fusgrusgr 29836 | A finite simple graph is a... |
| opfusgr 29837 | A finite simple graph repr... |
| usgredgffibi 29838 | The number of edges in a s... |
| fusgredgfi 29839 | In a finite simple graph t... |
| usgr1v0e 29840 | The size of a (finite) sim... |
| usgrfilem 29841 | In a finite simple graph, ... |
| fusgrfisbase 29842 | Induction base for ~ fusgr... |
| fusgrfisstep 29843 | Induction step in ~ fusgrf... |
| fusgrfis 29844 | A finite simple graph is o... |
| fusgrfupgrfs 29845 | A finite simple graph is a... |
| nbgrprc0 29848 | The set of neighbors is em... |
| nbgrcl 29849 | If a class ` X ` has at le... |
| nbgrval 29850 | The set of neighbors of a ... |
| dfnbgr2 29851 | Alternate definition of th... |
| dfnbgr3 29852 | Alternate definition of th... |
| nbgrnvtx0 29853 | If a class ` X ` is not a ... |
| nbgrel 29854 | Characterization of a neig... |
| nbgrisvtx 29855 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29856 | The neighbors of a vertex ... |
| nbuhgr 29857 | The set of neighbors of a ... |
| nbupgr 29858 | The set of neighbors of a ... |
| nbupgrel 29859 | A neighbor of a vertex in ... |
| nbumgrvtx 29860 | The set of neighbors of a ... |
| nbumgr 29861 | The set of neighbors of an... |
| nbusgrvtx 29862 | The set of neighbors of a ... |
| nbusgr 29863 | The set of neighbors of an... |
| nbgr2vtx1edg 29864 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29865 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29866 | If a hypergraph has two ve... |
| nbusgreledg 29867 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29868 | A vertex which is not endp... |
| nbgr0vtx 29869 | In a null graph (with no v... |
| nbgr0edglem 29870 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29871 | In an empty graph (with no... |
| nbgr1vtx 29872 | In a graph with one vertex... |
| nbgrnself 29873 | A vertex in a graph is not... |
| nbgrnself2 29874 | A class ` X ` is not a nei... |
| nbgrssovtx 29875 | The neighbors of a vertex ... |
| nbgrssvwo2 29876 | The neighbors of a vertex ... |
| nbgrsym 29877 | In a graph, the neighborho... |
| nbupgrres 29878 | The neighborhood of a vert... |
| usgrnbcnvfv 29879 | Applying the edge function... |
| nbusgredgeu 29880 | For each neighbor of a ver... |
| edgnbusgreu 29881 | For each edge incident to ... |
| nbusgredgeu0 29882 | For each neighbor of a ver... |
| nbusgrf1o0 29883 | The mapping of neighbors o... |
| nbusgrf1o1 29884 | The set of neighbors of a ... |
| nbusgrf1o 29885 | The set of neighbors of a ... |
| nbedgusgr 29886 | The number of neighbors of... |
| edgusgrnbfin 29887 | The number of neighbors of... |
| nbusgrfi 29888 | The class of neighbors of ... |
| nbfiusgrfi 29889 | The class of neighbors of ... |
| hashnbusgrnn0 29890 | The number of neighbors of... |
| nbfusgrlevtxm1 29891 | The number of neighbors of... |
| nbfusgrlevtxm2 29892 | If there is a vertex which... |
| nbusgrvtxm1 29893 | If the number of neighbors... |
| nb3grprlem1 29894 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29895 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29896 | The neighbors of a vertex ... |
| nb3grpr2 29897 | The neighbors of a vertex ... |
| nb3gr2nb 29898 | If the neighbors of two ve... |
| uvtxval 29901 | The set of all universal v... |
| uvtxel 29902 | A universal vertex, i.e. a... |
| uvtxisvtx 29903 | A universal vertex is a ve... |
| uvtxssvtx 29904 | The set of the universal v... |
| vtxnbuvtx 29905 | A universal vertex has all... |
| uvtxnbgrss 29906 | A universal vertex has all... |
| uvtxnbgrvtx 29907 | A universal vertex is neig... |
| uvtx0 29908 | There is no universal vert... |
| isuvtx 29909 | The set of all universal v... |
| uvtxel1 29910 | Characterization of a univ... |
| uvtx01vtx 29911 | If a graph/class has no ed... |
| uvtx2vtx1edg 29912 | If a graph has two vertice... |
| uvtx2vtx1edgb 29913 | If a hypergraph has two ve... |
| uvtxnbgr 29914 | A universal vertex has all... |
| uvtxnbgrb 29915 | A vertex is universal iff ... |
| uvtxusgr 29916 | The set of all universal v... |
| uvtxusgrel 29917 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29918 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29919 | If the number of neighbors... |
| uvtxnbvtxm1 29920 | A universal vertex has ` n... |
| nbupgruvtxres 29921 | The neighborhood of a univ... |
| uvtxupgrres 29922 | A universal vertex is univ... |
| cplgruvtxb 29927 | A graph ` G ` is complete ... |
| prcliscplgr 29928 | A proper class (representi... |
| iscplgr 29929 | The property of being a co... |
| iscplgrnb 29930 | A graph is complete iff al... |
| iscplgredg 29931 | A graph ` G ` is complete ... |
| iscusgr 29932 | The property of being a co... |
| cusgrusgr 29933 | A complete simple graph is... |
| cusgrcplgr 29934 | A complete simple graph is... |
| iscusgrvtx 29935 | A simple graph is complete... |
| cusgruvtxb 29936 | A simple graph is complete... |
| iscusgredg 29937 | A simple graph is complete... |
| cusgredg 29938 | In a complete simple graph... |
| cplgr0 29939 | The null graph (with no ve... |
| cusgr0 29940 | The null graph (with no ve... |
| cplgr0v 29941 | A null graph (with no vert... |
| cusgr0v 29942 | A graph with no vertices a... |
| cplgr1vlem 29943 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29944 | A graph with one vertex is... |
| cusgr1v 29945 | A graph with one vertex an... |
| cplgr2v 29946 | An undirected hypergraph w... |
| cplgr2vpr 29947 | An undirected hypergraph w... |
| nbcplgr 29948 | In a complete graph, each ... |
| cplgr3v 29949 | A pseudograph with three (... |
| cusgr3vnbpr 29950 | The neighbors of a vertex ... |
| cplgrop 29951 | A complete graph represent... |
| cusgrop 29952 | A complete simple graph re... |
| cusgrexilem1 29953 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29954 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29955 | An arbitrary set regarded ... |
| cusgrexilem2 29956 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29957 | An arbitrary set ` V ` reg... |
| cusgrexg 29958 | For each set there is a se... |
| structtousgr 29959 | Any (extensible) structure... |
| structtocusgr 29960 | Any (extensible) structure... |
| cffldtocusgr 29961 | The field of complex numbe... |
| cusgrres 29962 | Restricting a complete sim... |
| cusgrsizeindb0 29963 | Base case of the induction... |
| cusgrsizeindb1 29964 | Base case of the induction... |
| cusgrsizeindslem 29965 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29966 | Part 1 of induction step i... |
| cusgrsize2inds 29967 | Induction step in ~ cusgrs... |
| cusgrsize 29968 | The size of a finite compl... |
| cusgrfilem1 29969 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29970 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29971 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29972 | If the size of a complete ... |
| usgredgsscusgredg 29973 | A simple graph is a subgra... |
| usgrsscusgr 29974 | A simple graph is a subgra... |
| sizusglecusglem1 29975 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29976 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29977 | The size of a simple graph... |
| fusgrmaxsize 29978 | The maximum size of a fini... |
| vtxdgfval 29981 | The value of the vertex de... |
| vtxdgval 29982 | The degree of a vertex. (... |
| vtxdgfival 29983 | The degree of a vertex for... |
| vtxdgop 29984 | The vertex degree expresse... |
| vtxdgf 29985 | The vertex degree function... |
| vtxdgelxnn0 29986 | The degree of a vertex is ... |
| vtxdg0v 29987 | The degree of a vertex in ... |
| vtxdg0e 29988 | The degree of a vertex in ... |
| vtxdgfisnn0 29989 | The degree of a vertex in ... |
| vtxdgfisf 29990 | The vertex degree function... |
| vtxdeqd 29991 | Equality theorem for the v... |
| vtxduhgr0e 29992 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29993 | The degree of the vertex i... |
| vdumgr0 29994 | A vertex in a multigraph h... |
| vtxdun 29995 | The degree of a vertex in ... |
| vtxdfiun 29996 | The degree of a vertex in ... |
| vtxduhgrun 29997 | The degree of a vertex in ... |
| vtxduhgrfiun 29998 | The degree of a vertex in ... |
| vtxdlfgrval 29999 | The value of the vertex de... |
| vtxdumgrval 30000 | The value of the vertex de... |
| vtxdusgrval 30001 | The value of the vertex de... |
| vtxd0nedgb 30002 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 30003 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 30004 | The value of the vertex de... |
| vtxdusgrfvedg 30005 | The value of the vertex de... |
| vtxduhgr0nedg 30006 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 30007 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 30008 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 30009 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 30010 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 30011 | The vertex degree function... |
| vtxdgfusgr 30012 | In a finite simple graph, ... |
| fusgrn0degnn0 30013 | In a nonempty, finite grap... |
| 1loopgruspgr 30014 | A graph with one edge whic... |
| 1loopgredg 30015 | The set of edges in a grap... |
| 1loopgrnb0 30016 | In a graph (simple pseudog... |
| 1loopgrvd2 30017 | The vertex degree of a one... |
| 1loopgrvd0 30018 | The vertex degree of a one... |
| 1hevtxdg0 30019 | The vertex degree of verte... |
| 1hevtxdg1 30020 | The vertex degree of verte... |
| 1hegrvtxdg1 30021 | The vertex degree of a gra... |
| 1hegrvtxdg1r 30022 | The vertex degree of a gra... |
| 1egrvtxdg1 30023 | The vertex degree of a one... |
| 1egrvtxdg1r 30024 | The vertex degree of a one... |
| 1egrvtxdg0 30025 | The vertex degree of a one... |
| p1evtxdeqlem 30026 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 30027 | If an edge ` E ` which doe... |
| p1evtxdp1 30028 | If an edge ` E ` (not bein... |
| uspgrloopvtx 30029 | The set of vertices in a g... |
| uspgrloopvtxel 30030 | A vertex in a graph (simpl... |
| uspgrloopiedg 30031 | The set of edges in a grap... |
| uspgrloopedg 30032 | The set of edges in a grap... |
| uspgrloopnb0 30033 | In a graph (simple pseudog... |
| uspgrloopvd2 30034 | The vertex degree of a one... |
| umgr2v2evtx 30035 | The set of vertices in a m... |
| umgr2v2evtxel 30036 | A vertex in a multigraph w... |
| umgr2v2eiedg 30037 | The edge function in a mul... |
| umgr2v2eedg 30038 | The set of edges in a mult... |
| umgr2v2e 30039 | A multigraph with two edge... |
| umgr2v2enb1 30040 | In a multigraph with two e... |
| umgr2v2evd2 30041 | In a multigraph with two e... |
| hashnbusgrvd 30042 | In a simple graph, the num... |
| usgruvtxvdb 30043 | In a finite simple graph w... |
| vdiscusgrb 30044 | A finite simple graph with... |
| vdiscusgr 30045 | In a finite complete simpl... |
| vtxdusgradjvtx 30046 | The degree of a vertex in ... |
| usgrvd0nedg 30047 | If a vertex in a simple gr... |
| uhgrvd00 30048 | If every vertex in a hyper... |
| usgrvd00 30049 | If every vertex in a simpl... |
| vdegp1ai 30050 | The induction step for a v... |
| vdegp1bi 30051 | The induction step for a v... |
| vdegp1ci 30052 | The induction step for a v... |
| vtxdginducedm1lem1 30053 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 30054 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 30055 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 30056 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 30057 | The degree of a vertex ` v... |
| vtxdginducedm1fi 30058 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 30059 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 30060 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 30061 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 30062 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 30063 | Induction step of ~ finsum... |
| finsumvtxdg2size 30064 | The sum of the degrees of ... |
| fusgr1th 30065 | The sum of the degrees of ... |
| finsumvtxdgeven 30066 | The sum of the degrees of ... |
| vtxdgoddnumeven 30067 | The number of vertices of ... |
| fusgrvtxdgonume 30068 | The number of vertices of ... |
| isrgr 30073 | The property of a class be... |
| rgrprop 30074 | The properties of a k-regu... |
| isrusgr 30075 | The property of being a k-... |
| rusgrprop 30076 | The properties of a k-regu... |
| rusgrrgr 30077 | A k-regular simple graph i... |
| rusgrusgr 30078 | A k-regular simple graph i... |
| finrusgrfusgr 30079 | A finite regular simple gr... |
| isrusgr0 30080 | The property of being a k-... |
| rusgrprop0 30081 | The properties of a k-regu... |
| usgreqdrusgr 30082 | If all vertices in a simpl... |
| fusgrregdegfi 30083 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 30084 | If all vertices in a nonem... |
| frusgrnn0 30085 | In a nonempty finite k-reg... |
| 0edg0rgr 30086 | A graph is 0-regular if it... |
| uhgr0edg0rgr 30087 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 30088 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 30089 | A simple graph is 0-regula... |
| 0vtxrgr 30090 | A null graph (with no vert... |
| 0vtxrusgr 30091 | A graph with no vertices a... |
| 0uhgrrusgr 30092 | The null graph as hypergra... |
| 0grrusgr 30093 | The null graph represented... |
| 0grrgr 30094 | The null graph represented... |
| cusgrrusgr 30095 | A complete simple graph wi... |
| cusgrm1rusgr 30096 | A finite simple graph with... |
| rusgrpropnb 30097 | The properties of a k-regu... |
| rusgrpropedg 30098 | The properties of a k-regu... |
| rusgrpropadjvtx 30099 | The properties of a k-regu... |
| rusgrnumwrdl2 30100 | In a k-regular simple grap... |
| rusgr1vtxlem 30101 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 30102 | If a k-regular simple grap... |
| rgrusgrprc 30103 | The class of 0-regular sim... |
| rusgrprc 30104 | The class of 0-regular sim... |
| rgrprc 30105 | The class of 0-regular gra... |
| rgrprcx 30106 | The class of 0-regular gra... |
| rgrx0ndm 30107 | 0 is not in the domain of ... |
| rgrx0nd 30108 | The potentially alternativ... |
| ewlksfval 30115 | The set of s-walks of edge... |
| isewlk 30116 | Conditions for a function ... |
| ewlkprop 30117 | Properties of an s-walk of... |
| ewlkinedg 30118 | The intersection (common v... |
| ewlkle 30119 | An s-walk of edges is also... |
| upgrewlkle2 30120 | In a pseudograph, there is... |
| wkslem1 30121 | Lemma 1 for walks to subst... |
| wkslem2 30122 | Lemma 2 for walks to subst... |
| wksfval 30123 | The set of walks (in an un... |
| iswlk 30124 | Properties of a pair of fu... |
| wlkprop 30125 | Properties of a walk. (Co... |
| wlkv 30126 | The classes involved in a ... |
| iswlkg 30127 | Generalization of ~ iswlk ... |
| wlkf 30128 | The mapping enumerating th... |
| wlkcl 30129 | A walk has length ` # ( F ... |
| wlkp 30130 | The mapping enumerating th... |
| wlkpwrd 30131 | The sequence of vertices o... |
| wlklenvp1 30132 | The number of vertices of ... |
| wksv 30133 | The class of walks is a se... |
| wlkn0 30134 | The sequence of vertices o... |
| wlklenvm1 30135 | The number of edges of a w... |
| ifpsnprss 30136 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 30137 | Each pair of adjacent vert... |
| wlkvtxiedg 30138 | The vertices of a walk are... |
| relwlk 30139 | The set ` ( Walks `` G ) `... |
| wlkvv 30140 | If there is at least one w... |
| wlkop 30141 | A walk is an ordered pair.... |
| wlkcpr 30142 | A walk as class with two c... |
| wlk2f 30143 | If there is a walk ` W ` t... |
| wlkcomp 30144 | A walk expressed by proper... |
| wlkcompim 30145 | Implications for the prope... |
| wlkelwrd 30146 | The components of a walk a... |
| wlkeq 30147 | Conditions for two walks (... |
| edginwlk 30148 | The value of the edge func... |
| upgredginwlk 30149 | The value of the edge func... |
| iedginwlk 30150 | The value of the edge func... |
| wlkl1loop 30151 | A walk of length 1 from a ... |
| wlk1walk 30152 | A walk is a 1-walk "on the... |
| wlk1ewlk 30153 | A walk is an s-walk "on th... |
| upgriswlk 30154 | Properties of a pair of fu... |
| upgrwlkedg 30155 | The edges of a walk in a p... |
| upgrwlkcompim 30156 | Implications for the prope... |
| wlkvtxedg 30157 | The vertices of a walk are... |
| upgrwlkvtxedg 30158 | The pairs of connected ver... |
| uspgr2wlkeq 30159 | Conditions for two walks w... |
| uspgr2wlkeq2 30160 | Conditions for two walks w... |
| uspgr2wlkeqi 30161 | Conditions for two walks w... |
| umgrwlknloop 30162 | In a multigraph, each walk... |
| wlkv0 30163 | If there is a walk in the ... |
| g0wlk0 30164 | There is no walk in a null... |
| 0wlk0 30165 | There is no walk for the e... |
| wlk0prc 30166 | There is no walk in a null... |
| wlklenvclwlk 30167 | The number of vertices in ... |
| wlkson 30168 | The set of walks between t... |
| iswlkon 30169 | Properties of a pair of fu... |
| wlkonprop 30170 | Properties of a walk betwe... |
| wlkpvtx 30171 | A walk connects vertices. ... |
| wlkepvtx 30172 | The endpoints of a walk ar... |
| wlkoniswlk 30173 | A walk between two vertice... |
| wlkonwlk 30174 | A walk is a walk between i... |
| wlkonwlk1l 30175 | A walk is a walk from its ... |
| wlksoneq1eq2 30176 | Two walks with identical s... |
| wlkonl1iedg 30177 | If there is a walk between... |
| wlkon2n0 30178 | The length of a walk betwe... |
| 2wlklem 30179 | Lemma for theorems for wal... |
| upgr2wlk 30180 | Properties of a pair of fu... |
| wlkreslem 30181 | Lemma for ~ wlkres . (Con... |
| wlkres 30182 | The restriction ` <. H , Q... |
| redwlklem 30183 | Lemma for ~ redwlk . (Con... |
| redwlk 30184 | A walk ending at the last ... |
| wlkp1lem1 30185 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 30186 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 30187 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 30188 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 30189 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 30190 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 30191 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 30192 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 30193 | Append one path segment (e... |
| wlkdlem1 30194 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 30195 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 30196 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 30197 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 30198 | Two words representing a w... |
| pfxwlk 30199 | A prefix of a walk is a wa... |
| revwlk 30200 | The reverse of a walk is a... |
| swrdwlk 30201 | Two matching subwords of a... |
| subgrwlk 30202 | If a walk exists in a subg... |
| lfgrwlkprop 30203 | Two adjacent vertices in a... |
| lfgriswlk 30204 | Conditions for a pair of f... |
| lfgrwlknloop 30205 | In a loop-free graph, each... |
| reltrls 30210 | The set ` ( Trails `` G ) ... |
| trlsfval 30211 | The set of trails (in an u... |
| istrl 30212 | Conditions for a pair of c... |
| trliswlk 30213 | A trail is a walk. (Contr... |
| trlf1 30214 | The enumeration ` F ` of a... |
| trlreslem 30215 | Lemma for ~ trlres . Form... |
| trlres 30216 | The restriction ` <. H , Q... |
| upgrtrls 30217 | The set of trails in a pse... |
| upgristrl 30218 | Properties of a pair of fu... |
| upgrf1istrl 30219 | Properties of a pair of a ... |
| wksonproplem 30220 | Lemma for theorems for pro... |
| trlsonfval 30221 | The set of trails between ... |
| istrlson 30222 | Properties of a pair of fu... |
| trlsonprop 30223 | Properties of a trail betw... |
| trlsonistrl 30224 | A trail between two vertic... |
| trlsonwlkon 30225 | A trail between two vertic... |
| trlontrl 30226 | A trail is a trail between... |
| subgrtrl 30227 | If a trail exists in a sub... |
| relpths 30236 | The set ` ( Paths `` G ) `... |
| pthsfval 30237 | The set of paths (in an un... |
| spthsfval 30238 | The set of simple paths (i... |
| ispth 30239 | Conditions for a pair of c... |
| isspth 30240 | Conditions for a pair of c... |
| pthistrl 30241 | A path is a trail (in an u... |
| spthispth 30242 | A simple path is a path (i... |
| pthiswlk 30243 | A path is a walk (in an un... |
| spthiswlk 30244 | A simple path is a walk (i... |
| pthdivtx 30245 | The inner vertices of a pa... |
| pthdadjvtx 30246 | The adjacent vertices of a... |
| dfpth2 30247 | Alternate definition for a... |
| pthhashvtx 30248 | A graph containing a path ... |
| pthdifv 30249 | The vertices of a path are... |
| 2pthnloop 30250 | A path of length at least ... |
| upgr2pthnlp 30251 | A path of length at least ... |
| spthdifv 30252 | The vertices of a simple p... |
| spthdep 30253 | A simple path (at least of... |
| pthdepisspth 30254 | A path with different star... |
| upgrwlkdvdelem 30255 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 30256 | In a pseudograph, all edge... |
| upgrspthswlk 30257 | The set of simple paths in... |
| upgrwlkdvspth 30258 | A walk consisting of diffe... |
| pthsonfval 30259 | The set of paths between t... |
| spthson 30260 | The set of simple paths be... |
| ispthson 30261 | Properties of a pair of fu... |
| isspthson 30262 | Properties of a pair of fu... |
| pthsonprop 30263 | Properties of a path betwe... |
| spthonprop 30264 | Properties of a simple pat... |
| pthonispth 30265 | A path between two vertice... |
| pthontrlon 30266 | A path between two vertice... |
| pthonpth 30267 | A path is a path between i... |
| isspthonpth 30268 | A pair of functions is a s... |
| spthonisspth 30269 | A simple path between to v... |
| spthonpthon 30270 | A simple path between two ... |
| spthonepeq 30271 | The endpoints of a simple ... |
| uhgrwkspthlem1 30272 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 30273 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 30274 | Any walk of length 1 betwe... |
| usgr2wlkneq 30275 | The vertices and edges are... |
| usgr2wlkspthlem1 30276 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 30277 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 30278 | In a simple graph, any wal... |
| usgr2trlncl 30279 | In a simple graph, any tra... |
| usgr2trlspth 30280 | In a simple graph, any tra... |
| usgr2pthspth 30281 | In a simple graph, any pat... |
| usgr2pthlem 30282 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 30283 | In a simple graph, there i... |
| usgr2pth0 30284 | In a simply graph, there i... |
| pthdlem1 30285 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 30286 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 30287 | Lemma 2 for ~ pthd . (Con... |
| pthd 30288 | Two words representing a t... |
| subgrpth 30289 | If a path exists in a subg... |
| clwlks 30292 | The set of closed walks (i... |
| isclwlk 30293 | A pair of functions repres... |
| clwlkiswlk 30294 | A closed walk is a walk (i... |
| clwlkwlk 30295 | Closed walks are walks (in... |
| clwlkswks 30296 | Closed walks are walks (in... |
| isclwlke 30297 | Properties of a pair of fu... |
| isclwlkupgr 30298 | Properties of a pair of fu... |
| clwlkcomp 30299 | A closed walk expressed by... |
| clwlkcompim 30300 | Implications for the prope... |
| upgrclwlkcompim 30301 | Implications for the prope... |
| clwlkcompbp 30302 | Basic properties of the co... |
| clwlkl1loop 30303 | A closed walk of length 1 ... |
| crcts 30308 | The set of circuits (in an... |
| cycls 30309 | The set of cycles (in an u... |
| iscrct 30310 | Sufficient and necessary c... |
| iscycl 30311 | Sufficient and necessary c... |
| crctprop 30312 | The properties of a circui... |
| cyclprop 30313 | The properties of a cycle:... |
| crctisclwlk 30314 | A circuit is a closed walk... |
| crctistrl 30315 | A circuit is a trail. (Co... |
| crctiswlk 30316 | A circuit is a walk. (Con... |
| cyclispth 30317 | A cycle is a path. (Contr... |
| subgrcycl 30318 | If a cycle exists in a sub... |
| cycliswlk 30319 | A cycle is a walk. (Contr... |
| cycliscrct 30320 | A cycle is a circuit. (Co... |
| cyclnumvtx 30321 | The number of vertices of ... |
| cyclnspth 30322 | A (non-trivial) cycle is n... |
| pthisspthorcycl 30323 | A path is either a simple ... |
| pthspthcyc 30324 | A pair ` <. F , P >. ` rep... |
| spthcycl 30325 | A walk is a trivial path i... |
| cyclispthon 30326 | A cycle is a path starting... |
| lfgrn1cycl 30327 | In a loop-free graph there... |
| usgr2trlncrct 30328 | In a simple graph, any tra... |
| umgrn1cycl 30329 | In a multigraph graph (wit... |
| uspgrn2crct 30330 | In a simple pseudograph th... |
| usgrn2cycl 30331 | In a simple graph there ar... |
| crctcshwlkn0lem1 30332 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 30333 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 30334 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 30335 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 30336 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 30337 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 30338 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 30339 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 30340 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 30341 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 30342 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 30343 | Cyclically shifting the in... |
| crctcshwlk 30344 | Cyclically shifting the in... |
| crctcshtrl 30345 | Cyclically shifting the in... |
| crctcsh 30346 | Cyclically shifting the in... |
| wwlks 30357 | The set of walks (in an un... |
| iswwlks 30358 | A word over the set of ver... |
| wwlksn 30359 | The set of walks (in an un... |
| iswwlksn 30360 | A word over the set of ver... |
| wwlksnprcl 30361 | Derivation of the length o... |
| iswwlksnx 30362 | Properties of a word to re... |
| wwlkbp 30363 | Basic properties of a walk... |
| wwlknbp 30364 | Basic properties of a walk... |
| wwlknp 30365 | Properties of a set being ... |
| wwlknbp1 30366 | Other basic properties of ... |
| wwlknvtx 30367 | The symbols of a word ` W ... |
| wwlknllvtx 30368 | If a word ` W ` represents... |
| wwlknlsw 30369 | If a word represents a wal... |
| wspthsn 30370 | The set of simple paths of... |
| iswspthn 30371 | An element of the set of s... |
| wspthnp 30372 | Properties of a set being ... |
| wwlksnon 30373 | The set of walks of a fixe... |
| wspthsnon 30374 | The set of simple paths of... |
| iswwlksnon 30375 | The set of walks of a fixe... |
| wwlksnon0 30376 | Sufficient conditions for ... |
| wwlksonvtx 30377 | If a word ` W ` represents... |
| iswspthsnon 30378 | The set of simple paths of... |
| wwlknon 30379 | An element of the set of w... |
| wspthnon 30380 | An element of the set of s... |
| wspthnonp 30381 | Properties of a set being ... |
| wspthneq1eq2 30382 | Two simple paths with iden... |
| wwlksn0s 30383 | The set of all walks as wo... |
| wwlkssswrd 30384 | Walks (represented by word... |
| wwlksn0 30385 | A walk of length 0 is repr... |
| 0enwwlksnge1 30386 | In graphs without edges, t... |
| wwlkswwlksn 30387 | A walk of a fixed length a... |
| wwlkssswwlksn 30388 | The walks of a fixed lengt... |
| wlkiswwlks1 30389 | The sequence of vertices i... |
| wlklnwwlkln1 30390 | The sequence of vertices i... |
| wlkiswwlks2lem1 30391 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 30392 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 30393 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 30394 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 30395 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 30396 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 30397 | A walk as word corresponds... |
| wlkiswwlks 30398 | A walk as word corresponds... |
| wlkiswwlksupgr2 30399 | A walk as word corresponds... |
| wlkiswwlkupgr 30400 | A walk as word corresponds... |
| wlkswwlksf1o 30401 | The mapping of (ordinary) ... |
| wlkswwlksen 30402 | The set of walks as words ... |
| wwlksm1edg 30403 | Removing the trailing edge... |
| wlklnwwlkln2lem 30404 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 30405 | A walk of length ` N ` as ... |
| wlklnwwlkn 30406 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 30407 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 30408 | A walk of length ` N ` as ... |
| wlknewwlksn 30409 | If a walk in a pseudograph... |
| wlknwwlksnbij 30410 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 30411 | In a simple pseudograph, t... |
| wlknwwlksneqs 30412 | The set of walks of a fixe... |
| wwlkseq 30413 | Equality of two walks (as ... |
| wwlksnred 30414 | Reduction of a walk (as wo... |
| wwlksnext 30415 | Extension of a walk (as wo... |
| wwlksnextbi 30416 | Extension of a walk (as wo... |
| wwlksnredwwlkn 30417 | For each walk (as word) of... |
| wwlksnredwwlkn0 30418 | For each walk (as word) of... |
| wwlksnextwrd 30419 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 30420 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 30421 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 30422 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 30423 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 30424 | There is a bijection betwe... |
| wwlksnexthasheq 30425 | The number of the extensio... |
| disjxwwlksn 30426 | Sets of walks (as words) e... |
| wwlksnndef 30427 | Conditions for ` WWalksN `... |
| wwlksnfi 30428 | The number of walks repres... |
| wlksnfi 30429 | The number of walks of fix... |
| wlksnwwlknvbij 30430 | There is a bijection betwe... |
| wwlksnextproplem1 30431 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 30432 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 30433 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 30434 | Adding additional properti... |
| disjxwwlkn 30435 | Sets of walks (as words) e... |
| hashwwlksnext 30436 | Number of walks (as words)... |
| wwlksnwwlksnon 30437 | A walk of fixed length is ... |
| wspthsnwspthsnon 30438 | A simple path of fixed len... |
| wspthsnonn0vne 30439 | If the set of simple paths... |
| wspthsswwlkn 30440 | The set of simple paths of... |
| wspthnfi 30441 | In a finite graph, the set... |
| wwlksnonfi 30442 | In a finite graph, the set... |
| wspthsswwlknon 30443 | The set of simple paths of... |
| wspthnonfi 30444 | In a finite graph, the set... |
| wspniunwspnon 30445 | The set of nonempty simple... |
| wspn0 30446 | If there are no vertices, ... |
| 2wlkdlem1 30447 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 30448 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 30449 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 30450 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 30451 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 30452 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 30453 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 30454 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 30455 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 30456 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 30457 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 30458 | Construction of a walk fro... |
| 2wlkond 30459 | A walk of length 2 from on... |
| 2trld 30460 | Construction of a trail fr... |
| 2trlond 30461 | A trail of length 2 from o... |
| 2pthd 30462 | A path of length 2 from on... |
| 2spthd 30463 | A simple path of length 2 ... |
| 2pthond 30464 | A simple path of length 2 ... |
| 2pthon3v 30465 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30466 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30467 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30468 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30469 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30470 | In a multigraph, two adjac... |
| umgr2wlk 30471 | In a multigraph, there is ... |
| umgr2wlkon 30472 | For each pair of adjacent ... |
| elwwlks2s3 30473 | A walk of length 2 as word... |
| midwwlks2s3 30474 | There is a vertex between ... |
| wwlks2onv 30475 | If a length 3 string repre... |
| elwwlks2ons3im 30476 | A walk as word of length 2... |
| elwwlks2ons3 30477 | For each walk of length 2 ... |
| s3wwlks2on 30478 | A length 3 string which re... |
| sps3wwlks2on 30479 | A length 3 string which re... |
| usgrwwlks2on 30480 | A walk of length 2 between... |
| umgrwwlks2on 30481 | A walk of length 2 between... |
| wwlks2onsym 30482 | There is a walk of length ... |
| elwwlks2on 30483 | A walk of length 2 between... |
| elwspths2on 30484 | A simple path of length 2 ... |
| elwspths2onw 30485 | A simple path of length 2 ... |
| wpthswwlks2on 30486 | For two different vertices... |
| 2wspdisj 30487 | All simple paths of length... |
| 2wspiundisj 30488 | All simple paths of length... |
| usgr2wspthons3 30489 | A simple path of length 2 ... |
| usgr2wspthon 30490 | A simple path of length 2 ... |
| elwwlks2 30491 | A walk of length 2 between... |
| elwspths2spth 30492 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30493 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30494 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30495 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30496 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30497 | Induction base 1 for ~ rus... |
| rusgr0edg 30498 | Special case for graphs wi... |
| rusgrnumwwlks 30499 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30500 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30501 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30502 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30503 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30504 | In a ` K `-regular graph, ... |
| clwwlk 30507 | The set of closed walks (i... |
| isclwwlk 30508 | Properties of a word to re... |
| clwwlkbp 30509 | Basic properties of a clos... |
| clwwlkgt0 30510 | There is no empty closed w... |
| clwwlksswrd 30511 | Closed walks (represented ... |
| clwwlk1loop 30512 | A closed walk of length 1 ... |
| clwwlkccatlem 30513 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30514 | The concatenation of two w... |
| umgrclwwlkge2 30515 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30516 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30517 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30518 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30519 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30520 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30521 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30522 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30523 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30524 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30525 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30526 | A closed walk as word of l... |
| clwlkclwwlk2 30527 | A closed walk corresponds ... |
| clwlkclwwlkflem 30528 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30529 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30530 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30531 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30532 | ` F ` is a function from t... |
| clwlkclwwlkfo 30533 | ` F ` is a function from t... |
| clwlkclwwlkf1 30534 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30535 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30536 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30537 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30538 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30539 | Cyclically shifting a clos... |
| clwwisshclwwsn 30540 | Cyclically shifting a clos... |
| erclwwlkrel 30541 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30542 | Two classes are equivalent... |
| erclwwlkeqlen 30543 | If two classes are equival... |
| erclwwlkref 30544 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30545 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30546 | ` .~ ` is a transitive rel... |
| erclwwlk 30547 | ` .~ ` is an equivalence r... |
| clwwlkn 30550 | The set of closed walks of... |
| isclwwlkn 30551 | A word over the set of ver... |
| clwwlkn0 30552 | There is no closed walk of... |
| clwwlkneq0 30553 | Sufficient conditions for ... |
| clwwlkclwwlkn 30554 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30555 | The closed walks of a fixe... |
| clwwlknlen 30556 | The length of a word repre... |
| clwwlknnn 30557 | The length of a closed wal... |
| clwwlknwrd 30558 | A closed walk of a fixed l... |
| clwwlknbp 30559 | Basic properties of a clos... |
| isclwwlknx 30560 | Characterization of a word... |
| clwwlknp 30561 | Properties of a set being ... |
| clwwlknwwlksn 30562 | A word representing a clos... |
| clwwlknlbonbgr1 30563 | The last but one vertex in... |
| clwwlkinwwlk 30564 | If the initial vertex of a... |
| clwwlkn1 30565 | A closed walk of length 1 ... |
| loopclwwlkn1b 30566 | The singleton word consist... |
| clwwlkn1loopb 30567 | A word represents a closed... |
| clwwlkn2 30568 | A closed walk of length 2 ... |
| clwwlknfi 30569 | If there is only a finite ... |
| clwwlkel 30570 | Obtaining a closed walk (a... |
| clwwlkf 30571 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30572 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30573 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30574 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30575 | F is a 1-1 onto function, ... |
| clwwlken 30576 | The set of closed walks of... |
| clwwlknwwlkncl 30577 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30578 | A nonempty word over verti... |
| clwwlknwwlksnb 30579 | A word over vertices repre... |
| clwwlkext2edg 30580 | If a word concatenated wit... |
| wwlksext2clwwlk 30581 | If a word represents a wal... |
| wwlksubclwwlk 30582 | Any prefix of a word repre... |
| clwwnisshclwwsn 30583 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30584 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30585 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30586 | The set of cyclical shifts... |
| clwwlknccat 30587 | The concatenation of two w... |
| umgr2cwwk2dif 30588 | If a word represents a clo... |
| umgr2cwwkdifex 30589 | If a word represents a clo... |
| erclwwlknrel 30590 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30591 | Two classes are equivalent... |
| erclwwlkneqlen 30592 | If two classes are equival... |
| erclwwlknref 30593 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30594 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30595 | ` .~ ` is a transitive rel... |
| erclwwlkn 30596 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30597 | The quotient set of the se... |
| hashclwwlkn0 30598 | The number of closed walks... |
| eclclwwlkn1 30599 | An equivalence class accor... |
| eleclclwwlkn 30600 | A member of an equivalence... |
| hashecclwwlkn1 30601 | The size of every equivale... |
| umgrhashecclwwlk 30602 | The size of every equivale... |
| fusgrhashclwwlkn 30603 | The size of the set of clo... |
| clwwlkndivn 30604 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30605 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30606 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30607 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30608 | There is a one-to-one onto... |
| clwlkssizeeq 30609 | The size of the set of clo... |
| clwlksndivn 30610 | The size of the set of clo... |
| clwwlknonmpo 30613 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30614 | The set of closed walks on... |
| isclwwlknon 30615 | A word over the set of ver... |
| clwwlk0on0 30616 | There is no word over the ... |
| clwwlknon0 30617 | Sufficient conditions for ... |
| clwwlknonfin 30618 | In a finite graph ` G ` , ... |
| clwwlknonel 30619 | Characterization of a word... |
| clwwlknonccat 30620 | The concatenation of two w... |
| clwwlknon1 30621 | The set of closed walks on... |
| clwwlknon1loop 30622 | If there is a loop at vert... |
| clwwlknon1nloop 30623 | If there is no loop at ver... |
| clwwlknon1sn 30624 | The set of (closed) walks ... |
| clwwlknon1le1 30625 | There is at most one (clos... |
| clwwlknon2 30626 | The set of closed walks on... |
| clwwlknon2x 30627 | The set of closed walks on... |
| s2elclwwlknon2 30628 | Sufficient conditions of a... |
| clwwlknon2num 30629 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30630 | A word over vertices repre... |
| clwwlknonex2lem1 30631 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30632 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30633 | Extending a closed walk ` ... |
| clwwlknonex2e 30634 | Extending a closed walk ` ... |
| clwwlknondisj 30635 | The sets of closed walks o... |
| clwwlknun 30636 | The set of closed walks of... |
| clwwlkvbij 30637 | There is a bijection betwe... |
| 0ewlk 30638 | The empty set (empty seque... |
| 1ewlk 30639 | A sequence of 1 edge is an... |
| 0wlk 30640 | A pair of an empty set (of... |
| is0wlk 30641 | A pair of an empty set (of... |
| 0wlkonlem1 30642 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30643 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30644 | A walk of length 0 from a ... |
| 0wlkons1 30645 | A walk of length 0 from a ... |
| 0trl 30646 | A pair of an empty set (of... |
| is0trl 30647 | A pair of an empty set (of... |
| 0trlon 30648 | A trail of length 0 from a... |
| 0pth 30649 | A pair of an empty set (of... |
| 0spth 30650 | A pair of an empty set (of... |
| 0pthon 30651 | A path of length 0 from a ... |
| 0pthon1 30652 | A path of length 0 from a ... |
| 0pthonv 30653 | For each vertex there is a... |
| 0clwlk 30654 | A pair of an empty set (of... |
| 0clwlkv 30655 | Any vertex (more precisely... |
| 0clwlk0 30656 | There is no closed walk in... |
| 0crct 30657 | A pair of an empty set (of... |
| 0cycl 30658 | A pair of an empty set (of... |
| 1pthdlem1 30659 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30660 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30661 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30662 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30663 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30664 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30665 | In a graph with two vertic... |
| 1trld 30666 | In a graph with two vertic... |
| 1pthd 30667 | In a graph with two vertic... |
| 1pthond 30668 | In a graph with two vertic... |
| upgr1wlkdlem1 30669 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30670 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30671 | In a pseudograph with two ... |
| upgr1trld 30672 | In a pseudograph with two ... |
| upgr1pthd 30673 | In a pseudograph with two ... |
| upgr1pthond 30674 | In a pseudograph with two ... |
| lppthon 30675 | A loop (which is an edge a... |
| lp1cycl 30676 | A loop (which is an edge a... |
| loop1cycl 30677 | A hypergraph has a cycle o... |
| 2cycld 30678 | Construction of a 2-cycle ... |
| umgr2cycllem 30679 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 30680 | A multigraph with two dist... |
| dfacycgr1 30683 | An alternate definition of... |
| isacycgr 30684 | The property of being an a... |
| isacycgr1 30685 | The property of being an a... |
| acycgrcycl 30686 | Any cycle in an acyclic gr... |
| 1pthon2v 30687 | For each pair of adjacent ... |
| 1pthon2ve 30688 | For each pair of adjacent ... |
| wlk2v2elem1 30689 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30690 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30691 | In a graph with two vertic... |
| ntrl2v2e 30692 | A walk which is not a trai... |
| 3wlkdlem1 30693 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30694 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30695 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30696 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30697 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30698 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30699 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30700 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30701 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30702 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30703 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30704 | Construction of a walk fro... |
| 3wlkond 30705 | A walk of length 3 from on... |
| 3trld 30706 | Construction of a trail fr... |
| 3trlond 30707 | A trail of length 3 from o... |
| 3pthd 30708 | A path of length 3 from on... |
| 3pthond 30709 | A path of length 3 from on... |
| 3spthd 30710 | A simple path of length 3 ... |
| 3spthond 30711 | A simple path of length 3 ... |
| 3cycld 30712 | Construction of a 3-cycle ... |
| 3cyclpd 30713 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30714 | If there is a cycle of len... |
| uhgr3cyclexlem 30715 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30716 | If there are three differe... |
| umgr3cyclex 30717 | If there are three (differ... |
| umgr3v3e3cycl 30718 | If and only if there is a ... |
| upgr4cycl4dv4e 30719 | If there is a cycle of len... |
| dfconngr1 30722 | Alternative definition of ... |
| isconngr 30723 | The property of being a co... |
| isconngr1 30724 | The property of being a co... |
| cusconngr 30725 | A complete hypergraph is c... |
| 0conngr 30726 | A graph without vertices i... |
| 0vconngr 30727 | A graph without vertices i... |
| 1conngr 30728 | A graph with (at most) one... |
| conngrv2edg 30729 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30730 | A vertex in a connected mu... |
| releupth 30733 | The set ` ( EulerPaths `` ... |
| eupths 30734 | The Eulerian paths on the ... |
| iseupth 30735 | The property " ` <. F , P ... |
| iseupthf1o 30736 | The property " ` <. F , P ... |
| eupthi 30737 | Properties of an Eulerian ... |
| eupthf1o 30738 | The ` F ` function in an E... |
| eupthfi 30739 | Any graph with an Eulerian... |
| eupthseg 30740 | The ` N ` -th edge in an e... |
| upgriseupth 30741 | The property " ` <. F , P ... |
| upgreupthi 30742 | Properties of an Eulerian ... |
| upgreupthseg 30743 | The ` N ` -th edge in an e... |
| eupthcl 30744 | An Eulerian path has lengt... |
| eupthistrl 30745 | An Eulerian path is a trai... |
| eupthiswlk 30746 | An Eulerian path is a walk... |
| eupthpf 30747 | The ` P ` function in an E... |
| eupth0 30748 | There is an Eulerian path ... |
| eupthres 30749 | The restriction ` <. H , Q... |
| eupthp1 30750 | Append one path segment to... |
| eupth2eucrct 30751 | Append one path segment to... |
| eupth2lem1 30752 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30753 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30754 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30755 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30756 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30757 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30758 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30759 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30760 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30761 | The effect on vertex degre... |
| eupth2lem3lem1 30762 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30763 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30764 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30765 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30766 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30767 | Formerly part of proof of ... |
| eupth2lem3lem7 30768 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30769 | Formerly part of proof of ... |
| eupth2lem3 30770 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30771 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30772 | Lemma for ~ eupth2 (induct... |
| eupth2 30773 | The only vertices of odd d... |
| eulerpathpr 30774 | A graph with an Eulerian p... |
| eulerpath 30775 | A pseudograph with an Eule... |
| eulercrct 30776 | A pseudograph with an Eule... |
| eucrctshift 30777 | Cyclically shifting the in... |
| eucrct2eupth1 30778 | Removing one edge ` ( I ``... |
| eucrct2eupth 30779 | Removing one edge ` ( I ``... |
| konigsbergvtx 30780 | The set of vertices of the... |
| konigsbergiedg 30781 | The indexed edges of the K... |
| konigsbergiedgw 30782 | The indexed edges of the K... |
| konigsbergssiedgwpr 30783 | Each subset of the indexed... |
| konigsbergssiedgw 30784 | Each subset of the indexed... |
| konigsbergumgr 30785 | The Königsberg graph ... |
| konigsberglem1 30786 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30787 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30788 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30789 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30790 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30791 | The Königsberg Bridge... |
| isfrgr 30794 | The property of being a fr... |
| frgrusgr 30795 | A friendship graph is a si... |
| frgr0v 30796 | Any null graph (set with n... |
| frgr0vb 30797 | Any null graph (without ve... |
| frgruhgr0v 30798 | Any null graph (without ve... |
| frgr0 30799 | The null graph (graph with... |
| frcond1 30800 | The friendship condition: ... |
| frcond2 30801 | The friendship condition: ... |
| frgreu 30802 | Variant of ~ frcond2 : An... |
| frcond3 30803 | The friendship condition, ... |
| frcond4 30804 | The friendship condition, ... |
| frgr1v 30805 | Any graph with (at most) o... |
| nfrgr2v 30806 | Any graph with two (differ... |
| frgr3vlem1 30807 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30808 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30809 | Any graph with three verti... |
| 1vwmgr 30810 | Every graph with one verte... |
| 3vfriswmgrlem 30811 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30812 | Every friendship graph wit... |
| 1to2vfriswmgr 30813 | Every friendship graph wit... |
| 1to3vfriswmgr 30814 | Every friendship graph wit... |
| 1to3vfriendship 30815 | The friendship theorem for... |
| 2pthfrgrrn 30816 | Between any two (different... |
| 2pthfrgrrn2 30817 | Between any two (different... |
| 2pthfrgr 30818 | Between any two (different... |
| 3cyclfrgrrn1 30819 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30820 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30821 | Every vertex in a friendsh... |
| 3cyclfrgr 30822 | Every vertex in a friendsh... |
| 4cycl2v2nb 30823 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30824 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30825 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30826 | A graph with a 4-cycle is ... |
| frgrnbnb 30827 | If two neighbors ` U ` and... |
| frgrconngr 30828 | A friendship graph is conn... |
| vdgn0frgrv2 30829 | A vertex in a friendship g... |
| vdgn1frgrv2 30830 | Any vertex in a friendship... |
| vdgn1frgrv3 30831 | Any vertex in a friendship... |
| vdgfrgrgt2 30832 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30833 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30834 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30835 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30836 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30837 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30838 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30839 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30840 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30841 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30842 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30843 | In a friendship graph, two... |
| frgrwopreglem4a 30844 | In a friendship graph any ... |
| frgrwopreglem5a 30845 | If a friendship graph has ... |
| frgrwopreglem1 30846 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30847 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30848 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30849 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30850 | According to statement 5 i... |
| frgrwopregbsn 30851 | According to statement 5 i... |
| frgrwopreg1 30852 | According to statement 5 i... |
| frgrwopreg2 30853 | According to statement 5 i... |
| frgrwopreglem5lem 30854 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30855 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30856 | Alternate direct proof of ... |
| frgrwopreg 30857 | In a friendship graph ther... |
| frgrregorufr0 30858 | In a friendship graph ther... |
| frgrregorufr 30859 | If there is a vertex havin... |
| frgrregorufrg 30860 | If there is a vertex havin... |
| frgr2wwlkeu 30861 | For two different vertices... |
| frgr2wwlkn0 30862 | In a friendship graph, the... |
| frgr2wwlk1 30863 | In a friendship graph, the... |
| frgr2wsp1 30864 | In a friendship graph, the... |
| frgr2wwlkeqm 30865 | If there is a (simple) pat... |
| frgrhash2wsp 30866 | The number of simple paths... |
| fusgreg2wsplem 30867 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30868 | The set of paths of length... |
| fusgreghash2wspv 30869 | According to statement 7 i... |
| fusgreg2wsp 30870 | In a finite simple graph, ... |
| 2wspmdisj 30871 | The sets of paths of lengt... |
| fusgreghash2wsp 30872 | In a finite k-regular grap... |
| frrusgrord0lem 30873 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30874 | If a nonempty finite frien... |
| frrusgrord 30875 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30876 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30877 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30878 | If the initial vertex of a... |
| clwwnonrepclwwnon 30879 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30880 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30881 | Value of operation ` C ` ,... |
| 2clwwlk2 30882 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30883 | Characterization of an ele... |
| 2clwwlk2clwwlk 30884 | An element of the value of... |
| numclwwlk1lem2foalem 30885 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30886 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30887 | Characterization of an ele... |
| numclwwlk1lem2foa 30888 | Going forth and back from ... |
| numclwwlk1lem2f 30889 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30890 | Value of the function ` T ... |
| numclwwlk1lem2f1 30891 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30892 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30893 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30894 | The set of double loops of... |
| numclwwlk1 30895 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30896 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30897 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30898 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30899 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30900 | The sets of the two repres... |
| wlkl0 30901 | There is exactly one walk ... |
| clwlknon2num 30902 | There are k walks of lengt... |
| numclwlk1lem1 30903 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30904 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30905 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30906 | Value of operation ` H ` ,... |
| numclwwlkovh 30907 | Value of operation ` H ` ,... |
| numclwwlkovq 30908 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30909 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30910 | In a friendship graph, for... |
| numclwlk2lem2f 30911 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30912 | Value of the function ` R ... |
| numclwlk2lem2f1o 30913 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30914 | In a friendship graph, the... |
| numclwwlk2 30915 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30916 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30917 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30918 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30919 | Statement 12 in [Huneke] p... |
| numclwwlk4 30920 | The total number of closed... |
| numclwwlk5lem 30921 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30922 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30923 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30924 | For a prime divisor ` P ` ... |
| numclwwlk7 30925 | Statement 14 in [Huneke] p... |
| numclwwlk8 30926 | The size of the set of clo... |
| frgrreggt1 30927 | If a finite nonempty frien... |
| frgrreg 30928 | If a finite nonempty frien... |
| frgrregord013 30929 | If a finite friendship gra... |
| frgrregord13 30930 | If a nonempty finite frien... |
| frgrogt3nreg 30931 | If a finite friendship gra... |
| friendshipgt3 30932 | The friendship theorem for... |
| friendship 30933 | The friendship theorem: I... |
| conventions 30934 |
H... |
| conventions-labels 30935 |
... |
| conventions-comments 30936 |
... |
| natded 30937 | Here are typical n... |
| ex-natded5.2 30938 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30939 | A more efficient proof of ... |
| ex-natded5.2i 30940 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30941 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30942 | A more efficient proof of ... |
| ex-natded5.3i 30943 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30944 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30945 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30946 | A more efficient proof of ... |
| ex-natded5.8 30947 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30948 | A more efficient proof of ... |
| ex-natded5.13 30949 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30950 | A more efficient proof of ... |
| ex-natded9.20 30951 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30952 | A more efficient proof of ... |
| ex-natded9.26 30953 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30954 | A more efficient proof of ... |
| ex-or 30955 | Example for ~ df-or . Exa... |
| ex-an 30956 | Example for ~ df-an . Exa... |
| ex-dif 30957 | Example for ~ df-dif . Ex... |
| ex-un 30958 | Example for ~ df-un . Exa... |
| ex-in 30959 | Example for ~ df-in . Exa... |
| ex-uni 30960 | Example for ~ df-uni . Ex... |
| ex-ss 30961 | Example for ~ df-ss . Exa... |
| ex-pss 30962 | Example for ~ df-pss . Ex... |
| ex-pw 30963 | Example for ~ df-pw . Exa... |
| ex-pr 30964 | Example for ~ df-pr . (Co... |
| ex-br 30965 | Example for ~ df-br . Exa... |
| ex-opab 30966 | Example for ~ df-opab . E... |
| ex-eprel 30967 | Example for ~ df-eprel . ... |
| ex-id 30968 | Example for ~ df-id . Exa... |
| ex-po 30969 | Example for ~ df-po . Exa... |
| ex-xp 30970 | Example for ~ df-xp . Exa... |
| ex-cnv 30971 | Example for ~ df-cnv . Ex... |
| ex-co 30972 | Example for ~ df-co . Exa... |
| ex-dm 30973 | Example for ~ df-dm . Exa... |
| ex-rn 30974 | Example for ~ df-rn . Exa... |
| ex-res 30975 | Example for ~ df-res . Ex... |
| ex-ima 30976 | Example for ~ df-ima . Ex... |
| ex-fv 30977 | Example for ~ df-fv . Exa... |
| ex-1st 30978 | Example for ~ df-1st . Ex... |
| ex-2nd 30979 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30980 | Example for ~ df-dec , 100... |
| ex-fl 30981 | Example for ~ df-fl . Exa... |
| ex-ceil 30982 | Example for ~ df-ceil . (... |
| ex-mod 30983 | Example for ~ df-mod . (C... |
| ex-exp 30984 | Example for ~ df-exp . (C... |
| ex-fac 30985 | Example for ~ df-fac . (C... |
| ex-bc 30986 | Example for ~ df-bc . (Co... |
| ex-hash 30987 | Example for ~ df-hash . (... |
| ex-sqrt 30988 | Example for ~ df-sqrt . (... |
| ex-abs 30989 | Example for ~ df-abs . (C... |
| ex-dvds 30990 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30991 | Example for ~ df-gcd . (C... |
| ex-lcm 30992 | Example for ~ df-lcm . (C... |
| ex-prmo 30993 | Example for ~ df-prmo : ` ... |
| aevdemo 30994 | Proof illustrating the com... |
| ex-ind-dvds 30995 | Example of a proof by indu... |
| ex-fpar 30996 | Formalized example provide... |
| avril1 30997 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30998 | The law of excluded middle... |
| helloworld 30999 | The classic "Hello world" ... |
| 1p1e2apr1 31000 | One plus one equals two. ... |
| eqid1 31001 | Law of identity (reflexivi... |
| 1div0apr 31002 | Division by zero is forbid... |
| topnfbey 31003 | Nothing seems to be imposs... |
| 9p10ne21 31004 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 31005 | 9 + 10 equals 21. This as... |
| nrt2irr 31007 | The ` N ` -th root of 2 is... |
| nowisdomv 31008 | One's wisdom on matters of... |
| isplig 31011 | The predicate "is a planar... |
| ispligb 31012 | The predicate "is a planar... |
| tncp 31013 | In any planar incidence ge... |
| l2p 31014 | For any line in a planar i... |
| lpni 31015 | For any line in a planar i... |
| nsnlplig 31016 | There is no "one-point lin... |
| nsnlpligALT 31017 | Alternate version of ~ nsn... |
| n0lplig 31018 | There is no "empty line" i... |
| n0lpligALT 31019 | Alternate version of ~ n0l... |
| eulplig 31020 | Through two distinct point... |
| pliguhgr 31021 | Any planar incidence geome... |
| dummylink 31022 | Alias for ~ a1ii that may ... |
| id1 31023 | Alias for ~ idALT that may... |
| isgrpo 31032 | The predicate "is a group ... |
| isgrpoi 31033 | Properties that determine ... |
| grpofo 31034 | A group operation maps ont... |
| grpocl 31035 | Closure law for a group op... |
| grpolidinv 31036 | A group has a left identit... |
| grpon0 31037 | The base set of a group is... |
| grpoass 31038 | A group operation is assoc... |
| grpoidinvlem1 31039 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 31040 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 31041 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 31042 | Lemma for ~ grpoidinv . (... |
| grpoidinv 31043 | A group has a left and rig... |
| grpoideu 31044 | The left identity element ... |
| grporndm 31045 | A group's range in terms o... |
| 0ngrp 31046 | The empty set is not a gro... |
| gidval 31047 | The value of the identity ... |
| grpoidval 31048 | Lemma for ~ grpoidcl and o... |
| grpoidcl 31049 | The identity element of a ... |
| grpoidinv2 31050 | A group's properties using... |
| grpolid 31051 | The identity element of a ... |
| grporid 31052 | The identity element of a ... |
| grporcan 31053 | Right cancellation law for... |
| grpoinveu 31054 | The left inverse element o... |
| grpoid 31055 | Two ways of saying that an... |
| grporn 31056 | The range of a group opera... |
| grpoinvfval 31057 | The inverse function of a ... |
| grpoinvval 31058 | The inverse of a group ele... |
| grpoinvcl 31059 | A group element's inverse ... |
| grpoinv 31060 | The properties of a group ... |
| grpolinv 31061 | The left inverse of a grou... |
| grporinv 31062 | The right inverse of a gro... |
| grpoinvid1 31063 | The inverse of a group ele... |
| grpoinvid2 31064 | The inverse of a group ele... |
| grpolcan 31065 | Left cancellation law for ... |
| grpo2inv 31066 | Double inverse law for gro... |
| grpoinvf 31067 | Mapping of the inverse fun... |
| grpoinvop 31068 | The inverse of the group o... |
| grpodivfval 31069 | Group division (or subtrac... |
| grpodivval 31070 | Group division (or subtrac... |
| grpodivinv 31071 | Group division by an inver... |
| grpoinvdiv 31072 | Inverse of a group divisio... |
| grpodivf 31073 | Mapping for group division... |
| grpodivcl 31074 | Closure of group division ... |
| grpodivdiv 31075 | Double group division. (C... |
| grpomuldivass 31076 | Associative-type law for m... |
| grpodivid 31077 | Division of a group member... |
| grponpcan 31078 | Cancellation law for group... |
| isablo 31081 | The predicate "is an Abeli... |
| ablogrpo 31082 | An Abelian group operation... |
| ablocom 31083 | An Abelian group operation... |
| ablo32 31084 | Commutative/associative la... |
| ablo4 31085 | Commutative/associative la... |
| isabloi 31086 | Properties that determine ... |
| ablomuldiv 31087 | Law for group multiplicati... |
| ablodivdiv 31088 | Law for double group divis... |
| ablodivdiv4 31089 | Law for double group divis... |
| ablodiv32 31090 | Swap the second and third ... |
| ablonncan 31091 | Cancellation law for group... |
| ablonnncan1 31092 | Cancellation law for group... |
| vcrel 31095 | The class of all complex v... |
| vciOLD 31096 | Obsolete version of ~ cvsi... |
| vcsm 31097 | Functionality of th scalar... |
| vccl 31098 | Closure of the scalar prod... |
| vcidOLD 31099 | Identity element for the s... |
| vcdi 31100 | Distributive law for the s... |
| vcdir 31101 | Distributive law for the s... |
| vcass 31102 | Associative law for the sc... |
| vc2OLD 31103 | A vector plus itself is tw... |
| vcablo 31104 | Vector addition is an Abel... |
| vcgrp 31105 | Vector addition is a group... |
| vclcan 31106 | Left cancellation law for ... |
| vczcl 31107 | The zero vector is a vecto... |
| vc0rid 31108 | The zero vector is a right... |
| vc0 31109 | Zero times a vector is the... |
| vcz 31110 | Anything times the zero ve... |
| vcm 31111 | Minus 1 times a vector is ... |
| isvclem 31112 | Lemma for ~ isvcOLD . (Co... |
| vcex 31113 | The components of a comple... |
| isvcOLD 31114 | The predicate "is a comple... |
| isvciOLD 31115 | Properties that determine ... |
| cnaddabloOLD 31116 | Obsolete version of ~ cnad... |
| cnidOLD 31117 | Obsolete version of ~ cnad... |
| cncvcOLD 31118 | Obsolete version of ~ cncv... |
| nvss 31128 | Structure of the class of ... |
| nvvcop 31129 | A normed complex vector sp... |
| nvrel 31137 | The class of all normed co... |
| vafval 31138 | Value of the function for ... |
| bafval 31139 | Value of the function for ... |
| smfval 31140 | Value of the function for ... |
| 0vfval 31141 | Value of the function for ... |
| nmcvfval 31142 | Value of the norm function... |
| nvop2 31143 | A normed complex vector sp... |
| nvvop 31144 | The vector space component... |
| isnvlem 31145 | Lemma for ~ isnv . (Contr... |
| nvex 31146 | The components of a normed... |
| isnv 31147 | The predicate "is a normed... |
| isnvi 31148 | Properties that determine ... |
| nvi 31149 | The properties of a normed... |
| nvvc 31150 | The vector space component... |
| nvablo 31151 | The vector addition operat... |
| nvgrp 31152 | The vector addition operat... |
| nvgf 31153 | Mapping for the vector add... |
| nvsf 31154 | Mapping for the scalar mul... |
| nvgcl 31155 | Closure law for the vector... |
| nvcom 31156 | The vector addition (group... |
| nvass 31157 | The vector addition (group... |
| nvadd32 31158 | Commutative/associative la... |
| nvrcan 31159 | Right cancellation law for... |
| nvadd4 31160 | Rearrangement of 4 terms i... |
| nvscl 31161 | Closure law for the scalar... |
| nvsid 31162 | Identity element for the s... |
| nvsass 31163 | Associative law for the sc... |
| nvscom 31164 | Commutative law for the sc... |
| nvdi 31165 | Distributive law for the s... |
| nvdir 31166 | Distributive law for the s... |
| nv2 31167 | A vector plus itself is tw... |
| vsfval 31168 | Value of the function for ... |
| nvzcl 31169 | Closure law for the zero v... |
| nv0rid 31170 | The zero vector is a right... |
| nv0lid 31171 | The zero vector is a left ... |
| nv0 31172 | Zero times a vector is the... |
| nvsz 31173 | Anything times the zero ve... |
| nvinv 31174 | Minus 1 times a vector is ... |
| nvinvfval 31175 | Function for the negative ... |
| nvm 31176 | Vector subtraction in term... |
| nvmval 31177 | Value of vector subtractio... |
| nvmval2 31178 | Value of vector subtractio... |
| nvmfval 31179 | Value of the function for ... |
| nvmf 31180 | Mapping for the vector sub... |
| nvmcl 31181 | Closure law for the vector... |
| nvnnncan1 31182 | Cancellation law for vecto... |
| nvmdi 31183 | Distributive law for scala... |
| nvnegneg 31184 | Double negative of a vecto... |
| nvmul0or 31185 | If a scalar product is zer... |
| nvrinv 31186 | A vector minus itself. (C... |
| nvlinv 31187 | Minus a vector plus itself... |
| nvpncan2 31188 | Cancellation law for vecto... |
| nvpncan 31189 | Cancellation law for vecto... |
| nvaddsub 31190 | Commutative/associative la... |
| nvnpcan 31191 | Cancellation law for a nor... |
| nvaddsub4 31192 | Rearrangement of 4 terms i... |
| nvmeq0 31193 | The difference between two... |
| nvmid 31194 | A vector minus itself is t... |
| nvf 31195 | Mapping for the norm funct... |
| nvcl 31196 | The norm of a normed compl... |
| nvcli 31197 | The norm of a normed compl... |
| nvs 31198 | Proportionality property o... |
| nvsge0 31199 | The norm of a scalar produ... |
| nvm1 31200 | The norm of the negative o... |
| nvdif 31201 | The norm of the difference... |
| nvpi 31202 | The norm of a vector plus ... |
| nvz0 31203 | The norm of a zero vector ... |
| nvz 31204 | The norm of a vector is ze... |
| nvtri 31205 | Triangle inequality for th... |
| nvmtri 31206 | Triangle inequality for th... |
| nvabs 31207 | Norm difference property o... |
| nvge0 31208 | The norm of a normed compl... |
| nvgt0 31209 | A nonzero norm is positive... |
| nv1 31210 | From any nonzero vector, c... |
| nvop 31211 | A complex inner product sp... |
| cnnv 31212 | The set of complex numbers... |
| cnnvg 31213 | The vector addition (group... |
| cnnvba 31214 | The base set of the normed... |
| cnnvs 31215 | The scalar product operati... |
| cnnvnm 31216 | The norm operation of the ... |
| cnnvm 31217 | The vector subtraction ope... |
| elimnv 31218 | Hypothesis elimination lem... |
| elimnvu 31219 | Hypothesis elimination lem... |
| imsval 31220 | Value of the induced metri... |
| imsdval 31221 | Value of the induced metri... |
| imsdval2 31222 | Value of the distance func... |
| nvnd 31223 | The norm of a normed compl... |
| imsdf 31224 | Mapping for the induced me... |
| imsmetlem 31225 | Lemma for ~ imsmet . (Con... |
| imsmet 31226 | The induced metric of a no... |
| imsxmet 31227 | The induced metric of a no... |
| cnims 31228 | The metric induced on the ... |
| vacn 31229 | Vector addition is jointly... |
| nmcvcn 31230 | The norm of a normed compl... |
| nmcnc 31231 | The norm of a normed compl... |
| smcnlem 31232 | Lemma for ~ smcn . (Contr... |
| smcn 31233 | Scalar multiplication is j... |
| vmcn 31234 | Vector subtraction is join... |
| dipfval 31237 | The inner product function... |
| ipval 31238 | Value of the inner product... |
| ipval2lem2 31239 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 31240 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 31241 | Lemma for ~ ipval3 . (Con... |
| ipval2 31242 | Expansion of the inner pro... |
| 4ipval2 31243 | Four times the inner produ... |
| ipval3 31244 | Expansion of the inner pro... |
| ipidsq 31245 | The inner product of a vec... |
| ipnm 31246 | Norm expressed in terms of... |
| dipcl 31247 | An inner product is a comp... |
| ipf 31248 | Mapping for the inner prod... |
| dipcj 31249 | The complex conjugate of a... |
| ipipcj 31250 | An inner product times its... |
| diporthcom 31251 | Orthogonality (meaning inn... |
| dip0r 31252 | Inner product with a zero ... |
| dip0l 31253 | Inner product with a zero ... |
| ipz 31254 | The inner product of a vec... |
| dipcn 31255 | Inner product is jointly c... |
| sspval 31258 | The set of all subspaces o... |
| isssp 31259 | The predicate "is a subspa... |
| sspid 31260 | A normed complex vector sp... |
| sspnv 31261 | A subspace is a normed com... |
| sspba 31262 | The base set of a subspace... |
| sspg 31263 | Vector addition on a subsp... |
| sspgval 31264 | Vector addition on a subsp... |
| ssps 31265 | Scalar multiplication on a... |
| sspsval 31266 | Scalar multiplication on a... |
| sspmlem 31267 | Lemma for ~ sspm and other... |
| sspmval 31268 | Vector addition on a subsp... |
| sspm 31269 | Vector subtraction on a su... |
| sspz 31270 | The zero vector of a subsp... |
| sspn 31271 | The norm on a subspace is ... |
| sspnval 31272 | The norm on a subspace in ... |
| sspimsval 31273 | The induced metric on a su... |
| sspims 31274 | The induced metric on a su... |
| lnoval 31287 | The set of linear operator... |
| islno 31288 | The predicate "is a linear... |
| lnolin 31289 | Basic linearity property o... |
| lnof 31290 | A linear operator is a map... |
| lno0 31291 | The value of a linear oper... |
| lnocoi 31292 | The composition of two lin... |
| lnoadd 31293 | Addition property of a lin... |
| lnosub 31294 | Subtraction property of a ... |
| lnomul 31295 | Scalar multiplication prop... |
| nvo00 31296 | Two ways to express a zero... |
| nmoofval 31297 | The operator norm function... |
| nmooval 31298 | The operator norm function... |
| nmosetre 31299 | The set in the supremum of... |
| nmosetn0 31300 | The set in the supremum of... |
| nmoxr 31301 | The norm of an operator is... |
| nmooge0 31302 | The norm of an operator is... |
| nmorepnf 31303 | The norm of an operator is... |
| nmoreltpnf 31304 | The norm of any operator i... |
| nmogtmnf 31305 | The norm of an operator is... |
| nmoolb 31306 | A lower bound for an opera... |
| nmoubi 31307 | An upper bound for an oper... |
| nmoub3i 31308 | An upper bound for an oper... |
| nmoub2i 31309 | An upper bound for an oper... |
| nmobndi 31310 | Two ways to express that a... |
| nmounbi 31311 | Two ways two express that ... |
| nmounbseqi 31312 | An unbounded operator dete... |
| nmounbseqiALT 31313 | Alternate shorter proof of... |
| nmobndseqi 31314 | A bounded sequence determi... |
| nmobndseqiALT 31315 | Alternate shorter proof of... |
| bloval 31316 | The class of bounded linea... |
| isblo 31317 | The predicate "is a bounde... |
| isblo2 31318 | The predicate "is a bounde... |
| bloln 31319 | A bounded operator is a li... |
| blof 31320 | A bounded operator is an o... |
| nmblore 31321 | The norm of a bounded oper... |
| 0ofval 31322 | The zero operator between ... |
| 0oval 31323 | Value of the zero operator... |
| 0oo 31324 | The zero operator is an op... |
| 0lno 31325 | The zero operator is linea... |
| nmoo0 31326 | The operator norm of the z... |
| 0blo 31327 | The zero operator is a bou... |
| nmlno0lem 31328 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 31329 | The norm of a linear opera... |
| nmlno0 31330 | The norm of a linear opera... |
| nmlnoubi 31331 | An upper bound for the ope... |
| nmlnogt0 31332 | The norm of a nonzero line... |
| lnon0 31333 | The domain of a nonzero li... |
| nmblolbii 31334 | A lower bound for the norm... |
| nmblolbi 31335 | A lower bound for the norm... |
| isblo3i 31336 | The predicate "is a bounde... |
| blo3i 31337 | Properties that determine ... |
| blometi 31338 | Upper bound for the distan... |
| blocnilem 31339 | Lemma for ~ blocni and ~ l... |
| blocni 31340 | A linear operator is conti... |
| lnocni 31341 | If a linear operator is co... |
| blocn 31342 | A linear operator is conti... |
| blocn2 31343 | A bounded linear operator ... |
| ajfval 31344 | The adjoint function. (Co... |
| hmoval 31345 | The set of Hermitian (self... |
| ishmo 31346 | The predicate "is a hermit... |
| phnv 31349 | Every complex inner produc... |
| phrel 31350 | The class of all complex i... |
| phnvi 31351 | Every complex inner produc... |
| isphg 31352 | The predicate "is a comple... |
| phop 31353 | A complex inner product sp... |
| cncph 31354 | The set of complex numbers... |
| elimph 31355 | Hypothesis elimination lem... |
| elimphu 31356 | Hypothesis elimination lem... |
| isph 31357 | The predicate "is an inner... |
| phpar2 31358 | The parallelogram law for ... |
| phpar 31359 | The parallelogram law for ... |
| ip0i 31360 | A slight variant of Equati... |
| ip1ilem 31361 | Lemma for ~ ip1i . (Contr... |
| ip1i 31362 | Equation 6.47 of [Ponnusam... |
| ip2i 31363 | Equation 6.48 of [Ponnusam... |
| ipdirilem 31364 | Lemma for ~ ipdiri . (Con... |
| ipdiri 31365 | Distributive law for inner... |
| ipasslem1 31366 | Lemma for ~ ipassi . Show... |
| ipasslem2 31367 | Lemma for ~ ipassi . Show... |
| ipasslem3 31368 | Lemma for ~ ipassi . Show... |
| ipasslem4 31369 | Lemma for ~ ipassi . Show... |
| ipasslem5 31370 | Lemma for ~ ipassi . Show... |
| ipasslem7 31371 | Lemma for ~ ipassi . Show... |
| ipasslem8 31372 | Lemma for ~ ipassi . By ~... |
| ipasslem9 31373 | Lemma for ~ ipassi . Conc... |
| ipasslem10 31374 | Lemma for ~ ipassi . Show... |
| ipasslem11 31375 | Lemma for ~ ipassi . Show... |
| ipassi 31376 | Associative law for inner ... |
| dipdir 31377 | Distributive law for inner... |
| dipdi 31378 | Distributive law for inner... |
| ip2dii 31379 | Inner product of two sums.... |
| dipass 31380 | Associative law for inner ... |
| dipassr 31381 | "Associative" law for seco... |
| dipassr2 31382 | "Associative" law for inne... |
| dipsubdir 31383 | Distributive law for inner... |
| dipsubdi 31384 | Distributive law for inner... |
| pythi 31385 | The Pythagorean theorem fo... |
| siilem1 31386 | Lemma for ~ sii . (Contri... |
| siilem2 31387 | Lemma for ~ sii . (Contri... |
| siii 31388 | Inference from ~ sii . (C... |
| sii 31389 | Obsolete version of ~ ipca... |
| ipblnfi 31390 | A function ` F ` generated... |
| ip2eqi 31391 | Two vectors are equal iff ... |
| phoeqi 31392 | A condition implying that ... |
| ajmoi 31393 | Every operator has at most... |
| ajfuni 31394 | The adjoint function is a ... |
| ajfun 31395 | The adjoint function is a ... |
| ajval 31396 | Value of the adjoint funct... |
| iscbn 31399 | A complex Banach space is ... |
| cbncms 31400 | The induced metric on comp... |
| bnnv 31401 | Every complex Banach space... |
| bnrel 31402 | The class of all complex B... |
| bnsscmcl 31403 | A subspace of a Banach spa... |
| cnbn 31404 | The set of complex numbers... |
| ubthlem1 31405 | Lemma for ~ ubth . The fu... |
| ubthlem2 31406 | Lemma for ~ ubth . Given ... |
| ubthlem3 31407 | Lemma for ~ ubth . Prove ... |
| ubth 31408 | Uniform Boundedness Theore... |
| minvecolem1 31409 | Lemma for ~ minveco . The... |
| minvecolem2 31410 | Lemma for ~ minveco . Any... |
| minvecolem3 31411 | Lemma for ~ minveco . The... |
| minvecolem4a 31412 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 31413 | Lemma for ~ minveco . The... |
| minvecolem4c 31414 | Lemma for ~ minveco . The... |
| minvecolem4 31415 | Lemma for ~ minveco . The... |
| minvecolem5 31416 | Lemma for ~ minveco . Dis... |
| minvecolem6 31417 | Lemma for ~ minveco . Any... |
| minvecolem7 31418 | Lemma for ~ minveco . Sin... |
| minveco 31419 | Minimizing vector theorem,... |
| ishlo 31422 | The predicate "is a comple... |
| hlobn 31423 | Every complex Hilbert spac... |
| hlph 31424 | Every complex Hilbert spac... |
| hlrel 31425 | The class of all complex H... |
| hlnv 31426 | Every complex Hilbert spac... |
| hlnvi 31427 | Every complex Hilbert spac... |
| hlvc 31428 | Every complex Hilbert spac... |
| hlcmet 31429 | The induced metric on a co... |
| hlmet 31430 | The induced metric on a co... |
| hlpar2 31431 | The parallelogram law sati... |
| hlpar 31432 | The parallelogram law sati... |
| hlex 31433 | The base set of a Hilbert ... |
| hladdf 31434 | Mapping for Hilbert space ... |
| hlcom 31435 | Hilbert space vector addit... |
| hlass 31436 | Hilbert space vector addit... |
| hl0cl 31437 | The Hilbert space zero vec... |
| hladdid 31438 | Hilbert space addition wit... |
| hlmulf 31439 | Mapping for Hilbert space ... |
| hlmulid 31440 | Hilbert space scalar multi... |
| hlmulass 31441 | Hilbert space scalar multi... |
| hldi 31442 | Hilbert space scalar multi... |
| hldir 31443 | Hilbert space scalar multi... |
| hlmul0 31444 | Hilbert space scalar multi... |
| hlipf 31445 | Mapping for Hilbert space ... |
| hlipcj 31446 | Conjugate law for Hilbert ... |
| hlipdir 31447 | Distributive law for Hilbe... |
| hlipass 31448 | Associative law for Hilber... |
| hlipgt0 31449 | The inner product of a Hil... |
| hlcompl 31450 | Completeness of a Hilbert ... |
| cnchl 31451 | The set of complex numbers... |
| htthlem 31452 | Lemma for ~ htth . The co... |
| htth 31453 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31509 | The group (addition) opera... |
| h2hsm 31510 | The scalar product operati... |
| h2hnm 31511 | The norm function of Hilbe... |
| h2hvs 31512 | The vector subtraction ope... |
| h2hmetdval 31513 | Value of the distance func... |
| h2hcau 31514 | The Cauchy sequences of Hi... |
| h2hlm 31515 | The limit sequences of Hil... |
| axhilex-zf 31516 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31517 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31518 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31519 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31520 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31521 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31522 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31523 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31524 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31525 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31526 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31527 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31528 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31529 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31530 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31531 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31532 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31533 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31546 | The Hilbert space scalar p... |
| hvaddcl 31547 | Closure of vector addition... |
| hvmulcl 31548 | Closure of scalar multipli... |
| hvmulcli 31549 | Closure inference for scal... |
| hvsubf 31550 | Mapping domain and codomai... |
| hvsubval 31551 | Value of vector subtractio... |
| hvsubcl 31552 | Closure of vector subtract... |
| hvaddcli 31553 | Closure of vector addition... |
| hvcomi 31554 | Commutation of vector addi... |
| hvsubvali 31555 | Value of vector subtractio... |
| hvsubcli 31556 | Closure of vector subtract... |
| ifhvhv0 31557 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31558 | Addition with the zero vec... |
| hvmul0 31559 | Scalar multiplication with... |
| hvmul0or 31560 | If a scalar product is zer... |
| hvsubid 31561 | Subtraction of a vector fr... |
| hvnegid 31562 | Addition of negative of a ... |
| hv2neg 31563 | Two ways to express the ne... |
| hvaddlidi 31564 | Addition with the zero vec... |
| hvnegidi 31565 | Addition of negative of a ... |
| hv2negi 31566 | Two ways to express the ne... |
| hvm1neg 31567 | Convert minus one times a ... |
| hvaddsubval 31568 | Value of vector addition i... |
| hvadd32 31569 | Commutative/associative la... |
| hvadd12 31570 | Commutative/associative la... |
| hvadd4 31571 | Hilbert vector space addit... |
| hvsub4 31572 | Hilbert vector space addit... |
| hvaddsub12 31573 | Commutative/associative la... |
| hvpncan 31574 | Addition/subtraction cance... |
| hvpncan2 31575 | Addition/subtraction cance... |
| hvaddsubass 31576 | Associativity of sum and d... |
| hvpncan3 31577 | Subtraction and addition o... |
| hvmulcom 31578 | Scalar multiplication comm... |
| hvsubass 31579 | Hilbert vector space assoc... |
| hvsub32 31580 | Hilbert vector space commu... |
| hvmulassi 31581 | Scalar multiplication asso... |
| hvmulcomi 31582 | Scalar multiplication comm... |
| hvmul2negi 31583 | Double negative in scalar ... |
| hvsubdistr1 31584 | Scalar multiplication dist... |
| hvsubdistr2 31585 | Scalar multiplication dist... |
| hvdistr1i 31586 | Scalar multiplication dist... |
| hvsubdistr1i 31587 | Scalar multiplication dist... |
| hvassi 31588 | Hilbert vector space assoc... |
| hvadd32i 31589 | Hilbert vector space commu... |
| hvsubassi 31590 | Hilbert vector space assoc... |
| hvsub32i 31591 | Hilbert vector space commu... |
| hvadd12i 31592 | Hilbert vector space commu... |
| hvadd4i 31593 | Hilbert vector space addit... |
| hvsubsub4i 31594 | Hilbert vector space addit... |
| hvsubsub4 31595 | Hilbert vector space addit... |
| hv2times 31596 | Two times a vector. (Cont... |
| hvnegdii 31597 | Distribution of negative o... |
| hvsubeq0i 31598 | If the difference between ... |
| hvsubcan2i 31599 | Vector cancellation law. ... |
| hvaddcani 31600 | Cancellation law for vecto... |
| hvsubaddi 31601 | Relationship between vecto... |
| hvnegdi 31602 | Distribution of negative o... |
| hvsubeq0 31603 | If the difference between ... |
| hvaddeq0 31604 | If the sum of two vectors ... |
| hvaddcan 31605 | Cancellation law for vecto... |
| hvaddcan2 31606 | Cancellation law for vecto... |
| hvmulcan 31607 | Cancellation law for scala... |
| hvmulcan2 31608 | Cancellation law for scala... |
| hvsubcan 31609 | Cancellation law for vecto... |
| hvsubcan2 31610 | Cancellation law for vecto... |
| hvsub0 31611 | Subtraction of a zero vect... |
| hvsubadd 31612 | Relationship between vecto... |
| hvaddsub4 31613 | Hilbert vector space addit... |
| hicl 31615 | Closure of inner product. ... |
| hicli 31616 | Closure inference for inne... |
| his5 31621 | Associative law for inner ... |
| his52 31622 | Associative law for inner ... |
| his35 31623 | Move scalar multiplication... |
| his35i 31624 | Move scalar multiplication... |
| his7 31625 | Distributive law for inner... |
| hiassdi 31626 | Distributive/associative l... |
| his2sub 31627 | Distributive law for inner... |
| his2sub2 31628 | Distributive law for inner... |
| hire 31629 | A necessary and sufficient... |
| hiidrcl 31630 | Real closure of inner prod... |
| hi01 31631 | Inner product with the 0 v... |
| hi02 31632 | Inner product with the 0 v... |
| hiidge0 31633 | Inner product with self is... |
| his6 31634 | Zero inner product with se... |
| his1i 31635 | Conjugate law for inner pr... |
| abshicom 31636 | Commuted inner products ha... |
| hial0 31637 | A vector whose inner produ... |
| hial02 31638 | A vector whose inner produ... |
| hisubcomi 31639 | Two vector subtractions si... |
| hi2eq 31640 | Lemma used to prove equali... |
| hial2eq 31641 | Two vectors whose inner pr... |
| hial2eq2 31642 | Two vectors whose inner pr... |
| orthcom 31643 | Orthogonality commutes. (... |
| normlem0 31644 | Lemma used to derive prope... |
| normlem1 31645 | Lemma used to derive prope... |
| normlem2 31646 | Lemma used to derive prope... |
| normlem3 31647 | Lemma used to derive prope... |
| normlem4 31648 | Lemma used to derive prope... |
| normlem5 31649 | Lemma used to derive prope... |
| normlem6 31650 | Lemma used to derive prope... |
| normlem7 31651 | Lemma used to derive prope... |
| normlem8 31652 | Lemma used to derive prope... |
| normlem9 31653 | Lemma used to derive prope... |
| normlem7tALT 31654 | Lemma used to derive prope... |
| bcseqi 31655 | Equality case of Bunjakova... |
| normlem9at 31656 | Lemma used to derive prope... |
| dfhnorm2 31657 | Alternate definition of th... |
| normf 31658 | The norm function maps fro... |
| normval 31659 | The value of the norm of a... |
| normcl 31660 | Real closure of the norm o... |
| normge0 31661 | The norm of a vector is no... |
| normgt0 31662 | The norm of nonzero vector... |
| norm0 31663 | The norm of a zero vector.... |
| norm-i 31664 | Theorem 3.3(i) of [Beran] ... |
| normne0 31665 | A norm is nonzero iff its ... |
| normcli 31666 | Real closure of the norm o... |
| normsqi 31667 | The square of a norm. (Co... |
| norm-i-i 31668 | Theorem 3.3(i) of [Beran] ... |
| normsq 31669 | The square of a norm. (Co... |
| normsub0i 31670 | Two vectors are equal iff ... |
| normsub0 31671 | Two vectors are equal iff ... |
| norm-ii-i 31672 | Triangle inequality for no... |
| norm-ii 31673 | Triangle inequality for no... |
| norm-iii-i 31674 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31675 | Theorem 3.3(iii) of [Beran... |
| normsubi 31676 | Negative doesn't change th... |
| normpythi 31677 | Analogy to Pythagorean the... |
| normsub 31678 | Swapping order of subtract... |
| normneg 31679 | The norm of a vector equal... |
| normpyth 31680 | Analogy to Pythagorean the... |
| normpyc 31681 | Corollary to Pythagorean t... |
| norm3difi 31682 | Norm of differences around... |
| norm3adifii 31683 | Norm of differences around... |
| norm3lem 31684 | Lemma involving norm of di... |
| norm3dif 31685 | Norm of differences around... |
| norm3dif2 31686 | Norm of differences around... |
| norm3lemt 31687 | Lemma involving norm of di... |
| norm3adifi 31688 | Norm of differences around... |
| normpari 31689 | Parallelogram law for norm... |
| normpar 31690 | Parallelogram law for norm... |
| normpar2i 31691 | Corollary of parallelogram... |
| polid2i 31692 | Generalized polarization i... |
| polidi 31693 | Polarization identity. Re... |
| polid 31694 | Polarization identity. Re... |
| hilablo 31695 | Hilbert space vector addit... |
| hilid 31696 | The group identity element... |
| hilvc 31697 | Hilbert space is a complex... |
| hilnormi 31698 | Hilbert space norm in term... |
| hilhhi 31699 | Deduce the structure of Hi... |
| hhnv 31700 | Hilbert space is a normed ... |
| hhva 31701 | The group (addition) opera... |
| hhba 31702 | The base set of Hilbert sp... |
| hh0v 31703 | The zero vector of Hilbert... |
| hhsm 31704 | The scalar product operati... |
| hhvs 31705 | The vector subtraction ope... |
| hhnm 31706 | The norm function of Hilbe... |
| hhims 31707 | The induced metric of Hilb... |
| hhims2 31708 | Hilbert space distance met... |
| hhmet 31709 | The induced metric of Hilb... |
| hhxmet 31710 | The induced metric of Hilb... |
| hhmetdval 31711 | Value of the distance func... |
| hhip 31712 | The inner product operatio... |
| hhph 31713 | The Hilbert space of the H... |
| bcsiALT 31714 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31715 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31716 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31717 | Corollary of the Bunjakova... |
| bcs3 31718 | Corollary of the Bunjakova... |
| hcau 31719 | Member of the set of Cauch... |
| hcauseq 31720 | A Cauchy sequences on a Hi... |
| hcaucvg 31721 | A Cauchy sequence on a Hil... |
| seq1hcau 31722 | A sequence on a Hilbert sp... |
| hlimi 31723 | Express the predicate: Th... |
| hlimseqi 31724 | A sequence with a limit on... |
| hlimveci 31725 | Closure of the limit of a ... |
| hlimconvi 31726 | Convergence of a sequence ... |
| hlim2 31727 | The limit of a sequence on... |
| hlimadd 31728 | Limit of the sum of two se... |
| hilmet 31729 | The Hilbert space norm det... |
| hilxmet 31730 | The Hilbert space norm det... |
| hilmetdval 31731 | Value of the distance func... |
| hilims 31732 | Hilbert space distance met... |
| hhcau 31733 | The Cauchy sequences of Hi... |
| hhlm 31734 | The limit sequences of Hil... |
| hhcmpl 31735 | Lemma used for derivation ... |
| hilcompl 31736 | Lemma used for derivation ... |
| hhcms 31738 | The Hilbert space induced ... |
| hhhl 31739 | The Hilbert space structur... |
| hilcms 31740 | The Hilbert space norm det... |
| hilhl 31741 | The Hilbert space of the H... |
| issh 31743 | Subspace ` H ` of a Hilber... |
| issh2 31744 | Subspace ` H ` of a Hilber... |
| shss 31745 | A subspace is a subset of ... |
| shel 31746 | A member of a subspace of ... |
| shex 31747 | The set of subspaces of a ... |
| shssii 31748 | A closed subspace of a Hil... |
| sheli 31749 | A member of a subspace of ... |
| shelii 31750 | A member of a subspace of ... |
| sh0 31751 | The zero vector belongs to... |
| shaddcl 31752 | Closure of vector addition... |
| shmulcl 31753 | Closure of vector scalar m... |
| issh3 31754 | Subspace ` H ` of a Hilber... |
| shsubcl 31755 | Closure of vector subtract... |
| isch 31757 | Closed subspace ` H ` of a... |
| isch2 31758 | Closed subspace ` H ` of a... |
| chsh 31759 | A closed subspace is a sub... |
| chsssh 31760 | Closed subspaces are subsp... |
| chex 31761 | The set of closed subspace... |
| chshii 31762 | A closed subspace is a sub... |
| ch0 31763 | The zero vector belongs to... |
| chss 31764 | A closed subspace of a Hil... |
| chel 31765 | A member of a closed subsp... |
| chssii 31766 | A closed subspace of a Hil... |
| cheli 31767 | A member of a closed subsp... |
| chelii 31768 | A member of a closed subsp... |
| chlimi 31769 | The limit property of a cl... |
| hlim0 31770 | The zero sequence in Hilbe... |
| hlimcaui 31771 | If a sequence in Hilbert s... |
| hlimf 31772 | Function-like behavior of ... |
| hlimuni 31773 | A Hilbert space sequence c... |
| hlimreui 31774 | The limit of a Hilbert spa... |
| hlimeui 31775 | The limit of a Hilbert spa... |
| isch3 31776 | A Hilbert subspace is clos... |
| chcompl 31777 | Completeness of a closed s... |
| helch 31778 | The Hilbert lattice one (w... |
| ifchhv 31779 | Prove ` if ( A e. CH , A ,... |
| helsh 31780 | Hilbert space is a subspac... |
| shsspwh 31781 | Subspaces are subsets of H... |
| chsspwh 31782 | Closed subspaces are subse... |
| hsn0elch 31783 | The zero subspace belongs ... |
| norm1 31784 | From any nonzero Hilbert s... |
| norm1exi 31785 | A normalized vector exists... |
| norm1hex 31786 | A normalized vector can ex... |
| elch0 31789 | Membership in zero for clo... |
| h0elch 31790 | The zero subspace is a clo... |
| h0elsh 31791 | The zero subspace is a sub... |
| hhssva 31792 | The vector addition operat... |
| hhsssm 31793 | The scalar multiplication ... |
| hhssnm 31794 | The norm operation on a su... |
| issubgoilem 31795 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31796 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31797 | Abelian group property of ... |
| hhssablo 31798 | Abelian group property of ... |
| hhssnv 31799 | Normed complex vector spac... |
| hhssnvt 31800 | Normed complex vector spac... |
| hhsst 31801 | A member of ` SH ` is a su... |
| hhshsslem1 31802 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31803 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31804 | The predicate " ` H ` is a... |
| hhsssh2 31805 | The predicate " ` H ` is a... |
| hhssba 31806 | The base set of a subspace... |
| hhssvs 31807 | The vector subtraction ope... |
| hhssvsf 31808 | Mapping of the vector subt... |
| hhssims 31809 | Induced metric of a subspa... |
| hhssims2 31810 | Induced metric of a subspa... |
| hhssmet 31811 | Induced metric of a subspa... |
| hhssmetdval 31812 | Value of the distance func... |
| hhsscms 31813 | The induced metric of a cl... |
| hhssbnOLD 31814 | Obsolete version of ~ cssb... |
| ocval 31815 | Value of orthogonal comple... |
| ocel 31816 | Membership in orthogonal c... |
| shocel 31817 | Membership in orthogonal c... |
| ocsh 31818 | The orthogonal complement ... |
| shocsh 31819 | The orthogonal complement ... |
| ocss 31820 | An orthogonal complement i... |
| shocss 31821 | An orthogonal complement i... |
| occon 31822 | Contraposition law for ort... |
| occon2 31823 | Double contraposition for ... |
| occon2i 31824 | Double contraposition for ... |
| oc0 31825 | The zero vector belongs to... |
| ocorth 31826 | Members of a subset and it... |
| shocorth 31827 | Members of a subspace and ... |
| ococss 31828 | Inclusion in complement of... |
| shococss 31829 | Inclusion in complement of... |
| shorth 31830 | Members of orthogonal subs... |
| ocin 31831 | Intersection of a Hilbert ... |
| occon3 31832 | Hilbert lattice contraposi... |
| ocnel 31833 | A nonzero vector in the co... |
| chocvali 31834 | Value of the orthogonal co... |
| shuni 31835 | Two subspaces with trivial... |
| chocunii 31836 | Lemma for uniqueness part ... |
| pjhthmo 31837 | Projection Theorem, unique... |
| occllem 31838 | Lemma for ~ occl . (Contr... |
| occl 31839 | Closure of complement of H... |
| shoccl 31840 | Closure of complement of H... |
| choccl 31841 | Closure of complement of H... |
| choccli 31842 | Closure of ` CH ` orthocom... |
| shsval 31847 | Value of subspace sum of t... |
| shsss 31848 | The subspace sum is a subs... |
| shsel 31849 | Membership in the subspace... |
| shsel3 31850 | Membership in the subspace... |
| shseli 31851 | Membership in subspace sum... |
| shscli 31852 | Closure of subspace sum. ... |
| shscl 31853 | Closure of subspace sum. ... |
| shscom 31854 | Commutative law for subspa... |
| shsva 31855 | Vector sum belongs to subs... |
| shsel1 31856 | A subspace sum contains a ... |
| shsel2 31857 | A subspace sum contains a ... |
| shsvs 31858 | Vector subtraction belongs... |
| shsub1 31859 | Subspace sum is an upper b... |
| shsub2 31860 | Subspace sum is an upper b... |
| choc0 31861 | The orthocomplement of the... |
| choc1 31862 | The orthocomplement of the... |
| chocnul 31863 | Orthogonal complement of t... |
| shintcli 31864 | Closure of intersection of... |
| shintcl 31865 | The intersection of a none... |
| chintcli 31866 | The intersection of a none... |
| chintcl 31867 | The intersection (infimum)... |
| spanval 31868 | Value of the linear span o... |
| hsupval 31869 | Value of supremum of set o... |
| chsupval 31870 | The value of the supremum ... |
| spancl 31871 | The span of a subset of Hi... |
| elspancl 31872 | A member of a span is a ve... |
| shsupcl 31873 | Closure of the subspace su... |
| hsupcl 31874 | Closure of supremum of set... |
| chsupcl 31875 | Closure of supremum of sub... |
| hsupss 31876 | Subset relation for suprem... |
| chsupss 31877 | Subset relation for suprem... |
| hsupunss 31878 | The union of a set of Hilb... |
| chsupunss 31879 | The union of a set of clos... |
| spanss2 31880 | A subset of Hilbert space ... |
| shsupunss 31881 | The union of a set of subs... |
| spanid 31882 | A subspace of Hilbert spac... |
| spanss 31883 | Ordering relationship for ... |
| spanssoc 31884 | The span of a subset of Hi... |
| sshjval 31885 | Value of join for subsets ... |
| shjval 31886 | Value of join in ` SH ` . ... |
| chjval 31887 | Value of join in ` CH ` . ... |
| chjvali 31888 | Value of join in ` CH ` . ... |
| sshjval3 31889 | Value of join for subsets ... |
| sshjcl 31890 | Closure of join for subset... |
| shjcl 31891 | Closure of join in ` SH ` ... |
| chjcl 31892 | Closure of join in ` CH ` ... |
| shjcom 31893 | Commutative law for Hilber... |
| shless 31894 | Subset implies subset of s... |
| shlej1 31895 | Add disjunct to both sides... |
| shlej2 31896 | Add disjunct to both sides... |
| shincli 31897 | Closure of intersection of... |
| shscomi 31898 | Commutative law for subspa... |
| shsvai 31899 | Vector sum belongs to subs... |
| shsel1i 31900 | A subspace sum contains a ... |
| shsel2i 31901 | A subspace sum contains a ... |
| shsvsi 31902 | Vector subtraction belongs... |
| shunssi 31903 | Union is smaller than subs... |
| shunssji 31904 | Union is smaller than Hilb... |
| shsleji 31905 | Subspace sum is smaller th... |
| shjcomi 31906 | Commutative law for join i... |
| shsub1i 31907 | Subspace sum is an upper b... |
| shsub2i 31908 | Subspace sum is an upper b... |
| shub1i 31909 | Hilbert lattice join is an... |
| shjcli 31910 | Closure of ` CH ` join. (... |
| shjshcli 31911 | ` SH ` closure of join. (... |
| shlessi 31912 | Subset implies subset of s... |
| shlej1i 31913 | Add disjunct to both sides... |
| shlej2i 31914 | Add disjunct to both sides... |
| shslej 31915 | Subspace sum is smaller th... |
| shincl 31916 | Closure of intersection of... |
| shub1 31917 | Hilbert lattice join is an... |
| shub2 31918 | A subspace is a subset of ... |
| shsidmi 31919 | Idempotent law for Hilbert... |
| shslubi 31920 | The least upper bound law ... |
| shlesb1i 31921 | Hilbert lattice ordering i... |
| shsval2i 31922 | An alternate way to expres... |
| shsval3i 31923 | An alternate way to expres... |
| shmodsi 31924 | The modular law holds for ... |
| shmodi 31925 | The modular law is implied... |
| pjhthlem1 31926 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31927 | Lemma for ~ pjhth . (Cont... |
| pjhth 31928 | Projection Theorem: Any H... |
| pjhtheu 31929 | Projection Theorem: Any H... |
| pjhfval 31931 | The value of the projectio... |
| pjhval 31932 | Value of a projection. (C... |
| pjpreeq 31933 | Equality with a projection... |
| pjeq 31934 | Equality with a projection... |
| axpjcl 31935 | Closure of a projection in... |
| pjhcl 31936 | Closure of a projection in... |
| omlsilem 31937 | Lemma for orthomodular law... |
| omlsii 31938 | Subspace inference form of... |
| omlsi 31939 | Subspace form of orthomodu... |
| ococi 31940 | Complement of complement o... |
| ococ 31941 | Complement of complement o... |
| dfch2 31942 | Alternate definition of th... |
| ococin 31943 | The double complement is t... |
| hsupval2 31944 | Alternate definition of su... |
| chsupval2 31945 | The value of the supremum ... |
| sshjval2 31946 | Value of join in the set o... |
| chsupid 31947 | A subspace is the supremum... |
| chsupsn 31948 | Value of supremum of subse... |
| shlub 31949 | Hilbert lattice join is th... |
| shlubi 31950 | Hilbert lattice join is th... |
| pjhtheu2 31951 | Uniqueness of ` y ` for th... |
| pjcli 31952 | Closure of a projection in... |
| pjhcli 31953 | Closure of a projection in... |
| pjpjpre 31954 | Decomposition of a vector ... |
| axpjpj 31955 | Decomposition of a vector ... |
| pjclii 31956 | Closure of a projection in... |
| pjhclii 31957 | Closure of a projection in... |
| pjpj0i 31958 | Decomposition of a vector ... |
| pjpji 31959 | Decomposition of a vector ... |
| pjpjhth 31960 | Projection Theorem: Any H... |
| pjpjhthi 31961 | Projection Theorem: Any H... |
| pjop 31962 | Orthocomplement projection... |
| pjpo 31963 | Projection in terms of ort... |
| pjopi 31964 | Orthocomplement projection... |
| pjpoi 31965 | Projection in terms of ort... |
| pjoc1i 31966 | Projection of a vector in ... |
| pjchi 31967 | Projection of a vector in ... |
| pjoccl 31968 | The part of a vector that ... |
| pjoc1 31969 | Projection of a vector in ... |
| pjomli 31970 | Subspace form of orthomodu... |
| pjoml 31971 | Subspace form of orthomodu... |
| pjococi 31972 | Proof of orthocomplement t... |
| pjoc2i 31973 | Projection of a vector in ... |
| pjoc2 31974 | Projection of a vector in ... |
| sh0le 31975 | The zero subspace is the s... |
| ch0le 31976 | The zero subspace is the s... |
| shle0 31977 | No subspace is smaller tha... |
| chle0 31978 | No Hilbert lattice element... |
| chnlen0 31979 | A Hilbert lattice element ... |
| ch0pss 31980 | The zero subspace is a pro... |
| orthin 31981 | The intersection of orthog... |
| ssjo 31982 | The lattice join of a subs... |
| shne0i 31983 | A nonzero subspace has a n... |
| shs0i 31984 | Hilbert subspace sum with ... |
| shs00i 31985 | Two subspaces are zero iff... |
| ch0lei 31986 | The closed subspace zero i... |
| chle0i 31987 | No Hilbert closed subspace... |
| chne0i 31988 | A nonzero closed subspace ... |
| chocini 31989 | Intersection of a closed s... |
| chj0i 31990 | Join with lattice zero in ... |
| chm1i 31991 | Meet with lattice one in `... |
| chjcli 31992 | Closure of ` CH ` join. (... |
| chsleji 31993 | Subspace sum is smaller th... |
| chseli 31994 | Membership in subspace sum... |
| chincli 31995 | Closure of Hilbert lattice... |
| chsscon3i 31996 | Hilbert lattice contraposi... |
| chsscon1i 31997 | Hilbert lattice contraposi... |
| chsscon2i 31998 | Hilbert lattice contraposi... |
| chcon2i 31999 | Hilbert lattice contraposi... |
| chcon1i 32000 | Hilbert lattice contraposi... |
| chcon3i 32001 | Hilbert lattice contraposi... |
| chunssji 32002 | Union is smaller than ` CH... |
| chjcomi 32003 | Commutative law for join i... |
| chub1i 32004 | ` CH ` join is an upper bo... |
| chub2i 32005 | ` CH ` join is an upper bo... |
| chlubi 32006 | Hilbert lattice join is th... |
| chlubii 32007 | Hilbert lattice join is th... |
| chlej1i 32008 | Add join to both sides of ... |
| chlej2i 32009 | Add join to both sides of ... |
| chlej12i 32010 | Add join to both sides of ... |
| chlejb1i 32011 | Hilbert lattice ordering i... |
| chdmm1i 32012 | De Morgan's law for meet i... |
| chdmm2i 32013 | De Morgan's law for meet i... |
| chdmm3i 32014 | De Morgan's law for meet i... |
| chdmm4i 32015 | De Morgan's law for meet i... |
| chdmj1i 32016 | De Morgan's law for join i... |
| chdmj2i 32017 | De Morgan's law for join i... |
| chdmj3i 32018 | De Morgan's law for join i... |
| chdmj4i 32019 | De Morgan's law for join i... |
| chnlei 32020 | Equivalent expressions for... |
| chjassi 32021 | Associative law for Hilber... |
| chj00i 32022 | Two Hilbert lattice elemen... |
| chjoi 32023 | The join of a closed subsp... |
| chj1i 32024 | Join with Hilbert lattice ... |
| chm0i 32025 | Meet with Hilbert lattice ... |
| chm0 32026 | Meet with Hilbert lattice ... |
| shjshsi 32027 | Hilbert lattice join equal... |
| shjshseli 32028 | A closed subspace sum equa... |
| chne0 32029 | A nonzero closed subspace ... |
| chocin 32030 | Intersection of a closed s... |
| chssoc 32031 | A closed subspace less tha... |
| chj0 32032 | Join with Hilbert lattice ... |
| chslej 32033 | Subspace sum is smaller th... |
| chincl 32034 | Closure of Hilbert lattice... |
| chsscon3 32035 | Hilbert lattice contraposi... |
| chsscon1 32036 | Hilbert lattice contraposi... |
| chsscon2 32037 | Hilbert lattice contraposi... |
| chpsscon3 32038 | Hilbert lattice contraposi... |
| chpsscon1 32039 | Hilbert lattice contraposi... |
| chpsscon2 32040 | Hilbert lattice contraposi... |
| chjcom 32041 | Commutative law for Hilber... |
| chub1 32042 | Hilbert lattice join is gr... |
| chub2 32043 | Hilbert lattice join is gr... |
| chlub 32044 | Hilbert lattice join is th... |
| chlej1 32045 | Add join to both sides of ... |
| chlej2 32046 | Add join to both sides of ... |
| chlejb1 32047 | Hilbert lattice ordering i... |
| chlejb2 32048 | Hilbert lattice ordering i... |
| chnle 32049 | Equivalent expressions for... |
| chjo 32050 | The join of a closed subsp... |
| chabs1 32051 | Hilbert lattice absorption... |
| chabs2 32052 | Hilbert lattice absorption... |
| chabs1i 32053 | Hilbert lattice absorption... |
| chabs2i 32054 | Hilbert lattice absorption... |
| chjidm 32055 | Idempotent law for Hilbert... |
| chjidmi 32056 | Idempotent law for Hilbert... |
| chj12i 32057 | A rearrangement of Hilbert... |
| chj4i 32058 | Rearrangement of the join ... |
| chjjdiri 32059 | Hilbert lattice join distr... |
| chdmm1 32060 | De Morgan's law for meet i... |
| chdmm2 32061 | De Morgan's law for meet i... |
| chdmm3 32062 | De Morgan's law for meet i... |
| chdmm4 32063 | De Morgan's law for meet i... |
| chdmj1 32064 | De Morgan's law for join i... |
| chdmj2 32065 | De Morgan's law for join i... |
| chdmj3 32066 | De Morgan's law for join i... |
| chdmj4 32067 | De Morgan's law for join i... |
| chjass 32068 | Associative law for Hilber... |
| chj12 32069 | A rearrangement of Hilbert... |
| chj4 32070 | Rearrangement of the join ... |
| ledii 32071 | An ortholattice is distrib... |
| lediri 32072 | An ortholattice is distrib... |
| lejdii 32073 | An ortholattice is distrib... |
| lejdiri 32074 | An ortholattice is distrib... |
| ledi 32075 | An ortholattice is distrib... |
| spansn0 32076 | The span of the singleton ... |
| span0 32077 | The span of the empty set ... |
| elspani 32078 | Membership in the span of ... |
| spanuni 32079 | The span of a union is the... |
| spanun 32080 | The span of a union is the... |
| sshhococi 32081 | The join of two Hilbert sp... |
| hne0 32082 | Hilbert space has a nonzer... |
| chsup0 32083 | The supremum of the empty ... |
| h1deoi 32084 | Membership in orthocomplem... |
| h1dei 32085 | Membership in 1-dimensiona... |
| h1did 32086 | A generating vector belong... |
| h1dn0 32087 | A nonzero vector generates... |
| h1de2i 32088 | Membership in 1-dimensiona... |
| h1de2bi 32089 | Membership in 1-dimensiona... |
| h1de2ctlem 32090 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 32091 | Membership in 1-dimensiona... |
| spansni 32092 | The span of a singleton in... |
| elspansni 32093 | Membership in the span of ... |
| spansn 32094 | The span of a singleton in... |
| spansnch 32095 | The span of a Hilbert spac... |
| spansnsh 32096 | The span of a Hilbert spac... |
| spansnchi 32097 | The span of a singleton in... |
| spansnid 32098 | A vector belongs to the sp... |
| spansnmul 32099 | A scalar product with a ve... |
| elspansncl 32100 | A member of a span of a si... |
| elspansn 32101 | Membership in the span of ... |
| elspansn2 32102 | Membership in the span of ... |
| spansncol 32103 | The singletons of collinea... |
| spansneleqi 32104 | Membership relation implie... |
| spansneleq 32105 | Membership relation that i... |
| spansnss 32106 | The span of the singleton ... |
| elspansn3 32107 | A member of the span of th... |
| elspansn4 32108 | A span membership conditio... |
| elspansn5 32109 | A vector belonging to both... |
| spansnss2 32110 | The span of the singleton ... |
| normcan 32111 | Cancellation-type law that... |
| pjspansn 32112 | A projection on the span o... |
| spansnpji 32113 | A subset of Hilbert space ... |
| spanunsni 32114 | The span of the union of a... |
| spanpr 32115 | The span of a pair of vect... |
| h1datomi 32116 | A 1-dimensional subspace i... |
| h1datom 32117 | A 1-dimensional subspace i... |
| cmbr 32119 | Binary relation expressing... |
| pjoml2i 32120 | Variation of orthomodular ... |
| pjoml3i 32121 | Variation of orthomodular ... |
| pjoml4i 32122 | Variation of orthomodular ... |
| pjoml5i 32123 | The orthomodular law. Rem... |
| pjoml6i 32124 | An equivalent of the ortho... |
| cmbri 32125 | Binary relation expressing... |
| cmcmlem 32126 | Commutation is symmetric. ... |
| cmcmi 32127 | Commutation is symmetric. ... |
| cmcm2i 32128 | Commutation with orthocomp... |
| cmcm3i 32129 | Commutation with orthocomp... |
| cmcm4i 32130 | Commutation with orthocomp... |
| cmbr2i 32131 | Alternate definition of th... |
| cmcmii 32132 | Commutation is symmetric. ... |
| cmcm2ii 32133 | Commutation with orthocomp... |
| cmcm3ii 32134 | Commutation with orthocomp... |
| cmbr3i 32135 | Alternate definition for t... |
| cmbr4i 32136 | Alternate definition for t... |
| lecmi 32137 | Comparable Hilbert lattice... |
| lecmii 32138 | Comparable Hilbert lattice... |
| cmj1i 32139 | A Hilbert lattice element ... |
| cmj2i 32140 | A Hilbert lattice element ... |
| cmm1i 32141 | A Hilbert lattice element ... |
| cmm2i 32142 | A Hilbert lattice element ... |
| cmbr3 32143 | Alternate definition for t... |
| cm0 32144 | The zero Hilbert lattice e... |
| cmidi 32145 | The commutes relation is r... |
| pjoml2 32146 | Variation of orthomodular ... |
| pjoml3 32147 | Variation of orthomodular ... |
| pjoml5 32148 | The orthomodular law. Rem... |
| cmcm 32149 | Commutation is symmetric. ... |
| cmcm3 32150 | Commutation with orthocomp... |
| cmcm2 32151 | Commutation with orthocomp... |
| lecm 32152 | Comparable Hilbert lattice... |
| fh1 32153 | Foulis-Holland Theorem. I... |
| fh2 32154 | Foulis-Holland Theorem. I... |
| cm2j 32155 | A lattice element that com... |
| fh1i 32156 | Foulis-Holland Theorem. I... |
| fh2i 32157 | Foulis-Holland Theorem. I... |
| fh3i 32158 | Variation of the Foulis-Ho... |
| fh4i 32159 | Variation of the Foulis-Ho... |
| cm2ji 32160 | A lattice element that com... |
| cm2mi 32161 | A lattice element that com... |
| qlax1i 32162 | One of the equations showi... |
| qlax2i 32163 | One of the equations showi... |
| qlax3i 32164 | One of the equations showi... |
| qlax4i 32165 | One of the equations showi... |
| qlax5i 32166 | One of the equations showi... |
| qlaxr1i 32167 | One of the conditions show... |
| qlaxr2i 32168 | One of the conditions show... |
| qlaxr4i 32169 | One of the conditions show... |
| qlaxr5i 32170 | One of the conditions show... |
| qlaxr3i 32171 | A variation of the orthomo... |
| chscllem1 32172 | Lemma for ~ chscl . (Cont... |
| chscllem2 32173 | Lemma for ~ chscl . (Cont... |
| chscllem3 32174 | Lemma for ~ chscl . (Cont... |
| chscllem4 32175 | Lemma for ~ chscl . (Cont... |
| chscl 32176 | The subspace sum of two cl... |
| osumi 32177 | If two closed subspaces of... |
| osumcori 32178 | Corollary of ~ osumi . (C... |
| osumcor2i 32179 | Corollary of ~ osumi , sho... |
| osum 32180 | If two closed subspaces of... |
| spansnji 32181 | The subspace sum of a clos... |
| spansnj 32182 | The subspace sum of a clos... |
| spansnscl 32183 | The subspace sum of a clos... |
| sumspansn 32184 | The sum of two vectors bel... |
| spansnm0i 32185 | The meet of different one-... |
| nonbooli 32186 | A Hilbert lattice with two... |
| spansncvi 32187 | Hilbert space has the cove... |
| spansncv 32188 | Hilbert space has the cove... |
| 5oalem1 32189 | Lemma for orthoarguesian l... |
| 5oalem2 32190 | Lemma for orthoarguesian l... |
| 5oalem3 32191 | Lemma for orthoarguesian l... |
| 5oalem4 32192 | Lemma for orthoarguesian l... |
| 5oalem5 32193 | Lemma for orthoarguesian l... |
| 5oalem6 32194 | Lemma for orthoarguesian l... |
| 5oalem7 32195 | Lemma for orthoarguesian l... |
| 5oai 32196 | Orthoarguesian law 5OA. Th... |
| 3oalem1 32197 | Lemma for 3OA (weak) ortho... |
| 3oalem2 32198 | Lemma for 3OA (weak) ortho... |
| 3oalem3 32199 | Lemma for 3OA (weak) ortho... |
| 3oalem4 32200 | Lemma for 3OA (weak) ortho... |
| 3oalem5 32201 | Lemma for 3OA (weak) ortho... |
| 3oalem6 32202 | Lemma for 3OA (weak) ortho... |
| 3oai 32203 | 3OA (weak) orthoarguesian ... |
| pjorthi 32204 | Projection components on o... |
| pjch1 32205 | Property of identity proje... |
| pjo 32206 | The orthogonal projection.... |
| pjcompi 32207 | Component of a projection.... |
| pjidmi 32208 | A projection is idempotent... |
| pjadjii 32209 | A projection is self-adjoi... |
| pjaddii 32210 | Projection of vector sum i... |
| pjinormii 32211 | The inner product of a pro... |
| pjmulii 32212 | Projection of (scalar) pro... |
| pjsubii 32213 | Projection of vector diffe... |
| pjsslem 32214 | Lemma for subset relations... |
| pjss2i 32215 | Subset relationship for pr... |
| pjssmii 32216 | Projection meet property. ... |
| pjssge0ii 32217 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 32218 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 32219 | The projection on a subspa... |
| pjadji 32220 | A projection is self-adjoi... |
| pjaddi 32221 | Projection of vector sum i... |
| pjinormi 32222 | The inner product of a pro... |
| pjsubi 32223 | Projection of vector diffe... |
| pjmuli 32224 | Projection of scalar produ... |
| pjige0i 32225 | The inner product of a pro... |
| pjige0 32226 | The inner product of a pro... |
| pjcjt2 32227 | The projection on a subspa... |
| pj0i 32228 | The projection of the zero... |
| pjch 32229 | Projection of a vector in ... |
| pjid 32230 | The projection of a vector... |
| pjvec 32231 | The set of vectors belongi... |
| pjocvec 32232 | The set of vectors belongi... |
| pjocini 32233 | Membership of projection i... |
| pjini 32234 | Membership of projection i... |
| pjjsi 32235 | A sufficient condition for... |
| pjfni 32236 | Functionality of a project... |
| pjrni 32237 | The range of a projection.... |
| pjfoi 32238 | A projection maps onto its... |
| pjfi 32239 | The mapping of a projectio... |
| pjvi 32240 | The value of a projection ... |
| pjhfo 32241 | A projection maps onto its... |
| pjrn 32242 | The range of a projection.... |
| pjhf 32243 | The mapping of a projectio... |
| pjfn 32244 | Functionality of a project... |
| pjsumi 32245 | The projection on a subspa... |
| pj11i 32246 | One-to-one correspondence ... |
| pjdsi 32247 | Vector decomposition into ... |
| pjds3i 32248 | Vector decomposition into ... |
| pj11 32249 | One-to-one correspondence ... |
| pjmfn 32250 | Functionality of the proje... |
| pjmf1 32251 | The projector function map... |
| pjoi0 32252 | The inner product of proje... |
| pjoi0i 32253 | The inner product of proje... |
| pjopythi 32254 | Pythagorean theorem for pr... |
| pjopyth 32255 | Pythagorean theorem for pr... |
| pjnormi 32256 | The norm of the projection... |
| pjpythi 32257 | Pythagorean theorem for pr... |
| pjneli 32258 | If a vector does not belon... |
| pjnorm 32259 | The norm of the projection... |
| pjpyth 32260 | Pythagorean theorem for pr... |
| pjnel 32261 | If a vector does not belon... |
| pjnorm2 32262 | A vector belongs to the su... |
| mayete3i 32263 | Mayet's equation E_3. Par... |
| mayetes3i 32264 | Mayet's equation E^*_3, de... |
| hosmval 32270 | Value of the sum of two Hi... |
| hommval 32271 | Value of the scalar produc... |
| hodmval 32272 | Value of the difference of... |
| hfsmval 32273 | Value of the sum of two Hi... |
| hfmmval 32274 | Value of the scalar produc... |
| hosval 32275 | Value of the sum of two Hi... |
| homval 32276 | Value of the scalar produc... |
| hodval 32277 | Value of the difference of... |
| hfsval 32278 | Value of the sum of two Hi... |
| hfmval 32279 | Value of the scalar produc... |
| hoscl 32280 | Closure of the sum of two ... |
| homcl 32281 | Closure of the scalar prod... |
| hodcl 32282 | Closure of the difference ... |
| ho0val 32285 | Value of the zero Hilbert ... |
| ho0f 32286 | Functionality of the zero ... |
| df0op2 32287 | Alternate definition of Hi... |
| dfiop2 32288 | Alternate definition of Hi... |
| hoif 32289 | Functionality of the Hilbe... |
| hoival 32290 | The value of the Hilbert s... |
| hoico1 32291 | Composition with the Hilbe... |
| hoico2 32292 | Composition with the Hilbe... |
| hoaddcl 32293 | The sum of Hilbert space o... |
| homulcl 32294 | The scalar product of a Hi... |
| hoeq 32295 | Equality of Hilbert space ... |
| hoeqi 32296 | Equality of Hilbert space ... |
| hoscli 32297 | Closure of Hilbert space o... |
| hodcli 32298 | Closure of Hilbert space o... |
| hocoi 32299 | Composition of Hilbert spa... |
| hococli 32300 | Closure of composition of ... |
| hocofi 32301 | Mapping of composition of ... |
| hocofni 32302 | Functionality of compositi... |
| hoaddcli 32303 | Mapping of sum of Hilbert ... |
| hosubcli 32304 | Mapping of difference of H... |
| hoaddfni 32305 | Functionality of sum of Hi... |
| hosubfni 32306 | Functionality of differenc... |
| hoaddcomi 32307 | Commutativity of sum of Hi... |
| hosubcl 32308 | Mapping of difference of H... |
| hoaddcom 32309 | Commutativity of sum of Hi... |
| hodsi 32310 | Relationship between Hilbe... |
| hoaddassi 32311 | Associativity of sum of Hi... |
| hoadd12i 32312 | Commutative/associative la... |
| hoadd32i 32313 | Commutative/associative la... |
| hocadddiri 32314 | Distributive law for Hilbe... |
| hocsubdiri 32315 | Distributive law for Hilbe... |
| ho2coi 32316 | Double composition of Hilb... |
| hoaddass 32317 | Associativity of sum of Hi... |
| hoadd32 32318 | Commutative/associative la... |
| hoadd4 32319 | Rearrangement of 4 terms i... |
| hocsubdir 32320 | Distributive law for Hilbe... |
| hoaddridi 32321 | Sum of a Hilbert space ope... |
| hodidi 32322 | Difference of a Hilbert sp... |
| ho0coi 32323 | Composition of the zero op... |
| hoid1i 32324 | Composition of Hilbert spa... |
| hoid1ri 32325 | Composition of Hilbert spa... |
| hoaddrid 32326 | Sum of a Hilbert space ope... |
| hodid 32327 | Difference of a Hilbert sp... |
| hon0 32328 | A Hilbert space operator i... |
| hodseqi 32329 | Subtraction and addition o... |
| ho0subi 32330 | Subtraction of Hilbert spa... |
| honegsubi 32331 | Relationship between Hilbe... |
| ho0sub 32332 | Subtraction of Hilbert spa... |
| hosubid1 32333 | The zero operator subtract... |
| honegsub 32334 | Relationship between Hilbe... |
| homullid 32335 | An operator equals its sca... |
| homco1 32336 | Associative law for scalar... |
| homulass 32337 | Scalar product associative... |
| hoadddi 32338 | Scalar product distributiv... |
| hoadddir 32339 | Scalar product reverse dis... |
| homul12 32340 | Swap first and second fact... |
| honegneg 32341 | Double negative of a Hilbe... |
| hosubneg 32342 | Relationship between opera... |
| hosubdi 32343 | Scalar product distributiv... |
| honegdi 32344 | Distribution of negative o... |
| honegsubdi 32345 | Distribution of negative o... |
| honegsubdi2 32346 | Distribution of negative o... |
| hosubsub2 32347 | Law for double subtraction... |
| hosub4 32348 | Rearrangement of 4 terms i... |
| hosubadd4 32349 | Rearrangement of 4 terms i... |
| hoaddsubass 32350 | Associative-type law for a... |
| hoaddsub 32351 | Law for operator addition ... |
| hosubsub 32352 | Law for double subtraction... |
| hosubsub4 32353 | Law for double subtraction... |
| ho2times 32354 | Two times a Hilbert space ... |
| hoaddsubassi 32355 | Associativity of sum and d... |
| hoaddsubi 32356 | Law for sum and difference... |
| hosd1i 32357 | Hilbert space operator sum... |
| hosd2i 32358 | Hilbert space operator sum... |
| hopncani 32359 | Hilbert space operator can... |
| honpcani 32360 | Hilbert space operator can... |
| hosubeq0i 32361 | If the difference between ... |
| honpncani 32362 | Hilbert space operator can... |
| ho01i 32363 | A condition implying that ... |
| ho02i 32364 | A condition implying that ... |
| hoeq1 32365 | A condition implying that ... |
| hoeq2 32366 | A condition implying that ... |
| adjmo 32367 | Every Hilbert space operat... |
| adjsym 32368 | Symmetry property of an ad... |
| eigrei 32369 | A necessary and sufficient... |
| eigre 32370 | A necessary and sufficient... |
| eigposi 32371 | A sufficient condition (fi... |
| eigorthi 32372 | A necessary and sufficient... |
| eigorth 32373 | A necessary and sufficient... |
| nmopval 32391 | Value of the norm of a Hil... |
| elcnop 32392 | Property defining a contin... |
| ellnop 32393 | Property defining a linear... |
| lnopf 32394 | A linear Hilbert space ope... |
| elbdop 32395 | Property defining a bounde... |
| bdopln 32396 | A bounded linear Hilbert s... |
| bdopf 32397 | A bounded linear Hilbert s... |
| nmopsetretALT 32398 | The set in the supremum of... |
| nmopsetretHIL 32399 | The set in the supremum of... |
| nmopsetn0 32400 | The set in the supremum of... |
| nmopxr 32401 | The norm of a Hilbert spac... |
| nmoprepnf 32402 | The norm of a Hilbert spac... |
| nmopgtmnf 32403 | The norm of a Hilbert spac... |
| nmopreltpnf 32404 | The norm of a Hilbert spac... |
| nmopre 32405 | The norm of a bounded oper... |
| elbdop2 32406 | Property defining a bounde... |
| elunop 32407 | Property defining a unitar... |
| elhmop 32408 | Property defining a Hermit... |
| hmopf 32409 | A Hermitian operator is a ... |
| hmopex 32410 | The class of Hermitian ope... |
| nmfnval 32411 | Value of the norm of a Hil... |
| nmfnsetre 32412 | The set in the supremum of... |
| nmfnsetn0 32413 | The set in the supremum of... |
| nmfnxr 32414 | The norm of any Hilbert sp... |
| nmfnrepnf 32415 | The norm of a Hilbert spac... |
| nlfnval 32416 | Value of the null space of... |
| elcnfn 32417 | Property defining a contin... |
| ellnfn 32418 | Property defining a linear... |
| lnfnf 32419 | A linear Hilbert space fun... |
| dfadj2 32420 | Alternate definition of th... |
| funadj 32421 | Functionality of the adjoi... |
| dmadjss 32422 | The domain of the adjoint ... |
| dmadjop 32423 | A member of the domain of ... |
| adjeu 32424 | Elementhood in the domain ... |
| adjval 32425 | Value of the adjoint funct... |
| adjval2 32426 | Value of the adjoint funct... |
| cnvadj 32427 | The adjoint function equal... |
| funcnvadj 32428 | The converse of the adjoin... |
| adj1o 32429 | The adjoint function maps ... |
| dmadjrn 32430 | The adjoint of an operator... |
| eigvecval 32431 | The set of eigenvectors of... |
| eigvalfval 32432 | The eigenvalues of eigenve... |
| specval 32433 | The value of the spectrum ... |
| speccl 32434 | The spectrum of an operato... |
| hhlnoi 32435 | The linear operators of Hi... |
| hhnmoi 32436 | The norm of an operator in... |
| hhbloi 32437 | A bounded linear operator ... |
| hh0oi 32438 | The zero operator in Hilbe... |
| hhcno 32439 | The continuous operators o... |
| hhcnf 32440 | The continuous functionals... |
| dmadjrnb 32441 | The adjoint of an operator... |
| nmoplb 32442 | A lower bound for an opera... |
| nmopub 32443 | An upper bound for an oper... |
| nmopub2tALT 32444 | An upper bound for an oper... |
| nmopub2tHIL 32445 | An upper bound for an oper... |
| nmopge0 32446 | The norm of any Hilbert sp... |
| nmopgt0 32447 | A linear Hilbert space ope... |
| cnopc 32448 | Basic continuity property ... |
| lnopl 32449 | Basic linearity property o... |
| unop 32450 | Basic inner product proper... |
| unopf1o 32451 | A unitary operator in Hilb... |
| unopnorm 32452 | A unitary operator is idem... |
| cnvunop 32453 | The inverse (converse) of ... |
| unopadj 32454 | The inverse (converse) of ... |
| unoplin 32455 | A unitary operator is line... |
| counop 32456 | The composition of two uni... |
| hmop 32457 | Basic inner product proper... |
| hmopre 32458 | The inner product of the v... |
| nmfnlb 32459 | A lower bound for a functi... |
| nmfnleub 32460 | An upper bound for the nor... |
| nmfnleub2 32461 | An upper bound for the nor... |
| nmfnge0 32462 | The norm of any Hilbert sp... |
| elnlfn 32463 | Membership in the null spa... |
| elnlfn2 32464 | Membership in the null spa... |
| cnfnc 32465 | Basic continuity property ... |
| lnfnl 32466 | Basic linearity property o... |
| adjcl 32467 | Closure of the adjoint of ... |
| adj1 32468 | Property of an adjoint Hil... |
| adj2 32469 | Property of an adjoint Hil... |
| adjeq 32470 | A property that determines... |
| adjadj 32471 | Double adjoint. Theorem 3... |
| adjvalval 32472 | Value of the value of the ... |
| unopadj2 32473 | The adjoint of a unitary o... |
| hmopadj 32474 | A Hermitian operator is se... |
| hmdmadj 32475 | Every Hermitian operator h... |
| hmopadj2 32476 | An operator is Hermitian i... |
| hmoplin 32477 | A Hermitian operator is li... |
| brafval 32478 | The bra of a vector, expre... |
| braval 32479 | A bra-ket juxtaposition, e... |
| braadd 32480 | Linearity property of bra ... |
| bramul 32481 | Linearity property of bra ... |
| brafn 32482 | The bra function is a func... |
| bralnfn 32483 | The Dirac bra function is ... |
| bracl 32484 | Closure of the bra functio... |
| bra0 32485 | The Dirac bra of the zero ... |
| brafnmul 32486 | Anti-linearity property of... |
| kbfval 32487 | The outer product of two v... |
| kbop 32488 | The outer product of two v... |
| kbval 32489 | The value of the operator ... |
| kbmul 32490 | Multiplication property of... |
| kbpj 32491 | If a vector ` A ` has norm... |
| eleigvec 32492 | Membership in the set of e... |
| eleigvec2 32493 | Membership in the set of e... |
| eleigveccl 32494 | Closure of an eigenvector ... |
| eigvalval 32495 | The eigenvalue of an eigen... |
| eigvalcl 32496 | An eigenvalue is a complex... |
| eigvec1 32497 | Property of an eigenvector... |
| eighmre 32498 | The eigenvalues of a Hermi... |
| eighmorth 32499 | Eigenvectors of a Hermitia... |
| nmopnegi 32500 | Value of the norm of the n... |
| lnop0 32501 | The value of a linear Hilb... |
| lnopmul 32502 | Multiplicative property of... |
| lnopli 32503 | Basic scalar product prope... |
| lnopfi 32504 | A linear Hilbert space ope... |
| lnop0i 32505 | The value of a linear Hilb... |
| lnopaddi 32506 | Additive property of a lin... |
| lnopmuli 32507 | Multiplicative property of... |
| lnopaddmuli 32508 | Sum/product property of a ... |
| lnopsubi 32509 | Subtraction property for a... |
| lnopsubmuli 32510 | Subtraction/product proper... |
| lnopmulsubi 32511 | Product/subtraction proper... |
| homco2 32512 | Move a scalar product out ... |
| idunop 32513 | The identity function (res... |
| 0cnop 32514 | The identically zero funct... |
| 0cnfn 32515 | The identically zero funct... |
| idcnop 32516 | The identity function (res... |
| idhmop 32517 | The Hilbert space identity... |
| 0hmop 32518 | The identically zero funct... |
| 0lnop 32519 | The identically zero funct... |
| 0lnfn 32520 | The identically zero funct... |
| nmop0 32521 | The norm of the zero opera... |
| nmfn0 32522 | The norm of the identicall... |
| hmopbdoptHIL 32523 | A Hermitian operator is a ... |
| hoddii 32524 | Distributive law for Hilbe... |
| hoddi 32525 | Distributive law for Hilbe... |
| nmop0h 32526 | The norm of any operator o... |
| idlnop 32527 | The identity function (res... |
| 0bdop 32528 | The identically zero opera... |
| adj0 32529 | Adjoint of the zero operat... |
| nmlnop0iALT 32530 | A linear operator with a z... |
| nmlnop0iHIL 32531 | A linear operator with a z... |
| nmlnopgt0i 32532 | A linear Hilbert space ope... |
| nmlnop0 32533 | A linear operator with a z... |
| nmlnopne0 32534 | A linear operator with a n... |
| lnopmi 32535 | The scalar product of a li... |
| lnophsi 32536 | The sum of two linear oper... |
| lnophdi 32537 | The difference of two line... |
| lnopcoi 32538 | The composition of two lin... |
| lnopco0i 32539 | The composition of a linea... |
| lnopeq0lem1 32540 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32541 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32542 | A condition implying that ... |
| lnopeqi 32543 | Two linear Hilbert space o... |
| lnopeq 32544 | Two linear Hilbert space o... |
| lnopunilem1 32545 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32546 | Lemma for ~ lnopunii . (C... |
| lnopunii 32547 | If a linear operator (whos... |
| elunop2 32548 | An operator is unitary iff... |
| nmopun 32549 | Norm of a unitary Hilbert ... |
| unopbd 32550 | A unitary operator is a bo... |
| lnophmlem1 32551 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32552 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32553 | A linear operator is Hermi... |
| lnophm 32554 | A linear operator is Hermi... |
| hmops 32555 | The sum of two Hermitian o... |
| hmopm 32556 | The scalar product of a He... |
| hmopd 32557 | The difference of two Herm... |
| hmopco 32558 | The composition of two com... |
| nmbdoplbi 32559 | A lower bound for the norm... |
| nmbdoplb 32560 | A lower bound for the norm... |
| nmcexi 32561 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32562 | The norm of a continuous l... |
| nmcoplbi 32563 | A lower bound for the norm... |
| nmcopex 32564 | The norm of a continuous l... |
| nmcoplb 32565 | A lower bound for the norm... |
| nmophmi 32566 | The norm of the scalar pro... |
| bdophmi 32567 | The scalar product of a bo... |
| lnconi 32568 | Lemma for ~ lnopconi and ~... |
| lnopconi 32569 | A condition equivalent to ... |
| lnopcon 32570 | A condition equivalent to ... |
| lnopcnbd 32571 | A linear operator is conti... |
| lncnopbd 32572 | A continuous linear operat... |
| lncnbd 32573 | A continuous linear operat... |
| lnopcnre 32574 | A linear operator is conti... |
| lnfnli 32575 | Basic property of a linear... |
| lnfnfi 32576 | A linear Hilbert space fun... |
| lnfn0i 32577 | The value of a linear Hilb... |
| lnfnaddi 32578 | Additive property of a lin... |
| lnfnmuli 32579 | Multiplicative property of... |
| lnfnaddmuli 32580 | Sum/product property of a ... |
| lnfnsubi 32581 | Subtraction property for a... |
| lnfn0 32582 | The value of a linear Hilb... |
| lnfnmul 32583 | Multiplicative property of... |
| nmbdfnlbi 32584 | A lower bound for the norm... |
| nmbdfnlb 32585 | A lower bound for the norm... |
| nmcfnexi 32586 | The norm of a continuous l... |
| nmcfnlbi 32587 | A lower bound for the norm... |
| nmcfnex 32588 | The norm of a continuous l... |
| nmcfnlb 32589 | A lower bound of the norm ... |
| lnfnconi 32590 | A condition equivalent to ... |
| lnfncon 32591 | A condition equivalent to ... |
| lnfncnbd 32592 | A linear functional is con... |
| imaelshi 32593 | The image of a subspace un... |
| rnelshi 32594 | The range of a linear oper... |
| nlelshi 32595 | The null space of a linear... |
| nlelchi 32596 | The null space of a contin... |
| riesz3i 32597 | A continuous linear functi... |
| riesz4i 32598 | A continuous linear functi... |
| riesz4 32599 | A continuous linear functi... |
| riesz1 32600 | Part 1 of the Riesz repres... |
| riesz2 32601 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32602 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32603 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32604 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32605 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32606 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32607 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32608 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32609 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32610 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32611 | Every continuous linear op... |
| cnlnadjeui 32612 | Every continuous linear op... |
| cnlnadjeu 32613 | Every continuous linear op... |
| cnlnadj 32614 | Every continuous linear op... |
| cnlnssadj 32615 | Every continuous linear Hi... |
| bdopssadj 32616 | Every bounded linear Hilbe... |
| bdopadj 32617 | Every bounded linear Hilbe... |
| adjbdln 32618 | The adjoint of a bounded l... |
| adjbdlnb 32619 | An operator is bounded and... |
| adjbd1o 32620 | The mapping of adjoints of... |
| adjlnop 32621 | The adjoint of an operator... |
| adjsslnop 32622 | Every operator with an adj... |
| nmopadjlei 32623 | Property of the norm of an... |
| nmopadjlem 32624 | Lemma for ~ nmopadji . (C... |
| nmopadji 32625 | Property of the norm of an... |
| adjeq0 32626 | An operator is zero iff it... |
| adjmul 32627 | The adjoint of the scalar ... |
| adjadd 32628 | The adjoint of the sum of ... |
| nmoptrii 32629 | Triangle inequality for th... |
| nmopcoi 32630 | Upper bound for the norm o... |
| bdophsi 32631 | The sum of two bounded lin... |
| bdophdi 32632 | The difference between two... |
| bdopcoi 32633 | The composition of two bou... |
| nmoptri2i 32634 | Triangle-type inequality f... |
| adjcoi 32635 | The adjoint of a compositi... |
| nmopcoadji 32636 | The norm of an operator co... |
| nmopcoadj2i 32637 | The norm of an operator co... |
| nmopcoadj0i 32638 | An operator composed with ... |
| unierri 32639 | If we approximate a chain ... |
| branmfn 32640 | The norm of the bra functi... |
| brabn 32641 | The bra of a vector is a b... |
| rnbra 32642 | The set of bras equals the... |
| bra11 32643 | The bra function maps vect... |
| bracnln 32644 | A bra is a continuous line... |
| cnvbraval 32645 | Value of the converse of t... |
| cnvbracl 32646 | Closure of the converse of... |
| cnvbrabra 32647 | The converse bra of the br... |
| bracnvbra 32648 | The bra of the converse br... |
| bracnlnval 32649 | The vector that a continuo... |
| cnvbramul 32650 | Multiplication property of... |
| kbass1 32651 | Dirac bra-ket associative ... |
| kbass2 32652 | Dirac bra-ket associative ... |
| kbass3 32653 | Dirac bra-ket associative ... |
| kbass4 32654 | Dirac bra-ket associative ... |
| kbass5 32655 | Dirac bra-ket associative ... |
| kbass6 32656 | Dirac bra-ket associative ... |
| leopg 32657 | Ordering relation for posi... |
| leop 32658 | Ordering relation for oper... |
| leop2 32659 | Ordering relation for oper... |
| leop3 32660 | Operator ordering in terms... |
| leoppos 32661 | Binary relation defining a... |
| leoprf2 32662 | The ordering relation for ... |
| leoprf 32663 | The ordering relation for ... |
| leopsq 32664 | The square of a Hermitian ... |
| 0leop 32665 | The zero operator is a pos... |
| idleop 32666 | The identity operator is a... |
| leopadd 32667 | The sum of two positive op... |
| leopmuli 32668 | The scalar product of a no... |
| leopmul 32669 | The scalar product of a po... |
| leopmul2i 32670 | Scalar product applied to ... |
| leoptri 32671 | The positive operator orde... |
| leoptr 32672 | The positive operator orde... |
| leopnmid 32673 | A bounded Hermitian operat... |
| nmopleid 32674 | A nonzero, bounded Hermiti... |
| opsqrlem1 32675 | Lemma for opsqri . (Contr... |
| opsqrlem2 32676 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32677 | Lemma for opsqri . (Contr... |
| opsqrlem4 32678 | Lemma for opsqri . (Contr... |
| opsqrlem5 32679 | Lemma for opsqri . (Contr... |
| opsqrlem6 32680 | Lemma for opsqri . (Contr... |
| pjhmopi 32681 | A projector is a Hermitian... |
| pjlnopi 32682 | A projector is a linear op... |
| pjnmopi 32683 | The operator norm of a pro... |
| pjbdlni 32684 | A projector is a bounded l... |
| pjhmop 32685 | A projection is a Hermitia... |
| hmopidmchi 32686 | An idempotent Hermitian op... |
| hmopidmpji 32687 | An idempotent Hermitian op... |
| hmopidmch 32688 | An idempotent Hermitian op... |
| hmopidmpj 32689 | An idempotent Hermitian op... |
| pjsdii 32690 | Distributive law for Hilbe... |
| pjddii 32691 | Distributive law for Hilbe... |
| pjsdi2i 32692 | Chained distributive law f... |
| pjcoi 32693 | Composition of projections... |
| pjcocli 32694 | Closure of composition of ... |
| pjcohcli 32695 | Closure of composition of ... |
| pjadjcoi 32696 | Adjoint of composition of ... |
| pjcofni 32697 | Functionality of compositi... |
| pjss1coi 32698 | Subset relationship for pr... |
| pjss2coi 32699 | Subset relationship for pr... |
| pjssmi 32700 | Projection meet property. ... |
| pjssge0i 32701 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32702 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32703 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32704 | Composition of projections... |
| pjscji 32705 | The projection of orthogon... |
| pjssumi 32706 | The projection on a subspa... |
| pjssposi 32707 | Projector ordering can be ... |
| pjordi 32708 | The definition of projecto... |
| pjssdif2i 32709 | The projection subspace of... |
| pjssdif1i 32710 | A necessary and sufficient... |
| pjimai 32711 | The image of a projection.... |
| pjidmcoi 32712 | A projection is idempotent... |
| pjoccoi 32713 | Composition of projections... |
| pjtoi 32714 | Subspace sum of projection... |
| pjoci 32715 | Projection of orthocomplem... |
| pjidmco 32716 | A projection operator is i... |
| dfpjop 32717 | Definition of projection o... |
| pjhmopidm 32718 | Two ways to express the se... |
| elpjidm 32719 | A projection operator is i... |
| elpjhmop 32720 | A projection operator is H... |
| 0leopj 32721 | A projector is a positive ... |
| pjadj2 32722 | A projector is self-adjoin... |
| pjadj3 32723 | A projector is self-adjoin... |
| elpjch 32724 | Reconstruction of the subs... |
| elpjrn 32725 | Reconstruction of the subs... |
| pjinvari 32726 | A closed subspace ` H ` wi... |
| pjin1i 32727 | Lemma for Theorem 1.22 of ... |
| pjin2i 32728 | Lemma for Theorem 1.22 of ... |
| pjin3i 32729 | Lemma for Theorem 1.22 of ... |
| pjclem1 32730 | Lemma for projection commu... |
| pjclem2 32731 | Lemma for projection commu... |
| pjclem3 32732 | Lemma for projection commu... |
| pjclem4a 32733 | Lemma for projection commu... |
| pjclem4 32734 | Lemma for projection commu... |
| pjci 32735 | Two subspaces commute iff ... |
| pjcmul1i 32736 | A necessary and sufficient... |
| pjcmul2i 32737 | The projection subspace of... |
| pjcohocli 32738 | Closure of composition of ... |
| pjadj2coi 32739 | Adjoint of double composit... |
| pj2cocli 32740 | Closure of double composit... |
| pj3lem1 32741 | Lemma for projection tripl... |
| pj3si 32742 | Stronger projection triple... |
| pj3i 32743 | Projection triplet theorem... |
| pj3cor1i 32744 | Projection triplet corolla... |
| pjs14i 32745 | Theorem S-14 of Watanabe, ... |
| isst 32748 | Property of a state. (Con... |
| ishst 32749 | Property of a complex Hilb... |
| sticl 32750 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32751 | Real closure of the value ... |
| hstcl 32752 | Closure of the value of a ... |
| hst1a 32753 | Unit value of a Hilbert-sp... |
| hstel2 32754 | Properties of a Hilbert-sp... |
| hstorth 32755 | Orthogonality property of ... |
| hstosum 32756 | Orthogonal sum property of... |
| hstoc 32757 | Sum of a Hilbert-space-val... |
| hstnmoc 32758 | Sum of norms of a Hilbert-... |
| stge0 32759 | The value of a state is no... |
| stle1 32760 | The value of a state is le... |
| hstle1 32761 | The norm of the value of a... |
| hst1h 32762 | The norm of a Hilbert-spac... |
| hst0h 32763 | The norm of a Hilbert-spac... |
| hstpyth 32764 | Pythagorean property of a ... |
| hstle 32765 | Ordering property of a Hil... |
| hstles 32766 | Ordering property of a Hil... |
| hstoh 32767 | A Hilbert-space-valued sta... |
| hst0 32768 | A Hilbert-space-valued sta... |
| sthil 32769 | The value of a state at th... |
| stj 32770 | The value of a state on a ... |
| sto1i 32771 | The state of a subspace pl... |
| sto2i 32772 | The state of the orthocomp... |
| stge1i 32773 | If a state is greater than... |
| stle0i 32774 | If a state is less than or... |
| stlei 32775 | Ordering law for states. ... |
| stlesi 32776 | Ordering law for states. ... |
| stji1i 32777 | Join of components of Sasa... |
| stm1i 32778 | State of component of unit... |
| stm1ri 32779 | State of component of unit... |
| stm1addi 32780 | Sum of states whose meet i... |
| staddi 32781 | If the sum of 2 states is ... |
| stm1add3i 32782 | Sum of states whose meet i... |
| stadd3i 32783 | If the sum of 3 states is ... |
| st0 32784 | The state of the zero subs... |
| strlem1 32785 | Lemma for strong state the... |
| strlem2 32786 | Lemma for strong state the... |
| strlem3a 32787 | Lemma for strong state the... |
| strlem3 32788 | Lemma for strong state the... |
| strlem4 32789 | Lemma for strong state the... |
| strlem5 32790 | Lemma for strong state the... |
| strlem6 32791 | Lemma for strong state the... |
| stri 32792 | Strong state theorem. The... |
| strb 32793 | Strong state theorem (bidi... |
| hstrlem2 32794 | Lemma for strong set of CH... |
| hstrlem3a 32795 | Lemma for strong set of CH... |
| hstrlem3 32796 | Lemma for strong set of CH... |
| hstrlem4 32797 | Lemma for strong set of CH... |
| hstrlem5 32798 | Lemma for strong set of CH... |
| hstrlem6 32799 | Lemma for strong set of CH... |
| hstri 32800 | Hilbert space admits a str... |
| hstrbi 32801 | Strong CH-state theorem (b... |
| largei 32802 | A Hilbert lattice admits a... |
| jplem1 32803 | Lemma for Jauch-Piron theo... |
| jplem2 32804 | Lemma for Jauch-Piron theo... |
| jpi 32805 | The function ` S ` , that ... |
| golem1 32806 | Lemma for Godowski's equat... |
| golem2 32807 | Lemma for Godowski's equat... |
| goeqi 32808 | Godowski's equation, shown... |
| stcltr1i 32809 | Property of a strong class... |
| stcltr2i 32810 | Property of a strong class... |
| stcltrlem1 32811 | Lemma for strong classical... |
| stcltrlem2 32812 | Lemma for strong classical... |
| stcltrthi 32813 | Theorem for classically st... |
| cvbr 32817 | Binary relation expressing... |
| cvbr2 32818 | Binary relation expressing... |
| cvcon3 32819 | Contraposition law for the... |
| cvpss 32820 | The covers relation implie... |
| cvnbtwn 32821 | The covers relation implie... |
| cvnbtwn2 32822 | The covers relation implie... |
| cvnbtwn3 32823 | The covers relation implie... |
| cvnbtwn4 32824 | The covers relation implie... |
| cvnsym 32825 | The covers relation is not... |
| cvnref 32826 | The covers relation is not... |
| cvntr 32827 | The covers relation is not... |
| spansncv2 32828 | Hilbert space has the cove... |
| mdbr 32829 | Binary relation expressing... |
| mdi 32830 | Consequence of the modular... |
| mdbr2 32831 | Binary relation expressing... |
| mdbr3 32832 | Binary relation expressing... |
| mdbr4 32833 | Binary relation expressing... |
| dmdbr 32834 | Binary relation expressing... |
| dmdmd 32835 | The dual modular pair prop... |
| mddmd 32836 | The modular pair property ... |
| dmdi 32837 | Consequence of the dual mo... |
| dmdbr2 32838 | Binary relation expressing... |
| dmdi2 32839 | Consequence of the dual mo... |
| dmdbr3 32840 | Binary relation expressing... |
| dmdbr4 32841 | Binary relation expressing... |
| dmdi4 32842 | Consequence of the dual mo... |
| dmdbr5 32843 | Binary relation expressing... |
| mddmd2 32844 | Relationship between modul... |
| mdsl0 32845 | A sublattice condition tha... |
| ssmd1 32846 | Ordering implies the modul... |
| ssmd2 32847 | Ordering implies the modul... |
| ssdmd1 32848 | Ordering implies the dual ... |
| ssdmd2 32849 | Ordering implies the dual ... |
| dmdsl3 32850 | Sublattice mapping for a d... |
| mdsl3 32851 | Sublattice mapping for a m... |
| mdslle1i 32852 | Order preservation of the ... |
| mdslle2i 32853 | Order preservation of the ... |
| mdslj1i 32854 | Join preservation of the o... |
| mdslj2i 32855 | Meet preservation of the r... |
| mdsl1i 32856 | If the modular pair proper... |
| mdsl2i 32857 | If the modular pair proper... |
| mdsl2bi 32858 | If the modular pair proper... |
| cvmdi 32859 | The covering property impl... |
| mdslmd1lem1 32860 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32861 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32862 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32863 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32864 | Preservation of the modula... |
| mdslmd2i 32865 | Preservation of the modula... |
| mdsldmd1i 32866 | Preservation of the dual m... |
| mdslmd3i 32867 | Modular pair conditions th... |
| mdslmd4i 32868 | Modular pair condition tha... |
| csmdsymi 32869 | Cross-symmetry implies M-s... |
| mdexchi 32870 | An exchange lemma for modu... |
| cvmd 32871 | The covering property impl... |
| cvdmd 32872 | The covering property impl... |
| ela 32874 | Atoms in a Hilbert lattice... |
| elat2 32875 | Expanded membership relati... |
| elatcv0 32876 | A Hilbert lattice element ... |
| atcv0 32877 | An atom covers the zero su... |
| atssch 32878 | Atoms are a subset of the ... |
| atelch 32879 | An atom is a Hilbert latti... |
| atne0 32880 | An atom is not the Hilbert... |
| atss 32881 | A lattice element smaller ... |
| atsseq 32882 | Two atoms in a subset rela... |
| atcveq0 32883 | A Hilbert lattice element ... |
| h1da 32884 | A 1-dimensional subspace i... |
| spansna 32885 | The span of the singleton ... |
| sh1dle 32886 | A 1-dimensional subspace i... |
| ch1dle 32887 | A 1-dimensional subspace i... |
| atom1d 32888 | The 1-dimensional subspace... |
| superpos 32889 | Superposition Principle. ... |
| chcv1 32890 | The Hilbert lattice has th... |
| chcv2 32891 | The Hilbert lattice has th... |
| chjatom 32892 | The join of a closed subsp... |
| shatomici 32893 | The lattice of Hilbert sub... |
| hatomici 32894 | The Hilbert lattice is ato... |
| hatomic 32895 | A Hilbert lattice is atomi... |
| shatomistici 32896 | The lattice of Hilbert sub... |
| hatomistici 32897 | ` CH ` is atomistic, i.e. ... |
| chpssati 32898 | Two Hilbert lattice elemen... |
| chrelati 32899 | The Hilbert lattice is rel... |
| chrelat2i 32900 | A consequence of relative ... |
| cvati 32901 | If a Hilbert lattice eleme... |
| cvbr4i 32902 | An alternate way to expres... |
| cvexchlem 32903 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32904 | The Hilbert lattice satisf... |
| chrelat2 32905 | A consequence of relative ... |
| chrelat3 32906 | A consequence of relative ... |
| chrelat3i 32907 | A consequence of the relat... |
| chrelat4i 32908 | A consequence of relative ... |
| cvexch 32909 | The Hilbert lattice satisf... |
| cvp 32910 | The Hilbert lattice satisf... |
| atnssm0 32911 | The meet of a Hilbert latt... |
| atnemeq0 32912 | The meet of distinct atoms... |
| atssma 32913 | The meet with an atom's su... |
| atcv0eq 32914 | Two atoms covering the zer... |
| atcv1 32915 | Two atoms covering the zer... |
| atexch 32916 | The Hilbert lattice satisf... |
| atomli 32917 | An assertion holding in at... |
| atoml2i 32918 | An assertion holding in at... |
| atordi 32919 | An ordering law for a Hilb... |
| atcvatlem 32920 | Lemma for ~ atcvati . (Co... |
| atcvati 32921 | A nonzero Hilbert lattice ... |
| atcvat2i 32922 | A Hilbert lattice element ... |
| atord 32923 | An ordering law for a Hilb... |
| atcvat2 32924 | A Hilbert lattice element ... |
| chirredlem1 32925 | Lemma for ~ chirredi . (C... |
| chirredlem2 32926 | Lemma for ~ chirredi . (C... |
| chirredlem3 32927 | Lemma for ~ chirredi . (C... |
| chirredlem4 32928 | Lemma for ~ chirredi . (C... |
| chirredi 32929 | The Hilbert lattice is irr... |
| chirred 32930 | The Hilbert lattice is irr... |
| atcvat3i 32931 | A condition implying that ... |
| atcvat4i 32932 | A condition implying exist... |
| atdmd 32933 | Two Hilbert lattice elemen... |
| atmd 32934 | Two Hilbert lattice elemen... |
| atmd2 32935 | Two Hilbert lattice elemen... |
| atabsi 32936 | Absorption of an incompara... |
| atabs2i 32937 | Absorption of an incompara... |
| mdsymlem1 32938 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32939 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32940 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32941 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32942 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32943 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32944 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32945 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32946 | M-symmetry of the Hilbert ... |
| mdsym 32947 | M-symmetry of the Hilbert ... |
| dmdsym 32948 | Dual M-symmetry of the Hil... |
| atdmd2 32949 | Two Hilbert lattice elemen... |
| sumdmdii 32950 | If the subspace sum of two... |
| cmmdi 32951 | Commuting subspaces form a... |
| cmdmdi 32952 | Commuting subspaces form a... |
| sumdmdlem 32953 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32954 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32955 | The subspace sum of two Hi... |
| dmdbr4ati 32956 | Dual modular pair property... |
| dmdbr5ati 32957 | Dual modular pair property... |
| dmdbr6ati 32958 | Dual modular pair property... |
| dmdbr7ati 32959 | Dual modular pair property... |
| mdoc1i 32960 | Orthocomplements form a mo... |
| mdoc2i 32961 | Orthocomplements form a mo... |
| dmdoc1i 32962 | Orthocomplements form a du... |
| dmdoc2i 32963 | Orthocomplements form a du... |
| mdcompli 32964 | A condition equivalent to ... |
| dmdcompli 32965 | A condition equivalent to ... |
| mddmdin0i 32966 | If dual modular implies mo... |
| cdjreui 32967 | A member of the sum of dis... |
| cdj1i 32968 | Two ways to express " ` A ... |
| cdj3lem1 32969 | A property of " ` A ` and ... |
| cdj3lem2 32970 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32971 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32972 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32973 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32974 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32975 | Lemma for ~ cdj3i . The s... |
| cdj3i 32976 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32977 | (_This theorem is a dummy ... |
| sa-abvi 32978 | A theorem about the univer... |
| xfree 32979 | A partial converse to ~ 19... |
| xfree2 32980 | A partial converse to ~ 19... |
| addltmulALT 32981 | A proof readability experi... |
| ad11antr 32982 | Deduction adding 11 conjun... |
| simp-12l 32983 | Simplification of a conjun... |
| simp-12r 32984 | Simplification of a conjun... |
| an52ds 32985 | Inference exchanging the l... |
| an62ds 32986 | Inference exchanging the l... |
| an72ds 32987 | Inference exchanging the l... |
| an82ds 32988 | Inference exchanging the l... |
| syl22anbrc 32989 | Syllogism inference. (Con... |
| or3di 32990 | Distributive law for disju... |
| or3dir 32991 | Distributive law for disju... |
| 3o1cs 32992 | Deduction eliminating disj... |
| 3o2cs 32993 | Deduction eliminating disj... |
| 3o3cs 32994 | Deduction eliminating disj... |
| sbc2iedf 32995 | Conversion of implicit sub... |
| rspc2daf 32996 | Double restricted speciali... |
| ralcom4f 32997 | Commutation of restricted ... |
| rexcom4f 32998 | Commutation of restricted ... |
| 19.9d2rf 32999 | A deduction version of one... |
| 19.9d2r 33000 | A deduction version of one... |
| r19.29ffa 33001 | A commonly used pattern ba... |
| reu6dv 33002 | A condition which implies ... |
| eqtrb 33003 | A transposition of equalit... |
| eqelbid 33004 | A variable elimination law... |
| opsbc2ie 33005 | Conversion of implicit sub... |
| opreu2reuALT 33006 | Correspondence between uni... |
| 2reucom 33009 | Double restricted existent... |
| 2reu2rex1 33010 | Double restricted existent... |
| 2reureurex 33011 | Double restricted existent... |
| 2reu2reu2 33012 | Double restricted existent... |
| opreu2reu1 33013 | Equivalent definition of t... |
| sq2reunnltb 33014 | There exists a unique deco... |
| addsqnot2reu 33015 | For each complex number ` ... |
| sbceqbidf 33016 | Equality theorem for class... |
| sbcies 33017 | A special version of class... |
| mo5f 33018 | Alternate definition of "a... |
| nmo 33019 | Negation of "at most one".... |
| reuxfrdf 33020 | Transfer existential uniqu... |
| rexunirn 33021 | Restricted existential qua... |
| rmoxfrd 33022 | Transfer "at most one" res... |
| rmoun 33023 | "At most one" restricted e... |
| rmounid 33024 | A case where an "at most o... |
| riotaeqbidva 33025 | Equivalent wff's yield equ... |
| dmrab 33026 | Domain of a restricted cla... |
| difrab2 33027 | Difference of two restrict... |
| rabexgfGS 33028 | Separation Scheme in terms... |
| rabsnel 33029 | Truth implied by equality ... |
| rabsspr 33030 | Conditions for a restricte... |
| rabsstp 33031 | Conditions for a restricte... |
| 3unrab 33032 | Union of three restricted ... |
| foresf1o 33033 | From a surjective function... |
| rabfodom 33034 | Domination relation for re... |
| rabrexfi 33035 | Conditions for a class abs... |
| abrexdomjm 33036 | An indexed set is dominate... |
| abrexdom2jm 33037 | An indexed set is dominate... |
| abrexexd 33038 | Existence of a class abstr... |
| elabreximd 33039 | Class substitution in an i... |
| elabreximdv 33040 | Class substitution in an i... |
| abrexss 33041 | A necessary condition for ... |
| nelun 33042 | Negated membership for a u... |
| snsssng 33043 | If a singleton is a subset... |
| n0nsnel 33044 | If a class with one elemen... |
| inin 33045 | Intersection with an inter... |
| difininv 33046 | Condition for the intersec... |
| difeq 33047 | Rewriting an equation with... |
| eqdif 33048 | If both set differences of... |
| indifbi 33049 | Two ways to express equali... |
| diffib 33050 | Case where ~ diffi is a bi... |
| difxp1ss 33051 | Difference law for Cartesi... |
| difxp2ss 33052 | Difference law for Cartesi... |
| indifundif 33053 | A remarkable equation with... |
| elpwincl1 33054 | Closure of intersection wi... |
| elpwdifcl 33055 | Closure of class differenc... |
| elpwiuncl 33056 | Closure of indexed union w... |
| elpreq 33057 | Equality wihin a pair. (C... |
| prssad 33058 | If a pair is a subset of a... |
| prssbd 33059 | If a pair is a subset of a... |
| nelpr 33060 | A set ` A ` not in a pair ... |
| inpr0 33061 | Rewrite an empty intersect... |
| neldifpr1 33062 | The first element of a pai... |
| neldifpr2 33063 | The second element of a pa... |
| unidifsnel 33064 | The other element of a pai... |
| unidifsnne 33065 | The other element of a pai... |
| tpssg 33066 | An unordered triple of ele... |
| tpssd 33067 | Deduction version of tpssi... |
| tpssad 33068 | If an ordered triple is a ... |
| tpssbd 33069 | If an ordered triple is a ... |
| tpsscd 33070 | If an ordered triple is a ... |
| ifeqeqx 33071 | An equality theorem tailor... |
| elimifd 33072 | Elimination of a condition... |
| elim2if 33073 | Elimination of two conditi... |
| elim2ifim 33074 | Elimination of two conditi... |
| ifeq3da 33075 | Given an expression ` C ` ... |
| ifnetrue 33076 | Deduce truth from a condit... |
| ifnefals 33077 | Deduce falsehood from a co... |
| ifnebib 33078 | The converse of ~ ifbi hol... |
| ififcom 33079 | Commute two nested conditi... |
| uniinn0 33080 | Sufficient and necessary c... |
| difuncomp 33081 | Express a class difference... |
| elpwunicl 33082 | Closure of a set union wit... |
| cbviunf 33083 | Rule used to change the bo... |
| iuneq12daf 33084 | Equality deduction for ind... |
| iunin1f 33085 | Indexed union of intersect... |
| ssiun3 33086 | Subset equivalence for an ... |
| ssiun2sf 33087 | Subset relationship for an... |
| iuninc 33088 | The union of an increasing... |
| iundifdifd 33089 | The intersection of a set ... |
| iundifdif 33090 | The intersection of a set ... |
| iunrdx 33091 | Re-index an indexed union.... |
| iunrnmptss 33092 | A subset relation for an i... |
| iunxunsn 33093 | Appending a set to an inde... |
| iunxunpr 33094 | Appending two sets to an i... |
| iunxpssiun1 33095 | Provide an upper bound for... |
| iinabrex 33096 | Rewriting an indexed inter... |
| disjnf 33097 | In case ` x ` is not free ... |
| cbvdisjf 33098 | Change bound variables in ... |
| disjss1f 33099 | A subset of a disjoint col... |
| disjeq1f 33100 | Equality theorem for disjo... |
| disjxun0 33101 | Simplify a disjoint union.... |
| disjdifprg 33102 | A trivial partition into a... |
| disjdifprg2 33103 | A trivial partition of a s... |
| disji2f 33104 | Property of a disjoint col... |
| disjif 33105 | Property of a disjoint col... |
| disjorf 33106 | Two ways to say that a col... |
| disjorsf 33107 | Two ways to say that a col... |
| disjif2 33108 | Property of a disjoint col... |
| disjabrex 33109 | Rewriting a disjoint colle... |
| disjabrexf 33110 | Rewriting a disjoint colle... |
| disjpreima 33111 | A preimage of a disjoint s... |
| disjrnmpt 33112 | Rewriting a disjoint colle... |
| disjin 33113 | If a collection is disjoin... |
| disjin2 33114 | If a collection is disjoin... |
| disjxpin 33115 | Derive a disjunction over ... |
| iundisjf 33116 | Rewrite a countable union ... |
| iundisj2f 33117 | A disjoint union is disjoi... |
| disjrdx 33118 | Re-index a disjunct collec... |
| disjex 33119 | Two ways to say that two c... |
| disjexc 33120 | A variant of ~ disjex , ap... |
| disjunsn 33121 | Append an element to a dis... |
| disjun0 33122 | Adding the empty element p... |
| disjiunel 33123 | A set of elements B of a d... |
| disjuniel 33124 | A set of elements B of a d... |
| xpdisjres 33125 | Restriction of a constant ... |
| opeldifid 33126 | Ordered pair elementhood o... |
| difres 33127 | Case when class difference... |
| imadifxp 33128 | Image of the difference wi... |
| relfi 33129 | A relation (set) is finite... |
| 0res 33130 | Restriction of the empty f... |
| fcoinver 33131 | Build an equivalence relat... |
| fcoinvbr 33132 | Binary relation for the eq... |
| brabgaf 33133 | The law of concretion for ... |
| brelg 33134 | Two things in a binary rel... |
| br8d 33135 | Substitution for an eight-... |
| fnfvor 33136 | Relation between two funct... |
| ofrco 33137 | Function relation between ... |
| opabdm 33138 | Domain of an ordered-pair ... |
| opabrn 33139 | Range of an ordered-pair c... |
| opabssi 33140 | Sufficient condition for a... |
| opabid2ss 33141 | One direction of ~ opabid2... |
| ssrelf 33142 | A subclass relationship de... |
| eqrelrd2 33143 | A version of ~ eqrelrdv2 w... |
| erbr3b 33144 | Biconditional for equivale... |
| iunsnima 33145 | Image of a singleton by an... |
| iunsnima2 33146 | Version of ~ iunsnima with... |
| fconst7v 33147 | An alternative way to expr... |
| constcof 33148 | Composition with a constan... |
| ac6sf2 33149 | Alternate version of ~ ac6... |
| ac6mapd 33150 | Axiom of choice equivalent... |
| fnresin 33151 | Restriction of a function ... |
| fresunsn 33152 | Recover the original funct... |
| f1o3d 33153 | Describe an implicit one-t... |
| eldmne0 33154 | A function of nonempty dom... |
| f1rnen 33155 | Equinumerosity of the rang... |
| f1oeq3dd 33156 | Equality deduction for one... |
| rinvf1o 33157 | Sufficient conditions for ... |
| fresf1o 33158 | Conditions for a restricti... |
| nfpconfp 33159 | The set of fixed points of... |
| fmptco1f1o 33160 | The action of composing (t... |
| cofmpt2 33161 | Express composition of a m... |
| f1mptrn 33162 | Express injection for a ma... |
| dfimafnf 33163 | Alternate definition of th... |
| funimass4f 33164 | Membership relation for th... |
| suppss2f 33165 | Show that the support of a... |
| ofrn 33166 | The range of the function ... |
| ofrn2 33167 | The range of the function ... |
| off2 33168 | The function operation pro... |
| ofresid 33169 | Applying an operation rest... |
| unipreima 33170 | Preimage of a class union.... |
| opfv 33171 | Value of a function produc... |
| xppreima 33172 | The preimage of a Cartesia... |
| 2ndimaxp 33173 | Image of a cartesian produ... |
| dmdju 33174 | Domain of a disjoint union... |
| djussxp2 33175 | Stronger version of ~ djus... |
| 2ndresdju 33176 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 33177 | The ` 2nd ` function restr... |
| xppreima2 33178 | The preimage of a Cartesia... |
| abfmpunirn 33179 | Membership in a union of a... |
| rabfmpunirn 33180 | Membership in a union of a... |
| abfmpeld 33181 | Membership in an element o... |
| abfmpel 33182 | Membership in an element o... |
| fmptdf2 33183 | Domain and codomain of the... |
| fmptcof2 33184 | Composition of two functio... |
| fcomptf 33185 | Express composition of two... |
| acunirnmpt 33186 | Axiom of choice for the un... |
| acunirnmpt2 33187 | Axiom of choice for the un... |
| acunirnmpt2f 33188 | Axiom of choice for the un... |
| aciunf1lem 33189 | Choice in an index union. ... |
| aciunf1 33190 | Choice in an index union. ... |
| ofoprabco 33191 | Function operation as a co... |
| ofpreima 33192 | Express the preimage of a ... |
| ofpreima2 33193 | Express the preimage of a ... |
| funcnv5mpt 33194 | Two ways to say that a fun... |
| funcnv4mpt 33195 | Two ways to say that a fun... |
| preimane 33196 | Different elements have di... |
| fnpreimac 33197 | Choose a set ` x ` contain... |
| fgreu 33198 | Exactly one point of a fun... |
| fcnvgreu 33199 | If the converse of a relat... |
| rnmposs 33200 | The range of an operation ... |
| mptssALT 33201 | Deduce subset relation of ... |
| dfcnv2 33202 | Alternative definition of ... |
| partfun2 33203 | Rewrite a function defined... |
| rnressnsn 33204 | The range of a restriction... |
| mpomptxf 33205 | Express a two-argument fun... |
| of0r 33206 | Function operation with th... |
| suppovss 33207 | A bound for the support of... |
| elsuppfnd 33208 | Deduce membership in the s... |
| fisuppov1 33209 | Formula building theorem f... |
| suppun2 33210 | The support of a union is ... |
| fdifsupp 33211 | Express the support of a f... |
| suppiniseg 33212 | Relation between the suppo... |
| fsuppinisegfi 33213 | The initial segment ` ( ``... |
| fressupp 33214 | The restriction of a funct... |
| fdifsuppconst 33215 | A function is a zero const... |
| ressupprn 33216 | The range of a function re... |
| supppreima 33217 | Express the support of a f... |
| fsupprnfi 33218 | Finite support implies fin... |
| mptiffisupp 33219 | Conditions for a mapping f... |
| cosnopne 33220 | Composition of two ordered... |
| cosnop 33221 | Composition of two ordered... |
| cnvprop 33222 | Converse of a pair of orde... |
| brprop 33223 | Binary relation for a pair... |
| mptprop 33224 | Rewrite pairs of ordered p... |
| coprprop 33225 | Composition of two pairs o... |
| fmptunsnop 33226 | Two ways to express a func... |
| gtiso 33227 | Two ways to write a strict... |
| isoun 33228 | Infer an isomorphism from ... |
| disjdsct 33229 | A disjoint collection is d... |
| df1stres 33230 | Definition for a restricti... |
| df2ndres 33231 | Definition for a restricti... |
| 1stpreimas 33232 | The preimage of a singleto... |
| 1stpreima 33233 | The preimage by ` 1st ` is... |
| 2ndpreima 33234 | The preimage by ` 2nd ` is... |
| curry2ima 33235 | The image of a curried fun... |
| preiman0 33236 | The preimage of a nonempty... |
| intimafv 33237 | The intersection of an ima... |
| snct 33238 | A singleton is countable. ... |
| prct 33239 | An unordered pair is count... |
| mpocti 33240 | An operation is countable ... |
| mptctf 33241 | A countable mapping set is... |
| abrexctf 33242 | An image set of a countabl... |
| padct 33243 | Index a countable set with... |
| f1od2 33244 | Sufficient condition for a... |
| fcobij 33245 | Composing functions with a... |
| fcobijfs 33246 | Composing finitely support... |
| fcobijfs2 33247 | Composing finitely support... |
| suppss3 33248 | Deduce a function's suppor... |
| fsuppcurry1 33249 | Finite support of a currie... |
| fsuppcurry2 33250 | Finite support of a currie... |
| offinsupp1 33251 | Finite support for a funct... |
| ffs2 33252 | Rewrite a function's suppo... |
| ffsrn 33253 | The range of a finitely su... |
| cocnvf1o 33254 | Composing with the inverse... |
| resf1o 33255 | Restriction of functions t... |
| maprnin 33256 | Restricting the range of t... |
| fpwrelmapffslem 33257 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 33258 | Define a canonical mapping... |
| fpwrelmapffs 33259 | Define a canonical mapping... |
| sgnval2 33260 | Value of the signum of a r... |
| creq0 33261 | The real representation of... |
| 1nei 33262 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 33263 | An integer unit times itse... |
| nnmulge 33264 | Multiplying by a positive ... |
| submuladdd 33265 | The product of a differenc... |
| binom2subadd 33266 | The difference of the squa... |
| cjsubd 33267 | Complex conjugate distribu... |
| re0cj 33268 | The conjugate of a pure im... |
| receqid 33269 | Real numbers equal to thei... |
| pythagreim 33270 | A simplified version of th... |
| efiargd 33271 | The exponential of the "ar... |
| arginv 33272 | The argument of the invers... |
| argcj 33273 | The argument of the conjug... |
| quad3d 33274 | Variant of quadratic equat... |
| lt2addrd 33275 | If the right-hand side of ... |
| nn0mnfxrd 33276 | Nonnegative integers or mi... |
| xrlelttric 33277 | Trichotomy law for extende... |
| xaddeq0 33278 | Two extended reals which a... |
| rexmul2 33279 | If the result ` A ` of an ... |
| xrinfm 33280 | The extended real numbers ... |
| le2halvesd 33281 | A sum is less than the who... |
| xraddge02 33282 | A number is less than or e... |
| xrge0addge 33283 | A number is less than or e... |
| xlt2addrd 33284 | If the right-hand side of ... |
| xrge0infss 33285 | Any subset of nonnegative ... |
| xrge0infssd 33286 | Inequality deduction for i... |
| xrge0addcld 33287 | Nonnegative extended reals... |
| xrge0subcld 33288 | Condition for closure of n... |
| infxrge0lb 33289 | A member of a set of nonne... |
| infxrge0glb 33290 | The infimum of a set of no... |
| infxrge0gelb 33291 | The infimum of a set of no... |
| xrofsup 33292 | The supremum is preserved ... |
| supxrnemnf 33293 | The supremum of a nonempty... |
| xnn0gt0 33294 | Nonzero extended nonnegati... |
| xnn01gt 33295 | An extended nonnegative in... |
| nn0xmulclb 33296 | Finite multiplication in t... |
| xnn0nn0d 33297 | Conditions for an extended... |
| xnn0nnd 33298 | Conditions for an extended... |
| joiniooico 33299 | Disjoint joining an open i... |
| ubico 33300 | A right-open interval does... |
| xeqlelt 33301 | Equality in terms of 'less... |
| eliccelico 33302 | Relate elementhood to a cl... |
| elicoelioo 33303 | Relate elementhood to a cl... |
| iocinioc2 33304 | Intersection between two o... |
| xrdifh 33305 | Class difference of a half... |
| iocinif 33306 | Relate intersection of two... |
| difioo 33307 | The difference between two... |
| difico 33308 | The difference between two... |
| uzssico 33309 | Upper integer sets are a s... |
| fz2ssnn0 33310 | A finite set of sequential... |
| nndiffz1 33311 | Upper set of the positive ... |
| ssnnssfz 33312 | For any finite subset of `... |
| fzm1ne1 33313 | Elementhood of an integer ... |
| fzspl 33314 | Split the last element of ... |
| fzdif2 33315 | Split the last element of ... |
| fzodif2 33316 | Split the last element of ... |
| fzodif1 33317 | Set difference of two half... |
| fzsplit3 33318 | Split a finite interval of... |
| nn0diffz0 33319 | Upper set of the nonnegati... |
| bcm1n 33320 | The proportion of one bino... |
| iundisjfi 33321 | Rewrite a countable union ... |
| iundisj2fi 33322 | A disjoint union is disjoi... |
| iundisjcnt 33323 | Rewrite a countable union ... |
| iundisj2cnt 33324 | A countable disjoint union... |
| f1ocnt 33325 | Given a countable set ` A ... |
| fz1nnct 33326 | NN and integer ranges star... |
| fz1nntr 33327 | NN and integer ranges star... |
| fzo0opth 33328 | Equality for a half open i... |
| nn0difffzod 33329 | A nonnegative integer that... |
| suppssnn0 33330 | Show that the support of a... |
| hashunif 33331 | The cardinality of a disjo... |
| hashxpe 33332 | The size of the Cartesian ... |
| hashgt1 33333 | Restate "set contains at l... |
| hashne0 33334 | Deduce that the size of a ... |
| hashimaf1 33335 | Taking the image of a set ... |
| elq2 33336 | Elementhood in the rationa... |
| znumd 33337 | Numerator of an integer. ... |
| zdend 33338 | Denominator of an integer.... |
| numdenneg 33339 | Numerator and denominator ... |
| divnumden2 33340 | Calculate the reduced form... |
| expgt0b 33341 | A real number ` A ` raised... |
| nn0split01 33342 | Split 0 and 1 from the non... |
| nn0disj01 33343 | The pair ` { 0 , 1 } ` doe... |
| nnindf 33344 | Principle of Mathematical ... |
| nn0min 33345 | Extracting the minimum pos... |
| subne0nn 33346 | A nonnegative difference i... |
| ltesubnnd 33347 | Subtracting an integer num... |
| fprodeq02 33348 | If one of the factors is z... |
| fprodex01 33349 | A product of factors equal... |
| prodpr 33350 | A product over a pair is t... |
| prodtp 33351 | A product over a triple is... |
| fsumub 33352 | An upper bound for a term ... |
| fsumiunle 33353 | Upper bound for a sum of n... |
| dfdec100 33354 | Split the hundreds from a ... |
| sgnsgn 33355 | Signum is idempotent. (Co... |
| sgnmulsgp 33356 | If two real numbers are of... |
| nexple 33357 | A lower bound for an expon... |
| 2exple2exp 33358 | If a nonnegative integer `... |
| expevenpos 33359 | Even powers are positive. ... |
| oexpled 33360 | Odd power monomials are mo... |
| indsumin 33361 | Finite sum of a product wi... |
| prodindf 33362 | The product of indicators ... |
| indsn 33363 | The indicator function of ... |
| indf1o 33364 | The bijection between a po... |
| indpreima 33365 | A function with range ` { ... |
| indf1ofs 33366 | The bijection between fini... |
| indsupp 33367 | The support of the indicat... |
| indfsd 33368 | The indicator function of ... |
| indfsid 33369 | Conditions for a function ... |
| dp2eq1 33372 | Equality theorem for the d... |
| dp2eq2 33373 | Equality theorem for the d... |
| dp2eq1i 33374 | Equality theorem for the d... |
| dp2eq2i 33375 | Equality theorem for the d... |
| dp2eq12i 33376 | Equality theorem for the d... |
| dp20u 33377 | Add a zero in the tenths (... |
| dp20h 33378 | Add a zero in the unit pla... |
| dp2cl 33379 | Closure for the decimal fr... |
| dp2clq 33380 | Closure for a decimal frac... |
| rpdp2cl 33381 | Closure for a decimal frac... |
| rpdp2cl2 33382 | Closure for a decimal frac... |
| dp2lt10 33383 | Decimal fraction builds re... |
| dp2lt 33384 | Comparing two decimal frac... |
| dp2ltsuc 33385 | Comparing a decimal fracti... |
| dp2ltc 33386 | Comparing two decimal expa... |
| dpval 33389 | Define the value of the de... |
| dpcl 33390 | Prove that the closure of ... |
| dpfrac1 33391 | Prove a simple equivalence... |
| dpval2 33392 | Value of the decimal point... |
| dpval3 33393 | Value of the decimal point... |
| dpmul10 33394 | Multiply by 10 a decimal e... |
| decdiv10 33395 | Divide a decimal number by... |
| dpmul100 33396 | Multiply by 100 a decimal ... |
| dp3mul10 33397 | Multiply by 10 a decimal e... |
| dpmul1000 33398 | Multiply by 1000 a decimal... |
| dpval3rp 33399 | Value of the decimal point... |
| dp0u 33400 | Add a zero in the tenths p... |
| dp0h 33401 | Remove a zero in the units... |
| rpdpcl 33402 | Closure of the decimal poi... |
| dplt 33403 | Comparing two decimal expa... |
| dplti 33404 | Comparing a decimal expans... |
| dpgti 33405 | Comparing a decimal expans... |
| dpltc 33406 | Comparing two decimal inte... |
| dpexpp1 33407 | Add one zero to the mantis... |
| 0dp2dp 33408 | Multiply by 10 a decimal e... |
| dpadd2 33409 | Addition with one decimal,... |
| dpadd 33410 | Addition with one decimal.... |
| dpadd3 33411 | Addition with two decimals... |
| dpmul 33412 | Multiplication with one de... |
| dpmul4 33413 | An upper bound to multipli... |
| threehalves 33414 | Example theorem demonstrat... |
| 1mhdrd 33415 | Example theorem demonstrat... |
| xdivval 33418 | Value of division: the (un... |
| xrecex 33419 | Existence of reciprocal of... |
| xmulcand 33420 | Cancellation law for exten... |
| xreceu 33421 | Existential uniqueness of ... |
| xdivcld 33422 | Closure law for the extend... |
| xdivcl 33423 | Closure law for the extend... |
| xdivmul 33424 | Relationship between divis... |
| rexdiv 33425 | The extended real division... |
| xdivrec 33426 | Relationship between divis... |
| xdivid 33427 | A number divided by itself... |
| xdiv0 33428 | Division into zero is zero... |
| xdiv0rp 33429 | Division into zero is zero... |
| eliccioo 33430 | Membership in a closed int... |
| elxrge02 33431 | Elementhood in the set of ... |
| xdivpnfrp 33432 | Plus infinity divided by a... |
| rpxdivcld 33433 | Closure law for extended d... |
| xrpxdivcld 33434 | Closure law for extended d... |
| wrdres 33435 | Condition for the restrict... |
| wrdsplex 33436 | Existence of a split of a ... |
| wrdfsupp 33437 | A word has finite support.... |
| wrdpmcl 33438 | Closure of a word with per... |
| pfx1s2 33439 | The prefix of length 1 of ... |
| pfxrn2 33440 | The range of a prefix of a... |
| pfxrn3 33441 | Express the range of a pre... |
| pfxf1 33442 | Condition for a prefix to ... |
| s2f1 33443 | Conditions for a length 2 ... |
| s3f1 33444 | Conditions for a length 3 ... |
| s3clhash 33445 | Closure of the words of le... |
| pfxlsw2ccat 33446 | Reconstruct a word from it... |
| ccatws1f1o 33447 | Conditions for the concate... |
| ccatws1f1olast 33448 | Two ways to reorder symbol... |
| wrdt2ind 33449 | Perform an induction over ... |
| swrdrn2 33450 | The range of a subword is ... |
| swrdrndisj 33451 | Condition for the range of... |
| splfv3 33452 | Symbols to the right of a ... |
| 1cshid 33453 | Cyclically shifting a sing... |
| cshw1s2 33454 | Cyclically shifting a leng... |
| cshwrnid 33455 | Cyclically shifting a word... |
| cshf1o 33456 | Condition for the cyclic s... |
| ressplusf 33457 | The group operation functi... |
| ressnm 33458 | The norm in a restricted s... |
| abvpropd2 33459 | Weaker version of ~ abvpro... |
| ressprs 33460 | The restriction of a prose... |
| posrasymb 33461 | A poset ordering is asymme... |
| odutos 33462 | Being a toset is a self-du... |
| tlt2 33463 | In a Toset, two elements m... |
| tlt3 33464 | In a Toset, two elements m... |
| trleile 33465 | In a Toset, two elements m... |
| toslublem 33466 | Lemma for ~ toslub and ~ x... |
| toslub 33467 | In a toset, the lowest upp... |
| tosglblem 33468 | Lemma for ~ tosglb and ~ x... |
| tosglb 33469 | Same theorem as ~ toslub ,... |
| clatp0cl 33470 | The poset zero of a comple... |
| clatp1cl 33471 | The poset one of a complet... |
| mntoval 33476 | Operation value of the mon... |
| ismnt 33477 | Express the statement " ` ... |
| ismntd 33478 | Property of being a monoto... |
| mntf 33479 | A monotone function is a f... |
| mgcoval 33480 | Operation value of the mon... |
| mgcval 33481 | Monotone Galois connection... |
| mgcf1 33482 | The lower adjoint ` F ` of... |
| mgcf2 33483 | The upper adjoint ` G ` of... |
| mgccole1 33484 | An inequality for the kern... |
| mgccole2 33485 | Inequality for the closure... |
| mgcmnt1 33486 | The lower adjoint ` F ` of... |
| mgcmnt2 33487 | The upper adjoint ` G ` of... |
| mgcmntco 33488 | A Galois connection like s... |
| dfmgc2lem 33489 | Lemma for dfmgc2, backward... |
| dfmgc2 33490 | Alternate definition of th... |
| mgcmnt1d 33491 | Galois connection implies ... |
| mgcmnt2d 33492 | Galois connection implies ... |
| mgccnv 33493 | The inverse Galois connect... |
| pwrssmgc 33494 | Given a function ` F ` , e... |
| mgcf1olem1 33495 | Property of a Galois conne... |
| mgcf1olem2 33496 | Property of a Galois conne... |
| mgcf1o 33497 | Given a Galois connection,... |
| xrs0 33500 | The zero of the extended r... |
| xrslt 33501 | The "strictly less than" r... |
| xrsinvgval 33502 | The inversion operation in... |
| xrsmulgzz 33503 | The "multiple" function in... |
| xrstos 33504 | The extended real numbers ... |
| xrsclat 33505 | The extended real numbers ... |
| xrsp0 33506 | The poset 0 of the extende... |
| xrsp1 33507 | The poset 1 of the extende... |
| xrge00 33508 | The zero of the extended n... |
| xrge0mulgnn0 33509 | The group multiple functio... |
| xrge0addass 33510 | Associativity of extended ... |
| xrge0addgt0 33511 | The sum of nonnegative and... |
| xrge0adddir 33512 | Right-distributivity of ex... |
| xrge0adddi 33513 | Left-distributivity of ext... |
| xrge0npcan 33514 | Extended nonnegative real ... |
| fsumrp0cl 33515 | Closure of a finite sum of... |
| mndcld 33516 | Closure of the operation o... |
| mndassd 33517 | A monoid operation is asso... |
| mndlrinv 33518 | In a monoid, if an element... |
| mndlrinvb 33519 | In a monoid, if an element... |
| mndlactf1 33520 | If an element ` X ` of a m... |
| mndlactfo 33521 | An element ` X ` of a mono... |
| mndractf1 33522 | If an element ` X ` of a m... |
| mndractfo 33523 | An element ` X ` of a mono... |
| mndlactf1o 33524 | An element ` X ` of a mono... |
| mndractf1o 33525 | An element ` X ` of a mono... |
| cmn4d 33526 | Commutative/associative la... |
| cmn246135 33527 | Rearrange terms in a commu... |
| cmn145236 33528 | Rearrange terms in a commu... |
| abliso 33529 | The image of an Abelian gr... |
| lmhmghmd 33530 | A module homomorphism is a... |
| mhmimasplusg 33531 | Value of the operation of ... |
| lmhmimasvsca 33532 | Value of the scalar produc... |
| grpidcld 33533 | The identity element of a ... |
| grpinvinvd 33534 | Double inverse law for gro... |
| grpsubcld 33535 | Closure of group subtracti... |
| subgsubcld 33536 | A subgroup is closed under... |
| subgmulgcld 33537 | Closure of the group multi... |
| ressmulgnn0d 33538 | Values for the group multi... |
| ablcomd 33539 | An abelian group operation... |
| gsumsubg 33540 | The group sum in a subgrou... |
| gsumsra 33541 | The group sum in a subring... |
| gsummpt2co 33542 | Split a finite sum into a ... |
| gsummpt2d 33543 | Express a finite sum over ... |
| lmodvslmhm 33544 | Scalar multiplication in a... |
| gsumvsmul1 33545 | Pull a scalar multiplicati... |
| gsummptres 33546 | Extend a finite group sum ... |
| gsummptres2 33547 | Extend a finite group sum ... |
| gsummptfsres 33548 | Extend a finitely supporte... |
| gsummptf1od 33549 | Re-index a finite group su... |
| gsummptrev 33550 | Revert ordering in a group... |
| gsummptp1 33551 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33552 | Split a group sum expresse... |
| gsummptfzsplitla 33553 | Split a group sum expresse... |
| gsummptfsf1o 33554 | Re-index a finite group su... |
| gsumfs2d 33555 | Express a finite sum over ... |
| gsumzresunsn 33556 | Append an element to a fin... |
| gsumpart 33557 | Express a group sum as a d... |
| gsumtp 33558 | Group sum of an unordered ... |
| gsumzrsum 33559 | Relate a group sum on ` ZZ... |
| gsummulgc2 33560 | A finite group sum multipl... |
| gsumhashmul 33561 | Express a group sum by gro... |
| gsummulsubdishift1 33562 | Distribute a subtraction o... |
| gsummulsubdishift2 33563 | Distribute a subtraction o... |
| gsummulsubdishift1s 33564 | Distribute a subtraction o... |
| gsummulsubdishift2s 33565 | Distribute a subtraction o... |
| suppgsumssiun 33566 | The support of a function ... |
| xrge0tsmsd 33567 | Any finite or infinite sum... |
| xrge0tsmsbi 33568 | Any limit of a finite or i... |
| xrge0tsmseq 33569 | Any limit of a finite or i... |
| gsumwun 33570 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33571 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33572 | Rewrite a sum ranging over... |
| cntzun 33573 | The centralizer of a union... |
| cntzsnid 33574 | The centralizer of the ide... |
| cntrcrng 33575 | The center of a ring is a ... |
| symgfcoeu 33576 | Uniqueness property of per... |
| symgcom 33577 | Two permutations ` X ` and... |
| symgcom2 33578 | Two permutations ` X ` and... |
| symgcntz 33579 | All elements of a (finite)... |
| odpmco 33580 | The composition of two odd... |
| symgsubg 33581 | The value of the group sub... |
| pmtrprfv2 33582 | In a transposition of two ... |
| pmtrcnel 33583 | Composing a permutation ` ... |
| pmtrcnel2 33584 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33585 | Composing a permutation ` ... |
| fzo0pmtrlast 33586 | Reorder a half-open intege... |
| wrdpmtrlast 33587 | Reorder a word, so that th... |
| pmtridf1o 33588 | Transpositions of ` X ` an... |
| pmtridfv1 33589 | Value at X of the transpos... |
| pmtridfv2 33590 | Value at Y of the transpos... |
| psgnid 33591 | Permutation sign of the id... |
| psgndmfi 33592 | For a finite base set, the... |
| pmtrto1cl 33593 | Useful lemma for the follo... |
| psgnfzto1stlem 33594 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33595 | Value of our permutation `... |
| fzto1st1 33596 | Special case where the per... |
| fzto1st 33597 | The function moving one el... |
| fzto1stinvn 33598 | Value of the inverse of ou... |
| psgnfzto1st 33599 | The permutation sign for m... |
| tocycval 33602 | Value of the cycle builder... |
| tocycfv 33603 | Function value of a permut... |
| tocycfvres1 33604 | A cyclic permutation is a ... |
| tocycfvres2 33605 | A cyclic permutation is th... |
| cycpmfvlem 33606 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33607 | Value of a cycle function ... |
| cycpmfv2 33608 | Value of a cycle function ... |
| cycpmfv3 33609 | Values outside of the orbi... |
| cycpmcl 33610 | Cyclic permutations are pe... |
| tocycf 33611 | The permutation cycle buil... |
| tocyc01 33612 | Permutation cycles built f... |
| cycpm2tr 33613 | A cyclic permutation of 2 ... |
| cycpm2cl 33614 | Closure for the 2-cycles. ... |
| cyc2fv1 33615 | Function value of a 2-cycl... |
| cyc2fv2 33616 | Function value of a 2-cycl... |
| trsp2cyc 33617 | Exhibit the word a transpo... |
| cycpmco2f1 33618 | The word U used in ~ cycpm... |
| cycpmco2rn 33619 | The orbit of the compositi... |
| cycpmco2lem1 33620 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33621 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33622 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33623 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33624 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33625 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33626 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33627 | The composition of a cycli... |
| cyc2fvx 33628 | Function value of a 2-cycl... |
| cycpm3cl 33629 | Closure of the 3-cycles in... |
| cycpm3cl2 33630 | Closure of the 3-cycles in... |
| cyc3fv1 33631 | Function value of a 3-cycl... |
| cyc3fv2 33632 | Function value of a 3-cycl... |
| cyc3fv3 33633 | Function value of a 3-cycl... |
| cyc3co2 33634 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33635 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33636 | A formula for computing co... |
| cycpmrn 33637 | The range of the word used... |
| tocyccntz 33638 | All elements of a (finite)... |
| evpmval 33639 | Value of the set of even p... |
| cnmsgn0g 33640 | The neutral element of the... |
| evpmsubg 33641 | The alternating group is a... |
| evpmid 33642 | The identity is an even pe... |
| altgnsg 33643 | The alternating group ` ( ... |
| cyc3evpm 33644 | 3-Cycles are even permutat... |
| cyc3genpmlem 33645 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33646 | The alternating group ` A ... |
| cycpmgcl 33647 | Cyclic permutations are pe... |
| cycpmconjslem1 33648 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33649 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33650 | All cycles of the same len... |
| cyc3conja 33651 | All 3-cycles are conjugate... |
| sgnsv 33654 | The sign mapping. (Contri... |
| sgnsval 33655 | The sign value. (Contribu... |
| sgnsf 33656 | The sign function. (Contr... |
| fxpval 33659 | Value of the set of fixed ... |
| fxpss 33660 | The set of fixed points is... |
| fxpgaval 33661 | Value of the set of fixed ... |
| isfxp 33662 | Property of being a fixed ... |
| fxpgaeq 33663 | A fixed point ` X ` is inv... |
| conjga 33664 | Group conjugation induces ... |
| cntrval2 33665 | Express the center ` Z ` o... |
| fxpsubm 33666 | Provided the group action ... |
| fxpsubg 33667 | The fixed points of a grou... |
| fxpsubrg 33668 | The fixed points of a grou... |
| fxpsdrg 33669 | The fixed points of a grou... |
| inftmrel 33674 | The infinitesimal relation... |
| isinftm 33675 | Express ` x ` is infinites... |
| isarchi 33676 | Express the predicate " ` ... |
| pnfinf 33677 | Plus infinity is an infini... |
| xrnarchi 33678 | The completed real line is... |
| isarchi2 33679 | Alternative way to express... |
| submarchi 33680 | A submonoid is archimedean... |
| isarchi3 33681 | This is the usual definiti... |
| archirng 33682 | Property of Archimedean or... |
| archirngz 33683 | Property of Archimedean le... |
| archiexdiv 33684 | In an Archimedean group, g... |
| archiabllem1a 33685 | Lemma for ~ archiabl : In... |
| archiabllem1b 33686 | Lemma for ~ archiabl . (C... |
| archiabllem1 33687 | Archimedean ordered groups... |
| archiabllem2a 33688 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33689 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33690 | Lemma for ~ archiabl . (C... |
| archiabllem2 33691 | Archimedean ordered groups... |
| archiabl 33692 | Archimedean left- and righ... |
| isarchiofld 33693 | Axiom of Archimedes : a ch... |
| isslmd 33696 | The predicate "is a semimo... |
| slmdlema 33697 | Lemma for properties of a ... |
| lmodslmd 33698 | Left semimodules generaliz... |
| slmdcmn 33699 | A semimodule is a commutat... |
| slmdmnd 33700 | A semimodule is a monoid. ... |
| slmdsrg 33701 | The scalar component of a ... |
| slmdbn0 33702 | The base set of a semimodu... |
| slmdacl 33703 | Closure of ring addition f... |
| slmdmcl 33704 | Closure of ring multiplica... |
| slmdsn0 33705 | The set of scalars in a se... |
| slmdvacl 33706 | Closure of vector addition... |
| slmdass 33707 | Semiring left module vecto... |
| slmdvscl 33708 | Closure of scalar product ... |
| slmdvsdi 33709 | Distributive law for scala... |
| slmdvsdir 33710 | Distributive law for scala... |
| slmdvsass 33711 | Associative law for scalar... |
| slmd0cl 33712 | The ring zero in a semimod... |
| slmd1cl 33713 | The ring unity in a semiri... |
| slmdvs1 33714 | Scalar product with ring u... |
| slmd0vcl 33715 | The zero vector is a vecto... |
| slmd0vlid 33716 | Left identity law for the ... |
| slmd0vrid 33717 | Right identity law for the... |
| slmd0vs 33718 | Zero times a vector is the... |
| slmdvs0 33719 | Anything times the zero ve... |
| gsumvsca1 33720 | Scalar product of a finite... |
| gsumvsca2 33721 | Scalar product of a finite... |
| prmsimpcyc 33722 | A group of prime order is ... |
| ringrngd 33723 | A unital ring is a non-uni... |
| urpropd 33724 | Sufficient condition for r... |
| subrgmcld 33725 | A subring is closed under ... |
| ress1r 33726 | ` 1r ` is unaffected by re... |
| ringm1expp1 33727 | Ring exponentiation of min... |
| ringinvval 33728 | The ring inverse expressed... |
| dvrcan5 33729 | Cancellation law for commo... |
| subrgchr 33730 | If ` A ` is a subring of `... |
| rmfsupp2 33731 | A mapping of a multiplicat... |
| unitnz 33732 | In a nonzero ring, a unit ... |
| isunit2 33733 | Alternate definition of be... |
| isunit3 33734 | Alternate definition of be... |
| isunitc 33735 | Characterize units in a co... |
| elrgspnlem1 33736 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33737 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33738 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33739 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33740 | Membership in the subring ... |
| elrgspnsubrunlem1 33741 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33742 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33743 | Membership in the ring spa... |
| irrednzr 33744 | A ring with an irreducible... |
| 0ringsubrg 33745 | A subring of a zero ring i... |
| 0ringcring 33746 | The zero ring is commutati... |
| reldmrloc 33751 | Ring localization is a pro... |
| erlval 33752 | Value of the ring localiza... |
| rlocval 33753 | Expand the value of the ri... |
| erlcl1 33754 | Closure for the ring local... |
| erlcl2 33755 | Closure for the ring local... |
| erldi 33756 | Main property of the ring ... |
| erlbrd 33757 | Deduce the ring localizati... |
| erlbr2d 33758 | Deduce the ring localizati... |
| erler 33759 | The relation used to build... |
| erld2 33760 | Main property of the ring ... |
| elrlocbasi 33761 | Membership in the basis of... |
| rlocbas 33762 | The base set of a ring loc... |
| rlocaddval 33763 | Value of the addition in t... |
| rlocmulval 33764 | Value of the addition in t... |
| rloccring 33765 | The ring localization ` L ... |
| rloc0g 33766 | The zero of a ring localiz... |
| rloc1r 33767 | The multiplicative identit... |
| rlocf1 33768 | The embedding ` F ` of a r... |
| rlocinvunit 33769 | In the localization of a r... |
| rlocisunit 33770 | Characterize the units of ... |
| domnmuln0rd 33771 | In a domain, factors of a ... |
| domnprodn0 33772 | In a domain, a finite prod... |
| domnprodeq0 33773 | A product over a domain is... |
| domnpropd 33774 | If two structures have the... |
| idompropd 33775 | If two structures have the... |
| idomrcan 33776 | Right-cancellation law for... |
| 1rrg 33777 | The multiplicative identit... |
| rrgsubm 33778 | The left regular elements ... |
| subrdom 33779 | A subring of a domain is a... |
| subridom 33780 | A subring of an integral d... |
| subrfld 33781 | A subring of a field is an... |
| ricnzr1 33782 | A ring isomorphism maps a ... |
| ricdomn1 33783 | A ring isomorphism maps a ... |
| ricdomn 33784 | A ring is a domain if and ... |
| eufndx 33787 | Index value of the Euclide... |
| eufid 33788 | Utility theorem: index-ind... |
| rndrhmcl 33791 | The image of a division ri... |
| qfld 33792 | The field of rational numb... |
| subsdrg 33793 | A subring of a sub-divisio... |
| sdrgdvcl 33794 | A sub-division-ring is clo... |
| sdrginvcl 33795 | A sub-division-ring is clo... |
| primefldchr 33796 | The characteristic of a pr... |
| fracval 33799 | Value of the field of frac... |
| fracbas 33800 | The base of the field of f... |
| fracerl 33801 | Rewrite the ring localizat... |
| fracf1 33802 | The embedding of a commuta... |
| fracfld 33803 | The field of fractions of ... |
| idomsubr 33804 | Every integral domain is i... |
| fldgenval 33807 | Value of the field generat... |
| fldgenssid 33808 | The field generated by a s... |
| fldgensdrg 33809 | A generated subfield is a ... |
| fldgenssv 33810 | A generated subfield is a ... |
| fldgenss 33811 | Generated subfields preser... |
| fldgenidfld 33812 | The subfield generated by ... |
| fldgenssp 33813 | The field generated by a s... |
| fldgenid 33814 | The subfield of a field ` ... |
| fldgenfld 33815 | A generated subfield is a ... |
| primefldgen1 33816 | The prime field of a divis... |
| 1fldgenq 33817 | The field of rational numb... |
| rhmdvd 33818 | A ring homomorphism preser... |
| kerunit 33819 | If a unit element lies in ... |
| reldmresv 33822 | The scalar restriction is ... |
| resvval 33823 | Value of structure restric... |
| resvid2 33824 | General behavior of trivia... |
| resvval2 33825 | Value of nontrivial struct... |
| resvsca 33826 | Base set of a structure re... |
| resvlem 33827 | Other elements of a scalar... |
| resvbas 33828 | ` Base ` is unaffected by ... |
| resvplusg 33829 | ` +g ` is unaffected by sc... |
| resvvsca 33830 | ` .s ` is unaffected by sc... |
| resvmulr 33831 | ` .r ` is unaffected by sc... |
| resv0g 33832 | ` 0g ` is unaffected by sc... |
| resv1r 33833 | ` 1r ` is unaffected by sc... |
| resvcmn 33834 | Scalar restriction preserv... |
| gzcrng 33835 | The gaussian integers form... |
| cnfldfld 33836 | The complex numbers form a... |
| reofld 33837 | The real numbers form an o... |
| nn0omnd 33838 | The nonnegative integers f... |
| gsumind 33839 | The group sum of an indica... |
| rearchi 33840 | The field of the real numb... |
| nn0archi 33841 | The monoid of the nonnegat... |
| xrge0slmod 33842 | The extended nonnegative r... |
| qusker 33843 | The kernel of a quotient m... |
| eqgvscpbl 33844 | The left coset equivalence... |
| qusvscpbl 33845 | The quotient map distribut... |
| qusvsval 33846 | Value of the scalar multip... |
| imaslmod 33847 | The image structure of a l... |
| imasmhm 33848 | Given a function ` F ` wit... |
| imasghm 33849 | Given a function ` F ` wit... |
| imasrhm 33850 | Given a function ` F ` wit... |
| imaslmhm 33851 | Given a function ` F ` wit... |
| quslmod 33852 | If ` G ` is a submodule in... |
| quslmhm 33853 | If ` G ` is a submodule of... |
| quslvec 33854 | If ` S ` is a vector subsp... |
| znfermltl 33855 | Fermat's little theorem in... |
| islinds5 33856 | A set is linearly independ... |
| ellspds 33857 | Variation on ~ ellspd . (... |
| 0ellsp 33858 | Zero is in all spans. (Co... |
| 0nellinds 33859 | The group identity cannot ... |
| elrsp 33860 | Write the elements of a ri... |
| ellpi 33861 | Elementhood in a left prin... |
| lpirlidllpi 33862 | In a principal ideal ring,... |
| rspidlid 33863 | The ideal span of an ideal... |
| rsp2idlid 33864 | The ideal span of a two-si... |
| lbslsp 33865 | Any element of a left modu... |
| lindssn 33866 | Any singleton of a nonzero... |
| lindflbs 33867 | Conditions for an independ... |
| islbs5 33868 | An equivalent formulation ... |
| linds2eq 33869 | Deduce equality of element... |
| lindfpropd 33870 | Property deduction for lin... |
| lindspropd 33871 | Property deduction for lin... |
| dvdsruassoi 33872 | If two elements ` X ` and ... |
| dvdsruasso 33873 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33874 | A reformulation of ~ dvdsr... |
| dvdsrspss 33875 | In a ring, an element ` X ... |
| rspsnasso 33876 | Two elements ` X ` and ` Y... |
| unitprodclb 33877 | A finite product is a unit... |
| elgrplsmsn 33878 | Membership in a sumset wit... |
| lsmsnorb 33879 | The sumset of a group with... |
| lsmsnorb2 33880 | The sumset of a single ele... |
| ringlsmss 33881 | Closure of the product of ... |
| ringlsmss1 33882 | The product of an ideal ` ... |
| ringlsmss2 33883 | The product with an ideal ... |
| lsmsnpridl 33884 | The product of the ring wi... |
| lsmsnidl 33885 | The product of the ring wi... |
| lsmssass 33886 | Group sum is associative, ... |
| grplsm0l 33887 | Sumset with the identity s... |
| grplsmid 33888 | The direct sum of an eleme... |
| quslsm 33889 | Express the image by the q... |
| qusbas2 33890 | Alternate definition of th... |
| qus0g 33891 | The identity element of a ... |
| qusima 33892 | The image of a subgroup by... |
| qusrn 33893 | The natural map from eleme... |
| nsgqus0 33894 | A normal subgroup ` N ` is... |
| nsgmgclem 33895 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33896 | There is a monotone Galois... |
| nsgqusf1olem1 33897 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33898 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33899 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33900 | The canonical projection h... |
| lmhmqusker 33901 | A surjective module homomo... |
| lmicqusker 33902 | The image ` H ` of a modul... |
| intlidl 33903 | The intersection of a none... |
| inlidl 33904 | The intersection of two id... |
| pidlnzb 33905 | A principal ideal is nonze... |
| lidlunitel 33906 | If an ideal ` I ` contains... |
| unitpidl1 33907 | The ideal ` I ` generated ... |
| rhmquskerlem 33908 | The mapping ` J ` induced ... |
| rhmqusker 33909 | A surjective ring homomorp... |
| ricqusker 33910 | The image ` H ` of a ring ... |
| elrspunidl 33911 | Elementhood in the span of... |
| elrspunsn 33912 | Membership to the span of ... |
| lidlincl 33913 | Ideals are closed under in... |
| idlinsubrg 33914 | The intersection between a... |
| rhmimaidl 33915 | The image of an ideal ` I ... |
| drngidlhash 33916 | A ring is a division ring ... |
| mxidlval 33919 | The set of maximal ideals ... |
| ismxidl 33920 | The predicate "is a maxima... |
| mxidlidl 33921 | A maximal ideal is an idea... |
| mxidlnr 33922 | A maximal ideal is proper.... |
| mxidlmax 33923 | A maximal ideal is a maxim... |
| mxidln1 33924 | One is not contained in an... |
| mxidlnzr 33925 | A ring with a maximal idea... |
| mxidlmaxv 33926 | An ideal ` I ` strictly co... |
| crngmxidl 33927 | In a commutative ring, max... |
| mxidlprm 33928 | Every maximal ideal is pri... |
| mxidlirredi 33929 | In an integral domain, the... |
| mxidlirred 33930 | In a principal ideal domai... |
| ssmxidllem 33931 | The set ` P ` used in the ... |
| ssmxidl 33932 | Let ` R ` be a ring, and l... |
| drng0mxidl 33933 | In a division ring, the ze... |
| drngmxidl 33934 | The zero ideal is the only... |
| drngmxidlr 33935 | If a ring's only maximal i... |
| krull 33936 | Krull's theorem: Any nonz... |
| mxidlnzrb 33937 | A ring is nonzero if and o... |
| krullndrng 33938 | Krull's theorem for non-di... |
| opprabs 33939 | The opposite ring of the o... |
| oppreqg 33940 | Group coset equivalence re... |
| opprnsg 33941 | Normal subgroups of the op... |
| opprlidlabs 33942 | The ideals of the opposite... |
| oppr2idl 33943 | Two sided ideal of the opp... |
| opprmxidlabs 33944 | The maximal ideal of the o... |
| opprqusbas 33945 | The base of the quotient o... |
| opprqusplusg 33946 | The group operation of the... |
| opprqus0g 33947 | The group identity element... |
| opprqusmulr 33948 | The multiplication operati... |
| opprqus1r 33949 | The ring unity of the quot... |
| opprqusdrng 33950 | The quotient of the opposi... |
| qsdrngilem 33951 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33952 | A quotient by a maximal le... |
| qsdrnglem2 33953 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33954 | An ideal ` M ` is both lef... |
| qsfld 33955 | An ideal ` M ` in the comm... |
| mxidlprmALT 33956 | Every maximal ideal is pri... |
| drnglring 33957 | A division ring is a local... |
| dflring2 33958 | Alternate definition of a ... |
| dflringlem 33959 | Lemma for ~ dflring3 . If... |
| dflringlem2 33960 | Lemma for ~ dflring3 . In... |
| dflringlem3 33961 | Lemma for ~ dflring3 . In... |
| dflring3 33962 | Alternate definition of a ... |
| dflring4 33963 | Alternate definition of a ... |
| fldlring 33964 | A field is a local ring. ... |
| idlsrgstr 33967 | A constructed semiring of ... |
| idlsrgval 33968 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33969 | Base of the ideals of a ri... |
| idlsrgplusg 33970 | Additive operation of the ... |
| idlsrg0g 33971 | The zero ideal is the addi... |
| idlsrgmulr 33972 | Multiplicative operation o... |
| idlsrgtset 33973 | Topology component of the ... |
| idlsrgmulrval 33974 | Value of the ring multipli... |
| idlsrgmulrcl 33975 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33976 | In a commutative ring, the... |
| idlsrgmulrss2 33977 | The product of two ideals ... |
| idlsrgmulrssin 33978 | In a commutative ring, the... |
| idlsrgmnd 33979 | The ideals of a ring form ... |
| idlsrgcmnd 33980 | The ideals of a ring form ... |
| rprmval 33981 | The prime elements of a ri... |
| isrprm 33982 | Property for ` P ` to be a... |
| rprmcl 33983 | A ring prime is an element... |
| rprmdvds 33984 | If a ring prime ` Q ` divi... |
| rprmnz 33985 | A ring prime is nonzero. ... |
| rprmnunit 33986 | A ring prime is not a unit... |
| rsprprmprmidl 33987 | In a commutative ring, ide... |
| rsprprmprmidlb 33988 | An ideal generated by a si... |
| rprmndvdsr1 33989 | A ring prime element does ... |
| rprmasso 33990 | In an integral domain, the... |
| rprmasso2 33991 | In an integral domain, if ... |
| rprmasso3 33992 | In an integral domain, if ... |
| unitmulrprm 33993 | A ring unit multiplied by ... |
| rprmndvdsru 33994 | A ring prime element does ... |
| rprmirredlem 33995 | Lemma for ~ rprmirred . (... |
| rprmirred 33996 | In an integral domain, rin... |
| rprmirredb 33997 | In a principal ideal domai... |
| rprmdvdspow 33998 | If a prime element divides... |
| rprmdvdsprod 33999 | If a prime element ` Q ` d... |
| 1arithidomlem1 34000 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 34001 | Lemma for ~ 1arithidom : i... |
| 1arithidom 34002 | Uniqueness of prime factor... |
| isufd 34005 | The property of being a Un... |
| ufdprmidl 34006 | In a unique factorization ... |
| ufdidom 34007 | A nonzero unique factoriza... |
| pidufd 34008 | Every principal ideal doma... |
| 1arithufdlem1 34009 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 34010 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 34011 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 34012 | Lemma for ~ 1arithufd . N... |
| 1arithufd 34013 | Existence of a factorizati... |
| dfufd2lem 34014 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 34015 | Alternative definition of ... |
| zringidom 34016 | The ring of integers is an... |
| zringpid 34017 | The ring of integers is a ... |
| dfprm3 34018 | The (positive) prime eleme... |
| zringfrac 34019 | The field of fractions of ... |
| assaassd 34020 | Left-associative property ... |
| assaassrd 34021 | Right-associative property... |
| 0ringmon1p 34022 | There are no monic polynom... |
| fply1 34023 | Conditions for a function ... |
| ply1lvec 34024 | In a division ring, the un... |
| evls1fn 34025 | Functionality of the subri... |
| evls1dm 34026 | The domain of the subring ... |
| evls1fvf 34027 | The subring evaluation fun... |
| evl1fvf 34028 | The univariate polynomial ... |
| evl1fpws 34029 | Evaluation of a univariate... |
| ressply1evls1 34030 | Subring evaluation of a un... |
| ressdeg1 34031 | The degree of a univariate... |
| ressply10g 34032 | A restricted polynomial al... |
| ressply1mon1p 34033 | The monic polynomials of a... |
| ressply1invg 34034 | An element of a restricted... |
| ressply1sub 34035 | A restricted polynomial al... |
| ressasclcl 34036 | Closure of the univariate ... |
| evls1subd 34037 | Univariate polynomial eval... |
| deg1le0eq0 34038 | A polynomial with nonposit... |
| ply1asclunit 34039 | A nonzero scalar polynomia... |
| ply1unit 34040 | In a field ` F ` , a polyn... |
| evl1deg1 34041 | Evaluation of a univariate... |
| evl1deg2 34042 | Evaluation of a univariate... |
| evl1deg3 34043 | Evaluation of a univariate... |
| evls1monply1 34044 | Subring evaluation of a sc... |
| ply1dg1rt 34045 | Express the root ` - B / A... |
| ply1dg1rtn0 34046 | Polynomials of degree 1 ov... |
| ply1mulrtss 34047 | The roots of a factor ` F ... |
| deg1prod 34048 | Degree of a product of pol... |
| ply1dg3rt0irred 34049 | If a cubic polynomial over... |
| m1pmeq 34050 | If two monic polynomials `... |
| ply1fermltl 34051 | Fermat's little theorem fo... |
| coe1mon 34052 | Coefficient vector of a mo... |
| ply1moneq 34053 | Two monomials are equal if... |
| ply1coedeg 34054 | Decompose a univariate pol... |
| coe1zfv 34055 | The coefficients of the ze... |
| coe1vr1 34056 | Polynomial coefficient of ... |
| deg1vr 34057 | The degree of the variable... |
| vr1nz 34058 | A univariate polynomial va... |
| ply1degltel 34059 | Characterize elementhood i... |
| ply1degleel 34060 | Characterize elementhood i... |
| ply1degltlss 34061 | The space ` S ` of the uni... |
| gsummoncoe1fzo 34062 | A coefficient of the polyn... |
| gsummoncoe1fz 34063 | A coefficient of the polyn... |
| ply1gsumz 34064 | If a polynomial given as a... |
| deg1addlt 34065 | If both factors have degre... |
| ig1pnunit 34066 | The polynomial ideal gener... |
| ig1pmindeg 34067 | The polynomial ideal gener... |
| q1pdir 34068 | Distribution of univariate... |
| q1pvsca 34069 | Scalar multiplication prop... |
| r1pvsca 34070 | Scalar multiplication prop... |
| r1p0 34071 | Polynomial remainder opera... |
| r1pcyc 34072 | The polynomial remainder o... |
| r1padd1 34073 | Addition property of the p... |
| r1plmhm 34074 | The univariate polynomial ... |
| r1pquslmic 34075 | The univariate polynomial ... |
| psrbasfsupp 34076 | Rewrite a finite support f... |
| psrnzr 34077 | The ring of power series o... |
| mplnzr 34078 | The multivariate polynomia... |
| 0mplrim 34079 | Build a ring isomorphism b... |
| 0mplric 34080 | Multivariate polynomials w... |
| mplasclco 34081 | Case where composing an al... |
| selvascl 34082 | The "variable selection" f... |
| selvply1rhmlema 34083 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlemb 34084 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem1 34085 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem2 34086 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem3 34087 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem4 34088 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem5 34089 | Lemma for ~ selvply1rhm . ... |
| selvply1rhm 34090 | Build a ring homomorphism ... |
| selvply1rhm0 34091 | The ring homomorphism ` H ... |
| mplidomlem 34092 | Lemma for ~ mplidom . (Co... |
| mplidom 34093 | The multivariate polynomia... |
| extvval 34096 | Value of the "variable ext... |
| extvfval 34097 | The "variable extension" f... |
| extvfv 34098 | The "variable extension" f... |
| extvfvv 34099 | The "variable extension" f... |
| extvfvvcl 34100 | Closure for the "variable ... |
| extvfvcl 34101 | Closure for the "variable ... |
| extvfvalf 34102 | The "variable extension" f... |
| mvrvalind 34103 | Value of the generating el... |
| mplmulmvr 34104 | Multiply a polynomial ` F ... |
| evlscaval 34105 | Polynomial evaluation for ... |
| evlvarval 34106 | Polynomial evaluation buil... |
| evlextv 34107 | Evaluating a variable-exte... |
| mplvrpmlem 34108 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 34109 | Permuting variables in a m... |
| mplvrpmga 34110 | The action of permuting va... |
| mplvrpmmhm 34111 | The action of permuting va... |
| mplvrpmrhm 34112 | The action of permuting va... |
| psrgsum 34113 | Finite commutative sums of... |
| psrmon 34114 | A monomial is a power seri... |
| psrmonmul 34115 | The product of two power s... |
| psrmonmul2 34116 | The product of two power s... |
| psrmonprod 34117 | Finite product of bags of ... |
| mplgsum 34118 | Finite commutative sums of... |
| mplmonprod 34119 | Finite product of monomial... |
| splyval 34124 | The symmetric polynomials ... |
| splysubrg 34125 | The symmetric polynomials ... |
| issply 34126 | Conditions for being a sym... |
| esplyval 34127 | The elementary polynomials... |
| esplyfval 34128 | The ` K ` -th elementary p... |
| esplyfval0 34129 | The ` 0 ` -th elementary s... |
| esplyfval2 34130 | When ` K ` is out-of-bound... |
| esplylem 34131 | Lemma for ~ esplyfv and ot... |
| esplympl 34132 | Elementary symmetric polyn... |
| esplymhp 34133 | The ` K ` -th elementary s... |
| esplyfv1 34134 | Coefficient for the ` K ` ... |
| esplyfv 34135 | Coefficient for the ` K ` ... |
| esplysply 34136 | The ` K ` -th elementary s... |
| esplyfval3 34137 | Alternate expression for t... |
| esplyfval1 34138 | The first elementary symme... |
| esplyfvaln 34139 | The last elementary symmet... |
| esplyind 34140 | A recursive formula for th... |
| esplyindfv 34141 | A recursive formula for th... |
| esplyfvn 34142 | Express the last elementar... |
| vietadeg1 34143 | The degree of a product of... |
| vietalem 34144 | Lemma for ~ vieta : induct... |
| vieta 34145 | Vieta's Formulas: Coeffic... |
| sra1r 34146 | The unity element of a sub... |
| sradrng 34147 | Condition for a subring al... |
| sraidom 34148 | Condition for a subring al... |
| srasubrg 34149 | A subring of the original ... |
| sralvec 34150 | Given a sub division ring ... |
| srafldlvec 34151 | Given a subfield ` F ` of ... |
| resssra 34152 | The subring algebra of a r... |
| lsssra 34153 | A subring is a subspace of... |
| srapwov 34154 | The "power" operation on a... |
| drgext0g 34155 | The additive neutral eleme... |
| drgextvsca 34156 | The scalar multiplication ... |
| drgext0gsca 34157 | The additive neutral eleme... |
| drgextsubrg 34158 | The scalar field is a subr... |
| drgextlsp 34159 | The scalar field is a subs... |
| drgextgsum 34160 | Group sum in a division ri... |
| lvecdimfi 34161 | Finite version of ~ lvecdi... |
| exsslsb 34162 | Any finite generating set ... |
| lbslelsp 34163 | The size of a basis ` X ` ... |
| dimval 34166 | The dimension of a vector ... |
| dimvalfi 34167 | The dimension of a vector ... |
| dimcl 34168 | Closure of the vector spac... |
| lmimdim 34169 | Module isomorphisms preser... |
| lmicdim 34170 | Module isomorphisms preser... |
| lvecdim0i 34171 | A vector space of dimensio... |
| lvecdim0 34172 | A vector space of dimensio... |
| lssdimle 34173 | The dimension of a linear ... |
| dimpropd 34174 | If two structures have the... |
| rlmdim 34175 | The left vector space indu... |
| frlmdim 34176 | Dimension of a free left m... |
| tnglvec 34177 | Augmenting a structure wit... |
| tngdim 34178 | Dimension of a left vector... |
| rrxdim 34179 | Dimension of the generaliz... |
| matdim 34180 | Dimension of the space of ... |
| lbslsat 34181 | A nonzero vector ` X ` is ... |
| lsatdim 34182 | A line, spanned by a nonze... |
| drngdimgt0 34183 | The dimension of a vector ... |
| lmhmlvec2 34184 | A homomorphism of left vec... |
| kerlmhm 34185 | The kernel of a vector spa... |
| imlmhm 34186 | The image of a vector spac... |
| ply1degltdimlem 34187 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 34188 | The space ` S ` of the uni... |
| lindsunlem 34189 | Lemma for ~ lindsun . (Co... |
| lindsun 34190 | Condition for the union of... |
| lbsdiflsp0 34191 | The linear spans of two di... |
| dimkerim 34192 | Given a linear map ` F ` b... |
| qusdimsum 34193 | Let ` W ` be a vector spac... |
| fedgmullem1 34194 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 34195 | Lemma for ~ fedgmul . (Co... |
| fedgmul 34196 | The multiplicativity formu... |
| dimlssid 34197 | If the dimension of a line... |
| lvecendof1f1o 34198 | If an endomorphism ` U ` o... |
| lactlmhm 34199 | In an associative algebra ... |
| assalactf1o 34200 | In an associative algebra ... |
| assarrginv 34201 | If an element ` X ` of an ... |
| assafld 34202 | If an algebra ` A ` of fin... |
| relfldext 34209 | The field extension is a r... |
| brfldext 34210 | The field extension relati... |
| ccfldextrr 34211 | The field of the complex n... |
| fldextfld1 34212 | A field extension is only ... |
| fldextfld2 34213 | A field extension is only ... |
| fldextsubrg 34214 | Field extension implies a ... |
| sdrgfldext 34215 | A field ` E ` and any sub-... |
| fldextress 34216 | Field extension implies a ... |
| brfinext 34217 | The finite field extension... |
| extdgval 34218 | Value of the field extensi... |
| fldextsdrg 34219 | Deduce sub-division-ring f... |
| fldextsralvec 34220 | The subring algebra associ... |
| extdgcl 34221 | Closure of the field exten... |
| extdggt0 34222 | Degrees of field extension... |
| fldexttr 34223 | Field extension is a trans... |
| fldextid 34224 | The field extension relati... |
| extdgid 34225 | A trivial field extension ... |
| fldsdrgfldext 34226 | A sub-division-ring of a f... |
| fldsdrgfldext2 34227 | A sub-sub-division-ring of... |
| extdgmul 34228 | The multiplicativity formu... |
| finextfldext 34229 | A finite field extension i... |
| finexttrb 34230 | The extension ` E ` of ` K... |
| extdg1id 34231 | If the degree of the exten... |
| extdg1b 34232 | The degree of the extensio... |
| fldgenfldext 34233 | A subfield ` F ` extended ... |
| fldextchr 34234 | The characteristic of a su... |
| evls1fldgencl 34235 | Closure of the subring pol... |
| ccfldsrarelvec 34236 | The subring algebra of the... |
| ccfldextdgrr 34237 | The degree of the field ex... |
| fldextrspunlsplem 34238 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 34239 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 34240 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 34241 | The ring generated by the ... |
| fldextrspunlem2 34242 | Part of the proof of Propo... |
| fldextrspundgle 34243 | Inequality involving the d... |
| fldextrspundglemul 34244 | Given two field extensions... |
| fldextrspundgdvdslem 34245 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 34246 | Given two finite extension... |
| fldext2rspun 34247 | Given two field extensions... |
| irngval 34250 | The elements of a field ` ... |
| elirng 34251 | Property for an element ` ... |
| irngss 34252 | All elements of a subring ... |
| irngssv 34253 | An integral element is an ... |
| 0ringirng 34254 | A zero ring ` R ` has no i... |
| irngnzply1lem 34255 | In the case of a field ` E... |
| irngnzply1 34256 | In the case of a field ` E... |
| extdgfialglem1 34257 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 34258 | Lemma for ~ extdgfialg . ... |
| extdgfialg 34259 | A finite field extension `... |
| bralgext 34262 | Express the fact that a fi... |
| finextalg 34263 | A finite field extension i... |
| ply1annidllem 34266 | Write the set ` Q ` of pol... |
| ply1annidl 34267 | The set ` Q ` of polynomia... |
| ply1annnr 34268 | The set ` Q ` of polynomia... |
| ply1annig1p 34269 | The ideal ` Q ` of polynom... |
| minplyval 34270 | Expand the value of the mi... |
| minplycl 34271 | The minimal polynomial is ... |
| ply1annprmidl 34272 | The set ` Q ` of polynomia... |
| minplymindeg 34273 | The minimal polynomial of ... |
| minplyann 34274 | The minimal polynomial for... |
| minplyirredlem 34275 | Lemma for ~ minplyirred . ... |
| minplyirred 34276 | A nonzero minimal polynomi... |
| irngnminplynz 34277 | Integral elements have non... |
| minplym1p 34278 | A minimal polynomial is mo... |
| minplynzm1p 34279 | If a minimal polynomial is... |
| minplyelirng 34280 | If the minimal polynomial ... |
| irredminply 34281 | An irreducible, monic, ann... |
| algextdeglem1 34282 | Lemma for ~ algextdeg . (... |
| algextdeglem2 34283 | Lemma for ~ algextdeg . B... |
| algextdeglem3 34284 | Lemma for ~ algextdeg . T... |
| algextdeglem4 34285 | Lemma for ~ algextdeg . B... |
| algextdeglem5 34286 | Lemma for ~ algextdeg . T... |
| algextdeglem6 34287 | Lemma for ~ algextdeg . B... |
| algextdeglem7 34288 | Lemma for ~ algextdeg . T... |
| algextdeglem8 34289 | Lemma for ~ algextdeg . T... |
| algextdeg 34290 | The degree of an algebraic... |
| rtelextdg2lem 34291 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 34292 | If an element ` X ` is a s... |
| fldext2chn 34293 | In a non-empty chain ` T `... |
| constrrtll 34296 | In the construction of con... |
| constrrtlc1 34297 | In the construction of con... |
| constrrtlc2 34298 | In the construction of con... |
| constrrtcclem 34299 | In the construction of con... |
| constrrtcc 34300 | In the construction of con... |
| isconstr 34301 | Property of being a constr... |
| constr0 34302 | The first step of the cons... |
| constrsuc 34303 | Membership in the successo... |
| constrlim 34304 | Limit step of the construc... |
| constrsscn 34305 | Closure of the constructib... |
| constrsslem 34306 | Lemma for ~ constrss . Th... |
| constr01 34307 | ` 0 ` and ` 1 ` are in all... |
| constrss 34308 | Constructed points are in ... |
| constrmon 34309 | The construction of constr... |
| constrconj 34310 | If a point ` X ` of the co... |
| constrfin 34311 | Each step of the construct... |
| constrelextdg2 34312 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 34313 | Lemma for ~ constrextdg2 .... |
| constrextdg2 34314 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 34315 | Lemma for ~ constrext2chn ... |
| constrfiss 34316 | For any finite set ` A ` o... |
| constrllcllem 34317 | Constructible numbers are ... |
| constrlccllem 34318 | Constructible numbers are ... |
| constrcccllem 34319 | Constructible numbers are ... |
| constrcbvlem 34320 | Technical lemma for elimin... |
| constrllcl 34321 | Constructible numbers are ... |
| constrlccl 34322 | Constructible numbers are ... |
| constrcccl 34323 | Constructible numbers are ... |
| constrext2chn 34324 | If a constructible number ... |
| constrcn 34325 | Constructible numbers are ... |
| nn0constr 34326 | Nonnegative integers are c... |
| constraddcl 34327 | Constructive numbers are c... |
| constrnegcl 34328 | Constructible numbers are ... |
| zconstr 34329 | Integers are constructible... |
| constrdircl 34330 | Constructible numbers are ... |
| iconstr 34331 | The imaginary unit ` _i ` ... |
| constrremulcl 34332 | If two real numbers ` X ` ... |
| constrcjcl 34333 | Constructible numbers are ... |
| constrrecl 34334 | Constructible numbers are ... |
| constrimcl 34335 | Constructible numbers are ... |
| constrmulcl 34336 | Constructible numbers are ... |
| constrreinvcl 34337 | If a real number ` X ` is ... |
| constrinvcl 34338 | Constructible numbers are ... |
| constrcon 34339 | Contradiction of construct... |
| constrsdrg 34340 | Constructible numbers form... |
| constrfld 34341 | The constructible numbers ... |
| constrresqrtcl 34342 | If a positive real number ... |
| constrabscl 34343 | Constructible numbers are ... |
| constrsqrtcl 34344 | Constructible numbers are ... |
| 2sqr3minply 34345 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 34346 | Doubling the cube is an im... |
| cos9thpiminplylem1 34347 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 34348 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 34349 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 34350 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 34351 | The constructed complex nu... |
| cos9thpiminplylem6 34352 | Evaluation of the polynomi... |
| cos9thpiminply 34353 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 34354 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 34355 | The complex number ` A ` i... |
| cos9thpinconstr 34356 | Trisecting an angle is an ... |
| trisecnconstr 34357 | Not all angles can be tris... |
| smatfval 34360 | Value of the submatrix. (... |
| smatrcl 34361 | Closure of the rectangular... |
| smatlem 34362 | Lemma for the next theorem... |
| smattl 34363 | Entries of a submatrix, to... |
| smattr 34364 | Entries of a submatrix, to... |
| smatbl 34365 | Entries of a submatrix, bo... |
| smatbr 34366 | Entries of a submatrix, bo... |
| smatcl 34367 | Closure of the square subm... |
| matmpo 34368 | Write a square matrix as a... |
| 1smat1 34369 | The submatrix of the ident... |
| submat1n 34370 | One case where the submatr... |
| submatres 34371 | Special case where the sub... |
| submateqlem1 34372 | Lemma for ~ submateq . (C... |
| submateqlem2 34373 | Lemma for ~ submateq . (C... |
| submateq 34374 | Sufficient condition for t... |
| submatminr1 34375 | If we take a submatrix by ... |
| lmatval 34378 | Value of the literal matri... |
| lmatfval 34379 | Entries of a literal matri... |
| lmatfvlem 34380 | Useful lemma to extract li... |
| lmatcl 34381 | Closure of the literal mat... |
| lmat22lem 34382 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 34383 | Entry of a 2x2 literal mat... |
| lmat22e12 34384 | Entry of a 2x2 literal mat... |
| lmat22e21 34385 | Entry of a 2x2 literal mat... |
| lmat22e22 34386 | Entry of a 2x2 literal mat... |
| lmat22det 34387 | The determinant of a liter... |
| mdetpmtr1 34388 | The determinant of a matri... |
| mdetpmtr2 34389 | The determinant of a matri... |
| mdetpmtr12 34390 | The determinant of a matri... |
| mdetlap1 34391 | A Laplace expansion of the... |
| madjusmdetlem1 34392 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 34393 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 34394 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 34395 | Lemma for ~ madjusmdet . ... |
| madjusmdet 34396 | Express the cofactor of th... |
| mdetlap 34397 | Laplace expansion of the d... |
| ist0cld 34398 | The predicate "is a T_0 sp... |
| txomap 34399 | Given two open maps ` F ` ... |
| qtopt1 34400 | If every equivalence class... |
| qtophaus 34401 | If an open map's graph in ... |
| circtopn 34402 | The topology of the unit c... |
| circcn 34403 | The function gluing the re... |
| reff 34404 | For any cover refinement, ... |
| locfinreflem 34405 | A locally finite refinemen... |
| locfinref 34406 | A locally finite refinemen... |
| iscref 34409 | The property that every op... |
| crefeq 34410 | Equality theorem for the "... |
| creftop 34411 | A space where every open c... |
| crefi 34412 | The property that every op... |
| crefdf 34413 | A formulation of ~ crefi e... |
| crefss 34414 | The "every open cover has ... |
| cmpcref 34415 | Equivalent definition of c... |
| cmpfiref 34416 | Every open cover of a Comp... |
| ldlfcntref 34419 | Every open cover of a Lind... |
| ispcmp 34422 | The predicate "is a paraco... |
| cmppcmp 34423 | Every compact space is par... |
| dispcmp 34424 | Every discrete space is pa... |
| pcmplfin 34425 | Given a paracompact topolo... |
| pcmplfinf 34426 | Given a paracompact topolo... |
| rspecval 34429 | Value of the spectrum of t... |
| rspecbas 34430 | The prime ideals form the ... |
| rspectset 34431 | Topology component of the ... |
| rspectopn 34432 | The topology component of ... |
| zarcls0 34433 | The closure of the identit... |
| zarcls1 34434 | The unit ideal ` B ` is th... |
| zarclsun 34435 | The union of two closed se... |
| zarclsiin 34436 | In a Zariski topology, the... |
| zarclsint 34437 | The intersection of a fami... |
| zarclssn 34438 | The closed points of Zaris... |
| zarcls 34439 | The open sets of the Zaris... |
| zartopn 34440 | The Zariski topology is a ... |
| zartop 34441 | The Zariski topology is a ... |
| zartopon 34442 | The points of the Zariski ... |
| zar0ring 34443 | The Zariski Topology of th... |
| zart0 34444 | The Zariski topology is T_... |
| zarmxt1 34445 | The Zariski topology restr... |
| zarcmplem 34446 | Lemma for ~ zarcmp . (Con... |
| zarcmp 34447 | The Zariski topology is co... |
| rspectps 34448 | The spectrum of a ring ` R... |
| rhmpreimacnlem 34449 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 34450 | The function mapping a pri... |
| metidval 34455 | Value of the metric identi... |
| metidss 34456 | As a relation, the metric ... |
| metidv 34457 | ` A ` and ` B ` identify b... |
| metideq 34458 | Basic property of the metr... |
| metider 34459 | The metric identification ... |
| pstmval 34460 | Value of the metric induce... |
| pstmfval 34461 | Function value of the metr... |
| pstmxmet 34462 | The metric induced by a ps... |
| hauseqcn 34463 | In a Hausdorff topology, t... |
| elunitge0 34464 | An element of the closed u... |
| unitssxrge0 34465 | The closed unit interval i... |
| unitdivcld 34466 | Necessary conditions for a... |
| iistmd 34467 | The closed unit interval f... |
| unicls 34468 | The union of the closed se... |
| tpr2tp 34469 | The usual topology on ` ( ... |
| tpr2uni 34470 | The usual topology on ` ( ... |
| xpinpreima 34471 | Rewrite the cartesian prod... |
| xpinpreima2 34472 | Rewrite the cartesian prod... |
| sqsscirc1 34473 | The complex square of side... |
| sqsscirc2 34474 | The complex square of side... |
| cnre2csqlem 34475 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34476 | Image of a centered square... |
| tpr2rico 34477 | For any point of an open s... |
| cnvordtrestixx 34478 | The restriction of the 'gr... |
| prsdm 34479 | Domain of the relation of ... |
| prsrn 34480 | Range of the relation of a... |
| prsss 34481 | Relation of a subproset. ... |
| prsssdm 34482 | Domain of a subproset rela... |
| ordtprsval 34483 | Value of the order topolog... |
| ordtprsuni 34484 | Value of the order topolog... |
| ordtcnvNEW 34485 | The order dual generates t... |
| ordtrestNEW 34486 | The subspace topology of a... |
| ordtrest2NEWlem 34487 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34488 | An interval-closed set ` A... |
| ordtconnlem1 34489 | Connectedness in the order... |
| ordtconn 34490 | Connectedness in the order... |
| mndpluscn 34491 | A mapping that is both a h... |
| mhmhmeotmd 34492 | Deduce a Topological Monoi... |
| rmulccn 34493 | Multiplication by a real c... |
| raddcn 34494 | Addition in the real numbe... |
| xrmulc1cn 34495 | The operation multiplying ... |
| fmcncfil 34496 | The image of a Cauchy filt... |
| xrge0hmph 34497 | The extended nonnegative r... |
| xrge0iifcnv 34498 | Define a bijection from ` ... |
| xrge0iifcv 34499 | The defined function's val... |
| xrge0iifiso 34500 | The defined bijection from... |
| xrge0iifhmeo 34501 | Expose a homeomorphism fro... |
| xrge0iifhom 34502 | The defined function from ... |
| xrge0iif1 34503 | Condition for the defined ... |
| xrge0iifmhm 34504 | The defined function from ... |
| xrge0pluscn 34505 | The addition operation of ... |
| xrge0mulc1cn 34506 | The operation multiplying ... |
| xrge0tps 34507 | The extended nonnegative r... |
| xrge0topn 34508 | The topology of the extend... |
| xrge0haus 34509 | The topology of the extend... |
| xrge0tmd 34510 | The extended nonnegative r... |
| xrge0tmdALT 34511 | Alternate proof of ~ xrge0... |
| lmlim 34512 | Relate a limit in a given ... |
| lmlimxrge0 34513 | Relate a limit in the nonn... |
| rge0scvg 34514 | Implication of convergence... |
| fsumcvg4 34515 | A serie with finite suppor... |
| pnfneige0 34516 | A neighborhood of ` +oo ` ... |
| lmxrge0 34517 | Express "sequence ` F ` co... |
| lmdvg 34518 | If a monotonic sequence of... |
| lmdvglim 34519 | If a monotonic real number... |
| pl1cn 34520 | A univariate polynomial is... |
| zringnm 34523 | The norm (function) for a ... |
| zzsnm 34524 | The norm of the ring of th... |
| zlm0 34525 | Zero of a ` ZZ ` -module. ... |
| zlm1 34526 | Unity element of a ` ZZ ` ... |
| zlmds 34527 | Distance in a ` ZZ ` -modu... |
| zlmtset 34528 | Topology in a ` ZZ ` -modu... |
| zlmnm 34529 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34530 | The ` ZZ ` -module built f... |
| nmmulg 34531 | The norm of a group produc... |
| zrhnm 34532 | The norm of the image by `... |
| cnzh 34533 | The ` ZZ ` -module of ` CC... |
| rezh 34534 | The ` ZZ ` -module of ` RR... |
| qqhval 34537 | Value of the canonical hom... |
| zrhf1ker 34538 | The kernel of the homomorp... |
| zrhchr 34539 | The kernel of the homomorp... |
| zrhker 34540 | The kernel of the homomorp... |
| zrhunitpreima 34541 | The preimage by ` ZRHom ` ... |
| elzrhunit 34542 | Condition for the image by... |
| zrhneg 34543 | The canonical homomorphism... |
| zrhcntr 34544 | The canonical representati... |
| elzdif0 34545 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34546 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34547 | Value of the canonical hom... |
| qqhvval 34548 | Value of the canonical hom... |
| qqh0 34549 | The image of ` 0 ` by the ... |
| qqh1 34550 | The image of ` 1 ` by the ... |
| qqhf 34551 | ` QQHom ` as a function. ... |
| qqhvq 34552 | The image of a quotient by... |
| qqhghm 34553 | The ` QQHom ` homomorphism... |
| qqhrhm 34554 | The ` QQHom ` homomorphism... |
| qqhnm 34555 | The norm of the image by `... |
| qqhcn 34556 | The ` QQHom ` homomorphism... |
| qqhucn 34557 | The ` QQHom ` homomorphism... |
| rrhval 34561 | Value of the canonical hom... |
| rrhcn 34562 | If the topology of ` R ` i... |
| rrhf 34563 | If the topology of ` R ` i... |
| isrrext 34565 | Express the property " ` R... |
| rrextnrg 34566 | An extension of ` RR ` is ... |
| rrextdrg 34567 | An extension of ` RR ` is ... |
| rrextnlm 34568 | The norm of an extension o... |
| rrextchr 34569 | The ring characteristic of... |
| rrextcusp 34570 | An extension of ` RR ` is ... |
| rrexttps 34571 | An extension of ` RR ` is ... |
| rrexthaus 34572 | The topology of an extensi... |
| rrextust 34573 | The uniformity of an exten... |
| rerrext 34574 | The field of the real numb... |
| cnrrext 34575 | The field of the complex n... |
| qqtopn 34576 | The topology of the field ... |
| rrhfe 34577 | If ` R ` is an extension o... |
| rrhcne 34578 | If ` R ` is an extension o... |
| rrhqima 34579 | The ` RRHom ` homomorphism... |
| rrh0 34580 | The image of ` 0 ` by the ... |
| xrhval 34583 | The value of the embedding... |
| zrhre 34584 | The ` ZRHom ` homomorphism... |
| qqhre 34585 | The ` QQHom ` homomorphism... |
| rrhre 34586 | The ` RRHom ` homomorphism... |
| relmntop 34589 | Manifold is a relation. (... |
| ismntoplly 34590 | Property of being a manifo... |
| ismntop 34591 | Property of being a manifo... |
| esumex 34594 | An extended sum is a set b... |
| esumcl 34595 | Closure for extended sum i... |
| esumeq12dvaf 34596 | Equality deduction for ext... |
| esumeq12dva 34597 | Equality deduction for ext... |
| esumeq12d 34598 | Equality deduction for ext... |
| esumeq1 34599 | Equality theorem for an ex... |
| esumeq1d 34600 | Equality theorem for an ex... |
| esumeq2 34601 | Equality theorem for exten... |
| esumeq2d 34602 | Equality deduction for ext... |
| esumeq2dv 34603 | Equality deduction for ext... |
| esumeq2sdv 34604 | Equality deduction for ext... |
| nfesum1 34605 | Bound-variable hypothesis ... |
| nfesum2 34606 | Bound-variable hypothesis ... |
| cbvesum 34607 | Change bound variable in a... |
| cbvesumv 34608 | Change bound variable in a... |
| esumid 34609 | Identify the extended sum ... |
| esumgsum 34610 | A finite extended sum is t... |
| esumval 34611 | Develop the value of the e... |
| esumel 34612 | The extended sum is a limi... |
| esumnul 34613 | Extended sum over the empt... |
| esum0 34614 | Extended sum of zero. (Co... |
| esumf1o 34615 | Re-index an extended sum u... |
| esumc 34616 | Convert from the collectio... |
| esumrnmpt 34617 | Rewrite an extended sum in... |
| esumsplit 34618 | Split an extended sum into... |
| esummono 34619 | Extended sum is monotonic.... |
| esumpad 34620 | Extend an extended sum by ... |
| esumpad2 34621 | Remove zeroes from an exte... |
| esumadd 34622 | Addition of infinite sums.... |
| esumle 34623 | If all of the terms of an ... |
| gsumesum 34624 | Relate a group sum on ` ( ... |
| esumlub 34625 | The extended sum is the lo... |
| esumaddf 34626 | Addition of infinite sums.... |
| esumlef 34627 | If all of the terms of an ... |
| esumcst 34628 | The extended sum of a cons... |
| esumsnf 34629 | The extended sum of a sing... |
| esumsn 34630 | The extended sum of a sing... |
| esumpr 34631 | Extended sum over a pair. ... |
| esumpr2 34632 | Extended sum over a pair, ... |
| esumrnmpt2 34633 | Rewrite an extended sum in... |
| esumfzf 34634 | Formulating a partial exte... |
| esumfsup 34635 | Formulating an extended su... |
| esumfsupre 34636 | Formulating an extended su... |
| esumss 34637 | Change the index set to a ... |
| esumpinfval 34638 | The value of the extended ... |
| esumpfinvallem 34639 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34640 | The value of the extended ... |
| esumpfinvalf 34641 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34642 | The value of the extended ... |
| esumpcvgval 34643 | The value of the extended ... |
| esumpmono 34644 | The partial sums in an ext... |
| esumcocn 34645 | Lemma for ~ esummulc2 and ... |
| esummulc1 34646 | An extended sum multiplied... |
| esummulc2 34647 | An extended sum multiplied... |
| esumdivc 34648 | An extended sum divided by... |
| hashf2 34649 | Lemma for ~ hasheuni . (C... |
| hasheuni 34650 | The cardinality of a disjo... |
| esumcvg 34651 | The sequence of partial su... |
| esumcvg2 34652 | Simpler version of ~ esumc... |
| esumcvgsum 34653 | The value of the extended ... |
| esumsup 34654 | Express an extended sum as... |
| esumgect 34655 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34656 | All terms of a converging ... |
| esum2dlem 34657 | Lemma for ~ esum2d (finite... |
| esum2d 34658 | Write a double extended su... |
| esumiun 34659 | Sum over a nonnecessarily ... |
| ofceq 34662 | Equality theorem for funct... |
| ofcfval 34663 | Value of an operation appl... |
| ofcval 34664 | Evaluate a function/consta... |
| ofcfn 34665 | The function operation pro... |
| ofcfeqd2 34666 | Equality theorem for funct... |
| ofcfval3 34667 | General value of ` ( F oFC... |
| ofcf 34668 | The function/constant oper... |
| ofcfval2 34669 | The function operation exp... |
| ofcfval4 34670 | The function/constant oper... |
| ofcc 34671 | Left operation by a consta... |
| ofcof 34672 | Relate function operation ... |
| sigaex 34675 | Lemma for ~ issiga and ~ i... |
| sigaval 34676 | The set of sigma-algebra w... |
| issiga 34677 | An alternative definition ... |
| isrnsiga 34678 | The property of being a si... |
| 0elsiga 34679 | A sigma-algebra contains t... |
| baselsiga 34680 | A sigma-algebra contains i... |
| sigasspw 34681 | A sigma-algebra is a set o... |
| sigaclcu 34682 | A sigma-algebra is closed ... |
| sigaclcuni 34683 | A sigma-algebra is closed ... |
| sigaclfu 34684 | A sigma-algebra is closed ... |
| sigaclcu2 34685 | A sigma-algebra is closed ... |
| sigaclfu2 34686 | A sigma-algebra is closed ... |
| sigaclcu3 34687 | A sigma-algebra is closed ... |
| issgon 34688 | Property of being a sigma-... |
| sgon 34689 | A sigma-algebra is a sigma... |
| elsigass 34690 | An element of a sigma-alge... |
| elrnsiga 34691 | Dropping the base informat... |
| isrnsigau 34692 | The property of being a si... |
| unielsiga 34693 | A sigma-algebra contains i... |
| dmvlsiga 34694 | Lebesgue-measurable subset... |
| pwsiga 34695 | Any power set forms a sigm... |
| prsiga 34696 | The smallest possible sigm... |
| sigaclci 34697 | A sigma-algebra is closed ... |
| difunielsiga 34698 | A sigma-algebra is closed ... |
| unelsiga 34699 | A sigma-algebra is closed ... |
| difelsiga 34700 | A sigma-algebra is closed ... |
| inelsiga 34701 | A sigma-algebra is closed ... |
| sigainb 34702 | Building a sigma-algebra f... |
| insiga 34703 | The intersection of a coll... |
| sigagenval 34706 | Value of the generated sig... |
| sigagensiga 34707 | A generated sigma-algebra ... |
| sgsiga 34708 | A generated sigma-algebra ... |
| unisg 34709 | The sigma-algebra generate... |
| dmsigagen 34710 | A sigma-algebra can be gen... |
| sssigagen 34711 | A set is a subset of the s... |
| sssigagen2 34712 | A subset of the generating... |
| elsigagen 34713 | Any element of a set is al... |
| elsigagen2 34714 | Any countable union of ele... |
| sigagenss 34715 | The generated sigma-algebr... |
| sigagenss2 34716 | Sufficient condition for i... |
| sigagenid 34717 | The sigma-algebra generate... |
| ispisys 34718 | The property of being a pi... |
| ispisys2 34719 | The property of being a pi... |
| inelpisys 34720 | Pi-systems are closed unde... |
| sigapisys 34721 | All sigma-algebras are pi-... |
| isldsys 34722 | The property of being a la... |
| pwldsys 34723 | The power set of the unive... |
| unelldsys 34724 | Lambda-systems are closed ... |
| sigaldsys 34725 | All sigma-algebras are lam... |
| ldsysgenld 34726 | The intersection of all la... |
| sigapildsyslem 34727 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34728 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34729 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34730 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34731 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34732 | The lambda system ` E ` ge... |
| dynkin 34733 | Dynkin's lambda-pi theorem... |
| isros 34734 | The property of being a ri... |
| rossspw 34735 | A ring of sets is a collec... |
| 0elros 34736 | A ring of sets contains th... |
| unelros 34737 | A ring of sets is closed u... |
| difelros 34738 | A ring of sets is closed u... |
| inelros 34739 | A ring of sets is closed u... |
| fiunelros 34740 | A ring of sets is closed u... |
| issros 34741 | The property of being a se... |
| srossspw 34742 | A semiring of sets is a co... |
| 0elsros 34743 | A semiring of sets contain... |
| inelsros 34744 | A semiring of sets is clos... |
| diffiunisros 34745 | In semiring of sets, compl... |
| rossros 34746 | Rings of sets are semiring... |
| brsiga 34749 | The Borel Algebra on real ... |
| brsigarn 34750 | The Borel Algebra is a sig... |
| brsigasspwrn 34751 | The Borel Algebra is a set... |
| unibrsiga 34752 | The union of the Borel Alg... |
| cldssbrsiga 34753 | A Borel Algebra contains a... |
| sxval 34756 | Value of the product sigma... |
| sxsiga 34757 | A product sigma-algebra is... |
| sxsigon 34758 | A product sigma-algebra is... |
| sxuni 34759 | The base set of a product ... |
| elsx 34760 | The cartesian product of t... |
| measbase 34763 | The base set of a measure ... |
| measval 34764 | The value of the ` measure... |
| ismeas 34765 | The property of being a me... |
| isrnmeas 34766 | The property of being a me... |
| dmmeas 34767 | The domain of a measure is... |
| measbasedom 34768 | The base set of a measure ... |
| measfrge0 34769 | A measure is a function ov... |
| measfn 34770 | A measure is a function on... |
| measvxrge0 34771 | The values of a measure ar... |
| measvnul 34772 | The measure of the empty s... |
| measge0 34773 | A measure is nonnegative. ... |
| measle0 34774 | If the measure of a given ... |
| measvun 34775 | The measure of a countable... |
| measxun2 34776 | The measure the union of t... |
| measun 34777 | The measure the union of t... |
| measvunilem 34778 | Lemma for ~ measvuni . (C... |
| measvunilem0 34779 | Lemma for ~ measvuni . (C... |
| measvuni 34780 | The measure of a countable... |
| measssd 34781 | A measure is monotone with... |
| measunl 34782 | A measure is sub-additive ... |
| measiuns 34783 | The measure of the union o... |
| measiun 34784 | A measure is sub-additive.... |
| meascnbl 34785 | A measure is continuous fr... |
| measinblem 34786 | Lemma for ~ measinb . (Co... |
| measinb 34787 | Building a measure restric... |
| measres 34788 | Building a measure restric... |
| measinb2 34789 | Building a measure restric... |
| measdivcst 34790 | Division of a measure by a... |
| measdivcstALTV 34791 | Alternate version of ~ mea... |
| cntmeas 34792 | The Counting measure is a ... |
| pwcntmeas 34793 | The counting measure is a ... |
| cntnevol 34794 | Counting and Lebesgue meas... |
| voliune 34795 | The Lebesgue measure funct... |
| volfiniune 34796 | The Lebesgue measure funct... |
| volmeas 34797 | The Lebesgue measure is a ... |
| ddeval1 34800 | Value of the delta measure... |
| ddeval0 34801 | Value of the delta measure... |
| ddemeas 34802 | The Dirac delta measure is... |
| relae 34806 | 'almost everywhere' is a r... |
| brae 34807 | 'almost everywhere' relati... |
| braew 34808 | 'almost everywhere' relati... |
| truae 34809 | A truth holds almost every... |
| aean 34810 | A conjunction holds almost... |
| faeval 34812 | Value of the 'almost every... |
| relfae 34813 | The 'almost everywhere' bu... |
| brfae 34814 | 'almost everywhere' relati... |
| ismbfm 34817 | The predicate " ` F ` is a... |
| elunirnmbfm 34818 | The property of being a me... |
| mbfmfun 34819 | A measurable function is a... |
| mbfmf 34820 | A measurable function as a... |
| mbfmcnvima 34821 | The preimage by a measurab... |
| isanmbfm 34822 | The predicate to be a meas... |
| mbfmbfmOLD 34823 | A measurable function to a... |
| mbfmbfm 34824 | A measurable function to a... |
| mbfmcst 34825 | A constant function is mea... |
| 1stmbfm 34826 | The first projection map i... |
| 2ndmbfm 34827 | The second projection map ... |
| imambfm 34828 | If the sigma-algebra in th... |
| cnmbfm 34829 | A continuous function is m... |
| mbfmco 34830 | The composition of two mea... |
| mbfmco2 34831 | The pair building of two m... |
| mbfmvolf 34832 | Measurable functions with ... |
| elmbfmvol2 34833 | Measurable functions with ... |
| mbfmcnt 34834 | All functions are measurab... |
| br2base 34835 | The base set for the gener... |
| dya2ub 34836 | An upper bound for a dyadi... |
| sxbrsigalem0 34837 | The closed half-spaces of ... |
| sxbrsigalem3 34838 | The sigma-algebra generate... |
| dya2iocival 34839 | The function ` I ` returns... |
| dya2iocress 34840 | Dyadic intervals are subse... |
| dya2iocbrsiga 34841 | Dyadic intervals are Borel... |
| dya2icobrsiga 34842 | Dyadic intervals are Borel... |
| dya2icoseg 34843 | For any point and any clos... |
| dya2icoseg2 34844 | For any point and any open... |
| dya2iocrfn 34845 | The function returning dya... |
| dya2iocct 34846 | The dyadic rectangle set i... |
| dya2iocnrect 34847 | For any point of an open r... |
| dya2iocnei 34848 | For any point of an open s... |
| dya2iocuni 34849 | Every open set of ` ( RR X... |
| dya2iocucvr 34850 | The dyadic rectangular set... |
| sxbrsigalem1 34851 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34852 | The sigma-algebra generate... |
| sxbrsigalem4 34853 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34854 | First direction for ~ sxbr... |
| sxbrsigalem6 34855 | First direction for ~ sxbr... |
| sxbrsiga 34856 | The product sigma-algebra ... |
| omsval 34859 | Value of the function mapp... |
| omsfval 34860 | Value of the outer measure... |
| omscl 34861 | A closure lemma for the co... |
| omsf 34862 | A constructed outer measur... |
| oms0 34863 | A constructed outer measur... |
| omsmon 34864 | A constructed outer measur... |
| omssubaddlem 34865 | For any small margin ` E `... |
| omssubadd 34866 | A constructed outer measur... |
| carsgval 34869 | Value of the Caratheodory ... |
| carsgcl 34870 | Closure of the Caratheodor... |
| elcarsg 34871 | Property of being a Carath... |
| baselcarsg 34872 | The universe set, ` O ` , ... |
| 0elcarsg 34873 | The empty set is Caratheod... |
| carsguni 34874 | The union of all Caratheod... |
| elcarsgss 34875 | Caratheodory measurable se... |
| difelcarsg 34876 | The Caratheodory measurabl... |
| inelcarsg 34877 | The Caratheodory measurabl... |
| unelcarsg 34878 | The Caratheodory-measurabl... |
| difelcarsg2 34879 | The Caratheodory-measurabl... |
| carsgmon 34880 | Utility lemma: Apply mono... |
| carsgsigalem 34881 | Lemma for the following th... |
| fiunelcarsg 34882 | The Caratheodory measurabl... |
| carsgclctunlem1 34883 | Lemma for ~ carsgclctun . ... |
| carsggect 34884 | The outer measure is count... |
| carsgclctunlem2 34885 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34886 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34887 | The Caratheodory measurabl... |
| carsgsiga 34888 | The Caratheodory measurabl... |
| omsmeas 34889 | The restriction of a const... |
| pmeasmono 34890 | This theorem's hypotheses ... |
| pmeasadd 34891 | A premeasure on a ring of ... |
| itgeq12dv 34892 | Equality theorem for an in... |
| sitgval 34898 | Value of the simple functi... |
| issibf 34899 | The predicate " ` F ` is a... |
| sibf0 34900 | The constant zero function... |
| sibfmbl 34901 | A simple function is measu... |
| sibff 34902 | A simple function is a fun... |
| sibfrn 34903 | A simple function has fini... |
| sibfima 34904 | Any preimage of a singleto... |
| sibfinima 34905 | The measure of the interse... |
| sibfof 34906 | Applying function operatio... |
| sitgfval 34907 | Value of the Bochner integ... |
| sitgclg 34908 | Closure of the Bochner int... |
| sitgclbn 34909 | Closure of the Bochner int... |
| sitgclcn 34910 | Closure of the Bochner int... |
| sitgclre 34911 | Closure of the Bochner int... |
| sitg0 34912 | The integral of the consta... |
| sitgf 34913 | The integral for simple fu... |
| sitgaddlemb 34914 | Lemma for * sitgadd . (Co... |
| sitmval 34915 | Value of the simple functi... |
| sitmfval 34916 | Value of the integral dist... |
| sitmcl 34917 | Closure of the integral di... |
| sitmf 34918 | The integral metric as a f... |
| oddpwdc 34920 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34921 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34922 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34923 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34924 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34925 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34926 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34927 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34928 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34929 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34930 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34931 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34932 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34933 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34934 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34935 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34936 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34937 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34938 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34939 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34940 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34941 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34942 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34943 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34944 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34945 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34946 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34947 | Lemma for ~ eulerpart . (... |
| eulerpart 34948 | Euler's theorem on partiti... |
| subiwrd 34951 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34952 | Length of a subword of an ... |
| iwrdsplit 34953 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34954 | Value of the strong sequen... |
| sseqfv1 34955 | Value of the strong sequen... |
| sseqfn 34956 | A strong recursive sequenc... |
| sseqmw 34957 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34958 | A strong recursive sequenc... |
| sseqfres 34959 | The first elements in the ... |
| sseqfv2 34960 | Value of the strong sequen... |
| sseqp1 34961 | Value of the strong sequen... |
| fiblem 34964 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34965 | Value of the Fibonacci seq... |
| fib1 34966 | Value of the Fibonacci seq... |
| fibp1 34967 | Value of the Fibonacci seq... |
| fib2 34968 | Value of the Fibonacci seq... |
| fib3 34969 | Value of the Fibonacci seq... |
| fib4 34970 | Value of the Fibonacci seq... |
| fib5 34971 | Value of the Fibonacci seq... |
| fib6 34972 | Value of the Fibonacci seq... |
| elprob 34975 | The property of being a pr... |
| domprobmeas 34976 | A probability measure is a... |
| domprobsiga 34977 | The domain of a probabilit... |
| probtot 34978 | The probability of the uni... |
| prob01 34979 | A probability is an elemen... |
| probnul 34980 | The probability of the emp... |
| unveldomd 34981 | The universe is an element... |
| unveldom 34982 | The universe is an element... |
| nuleldmp 34983 | The empty set is an elemen... |
| probcun 34984 | The probability of the uni... |
| probun 34985 | The probability of the uni... |
| probdif 34986 | The probability of the dif... |
| probinc 34987 | A probability law is incre... |
| probdsb 34988 | The probability of the com... |
| probmeasd 34989 | A probability measure is a... |
| probvalrnd 34990 | The value of a probability... |
| probtotrnd 34991 | The probability of the uni... |
| totprobd 34992 | Law of total probability, ... |
| totprob 34993 | Law of total probability. ... |
| probfinmeasb 34994 | Build a probability measur... |
| probfinmeasbALTV 34995 | Alternate version of ~ pro... |
| probmeasb 34996 | Build a probability from a... |
| cndprobval 34999 | The value of the condition... |
| cndprobin 35000 | An identity linking condit... |
| cndprob01 35001 | The conditional probabilit... |
| cndprobtot 35002 | The conditional probabilit... |
| cndprobnul 35003 | The conditional probabilit... |
| cndprobprob 35004 | The conditional probabilit... |
| bayesth 35005 | Bayes Theorem. (Contribut... |
| rrvmbfm 35008 | A real-valued random varia... |
| isrrvv 35009 | Elementhood to the set of ... |
| rrvvf 35010 | A real-valued random varia... |
| rrvfn 35011 | A real-valued random varia... |
| rrvdm 35012 | The domain of a random var... |
| rrvrnss 35013 | The range of a random vari... |
| rrvf2 35014 | A real-valued random varia... |
| rrvdmss 35015 | The domain of a random var... |
| rrvfinvima 35016 | For a real-value random va... |
| 0rrv 35017 | The constant function equa... |
| rrvadd 35018 | The sum of two random vari... |
| rrvmulc 35019 | A random variable multipli... |
| rrvsum 35020 | An indexed sum of random v... |
| boolesineq 35021 | Boole's inequality (union ... |
| orvcval 35024 | Value of the preimage mapp... |
| orvcval2 35025 | Another way to express the... |
| elorvc 35026 | Elementhood of a preimage.... |
| orvcval4 35027 | The value of the preimage ... |
| orvcoel 35028 | If the relation produces o... |
| orvccel 35029 | If the relation produces c... |
| elorrvc 35030 | Elementhood of a preimage ... |
| orrvcval4 35031 | The value of the preimage ... |
| orrvcoel 35032 | If the relation produces o... |
| orrvccel 35033 | If the relation produces c... |
| orvcgteel 35034 | Preimage maps produced by ... |
| orvcelval 35035 | Preimage maps produced by ... |
| orvcelel 35036 | Preimage maps produced by ... |
| dstrvval 35037 | The value of the distribut... |
| dstrvprob 35038 | The distribution of a rand... |
| orvclteel 35039 | Preimage maps produced by ... |
| dstfrvel 35040 | Elementhood of preimage ma... |
| dstfrvunirn 35041 | The limit of all preimage ... |
| orvclteinc 35042 | Preimage maps produced by ... |
| dstfrvinc 35043 | A cumulative distribution ... |
| dstfrvclim1 35044 | The limit of the cumulativ... |
| coinfliplem 35045 | Division in the extended r... |
| coinflipprob 35046 | The ` P ` we defined for c... |
| coinflipspace 35047 | The space of our coin-flip... |
| coinflipuniv 35048 | The universe of our coin-f... |
| coinfliprv 35049 | The ` X ` we defined for c... |
| coinflippv 35050 | The probability of heads i... |
| coinflippvt 35051 | The probability of tails i... |
| ballotlemoex 35052 | ` O ` is a set. (Contribu... |
| ballotlem1 35053 | The size of the universe i... |
| ballotlemelo 35054 | Elementhood in ` O ` . (C... |
| ballotlem2 35055 | The probability that the f... |
| ballotlemfval 35056 | The value of ` F ` . (Con... |
| ballotlemfelz 35057 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 35058 | If the ` J ` th ballot is ... |
| ballotlemfc0 35059 | ` F ` takes value 0 betwee... |
| ballotlemfcc 35060 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 35061 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 35062 | ` ( F `` C ) ` always star... |
| ballotleme 35063 | Elements of ` E ` . (Cont... |
| ballotlemodife 35064 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 35065 | If the first pick is a vot... |
| ballotlem5 35066 | If A is not ahead througho... |
| ballotlemi 35067 | Value of ` I ` for a given... |
| ballotlemiex 35068 | Properties of ` ( I `` C )... |
| ballotlemi1 35069 | The first tie cannot be re... |
| ballotlemii 35070 | The first tie cannot be re... |
| ballotlemsup 35071 | The set of zeroes of ` F `... |
| ballotlemimin 35072 | ` ( I `` C ) ` is the firs... |
| ballotlemic 35073 | If the first vote is for B... |
| ballotlem1c 35074 | If the first vote is for A... |
| ballotlemsval 35075 | Value of ` S ` . (Contrib... |
| ballotlemsv 35076 | Value of ` S ` evaluated a... |
| ballotlemsgt1 35077 | ` S ` maps values less tha... |
| ballotlemsdom 35078 | Domain of ` S ` for a give... |
| ballotlemsel1i 35079 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 35080 | The defined ` S ` is a bij... |
| ballotlemsi 35081 | The image by ` S ` of the ... |
| ballotlemsima 35082 | The image by ` S ` of an i... |
| ballotlemieq 35083 | If two countings share the... |
| ballotlemrval 35084 | Value of ` R ` . (Contrib... |
| ballotlemscr 35085 | The image of ` ( R `` C ) ... |
| ballotlemrv 35086 | Value of ` R ` evaluated a... |
| ballotlemrv1 35087 | Value of ` R ` before the ... |
| ballotlemrv2 35088 | Value of ` R ` after the t... |
| ballotlemro 35089 | Range of ` R ` is included... |
| ballotlemgval 35090 | Expand the value of ` .^ `... |
| ballotlemgun 35091 | A property of the defined ... |
| ballotlemfg 35092 | Express the value of ` ( F... |
| ballotlemfrc 35093 | Express the value of ` ( F... |
| ballotlemfrci 35094 | Reverse counting preserves... |
| ballotlemfrceq 35095 | Value of ` F ` for a rever... |
| ballotlemfrcn0 35096 | Value of ` F ` for a rever... |
| ballotlemrc 35097 | Range of ` R ` . (Contrib... |
| ballotlemirc 35098 | Applying ` R ` does not ch... |
| ballotlemrinv0 35099 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 35100 | ` R ` is its own inverse :... |
| ballotlem1ri 35101 | When the vote on the first... |
| ballotlem7 35102 | ` R ` is a bijection betwe... |
| ballotlem8 35103 | There are as many counting... |
| ballotth 35104 | Bertrand's ballot problem ... |
| fzssfzo 35105 | Condition for an integer i... |
| gsumncl 35106 | Closure of a group sum in ... |
| gsumnunsn 35107 | Closure of a group sum in ... |
| ccatmulgnn0dir 35108 | Concatenation of words fol... |
| ofcccat 35109 | Letterwise operations on w... |
| ofcs1 35110 | Letterwise operations on a... |
| ofcs2 35111 | Letterwise operations on a... |
| plyrecld 35112 | Closure of a polynomial wi... |
| signsplypnf 35113 | The quotient of a polynomi... |
| signsply0 35114 | Lemma for the rule of sign... |
| signspval 35115 | The value of the skipping ... |
| signsw0glem 35116 | Neutral element property o... |
| signswbase 35117 | The base of ` W ` is the u... |
| signswplusg 35118 | The operation of ` W ` . ... |
| signsw0g 35119 | The neutral element of ` W... |
| signswmnd 35120 | ` W ` is a monoid structur... |
| signswrid 35121 | The zero-skipping operatio... |
| signswlid 35122 | The zero-skipping operatio... |
| signswn0 35123 | The zero-skipping operatio... |
| signswch 35124 | The zero-skipping operatio... |
| signslema 35125 | Computational part of ~~? ... |
| signstfv 35126 | Value of the zero-skipping... |
| signstfval 35127 | Value of the zero-skipping... |
| signstcl 35128 | Closure of the zero skippi... |
| signstf 35129 | The zero skipping sign wor... |
| signstlen 35130 | Length of the zero skippin... |
| signstf0 35131 | Sign of a single letter wo... |
| signstfvn 35132 | Zero-skipping sign in a wo... |
| signsvtn0 35133 | If the last letter is nonz... |
| signstfvp 35134 | Zero-skipping sign in a wo... |
| signstfvneq0 35135 | In case the first letter i... |
| signstfvcl 35136 | Closure of the zero skippi... |
| signstfvc 35137 | Zero-skipping sign in a wo... |
| signstres 35138 | Restriction of a zero skip... |
| signstfveq0a 35139 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 35140 | In case the last letter is... |
| signsvvfval 35141 | The value of ` V ` , which... |
| signsvvf 35142 | ` V ` is a function. (Con... |
| signsvf0 35143 | There is no change of sign... |
| signsvf1 35144 | In a single-letter word, w... |
| signsvfn 35145 | Number of changes in a wor... |
| signsvtp 35146 | Adding a letter of the sam... |
| signsvtn 35147 | Adding a letter of a diffe... |
| signsvfpn 35148 | Adding a letter of the sam... |
| signsvfnn 35149 | Adding a letter of a diffe... |
| signlem0 35150 | Adding a zero as the highe... |
| signshf 35151 | ` H ` , corresponding to t... |
| signshwrd 35152 | ` H ` , corresponding to t... |
| signshlen 35153 | Length of ` H ` , correspo... |
| signshnz 35154 | ` H ` is not the empty wor... |
| iblidicc 35155 | The identity function is i... |
| rpsqrtcn 35156 | Continuity of the real pos... |
| divsqrtid 35157 | A real number divided by i... |
| cxpcncf1 35158 | The power function on comp... |
| efmul2picn 35159 | Multiplying by ` ( _i x. (... |
| fct2relem 35160 | Lemma for ~ ftc2re . (Con... |
| ftc2re 35161 | The Fundamental Theorem of... |
| fdvposlt 35162 | Functions with a positive ... |
| fdvneggt 35163 | Functions with a negative ... |
| fdvposle 35164 | Functions with a nonnegati... |
| fdvnegge 35165 | Functions with a nonpositi... |
| prodfzo03 35166 | A product of three factors... |
| actfunsnf1o 35167 | The action ` F ` of extend... |
| actfunsnrndisj 35168 | The action ` F ` of extend... |
| itgexpif 35169 | The basis for the circle m... |
| fsum2dsub 35170 | Lemma for ~ breprexp - Re-... |
| reprval 35173 | Value of the representatio... |
| repr0 35174 | There is exactly one repre... |
| reprf 35175 | Members of the representat... |
| reprsum 35176 | Sums of values of the memb... |
| reprle 35177 | Upper bound to the terms i... |
| reprsuc 35178 | Express the representation... |
| reprfi 35179 | Bounded representations ar... |
| reprss 35180 | Representations with terms... |
| reprinrn 35181 | Representations with term ... |
| reprlt 35182 | There are no representatio... |
| hashreprin 35183 | Express a sum of represent... |
| reprgt 35184 | There are no representatio... |
| reprinfz1 35185 | For the representation of ... |
| reprfi2 35186 | Corollary of ~ reprinfz1 .... |
| reprfz1 35187 | Corollary of ~ reprinfz1 .... |
| hashrepr 35188 | Develop the number of repr... |
| reprpmtf1o 35189 | Transposing ` 0 ` and ` X ... |
| reprdifc 35190 | Express the representation... |
| chpvalz 35191 | Value of the second Chebys... |
| chtvalz 35192 | Value of the Chebyshev fun... |
| breprexplema 35193 | Lemma for ~ breprexp (indu... |
| breprexplemb 35194 | Lemma for ~ breprexp (clos... |
| breprexplemc 35195 | Lemma for ~ breprexp (indu... |
| breprexp 35196 | Express the ` S ` th power... |
| breprexpnat 35197 | Express the ` S ` th power... |
| vtsval 35200 | Value of the Vinogradov tr... |
| vtscl 35201 | Closure of the Vinogradov ... |
| vtsprod 35202 | Express the Vinogradov tri... |
| circlemeth 35203 | The Hardy, Littlewood and ... |
| circlemethnat 35204 | The Hardy, Littlewood and ... |
| circlevma 35205 | The Circle Method, where t... |
| circlemethhgt 35206 | The circle method, where t... |
| hgt750lemc 35210 | An upper bound to the summ... |
| hgt750lemd 35211 | An upper bound to the summ... |
| hgt749d 35212 | A deduction version of ~ a... |
| logdivsqrle 35213 | Conditions for ` ( ( log `... |
| hgt750lem 35214 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 35215 | Decimal multiplication gal... |
| hgt750lemf 35216 | Lemma for the statement 7.... |
| hgt750lemg 35217 | Lemma for the statement 7.... |
| oddprm2 35218 | Two ways to write the set ... |
| hgt750lemb 35219 | An upper bound on the cont... |
| hgt750lema 35220 | An upper bound on the cont... |
| hgt750leme 35221 | An upper bound on the cont... |
| tgoldbachgnn 35222 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 35223 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 35224 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 35225 | Odd integers greater than ... |
| tgoldbachgt 35226 | Odd integers greater than ... |
| istrkg2d 35229 | Property of fulfilling dim... |
| axtglowdim2ALTV 35230 | Alternate version of ~ axt... |
| axtgupdim2ALTV 35231 | Alternate version of ~ axt... |
| cgranbtwn 35232 | Null angle implies between... |
| btwnlng13 35233 | If ` Z ` is between ` X ` ... |
| morleylemrneab 35234 | Lemma for morley . (Contr... |
| afsval 35237 | Value of the AFS relation ... |
| brafs 35238 | Binary relation form of th... |
| tg5segofs 35239 | Rephrase ~ axtg5seg using ... |
| lpadval 35242 | Value of the ` leftpad ` f... |
| lpadlem1 35243 | Lemma for the ` leftpad ` ... |
| lpadlem3 35244 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 35245 | Length of a left-padded wo... |
| lpadlem2 35246 | Lemma for the ` leftpad ` ... |
| lpadlen2 35247 | Length of a left-padded wo... |
| lpadmax 35248 | Length of a left-padded wo... |
| lpadleft 35249 | The contents of prefix of ... |
| lpadright 35250 | The suffix of a left-padde... |
| bnj170 35263 | ` /\ ` -manipulation. (Co... |
| bnj240 35264 | ` /\ ` -manipulation. (Co... |
| bnj248 35265 | ` /\ ` -manipulation. (Co... |
| bnj250 35266 | ` /\ ` -manipulation. (Co... |
| bnj251 35267 | ` /\ ` -manipulation. (Co... |
| bnj252 35268 | ` /\ ` -manipulation. (Co... |
| bnj253 35269 | ` /\ ` -manipulation. (Co... |
| bnj255 35270 | ` /\ ` -manipulation. (Co... |
| bnj256 35271 | ` /\ ` -manipulation. (Co... |
| bnj257 35272 | ` /\ ` -manipulation. (Co... |
| bnj258 35273 | ` /\ ` -manipulation. (Co... |
| bnj268 35274 | ` /\ ` -manipulation. (Co... |
| bnj290 35275 | ` /\ ` -manipulation. (Co... |
| bnj291 35276 | ` /\ ` -manipulation. (Co... |
| bnj312 35277 | ` /\ ` -manipulation. (Co... |
| bnj334 35278 | ` /\ ` -manipulation. (Co... |
| bnj345 35279 | ` /\ ` -manipulation. (Co... |
| bnj422 35280 | ` /\ ` -manipulation. (Co... |
| bnj432 35281 | ` /\ ` -manipulation. (Co... |
| bnj446 35282 | ` /\ ` -manipulation. (Co... |
| bnj23 35283 | First-order logic and set ... |
| bnj31 35284 | First-order logic and set ... |
| bnj62 35285 | First-order logic and set ... |
| bnj89 35286 | First-order logic and set ... |
| bnj90 35287 | First-order logic and set ... |
| bnj101 35288 | First-order logic and set ... |
| bnj105 35289 | First-order logic and set ... |
| bnj115 35290 | First-order logic and set ... |
| bnj132 35291 | First-order logic and set ... |
| bnj133 35292 | First-order logic and set ... |
| bnj156 35293 | First-order logic and set ... |
| bnj158 35294 | First-order logic and set ... |
| bnj168 35295 | First-order logic and set ... |
| bnj206 35296 | First-order logic and set ... |
| bnj216 35297 | First-order logic and set ... |
| bnj219 35298 | First-order logic and set ... |
| bnj226 35299 | First-order logic and set ... |
| bnj228 35300 | First-order logic and set ... |
| bnj519 35301 | First-order logic and set ... |
| bnj524 35302 | First-order logic and set ... |
| bnj525 35303 | First-order logic and set ... |
| bnj534 35304 | First-order logic and set ... |
| bnj538 35305 | First-order logic and set ... |
| bnj529 35306 | First-order logic and set ... |
| bnj551 35307 | First-order logic and set ... |
| bnj563 35308 | First-order logic and set ... |
| bnj564 35309 | First-order logic and set ... |
| bnj593 35310 | First-order logic and set ... |
| bnj596 35311 | First-order logic and set ... |
| bnj610 35312 | Pass from equality ( ` x =... |
| bnj642 35313 | ` /\ ` -manipulation. (Co... |
| bnj643 35314 | ` /\ ` -manipulation. (Co... |
| bnj645 35315 | ` /\ ` -manipulation. (Co... |
| bnj658 35316 | ` /\ ` -manipulation. (Co... |
| bnj667 35317 | ` /\ ` -manipulation. (Co... |
| bnj705 35318 | ` /\ ` -manipulation. (Co... |
| bnj706 35319 | ` /\ ` -manipulation. (Co... |
| bnj707 35320 | ` /\ ` -manipulation. (Co... |
| bnj708 35321 | ` /\ ` -manipulation. (Co... |
| bnj721 35322 | ` /\ ` -manipulation. (Co... |
| bnj832 35323 | ` /\ ` -manipulation. (Co... |
| bnj835 35324 | ` /\ ` -manipulation. (Co... |
| bnj836 35325 | ` /\ ` -manipulation. (Co... |
| bnj837 35326 | ` /\ ` -manipulation. (Co... |
| bnj769 35327 | ` /\ ` -manipulation. (Co... |
| bnj770 35328 | ` /\ ` -manipulation. (Co... |
| bnj771 35329 | ` /\ ` -manipulation. (Co... |
| bnj887 35330 | ` /\ ` -manipulation. (Co... |
| bnj918 35331 | First-order logic and set ... |
| bnj919 35332 | First-order logic and set ... |
| bnj923 35333 | First-order logic and set ... |
| bnj927 35334 | First-order logic and set ... |
| bnj931 35335 | First-order logic and set ... |
| bnj937 35336 | First-order logic and set ... |
| bnj941 35337 | First-order logic and set ... |
| bnj945 35338 | Technical lemma for ~ bnj6... |
| bnj946 35339 | First-order logic and set ... |
| bnj951 35340 | ` /\ ` -manipulation. (Co... |
| bnj956 35341 | First-order logic and set ... |
| bnj976 35342 | First-order logic and set ... |
| bnj982 35343 | First-order logic and set ... |
| bnj1019 35344 | First-order logic and set ... |
| bnj1023 35345 | First-order logic and set ... |
| bnj1095 35346 | First-order logic and set ... |
| bnj1096 35347 | First-order logic and set ... |
| bnj1098 35348 | First-order logic and set ... |
| bnj1101 35349 | First-order logic and set ... |
| bnj1113 35350 | First-order logic and set ... |
| bnj1109 35351 | First-order logic and set ... |
| bnj1131 35352 | First-order logic and set ... |
| bnj1138 35353 | First-order logic and set ... |
| bnj1143 35354 | First-order logic and set ... |
| bnj1146 35355 | First-order logic and set ... |
| bnj1149 35356 | First-order logic and set ... |
| bnj1185 35357 | First-order logic and set ... |
| bnj1196 35358 | First-order logic and set ... |
| bnj1198 35359 | First-order logic and set ... |
| bnj1209 35360 | First-order logic and set ... |
| bnj1211 35361 | First-order logic and set ... |
| bnj1213 35362 | First-order logic and set ... |
| bnj1212 35363 | First-order logic and set ... |
| bnj1219 35364 | First-order logic and set ... |
| bnj1224 35365 | First-order logic and set ... |
| bnj1230 35366 | First-order logic and set ... |
| bnj1232 35367 | First-order logic and set ... |
| bnj1235 35368 | First-order logic and set ... |
| bnj1239 35369 | First-order logic and set ... |
| bnj1238 35370 | First-order logic and set ... |
| bnj1241 35371 | First-order logic and set ... |
| bnj1247 35372 | First-order logic and set ... |
| bnj1254 35373 | First-order logic and set ... |
| bnj1262 35374 | First-order logic and set ... |
| bnj1266 35375 | First-order logic and set ... |
| bnj1265 35376 | First-order logic and set ... |
| bnj1275 35377 | First-order logic and set ... |
| bnj1276 35378 | First-order logic and set ... |
| bnj1292 35379 | First-order logic and set ... |
| bnj1293 35380 | First-order logic and set ... |
| bnj1294 35381 | First-order logic and set ... |
| bnj1299 35382 | First-order logic and set ... |
| bnj1304 35383 | First-order logic and set ... |
| bnj1316 35384 | First-order logic and set ... |
| bnj1317 35385 | First-order logic and set ... |
| bnj1322 35386 | First-order logic and set ... |
| bnj1340 35387 | First-order logic and set ... |
| bnj1345 35388 | First-order logic and set ... |
| bnj1350 35389 | First-order logic and set ... |
| bnj1351 35390 | First-order logic and set ... |
| bnj1352 35391 | First-order logic and set ... |
| bnj1361 35392 | First-order logic and set ... |
| bnj1366 35393 | First-order logic and set ... |
| bnj1379 35394 | First-order logic and set ... |
| bnj1383 35395 | First-order logic and set ... |
| bnj1385 35396 | First-order logic and set ... |
| bnj1386 35397 | First-order logic and set ... |
| bnj1397 35398 | First-order logic and set ... |
| bnj1400 35399 | First-order logic and set ... |
| bnj1405 35400 | First-order logic and set ... |
| bnj1422 35401 | First-order logic and set ... |
| bnj1424 35402 | First-order logic and set ... |
| bnj1436 35403 | First-order logic and set ... |
| bnj1441 35404 | First-order logic and set ... |
| bnj1441g 35405 | First-order logic and set ... |
| bnj1454 35406 | First-order logic and set ... |
| bnj1459 35407 | First-order logic and set ... |
| bnj1464 35408 | Conversion of implicit sub... |
| bnj1465 35409 | First-order logic and set ... |
| bnj1468 35410 | Conversion of implicit sub... |
| bnj1476 35411 | First-order logic and set ... |
| bnj1502 35412 | First-order logic and set ... |
| bnj1503 35413 | First-order logic and set ... |
| bnj1517 35414 | First-order logic and set ... |
| bnj1521 35415 | First-order logic and set ... |
| bnj1533 35416 | First-order logic and set ... |
| bnj1534 35417 | First-order logic and set ... |
| bnj1536 35418 | First-order logic and set ... |
| bnj1538 35419 | First-order logic and set ... |
| bnj1541 35420 | First-order logic and set ... |
| bnj1542 35421 | First-order logic and set ... |
| bnj110 35422 | Well-founded induction res... |
| bnj157 35423 | Well-founded induction res... |
| bnj66 35424 | Technical lemma for ~ bnj6... |
| bnj91 35425 | First-order logic and set ... |
| bnj92 35426 | First-order logic and set ... |
| bnj93 35427 | Technical lemma for ~ bnj9... |
| bnj95 35428 | Technical lemma for ~ bnj1... |
| bnj96 35429 | Technical lemma for ~ bnj1... |
| bnj97 35430 | Technical lemma for ~ bnj1... |
| bnj98 35431 | Technical lemma for ~ bnj1... |
| bnj106 35432 | First-order logic and set ... |
| bnj118 35433 | First-order logic and set ... |
| bnj121 35434 | First-order logic and set ... |
| bnj124 35435 | Technical lemma for ~ bnj1... |
| bnj125 35436 | Technical lemma for ~ bnj1... |
| bnj126 35437 | Technical lemma for ~ bnj1... |
| bnj130 35438 | Technical lemma for ~ bnj1... |
| bnj149 35439 | Technical lemma for ~ bnj1... |
| bnj150 35440 | Technical lemma for ~ bnj1... |
| bnj151 35441 | Technical lemma for ~ bnj1... |
| bnj154 35442 | Technical lemma for ~ bnj1... |
| bnj155 35443 | Technical lemma for ~ bnj1... |
| bnj153 35444 | Technical lemma for ~ bnj8... |
| bnj207 35445 | Technical lemma for ~ bnj8... |
| bnj213 35446 | First-order logic and set ... |
| bnj222 35447 | Technical lemma for ~ bnj2... |
| bnj229 35448 | Technical lemma for ~ bnj5... |
| bnj517 35449 | Technical lemma for ~ bnj5... |
| bnj518 35450 | Technical lemma for ~ bnj8... |
| bnj523 35451 | Technical lemma for ~ bnj8... |
| bnj526 35452 | Technical lemma for ~ bnj8... |
| bnj528 35453 | Technical lemma for ~ bnj8... |
| bnj535 35454 | Technical lemma for ~ bnj8... |
| bnj539 35455 | Technical lemma for ~ bnj8... |
| bnj540 35456 | Technical lemma for ~ bnj8... |
| bnj543 35457 | Technical lemma for ~ bnj8... |
| bnj544 35458 | Technical lemma for ~ bnj8... |
| bnj545 35459 | Technical lemma for ~ bnj8... |
| bnj546 35460 | Technical lemma for ~ bnj8... |
| bnj548 35461 | Technical lemma for ~ bnj8... |
| bnj553 35462 | Technical lemma for ~ bnj8... |
| bnj554 35463 | Technical lemma for ~ bnj8... |
| bnj556 35464 | Technical lemma for ~ bnj8... |
| bnj557 35465 | Technical lemma for ~ bnj8... |
| bnj558 35466 | Technical lemma for ~ bnj8... |
| bnj561 35467 | Technical lemma for ~ bnj8... |
| bnj562 35468 | Technical lemma for ~ bnj8... |
| bnj570 35469 | Technical lemma for ~ bnj8... |
| bnj571 35470 | Technical lemma for ~ bnj8... |
| bnj605 35471 | Technical lemma. This lem... |
| bnj581 35472 | Technical lemma for ~ bnj5... |
| bnj589 35473 | Technical lemma for ~ bnj8... |
| bnj590 35474 | Technical lemma for ~ bnj8... |
| bnj591 35475 | Technical lemma for ~ bnj8... |
| bnj594 35476 | Technical lemma for ~ bnj8... |
| bnj580 35477 | Technical lemma for ~ bnj5... |
| bnj579 35478 | Technical lemma for ~ bnj8... |
| bnj602 35479 | Equality theorem for the `... |
| bnj607 35480 | Technical lemma for ~ bnj8... |
| bnj609 35481 | Technical lemma for ~ bnj8... |
| bnj611 35482 | Technical lemma for ~ bnj8... |
| bnj600 35483 | Technical lemma for ~ bnj8... |
| bnj601 35484 | Technical lemma for ~ bnj8... |
| bnj852 35485 | Technical lemma for ~ bnj6... |
| bnj864 35486 | Technical lemma for ~ bnj6... |
| bnj865 35487 | Technical lemma for ~ bnj6... |
| bnj873 35488 | Technical lemma for ~ bnj6... |
| bnj849 35489 | Technical lemma for ~ bnj6... |
| bnj882 35490 | Definition (using hypothes... |
| bnj18eq1 35491 | Equality theorem for trans... |
| bnj893 35492 | Property of ` _trCl ` . U... |
| bnj900 35493 | Technical lemma for ~ bnj6... |
| bnj906 35494 | Property of ` _trCl ` . (... |
| bnj908 35495 | Technical lemma for ~ bnj6... |
| bnj911 35496 | Technical lemma for ~ bnj6... |
| bnj916 35497 | Technical lemma for ~ bnj6... |
| bnj917 35498 | Technical lemma for ~ bnj6... |
| bnj934 35499 | Technical lemma for ~ bnj6... |
| bnj929 35500 | Technical lemma for ~ bnj6... |
| bnj938 35501 | Technical lemma for ~ bnj6... |
| bnj944 35502 | Technical lemma for ~ bnj6... |
| bnj953 35503 | Technical lemma for ~ bnj6... |
| bnj958 35504 | Technical lemma for ~ bnj6... |
| bnj1000 35505 | Technical lemma for ~ bnj8... |
| bnj965 35506 | Technical lemma for ~ bnj8... |
| bnj964 35507 | Technical lemma for ~ bnj6... |
| bnj966 35508 | Technical lemma for ~ bnj6... |
| bnj967 35509 | Technical lemma for ~ bnj6... |
| bnj969 35510 | Technical lemma for ~ bnj6... |
| bnj970 35511 | Technical lemma for ~ bnj6... |
| bnj910 35512 | Technical lemma for ~ bnj6... |
| bnj978 35513 | Technical lemma for ~ bnj6... |
| bnj981 35514 | Technical lemma for ~ bnj6... |
| bnj983 35515 | Technical lemma for ~ bnj6... |
| bnj984 35516 | Technical lemma for ~ bnj6... |
| bnj985v 35517 | Version of ~ bnj985 with a... |
| bnj985 35518 | Technical lemma for ~ bnj6... |
| bnj986 35519 | Technical lemma for ~ bnj6... |
| bnj996 35520 | Technical lemma for ~ bnj6... |
| bnj998 35521 | Technical lemma for ~ bnj6... |
| bnj999 35522 | Technical lemma for ~ bnj6... |
| bnj1001 35523 | Technical lemma for ~ bnj6... |
| bnj1006 35524 | Technical lemma for ~ bnj6... |
| bnj1014 35525 | Technical lemma for ~ bnj6... |
| bnj1015 35526 | Technical lemma for ~ bnj6... |
| bnj1018g 35527 | Version of ~ bnj1018 with ... |
| bnj1018 35528 | Technical lemma for ~ bnj6... |
| bnj1020 35529 | Technical lemma for ~ bnj6... |
| bnj1021 35530 | Technical lemma for ~ bnj6... |
| bnj907 35531 | Technical lemma for ~ bnj6... |
| bnj1029 35532 | Property of ` _trCl ` . (... |
| bnj1033 35533 | Technical lemma for ~ bnj6... |
| bnj1034 35534 | Technical lemma for ~ bnj6... |
| bnj1039 35535 | Technical lemma for ~ bnj6... |
| bnj1040 35536 | Technical lemma for ~ bnj6... |
| bnj1047 35537 | Technical lemma for ~ bnj6... |
| bnj1049 35538 | Technical lemma for ~ bnj6... |
| bnj1052 35539 | Technical lemma for ~ bnj6... |
| bnj1053 35540 | Technical lemma for ~ bnj6... |
| bnj1071 35541 | Technical lemma for ~ bnj6... |
| bnj1083 35542 | Technical lemma for ~ bnj6... |
| bnj1090 35543 | Technical lemma for ~ bnj6... |
| bnj1093 35544 | Technical lemma for ~ bnj6... |
| bnj1097 35545 | Technical lemma for ~ bnj6... |
| bnj1110 35546 | Technical lemma for ~ bnj6... |
| bnj1112 35547 | Technical lemma for ~ bnj6... |
| bnj1118 35548 | Technical lemma for ~ bnj6... |
| bnj1121 35549 | Technical lemma for ~ bnj6... |
| bnj1123 35550 | Technical lemma for ~ bnj6... |
| bnj1030 35551 | Technical lemma for ~ bnj6... |
| bnj1124 35552 | Property of ` _trCl ` . (... |
| bnj1133 35553 | Technical lemma for ~ bnj6... |
| bnj1128 35554 | Technical lemma for ~ bnj6... |
| bnj1127 35555 | Property of ` _trCl ` . (... |
| bnj1125 35556 | Property of ` _trCl ` . (... |
| bnj1145 35557 | Technical lemma for ~ bnj6... |
| bnj1147 35558 | Property of ` _trCl ` . (... |
| bnj1137 35559 | Property of ` _trCl ` . (... |
| bnj1148 35560 | Property of ` _pred ` . (... |
| bnj1136 35561 | Technical lemma for ~ bnj6... |
| bnj1152 35562 | Technical lemma for ~ bnj6... |
| bnj1154 35563 | Property of ` Fr ` . (Con... |
| bnj1171 35564 | Technical lemma for ~ bnj6... |
| bnj1172 35565 | Technical lemma for ~ bnj6... |
| bnj1173 35566 | Technical lemma for ~ bnj6... |
| bnj1174 35567 | Technical lemma for ~ bnj6... |
| bnj1175 35568 | Technical lemma for ~ bnj6... |
| bnj1176 35569 | Technical lemma for ~ bnj6... |
| bnj1177 35570 | Technical lemma for ~ bnj6... |
| bnj1186 35571 | Technical lemma for ~ bnj6... |
| bnj1190 35572 | Technical lemma for ~ bnj6... |
| bnj1189 35573 | Technical lemma for ~ bnj6... |
| bnj69 35574 | Existence of a minimal ele... |
| bnj1228 35575 | Existence of a minimal ele... |
| bnj1204 35576 | Well-founded induction. T... |
| bnj1234 35577 | Technical lemma for ~ bnj6... |
| bnj1245 35578 | Technical lemma for ~ bnj6... |
| bnj1256 35579 | Technical lemma for ~ bnj6... |
| bnj1259 35580 | Technical lemma for ~ bnj6... |
| bnj1253 35581 | Technical lemma for ~ bnj6... |
| bnj1279 35582 | Technical lemma for ~ bnj6... |
| bnj1286 35583 | Technical lemma for ~ bnj6... |
| bnj1280 35584 | Technical lemma for ~ bnj6... |
| bnj1296 35585 | Technical lemma for ~ bnj6... |
| bnj1309 35586 | Technical lemma for ~ bnj6... |
| bnj1307 35587 | Technical lemma for ~ bnj6... |
| bnj1311 35588 | Technical lemma for ~ bnj6... |
| bnj1318 35589 | Technical lemma for ~ bnj6... |
| bnj1326 35590 | Technical lemma for ~ bnj6... |
| bnj1321 35591 | Technical lemma for ~ bnj6... |
| bnj1364 35592 | Property of ` _FrSe ` . (... |
| bnj1371 35593 | Technical lemma for ~ bnj6... |
| bnj1373 35594 | Technical lemma for ~ bnj6... |
| bnj1374 35595 | Technical lemma for ~ bnj6... |
| bnj1384 35596 | Technical lemma for ~ bnj6... |
| bnj1388 35597 | Technical lemma for ~ bnj6... |
| bnj1398 35598 | Technical lemma for ~ bnj6... |
| bnj1413 35599 | Property of ` _trCl ` . (... |
| bnj1408 35600 | Technical lemma for ~ bnj1... |
| bnj1414 35601 | Property of ` _trCl ` . (... |
| bnj1415 35602 | Technical lemma for ~ bnj6... |
| bnj1416 35603 | Technical lemma for ~ bnj6... |
| bnj1418 35604 | Property of ` _pred ` . (... |
| bnj1417 35605 | Technical lemma for ~ bnj6... |
| bnj1421 35606 | Technical lemma for ~ bnj6... |
| bnj1444 35607 | Technical lemma for ~ bnj6... |
| bnj1445 35608 | Technical lemma for ~ bnj6... |
| bnj1446 35609 | Technical lemma for ~ bnj6... |
| bnj1447 35610 | Technical lemma for ~ bnj6... |
| bnj1448 35611 | Technical lemma for ~ bnj6... |
| bnj1449 35612 | Technical lemma for ~ bnj6... |
| bnj1442 35613 | Technical lemma for ~ bnj6... |
| bnj1450 35614 | Technical lemma for ~ bnj6... |
| bnj1423 35615 | Technical lemma for ~ bnj6... |
| bnj1452 35616 | Technical lemma for ~ bnj6... |
| bnj1466 35617 | Technical lemma for ~ bnj6... |
| bnj1467 35618 | Technical lemma for ~ bnj6... |
| bnj1463 35619 | Technical lemma for ~ bnj6... |
| bnj1489 35620 | Technical lemma for ~ bnj6... |
| bnj1491 35621 | Technical lemma for ~ bnj6... |
| bnj1312 35622 | Technical lemma for ~ bnj6... |
| bnj1493 35623 | Technical lemma for ~ bnj6... |
| bnj1497 35624 | Technical lemma for ~ bnj6... |
| bnj1498 35625 | Technical lemma for ~ bnj6... |
| bnj60 35626 | Well-founded recursion, pa... |
| bnj1514 35627 | Technical lemma for ~ bnj1... |
| bnj1518 35628 | Technical lemma for ~ bnj1... |
| bnj1519 35629 | Technical lemma for ~ bnj1... |
| bnj1520 35630 | Technical lemma for ~ bnj1... |
| bnj1501 35631 | Technical lemma for ~ bnj1... |
| bnj1500 35632 | Well-founded recursion, pa... |
| bnj1525 35633 | Technical lemma for ~ bnj1... |
| bnj1529 35634 | Technical lemma for ~ bnj1... |
| bnj1523 35635 | Technical lemma for ~ bnj1... |
| bnj1522 35636 | Well-founded recursion, pa... |
| nfan1c 35637 | Variant of ~ nfan and comm... |
| cbvex1v 35638 | Rule used to change bound ... |
| dvelimalcased 35639 | Eliminate a disjoint varia... |
| dvelimalcasei 35640 | Eliminate a disjoint varia... |
| dvelimexcased 35641 | Eliminate a disjoint varia... |
| dvelimexcasei 35642 | Eliminate a disjoint varia... |
| inv2 35643 | The intersection of the un... |
| exdifsn 35644 | There exists an element in... |
| axnulALT2 35645 | Alternate proof of ~ axnul... |
| fnfvintima 35646 | Condition for a function v... |
| ordprcon 35647 | If an ordinal class is not... |
| xoromon 35648 | ` _om ` is either an ordin... |
| ordtypeon 35649 | A proper class with a set-... |
| fnrelpredd 35650 | A function that preserves ... |
| cardpred 35651 | The cardinality function p... |
| nummin 35652 | Every nonempty class of nu... |
| 1enumen 35653 | The Fundamental Theorem of... |
| 1enumcard 35654 | The Fundamental Theorem of... |
| r11 35655 | Value of the cumulative hi... |
| r12 35656 | Value of the cumulative hi... |
| onrankid 35657 | The rank of an ordinal num... |
| rankfilimb 35658 | The rank of a finite well-... |
| r1filim 35659 | A finite set appears in th... |
| r1omfi 35660 | Obsolete theorem, use ~ hf... |
| r1omhf 35661 | A set is hereditarily fini... |
| r1ssel 35662 | A set is a subset of the v... |
| axnulALT3 35663 | Alternate proof of ~ axnul... |
| axprALT2 35664 | Alternate proof of ~ axpr ... |
| r1omfv 35665 | Value of the cumulative hi... |
| rankfo 35666 | The rank function maps the... |
| rankfn 35667 | The rank function is a fun... |
| trssfir1om 35668 | If every element in a tran... |
| r1omhfb 35669 | The class of all hereditar... |
| scotteqi 35670 | Equality theorem for the S... |
| elscott 35671 | Membership in a Scott's tr... |
| dfscott2 35672 | Alternate definition of a ... |
| dfscott3 35673 | Alternate definition of a ... |
| elscott2 35674 | Membership in a Scott's tr... |
| elscottrank 35675 | The rank of an element in ... |
| elscottrankeq 35676 | Elements in a Scott's tric... |
| elscottrankss 35677 | Relationship between the r... |
| scottrankeqel 35678 | If a member of the input s... |
| nelscottrankgt 35679 | If a member of the input s... |
| scottsn 35680 | Applying Scott's trick to ... |
| scott0bOLD 35681 | Obsolete version of ~ scot... |
| rankscott 35682 | The rank of a nonempty Sco... |
| rankscottu 35683 | An upper bound on the rank... |
| scottssr1 35684 | Relationship between a Sco... |
| acnum 35685 | The Axiom of Choice implie... |
| 5on 35696 | Ordinal 5 is an ordinal nu... |
| 6on 35697 | Ordinal 6 is an ordinal nu... |
| 7on 35698 | Ordinal 7 is an ordinal nu... |
| 8on 35699 | Ordinal 8 is an ordinal nu... |
| 9on 35700 | Ordinal 9 is an ordinal nu... |
| 5onn 35701 | The ordinal 5 is a natural... |
| 6onn 35702 | The ordinal 6 is a natural... |
| 7onn 35703 | The ordinal 7 is a natural... |
| 8onn 35704 | The ordinal 8 is a natural... |
| 9onn 35705 | The ordinal 9 is a natural... |
| prcinf 35706 | Any proper class is litera... |
| fineqvrep 35707 | If all sets are finite, th... |
| fineqvpow 35708 | If all sets are finite, th... |
| fineqvac 35709 | If all sets are finite, th... |
| fineqvacALT 35710 | Shorter proof of ~ fineqva... |
| fineqvomon 35711 | If all sets are finite, th... |
| fineqvomonb 35712 | All sets are finite iff al... |
| omprcomonb 35713 | The class of all finite or... |
| fineqvnttrclselem1 35714 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35715 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35716 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35717 | A counterexample demonstra... |
| fineqvinfep 35718 | A counterexample demonstra... |
| axreg 35720 | Derivation of ~ ax-reg fro... |
| axregscl 35721 | A version of ~ ax-regs wit... |
| axregszf 35722 | Derivation of ~ zfregs usi... |
| setindregs 35723 | Set (epsilon) induction. ... |
| setinds2regs 35724 | Principle of set induction... |
| noinfepfnregs 35725 | There are no infinite desc... |
| noinfepregs 35726 | There are no infinite desc... |
| tz9.1regs 35727 | Every set has a transitive... |
| unir1regs 35728 | The cumulative hierarchy o... |
| trssfir1omregs 35729 | If every element in a tran... |
| r1omhfbregs 35730 | The class of all hereditar... |
| fineqvr1ombregs 35731 | All sets are finite iff al... |
| axregs 35732 | Derivation of ~ ax-regs fr... |
| axsepg2 35733 | A generalization of ~ ax-s... |
| axsepg3 35734 | A generalization of ~ ax-s... |
| axsepg3ALT 35735 | Alternate proof of ~ axsep... |
| axsepg4 35736 | A generalization of ~ ax-s... |
| axsepg5 35737 | A generalization of ~ ax-s... |
| axnulg 35738 | A generalization of ~ ax-n... |
| axpowg 35739 | A generalization of ~ ax-p... |
| axpowg2 35740 | A generalization of ~ ax-p... |
| axpowg3 35741 | A generalization of ~ ax-p... |
| kardfn 35744 | The ` kard ` class is a fu... |
| kardval 35745 | The value of the ` kard ` ... |
| kardval2 35746 | The value of the ` kard ` ... |
| kard0 35747 | The ` kard ` cardinality o... |
| elkarden 35748 | Any member of the ` kard `... |
| kardeq0 35749 | Applying ` kard ` to a cla... |
| kardeng 35750 | Two sets are equinumerous ... |
| kardenir 35751 | If two sets are equinumero... |
| kard0b 35752 | The empty set is the only ... |
| kardsn 35753 | A singleton has cardinalit... |
| karddom 35754 | One set dominates another ... |
| kardsdom 35755 | One set strictly dominates... |
| kardexen 35756 | One set is equinumerous to... |
| kardcard2a 35757 | If two sets have equal non... |
| kardcard2b 35758 | If two sets have equal ` k... |
| kardcard2 35759 | Two numerable sets have eq... |
| ackardcard 35760 | The Axiom of Choice implie... |
| kardcard 35761 | Two sets have equal ` kard... |
| kardnnfi 35762 | The ` kard ` cardinal numb... |
| kardfi 35763 | The ` kard ` cardinal numb... |
| rankkardu 35764 | An upper bound on the rank... |
| 1enumkard 35765 | The Fundamental Theorem of... |
| gblacfnacd 35806 | If ` G ` is a global choic... |
| onvf1odlem1 35807 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35808 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35809 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35810 | Lemma for ~ onvf1od . If ... |
| onvf1od 35811 | If ` G ` is a global choic... |
| vonf1wev 35812 | If ` F ` maps the universe... |
| vonf1owev 35813 | If ` F ` is a bijection fr... |
| vonf1owevOLD 35814 | Obsolete version of ~ vonf... |
| wevgblacfn 35815 | If ` R ` is a well-orderin... |
| vonf1osev 35816 | If ` F ` is a bijection fr... |
| wevonprcf1o 35817 | If ` R ` is a set-like wel... |
| vonf1oonf1 35818 | If ` F ` is a bijection fr... |
| vonf1oonfo 35819 | If ` F ` is a bijection fr... |
| onvfowev 35820 | If ` F ` maps the ordinals... |
| zltp1ne 35821 | Integer ordering relation.... |
| nnltp1ne 35822 | Positive integer ordering ... |
| nn0ltp1ne 35823 | Nonnegative integer orderi... |
| fisshasheq 35824 | A finite set is equal to i... |
| 1enum 35825 | The Fundamental Theorem of... |
| cplgredgex 35826 | Any two (distinct) vertice... |
| cusgredgex 35827 | Any two (distinct) vertice... |
| cusgredgex2 35828 | Any two distinct vertices ... |
| revwlkb 35829 | Two words represent a walk... |
| usgrgt2cycl 35830 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35831 | A simple graph with a non-... |
| cusgr3cyclex 35832 | Every complete simple grap... |
| 2cycl2d 35833 | Construction of a 2-cycle ... |
| acycgr0v 35834 | A null graph (with no vert... |
| acycgr1v 35835 | A multigraph with one vert... |
| acycgr2v 35836 | A simple graph with two ve... |
| prclisacycgr 35837 | A proper class (representi... |
| acycgrislfgr 35838 | An acyclic hypergraph is a... |
| upgracycumgr 35839 | An acyclic pseudograph is ... |
| umgracycusgr 35840 | An acyclic multigraph is a... |
| upgracycusgr 35841 | An acyclic pseudograph is ... |
| cusgracyclt3v 35842 | A complete simple graph is... |
| pthacycspth 35843 | A path in an acyclic graph... |
| acycgrsubgr 35844 | The subgraph of an acyclic... |
| quartfull 35851 | The quartic equation, writ... |
| deranglem 35852 | Lemma for derangements. (... |
| derangval 35853 | Define the derangement fun... |
| derangf 35854 | The derangement number is ... |
| derang0 35855 | The derangement number of ... |
| derangsn 35856 | The derangement number of ... |
| derangenlem 35857 | One half of ~ derangen . ... |
| derangen 35858 | The derangement number is ... |
| subfacval 35859 | The subfactorial is define... |
| derangen2 35860 | Write the derangement numb... |
| subfacf 35861 | The subfactorial is a func... |
| subfaclefac 35862 | The subfactorial is less t... |
| subfac0 35863 | The subfactorial at zero. ... |
| subfac1 35864 | The subfactorial at one. ... |
| subfacp1lem1 35865 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35866 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35867 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35868 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35869 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35870 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35871 | Lemma for ~ subfacp1 . By... |
| subfacp1 35872 | A two-term recurrence for ... |
| subfacval2 35873 | A closed-form expression f... |
| subfaclim 35874 | The subfactorial converges... |
| subfacval3 35875 | Another closed form expres... |
| derangfmla 35876 | The derangements formula, ... |
| erdszelem1 35877 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35878 | Lemma for ~ erdsze . (Con... |
| erdszelem3 35879 | Lemma for ~ erdsze . (Con... |
| erdszelem4 35880 | Lemma for ~ erdsze . (Con... |
| erdszelem5 35881 | Lemma for ~ erdsze . (Con... |
| erdszelem6 35882 | Lemma for ~ erdsze . (Con... |
| erdszelem7 35883 | Lemma for ~ erdsze . (Con... |
| erdszelem8 35884 | Lemma for ~ erdsze . (Con... |
| erdszelem9 35885 | Lemma for ~ erdsze . (Con... |
| erdszelem10 35886 | Lemma for ~ erdsze . (Con... |
| erdszelem11 35887 | Lemma for ~ erdsze . (Con... |
| erdsze 35888 | The Erdős-Szekeres th... |
| erdsze2lem1 35889 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 35890 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 35891 | Generalize the statement o... |
| kur14lem1 35892 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 35893 | Lemma for ~ kur14 . Write... |
| kur14lem3 35894 | Lemma for ~ kur14 . A clo... |
| kur14lem4 35895 | Lemma for ~ kur14 . Compl... |
| kur14lem5 35896 | Lemma for ~ kur14 . Closu... |
| kur14lem6 35897 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 35898 | Lemma for ~ kur14 : main p... |
| kur14lem8 35899 | Lemma for ~ kur14 . Show ... |
| kur14lem9 35900 | Lemma for ~ kur14 . Since... |
| kur14lem10 35901 | Lemma for ~ kur14 . Disch... |
| kur14 35902 | Kuratowski's closure-compl... |
| ispconn 35909 | The property of being a pa... |
| pconncn 35910 | The property of being a pa... |
| pconntop 35911 | A simply connected space i... |
| issconn 35912 | The property of being a si... |
| sconnpconn 35913 | A simply connected space i... |
| sconntop 35914 | A simply connected space i... |
| sconnpht 35915 | A closed path in a simply ... |
| cnpconn 35916 | An image of a path-connect... |
| pconnconn 35917 | A path-connected space is ... |
| txpconn 35918 | The topological product of... |
| ptpconn 35919 | The topological product of... |
| indispconn 35920 | The indiscrete topology (o... |
| connpconn 35921 | A connected and locally pa... |
| qtoppconn 35922 | A quotient of a path-conne... |
| pconnpi1 35923 | All fundamental groups in ... |
| sconnpht2 35924 | Any two paths in a simply ... |
| sconnpi1 35925 | A path-connected topologic... |
| txsconnlem 35926 | Lemma for ~ txsconn . (Co... |
| txsconn 35927 | The topological product of... |
| cvxpconn 35928 | A convex subset of the com... |
| cvxsconn 35929 | A convex subset of the com... |
| blsconn 35930 | An open ball in the comple... |
| cnllysconn 35931 | The topology of the comple... |
| resconn 35932 | A subset of ` RR ` is simp... |
| ioosconn 35933 | An open interval is simply... |
| iccsconn 35934 | A closed interval is simpl... |
| retopsconn 35935 | The real numbers are simpl... |
| iccllysconn 35936 | A closed interval is local... |
| rellysconn 35937 | The real numbers are local... |
| iisconn 35938 | The unit interval is simpl... |
| iillysconn 35939 | The unit interval is local... |
| iinllyconn 35940 | The unit interval is local... |
| fncvm 35943 | Lemma for covering maps. ... |
| cvmscbv 35944 | Change bound variables in ... |
| iscvm 35945 | The property of being a co... |
| cvmtop1 35946 | Reverse closure for a cove... |
| cvmtop2 35947 | Reverse closure for a cove... |
| cvmcn 35948 | A covering map is a contin... |
| cvmcov 35949 | Property of a covering map... |
| cvmsrcl 35950 | Reverse closure for an eve... |
| cvmsi 35951 | One direction of ~ cvmsval... |
| cvmsval 35952 | Elementhood in the set ` S... |
| cvmsss 35953 | An even covering is a subs... |
| cvmsn0 35954 | An even covering is nonemp... |
| cvmsuni 35955 | An even covering of ` U ` ... |
| cvmsdisj 35956 | An even covering of ` U ` ... |
| cvmshmeo 35957 | Every element of an even c... |
| cvmsf1o 35958 | ` F ` , localized to an el... |
| cvmscld 35959 | The sets of an even coveri... |
| cvmsss2 35960 | An open subset of an evenl... |
| cvmcov2 35961 | The covering map property ... |
| cvmseu 35962 | Every element in ` U. T ` ... |
| cvmsiota 35963 | Identify the unique elemen... |
| cvmopnlem 35964 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 35965 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 35966 | A covering map is an open ... |
| cvmliftmolem1 35967 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 35968 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 35969 | A lift of a continuous fun... |
| cvmliftmo 35970 | A lift of a continuous fun... |
| cvmliftlem1 35971 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 35972 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 35973 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 35974 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 35975 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 35976 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 35977 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 35978 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 35979 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 35980 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 35981 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 35982 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 35983 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 35984 | Lemma for ~ cvmlift . Dis... |
| cvmlift 35985 | One of the important prope... |
| cvmfo 35986 | A covering map is an onto ... |
| cvmliftiota 35987 | Write out a function ` H `... |
| cvmlift2lem1 35988 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 35989 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 35990 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 35991 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 35992 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 35993 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 35994 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 35995 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 35996 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 35997 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 35998 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 35999 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 36000 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 36001 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 36002 | A two-dimensional version ... |
| cvmliftphtlem 36003 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 36004 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 36005 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 36006 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 36007 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 36008 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 36009 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 36010 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 36011 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 36012 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 36013 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 36014 | A general version of ~ cvm... |
| snmlff 36015 | The function ` F ` from ~ ... |
| snmlfval 36016 | The function ` F ` from ~ ... |
| snmlval 36017 | The property " ` A ` is si... |
| snmlflim 36018 | If ` A ` is simply normal,... |
| goel 36033 | A "Godel-set of membership... |
| goelel3xp 36034 | A "Godel-set of membership... |
| goeleq12bg 36035 | Two "Godel-set of membersh... |
| gonafv 36036 | The "Godel-set for the She... |
| goaleq12d 36037 | Equality of the "Godel-set... |
| gonanegoal 36038 | The Godel-set for the Shef... |
| satf 36039 | The satisfaction predicate... |
| satfsucom 36040 | The satisfaction predicate... |
| satfn 36041 | The satisfaction predicate... |
| satom 36042 | The satisfaction predicate... |
| satfvsucom 36043 | The satisfaction predicate... |
| satfv0 36044 | The value of the satisfact... |
| satfvsuclem1 36045 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 36046 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 36047 | The value of the satisfact... |
| satfv1lem 36048 | Lemma for ~ satfv1 . (Con... |
| satfv1 36049 | The value of the satisfact... |
| satfsschain 36050 | The binary relation of a s... |
| satfvsucsuc 36051 | The satisfaction predicate... |
| satfbrsuc 36052 | The binary relation of a s... |
| satfrel 36053 | The value of the satisfact... |
| satfdmlem 36054 | Lemma for ~ satfdm . (Con... |
| satfdm 36055 | The domain of the satisfac... |
| satfrnmapom 36056 | The range of the satisfact... |
| satfv0fun 36057 | The value of the satisfact... |
| satf0 36058 | The satisfaction predicate... |
| satf0sucom 36059 | The satisfaction predicate... |
| satf00 36060 | The value of the satisfact... |
| satf0suclem 36061 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 36062 | The value of the satisfact... |
| satf0op 36063 | An element of a value of t... |
| satf0n0 36064 | The value of the satisfact... |
| sat1el2xp 36065 | The first component of an ... |
| fmlafv 36066 | The valid Godel formulas o... |
| fmla 36067 | The set of all valid Godel... |
| fmla0 36068 | The valid Godel formulas o... |
| fmla0xp 36069 | The valid Godel formulas o... |
| fmlasuc0 36070 | The valid Godel formulas o... |
| fmlafvel 36071 | A class is a valid Godel f... |
| fmlasuc 36072 | The valid Godel formulas o... |
| fmla1 36073 | The valid Godel formulas o... |
| isfmlasuc 36074 | The characterization of a ... |
| fmlasssuc 36075 | The Godel formulas of heig... |
| fmlaomn0 36076 | The empty set is not a God... |
| fmlan0 36077 | The empty set is not a God... |
| gonan0 36078 | The "Godel-set of NAND" is... |
| goaln0 36079 | The "Godel-set of universa... |
| gonarlem 36080 | Lemma for ~ gonar (inducti... |
| gonar 36081 | If the "Godel-set of NAND"... |
| goalrlem 36082 | Lemma for ~ goalr (inducti... |
| goalr 36083 | If the "Godel-set of unive... |
| fmla0disjsuc 36084 | The set of valid Godel for... |
| fmlasucdisj 36085 | The valid Godel formulas o... |
| satfdmfmla 36086 | The domain of the satisfac... |
| satffunlem 36087 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 36088 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 36089 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 36090 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 36091 | The domain of an ordered p... |
| satffunlem2lem2 36092 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 36093 | Lemma 1 for ~ satffun : in... |
| satffunlem2 36094 | Lemma 2 for ~ satffun : in... |
| satffun 36095 | The value of the satisfact... |
| satff 36096 | The satisfaction predicate... |
| satfun 36097 | The satisfaction predicate... |
| satfvel 36098 | An element of the value of... |
| satfv0fvfmla0 36099 | The value of the satisfact... |
| satefv 36100 | The simplified satisfactio... |
| sate0 36101 | The simplified satisfactio... |
| satef 36102 | The simplified satisfactio... |
| sate0fv0 36103 | A simplified satisfaction ... |
| satefvfmla0 36104 | The simplified satisfactio... |
| sategoelfvb 36105 | Characterization of a valu... |
| sategoelfv 36106 | Condition of a valuation `... |
| ex-sategoelel 36107 | Example of a valuation of ... |
| ex-sategoel 36108 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 36109 | The value of the satisfact... |
| 2goelgoanfmla1 36110 | Two Godel-sets of membersh... |
| satefvfmla1 36111 | The simplified satisfactio... |
| ex-sategoelelomsuc 36112 | Example of a valuation of ... |
| ex-sategoelel12 36113 | Example of a valuation of ... |
| prv 36114 | The "proves" relation on a... |
| elnanelprv 36115 | The wff ` ( A e. B -/\ B e... |
| prv0 36116 | Every wff encoded as ` U `... |
| prv1n 36117 | No wff encoded as a Godel-... |
| mvtval 36186 | The set of variable typeco... |
| mrexval 36187 | The set of "raw expression... |
| mexval 36188 | The set of expressions, wh... |
| mexval2 36189 | The set of expressions, wh... |
| mdvval 36190 | The set of disjoint variab... |
| mvrsval 36191 | The set of variables in an... |
| mvrsfpw 36192 | The set of variables in an... |
| mrsubffval 36193 | The substitution of some v... |
| mrsubfval 36194 | The substitution of some v... |
| mrsubval 36195 | The substitution of some v... |
| mrsubcv 36196 | The value of a substituted... |
| mrsubvr 36197 | The value of a substituted... |
| mrsubff 36198 | A substitution is a functi... |
| mrsubrn 36199 | Although it is defined for... |
| mrsubff1 36200 | When restricted to complet... |
| mrsubff1o 36201 | When restricted to complet... |
| mrsub0 36202 | The value of the substitut... |
| mrsubf 36203 | A substitution is a functi... |
| mrsubccat 36204 | Substitution distributes o... |
| mrsubcn 36205 | A substitution does not ch... |
| elmrsubrn 36206 | Characterization of the su... |
| mrsubco 36207 | The composition of two sub... |
| mrsubvrs 36208 | The set of variables in a ... |
| msubffval 36209 | A substitution applied to ... |
| msubfval 36210 | A substitution applied to ... |
| msubval 36211 | A substitution applied to ... |
| msubrsub 36212 | A substitution applied to ... |
| msubty 36213 | The type of a substituted ... |
| elmsubrn 36214 | Characterization of substi... |
| msubrn 36215 | Although it is defined for... |
| msubff 36216 | A substitution is a functi... |
| msubco 36217 | The composition of two sub... |
| msubf 36218 | A substitution is a functi... |
| mvhfval 36219 | Value of the function mapp... |
| mvhval 36220 | Value of the function mapp... |
| mpstval 36221 | A pre-statement is an orde... |
| elmpst 36222 | Property of being a pre-st... |
| msrfval 36223 | Value of the reduct of a p... |
| msrval 36224 | Value of the reduct of a p... |
| mpstssv 36225 | A pre-statement is an orde... |
| mpst123 36226 | Decompose a pre-statement ... |
| mpstrcl 36227 | The elements of a pre-stat... |
| msrf 36228 | The reduct of a pre-statem... |
| msrrcl 36229 | If ` X ` and ` Y ` have th... |
| mstaval 36230 | Value of the set of statem... |
| msrid 36231 | The reduct of a statement ... |
| msrfo 36232 | The reduct of a pre-statem... |
| mstapst 36233 | A statement is a pre-state... |
| elmsta 36234 | Property of being a statem... |
| ismfs 36235 | A formal system is a tuple... |
| mfsdisj 36236 | The constants and variable... |
| mtyf2 36237 | The type function maps var... |
| mtyf 36238 | The type function maps var... |
| mvtss 36239 | The set of variable typeco... |
| maxsta 36240 | An axiom is a statement. ... |
| mvtinf 36241 | Each variable typecode has... |
| msubff1 36242 | When restricted to complet... |
| msubff1o 36243 | When restricted to complet... |
| mvhf 36244 | The function mapping varia... |
| mvhf1 36245 | The function mapping varia... |
| msubvrs 36246 | The set of variables in a ... |
| mclsrcl 36247 | Reverse closure for the cl... |
| mclsssvlem 36248 | Lemma for ~ mclsssv . (Co... |
| mclsval 36249 | The function mapping varia... |
| mclsssv 36250 | The closure of a set of ex... |
| ssmclslem 36251 | Lemma for ~ ssmcls . (Con... |
| vhmcls 36252 | All variable hypotheses ar... |
| ssmcls 36253 | The original expressions a... |
| ss2mcls 36254 | The closure is monotonic u... |
| mclsax 36255 | The closure is closed unde... |
| mclsind 36256 | Induction theorem for clos... |
| mppspstlem 36257 | Lemma for ~ mppspst . (Co... |
| mppsval 36258 | Definition of a provable p... |
| elmpps 36259 | Definition of a provable p... |
| mppspst 36260 | A provable pre-statement i... |
| mthmval 36261 | A theorem is a pre-stateme... |
| elmthm 36262 | A theorem is a pre-stateme... |
| mthmi 36263 | A statement whose reduct i... |
| mthmsta 36264 | A theorem is a pre-stateme... |
| mppsthm 36265 | A provable pre-statement i... |
| mthmblem 36266 | Lemma for ~ mthmb . (Cont... |
| mthmb 36267 | If two statements have the... |
| mthmpps 36268 | Given a theorem, there is ... |
| mclsppslem 36269 | The closure is closed unde... |
| mclspps 36270 | The closure is closed unde... |
| rexxfr3d 36324 | Transfer existential quant... |
| rexxfr3dALT 36325 | Longer proof of ~ rexxfr3d... |
| rspssbasd 36326 | The span of a set of ring ... |
| ellcsrspsn 36327 | Membership in a left coset... |
| ply1divalg3 36328 | Uniqueness of polynomial r... |
| r1peuqusdeg1 36329 | Uniqueness of polynomial r... |
| problem1 36351 | Practice problem 1. Clues... |
| problem2 36352 | Practice problem 2. Clues... |
| problem3 36353 | Practice problem 3. Clues... |
| problem4 36354 | Practice problem 4. Clues... |
| problem5 36355 | Practice problem 5. Clues... |
| quad3 36356 | Variant of quadratic equat... |
| climuzcnv 36357 | Utility lemma to convert b... |
| sinccvglem 36358 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 36359 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 36360 | The circumference of a cir... |
| elfzm12 36361 | Membership in a curtailed ... |
| nn0seqcvg 36362 | A strictly-decreasing nonn... |
| lediv2aALT 36363 | Division of both sides of ... |
| abs2sqlei 36364 | The absolute values of two... |
| abs2sqlti 36365 | The absolute values of two... |
| abs2sqle 36366 | The absolute values of two... |
| abs2sqlt 36367 | The absolute values of two... |
| abs2difi 36368 | Difference of absolute val... |
| abs2difabsi 36369 | Absolute value of differen... |
| 2thALT 36370 | Alternate proof of ~ 2th .... |
| orbi2iALT 36371 | Alternate proof of ~ orbi2... |
| pm3.48ALT 36372 | Alternate proof of ~ pm3.4... |
| 3jcadALT 36373 | Alternate proof of ~ 3jcad... |
| currybi 36374 | Biconditional version of C... |
| antnest 36375 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 36376 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 36377 | A law of nested antecedent... |
| antnestlaw2 36378 | A law of nested antecedent... |
| antnestlaw3 36379 | A law of nested antecedent... |
| antnestALT 36380 | Alternative proof of ~ ant... |
| axextprim 36387 | ~ ax-ext without distinct ... |
| axrepprim 36388 | ~ ax-rep without distinct ... |
| axunprim 36389 | ~ ax-un without distinct v... |
| axpowprim 36390 | ~ ax-pow without distinct ... |
| axregprim 36391 | ~ ax-reg without distinct ... |
| axinfprim 36392 | ~ ax-inf without distinct ... |
| axacprim 36393 | ~ ax-ac without distinct v... |
| untelirr 36394 | We call a class "untanged"... |
| untuni 36395 | The union of a class is un... |
| untsucf 36396 | If a class is untangled, t... |
| unt0 36397 | The empty set is untangled... |
| untint 36398 | If there is an untangled e... |
| efrunt 36399 | If ` A ` is well-founded b... |
| untangtr 36400 | A transitive class is unta... |
| 3jaodd 36401 | Double deduction form of ~... |
| 3orit 36402 | Closed form of ~ 3ori . (... |
| biimpexp 36403 | A biconditional in the ant... |
| nepss 36404 | Two classes are unequal if... |
| 3ccased 36405 | Triple disjunction form of... |
| dfso3 36406 | Expansion of the definitio... |
| brtpid1 36407 | A binary relation involvin... |
| brtpid2 36408 | A binary relation involvin... |
| brtpid3 36409 | A binary relation involvin... |
| iota5f 36410 | A method for computing iot... |
| jath 36411 | Closed form of ~ ja . Pro... |
| xpab 36412 | Cartesian product of two c... |
| nnuni 36413 | The union of a finite ordi... |
| sqdivzi 36414 | Distribution of square ove... |
| supfz 36415 | The supremum of a finite s... |
| inffz 36416 | The infimum of a finite se... |
| fz0n 36417 | The sequence ` ( 0 ... ( N... |
| shftvalg 36418 | Value of a sequence shifte... |
| divcnvlin 36419 | Limit of the ratio of two ... |
| climlec3 36420 | Comparison of a constant t... |
| iexpire 36421 | ` _i ` raised to itself is... |
| bcneg1 36422 | The binomial coefficient o... |
| bcm1nt 36423 | The proportion of one bino... |
| bcprod 36424 | A product identity for bin... |
| bccolsum 36425 | A column-sum rule for bino... |
| iprodefisumlem 36426 | Lemma for ~ iprodefisum . ... |
| iprodefisum 36427 | Applying the exponential f... |
| iprodgam 36428 | An infinite product versio... |
| faclimlem1 36429 | Lemma for ~ faclim . Clos... |
| faclimlem2 36430 | Lemma for ~ faclim . Show... |
| faclimlem3 36431 | Lemma for ~ faclim . Alge... |
| faclim 36432 | An infinite product expres... |
| iprodfac 36433 | An infinite product expres... |
| faclim2 36434 | Another factorial limit du... |
| gcd32 36435 | Swap the second and third ... |
| gcdabsorb 36436 | Absorption law for gcd. (... |
| dftr6 36437 | A potential definition of ... |
| coep 36438 | Composition with the membe... |
| coepr 36439 | Composition with the conve... |
| dffr5 36440 | A quantifier-free definiti... |
| dfso2 36441 | Quantifier-free definition... |
| br8 36442 | Substitution for an eight-... |
| br6 36443 | Substitution for a six-pla... |
| br4 36444 | Substitution for a four-pl... |
| cnvco1 36445 | Another distributive law o... |
| cnvco2 36446 | Another distributive law o... |
| eldm3 36447 | Quantifier-free definition... |
| elrn3 36448 | Quantifier-free definition... |
| pocnv 36449 | The converse of a partial ... |
| socnv 36450 | The converse of a strict o... |
| elintfv 36451 | Membership in an intersect... |
| funpsstri 36452 | A condition for subset tri... |
| fundmpss 36453 | If a class ` F ` is a prop... |
| funsseq 36454 | Given two functions with e... |
| fununiq 36455 | The uniqueness condition o... |
| funbreq 36456 | An equality condition for ... |
| br1steq 36457 | Uniqueness condition for t... |
| br2ndeq 36458 | Uniqueness condition for t... |
| dfdm5 36459 | Definition of domain in te... |
| dfrn5 36460 | Definition of range in ter... |
| opelco3 36461 | Alternate way of saying th... |
| elima4 36462 | Quantifier-free expression... |
| fv1stcnv 36463 | The value of the converse ... |
| fv2ndcnv 36464 | The value of the converse ... |
| elpotr 36465 | A class of transitive sets... |
| dford5reg 36466 | Given ~ ax-reg , an ordina... |
| dfon2lem1 36467 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 36468 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 36469 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 36470 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 36471 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 36472 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 36473 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 36474 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 36475 | Lemma for ~ dfon2 . A cla... |
| dfon2 36476 | ` On ` consists of all set... |
| rdgprc0 36477 | The value of the recursive... |
| rdgprc 36478 | The value of the recursive... |
| dfrdg2 36479 | Alternate definition of th... |
| dfrdg3 36480 | Generalization of ~ dfrdg2... |
| axextdfeq 36481 | A version of ~ ax-ext for ... |
| ax8dfeq 36482 | A version of ~ ax-8 for us... |
| axextdist 36483 | ~ ax-ext with distinctors ... |
| axextbdist 36484 | ~ axextb with distinctors ... |
| 19.12b 36485 | Version of ~ 19.12vv with ... |
| exnel 36486 | There is always a set not ... |
| distel 36487 | Distinctors in terms of me... |
| axextndbi 36488 | ~ axextnd as a bicondition... |
| hbntg 36489 | A more general form of ~ h... |
| hbimtg 36490 | A more general and closed ... |
| hbaltg 36491 | A more general and closed ... |
| hbng 36492 | A more general form of ~ h... |
| hbimg 36493 | A more general form of ~ h... |
| wsuceq123 36498 | Equality theorem for well-... |
| wsuceq1 36499 | Equality theorem for well-... |
| wsuceq2 36500 | Equality theorem for well-... |
| wsuceq3 36501 | Equality theorem for well-... |
| nfwsuc 36502 | Bound-variable hypothesis ... |
| wlimeq12 36503 | Equality theorem for the l... |
| wlimeq1 36504 | Equality theorem for the l... |
| wlimeq2 36505 | Equality theorem for the l... |
| nfwlim 36506 | Bound-variable hypothesis ... |
| elwlim 36507 | Membership in the limit cl... |
| wzel 36508 | The zero of a well-founded... |
| wsuclem 36509 | Lemma for the supremum pro... |
| wsucex 36510 | Existence theorem for well... |
| wsuccl 36511 | If ` X ` is a set with an ... |
| wsuclb 36512 | A well-founded successor i... |
| wlimss 36513 | The class of limit points ... |
| txpss3v 36562 | A tail Cartesian product i... |
| txprel 36563 | A tail Cartesian product i... |
| brtxp 36564 | Characterize a ternary rel... |
| brtxp2 36565 | The binary relation over a... |
| dfpprod2 36566 | Expanded definition of par... |
| pprodcnveq 36567 | A converse law for paralle... |
| pprodss4v 36568 | The parallel product is a ... |
| brpprod 36569 | Characterize a quaternary ... |
| brpprod3a 36570 | Condition for parallel pro... |
| brpprod3b 36571 | Condition for parallel pro... |
| relsset 36572 | The subset class is a bina... |
| brsset 36573 | For sets, the ` SSet ` bin... |
| idsset 36574 | ` _I ` is equal to the int... |
| eltrans 36575 | Membership in the class of... |
| dfon3 36576 | A quantifier-free definiti... |
| dfon4 36577 | Another quantifier-free de... |
| brtxpsd 36578 | Expansion of a common form... |
| brtxpsd2 36579 | Another common abbreviatio... |
| brtxpsd3 36580 | A third common abbreviatio... |
| relbigcup 36581 | The ` Bigcup ` relationshi... |
| brbigcup 36582 | Binary relation over ` Big... |
| dfbigcup2 36583 | ` Bigcup ` using maps-to n... |
| fobigcup 36584 | ` Bigcup ` maps the univer... |
| fnbigcup 36585 | ` Bigcup ` is a function o... |
| fvbigcup 36586 | For sets, ` Bigcup ` yield... |
| elfix 36587 | Membership in the fixpoint... |
| elfix2 36588 | Alternative membership in ... |
| dffix2 36589 | The fixpoints of a class i... |
| fixssdm 36590 | The fixpoints of a class a... |
| fixssrn 36591 | The fixpoints of a class a... |
| fixcnv 36592 | The fixpoints of a class a... |
| fixun 36593 | The fixpoint operator dist... |
| ellimits 36594 | Membership in the class of... |
| limitssson 36595 | The class of all limit ord... |
| dfom5b 36596 | A quantifier-free definiti... |
| sscoid 36597 | A condition for subset and... |
| dffun10 36598 | Another potential definiti... |
| elfuns 36599 | Membership in the class of... |
| elfunsg 36600 | Closed form of ~ elfuns . ... |
| brsingle 36601 | The binary relation form o... |
| elsingles 36602 | Membership in the class of... |
| fnsingle 36603 | The singleton relationship... |
| fvsingle 36604 | The value of the singleton... |
| dfsingles2 36605 | Alternate definition of th... |
| snelsingles 36606 | A singleton is a member of... |
| dfiota3 36607 | A definition of iota using... |
| dffv5 36608 | Another quantifier-free de... |
| unisnif 36609 | Express union of singleton... |
| brimage 36610 | Binary relation form of th... |
| brimageg 36611 | Closed form of ~ brimage .... |
| funimage 36612 | ` Image A ` is a function.... |
| fnimage 36613 | ` Image R ` is a function ... |
| imageval 36614 | The image functor in maps-... |
| fvimage 36615 | Value of the image functor... |
| brcart 36616 | Binary relation form of th... |
| brdomain 36617 | Binary relation form of th... |
| brrange 36618 | Binary relation form of th... |
| brdomaing 36619 | Closed form of ~ brdomain ... |
| brrangeg 36620 | Closed form of ~ brrange .... |
| brimg 36621 | Binary relation form of th... |
| brapply 36622 | Binary relation form of th... |
| brcup 36623 | Binary relation form of th... |
| brcap 36624 | Binary relation form of th... |
| lemsuccf 36625 | Lemma for unfolding differ... |
| brsuccf 36626 | Binary relation form of th... |
| dfsuccf2 36627 | Alternate definition of Sc... |
| funpartlem 36628 | Lemma for ~ funpartfun . ... |
| funpartfun 36629 | The functional part of ` F... |
| funpartss 36630 | The functional part of ` F... |
| funpartfv 36631 | The function value of the ... |
| fullfunfnv 36632 | The full functional part o... |
| fullfunfv 36633 | The function value of the ... |
| brfullfun 36634 | A binary relation form con... |
| brrestrict 36635 | Binary relation form of th... |
| dfrecs2 36636 | A quantifier-free definiti... |
| dfrdg4 36637 | A quantifier-free definiti... |
| dfint3 36638 | Quantifier-free definition... |
| imagesset 36639 | The Image functor applied ... |
| brub 36640 | Binary relation form of th... |
| brlb 36641 | Binary relation form of th... |
| dffr7 36642 | Alternate quantifier-free ... |
| altopex 36647 | Alternative ordered pairs ... |
| altopthsn 36648 | Two alternate ordered pair... |
| altopeq12 36649 | Equality for alternate ord... |
| altopeq1 36650 | Equality for alternate ord... |
| altopeq2 36651 | Equality for alternate ord... |
| altopth1 36652 | Equality of the first memb... |
| altopth2 36653 | Equality of the second mem... |
| altopthg 36654 | Alternate ordered pair the... |
| altopthbg 36655 | Alternate ordered pair the... |
| altopth 36656 | The alternate ordered pair... |
| altopthb 36657 | Alternate ordered pair the... |
| altopthc 36658 | Alternate ordered pair the... |
| altopthd 36659 | Alternate ordered pair the... |
| altxpeq1 36660 | Equality for alternate Car... |
| altxpeq2 36661 | Equality for alternate Car... |
| elaltxp 36662 | Membership in alternate Ca... |
| altopelaltxp 36663 | Alternate ordered pair mem... |
| altxpsspw 36664 | An inclusion rule for alte... |
| altxpexg 36665 | The alternate Cartesian pr... |
| rankaltopb 36666 | Compute the rank of an alt... |
| nfaltop 36667 | Bound-variable hypothesis ... |
| sbcaltop 36668 | Distribution of class subs... |
| cgrrflx2d 36671 | Deduction form of ~ axcgrr... |
| cgrtr4d 36672 | Deduction form of ~ axcgrt... |
| cgrtr4and 36673 | Deduction form of ~ axcgrt... |
| cgrrflx 36674 | Reflexivity law for congru... |
| cgrrflxd 36675 | Deduction form of ~ cgrrfl... |
| cgrcomim 36676 | Congruence commutes on the... |
| cgrcom 36677 | Congruence commutes betwee... |
| cgrcomand 36678 | Deduction form of ~ cgrcom... |
| cgrtr 36679 | Transitivity law for congr... |
| cgrtrand 36680 | Deduction form of ~ cgrtr ... |
| cgrtr3 36681 | Transitivity law for congr... |
| cgrtr3and 36682 | Deduction form of ~ cgrtr3... |
| cgrcoml 36683 | Congruence commutes on the... |
| cgrcomr 36684 | Congruence commutes on the... |
| cgrcomlr 36685 | Congruence commutes on bot... |
| cgrcomland 36686 | Deduction form of ~ cgrcom... |
| cgrcomrand 36687 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36688 | Deduction form of ~ cgrcom... |
| cgrtriv 36689 | Degenerate segments are co... |
| cgrid2 36690 | Identity law for congruenc... |
| cgrdegen 36691 | Two congruent segments are... |
| brofs 36692 | Binary relation form of th... |
| 5segofs 36693 | Rephrase ~ ax5seg using th... |
| ofscom 36694 | The outer five segment pre... |
| cgrextend 36695 | Link congruence over a pai... |
| cgrextendand 36696 | Deduction form of ~ cgrext... |
| segconeq 36697 | Two points that satisfy th... |
| segconeu 36698 | Existential uniqueness ver... |
| btwntriv2 36699 | Betweenness always holds f... |
| btwncomim 36700 | Betweenness commutes. Imp... |
| btwncom 36701 | Betweenness commutes. (Co... |
| btwncomand 36702 | Deduction form of ~ btwnco... |
| btwntriv1 36703 | Betweenness always holds f... |
| btwnswapid 36704 | If you can swap the first ... |
| btwnswapid2 36705 | If you can swap arguments ... |
| btwnintr 36706 | Inner transitivity law for... |
| btwnexch3 36707 | Exchange the first endpoin... |
| btwnexch3and 36708 | Deduction form of ~ btwnex... |
| btwnouttr2 36709 | Outer transitivity law for... |
| btwnexch2 36710 | Exchange the outer point o... |
| btwnouttr 36711 | Outer transitivity law for... |
| btwnexch 36712 | Outer transitivity law for... |
| btwnexchand 36713 | Deduction form of ~ btwnex... |
| btwndiff 36714 | There is always a ` c ` di... |
| trisegint 36715 | A line segment between two... |
| funtransport 36718 | The ` TransportTo ` relati... |
| fvtransport 36719 | Calculate the value of the... |
| transportcl 36720 | Closure law for segment tr... |
| transportprops 36721 | Calculate the defining pro... |
| brifs 36730 | Binary relation form of th... |
| ifscgr 36731 | Inner five segment congrue... |
| cgrsub 36732 | Removing identical parts f... |
| brcgr3 36733 | Binary relation form of th... |
| cgr3permute3 36734 | Permutation law for three-... |
| cgr3permute1 36735 | Permutation law for three-... |
| cgr3permute2 36736 | Permutation law for three-... |
| cgr3permute4 36737 | Permutation law for three-... |
| cgr3permute5 36738 | Permutation law for three-... |
| cgr3tr4 36739 | Transitivity law for three... |
| cgr3com 36740 | Commutativity law for thre... |
| cgr3rflx 36741 | Identity law for three-pla... |
| cgrxfr 36742 | A line segment can be divi... |
| btwnxfr 36743 | A condition for extending ... |
| colinrel 36744 | Colinearity is a relations... |
| brcolinear2 36745 | Alternate colinearity bina... |
| brcolinear 36746 | The binary relation form o... |
| colinearex 36747 | The colinear predicate exi... |
| colineardim1 36748 | If ` A ` is colinear with ... |
| colinearperm1 36749 | Permutation law for coline... |
| colinearperm3 36750 | Permutation law for coline... |
| colinearperm2 36751 | Permutation law for coline... |
| colinearperm4 36752 | Permutation law for coline... |
| colinearperm5 36753 | Permutation law for coline... |
| colineartriv1 36754 | Trivial case of colinearit... |
| colineartriv2 36755 | Trivial case of colinearit... |
| btwncolinear1 36756 | Betweenness implies coline... |
| btwncolinear2 36757 | Betweenness implies coline... |
| btwncolinear3 36758 | Betweenness implies coline... |
| btwncolinear4 36759 | Betweenness implies coline... |
| btwncolinear5 36760 | Betweenness implies coline... |
| btwncolinear6 36761 | Betweenness implies coline... |
| colinearxfr 36762 | Transfer law for colineari... |
| lineext 36763 | Extend a line with a missi... |
| brofs2 36764 | Change some conditions for... |
| brifs2 36765 | Change some conditions for... |
| brfs 36766 | Binary relation form of th... |
| fscgr 36767 | Congruence law for the gen... |
| linecgr 36768 | Congruence rule for lines.... |
| linecgrand 36769 | Deduction form of ~ linecg... |
| lineid 36770 | Identity law for points on... |
| idinside 36771 | Law for finding a point in... |
| endofsegid 36772 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36773 | Deduction form of ~ endofs... |
| btwnconn1lem1 36774 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36775 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36776 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36777 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36778 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36779 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36780 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36781 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36782 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36783 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36784 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36785 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36786 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36787 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36788 | Connectitivy law for betwe... |
| btwnconn2 36789 | Another connectivity law f... |
| btwnconn3 36790 | Inner connectivity law for... |
| midofsegid 36791 | If two points fall in the ... |
| segcon2 36792 | Generalization of ~ axsegc... |
| brsegle 36795 | Binary relation form of th... |
| brsegle2 36796 | Alternate characterization... |
| seglecgr12im 36797 | Substitution law for segme... |
| seglecgr12 36798 | Substitution law for segme... |
| seglerflx 36799 | Segment comparison is refl... |
| seglemin 36800 | Any segment is at least as... |
| segletr 36801 | Segment less than is trans... |
| segleantisym 36802 | Antisymmetry law for segme... |
| seglelin 36803 | Linearity law for segment ... |
| btwnsegle 36804 | If ` B ` falls between ` A... |
| colinbtwnle 36805 | Given three colinear point... |
| broutsideof 36808 | Binary relation form of ` ... |
| broutsideof2 36809 | Alternate form of ` Outsid... |
| outsidene1 36810 | Outsideness implies inequa... |
| outsidene2 36811 | Outsideness implies inequa... |
| btwnoutside 36812 | A principle linking outsid... |
| broutsideof3 36813 | Characterization of outsid... |
| outsideofrflx 36814 | Reflexivity of outsideness... |
| outsideofcom 36815 | Commutativity law for outs... |
| outsideoftr 36816 | Transitivity law for outsi... |
| outsideofeq 36817 | Uniqueness law for ` Outsi... |
| outsideofeu 36818 | Given a nondegenerate ray,... |
| outsidele 36819 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36820 | Outside of implies colinea... |
| funray 36827 | Show that the ` Ray ` rela... |
| fvray 36828 | Calculate the value of the... |
| funline 36829 | Show that the ` Line ` rel... |
| linedegen 36830 | When ` Line ` is applied w... |
| fvline 36831 | Calculate the value of the... |
| liness 36832 | A line is a subset of the ... |
| fvline2 36833 | Alternate definition of a ... |
| lineunray 36834 | A line is composed of a po... |
| lineelsb2 36835 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36836 | Reflexivity law for line m... |
| linecom 36837 | Commutativity law for line... |
| linerflx2 36838 | Reflexivity law for line m... |
| ellines 36839 | Membership in the set of a... |
| linethru 36840 | If ` A ` is a line contain... |
| hilbert1.1 36841 | There is a line through an... |
| hilbert1.2 36842 | There is at most one line ... |
| linethrueu 36843 | There is a unique line goi... |
| lineintmo 36844 | Two distinct lines interse... |
| fwddifval 36849 | Calculate the value of the... |
| fwddifnval 36850 | The value of the forward d... |
| fwddifn0 36851 | The value of the n-iterate... |
| fwddifnp1 36852 | The value of the n-iterate... |
| rank0 36853 | The rank of the empty set ... |
| rankeq1o 36854 | The only set with rank ` 1... |
| hftr 36855 | The class of all hereditar... |
| hfext 36856 | Extensionality for HF sets... |
| hfninf 36857 | ` _om ` is not hereditaril... |
| nmulfn 36860 | Natural multiplication is ... |
| nmulprop 36861 | Show closure and value of ... |
| nmulcl 36862 | Closure law for natural mu... |
| nmulval 36863 | Show the value of natural ... |
| nmulcld 36864 | Closure law for natural mu... |
| nmulcom 36865 | Natural multiplication is ... |
| nmulr0 36866 | Natural multiplication by ... |
| nmull0 36867 | Natural multiplication by ... |
| nmulrid 36868 | Identity law for natural m... |
| nmullid 36869 | Identity law for natural m... |
| onelond 36870 | An element of an ordinal n... |
| ontr2d 36871 | Transitive law for ordinal... |
| onelssd 36872 | An element of an ordinal n... |
| nmulr0d 36873 | Natural multiplication by ... |
| nmull0d 36874 | Natural multiplication by ... |
| nmulridd 36875 | Identity law for natural m... |
| nmullidd 36876 | Identity law for natural m... |
| nmulcomd 36877 | Natural multiplication com... |
| naddridd 36878 | Identity law for natural a... |
| naddlidd 36879 | Identity law for natural a... |
| naddcomd 36880 | Natural addition commutes.... |
| naddassd 36881 | Natural addition associate... |
| nadd32d 36882 | Commutative/associative la... |
| nmuladdel 36883 | Ordering relationship for ... |
| nmuladdss 36884 | Ordering relationship for ... |
| nmulss1 36885 | Natural multiplication pre... |
| nmulel1 36886 | Natural multiplication by ... |
| ltnmul 36887 | Characterize less-than a n... |
| nmulle 36888 | A condition for bounding a... |
| ltnadd 36889 | Condition for bounding a n... |
| naddle 36890 | Condition for bounding nat... |
| nadddilem1 36891 | Lemma for ~ nadddi . Prov... |
| nadddilem2 36892 | Lemma for ~ nadddi . Prov... |
| nadddilem3 36893 | Lemma for ~ nadddi . Prov... |
| nadddilem4 36894 | Lemma for ~ nadddi . Prov... |
| nadddi 36895 | Natural multiplication dis... |
| nadddid 36896 | Natural multiplication dis... |
| nadddird 36897 | Natural multiplication dis... |
| rmoeqi 36898 | Equality inference for res... |
| rmoeqbii 36899 | Equality inference for res... |
| reueqi 36900 | Equality inference for res... |
| reueqbii 36901 | Equality inference for res... |
| sbceqbii 36902 | Formula-building inference... |
| disjeq1i 36903 | Equality theorem for disjo... |
| disjeq12i 36904 | Equality theorem for disjo... |
| rabeqbii 36905 | Equality theorem for restr... |
| iuneq12i 36906 | Equality theorem for index... |
| iineq1i 36907 | Equality theorem for index... |
| iineq12i 36908 | Equality theorem for index... |
| riotaeqbii 36909 | Equivalent wff's and equal... |
| riotaeqi 36910 | Equal domains yield equal ... |
| ixpeq1i 36911 | Equality inference for inf... |
| ixpeq12i 36912 | Equality inference for inf... |
| sumeq2si 36913 | Equality inference for sum... |
| sumeq12si 36914 | Equality inference for sum... |
| prodeq2si 36915 | Equality inference for pro... |
| prodeq12si 36916 | Equality inference for pro... |
| itgeq12i 36917 | Equality inference for an ... |
| itgeq1i 36918 | Equality inference for an ... |
| itgeq2i 36919 | Equality inference for an ... |
| ditgeq123i 36920 | Equality inference for the... |
| ditgeq12i 36921 | Equality inference for the... |
| ditgeq3i 36922 | Equality inference for the... |
| rmoeqdv 36923 | Formula-building rule for ... |
| rmoeqbidv 36924 | Formula-building rule for ... |
| sbequbidv 36925 | Deduction substituting bot... |
| disjeq12dv 36926 | Equality theorem for disjo... |
| ixpeq12dv 36927 | Equality theorem for infin... |
| sumeq12sdv 36928 | Equality deduction for sum... |
| prodeq12sdv 36929 | Equality deduction for pro... |
| itgeq12sdv 36930 | Equality theorem for an in... |
| itgeq2sdv 36931 | Equality theorem for an in... |
| ditgeq123dv 36932 | Equality theorem for the d... |
| ditgeq12d 36933 | Equality theorem for the d... |
| ditgeq3sdv 36934 | Equality theorem for the d... |
| in-ax8 36935 | A proof of ~ ax-8 that doe... |
| ss-ax8 36936 | A proof of ~ ax-8 that doe... |
| cbvralvw2 36937 | Change bound variable and ... |
| cbvrexvw2 36938 | Change bound variable and ... |
| cbvrmovw2 36939 | Change bound variable and ... |
| cbvreuvw2 36940 | Change bound variable and ... |
| cbvsbcvw2 36941 | Change bound variable of a... |
| cbvcsbvw2 36942 | Change bound variable of a... |
| cbviunvw2 36943 | Change bound variable and ... |
| cbviinvw2 36944 | Change bound variable and ... |
| cbvmptvw2 36945 | Change bound variable and ... |
| cbvdisjvw2 36946 | Change bound variable and ... |
| cbvriotavw2 36947 | Change bound variable and ... |
| cbvoprab1vw 36948 | Change the first bound var... |
| cbvoprab2vw 36949 | Change the second bound va... |
| cbvoprab123vw 36950 | Change all bound variables... |
| cbvoprab23vw 36951 | Change the second and thir... |
| cbvoprab13vw 36952 | Change the first and third... |
| cbvmpovw2 36953 | Change bound variables and... |
| cbvmpo1vw2 36954 | Change domains and the fir... |
| cbvmpo2vw2 36955 | Change domains and the sec... |
| cbvixpvw2 36956 | Change bound variable and ... |
| cbvsumvw2 36957 | Change bound variable and ... |
| cbvprodvw2 36958 | Change bound variable and ... |
| cbvitgvw2 36959 | Change bound variable and ... |
| cbvditgvw2 36960 | Change bound variable and ... |
| cbvmodavw 36961 | Change bound variable in t... |
| cbveudavw 36962 | Change bound variable in t... |
| cbvrmodavw 36963 | Change bound variable in t... |
| cbvreudavw 36964 | Change bound variable in t... |
| cbvsbdavw 36965 | Change bound variable in p... |
| cbvsbdavw2 36966 | Change bound variable in p... |
| cbvabdavw 36967 | Change bound variable in c... |
| cbvsbcdavw 36968 | Change bound variable of a... |
| cbvsbcdavw2 36969 | Change bound variable of a... |
| cbvcsbdavw 36970 | Change bound variable of a... |
| cbvcsbdavw2 36971 | Change bound variable of a... |
| cbvrabdavw 36972 | Change bound variable in r... |
| cbviundavw 36973 | Change bound variable in i... |
| cbviindavw 36974 | Change bound variable in i... |
| cbvopab1davw 36975 | Change the first bound var... |
| cbvopab2davw 36976 | Change the second bound va... |
| cbvopabdavw 36977 | Change bound variables in ... |
| cbvmptdavw 36978 | Change bound variable in a... |
| cbvdisjdavw 36979 | Change bound variable in a... |
| cbviotadavw 36980 | Change bound variable in a... |
| cbvriotadavw 36981 | Change bound variable in a... |
| cbvoprab1davw 36982 | Change the first bound var... |
| cbvoprab2davw 36983 | Change the second bound va... |
| cbvoprab3davw 36984 | Change the third bound var... |
| cbvoprab123davw 36985 | Change all bound variables... |
| cbvoprab12davw 36986 | Change the first and secon... |
| cbvoprab23davw 36987 | Change the second and thir... |
| cbvoprab13davw 36988 | Change the first and third... |
| cbvixpdavw 36989 | Change bound variable in a... |
| cbvsumdavw 36990 | Change bound variable in a... |
| cbvproddavw 36991 | Change bound variable in a... |
| cbvitgdavw 36992 | Change bound variable in a... |
| cbvditgdavw 36993 | Change bound variable in a... |
| cbvrmodavw2 36994 | Change bound variable and ... |
| cbvreudavw2 36995 | Change bound variable and ... |
| cbvrabdavw2 36996 | Change bound variable and ... |
| cbviundavw2 36997 | Change bound variable and ... |
| cbviindavw2 36998 | Change bound variable and ... |
| cbvmptdavw2 36999 | Change bound variable and ... |
| cbvdisjdavw2 37000 | Change bound variable and ... |
| cbvriotadavw2 37001 | Change bound variable and ... |
| cbvmpodavw2 37002 | Change bound variable and ... |
| cbvmpo1davw2 37003 | Change first bound variabl... |
| cbvmpo2davw2 37004 | Change second bound variab... |
| cbvixpdavw2 37005 | Change bound variable and ... |
| cbvsumdavw2 37006 | Change bound variable and ... |
| cbvproddavw2 37007 | Change bound variable and ... |
| cbvitgdavw2 37008 | Change bound variable and ... |
| cbvditgdavw2 37009 | Change bound variable and ... |
| mpomulnzcnf 37010 | Multiplication maps nonzer... |
| a1i14 37011 | Add two antecedents to a w... |
| a1i24 37012 | Add two antecedents to a w... |
| exp5d 37013 | An exportation inference. ... |
| exp5g 37014 | An exportation inference. ... |
| exp5k 37015 | An exportation inference. ... |
| exp56 37016 | An exportation inference. ... |
| exp58 37017 | An exportation inference. ... |
| exp510 37018 | An exportation inference. ... |
| exp511 37019 | An exportation inference. ... |
| exp512 37020 | An exportation inference. ... |
| 3com12d 37021 | Commutation in consequent.... |
| imp5p 37022 | A triple importation infer... |
| imp5q 37023 | A triple importation infer... |
| subtr 37024 | Transitivity of implicit s... |
| subtr2 37025 | Transitivity of implicit s... |
| trer 37026 | A relation intersected wit... |
| elicc3 37027 | An equivalent membership c... |
| finminlem 37028 | A useful lemma about finit... |
| gtinf 37029 | Any number greater than an... |
| opnrebl 37030 | A set is open in the stand... |
| opnrebl2 37031 | A set is open in the stand... |
| nn0prpwlem 37032 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 37033 | Two nonnegative integers a... |
| topbnd 37034 | Two equivalent expressions... |
| opnbnd 37035 | A set is open iff it is di... |
| cldbnd 37036 | A set is closed iff it con... |
| ntruni 37037 | A union of interiors is a ... |
| clsun 37038 | A pairwise union of closur... |
| clsint2 37039 | The closure of an intersec... |
| opnregcld 37040 | A set is regularly closed ... |
| cldregopn 37041 | A set if regularly open if... |
| neiin 37042 | Two neighborhoods intersec... |
| hmeoclda 37043 | Homeomorphisms preserve cl... |
| hmeocldb 37044 | Homeomorphisms preserve cl... |
| ivthALT 37045 | An alternate proof of the ... |
| fnerel 37048 | Fineness is a relation. (... |
| isfne 37049 | The predicate " ` B ` is f... |
| isfne4 37050 | The predicate " ` B ` is f... |
| isfne4b 37051 | A condition for a topology... |
| isfne2 37052 | The predicate " ` B ` is f... |
| isfne3 37053 | The predicate " ` B ` is f... |
| fnebas 37054 | A finer cover covers the s... |
| fnetg 37055 | A finer cover generates a ... |
| fnessex 37056 | If ` B ` is finer than ` A... |
| fneuni 37057 | If ` B ` is finer than ` A... |
| fneint 37058 | If a cover is finer than a... |
| fness 37059 | A cover is finer than its ... |
| fneref 37060 | Reflexivity of the finenes... |
| fnetr 37061 | Transitivity of the finene... |
| fneval 37062 | Two covers are finer than ... |
| fneer 37063 | Fineness intersected with ... |
| topfne 37064 | Fineness for covers corres... |
| topfneec 37065 | A cover is equivalent to a... |
| topfneec2 37066 | A topology is precisely id... |
| fnessref 37067 | A cover is finer iff it ha... |
| refssfne 37068 | A cover is a refinement if... |
| neibastop1 37069 | A collection of neighborho... |
| neibastop2lem 37070 | Lemma for ~ neibastop2 . ... |
| neibastop2 37071 | In the topology generated ... |
| neibastop3 37072 | The topology generated by ... |
| topmtcl 37073 | The meet of a collection o... |
| topmeet 37074 | Two equivalent formulation... |
| topjoin 37075 | Two equivalent formulation... |
| fnemeet1 37076 | The meet of a collection o... |
| fnemeet2 37077 | The meet of equivalence cl... |
| fnejoin1 37078 | Join of equivalence classe... |
| fnejoin2 37079 | Join of equivalence classe... |
| fgmin 37080 | Minimality property of a g... |
| neifg 37081 | The neighborhood filter of... |
| tailfval 37082 | The tail function for a di... |
| tailval 37083 | The tail of an element in ... |
| eltail 37084 | An element of a tail. (Co... |
| tailf 37085 | The tail function of a dir... |
| tailini 37086 | A tail contains its initia... |
| tailfb 37087 | The collection of tails of... |
| filnetlem1 37088 | Lemma for ~ filnet . Chan... |
| filnetlem2 37089 | Lemma for ~ filnet . The ... |
| filnetlem3 37090 | Lemma for ~ filnet . (Con... |
| filnetlem4 37091 | Lemma for ~ filnet . (Con... |
| filnet 37092 | A filter has the same conv... |
| tb-ax1 37093 | The first of three axioms ... |
| tb-ax2 37094 | The second of three axioms... |
| tb-ax3 37095 | The third of three axioms ... |
| tbsyl 37096 | The weak syllogism from Ta... |
| re1ax2lem 37097 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 37098 | ~ ax-2 rederived from the ... |
| naim1 37099 | Constructor theorem for ` ... |
| naim2 37100 | Constructor theorem for ` ... |
| naim1i 37101 | Constructor rule for ` -/\... |
| naim2i 37102 | Constructor rule for ` -/\... |
| naim12i 37103 | Constructor rule for ` -/\... |
| nabi1i 37104 | Constructor rule for ` -/\... |
| nabi2i 37105 | Constructor rule for ` -/\... |
| nabi12i 37106 | Constructor rule for ` -/\... |
| df3nandALT1 37109 | The double nand expressed ... |
| df3nandALT2 37110 | The double nand expressed ... |
| andnand1 37111 | Double and in terms of dou... |
| imnand2 37112 | An ` -> ` nand relation. ... |
| nalfal 37113 | Not all sets hold ` F. ` a... |
| nexntru 37114 | There does not exist a set... |
| nexfal 37115 | There does not exist a set... |
| neufal 37116 | There does not exist exact... |
| neutru 37117 | There does not exist exact... |
| nmotru 37118 | There does not exist at mo... |
| mofal 37119 | There exist at most one se... |
| nrmo 37120 | "At most one" restricted e... |
| meran1 37121 | A single axiom for proposi... |
| meran2 37122 | A single axiom for proposi... |
| meran3 37123 | A single axiom for proposi... |
| waj-ax 37124 | A single axiom for proposi... |
| lukshef-ax2 37125 | A single axiom for proposi... |
| arg-ax 37126 | A single axiom for proposi... |
| negsym1 37127 | In the paper "On Variable ... |
| imsym1 37128 | A symmetry with ` -> ` . ... |
| bisym1 37129 | A symmetry with ` <-> ` . ... |
| consym1 37130 | A symmetry with ` /\ ` . ... |
| dissym1 37131 | A symmetry with ` \/ ` . ... |
| nandsym1 37132 | A symmetry with ` -/\ ` . ... |
| unisym1 37133 | A symmetry with ` A. ` . ... |
| exisym1 37134 | A symmetry with ` E. ` . ... |
| unqsym1 37135 | A symmetry with ` E! ` . ... |
| amosym1 37136 | A symmetry with ` E* ` . ... |
| subsym1 37137 | A symmetry with ` [ x / y ... |
| ontopbas 37138 | An ordinal number is a top... |
| onsstopbas 37139 | The class of ordinal numbe... |
| onpsstopbas 37140 | The class of ordinal numbe... |
| ontgval 37141 | The topology generated fro... |
| ontgsucval 37142 | The topology generated fro... |
| onsuctop 37143 | A successor ordinal number... |
| onsuctopon 37144 | One of the topologies on a... |
| ordtoplem 37145 | Membership of the class of... |
| ordtop 37146 | An ordinal is a topology i... |
| onsucconni 37147 | A successor ordinal number... |
| onsucconn 37148 | A successor ordinal number... |
| ordtopconn 37149 | An ordinal topology is con... |
| onintopssconn 37150 | An ordinal topology is con... |
| onsuct0 37151 | A successor ordinal number... |
| ordtopt0 37152 | An ordinal topology is T_0... |
| onsucsuccmpi 37153 | The successor of a success... |
| onsucsuccmp 37154 | The successor of a success... |
| limsucncmpi 37155 | The successor of a limit o... |
| limsucncmp 37156 | The successor of a limit o... |
| ordcmp 37157 | An ordinal topology is com... |
| ssoninhaus 37158 | The ordinal topologies ` 1... |
| onint1 37159 | The ordinal T_1 spaces are... |
| oninhaus 37160 | The ordinal Hausdorff spac... |
| fveleq 37161 | Please add description her... |
| findfvcl 37162 | Please add description her... |
| findreccl 37163 | Please add description her... |
| findabrcl 37164 | Please add description her... |
| nnssi2 37165 | Convert a theorem for real... |
| nnssi3 37166 | Convert a theorem for real... |
| nndivsub 37167 | Please add description her... |
| nndivlub 37168 | A factor of a positive int... |
| ee7.2aOLD 37171 | Lemma for Euclid's Element... |
| weiunval 37172 | Value of the relation cons... |
| weiunlem 37173 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 37174 | Lemma for ~ weiunfr . (Co... |
| weiunpo 37175 | A partial ordering on an i... |
| weiunso 37176 | A strict ordering on an in... |
| weiunfr 37177 | A well-founded relation on... |
| weiunse 37178 | The relation constructed i... |
| weiunwe 37179 | A well-ordering on an inde... |
| numiunnum 37180 | An indexed union of sets i... |
| axtco 37181 | Axiom of Transitive Contai... |
| axtco1 37183 | Strong form of the Axiom o... |
| axtco2 37184 | Weak form of the Axiom of ... |
| axtco1from2 37185 | Strong form ~ axtco1 of th... |
| axtco1g 37186 | Strong form of the Axiom o... |
| axtco2g 37187 | Weak form of the Axiom of ... |
| axtcond 37188 | A version of the Axiom of ... |
| axuntco 37189 | Derivation of ~ ax-un from... |
| axnulregtco 37190 | Derivation of ~ ax-nul fro... |
| elALTtco 37191 | Derivation of ~ el from ~ ... |
| tz9.1ctco 37192 | Version of ~ tz9.1c derive... |
| tz9.1tco 37193 | Version of ~ tz9.1 derived... |
| tr0elw 37194 | Every nonempty transitive ... |
| tr0el 37195 | Every nonempty transitive ... |
| ttceq 37198 | Equality theorem for trans... |
| ttceqi 37199 | Equality inference for tra... |
| ttceqd 37200 | Equality deduction for tra... |
| nfttc 37201 | Bound-variable hypothesis ... |
| ttcid 37202 | The transitive closure con... |
| ttctr 37203 | The transitive closure of ... |
| ttctr2 37204 | The transitive closure of ... |
| ttctr3 37205 | The transitive closure of ... |
| ttcmin 37206 | The transitive closure of ... |
| ttcexrg 37207 | If the transitive closure ... |
| ttcss 37208 | A transitive closure conta... |
| ttcss2 37209 | The subclass relationship ... |
| ttcel 37210 | A transitive closure conta... |
| ttcel2 37211 | Elements turn into subclas... |
| ttctrid 37212 | The transitive closure of ... |
| ttcidm 37213 | The transitive closure ope... |
| ssttctr 37214 | Transitivity of ` A C_ TC+... |
| elttctr 37215 | Transitivity of ` A e. TC+... |
| dfttc2g 37216 | A shorter expression for t... |
| ttc0 37217 | The transitive closure of ... |
| ttc00 37218 | A class has an empty trans... |
| csbttc 37219 | Distribute proper substitu... |
| ttcuniun 37220 | Relationship between ` TC+... |
| ttciunun 37221 | Relationship between ` TC+... |
| ttcun 37222 | Distribute union of two cl... |
| ttcuni 37223 | Distribute union of a clas... |
| ttciun 37224 | Distribute indexed union t... |
| ttcpwss 37225 | The transitive closure of ... |
| ttcsnssg 37226 | The transitive closure is ... |
| ttcsnidg 37227 | The singleton transitive c... |
| ttcsnmin 37228 | The singleton transitive c... |
| ttcsng 37229 | Relationship between ` TC+... |
| ttcsnexg 37230 | If the transitive closure ... |
| ttcsnexbig 37231 | The transitive closure of ... |
| ttcsntrsucg 37232 | The singleton transitive c... |
| dfttc3gw 37233 | If the transitive closure ... |
| ttcwf 37234 | A set is well-founded iff ... |
| ttcwf2 37235 | If a transitive closure cl... |
| ttcwf3 37236 | The sets whose transitive ... |
| ttc0elw 37237 | If a transitive closure is... |
| dfttc4lem1 37238 | Lemma for ~ dfttc4 . (Con... |
| dfttc4lem2 37239 | Lemma for ~ dfttc4 . (Con... |
| dfttc4 37240 | An alternative expression ... |
| elttcirr 37241 | Irreflexivity of ` A e. TC... |
| ttcexg 37242 | The transitive closure of ... |
| ttcexbi 37243 | A class is a set iff its t... |
| dfttc3g 37244 | The transitive closure of ... |
| ttc0el 37245 | A transitive closure conta... |
| mh-setind 37246 | Principle of set induction... |
| mh-setindnd 37247 | A version of ~ mh-setind w... |
| regsfromregtco 37248 | Derivation of ~ ax-regs fr... |
| regsfromsetind 37249 | Derivation of ~ ax-regs fr... |
| regsfromunir1 37250 | Derivation of ~ ax-regs fr... |
| mh-inf3f1 37251 | A variant of ~ inf3 . If ... |
| mh-inf3sn 37252 | Version of ~ inf3 for the ... |
| mh-prprimbi 37253 | Shortest possible version ... |
| mh-unprimbi 37254 | Shortest possible version ... |
| mh-regprimbi 37255 | Shortest possible version ... |
| mh-infprim1bi 37256 | Shortest possible axiom of... |
| mh-infprim2bi 37257 | Shortest possible axiom of... |
| mh-infprim3bi 37258 | An axiom of infinity in pr... |
| dnival 37259 | Value of the "distance to ... |
| dnicld1 37260 | Closure theorem for the "d... |
| dnicld2 37261 | Closure theorem for the "d... |
| dnif 37262 | The "distance to nearest i... |
| dnizeq0 37263 | The distance to nearest in... |
| dnizphlfeqhlf 37264 | The distance to nearest in... |
| rddif2 37265 | Variant of ~ rddif . (Con... |
| dnibndlem1 37266 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 37267 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 37268 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 37269 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 37270 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 37271 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 37272 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 37273 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 37274 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 37275 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 37276 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 37277 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 37278 | Lemma for ~ dnibnd . (Con... |
| dnibnd 37279 | The "distance to nearest i... |
| dnicn 37280 | The "distance to nearest i... |
| knoppcnlem1 37281 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 37282 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 37283 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 37284 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 37285 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 37286 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 37287 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 37288 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 37289 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 37290 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 37291 | Lemma for ~ knoppcn . (Co... |
| knoppcn 37292 | The continuous nowhere dif... |
| knoppcld 37293 | Closure theorem for Knopp'... |
| unblimceq0lem 37294 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 37295 | If ` F ` is unbounded near... |
| unbdqndv1 37296 | If the difference quotient... |
| unbdqndv2lem1 37297 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 37298 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 37299 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 37300 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 37301 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 37302 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 37303 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 37304 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 37305 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 37306 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 37307 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 37308 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 37309 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 37310 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 37311 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 37312 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 37313 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 37314 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 37315 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 37316 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 37317 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 37318 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 37319 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 37320 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 37321 | Lemma for ~ knoppndv . (C... |
| knoppndv 37322 | The continuous nowhere dif... |
| knoppf 37323 | Knopp's function is a func... |
| knoppcn2 37324 | Variant of ~ knoppcn with ... |
| cnndvlem1 37325 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 37326 | Lemma for ~ cnndv . (Cont... |
| cnndv 37327 | There exists a continuous ... |
| bj-mp2c 37328 | A double _modus ponens_ in... |
| bj-mp2d 37329 | A double _modus ponens_ in... |
| bj-0 37330 | A syntactic theorem. See ... |
| bj-1 37331 | In this proof, the use of ... |
| bj-poni 37332 | Inference associated with ... |
| bj-nnclav 37333 | When ` F. ` is substituted... |
| bj-nnclavi 37334 | Inference associated with ... |
| bj-nnclavc 37335 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 37336 | Inference associated with ... |
| bj-jarrii 37337 | Inference associated with ... |
| bj-imim21 37338 | The propositional function... |
| bj-imim21i 37339 | The propositional function... |
| bj-imim11 37340 | The propositional function... |
| bj-imim11i 37341 | The propositional function... |
| bj-peircestab 37342 | Over minimal implicational... |
| bj-stabpeirce 37343 | This minimal implicational... |
| bj-bisimpl 37344 | Implication from equivalen... |
| bj-bisimpr 37345 | Implication from equivalen... |
| bj-syl66ib 37346 | A mixed syllogism inferenc... |
| bj-orim2 37347 | Proof of ~ orim2 from the ... |
| bj-currypeirce 37348 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 37349 | Peirce's axiom ~ peirce im... |
| bj-animbi 37350 | Conjunction in terms of im... |
| bj-currypara 37351 | Curry's paradox. Note tha... |
| bj-con2com 37352 | A commuted form of the con... |
| bj-con2comi 37353 | Inference associated with ... |
| bj-nimn 37354 | If a formula is true, then... |
| bj-nimni 37355 | Inference associated with ... |
| bj-peircei 37356 | Inference associated with ... |
| bj-looinvi 37357 | Inference associated with ... |
| bj-looinvii 37358 | Inference associated with ... |
| bj-mt2bi 37359 | Version of ~ mt2 where the... |
| bj-fal 37360 | Shortening of ~ fal using ... |
| bj-ntrufal 37361 | The negation of a theorem ... |
| bj-dfnul2 37362 | Alternate definition of th... |
| bj-jaoi1 37363 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 37364 | Shortens ~ consensus (110>... |
| bj-dfbi4 37365 | Alternate definition of th... |
| bj-dfbi5 37366 | Alternate definition of th... |
| bj-dfbi6 37367 | Alternate definition of th... |
| bj-bijust0ALT 37368 | Alternate proof of ~ bijus... |
| bj-bijust00 37369 | A self-implication does no... |
| bj-consensus 37370 | Version of ~ consensus exp... |
| bj-consensusALT 37371 | Alternate proof of ~ bj-co... |
| bj-df-ifc 37372 | Candidate definition for t... |
| bj-dfif 37373 | Alternate definition of th... |
| bj-ififc 37374 | A biconditional connecting... |
| bj-imbi12 37375 | Uncurried (imported) form ... |
| bj-falor 37376 | Dual of ~ truan (which has... |
| bj-falor2 37377 | Dual of ~ truan . (Contri... |
| bj-bibibi 37378 | A property of the bicondit... |
| bj-imn3ani 37379 | Duplication of ~ bnj1224 .... |
| bj-andnotim 37380 | Two ways of expressing a c... |
| bj-bi3ant 37381 | This used to be in the mai... |
| bj-bisym 37382 | This used to be in the mai... |
| bj-bixor 37383 | Equivalence of two ternary... |
| bj-axdd2 37384 | This implication, proved u... |
| bj-axd2d 37385 | This implication, proved u... |
| bj-axtd 37386 | This implication, proved f... |
| bj-gl4 37387 | In a normal modal logic, t... |
| bj-axc4 37388 | Over minimal calculus, the... |
| prvlem1 37393 | An elementary property of ... |
| prvlem2 37394 | An elementary property of ... |
| bj-babygodel 37395 | See the section header com... |
| bj-babylob 37396 | See the section header com... |
| bj-godellob 37397 | Proof of Gödel's theo... |
| bj-exexalal 37398 | A lemma for changing bound... |
| bj-genr 37399 | Generalization rule on the... |
| bj-genl 37400 | Generalization rule on the... |
| bj-genan 37401 | Generalization rule on a c... |
| bj-mpgs 37402 | From a closed form theorem... |
| bj-almp 37403 | A quantified form of ~ ax-... |
| bj-sylggt 37404 | Stronger form of ~ sylgt ,... |
| bj-alrimg 37405 | The general form of the *a... |
| bj-sylgt2 37406 | Uncurried (imported) form ... |
| bj-nexdh 37407 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 37408 | Uncurried (imported) form ... |
| bj-alimii 37409 | Inference associated with ... |
| bj-ala1i 37410 | Add an antecedent in a uni... |
| bj-almpi 37411 | A quantified form of ~ mpi... |
| bj-almpig 37412 | A partially quantified for... |
| bj-alsyl 37413 | Syllogism under the univer... |
| bj-2alim 37414 | Closed form of ~ 2alimi . ... |
| bj-alimdh 37415 | General instance of ~ alim... |
| bj-alrimdh 37416 | Deduction form of Theorem ... |
| bj-alrimd 37417 | A slightly more general ~ ... |
| bj-exa1i 37418 | Add an antecedent in an ex... |
| bj-alanim 37419 | Closed form of ~ alanimi .... |
| bj-2albi 37420 | Closed form of ~ 2albii . ... |
| bj-notalbii 37421 | Equivalence of universal q... |
| bj-2exim 37422 | Closed form of ~ 2eximi . ... |
| bj-2exbi 37423 | Closed form of ~ 2exbii . ... |
| bj-3exbi 37424 | Closed form of ~ 3exbii . ... |
| bj-sylget 37425 | Dual statement of ~ sylgt ... |
| bj-sylget2 37426 | Uncurried (imported) form ... |
| bj-exlimg 37427 | The general form of the *e... |
| bj-sylge 37428 | Dual statement of ~ sylg (... |
| bj-exlimd 37429 | A slightly more general ~ ... |
| bj-nfimexal 37430 | A weak from of nonfreeness... |
| bj-exim 37431 | Theorem 19.22 of [Margaris... |
| bj-alexim 37432 | Closed form of ~ aleximi .... |
| bj-aleximiALT 37433 | Alternate proof of ~ alexi... |
| bj-hbxfrbi 37434 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 37435 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 37436 | Distribute quantifiers ove... |
| bj-exalimi 37437 | An inference for distribut... |
| bj-eximcom 37438 | A commuted form of ~ exim ... |
| bj-exalims 37439 | Distributing quantifiers o... |
| bj-exalimsi 37440 | An inference for distribut... |
| bj-axdd2ALT 37441 | Alternate proof of ~ bj-ax... |
| bj-ax12ig 37442 | A lemma used to prove a we... |
| bj-ax12i 37443 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 37444 | Closed form of ~ nfim and ... |
| bj-spimnfe 37445 | A universal specification ... |
| bj-spimenfa 37446 | An existential generalizat... |
| bj-spim 37447 | A lemma for universal spec... |
| bj-spime 37448 | A lemma for existential ge... |
| bj-cbvalimd0 37449 | A lemma for alpha-renaming... |
| bj-cbvalimdlem 37450 | A lemma for alpha-renaming... |
| bj-cbveximdlem 37451 | A lemma for alpha-renaming... |
| bj-cbvalimd 37452 | A lemma for alpha-renaming... |
| bj-cbveximd 37453 | A lemma for alpha-renaming... |
| bj-cbvalimdv 37454 | A lemma for alpha-renaming... |
| bj-cbveximdv 37455 | A lemma for alpha-renaming... |
| bj-spvw 37456 | Version of ~ spvw and ~ 19... |
| bj-spvew 37457 | Version of ~ 19.8v and ~ 1... |
| bj-alextruim 37458 | An equivalent expression f... |
| bj-exextruan 37459 | An equivalent expression f... |
| bj-cbvalvv 37460 | Universally quantifying ov... |
| bj-cbvexvv 37461 | Existentially quantifying ... |
| bj-cbvaw 37462 | Universally quantifying ov... |
| bj-cbvew 37463 | Existentially quantifying ... |
| bj-cbveaw 37464 | Universally quantifying ov... |
| bj-cbvaew 37465 | Exixtentially quantifying ... |
| bj-ax12wlem 37466 | A lemma used to prove a we... |
| bj-cbval 37467 | Changing a bound variable ... |
| bj-cbvex 37468 | Changing a bound variable ... |
| bj-df-sb 37471 | Proposed definition to rep... |
| bj-sbcex 37472 | Proof of ~ sbcex when taki... |
| bj-dfsbc 37473 | Proof of ~ df-sbc when tak... |
| bj-ssbeq 37474 | Substitution in an equalit... |
| bj-ssblem1 37475 | A lemma for the definiens ... |
| bj-ssblem2 37476 | An instance of ~ ax-11 pro... |
| bj-ax12v 37477 | A weaker form of ~ ax-12 a... |
| bj-ax12 37478 | Remove a DV condition from... |
| bj-ax12ssb 37479 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 37480 | Special case of ~ 19.41 pr... |
| bj-equsexval 37481 | Special case of ~ equsexv ... |
| bj-subst 37482 | Proof of ~ sbalex from cor... |
| bj-ssbid2 37483 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 37484 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 37485 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 37486 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 37487 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 37488 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 37489 | Proof of ~ ax6e (hence ~ a... |
| bj-spim0 37490 | A universal specialization... |
| bj-spimvwt 37491 | Closed form of ~ spimvw . ... |
| bj-spnfw 37492 | Theorem close to a closed ... |
| bj-cbvexiw 37493 | Change bound variable. Th... |
| bj-cbvexivw 37494 | Change bound variable. Th... |
| bj-modald 37495 | A short form of the axiom ... |
| bj-denot 37496 | A weakening of ~ ax-6 and ... |
| bj-eqs 37497 | A lemma for substitutions,... |
| bj-cbvexw 37498 | Change bound variable. Th... |
| bj-ax12w 37499 | The general statement that... |
| bj-ax89 37500 | A theorem which could be u... |
| bj-cleljusti 37501 | One direction of ~ cleljus... |
| bj-alcomexcom 37502 | Commutation of two existen... |
| bj-hbald 37503 | General statement that ~ h... |
| bj-hbalt 37504 | Closed form of (general in... |
| bj-hbal 37505 | More general instance of ~... |
| axc11n11 37506 | Proof of ~ axc11n from { ~... |
| axc11n11r 37507 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 37508 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 37509 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 37510 | Alternate proof of ~ bj-ax... |
| bj-sb 37511 | A weak variant of ~ sbid2 ... |
| bj-modalbe 37512 | The predicate-calculus ver... |
| bj-spst 37513 | Closed form of ~ sps . On... |
| bj-19.21bit 37514 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 37515 | Closed form of ~ 19.23bi .... |
| bj-nexrt 37516 | Closed form of ~ nexr . C... |
| bj-alrim 37517 | Closed form of ~ alrimi . ... |
| bj-alrim2 37518 | Uncurried (imported) form ... |
| bj-nfdt0 37519 | A theorem close to a close... |
| bj-nfdt 37520 | Closed form of ~ nf5d and ... |
| bj-nexdt 37521 | Closed form of ~ nexd . (... |
| bj-nexdvt 37522 | Closed form of ~ nexdv . ... |
| bj-alexbiex 37523 | Adding a second quantifier... |
| bj-exexbiex 37524 | Adding a second quantifier... |
| bj-alalbial 37525 | Adding a second quantifier... |
| bj-exalbial 37526 | Adding a second quantifier... |
| bj-19.9htbi 37527 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 37528 | Strengthening ~ hbnt by re... |
| bj-biexal1 37529 | A general FOL biconditiona... |
| bj-biexal2 37530 | When ` ph ` is substituted... |
| bj-biexal3 37531 | When ` ph ` is substituted... |
| bj-bialal 37532 | When ` ph ` is substituted... |
| bj-biexex 37533 | When ` ph ` is substituted... |
| bj-hbexd 37534 | A more general instance of... |
| bj-hbext 37535 | Closed form of ~ bj-hbex a... |
| bj-hbex 37536 | A more general instance of... |
| bj-nfalt 37537 | Closed form of ~ nfal . (... |
| bj-nfext 37538 | Closed form of ~ nfex . (... |
| bj-eeanvw 37539 | Version of ~ exdistrv with... |
| bj-modal4 37540 | First-order logic form of ... |
| bj-modal4e 37541 | First-order logic form of ... |
| bj-modalb 37542 | A short form of the axiom ... |
| bj-wnf1 37543 | When ` ph ` is substituted... |
| bj-wnf2 37544 | When ` ph ` is substituted... |
| bj-wnfanf 37545 | When ` ph ` is substituted... |
| bj-wnfenf 37546 | When ` ph ` is substituted... |
| bj-19.12 37547 | See ~ 19.12 . Could be la... |
| bj-substax12 37548 | Equivalent form of the axi... |
| bj-substw 37549 | Weak form of the LHS of ~ ... |
| bj-nnfa 37552 | Nonfreeness implies the eq... |
| bj-nnfad 37553 | Nonfreeness implies the eq... |
| bj-nnfai 37554 | Nonfreeness implies the eq... |
| bj-nnfe 37555 | Nonfreeness implies the eq... |
| bj-nnfed 37556 | Nonfreeness implies the eq... |
| bj-nnfei 37557 | Nonfreeness implies the eq... |
| bj-nnfea 37558 | Nonfreeness implies the eq... |
| bj-nnfead 37559 | Nonfreeness implies the eq... |
| bj-nnfeai 37560 | Nonfreeness implies the eq... |
| bj-alnnf 37561 | In deduction-style proofs,... |
| bj-alnnf2 37562 | If a proposition holds, th... |
| bj-dfnnf2 37563 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 37564 | New nonfreeness implies ol... |
| bj-nnfim1 37565 | A consequence of nonfreene... |
| bj-nnfim2 37566 | A consequence of nonfreene... |
| bj-nnftht 37567 | A variable is nonfree in a... |
| bj-nnfth 37568 | A variable is nonfree in a... |
| bj-nnf-alrim 37569 | Proof of the closed form o... |
| bj-stdpc5t 37570 | Alias of ~ bj-nnf-alrim fo... |
| bj-nnfbi 37571 | If two formulas are equiva... |
| bj-nnfbd0 37572 | If two formulas are equiva... |
| bj-nnfbii 37573 | If two formulas are equiva... |
| bj-nnfnt 37574 | A variable is nonfree in a... |
| bj-nnfnth 37575 | A variable is nonfree in t... |
| bj-nnfim 37576 | Nonfreeness in the anteced... |
| bj-nnfimd 37577 | Nonfreeness in the anteced... |
| bj-nnfan 37578 | Nonfreeness in both conjun... |
| bj-nnfand 37579 | Nonfreeness in both conjun... |
| bj-nnfor 37580 | Nonfreeness in both disjun... |
| bj-nnford 37581 | Nonfreeness in both disjun... |
| bj-nnfbit 37582 | Nonfreeness in both sides ... |
| bj-nnfbid 37583 | Nonfreeness in both sides ... |
| bj-nnf-exlim 37584 | Proof of the closed form o... |
| bj-19.21t 37585 | Statement ~ 19.21t proved ... |
| bj-19.23t 37586 | Statement ~ 19.23t proved ... |
| bj-19.36im 37587 | One direction of ~ 19.36 f... |
| bj-19.37im 37588 | One direction of ~ 19.37 f... |
| bj-19.42t 37589 | Closed form of ~ 19.42 fro... |
| bj-19.41t 37590 | Closed form of ~ 19.41 fro... |
| bj-pm11.53vw 37591 | Version of ~ pm11.53v with... |
| bj-nnfv 37592 | A non-occurring variable i... |
| bj-nnfbd 37593 | If two formulas are equiva... |
| bj-pm11.53a 37594 | A variant of ~ pm11.53v . ... |
| bj-equsvt 37595 | A variant of ~ equsv . (C... |
| bj-equsalvwd 37596 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 37597 | Variant of ~ equsexvw . (... |
| bj-nnf-spim 37598 | A universal specialization... |
| bj-nnf-spime 37599 | An existential generalizat... |
| bj-nnf-cbvaliv 37600 | The only DV conditions are... |
| bj-sbievwd 37601 | Variant of ~ sbievw . (Co... |
| bj-sbft 37602 | Version of ~ sbft using ` ... |
| bj-nnf-cbvali 37603 | Compared with ~ bj-nnf-cbv... |
| bj-nnf-cbval 37604 | Compared with ~ cbvalv1 , ... |
| bj-dfnnf3 37605 | Alternate definition of no... |
| bj-nfnnfTEMP 37606 | New nonfreeness is equival... |
| bj-wnfnf 37607 | When ` ph ` is substituted... |
| bj-nnfa1 37608 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 37609 | See ~ nfe1 . (Contributed... |
| bj-nnflemaa 37610 | One of four lemmas for non... |
| bj-nnflemee 37611 | One of four lemmas for non... |
| bj-nnflemae 37612 | One of four lemmas for non... |
| bj-nnflemea 37613 | One of four lemmas for non... |
| bj-nnfalt 37614 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 37615 | See ~ nfex and ~ bj-nfext ... |
| bj-pm11.53v 37616 | Version of ~ pm11.53v with... |
| bj-axc10 37617 | Alternate proof of ~ axc10... |
| bj-alequex 37618 | A fol lemma. See ~ aleque... |
| bj-spimt2 37619 | A step in the proof of ~ s... |
| bj-cbv3ta 37620 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 37621 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 37622 | A theorem close to a close... |
| bj-hbsb3 37623 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 37624 | A theorem close to a close... |
| bj-nfs1t2 37625 | A theorem close to a close... |
| bj-nfs1 37626 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 37627 | Version of ~ axc10 with a ... |
| bj-spimtv 37628 | Version of ~ spimt with a ... |
| bj-cbv3hv2 37629 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 37630 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 37631 | Version of ~ cbv2h with a ... |
| bj-cbv2v 37632 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 37633 | Version of ~ cbvald with a... |
| bj-cbvexdv 37634 | Version of ~ cbvexd with a... |
| bj-cbval2vv 37635 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 37636 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 37637 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 37638 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 37639 | Version of ~ cbvex4v with ... |
| bj-equsalhv 37640 | Version of ~ equsalh with ... |
| bj-axc11nv 37641 | Version of ~ axc11n with a... |
| bj-aecomsv 37642 | Version of ~ aecoms with a... |
| bj-axc11v 37643 | Version of ~ axc11 with a ... |
| bj-drnf2v 37644 | Version of ~ drnf2 with a ... |
| bj-equs45fv 37645 | Version of ~ equs45f with ... |
| bj-hbs1 37646 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 37647 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 37648 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 37649 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 37650 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 37651 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 37652 | Biconditional version of a... |
| bj-hbaeb 37653 | Biconditional version of ~... |
| bj-hbnaeb 37654 | Biconditional version of ~... |
| bj-dvv 37655 | A special instance of ~ bj... |
| bj-equsal1t 37656 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 37657 | Inference associated with ... |
| bj-equsal1 37658 | One direction of ~ equsal ... |
| bj-equsal2 37659 | One direction of ~ equsal ... |
| bj-equsal 37660 | Shorter proof of ~ equsal ... |
| stdpc5t 37661 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 37662 | More direct proof of ~ std... |
| 2stdpc5 37663 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 37664 | Proof of ~ 19.21t from ~ s... |
| exlimii 37665 | Inference associated with ... |
| ax11-pm 37666 | Proof of ~ ax-11 similar t... |
| ax6er 37667 | Commuted form of ~ ax6e . ... |
| exlimiieq1 37668 | Inferring a theorem when i... |
| exlimiieq2 37669 | Inferring a theorem when i... |
| ax11-pm2 37670 | Proof of ~ ax-11 from the ... |
| bj-sbsb 37671 | Biconditional showing two ... |
| bj-dfsb2 37672 | Alternate (dual) definitio... |
| bj-sbf3 37673 | Substitution has no effect... |
| bj-sbf4 37674 | Substitution has no effect... |
| bj-eu3f 37675 | Version of ~ eu3v where th... |
| bj-sblem1 37676 | Lemma for substitution. (... |
| bj-sblem2 37677 | Lemma for substitution. (... |
| bj-sblem 37678 | Lemma for substitution. (... |
| bj-sbievw1 37679 | Lemma for substitution. (... |
| bj-sbievw2 37680 | Lemma for substitution. (... |
| bj-sbievw 37681 | Lemma for substitution. C... |
| bj-sbievv 37682 | Version of ~ sbie with a s... |
| bj-moeub 37683 | Uniqueness is equivalent t... |
| bj-sbidmOLD 37684 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 37685 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 37686 | Curried (exported) form of... |
| bj-dvelimv 37687 | A version of ~ dvelim usin... |
| bj-nfeel2 37688 | Nonfreeness in a membershi... |
| bj-axc14nf 37689 | Proof of a version of ~ ax... |
| bj-axc14 37690 | Alternate proof of ~ axc14... |
| mobidvALT 37691 | Alternate proof of ~ mobid... |
| sbn1ALT 37692 | Alternate proof of ~ sbn1 ... |
| eliminable1 37693 | A theorem used to prove th... |
| eliminable2a 37694 | A theorem used to prove th... |
| eliminable2b 37695 | A theorem used to prove th... |
| eliminable2c 37696 | A theorem used to prove th... |
| eliminable3a 37697 | A theorem used to prove th... |
| eliminable3b 37698 | A theorem used to prove th... |
| eliminable-velab 37699 | A theorem used to prove th... |
| eliminable-veqab 37700 | A theorem used to prove th... |
| eliminable-abeqv 37701 | A theorem used to prove th... |
| eliminable-abeqab 37702 | A theorem used to prove th... |
| eliminable-abelv 37703 | A theorem used to prove th... |
| eliminable-abelab 37704 | A theorem used to prove th... |
| bj-denoteslem 37705 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37706 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37707 | Moved to main as ~ issettr... |
| bj-elabtru 37708 | This is as close as we can... |
| bj-issetwt 37709 | Closed form of ~ bj-issetw... |
| bj-issetw 37710 | The closest one can get to... |
| bj-issetiv 37711 | Version of ~ bj-isseti wit... |
| bj-isseti 37712 | Version of ~ isseti with a... |
| bj-ralvw 37713 | A weak version of ~ ralv n... |
| bj-rexvw 37714 | A weak version of ~ rexv n... |
| bj-rababw 37715 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37716 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37717 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37718 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37719 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37720 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37721 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37722 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37723 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37724 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37725 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37726 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37727 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37728 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37729 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37730 | Remove from ~ spcimdv depe... |
| elelb 37731 | Equivalence between two co... |
| bj-pwvrelb 37732 | Characterization of the el... |
| bj-nfcsym 37733 | The nonfreeness quantifier... |
| bj-sbeqALT 37734 | Substitution in an equalit... |
| bj-sbeq 37735 | Distribute proper substitu... |
| bj-sbceqgALT 37736 | Distribute proper substitu... |
| bj-csbsnlem 37737 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37738 | Substitution in a singleto... |
| bj-sbel1 37739 | Version of ~ sbcel1g when ... |
| bj-abv 37740 | The class of sets verifyin... |
| bj-abvALT 37741 | Alternate version of ~ bj-... |
| bj-ab0 37742 | The class of sets verifyin... |
| bj-abf 37743 | Shorter proof of ~ abf (wh... |
| bj-csbprc 37744 | More direct proof of ~ csb... |
| bj-exlimvmpi 37745 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 37746 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 37747 | Lemma for theorems of the ... |
| bj-exlimmpbir 37748 | Lemma for theorems of the ... |
| bj-vtoclf 37749 | Remove dependency on ~ ax-... |
| bj-vtocl 37750 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 37751 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 37752 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 37753 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 37754 | A version of ~ vtoclg with... |
| bj-rabeqbid 37755 | Version of ~ rabeqbidv wit... |
| bj-seex 37756 | Version of ~ seex with a d... |
| bj-nfcf 37757 | Version of ~ df-nfc with a... |
| bj-sepg 37758 | Version of ~ sepg which do... |
| bj-inex1gALT 37759 | Proof of ~ inex1g from ~ s... |
| bj-elabd2ALT 37760 | Alternate proof of ~ elabd... |
| bj-unrab 37761 | Generalization of ~ unrab ... |
| bj-inrab 37762 | Generalization of ~ inrab ... |
| bj-inrab2 37763 | Shorter proof of ~ inrab .... |
| bj-inrab3 37764 | Generalization of ~ dfrab3... |
| bj-rabtr 37765 | Restricted class abstracti... |
| bj-rabtrALT 37766 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 37767 | Proof of ~ bj-rabtr found ... |
| bj-gabss 37770 | Inclusion of generalized c... |
| bj-gabssd 37771 | Inclusion of generalized c... |
| bj-gabeqd 37772 | Equality of generalized cl... |
| bj-gabeqis 37773 | Equality of generalized cl... |
| bj-elgab 37774 | Elements of a generalized ... |
| bj-gabima 37775 | Generalized class abstract... |
| bj-ru1 37778 | A version of Russell's par... |
| bj-ru 37779 | Remove dependency on ~ ax-... |
| currysetlem 37780 | Lemma for ~ currysetlem , ... |
| curryset 37781 | Curry's paradox in set the... |
| currysetlem1 37782 | Lemma for ~ currysetALT . ... |
| currysetlem2 37783 | Lemma for ~ currysetALT . ... |
| currysetlem3 37784 | Lemma for ~ currysetALT . ... |
| currysetALT 37785 | Alternate proof of ~ curry... |
| bj-n0i 37786 | Inference associated with ... |
| bj-disjsn01 37787 | Disjointness of the single... |
| bj-0nel1 37788 | The empty set does not bel... |
| bj-1nel0 37789 | ` 1o ` does not belong to ... |
| bj-xpimasn 37790 | The image of a singleton, ... |
| bj-xpima1sn 37791 | The image of a singleton b... |
| bj-xpima1snALT 37792 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 37793 | The image of a singleton b... |
| bj-xpnzex 37794 | If the first factor of a p... |
| bj-xpexg2 37795 | Curried (exported) form of... |
| bj-xpnzexb 37796 | If the first factor of a p... |
| bj-cleq 37797 | Substitution property for ... |
| bj-snsetex 37798 | The class of sets "whose s... |
| bj-clexab 37799 | Sethood of certain classes... |
| bj-sngleq 37802 | Substitution property for ... |
| bj-elsngl 37803 | Characterization of the el... |
| bj-snglc 37804 | Characterization of the el... |
| bj-snglss 37805 | The singletonization of a ... |
| bj-0nelsngl 37806 | The empty set is not a mem... |
| bj-snglinv 37807 | Inverse of singletonizatio... |
| bj-snglex 37808 | A class is a set if and on... |
| bj-tageq 37811 | Substitution property for ... |
| bj-eltag 37812 | Characterization of the el... |
| bj-0eltag 37813 | The empty set belongs to t... |
| bj-tagn0 37814 | The tagging of a class is ... |
| bj-tagss 37815 | The tagging of a class is ... |
| bj-snglsstag 37816 | The singletonization is in... |
| bj-sngltagi 37817 | The singletonization is in... |
| bj-sngltag 37818 | The singletonization and t... |
| bj-tagci 37819 | Characterization of the el... |
| bj-tagcg 37820 | Characterization of the el... |
| bj-taginv 37821 | Inverse of tagging. (Cont... |
| bj-tagex 37822 | A class is a set if and on... |
| bj-xtageq 37823 | The products of a given cl... |
| bj-xtagex 37824 | The product of a set and t... |
| bj-projeq 37827 | Substitution property for ... |
| bj-projeq2 37828 | Substitution property for ... |
| bj-projun 37829 | The class projection on a ... |
| bj-projex 37830 | Sethood of the class proje... |
| bj-projval 37831 | Value of the class project... |
| bj-1upleq 37834 | Substitution property for ... |
| bj-pr1eq 37837 | Substitution property for ... |
| bj-pr1un 37838 | The first projection prese... |
| bj-pr1val 37839 | Value of the first project... |
| bj-pr11val 37840 | Value of the first project... |
| bj-pr1ex 37841 | Sethood of the first proje... |
| bj-1uplth 37842 | The characteristic propert... |
| bj-1uplex 37843 | A monuple is a set if and ... |
| bj-1upln0 37844 | A monuple is nonempty. (C... |
| bj-2upleq 37847 | Substitution property for ... |
| bj-pr21val 37848 | Value of the first project... |
| bj-pr2eq 37851 | Substitution property for ... |
| bj-pr2un 37852 | The second projection pres... |
| bj-pr2val 37853 | Value of the second projec... |
| bj-pr22val 37854 | Value of the second projec... |
| bj-pr2ex 37855 | Sethood of the second proj... |
| bj-2uplth 37856 | The characteristic propert... |
| bj-2uplex 37857 | A couple is a set if and o... |
| bj-2upln0 37858 | A couple is nonempty. (Co... |
| bj-2upln1upl 37859 | A couple is never equal to... |
| bj-rcleqf 37860 | Relative version of ~ cleq... |
| bj-rcleq 37861 | Relative version of ~ dfcl... |
| bj-reabeq 37862 | Relative form of ~ eqabb .... |
| bj-disj2r 37863 | Relative version of ~ ssdi... |
| bj-sscon 37864 | Contraposition law for rel... |
| bj-abex 37865 | Two ways of stating that t... |
| bj-clex 37866 | Two ways of stating that a... |
| bj-axsn 37867 | Two ways of stating the ax... |
| bj-snexg 37869 | A singleton built on a set... |
| bj-snex 37870 | A singleton is a set. See... |
| bj-axbun 37871 | Two ways of stating the ax... |
| bj-unexg 37873 | Existence of binary unions... |
| bj-prexg 37874 | Existence of unordered pai... |
| bj-prex 37875 | Existence of unordered pai... |
| bj-axadj 37876 | Two ways of stating the ax... |
| bj-adjg1 37878 | Existence of the result of... |
| bj-snfromadj 37879 | Singleton from adjunction ... |
| bj-prfromadj 37880 | Unordered pair from adjunc... |
| bj-adjfrombun 37881 | Adjunction from singleton ... |
| eleq2w2ALT 37882 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 37883 | Alternate proof of ~ clel3... |
| bj-pw0ALT 37884 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 37885 | Quantitative version of ~ ... |
| bj-unirel 37886 | Quantitative version of ~ ... |
| bj-elpwg 37887 | If the intersection of two... |
| bj-velpwALT 37888 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 37889 | Alternate proof of ~ elpwg... |
| bj-vjust 37890 | Justification theorem for ... |
| bj-nul 37891 | Two formulations of the ax... |
| bj-nuliota 37892 | Definition of the empty se... |
| bj-nuliotaALT 37893 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 37894 | Alternate proof of ~ vtocl... |
| bj-elsn12g 37895 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 37896 | Biconditional version of ~... |
| bj-pwcfsdom 37897 | Remove hypothesis from ~ p... |
| bj-grur1 37898 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 37899 | The extension of a predica... |
| bj-dfid2ALT 37900 | Alternate version of ~ dfi... |
| bj-0nelopab 37901 | The empty set is never an ... |
| bj-brrelex12ALT 37902 | Two classes related by a b... |
| bj-epelg 37903 | The membership relation an... |
| bj-epelb 37904 | Two classes are related by... |
| bj-nsnid 37905 | A set does not contain the... |
| bj-rdg0gALT 37906 | Alternate proof of ~ rdg0g... |
| bj-vn0ALT 37907 | Alternate proof of ~ vn0 w... |
| bj-axnul 37908 | Over the base theory ~ ax-... |
| bj-rep 37909 | Version of the axiom of re... |
| bj-axseprep 37910 | Axiom of separation (unive... |
| bj-axreprepsep 37911 | Strong axiom of replacemen... |
| bj-evaleq 37912 | Equality theorem for the `... |
| bj-evalfun 37913 | The evaluation at a class ... |
| bj-evalfn 37914 | The evaluation at a class ... |
| bj-evalf 37915 | The evaluation at a class ... |
| bj-evalval 37916 | Value of the evaluation at... |
| bj-evalid 37917 | The evaluation at a set of... |
| bj-ndxarg 37918 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 37919 | Closed general form of ~ s... |
| bj-rest00 37922 | An elementwise intersectio... |
| bj-restsn 37923 | An elementwise intersectio... |
| bj-restsnss 37924 | Special case of ~ bj-rests... |
| bj-restsnss2 37925 | Special case of ~ bj-rests... |
| bj-restsn0 37926 | An elementwise intersectio... |
| bj-restsn10 37927 | Special case of ~ bj-rests... |
| bj-restsnid 37928 | The elementwise intersecti... |
| bj-rest10 37929 | An elementwise intersectio... |
| bj-rest10b 37930 | Alternate version of ~ bj-... |
| bj-restn0 37931 | An elementwise intersectio... |
| bj-restn0b 37932 | Alternate version of ~ bj-... |
| bj-restpw 37933 | The elementwise intersecti... |
| bj-rest0 37934 | An elementwise intersectio... |
| bj-restb 37935 | An elementwise intersectio... |
| bj-restv 37936 | An elementwise intersectio... |
| bj-resta 37937 | An elementwise intersectio... |
| bj-restuni 37938 | The union of an elementwis... |
| bj-restuni2 37939 | The union of an elementwis... |
| bj-restreg 37940 | A reformulation of the axi... |
| bj-raldifsn 37941 | All elements in a set sati... |
| bj-0int 37942 | If ` A ` is a collection o... |
| bj-mooreset 37943 | A Moore collection is a se... |
| bj-ismoore 37946 | Characterization of Moore ... |
| bj-ismoored0 37947 | Necessary condition to be ... |
| bj-ismoored 37948 | Necessary condition to be ... |
| bj-ismoored2 37949 | Necessary condition to be ... |
| bj-ismooredr 37950 | Sufficient condition to be... |
| bj-ismooredr2 37951 | Sufficient condition to be... |
| bj-discrmoore 37952 | The powerclass ` ~P A ` is... |
| bj-0nmoore 37953 | The empty set is not a Moo... |
| bj-snmoore 37954 | A singleton is a Moore col... |
| bj-snmooreb 37955 | A singleton is a Moore col... |
| bj-prmoore 37956 | A pair formed of two neste... |
| bj-0nelmpt 37957 | The empty set is not an el... |
| bj-mptval 37958 | Value of a function given ... |
| bj-dfmpoa 37959 | An equivalent definition o... |
| bj-mpomptALT 37960 | Alternate proof of ~ mpomp... |
| setsstrset 37975 | Relation between ~ df-sets... |
| bj-nfald 37976 | Variant of ~ nfald . (Con... |
| bj-nfexd 37977 | Variant of ~ nfexd . (Con... |
| cgsex2gd 37978 | Implicit substitution infe... |
| copsex2gd 37979 | Implicit substitution infe... |
| copsex2d 37980 | Implicit substitution dedu... |
| copsex2b 37981 | Biconditional form of ~ co... |
| opelopabd 37982 | Membership of an ordered p... |
| opelopabb 37983 | Membership of an ordered p... |
| opelopabbv 37984 | Membership of an ordered p... |
| bj-opelrelex 37985 | The coordinates of an orde... |
| bj-opelresdm 37986 | If an ordered pair is in a... |
| bj-brresdm 37987 | If two classes are related... |
| brabd0 37988 | Expressing that two sets a... |
| brabd 37989 | Expressing that two sets a... |
| bj-brab2a1 37990 | "Unbounded" version of ~ b... |
| bj-opabssvv 37991 | A variant of ~ relopabiv (... |
| bj-funidres 37992 | The restricted identity re... |
| bj-opelidb 37993 | Characterization of the or... |
| bj-opelidb1 37994 | Characterization of the or... |
| bj-inexeqex 37995 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 37996 | If the intersection of two... |
| bj-opelid 37997 | Characterization of the or... |
| bj-ideqg 37998 | Characterization of the cl... |
| bj-ideqgALT 37999 | Alternate proof of ~ bj-id... |
| bj-ideqb 38000 | Characterization of classe... |
| bj-idres 38001 | Alternate expression for t... |
| bj-opelidres 38002 | Characterization of the or... |
| bj-idreseq 38003 | Sufficient condition for t... |
| bj-idreseqb 38004 | Characterization for two c... |
| bj-ideqg1 38005 | For sets, the identity rel... |
| bj-ideqg1ALT 38006 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 38007 | Characterization of the co... |
| bj-elid3 38008 | Characterization of the co... |
| bj-elid4 38009 | Characterization of the el... |
| bj-elid5 38010 | Characterization of the el... |
| bj-elid6 38011 | Characterization of the el... |
| bj-elid7 38012 | Characterization of the el... |
| bj-diagval 38015 | Value of the functionalize... |
| bj-diagval2 38016 | Value of the functionalize... |
| bj-eldiag 38017 | Characterization of the el... |
| bj-eldiag2 38018 | Characterization of the el... |
| bj-imdirvallem 38021 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 38022 | Value of the functionalize... |
| bj-imdirval2lem 38023 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 38024 | Value of the functionalize... |
| bj-imdirval3 38025 | Value of the functionalize... |
| bj-imdiridlem 38026 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 38027 | Functorial property of the... |
| bj-opelopabid 38028 | Membership in an ordered-p... |
| bj-opabco 38029 | Composition of ordered-pai... |
| bj-xpcossxp 38030 | The composition of two Car... |
| bj-imdirco 38031 | Functorial property of the... |
| bj-iminvval 38034 | Value of the functionalize... |
| bj-iminvval2 38035 | Value of the functionalize... |
| bj-iminvid 38036 | Functorial property of the... |
| bj-inftyexpitaufo 38043 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 38046 | An element of the circle a... |
| bj-inftyexpiinv 38049 | Utility theorem for the in... |
| bj-inftyexpiinj 38050 | Injectivity of the paramet... |
| bj-inftyexpidisj 38051 | An element of the circle a... |
| bj-ccinftydisj 38054 | The circle at infinity is ... |
| bj-elccinfty 38055 | A lemma for infinite exten... |
| bj-ccssccbar 38058 | Complex numbers are extend... |
| bj-ccinftyssccbar 38059 | Infinite extended complex ... |
| bj-pinftyccb 38062 | The class ` pinfty ` is an... |
| bj-pinftynrr 38063 | The extended complex numbe... |
| bj-minftyccb 38066 | The class ` minfty ` is an... |
| bj-minftynrr 38067 | The extended complex numbe... |
| bj-pinftynminfty 38068 | The extended complex numbe... |
| bj-rrhatsscchat 38077 | The real projective line i... |
| bj-imafv 38092 | If the direct image of a s... |
| bj-funun 38093 | Value of a function expres... |
| bj-fununsn1 38094 | Value of a function expres... |
| bj-fununsn2 38095 | Value of a function expres... |
| bj-fvsnun1 38096 | The value of a function wi... |
| bj-fvsnun2 38097 | The value of a function wi... |
| bj-fvmptunsn1 38098 | Value of a function expres... |
| bj-fvmptunsn2 38099 | Value of a function expres... |
| bj-iomnnom 38100 | The canonical bijection fr... |
| bj-smgrpssmgm 38109 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 38110 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 38111 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 38112 | Monoids are semigroups (el... |
| bj-cmnssmnd 38113 | Commutative monoids are mo... |
| bj-cmnssmndel 38114 | Commutative monoids are mo... |
| bj-grpssmnd 38115 | Groups are monoids. (Cont... |
| bj-grpssmndel 38116 | Groups are monoids (elemen... |
| bj-ablssgrp 38117 | Abelian groups are groups.... |
| bj-ablssgrpel 38118 | Abelian groups are groups ... |
| bj-ablsscmn 38119 | Abelian groups are commuta... |
| bj-ablsscmnel 38120 | Abelian groups are commuta... |
| bj-modssabl 38121 | (The additive groups of) m... |
| bj-vecssmod 38122 | Vector spaces are modules.... |
| bj-vecssmodel 38123 | Vector spaces are modules ... |
| bj-finsumval0 38126 | Value of a finite sum. (C... |
| bj-fvimacnv0 38127 | Variant of ~ fvimacnv wher... |
| bj-isvec 38128 | The predicate "is a vector... |
| bj-fldssdrng 38129 | Fields are division rings.... |
| bj-flddrng 38130 | Fields are division rings ... |
| bj-rrdrg 38131 | The field of real numbers ... |
| bj-isclm 38132 | The predicate "is a subcom... |
| bj-isrvec 38135 | The predicate "is a real v... |
| bj-rvecmod 38136 | Real vector spaces are mod... |
| bj-rvecssmod 38137 | Real vector spaces are mod... |
| bj-rvecrr 38138 | The field of scalars of a ... |
| bj-isrvecd 38139 | The predicate "is a real v... |
| bj-rvecvec 38140 | Real vector spaces are vec... |
| bj-isrvec2 38141 | The predicate "is a real v... |
| bj-rvecssvec 38142 | Real vector spaces are vec... |
| bj-rveccmod 38143 | Real vector spaces are sub... |
| bj-rvecsscmod 38144 | Real vector spaces are sub... |
| bj-rvecsscvec 38145 | Real vector spaces are sub... |
| bj-rveccvec 38146 | Real vector spaces are sub... |
| bj-rvecssabl 38147 | (The additive groups of) r... |
| bj-rvecabl 38148 | (The additive groups of) r... |
| bj-subcom 38149 | A consequence of commutati... |
| bj-lineqi 38150 | Solution of a (scalar) lin... |
| bj-bary1lem 38151 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 38152 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 38153 | Barycentric coordinates in... |
| bj-endval 38156 | Value of the monoid of end... |
| bj-endbase 38157 | Base set of the monoid of ... |
| bj-endcomp 38158 | Composition law of the mon... |
| bj-endmnd 38159 | The monoid of endomorphism... |
| taupilem3 38160 | Lemma for tau-related theo... |
| taupilemrplb 38161 | A set of positive reals ha... |
| taupilem1 38162 | Lemma for ~ taupi . A pos... |
| taupilem2 38163 | Lemma for ~ taupi . The s... |
| taupi 38164 | Relationship between ` _ta... |
| dfgcd3 38165 | Alternate definition of th... |
| irrdifflemf 38166 | Lemma for ~ irrdiff . The... |
| irrdiff 38167 | The irrationals are exactl... |
| qdiff 38168 | The rationals are exactly ... |
| qdiffALT 38169 | Alternate proof of ~ qdiff... |
| iccioo01 38170 | The closed unit interval i... |
| csbrecsg 38171 | Move class substitution in... |
| csbrdgg 38172 | Move class substitution in... |
| csboprabg 38173 | Move class substitution in... |
| csbmpo123 38174 | Move class substitution in... |
| con1bii2 38175 | A contraposition inference... |
| con2bii2 38176 | A contraposition inference... |
| vtoclefex 38177 | Implicit substitution of a... |
| rnmptsn 38178 | The range of a function ma... |
| f1omptsnlem 38179 | This is the core of the pr... |
| f1omptsn 38180 | A function mapping to sing... |
| mptsnunlem 38181 | This is the core of the pr... |
| mptsnun 38182 | A class ` B ` is equal to ... |
| dissneqlem 38183 | This is the core of the pr... |
| dissneq 38184 | Any topology that contains... |
| exlimim 38185 | Closed form of ~ exlimimd ... |
| exlimimd 38186 | Existential elimination ru... |
| exellim 38187 | Closed form of ~ exellimdd... |
| exellimddv 38188 | Eliminate an antecedent wh... |
| topdifinfindis 38189 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 38190 | This is the core of the pr... |
| topdifinffin 38191 | Part of Exercise 3 of [Mun... |
| topdifinf 38192 | Part of Exercise 3 of [Mun... |
| topdifinfeq 38193 | Two different ways of defi... |
| icorempo 38194 | Closed-below, open-above i... |
| icoreresf 38195 | Closed-below, open-above i... |
| icoreval 38196 | Value of the closed-below,... |
| icoreelrnab 38197 | Elementhood in the set of ... |
| isbasisrelowllem1 38198 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 38199 | Lemma for ~ isbasisrelowl ... |
| icoreclin 38200 | The set of closed-below, o... |
| isbasisrelowl 38201 | The set of all closed-belo... |
| icoreunrn 38202 | The union of all closed-be... |
| istoprelowl 38203 | The set of all closed-belo... |
| icoreelrn 38204 | A class abstraction which ... |
| iooelexlt 38205 | An element of an open inte... |
| relowlssretop 38206 | The lower limit topology o... |
| relowlpssretop 38207 | The lower limit topology o... |
| sucneqond 38208 | Inequality of an ordinal s... |
| sucneqoni 38209 | Inequality of an ordinal s... |
| onsucuni3 38210 | If an ordinal number has a... |
| 1oequni2o 38211 | The ordinal number ` 1o ` ... |
| rdgsucuni 38212 | If an ordinal number has a... |
| rdgeqoa 38213 | If a recursive function wi... |
| elxp8 38214 | Membership in a Cartesian ... |
| cbveud 38215 | Deduction used to change b... |
| cbvreud 38216 | Deduction used to change b... |
| difunieq 38217 | The difference of unions i... |
| inunissunidif 38218 | Theorem about subsets of t... |
| rdgellim 38219 | Elementhood in a recursive... |
| rdglimss 38220 | A recursive definition at ... |
| rdgssun 38221 | In a recursive definition ... |
| exrecfnlem 38222 | Lemma for ~ exrecfn . (Co... |
| exrecfn 38223 | Theorem about the existenc... |
| exrecfnpw 38224 | For any base set, a set wh... |
| finorwe 38225 | If the Axiom of Infinity i... |
| dffinxpf 38228 | This theorem is the same a... |
| finxpeq1 38229 | Equality theorem for Carte... |
| finxpeq2 38230 | Equality theorem for Carte... |
| csbfinxpg 38231 | Distribute proper substitu... |
| finxpreclem1 38232 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 38233 | Lemma for ` ^^ ` recursion... |
| finxp0 38234 | The value of Cartesian exp... |
| finxp1o 38235 | The value of Cartesian exp... |
| finxpreclem3 38236 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 38237 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 38238 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 38239 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 38240 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 38241 | The value of Cartesian exp... |
| finxp2o 38242 | The value of Cartesian exp... |
| finxp3o 38243 | The value of Cartesian exp... |
| finxpnom 38244 | Cartesian exponentiation w... |
| finxp00 38245 | Cartesian exponentiation o... |
| iunctb2 38246 | Using the axiom of countab... |
| domalom 38247 | A class which dominates ev... |
| isinf2 38248 | The converse of ~ isinf . ... |
| ctbssinf 38249 | Using the axiom of choice,... |
| ralssiun 38250 | The index set of an indexe... |
| nlpineqsn 38251 | For every point ` p ` of a... |
| nlpfvineqsn 38252 | Given a subset ` A ` of ` ... |
| fvineqsnf1 38253 | A theorem about functions ... |
| fvineqsneu 38254 | A theorem about functions ... |
| fvineqsneq 38255 | A theorem about functions ... |
| pibp16 38256 | Property P000016 of pi-bas... |
| pibp19 38257 | Property P000019 of pi-bas... |
| pibp21 38258 | Property P000021 of pi-bas... |
| pibt1 38259 | Theorem T000001 of pi-base... |
| pibt2 38260 | Theorem T000002 of pi-base... |
| wl-section-prop 38261 | Intuitionistic logic is no... |
| wl-section-boot 38265 | In this section, I provide... |
| wl-luk-imim1i 38266 | Inference adding common co... |
| wl-luk-syl 38267 | An inference version of th... |
| wl-luk-imtrid 38268 | A syllogism rule of infere... |
| wl-luk-pm2.18d 38269 | Deduction based on reducti... |
| wl-luk-con4i 38270 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 38271 | Inference rule. Copy of ~... |
| wl-luk-a1i 38272 | Inference rule. Copy of ~... |
| wl-luk-mpi 38273 | A nested _modus ponens_ in... |
| wl-luk-imim2i 38274 | Inference adding common an... |
| wl-luk-imtrdi 38275 | A syllogism rule of infere... |
| wl-luk-ax3 38276 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 38277 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 38278 | This theorem, called "Asse... |
| wl-luk-com12 38279 | Inference that swaps (comm... |
| wl-luk-pm2.21 38280 | From a wff and its negatio... |
| wl-luk-con1i 38281 | A contraposition inference... |
| wl-luk-ja 38282 | Inference joining the ante... |
| wl-luk-imim2 38283 | A closed form of syllogism... |
| wl-luk-a1d 38284 | Deduction introducing an e... |
| wl-luk-ax2 38285 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 38286 | Principle of identity. Th... |
| wl-luk-notnotr 38287 | Converse of double negatio... |
| wl-luk-pm2.04 38288 | Swap antecedents. Theorem... |
| wl-section-impchain 38289 | An implication like ` ( ps... |
| wl-impchain-mp-x 38290 | This series of theorems pr... |
| wl-impchain-mp-0 38291 | This theorem is the start ... |
| wl-impchain-mp-1 38292 | This theorem is in fact a ... |
| wl-impchain-mp-2 38293 | This theorem is in fact a ... |
| wl-impchain-com-1.x 38294 | It is often convenient to ... |
| wl-impchain-com-1.1 38295 | A degenerate form of antec... |
| wl-impchain-com-1.2 38296 | This theorem is in fact a ... |
| wl-impchain-com-1.3 38297 | This theorem is in fact a ... |
| wl-impchain-com-1.4 38298 | This theorem is in fact a ... |
| wl-impchain-com-n.m 38299 | This series of theorems al... |
| wl-impchain-com-2.3 38300 | This theorem is in fact a ... |
| wl-impchain-com-2.4 38301 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 38302 | This theorem is in fact a ... |
| wl-impchain-a1-x 38303 | If an implication chain is... |
| wl-impchain-a1-1 38304 | Inference rule, a copy of ... |
| wl-impchain-a1-2 38305 | Inference rule, a copy of ... |
| wl-impchain-a1-3 38306 | Inference rule, a copy of ... |
| wl-ifp-ncond1 38307 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 38308 | If one case of an ` if- ` ... |
| wl-ifpimpr 38309 | If one case of an ` if- ` ... |
| wl-ifp4impr 38310 | If one case of an ` if- ` ... |
| wl-df-3xor 38311 | Alternative definition of ... |
| wl-df3xor2 38312 | Alternative definition of ... |
| wl-df3xor3 38313 | Alternative form of ~ wl-d... |
| wl-3xortru 38314 | If the first input is true... |
| wl-3xorfal 38315 | If the first input is fals... |
| wl-3xorbi 38316 | Triple xor can be replaced... |
| wl-3xorbi2 38317 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 38318 | Equivalence theorem for tr... |
| wl-3xorbi123i 38319 | Equivalence theorem for tr... |
| wl-3xorrot 38320 | Rotation law for triple xo... |
| wl-3xorcoma 38321 | Commutative law for triple... |
| wl-3xorcomb 38322 | Commutative law for triple... |
| wl-3xornot1 38323 | Flipping the first input f... |
| wl-3xornot 38324 | Triple xor distributes ove... |
| wl-1xor 38325 | In the recursive scheme ... |
| wl-2xor 38326 | In the recursive scheme ... |
| wl-df-3mintru2 38327 | Alternative definition of ... |
| wl-df2-3mintru2 38328 | The adder carry in disjunc... |
| wl-df3-3mintru2 38329 | The adder carry in conjunc... |
| wl-df4-3mintru2 38330 | An alternative definition ... |
| wl-1mintru1 38331 | Using the recursion formul... |
| wl-1mintru2 38332 | Using the recursion formul... |
| wl-2mintru1 38333 | Using the recursion formul... |
| wl-2mintru2 38334 | Using the recursion formul... |
| wl-df3maxtru1 38335 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 38337 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 38338 |
Disclaimer: |
| wl-cleq-1 38339 |
Disclaimer: |
| wl-cleq-2 38340 |
Disclaimer: |
| wl-cleq-3 38341 |
Disclaimer: |
| wl-cleq-4 38342 |
Disclaimer: |
| wl-cleq-5 38343 |
Disclaimer: |
| wl-cleq-6 38344 |
Disclaimer: |
| wl-df-clab 38347 | Disclaimer: The material ... |
| wl-isseteq 38348 | A class equal to a set var... |
| wl-ax12v2cl 38349 | The class version of ~ ax1... |
| wl-df.clab 38350 | Define class abstractions,... |
| wl-df.cleq 38351 | Define the equality connec... |
| wl-dfcleq.basic 38352 | This theorem is a conserva... |
| wl-dfcleq.just 38353 | The hypotheses added to th... |
| wl-df.clel 38354 | Define the membership conn... |
| wl-dfclel.basic 38355 | This theorem gives a conse... |
| wl-dfclel.just 38356 | Add a hypothesis to ~ wl-d... |
| wl-dfcleq 38357 | The defining characterizat... |
| wl-dfclel 38358 | The defining characterizat... |
| wl-mps 38359 | Replacing a nested consequ... |
| wl-syls1 38360 | Replacing a nested consequ... |
| wl-syls2 38361 | Replacing a nested anteced... |
| wl-embant 38362 | A true wff can always be a... |
| wl-orel12 38363 | In a conjunctive normal fo... |
| wl-cases2-dnf 38364 | A particular instance of ~... |
| wl-cbvmotv 38365 | Change bound variable. Us... |
| wl-moteq 38366 | Change bound variable. Us... |
| wl-motae 38367 | Change bound variable. Us... |
| wl-moae 38368 | Two ways to express "at mo... |
| wl-euae 38369 | Two ways to express "exact... |
| wl-nax6im 38370 | The following series of th... |
| wl-hbae1 38371 | This specialization of ~ h... |
| wl-naevhba1v 38372 | An instance of ~ hbn1w app... |
| wl-spae 38373 | Prove an instance of ~ sp ... |
| wl-speqv 38374 | Under the assumption ` -. ... |
| wl-19.8eqv 38375 | Under the assumption ` -. ... |
| wl-19.2reqv 38376 | Under the assumption ` -. ... |
| wl-nfalv 38377 | If ` x ` is not present in... |
| wl-nfimf1 38378 | An antecedent is irrelevan... |
| wl-nfae1 38379 | Unlike ~ nfae , this speci... |
| wl-nfnae1 38380 | Unlike ~ nfnae , this spec... |
| wl-aetr 38381 | A transitive law for varia... |
| wl-axc11r 38382 | Same as ~ axc11r , but usi... |
| wl-dral1d 38383 | A version of ~ dral1 with ... |
| wl-cbvalnaed 38384 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 38385 | A more general version of ... |
| wl-exeq 38386 | The semantics of ` E. x y ... |
| wl-aleq 38387 | The semantics of ` A. x y ... |
| wl-nfeqfb 38388 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 38389 | If ` y ` is not free in ` ... |
| wl-equsalvw 38390 | Version of ~ equsalv with ... |
| wl-equsald 38391 | Deduction version of ~ equ... |
| wl-equsaldv 38392 | Deduction version of ~ equ... |
| wl-equsal 38393 | A useful equivalence relat... |
| wl-equsal1t 38394 | The expression ` x = y ` i... |
| wl-equsalcom 38395 | This simple equivalence ea... |
| wl-equsal1i 38396 | The antecedent ` x = y ` i... |
| wl-sbid2ft 38397 | A more general version of ... |
| wl-cbvalsbi 38398 | Change bounded variables i... |
| wl-sbrimt 38399 | Substitution with a variab... |
| wl-sblimt 38400 | Substitution with a variab... |
| wl-sb9v 38401 | Commutation of quantificat... |
| wl-sb8ft 38402 | Substitution of variable i... |
| wl-sb8eft 38403 | Substitution of variable i... |
| wl-sb8t 38404 | Substitution of variable i... |
| wl-sb8et 38405 | Substitution of variable i... |
| wl-sbhbt 38406 | Closed form of ~ sbhb . C... |
| wl-sbnf1 38407 | Two ways expressing that `... |
| wl-equsb3 38408 | ~ equsb3 with a distinctor... |
| wl-equsb4 38409 | Substitution applied to an... |
| wl-2sb6d 38410 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 38411 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 38412 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 38413 | Version of ~ sbcom2 with a... |
| wl-sbalnae 38414 | A theorem used in eliminat... |
| wl-sbal1 38415 | A theorem used in eliminat... |
| wl-sbal2 38416 | Move quantifier in and out... |
| wl-2spsbbi 38417 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 38418 | This theorem provides a ba... |
| wl-lem-nexmo 38419 | This theorem provides a ba... |
| wl-lem-moexsb 38420 | The antecedent ` A. x ( ph... |
| wl-alanbii 38421 | This theorem extends ~ ala... |
| wl-mo2df 38422 | Version of ~ mof with a co... |
| wl-mo2tf 38423 | Closed form of ~ mof with ... |
| wl-eudf 38424 | Version of ~ eu6 with a co... |
| wl-eutf 38425 | Closed form of ~ eu6 with ... |
| wl-euequf 38426 | ~ euequ proved with a dist... |
| wl-mo2t 38427 | Closed form of ~ mof . (C... |
| wl-mo3t 38428 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 38429 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 38430 | Substitution of variable i... |
| wl-sb8eutv 38431 | Substitution of variable i... |
| wl-sb8mot 38432 | Substitution of variable i... |
| wl-sb8motv 38433 | Substitution of variable i... |
| wl-issetft 38434 | A closed form of ~ issetf ... |
| wl-axc11rc11 38435 | Proving ~ axc11r from ~ ax... |
| wl-clabv 38436 | Variant of ~ df-clab , whe... |
| wl-dfclab 38437 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 38438 | Using class abstraction in... |
| wl-clabt 38439 | Using class abstraction in... |
| wl-eujustlem1 38440 | Version of ~ cbvexvw with ... |
| rabiun 38441 | Abstraction restricted to ... |
| iundif1 38442 | Indexed union of class dif... |
| imadifss 38443 | The difference of images i... |
| unceq 38444 | Equality theorem for uncur... |
| curunc 38445 | Currying of uncurrying. (... |
| unccur 38446 | Uncurrying of currying. (... |
| phpreu 38447 | Theorem related to pigeonh... |
| finixpnum 38448 | A finite Cartesian product... |
| fin2solem 38449 | Lemma for ~ fin2so . (Con... |
| fin2so 38450 | Any totally ordered Tarski... |
| ltflcei 38451 | Theorem to move the floor ... |
| leceifl 38452 | Theorem to move the floor ... |
| sin2h 38453 | Half-angle rule for sine. ... |
| cos2h 38454 | Half-angle rule for cosine... |
| tan2h 38455 | Half-angle rule for tangen... |
| lindsadd 38456 | In a vector space, the uni... |
| ptrest 38457 | Expressing a restriction o... |
| ptrecube 38458 | Any point in an open set o... |
| poimirlem1 38459 | Lemma for ~ poimir - the v... |
| poimirlem2 38460 | Lemma for ~ poimir - conse... |
| poimirlem3 38461 | Lemma for ~ poimir to add ... |
| poimirlem4 38462 | Lemma for ~ poimir connect... |
| poimirlem5 38463 | Lemma for ~ poimir to esta... |
| poimirlem6 38464 | Lemma for ~ poimir establi... |
| poimirlem7 38465 | Lemma for ~ poimir , simil... |
| poimirlem8 38466 | Lemma for ~ poimir , estab... |
| poimirlem9 38467 | Lemma for ~ poimir , estab... |
| poimirlem10 38468 | Lemma for ~ poimir establi... |
| poimirlem11 38469 | Lemma for ~ poimir connect... |
| poimirlem12 38470 | Lemma for ~ poimir connect... |
| poimirlem13 38471 | Lemma for ~ poimir - for a... |
| poimirlem14 38472 | Lemma for ~ poimir - for a... |
| poimirlem15 38473 | Lemma for ~ poimir , that ... |
| poimirlem16 38474 | Lemma for ~ poimir establi... |
| poimirlem17 38475 | Lemma for ~ poimir establi... |
| poimirlem18 38476 | Lemma for ~ poimir stating... |
| poimirlem19 38477 | Lemma for ~ poimir establi... |
| poimirlem20 38478 | Lemma for ~ poimir establi... |
| poimirlem21 38479 | Lemma for ~ poimir stating... |
| poimirlem22 38480 | Lemma for ~ poimir , that ... |
| poimirlem23 38481 | Lemma for ~ poimir , two w... |
| poimirlem24 38482 | Lemma for ~ poimir , two w... |
| poimirlem25 38483 | Lemma for ~ poimir stating... |
| poimirlem26 38484 | Lemma for ~ poimir showing... |
| poimirlem27 38485 | Lemma for ~ poimir showing... |
| poimirlem28 38486 | Lemma for ~ poimir , a var... |
| poimirlem29 38487 | Lemma for ~ poimir connect... |
| poimirlem30 38488 | Lemma for ~ poimir combini... |
| poimirlem31 38489 | Lemma for ~ poimir , assig... |
| poimirlem32 38490 | Lemma for ~ poimir , combi... |
| poimir 38491 | Poincare-Miranda theorem. ... |
| broucube 38492 | Brouwer - or as Kulpa call... |
| heicant 38493 | Heine-Cantor theorem: a co... |
| opnmbllem0 38494 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 38495 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 38496 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 38497 | The difference between two... |
| mblfinlem4 38498 | Backward direction of ~ is... |
| ismblfin 38499 | Measurability in terms of ... |
| ovoliunnfl 38500 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 38501 | Demonstration of ~ ovoliun... |
| voliunnfl 38502 | ~ voliun is incompatible w... |
| volsupnfl 38503 | ~ volsup is incompatible w... |
| mbfresfi 38504 | Measurability of a piecewi... |
| mbfposadd 38505 | If the sum of two measurab... |
| cnambfre 38506 | A real-valued, a.e. contin... |
| dvtanlem 38507 | Lemma for ~ dvtan - the do... |
| dvtan 38508 | Derivative of tangent. (C... |
| itg2addnclem 38509 | An alternate expression fo... |
| itg2addnclem2 38510 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 38511 | Lemma incomprehensible in ... |
| itg2addnc 38512 | Alternate proof of ~ itg2a... |
| itg2gt0cn 38513 | ~ itg2gt0 holds on functio... |
| ibladdnclem 38514 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 38515 | Choice-free analogue of ~ ... |
| itgaddnclem1 38516 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 38517 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 38518 | Choice-free analogue of ~ ... |
| iblsubnc 38519 | Choice-free analogue of ~ ... |
| itgsubnc 38520 | Choice-free analogue of ~ ... |
| iblabsnclem 38521 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 38522 | Choice-free analogue of ~ ... |
| iblmulc2nc 38523 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 38524 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 38525 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 38526 | Choice-free analogue of ~ ... |
| itgabsnc 38527 | Choice-free analogue of ~ ... |
| itggt0cn 38528 | ~ itggt0 holds for continu... |
| ftc1cnnclem 38529 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 38530 | Choice-free proof of ~ ftc... |
| ftc1anclem1 38531 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 38532 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 38533 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 38534 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 38535 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 38536 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 38537 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 38538 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 38539 | ~ ftc1a holds for function... |
| ftc2nc 38540 | Choice-free proof of ~ ftc... |
| asindmre 38541 | Real part of domain of dif... |
| dvasin 38542 | Derivative of arcsine. (C... |
| dvacos 38543 | Derivative of arccosine. ... |
| dvreasin 38544 | Real derivative of arcsine... |
| dvreacos 38545 | Real derivative of arccosi... |
| areacirclem1 38546 | Antiderivative of cross-se... |
| areacirclem2 38547 | Endpoint-inclusive continu... |
| areacirclem3 38548 | Integrability of cross-sec... |
| areacirclem4 38549 | Endpoint-inclusive continu... |
| areacirclem5 38550 | Finding the cross-section ... |
| areacirc 38551 | The area of a circle of ra... |
| findcard4 38552 | Schema for strong inductio... |
| dfproplem 38561 | Given a set A, the set of ... |
| varprop 38562 | Variables encoded as natur... |
| negprop 38563 | The negation of a sentence... |
| impprop 38564 | The implication between tw... |
| dfprop1 38565 | The set of variables encod... |
| dfprop2 38566 | Every sentence of proposit... |
| dfprop 38567 | The set of sentences of pr... |
| unirep 38568 | Define a quantity whose de... |
| cover2 38569 | Two ways of expressing the... |
| cover2g 38570 | Two ways of expressing the... |
| brabg2 38571 | Relation by a binary relat... |
| opelopab3 38572 | Ordered pair membership in... |
| cocanfo 38573 | Cancellation of a surjecti... |
| brresi2 38574 | Restriction of a binary re... |
| fnopabeqd 38575 | Equality deduction for fun... |
| fvopabf4g 38576 | Function value of an opera... |
| fnopabco 38577 | Composition of a function ... |
| opropabco 38578 | Composition of an operator... |
| cocnv 38579 | Composition with a functio... |
| f1ocan1fv 38580 | Cancel a composition by a ... |
| f1ocan2fv 38581 | Cancel a composition by th... |
| inixp 38582 | Intersection of Cartesian ... |
| upixp 38583 | Universal property of the ... |
| abrexdom 38584 | An indexed set is dominate... |
| abrexdom2 38585 | An indexed set is dominate... |
| ac6gf 38586 | Axiom of Choice. (Contrib... |
| indexa 38587 | If for every element of an... |
| indexdom 38588 | If for every element of an... |
| frinfm 38589 | A subset of a well-founded... |
| welb 38590 | A nonempty subset of a wel... |
| supex2g 38591 | Existence of supremum. (C... |
| supclt 38592 | Closure of supremum. (Con... |
| supubt 38593 | Upper bound property of su... |
| filbcmb 38594 | Combine a finite set of lo... |
| fzmul 38595 | Membership of a product in... |
| sdclem2 38596 | Lemma for ~ sdc . (Contri... |
| sdclem1 38597 | Lemma for ~ sdc . (Contri... |
| sdc 38598 | Strong dependent choice. ... |
| fdc 38599 | Finite version of dependen... |
| fdc1 38600 | Variant of ~ fdc with no s... |
| seqpo 38601 | Two ways to say that a seq... |
| incsequz 38602 | An increasing sequence of ... |
| incsequz2 38603 | An increasing sequence of ... |
| nnubfi 38604 | A bounded above set of pos... |
| nninfnub 38605 | An infinite set of positiv... |
| subspopn 38606 | An open set is open in the... |
| neificl 38607 | Neighborhoods are closed u... |
| lpss2 38608 | Limit points of a subset a... |
| metf1o 38609 | Use a bijection with a met... |
| blssp 38610 | A ball in the subspace met... |
| mettrifi 38611 | Generalized triangle inequ... |
| lmclim2 38612 | A sequence in a metric spa... |
| geomcau 38613 | If the distance between co... |
| caures 38614 | The restriction of a Cauch... |
| caushft 38615 | A shifted Cauchy sequence ... |
| constcncf 38616 | A constant function is a c... |
| cnres2 38617 | The restriction of a conti... |
| cnresima 38618 | A continuous function is c... |
| cncfres 38619 | A continuous function on c... |
| istotbnd 38623 | The predicate "is a totall... |
| istotbnd2 38624 | The predicate "is a totall... |
| istotbnd3 38625 | A metric space is totally ... |
| totbndmet 38626 | The predicate "totally bou... |
| 0totbnd 38627 | The metric (there is only ... |
| sstotbnd2 38628 | Condition for a subset of ... |
| sstotbnd 38629 | Condition for a subset of ... |
| sstotbnd3 38630 | Use a net that is not nece... |
| totbndss 38631 | A subset of a totally boun... |
| equivtotbnd 38632 | If the metric ` M ` is "st... |
| isbnd 38634 | The predicate "is a bounde... |
| bndmet 38635 | A bounded metric space is ... |
| isbndx 38636 | A "bounded extended metric... |
| isbnd2 38637 | The predicate "is a bounde... |
| isbnd3 38638 | A metric space is bounded ... |
| isbnd3b 38639 | A metric space is bounded ... |
| bndss 38640 | A subset of a bounded metr... |
| blbnd 38641 | A ball is bounded. (Contr... |
| ssbnd 38642 | A subset of a metric space... |
| totbndbnd 38643 | A totally bounded metric s... |
| equivbnd 38644 | If the metric ` M ` is "st... |
| bnd2lem 38645 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 38646 | If balls are totally bound... |
| prdsbnd 38647 | The product metric over fi... |
| prdstotbnd 38648 | The product metric over fi... |
| prdsbnd2 38649 | If balls are totally bound... |
| cntotbnd 38650 | A subset of the complex nu... |
| cnpwstotbnd 38651 | A subset of ` A ^ I ` , wh... |
| ismtyval 38654 | The set of isometries betw... |
| isismty 38655 | The condition "is an isome... |
| ismtycnv 38656 | The inverse of an isometry... |
| ismtyima 38657 | The image of a ball under ... |
| ismtyhmeolem 38658 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 38659 | An isometry is a homeomorp... |
| ismtybndlem 38660 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 38661 | Isometries preserve bounde... |
| ismtyres 38662 | A restriction of an isomet... |
| heibor1lem 38663 | Lemma for ~ heibor1 . A c... |
| heibor1 38664 | One half of ~ heibor , tha... |
| heiborlem1 38665 | Lemma for ~ heibor . We w... |
| heiborlem2 38666 | Lemma for ~ heibor . Subs... |
| heiborlem3 38667 | Lemma for ~ heibor . Usin... |
| heiborlem4 38668 | Lemma for ~ heibor . Usin... |
| heiborlem5 38669 | Lemma for ~ heibor . The ... |
| heiborlem6 38670 | Lemma for ~ heibor . Sinc... |
| heiborlem7 38671 | Lemma for ~ heibor . Sinc... |
| heiborlem8 38672 | Lemma for ~ heibor . The ... |
| heiborlem9 38673 | Lemma for ~ heibor . Disc... |
| heiborlem10 38674 | Lemma for ~ heibor . The ... |
| heibor 38675 | Generalized Heine-Borel Th... |
| bfplem1 38676 | Lemma for ~ bfp . The seq... |
| bfplem2 38677 | Lemma for ~ bfp . Using t... |
| bfp 38678 | Banach fixed point theorem... |
| rrnval 38681 | The n-dimensional Euclidea... |
| rrnmval 38682 | The value of the Euclidean... |
| rrnmet 38683 | Euclidean space is a metri... |
| rrndstprj1 38684 | The distance between two p... |
| rrndstprj2 38685 | Bound on the distance betw... |
| rrncmslem 38686 | Lemma for ~ rrncms . (Con... |
| rrncms 38687 | Euclidean space is complet... |
| repwsmet 38688 | The supremum metric on ` R... |
| rrnequiv 38689 | The supremum metric on ` R... |
| rrntotbnd 38690 | A set in Euclidean space i... |
| rrnheibor 38691 | Heine-Borel theorem for Eu... |
| ismrer1 38692 | An isometry between ` RR `... |
| reheibor 38693 | Heine-Borel theorem for re... |
| iccbnd 38694 | A closed interval in ` RR ... |
| icccmpALT 38695 | A closed interval in ` RR ... |
| isass 38700 | The predicate "is an assoc... |
| isexid 38701 | The predicate ` G ` has a ... |
| ismgmOLD 38704 | Obsolete version of ~ ismg... |
| clmgmOLD 38705 | Obsolete version of ~ mgmc... |
| opidonOLD 38706 | Obsolete version of ~ mgmi... |
| rngopidOLD 38707 | Obsolete version of ~ mgmi... |
| opidon2OLD 38708 | Obsolete version of ~ mgmf... |
| isexid2 38709 | Obsolete theorem. If ` G ... |
| exidu1 38710 | Obsolete theorem, use ~ mg... |
| idrval 38711 | Obsolete theorem, use ~ id... |
| iorlid 38712 | Obsolete theorem, use ~ mg... |
| cmpidelt 38713 | Obsolete theorem, use ~ mg... |
| smgrpismgmOLD 38716 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38717 | Obsolete version of ~ issg... |
| smgrpmgm 38718 | Obsolete theorem, use ~ sg... |
| smgrpassOLD 38719 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38722 | Obsolete version of ~ mnds... |
| mndoisexid 38723 | Obsolete theorem, use ~ mn... |
| mndoismgmOLD 38724 | Obsolete version of ~ mndm... |
| mndomgmid 38725 | Obsolete theorem, use ~ mn... |
| ismndo 38726 | Obsolete theorem, use ~ is... |
| ismndo1 38727 | Obsolete theorem, use ~ is... |
| ismndo2 38728 | Obsolete theorem, use ~ is... |
| grpomndo 38729 | Obsolete theorem, use ~ gr... |
| exidcl 38730 | Obsolete theorem, use ~ mg... |
| exidreslem 38731 | Obsolete theorem, use ~ 0g... |
| exidres 38732 | Obsolete theorem, use ~ id... |
| exidresid 38733 | Obsolete theorem, use ~ id... |
| ablo4pnp 38734 | Obsolete theorem, use ~ ab... |
| grpoeqdivid 38735 | Obsolete theorem, use ~ gr... |
| grposnOLD 38736 | The group operation for th... |
| elghomlem1OLD 38739 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 38740 | Obsolete as of 15-Mar-2020... |
| elghomOLD 38741 | Obsolete version of ~ isgh... |
| ghomlinOLD 38742 | Obsolete version of ~ ghml... |
| ghomidOLD 38743 | Obsolete version of ~ ghmi... |
| ghomf 38744 | Obsolete theorem, use ~ gh... |
| ghomco 38745 | Obsolete theorem, use ~ gh... |
| ghomdiv 38746 | Obsolete theorem, use ~ gh... |
| grpokerinj 38747 | Obsolete theorem, use ~ ke... |
| relrngo 38750 | Obsolete theorem. The cla... |
| isrngo 38751 | Obsolete theorem, use ~ df... |
| isrngod 38752 | Obsolete theorem, use ~ is... |
| rngoi 38753 | Obsolete theorem, use ~ df... |
| rngosm 38754 | Obsolete theorem. Functio... |
| rngocl 38755 | Obsolete theorem, use ~ ri... |
| rngoid 38756 | Obsolete theorem, use ~ ri... |
| rngoideu 38757 | Obsolete theorem, use ~ ri... |
| rngodi 38758 | Obsolete theorem, use ~ ri... |
| rngodir 38759 | Obsolete theorem, use ~ ri... |
| rngoass 38760 | Obsolete theorem, use ~ ri... |
| rngo2 38761 | Obsolete theorem, use ~ ri... |
| rngoablo 38762 | Obsolete theorem, use ~ ri... |
| rngoablo2 38763 | Obsolete theorem, use ~ ri... |
| rngogrpo 38764 | Obsolete theorem, use ~ ri... |
| rngone0 38765 | Obsolete theorem, use ~ ri... |
| rngogcl 38766 | Obsolete theorem, use ~ ri... |
| rngocom 38767 | Obsolete theorem, use ~ ri... |
| rngoaass 38768 | Obsolete theorem, use ~ ri... |
| rngoa32 38769 | Obsolete theorem, use ~ ri... |
| rngoa4 38770 | Obsolete theorem, use ~ ri... |
| rngorcan 38771 | Obsolete theorem, use ~ ri... |
| rngolcan 38772 | Obsolete theorem, use ~ ri... |
| rngo0cl 38773 | Obsolete theorem, use ~ ri... |
| rngo0rid 38774 | Obsolete theorem, use ~ ri... |
| rngo0lid 38775 | Obsolete theorem, use ~ ri... |
| rngolz 38776 | Obsolete theorem, use ~ ri... |
| rngorz 38777 | Obsolete theorem, use ~ ri... |
| rngosn3 38778 | Obsolete as of 25-Jan-2020... |
| rngosn4 38779 | Obsolete as of 25-Jan-2020... |
| rngosn6 38780 | Obsolete as of 25-Jan-2020... |
| rngonegcl 38781 | Obsolete theorem, use ~ ri... |
| rngoaddneg1 38782 | Obsolete theorem, use ~ ri... |
| rngoaddneg2 38783 | Obsolete theorem, use ~ ri... |
| rngosub 38784 | Obsolete theorem, use ~ ri... |
| rngmgmbs4 38785 | Obsolete theorem. The ran... |
| rngodm1dm2 38786 | Obsolete theorem. In a un... |
| rngorn1 38787 | Obsolete theorem. In a un... |
| rngorn1eq 38788 | Obsolete theorem. In a un... |
| rngomndo 38789 | Obsolete theorem, use ~ ri... |
| rngoidmlem 38790 | Obsolete theorem, use ~ ri... |
| rngolidm 38791 | Obsolete theorem, use ~ ri... |
| rngoridm 38792 | Obsolete theorem, use ~ ri... |
| rngo1cl 38793 | Obsolete theorem, use ~ ri... |
| rngoueqz 38794 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 38795 | Obsolete theorem, use ~ ri... |
| rngonegmn1r 38796 | Obsolete theorem, use ~ ri... |
| rngoneglmul 38797 | Obsolete theorem, use ~ ri... |
| rngonegrmul 38798 | Obsolete theorem, use ~ ri... |
| rngosubdi 38799 | Obsolete theorem, use ~ ri... |
| rngosubdir 38800 | Obsolete theorem, use ~ ri... |
| zerdivemp1x 38801 | Obsolete theorem, use ~ ri... |
| isdivrngo 38804 | Obsolete theorem, use ~ is... |
| drngoi 38805 | Obsolete theorem, use ~ dr... |
| gidsn 38806 | Obsolete as of 23-Jan-2020... |
| zrdivrng 38807 | Obsolete theorem, use ~ zr... |
| dvrunz 38808 | Obsolete theorem, use ~ dr... |
| isgrpda 38809 | Obsolete theorem, use ~ is... |
| isdrngo1 38810 | Obsolete theorem, use ~ is... |
| divrngcl 38811 | Obsolete theorem, use ~ dr... |
| isdrngo2 38812 | Obsolete theorem, use ~ is... |
| isdrngo3 38813 | Obsolete theorem, use ~ is... |
| rngohomval 38818 | Obsolete theorem, use ~ rh... |
| isrngohom 38819 | Obsolete theorem, use ~ is... |
| rngohomf 38820 | Obsolete theorem, use ~ rh... |
| rngohomcl 38821 | Obsolete theorem, use ~ rh... |
| rngohom1 38822 | Obsolete theorem, use ~ rh... |
| rngohomadd 38823 | Obsolete theorem, use ~ rh... |
| rngohommul 38824 | Obsolete theorem, use ~ rh... |
| rngogrphom 38825 | Obsolete theorem, use ~ rh... |
| rngohom0 38826 | Obsolete theorem, use ~ rh... |
| rngohomsub 38827 | Obsolete theorem, use ~ rh... |
| rngohomco 38828 | Obsolete theorem, use ~ rh... |
| rngokerinj 38829 | Obsolete theorem, use ~ rh... |
| rngoisoval 38831 | Obsolete theorem, use ~ ri... |
| isrngoiso 38832 | Obsolete theorem, use ~ is... |
| rngoiso1o 38833 | Obsolete theorem, use ~ ri... |
| rngoisohom 38834 | Obsolete theorem, use ~ ri... |
| rngoisocnv 38835 | Obsolete theorem, use ~ ri... |
| rngoisoco 38836 | Obsolete theorem, use ~ ri... |
| isriscg 38838 | Obsolete theorem, use ~ is... |
| isrisc 38839 | Obsolete theorem, use ~ is... |
| risc 38840 | Obsolete theorem, use ~ br... |
| risci 38841 | Obsolete theorem, use ~ br... |
| riscer 38842 | Obsolete theorem, use ~ ri... |
| iscom2 38849 | Obsolete theorem, used (as... |
| iscrngo 38850 | Obsolete theorem, use ~ is... |
| iscrngo2 38851 | Obsolete theorem, use ~ is... |
| iscringd 38852 | Obsolete theorem, use ~ is... |
| flddivrng 38853 | Obsolete theorem, use ~ fl... |
| crngorngo 38854 | Obsolete theorem, use ~ cr... |
| crngocom 38855 | Obsolete theorem, use ~ cr... |
| crngm23 38856 | Obsolete theorem, use ~ cr... |
| crngm4 38857 | Obsolete theorem, use ~ cr... |
| fldcrngo 38858 | Obsolete theorem, use ~ fl... |
| isfld2 38859 | Obsolete theorem, use ~ is... |
| crngohomfo 38860 | Obsolete theorem, use ~ cr... |
| idlval 38867 | Obsolete theorem, use ~ 2i... |
| isidl 38868 | Obsolete theorem, use ~ df... |
| isidlc 38869 | Obsolete theorem, use ~ df... |
| idlss 38870 | Obsolete theorem, use ~ 2i... |
| idlcl 38871 | Obsolete theorem, use ~ 2i... |
| idl0cl 38872 | Obsolete theorem, use ~ ri... |
| idladdcl 38873 | Obsolete theorem, use ~ 2i... |
| idllmulcl 38874 | Obsolete theorem, use ~ 2i... |
| idlrmulcl 38875 | Obsolete theorem, use ~ 2i... |
| idlnegcl 38876 | Obsolete theorem, use ~ 2i... |
| idlsubcl 38877 | Obsolete theorem, use ~ 2i... |
| rngoidl 38878 | Obsolete theorem, use ~ 2i... |
| 0idl 38879 | Obsolete theorem, use ~ 2i... |
| 1idl 38880 | Obsolete theorem, use ~ 2i... |
| 0rngo 38881 | Obsolete theorem, use ~ 0r... |
| divrngidl 38882 | Obsolete theorem, use ~ dr... |
| intidl 38883 | Obsolete theorem, use ~ in... |
| inidl 38884 | Obsolete theorem, use ~ in... |
| unichnidl 38885 | Obsolete theorem, use ~ un... |
| keridl 38886 | Obsolete theorem, use ~ ke... |
| pridlval 38887 | Obsolete theorem, use ~ pr... |
| ispridl 38888 | Obsolete theorem, use ~ is... |
| pridlidl 38889 | Obsolete theorem, use ~ pr... |
| pridlnr 38890 | Obsolete theorem, use ~ pr... |
| pridl 38891 | Obsolete theorem, use ~ is... |
| ispridl2 38892 | Obsolete theorem, use ~ pr... |
| maxidlval 38893 | Obsolete theorem, use ~ mx... |
| ismaxidl 38894 | Obsolete theorem, use ~ is... |
| maxidlidl 38895 | Obsolete theorem, use ~ mx... |
| maxidlnr 38896 | Obsolete theorem, use ~ mx... |
| maxidlmax 38897 | Obsolete theorem, use ~ mx... |
| maxidln1 38898 | Obsolete theorem, use ~ mx... |
| maxidln0 38899 | Obsolete theorem, use ~ mx... |
| isprrngo 38904 | Obsolete theorem, use ~ is... |
| prrngorngo 38905 | Obsolete theorem, use ~ pr... |
| smprngopr 38906 | Obsolete theorem, use ~ sm... |
| divrngpr 38907 | Obsolete theorem, use ~ dr... |
| isdmn 38908 | Obsolete theorem, use ~ is... |
| isdmn2 38909 | Obsolete theorem, use ~ is... |
| dmncrng 38910 | Obsolete theorem, use ~ id... |
| dmnrngo 38911 | Obsolete theorem, use ~ id... |
| flddmn 38912 | Obsolete theorem, use ~ fl... |
| igenval 38915 | Obsolete theorem, use ~ rs... |
| igenss 38916 | Obsolete theorem, use ~ rs... |
| igenidl 38917 | Obsolete theorem, use ~ rs... |
| igenmin 38918 | Obsolete theorem, use ~ rs... |
| igenidl2 38919 | Obsolete theorem, use ~ rs... |
| igenval2 38920 | Obsolete theorem, use ~ rs... |
| prnc 38921 | Obsolete theorem, use ~ rs... |
| isfldidl 38922 | Obsolete theorem, use ~ is... |
| isfldidl2 38923 | Obsolete theorem, use ~ is... |
| ispridlc 38924 | Obsolete theorem, use ~ is... |
| pridlc 38925 | Obsolete theorem, use ~ pr... |
| pridlc2 38926 | Obsolete theorem, use ~ pr... |
| pridlc3 38927 | Obsolete theorem, use ~ cm... |
| isdmn3 38928 | Obsolete theorem, use ~ is... |
| dmnnzd 38929 | Obsolete theorem, use ~ id... |
| dmncan1 38930 | Obsolete theorem, use ~ id... |
| dmncan2 38931 | Obsolete theorem, use ~ id... |
| efald2 38932 | A proof by contradiction. ... |
| notbinot1 38933 | Simplification rule of neg... |
| bicontr 38934 | Biconditional of its own n... |
| impor 38935 | An equivalent formula for ... |
| orfa 38936 | The falsum ` F. ` can be r... |
| notbinot2 38937 | Commutation rule between n... |
| biimpor 38938 | A rewriting rule for bicon... |
| orfa1 38939 | Add a contradicting disjun... |
| orfa2 38940 | Remove a contradicting dis... |
| bifald 38941 | Infer the equivalence to a... |
| cnf1dd 38942 | A lemma for Conjunctive No... |
| cnf2dd 38943 | A lemma for Conjunctive No... |
| cnfn1dd 38944 | A lemma for Conjunctive No... |
| cnfn2dd 38945 | A lemma for Conjunctive No... |
| or32dd 38946 | A rearrangement of disjunc... |
| notornotel1 38947 | A lemma for not-or-not eli... |
| notornotel2 38948 | A lemma for not-or-not eli... |
| contrd 38949 | A proof by contradiction, ... |
| an12i 38950 | An inference from commutin... |
| exmid2 38951 | An excluded middle law. (... |
| selconj 38952 | An inference for selecting... |
| truconj 38953 | Add true as a conjunct. (... |
| orel 38954 | An inference for disjuncti... |
| negel 38955 | An inference for negation ... |
| botel 38956 | An inference for bottom el... |
| tradd 38957 | Add top ad a conjunct. (C... |
| gm-sbtru 38958 | Substitution does not chan... |
| sbfal 38959 | Substitution does not chan... |
| sbcani 38960 | Distribution of class subs... |
| sbcori 38961 | Distribution of class subs... |
| sbcimi 38962 | Distribution of class subs... |
| sbcni 38963 | Move class substitution in... |
| sbali 38964 | Discard class substitution... |
| sbexi 38965 | Discard class substitution... |
| sbcalf 38966 | Move universal quantifier ... |
| sbcexf 38967 | Move existential quantifie... |
| sbcalfi 38968 | Move universal quantifier ... |
| sbcexfi 38969 | Move existential quantifie... |
| spsbcdi 38970 | A lemma for eliminating a ... |
| alrimii 38971 | A lemma for introducing a ... |
| spesbcdi 38972 | A lemma for introducing an... |
| exlimddvf 38973 | A lemma for eliminating an... |
| exlimddvfi 38974 | A lemma for eliminating an... |
| sbceq1ddi 38975 | A lemma for eliminating in... |
| sbccom2lem 38976 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 38977 | Commutative law for double... |
| sbccom2f 38978 | Commutative law for double... |
| sbccom2fi 38979 | Commutative law for double... |
| csbcom2fi 38980 | Commutative law for double... |
| fald 38981 | Refutation of falsity, in ... |
| tsim1 38982 | A Tseitin axiom for logica... |
| tsim2 38983 | A Tseitin axiom for logica... |
| tsim3 38984 | A Tseitin axiom for logica... |
| tsbi1 38985 | A Tseitin axiom for logica... |
| tsbi2 38986 | A Tseitin axiom for logica... |
| tsbi3 38987 | A Tseitin axiom for logica... |
| tsbi4 38988 | A Tseitin axiom for logica... |
| tsxo1 38989 | A Tseitin axiom for logica... |
| tsxo2 38990 | A Tseitin axiom for logica... |
| tsxo3 38991 | A Tseitin axiom for logica... |
| tsxo4 38992 | A Tseitin axiom for logica... |
| tsan1 38993 | A Tseitin axiom for logica... |
| tsan2 38994 | A Tseitin axiom for logica... |
| tsan3 38995 | A Tseitin axiom for logica... |
| tsna1 38996 | A Tseitin axiom for logica... |
| tsna2 38997 | A Tseitin axiom for logica... |
| tsna3 38998 | A Tseitin axiom for logica... |
| tsor1 38999 | A Tseitin axiom for logica... |
| tsor2 39000 | A Tseitin axiom for logica... |
| tsor3 39001 | A Tseitin axiom for logica... |
| ts3an1 39002 | A Tseitin axiom for triple... |
| ts3an2 39003 | A Tseitin axiom for triple... |
| ts3an3 39004 | A Tseitin axiom for triple... |
| ts3or1 39005 | A Tseitin axiom for triple... |
| ts3or2 39006 | A Tseitin axiom for triple... |
| ts3or3 39007 | A Tseitin axiom for triple... |
| iuneq2f 39008 | Equality deduction for ind... |
| rabeq12f 39009 | Equality deduction for res... |
| csbeq12 39010 | Equality deduction for sub... |
| sbeqi 39011 | Equality deduction for sub... |
| ralbi12f 39012 | Equality deduction for res... |
| oprabbi 39013 | Equality deduction for cla... |
| mpobi123f 39014 | Equality deduction for map... |
| iuneq12f 39015 | Equality deduction for ind... |
| iineq12f 39016 | Equality deduction for ind... |
| opabbi 39017 | Equality deduction for cla... |
| mptbi12f 39018 | Equality deduction for map... |
| orcomdd 39019 | Commutativity of logic dis... |
| scottexf 39020 | A version of ~ scottex wit... |
| scott0f 39021 | A version of ~ scott0b wit... |
| scottn0f 39022 | A version of ~ scott0f wit... |
| ac6s3f 39023 | Generalization of the Axio... |
| ac6s6 39024 | Generalization of the Axio... |
| ac6s6f 39025 | Generalization of the Axio... |
| el2v1 39081 | New way ( ~ elv , and the ... |
| el3v1 39082 | New way ( ~ elv , and the ... |
| el3v2 39083 | New way ( ~ elv , and the ... |
| el3v12 39084 | New way ( ~ elv , and the ... |
| el3v13 39085 | New way ( ~ elv , and the ... |
| el3v23 39086 | New way ( ~ elv , and the ... |
| anan 39087 | Multiple commutations in c... |
| triantru3 39088 | A wff is equivalent to its... |
| biorfd 39089 | A wff is equivalent to its... |
| eqbrtr 39090 | Substitution of equal clas... |
| eqbrb 39091 | Substitution of equal clas... |
| eqeltr 39092 | Substitution of equal clas... |
| eqelb 39093 | Substitution of equal clas... |
| eqeqan2d 39094 | Implication of introducing... |
| disjresin 39095 | The restriction to a disjo... |
| disjresdisj 39096 | The intersection of restri... |
| disjresdif 39097 | The difference between res... |
| disjresundif 39098 | Lemma for ~ ressucdifsn2 .... |
| inres2 39099 | Two ways of expressing the... |
| coideq 39100 | Equality theorem for compo... |
| nexmo1 39101 | If there is no case where ... |
| eqab2 39102 | Implication of a class abs... |
| r2alan 39103 | Double restricted universa... |
| ssrabi 39104 | Inference of restricted ab... |
| rabimbieq 39105 | Restricted equivalent wff'... |
| abeqin 39106 | Intersection with class ab... |
| abeqinbi 39107 | Intersection with class ab... |
| eqrabi 39108 | Class element of a restric... |
| rabeqel 39109 | Class element of a restric... |
| eqrelf 39110 | The equality connective be... |
| br1cnvinxp 39111 | Binary relation on the con... |
| releleccnv 39112 | Elementhood in a converse ... |
| releccnveq 39113 | Equality of converse ` R `... |
| xpv 39114 | Cartesian product of a cla... |
| vxp 39115 | Cartesian product of the u... |
| opelvvdif 39116 | Negated elementhood of ord... |
| vvdifopab 39117 | Ordered-pair class abstrac... |
| brvdif 39118 | Binary relation with unive... |
| brvdif2 39119 | Binary relation with unive... |
| brvvdif 39120 | Binary relation with the c... |
| brvbrvvdif 39121 | Binary relation with the c... |
| brcnvep 39122 | The converse of the binary... |
| elecALTV 39123 | Elementhood in the ` R ` -... |
| brcnvepres 39124 | Restricted converse epsilo... |
| brres2 39125 | Binary relation on a restr... |
| br1cnvres 39126 | Binary relation on the con... |
| elec1cnvres 39127 | Elementhood in the convers... |
| ec1cnvres 39128 | Converse restricted coset ... |
| eldmres 39129 | Elementhood in the domain ... |
| elrnres 39130 | Element of the range of a ... |
| eldmressnALTV 39131 | Element of the domain of a... |
| elrnressn 39132 | Element of the range of a ... |
| eldm4 39133 | Elementhood in a domain. ... |
| eldmres2 39134 | Elementhood in the domain ... |
| eldmres3 39135 | Elementhood in the domain ... |
| eceq1i 39136 | Equality theorem for ` C `... |
| ecres 39137 | Restricted coset of ` B ` ... |
| eccnvepres 39138 | Restricted converse epsilo... |
| eleccnvep 39139 | Elementhood in the convers... |
| eccnvep 39140 | The converse epsilon coset... |
| extep 39141 | Property of epsilon relati... |
| disjeccnvep 39142 | Property of the epsilon re... |
| eccnvepres2 39143 | The restricted converse ep... |
| eccnvepres3 39144 | Condition for a restricted... |
| eldmqsres 39145 | Elementhood in a restricte... |
| eldmqsres2 39146 | Elementhood in a restricte... |
| qsss1 39147 | Subclass theorem for quoti... |
| qseq1i 39148 | Equality theorem for quoti... |
| brinxprnres 39149 | Binary relation on a restr... |
| inxprnres 39150 | Restriction of a class as ... |
| dfres4 39151 | Alternate definition of th... |
| exan3 39152 | Equivalent expressions wit... |
| exanres 39153 | Equivalent expressions wit... |
| exanres3 39154 | Equivalent expressions wit... |
| exanres2 39155 | Equivalent expressions wit... |
| cnvepres 39156 | Restricted converse epsilo... |
| eqrel2 39157 | Equality of relations. (C... |
| rncnv 39158 | Range of converse is the d... |
| dfdm6 39159 | Alternate definition of do... |
| dfrn6 39160 | Alternate definition of ra... |
| rncnvepres 39161 | The range of the restricte... |
| dmecd 39162 | Equality of the coset of `... |
| dmec2d 39163 | Equality of the coset of `... |
| brid 39164 | Property of the identity b... |
| ideq2 39165 | For sets, the identity bin... |
| idresssidinxp 39166 | Condition for the identity... |
| idreseqidinxp 39167 | Condition for the identity... |
| extid 39168 | Property of identity relat... |
| inxpss 39169 | Two ways to say that an in... |
| idinxpss 39170 | Two ways to say that an in... |
| ref5 39171 | Two ways to say that an in... |
| inxpss3 39172 | Two ways to say that an in... |
| inxpss2 39173 | Two ways to say that inter... |
| inxpssidinxp 39174 | Two ways to say that inter... |
| idinxpssinxp 39175 | Two ways to say that inter... |
| idinxpssinxp2 39176 | Identity intersection with... |
| idinxpssinxp3 39177 | Identity intersection with... |
| idinxpssinxp4 39178 | Identity intersection with... |
| relcnveq3 39179 | Two ways of saying a relat... |
| relcnveq 39180 | Two ways of saying a relat... |
| relcnveq2 39181 | Two ways of saying a relat... |
| relcnveq4 39182 | Two ways of saying a relat... |
| qsresid 39183 | Simplification of a specia... |
| n0elqs 39184 | Two ways of expressing tha... |
| n0elqs2 39185 | Two ways of expressing tha... |
| rnresequniqs 39186 | The range of a restriction... |
| n0el2 39187 | Two ways of expressing tha... |
| cnvepresex 39188 | Sethood condition for the ... |
| cnvepima 39189 | The image of converse epsi... |
| inex3 39190 | Sufficient condition for t... |
| inxpex 39191 | Sufficient condition for a... |
| eqres 39192 | Converting a class constan... |
| brrabga 39193 | The law of concretion for ... |
| brcnvrabga 39194 | The law of concretion for ... |
| opideq 39195 | Equality conditions for or... |
| iss2 39196 | A subclass of the identity... |
| eldmcnv 39197 | Elementhood in a domain of... |
| dfrel5 39198 | Alternate definition of th... |
| dfrel6 39199 | Alternate definition of th... |
| cnvresrn 39200 | Converse restricted to ran... |
| relssinxpdmrn 39201 | Subset of restriction, spe... |
| cnvref4 39202 | Two ways to say that a rel... |
| cnvref5 39203 | Two ways to say that a rel... |
| ecin0 39204 | Two ways of saying that th... |
| ecinn0 39205 | Two ways of saying that th... |
| ineleq 39206 | Equivalence of restricted ... |
| inecmo 39207 | Equivalence of a double re... |
| inecmo2 39208 | Equivalence of a double re... |
| ineccnvmo 39209 | Equivalence of a double re... |
| alrmomorn 39210 | Equivalence of an "at most... |
| alrmomodm 39211 | Equivalence of an "at most... |
| ralmo 39212 | "At most one" can be restr... |
| ralrnmo 39213 | On the range, "at most one... |
| dmqsex 39214 | Sethood of the domain quot... |
| raldmqsmo 39215 | On the quotient carrier, "... |
| ralrmo3 39216 | Pull a restricted universa... |
| raldmqseu 39217 | Equivalence between "exact... |
| rsp3 39218 | From a restricted universa... |
| rsp3eq 39219 | From a restricted universa... |
| ineccnvmo2 39220 | Equivalence of a double un... |
| inecmo3 39221 | Equivalence of a double un... |
| moeu2 39222 | Uniqueness is equivalent t... |
| mopickr 39223 | "At most one" picks a vari... |
| moantr 39224 | Sufficient condition for t... |
| brabidgaw 39225 | The law of concretion for ... |
| brabidga 39226 | The law of concretion for ... |
| inxp2 39227 | Intersection with a Cartes... |
| opabf 39228 | A class abstraction of a c... |
| ec0 39229 | The empty-coset of a class... |
| brcnvin 39230 | Intersection with a conver... |
| ssdmral 39231 | Subclass of a domain. (Co... |
| xrnss3v 39233 | A range Cartesian product ... |
| xrnrel 39234 | A range Cartesian product ... |
| brxrn 39235 | Characterize a ternary rel... |
| brxrn2 39236 | A characterization of the ... |
| dfxrn2 39237 | Alternate definition of th... |
| brxrncnvep 39238 | The range product with con... |
| dmxrn 39239 | Domain of the range produc... |
| dmcnvep 39240 | Domain of converse epsilon... |
| dmxrncnvep 39241 | Domain of the range produc... |
| dmcnvepres 39242 | Domain of the restricted c... |
| dmuncnvepres 39243 | Domain of the union with t... |
| dmxrnuncnvepres 39244 | Domain of the combined rel... |
| ecun 39245 | The union coset of ` A ` .... |
| ecunres 39246 | The restricted union coset... |
| ecuncnvepres 39247 | The restricted union with ... |
| xrneq1 39248 | Equality theorem for the r... |
| xrneq1i 39249 | Equality theorem for the r... |
| xrneq1d 39250 | Equality theorem for the r... |
| xrneq2 39251 | Equality theorem for the r... |
| xrneq2i 39252 | Equality theorem for the r... |
| xrneq2d 39253 | Equality theorem for the r... |
| xrneq12 39254 | Equality theorem for the r... |
| xrneq12i 39255 | Equality theorem for the r... |
| xrneq12d 39256 | Equality theorem for the r... |
| elecxrn 39257 | Elementhood in the ` ( R |... |
| ecxrn 39258 | The ` ( R |X. S ) ` -coset... |
| relecxrn 39259 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 39260 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 39261 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 39262 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 39263 | Double restricted quantifi... |
| disjecxrn 39264 | Two ways of saying that ` ... |
| disjecxrncnvep 39265 | Two ways of saying that co... |
| disjsuc2 39266 | Double restricted quantifi... |
| xrninxp 39267 | Intersection of a range Ca... |
| xrninxp2 39268 | Intersection of a range Ca... |
| xrninxpex 39269 | Sufficient condition for t... |
| inxpxrn 39270 | Two ways to express the in... |
| br1cnvxrn2 39271 | The converse of a binary r... |
| elec1cnvxrn2 39272 | Elementhood in the convers... |
| rnxrn 39273 | Range of the range Cartesi... |
| rnxrnres 39274 | Range of a range Cartesian... |
| rnxrncnvepres 39275 | Range of a range Cartesian... |
| rnxrnidres 39276 | Range of a range Cartesian... |
| xrnres 39277 | Two ways to express restri... |
| xrnres2 39278 | Two ways to express restri... |
| xrnres3 39279 | Two ways to express restri... |
| xrnres4 39280 | Two ways to express restri... |
| xrnresex 39281 | Sufficient condition for a... |
| xrnidresex 39282 | Sufficient condition for a... |
| xrncnvepresex 39283 | Sufficient condition for a... |
| dmxrncnvepres 39284 | Domain of the range produc... |
| dmxrncnvepres2 39285 | Domain of the range produc... |
| eldmxrncnvepres 39286 | Element of the domain of t... |
| eldmxrncnvepres2 39287 | Element of the domain of t... |
| eceldmqsxrncnvepres 39288 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 39289 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 39290 | Binary relation on an inte... |
| brin3 39291 | Binary relation on an inte... |
| elrels2 39293 | The element of the relatio... |
| elrelsrel 39294 | The element of the relatio... |
| elrelsrelim 39295 | The element of the relatio... |
| elrels5 39296 | Equivalent expressions for... |
| elrels6 39297 | Equivalent expressions for... |
| dfqmap2 39299 | Alternate definition of th... |
| dfqmap3 39300 | Alternate definition of th... |
| ecqmap 39301 | ` QMap ` fibers are single... |
| ecqmap2 39302 | Fiber of ` QMap ` equals s... |
| qmapex 39303 | Quotient map exists if ` R... |
| relqmap 39304 | Quotient map is a relation... |
| dmqmap 39305 | ` QMap ` preserves the dom... |
| rnqmap 39306 | The range of the quotient ... |
| dfadjliftmap 39308 | Alternate (expanded) defin... |
| dfadjliftmap2 39309 | Alternate definition of th... |
| blockadjliftmap 39310 | A "two-stage" construction... |
| dfblockliftmap 39312 | Alternate definition of th... |
| dfblockliftmap2 39313 | Alternate definition of th... |
| dfsucmap3 39315 | Alternate definition of th... |
| dfsucmap2 39316 | Alternate definition of th... |
| dfsucmap4 39317 | Alternate definition of th... |
| brsucmap 39318 | Binary relation form of th... |
| relsucmap 39319 | The successor map is a rel... |
| dmsucmap 39320 | The domain of the successo... |
| dfsuccl2 39322 | Alternate definition of th... |
| mopre 39323 | There is at most one prede... |
| exeupre2 39324 | Whenever a predecessor exi... |
| dfsuccl3 39325 | Alternate definition of th... |
| dfsuccl4 39326 | Alternate definition that ... |
| dfpre 39328 | Alternate definition of th... |
| dfpre2 39329 | Alternate definition of th... |
| dfpre3 39330 | Alternate definition of th... |
| dfpred4 39331 | Alternate definition of th... |
| dfpre4 39332 | Alternate definition of th... |
| shiftstableeq2 39335 | Equality theorem for shift... |
| suceqsneq 39336 | One-to-one relationship be... |
| sucdifsn2 39337 | Absorption of union with a... |
| sucdifsn 39338 | The difference between the... |
| ressucdifsn2 39339 | The difference between res... |
| ressucdifsn 39340 | The difference between res... |
| sucmapsuc 39341 | A set is succeeded by its ... |
| sucmapleftuniq 39342 | Left uniqueness of the suc... |
| exeupre 39343 | Whenever a predecessor exi... |
| preex 39344 | The successor-predecessor ... |
| eupre2 39345 | Unique predecessor exists ... |
| eupre 39346 | Unique predecessor exists ... |
| presucmap 39347 | ` pre ` is really a predec... |
| preuniqval 39348 | Uniqueness/canonicity of `... |
| sucpre 39349 | ` suc ` is a right-inverse... |
| presuc 39350 | ` pre ` is a left-inverse ... |
| press 39351 | Predecessor is a subset of... |
| preel 39352 | Predecessor is a subset of... |
| dfcoss2 39355 | Alternate definition of th... |
| dfcoss3 39356 | Alternate definition of th... |
| dfcoss4 39357 | Alternate definition of th... |
| cosscnv 39358 | Class of cosets by the con... |
| coss1cnvres 39359 | Class of cosets by the con... |
| coss2cnvepres 39360 | Special case of ~ coss1cnv... |
| cossex 39361 | If ` A ` is a set then the... |
| cosscnvex 39362 | If ` A ` is a set then the... |
| 1cosscnvepresex 39363 | Sufficient condition for a... |
| 1cossxrncnvepresex 39364 | Sufficient condition for a... |
| relcoss 39365 | Cosets by ` R ` is a relat... |
| relcoels 39366 | Coelements on ` A ` is a r... |
| cossss 39367 | Subclass theorem for the c... |
| cosseq 39368 | Equality theorem for the c... |
| cosseqi 39369 | Equality theorem for the c... |
| cosseqd 39370 | Equality theorem for the c... |
| 1cossres 39371 | The class of cosets by a r... |
| dfcoels 39372 | Alternate definition of th... |
| brcoss 39373 | ` A ` and ` B ` are cosets... |
| brcoss2 39374 | Alternate form of the ` A ... |
| brcoss3 39375 | Alternate form of the ` A ... |
| brcosscnvcoss 39376 | For sets, the ` A ` and ` ... |
| brcoels 39377 | ` B ` and ` C ` are coelem... |
| cocossss 39378 | Two ways of saying that co... |
| cnvcosseq 39379 | The converse of cosets by ... |
| br2coss 39380 | Cosets by ` ,~ R ` binary ... |
| br1cossres 39381 | ` B ` and ` C ` are cosets... |
| br1cossres2 39382 | ` B ` and ` C ` are cosets... |
| brressn 39383 | Binary relation on a restr... |
| ressn2 39384 | A class ' R ' restricted t... |
| refressn 39385 | Any class ' R ' restricted... |
| antisymressn 39386 | Every class ' R ' restrict... |
| trressn 39387 | Any class ' R ' restricted... |
| relbrcoss 39388 | ` A ` and ` B ` are cosets... |
| br1cossinres 39389 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 39390 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 39391 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 39392 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 39393 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 39394 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 39395 | The domain of cosets is th... |
| dmcoss2 39396 | The domain of cosets is th... |
| rncossdmcoss 39397 | The range of cosets is the... |
| dm1cosscnvepres 39398 | The domain of cosets of th... |
| dmcoels 39399 | The domain of coelements i... |
| eldmcoss 39400 | Elementhood in the domain ... |
| eldmcoss2 39401 | Elementhood in the domain ... |
| eldm1cossres 39402 | Elementhood in the domain ... |
| eldm1cossres2 39403 | Elementhood in the domain ... |
| refrelcosslem 39404 | Lemma for the left side of... |
| refrelcoss3 39405 | The class of cosets by ` R... |
| refrelcoss2 39406 | The class of cosets by ` R... |
| symrelcoss3 39407 | The class of cosets by ` R... |
| symrelcoss2 39408 | The class of cosets by ` R... |
| cossssid 39409 | Equivalent expressions for... |
| cossssid2 39410 | Equivalent expressions for... |
| cossssid3 39411 | Equivalent expressions for... |
| cossssid4 39412 | Equivalent expressions for... |
| cossssid5 39413 | Equivalent expressions for... |
| brcosscnv 39414 | ` A ` and ` B ` are cosets... |
| brcosscnv2 39415 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 39416 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 39417 | Cosets by the converse ran... |
| cosscnvssid3 39418 | Equivalent expressions for... |
| cosscnvssid4 39419 | Equivalent expressions for... |
| cosscnvssid5 39420 | Equivalent expressions for... |
| coss0 39421 | Cosets by the empty set ar... |
| cossid 39422 | Cosets by the identity rel... |
| cosscnvid 39423 | Cosets by the converse ide... |
| trcoss 39424 | Sufficient condition for t... |
| eleccossin 39425 | Two ways of saying that th... |
| trcoss2 39426 | Equivalent expressions for... |
| cosselrels 39427 | Cosets of sets are element... |
| cnvelrels 39428 | The converse of a set is a... |
| cosscnvelrels 39429 | Cosets of converse sets ar... |
| dfssr2 39431 | Alternate definition of th... |
| relssr 39432 | The subset relation is a r... |
| brssr 39433 | The subset relation and su... |
| brssrid 39434 | Any set is a subset of its... |
| issetssr 39435 | Two ways of expressing set... |
| brssrres 39436 | Restricted subset binary r... |
| br1cnvssrres 39437 | Restricted converse subset... |
| brcnvssr 39438 | The converse of a subset r... |
| brcnvssrid 39439 | Any set is a converse subs... |
| br1cossxrncnvssrres 39440 | ` <. B , C >. ` and ` <. D... |
| extssr 39441 | Property of subset relatio... |
| dfrefrels2 39445 | Alternate definition of th... |
| dfrefrels3 39446 | Alternate definition of th... |
| dfrefrel2 39447 | Alternate definition of th... |
| dfrefrel3 39448 | Alternate definition of th... |
| dfrefrel5 39449 | Alternate definition of th... |
| elrefrels2 39450 | Element of the class of re... |
| elrefrels3 39451 | Element of the class of re... |
| elrefrelsrel 39452 | For sets, being an element... |
| refreleq 39453 | Equality theorem for refle... |
| refrelid 39454 | Identity relation is refle... |
| refrelcoss 39455 | The class of cosets by ` R... |
| refrelressn 39456 | Any class ' R ' restricted... |
| dfcnvrefrels2 39460 | Alternate definition of th... |
| dfcnvrefrels3 39461 | Alternate definition of th... |
| dfcnvrefrel2 39462 | Alternate definition of th... |
| dfcnvrefrel3 39463 | Alternate definition of th... |
| dfcnvrefrel4 39464 | Alternate definition of th... |
| dfcnvrefrel5 39465 | Alternate definition of th... |
| elcnvrefrels2 39466 | Element of the class of co... |
| elcnvrefrels3 39467 | Element of the class of co... |
| elcnvrefrelsrel 39468 | For sets, being an element... |
| cnvrefrelcoss2 39469 | Necessary and sufficient c... |
| cosselcnvrefrels2 39470 | Necessary and sufficient c... |
| cosselcnvrefrels3 39471 | Necessary and sufficient c... |
| cosselcnvrefrels4 39472 | Necessary and sufficient c... |
| cosselcnvrefrels5 39473 | Necessary and sufficient c... |
| dfsymrels2 39477 | Alternate definition of th... |
| dfsymrels3 39478 | Alternate definition of th... |
| elrelscnveq3 39479 | Two ways of saying a relat... |
| elrelscnveq 39480 | Two ways of saying a relat... |
| elrelscnveq2 39481 | Two ways of saying a relat... |
| elrelscnveq4 39482 | Two ways of saying a relat... |
| dfsymrels4 39483 | Alternate definition of th... |
| dfsymrels5 39484 | Alternate definition of th... |
| dfsymrel2 39485 | Alternate definition of th... |
| dfsymrel3 39486 | Alternate definition of th... |
| dfsymrel4 39487 | Alternate definition of th... |
| dfsymrel5 39488 | Alternate definition of th... |
| elsymrels2 39489 | Element of the class of sy... |
| elsymrels3 39490 | Element of the class of sy... |
| elsymrels4 39491 | Element of the class of sy... |
| elsymrels5 39492 | Element of the class of sy... |
| elsymrelsrel 39493 | For sets, being an element... |
| symreleq 39494 | Equality theorem for symme... |
| symrelim 39495 | Symmetric relation implies... |
| symrelcoss 39496 | The class of cosets by ` R... |
| idsymrel 39497 | The identity relation is s... |
| epnsymrel 39498 | The membership (epsilon) r... |
| symrefref2 39499 | Symmetry is a sufficient c... |
| symrefref3 39500 | Symmetry is a sufficient c... |
| refsymrels2 39501 | Elements of the class of r... |
| refsymrels3 39502 | Elements of the class of r... |
| refsymrel2 39503 | A relation which is reflex... |
| refsymrel3 39504 | A relation which is reflex... |
| elrefsymrels2 39505 | Elements of the class of r... |
| elrefsymrels3 39506 | Elements of the class of r... |
| elrefsymrelsrel 39507 | For sets, being an element... |
| dftrrels2 39511 | Alternate definition of th... |
| dftrrels3 39512 | Alternate definition of th... |
| dftrrel2 39513 | Alternate definition of th... |
| dftrrel3 39514 | Alternate definition of th... |
| eltrrels2 39515 | Element of the class of tr... |
| eltrrels3 39516 | Element of the class of tr... |
| eltrrelsrel 39517 | For sets, being an element... |
| trreleq 39518 | Equality theorem for the t... |
| trrelressn 39519 | Any class ' R ' restricted... |
| dfeqvrels2 39524 | Alternate definition of th... |
| dfeqvrels3 39525 | Alternate definition of th... |
| dfeqvrel2 39526 | Alternate definition of th... |
| dfeqvrel3 39527 | Alternate definition of th... |
| eleqvrels2 39528 | Element of the class of eq... |
| eleqvrels3 39529 | Element of the class of eq... |
| eleqvrelsrel 39530 | For sets, being an element... |
| elcoeleqvrels 39531 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 39532 | For sets, being an element... |
| eqvrelrel 39533 | An equivalence relation is... |
| eqvrelrefrel 39534 | An equivalence relation is... |
| eqvrelsymrel 39535 | An equivalence relation is... |
| eqvreltrrel 39536 | An equivalence relation is... |
| eqvrelim 39537 | Equivalence relation impli... |
| eqvreleq 39538 | Equality theorem for equiv... |
| eqvreleqi 39539 | Equality theorem for equiv... |
| eqvreleqd 39540 | Equality theorem for equiv... |
| eqvrelsym 39541 | An equivalence relation is... |
| eqvrelsymb 39542 | An equivalence relation is... |
| eqvreltr 39543 | An equivalence relation is... |
| eqvreltrd 39544 | A transitivity relation fo... |
| eqvreltr4d 39545 | A transitivity relation fo... |
| eqvrelref 39546 | An equivalence relation is... |
| eqvrelth 39547 | Basic property of equivale... |
| eqvrelcl 39548 | Elementhood in the field o... |
| eqvrelthi 39549 | Basic property of equivale... |
| eqvreldisj 39550 | Equivalence classes do not... |
| qsdisjALTV 39551 | Elements of a quotient set... |
| eqvrelqsel 39552 | If an element of a quotien... |
| eqvrelcoss 39553 | Two ways to express equiva... |
| eqvrelcoss3 39554 | Two ways to express equiva... |
| eqvrelcoss2 39555 | Two ways to express equiva... |
| eqvrelcoss4 39556 | Two ways to express equiva... |
| dfcoeleqvrels 39557 | Alternate definition of th... |
| dfcoeleqvrel 39558 | Alternate definition of th... |
| brredunds 39562 | Binary relation on the cla... |
| brredundsredund 39563 | For sets, binary relation ... |
| redundss3 39564 | Implication of redundancy ... |
| redundeq1 39565 | Equivalence of redundancy ... |
| redundpim3 39566 | Implication of redundancy ... |
| redundpbi1 39567 | Equivalence of redundancy ... |
| refrelsredund4 39568 | The naive version of the c... |
| refrelsredund2 39569 | The naive version of the c... |
| refrelsredund3 39570 | The naive version of the c... |
| refrelredund4 39571 | The naive version of the d... |
| refrelredund2 39572 | The naive version of the d... |
| refrelredund3 39573 | The naive version of the d... |
| dmqseq 39576 | Equality theorem for domai... |
| dmqseqi 39577 | Equality theorem for domai... |
| dmqseqd 39578 | Equality theorem for domai... |
| dmqseqeq1 39579 | Equality theorem for domai... |
| dmqseqeq1i 39580 | Equality theorem for domai... |
| dmqseqeq1d 39581 | Equality theorem for domai... |
| brdmqss 39582 | The domain quotient binary... |
| brdmqssqs 39583 | If ` A ` and ` R ` are set... |
| n0eldmqs 39584 | The empty set is not an el... |
| qseq 39585 | The quotient set equal to ... |
| n0eldmqseq 39586 | The empty set is not an el... |
| n0elim 39587 | Implication of that the em... |
| n0el3 39588 | Two ways of expressing tha... |
| cnvepresdmqss 39589 | The domain quotient binary... |
| cnvepresdmqs 39590 | The domain quotient predic... |
| unidmqs 39591 | The range of a relation is... |
| unidmqseq 39592 | The union of the domain qu... |
| dmqseqim 39593 | If the domain quotient of ... |
| dmqseqim2 39594 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 39595 | Elementhood in the domain ... |
| eldmqs1cossres 39596 | Elementhood in the domain ... |
| releldmqscoss 39597 | Elementhood in the domain ... |
| dmqscoelseq 39598 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 39599 | Two ways to express the eq... |
| brers 39604 | Binary equivalence relatio... |
| dferALTV2 39605 | Equivalence relation with ... |
| erALTVeq1 39606 | Equality theorem for equiv... |
| erALTVeq1i 39607 | Equality theorem for equiv... |
| erALTVeq1d 39608 | Equality theorem for equiv... |
| dfcomember 39609 | Alternate definition of th... |
| dfcomember2 39610 | Alternate definition of th... |
| dfcomember3 39611 | Alternate definition of th... |
| eqvreldmqs 39612 | Two ways to express comemb... |
| eqvreldmqs2 39613 | Two ways to express comemb... |
| brerser 39614 | Binary equivalence relatio... |
| erimeq2 39615 | Equivalence relation on it... |
| erimeq 39616 | Equivalence relation on it... |
| dffunsALTV 39620 | Alternate definition of th... |
| dffunsALTV2 39621 | Alternate definition of th... |
| dffunsALTV3 39622 | Alternate definition of th... |
| dffunsALTV4 39623 | Alternate definition of th... |
| dffunsALTV5 39624 | Alternate definition of th... |
| dffunALTV2 39625 | Alternate definition of th... |
| dffunALTV3 39626 | Alternate definition of th... |
| dffunALTV4 39627 | Alternate definition of th... |
| dffunALTV5 39628 | Alternate definition of th... |
| elfunsALTV 39629 | Elementhood in the class o... |
| elfunsALTV2 39630 | Elementhood in the class o... |
| elfunsALTV3 39631 | Elementhood in the class o... |
| elfunsALTV4 39632 | Elementhood in the class o... |
| elfunsALTV5 39633 | Elementhood in the class o... |
| elfunsALTVfunALTV 39634 | The element of the class o... |
| funALTVfun 39635 | Our definition of the func... |
| funALTVss 39636 | Subclass theorem for funct... |
| funALTVeq 39637 | Equality theorem for funct... |
| funALTVeqi 39638 | Equality inference for the... |
| funALTVeqd 39639 | Equality deduction for the... |
| dfdisjs 39645 | Alternate definition of th... |
| dfdisjs2 39646 | Alternate definition of th... |
| dfdisjs3 39647 | Alternate definition of th... |
| dfdisjs4 39648 | Alternate definition of th... |
| dfdisjs5 39649 | Alternate definition of th... |
| dfdisjALTV 39650 | Alternate definition of th... |
| dfdisjALTV2 39651 | Alternate definition of th... |
| dfdisjALTV3 39652 | Alternate definition of th... |
| dfdisjALTV4 39653 | Alternate definition of th... |
| dfdisjALTV5 39654 | Alternate definition of th... |
| dfdisjALTV5a 39655 | Alternate definition of th... |
| disjimeceqim 39656 | ` Disj ` implies coset-equ... |
| disjimeceqim2 39657 | ` Disj ` implies injectivi... |
| disjimeceqbi 39658 | ` Disj ` gives bicondition... |
| disjimeceqbi2 39659 | Injectivity of the block c... |
| disjimrmoeqec 39660 | Under ` Disj ` , every blo... |
| disjimdmqseq 39661 | Disjointness implies uniqu... |
| dfeldisj2 39662 | Alternate definition of th... |
| dfeldisj3 39663 | Alternate definition of th... |
| dfeldisj4 39664 | Alternate definition of th... |
| dfeldisj5 39665 | Alternate definition of th... |
| dfeldisj5a 39666 | Alternate definition of th... |
| eldisjim3 39667 | ` ElDisj ` elimination (tw... |
| eldisjdmqsim2 39668 | ElDisj of quotient implies... |
| eldisjdmqsim 39669 | Shared output implies equa... |
| suceldisj 39670 | Disjointness of successor ... |
| eldisjs 39671 | Elementhood in the class o... |
| eldisjs2 39672 | Elementhood in the class o... |
| eldisjs3 39673 | Elementhood in the class o... |
| eldisjs4 39674 | Elementhood in the class o... |
| eldisjs5 39675 | Elementhood in the class o... |
| eldisjsdisj 39676 | The element of the class o... |
| qmapeldisjs 39677 | When ` R ` is a set (e.g.,... |
| disjqmap2 39678 | Disjointness of ` QMap ` e... |
| disjqmap 39679 | Disjointness of ` QMap ` e... |
| eleldisjs 39680 | Elementhood in the disjoin... |
| eleldisjseldisj 39681 | The element of the disjoin... |
| disjrel 39682 | Disjoint relation is a rel... |
| disjss 39683 | Subclass theorem for disjo... |
| disjssi 39684 | Subclass theorem for disjo... |
| disjssd 39685 | Subclass theorem for disjo... |
| disjeq 39686 | Equality theorem for disjo... |
| disjeqi 39687 | Equality theorem for disjo... |
| disjeqd 39688 | Equality theorem for disjo... |
| disjdmqseqeq1 39689 | Lemma for the equality the... |
| eldisjss 39690 | Subclass theorem for disjo... |
| eldisjssi 39691 | Subclass theorem for disjo... |
| eldisjssd 39692 | Subclass theorem for disjo... |
| eldisjeq 39693 | Equality theorem for disjo... |
| eldisjeqi 39694 | Equality theorem for disjo... |
| eldisjeqd 39695 | Equality theorem for disjo... |
| disjres 39696 | Disjoint restriction. (Co... |
| eldisjn0elb 39697 | Two forms of disjoint elem... |
| disjxrn 39698 | Two ways of saying that a ... |
| disjxrnres5 39699 | Disjoint range Cartesian p... |
| disjorimxrn 39700 | Disjointness condition for... |
| disjimxrn 39701 | Disjointness condition for... |
| disjimres 39702 | Disjointness condition for... |
| disjimin 39703 | Disjointness condition for... |
| disjiminres 39704 | Disjointness condition for... |
| disjimxrnres 39705 | Disjointness condition for... |
| disjALTV0 39706 | The null class is disjoint... |
| disjALTVid 39707 | The class of identity rela... |
| disjALTVidres 39708 | The class of identity rela... |
| disjALTVinidres 39709 | The intersection with rest... |
| disjALTVxrnidres 39710 | The class of range Cartesi... |
| disjsuc 39711 | Disjoint range Cartesian p... |
| qmapeldisjsim 39712 | Injectivity of coset map f... |
| qmapeldisjsbi 39713 | Injectivity of coset map f... |
| rnqmapeleldisjsim 39714 | Element-disjointness of th... |
| dfantisymrel4 39716 | Alternate definition of th... |
| dfantisymrel5 39717 | Alternate definition of th... |
| antisymrelres 39718 | (Contributed by Peter Mazs... |
| antisymrelressn 39719 | (Contributed by Peter Mazs... |
| dfpart2 39724 | Alternate definition of th... |
| dfmembpart2 39725 | Alternate definition of th... |
| brparts 39726 | Binary partitions relation... |
| brparts2 39727 | Binary partitions relation... |
| brpartspart 39728 | Binary partition and the p... |
| parteq1 39729 | Equality theorem for parti... |
| parteq2 39730 | Equality theorem for parti... |
| parteq12 39731 | Equality theorem for parti... |
| parteq1i 39732 | Equality theorem for parti... |
| parteq1d 39733 | Equality theorem for parti... |
| partsuc2 39734 | Property of the partition.... |
| partsuc 39735 | Property of the partition.... |
| disjim 39736 | The "Divide et Aequivalere... |
| disjimi 39737 | Every disjoint relation ge... |
| detlem 39738 | If a relation is disjoint,... |
| eldisjim 39739 | If the elements of ` A ` a... |
| eldisjim2 39740 | Alternate form of ~ eldisj... |
| eqvrel0 39741 | The null class is an equiv... |
| det0 39742 | The cosets by the null cla... |
| eqvrelcoss0 39743 | The cosets by the null cla... |
| eqvrelid 39744 | The identity relation is a... |
| eqvrel1cossidres 39745 | The cosets by a restricted... |
| eqvrel1cossinidres 39746 | The cosets by an intersect... |
| eqvrel1cossxrnidres 39747 | The cosets by a range Cart... |
| detid 39748 | The cosets by the identity... |
| eqvrelcossid 39749 | The cosets by the identity... |
| detidres 39750 | The cosets by the restrict... |
| detinidres 39751 | The cosets by the intersec... |
| detxrnidres 39752 | The cosets by the range Ca... |
| disjlem14 39753 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 39754 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 39755 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 39756 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 39757 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 39758 | Lemma for ~ disjdmqseq via... |
| disjdmqs 39759 | If a relation is disjoint,... |
| disjdmqseq 39760 | If a relation is disjoint,... |
| eldisjn0el 39761 | Special case of ~ disjdmqs... |
| partim2 39762 | Disjoint relation on its n... |
| partim 39763 | Partition implies equivale... |
| partimeq 39764 | Partition implies that the... |
| eldisjlem19 39765 | Special case of ~ disjlem1... |
| membpartlem19 39766 | Together with ~ disjlem19 ... |
| petlem 39767 | If you can prove that the ... |
| petlemi 39768 | If you can prove disjointn... |
| pet02 39769 | Class ` A ` is a partition... |
| pet0 39770 | Class ` A ` is a partition... |
| petid2 39771 | Class ` A ` is a partition... |
| petid 39772 | A class is a partition by ... |
| petidres2 39773 | Class ` A ` is a partition... |
| petidres 39774 | A class is a partition by ... |
| petinidres2 39775 | Class ` A ` is a partition... |
| petinidres 39776 | A class is a partition by ... |
| petxrnidres2 39777 | Class ` A ` is a partition... |
| petxrnidres 39778 | A class is a partition by ... |
| eqvreldisj1 39779 | The elements of the quotie... |
| eqvreldisj2 39780 | The elements of the quotie... |
| eqvreldisj3 39781 | The elements of the quotie... |
| eqvreldisj4 39782 | Intersection with the conv... |
| eqvreldisj5 39783 | Range Cartesian product wi... |
| eqvrelqseqdisj2 39784 | Implication of ~ eqvreldis... |
| disjimeldisjdmqs 39785 | ` Disj ` implies element-d... |
| eldisjsim1 39786 | An element of the class of... |
| eldisjsim2 39787 | An element of the class of... |
| disjsssrels 39788 | The class of disjoint rela... |
| eldisjsim3 39789 | ` Disjs ` implies element-... |
| eldisjsim4 39790 | ` Disjs ` implies element-... |
| eldisjsim5 39791 | ` Disjs ` is closed under ... |
| eldisjs6 39792 | Elementhood in the class o... |
| eldisjs7 39793 | Elementhood in the class o... |
| dfdisjs6 39794 | Alternate definition of th... |
| dfdisjs7 39795 | Alternate definition of th... |
| fences3 39796 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 39797 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 39798 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 39799 | Lemma for the Partition-Eq... |
| mainer 39800 | The Main Theorem of Equiva... |
| partimcomember 39801 | Partition with general ` R... |
| mpet3 39802 | Member Partition-Equivalen... |
| cpet2 39803 | The conventional form of t... |
| cpet 39804 | The conventional form of M... |
| mpet 39805 | Member Partition-Equivalen... |
| mpet2 39806 | Member Partition-Equivalen... |
| mpets2 39807 | Member Partition-Equivalen... |
| mpets 39808 | Member Partition-Equivalen... |
| mainpart 39809 | Partition with general ` R... |
| fences 39810 | The Theorem of Fences by E... |
| fences2 39811 | The Theorem of Fences by E... |
| mainer2 39812 | The Main Theorem of Equiva... |
| mainerim 39813 | Every equivalence relation... |
| petincnvepres2 39814 | A partition-equivalence th... |
| petincnvepres 39815 | The shortest form of a par... |
| pet2 39816 | Partition-Equivalence Theo... |
| pet 39817 | Partition-Equivalence Theo... |
| pets 39818 | Partition-Equivalence Theo... |
| dmqsblocks 39819 | If the ~ pet span ` ( R |X... |
| dfpetparts2 39824 | Alternate definition of ` ... |
| dfpet2parts2 39825 | Grade stability applied to... |
| dfpeters2 39826 | Alternate definition of ` ... |
| typesafepets 39827 | Type-safe ~ pets scheme. ... |
| petseq 39828 | Generalized partition-equi... |
| pets2eq 39829 | Grade-stable generalized p... |
| prtlem60 39830 | Lemma for ~ prter3 . (Con... |
| bicomdd 39831 | Commute two sides of a bic... |
| jca2r 39832 | Inference conjoining the c... |
| jca3 39833 | Inference conjoining the c... |
| prtlem70 39834 | Lemma for ~ prter3 : a rea... |
| ibdr 39835 | Reverse of ~ ibd . (Contr... |
| prtlem100 39836 | Lemma for ~ prter3 . (Con... |
| prtlem5 39837 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 39838 | Lemma for ~ prter2 . (Con... |
| brabsb2 39839 | A closed form of ~ brabsb ... |
| eqbrrdv2 39840 | Other version of ~ eqbrrdi... |
| prtlem9 39841 | Lemma for ~ prter3 . (Con... |
| prtlem10 39842 | Lemma for ~ prter3 . (Con... |
| prtlem11 39843 | Lemma for ~ prter2 . (Con... |
| prtlem12 39844 | Lemma for ~ prtex and ~ pr... |
| prtlem13 39845 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 39846 | Lemma for ~ prtex , ~ prte... |
| prtlem400 39847 | Lemma for ~ prter2 and als... |
| erprt 39850 | The quotient set of an equ... |
| prtlem14 39851 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 39852 | Lemma for ~ prter1 and ~ p... |
| prtlem17 39853 | Lemma for ~ prter2 . (Con... |
| prtlem18 39854 | Lemma for ~ prter2 . (Con... |
| prtlem19 39855 | Lemma for ~ prter2 . (Con... |
| prter1 39856 | Every partition generates ... |
| prtex 39857 | The equivalence relation g... |
| prter2 39858 | The quotient set of the eq... |
| prter3 39859 | For every partition there ... |
| axc5 39870 | This theorem repeats ~ sp ... |
| ax4fromc4 39871 | Rederivation of Axiom ~ ax... |
| ax10fromc7 39872 | Rederivation of Axiom ~ ax... |
| ax6fromc10 39873 | Rederivation of Axiom ~ ax... |
| hba1-o 39874 | The setvar ` x ` is not fr... |
| axc4i-o 39875 | Inference version of ~ ax-... |
| equid1 39876 | Proof of ~ equid from our ... |
| equcomi1 39877 | Proof of ~ equcomi from ~ ... |
| aecom-o 39878 | Commutation law for identi... |
| aecoms-o 39879 | A commutation rule for ide... |
| hbae-o 39880 | All variables are effectiv... |
| dral1-o 39881 | Formula-building lemma for... |
| ax12fromc15 39882 | Rederivation of Axiom ~ ax... |
| ax13fromc9 39883 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 39884 | Axiom to quantify a variab... |
| sps-o 39885 | Generalization of antecede... |
| hbequid 39886 | Bound-variable hypothesis ... |
| nfequid-o 39887 | Bound-variable hypothesis ... |
| axc5c7 39888 | Proof of a single axiom th... |
| axc5c7toc5 39889 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 39890 | Rederivation of ~ ax-c7 fr... |
| axc711 39891 | Proof of a single axiom th... |
| nfa1-o 39892 | ` x ` is not free in ` A. ... |
| axc711toc7 39893 | Rederivation of ~ ax-c7 fr... |
| axc711to11 39894 | Rederivation of ~ ax-11 fr... |
| axc5c711 39895 | Proof of a single axiom th... |
| axc5c711toc5 39896 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 39897 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 39898 | Rederivation of ~ ax-11 fr... |
| equidqe 39899 | ~ equid with existential q... |
| axc5sp1 39900 | A special case of ~ ax-c5 ... |
| equidq 39901 | ~ equid with universal qua... |
| equid1ALT 39902 | Alternate proof of ~ equid... |
| axc11nfromc11 39903 | Rederivation of ~ ax-c11n ... |
| naecoms-o 39904 | A commutation rule for dis... |
| hbnae-o 39905 | All variables are effectiv... |
| dvelimf-o 39906 | Proof of ~ dvelimh that us... |
| dral2-o 39907 | Formula-building lemma for... |
| aev-o 39908 | A "distinctor elimination"... |
| ax5eq 39909 | Theorem to add distinct qu... |
| dveeq2-o 39910 | Quantifier introduction wh... |
| axc16g-o 39911 | A generalization of Axiom ... |
| dveeq1-o 39912 | Quantifier introduction wh... |
| dveeq1-o16 39913 | Version of ~ dveeq1 using ... |
| ax5el 39914 | Theorem to add distinct qu... |
| axc11n-16 39915 | This theorem shows that, g... |
| dveel2ALT 39916 | Alternate proof of ~ dveel... |
| ax12f 39917 | Basis step for constructin... |
| ax12eq 39918 | Basis step for constructin... |
| ax12el 39919 | Basis step for constructin... |
| ax12indn 39920 | Induction step for constru... |
| ax12indi 39921 | Induction step for constru... |
| ax12indalem 39922 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 39923 | Alternate proof of ~ ax12i... |
| ax12inda2 39924 | Induction step for constru... |
| ax12inda 39925 | Induction step for constru... |
| ax12v2-o 39926 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 39927 | Derive ~ ax-c15 from a hyp... |
| axc11-o 39928 | Show that ~ ax-c11 can be ... |
| fsumshftd 39929 | Index shift of a finite su... |
| riotaclbgBAD 39931 | Closure of restricted iota... |
| riotaclbBAD 39932 | Closure of restricted iota... |
| riotasvd 39933 | Deduction version of ~ rio... |
| riotasv2d 39934 | Value of description binde... |
| riotasv2s 39935 | The value of description b... |
| riotasv 39936 | Value of description binde... |
| riotasv3d 39937 | A property ` ch ` holding ... |
| elimhyps 39938 | A version of ~ elimhyp usi... |
| dedths 39939 | A version of weak deductio... |
| renegclALT 39940 | Closure law for negative o... |
| elimhyps2 39941 | Generalization of ~ elimhy... |
| dedths2 39942 | Generalization of ~ dedths... |
| nfcxfrdf 39943 | A utility lemma to transfe... |
| nfded 39944 | A deduction theorem that c... |
| nfded2 39945 | A deduction theorem that c... |
| nfunidALT2 39946 | Deduction version of ~ nfu... |
| nfunidALT 39947 | Deduction version of ~ nfu... |
| nfopdALT 39948 | Deduction version of bound... |
| cnaddcom 39949 | Recover the commutative la... |
| toycom 39950 | Show the commutative law f... |
| lshpset 39955 | The set of all hyperplanes... |
| islshp 39956 | The predicate "is a hyperp... |
| islshpsm 39957 | Hyperplane properties expr... |
| lshplss 39958 | A hyperplane is a subspace... |
| lshpne 39959 | A hyperplane is not equal ... |
| lshpnel 39960 | A hyperplane's generating ... |
| lshpnelb 39961 | The subspace sum of a hype... |
| lshpnel2N 39962 | Condition that determines ... |
| lshpne0 39963 | The member of the span in ... |
| lshpdisj 39964 | A hyperplane and the span ... |
| lshpcmp 39965 | If two hyperplanes are com... |
| lshpinN 39966 | The intersection of two di... |
| lsatset 39967 | The set of all 1-dim subsp... |
| islsat 39968 | The predicate "is a 1-dim ... |
| lsatlspsn2 39969 | The span of a nonzero sing... |
| lsatlspsn 39970 | The span of a nonzero sing... |
| islsati 39971 | A 1-dim subspace (atom) (o... |
| lsateln0 39972 | A 1-dim subspace (atom) (o... |
| lsatlss 39973 | The set of 1-dim subspaces... |
| lsatlssel 39974 | An atom is a subspace. (C... |
| lsatssv 39975 | An atom is a set of vector... |
| lsatn0 39976 | A 1-dim subspace (atom) of... |
| lsatspn0 39977 | The span of a vector is an... |
| lsator0sp 39978 | The span of a vector is ei... |
| lsatssn0 39979 | A subspace (or any class) ... |
| lsatcmp 39980 | If two atoms are comparabl... |
| lsatcmp2 39981 | If an atom is included in ... |
| lsatel 39982 | A nonzero vector in an ato... |
| lsatelbN 39983 | A nonzero vector in an ato... |
| lsat2el 39984 | Two atoms sharing a nonzer... |
| lsmsat 39985 | Convert comparison of atom... |
| lsatfixedN 39986 | Show equality with the spa... |
| lsmsatcv 39987 | Subspace sum has the cover... |
| lssatomic 39988 | The lattice of subspaces i... |
| lssats 39989 | The lattice of subspaces i... |
| lpssat 39990 | Two subspaces in a proper ... |
| lrelat 39991 | Subspaces are relatively a... |
| lssatle 39992 | The ordering of two subspa... |
| lssat 39993 | Two subspaces in a proper ... |
| islshpat 39994 | Hyperplane properties expr... |
| lcvfbr 39997 | The covers relation for a ... |
| lcvbr 39998 | The covers relation for a ... |
| lcvbr2 39999 | The covers relation for a ... |
| lcvbr3 40000 | The covers relation for a ... |
| lcvpss 40001 | The covers relation implie... |
| lcvnbtwn 40002 | The covers relation implie... |
| lcvntr 40003 | The covers relation is not... |
| lcvnbtwn2 40004 | The covers relation implie... |
| lcvnbtwn3 40005 | The covers relation implie... |
| lsmcv2 40006 | Subspace sum has the cover... |
| lcvat 40007 | If a subspace covers anoth... |
| lsatcv0 40008 | An atom covers the zero su... |
| lsatcveq0 40009 | A subspace covered by an a... |
| lsat0cv 40010 | A subspace is an atom iff ... |
| lcvexchlem1 40011 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 40012 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 40013 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 40014 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 40015 | Lemma for ~ lcvexch . (Co... |
| lcvexch 40016 | Subspaces satisfy the exch... |
| lcvp 40017 | Covering property of Defin... |
| lcv1 40018 | Covering property of a sub... |
| lcv2 40019 | Covering property of a sub... |
| lsatexch 40020 | The atom exchange property... |
| lsatnle 40021 | The meet of a subspace and... |
| lsatnem0 40022 | The meet of distinct atoms... |
| lsatexch1 40023 | The atom exch1ange propert... |
| lsatcv0eq 40024 | If the sum of two atoms co... |
| lsatcv1 40025 | Two atoms covering the zer... |
| lsatcvatlem 40026 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 40027 | A nonzero subspace less th... |
| lsatcvat2 40028 | A subspace covered by the ... |
| lsatcvat3 40029 | A condition implying that ... |
| islshpcv 40030 | Hyperplane properties expr... |
| l1cvpat 40031 | A subspace covered by the ... |
| l1cvat 40032 | Create an atom under an el... |
| lshpat 40033 | Create an atom under a hyp... |
| lflset 40036 | The set of linear function... |
| islfl 40037 | The predicate "is a linear... |
| lfli 40038 | Property of a linear funct... |
| islfld 40039 | Properties that determine ... |
| lflf 40040 | A linear functional is a f... |
| lflcl 40041 | A linear functional value ... |
| lfl0 40042 | A linear functional is zer... |
| lfladd 40043 | Property of a linear funct... |
| lflsub 40044 | Property of a linear funct... |
| lflmul 40045 | Property of a linear funct... |
| lfl0f 40046 | The zero function is a fun... |
| lfl1 40047 | A nonzero functional has a... |
| lfladdcl 40048 | Closure of addition of two... |
| lfladdcom 40049 | Commutativity of functiona... |
| lfladdass 40050 | Associativity of functiona... |
| lfladd0l 40051 | Functional addition with t... |
| lflnegcl 40052 | Closure of the negative of... |
| lflnegl 40053 | A functional plus its nega... |
| lflvscl 40054 | Closure of a scalar produc... |
| lflvsdi1 40055 | Distributive law for (righ... |
| lflvsdi2 40056 | Reverse distributive law f... |
| lflvsdi2a 40057 | Reverse distributive law f... |
| lflvsass 40058 | Associative law for (right... |
| lfl0sc 40059 | The (right vector space) s... |
| lflsc0N 40060 | The scalar product with th... |
| lfl1sc 40061 | The (right vector space) s... |
| lkrfval 40064 | The kernel of a functional... |
| lkrval 40065 | Value of the kernel of a f... |
| ellkr 40066 | Membership in the kernel o... |
| lkrval2 40067 | Value of the kernel of a f... |
| ellkr2 40068 | Membership in the kernel o... |
| lkrcl 40069 | A member of the kernel of ... |
| lkrf0 40070 | The value of a functional ... |
| lkr0f 40071 | The kernel of the zero fun... |
| lkrlss 40072 | The kernel of a linear fun... |
| lkrssv 40073 | The kernel of a linear fun... |
| lkrsc 40074 | The kernel of a nonzero sc... |
| lkrscss 40075 | The kernel of a scalar pro... |
| eqlkr 40076 | Two functionals with the s... |
| eqlkr2 40077 | Two functionals with the s... |
| eqlkr3 40078 | Two functionals with the s... |
| lkrlsp 40079 | The subspace sum of a kern... |
| lkrlsp2 40080 | The subspace sum of a kern... |
| lkrlsp3 40081 | The subspace sum of a kern... |
| lkrshp 40082 | The kernel of a nonzero fu... |
| lkrshp3 40083 | The kernels of nonzero fun... |
| lkrshpor 40084 | The kernel of a functional... |
| lkrshp4 40085 | A kernel is a hyperplane i... |
| lshpsmreu 40086 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 40087 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 40088 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 40089 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 40090 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 40091 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 40092 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 40093 | The set ` G ` defined by h... |
| lshpkr 40094 | The kernel of functional `... |
| lshpkrex 40095 | There exists a functional ... |
| lshpset2N 40096 | The set of all hyperplanes... |
| islshpkrN 40097 | The predicate "is a hyperp... |
| lfl1dim 40098 | Equivalent expressions for... |
| lfl1dim2N 40099 | Equivalent expressions for... |
| ldualset 40102 | Define the (left) dual of ... |
| ldualvbase 40103 | The vectors of a dual spac... |
| ldualelvbase 40104 | Utility theorem for conver... |
| ldualfvadd 40105 | Vector addition in the dua... |
| ldualvadd 40106 | Vector addition in the dua... |
| ldualvaddcl 40107 | The value of vector additi... |
| ldualvaddval 40108 | The value of the value of ... |
| ldualsca 40109 | The ring of scalars of the... |
| ldualsbase 40110 | Base set of scalar ring fo... |
| ldualsaddN 40111 | Scalar addition for the du... |
| ldualsmul 40112 | Scalar multiplication for ... |
| ldualfvs 40113 | Scalar product operation f... |
| ldualvs 40114 | Scalar product operation v... |
| ldualvsval 40115 | Value of scalar product op... |
| ldualvscl 40116 | The scalar product operati... |
| ldualvaddcom 40117 | Commutative law for vector... |
| ldualvsass 40118 | Associative law for scalar... |
| ldualvsass2 40119 | Associative law for scalar... |
| ldualvsdi1 40120 | Distributive law for scala... |
| ldualvsdi2 40121 | Reverse distributive law f... |
| ldualgrplem 40122 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 40123 | The dual of a vector space... |
| ldual0 40124 | The zero scalar of the dua... |
| ldual1 40125 | The unit scalar of the dua... |
| ldualneg 40126 | The negative of a scalar o... |
| ldual0v 40127 | The zero vector of the dua... |
| ldual0vcl 40128 | The dual zero vector is a ... |
| lduallmodlem 40129 | Lemma for ~ lduallmod . (... |
| lduallmod 40130 | The dual of a left module ... |
| lduallvec 40131 | The dual of a left vector ... |
| ldualvsub 40132 | The value of vector subtra... |
| ldualvsubcl 40133 | Closure of vector subtract... |
| ldualvsubval 40134 | The value of the value of ... |
| ldualssvscl 40135 | Closure of scalar product ... |
| ldualssvsubcl 40136 | Closure of vector subtract... |
| ldual0vs 40137 | Scalar zero times a functi... |
| lkr0f2 40138 | The kernel of the zero fun... |
| lduallkr3 40139 | The kernels of nonzero fun... |
| lkrpssN 40140 | Proper subset relation bet... |
| lkrin 40141 | Intersection of the kernel... |
| eqlkr4 40142 | Two functionals with the s... |
| ldual1dim 40143 | Equivalent expressions for... |
| ldualkrsc 40144 | The kernel of a nonzero sc... |
| lkrss 40145 | The kernel of a scalar pro... |
| lkrss2N 40146 | Two functionals with kerne... |
| lkreqN 40147 | Proportional functionals h... |
| lkrlspeqN 40148 | Condition for colinear fun... |
| isopos 40157 | The predicate "is an ortho... |
| opposet 40158 | Every orthoposet is a pose... |
| oposlem 40159 | Lemma for orthoposet prope... |
| op01dm 40160 | Conditions necessary for z... |
| op0cl 40161 | An orthoposet has a zero e... |
| op1cl 40162 | An orthoposet has a unity ... |
| op0le 40163 | Orthoposet zero is less th... |
| ople0 40164 | An element less than or eq... |
| opnlen0 40165 | An element not less than a... |
| lub0N 40166 | The least upper bound of t... |
| opltn0 40167 | A lattice element greater ... |
| ople1 40168 | Any element is less than t... |
| op1le 40169 | If the orthoposet unity is... |
| glb0N 40170 | The greatest lower bound o... |
| opoccl 40171 | Closure of orthocomplement... |
| opococ 40172 | Double negative law for or... |
| opcon3b 40173 | Contraposition law for ort... |
| opcon2b 40174 | Orthocomplement contraposi... |
| opcon1b 40175 | Orthocomplement contraposi... |
| oplecon3 40176 | Contraposition law for ort... |
| oplecon3b 40177 | Contraposition law for ort... |
| oplecon1b 40178 | Contraposition law for str... |
| opoc1 40179 | Orthocomplement of orthopo... |
| opoc0 40180 | Orthocomplement of orthopo... |
| opltcon3b 40181 | Contraposition law for str... |
| opltcon1b 40182 | Contraposition law for str... |
| opltcon2b 40183 | Contraposition law for str... |
| opexmid 40184 | Law of excluded middle for... |
| opnoncon 40185 | Law of contradiction for o... |
| riotaocN 40186 | The orthocomplement of the... |
| cmtfvalN 40187 | Value of commutes relation... |
| cmtvalN 40188 | Equivalence for commutes r... |
| isolat 40189 | The predicate "is an ortho... |
| ollat 40190 | An ortholattice is a latti... |
| olop 40191 | An ortholattice is an orth... |
| olposN 40192 | An ortholattice is a poset... |
| isolatiN 40193 | Properties that determine ... |
| oldmm1 40194 | De Morgan's law for meet i... |
| oldmm2 40195 | De Morgan's law for meet i... |
| oldmm3N 40196 | De Morgan's law for meet i... |
| oldmm4 40197 | De Morgan's law for meet i... |
| oldmj1 40198 | De Morgan's law for join i... |
| oldmj2 40199 | De Morgan's law for join i... |
| oldmj3 40200 | De Morgan's law for join i... |
| oldmj4 40201 | De Morgan's law for join i... |
| olj01 40202 | An ortholattice element jo... |
| olj02 40203 | An ortholattice element jo... |
| olm11 40204 | The meet of an ortholattic... |
| olm12 40205 | The meet of an ortholattic... |
| latmassOLD 40206 | Ortholattice meet is assoc... |
| latm12 40207 | A rearrangement of lattice... |
| latm32 40208 | A rearrangement of lattice... |
| latmrot 40209 | Rotate lattice meet of 3 c... |
| latm4 40210 | Rearrangement of lattice m... |
| latmmdiN 40211 | Lattice meet distributes o... |
| latmmdir 40212 | Lattice meet distributes o... |
| olm01 40213 | Meet with lattice zero is ... |
| olm02 40214 | Meet with lattice zero is ... |
| isoml 40215 | The predicate "is an ortho... |
| isomliN 40216 | Properties that determine ... |
| omlol 40217 | An orthomodular lattice is... |
| omlop 40218 | An orthomodular lattice is... |
| omllat 40219 | An orthomodular lattice is... |
| omllaw 40220 | The orthomodular law. (Co... |
| omllaw2N 40221 | Variation of orthomodular ... |
| omllaw3 40222 | Orthomodular law equivalen... |
| omllaw4 40223 | Orthomodular law equivalen... |
| omllaw5N 40224 | The orthomodular law. Rem... |
| cmtcomlemN 40225 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 40226 | Commutation is symmetric. ... |
| cmt2N 40227 | Commutation with orthocomp... |
| cmt3N 40228 | Commutation with orthocomp... |
| cmt4N 40229 | Commutation with orthocomp... |
| cmtbr2N 40230 | Alternate definition of th... |
| cmtbr3N 40231 | Alternate definition for t... |
| cmtbr4N 40232 | Alternate definition for t... |
| lecmtN 40233 | Ordered elements commute. ... |
| cmtidN 40234 | Any element commutes with ... |
| omlfh1N 40235 | Foulis-Holland Theorem, pa... |
| omlfh3N 40236 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 40237 | Analogue of modular law ~ ... |
| omlspjN 40238 | Contraction of a Sasaki pr... |
| cvrfval 40245 | Value of covers relation "... |
| cvrval 40246 | Binary relation expressing... |
| cvrlt 40247 | The covers relation implie... |
| cvrnbtwn 40248 | There is no element betwee... |
| ncvr1 40249 | No element covers the latt... |
| cvrletrN 40250 | Property of an element abo... |
| cvrval2 40251 | Binary relation expressing... |
| cvrnbtwn2 40252 | The covers relation implie... |
| cvrnbtwn3 40253 | The covers relation implie... |
| cvrcon3b 40254 | Contraposition law for the... |
| cvrle 40255 | The covers relation implie... |
| cvrnbtwn4 40256 | The covers relation implie... |
| cvrnle 40257 | The covers relation implie... |
| cvrne 40258 | The covers relation implie... |
| cvrnrefN 40259 | The covers relation is not... |
| cvrcmp 40260 | If two lattice elements th... |
| cvrcmp2 40261 | If two lattice elements co... |
| pats 40262 | The set of atoms in a pose... |
| isat 40263 | The predicate "is an atom"... |
| isat2 40264 | The predicate "is an atom"... |
| atcvr0 40265 | An atom covers zero. ( ~ ... |
| atbase 40266 | An atom is a member of the... |
| atssbase 40267 | The set of atoms is a subs... |
| 0ltat 40268 | An atom is greater than ze... |
| leatb 40269 | A poset element less than ... |
| leat 40270 | A poset element less than ... |
| leat2 40271 | A nonzero poset element le... |
| leat3 40272 | A poset element less than ... |
| meetat 40273 | The meet of any element wi... |
| meetat2 40274 | The meet of any element wi... |
| isatl 40276 | The predicate "is an atomi... |
| atllat 40277 | An atomic lattice is a lat... |
| atlpos 40278 | An atomic lattice is a pos... |
| atl0dm 40279 | Condition necessary for ze... |
| atl0cl 40280 | An atomic lattice has a ze... |
| atl0le 40281 | Orthoposet zero is less th... |
| atlle0 40282 | An element less than or eq... |
| atlltn0 40283 | A lattice element greater ... |
| isat3 40284 | The predicate "is an atom"... |
| atn0 40285 | An atom is not zero. ( ~ ... |
| atnle0 40286 | An atom is not less than o... |
| atlen0 40287 | A lattice element is nonze... |
| atcmp 40288 | If two atoms are comparabl... |
| atncmp 40289 | Frequently-used variation ... |
| atnlt 40290 | Two atoms cannot satisfy t... |
| atcvreq0 40291 | An element covered by an a... |
| atncvrN 40292 | Two atoms cannot satisfy t... |
| atlex 40293 | Every nonzero element of a... |
| atnle 40294 | Two ways of expressing "an... |
| atnem0 40295 | The meet of distinct atoms... |
| atlatmstc 40296 | An atomic, complete, ortho... |
| atlatle 40297 | The ordering of two Hilber... |
| atlrelat1 40298 | An atomistic lattice with ... |
| iscvlat 40300 | The predicate "is an atomi... |
| iscvlat2N 40301 | The predicate "is an atomi... |
| cvlatl 40302 | An atomic lattice with the... |
| cvllat 40303 | An atomic lattice with the... |
| cvlposN 40304 | An atomic lattice with the... |
| cvlexch1 40305 | An atomic covering lattice... |
| cvlexch2 40306 | An atomic covering lattice... |
| cvlexchb1 40307 | An atomic covering lattice... |
| cvlexchb2 40308 | An atomic covering lattice... |
| cvlexch3 40309 | An atomic covering lattice... |
| cvlexch4N 40310 | An atomic covering lattice... |
| cvlatexchb1 40311 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 40312 | A version of ~ cvlexchb2 f... |
| cvlatexch1 40313 | Atom exchange property. (... |
| cvlatexch2 40314 | Atom exchange property. (... |
| cvlatexch3 40315 | Atom exchange property. (... |
| cvlcvr1 40316 | The covering property. Pr... |
| cvlcvrp 40317 | A Hilbert lattice satisfie... |
| cvlatcvr1 40318 | An atom is covered by its ... |
| cvlatcvr2 40319 | An atom is covered by its ... |
| cvlsupr2 40320 | Two equivalent ways of exp... |
| cvlsupr3 40321 | Two equivalent ways of exp... |
| cvlsupr4 40322 | Consequence of superpositi... |
| cvlsupr5 40323 | Consequence of superpositi... |
| cvlsupr6 40324 | Consequence of superpositi... |
| cvlsupr7 40325 | Consequence of superpositi... |
| cvlsupr8 40326 | Consequence of superpositi... |
| ishlat1 40329 | The predicate "is a Hilber... |
| ishlat2 40330 | The predicate "is a Hilber... |
| ishlat3N 40331 | The predicate "is a Hilber... |
| ishlatiN 40332 | Properties that determine ... |
| hlomcmcv 40333 | A Hilbert lattice is ortho... |
| hloml 40334 | A Hilbert lattice is ortho... |
| hlclat 40335 | A Hilbert lattice is compl... |
| hlcvl 40336 | A Hilbert lattice is an at... |
| hlatl 40337 | A Hilbert lattice is atomi... |
| hlol 40338 | A Hilbert lattice is an or... |
| hlop 40339 | A Hilbert lattice is an or... |
| hllat 40340 | A Hilbert lattice is a lat... |
| hllatd 40341 | Deduction form of ~ hllat ... |
| hlomcmat 40342 | A Hilbert lattice is ortho... |
| hlpos 40343 | A Hilbert lattice is a pos... |
| hlatjcl 40344 | Closure of join operation.... |
| hlatjcom 40345 | Commutatitivity of join op... |
| hlatjidm 40346 | Idempotence of join operat... |
| hlatjass 40347 | Lattice join is associativ... |
| hlatj12 40348 | Swap 1st and 2nd members o... |
| hlatj32 40349 | Swap 2nd and 3rd members o... |
| hlatjrot 40350 | Rotate lattice join of 3 c... |
| hlatj4 40351 | Rearrangement of lattice j... |
| hlatlej1 40352 | A join's first argument is... |
| hlatlej2 40353 | A join's second argument i... |
| glbconN 40354 | De Morgan's law for GLB an... |
| glbconxN 40355 | De Morgan's law for GLB an... |
| atnlej1 40356 | If an atom is not less tha... |
| atnlej2 40357 | If an atom is not less tha... |
| hlsuprexch 40358 | A Hilbert lattice has the ... |
| hlexch1 40359 | A Hilbert lattice has the ... |
| hlexch2 40360 | A Hilbert lattice has the ... |
| hlexchb1 40361 | A Hilbert lattice has the ... |
| hlexchb2 40362 | A Hilbert lattice has the ... |
| hlsupr 40363 | A Hilbert lattice has the ... |
| hlsupr2 40364 | A Hilbert lattice has the ... |
| hlhgt4 40365 | A Hilbert lattice has a he... |
| hlhgt2 40366 | A Hilbert lattice has a he... |
| hl0lt1N 40367 | Lattice 0 is less than lat... |
| hlexch3 40368 | A Hilbert lattice has the ... |
| hlexch4N 40369 | A Hilbert lattice has the ... |
| hlatexchb1 40370 | A version of ~ hlexchb1 fo... |
| hlatexchb2 40371 | A version of ~ hlexchb2 fo... |
| hlatexch1 40372 | Atom exchange property. (... |
| hlatexch2 40373 | Atom exchange property. (... |
| hlatmstcOLDN 40374 | An atomic, complete, ortho... |
| hlatle 40375 | The ordering of two Hilber... |
| hlateq 40376 | The equality of two Hilber... |
| hlrelat1 40377 | An atomistic lattice with ... |
| hlrelat5N 40378 | An atomistic lattice with ... |
| hlrelat 40379 | A Hilbert lattice is relat... |
| hlrelat2 40380 | A consequence of relative ... |
| exatleN 40381 | A condition for an atom to... |
| hl2at 40382 | A Hilbert lattice has at l... |
| atex 40383 | At least one atom exists. ... |
| intnatN 40384 | If the intersection with a... |
| 2llnne2N 40385 | Condition implying that tw... |
| 2llnneN 40386 | Condition implying that tw... |
| cvr1 40387 | A Hilbert lattice has the ... |
| cvr2N 40388 | Less-than and covers equiv... |
| hlrelat3 40389 | The Hilbert lattice is rel... |
| cvrval3 40390 | Binary relation expressing... |
| cvrval4N 40391 | Binary relation expressing... |
| cvrval5 40392 | Binary relation expressing... |
| cvrp 40393 | A Hilbert lattice satisfie... |
| atcvr1 40394 | An atom is covered by its ... |
| atcvr2 40395 | An atom is covered by its ... |
| cvrexchlem 40396 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 40397 | A Hilbert lattice satisfie... |
| cvratlem 40398 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 40399 | A nonzero Hilbert lattice ... |
| ltltncvr 40400 | A chained strong ordering ... |
| ltcvrntr 40401 | Non-transitive condition f... |
| cvrntr 40402 | The covers relation is not... |
| atcvr0eq 40403 | The covers relation is not... |
| lnnat 40404 | A line (the join of two di... |
| atcvrj0 40405 | Two atoms covering the zer... |
| cvrat2 40406 | A Hilbert lattice element ... |
| atcvrneN 40407 | Inequality derived from at... |
| atcvrj1 40408 | Condition for an atom to b... |
| atcvrj2b 40409 | Condition for an atom to b... |
| atcvrj2 40410 | Condition for an atom to b... |
| atleneN 40411 | Inequality derived from at... |
| atltcvr 40412 | An equivalence of less-tha... |
| atle 40413 | Any nonzero element has an... |
| atlt 40414 | Two atoms are unequal iff ... |
| atlelt 40415 | Transfer less-than relatio... |
| 2atlt 40416 | Given an atom less than an... |
| atexchcvrN 40417 | Atom exchange property. V... |
| atexchltN 40418 | Atom exchange property. V... |
| cvrat3 40419 | A condition implying that ... |
| cvrat4 40420 | A condition implying exist... |
| cvrat42 40421 | Commuted version of ~ cvra... |
| 2atjm 40422 | The meet of a line (expres... |
| atbtwn 40423 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 40424 | There exists a 3rd atom ` ... |
| atbtwnex 40425 | Given atoms ` P ` in ` X `... |
| 3noncolr2 40426 | Two ways to express 3 non-... |
| 3noncolr1N 40427 | Two ways to express 3 non-... |
| hlatcon3 40428 | Atom exchange combined wit... |
| hlatcon2 40429 | Atom exchange combined wit... |
| 4noncolr3 40430 | A way to express 4 non-col... |
| 4noncolr2 40431 | A way to express 4 non-col... |
| 4noncolr1 40432 | A way to express 4 non-col... |
| athgt 40433 | A Hilbert lattice, whose h... |
| 3dim0 40434 | There exists a 3-dimension... |
| 3dimlem1 40435 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 40436 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 40437 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 40438 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 40439 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 40440 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 40441 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 40442 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 40443 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 40444 | Construct a 3-dimensional ... |
| 3dim2 40445 | Construct 2 new layers on ... |
| 3dim3 40446 | Construct a new layer on t... |
| 2dim 40447 | Generate a height-3 elemen... |
| 1dimN 40448 | An atom is covered by a he... |
| 1cvrco 40449 | The orthocomplement of an ... |
| 1cvratex 40450 | There exists an atom less ... |
| 1cvratlt 40451 | An atom less than or equal... |
| 1cvrjat 40452 | An element covered by the ... |
| 1cvrat 40453 | Create an atom under an el... |
| ps-1 40454 | The join of two atoms ` R ... |
| ps-2 40455 | Lattice analogue for the p... |
| 2atjlej 40456 | Two atoms are different if... |
| hlatexch3N 40457 | Rearrange join of atoms in... |
| hlatexch4 40458 | Exchange 2 atoms. (Contri... |
| ps-2b 40459 | Variation of projective ge... |
| 3atlem1 40460 | Lemma for ~ 3at . (Contri... |
| 3atlem2 40461 | Lemma for ~ 3at . (Contri... |
| 3atlem3 40462 | Lemma for ~ 3at . (Contri... |
| 3atlem4 40463 | Lemma for ~ 3at . (Contri... |
| 3atlem5 40464 | Lemma for ~ 3at . (Contri... |
| 3atlem6 40465 | Lemma for ~ 3at . (Contri... |
| 3atlem7 40466 | Lemma for ~ 3at . (Contri... |
| 3at 40467 | Any three non-colinear ato... |
| llnset 40482 | The set of lattice lines i... |
| islln 40483 | The predicate "is a lattic... |
| islln4 40484 | The predicate "is a lattic... |
| llni 40485 | Condition implying a latti... |
| llnbase 40486 | A lattice line is a lattic... |
| islln3 40487 | The predicate "is a lattic... |
| islln2 40488 | The predicate "is a lattic... |
| llni2 40489 | The join of two different ... |
| llnnleat 40490 | An atom cannot majorize a ... |
| llnneat 40491 | A lattice line is not an a... |
| 2atneat 40492 | The join of two distinct a... |
| llnn0 40493 | A lattice line is nonzero.... |
| islln2a 40494 | The predicate "is a lattic... |
| llnle 40495 | Any element greater than 0... |
| atcvrlln2 40496 | An atom under a line is co... |
| atcvrlln 40497 | An element covering an ato... |
| llnexatN 40498 | Given an atom on a line, t... |
| llncmp 40499 | If two lattice lines are c... |
| llnnlt 40500 | Two lattice lines cannot s... |
| 2llnmat 40501 | Two intersecting lines int... |
| 2at0mat0 40502 | Special case of ~ 2atmat0 ... |
| 2atmat0 40503 | The meet of two unequal li... |
| 2atm 40504 | An atom majorized by two d... |
| ps-2c 40505 | Variation of projective ge... |
| lplnset 40506 | The set of lattice planes ... |
| islpln 40507 | The predicate "is a lattic... |
| islpln4 40508 | The predicate "is a lattic... |
| lplni 40509 | Condition implying a latti... |
| islpln3 40510 | The predicate "is a lattic... |
| lplnbase 40511 | A lattice plane is a latti... |
| islpln5 40512 | The predicate "is a lattic... |
| islpln2 40513 | The predicate "is a lattic... |
| lplni2 40514 | The join of 3 different at... |
| lvolex3N 40515 | There is an atom outside o... |
| llnmlplnN 40516 | The intersection of a line... |
| lplnle 40517 | Any element greater than 0... |
| lplnnle2at 40518 | A lattice line (or atom) c... |
| lplnnleat 40519 | A lattice plane cannot maj... |
| lplnnlelln 40520 | A lattice plane is not les... |
| 2atnelpln 40521 | The join of two atoms is n... |
| lplnneat 40522 | No lattice plane is an ato... |
| lplnnelln 40523 | No lattice plane is a latt... |
| lplnn0N 40524 | A lattice plane is nonzero... |
| islpln2a 40525 | The predicate "is a lattic... |
| islpln2ah 40526 | The predicate "is a lattic... |
| lplnriaN 40527 | Property of a lattice plan... |
| lplnribN 40528 | Property of a lattice plan... |
| lplnric 40529 | Property of a lattice plan... |
| lplnri1 40530 | Property of a lattice plan... |
| lplnri2N 40531 | Property of a lattice plan... |
| lplnri3N 40532 | Property of a lattice plan... |
| lplnllnneN 40533 | Two lattice lines defined ... |
| llncvrlpln2 40534 | A lattice line under a lat... |
| llncvrlpln 40535 | An element covering a latt... |
| 2lplnmN 40536 | If the join of two lattice... |
| 2llnmj 40537 | The meet of two lattice li... |
| 2atmat 40538 | The meet of two intersecti... |
| lplncmp 40539 | If two lattice planes are ... |
| lplnexatN 40540 | Given a lattice line on a ... |
| lplnexllnN 40541 | Given an atom on a lattice... |
| lplnnlt 40542 | Two lattice planes cannot ... |
| 2llnjaN 40543 | The join of two different ... |
| 2llnjN 40544 | The join of two different ... |
| 2llnm2N 40545 | The meet of two different ... |
| 2llnm3N 40546 | Two lattice lines in a lat... |
| 2llnm4 40547 | Two lattice lines that maj... |
| 2llnmeqat 40548 | An atom equals the interse... |
| lvolset 40549 | The set of 3-dim lattice v... |
| islvol 40550 | The predicate "is a 3-dim ... |
| islvol4 40551 | The predicate "is a 3-dim ... |
| lvoli 40552 | Condition implying a 3-dim... |
| islvol3 40553 | The predicate "is a 3-dim ... |
| lvoli3 40554 | Condition implying a 3-dim... |
| lvolbase 40555 | A 3-dim lattice volume is ... |
| islvol5 40556 | The predicate "is a 3-dim ... |
| islvol2 40557 | The predicate "is a 3-dim ... |
| lvoli2 40558 | The join of 4 different at... |
| lvolnle3at 40559 | A lattice plane (or lattic... |
| lvolnleat 40560 | An atom cannot majorize a ... |
| lvolnlelln 40561 | A lattice line cannot majo... |
| lvolnlelpln 40562 | A lattice plane cannot maj... |
| 3atnelvolN 40563 | The join of 3 atoms is not... |
| 2atnelvolN 40564 | The join of two atoms is n... |
| lvolneatN 40565 | No lattice volume is an at... |
| lvolnelln 40566 | No lattice volume is a lat... |
| lvolnelpln 40567 | No lattice volume is a lat... |
| lvoln0N 40568 | A lattice volume is nonzer... |
| islvol2aN 40569 | The predicate "is a lattic... |
| 4atlem0a 40570 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 40571 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 40572 | Lemma for ~ 4at . (Contri... |
| 4atlem3 40573 | Lemma for ~ 4at . Break i... |
| 4atlem3a 40574 | Lemma for ~ 4at . Break i... |
| 4atlem3b 40575 | Lemma for ~ 4at . Break i... |
| 4atlem4a 40576 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 40577 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 40578 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 40579 | Lemma for ~ 4at . Frequen... |
| 4atlem9 40580 | Lemma for ~ 4at . Substit... |
| 4atlem10a 40581 | Lemma for ~ 4at . Substit... |
| 4atlem10b 40582 | Lemma for ~ 4at . Substit... |
| 4atlem10 40583 | Lemma for ~ 4at . Combine... |
| 4atlem11a 40584 | Lemma for ~ 4at . Substit... |
| 4atlem11b 40585 | Lemma for ~ 4at . Substit... |
| 4atlem11 40586 | Lemma for ~ 4at . Combine... |
| 4atlem12a 40587 | Lemma for ~ 4at . Substit... |
| 4atlem12b 40588 | Lemma for ~ 4at . Substit... |
| 4atlem12 40589 | Lemma for ~ 4at . Combine... |
| 4at 40590 | Four atoms determine a lat... |
| 4at2 40591 | Four atoms determine a lat... |
| lplncvrlvol2 40592 | A lattice line under a lat... |
| lplncvrlvol 40593 | An element covering a latt... |
| lvolcmp 40594 | If two lattice planes are ... |
| lvolnltN 40595 | Two lattice volumes cannot... |
| 2lplnja 40596 | The join of two different ... |
| 2lplnj 40597 | The join of two different ... |
| 2lplnm2N 40598 | The meet of two different ... |
| 2lplnmj 40599 | The meet of two lattice pl... |
| dalemkehl 40600 | Lemma for ~ dath . Freque... |
| dalemkelat 40601 | Lemma for ~ dath . Freque... |
| dalemkeop 40602 | Lemma for ~ dath . Freque... |
| dalempea 40603 | Lemma for ~ dath . Freque... |
| dalemqea 40604 | Lemma for ~ dath . Freque... |
| dalemrea 40605 | Lemma for ~ dath . Freque... |
| dalemsea 40606 | Lemma for ~ dath . Freque... |
| dalemtea 40607 | Lemma for ~ dath . Freque... |
| dalemuea 40608 | Lemma for ~ dath . Freque... |
| dalemyeo 40609 | Lemma for ~ dath . Freque... |
| dalemzeo 40610 | Lemma for ~ dath . Freque... |
| dalemclpjs 40611 | Lemma for ~ dath . Freque... |
| dalemclqjt 40612 | Lemma for ~ dath . Freque... |
| dalemclrju 40613 | Lemma for ~ dath . Freque... |
| dalem-clpjq 40614 | Lemma for ~ dath . Freque... |
| dalemceb 40615 | Lemma for ~ dath . Freque... |
| dalempeb 40616 | Lemma for ~ dath . Freque... |
| dalemqeb 40617 | Lemma for ~ dath . Freque... |
| dalemreb 40618 | Lemma for ~ dath . Freque... |
| dalemseb 40619 | Lemma for ~ dath . Freque... |
| dalemteb 40620 | Lemma for ~ dath . Freque... |
| dalemueb 40621 | Lemma for ~ dath . Freque... |
| dalempjqeb 40622 | Lemma for ~ dath . Freque... |
| dalemsjteb 40623 | Lemma for ~ dath . Freque... |
| dalemtjueb 40624 | Lemma for ~ dath . Freque... |
| dalemqrprot 40625 | Lemma for ~ dath . Freque... |
| dalemyeb 40626 | Lemma for ~ dath . Freque... |
| dalemcnes 40627 | Lemma for ~ dath . Freque... |
| dalempnes 40628 | Lemma for ~ dath . Freque... |
| dalemqnet 40629 | Lemma for ~ dath . Freque... |
| dalempjsen 40630 | Lemma for ~ dath . Freque... |
| dalemply 40631 | Lemma for ~ dath . Freque... |
| dalemsly 40632 | Lemma for ~ dath . Freque... |
| dalemswapyz 40633 | Lemma for ~ dath . Swap t... |
| dalemrot 40634 | Lemma for ~ dath . Rotate... |
| dalemrotyz 40635 | Lemma for ~ dath . Rotate... |
| dalem1 40636 | Lemma for ~ dath . Show t... |
| dalemcea 40637 | Lemma for ~ dath . Freque... |
| dalem2 40638 | Lemma for ~ dath . Show t... |
| dalemdea 40639 | Lemma for ~ dath . Freque... |
| dalemeea 40640 | Lemma for ~ dath . Freque... |
| dalem3 40641 | Lemma for ~ dalemdnee . (... |
| dalem4 40642 | Lemma for ~ dalemdnee . (... |
| dalemdnee 40643 | Lemma for ~ dath . Axis o... |
| dalem5 40644 | Lemma for ~ dath . Atom `... |
| dalem6 40645 | Lemma for ~ dath . Analog... |
| dalem7 40646 | Lemma for ~ dath . Analog... |
| dalem8 40647 | Lemma for ~ dath . Plane ... |
| dalem-cly 40648 | Lemma for ~ dalem9 . Cent... |
| dalem9 40649 | Lemma for ~ dath . Since ... |
| dalem10 40650 | Lemma for ~ dath . Atom `... |
| dalem11 40651 | Lemma for ~ dath . Analog... |
| dalem12 40652 | Lemma for ~ dath . Analog... |
| dalem13 40653 | Lemma for ~ dalem14 . (Co... |
| dalem14 40654 | Lemma for ~ dath . Planes... |
| dalem15 40655 | Lemma for ~ dath . The ax... |
| dalem16 40656 | Lemma for ~ dath . The at... |
| dalem17 40657 | Lemma for ~ dath . When p... |
| dalem18 40658 | Lemma for ~ dath . Show t... |
| dalem19 40659 | Lemma for ~ dath . Show t... |
| dalemccea 40660 | Lemma for ~ dath . Freque... |
| dalemddea 40661 | Lemma for ~ dath . Freque... |
| dalem-ccly 40662 | Lemma for ~ dath . Freque... |
| dalem-ddly 40663 | Lemma for ~ dath . Freque... |
| dalemccnedd 40664 | Lemma for ~ dath . Freque... |
| dalemclccjdd 40665 | Lemma for ~ dath . Freque... |
| dalemcceb 40666 | Lemma for ~ dath . Freque... |
| dalemswapyzps 40667 | Lemma for ~ dath . Swap t... |
| dalemrotps 40668 | Lemma for ~ dath . Rotate... |
| dalemcjden 40669 | Lemma for ~ dath . Show t... |
| dalem20 40670 | Lemma for ~ dath . Show t... |
| dalem21 40671 | Lemma for ~ dath . Show t... |
| dalem22 40672 | Lemma for ~ dath . Show t... |
| dalem23 40673 | Lemma for ~ dath . Show t... |
| dalem24 40674 | Lemma for ~ dath . Show t... |
| dalem25 40675 | Lemma for ~ dath . Show t... |
| dalem27 40676 | Lemma for ~ dath . Show t... |
| dalem28 40677 | Lemma for ~ dath . Lemma ... |
| dalem29 40678 | Lemma for ~ dath . Analog... |
| dalem30 40679 | Lemma for ~ dath . Analog... |
| dalem31N 40680 | Lemma for ~ dath . Analog... |
| dalem32 40681 | Lemma for ~ dath . Analog... |
| dalem33 40682 | Lemma for ~ dath . Analog... |
| dalem34 40683 | Lemma for ~ dath . Analog... |
| dalem35 40684 | Lemma for ~ dath . Analog... |
| dalem36 40685 | Lemma for ~ dath . Analog... |
| dalem37 40686 | Lemma for ~ dath . Analog... |
| dalem38 40687 | Lemma for ~ dath . Plane ... |
| dalem39 40688 | Lemma for ~ dath . Auxili... |
| dalem40 40689 | Lemma for ~ dath . Analog... |
| dalem41 40690 | Lemma for ~ dath . (Contr... |
| dalem42 40691 | Lemma for ~ dath . Auxili... |
| dalem43 40692 | Lemma for ~ dath . Planes... |
| dalem44 40693 | Lemma for ~ dath . Dummy ... |
| dalem45 40694 | Lemma for ~ dath . Dummy ... |
| dalem46 40695 | Lemma for ~ dath . Analog... |
| dalem47 40696 | Lemma for ~ dath . Analog... |
| dalem48 40697 | Lemma for ~ dath . Analog... |
| dalem49 40698 | Lemma for ~ dath . Analog... |
| dalem50 40699 | Lemma for ~ dath . Analog... |
| dalem51 40700 | Lemma for ~ dath . Constr... |
| dalem52 40701 | Lemma for ~ dath . Lines ... |
| dalem53 40702 | Lemma for ~ dath . The au... |
| dalem54 40703 | Lemma for ~ dath . Line `... |
| dalem55 40704 | Lemma for ~ dath . Lines ... |
| dalem56 40705 | Lemma for ~ dath . Analog... |
| dalem57 40706 | Lemma for ~ dath . Axis o... |
| dalem58 40707 | Lemma for ~ dath . Analog... |
| dalem59 40708 | Lemma for ~ dath . Analog... |
| dalem60 40709 | Lemma for ~ dath . ` B ` i... |
| dalem61 40710 | Lemma for ~ dath . Show t... |
| dalem62 40711 | Lemma for ~ dath . Elimin... |
| dalem63 40712 | Lemma for ~ dath . Combin... |
| dath 40713 | Desargues's theorem of pro... |
| dath2 40714 | Version of Desargues's the... |
| lineset 40715 | The set of lines in a Hilb... |
| isline 40716 | The predicate "is a line".... |
| islinei 40717 | Condition implying "is a l... |
| pointsetN 40718 | The set of points in a Hil... |
| ispointN 40719 | The predicate "is a point"... |
| atpointN 40720 | The singleton of an atom i... |
| psubspset 40721 | The set of projective subs... |
| ispsubsp 40722 | The predicate "is a projec... |
| ispsubsp2 40723 | The predicate "is a projec... |
| psubspi 40724 | Property of a projective s... |
| psubspi2N 40725 | Property of a projective s... |
| 0psubN 40726 | The empty set is a project... |
| snatpsubN 40727 | The singleton of an atom i... |
| pointpsubN 40728 | A point (singleton of an a... |
| linepsubN 40729 | A line is a projective sub... |
| atpsubN 40730 | The set of all atoms is a ... |
| psubssat 40731 | A projective subspace cons... |
| psubatN 40732 | A member of a projective s... |
| pmapfval 40733 | The projective map of a Hi... |
| pmapval 40734 | Value of the projective ma... |
| elpmap 40735 | Member of a projective map... |
| pmapssat 40736 | The projective map of a Hi... |
| pmapssbaN 40737 | A weakening of ~ pmapssat ... |
| pmaple 40738 | The projective map of a Hi... |
| pmap11 40739 | The projective map of a Hi... |
| pmapat 40740 | The projective map of an a... |
| elpmapat 40741 | Member of the projective m... |
| pmap0 40742 | Value of the projective ma... |
| pmapeq0 40743 | A projective map value is ... |
| pmap1N 40744 | Value of the projective ma... |
| pmapsub 40745 | The projective map of a Hi... |
| pmapglbx 40746 | The projective map of the ... |
| pmapglb 40747 | The projective map of the ... |
| pmapglb2N 40748 | The projective map of the ... |
| pmapglb2xN 40749 | The projective map of the ... |
| pmapmeet 40750 | The projective map of a me... |
| isline2 40751 | Definition of line in term... |
| linepmap 40752 | A line described with a pr... |
| isline3 40753 | Definition of line in term... |
| isline4N 40754 | Definition of line in term... |
| lneq2at 40755 | A line equals the join of ... |
| lnatexN 40756 | There is an atom in a line... |
| lnjatN 40757 | Given an atom in a line, t... |
| lncvrelatN 40758 | A lattice element covered ... |
| lncvrat 40759 | A line covers the atoms it... |
| lncmp 40760 | If two lines are comparabl... |
| 2lnat 40761 | Two intersecting lines int... |
| 2atm2atN 40762 | Two joins with a common at... |
| 2llnma1b 40763 | Generalization of ~ 2llnma... |
| 2llnma1 40764 | Two different intersecting... |
| 2llnma3r 40765 | Two different intersecting... |
| 2llnma2 40766 | Two different intersecting... |
| 2llnma2rN 40767 | Two different intersecting... |
| cdlema1N 40768 | A condition for required f... |
| cdlema2N 40769 | A condition for required f... |
| cdlemblem 40770 | Lemma for ~ cdlemb . (Con... |
| cdlemb 40771 | Given two atoms not less t... |
| paddfval 40774 | Projective subspace sum op... |
| paddval 40775 | Projective subspace sum op... |
| elpadd 40776 | Member of a projective sub... |
| elpaddn0 40777 | Member of projective subsp... |
| paddvaln0N 40778 | Projective subspace sum op... |
| elpaddri 40779 | Condition implying members... |
| elpaddatriN 40780 | Condition implying members... |
| elpaddat 40781 | Membership in a projective... |
| elpaddatiN 40782 | Consequence of membership ... |
| elpadd2at 40783 | Membership in a projective... |
| elpadd2at2 40784 | Membership in a projective... |
| paddunssN 40785 | Projective subspace sum in... |
| elpadd0 40786 | Member of projective subsp... |
| paddval0 40787 | Projective subspace sum wi... |
| padd01 40788 | Projective subspace sum wi... |
| padd02 40789 | Projective subspace sum wi... |
| paddcom 40790 | Projective subspace sum co... |
| paddssat 40791 | A projective subspace sum ... |
| sspadd1 40792 | A projective subspace sum ... |
| sspadd2 40793 | A projective subspace sum ... |
| paddss1 40794 | Subset law for projective ... |
| paddss2 40795 | Subset law for projective ... |
| paddss12 40796 | Subset law for projective ... |
| paddasslem1 40797 | Lemma for ~ paddass . (Co... |
| paddasslem2 40798 | Lemma for ~ paddass . (Co... |
| paddasslem3 40799 | Lemma for ~ paddass . Res... |
| paddasslem4 40800 | Lemma for ~ paddass . Com... |
| paddasslem5 40801 | Lemma for ~ paddass . Sho... |
| paddasslem6 40802 | Lemma for ~ paddass . (Co... |
| paddasslem7 40803 | Lemma for ~ paddass . Com... |
| paddasslem8 40804 | Lemma for ~ paddass . (Co... |
| paddasslem9 40805 | Lemma for ~ paddass . Com... |
| paddasslem10 40806 | Lemma for ~ paddass . Use... |
| paddasslem11 40807 | Lemma for ~ paddass . The... |
| paddasslem12 40808 | Lemma for ~ paddass . The... |
| paddasslem13 40809 | Lemma for ~ paddass . The... |
| paddasslem14 40810 | Lemma for ~ paddass . Rem... |
| paddasslem15 40811 | Lemma for ~ paddass . Use... |
| paddasslem16 40812 | Lemma for ~ paddass . Use... |
| paddasslem17 40813 | Lemma for ~ paddass . The... |
| paddasslem18 40814 | Lemma for ~ paddass . Com... |
| paddass 40815 | Projective subspace sum is... |
| padd12N 40816 | Commutative/associative la... |
| padd4N 40817 | Rearrangement of 4 terms i... |
| paddidm 40818 | Projective subspace sum is... |
| paddclN 40819 | The projective sum of two ... |
| paddssw1 40820 | Subset law for projective ... |
| paddssw2 40821 | Subset law for projective ... |
| paddss 40822 | Subset law for projective ... |
| pmodlem1 40823 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 40824 | Lemma for ~ pmod1i . (Con... |
| pmod1i 40825 | The modular law holds in a... |
| pmod2iN 40826 | Dual of the modular law. ... |
| pmodN 40827 | The modular law for projec... |
| pmodl42N 40828 | Lemma derived from modular... |
| pmapjoin 40829 | The projective map of the ... |
| pmapjat1 40830 | The projective map of the ... |
| pmapjat2 40831 | The projective map of the ... |
| pmapjlln1 40832 | The projective map of the ... |
| hlmod1i 40833 | A version of the modular l... |
| atmod1i1 40834 | Version of modular law ~ p... |
| atmod1i1m 40835 | Version of modular law ~ p... |
| atmod1i2 40836 | Version of modular law ~ p... |
| llnmod1i2 40837 | Version of modular law ~ p... |
| atmod2i1 40838 | Version of modular law ~ p... |
| atmod2i2 40839 | Version of modular law ~ p... |
| llnmod2i2 40840 | Version of modular law ~ p... |
| atmod3i1 40841 | Version of modular law tha... |
| atmod3i2 40842 | Version of modular law tha... |
| atmod4i1 40843 | Version of modular law tha... |
| atmod4i2 40844 | Version of modular law tha... |
| llnexchb2lem 40845 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 40846 | Line exchange property (co... |
| llnexch2N 40847 | Line exchange property (co... |
| dalawlem1 40848 | Lemma for ~ dalaw . Speci... |
| dalawlem2 40849 | Lemma for ~ dalaw . Utili... |
| dalawlem3 40850 | Lemma for ~ dalaw . First... |
| dalawlem4 40851 | Lemma for ~ dalaw . Secon... |
| dalawlem5 40852 | Lemma for ~ dalaw . Speci... |
| dalawlem6 40853 | Lemma for ~ dalaw . First... |
| dalawlem7 40854 | Lemma for ~ dalaw . Secon... |
| dalawlem8 40855 | Lemma for ~ dalaw . Speci... |
| dalawlem9 40856 | Lemma for ~ dalaw . Speci... |
| dalawlem10 40857 | Lemma for ~ dalaw . Combi... |
| dalawlem11 40858 | Lemma for ~ dalaw . First... |
| dalawlem12 40859 | Lemma for ~ dalaw . Secon... |
| dalawlem13 40860 | Lemma for ~ dalaw . Speci... |
| dalawlem14 40861 | Lemma for ~ dalaw . Combi... |
| dalawlem15 40862 | Lemma for ~ dalaw . Swap ... |
| dalaw 40863 | Desargues's law, derived f... |
| pclfvalN 40866 | The projective subspace cl... |
| pclvalN 40867 | Value of the projective su... |
| pclclN 40868 | Closure of the projective ... |
| elpclN 40869 | Membership in the projecti... |
| elpcliN 40870 | Implication of membership ... |
| pclssN 40871 | Ordering is preserved by s... |
| pclssidN 40872 | A set of atoms is included... |
| pclidN 40873 | The projective subspace cl... |
| pclbtwnN 40874 | A projective subspace sand... |
| pclunN 40875 | The projective subspace cl... |
| pclun2N 40876 | The projective subspace cl... |
| pclfinN 40877 | The projective subspace cl... |
| pclcmpatN 40878 | The set of projective subs... |
| polfvalN 40881 | The projective subspace po... |
| polvalN 40882 | Value of the projective su... |
| polval2N 40883 | Alternate expression for v... |
| polsubN 40884 | The polarity of a set of a... |
| polssatN 40885 | The polarity of a set of a... |
| pol0N 40886 | The polarity of the empty ... |
| pol1N 40887 | The polarity of the whole ... |
| 2pol0N 40888 | The closed subspace closur... |
| polpmapN 40889 | The polarity of a projecti... |
| 2polpmapN 40890 | Double polarity of a proje... |
| 2polvalN 40891 | Value of double polarity. ... |
| 2polssN 40892 | A set of atoms is a subset... |
| 3polN 40893 | Triple polarity cancels to... |
| polcon3N 40894 | Contraposition law for pol... |
| 2polcon4bN 40895 | Contraposition law for pol... |
| polcon2N 40896 | Contraposition law for pol... |
| polcon2bN 40897 | Contraposition law for pol... |
| pclss2polN 40898 | The projective subspace cl... |
| pcl0N 40899 | The projective subspace cl... |
| pcl0bN 40900 | The projective subspace cl... |
| pmaplubN 40901 | The LUB of a projective ma... |
| sspmaplubN 40902 | A set of atoms is a subset... |
| 2pmaplubN 40903 | Double projective map of a... |
| paddunN 40904 | The closure of the project... |
| poldmj1N 40905 | De Morgan's law for polari... |
| pmapj2N 40906 | The projective map of the ... |
| pmapocjN 40907 | The projective map of the ... |
| polatN 40908 | The polarity of the single... |
| 2polatN 40909 | Double polarity of the sin... |
| pnonsingN 40910 | The intersection of a set ... |
| psubclsetN 40913 | The set of closed projecti... |
| ispsubclN 40914 | The predicate "is a closed... |
| psubcliN 40915 | Property of a closed proje... |
| psubcli2N 40916 | Property of a closed proje... |
| psubclsubN 40917 | A closed projective subspa... |
| psubclssatN 40918 | A closed projective subspa... |
| pmapidclN 40919 | Projective map of the LUB ... |
| 0psubclN 40920 | The empty set is a closed ... |
| 1psubclN 40921 | The set of all atoms is a ... |
| atpsubclN 40922 | A point (singleton of an a... |
| pmapsubclN 40923 | A projective map value is ... |
| ispsubcl2N 40924 | Alternate predicate for "i... |
| psubclinN 40925 | The intersection of two cl... |
| paddatclN 40926 | The projective sum of a cl... |
| pclfinclN 40927 | The projective subspace cl... |
| linepsubclN 40928 | A line is a closed project... |
| polsubclN 40929 | A polarity is a closed pro... |
| poml4N 40930 | Orthomodular law for proje... |
| poml5N 40931 | Orthomodular law for proje... |
| poml6N 40932 | Orthomodular law for proje... |
| osumcllem1N 40933 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 40934 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 40935 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 40936 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 40937 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 40938 | Lemma for ~ osumclN . Use... |
| osumcllem7N 40939 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 40940 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 40941 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 40942 | Lemma for ~ osumclN . Con... |
| osumcllem11N 40943 | Lemma for ~ osumclN . (Co... |
| osumclN 40944 | Closure of orthogonal sum.... |
| pmapojoinN 40945 | For orthogonal elements, p... |
| pexmidN 40946 | Excluded middle law for cl... |
| pexmidlem1N 40947 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 40948 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 40949 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 40950 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 40951 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 40952 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 40953 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 40954 | Lemma for ~ pexmidN . The... |
| pexmidALTN 40955 | Excluded middle law for cl... |
| pl42lem1N 40956 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 40957 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 40958 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 40959 | Lemma for ~ pl42N . (Cont... |
| pl42N 40960 | Law holding in a Hilbert l... |
| watfvalN 40969 | The W atoms function. (Co... |
| watvalN 40970 | Value of the W atoms funct... |
| iswatN 40971 | The predicate "is a W atom... |
| lhpset 40972 | The set of co-atoms (latti... |
| islhp 40973 | The predicate "is a co-ato... |
| islhp2 40974 | The predicate "is a co-ato... |
| lhpbase 40975 | A co-atom is a member of t... |
| lhp1cvr 40976 | The lattice unity covers a... |
| lhplt 40977 | An atom under a co-atom is... |
| lhp2lt 40978 | The join of two atoms unde... |
| lhpexlt 40979 | There exists an atom less ... |
| lhp0lt 40980 | A co-atom is greater than ... |
| lhpn0 40981 | A co-atom is nonzero. TOD... |
| lhpexle 40982 | There exists an atom under... |
| lhpexnle 40983 | There exists an atom not u... |
| lhpexle1lem 40984 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 40985 | There exists an atom under... |
| lhpexle2lem 40986 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 40987 | There exists atom under a ... |
| lhpexle3lem 40988 | There exists atom under a ... |
| lhpexle3 40989 | There exists atom under a ... |
| lhpex2leN 40990 | There exist at least two d... |
| lhpoc 40991 | The orthocomplement of a c... |
| lhpoc2N 40992 | The orthocomplement of an ... |
| lhpocnle 40993 | The orthocomplement of a c... |
| lhpocat 40994 | The orthocomplement of a c... |
| lhpocnel 40995 | The orthocomplement of a c... |
| lhpocnel2 40996 | The orthocomplement of a c... |
| lhpjat1 40997 | The join of a co-atom (hyp... |
| lhpjat2 40998 | The join of a co-atom (hyp... |
| lhpj1 40999 | The join of a co-atom (hyp... |
| lhpmcvr 41000 | The meet of a lattice hype... |
| lhpmcvr2 41001 | Alternate way to express t... |
| lhpmcvr3 41002 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 41003 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 41004 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 41005 | Specialization of ~ lhpmcv... |
| lhpm0atN 41006 | If the meet of a lattice h... |
| lhpmat 41007 | An element covered by the ... |
| lhpmatb 41008 | An element covered by the ... |
| lhp2at0 41009 | Join and meet with differe... |
| lhp2atnle 41010 | Inequality for 2 different... |
| lhp2atne 41011 | Inequality for joins with ... |
| lhp2at0nle 41012 | Inequality for 2 different... |
| lhp2at0ne 41013 | Inequality for joins with ... |
| lhpelim 41014 | Eliminate an atom not unde... |
| lhpmod2i2 41015 | Modular law for hyperplane... |
| lhpmod6i1 41016 | Modular law for hyperplane... |
| lhprelat3N 41017 | The Hilbert lattice is rel... |
| cdlemb2 41018 | Given two atoms not under ... |
| lhple 41019 | Property of a lattice elem... |
| lhpat 41020 | Create an atom under a co-... |
| lhpat4N 41021 | Property of an atom under ... |
| lhpat2 41022 | Create an atom under a co-... |
| lhpat3 41023 | There is only one atom und... |
| 4atexlemk 41024 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 41025 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 41026 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 41027 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 41028 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 41029 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 41030 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 41031 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 41032 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 41033 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 41034 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 41035 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 41036 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 41037 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 41038 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 41039 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 41040 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 41041 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 41042 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 41043 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 41044 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 41045 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 41046 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 41047 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 41048 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 41049 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 41050 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 41051 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 41052 | Whenever there are at leas... |
| 4atex 41053 | Whenever there are at leas... |
| 4atex2 41054 | More general version of ~ ... |
| 4atex2-0aOLDN 41055 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 41056 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 41057 | Same as ~ 4atex2 except th... |
| 4atex3 41058 | More general version of ~ ... |
| lautset 41059 | The set of lattice automor... |
| islaut 41060 | The predicate "is a lattic... |
| lautle 41061 | Less-than or equal propert... |
| laut1o 41062 | A lattice automorphism is ... |
| laut11 41063 | One-to-one property of a l... |
| lautcl 41064 | A lattice automorphism val... |
| lautcnvclN 41065 | Reverse closure of a latti... |
| lautcnvle 41066 | Less-than or equal propert... |
| lautcnv 41067 | The converse of a lattice ... |
| lautlt 41068 | Less-than property of a la... |
| lautcvr 41069 | Covering property of a lat... |
| lautj 41070 | Meet property of a lattice... |
| lautm 41071 | Meet property of a lattice... |
| lauteq 41072 | A lattice automorphism arg... |
| idlaut 41073 | The identity function is a... |
| lautco 41074 | The composition of two lat... |
| pautsetN 41075 | The set of projective auto... |
| ispautN 41076 | The predicate "is a projec... |
| ldilfset 41085 | The mapping from fiducial ... |
| ldilset 41086 | The set of lattice dilatio... |
| isldil 41087 | The predicate "is a lattic... |
| ldillaut 41088 | A lattice dilation is an a... |
| ldil1o 41089 | A lattice dilation is a on... |
| ldilval 41090 | Value of a lattice dilatio... |
| idldil 41091 | The identity function is a... |
| ldilcnv 41092 | The converse of a lattice ... |
| ldilco 41093 | The composition of two lat... |
| ltrnfset 41094 | The set of all lattice tra... |
| ltrnset 41095 | The set of lattice transla... |
| isltrn 41096 | The predicate "is a lattic... |
| isltrn2N 41097 | The predicate "is a lattic... |
| ltrnu 41098 | Uniqueness property of a l... |
| ltrnldil 41099 | A lattice translation is a... |
| ltrnlaut 41100 | A lattice translation is a... |
| ltrn1o 41101 | A lattice translation is a... |
| ltrncl 41102 | Closure of a lattice trans... |
| ltrn11 41103 | One-to-one property of a l... |
| ltrncnvnid 41104 | If a translation is differ... |
| ltrncoidN 41105 | Two translations are equal... |
| ltrnle 41106 | Less-than or equal propert... |
| ltrncnvleN 41107 | Less-than or equal propert... |
| ltrnm 41108 | Lattice translation of a m... |
| ltrnj 41109 | Lattice translation of a m... |
| ltrncvr 41110 | Covering property of a lat... |
| ltrnval1 41111 | Value of a lattice transla... |
| ltrnid 41112 | A lattice translation is t... |
| ltrnnid 41113 | If a lattice translation i... |
| ltrnatb 41114 | The lattice translation of... |
| ltrncnvatb 41115 | The converse of the lattic... |
| ltrnel 41116 | The lattice translation of... |
| ltrnat 41117 | The lattice translation of... |
| ltrncnvat 41118 | The converse of the lattic... |
| ltrncnvel 41119 | The converse of the lattic... |
| ltrncoelN 41120 | Composition of lattice tra... |
| ltrncoat 41121 | Composition of lattice tra... |
| ltrncoval 41122 | Two ways to express value ... |
| ltrncnv 41123 | The converse of a lattice ... |
| ltrn11at 41124 | Frequently used one-to-one... |
| ltrneq2 41125 | The equality of two transl... |
| ltrneq 41126 | The equality of two transl... |
| idltrn 41127 | The identity function is a... |
| ltrnmw 41128 | Property of lattice transl... |
| dilfsetN 41129 | The mapping from fiducial ... |
| dilsetN 41130 | The set of dilations for a... |
| isdilN 41131 | The predicate "is a dilati... |
| trnfsetN 41132 | The mapping from fiducial ... |
| trnsetN 41133 | The set of translations fo... |
| istrnN 41134 | The predicate "is a transl... |
| trlfset 41137 | The set of all traces of l... |
| trlset 41138 | The set of traces of latti... |
| trlval 41139 | The value of the trace of ... |
| trlval2 41140 | The value of the trace of ... |
| trlcl 41141 | Closure of the trace of a ... |
| trlcnv 41142 | The trace of the converse ... |
| trljat1 41143 | The value of a translation... |
| trljat2 41144 | The value of a translation... |
| trljat3 41145 | The value of a translation... |
| trlat 41146 | If an atom differs from it... |
| trl0 41147 | If an atom not under the f... |
| trlator0 41148 | The trace of a lattice tra... |
| trlatn0 41149 | The trace of a lattice tra... |
| trlnidat 41150 | The trace of a lattice tra... |
| ltrnnidn 41151 | If a lattice translation i... |
| ltrnideq 41152 | Property of the identity l... |
| trlid0 41153 | The trace of the identity ... |
| trlnidatb 41154 | A lattice translation is n... |
| trlid0b 41155 | A lattice translation is t... |
| trlnid 41156 | Different translations wit... |
| ltrn2ateq 41157 | Property of the equality o... |
| ltrnateq 41158 | If any atom (under ` W ` )... |
| ltrnatneq 41159 | If any atom (under ` W ` )... |
| ltrnatlw 41160 | If the value of an atom eq... |
| trlle 41161 | The trace of a lattice tra... |
| trlne 41162 | The trace of a lattice tra... |
| trlnle 41163 | The atom not under the fid... |
| trlval3 41164 | The value of the trace of ... |
| trlval4 41165 | The value of the trace of ... |
| trlval5 41166 | The value of the trace of ... |
| arglem1N 41167 | Lemma for Desargues's law.... |
| cdlemc1 41168 | Part of proof of Lemma C i... |
| cdlemc2 41169 | Part of proof of Lemma C i... |
| cdlemc3 41170 | Part of proof of Lemma C i... |
| cdlemc4 41171 | Part of proof of Lemma C i... |
| cdlemc5 41172 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 41173 | Lemma for ~ cdlemc . (Con... |
| cdlemc 41174 | Lemma C in [Crawley] p. 11... |
| cdlemd1 41175 | Part of proof of Lemma D i... |
| cdlemd2 41176 | Part of proof of Lemma D i... |
| cdlemd3 41177 | Part of proof of Lemma D i... |
| cdlemd4 41178 | Part of proof of Lemma D i... |
| cdlemd5 41179 | Part of proof of Lemma D i... |
| cdlemd6 41180 | Part of proof of Lemma D i... |
| cdlemd7 41181 | Part of proof of Lemma D i... |
| cdlemd8 41182 | Part of proof of Lemma D i... |
| cdlemd9 41183 | Part of proof of Lemma D i... |
| cdlemd 41184 | If two translations agree ... |
| ltrneq3 41185 | Two translations agree at ... |
| cdleme00a 41186 | Part of proof of Lemma E i... |
| cdleme0aa 41187 | Part of proof of Lemma E i... |
| cdleme0a 41188 | Part of proof of Lemma E i... |
| cdleme0b 41189 | Part of proof of Lemma E i... |
| cdleme0c 41190 | Part of proof of Lemma E i... |
| cdleme0cp 41191 | Part of proof of Lemma E i... |
| cdleme0cq 41192 | Part of proof of Lemma E i... |
| cdleme0dN 41193 | Part of proof of Lemma E i... |
| cdleme0e 41194 | Part of proof of Lemma E i... |
| cdleme0fN 41195 | Part of proof of Lemma E i... |
| cdleme0gN 41196 | Part of proof of Lemma E i... |
| cdlemeulpq 41197 | Part of proof of Lemma E i... |
| cdleme01N 41198 | Part of proof of Lemma E i... |
| cdleme02N 41199 | Part of proof of Lemma E i... |
| cdleme0ex1N 41200 | Part of proof of Lemma E i... |
| cdleme0ex2N 41201 | Part of proof of Lemma E i... |
| cdleme0moN 41202 | Part of proof of Lemma E i... |
| cdleme1b 41203 | Part of proof of Lemma E i... |
| cdleme1 41204 | Part of proof of Lemma E i... |
| cdleme2 41205 | Part of proof of Lemma E i... |
| cdleme3b 41206 | Part of proof of Lemma E i... |
| cdleme3c 41207 | Part of proof of Lemma E i... |
| cdleme3d 41208 | Part of proof of Lemma E i... |
| cdleme3e 41209 | Part of proof of Lemma E i... |
| cdleme3fN 41210 | Part of proof of Lemma E i... |
| cdleme3g 41211 | Part of proof of Lemma E i... |
| cdleme3h 41212 | Part of proof of Lemma E i... |
| cdleme3fa 41213 | Part of proof of Lemma E i... |
| cdleme3 41214 | Part of proof of Lemma E i... |
| cdleme4 41215 | Part of proof of Lemma E i... |
| cdleme4a 41216 | Part of proof of Lemma E i... |
| cdleme5 41217 | Part of proof of Lemma E i... |
| cdleme6 41218 | Part of proof of Lemma E i... |
| cdleme7aa 41219 | Part of proof of Lemma E i... |
| cdleme7a 41220 | Part of proof of Lemma E i... |
| cdleme7b 41221 | Part of proof of Lemma E i... |
| cdleme7c 41222 | Part of proof of Lemma E i... |
| cdleme7d 41223 | Part of proof of Lemma E i... |
| cdleme7e 41224 | Part of proof of Lemma E i... |
| cdleme7ga 41225 | Part of proof of Lemma E i... |
| cdleme7 41226 | Part of proof of Lemma E i... |
| cdleme8 41227 | Part of proof of Lemma E i... |
| cdleme9a 41228 | Part of proof of Lemma E i... |
| cdleme9b 41229 | Utility lemma for Lemma E ... |
| cdleme9 41230 | Part of proof of Lemma E i... |
| cdleme10 41231 | Part of proof of Lemma E i... |
| cdleme8tN 41232 | Part of proof of Lemma E i... |
| cdleme9taN 41233 | Part of proof of Lemma E i... |
| cdleme9tN 41234 | Part of proof of Lemma E i... |
| cdleme10tN 41235 | Part of proof of Lemma E i... |
| cdleme16aN 41236 | Part of proof of Lemma E i... |
| cdleme11a 41237 | Part of proof of Lemma E i... |
| cdleme11c 41238 | Part of proof of Lemma E i... |
| cdleme11dN 41239 | Part of proof of Lemma E i... |
| cdleme11e 41240 | Part of proof of Lemma E i... |
| cdleme11fN 41241 | Part of proof of Lemma E i... |
| cdleme11g 41242 | Part of proof of Lemma E i... |
| cdleme11h 41243 | Part of proof of Lemma E i... |
| cdleme11j 41244 | Part of proof of Lemma E i... |
| cdleme11k 41245 | Part of proof of Lemma E i... |
| cdleme11l 41246 | Part of proof of Lemma E i... |
| cdleme11 41247 | Part of proof of Lemma E i... |
| cdleme12 41248 | Part of proof of Lemma E i... |
| cdleme13 41249 | Part of proof of Lemma E i... |
| cdleme14 41250 | Part of proof of Lemma E i... |
| cdleme15a 41251 | Part of proof of Lemma E i... |
| cdleme15b 41252 | Part of proof of Lemma E i... |
| cdleme15c 41253 | Part of proof of Lemma E i... |
| cdleme15d 41254 | Part of proof of Lemma E i... |
| cdleme15 41255 | Part of proof of Lemma E i... |
| cdleme16b 41256 | Part of proof of Lemma E i... |
| cdleme16c 41257 | Part of proof of Lemma E i... |
| cdleme16d 41258 | Part of proof of Lemma E i... |
| cdleme16e 41259 | Part of proof of Lemma E i... |
| cdleme16f 41260 | Part of proof of Lemma E i... |
| cdleme16g 41261 | Part of proof of Lemma E i... |
| cdleme16 41262 | Part of proof of Lemma E i... |
| cdleme17a 41263 | Part of proof of Lemma E i... |
| cdleme17b 41264 | Lemma leading to ~ cdleme1... |
| cdleme17c 41265 | Part of proof of Lemma E i... |
| cdleme17d1 41266 | Part of proof of Lemma E i... |
| cdleme0nex 41267 | Part of proof of Lemma E i... |
| cdleme18a 41268 | Part of proof of Lemma E i... |
| cdleme18b 41269 | Part of proof of Lemma E i... |
| cdleme18c 41270 | Part of proof of Lemma E i... |
| cdleme22gb 41271 | Utility lemma for Lemma E ... |
| cdleme18d 41272 | Part of proof of Lemma E i... |
| cdlemesner 41273 | Part of proof of Lemma E i... |
| cdlemedb 41274 | Part of proof of Lemma E i... |
| cdlemeda 41275 | Part of proof of Lemma E i... |
| cdlemednpq 41276 | Part of proof of Lemma E i... |
| cdlemednuN 41277 | Part of proof of Lemma E i... |
| cdleme20zN 41278 | Part of proof of Lemma E i... |
| cdleme20y 41279 | Part of proof of Lemma E i... |
| cdleme19a 41280 | Part of proof of Lemma E i... |
| cdleme19b 41281 | Part of proof of Lemma E i... |
| cdleme19c 41282 | Part of proof of Lemma E i... |
| cdleme19d 41283 | Part of proof of Lemma E i... |
| cdleme19e 41284 | Part of proof of Lemma E i... |
| cdleme19f 41285 | Part of proof of Lemma E i... |
| cdleme20aN 41286 | Part of proof of Lemma E i... |
| cdleme20bN 41287 | Part of proof of Lemma E i... |
| cdleme20c 41288 | Part of proof of Lemma E i... |
| cdleme20d 41289 | Part of proof of Lemma E i... |
| cdleme20e 41290 | Part of proof of Lemma E i... |
| cdleme20f 41291 | Part of proof of Lemma E i... |
| cdleme20g 41292 | Part of proof of Lemma E i... |
| cdleme20h 41293 | Part of proof of Lemma E i... |
| cdleme20i 41294 | Part of proof of Lemma E i... |
| cdleme20j 41295 | Part of proof of Lemma E i... |
| cdleme20k 41296 | Part of proof of Lemma E i... |
| cdleme20l1 41297 | Part of proof of Lemma E i... |
| cdleme20l2 41298 | Part of proof of Lemma E i... |
| cdleme20l 41299 | Part of proof of Lemma E i... |
| cdleme20m 41300 | Part of proof of Lemma E i... |
| cdleme20 41301 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 41302 | Part of proof of Lemma E i... |
| cdleme21b 41303 | Part of proof of Lemma E i... |
| cdleme21c 41304 | Part of proof of Lemma E i... |
| cdleme21at 41305 | Part of proof of Lemma E i... |
| cdleme21ct 41306 | Part of proof of Lemma E i... |
| cdleme21d 41307 | Part of proof of Lemma E i... |
| cdleme21e 41308 | Part of proof of Lemma E i... |
| cdleme21f 41309 | Part of proof of Lemma E i... |
| cdleme21g 41310 | Part of proof of Lemma E i... |
| cdleme21h 41311 | Part of proof of Lemma E i... |
| cdleme21i 41312 | Part of proof of Lemma E i... |
| cdleme21j 41313 | Combine ~ cdleme20 and ~ c... |
| cdleme21 41314 | Part of proof of Lemma E i... |
| cdleme21k 41315 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 41316 | Part of proof of Lemma E i... |
| cdleme22a 41317 | Part of proof of Lemma E i... |
| cdleme22b 41318 | Part of proof of Lemma E i... |
| cdleme22cN 41319 | Part of proof of Lemma E i... |
| cdleme22d 41320 | Part of proof of Lemma E i... |
| cdleme22e 41321 | Part of proof of Lemma E i... |
| cdleme22eALTN 41322 | Part of proof of Lemma E i... |
| cdleme22f 41323 | Part of proof of Lemma E i... |
| cdleme22f2 41324 | Part of proof of Lemma E i... |
| cdleme22g 41325 | Part of proof of Lemma E i... |
| cdleme23a 41326 | Part of proof of Lemma E i... |
| cdleme23b 41327 | Part of proof of Lemma E i... |
| cdleme23c 41328 | Part of proof of Lemma E i... |
| cdleme24 41329 | Quantified version of ~ cd... |
| cdleme25a 41330 | Lemma for ~ cdleme25b . (... |
| cdleme25b 41331 | Transform ~ cdleme24 . TO... |
| cdleme25c 41332 | Transform ~ cdleme25b . (... |
| cdleme25dN 41333 | Transform ~ cdleme25c . (... |
| cdleme25cl 41334 | Show closure of the unique... |
| cdleme25cv 41335 | Change bound variables in ... |
| cdleme26e 41336 | Part of proof of Lemma E i... |
| cdleme26ee 41337 | Part of proof of Lemma E i... |
| cdleme26eALTN 41338 | Part of proof of Lemma E i... |
| cdleme26fALTN 41339 | Part of proof of Lemma E i... |
| cdleme26f 41340 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 41341 | Part of proof of Lemma E i... |
| cdleme26f2 41342 | Part of proof of Lemma E i... |
| cdleme27cl 41343 | Part of proof of Lemma E i... |
| cdleme27a 41344 | Part of proof of Lemma E i... |
| cdleme27b 41345 | Lemma for ~ cdleme27N . (... |
| cdleme27N 41346 | Part of proof of Lemma E i... |
| cdleme28a 41347 | Lemma for ~ cdleme25b . T... |
| cdleme28b 41348 | Lemma for ~ cdleme25b . T... |
| cdleme28c 41349 | Part of proof of Lemma E i... |
| cdleme28 41350 | Quantified version of ~ cd... |
| cdleme29ex 41351 | Lemma for ~ cdleme29b . (... |
| cdleme29b 41352 | Transform ~ cdleme28 . (C... |
| cdleme29c 41353 | Transform ~ cdleme28b . (... |
| cdleme29cl 41354 | Show closure of the unique... |
| cdleme30a 41355 | Part of proof of Lemma E i... |
| cdleme31so 41356 | Part of proof of Lemma E i... |
| cdleme31sn 41357 | Part of proof of Lemma E i... |
| cdleme31sn1 41358 | Part of proof of Lemma E i... |
| cdleme31se 41359 | Part of proof of Lemma D i... |
| cdleme31se2 41360 | Part of proof of Lemma D i... |
| cdleme31sc 41361 | Part of proof of Lemma E i... |
| cdleme31sde 41362 | Part of proof of Lemma D i... |
| cdleme31snd 41363 | Part of proof of Lemma D i... |
| cdleme31sdnN 41364 | Part of proof of Lemma E i... |
| cdleme31sn1c 41365 | Part of proof of Lemma E i... |
| cdleme31sn2 41366 | Part of proof of Lemma E i... |
| cdleme31fv 41367 | Part of proof of Lemma E i... |
| cdleme31fv1 41368 | Part of proof of Lemma E i... |
| cdleme31fv1s 41369 | Part of proof of Lemma E i... |
| cdleme31fv2 41370 | Part of proof of Lemma E i... |
| cdleme31id 41371 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 41372 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 41373 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 41374 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 41375 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 41376 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 41377 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 41378 | Part of proof of Lemma E i... |
| cdlemefr29exN 41379 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 41380 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 41381 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 41382 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 41383 | TODO fix comment. (Contri... |
| cdlemefr29clN 41384 | Show closure of the unique... |
| cdleme43frv1snN 41385 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 41386 | Part of proof of Lemma E i... |
| cdlemefr32fva1 41387 | Part of proof of Lemma E i... |
| cdlemefr31fv1 41388 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 41389 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 41390 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 41391 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 41392 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 41393 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 41394 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 41395 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 41396 | Show closure of the unique... |
| cdleme43fsv1snlem 41397 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 41398 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 41399 | Part of proof of Lemma E i... |
| cdlemefs32fva1 41400 | Part of proof of Lemma E i... |
| cdlemefs31fv1 41401 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 41402 | Value of f(r) when r is an... |
| cdlemefs44 41403 | Value of f_s(r) when r is ... |
| cdlemefr45 41404 | Value of f(r) when r is an... |
| cdlemefr45e 41405 | Explicit expansion of ~ cd... |
| cdlemefs45 41406 | Value of f_s(r) when r is ... |
| cdlemefs45ee 41407 | Explicit expansion of ~ cd... |
| cdlemefs45eN 41408 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 41409 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 41410 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 41411 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 41412 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 41413 | Show closure of ` [_ R / s... |
| cdleme32fva 41414 | Part of proof of Lemma D i... |
| cdleme32fva1 41415 | Part of proof of Lemma D i... |
| cdleme32fvaw 41416 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 41417 | Part of proof of Lemma D i... |
| cdleme32a 41418 | Part of proof of Lemma D i... |
| cdleme32b 41419 | Part of proof of Lemma D i... |
| cdleme32c 41420 | Part of proof of Lemma D i... |
| cdleme32d 41421 | Part of proof of Lemma D i... |
| cdleme32e 41422 | Part of proof of Lemma D i... |
| cdleme32f 41423 | Part of proof of Lemma D i... |
| cdleme32le 41424 | Part of proof of Lemma D i... |
| cdleme35a 41425 | Part of proof of Lemma E i... |
| cdleme35fnpq 41426 | Part of proof of Lemma E i... |
| cdleme35b 41427 | Part of proof of Lemma E i... |
| cdleme35c 41428 | Part of proof of Lemma E i... |
| cdleme35d 41429 | Part of proof of Lemma E i... |
| cdleme35e 41430 | Part of proof of Lemma E i... |
| cdleme35f 41431 | Part of proof of Lemma E i... |
| cdleme35g 41432 | Part of proof of Lemma E i... |
| cdleme35h 41433 | Part of proof of Lemma E i... |
| cdleme35h2 41434 | Part of proof of Lemma E i... |
| cdleme35sn2aw 41435 | Part of proof of Lemma E i... |
| cdleme35sn3a 41436 | Part of proof of Lemma E i... |
| cdleme36a 41437 | Part of proof of Lemma E i... |
| cdleme36m 41438 | Part of proof of Lemma E i... |
| cdleme37m 41439 | Part of proof of Lemma E i... |
| cdleme38m 41440 | Part of proof of Lemma E i... |
| cdleme38n 41441 | Part of proof of Lemma E i... |
| cdleme39a 41442 | Part of proof of Lemma E i... |
| cdleme39n 41443 | Part of proof of Lemma E i... |
| cdleme40m 41444 | Part of proof of Lemma E i... |
| cdleme40n 41445 | Part of proof of Lemma E i... |
| cdleme40v 41446 | Part of proof of Lemma E i... |
| cdleme40w 41447 | Part of proof of Lemma E i... |
| cdleme42a 41448 | Part of proof of Lemma E i... |
| cdleme42c 41449 | Part of proof of Lemma E i... |
| cdleme42d 41450 | Part of proof of Lemma E i... |
| cdleme41sn3aw 41451 | Part of proof of Lemma E i... |
| cdleme41sn4aw 41452 | Part of proof of Lemma E i... |
| cdleme41snaw 41453 | Part of proof of Lemma E i... |
| cdleme41fva11 41454 | Part of proof of Lemma E i... |
| cdleme42b 41455 | Part of proof of Lemma E i... |
| cdleme42e 41456 | Part of proof of Lemma E i... |
| cdleme42f 41457 | Part of proof of Lemma E i... |
| cdleme42g 41458 | Part of proof of Lemma E i... |
| cdleme42h 41459 | Part of proof of Lemma E i... |
| cdleme42i 41460 | Part of proof of Lemma E i... |
| cdleme42k 41461 | Part of proof of Lemma E i... |
| cdleme42ke 41462 | Part of proof of Lemma E i... |
| cdleme42keg 41463 | Part of proof of Lemma E i... |
| cdleme42mN 41464 | Part of proof of Lemma E i... |
| cdleme42mgN 41465 | Part of proof of Lemma E i... |
| cdleme43aN 41466 | Part of proof of Lemma E i... |
| cdleme43bN 41467 | Lemma for Lemma E in [Craw... |
| cdleme43cN 41468 | Part of proof of Lemma E i... |
| cdleme43dN 41469 | Part of proof of Lemma E i... |
| cdleme46f2g2 41470 | Conversion for ` G ` to re... |
| cdleme46f2g1 41471 | Conversion for ` G ` to re... |
| cdleme17d2 41472 | Part of proof of Lemma E i... |
| cdleme17d3 41473 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 41474 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 41475 | Part of proof of Lemma E i... |
| cdleme48fv 41476 | Part of proof of Lemma D i... |
| cdleme48fvg 41477 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 41478 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 41479 | TODO: fix comment. TODO: ... |
| cdleme48b 41480 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 41481 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 41482 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 41483 | TODO: fix comment. (Contr... |
| cdleme4gfv 41484 | Part of proof of Lemma D i... |
| cdlemeg47b 41485 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 41486 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 41487 | Value of g_s(r) when r is ... |
| cdlemeg49le 41488 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 41489 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 41490 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 41491 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 41492 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 41493 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 41494 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 41495 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 41496 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 41497 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 41498 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 41499 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 41500 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 41501 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 41502 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 41503 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 41504 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 41505 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 41506 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 41507 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 41508 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 41509 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 41510 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 41511 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 41512 | TODO: fix comment. (Contr... |
| cdleme48gfv1 41513 | TODO: fix comment. (Contr... |
| cdleme48gfv 41514 | TODO: fix comment. (Contr... |
| cdleme48fgv 41515 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 41516 | Part of proof of Lemma D i... |
| cdleme50lebi 41517 | Part of proof of Lemma D i... |
| cdleme50eq 41518 | Part of proof of Lemma D i... |
| cdleme50f 41519 | Part of proof of Lemma D i... |
| cdleme50f1 41520 | Part of proof of Lemma D i... |
| cdleme50rnlem 41521 | Part of proof of Lemma D i... |
| cdleme50rn 41522 | Part of proof of Lemma D i... |
| cdleme50f1o 41523 | Part of proof of Lemma D i... |
| cdleme50laut 41524 | Part of proof of Lemma D i... |
| cdleme50ldil 41525 | Part of proof of Lemma D i... |
| cdleme50trn1 41526 | Part of proof that ` F ` i... |
| cdleme50trn2a 41527 | Part of proof that ` F ` i... |
| cdleme50trn2 41528 | Part of proof that ` F ` i... |
| cdleme50trn12 41529 | Part of proof that ` F ` i... |
| cdleme50trn3 41530 | Part of proof that ` F ` i... |
| cdleme50trn123 41531 | Part of proof that ` F ` i... |
| cdleme51finvfvN 41532 | Part of proof of Lemma E i... |
| cdleme51finvN 41533 | Part of proof of Lemma E i... |
| cdleme50ltrn 41534 | Part of proof of Lemma E i... |
| cdleme51finvtrN 41535 | Part of proof of Lemma E i... |
| cdleme50ex 41536 | Part of Lemma E in [Crawle... |
| cdleme 41537 | Lemma E in [Crawley] p. 11... |
| cdlemf1 41538 | Part of Lemma F in [Crawle... |
| cdlemf2 41539 | Part of Lemma F in [Crawle... |
| cdlemf 41540 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 41541 | ~ cdlemf with additional c... |
| cdlemftr3 41542 | Special case of ~ cdlemf s... |
| cdlemftr2 41543 | Special case of ~ cdlemf s... |
| cdlemftr1 41544 | Part of proof of Lemma G o... |
| cdlemftr0 41545 | Special case of ~ cdlemf s... |
| trlord 41546 | The ordering of two Hilber... |
| cdlemg1a 41547 | Shorter expression for ` G... |
| cdlemg1b2 41548 | This theorem can be used t... |
| cdlemg1idlemN 41549 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 41550 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 41551 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 41552 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 41553 | This theorem can be used t... |
| cdlemg1idN 41554 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 41555 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 41556 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 41557 | Version of ~ cdleme51finvt... |
| ltrniotaval 41558 | Value of the unique transl... |
| ltrniotacnvval 41559 | Converse value of the uniq... |
| ltrniotaidvalN 41560 | Value of the unique transl... |
| ltrniotavalbN 41561 | Value of the unique transl... |
| cdlemeiota 41562 | A translation is uniquely ... |
| cdlemg1ci2 41563 | Any function of the form o... |
| cdlemg1cN 41564 | Any translation belongs to... |
| cdlemg1cex 41565 | Any translation is one of ... |
| cdlemg2cN 41566 | Any translation belongs to... |
| cdlemg2dN 41567 | This theorem can be used t... |
| cdlemg2cex 41568 | Any translation is one of ... |
| cdlemg2ce 41569 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 41570 | Part of proof of Lemma E i... |
| cdlemg2fvlem 41571 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 41572 | ~ cdleme42keg with simpler... |
| cdlemg2idN 41573 | Version of ~ cdleme31id wi... |
| cdlemg3a 41574 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 41575 | TODO: Replace this with ~... |
| cdlemg2fv 41576 | Value of a translation in ... |
| cdlemg2fv2 41577 | Value of a translation in ... |
| cdlemg2k 41578 | ~ cdleme42keg with simpler... |
| cdlemg2kq 41579 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 41580 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 41581 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 41582 | TODO: Is there a simpler ... |
| cdlemb3 41583 | Given two atoms not under ... |
| cdlemg7fvbwN 41584 | Properties of a translatio... |
| cdlemg4a 41585 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 41586 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 41587 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 41588 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 41589 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 41590 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 41591 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 41592 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 41593 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 41594 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 41595 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 41596 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 41597 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 41598 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 41599 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 41600 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 41601 | Value of a translation com... |
| cdlemg7aN 41602 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 41603 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 41604 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 41605 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 41606 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 41607 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 41608 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 41609 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 41610 | The triples ` <. P , ( F `... |
| cdlemg9 41611 | The triples ` <. P , ( F `... |
| cdlemg10b 41612 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 41613 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 41614 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 41615 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 41616 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 41617 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 41618 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 41619 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 41620 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 41621 | The triples ` <. P , ( F `... |
| cdlemg12c 41622 | The triples ` <. P , ( F `... |
| cdlemg12d 41623 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 41624 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 41625 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 41626 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 41627 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 41628 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 41629 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 41630 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 41631 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 41632 | Eliminate the ` ( F `` P )... |
| cdlemg15 41633 | Eliminate the ` ( (... |
| cdlemg16 41634 | Part of proof of Lemma G o... |
| cdlemg16ALTN 41635 | This version of ~ cdlemg16... |
| cdlemg16z 41636 | Eliminate ` ( ( F `... |
| cdlemg16zz 41637 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 41638 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 41639 | Part of proof of Lemma G i... |
| cdlemg17dN 41640 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 41641 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 41642 | TODO: fix comment. (Contr... |
| cdlemg17f 41643 | TODO: fix comment. (Contr... |
| cdlemg17g 41644 | TODO: fix comment. (Contr... |
| cdlemg17h 41645 | TODO: fix comment. (Contr... |
| cdlemg17i 41646 | TODO: fix comment. (Contr... |
| cdlemg17ir 41647 | TODO: fix comment. (Contr... |
| cdlemg17j 41648 | TODO: fix comment. (Contr... |
| cdlemg17pq 41649 | Utility theorem for swappi... |
| cdlemg17bq 41650 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 41651 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 41652 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 41653 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 41654 | Part of Lemma G of [Crawle... |
| cdlemg18a 41655 | Show two lines are differe... |
| cdlemg18b 41656 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 41657 | Show two lines intersect a... |
| cdlemg18d 41658 | Show two lines intersect a... |
| cdlemg18 41659 | Show two lines intersect a... |
| cdlemg19a 41660 | Show two lines intersect a... |
| cdlemg19 41661 | Show two lines intersect a... |
| cdlemg20 41662 | Show two lines intersect a... |
| cdlemg21 41663 | Version of cdlemg19 with `... |
| cdlemg22 41664 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 41665 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 41666 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 41667 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 41668 | ~ cdlemg16zz restated for ... |
| cdlemg27a 41669 | For use with case when ` (... |
| cdlemg28a 41670 | Part of proof of Lemma G o... |
| cdlemg31b0N 41671 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 41672 | TODO: Fix comment. (Cont... |
| cdlemg27b 41673 | TODO: Fix comment. (Cont... |
| cdlemg31a 41674 | TODO: fix comment. (Contr... |
| cdlemg31b 41675 | TODO: fix comment. (Contr... |
| cdlemg31c 41676 | Show that when ` N ` is an... |
| cdlemg31d 41677 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 41678 | TODO: Fix comment. (Cont... |
| cdlemg33c0 41679 | TODO: Fix comment. (Cont... |
| cdlemg28b 41680 | Part of proof of Lemma G o... |
| cdlemg28 41681 | Part of proof of Lemma G o... |
| cdlemg29 41682 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 41683 | TODO: Fix comment. (Cont... |
| cdlemg33b 41684 | TODO: Fix comment. (Cont... |
| cdlemg33c 41685 | TODO: Fix comment. (Cont... |
| cdlemg33d 41686 | TODO: Fix comment. (Cont... |
| cdlemg33e 41687 | TODO: Fix comment. (Cont... |
| cdlemg33 41688 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 41689 | Use cdlemg33 to eliminate ... |
| cdlemg35 41690 | TODO: Fix comment. TODO:... |
| cdlemg36 41691 | Use cdlemg35 to eliminate ... |
| cdlemg38 41692 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 41693 | Eliminate ` =/= ` conditio... |
| cdlemg40 41694 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 41695 | Convert ~ cdlemg40 to func... |
| ltrnco 41696 | The composition of two tra... |
| trlcocnv 41697 | Swap the arguments of the ... |
| trlcoabs 41698 | Absorption into a composit... |
| trlcoabs2N 41699 | Absorption of the trace of... |
| trlcoat 41700 | The trace of a composition... |
| trlcocnvat 41701 | Commonly used special case... |
| trlconid 41702 | The composition of two dif... |
| trlcolem 41703 | Lemma for ~ trlco . (Cont... |
| trlco 41704 | The trace of a composition... |
| trlcone 41705 | If two translations have d... |
| cdlemg42 41706 | Part of proof of Lemma G o... |
| cdlemg43 41707 | Part of proof of Lemma G o... |
| cdlemg44a 41708 | Part of proof of Lemma G o... |
| cdlemg44b 41709 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 41710 | Part of proof of Lemma G o... |
| cdlemg47a 41711 | TODO: fix comment. TODO: ... |
| cdlemg46 41712 | Part of proof of Lemma G o... |
| cdlemg47 41713 | Part of proof of Lemma G o... |
| cdlemg48 41714 | Eliminate ` h ` from ~ cdl... |
| ltrncom 41715 | Composition is commutative... |
| ltrnco4 41716 | Rearrange a composition of... |
| trljco 41717 | Trace joined with trace of... |
| trljco2 41718 | Trace joined with trace of... |
| tgrpfset 41721 | The translation group maps... |
| tgrpset 41722 | The translation group for ... |
| tgrpbase 41723 | The base set of the transl... |
| tgrpopr 41724 | The group operation of the... |
| tgrpov 41725 | The group operation value ... |
| tgrpgrplem 41726 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 41727 | The translation group is a... |
| tgrpabl 41728 | The translation group is a... |
| tendofset 41735 | The set of all trace-prese... |
| tendoset 41736 | The set of trace-preservin... |
| istendo 41737 | The predicate "is a trace-... |
| tendotp 41738 | Trace-preserving property ... |
| istendod 41739 | Deduce the predicate "is a... |
| tendof 41740 | Functionality of a trace-p... |
| tendoeq1 41741 | Condition determining equa... |
| tendovalco 41742 | Value of composition of tr... |
| tendocoval 41743 | Value of composition of en... |
| tendocl 41744 | Closure of a trace-preserv... |
| tendoco2 41745 | Distribution of compositio... |
| tendoidcl 41746 | The identity is a trace-pr... |
| tendo1mul 41747 | Multiplicative identity mu... |
| tendo1mulr 41748 | Multiplicative identity mu... |
| tendococl 41749 | The composition of two tra... |
| tendoid 41750 | The identity value of a tr... |
| tendoeq2 41751 | Condition determining equa... |
| tendoplcbv 41752 | Define sum operation for t... |
| tendopl 41753 | Value of endomorphism sum ... |
| tendopl2 41754 | Value of result of endomor... |
| tendoplcl2 41755 | Value of result of endomor... |
| tendoplco2 41756 | Value of result of endomor... |
| tendopltp 41757 | Trace-preserving property ... |
| tendoplcl 41758 | Endomorphism sum is a trac... |
| tendoplcom 41759 | The endomorphism sum opera... |
| tendoplass 41760 | The endomorphism sum opera... |
| tendodi1 41761 | Endomorphism composition d... |
| tendodi2 41762 | Endomorphism composition d... |
| tendo0cbv 41763 | Define additive identity f... |
| tendo02 41764 | Value of additive identity... |
| tendo0co2 41765 | The additive identity trac... |
| tendo0tp 41766 | Trace-preserving property ... |
| tendo0cl 41767 | The additive identity is a... |
| tendo0pl 41768 | Property of the additive i... |
| tendo0plr 41769 | Property of the additive i... |
| tendoicbv 41770 | Define inverse function fo... |
| tendoi 41771 | Value of inverse endomorph... |
| tendoi2 41772 | Value of additive inverse ... |
| tendoicl 41773 | Closure of the additive in... |
| tendoipl 41774 | Property of the additive i... |
| tendoipl2 41775 | Property of the additive i... |
| erngfset 41776 | The division rings on trac... |
| erngset 41777 | The division ring on trace... |
| erngbase 41778 | The base set of the divisi... |
| erngfplus 41779 | Ring addition operation. ... |
| erngplus 41780 | Ring addition operation. ... |
| erngplus2 41781 | Ring addition operation. ... |
| erngfmul 41782 | Ring multiplication operat... |
| erngmul 41783 | Ring addition operation. ... |
| erngfset-rN 41784 | The division rings on trac... |
| erngset-rN 41785 | The division ring on trace... |
| erngbase-rN 41786 | The base set of the divisi... |
| erngfplus-rN 41787 | Ring addition operation. ... |
| erngplus-rN 41788 | Ring addition operation. ... |
| erngplus2-rN 41789 | Ring addition operation. ... |
| erngfmul-rN 41790 | Ring multiplication operat... |
| erngmul-rN 41791 | Ring addition operation. ... |
| cdlemh1 41792 | Part of proof of Lemma H o... |
| cdlemh2 41793 | Part of proof of Lemma H o... |
| cdlemh 41794 | Lemma H of [Crawley] p. 11... |
| cdlemi1 41795 | Part of proof of Lemma I o... |
| cdlemi2 41796 | Part of proof of Lemma I o... |
| cdlemi 41797 | Lemma I of [Crawley] p. 11... |
| cdlemj1 41798 | Part of proof of Lemma J o... |
| cdlemj2 41799 | Part of proof of Lemma J o... |
| cdlemj3 41800 | Part of proof of Lemma J o... |
| tendocan 41801 | Cancellation law: if the v... |
| tendoid0 41802 | A trace-preserving endomor... |
| tendo0mul 41803 | Additive identity multipli... |
| tendo0mulr 41804 | Additive identity multipli... |
| tendo1ne0 41805 | The identity (unity) is no... |
| tendoconid 41806 | The composition (product) ... |
| tendotr 41807 | The trace of the value of ... |
| cdlemk1 41808 | Part of proof of Lemma K o... |
| cdlemk2 41809 | Part of proof of Lemma K o... |
| cdlemk3 41810 | Part of proof of Lemma K o... |
| cdlemk4 41811 | Part of proof of Lemma K o... |
| cdlemk5a 41812 | Part of proof of Lemma K o... |
| cdlemk5 41813 | Part of proof of Lemma K o... |
| cdlemk6 41814 | Part of proof of Lemma K o... |
| cdlemk8 41815 | Part of proof of Lemma K o... |
| cdlemk9 41816 | Part of proof of Lemma K o... |
| cdlemk9bN 41817 | Part of proof of Lemma K o... |
| cdlemki 41818 | Part of proof of Lemma K o... |
| cdlemkvcl 41819 | Part of proof of Lemma K o... |
| cdlemk10 41820 | Part of proof of Lemma K o... |
| cdlemksv 41821 | Part of proof of Lemma K o... |
| cdlemksel 41822 | Part of proof of Lemma K o... |
| cdlemksat 41823 | Part of proof of Lemma K o... |
| cdlemksv2 41824 | Part of proof of Lemma K o... |
| cdlemk7 41825 | Part of proof of Lemma K o... |
| cdlemk11 41826 | Part of proof of Lemma K o... |
| cdlemk12 41827 | Part of proof of Lemma K o... |
| cdlemkoatnle 41828 | Utility lemma. (Contribut... |
| cdlemk13 41829 | Part of proof of Lemma K o... |
| cdlemkole 41830 | Utility lemma. (Contribut... |
| cdlemk14 41831 | Part of proof of Lemma K o... |
| cdlemk15 41832 | Part of proof of Lemma K o... |
| cdlemk16a 41833 | Part of proof of Lemma K o... |
| cdlemk16 41834 | Part of proof of Lemma K o... |
| cdlemk17 41835 | Part of proof of Lemma K o... |
| cdlemk1u 41836 | Part of proof of Lemma K o... |
| cdlemk5auN 41837 | Part of proof of Lemma K o... |
| cdlemk5u 41838 | Part of proof of Lemma K o... |
| cdlemk6u 41839 | Part of proof of Lemma K o... |
| cdlemkj 41840 | Part of proof of Lemma K o... |
| cdlemkuvN 41841 | Part of proof of Lemma K o... |
| cdlemkuel 41842 | Part of proof of Lemma K o... |
| cdlemkuat 41843 | Part of proof of Lemma K o... |
| cdlemkuv2 41844 | Part of proof of Lemma K o... |
| cdlemk18 41845 | Part of proof of Lemma K o... |
| cdlemk19 41846 | Part of proof of Lemma K o... |
| cdlemk7u 41847 | Part of proof of Lemma K o... |
| cdlemk11u 41848 | Part of proof of Lemma K o... |
| cdlemk12u 41849 | Part of proof of Lemma K o... |
| cdlemk21N 41850 | Part of proof of Lemma K o... |
| cdlemk20 41851 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 41852 | Utility lemma. (Contribut... |
| cdlemk13-2N 41853 | Part of proof of Lemma K o... |
| cdlemkole-2N 41854 | Utility lemma. (Contribut... |
| cdlemk14-2N 41855 | Part of proof of Lemma K o... |
| cdlemk15-2N 41856 | Part of proof of Lemma K o... |
| cdlemk16-2N 41857 | Part of proof of Lemma K o... |
| cdlemk17-2N 41858 | Part of proof of Lemma K o... |
| cdlemkj-2N 41859 | Part of proof of Lemma K o... |
| cdlemkuv-2N 41860 | Part of proof of Lemma K o... |
| cdlemkuel-2N 41861 | Part of proof of Lemma K o... |
| cdlemkuv2-2 41862 | Part of proof of Lemma K o... |
| cdlemk18-2N 41863 | Part of proof of Lemma K o... |
| cdlemk19-2N 41864 | Part of proof of Lemma K o... |
| cdlemk7u-2N 41865 | Part of proof of Lemma K o... |
| cdlemk11u-2N 41866 | Part of proof of Lemma K o... |
| cdlemk12u-2N 41867 | Part of proof of Lemma K o... |
| cdlemk21-2N 41868 | Part of proof of Lemma K o... |
| cdlemk20-2N 41869 | Part of proof of Lemma K o... |
| cdlemk22 41870 | Part of proof of Lemma K o... |
| cdlemk30 41871 | Part of proof of Lemma K o... |
| cdlemkuu 41872 | Convert between function a... |
| cdlemk31 41873 | Part of proof of Lemma K o... |
| cdlemk32 41874 | Part of proof of Lemma K o... |
| cdlemkuel-3 41875 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 41876 | Part of proof of Lemma K o... |
| cdlemk18-3N 41877 | Part of proof of Lemma K o... |
| cdlemk22-3 41878 | Part of proof of Lemma K o... |
| cdlemk23-3 41879 | Part of proof of Lemma K o... |
| cdlemk24-3 41880 | Part of proof of Lemma K o... |
| cdlemk25-3 41881 | Part of proof of Lemma K o... |
| cdlemk26b-3 41882 | Part of proof of Lemma K o... |
| cdlemk26-3 41883 | Part of proof of Lemma K o... |
| cdlemk27-3 41884 | Part of proof of Lemma K o... |
| cdlemk28-3 41885 | Part of proof of Lemma K o... |
| cdlemk33N 41886 | Part of proof of Lemma K o... |
| cdlemk34 41887 | Part of proof of Lemma K o... |
| cdlemk29-3 41888 | Part of proof of Lemma K o... |
| cdlemk35 41889 | Part of proof of Lemma K o... |
| cdlemk36 41890 | Part of proof of Lemma K o... |
| cdlemk37 41891 | Part of proof of Lemma K o... |
| cdlemk38 41892 | Part of proof of Lemma K o... |
| cdlemk39 41893 | Part of proof of Lemma K o... |
| cdlemk40 41894 | TODO: fix comment. (Contr... |
| cdlemk40t 41895 | TODO: fix comment. (Contr... |
| cdlemk40f 41896 | TODO: fix comment. (Contr... |
| cdlemk41 41897 | Part of proof of Lemma K o... |
| cdlemkfid1N 41898 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 41899 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 41900 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 41901 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 41902 | TODO: is this useful or sh... |
| cdlemky 41903 | Part of proof of Lemma K o... |
| cdlemkyu 41904 | Convert between function a... |
| cdlemkyuu 41905 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 41906 | Part of proof of Lemma K o... |
| cdlemk19ylem 41907 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 41908 | Part of proof of Lemma K o... |
| cdlemk19y 41909 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 41910 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 41911 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 41912 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 41913 | The value of the tau funct... |
| cdlemk35s 41914 | Substitution version of ~ ... |
| cdlemk35s-id 41915 | Substitution version of ~ ... |
| cdlemk39s 41916 | Substitution version of ~ ... |
| cdlemk39s-id 41917 | Substitution version of ~ ... |
| cdlemk42 41918 | Part of proof of Lemma K o... |
| cdlemk19xlem 41919 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 41920 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 41921 | Part of proof of Lemma K o... |
| cdlemk11tc 41922 | Part of proof of Lemma K o... |
| cdlemk11t 41923 | Part of proof of Lemma K o... |
| cdlemk45 41924 | Part of proof of Lemma K o... |
| cdlemk46 41925 | Part of proof of Lemma K o... |
| cdlemk47 41926 | Part of proof of Lemma K o... |
| cdlemk48 41927 | Part of proof of Lemma K o... |
| cdlemk49 41928 | Part of proof of Lemma K o... |
| cdlemk50 41929 | Part of proof of Lemma K o... |
| cdlemk51 41930 | Part of proof of Lemma K o... |
| cdlemk52 41931 | Part of proof of Lemma K o... |
| cdlemk53a 41932 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 41933 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 41934 | Part of proof of Lemma K o... |
| cdlemk54 41935 | Part of proof of Lemma K o... |
| cdlemk55a 41936 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 41937 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 41938 | Part of proof of Lemma K o... |
| cdlemkyyN 41939 | Part of proof of Lemma K o... |
| cdlemk43N 41940 | Part of proof of Lemma K o... |
| cdlemk35u 41941 | Substitution version of ~ ... |
| cdlemk55u1 41942 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 41943 | Part of proof of Lemma K o... |
| cdlemk39u1 41944 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 41945 | Part of proof of Lemma K o... |
| cdlemk19u1 41946 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 41947 | Part of Lemma K of [Crawle... |
| cdlemk56 41948 | Part of Lemma K of [Crawle... |
| cdlemk19w 41949 | Use a fixed element to eli... |
| cdlemk56w 41950 | Use a fixed element to eli... |
| cdlemk 41951 | Lemma K of [Crawley] p. 11... |
| tendoex 41952 | Generalization of Lemma K ... |
| cdleml1N 41953 | Part of proof of Lemma L o... |
| cdleml2N 41954 | Part of proof of Lemma L o... |
| cdleml3N 41955 | Part of proof of Lemma L o... |
| cdleml4N 41956 | Part of proof of Lemma L o... |
| cdleml5N 41957 | Part of proof of Lemma L o... |
| cdleml6 41958 | Part of proof of Lemma L o... |
| cdleml7 41959 | Part of proof of Lemma L o... |
| cdleml8 41960 | Part of proof of Lemma L o... |
| cdleml9 41961 | Part of proof of Lemma L o... |
| dva1dim 41962 | Two expressions for the 1-... |
| dvhb1dimN 41963 | Two expressions for the 1-... |
| erng1lem 41964 | Value of the endomorphism ... |
| erngdvlem1 41965 | Lemma for ~ eringring . (... |
| erngdvlem2N 41966 | Lemma for ~ eringring . (... |
| erngdvlem3 41967 | Lemma for ~ eringring . (... |
| erngdvlem4 41968 | Lemma for ~ erngdv . (Con... |
| eringring 41969 | An endomorphism ring is a ... |
| erngdv 41970 | An endomorphism ring is a ... |
| erng0g 41971 | The division ring zero of ... |
| erng1r 41972 | The division ring unity of... |
| erngdvlem1-rN 41973 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 41974 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 41975 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 41976 | Lemma for ~ erngdv . (Con... |
| erngring-rN 41977 | An endomorphism ring is a ... |
| erngdv-rN 41978 | An endomorphism ring is a ... |
| dvafset 41981 | The constructed partial ve... |
| dvaset 41982 | The constructed partial ve... |
| dvasca 41983 | The ring base set of the c... |
| dvabase 41984 | The ring base set of the c... |
| dvafplusg 41985 | Ring addition operation fo... |
| dvaplusg 41986 | Ring addition operation fo... |
| dvaplusgv 41987 | Ring addition operation fo... |
| dvafmulr 41988 | Ring multiplication operat... |
| dvamulr 41989 | Ring multiplication operat... |
| dvavbase 41990 | The vectors (vector base s... |
| dvafvadd 41991 | The vector sum operation f... |
| dvavadd 41992 | Ring addition operation fo... |
| dvafvsca 41993 | Ring addition operation fo... |
| dvavsca 41994 | Ring addition operation fo... |
| tendospcl 41995 | Closure of endomorphism sc... |
| tendospass 41996 | Associative law for endomo... |
| tendospdi1 41997 | Forward distributive law f... |
| tendocnv 41998 | Converse of a trace-preser... |
| tendospdi2 41999 | Reverse distributive law f... |
| tendospcanN 42000 | Cancellation law for trace... |
| dvaabl 42001 | The constructed partial ve... |
| dvalveclem 42002 | Lemma for ~ dvalvec . (Co... |
| dvalvec 42003 | The constructed partial ve... |
| dva0g 42004 | The zero vector of partial... |
| diaffval 42007 | The partial isomorphism A ... |
| diafval 42008 | The partial isomorphism A ... |
| diaval 42009 | The partial isomorphism A ... |
| diaelval 42010 | Member of the partial isom... |
| diafn 42011 | Functionality and domain o... |
| diadm 42012 | Domain of the partial isom... |
| diaeldm 42013 | Member of domain of the pa... |
| diadmclN 42014 | A member of domain of the ... |
| diadmleN 42015 | A member of domain of the ... |
| dian0 42016 | The value of the partial i... |
| dia0eldmN 42017 | The lattice zero belongs t... |
| dia1eldmN 42018 | The fiducial hyperplane (t... |
| diass 42019 | The value of the partial i... |
| diael 42020 | A member of the value of t... |
| diatrl 42021 | Trace of a member of the p... |
| diaelrnN 42022 | Any value of the partial i... |
| dialss 42023 | The value of partial isomo... |
| diaord 42024 | The partial isomorphism A ... |
| dia11N 42025 | The partial isomorphism A ... |
| diaf11N 42026 | The partial isomorphism A ... |
| diaclN 42027 | Closure of partial isomorp... |
| diacnvclN 42028 | Closure of partial isomorp... |
| dia0 42029 | The value of the partial i... |
| dia1N 42030 | The value of the partial i... |
| dia1elN 42031 | The largest subspace in th... |
| diaglbN 42032 | Partial isomorphism A of a... |
| diameetN 42033 | Partial isomorphism A of a... |
| diainN 42034 | Inverse partial isomorphis... |
| diaintclN 42035 | The intersection of partia... |
| diasslssN 42036 | The partial isomorphism A ... |
| diassdvaN 42037 | The partial isomorphism A ... |
| dia1dim 42038 | Two expressions for the 1-... |
| dia1dim2 42039 | Two expressions for a 1-di... |
| dia1dimid 42040 | A vector (translation) bel... |
| dia2dimlem1 42041 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 42042 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 42043 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 42044 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 42045 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 42046 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 42047 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 42048 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 42049 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 42050 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 42051 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 42052 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 42053 | Lemma for ~ dia2dim . Eli... |
| dia2dim 42054 | A two-dimensional subspace... |
| dvhfset 42057 | The constructed full vecto... |
| dvhset 42058 | The constructed full vecto... |
| dvhsca 42059 | The ring of scalars of the... |
| dvhbase 42060 | The ring base set of the c... |
| dvhfplusr 42061 | Ring addition operation fo... |
| dvhfmulr 42062 | Ring multiplication operat... |
| dvhmulr 42063 | Ring multiplication operat... |
| dvhvbase 42064 | The vectors (vector base s... |
| dvhelvbasei 42065 | Vector membership in the c... |
| dvhvaddcbv 42066 | Change bound variables to ... |
| dvhvaddval 42067 | The vector sum operation f... |
| dvhfvadd 42068 | The vector sum operation f... |
| dvhvadd 42069 | The vector sum operation f... |
| dvhopvadd 42070 | The vector sum operation f... |
| dvhopvadd2 42071 | The vector sum operation f... |
| dvhvaddcl 42072 | Closure of the vector sum ... |
| dvhvaddcomN 42073 | Commutativity of vector su... |
| dvhvaddass 42074 | Associativity of vector su... |
| dvhvscacbv 42075 | Change bound variables to ... |
| dvhvscaval 42076 | The scalar product operati... |
| dvhfvsca 42077 | Scalar product operation f... |
| dvhvsca 42078 | Scalar product operation f... |
| dvhopvsca 42079 | Scalar product operation f... |
| dvhvscacl 42080 | Closure of the scalar prod... |
| tendoinvcl 42081 | Closure of multiplicative ... |
| tendolinv 42082 | Left multiplicative invers... |
| tendorinv 42083 | Right multiplicative inver... |
| dvhgrp 42084 | The full vector space ` U ... |
| dvhlveclem 42085 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 42086 | The full vector space ` U ... |
| dvhlmod 42087 | The full vector space ` U ... |
| dvh0g 42088 | The zero vector of vector ... |
| dvheveccl 42089 | Properties of a unit vecto... |
| dvhopclN 42090 | Closure of a ` DVecH ` vec... |
| dvhopaddN 42091 | Sum of ` DVecH ` vectors e... |
| dvhopspN 42092 | Scalar product of ` DVecH ... |
| dvhopN 42093 | Decompose a ` DVecH ` vect... |
| dvhopellsm 42094 | Ordered pair membership in... |
| cdlemm10N 42095 | The image of the map ` G `... |
| docaffvalN 42098 | Subspace orthocomplement f... |
| docafvalN 42099 | Subspace orthocomplement f... |
| docavalN 42100 | Subspace orthocomplement f... |
| docaclN 42101 | Closure of subspace orthoc... |
| diaocN 42102 | Value of partial isomorphi... |
| doca2N 42103 | Double orthocomplement of ... |
| doca3N 42104 | Double orthocomplement of ... |
| dvadiaN 42105 | Any closed subspace is a m... |
| diarnN 42106 | Partial isomorphism A maps... |
| diaf1oN 42107 | The partial isomorphism A ... |
| djaffvalN 42110 | Subspace join for ` DVecA ... |
| djafvalN 42111 | Subspace join for ` DVecA ... |
| djavalN 42112 | Subspace join for ` DVecA ... |
| djaclN 42113 | Closure of subspace join f... |
| djajN 42114 | Transfer lattice join to `... |
| dibffval 42117 | The partial isomorphism B ... |
| dibfval 42118 | The partial isomorphism B ... |
| dibval 42119 | The partial isomorphism B ... |
| dibopelvalN 42120 | Member of the partial isom... |
| dibval2 42121 | Value of the partial isomo... |
| dibopelval2 42122 | Member of the partial isom... |
| dibval3N 42123 | Value of the partial isomo... |
| dibelval3 42124 | Member of the partial isom... |
| dibopelval3 42125 | Member of the partial isom... |
| dibelval1st 42126 | Membership in value of the... |
| dibelval1st1 42127 | Membership in value of the... |
| dibelval1st2N 42128 | Membership in value of the... |
| dibelval2nd 42129 | Membership in value of the... |
| dibn0 42130 | The value of the partial i... |
| dibfna 42131 | Functionality and domain o... |
| dibdiadm 42132 | Domain of the partial isom... |
| dibfnN 42133 | Functionality and domain o... |
| dibdmN 42134 | Domain of the partial isom... |
| dibeldmN 42135 | Member of domain of the pa... |
| dibord 42136 | The isomorphism B for a la... |
| dib11N 42137 | The isomorphism B for a la... |
| dibf11N 42138 | The partial isomorphism A ... |
| dibclN 42139 | Closure of partial isomorp... |
| dibvalrel 42140 | The value of partial isomo... |
| dib0 42141 | The value of partial isomo... |
| dib1dim 42142 | Two expressions for the 1-... |
| dibglbN 42143 | Partial isomorphism B of a... |
| dibintclN 42144 | The intersection of partia... |
| dib1dim2 42145 | Two expressions for a 1-di... |
| dibss 42146 | The partial isomorphism B ... |
| diblss 42147 | The value of partial isomo... |
| diblsmopel 42148 | Membership in subspace sum... |
| dicffval 42151 | The partial isomorphism C ... |
| dicfval 42152 | The partial isomorphism C ... |
| dicval 42153 | The partial isomorphism C ... |
| dicopelval 42154 | Membership in value of the... |
| dicelvalN 42155 | Membership in value of the... |
| dicval2 42156 | The partial isomorphism C ... |
| dicelval3 42157 | Member of the partial isom... |
| dicopelval2 42158 | Membership in value of the... |
| dicelval2N 42159 | Membership in value of the... |
| dicfnN 42160 | Functionality and domain o... |
| dicdmN 42161 | Domain of the partial isom... |
| dicvalrelN 42162 | The value of partial isomo... |
| dicssdvh 42163 | The partial isomorphism C ... |
| dicelval1sta 42164 | Membership in value of the... |
| dicelval1stN 42165 | Membership in value of the... |
| dicelval2nd 42166 | Membership in value of the... |
| dicvaddcl 42167 | Membership in value of the... |
| dicvscacl 42168 | Membership in value of the... |
| dicn0 42169 | The value of the partial i... |
| diclss 42170 | The value of partial isomo... |
| diclspsn 42171 | The value of isomorphism C... |
| cdlemn2 42172 | Part of proof of Lemma N o... |
| cdlemn2a 42173 | Part of proof of Lemma N o... |
| cdlemn3 42174 | Part of proof of Lemma N o... |
| cdlemn4 42175 | Part of proof of Lemma N o... |
| cdlemn4a 42176 | Part of proof of Lemma N o... |
| cdlemn5pre 42177 | Part of proof of Lemma N o... |
| cdlemn5 42178 | Part of proof of Lemma N o... |
| cdlemn6 42179 | Part of proof of Lemma N o... |
| cdlemn7 42180 | Part of proof of Lemma N o... |
| cdlemn8 42181 | Part of proof of Lemma N o... |
| cdlemn9 42182 | Part of proof of Lemma N o... |
| cdlemn10 42183 | Part of proof of Lemma N o... |
| cdlemn11a 42184 | Part of proof of Lemma N o... |
| cdlemn11b 42185 | Part of proof of Lemma N o... |
| cdlemn11c 42186 | Part of proof of Lemma N o... |
| cdlemn11pre 42187 | Part of proof of Lemma N o... |
| cdlemn11 42188 | Part of proof of Lemma N o... |
| cdlemn 42189 | Lemma N of [Crawley] p. 12... |
| dihordlem6 42190 | Part of proof of Lemma N o... |
| dihordlem7 42191 | Part of proof of Lemma N o... |
| dihordlem7b 42192 | Part of proof of Lemma N o... |
| dihjustlem 42193 | Part of proof after Lemma ... |
| dihjust 42194 | Part of proof after Lemma ... |
| dihord1 42195 | Part of proof after Lemma ... |
| dihord2a 42196 | Part of proof after Lemma ... |
| dihord2b 42197 | Part of proof after Lemma ... |
| dihord2cN 42198 | Part of proof after Lemma ... |
| dihord11b 42199 | Part of proof after Lemma ... |
| dihord10 42200 | Part of proof after Lemma ... |
| dihord11c 42201 | Part of proof after Lemma ... |
| dihord2pre 42202 | Part of proof after Lemma ... |
| dihord2pre2 42203 | Part of proof after Lemma ... |
| dihord2 42204 | Part of proof after Lemma ... |
| dihffval 42207 | The isomorphism H for a la... |
| dihfval 42208 | Isomorphism H for a lattic... |
| dihval 42209 | Value of isomorphism H for... |
| dihvalc 42210 | Value of isomorphism H for... |
| dihlsscpre 42211 | Closure of isomorphism H f... |
| dihvalcqpre 42212 | Value of isomorphism H for... |
| dihvalcq 42213 | Value of isomorphism H for... |
| dihvalb 42214 | Value of isomorphism H for... |
| dihopelvalbN 42215 | Ordered pair member of the... |
| dihvalcqat 42216 | Value of isomorphism H for... |
| dih1dimb 42217 | Two expressions for a 1-di... |
| dih1dimb2 42218 | Isomorphism H at an atom u... |
| dih1dimc 42219 | Isomorphism H at an atom n... |
| dib2dim 42220 | Extend ~ dia2dim to partia... |
| dih2dimb 42221 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 42222 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 42223 | Ordered pair member of the... |
| dihvalcq2 42224 | Value of isomorphism H for... |
| dihopelvalcpre 42225 | Member of value of isomorp... |
| dihopelvalc 42226 | Member of value of isomorp... |
| dihlss 42227 | The value of isomorphism H... |
| dihss 42228 | The value of isomorphism H... |
| dihssxp 42229 | An isomorphism H value is ... |
| dihopcl 42230 | Closure of an ordered pair... |
| xihopellsmN 42231 | Ordered pair membership in... |
| dihopellsm 42232 | Ordered pair membership in... |
| dihord6apre 42233 | Part of proof that isomorp... |
| dihord3 42234 | The isomorphism H for a la... |
| dihord4 42235 | The isomorphism H for a la... |
| dihord5b 42236 | Part of proof that isomorp... |
| dihord6b 42237 | Part of proof that isomorp... |
| dihord6a 42238 | Part of proof that isomorp... |
| dihord5apre 42239 | Part of proof that isomorp... |
| dihord5a 42240 | Part of proof that isomorp... |
| dihord 42241 | The isomorphism H is order... |
| dih11 42242 | The isomorphism H is one-t... |
| dihf11lem 42243 | Functionality of the isomo... |
| dihf11 42244 | The isomorphism H for a la... |
| dihfn 42245 | Functionality and domain o... |
| dihdm 42246 | Domain of isomorphism H. (... |
| dihcl 42247 | Closure of isomorphism H. ... |
| dihcnvcl 42248 | Closure of isomorphism H c... |
| dihcnvid1 42249 | The converse isomorphism o... |
| dihcnvid2 42250 | The isomorphism of a conve... |
| dihcnvord 42251 | Ordering property for conv... |
| dihcnv11 42252 | The converse of isomorphis... |
| dihsslss 42253 | The isomorphism H maps to ... |
| dihrnlss 42254 | The isomorphism H maps to ... |
| dihrnss 42255 | The isomorphism H maps to ... |
| dihvalrel 42256 | The value of isomorphism H... |
| dih0 42257 | The value of isomorphism H... |
| dih0bN 42258 | A lattice element is zero ... |
| dih0vbN 42259 | A vector is zero iff its s... |
| dih0cnv 42260 | The isomorphism H converse... |
| dih0rn 42261 | The zero subspace belongs ... |
| dih0sb 42262 | A subspace is zero iff the... |
| dih1 42263 | The value of isomorphism H... |
| dih1rn 42264 | The full vector space belo... |
| dih1cnv 42265 | The isomorphism H converse... |
| dihwN 42266 | Value of isomorphism H at ... |
| dihmeetlem1N 42267 | Isomorphism H of a conjunc... |
| dihglblem5apreN 42268 | A conjunction property of ... |
| dihglblem5aN 42269 | A conjunction property of ... |
| dihglblem2aN 42270 | Lemma for isomorphism H of... |
| dihglblem2N 42271 | The GLB of a set of lattic... |
| dihglblem3N 42272 | Isomorphism H of a lattice... |
| dihglblem3aN 42273 | Isomorphism H of a lattice... |
| dihglblem4 42274 | Isomorphism H of a lattice... |
| dihglblem5 42275 | Isomorphism H of a lattice... |
| dihmeetlem2N 42276 | Isomorphism H of a conjunc... |
| dihglbcpreN 42277 | Isomorphism H of a lattice... |
| dihglbcN 42278 | Isomorphism H of a lattice... |
| dihmeetcN 42279 | Isomorphism H of a lattice... |
| dihmeetbN 42280 | Isomorphism H of a lattice... |
| dihmeetbclemN 42281 | Lemma for isomorphism H of... |
| dihmeetlem3N 42282 | Lemma for isomorphism H of... |
| dihmeetlem4preN 42283 | Lemma for isomorphism H of... |
| dihmeetlem4N 42284 | Lemma for isomorphism H of... |
| dihmeetlem5 42285 | Part of proof that isomorp... |
| dihmeetlem6 42286 | Lemma for isomorphism H of... |
| dihmeetlem7N 42287 | Lemma for isomorphism H of... |
| dihjatc1 42288 | Lemma for isomorphism H of... |
| dihjatc2N 42289 | Isomorphism H of join with... |
| dihjatc3 42290 | Isomorphism H of join with... |
| dihmeetlem8N 42291 | Lemma for isomorphism H of... |
| dihmeetlem9N 42292 | Lemma for isomorphism H of... |
| dihmeetlem10N 42293 | Lemma for isomorphism H of... |
| dihmeetlem11N 42294 | Lemma for isomorphism H of... |
| dihmeetlem12N 42295 | Lemma for isomorphism H of... |
| dihmeetlem13N 42296 | Lemma for isomorphism H of... |
| dihmeetlem14N 42297 | Lemma for isomorphism H of... |
| dihmeetlem15N 42298 | Lemma for isomorphism H of... |
| dihmeetlem16N 42299 | Lemma for isomorphism H of... |
| dihmeetlem17N 42300 | Lemma for isomorphism H of... |
| dihmeetlem18N 42301 | Lemma for isomorphism H of... |
| dihmeetlem19N 42302 | Lemma for isomorphism H of... |
| dihmeetlem20N 42303 | Lemma for isomorphism H of... |
| dihmeetALTN 42304 | Isomorphism H of a lattice... |
| dih1dimatlem0 42305 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 42306 | Lemma for ~ dih1dimat . (... |
| dih1dimat 42307 | Any 1-dimensional subspace... |
| dihlsprn 42308 | The span of a vector belon... |
| dihlspsnssN 42309 | A subspace included in a 1... |
| dihlspsnat 42310 | The inverse isomorphism H ... |
| dihatlat 42311 | The isomorphism H of an at... |
| dihat 42312 | There exists at least one ... |
| dihpN 42313 | The value of isomorphism H... |
| dihlatat 42314 | The reverse isomorphism H ... |
| dihatexv 42315 | There is a nonzero vector ... |
| dihatexv2 42316 | There is a nonzero vector ... |
| dihglblem6 42317 | Isomorphism H of a lattice... |
| dihglb 42318 | Isomorphism H of a lattice... |
| dihglb2 42319 | Isomorphism H of a lattice... |
| dihmeet 42320 | Isomorphism H of a lattice... |
| dihintcl 42321 | The intersection of closed... |
| dihmeetcl 42322 | Closure of closed subspace... |
| dihmeet2 42323 | Reverse isomorphism H of a... |
| dochffval 42326 | Subspace orthocomplement f... |
| dochfval 42327 | Subspace orthocomplement f... |
| dochval 42328 | Subspace orthocomplement f... |
| dochval2 42329 | Subspace orthocomplement f... |
| dochcl 42330 | Closure of subspace orthoc... |
| dochlss 42331 | A subspace orthocomplement... |
| dochssv 42332 | A subspace orthocomplement... |
| dochfN 42333 | Domain and codomain of the... |
| dochvalr 42334 | Orthocomplement of a close... |
| doch0 42335 | Orthocomplement of the zer... |
| doch1 42336 | Orthocomplement of the uni... |
| dochoc0 42337 | The zero subspace is close... |
| dochoc1 42338 | The unit subspace (all vec... |
| dochvalr2 42339 | Orthocomplement of a close... |
| dochvalr3 42340 | Orthocomplement of a close... |
| doch2val2 42341 | Double orthocomplement for... |
| dochss 42342 | Subset law for orthocomple... |
| dochocss 42343 | Double negative law for or... |
| dochoc 42344 | Double negative law for or... |
| dochsscl 42345 | If a set of vectors is inc... |
| dochoccl 42346 | A set of vectors is closed... |
| dochord 42347 | Ordering law for orthocomp... |
| dochord2N 42348 | Ordering law for orthocomp... |
| dochord3 42349 | Ordering law for orthocomp... |
| doch11 42350 | Orthocomplement is one-to-... |
| dochsordN 42351 | Strict ordering law for or... |
| dochn0nv 42352 | An orthocomplement is nonz... |
| dihoml4c 42353 | Version of ~ dihoml4 with ... |
| dihoml4 42354 | Orthomodular law for const... |
| dochspss 42355 | The span of a set of vecto... |
| dochocsp 42356 | The span of an orthocomple... |
| dochspocN 42357 | The span of an orthocomple... |
| dochocsn 42358 | The double orthocomplement... |
| dochsncom 42359 | Swap vectors in an orthoco... |
| dochsat 42360 | The double orthocomplement... |
| dochshpncl 42361 | If a hyperplane is not clo... |
| dochlkr 42362 | Equivalent conditions for ... |
| dochkrshp 42363 | The closure of a kernel is... |
| dochkrshp2 42364 | Properties of the closure ... |
| dochkrshp3 42365 | Properties of the closure ... |
| dochkrshp4 42366 | Properties of the closure ... |
| dochdmj1 42367 | De Morgan-like law for sub... |
| dochnoncon 42368 | Law of noncontradiction. ... |
| dochnel2 42369 | A nonzero member of a subs... |
| dochnel 42370 | A nonzero vector doesn't b... |
| djhffval 42373 | Subspace join for ` DVecH ... |
| djhfval 42374 | Subspace join for ` DVecH ... |
| djhval 42375 | Subspace join for ` DVecH ... |
| djhval2 42376 | Value of subspace join for... |
| djhcl 42377 | Closure of subspace join f... |
| djhlj 42378 | Transfer lattice join to `... |
| djhljjN 42379 | Lattice join in terms of `... |
| djhjlj 42380 | ` DVecH ` vector space clo... |
| djhj 42381 | ` DVecH ` vector space clo... |
| djhcom 42382 | Subspace join is commutati... |
| djhspss 42383 | Subspace span of union is ... |
| djhsumss 42384 | Subspace sum is a subset o... |
| dihsumssj 42385 | The subspace sum of two is... |
| djhunssN 42386 | Subspace union is a subset... |
| dochdmm1 42387 | De Morgan-like law for clo... |
| djhexmid 42388 | Excluded middle property o... |
| djh01 42389 | Closed subspace join with ... |
| djh02 42390 | Closed subspace join with ... |
| djhlsmcl 42391 | A closed subspace sum equa... |
| djhcvat42 42392 | A covering property. ( ~ ... |
| dihjatb 42393 | Isomorphism H of lattice j... |
| dihjatc 42394 | Isomorphism H of lattice j... |
| dihjatcclem1 42395 | Lemma for isomorphism H of... |
| dihjatcclem2 42396 | Lemma for isomorphism H of... |
| dihjatcclem3 42397 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 42398 | Lemma for isomorphism H of... |
| dihjatcc 42399 | Isomorphism H of lattice j... |
| dihjat 42400 | Isomorphism H of lattice j... |
| dihprrnlem1N 42401 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 42402 | Lemma for ~ dihprrn . (Co... |
| dihprrn 42403 | The span of a vector pair ... |
| djhlsmat 42404 | The sum of two subspace at... |
| dihjat1lem 42405 | Subspace sum of a closed s... |
| dihjat1 42406 | Subspace sum of a closed s... |
| dihsmsprn 42407 | Subspace sum of a closed s... |
| dihjat2 42408 | The subspace sum of a clos... |
| dihjat3 42409 | Isomorphism H of lattice j... |
| dihjat4 42410 | Transfer the subspace sum ... |
| dihjat6 42411 | Transfer the subspace sum ... |
| dihsmsnrn 42412 | The subspace sum of two si... |
| dihsmatrn 42413 | The subspace sum of a clos... |
| dihjat5N 42414 | Transfer lattice join with... |
| dvh4dimat 42415 | There is an atom that is o... |
| dvh3dimatN 42416 | There is an atom that is o... |
| dvh2dimatN 42417 | Given an atom, there exist... |
| dvh1dimat 42418 | There exists an atom. (Co... |
| dvh1dim 42419 | There exists a nonzero vec... |
| dvh4dimlem 42420 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 42421 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 42422 | There is a vector that is ... |
| dvh3dim 42423 | There is a vector that is ... |
| dvh4dimN 42424 | There is a vector that is ... |
| dvh3dim2 42425 | There is a vector that is ... |
| dvh3dim3N 42426 | There is a vector that is ... |
| dochsnnz 42427 | The orthocomplement of a s... |
| dochsatshp 42428 | The orthocomplement of a s... |
| dochsatshpb 42429 | The orthocomplement of a s... |
| dochsnshp 42430 | The orthocomplement of a n... |
| dochshpsat 42431 | A hyperplane is closed iff... |
| dochkrsat 42432 | The orthocomplement of a k... |
| dochkrsat2 42433 | The orthocomplement of a k... |
| dochsat0 42434 | The orthocomplement of a k... |
| dochkrsm 42435 | The subspace sum of a clos... |
| dochexmidat 42436 | Special case of excluded m... |
| dochexmidlem1 42437 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 42438 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 42439 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 42440 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 42441 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 42442 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 42443 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 42444 | Lemma for ~ dochexmid . T... |
| dochexmid 42445 | Excluded middle law for cl... |
| dochsnkrlem1 42446 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 42447 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 42448 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 42449 | A (closed) kernel expresse... |
| dochsnkr2 42450 | Kernel of the explicit fun... |
| dochsnkr2cl 42451 | The ` X ` determining func... |
| dochflcl 42452 | Closure of the explicit fu... |
| dochfl1 42453 | The value of the explicit ... |
| dochfln0 42454 | The value of a functional ... |
| dochkr1 42455 | A nonzero functional has a... |
| dochkr1OLDN 42456 | A nonzero functional has a... |
| lpolsetN 42459 | The set of polarities of a... |
| islpolN 42460 | The predicate "is a polari... |
| islpoldN 42461 | Properties that determine ... |
| lpolfN 42462 | Functionality of a polarit... |
| lpolvN 42463 | The polarity of the whole ... |
| lpolconN 42464 | Contraposition property of... |
| lpolsatN 42465 | The polarity of an atomic ... |
| lpolpolsatN 42466 | Property of a polarity. (... |
| dochpolN 42467 | The subspace orthocompleme... |
| lcfl1lem 42468 | Property of a functional w... |
| lcfl1 42469 | Property of a functional w... |
| lcfl2 42470 | Property of a functional w... |
| lcfl3 42471 | Property of a functional w... |
| lcfl4N 42472 | Property of a functional w... |
| lcfl5 42473 | Property of a functional w... |
| lcfl5a 42474 | Property of a functional w... |
| lcfl6lem 42475 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 42476 | Lemma for ~ lcfl7N . If t... |
| lcfl6 42477 | Property of a functional w... |
| lcfl7N 42478 | Property of a functional w... |
| lcfl8 42479 | Property of a functional w... |
| lcfl8a 42480 | Property of a functional w... |
| lcfl8b 42481 | Property of a nonzero func... |
| lcfl9a 42482 | Property implying that a f... |
| lclkrlem1 42483 | The set of functionals hav... |
| lclkrlem2a 42484 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 42485 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 42486 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 42487 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 42488 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 42489 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 42490 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 42491 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 42492 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 42493 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 42494 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 42495 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 42496 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 42497 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 42498 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 42499 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 42500 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 42501 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 42502 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 42503 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 42504 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 42505 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 42506 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 42507 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 42508 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 42509 | The set of functionals hav... |
| lclkr 42510 | The set of functionals wit... |
| lcfls1lem 42511 | Property of a functional w... |
| lcfls1N 42512 | Property of a functional w... |
| lcfls1c 42513 | Property of a functional w... |
| lclkrslem1 42514 | The set of functionals hav... |
| lclkrslem2 42515 | The set of functionals hav... |
| lclkrs 42516 | The set of functionals hav... |
| lclkrs2 42517 | The set of functionals wit... |
| lcfrvalsnN 42518 | Reconstruction from the du... |
| lcfrlem1 42519 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 42520 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 42521 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 42522 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 42523 | Lemma for ~ lcfr . The se... |
| lcfrlem6 42524 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 42525 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 42526 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 42527 | Lemma for ~ lcf1o . (This... |
| lcf1o 42528 | Define a function ` J ` th... |
| lcfrlem10 42529 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 42530 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 42531 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 42532 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 42533 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 42534 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 42535 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 42536 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 42537 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 42538 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 42539 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 42540 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 42541 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 42542 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 42543 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 42544 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 42545 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 42546 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 42547 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 42548 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 42549 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 42550 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 42551 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 42552 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 42553 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 42554 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 42555 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 42556 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 42557 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 42558 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 42559 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 42560 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 42561 | Lemma for ~ lcfr . Elimin... |
| lcfr 42562 | Reconstruction of a subspa... |
| lcdfval 42565 | Dual vector space of funct... |
| lcdval 42566 | Dual vector space of funct... |
| lcdval2 42567 | Dual vector space of funct... |
| lcdlvec 42568 | The dual vector space of f... |
| lcdlmod 42569 | The dual vector space of f... |
| lcdvbase 42570 | Vector base set of a dual ... |
| lcdvbasess 42571 | The vector base set of the... |
| lcdvbaselfl 42572 | A vector in the base set o... |
| lcdvbasecl 42573 | Closure of the value of a ... |
| lcdvadd 42574 | Vector addition for the cl... |
| lcdvaddval 42575 | The value of the value of ... |
| lcdsca 42576 | The ring of scalars of the... |
| lcdsbase 42577 | Base set of scalar ring fo... |
| lcdsadd 42578 | Scalar addition for the cl... |
| lcdsmul 42579 | Scalar multiplication for ... |
| lcdvs 42580 | Scalar product for the clo... |
| lcdvsval 42581 | Value of scalar product op... |
| lcdvscl 42582 | The scalar product operati... |
| lcdlssvscl 42583 | Closure of scalar product ... |
| lcdvsass 42584 | Associative law for scalar... |
| lcd0 42585 | The zero scalar of the clo... |
| lcd1 42586 | The unit scalar of the clo... |
| lcdneg 42587 | The unit scalar of the clo... |
| lcd0v 42588 | The zero functional in the... |
| lcd0v2 42589 | The zero functional in the... |
| lcd0vvalN 42590 | Value of the zero function... |
| lcd0vcl 42591 | Closure of the zero functi... |
| lcd0vs 42592 | A scalar zero times a func... |
| lcdvs0N 42593 | A scalar times the zero fu... |
| lcdvsub 42594 | The value of vector subtra... |
| lcdvsubval 42595 | The value of the value of ... |
| lcdlss 42596 | Subspaces of a dual vector... |
| lcdlss2N 42597 | Subspaces of a dual vector... |
| lcdlsp 42598 | Span in the set of functio... |
| lcdlkreqN 42599 | Colinear functionals have ... |
| lcdlkreq2N 42600 | Colinear functionals have ... |
| mapdffval 42603 | Projectivity from vector s... |
| mapdfval 42604 | Projectivity from vector s... |
| mapdval 42605 | Value of projectivity from... |
| mapdvalc 42606 | Value of projectivity from... |
| mapdval2N 42607 | Value of projectivity from... |
| mapdval3N 42608 | Value of projectivity from... |
| mapdval4N 42609 | Value of projectivity from... |
| mapdval5N 42610 | Value of projectivity from... |
| mapdordlem1a 42611 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 42612 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 42613 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 42614 | Lemma for ~ mapdord . Ord... |
| mapdord 42615 | Ordering property of the m... |
| mapd11 42616 | The map defined by ~ df-ma... |
| mapddlssN 42617 | The mapping of a subspace ... |
| mapdsn 42618 | Value of the map defined b... |
| mapdsn2 42619 | Value of the map defined b... |
| mapdsn3 42620 | Value of the map defined b... |
| mapd1dim2lem1N 42621 | Value of the map defined b... |
| mapdrvallem2 42622 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 42623 | Lemma for ~ mapdrval . (C... |
| mapdrval 42624 | Given a dual subspace ` R ... |
| mapd1o 42625 | The map defined by ~ df-ma... |
| mapdrn 42626 | Range of the map defined b... |
| mapdunirnN 42627 | Union of the range of the ... |
| mapdrn2 42628 | Range of the map defined b... |
| mapdcnvcl 42629 | Closure of the converse of... |
| mapdcl 42630 | Closure the value of the m... |
| mapdcnvid1N 42631 | Converse of the value of t... |
| mapdsord 42632 | Strong ordering property o... |
| mapdcl2 42633 | The mapping of a subspace ... |
| mapdcnvid2 42634 | Value of the converse of t... |
| mapdcnvordN 42635 | Ordering property of the c... |
| mapdcnv11N 42636 | The converse of the map de... |
| mapdcv 42637 | Covering property of the c... |
| mapdincl 42638 | Closure of dual subspace i... |
| mapdin 42639 | Subspace intersection is p... |
| mapdlsmcl 42640 | Closure of dual subspace s... |
| mapdlsm 42641 | Subspace sum is preserved ... |
| mapd0 42642 | Projectivity map of the ze... |
| mapdcnvatN 42643 | Atoms are preserved by the... |
| mapdat 42644 | Atoms are preserved by the... |
| mapdspex 42645 | The map of a span equals t... |
| mapdn0 42646 | Transfer nonzero property ... |
| mapdncol 42647 | Transfer non-colinearity f... |
| mapdindp 42648 | Transfer (part of) vector ... |
| mapdpglem1 42649 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 42650 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 42651 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 42652 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 42653 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 42654 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 42655 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 42656 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 42657 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 42658 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 42659 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 42660 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 42661 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 42662 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 42663 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 42664 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 42665 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 42666 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 42667 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 42668 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 42669 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 42670 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 42671 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 42672 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 42673 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 42674 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 42675 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 42676 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 42677 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 42678 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 42679 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 42680 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 42681 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 42682 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 42683 | Part 1 of proof of the fir... |
| baerlem3lem1 42684 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 42685 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 42686 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 42687 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 42688 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 42689 | Lemma for ~ baerlem5b . (... |
| baerlem3 42690 | An equality that holds whe... |
| baerlem5a 42691 | An equality that holds whe... |
| baerlem5b 42692 | An equality that holds whe... |
| baerlem5amN 42693 | An equality that holds whe... |
| baerlem5bmN 42694 | An equality that holds whe... |
| baerlem5abmN 42695 | An equality that holds whe... |
| mapdindp0 42696 | Vector independence lemma.... |
| mapdindp1 42697 | Vector independence lemma.... |
| mapdindp2 42698 | Vector independence lemma.... |
| mapdindp3 42699 | Vector independence lemma.... |
| mapdindp4 42700 | Vector independence lemma.... |
| mapdhval 42701 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 42702 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 42703 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 42704 | Lemmma for ~~? mapdh . (C... |
| mapdheq 42705 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 42706 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 42707 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 42708 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 42709 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 42710 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 42711 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 42712 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 42713 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 42714 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 42715 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 42716 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 42717 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 42718 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 42719 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 42720 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 42721 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 42722 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 42723 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 42724 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 42725 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 42726 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 42727 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 42728 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 42729 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 42730 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 42731 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 42732 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 42735 | Map from nonzero vectors t... |
| hvmapfval 42736 | Map from nonzero vectors t... |
| hvmapval 42737 | Value of map from nonzero ... |
| hvmapvalvalN 42738 | Value of value of map (i.e... |
| hvmapidN 42739 | The value of the vector to... |
| hvmap1o 42740 | The vector to functional m... |
| hvmapclN 42741 | Closure of the vector to f... |
| hvmap1o2 42742 | The vector to functional m... |
| hvmapcl2 42743 | Closure of the vector to f... |
| hvmaplfl 42744 | The vector to functional m... |
| hvmaplkr 42745 | Kernel of the vector to fu... |
| mapdhvmap 42746 | Relationship between ` map... |
| lspindp5 42747 | Obtain an independent vect... |
| hdmaplem1 42748 | Lemma to convert a frequen... |
| hdmaplem2N 42749 | Lemma to convert a frequen... |
| hdmaplem3 42750 | Lemma to convert a frequen... |
| hdmaplem4 42751 | Lemma to convert a frequen... |
| mapdh8a 42752 | Part of Part (8) in [Baer]... |
| mapdh8aa 42753 | Part of Part (8) in [Baer]... |
| mapdh8ab 42754 | Part of Part (8) in [Baer]... |
| mapdh8ac 42755 | Part of Part (8) in [Baer]... |
| mapdh8ad 42756 | Part of Part (8) in [Baer]... |
| mapdh8b 42757 | Part of Part (8) in [Baer]... |
| mapdh8c 42758 | Part of Part (8) in [Baer]... |
| mapdh8d0N 42759 | Part of Part (8) in [Baer]... |
| mapdh8d 42760 | Part of Part (8) in [Baer]... |
| mapdh8e 42761 | Part of Part (8) in [Baer]... |
| mapdh8g 42762 | Part of Part (8) in [Baer]... |
| mapdh8i 42763 | Part of Part (8) in [Baer]... |
| mapdh8j 42764 | Part of Part (8) in [Baer]... |
| mapdh8 42765 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 42766 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 42767 | Lemma for part (9) in [Bae... |
| hdmap1ffval 42772 | Preliminary map from vecto... |
| hdmap1fval 42773 | Preliminary map from vecto... |
| hdmap1vallem 42774 | Value of preliminary map f... |
| hdmap1val 42775 | Value of preliminary map f... |
| hdmap1val0 42776 | Value of preliminary map f... |
| hdmap1val2 42777 | Value of preliminary map f... |
| hdmap1eq 42778 | The defining equation for ... |
| hdmap1cbv 42779 | Frequently used lemma to c... |
| hdmap1valc 42780 | Connect the value of the p... |
| hdmap1cl 42781 | Convert closure theorem ~ ... |
| hdmap1eq2 42782 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 42783 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 42784 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 42785 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 42786 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 42787 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 42788 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 42789 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 42790 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 42791 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 42792 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 42793 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 42794 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 42795 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 42796 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 42797 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 42798 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 42799 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 42800 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 42801 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 42802 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 42803 | Map from vectors to functi... |
| hdmapfval 42804 | Map from vectors to functi... |
| hdmapval 42805 | Value of map from vectors ... |
| hdmapfnN 42806 | Functionality of map from ... |
| hdmapcl 42807 | Closure of map from vector... |
| hdmapval2lem 42808 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 42809 | Value of map from vectors ... |
| hdmapval0 42810 | Value of map from vectors ... |
| hdmapeveclem 42811 | Lemma for ~ hdmapevec . T... |
| hdmapevec 42812 | Value of map from vectors ... |
| hdmapevec2 42813 | The inner product of the r... |
| hdmapval3lemN 42814 | Value of map from vectors ... |
| hdmapval3N 42815 | Value of map from vectors ... |
| hdmap10lem 42816 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 42817 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 42818 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 42819 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 42820 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 42821 | Part of proof of part 12 i... |
| hdmapnzcl 42822 | Nonzero vector closure of ... |
| hdmapneg 42823 | Part of proof of part 12 i... |
| hdmapsub 42824 | Part of proof of part 12 i... |
| hdmap11 42825 | Part of proof of part 12 i... |
| hdmaprnlem1N 42826 | Part of proof of part 12 i... |
| hdmaprnlem3N 42827 | Part of proof of part 12 i... |
| hdmaprnlem3uN 42828 | Part of proof of part 12 i... |
| hdmaprnlem4tN 42829 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 42830 | Part of proof of part 12 i... |
| hdmaprnlem6N 42831 | Part of proof of part 12 i... |
| hdmaprnlem7N 42832 | Part of proof of part 12 i... |
| hdmaprnlem8N 42833 | Part of proof of part 12 i... |
| hdmaprnlem9N 42834 | Part of proof of part 12 i... |
| hdmaprnlem3eN 42835 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 42836 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 42837 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 42838 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 42839 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 42840 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 42841 | Part of proof of part 12 i... |
| hdmapf1oN 42842 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 42843 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 42844 | Prior to part 14 in [Baer]... |
| hdmap14lem1 42845 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 42846 | Prior to part 14 in [Baer]... |
| hdmap14lem3 42847 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 42848 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 42849 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 42850 | Case where ` F ` is zero. ... |
| hdmap14lem7 42851 | Combine cases of ` F ` . ... |
| hdmap14lem8 42852 | Part of proof of part 14 i... |
| hdmap14lem9 42853 | Part of proof of part 14 i... |
| hdmap14lem10 42854 | Part of proof of part 14 i... |
| hdmap14lem11 42855 | Part of proof of part 14 i... |
| hdmap14lem12 42856 | Lemma for proof of part 14... |
| hdmap14lem13 42857 | Lemma for proof of part 14... |
| hdmap14lem14 42858 | Part of proof of part 14 i... |
| hdmap14lem15 42859 | Part of proof of part 14 i... |
| hgmapffval 42862 | Map from the scalar divisi... |
| hgmapfval 42863 | Map from the scalar divisi... |
| hgmapval 42864 | Value of map from the scal... |
| hgmapfnN 42865 | Functionality of scalar si... |
| hgmapcl 42866 | Closure of scalar sigma ma... |
| hgmapdcl 42867 | Closure of the vector spac... |
| hgmapvs 42868 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 42869 | Value of the scalar sigma ... |
| hgmapval1 42870 | Value of the scalar sigma ... |
| hgmapadd 42871 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 42872 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 42873 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 42874 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 42875 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 42876 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 42877 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 42878 | Part of proof of part 16 i... |
| hgmap11 42879 | The scalar sigma map is on... |
| hgmapf1oN 42880 | The scalar sigma map is a ... |
| hgmapeq0 42881 | The scalar sigma map is ze... |
| hdmapipcl 42882 | The inner product (Hermiti... |
| hdmapln1 42883 | Linearity property that wi... |
| hdmaplna1 42884 | Additive property of first... |
| hdmaplns1 42885 | Subtraction property of fi... |
| hdmaplnm1 42886 | Multiplicative property of... |
| hdmaplna2 42887 | Additive property of secon... |
| hdmapglnm2 42888 | g-linear property of secon... |
| hdmapgln2 42889 | g-linear property that wil... |
| hdmaplkr 42890 | Kernel of the vector to du... |
| hdmapellkr 42891 | Membership in the kernel (... |
| hdmapip0 42892 | Zero property that will be... |
| hdmapip1 42893 | Construct a proportional v... |
| hdmapip0com 42894 | Commutation property of Ba... |
| hdmapinvlem1 42895 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 42896 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 42897 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 42898 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 42899 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 42900 | Involution property of sca... |
| hgmapvvlem2 42901 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 42902 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 42903 | Value of a double involuti... |
| hdmapglem7a 42904 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 42905 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 42906 | Lemma for ~ hdmapg . Line... |
| hdmapg 42907 | Apply the scalar sigma fun... |
| hdmapoc 42908 | Express our constructed or... |
| hlhilset 42911 | The final Hilbert space co... |
| hlhilsca 42912 | The scalar of the final co... |
| hlhilbase 42913 | The base set of the final ... |
| hlhilplus 42914 | The vector addition for th... |
| hlhilslem 42915 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 42916 | The scalar base set of the... |
| hlhilsplus 42917 | Scalar addition for the fi... |
| hlhilsmul 42918 | Scalar multiplication for ... |
| hlhilsbase2 42919 | The scalar base set of the... |
| hlhilsplus2 42920 | Scalar addition for the fi... |
| hlhilsmul2 42921 | Scalar multiplication for ... |
| hlhils0 42922 | The scalar ring zero for t... |
| hlhils1N 42923 | The scalar ring unity for ... |
| hlhilvsca 42924 | The scalar product for the... |
| hlhilip 42925 | Inner product operation fo... |
| hlhilipval 42926 | Value of inner product ope... |
| hlhilnvl 42927 | The involution operation o... |
| hlhillvec 42928 | The final constructed Hilb... |
| hlhildrng 42929 | The star division ring for... |
| hlhilsrnglem 42930 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 42931 | The star division ring for... |
| hlhil0 42932 | The zero vector for the fi... |
| hlhillsm 42933 | The vector sum operation f... |
| hlhilocv 42934 | The orthocomplement for th... |
| hlhillcs 42935 | The closed subspaces of th... |
| hlhilphllem 42936 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 42937 | Lemma for ~ hlhil . (Cont... |
| hlathil 42938 | Construction of a Hilbert ... |
| iscsrg 42941 | A commutative semiring is ... |
| rhmzrhval 42942 | Evaluation of integers acr... |
| zndvdchrrhm 42943 | Construction of a ring hom... |
| relogbcld 42944 | Closure of the general log... |
| relogbexpd 42945 | Identity law for general l... |
| relogbzexpd 42946 | Power law for the general ... |
| logblebd 42947 | The general logarithm is m... |
| uzindd 42948 | Induction on the upper int... |
| fzadd2d 42949 | Membership of a sum in a f... |
| fzne2d 42950 | Elementhood in a finite se... |
| eqfnfv2d2 42951 | Equality of functions is d... |
| fzsplitnd 42952 | Split a finite interval of... |
| fzsplitnr 42953 | Split a finite interval of... |
| addassnni 42954 | Associative law for additi... |
| addcomnni 42955 | Commutative law for additi... |
| mulassnni 42956 | Associative law for multip... |
| mulcomnni 42957 | Commutative law for multip... |
| gcdcomnni 42958 | Commutative law for gcd. ... |
| gcdnegnni 42959 | Negation invariance for gc... |
| neggcdnni 42960 | Negation invariance for gc... |
| bccl2d 42961 | Closure of the binomial co... |
| recbothd 42962 | Take reciprocal on both si... |
| gcdmultiplei 42963 | The GCD of a multiple of a... |
| gcdaddmzz2nni 42964 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 42965 | Adding a multiple of one o... |
| gcdnncli 42966 | Closure of the gcd operato... |
| muldvds1d 42967 | If a product divides an in... |
| muldvds2d 42968 | If a product divides an in... |
| nndivdvdsd 42969 | A positive integer divides... |
| nnproddivdvdsd 42970 | A product of natural numbe... |
| coprmdvds2d 42971 | If an integer is divisible... |
| imadomfi 42972 | An image of a function und... |
| 12gcd5e1 42973 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 42974 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 42975 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 42976 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 42977 | Calculate the least common... |
| 12lcm5e60 42978 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 42979 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 42980 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 42981 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 42982 | Useful equation to calcula... |
| lcm1un 42983 | Least common multiple of n... |
| lcm2un 42984 | Least common multiple of n... |
| lcm3un 42985 | Least common multiple of n... |
| lcm4un 42986 | Least common multiple of n... |
| lcm5un 42987 | Least common multiple of n... |
| lcm6un 42988 | Least common multiple of n... |
| lcm7un 42989 | Least common multiple of n... |
| lcm8un 42990 | Least common multiple of n... |
| 3factsumint1 42991 | Move constants out of inte... |
| 3factsumint2 42992 | Move constants out of inte... |
| 3factsumint3 42993 | Move constants out of inte... |
| 3factsumint4 42994 | Move constants out of inte... |
| 3factsumint 42995 | Helpful equation for lcm i... |
| resopunitintvd 42996 | Restrict continuous functi... |
| resclunitintvd 42997 | Restrict continuous functi... |
| resdvopclptsd 42998 | Restrict derivative on uni... |
| lcmineqlem1 42999 | Part of lcm inequality lem... |
| lcmineqlem2 43000 | Part of lcm inequality lem... |
| lcmineqlem3 43001 | Part of lcm inequality lem... |
| lcmineqlem4 43002 | Part of lcm inequality lem... |
| lcmineqlem5 43003 | Technical lemma for recipr... |
| lcmineqlem6 43004 | Part of lcm inequality lem... |
| lcmineqlem7 43005 | Derivative of 1-x for chai... |
| lcmineqlem8 43006 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 43007 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 43008 | Induction step of ~ lcmine... |
| lcmineqlem11 43009 | Induction step, continuati... |
| lcmineqlem12 43010 | Base case for induction. ... |
| lcmineqlem13 43011 | Induction proof for lcm in... |
| lcmineqlem14 43012 | Technical lemma for inequa... |
| lcmineqlem15 43013 | F times the least common m... |
| lcmineqlem16 43014 | Technical divisibility lem... |
| lcmineqlem17 43015 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 43016 | Technical lemma to shift f... |
| lcmineqlem19 43017 | Dividing implies inequalit... |
| lcmineqlem20 43018 | Inequality for lcm lemma. ... |
| lcmineqlem21 43019 | The lcm inequality lemma w... |
| lcmineqlem22 43020 | The lcm inequality lemma w... |
| lcmineqlem23 43021 | Penultimate step to the lc... |
| lcmineqlem 43022 | The least common multiple ... |
| 3exp7 43023 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 43024 | First inequality in inequa... |
| 3lexlogpow5ineq2 43025 | Second inequality in inequ... |
| 3lexlogpow5ineq4 43026 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 43027 | Combined inequality chain ... |
| 3lexlogpow2ineq1 43028 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 43029 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 43030 | Result for bound in AKS in... |
| intlewftc 43031 | Inequality inference by in... |
| aks4d1lem1 43032 | Technical lemma to reduce ... |
| aks4d1p1p1 43033 | Exponential law for finite... |
| dvrelog2 43034 | The derivative of the loga... |
| dvrelog3 43035 | The derivative of the loga... |
| dvrelog2b 43036 | Derivative of the binary l... |
| 0nonelalab 43037 | Technical lemma for open i... |
| dvrelogpow2b 43038 | Derivative of the power of... |
| aks4d1p1p3 43039 | Bound of a ceiling of the ... |
| aks4d1p1p2 43040 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 43041 | Technical step for inequal... |
| dvle2 43042 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 43043 | Inequality lift to differe... |
| aks4d1p1p7 43044 | Bound of intermediary of i... |
| aks4d1p1p5 43045 | Show inequality for existe... |
| aks4d1p1 43046 | Show inequality for existe... |
| aks4d1p2 43047 | Technical lemma for existe... |
| aks4d1p3 43048 | There exists a small enoug... |
| aks4d1p4 43049 | There exists a small enoug... |
| aks4d1p5 43050 | Show that ` N ` and ` R ` ... |
| aks4d1p6 43051 | The maximal prime power ex... |
| aks4d1p7d1 43052 | Technical step in AKS lemm... |
| aks4d1p7 43053 | Technical step in AKS lemm... |
| aks4d1p8d1 43054 | If a prime divides one num... |
| aks4d1p8d2 43055 | Any prime power dividing a... |
| aks4d1p8d3 43056 | The remainder of a divisio... |
| aks4d1p8 43057 | Show that ` N ` and ` R ` ... |
| aks4d1p9 43058 | Show that the order is bou... |
| aks4d1 43059 | Lemma 4.1 from ~ https://w... |
| fldhmf1 43060 | A field homomorphism is in... |
| isprimroot 43063 | The value of a primitive r... |
| isprimroot2 43064 | Alternative way of creatin... |
| mndmolinv 43065 | An element of a monoid tha... |
| linvh 43066 | If an element has a unique... |
| primrootsunit1 43067 | Primitive roots have left ... |
| primrootsunit 43068 | Primitive roots have left ... |
| primrootscoprmpow 43069 | Coprime powers of primitiv... |
| posbezout 43070 | Bezout's identity restrict... |
| primrootscoprf 43071 | Coprime powers of primitiv... |
| primrootscoprbij 43072 | A bijection between coprim... |
| primrootscoprbij2 43073 | A bijection between coprim... |
| remexz 43074 | Division with rest. (Cont... |
| primrootlekpowne0 43075 | There is no smaller power ... |
| primrootspoweq0 43076 | The power of a ` R ` -th p... |
| aks6d1c1p1 43077 | Definition of the introspe... |
| aks6d1c1p1rcl 43078 | Reverse closure of the int... |
| aks6d1c1p2 43079 | ` P ` and linear factors a... |
| aks6d1c1p3 43080 | In a field with a Frobeniu... |
| aks6d1c1p4 43081 | The product of polynomials... |
| aks6d1c1p5 43082 | The product of exponents i... |
| aks6d1c1p7 43083 | ` X ` is introspective to ... |
| aks6d1c1p6 43084 | If a polynomials ` F ` is ... |
| aks6d1c1p8 43085 | If a number ` E ` is intro... |
| aks6d1c1 43086 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 43087 | Polynomial evaluation buil... |
| aks6d1c2p1 43088 | In the AKS-theorem the sub... |
| aks6d1c2p2 43089 | Injective condition for co... |
| hashscontpowcl 43090 | Closure of E for ~ https:/... |
| hashscontpow1 43091 | Helper lemma for to prove ... |
| hashscontpow 43092 | If a set contains all ` N ... |
| aks6d1c3 43093 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 43094 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 43095 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 43096 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 43097 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 43098 | If the number of elements ... |
| hashnexinjle 43099 | If the number of elements ... |
| aks6d1c2 43100 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 43101 | Special case related to ~ ... |
| idomnnzpownz 43102 | A nonzero power in an inte... |
| idomnnzgmulnz 43103 | A finite product of nonzer... |
| ringexp0nn 43104 | Zero to the power of a pos... |
| aks6d1c5lem0 43105 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 43106 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 43107 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 43108 | Lemma for Claim 5, contrad... |
| aks6d1c5 43109 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 43110 | Degree multiplication is a... |
| deg1pow 43111 | Exact degree of a power of... |
| 5bc2eq10 43112 | The value of 5 choose 2. ... |
| facp2 43113 | The factorial of a success... |
| 2np3bcnp1 43114 | Part of induction step for... |
| 2ap1caineq 43115 | Inequality for Theorem 6.6... |
| sticksstones1 43116 | Different strictly monoton... |
| sticksstones2 43117 | The range function on stri... |
| sticksstones3 43118 | The range function on stri... |
| sticksstones4 43119 | Equinumerosity lemma for s... |
| sticksstones5 43120 | Count the number of strict... |
| sticksstones6 43121 | Function induces an order ... |
| sticksstones7 43122 | Closure property of sticks... |
| sticksstones8 43123 | Establish mapping between ... |
| sticksstones9 43124 | Establish mapping between ... |
| sticksstones10 43125 | Establish mapping between ... |
| sticksstones11 43126 | Establish bijective mappin... |
| sticksstones12a 43127 | Establish bijective mappin... |
| sticksstones12 43128 | Establish bijective mappin... |
| sticksstones13 43129 | Establish bijective mappin... |
| sticksstones14 43130 | Sticks and stones with def... |
| sticksstones15 43131 | Sticks and stones with alm... |
| sticksstones16 43132 | Sticks and stones with col... |
| sticksstones17 43133 | Extend sticks and stones t... |
| sticksstones18 43134 | Extend sticks and stones t... |
| sticksstones19 43135 | Extend sticks and stones t... |
| sticksstones20 43136 | Lift sticks and stones to ... |
| sticksstones21 43137 | Lift sticks and stones to ... |
| sticksstones22 43138 | Non-exhaustive sticks and ... |
| sticksstones23 43139 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 43140 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 43141 | Every primitive root is ro... |
| aks6d1c6lem3 43142 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 43143 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 43144 | Lemma to construct the map... |
| aks6d1c6isolem2 43145 | Lemma to construct the gro... |
| aks6d1c6isolem3 43146 | The preimage of a map send... |
| aks6d1c6lem5 43147 | Eliminate the size hypothe... |
| bcled 43148 | Inequality for binomial co... |
| bcle2d 43149 | Inequality for binomial co... |
| aks6d1c7lem1 43150 | The last set of inequaliti... |
| aks6d1c7lem2 43151 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 43152 | Remove lots of hypotheses ... |
| aks6d1c7lem4 43153 | In the AKS algorithm there... |
| aks6d1c7 43154 | ` N ` is a prime power if ... |
| rhmqusspan 43155 | Ring homomorphism out of a... |
| aks5lem1 43156 | Section 5 of ~ https://www... |
| aks5lem2 43157 | Lemma for section 5 ~ http... |
| ply1asclzrhval 43158 | Transfer results from alge... |
| aks5lem3a 43159 | Lemma for AKS section 5. ... |
| aks5lem4a 43160 | Lemma for AKS section 5, r... |
| aks5lem5a 43161 | Lemma for AKS, section 5, ... |
| aks5lem6 43162 | Connect results of section... |
| indstrd 43163 | Strong induction, deductio... |
| grpods 43164 | Relate sums of elements of... |
| unitscyglem1 43165 | Lemma for unitscyg . (Con... |
| unitscyglem2 43166 | Lemma for unitscyg . (Con... |
| unitscyglem3 43167 | Lemma for unitscyg . (Con... |
| unitscyglem4 43168 | Lemma for unitscyg . (Con... |
| unitscyglem5 43169 | Lemma for unitscyg . (Con... |
| aks5lem7 43170 | Lemma for aks5. We clean ... |
| aks5lem8 43171 | Lemma for aks5. Clean up ... |
| exfinfldd 43173 | For any prime ` P ` and an... |
| aks5 43174 | The AKS Primality test, gi... |
| quadfac 43175 | The solution of a quadrati... |
| 25or6to4 43176 | Question 67 of 68 from a l... |
| jarrii 43177 | Inference associated with ... |
| intnanrt 43178 | Introduction of conjunct i... |
| ioin9i8 43179 | Miscellaneous inference cr... |
| jaodd 43180 | Double deduction form of ~... |
| syl3an12 43181 | A double syllogism inferen... |
| exbiii 43182 | Inference associated with ... |
| sbtd 43183 | A true statement is true u... |
| sbor2 43184 | One direction of ~ sbor , ... |
| sbalexi 43185 | Inference form of ~ sbalex... |
| nfalh 43186 | Version of ~ nfal with an ... |
| nfe2 43187 | An inner existential quant... |
| nfale2 43188 | An inner existential quant... |
| 19.9dev 43189 | ~ 19.9d in the case of an ... |
| 3rspcedvd 43190 | Triple application of ~ rs... |
| sn-axrep5v 43191 | A condensed form of ~ axre... |
| sn-axprlem3 43192 | ~ axprlem3 using only Tars... |
| sn-exelALT 43193 | Alternate proof of ~ exel ... |
| ssabdv 43194 | Deduction of abstraction s... |
| sn-iotalem 43195 | An unused lemma showing th... |
| sn-iotalemcor 43196 | Corollary of ~ sn-iotalem ... |
| abbi1sn 43197 | Originally part of ~ uniab... |
| brif2 43198 | Move a relation inside and... |
| brif12 43199 | Move a relation inside and... |
| pssexg 43200 | The proper subset of a set... |
| pssn0 43201 | A proper superset is nonem... |
| psspwb 43202 | Classes are proper subclas... |
| xppss12 43203 | Proper subset theorem for ... |
| elpwbi 43204 | Membership in a power set,... |
| imaopab 43205 | The image of a class of or... |
| eqresfnbd 43206 | Property of being the rest... |
| fmpocos 43207 | Composition of two functio... |
| ovmpogad 43208 | Value of an operation give... |
| ofun 43209 | A function operation of un... |
| dfqs3 43210 | Alternate definition of qu... |
| qseq12d 43211 | Equality theorem for quoti... |
| qsalrel 43212 | The quotient set is equal ... |
| supinf 43213 | The supremum is the infimu... |
| mapcod 43214 | Compose two mappings. (Co... |
| fisdomnn 43215 | A finite set is dominated ... |
| ltex 43216 | The less-than relation is ... |
| leex 43217 | The less-than-or-equal-to ... |
| subex 43218 | The subtraction operation ... |
| absex 43219 | The absolute value functio... |
| cjex 43220 | The conjugate function is ... |
| fzosumm1 43221 | Separate out the last term... |
| ccatcan2d 43222 | Cancellation law for conca... |
| c0exALT 43223 | Alternate proof of ~ c0ex ... |
| 0cnALT3 43224 | Alternate proof of ~ 0cn u... |
| elre0re 43225 | Specialized version of ~ 0... |
| lttrii 43226 | 'Less than' is transitive.... |
| remulcan2d 43227 | ~ mulcan2d for real number... |
| readdridaddlidd 43228 | Given some real number ` B... |
| 4p4e8ALT 43229 | A shorter proof of ~ 4p4e8... |
| 1p3e4 43230 | 1 + 3 = 4. (Contributed b... |
| 1p4e5 43231 | 1 + 4 = 5. (Contributed b... |
| 1p5e6 43232 | 1 + 5 = 6. (Contributed b... |
| 1p6e7 43233 | 1 + 6 = 7. (Contributed b... |
| 1p7e8 43234 | 1 + 7 = 8. (Contributed b... |
| 1p8e9 43235 | 1 + 8 = 9. (Contributed b... |
| 2p3e5 43236 | 2 + 3 = 5. (Contributed b... |
| 2p4e6 43237 | 2 + 4 = 6. (Contributed b... |
| 2p5e7 43238 | 2 + 5 = 7. (Contributed b... |
| 2p6e8 43239 | 2 + 6 = 8. (Contributed b... |
| 2p7e9 43240 | 2 + 7 = 9. (Contributed b... |
| 3p4e7 43241 | 3 + 4 = 7. (Contributed b... |
| 3p5e8 43242 | 3 + 5 = 8. (Contributed b... |
| 3p6e9 43243 | 3 + 6 = 9. (Contributed b... |
| 4p5e9 43244 | 4 + 5 = 9. (Contributed b... |
| 5ne0 43245 | The number 5 is nonzero. ... |
| 6ne0 43246 | The number 6 is nonzero. ... |
| 7ne0 43247 | The number 7 is nonzero. ... |
| 8ne0 43248 | The number 8 is nonzero. ... |
| 9ne0 43249 | The number 9 is nonzero. ... |
| sn-1ne2 43250 | A proof of ~ 1ne2 without ... |
| nnn1suc 43251 | A positive integer that is... |
| readdrcl2d 43252 | Reverse closure for additi... |
| mvrrsubd 43253 | Move a subtraction in the ... |
| laddrotrd 43254 | Rotate the variables right... |
| raddswap12d 43255 | Swap the first two variabl... |
| lsubrotld 43256 | Rotate the variables left ... |
| rsubrotld 43257 | Rotate the variables left ... |
| lsubswap23d 43258 | Swap the second and third ... |
| addsubeq4com 43259 | Relation between sums and ... |
| sqsumi 43260 | A sum squared. (Contribut... |
| negn0nposznnd 43261 | Lemma for ~ dffltz . (Con... |
| sqmid3api 43262 | Value of the square of the... |
| decaddcom 43263 | Commute ones place in addi... |
| sqn5i 43264 | The square of a number end... |
| sqn5ii 43265 | The square of a number end... |
| decpmulnc 43266 | Partial products algorithm... |
| decpmul 43267 | Partial products algorithm... |
| sqdeccom12 43268 | The square of a number in ... |
| sq3deccom12 43269 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 43270 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 43271 | A third of a number plus t... |
| sq4 43272 | The square of 4 is 16. (C... |
| sq5 43273 | The square of 5 is 25. (C... |
| sq6 43274 | The square of 6 is 36. (C... |
| sq7 43275 | The square of 7 is 49. (C... |
| sq8 43276 | The square of 8 is 64. (C... |
| sq9 43277 | The square of 9 is 81. (C... |
| rpsscn 43278 | The positive reals are a s... |
| 4rp 43279 | 4 is a positive real. (Co... |
| 6rp 43280 | 6 is a positive real. (Co... |
| 7rp 43281 | 7 is a positive real. (Co... |
| 8rp 43282 | 8 is a positive real. (Co... |
| 9rp 43283 | 9 is a positive real. (Co... |
| 235t711 43284 | Calculate a product by lon... |
| ex-decpmul 43285 | Example usage of ~ decpmul... |
| eluzp1 43286 | Membership in a successor ... |
| sn-eluzp1l 43287 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 43288 | The sum of ` N ` constant ... |
| fz1sump1 43289 | Add one more term to a sum... |
| oddnumth 43290 | The Odd Number Theorem. T... |
| nicomachus 43291 | Nicomachus's Theorem. The... |
| sumcubes 43292 | The sum of the first ` N `... |
| ine1 43293 | ` _i ` is not 1. (Contrib... |
| 0tie0 43294 | 0 times ` _i ` equals 0. ... |
| it1ei 43295 | ` _i ` times 1 equals ` _i... |
| 1tiei 43296 | 1 times ` _i ` equals ` _i... |
| itrere 43297 | ` _i ` times a real is rea... |
| retire 43298 | A real times ` _i ` is rea... |
| iocioodisjd 43299 | Adjacent intervals where t... |
| rpabsid 43300 | A positive real is its own... |
| oexpreposd 43301 | Lemma for ~ dffltz . For ... |
| explt1d 43302 | A nonnegative real number ... |
| expeq1d 43303 | A nonnegative real number ... |
| expeqidd 43304 | A nonnegative real number ... |
| exp11d 43305 | ~ exp11nnd for nonzero int... |
| 0dvds0 43306 | 0 divides 0. (Contributed... |
| absdvdsabsb 43307 | Divisibility is invariant ... |
| gcdnn0id 43308 | The ` gcd ` of a nonnegati... |
| gcdle1d 43309 | The greatest common diviso... |
| gcdle2d 43310 | The greatest common diviso... |
| dvdsexpad 43311 | Deduction associated with ... |
| dvdsexpnn 43312 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 43313 | ~ dvdsexpnn generalized to... |
| dvdsexpb 43314 | ~ dvdssq generalized to po... |
| posqsqznn 43315 | When a positive rational s... |
| zdivgd 43316 | Two ways to express " ` N ... |
| efsubd 43317 | Difference of exponents la... |
| ef11d 43318 | General condition for the ... |
| logccne0d 43319 | The logarithm isn't 0 if i... |
| cxp112d 43320 | General condition for comp... |
| cxp111d 43321 | General condition for comp... |
| cxpi11d 43322 | ` _i ` to the powers of ` ... |
| logne0d 43323 | Deduction form of ~ logne0... |
| rxp112d 43324 | Real exponentiation is one... |
| log11d 43325 | The natural logarithm is o... |
| rplog11d 43326 | The natural logarithm is o... |
| rxp11d 43327 | Real exponentiation is one... |
| tanhalfpim 43328 | The tangent of ` _pi / 2 `... |
| sinpim 43329 | Sine of a number subtracte... |
| cospim 43330 | Cosine of a number subtrac... |
| tan3rdpi 43331 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 43332 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 43333 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 43334 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 43335 | The cosine of ` 4 x. ( _pi... |
| asin1half 43336 | The arcsine of ` 1 / 2 ` i... |
| acos1half 43337 | The arccosine of ` 1 / 2 `... |
| dvun 43338 | Condition for the union of... |
| redvmptabs 43339 | The derivative of the abso... |
| readvrec2 43340 | The antiderivative of 1/x ... |
| readvrec 43341 | For real numbers, the anti... |
| resuppsinopn 43342 | The support of sin ( ~ df-... |
| readvcot 43343 | Real antiderivative of cot... |
| resubval 43346 | Value of real subtraction,... |
| renegeulemv 43347 | Lemma for ~ renegeu and si... |
| renegeulem 43348 | Lemma for ~ renegeu and si... |
| renegeu 43349 | Existential uniqueness of ... |
| rernegcl 43350 | Closure law for negative r... |
| renegadd 43351 | Relationship between real ... |
| renegid 43352 | Addition of a real number ... |
| reneg0addlid 43353 | Negative zero is a left ad... |
| resubeulem1 43354 | Lemma for ~ resubeu . A v... |
| resubeulem2 43355 | Lemma for ~ resubeu . A v... |
| resubeu 43356 | Existential uniqueness of ... |
| rersubcl 43357 | Closure for real subtracti... |
| resubadd 43358 | Relation between real subt... |
| resubaddd 43359 | Relationship between subtr... |
| resubf 43360 | Real subtraction is an ope... |
| repncan2 43361 | Addition and subtraction o... |
| repncan3 43362 | Addition and subtraction o... |
| readdsub 43363 | Law for addition and subtr... |
| reladdrsub 43364 | Move LHS of a sum into RHS... |
| reltsub1 43365 | Subtraction from both side... |
| reltsubadd2 43366 | 'Less than' relationship b... |
| resubcan2 43367 | Cancellation law for real ... |
| resubsub4 43368 | Law for double subtraction... |
| rennncan2 43369 | Cancellation law for real ... |
| renpncan3 43370 | Cancellation law for real ... |
| repnpcan 43371 | Cancellation law for addit... |
| reppncan 43372 | Cancellation law for mixed... |
| resubidaddlidlem 43373 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 43374 | Any real number subtracted... |
| resubdi 43375 | Distribution of multiplica... |
| re1m1e0m0 43376 | Equality of two left-addit... |
| sn-00idlem1 43377 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 43378 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 43379 | Lemma for ~ sn-00id . (Co... |
| sn-00id 43380 | ~ 00id proven without ~ ax... |
| re0m0e0 43381 | Real number version of ~ 0... |
| readdlid 43382 | Real number version of ~ a... |
| sn-addlid 43383 | ~ addlid without ~ ax-mulc... |
| remul02 43384 | Real number version of ~ m... |
| sn-0ne2 43385 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 43386 | Real number version of ~ m... |
| sn-remul0ord 43387 | A product is zero iff one ... |
| resubid 43388 | Subtraction of a real numb... |
| readdrid 43389 | Real number version of ~ a... |
| resubid1 43390 | Real number version of ~ s... |
| renegneg 43391 | A real number is equal to ... |
| readdcan2 43392 | Commuted version of ~ read... |
| renegid2 43393 | Commuted version of ~ rene... |
| remulneg2d 43394 | Product with negative is n... |
| sn-it0e0 43395 | Proof of ~ it0e0 without ~... |
| sn-negex12 43396 | A combination of ~ cnegex ... |
| sn-negex 43397 | Proof of ~ cnegex without ... |
| sn-negex2 43398 | Proof of ~ cnegex2 without... |
| sn-addcand 43399 | ~ addcand without ~ ax-mul... |
| sn-addrid 43400 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 43401 | ~ addcan2d without ~ ax-mu... |
| reixi 43402 | ~ ixi without ~ ax-mulcom ... |
| rei4 43403 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 43404 | A number that sums to itse... |
| sn-mul01 43405 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 43406 | ~ negeu without ~ ax-mulco... |
| sn-subcl 43407 | ~ subcl without ~ ax-mulco... |
| sn-subf 43408 | ~ subf without ~ ax-mulcom... |
| resubeqsub 43409 | Equivalence between real s... |
| subresre 43410 | Subtraction restricted to ... |
| addinvcom 43411 | A number commutes with its... |
| remulinvcom 43412 | A left multiplicative inve... |
| remullid 43413 | Commuted version of ~ ax-1... |
| sn-1ticom 43414 | Lemma for ~ sn-mullid and ... |
| sn-mullid 43415 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 43416 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 43417 | The multiplicative inverse... |
| remulcand 43418 | Commuted version of ~ remu... |
| redivvald 43421 | Value of real division, wh... |
| rediveud 43422 | Existential uniqueness of ... |
| sn-redivcld 43423 | Closure law for real divis... |
| redivmuld 43424 | Relationship between divis... |
| redivmul2d 43425 | Relationship between divis... |
| redivcan2d 43426 | A cancellation law for div... |
| redivcan3d 43427 | A cancellation law for div... |
| rediveq0d 43428 | A ratio is zero iff the nu... |
| redivne0bd 43429 | The ratio of nonzero numbe... |
| rediveq1d 43430 | Equality in terms of unit ... |
| sn-rediv1d 43431 | A number divided by 1 is i... |
| sn-rediv0d 43432 | Division into zero is zero... |
| sn-redividd 43433 | A number divided by itself... |
| sn-rereccld 43434 | Closure law for reciprocal... |
| rerecne0d 43435 | The reciprocal of a nonzer... |
| rerecidd 43436 | Multiplication of a number... |
| rerecid2d 43437 | Multiplication of a number... |
| rerecrecd 43438 | A number is equal to the r... |
| redivrec2d 43439 | Relationship between divis... |
| rediv23d 43440 | A "commutative"/associativ... |
| redivdird 43441 | Distribution of division o... |
| rediv11d 43442 | One-to-one relationship fo... |
| sn-0tie0 43443 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 43444 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 43445 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 43446 | ~ ltaddneg without ~ ax-mu... |
| reposdif 43447 | Comparison of two numbers ... |
| relt0neg1 43448 | Comparison of a real and i... |
| relt0neg2 43449 | Comparison of a real and i... |
| sn-addlt0d 43450 | The sum of negative number... |
| sn-addgt0d 43451 | The sum of positive number... |
| sn-nnne0 43452 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 43453 | ~ elznn0nn restated using ... |
| nn0addcom 43454 | Addition is commutative fo... |
| zaddcomlem 43455 | Lemma for ~ zaddcom . (Co... |
| zaddcom 43456 | Addition is commutative fo... |
| renegmulnnass 43457 | Move multiplication by a n... |
| nn0mulcom 43458 | Multiplication is commutat... |
| zmulcomlem 43459 | Lemma for ~ zmulcom . (Co... |
| zmulcom 43460 | Multiplication is commutat... |
| mulgt0con1dlem 43461 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 43462 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 43463 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 43464 | Biconditional, deductive f... |
| sn-ltmul2d 43465 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 43466 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 43467 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 43468 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 43469 | The reciprocal of a positi... |
| mulgt0b2d 43470 | Biconditional, deductive f... |
| sn-mulgt1d 43471 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 43472 | Negative one is a negative... |
| sn-reclt0d 43473 | The reciprocal of a negati... |
| mulltgt0d 43474 | Negative times positive is... |
| mullt0b1d 43475 | When the first term is neg... |
| mullt0b2d 43476 | When the second term is ne... |
| sn-mullt0d 43477 | The product of two negativ... |
| sn-msqgt0d 43478 | A nonzero square is positi... |
| sn-inelr 43479 | ~ inelr without ~ ax-mulco... |
| sn-itrere 43480 | ` _i ` times a real is rea... |
| sn-retire 43481 | Commuted version of ~ sn-i... |
| cnreeu 43482 | The reals in the expressio... |
| sn-sup2 43483 | ~ sup2 with exactly the sa... |
| sn-sup3d 43484 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 43485 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 43486 | ~ suprubd without ~ ax-mul... |
| sn-base0 43487 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 43488 | The inverse of a non-subgr... |
| nelsubgcld 43489 | A non-subgroup-member plus... |
| nelsubgsubcld 43490 | A non-subgroup-member minu... |
| rnasclg 43491 | The set of injected scalar... |
| frlmfielbas 43492 | The vectors of a finite fr... |
| frlmfzwrd 43493 | A vector of a module with ... |
| frlmfzowrd 43494 | A vector of a module with ... |
| frlmfzolen 43495 | The dimension of a vector ... |
| frlmfzowrdb 43496 | The vectors of a module wi... |
| frlmfzoccat 43497 | The concatenation of two v... |
| frlmvscadiccat 43498 | Scalar multiplication dist... |
| grpasscan2d 43499 | An associative cancellatio... |
| grpcominv1 43500 | If two elements commute, t... |
| grpcominv2 43501 | If two elements commute, t... |
| finsubmsubg 43502 | A submonoid of a finite gr... |
| opprmndb 43503 | A class is a monoid if and... |
| opprgrpb 43504 | A class is a group if and ... |
| opprablb 43505 | A class is an Abelian grou... |
| imacrhmcl 43506 | The image of a commutative... |
| riccrng1 43507 | Ring isomorphism preserves... |
| riccrng 43508 | A ring is commutative if a... |
| domnexpgn0cl 43509 | In a domain, a (nonnegativ... |
| drnginvrn0d 43510 | A multiplicative inverse i... |
| drngmullcan 43511 | Cancellation of a nonzero ... |
| drngmulrcan 43512 | Cancellation of a nonzero ... |
| drnginvmuld 43513 | Inverse of a nonzero produ... |
| ricdrng1 43514 | A ring isomorphism maps a ... |
| ricdrng 43515 | A ring is a division ring ... |
| ricfld 43516 | A ring is a field if and o... |
| asclf1 43517 | Two ways of saying the sca... |
| abvexp 43518 | Move exponentiation in and... |
| fimgmcyclem 43519 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 43520 | Version of ~ odcl2 for fin... |
| fidomncyc 43521 | Version of ~ odcl2 for mul... |
| fiabv 43522 | In a finite domain (a fini... |
| lvecgrp 43523 | A vector space is a group.... |
| lvecring 43524 | The scalar component of a ... |
| frlm0vald 43525 | All coordinates of the zer... |
| frlmsnic 43526 | Given a free module with a... |
| uvccl 43527 | A unit vector is a vector.... |
| uvcn0 43528 | A unit vector is nonzero. ... |
| psrmnd 43529 | The ring of power series i... |
| mhmcopsr 43530 | The composition of a monoi... |
| mhmcoaddpsr 43531 | Show that the ring homomor... |
| rhmcomulpsr 43532 | Show that the ring homomor... |
| rhmpsr 43533 | Provide a ring homomorphis... |
| rhmpsr1 43534 | Provide a ring homomorphis... |
| evl0 43535 | The zero polynomial evalua... |
| evlsbagval 43536 | Polynomial evaluation buil... |
| evlvvvallem 43537 | Lemma for theorems using ~... |
| evlselvlem 43538 | Lemma for ~ evlselv . Use... |
| evlselv 43539 | Evaluating a selection of ... |
| fsuppind 43540 | Induction on functions ` F... |
| fsuppssindlem1 43541 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 43542 | Lemma for ~ fsuppssind . ... |
| fsuppssind 43543 | Induction on functions ` F... |
| mhpind 43544 | The homogeneous polynomial... |
| evlsmhpvvval 43545 | Give a formula for the eva... |
| mhphflem 43546 | Lemma for ~ mhphf . Add s... |
| mhphf 43547 | A homogeneous polynomial d... |
| mhphf2 43548 | A homogeneous polynomial d... |
| mhphf3 43549 | A homogeneous polynomial d... |
| mhphf4 43550 | A homogeneous polynomial d... |
| prjspval 43553 | Value of the projective sp... |
| prjsprel 43554 | Utility theorem regarding ... |
| prjspertr 43555 | The relation in ` PrjSp ` ... |
| prjsperref 43556 | The relation in ` PrjSp ` ... |
| prjspersym 43557 | The relation in ` PrjSp ` ... |
| prjsper 43558 | The relation used to defin... |
| prjspreln0 43559 | Two nonzero vectors are eq... |
| prjspvs 43560 | A nonzero multiple of a ve... |
| prjsprellsp 43561 | Two vectors are equivalent... |
| prjspeclsp 43562 | The vectors equivalent to ... |
| prjspval2 43563 | Alternate definition of pr... |
| prjspnval 43566 | Value of the n-dimensional... |
| prjspnerlem 43567 | A lemma showing that the e... |
| prjspnval2 43568 | Value of the n-dimensional... |
| prjspner 43569 | The relation used to defin... |
| prjspnvs 43570 | A nonzero multiple of a ve... |
| prjspnssbas 43571 | A projective point spans a... |
| prjspnn0 43572 | A projective point is none... |
| 0prjspnlem 43573 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 43574 | Any vector is equivalent t... |
| prjspner01 43575 | Any vector is equivalent t... |
| prjspner1 43576 | Two vectors whose zeroth c... |
| 0prjspnrel 43577 | In the zero-dimensional pr... |
| 0prjspn 43578 | A zero-dimensional project... |
| prjcrvfval 43581 | Value of the projective cu... |
| prjcrvval 43582 | Value of the projective cu... |
| prjcrv0 43583 | The "curve" (zero set) cor... |
| dffltz 43584 | Fermat's Last Theorem (FLT... |
| fltmul 43585 | A counterexample to FLT st... |
| fltdiv 43586 | A counterexample to FLT st... |
| flt0 43587 | A counterexample for FLT d... |
| fltdvdsabdvdsc 43588 | Any factor of both ` A ` a... |
| fltabcoprmex 43589 | A counterexample to FLT im... |
| fltaccoprm 43590 | A counterexample to FLT wi... |
| fltbccoprm 43591 | A counterexample to FLT wi... |
| fltabcoprm 43592 | A counterexample to FLT wi... |
| infdesc 43593 | Infinite descent. The hyp... |
| fltne 43594 | If a counterexample to FLT... |
| flt4lem 43595 | Raising a number to the fo... |
| flt4lem1 43596 | Satisfy the antecedent use... |
| flt4lem2 43597 | If ` A ` is even, ` B ` is... |
| flt4lem3 43598 | Equivalent to ~ pythagtrip... |
| flt4lem4 43599 | If the product of two copr... |
| flt4lem5 43600 | In the context of the lemm... |
| flt4lem5elem 43601 | Version of ~ fltaccoprm an... |
| flt4lem5a 43602 | Part 1 of Equation 1 of ... |
| flt4lem5b 43603 | Part 2 of Equation 1 of ... |
| flt4lem5c 43604 | Part 2 of Equation 2 of ... |
| flt4lem5d 43605 | Part 3 of Equation 2 of ... |
| flt4lem5e 43606 | Satisfy the hypotheses of ... |
| flt4lem5f 43607 | Final equation of ~... |
| flt4lem6 43608 | Remove shared factors in a... |
| flt4lem7 43609 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 43610 | Strengthening of Fermat's ... |
| fltltc 43611 | ` ( C ^ N ) ` is the large... |
| fltnltalem 43612 | Lemma for ~ fltnlta . A l... |
| fltnlta 43613 | In a Fermat counterexample... |
| iddii 43614 | Version of ~ a1ii with the... |
| bicomdALT 43615 | Alternate proof of ~ bicom... |
| alan 43616 | Alias for ~ 19.26 for easi... |
| exor 43617 | Alias for ~ 19.43 for easi... |
| rexor 43618 | Alias for ~ r19.43 for eas... |
| ruvALT 43619 | Alternate proof of ~ ruv w... |
| sn-wcdeq 43620 | Alternative to ~ wcdeq and... |
| sq45 43621 | 45 squared is 2025. (Cont... |
| sum9cubes 43622 | The sum of the first nine ... |
| sn-isghm 43623 | Longer proof of ~ isghm , ... |
| aprilfools2025 43624 | An abuse of notation. (Co... |
| nfa1w 43625 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 43626 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 43627 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 43628 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 43629 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 43630 | Cube of sum of three numbe... |
| negexpidd 43631 | The sum of a real number t... |
| rexlimdv3d 43632 | An extended version of ~ r... |
| 3cubeslem1 43633 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 43634 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 43635 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 43636 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 43637 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 43638 | Lemma for ~ 3cubes . This... |
| 3cubes 43639 | Every rational number is a... |
| rntrclfvOAI 43640 | The range of the transitiv... |
| moxfr 43641 | Transfer at-most-one betwe... |
| imaiinfv 43642 | Indexed intersection of an... |
| elrfi 43643 | Elementhood in a set of re... |
| elrfirn 43644 | Elementhood in a set of re... |
| elrfirn2 43645 | Elementhood in a set of re... |
| cmpfiiin 43646 | In a compact topology, a s... |
| ismrcd1 43647 | Any function from the subs... |
| ismrcd2 43648 | Second half of ~ ismrcd1 .... |
| istopclsd 43649 | A closure function which s... |
| ismrc 43650 | A function is a Moore clos... |
| isnacs 43653 | Expand definition of Noeth... |
| nacsfg 43654 | In a Noetherian-type closu... |
| isnacs2 43655 | Express Noetherian-type cl... |
| mrefg2 43656 | Slight variation on finite... |
| mrefg3 43657 | Slight variation on finite... |
| nacsacs 43658 | A closure system of Noethe... |
| isnacs3 43659 | A choice-free order equiva... |
| incssnn0 43660 | Transitivity induction of ... |
| nacsfix 43661 | An increasing sequence of ... |
| constmap 43662 | A constant (represented wi... |
| mapco2g 43663 | Renaming indices in a tupl... |
| mapco2 43664 | Post-composition (renaming... |
| mapfzcons 43665 | Extending a one-based mapp... |
| mapfzcons1 43666 | Recover prefix mapping fro... |
| mapfzcons1cl 43667 | A nonempty mapping has a p... |
| mapfzcons2 43668 | Recover added element from... |
| mptfcl 43669 | Interpret range of a maps-... |
| mzpclval 43674 | Substitution lemma for ` m... |
| elmzpcl 43675 | Double substitution lemma ... |
| mzpclall 43676 | The set of all functions w... |
| mzpcln0 43677 | Corollary of ~ mzpclall : ... |
| mzpcl1 43678 | Defining property 1 of a p... |
| mzpcl2 43679 | Defining property 2 of a p... |
| mzpcl34 43680 | Defining properties 3 and ... |
| mzpval 43681 | Value of the ` mzPoly ` fu... |
| dmmzp 43682 | ` mzPoly ` is defined for ... |
| mzpincl 43683 | Polynomial closedness is a... |
| mzpconst 43684 | Constant functions are pol... |
| mzpf 43685 | A polynomial function is a... |
| mzpproj 43686 | A projection function is p... |
| mzpadd 43687 | The pointwise sum of two p... |
| mzpmul 43688 | The pointwise product of t... |
| mzpconstmpt 43689 | A constant function expres... |
| mzpaddmpt 43690 | Sum of polynomial function... |
| mzpmulmpt 43691 | Product of polynomial func... |
| mzpsubmpt 43692 | The difference of two poly... |
| mzpnegmpt 43693 | Negation of a polynomial f... |
| mzpexpmpt 43694 | Raise a polynomial functio... |
| mzpindd 43695 | "Structural" induction to ... |
| mzpmfp 43696 | Relationship between multi... |
| mzpsubst 43697 | Substituting polynomials f... |
| mzprename 43698 | Simplified version of ~ mz... |
| mzpresrename 43699 | A polynomial is a polynomi... |
| mzpcompact2lem 43700 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 43701 | Polynomials are finitary o... |
| coeq0i 43702 | ~ coeq0 but without explic... |
| fzsplit1nn0 43703 | Split a finite 1-based set... |
| eldiophb 43706 | Initial expression of Diop... |
| eldioph 43707 | Condition for a set to be ... |
| diophrw 43708 | Renaming and adding unused... |
| eldioph2lem1 43709 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 43710 | Lemma for ~ eldioph2 . Co... |
| eldioph2 43711 | Construct a Diophantine se... |
| eldioph2b 43712 | While Diophantine sets wer... |
| eldiophelnn0 43713 | Remove antecedent on ` B `... |
| eldioph3b 43714 | Define Diophantine sets in... |
| eldioph3 43715 | Inference version of ~ eld... |
| ellz1 43716 | Membership in a lower set ... |
| lzunuz 43717 | The union of a lower set o... |
| fz1eqin 43718 | Express a one-based finite... |
| lzenom 43719 | Lower integers are countab... |
| elmapresaunres2 43720 | ~ fresaunres2 transposed t... |
| diophin 43721 | If two sets are Diophantin... |
| diophun 43722 | If two sets are Diophantin... |
| eldiophss 43723 | Diophantine sets are sets ... |
| diophrex 43724 | Projecting a Diophantine s... |
| eq0rabdioph 43725 | This is the first of a num... |
| eqrabdioph 43726 | Diophantine set builder fo... |
| 0dioph 43727 | The empty set is Diophanti... |
| vdioph 43728 | The "universal" set (as la... |
| anrabdioph 43729 | Diophantine set builder fo... |
| orrabdioph 43730 | Diophantine set builder fo... |
| 3anrabdioph 43731 | Diophantine set builder fo... |
| 3orrabdioph 43732 | Diophantine set builder fo... |
| 2sbcrex 43733 | Exchange an existential qu... |
| sbc2rex 43734 | Exchange a substitution wi... |
| sbc4rex 43735 | Exchange a substitution wi... |
| sbcrot3 43736 | Rotate a sequence of three... |
| sbcrot5 43737 | Rotate a sequence of five ... |
| sbccomieg 43738 | Commute two explicit subst... |
| rexrabdioph 43739 | Diophantine set builder fo... |
| rexfrabdioph 43740 | Diophantine set builder fo... |
| 2rexfrabdioph 43741 | Diophantine set builder fo... |
| 3rexfrabdioph 43742 | Diophantine set builder fo... |
| 4rexfrabdioph 43743 | Diophantine set builder fo... |
| 6rexfrabdioph 43744 | Diophantine set builder fo... |
| 7rexfrabdioph 43745 | Diophantine set builder fo... |
| rabdiophlem1 43746 | Lemma for arithmetic dioph... |
| rabdiophlem2 43747 | Lemma for arithmetic dioph... |
| elnn0rabdioph 43748 | Diophantine set builder fo... |
| rexzrexnn0 43749 | Rewrite an existential qua... |
| lerabdioph 43750 | Diophantine set builder fo... |
| eluzrabdioph 43751 | Diophantine set builder fo... |
| elnnrabdioph 43752 | Diophantine set builder fo... |
| ltrabdioph 43753 | Diophantine set builder fo... |
| nerabdioph 43754 | Diophantine set builder fo... |
| dvdsrabdioph 43755 | Divisibility is a Diophant... |
| eldioph4b 43756 | Membership in ` Dioph ` ex... |
| eldioph4i 43757 | Forward-only version of ~ ... |
| diophren 43758 | Change variables in a Diop... |
| rabrenfdioph 43759 | Change variable numbers in... |
| rabren3dioph 43760 | Change variable numbers in... |
| fphpd 43761 | Pigeonhole principle expre... |
| fphpdo 43762 | Pigeonhole principle for s... |
| ctbnfien 43763 | An infinite subset of a co... |
| fiphp3d 43764 | Infinite pigeonhole princi... |
| rencldnfilem 43765 | Lemma for ~ rencldnfi . (... |
| rencldnfi 43766 | A set of real numbers whic... |
| irrapxlem1 43767 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 43768 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 43769 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 43770 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 43771 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 43772 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 43773 | Dirichlet's approximation ... |
| pellexlem1 43774 | Lemma for ~ pellex . Arit... |
| pellexlem2 43775 | Lemma for ~ pellex . Arit... |
| pellexlem3 43776 | Lemma for ~ pellex . To e... |
| pellexlem4 43777 | Lemma for ~ pellex . Invo... |
| pellexlem5 43778 | Lemma for ~ pellex . Invo... |
| pellexlem6 43779 | Lemma for ~ pellex . Doin... |
| pellex 43780 | Every Pell equation has a ... |
| pell1qrval 43791 | Value of the set of first-... |
| elpell1qr 43792 | Membership in a first-quad... |
| pell14qrval 43793 | Value of the set of positi... |
| elpell14qr 43794 | Membership in the set of p... |
| pell1234qrval 43795 | Value of the set of genera... |
| elpell1234qr 43796 | Membership in the set of g... |
| pell1234qrre 43797 | General Pell solutions are... |
| pell1234qrne0 43798 | No solution to a Pell equa... |
| pell1234qrreccl 43799 | General solutions of the P... |
| pell1234qrmulcl 43800 | General solutions of the P... |
| pell14qrss1234 43801 | A positive Pell solution i... |
| pell14qrre 43802 | A positive Pell solution i... |
| pell14qrne0 43803 | A positive Pell solution i... |
| pell14qrgt0 43804 | A positive Pell solution i... |
| pell14qrrp 43805 | A positive Pell solution i... |
| pell1234qrdich 43806 | A general Pell solution is... |
| elpell14qr2 43807 | A number is a positive Pel... |
| pell14qrmulcl 43808 | Positive Pell solutions ar... |
| pell14qrreccl 43809 | Positive Pell solutions ar... |
| pell14qrdivcl 43810 | Positive Pell solutions ar... |
| pell14qrexpclnn0 43811 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 43812 | Positive Pell solutions ar... |
| pell1qrss14 43813 | First-quadrant Pell soluti... |
| pell14qrdich 43814 | A positive Pell solution i... |
| pell1qrge1 43815 | A Pell solution in the fir... |
| pell1qr1 43816 | 1 is a Pell solution and i... |
| elpell1qr2 43817 | The first quadrant solutio... |
| pell1qrgaplem 43818 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 43819 | First-quadrant Pell soluti... |
| pell14qrgap 43820 | Positive Pell solutions ar... |
| pell14qrgapw 43821 | Positive Pell solutions ar... |
| pellqrexplicit 43822 | Condition for a calculated... |
| infmrgelbi 43823 | Any lower bound of a nonem... |
| pellqrex 43824 | There is a nontrivial solu... |
| pellfundval 43825 | Value of the fundamental s... |
| pellfundre 43826 | The fundamental solution o... |
| pellfundge 43827 | Lower bound on the fundame... |
| pellfundgt1 43828 | Weak lower bound on the Pe... |
| pellfundlb 43829 | A nontrivial first quadran... |
| pellfundglb 43830 | If a real is larger than t... |
| pellfundex 43831 | The fundamental solution a... |
| pellfund14gap 43832 | There are no solutions bet... |
| pellfundrp 43833 | The fundamental Pell solut... |
| pellfundne1 43834 | The fundamental Pell solut... |
| reglogcl 43835 | General logarithm is a rea... |
| reglogltb 43836 | General logarithm preserve... |
| reglogleb 43837 | General logarithm preserve... |
| reglogmul 43838 | Multiplication law for gen... |
| reglogexp 43839 | Power law for general log.... |
| reglogbas 43840 | General log of the base is... |
| reglog1 43841 | General log of 1 is 0. (C... |
| reglogexpbas 43842 | General log of a power of ... |
| pellfund14 43843 | Every positive Pell soluti... |
| pellfund14b 43844 | The positive Pell solution... |
| rmxfval 43849 | Value of the X sequence. ... |
| rmyfval 43850 | Value of the Y sequence. ... |
| rmspecsqrtnq 43851 | The discriminant used to d... |
| rmspecnonsq 43852 | The discriminant used to d... |
| qirropth 43853 | This lemma implements the ... |
| rmspecfund 43854 | The base of exponent used ... |
| rmxyelqirr 43855 | The solutions used to cons... |
| rmxypairf1o 43856 | The function used to extra... |
| rmxyelxp 43857 | Lemma for ~ frmx and ~ frm... |
| frmx 43858 | The X sequence is a nonneg... |
| frmy 43859 | The Y sequence is an integ... |
| rmxyval 43860 | Main definition of the X a... |
| rmspecpos 43861 | The discriminant used to d... |
| rmxycomplete 43862 | The X and Y sequences take... |
| rmxynorm 43863 | The X and Y sequences defi... |
| rmbaserp 43864 | The base of exponentiation... |
| rmxyneg 43865 | Negation law for X and Y s... |
| rmxyadd 43866 | Addition formula for X and... |
| rmxy1 43867 | Value of the X and Y seque... |
| rmxy0 43868 | Value of the X and Y seque... |
| rmxneg 43869 | Negation law (even functio... |
| rmx0 43870 | Value of X sequence at 0. ... |
| rmx1 43871 | Value of X sequence at 1. ... |
| rmxadd 43872 | Addition formula for X seq... |
| rmyneg 43873 | Negation formula for Y seq... |
| rmy0 43874 | Value of Y sequence at 0. ... |
| rmy1 43875 | Value of Y sequence at 1. ... |
| rmyadd 43876 | Addition formula for Y seq... |
| rmxp1 43877 | Special addition-of-1 form... |
| rmyp1 43878 | Special addition of 1 form... |
| rmxm1 43879 | Subtraction of 1 formula f... |
| rmym1 43880 | Subtraction of 1 formula f... |
| rmxluc 43881 | The X sequence is a Lucas ... |
| rmyluc 43882 | The Y sequence is a Lucas ... |
| rmyluc2 43883 | Lucas sequence property of... |
| rmxdbl 43884 | "Double-angle formula" for... |
| rmydbl 43885 | "Double-angle formula" for... |
| monotuz 43886 | A function defined on an u... |
| monotoddzzfi 43887 | A function which is odd an... |
| monotoddzz 43888 | A function (given implicit... |
| oddcomabszz 43889 | An odd function which take... |
| 2nn0ind 43890 | Induction on nonnegative i... |
| zindbi 43891 | Inductively transfer a pro... |
| rmxypos 43892 | For all nonnegative indice... |
| ltrmynn0 43893 | The Y-sequence is strictly... |
| ltrmxnn0 43894 | The X-sequence is strictly... |
| lermxnn0 43895 | The X-sequence is monotoni... |
| rmxnn 43896 | The X-sequence is defined ... |
| ltrmy 43897 | The Y-sequence is strictly... |
| rmyeq0 43898 | Y is zero only at zero. (... |
| rmyeq 43899 | Y is one-to-one. (Contrib... |
| lermy 43900 | Y is monotonic (non-strict... |
| rmynn 43901 | ` rmY ` is positive for po... |
| rmynn0 43902 | ` rmY ` is nonnegative for... |
| rmyabs 43903 | ` rmY ` commutes with ` ab... |
| jm2.24nn 43904 | X(n) is strictly greater t... |
| jm2.17a 43905 | First half of lemma 2.17 o... |
| jm2.17b 43906 | Weak form of the second ha... |
| jm2.17c 43907 | Second half of lemma 2.17 ... |
| jm2.24 43908 | Lemma 2.24 of [JonesMatija... |
| rmygeid 43909 | Y(n) increases faster than... |
| congtr 43910 | A wff of the form ` A || (... |
| congadd 43911 | If two pairs of numbers ar... |
| congmul 43912 | If two pairs of numbers ar... |
| congsym 43913 | Congruence mod ` A ` is a ... |
| congneg 43914 | If two integers are congru... |
| congsub 43915 | If two pairs of numbers ar... |
| congid 43916 | Every integer is congruent... |
| mzpcong 43917 | Polynomials commute with c... |
| congrep 43918 | Every integer is congruent... |
| congabseq 43919 | If two integers are congru... |
| acongid 43920 | A wff like that in this th... |
| acongsym 43921 | Symmetry of alternating co... |
| acongneg2 43922 | Negate right side of alter... |
| acongtr 43923 | Transitivity of alternatin... |
| acongeq12d 43924 | Substitution deduction for... |
| acongrep 43925 | Every integer is alternati... |
| fzmaxdif 43926 | Bound on the difference be... |
| fzneg 43927 | Reflection of a finite ran... |
| acongeq 43928 | Two numbers in the fundame... |
| dvdsacongtr 43929 | Alternating congruence pas... |
| coprmdvdsb 43930 | Multiplication by a coprim... |
| modabsdifz 43931 | Divisibility in terms of m... |
| dvdsabsmod0 43932 | Divisibility in terms of m... |
| jm2.18 43933 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 43934 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 43935 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 43936 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 43937 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 43938 | Lemma 2.19 of [JonesMatija... |
| jm2.21 43939 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 43940 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 43941 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 43942 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 43943 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 43944 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 43945 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 43946 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 43947 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 43948 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 43949 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 43950 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 43951 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 43952 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 43953 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 43954 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 43955 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 43956 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 43957 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 43958 | Lemma for ~ rmydioph . Us... |
| rmydioph 43959 | ~ jm2.27 restated in terms... |
| rmxdiophlem 43960 | X can be expressed in term... |
| rmxdioph 43961 | X is a Diophantine functio... |
| jm3.1lem1 43962 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 43963 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 43964 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 43965 | Diophantine expression for... |
| expdiophlem1 43966 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 43967 | Lemma for ~ expdioph . Ex... |
| expdioph 43968 | The exponential function i... |
| setindtr 43969 | Set induction for sets con... |
| setindtrs 43970 | Set induction scheme witho... |
| dford3lem1 43971 | Lemma for ~ dford3 . (Con... |
| dford3lem2 43972 | Lemma for ~ dford3 . (Con... |
| dford3 43973 | Ordinals are precisely the... |
| dford4 43974 | ~ dford3 expressed in prim... |
| wopprc 43975 | Unrelated: Wiener pairs t... |
| rpnnen3lem 43976 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 43977 | Dedekind cut injection of ... |
| axac10 43978 | Characterization of choice... |
| harinf 43979 | The Hartogs number of an i... |
| wdom2d2 43980 | Deduction for weak dominan... |
| ttac 43981 | Tarski's theorem about cho... |
| pw2f1ocnv 43982 | Define a bijection between... |
| pw2f1o2 43983 | Define a bijection between... |
| pw2f1o2val 43984 | Function value of the ~ pw... |
| pw2f1o2val2 43985 | Membership in a mapped set... |
| limsuc2 43986 | Limit ordinals in the sens... |
| wepwsolem 43987 | Transfer an ordering on ch... |
| wepwso 43988 | A well-ordering induces a ... |
| dnnumch1 43989 | Define an enumeration of a... |
| dnnumch2 43990 | Define an enumeration (wea... |
| dnnumch3lem 43991 | Value of the ordinal injec... |
| dnnumch3 43992 | Define an injection from a... |
| dnwech 43993 | Define a well-ordering fro... |
| fnwe2val 43994 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 43995 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 43996 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 43997 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 43998 | A well-ordering can be con... |
| aomclem1 43999 | Lemma for ~ dfac11 . This... |
| aomclem2 44000 | Lemma for ~ dfac11 . Succ... |
| aomclem3 44001 | Lemma for ~ dfac11 . Succ... |
| aomclem4 44002 | Lemma for ~ dfac11 . Limi... |
| aomclem5 44003 | Lemma for ~ dfac11 . Comb... |
| aomclem6 44004 | Lemma for ~ dfac11 . Tran... |
| aomclem7 44005 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 44006 | Lemma for ~ dfac11 . Perf... |
| dfac11 44007 | The right-hand side of thi... |
| kelac1 44008 | Kelley's choice, basic for... |
| kelac2lem 44009 | Lemma for ~ kelac2 and ~ d... |
| kelac2 44010 | Kelley's choice, most comm... |
| dfac21 44011 | Tychonoff's theorem is a c... |
| islmodfg 44014 | Property of a finitely gen... |
| islssfg 44015 | Property of a finitely gen... |
| islssfg2 44016 | Property of a finitely gen... |
| islssfgi 44017 | Finitely spanned subspaces... |
| fglmod 44018 | Finitely generated left mo... |
| lsmfgcl 44019 | The sum of two finitely ge... |
| islnm 44022 | Property of being a Noethe... |
| islnm2 44023 | Property of being a Noethe... |
| lnmlmod 44024 | A Noetherian left module i... |
| lnmlssfg 44025 | A submodule of Noetherian ... |
| lnmlsslnm 44026 | All submodules of a Noethe... |
| lnmfg 44027 | A Noetherian left module i... |
| kercvrlsm 44028 | The domain of a linear fun... |
| lmhmfgima 44029 | A homomorphism maps finite... |
| lnmepi 44030 | Epimorphic images of Noeth... |
| lmhmfgsplit 44031 | If the kernel and range of... |
| lmhmlnmsplit 44032 | If the kernel and range of... |
| lnmlmic 44033 | Noetherian is an invariant... |
| pwssplit4 44034 | Splitting for structure po... |
| filnm 44035 | Finite left modules are No... |
| pwslnmlem0 44036 | Zeroeth powers are Noether... |
| pwslnmlem1 44037 | First powers are Noetheria... |
| pwslnmlem2 44038 | A sum of powers is Noether... |
| pwslnm 44039 | Finite powers of Noetheria... |
| unxpwdom3 44040 | Weaker version of ~ unxpwd... |
| pwfi2f1o 44041 | The ~ pw2f1o bijection rel... |
| pwfi2en 44042 | Finitely supported indicat... |
| frlmpwfi 44043 | Formal linear combinations... |
| gicabl 44044 | Being Abelian is a group i... |
| imasgim 44045 | A relabeling of the elemen... |
| isnumbasgrplem1 44046 | A set which is equipollent... |
| harn0 44047 | The Hartogs number of a se... |
| numinfctb 44048 | A numerable infinite set c... |
| isnumbasgrplem2 44049 | If the (to be thought of a... |
| isnumbasgrplem3 44050 | Every nonempty numerable s... |
| isnumbasabl 44051 | A set is numerable iff it ... |
| isnumbasgrp 44052 | A set is numerable iff it ... |
| dfacbasgrp 44053 | A choice equivalent in abs... |
| islnr 44056 | Property of a left-Noether... |
| lnrring 44057 | Left-Noetherian rings are ... |
| lnrlnm 44058 | Left-Noetherian rings have... |
| islnr2 44059 | Property of being a left-N... |
| islnr3 44060 | Relate left-Noetherian rin... |
| lnr2i 44061 | Given an ideal in a left-N... |
| lpirlnr 44062 | Left principal ideal rings... |
| lnrfrlm 44063 | Finite-dimensional free mo... |
| lnrfg 44064 | Finitely-generated modules... |
| lnrfgtr 44065 | A submodule of a finitely ... |
| hbtlem1 44068 | Value of the leading coeff... |
| hbtlem2 44069 | Leading coefficient ideals... |
| hbtlem7 44070 | Functionality of leading c... |
| hbtlem4 44071 | The leading ideal function... |
| hbtlem3 44072 | The leading ideal function... |
| hbtlem5 44073 | The leading ideal function... |
| hbtlem6 44074 | There is a finite set of p... |
| hbt 44075 | The Hilbert Basis Theorem ... |
| dgrsub2 44080 | Subtracting two polynomial... |
| elmnc 44081 | Property of a monic polyno... |
| mncply 44082 | A monic polynomial is a po... |
| mnccoe 44083 | A monic polynomial has lea... |
| mncn0 44084 | A monic polynomial is not ... |
| dgraaval 44089 | Value of the degree functi... |
| dgraalem 44090 | Properties of the degree o... |
| dgraacl 44091 | Closure of the degree func... |
| dgraaf 44092 | Degree function on algebra... |
| dgraaub 44093 | Upper bound on degree of a... |
| dgraa0p 44094 | A rational polynomial of d... |
| mpaaeu 44095 | An algebraic number has ex... |
| mpaaval 44096 | Value of the minimal polyn... |
| mpaalem 44097 | Properties of the minimal ... |
| mpaacl 44098 | Minimal polynomial is a po... |
| mpaadgr 44099 | Minimal polynomial has deg... |
| mpaaroot 44100 | The minimal polynomial of ... |
| mpaamn 44101 | Minimal polynomial is moni... |
| itgoval 44106 | Value of the integral-over... |
| aaitgo 44107 | The standard algebraic num... |
| itgoss 44108 | An integral element is int... |
| itgocn 44109 | All integral elements are ... |
| cnsrexpcl 44110 | Exponentiation is closed i... |
| fsumcnsrcl 44111 | Finite sums are closed in ... |
| cnsrplycl 44112 | Polynomials are closed in ... |
| rgspnid 44113 | The span of a subring is i... |
| rngunsnply 44114 | Adjoining one element to a... |
| flcidc 44115 | Finite linear combinations... |
| algstr 44118 | Lemma to shorten proofs of... |
| algbase 44119 | The base set of a construc... |
| algaddg 44120 | The additive operation of ... |
| algmulr 44121 | The multiplicative operati... |
| algsca 44122 | The set of scalars of a co... |
| algvsca 44123 | The scalar product operati... |
| mendval 44124 | Value of the module endomo... |
| mendbas 44125 | Base set of the module end... |
| mendplusgfval 44126 | Addition in the module end... |
| mendplusg 44127 | A specific addition in the... |
| mendmulrfval 44128 | Multiplication in the modu... |
| mendmulr 44129 | A specific multiplication ... |
| mendsca 44130 | The module endomorphism al... |
| mendvscafval 44131 | Scalar multiplication in t... |
| mendvsca 44132 | A specific scalar multipli... |
| mendring 44133 | The module endomorphism al... |
| mendlmod 44134 | The module endomorphism al... |
| mendassa 44135 | The module endomorphism al... |
| idomodle 44136 | Limit on the number of ` N... |
| fiuneneq 44137 | Two finite sets of equal s... |
| idomsubgmo 44138 | The units of an integral d... |
| proot1mul 44139 | Any primitive ` N ` -th ro... |
| proot1hash 44140 | If an integral domain has ... |
| proot1ex 44141 | The complex field has prim... |
| mon1psubm 44144 | Monic polynomials are a mu... |
| deg1mhm 44145 | Homomorphic property of th... |
| cytpfn 44146 | Functionality of the cyclo... |
| cytpval 44147 | Substitutions for the Nth ... |
| fgraphopab 44148 | Express a function as a su... |
| fgraphxp 44149 | Express a function as a su... |
| hausgraph 44150 | The graph of a continuous ... |
| r1sssucd 44155 | Deductive form of ~ r1sssu... |
| iocunico 44156 | Split an open interval int... |
| iocinico 44157 | The intersection of two se... |
| iocmbl 44158 | An open-below, closed-abov... |
| cnioobibld 44159 | A bounded, continuous func... |
| arearect 44160 | The area of a rectangle wh... |
| areaquad 44161 | The area of a quadrilatera... |
| uniel 44162 | Two ways to say a union is... |
| unielss 44163 | Two ways to say the union ... |
| unielid 44164 | Two ways to say the union ... |
| ssunib 44165 | Two ways to say a class is... |
| rp-intrabeq 44166 | Equality theorem for supre... |
| rp-unirabeq 44167 | Equality theorem for infim... |
| onmaxnelsup 44168 | Two ways to say the maximu... |
| onsupneqmaxlim0 44169 | If the supremum of a class... |
| onsupcl2 44170 | The supremum of a set of o... |
| onuniintrab 44171 | The union of a set of ordi... |
| onintunirab 44172 | The intersection of a non-... |
| onsupnmax 44173 | If the union of a class of... |
| onsupuni 44174 | The supremum of a set of o... |
| onsupuni2 44175 | The supremum of a set of o... |
| onsupintrab 44176 | The supremum of a set of o... |
| onsupintrab2 44177 | The supremum of a set of o... |
| onsupcl3 44178 | The supremum of a set of o... |
| onsupex3 44179 | The supremum of a set of o... |
| onuniintrab2 44180 | The union of a set of ordi... |
| oninfint 44181 | The infimum of a non-empty... |
| oninfunirab 44182 | The infimum of a non-empty... |
| oninfcl2 44183 | The infimum of a non-empty... |
| onsupmaxb 44184 | The union of a class of or... |
| onexgt 44185 | For any ordinal, there is ... |
| onexomgt 44186 | For any ordinal, there is ... |
| omlimcl2 44187 | The product of a limit ord... |
| onexlimgt 44188 | For any ordinal, there is ... |
| onexoegt 44189 | For any ordinal, there is ... |
| oninfex2 44190 | The infimum of a non-empty... |
| onsupeqmax 44191 | Condition when the supremu... |
| onsupeqnmax 44192 | Condition when the supremu... |
| onsuplub 44193 | The supremum of a set of o... |
| onsupnub 44194 | An upper bound of a set of... |
| onfisupcl 44195 | Sufficient condition when ... |
| onelord 44196 | Every element of a ordinal... |
| onepsuc 44197 | Every ordinal is less than... |
| epsoon 44198 | The ordinals are strictly ... |
| epirron 44199 | The strict order on the or... |
| oneptr 44200 | The strict order on the or... |
| oneltr 44201 | The elementhood relation o... |
| oneptri 44202 | The strict, complete (line... |
| ordeldif 44203 | Membership in the differen... |
| ordeldifsucon 44204 | Membership in the differen... |
| ordeldif1o 44205 | Membership in the differen... |
| ordne0gt0 44206 | Ordinal zero is less than ... |
| ondif1i 44207 | Ordinal zero is less than ... |
| onsucelab 44208 | The successor of every ord... |
| dflim6 44209 | A limit ordinal is a nonze... |
| limnsuc 44210 | A limit ordinal is not an ... |
| onsucss 44211 | If one ordinal is less tha... |
| ordnexbtwnsuc 44212 | For any distinct pair of o... |
| orddif0suc 44213 | For any distinct pair of o... |
| onsucf1lem 44214 | For ordinals, the successo... |
| onsucf1olem 44215 | The successor operation is... |
| onsucrn 44216 | The successor operation is... |
| onsucf1o 44217 | The successor operation is... |
| dflim7 44218 | A limit ordinal is a nonze... |
| onov0suclim 44219 | Compactly express rules fo... |
| oa0suclim 44220 | Closed form expression of ... |
| om0suclim 44221 | Closed form expression of ... |
| oe0suclim 44222 | Closed form expression of ... |
| oaomoecl 44223 | The operations of addition... |
| onsupsucismax 44224 | If the union of a set of o... |
| onsssupeqcond 44225 | If for every element of a ... |
| limexissup 44226 | An ordinal which is a limi... |
| limiun 44227 | A limit ordinal is the uni... |
| limexissupab 44228 | An ordinal which is a limi... |
| om1om1r 44229 | Ordinal one is both a left... |
| oe0rif 44230 | Ordinal zero raised to any... |
| oasubex 44231 | While subtraction can't be... |
| nnamecl 44232 | Natural numbers are closed... |
| onsucwordi 44233 | The successor operation pr... |
| oalim2cl 44234 | The ordinal sum of any ord... |
| oaltublim 44235 | Given ` C ` is a limit ord... |
| oaordi3 44236 | Ordinal addition of the sa... |
| oaord3 44237 | When the same ordinal is a... |
| 1oaomeqom 44238 | Ordinal one plus omega is ... |
| oaabsb 44239 | The right addend absorbs t... |
| oaordnrex 44240 | When omega is added on the... |
| oaordnr 44241 | When the same ordinal is a... |
| omge1 44242 | Any nonzero ordinal produc... |
| omge2 44243 | Any nonzero ordinal produc... |
| omlim2 44244 | The nonzero product with a... |
| omord2lim 44245 | Given a limit ordinal, the... |
| omord2i 44246 | Ordinal multiplication of ... |
| omord2com 44247 | When the same nonzero ordi... |
| 2omomeqom 44248 | Ordinal two times omega is... |
| omnord1ex 44249 | When omega is multiplied o... |
| omnord1 44250 | When the same nonzero ordi... |
| oege1 44251 | Any nonzero ordinal power ... |
| oege2 44252 | Any power of an ordinal at... |
| rp-oelim2 44253 | The power of an ordinal at... |
| oeord2lim 44254 | Given a limit ordinal, the... |
| oeord2i 44255 | Ordinal exponentiation of ... |
| oeord2com 44256 | When the same base at leas... |
| nnoeomeqom 44257 | Any natural number at leas... |
| df3o2 44258 | Ordinal 3 is the unordered... |
| df3o3 44259 | Ordinal 3, fully expanded.... |
| oenord1ex 44260 | When ordinals two and thre... |
| oenord1 44261 | When two ordinals (both at... |
| oaomoencom 44262 | Ordinal addition, multipli... |
| oenassex 44263 | Ordinal two raised to two ... |
| oenass 44264 | Ordinal exponentiation is ... |
| cantnftermord 44265 | For terms of the form of a... |
| cantnfub 44266 | Given a finite number of t... |
| cantnfub2 44267 | Given a finite number of t... |
| bropabg 44268 | Equivalence for two classe... |
| cantnfresb 44269 | A Cantor normal form which... |
| cantnf2 44270 | For every ordinal, ` A ` ,... |
| oawordex2 44271 | If ` C ` is between ` A ` ... |
| nnawordexg 44272 | If an ordinal, ` B ` , is ... |
| succlg 44273 | Closure law for ordinal su... |
| dflim5 44274 | A limit ordinal is either ... |
| oacl2g 44275 | Closure law for ordinal ad... |
| onmcl 44276 | If an ordinal is less than... |
| omabs2 44277 | Ordinal multiplication by ... |
| omcl2 44278 | Closure law for ordinal mu... |
| omcl3g 44279 | Closure law for ordinal mu... |
| ordsssucb 44280 | An ordinal number is less ... |
| tfsconcatlem 44281 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 44282 | The concatenation of two t... |
| tfsconcatfn 44283 | The concatenation of two t... |
| tfsconcatfv1 44284 | An early value of the conc... |
| tfsconcatfv2 44285 | A latter value of the conc... |
| tfsconcatfv 44286 | The value of the concatena... |
| tfsconcatrn 44287 | The range of the concatena... |
| tfsconcatfo 44288 | The concatenation of two t... |
| tfsconcatb0 44289 | The concatentation with th... |
| tfsconcat0i 44290 | The concatentation with th... |
| tfsconcat0b 44291 | The concatentation with th... |
| tfsconcat00 44292 | The concatentation of two ... |
| tfsconcatrev 44293 | If the domain of a transfi... |
| tfsconcatrnss12 44294 | The range of the concatena... |
| tfsconcatrnss 44295 | The concatenation of trans... |
| tfsconcatrnsson 44296 | The concatenation of trans... |
| tfsnfin 44297 | A transfinite sequence is ... |
| rp-tfslim 44298 | The limit of a sequence of... |
| ofoafg 44299 | Addition operator for func... |
| ofoaf 44300 | Addition operator for func... |
| ofoafo 44301 | Addition operator for func... |
| ofoacl 44302 | Closure law for component ... |
| ofoaid1 44303 | Identity law for component... |
| ofoaid2 44304 | Identity law for component... |
| ofoaass 44305 | Component-wise addition of... |
| ofoacom 44306 | Component-wise addition of... |
| naddcnff 44307 | Addition operator for Cant... |
| naddcnffn 44308 | Addition operator for Cant... |
| naddcnffo 44309 | Addition of Cantor normal ... |
| naddcnfcl 44310 | Closure law for component-... |
| naddcnfcom 44311 | Component-wise ordinal add... |
| naddcnfid1 44312 | Identity law for component... |
| naddcnfid2 44313 | Identity law for component... |
| naddcnfass 44314 | Component-wise addition of... |
| onsucunifi 44315 | The successor to the union... |
| sucunisn 44316 | The successor to the union... |
| onsucunipr 44317 | The successor to the union... |
| onsucunitp 44318 | The successor to the union... |
| oaun3lem1 44319 | The class of all ordinal s... |
| oaun3lem2 44320 | The class of all ordinal s... |
| oaun3lem3 44321 | The class of all ordinal s... |
| oaun3lem4 44322 | The class of all ordinal s... |
| rp-abid 44323 | Two ways to express a clas... |
| oadif1lem 44324 | Express the set difference... |
| oadif1 44325 | Express the set difference... |
| oaun2 44326 | Ordinal addition as a unio... |
| oaun3 44327 | Ordinal addition as a unio... |
| naddov4 44328 | Alternate expression for n... |
| nadd2rabtr 44329 | The set of ordinals which ... |
| nadd2rabord 44330 | The set of ordinals which ... |
| nadd2rabex 44331 | The class of ordinals whic... |
| nadd2rabon 44332 | The set of ordinals which ... |
| nadd1rabtr 44333 | The set of ordinals which ... |
| nadd1rabord 44334 | The set of ordinals which ... |
| nadd1rabex 44335 | The class of ordinals whic... |
| nadd1rabon 44336 | The set of ordinals which ... |
| nadd1suc 44337 | Natural addition with 1 is... |
| naddass1 44338 | Natural addition of ordina... |
| naddgeoa 44339 | Natural addition results i... |
| naddonnn 44340 | Natural addition with a na... |
| naddwordnexlem0 44341 | When ` A ` is the sum of a... |
| naddwordnexlem1 44342 | When ` A ` is the sum of a... |
| naddwordnexlem2 44343 | When ` A ` is the sum of a... |
| naddwordnexlem3 44344 | When ` A ` is the sum of a... |
| oawordex3 44345 | When ` A ` is the sum of a... |
| naddwordnexlem4 44346 | When ` A ` is the sum of a... |
| ordsssucim 44347 | If an ordinal is less than... |
| insucid 44348 | The intersection of a clas... |
| oaltom 44349 | Multiplication eventually ... |
| oe2 44350 | Two ways to square an ordi... |
| omltoe 44351 | Exponentiation eventually ... |
| abeqabi 44352 | Generalized condition for ... |
| abpr 44353 | Condition for a class abst... |
| abtp 44354 | Condition for a class abst... |
| ralopabb 44355 | Restricted universal quant... |
| fpwfvss 44356 | Functions into a powerset ... |
| sdomne0 44357 | A class that strictly domi... |
| sdomne0d 44358 | A class that strictly domi... |
| safesnsupfiss 44359 | If ` B ` is a finite subse... |
| safesnsupfiub 44360 | If ` B ` is a finite subse... |
| safesnsupfidom1o 44361 | If ` B ` is a finite subse... |
| safesnsupfilb 44362 | If ` B ` is a finite subse... |
| isoeq145d 44363 | Equality deduction for iso... |
| resisoeq45d 44364 | Equality deduction for equ... |
| negslem1 44365 | An equivalence between ide... |
| nvocnvb 44366 | Equivalence to saying the ... |
| rp-brsslt 44367 | Binary relation form of a ... |
| nla0002 44368 | Extending a linear order t... |
| nla0003 44369 | Extending a linear order t... |
| nla0001 44370 | Extending a linear order t... |
| faosnf0.11b 44371 | ` B ` is called a non-limi... |
| dfno2 44372 | A surreal number, in the f... |
| onnoxpg 44373 | Every ordinal maps to a su... |
| onnobdayg 44374 | Every ordinal maps to a su... |
| bdaybndex 44375 | Bounds formed from the bir... |
| bdaybndbday 44376 | Bounds formed from the bir... |
| onnoxp 44377 | Every ordinal maps to a su... |
| onnoxpi 44378 | Every ordinal maps to a su... |
| 0fno 44379 | Ordinal zero maps to a sur... |
| 1fno 44380 | Ordinal one maps to a surr... |
| 2fno 44381 | Ordinal two maps to a surr... |
| 3fno 44382 | Ordinal three maps to a su... |
| 4fno 44383 | Ordinal four maps to a sur... |
| fnimafnex 44384 | The functional image of a ... |
| nlimsuc 44385 | A successor is not a limit... |
| nlim1NEW 44386 | 1 is not a limit ordinal. ... |
| nlim2NEW 44387 | 2 is not a limit ordinal. ... |
| nlim3 44388 | 3 is not a limit ordinal. ... |
| nlim4 44389 | 4 is not a limit ordinal. ... |
| oa1un 44390 | Given ` A e. On ` , let ` ... |
| oa1cl 44391 | ` A +o 1o ` is in ` On ` .... |
| 0finon 44392 | 0 is a finite ordinal. Se... |
| 1finon 44393 | 1 is a finite ordinal. Se... |
| 2finon 44394 | 2 is a finite ordinal. Se... |
| 3finon 44395 | 3 is a finite ordinal. Se... |
| 4finon 44396 | 4 is a finite ordinal. Se... |
| finona1cl 44397 | The finite ordinals are cl... |
| finonex 44398 | The finite ordinals are a ... |
| fzunt 44399 | Union of two adjacent fini... |
| fzuntd 44400 | Union of two adjacent fini... |
| fzunt1d 44401 | Union of two overlapping f... |
| fzuntgd 44402 | Union of two adjacent or o... |
| ifpan123g 44403 | Conjunction of conditional... |
| ifpan23 44404 | Conjunction of conditional... |
| ifpdfor2 44405 | Define or in terms of cond... |
| ifporcor 44406 | Corollary of commutation o... |
| ifpdfan2 44407 | Define and with conditiona... |
| ifpancor 44408 | Corollary of commutation o... |
| ifpdfor 44409 | Define or in terms of cond... |
| ifpdfan 44410 | Define and with conditiona... |
| ifpbi2 44411 | Equivalence theorem for co... |
| ifpbi3 44412 | Equivalence theorem for co... |
| ifpim1 44413 | Restate implication as con... |
| ifpnot 44414 | Restate negated wff as con... |
| ifpid2 44415 | Restate wff as conditional... |
| ifpim2 44416 | Restate implication as con... |
| ifpbi23 44417 | Equivalence theorem for co... |
| ifpbiidcor 44418 | Restatement of ~ biid . (... |
| ifpbicor 44419 | Corollary of commutation o... |
| ifpxorcor 44420 | Corollary of commutation o... |
| ifpbi1 44421 | Equivalence theorem for co... |
| ifpnot23 44422 | Negation of conditional lo... |
| ifpnotnotb 44423 | Factor conditional logic o... |
| ifpnorcor 44424 | Corollary of commutation o... |
| ifpnancor 44425 | Corollary of commutation o... |
| ifpnot23b 44426 | Negation of conditional lo... |
| ifpbiidcor2 44427 | Restatement of ~ biid . (... |
| ifpnot23c 44428 | Negation of conditional lo... |
| ifpnot23d 44429 | Negation of conditional lo... |
| ifpdfnan 44430 | Define nand as conditional... |
| ifpdfxor 44431 | Define xor as conditional ... |
| ifpbi12 44432 | Equivalence theorem for co... |
| ifpbi13 44433 | Equivalence theorem for co... |
| ifpbi123 44434 | Equivalence theorem for co... |
| ifpidg 44435 | Restate wff as conditional... |
| ifpid3g 44436 | Restate wff as conditional... |
| ifpid2g 44437 | Restate wff as conditional... |
| ifpid1g 44438 | Restate wff as conditional... |
| ifpim23g 44439 | Restate implication as con... |
| ifpim3 44440 | Restate implication as con... |
| ifpnim1 44441 | Restate negated implicatio... |
| ifpim4 44442 | Restate implication as con... |
| ifpnim2 44443 | Restate negated implicatio... |
| ifpim123g 44444 | Implication of conditional... |
| ifpim1g 44445 | Implication of conditional... |
| ifp1bi 44446 | Substitute the first eleme... |
| ifpbi1b 44447 | When the first variable is... |
| ifpimimb 44448 | Factor conditional logic o... |
| ifpororb 44449 | Factor conditional logic o... |
| ifpananb 44450 | Factor conditional logic o... |
| ifpnannanb 44451 | Factor conditional logic o... |
| ifpor123g 44452 | Disjunction of conditional... |
| ifpimim 44453 | Consequnce of implication.... |
| ifpbibib 44454 | Factor conditional logic o... |
| ifpxorxorb 44455 | Factor conditional logic o... |
| rp-fakeimass 44456 | A special case where impli... |
| rp-fakeanorass 44457 | A special case where a mix... |
| rp-fakeoranass 44458 | A special case where a mix... |
| rp-fakeinunass 44459 | A special case where a mix... |
| rp-fakeuninass 44460 | A special case where a mix... |
| rp-isfinite5 44461 | A set is said to be finite... |
| rp-isfinite6 44462 | A set is said to be finite... |
| intabssd 44463 | When for each element ` y ... |
| eu0 44464 | There is only one empty se... |
| epelon2 44465 | Over the ordinal numbers, ... |
| ontric3g 44466 | For all ` x , y e. On ` , ... |
| dfsucon 44467 | ` A ` is called a successo... |
| snen1g 44468 | A singleton is equinumerou... |
| snen1el 44469 | A singleton is equinumerou... |
| sn1dom 44470 | A singleton is dominated b... |
| pr2dom 44471 | An unordered pair is domin... |
| tr3dom 44472 | An unordered triple is dom... |
| ensucne0 44473 | A class equinumerous to a ... |
| ensucne0OLD 44474 | A class equinumerous to a ... |
| dfom6 44475 | Let ` _om ` be defined to ... |
| infordmin 44476 | ` _om ` is the smallest in... |
| iscard4 44477 | Two ways to express the pr... |
| minregex 44478 | Given any cardinal number ... |
| minregex2 44479 | Given any cardinal number ... |
| iscard5 44480 | Two ways to express the pr... |
| elrncard 44481 | Let us define a cardinal n... |
| harval3 44482 | ` ( har `` A ) ` is the le... |
| harval3on 44483 | For any ordinal number ` A... |
| omssrncard 44484 | All natural numbers are ca... |
| 0iscard 44485 | 0 is a cardinal number. (... |
| 1iscard 44486 | 1 is a cardinal number. (... |
| omiscard 44487 | ` _om ` is a cardinal numb... |
| sucomisnotcard 44488 | ` _om +o 1o ` is not a car... |
| nna1iscard 44489 | For any natural number, th... |
| har2o 44490 | The least cardinal greater... |
| en2pr 44491 | A class is equinumerous to... |
| pr2cv 44492 | If an unordered pair is eq... |
| pr2el1 44493 | If an unordered pair is eq... |
| pr2cv1 44494 | If an unordered pair is eq... |
| pr2el2 44495 | If an unordered pair is eq... |
| pr2cv2 44496 | If an unordered pair is eq... |
| pren2 44497 | An unordered pair is equin... |
| pr2eldif1 44498 | If an unordered pair is eq... |
| pr2eldif2 44499 | If an unordered pair is eq... |
| pren2d 44500 | A pair of two distinct set... |
| aleph1min 44501 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 44502 | ` aleph ` is a strictly or... |
| alephiso3 44503 | ` aleph ` is a strictly or... |
| pwelg 44504 | The powerclass is an eleme... |
| pwinfig 44505 | The powerclass of an infin... |
| pwinfi2 44506 | The powerclass of an infin... |
| pwinfi3 44507 | The powerclass of an infin... |
| pwinfi 44508 | The powerclass of an infin... |
| fipjust 44509 | A definition of the finite... |
| cllem0 44510 | The class of all sets with... |
| superficl 44511 | The class of all supersets... |
| superuncl 44512 | The class of all supersets... |
| ssficl 44513 | The class of all subsets o... |
| ssuncl 44514 | The class of all subsets o... |
| ssdifcl 44515 | The class of all subsets o... |
| sssymdifcl 44516 | The class of all subsets o... |
| fiinfi 44517 | If two classes have the fi... |
| rababg 44518 | Condition when restricted ... |
| elinintab 44519 | Two ways of saying a set i... |
| elmapintrab 44520 | Two ways to say a set is a... |
| elinintrab 44521 | Two ways of saying a set i... |
| inintabss 44522 | Upper bound on intersectio... |
| inintabd 44523 | Value of the intersection ... |
| xpinintabd 44524 | Value of the intersection ... |
| relintabex 44525 | If the intersection of a c... |
| elcnvcnvintab 44526 | Two ways of saying a set i... |
| relintab 44527 | Value of the intersection ... |
| nonrel 44528 | A non-relation is equal to... |
| elnonrel 44529 | Only an ordered pair where... |
| cnvssb 44530 | Subclass theorem for conve... |
| relnonrel 44531 | The non-relation part of a... |
| cnvnonrel 44532 | The converse of the non-re... |
| brnonrel 44533 | A non-relation cannot rela... |
| dmnonrel 44534 | The domain of the non-rela... |
| rnnonrel 44535 | The range of the non-relat... |
| resnonrel 44536 | A restriction of the non-r... |
| imanonrel 44537 | An image under the non-rel... |
| cononrel1 44538 | Composition with the non-r... |
| cononrel2 44539 | Composition with the non-r... |
| elmapintab 44540 | Two ways to say a set is a... |
| fvnonrel 44541 | The function value of any ... |
| elinlem 44542 | Two ways to say a set is a... |
| elcnvcnvlem 44543 | Two ways to say a set is a... |
| cnvcnvintabd 44544 | Value of the relationship ... |
| elcnvlem 44545 | Two ways to say a set is a... |
| elcnvintab 44546 | Two ways of saying a set i... |
| cnvintabd 44547 | Value of the converse of t... |
| undmrnresiss 44548 | Two ways of saying the ide... |
| reflexg 44549 | Two ways of saying a relat... |
| cnvssco 44550 | A condition weaker than re... |
| refimssco 44551 | Reflexive relations are su... |
| cleq2lem 44552 | Equality implies bijection... |
| cbvcllem 44553 | Change of bound variable i... |
| clublem 44554 | If a superset ` Y ` of ` X... |
| clss2lem 44555 | The closure of a property ... |
| dfid7 44556 | Definition of identity rel... |
| mptrcllem 44557 | Show two versions of a clo... |
| cotrintab 44558 | The intersection of a clas... |
| rclexi 44559 | The reflexive closure of a... |
| rtrclexlem 44560 | Existence of relation impl... |
| rtrclex 44561 | The reflexive-transitive c... |
| trclubgNEW 44562 | If a relation exists then ... |
| trclubNEW 44563 | If a relation exists then ... |
| trclexi 44564 | The transitive closure of ... |
| rtrclexi 44565 | The reflexive-transitive c... |
| clrellem 44566 | When the property ` ps ` h... |
| clcnvlem 44567 | When ` A ` , an upper boun... |
| cnvtrucl0 44568 | The converse of the trivia... |
| cnvrcl0 44569 | The converse of the reflex... |
| cnvtrcl0 44570 | The converse of the transi... |
| dmtrcl 44571 | The domain of the transiti... |
| rntrcl 44572 | The range of the transitiv... |
| dfrtrcl5 44573 | Definition of reflexive-tr... |
| trcleq2lemRP 44574 | Equality implies bijection... |
| sqrtcvallem1 44575 | Two ways of saying a compl... |
| reabsifneg 44576 | Alternate expression for t... |
| reabsifnpos 44577 | Alternate expression for t... |
| reabsifpos 44578 | Alternate expression for t... |
| reabsifnneg 44579 | Alternate expression for t... |
| reabssgn 44580 | Alternate expression for t... |
| sqrtcvallem2 44581 | Equivalent to saying that ... |
| sqrtcvallem3 44582 | Equivalent to saying that ... |
| sqrtcvallem4 44583 | Equivalent to saying that ... |
| sqrtcvallem5 44584 | Equivalent to saying that ... |
| sqrtcval 44585 | Explicit formula for the c... |
| sqrtcval2 44586 | Explicit formula for the c... |
| resqrtval 44587 | Real part of the complex s... |
| imsqrtval 44588 | Imaginary part of the comp... |
| resqrtvalex 44589 | Example for ~ resqrtval . ... |
| imsqrtvalex 44590 | Example for ~ imsqrtval . ... |
| al3im 44591 | Version of ~ ax-4 for a ne... |
| intima0 44592 | Two ways of expressing the... |
| elimaint 44593 | Element of image of inters... |
| cnviun 44594 | Converse of indexed union.... |
| imaiun1 44595 | The image of an indexed un... |
| coiun1 44596 | Composition with an indexe... |
| elintima 44597 | Element of intersection of... |
| intimass 44598 | The image under the inters... |
| intimass2 44599 | The image under the inters... |
| intimag 44600 | Requirement for the image ... |
| intimasn 44601 | Two ways to express the im... |
| intimasn2 44602 | Two ways to express the im... |
| ss2iundf 44603 | Subclass theorem for index... |
| ss2iundv 44604 | Subclass theorem for index... |
| cbviuneq12df 44605 | Rule used to change the bo... |
| cbviuneq12dv 44606 | Rule used to change the bo... |
| conrel1d 44607 | Deduction about compositio... |
| conrel2d 44608 | Deduction about compositio... |
| trrelind 44609 | The intersection of transi... |
| xpintrreld 44610 | The intersection of a tran... |
| restrreld 44611 | The restriction of a trans... |
| trrelsuperreldg 44612 | Concrete construction of a... |
| trficl 44613 | The class of all transitiv... |
| cnvtrrel 44614 | The converse of a transiti... |
| trrelsuperrel2dg 44615 | Concrete construction of a... |
| dfrcl2 44618 | Reflexive closure of a rel... |
| dfrcl3 44619 | Reflexive closure of a rel... |
| dfrcl4 44620 | Reflexive closure of a rel... |
| relexp2 44621 | A set operated on by the r... |
| relexpnul 44622 | If the domain and range of... |
| eliunov2 44623 | Membership in the indexed ... |
| eltrclrec 44624 | Membership in the indexed ... |
| elrtrclrec 44625 | Membership in the indexed ... |
| briunov2 44626 | Two classes related by the... |
| brmptiunrelexpd 44627 | If two elements are connec... |
| fvmptiunrelexplb0d 44628 | If the indexed union range... |
| fvmptiunrelexplb0da 44629 | If the indexed union range... |
| fvmptiunrelexplb1d 44630 | If the indexed union range... |
| brfvid 44631 | If two elements are connec... |
| brfvidRP 44632 | If two elements are connec... |
| fvilbd 44633 | A set is a subset of its i... |
| fvilbdRP 44634 | A set is a subset of its i... |
| brfvrcld 44635 | If two elements are connec... |
| brfvrcld2 44636 | If two elements are connec... |
| fvrcllb0d 44637 | A restriction of the ident... |
| fvrcllb0da 44638 | A restriction of the ident... |
| fvrcllb1d 44639 | A set is a subset of its i... |
| brtrclrec 44640 | Two classes related by the... |
| brrtrclrec 44641 | Two classes related by the... |
| briunov2uz 44642 | Two classes related by the... |
| eliunov2uz 44643 | Membership in the indexed ... |
| ov2ssiunov2 44644 | Any particular operator va... |
| relexp0eq 44645 | The zeroth power of relati... |
| iunrelexp0 44646 | Simplification of zeroth p... |
| relexpxpnnidm 44647 | Any positive power of a Ca... |
| relexpiidm 44648 | Any power of any restricti... |
| relexpss1d 44649 | The relational power of a ... |
| comptiunov2i 44650 | The composition two indexe... |
| corclrcl 44651 | The reflexive closure is i... |
| iunrelexpmin1 44652 | The indexed union of relat... |
| relexpmulnn 44653 | With exponents limited to ... |
| relexpmulg 44654 | With ordered exponents, th... |
| trclrelexplem 44655 | The union of relational po... |
| iunrelexpmin2 44656 | The indexed union of relat... |
| relexp01min 44657 | With exponents limited to ... |
| relexp1idm 44658 | Repeated raising a relatio... |
| relexp0idm 44659 | Repeated raising a relatio... |
| relexp0a 44660 | Absorption law for zeroth ... |
| relexpxpmin 44661 | The composition of powers ... |
| relexpaddss 44662 | The composition of two pow... |
| iunrelexpuztr 44663 | The indexed union of relat... |
| dftrcl3 44664 | Transitive closure of a re... |
| brfvtrcld 44665 | If two elements are connec... |
| fvtrcllb1d 44666 | A set is a subset of its i... |
| trclfvcom 44667 | The transitive closure of ... |
| cnvtrclfv 44668 | The converse of the transi... |
| cotrcltrcl 44669 | The transitive closure is ... |
| trclimalb2 44670 | Lower bound for image unde... |
| brtrclfv2 44671 | Two ways to indicate two e... |
| trclfvdecomr 44672 | The transitive closure of ... |
| trclfvdecoml 44673 | The transitive closure of ... |
| dmtrclfvRP 44674 | The domain of the transiti... |
| rntrclfvRP 44675 | The range of the transitiv... |
| rntrclfv 44676 | The range of the transitiv... |
| dfrtrcl3 44677 | Reflexive-transitive closu... |
| brfvrtrcld 44678 | If two elements are connec... |
| fvrtrcllb0d 44679 | A restriction of the ident... |
| fvrtrcllb0da 44680 | A restriction of the ident... |
| fvrtrcllb1d 44681 | A set is a subset of its i... |
| dfrtrcl4 44682 | Reflexive-transitive closu... |
| corcltrcl 44683 | The composition of the ref... |
| cortrcltrcl 44684 | Composition with the refle... |
| corclrtrcl 44685 | Composition with the refle... |
| cotrclrcl 44686 | The composition of the ref... |
| cortrclrcl 44687 | Composition with the refle... |
| cotrclrtrcl 44688 | Composition with the refle... |
| cortrclrtrcl 44689 | The reflexive-transitive c... |
| frege77d 44690 | If the images of both ` { ... |
| frege81d 44691 | If the image of ` U ` is a... |
| frege83d 44692 | If the image of the union ... |
| frege96d 44693 | If ` C ` follows ` A ` in ... |
| frege87d 44694 | If the images of both ` { ... |
| frege91d 44695 | If ` B ` follows ` A ` in ... |
| frege97d 44696 | If ` A ` contains all elem... |
| frege98d 44697 | If ` C ` follows ` A ` and... |
| frege102d 44698 | If either ` A ` and ` C ` ... |
| frege106d 44699 | If ` B ` follows ` A ` in ... |
| frege108d 44700 | If either ` A ` and ` C ` ... |
| frege109d 44701 | If ` A ` contains all elem... |
| frege114d 44702 | If either ` R ` relates ` ... |
| frege111d 44703 | If either ` A ` and ` C ` ... |
| frege122d 44704 | If ` F ` is a function, ` ... |
| frege124d 44705 | If ` F ` is a function, ` ... |
| frege126d 44706 | If ` F ` is a function, ` ... |
| frege129d 44707 | If ` F ` is a function and... |
| frege131d 44708 | If ` F ` is a function and... |
| frege133d 44709 | If ` F ` is a function and... |
| dfxor4 44710 | Express exclusive-or in te... |
| dfxor5 44711 | Express exclusive-or in te... |
| df3or2 44712 | Express triple-or in terms... |
| df3an2 44713 | Express triple-and in term... |
| nev 44714 | Express that not every set... |
| 0pssin 44715 | Express that an intersecti... |
| dfhe2 44718 | The property of relation `... |
| dfhe3 44719 | The property of relation `... |
| heeq12 44720 | Equality law for relations... |
| heeq1 44721 | Equality law for relations... |
| heeq2 44722 | Equality law for relations... |
| sbcheg 44723 | Distribute proper substitu... |
| hess 44724 | Subclass law for relations... |
| xphe 44725 | Any Cartesian product is h... |
| 0he 44726 | The empty relation is here... |
| 0heALT 44727 | The empty relation is here... |
| he0 44728 | Any relation is hereditary... |
| unhe1 44729 | The union of two relations... |
| snhesn 44730 | Any singleton is hereditar... |
| idhe 44731 | The identity relation is h... |
| psshepw 44732 | The relation between sets ... |
| sshepw 44733 | The relation between sets ... |
| rp-simp2-frege 44736 | Simplification of triple c... |
| rp-simp2 44737 | Simplification of triple c... |
| rp-frege3g 44738 | Add antecedent to ~ ax-fre... |
| frege3 44739 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 44740 | Double-use of ~ ax-frege2 ... |
| rp-frege24 44741 | Introducing an embedded an... |
| rp-frege4g 44742 | Deduction related to distr... |
| frege4 44743 | Special case of closed for... |
| frege5 44744 | A closed form of ~ syl . ... |
| rp-7frege 44745 | Distribute antecedent and ... |
| rp-4frege 44746 | Elimination of a nested an... |
| rp-6frege 44747 | Elimination of a nested an... |
| rp-8frege 44748 | Eliminate antecedent when ... |
| rp-frege25 44749 | Closed form for ~ a1dd . ... |
| frege6 44750 | A closed form of ~ imim2d ... |
| axfrege8 44751 | Swap antecedents. Identic... |
| frege7 44752 | A closed form of ~ syl6 . ... |
| frege26 44754 | Identical to ~ idd . Prop... |
| frege27 44755 | We cannot (at the same tim... |
| frege9 44756 | Closed form of ~ syl with ... |
| frege12 44757 | A closed form of ~ com23 .... |
| frege11 44758 | Elimination of a nested an... |
| frege24 44759 | Closed form for ~ a1d . D... |
| frege16 44760 | A closed form of ~ com34 .... |
| frege25 44761 | Closed form for ~ a1dd . ... |
| frege18 44762 | Closed form of a syllogism... |
| frege22 44763 | A closed form of ~ com45 .... |
| frege10 44764 | Result commuting anteceden... |
| frege17 44765 | A closed form of ~ com3l .... |
| frege13 44766 | A closed form of ~ com3r .... |
| frege14 44767 | Closed form of a deduction... |
| frege19 44768 | A closed form of ~ syl6 . ... |
| frege23 44769 | Syllogism followed by rota... |
| frege15 44770 | A closed form of ~ com4r .... |
| frege21 44771 | Replace antecedent in ante... |
| frege20 44772 | A closed form of ~ syl8 . ... |
| axfrege28 44773 | Contraposition. Identical... |
| frege29 44775 | Closed form of ~ con3d . ... |
| frege30 44776 | Commuted, closed form of ~... |
| axfrege31 44777 | Identical to ~ notnotr . ... |
| frege32 44779 | Deduce ~ con1 from ~ con3 ... |
| frege33 44780 | If ` ph ` or ` ps ` takes ... |
| frege34 44781 | If as a consequence of the... |
| frege35 44782 | Commuted, closed form of ~... |
| frege36 44783 | The case in which ` ps ` i... |
| frege37 44784 | If ` ch ` is a necessary c... |
| frege38 44785 | Identical to ~ pm2.21 . P... |
| frege39 44786 | Syllogism between ~ pm2.18... |
| frege40 44787 | Anything implies ~ pm2.18 ... |
| axfrege41 44788 | Identical to ~ notnot . A... |
| frege42 44790 | Not not ~ id . Propositio... |
| frege43 44791 | If there is a choice only ... |
| frege44 44792 | Similar to a commuted ~ pm... |
| frege45 44793 | Deduce ~ pm2.6 from ~ con1... |
| frege46 44794 | If ` ps ` holds when ` ph ... |
| frege47 44795 | Deduce consequence follows... |
| frege48 44796 | Closed form of syllogism w... |
| frege49 44797 | Closed form of deduction w... |
| frege50 44798 | Closed form of ~ jaoi . P... |
| frege51 44799 | Compare with ~ jaod . Pro... |
| axfrege52a 44800 | Justification for ~ ax-fre... |
| frege52aid 44802 | The case when the content ... |
| frege53aid 44803 | Specialization of ~ frege5... |
| frege53a 44804 | Lemma for ~ frege55a . Pr... |
| axfrege54a 44805 | Justification for ~ ax-fre... |
| frege54cor0a 44807 | Synonym for logical equiva... |
| frege54cor1a 44808 | Reflexive equality. (Cont... |
| frege55aid 44809 | Lemma for ~ frege57aid . ... |
| frege55lem1a 44810 | Necessary deduction regard... |
| frege55lem2a 44811 | Core proof of Proposition ... |
| frege55a 44812 | Proposition 55 of [Frege18... |
| frege55cor1a 44813 | Proposition 55 of [Frege18... |
| frege56aid 44814 | Lemma for ~ frege57aid . ... |
| frege56a 44815 | Proposition 56 of [Frege18... |
| frege57aid 44816 | This is the all important ... |
| frege57a 44817 | Analogue of ~ frege57aid .... |
| axfrege58a 44818 | Identical to ~ anifp . Ju... |
| frege58acor 44820 | Lemma for ~ frege59a . (C... |
| frege59a 44821 | A kind of Aristotelian inf... |
| frege60a 44822 | Swap antecedents of ~ ax-f... |
| frege61a 44823 | Lemma for ~ frege65a . Pr... |
| frege62a 44824 | A kind of Aristotelian inf... |
| frege63a 44825 | Proposition 63 of [Frege18... |
| frege64a 44826 | Lemma for ~ frege65a . Pr... |
| frege65a 44827 | A kind of Aristotelian inf... |
| frege66a 44828 | Swap antecedents of ~ freg... |
| frege67a 44829 | Lemma for ~ frege68a . Pr... |
| frege68a 44830 | Combination of applying a ... |
| axfrege52c 44831 | Justification for ~ ax-fre... |
| frege52b 44833 | The case when the content ... |
| frege53b 44834 | Lemma for frege102 (via ~ ... |
| axfrege54c 44835 | Reflexive equality of clas... |
| frege54b 44837 | Reflexive equality of sets... |
| frege54cor1b 44838 | Reflexive equality. (Cont... |
| frege55lem1b 44839 | Necessary deduction regard... |
| frege55lem2b 44840 | Lemma for ~ frege55b . Co... |
| frege55b 44841 | Lemma for ~ frege57b . Pr... |
| frege56b 44842 | Lemma for ~ frege57b . Pr... |
| frege57b 44843 | Analogue of ~ frege57aid .... |
| axfrege58b 44844 | If ` A. x ph ` is affirmed... |
| frege58bid 44846 | If ` A. x ph ` is affirmed... |
| frege58bcor 44847 | Lemma for ~ frege59b . (C... |
| frege59b 44848 | A kind of Aristotelian inf... |
| frege60b 44849 | Swap antecedents of ~ ax-f... |
| frege61b 44850 | Lemma for ~ frege65b . Pr... |
| frege62b 44851 | A kind of Aristotelian inf... |
| frege63b 44852 | Lemma for ~ frege91 . Pro... |
| frege64b 44853 | Lemma for ~ frege65b . Pr... |
| frege65b 44854 | A kind of Aristotelian inf... |
| frege66b 44855 | Swap antecedents of ~ freg... |
| frege67b 44856 | Lemma for ~ frege68b . Pr... |
| frege68b 44857 | Combination of applying a ... |
| frege53c 44858 | Proposition 53 of [Frege18... |
| frege54cor1c 44859 | Reflexive equality. (Cont... |
| frege55lem1c 44860 | Necessary deduction regard... |
| frege55lem2c 44861 | Core proof of Proposition ... |
| frege55c 44862 | Proposition 55 of [Frege18... |
| frege56c 44863 | Lemma for ~ frege57c . Pr... |
| frege57c 44864 | Swap order of implication ... |
| frege58c 44865 | Principle related to ~ sp ... |
| frege59c 44866 | A kind of Aristotelian inf... |
| frege60c 44867 | Swap antecedents of ~ freg... |
| frege61c 44868 | Lemma for ~ frege65c . Pr... |
| frege62c 44869 | A kind of Aristotelian inf... |
| frege63c 44870 | Analogue of ~ frege63b . ... |
| frege64c 44871 | Lemma for ~ frege65c . Pr... |
| frege65c 44872 | A kind of Aristotelian inf... |
| frege66c 44873 | Swap antecedents of ~ freg... |
| frege67c 44874 | Lemma for ~ frege68c . Pr... |
| frege68c 44875 | Combination of applying a ... |
| dffrege69 44876 | If from the proposition th... |
| frege70 44877 | Lemma for ~ frege72 . Pro... |
| frege71 44878 | Lemma for ~ frege72 . Pro... |
| frege72 44879 | If property ` A ` is hered... |
| frege73 44880 | Lemma for ~ frege87 . Pro... |
| frege74 44881 | If ` X ` has a property ` ... |
| frege75 44882 | If from the proposition th... |
| dffrege76 44883 | If from the two propositio... |
| frege77 44884 | If ` Y ` follows ` X ` in ... |
| frege78 44885 | Commuted form of ~ frege77... |
| frege79 44886 | Distributed form of ~ freg... |
| frege80 44887 | Add additional condition t... |
| frege81 44888 | If ` X ` has a property ` ... |
| frege82 44889 | Closed-form deduction base... |
| frege83 44890 | Apply commuted form of ~ f... |
| frege84 44891 | Commuted form of ~ frege81... |
| frege85 44892 | Commuted form of ~ frege77... |
| frege86 44893 | Conclusion about element o... |
| frege87 44894 | If ` Z ` is a result of an... |
| frege88 44895 | Commuted form of ~ frege87... |
| frege89 44896 | One direction of ~ dffrege... |
| frege90 44897 | Add antecedent to ~ frege8... |
| frege91 44898 | Every result of an applica... |
| frege92 44899 | Inference from ~ frege91 .... |
| frege93 44900 | Necessary condition for tw... |
| frege94 44901 | Looking one past a pair re... |
| frege95 44902 | Looking one past a pair re... |
| frege96 44903 | Every result of an applica... |
| frege97 44904 | The property of following ... |
| frege98 44905 | If ` Y ` follows ` X ` and... |
| dffrege99 44906 | If ` Z ` is identical with... |
| frege100 44907 | One direction of ~ dffrege... |
| frege101 44908 | Lemma for ~ frege102 . Pr... |
| frege102 44909 | If ` Z ` belongs to the ` ... |
| frege103 44910 | Proposition 103 of [Frege1... |
| frege104 44911 | Proposition 104 of [Frege1... |
| frege105 44912 | Proposition 105 of [Frege1... |
| frege106 44913 | Whatever follows ` X ` in ... |
| frege107 44914 | Proposition 107 of [Frege1... |
| frege108 44915 | If ` Y ` belongs to the ` ... |
| frege109 44916 | The property of belonging ... |
| frege110 44917 | Proposition 110 of [Frege1... |
| frege111 44918 | If ` Y ` belongs to the ` ... |
| frege112 44919 | Identity implies belonging... |
| frege113 44920 | Proposition 113 of [Frege1... |
| frege114 44921 | If ` X ` belongs to the ` ... |
| dffrege115 44922 | If from the circumstance t... |
| frege116 44923 | One direction of ~ dffrege... |
| frege117 44924 | Lemma for ~ frege118 . Pr... |
| frege118 44925 | Simplified application of ... |
| frege119 44926 | Lemma for ~ frege120 . Pr... |
| frege120 44927 | Simplified application of ... |
| frege121 44928 | Lemma for ~ frege122 . Pr... |
| frege122 44929 | If ` X ` is a result of an... |
| frege123 44930 | Lemma for ~ frege124 . Pr... |
| frege124 44931 | If ` X ` is a result of an... |
| frege125 44932 | Lemma for ~ frege126 . Pr... |
| frege126 44933 | If ` M ` follows ` Y ` in ... |
| frege127 44934 | Communte antecedents of ~ ... |
| frege128 44935 | Lemma for ~ frege129 . Pr... |
| frege129 44936 | If the procedure ` R ` is ... |
| frege130 44937 | Lemma for ~ frege131 . Pr... |
| frege131 44938 | If the procedure ` R ` is ... |
| frege132 44939 | Lemma for ~ frege133 . Pr... |
| frege133 44940 | If the procedure ` R ` is ... |
| enrelmap 44941 | The set of all possible re... |
| enrelmapr 44942 | The set of all possible re... |
| enmappw 44943 | The set of all mappings fr... |
| enmappwid 44944 | The set of all mappings fr... |
| rfovd 44945 | Value of the operator, ` (... |
| rfovfvd 44946 | Value of the operator, ` (... |
| rfovfvfvd 44947 | Value of the operator, ` (... |
| rfovcnvf1od 44948 | Properties of the operator... |
| rfovcnvd 44949 | Value of the converse of t... |
| rfovf1od 44950 | The value of the operator,... |
| rfovcnvfvd 44951 | Value of the converse of t... |
| fsovd 44952 | Value of the operator, ` (... |
| fsovrfovd 44953 | The operator which gives a... |
| fsovfvd 44954 | Value of the operator, ` (... |
| fsovfvfvd 44955 | Value of the operator, ` (... |
| fsovfd 44956 | The operator, ` ( A O B ) ... |
| fsovcnvlem 44957 | The ` O ` operator, which ... |
| fsovcnvd 44958 | The value of the converse ... |
| fsovcnvfvd 44959 | The value of the converse ... |
| fsovf1od 44960 | The value of ` ( A O B ) `... |
| dssmapfvd 44961 | Value of the duality opera... |
| dssmapfv2d 44962 | Value of the duality opera... |
| dssmapfv3d 44963 | Value of the duality opera... |
| dssmapnvod 44964 | For any base set ` B ` the... |
| dssmapf1od 44965 | For any base set ` B ` the... |
| dssmap2d 44966 | For any base set ` B ` the... |
| or3or 44967 | Decompose disjunction into... |
| andi3or 44968 | Distribute over triple dis... |
| uneqsn 44969 | If a union of classes is e... |
| brfvimex 44970 | If a binary relation holds... |
| brovmptimex 44971 | If a binary relation holds... |
| brovmptimex1 44972 | If a binary relation holds... |
| brovmptimex2 44973 | If a binary relation holds... |
| brcoffn 44974 | Conditions allowing the de... |
| brcofffn 44975 | Conditions allowing the de... |
| brco2f1o 44976 | Conditions allowing the de... |
| brco3f1o 44977 | Conditions allowing the de... |
| ntrclsbex 44978 | If (pseudo-)interior and (... |
| ntrclsrcomplex 44979 | The relative complement of... |
| neik0imk0p 44980 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 44981 | If an interior function is... |
| ntrkbimka 44982 | If the interiors of disjoi... |
| ntrk0kbimka 44983 | If the interiors of disjoi... |
| clsk3nimkb 44984 | If the base set is not emp... |
| clsk1indlem0 44985 | The ansatz closure functio... |
| clsk1indlem2 44986 | The ansatz closure functio... |
| clsk1indlem3 44987 | The ansatz closure functio... |
| clsk1indlem4 44988 | The ansatz closure functio... |
| clsk1indlem1 44989 | The ansatz closure functio... |
| clsk1independent 44990 | For generalized closure fu... |
| neik0pk1imk0 44991 | Kuratowski's K0' and K1 ax... |
| isotone1 44992 | Two different ways to say ... |
| isotone2 44993 | Two different ways to say ... |
| ntrk1k3eqk13 44994 | An interior function is bo... |
| ntrclsf1o 44995 | If (pseudo-)interior and (... |
| ntrclsnvobr 44996 | If (pseudo-)interior and (... |
| ntrclsiex 44997 | If (pseudo-)interior and (... |
| ntrclskex 44998 | If (pseudo-)interior and (... |
| ntrclsfv1 44999 | If (pseudo-)interior and (... |
| ntrclsfv2 45000 | If (pseudo-)interior and (... |
| ntrclselnel1 45001 | If (pseudo-)interior and (... |
| ntrclselnel2 45002 | If (pseudo-)interior and (... |
| ntrclsfv 45003 | The value of the interior ... |
| ntrclsfveq1 45004 | If interior and closure fu... |
| ntrclsfveq2 45005 | If interior and closure fu... |
| ntrclsfveq 45006 | If interior and closure fu... |
| ntrclsss 45007 | If interior and closure fu... |
| ntrclsneine0lem 45008 | If (pseudo-)interior and (... |
| ntrclsneine0 45009 | If (pseudo-)interior and (... |
| ntrclscls00 45010 | If (pseudo-)interior and (... |
| ntrclsiso 45011 | If (pseudo-)interior and (... |
| ntrclsk2 45012 | An interior function is co... |
| ntrclskb 45013 | The interiors of disjoint ... |
| ntrclsk3 45014 | The intersection of interi... |
| ntrclsk13 45015 | The interior of the inters... |
| ntrclsk4 45016 | Idempotence of the interio... |
| ntrneibex 45017 | If (pseudo-)interior and (... |
| ntrneircomplex 45018 | The relative complement of... |
| ntrneif1o 45019 | If (pseudo-)interior and (... |
| ntrneiiex 45020 | If (pseudo-)interior and (... |
| ntrneinex 45021 | If (pseudo-)interior and (... |
| ntrneicnv 45022 | If (pseudo-)interior and (... |
| ntrneifv1 45023 | If (pseudo-)interior and (... |
| ntrneifv2 45024 | If (pseudo-)interior and (... |
| ntrneiel 45025 | If (pseudo-)interior and (... |
| ntrneifv3 45026 | The value of the neighbors... |
| ntrneineine0lem 45027 | If (pseudo-)interior and (... |
| ntrneineine1lem 45028 | If (pseudo-)interior and (... |
| ntrneifv4 45029 | The value of the interior ... |
| ntrneiel2 45030 | Membership in iterated int... |
| ntrneineine0 45031 | If (pseudo-)interior and (... |
| ntrneineine1 45032 | If (pseudo-)interior and (... |
| ntrneicls00 45033 | If (pseudo-)interior and (... |
| ntrneicls11 45034 | If (pseudo-)interior and (... |
| ntrneiiso 45035 | If (pseudo-)interior and (... |
| ntrneik2 45036 | An interior function is co... |
| ntrneix2 45037 | An interior (closure) func... |
| ntrneikb 45038 | The interiors of disjoint ... |
| ntrneixb 45039 | The interiors (closures) o... |
| ntrneik3 45040 | The intersection of interi... |
| ntrneix3 45041 | The closure of the union o... |
| ntrneik13 45042 | The interior of the inters... |
| ntrneix13 45043 | The closure of the union o... |
| ntrneik4w 45044 | Idempotence of the interio... |
| ntrneik4 45045 | Idempotence of the interio... |
| clsneibex 45046 | If (pseudo-)closure and (p... |
| clsneircomplex 45047 | The relative complement of... |
| clsneif1o 45048 | If a (pseudo-)closure func... |
| clsneicnv 45049 | If a (pseudo-)closure func... |
| clsneikex 45050 | If closure and neighborhoo... |
| clsneinex 45051 | If closure and neighborhoo... |
| clsneiel1 45052 | If a (pseudo-)closure func... |
| clsneiel2 45053 | If a (pseudo-)closure func... |
| clsneifv3 45054 | Value of the neighborhoods... |
| clsneifv4 45055 | Value of the closure (inte... |
| neicvgbex 45056 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 45057 | The relative complement of... |
| neicvgf1o 45058 | If neighborhood and conver... |
| neicvgnvo 45059 | If neighborhood and conver... |
| neicvgnvor 45060 | If neighborhood and conver... |
| neicvgmex 45061 | If the neighborhoods and c... |
| neicvgnex 45062 | If the neighborhoods and c... |
| neicvgel1 45063 | A subset being an element ... |
| neicvgel2 45064 | The complement of a subset... |
| neicvgfv 45065 | The value of the neighborh... |
| ntrrn 45066 | The range of the interior ... |
| ntrf 45067 | The interior function of a... |
| ntrf2 45068 | The interior function is a... |
| ntrelmap 45069 | The interior function is a... |
| clsf2 45070 | The closure function is a ... |
| clselmap 45071 | The closure function is a ... |
| dssmapntrcls 45072 | The interior and closure o... |
| dssmapclsntr 45073 | The closure and interior o... |
| gneispa 45074 | Each point ` p ` of the ne... |
| gneispb 45075 | Given a neighborhood ` N `... |
| gneispace2 45076 | The predicate that ` F ` i... |
| gneispace3 45077 | The predicate that ` F ` i... |
| gneispace 45078 | The predicate that ` F ` i... |
| gneispacef 45079 | A generic neighborhood spa... |
| gneispacef2 45080 | A generic neighborhood spa... |
| gneispacefun 45081 | A generic neighborhood spa... |
| gneispacern 45082 | A generic neighborhood spa... |
| gneispacern2 45083 | A generic neighborhood spa... |
| gneispace0nelrn 45084 | A generic neighborhood spa... |
| gneispace0nelrn2 45085 | A generic neighborhood spa... |
| gneispace0nelrn3 45086 | A generic neighborhood spa... |
| gneispaceel 45087 | Every neighborhood of a po... |
| gneispaceel2 45088 | Every neighborhood of a po... |
| gneispacess 45089 | All supersets of a neighbo... |
| gneispacess2 45090 | All supersets of a neighbo... |
| k0004lem1 45091 | Application of ~ ssin to r... |
| k0004lem2 45092 | A mapping with a particula... |
| k0004lem3 45093 | When the value of a mappin... |
| k0004val 45094 | The topological simplex of... |
| k0004ss1 45095 | The topological simplex of... |
| k0004ss2 45096 | The topological simplex of... |
| k0004ss3 45097 | The topological simplex of... |
| k0004val0 45098 | The topological simplex of... |
| inductionexd 45099 | Simple induction example. ... |
| wwlemuld 45100 | Natural deduction form of ... |
| leeq1d 45101 | Specialization of ~ breq1d... |
| leeq2d 45102 | Specialization of ~ breq2d... |
| absmulrposd 45103 | Specialization of absmuld ... |
| imadisjld 45104 | Natural dduction form of o... |
| wnefimgd 45105 | The image of a mapping fro... |
| fco2d 45106 | Natural deduction form of ... |
| wfximgfd 45107 | The value of a function on... |
| extoimad 45108 | If |f(x)| <= C for all x t... |
| imo72b2lem0 45109 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 45110 | Natural deduction form of ... |
| imo72b2lem2 45111 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 45112 | Natural deduction form of ... |
| imo72b2lem1 45113 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 45114 | 'Less than or equal to' re... |
| lemuldiv4d 45115 | 'Less than or equal to' re... |
| imo72b2 45116 | IMO 1972 B2. (14th Intern... |
| int-addcomd 45117 | AdditionCommutativity gene... |
| int-addassocd 45118 | AdditionAssociativity gene... |
| int-addsimpd 45119 | AdditionSimplification gen... |
| int-mulcomd 45120 | MultiplicationCommutativit... |
| int-mulassocd 45121 | MultiplicationAssociativit... |
| int-mulsimpd 45122 | MultiplicationSimplificati... |
| int-leftdistd 45123 | AdditionMultiplicationLeft... |
| int-rightdistd 45124 | AdditionMultiplicationRigh... |
| int-sqdefd 45125 | SquareDefinition generator... |
| int-mul11d 45126 | First MultiplicationOne ge... |
| int-mul12d 45127 | Second MultiplicationOne g... |
| int-add01d 45128 | First AdditionZero generat... |
| int-add02d 45129 | Second AdditionZero genera... |
| int-sqgeq0d 45130 | SquareGEQZero generator ru... |
| int-eqprincd 45131 | PrincipleOfEquality genera... |
| int-eqtransd 45132 | EqualityTransitivity gener... |
| int-eqmvtd 45133 | EquMoveTerm generator rule... |
| int-eqineqd 45134 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 45135 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 45136 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 45137 | SecondPrincipleOfInequalit... |
| int-ineqtransd 45138 | InequalityTransitivity gen... |
| unitadd 45139 | Theorem used in conjunctio... |
| gsumws3 45140 | Valuation of a length 3 wo... |
| gsumws4 45141 | Valuation of a length 4 wo... |
| amgm2d 45142 | Arithmetic-geometric mean ... |
| amgm3d 45143 | Arithmetic-geometric mean ... |
| amgm4d 45144 | Arithmetic-geometric mean ... |
| spALT 45145 | ~ sp can be proven from th... |
| rr-spce 45146 | Prove an existential. (Co... |
| rexlimdvaacbv 45147 | Unpack a restricted existe... |
| rexlimddvcbvw 45148 | Unpack a restricted existe... |
| rexlimddvcbv 45149 | Unpack a restricted existe... |
| rr-elrnmpt3d 45150 | Elementhood in an image se... |
| rr-phpd 45151 | Equivalent of ~ php withou... |
| tfindsd 45152 | Deduction associated with ... |
| mnringvald 45155 | Value of the monoid ring f... |
| mnringnmulrd 45156 | Components of a monoid rin... |
| mnringbased 45157 | The base set of a monoid r... |
| mnringbaserd 45158 | The base set of a monoid r... |
| mnringelbased 45159 | Membership in the base set... |
| mnringbasefd 45160 | Elements of a monoid ring ... |
| mnringbasefsuppd 45161 | Elements of a monoid ring ... |
| mnringaddgd 45162 | The additive operation of ... |
| mnring0gd 45163 | The additive identity of a... |
| mnring0g2d 45164 | The additive identity of a... |
| mnringmulrd 45165 | The ring product of a mono... |
| mnringscad 45166 | The scalar ring of a monoi... |
| mnringvscad 45167 | The scalar product of a mo... |
| mnringlmodd 45168 | Monoid rings are left modu... |
| mnringmulrvald 45169 | Value of multiplication in... |
| mnringmulrcld 45170 | Monoid rings are closed un... |
| gru0eld 45171 | A nonempty Grothendieck un... |
| grusucd 45172 | Grothendieck universes are... |
| r1rankcld 45173 | Any rank of the cumulative... |
| grur1cld 45174 | Grothendieck universes are... |
| grurankcld 45175 | Grothendieck universes are... |
| grurankrcld 45176 | If a Grothendieck universe... |
| gruscottcld 45177 | If a Grothendieck universe... |
| dfcoll2 45180 | Alternate definition of th... |
| colleq12d 45181 | Equality theorem for the c... |
| colleq1 45182 | Equality theorem for the c... |
| colleq2 45183 | Equality theorem for the c... |
| nfcoll 45184 | Bound-variable hypothesis ... |
| collexd 45185 | The output of the collecti... |
| cpcolld 45186 | Property of the collection... |
| cpcoll2d 45187 | ~ cpcolld with an extra ex... |
| grucollcld 45188 | A Grothendieck universe co... |
| ismnu 45189 | The hypothesis of this the... |
| mnuop123d 45190 | Operations of a minimal un... |
| mnussd 45191 | Minimal universes are clos... |
| mnuss2d 45192 | ~ mnussd with arguments pr... |
| mnu0eld 45193 | A nonempty minimal univers... |
| mnuop23d 45194 | Second and third operation... |
| mnupwd 45195 | Minimal universes are clos... |
| mnusnd 45196 | Minimal universes are clos... |
| mnuprssd 45197 | A minimal universe contain... |
| mnuprss2d 45198 | Special case of ~ mnuprssd... |
| mnuop3d 45199 | Third operation of a minim... |
| mnuprdlem1 45200 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 45201 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 45202 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 45203 | Lemma for ~ mnuprd . Gene... |
| mnuprd 45204 | Minimal universes are clos... |
| mnuunid 45205 | Minimal universes are clos... |
| mnuund 45206 | Minimal universes are clos... |
| mnutrcld 45207 | Minimal universes contain ... |
| mnutrd 45208 | Minimal universes are tran... |
| mnurndlem1 45209 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 45210 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 45211 | Minimal universes contain ... |
| mnugrud 45212 | Minimal universes are Grot... |
| grumnudlem 45213 | Lemma for ~ grumnud . (Co... |
| grumnud 45214 | Grothendieck universes are... |
| grumnueq 45215 | The class of Grothendieck ... |
| expandan 45216 | Expand conjunction to prim... |
| expandexn 45217 | Expand an existential quan... |
| expandral 45218 | Expand a restricted univer... |
| expandrexn 45219 | Expand a restricted existe... |
| expandrex 45220 | Expand a restricted existe... |
| expanduniss 45221 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 45222 | Express the predicate on `... |
| rr-grothprimbi 45223 | Express "every set is cont... |
| inagrud 45224 | Inaccessible levels of the... |
| inaex 45225 | Assuming the Tarski-Grothe... |
| gruex 45226 | Assuming the Tarski-Grothe... |
| rr-groth 45227 | An equivalent of ~ ax-grot... |
| rr-grothprim 45228 | An equivalent of ~ ax-grot... |
| ismnushort 45229 | Express the predicate on `... |
| dfuniv2 45230 | Alternative definition of ... |
| rr-grothshortbi 45231 | Express "every set is cont... |
| rr-grothshort 45232 | A shorter equivalent of ~ ... |
| nanorxor 45233 | 'nand' is equivalent to th... |
| undisjrab 45234 | Union of two disjoint rest... |
| iso0 45235 | The empty set is an ` R , ... |
| ssrecnpr 45236 | ` RR ` is a subset of both... |
| seff 45237 | Let set ` S ` be the real ... |
| sblpnf 45238 | The infinity ball in the a... |
| prmunb2 45239 | The primes are unbounded. ... |
| dvgrat 45240 | Ratio test for divergence ... |
| cvgdvgrat 45241 | Ratio test for convergence... |
| radcnvrat 45242 | Let ` L ` be the limit, if... |
| reldvds 45243 | The divides relation is in... |
| nznngen 45244 | All positive integers in t... |
| nzss 45245 | The set of multiples of _m... |
| nzin 45246 | The intersection of the se... |
| nzprmdif 45247 | Subtract one prime's multi... |
| hashnzfz 45248 | Special case of ~ hashdvds... |
| hashnzfz2 45249 | Special case of ~ hashnzfz... |
| hashnzfzclim 45250 | As the upper bound ` K ` o... |
| caofcan 45251 | Transfer a cancellation la... |
| ofsubid 45252 | Function analogue of ~ sub... |
| ofmul12 45253 | Function analogue of ~ mul... |
| ofdivrec 45254 | Function analogue of ~ div... |
| ofdivcan4 45255 | Function analogue of ~ div... |
| ofdivdiv2 45256 | Function analogue of ~ div... |
| lhe4.4ex1a 45257 | Example of the Fundamental... |
| dvsconst 45258 | Derivative of a constant f... |
| dvsid 45259 | Derivative of the identity... |
| dvsef 45260 | Derivative of the exponent... |
| expgrowthi 45261 | Exponential growth and dec... |
| dvconstbi 45262 | The derivative of a functi... |
| expgrowth 45263 | Exponential growth and dec... |
| bccval 45266 | Value of the generalized b... |
| bcccl 45267 | Closure of the generalized... |
| bcc0 45268 | The generalized binomial c... |
| bccp1k 45269 | Generalized binomial coeff... |
| bccm1k 45270 | Generalized binomial coeff... |
| bccn0 45271 | Generalized binomial coeff... |
| bccn1 45272 | Generalized binomial coeff... |
| bccbc 45273 | The binomial coefficient a... |
| uzmptshftfval 45274 | When ` F ` is a maps-to fu... |
| dvradcnv2 45275 | The radius of convergence ... |
| binomcxplemwb 45276 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 45277 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 45278 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 45279 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 45280 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 45281 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 45282 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 45283 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 45284 | Lemma for ~ binomcxp . Wh... |
| binomcxp 45285 | Generalize the binomial th... |
| pm10.12 45286 | Theorem *10.12 in [Whitehe... |
| pm10.14 45287 | Theorem *10.14 in [Whitehe... |
| pm10.251 45288 | Theorem *10.251 in [Whiteh... |
| pm10.252 45289 | Theorem *10.252 in [Whiteh... |
| pm10.253 45290 | Theorem *10.253 in [Whiteh... |
| albitr 45291 | Theorem *10.301 in [Whiteh... |
| pm10.42 45292 | Theorem *10.42 in [Whitehe... |
| pm10.52 45293 | Theorem *10.52 in [Whitehe... |
| pm10.53 45294 | Theorem *10.53 in [Whitehe... |
| pm10.541 45295 | Theorem *10.541 in [Whiteh... |
| pm10.542 45296 | Theorem *10.542 in [Whiteh... |
| pm10.55 45297 | Theorem *10.55 in [Whitehe... |
| pm10.56 45298 | Theorem *10.56 in [Whitehe... |
| pm10.57 45299 | Theorem *10.57 in [Whitehe... |
| 2alanimi 45300 | Removes two universal quan... |
| 2al2imi 45301 | Removes two universal quan... |
| pm11.11 45302 | Theorem *11.11 in [Whitehe... |
| pm11.12 45303 | Theorem *11.12 in [Whitehe... |
| 19.21vv 45304 | Compare Theorem *11.3 in [... |
| 2alim 45305 | Theorem *11.32 in [Whitehe... |
| 2albi 45306 | Theorem *11.33 in [Whitehe... |
| 2exim 45307 | Theorem *11.34 in [Whitehe... |
| 2exbi 45308 | Theorem *11.341 in [Whiteh... |
| spsbce-2 45309 | Theorem *11.36 in [Whitehe... |
| 19.33-2 45310 | Theorem *11.421 in [Whiteh... |
| 19.36vv 45311 | Theorem *11.43 in [Whitehe... |
| 19.31vv 45312 | Theorem *11.44 in [Whitehe... |
| 19.37vv 45313 | Theorem *11.46 in [Whitehe... |
| 19.28vv 45314 | Theorem *11.47 in [Whitehe... |
| pm11.52 45315 | Theorem *11.52 in [Whitehe... |
| aaanv 45316 | Theorem *11.56 in [Whitehe... |
| pm11.57 45317 | Theorem *11.57 in [Whitehe... |
| pm11.58 45318 | Theorem *11.58 in [Whitehe... |
| pm11.59 45319 | Theorem *11.59 in [Whitehe... |
| pm11.6 45320 | Theorem *11.6 in [Whitehea... |
| pm11.61 45321 | Theorem *11.61 in [Whitehe... |
| pm11.62 45322 | Theorem *11.62 in [Whitehe... |
| pm11.63 45323 | Theorem *11.63 in [Whitehe... |
| pm11.7 45324 | Theorem *11.7 in [Whitehea... |
| pm11.71 45325 | Theorem *11.71 in [Whitehe... |
| sbeqal1 45326 | If ` x = y ` always implie... |
| sbeqal1i 45327 | Suppose you know ` x = y `... |
| sbeqal2i 45328 | If ` x = y ` implies ` x =... |
| axc5c4c711 45329 | Proof of a theorem that ca... |
| axc5c4c711toc5 45330 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 45331 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 45332 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 45333 | Rederivation of ~ ax-11 fr... |
| axc11next 45334 | This theorem shows that, g... |
| pm13.13a 45335 | One result of theorem *13.... |
| pm13.13b 45336 | Theorem *13.13 in [Whitehe... |
| pm13.14 45337 | Theorem *13.14 in [Whitehe... |
| pm13.192 45338 | Theorem *13.192 in [Whiteh... |
| pm13.193 45339 | Theorem *13.193 in [Whiteh... |
| pm13.194 45340 | Theorem *13.194 in [Whiteh... |
| pm13.195 45341 | Theorem *13.195 in [Whiteh... |
| pm13.196a 45342 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 45343 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 45344 | Theorem *13.22 in [Whitehe... |
| iotain 45345 | Equivalence between two di... |
| iotaexeu 45346 | The iota class exists. Th... |
| iotasbc 45347 | Definition *14.01 in [Whit... |
| iotasbc2 45348 | Theorem *14.111 in [Whiteh... |
| pm14.12 45349 | Theorem *14.12 in [Whitehe... |
| pm14.122a 45350 | Theorem *14.122 in [Whiteh... |
| pm14.122b 45351 | Theorem *14.122 in [Whiteh... |
| pm14.122c 45352 | Theorem *14.122 in [Whiteh... |
| pm14.123a 45353 | Theorem *14.123 in [Whiteh... |
| pm14.123b 45354 | Theorem *14.123 in [Whiteh... |
| pm14.123c 45355 | Theorem *14.123 in [Whiteh... |
| pm14.18 45356 | Theorem *14.18 in [Whitehe... |
| iotaequ 45357 | Theorem *14.2 in [Whitehea... |
| iotavalb 45358 | Theorem *14.202 in [Whiteh... |
| iotasbc5 45359 | Theorem *14.205 in [Whiteh... |
| pm14.24 45360 | Theorem *14.24 in [Whitehe... |
| iotavalsb 45361 | Theorem *14.242 in [Whiteh... |
| sbiota1 45362 | Theorem *14.25 in [Whitehe... |
| sbaniota 45363 | Theorem *14.26 in [Whitehe... |
| iotasbcq 45364 | Theorem *14.272 in [Whiteh... |
| elnev 45365 | Any set that contains one ... |
| rusbcALT 45366 | A version of Russell's par... |
| compeq 45367 | Equality between two ways ... |
| compne 45368 | The complement of ` A ` is... |
| compab 45369 | Two ways of saying "the co... |
| conss2 45370 | Contrapositive law for sub... |
| conss1 45371 | Contrapositive law for sub... |
| ralbidar 45372 | More general form of ~ ral... |
| rexbidar 45373 | More general form of ~ rex... |
| dropab1 45374 | Theorem to aid use of the ... |
| dropab2 45375 | Theorem to aid use of the ... |
| ipo0 45376 | If the identity relation p... |
| ifr0 45377 | A class that is founded by... |
| fvsb 45378 | Explicit substitution of a... |
| fveqsb 45379 | Implicit substitution of a... |
| xpexb 45380 | A Cartesian product exists... |
| trelpss 45381 | An element of a transitive... |
| addcomgi 45382 | Generalization of commutat... |
| addrval 45392 | Value of the operation of ... |
| subrval 45393 | Value of the operation of ... |
| mulvval 45394 | Value of the operation of ... |
| addrfv 45395 | Vector addition at a value... |
| subrfv 45396 | Vector subtraction at a va... |
| mulvfv 45397 | Scalar multiplication at a... |
| addrfn 45398 | Vector addition produces a... |
| subrfn 45399 | Vector subtraction produce... |
| mulvfn 45400 | Scalar multiplication prod... |
| addrcom 45401 | Vector addition is commuta... |
| idiALT 45405 | Placeholder for ~ idi . T... |
| exbir 45406 | Exportation implication al... |
| 3impexpbicom 45407 | Version of ~ 3impexp where... |
| 3impexpbicomi 45408 | Inference associated with ... |
| bi1imp 45409 | Importation inference simi... |
| bi2imp 45410 | Importation inference simi... |
| bi3impb 45411 | Similar to ~ 3impb with im... |
| bi3impa 45412 | Similar to ~ 3impa with im... |
| bi23impib 45413 | ~ 3impib with the inner im... |
| bi13impib 45414 | ~ 3impib with the outer im... |
| bi123impib 45415 | ~ 3impib with the implicat... |
| bi13impia 45416 | ~ 3impia with the outer im... |
| bi123impia 45417 | ~ 3impia with the implicat... |
| bi33imp12 45418 | ~ 3imp with innermost impl... |
| bi13imp23 45419 | ~ 3imp with outermost impl... |
| bi13imp2 45420 | Similar to ~ 3imp except t... |
| bi12imp3 45421 | Similar to ~ 3imp except a... |
| bi23imp1 45422 | Similar to ~ 3imp except a... |
| bi123imp0 45423 | Similar to ~ 3imp except a... |
| 4animp1 45424 | A single hypothesis unific... |
| 4an31 45425 | A rearrangement of conjunc... |
| 4an4132 45426 | A rearrangement of conjunc... |
| expcomdg 45427 | Biconditional form of ~ ex... |
| iidn3 45428 | ~ idn3 without virtual ded... |
| ee222 45429 | ~ e222 without virtual ded... |
| ee3bir 45430 | Right-biconditional form o... |
| ee13 45431 | ~ e13 without virtual dedu... |
| ee121 45432 | ~ e121 without virtual ded... |
| ee122 45433 | ~ e122 without virtual ded... |
| ee333 45434 | ~ e333 without virtual ded... |
| ee323 45435 | ~ e323 without virtual ded... |
| 3ornot23 45436 | If the second and third di... |
| orbi1r 45437 | ~ orbi1 with order of disj... |
| 3orbi123 45438 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 45439 | Closed form of ~ syl5 . D... |
| impexpd 45440 | The following User's Proof... |
| com3rgbi 45441 | The following User's Proof... |
| impexpdcom 45442 | The following User's Proof... |
| ee1111 45443 | Non-virtual deduction form... |
| pm2.43bgbi 45444 | Logical equivalence of a 2... |
| pm2.43cbi 45445 | Logical equivalence of a 3... |
| ee233 45446 | Non-virtual deduction form... |
| imbi13 45447 | Join three logical equival... |
| ee33 45448 | Non-virtual deduction form... |
| con5 45449 | Biconditional contrapositi... |
| con5i 45450 | Inference form of ~ con5 .... |
| exlimexi 45451 | Inference similar to Theor... |
| sb5ALT 45452 | Equivalence for substituti... |
| eexinst01 45453 | ~ exinst01 without virtual... |
| eexinst11 45454 | ~ exinst11 without virtual... |
| vk15.4j 45455 | Excercise 4j of Unit 15 of... |
| notnotrALT 45456 | Converse of double negatio... |
| con3ALT2 45457 | Contraposition. Alternate... |
| ssralv2 45458 | Quantification restricted ... |
| sbc3or 45459 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 45460 | Closed form of ~ alrimi wi... |
| rspsbc2 45461 | ~ rspsbc with two quantify... |
| sbcoreleleq 45462 | Substitution of a setvar v... |
| tratrb 45463 | If a class is transitive a... |
| ordelordALT 45464 | An element of an ordinal c... |
| sbcim2g 45465 | Distribution of class subs... |
| sbcbi 45466 | Implication form of ~ sbcb... |
| trsbc 45467 | Formula-building inference... |
| truniALT 45468 | The union of a class of tr... |
| onfrALTlem5 45469 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 45470 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 45471 | Lemma for ~ onfrALT . (Co... |
| ggen31 45472 | ~ gen31 without virtual de... |
| onfrALTlem2 45473 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 45474 | A theorem pertaining to th... |
| onfrALTlem1 45475 | Lemma for ~ onfrALT . (Co... |
| onfrALT 45476 | The membership relation is... |
| 19.41rg 45477 | Closed form of right-to-le... |
| opelopab4 45478 | Ordered pair membership in... |
| 2pm13.193 45479 | ~ pm13.193 for two variabl... |
| hbntal 45480 | A closed form of ~ hbn . ~... |
| hbimpg 45481 | A closed form of ~ hbim . ... |
| hbalg 45482 | Closed form of ~ hbal . D... |
| hbexg 45483 | Closed form of ~ nfex . D... |
| ax6e2eq 45484 | Alternate form of ~ ax6e f... |
| ax6e2nd 45485 | If at least two sets exist... |
| ax6e2ndeq 45486 | "At least two sets exist" ... |
| 2sb5nd 45487 | Equivalence for double sub... |
| 2uasbanh 45488 | Distribute the unabbreviat... |
| 2uasban 45489 | Distribute the unabbreviat... |
| e2ebind 45490 | Absorption of an existenti... |
| elpwgded 45491 | ~ elpwgdedVD in convention... |
| trelded 45492 | Deduction form of ~ trel .... |
| jaoded 45493 | Deduction form of ~ jao . ... |
| sbtT 45494 | A substitution into a theo... |
| not12an2impnot1 45495 | If a double conjunction is... |
| in1 45498 | Inference form of ~ df-vd1... |
| iin1 45499 | ~ in1 without virtual dedu... |
| dfvd1ir 45500 | Inference form of ~ df-vd1... |
| idn1 45501 | Virtual deduction identity... |
| dfvd1imp 45502 | Left-to-right part of defi... |
| dfvd1impr 45503 | Right-to-left part of defi... |
| dfvd2 45506 | Definition of a 2-hypothes... |
| dfvd2an 45509 | Definition of a 2-hypothes... |
| dfvd2ani 45510 | Inference form of ~ dfvd2a... |
| dfvd2anir 45511 | Right-to-left inference fo... |
| dfvd2i 45512 | Inference form of ~ dfvd2 ... |
| dfvd2ir 45513 | Right-to-left inference fo... |
| dfvd3 45518 | Definition of a 3-hypothes... |
| dfvd3i 45519 | Inference form of ~ dfvd3 ... |
| dfvd3ir 45520 | Right-to-left inference fo... |
| dfvd3an 45521 | Definition of a 3-hypothes... |
| dfvd3ani 45522 | Inference form of ~ dfvd3a... |
| dfvd3anir 45523 | Right-to-left inference fo... |
| vd01 45524 | A virtual hypothesis virtu... |
| vd02 45525 | Two virtual hypotheses vir... |
| vd03 45526 | A theorem is virtually inf... |
| vd12 45527 | A virtual deduction with 1... |
| vd13 45528 | A virtual deduction with 1... |
| vd23 45529 | A virtual deduction with 2... |
| dfvd2imp 45530 | The virtual deduction form... |
| dfvd2impr 45531 | A 2-antecedent nested impl... |
| in2 45532 | The virtual deduction intr... |
| int2 45533 | The virtual deduction intr... |
| iin2 45534 | ~ in2 without virtual dedu... |
| in2an 45535 | The virtual deduction intr... |
| in3 45536 | The virtual deduction intr... |
| iin3 45537 | ~ in3 without virtual dedu... |
| in3an 45538 | The virtual deduction intr... |
| int3 45539 | The virtual deduction intr... |
| idn2 45540 | Virtual deduction identity... |
| iden2 45541 | Virtual deduction identity... |
| idn3 45542 | Virtual deduction identity... |
| gen11 45543 | Virtual deduction generali... |
| gen11nv 45544 | Virtual deduction generali... |
| gen12 45545 | Virtual deduction generali... |
| gen21 45546 | Virtual deduction generali... |
| gen21nv 45547 | Virtual deduction form of ... |
| gen31 45548 | Virtual deduction generali... |
| gen22 45549 | Virtual deduction generali... |
| ggen22 45550 | ~ gen22 without virtual de... |
| exinst 45551 | Existential Instantiation.... |
| exinst01 45552 | Existential Instantiation.... |
| exinst11 45553 | Existential Instantiation.... |
| e1a 45554 | A Virtual deduction elimin... |
| el1 45555 | A Virtual deduction elimin... |
| e1bi 45556 | Biconditional form of ~ e1... |
| e1bir 45557 | Right biconditional form o... |
| e2 45558 | A virtual deduction elimin... |
| e2bi 45559 | Biconditional form of ~ e2... |
| e2bir 45560 | Right biconditional form o... |
| ee223 45561 | ~ e223 without virtual ded... |
| e223 45562 | A virtual deduction elimin... |
| e222 45563 | A virtual deduction elimin... |
| e220 45564 | A virtual deduction elimin... |
| ee220 45565 | ~ e220 without virtual ded... |
| e202 45566 | A virtual deduction elimin... |
| ee202 45567 | ~ e202 without virtual ded... |
| e022 45568 | A virtual deduction elimin... |
| ee022 45569 | ~ e022 without virtual ded... |
| e002 45570 | A virtual deduction elimin... |
| ee002 45571 | ~ e002 without virtual ded... |
| e020 45572 | A virtual deduction elimin... |
| ee020 45573 | ~ e020 without virtual ded... |
| e200 45574 | A virtual deduction elimin... |
| ee200 45575 | ~ e200 without virtual ded... |
| e221 45576 | A virtual deduction elimin... |
| ee221 45577 | ~ e221 without virtual ded... |
| e212 45578 | A virtual deduction elimin... |
| ee212 45579 | ~ e212 without virtual ded... |
| e122 45580 | A virtual deduction elimin... |
| e112 45581 | A virtual deduction elimin... |
| ee112 45582 | ~ e112 without virtual ded... |
| e121 45583 | A virtual deduction elimin... |
| e211 45584 | A virtual deduction elimin... |
| ee211 45585 | ~ e211 without virtual ded... |
| e210 45586 | A virtual deduction elimin... |
| ee210 45587 | ~ e210 without virtual ded... |
| e201 45588 | A virtual deduction elimin... |
| ee201 45589 | ~ e201 without virtual ded... |
| e120 45590 | A virtual deduction elimin... |
| ee120 45591 | Virtual deduction rule ~ e... |
| e021 45592 | A virtual deduction elimin... |
| ee021 45593 | ~ e021 without virtual ded... |
| e012 45594 | A virtual deduction elimin... |
| ee012 45595 | ~ e012 without virtual ded... |
| e102 45596 | A virtual deduction elimin... |
| ee102 45597 | ~ e102 without virtual ded... |
| e22 45598 | A virtual deduction elimin... |
| e22an 45599 | Conjunction form of ~ e22 ... |
| ee22an 45600 | ~ e22an without virtual de... |
| e111 45601 | A virtual deduction elimin... |
| e1111 45602 | A virtual deduction elimin... |
| e110 45603 | A virtual deduction elimin... |
| ee110 45604 | ~ e110 without virtual ded... |
| e101 45605 | A virtual deduction elimin... |
| ee101 45606 | ~ e101 without virtual ded... |
| e011 45607 | A virtual deduction elimin... |
| ee011 45608 | ~ e011 without virtual ded... |
| e100 45609 | A virtual deduction elimin... |
| ee100 45610 | ~ e100 without virtual ded... |
| e010 45611 | A virtual deduction elimin... |
| ee010 45612 | ~ e010 without virtual ded... |
| e001 45613 | A virtual deduction elimin... |
| ee001 45614 | ~ e001 without virtual ded... |
| e11 45615 | A virtual deduction elimin... |
| e11an 45616 | Conjunction form of ~ e11 ... |
| ee11an 45617 | ~ e11an without virtual de... |
| e01 45618 | A virtual deduction elimin... |
| e01an 45619 | Conjunction form of ~ e01 ... |
| ee01an 45620 | ~ e01an without virtual de... |
| e10 45621 | A virtual deduction elimin... |
| e10an 45622 | Conjunction form of ~ e10 ... |
| ee10an 45623 | ~ e10an without virtual de... |
| e02 45624 | A virtual deduction elimin... |
| e02an 45625 | Conjunction form of ~ e02 ... |
| ee02an 45626 | ~ e02an without virtual de... |
| eel021old 45627 | ~ el021old without virtual... |
| el021old 45628 | A virtual deduction elimin... |
| eel000cT 45629 | An elimination deduction. ... |
| eel0TT 45630 | An elimination deduction. ... |
| eelT00 45631 | An elimination deduction. ... |
| eelTTT 45632 | An elimination deduction. ... |
| eelT11 45633 | An elimination deduction. ... |
| eelT1 45634 | Syllogism inference combin... |
| eelT12 45635 | An elimination deduction. ... |
| eelTT1 45636 | An elimination deduction. ... |
| eelT01 45637 | An elimination deduction. ... |
| eel0T1 45638 | An elimination deduction. ... |
| eel12131 45639 | An elimination deduction. ... |
| eel2131 45640 | ~ syl2an with antecedents ... |
| eel3132 45641 | ~ syl2an with antecedents ... |
| eel0321old 45642 | ~ el0321old without virtua... |
| el0321old 45643 | A virtual deduction elimin... |
| eel2122old 45644 | ~ el2122old without virtua... |
| el2122old 45645 | A virtual deduction elimin... |
| eel0000 45646 | Elimination rule similar t... |
| eel00001 45647 | An elimination deduction. ... |
| eel00000 45648 | Elimination rule similar ~... |
| eel11111 45649 | Five-hypothesis eliminatio... |
| e12 45650 | A virtual deduction elimin... |
| e12an 45651 | Conjunction form of ~ e12 ... |
| el12 45652 | Virtual deduction form of ... |
| e20 45653 | A virtual deduction elimin... |
| e20an 45654 | Conjunction form of ~ e20 ... |
| ee20an 45655 | ~ e20an without virtual de... |
| e21 45656 | A virtual deduction elimin... |
| e21an 45657 | Conjunction form of ~ e21 ... |
| ee21an 45658 | ~ e21an without virtual de... |
| e333 45659 | A virtual deduction elimin... |
| e33 45660 | A virtual deduction elimin... |
| e33an 45661 | Conjunction form of ~ e33 ... |
| ee33an 45662 | ~ e33an without virtual de... |
| e3 45663 | Meta-connective form of ~ ... |
| e3bi 45664 | Biconditional form of ~ e3... |
| e3bir 45665 | Right biconditional form o... |
| e03 45666 | A virtual deduction elimin... |
| ee03 45667 | ~ e03 without virtual dedu... |
| e03an 45668 | Conjunction form of ~ e03 ... |
| ee03an 45669 | Conjunction form of ~ ee03... |
| e30 45670 | A virtual deduction elimin... |
| ee30 45671 | ~ e30 without virtual dedu... |
| e30an 45672 | A virtual deduction elimin... |
| ee30an 45673 | Conjunction form of ~ ee30... |
| e13 45674 | A virtual deduction elimin... |
| e13an 45675 | A virtual deduction elimin... |
| ee13an 45676 | ~ e13an without virtual de... |
| e31 45677 | A virtual deduction elimin... |
| ee31 45678 | ~ e31 without virtual dedu... |
| e31an 45679 | A virtual deduction elimin... |
| ee31an 45680 | ~ e31an without virtual de... |
| e23 45681 | A virtual deduction elimin... |
| e23an 45682 | A virtual deduction elimin... |
| ee23an 45683 | ~ e23an without virtual de... |
| e32 45684 | A virtual deduction elimin... |
| ee32 45685 | ~ e32 without virtual dedu... |
| e32an 45686 | A virtual deduction elimin... |
| ee32an 45687 | ~ e33an without virtual de... |
| e123 45688 | A virtual deduction elimin... |
| ee123 45689 | ~ e123 without virtual ded... |
| el123 45690 | A virtual deduction elimin... |
| e233 45691 | A virtual deduction elimin... |
| e323 45692 | A virtual deduction elimin... |
| e000 45693 | A virtual deduction elimin... |
| e00 45694 | Elimination rule identical... |
| e00an 45695 | Elimination rule identical... |
| eel00cT 45696 | An elimination deduction. ... |
| eelTT 45697 | An elimination deduction. ... |
| e0a 45698 | Elimination rule identical... |
| eelT 45699 | An elimination deduction. ... |
| eel0cT 45700 | An elimination deduction. ... |
| eelT0 45701 | An elimination deduction. ... |
| e0bi 45702 | Elimination rule identical... |
| e0bir 45703 | Elimination rule identical... |
| uun0.1 45704 | Convention notation form o... |
| un0.1 45705 | ` T. ` is the constant tru... |
| uunT1 45706 | A deduction unionizing a n... |
| uunT1p1 45707 | A deduction unionizing a n... |
| uunT21 45708 | A deduction unionizing a n... |
| uun121 45709 | A deduction unionizing a n... |
| uun121p1 45710 | A deduction unionizing a n... |
| uun132 45711 | A deduction unionizing a n... |
| uun132p1 45712 | A deduction unionizing a n... |
| anabss7p1 45713 | A deduction unionizing a n... |
| un10 45714 | A unionizing deduction. (... |
| un01 45715 | A unionizing deduction. (... |
| un2122 45716 | A deduction unionizing a n... |
| uun2131 45717 | A deduction unionizing a n... |
| uun2131p1 45718 | A deduction unionizing a n... |
| uunTT1 45719 | A deduction unionizing a n... |
| uunTT1p1 45720 | A deduction unionizing a n... |
| uunTT1p2 45721 | A deduction unionizing a n... |
| uunT11 45722 | A deduction unionizing a n... |
| uunT11p1 45723 | A deduction unionizing a n... |
| uunT11p2 45724 | A deduction unionizing a n... |
| uunT12 45725 | A deduction unionizing a n... |
| uunT12p1 45726 | A deduction unionizing a n... |
| uunT12p2 45727 | A deduction unionizing a n... |
| uunT12p3 45728 | A deduction unionizing a n... |
| uunT12p4 45729 | A deduction unionizing a n... |
| uunT12p5 45730 | A deduction unionizing a n... |
| uun111 45731 | A deduction unionizing a n... |
| 3anidm12p1 45732 | A deduction unionizing a n... |
| 3anidm12p2 45733 | A deduction unionizing a n... |
| uun123 45734 | A deduction unionizing a n... |
| uun123p1 45735 | A deduction unionizing a n... |
| uun123p2 45736 | A deduction unionizing a n... |
| uun123p3 45737 | A deduction unionizing a n... |
| uun123p4 45738 | A deduction unionizing a n... |
| uun2221 45739 | A deduction unionizing a n... |
| uun2221p1 45740 | A deduction unionizing a n... |
| uun2221p2 45741 | A deduction unionizing a n... |
| 3impdirp1 45742 | A deduction unionizing a n... |
| 3impcombi 45743 | A 1-hypothesis proposition... |
| trsspwALT 45744 | Virtual deduction proof of... |
| trsspwALT2 45745 | Virtual deduction proof of... |
| trsspwALT3 45746 | Short predicate calculus p... |
| sspwtr 45747 | Virtual deduction proof of... |
| sspwtrALT 45748 | Virtual deduction proof of... |
| sspwtrALT2 45749 | Short predicate calculus p... |
| pwtrVD 45750 | Virtual deduction proof of... |
| pwtrrVD 45751 | Virtual deduction proof of... |
| suctrALT 45752 | The successor of a transit... |
| snssiALTVD 45753 | Virtual deduction proof of... |
| snssiALT 45754 | If a class is an element o... |
| snsslVD 45755 | Virtual deduction proof of... |
| snssl 45756 | If a singleton is a subcla... |
| snelpwrVD 45757 | Virtual deduction proof of... |
| unipwrVD 45758 | Virtual deduction proof of... |
| unipwr 45759 | A class is a subclass of t... |
| sstrALT2VD 45760 | Virtual deduction proof of... |
| sstrALT2 45761 | Virtual deduction proof of... |
| suctrALT2VD 45762 | Virtual deduction proof of... |
| suctrALT2 45763 | Virtual deduction proof of... |
| elex2VD 45764 | Virtual deduction proof of... |
| elex22VD 45765 | Virtual deduction proof of... |
| eqsbc2VD 45766 | Virtual deduction proof of... |
| zfregs2VD 45767 | Virtual deduction proof of... |
| tpid3gVD 45768 | Virtual deduction proof of... |
| en3lplem1VD 45769 | Virtual deduction proof of... |
| en3lplem2VD 45770 | Virtual deduction proof of... |
| en3lpVD 45771 | Virtual deduction proof of... |
| simplbi2VD 45772 | Virtual deduction proof of... |
| 3ornot23VD 45773 | Virtual deduction proof of... |
| orbi1rVD 45774 | Virtual deduction proof of... |
| bitr3VD 45775 | Virtual deduction proof of... |
| 3orbi123VD 45776 | Virtual deduction proof of... |
| sbc3orgVD 45777 | Virtual deduction proof of... |
| 19.21a3con13vVD 45778 | Virtual deduction proof of... |
| exbirVD 45779 | Virtual deduction proof of... |
| exbiriVD 45780 | Virtual deduction proof of... |
| rspsbc2VD 45781 | Virtual deduction proof of... |
| 3impexpVD 45782 | Virtual deduction proof of... |
| 3impexpbicomVD 45783 | Virtual deduction proof of... |
| 3impexpbicomiVD 45784 | Virtual deduction proof of... |
| sbcoreleleqVD 45785 | Virtual deduction proof of... |
| hbra2VD 45786 | Virtual deduction proof of... |
| tratrbVD 45787 | Virtual deduction proof of... |
| al2imVD 45788 | Virtual deduction proof of... |
| syl5impVD 45789 | Virtual deduction proof of... |
| idiVD 45790 | Virtual deduction proof of... |
| ancomstVD 45791 | Closed form of ~ ancoms . ... |
| ssralv2VD 45792 | Quantification restricted ... |
| ordelordALTVD 45793 | An element of an ordinal c... |
| equncomVD 45794 | If a class equals the unio... |
| equncomiVD 45795 | Inference form of ~ equnco... |
| sucidALTVD 45796 | A set belongs to its succe... |
| sucidALT 45797 | A set belongs to its succe... |
| sucidVD 45798 | A set belongs to its succe... |
| imbi12VD 45799 | Implication form of ~ imbi... |
| imbi13VD 45800 | Join three logical equival... |
| sbcim2gVD 45801 | Distribution of class subs... |
| sbcbiVD 45802 | Implication form of ~ sbcb... |
| trsbcVD 45803 | Formula-building inference... |
| truniALTVD 45804 | The union of a class of tr... |
| ee33VD 45805 | Non-virtual deduction form... |
| trintALTVD 45806 | The intersection of a clas... |
| trintALT 45807 | The intersection of a clas... |
| undif3VD 45808 | The first equality of Exer... |
| sbcssgVD 45809 | Virtual deduction proof of... |
| csbingVD 45810 | Virtual deduction proof of... |
| onfrALTlem5VD 45811 | Virtual deduction proof of... |
| onfrALTlem4VD 45812 | Virtual deduction proof of... |
| onfrALTlem3VD 45813 | Virtual deduction proof of... |
| simplbi2comtVD 45814 | Virtual deduction proof of... |
| onfrALTlem2VD 45815 | Virtual deduction proof of... |
| onfrALTlem1VD 45816 | Virtual deduction proof of... |
| onfrALTVD 45817 | Virtual deduction proof of... |
| csbeq2gVD 45818 | Virtual deduction proof of... |
| csbsngVD 45819 | Virtual deduction proof of... |
| csbxpgVD 45820 | Virtual deduction proof of... |
| csbresgVD 45821 | Virtual deduction proof of... |
| csbrngVD 45822 | Virtual deduction proof of... |
| csbima12gALTVD 45823 | Virtual deduction proof of... |
| csbunigVD 45824 | Virtual deduction proof of... |
| csbfv12gALTVD 45825 | Virtual deduction proof of... |
| con5VD 45826 | Virtual deduction proof of... |
| relopabVD 45827 | Virtual deduction proof of... |
| 19.41rgVD 45828 | Virtual deduction proof of... |
| 2pm13.193VD 45829 | Virtual deduction proof of... |
| hbimpgVD 45830 | Virtual deduction proof of... |
| hbalgVD 45831 | Virtual deduction proof of... |
| hbexgVD 45832 | Virtual deduction proof of... |
| ax6e2eqVD 45833 | The following User's Proof... |
| ax6e2ndVD 45834 | The following User's Proof... |
| ax6e2ndeqVD 45835 | The following User's Proof... |
| 2sb5ndVD 45836 | The following User's Proof... |
| 2uasbanhVD 45837 | The following User's Proof... |
| e2ebindVD 45838 | The following User's Proof... |
| sb5ALTVD 45839 | The following User's Proof... |
| vk15.4jVD 45840 | The following User's Proof... |
| notnotrALTVD 45841 | The following User's Proof... |
| con3ALTVD 45842 | The following User's Proof... |
| elpwgdedVD 45843 | Membership in a power clas... |
| sspwimp 45844 | If a class is a subclass o... |
| sspwimpVD 45845 | The following User's Proof... |
| sspwimpcf 45846 | If a class is a subclass o... |
| sspwimpcfVD 45847 | The following User's Proof... |
| suctrALTcf 45848 | The successor of a transit... |
| suctrALTcfVD 45849 | The following User's Proof... |
| suctrALT3 45850 | The successor of a transit... |
| sspwimpALT 45851 | If a class is a subclass o... |
| unisnALT 45852 | A set equals the union of ... |
| notnotrALT2 45853 | Converse of double negatio... |
| sspwimpALT2 45854 | If a class is a subclass o... |
| e2ebindALT 45855 | Absorption of an existenti... |
| ax6e2ndALT 45856 | If at least two sets exist... |
| ax6e2ndeqALT 45857 | "At least two sets exist" ... |
| 2sb5ndALT 45858 | Equivalence for double sub... |
| chordthmALT 45859 | The intersecting chords th... |
| isosctrlem1ALT 45860 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 45861 | The indexed union of conne... |
| iunconnALT 45862 | The indexed union of conne... |
| sineq0ALT 45863 | A complex number whose sin... |
| rspesbcd 45864 | Restricted quantifier vers... |
| rext0 45865 | Nonempty existential quant... |
| dfbi1ALTa 45866 | Version of ~ dfbi1ALT usin... |
| simprimi 45867 | Inference associated with ... |
| dfbi1ALTb 45868 | Further shorten ~ dfbi1ALT... |
| relpeq1 45871 | Equality theorem for relat... |
| relpeq2 45872 | Equality theorem for relat... |
| relpeq3 45873 | Equality theorem for relat... |
| relpeq4 45874 | Equality theorem for relat... |
| relpeq5 45875 | Equality theorem for relat... |
| nfrelp 45876 | Bound-variable hypothesis ... |
| relpf 45877 | A relation-preserving func... |
| relprel 45878 | A relation-preserving func... |
| relpmin 45879 | A preimage of a minimal el... |
| relpfrlem 45880 | Lemma for ~ relpfr . Prov... |
| relpfr 45881 | If the image of a set unde... |
| orbitex 45882 | Orbits exist. Given a set... |
| orbitinit 45883 | A set is contained in its ... |
| orbitcl 45884 | The orbit under a function... |
| orbitclmpt 45885 | Version of ~ orbitcl using... |
| trwf 45886 | The class of well-founded ... |
| rankrelp 45887 | The rank function preserve... |
| wffr 45888 | The class of well-founded ... |
| trfr 45889 | A transitive class well-fo... |
| tcfr 45890 | A set is well-founded if a... |
| xpwf 45891 | The Cartesian product of t... |
| dmwf 45892 | The domain of a well-found... |
| rnwf 45893 | The range of a well-founde... |
| relwf 45894 | A relation is a well-found... |
| ralabso 45895 | Simplification of restrict... |
| rexabso 45896 | Simplification of restrict... |
| ralabsod 45897 | Deduction form of ~ ralabs... |
| rexabsod 45898 | Deduction form of ~ rexabs... |
| ralabsobidv 45899 | Formula-building lemma for... |
| rexabsobidv 45900 | Formula-building lemma for... |
| ssabso 45901 | The notion " ` x ` is a su... |
| disjabso 45902 | Disjointness is absolute f... |
| n0abso 45903 | Nonemptiness is absolute f... |
| traxext 45904 | A transitive class models ... |
| modelaxreplem1 45905 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 45906 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 45907 | Lemma for ~ modelaxrep . ... |
| modelaxrep 45908 | Conditions which guarantee... |
| ssclaxsep 45909 | A class that is closed und... |
| 0elaxnul 45910 | A class that contains the ... |
| pwclaxpow 45911 | Suppose ` M ` is a transit... |
| prclaxpr 45912 | A class that is closed und... |
| uniclaxun 45913 | A class that is closed und... |
| sswfaxreg 45914 | A subclass of the class of... |
| omssaxinf2 45915 | A class that contains all ... |
| omelaxinf2 45916 | A transitive class that co... |
| dfac5prim 45917 | ~ dfac5 expanded into prim... |
| ac8prim 45918 | ~ ac8 expanded into primit... |
| modelac8prim 45919 | If ` M ` is a transitive c... |
| wfaxext 45920 | The class of well-founded ... |
| wfaxrep 45921 | The class of well-founded ... |
| wfaxsep 45922 | The class of well-founded ... |
| wfaxnul 45923 | The class of well-founded ... |
| wfaxpow 45924 | The class of well-founded ... |
| wfaxpr 45925 | The class of well-founded ... |
| wfaxun 45926 | The class of well-founded ... |
| wfaxreg 45927 | The class of well-founded ... |
| wfaxinf2 45928 | The class of well-founded ... |
| wfac8prim 45929 | The class of well-founded ... |
| brpermmodel 45930 | The membership relation in... |
| brpermmodelcnv 45931 | Ordinary membership expres... |
| permaxext 45932 | The Axiom of Extensionalit... |
| permaxrep 45933 | The Axiom of Replacement ~... |
| permaxsep 45934 | The Axiom of Separation ~ ... |
| permaxnul 45935 | The Null Set Axiom ~ ax-nu... |
| permaxpow 45936 | The Axiom of Power Sets ~ ... |
| permaxpr 45937 | The Axiom of Pairing ~ ax-... |
| permaxun 45938 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 45939 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 45940 | The Axiom of Infinity ~ ax... |
| permac8prim 45941 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 45942 | Define a permutation ` F `... |
| nregmodellem 45943 | Lemma for ~ nregmodel . (... |
| nregmodel 45944 | The Axiom of Regularity ~ ... |
| nregmodelaxext 45945 | The Axiom of Extensionalit... |
| hashnna 45946 | The ` # ` function on ` _o... |
| hashnnsuc 45947 | The ` # ` function on ` _o... |
| hashnnm 45948 | The ` # ` function on ` _o... |
| hashnnlt 45949 | The ` # ` function on ` _o... |
| hashnnltb 45950 | The ` # ` function on ` _o... |
| hashomf1o 45951 | The ` # ` function yields ... |
| hashomiso 45952 | The ` # ` function yields ... |
| evth2f 45953 | A version of ~ evth2 using... |
| elunif 45954 | A version of ~ eluni using... |
| rzalf 45955 | A version of ~ rzal using ... |
| fvelrnbf 45956 | A version of ~ fvelrnb usi... |
| rfcnpre1 45957 | If F is a continuous funct... |
| ubelsupr 45958 | If U belongs to A and U is... |
| fsumcnf 45959 | A finite sum of functions ... |
| mulltgt0 45960 | The product of a negative ... |
| rspcegf 45961 | A version of ~ rspcev usin... |
| rabexgf 45962 | A version of ~ rabexg usin... |
| fcnre 45963 | A function continuous with... |
| sumsnd 45964 | A sum of a singleton is th... |
| evthf 45965 | A version of ~ evth using ... |
| cnfex 45966 | The class of continuous fu... |
| fnchoice 45967 | For a finite set, a choice... |
| refsumcn 45968 | A finite sum of continuous... |
| rfcnpre2 45969 | If ` F ` is a continuous f... |
| cncmpmax 45970 | When the hypothesis for th... |
| rfcnpre3 45971 | If F is a continuous funct... |
| rfcnpre4 45972 | If F is a continuous funct... |
| sumpair 45973 | Sum of two distinct comple... |
| rfcnnnub 45974 | Given a real continuous fu... |
| refsum2cnlem1 45975 | This is the core Lemma for... |
| refsum2cn 45976 | The sum of two continuus r... |
| adantlllr 45977 | Deduction adding a conjunc... |
| 3adantlr3 45978 | Deduction adding a conjunc... |
| 3adantll2 45979 | Deduction adding a conjunc... |
| 3adantll3 45980 | Deduction adding a conjunc... |
| ssnel 45981 | If not element of a set, t... |
| sncldre 45982 | A singleton is closed w.r.... |
| n0p 45983 | A polynomial with a nonzer... |
| pm2.65ni 45984 | Inference rule for proof b... |
| iuneq2df 45985 | Equality deduction for ind... |
| nnfoctb 45986 | There exists a mapping fro... |
| elpwinss 45987 | An element of the powerset... |
| unidmex 45988 | If ` F ` is a set, then ` ... |
| ndisj2 45989 | A non-disjointness conditi... |
| zenom 45990 | The set of integer numbers... |
| uzwo4 45991 | Well-ordering principle: a... |
| unisn0 45992 | The union of the singleton... |
| ssin0 45993 | If two classes are disjoin... |
| inabs3 45994 | Absorption law for interse... |
| pwpwuni 45995 | Relationship between power... |
| disjiun2 45996 | In a disjoint collection, ... |
| 0pwfi 45997 | The empty set is in any po... |
| ssinss2d 45998 | Intersection preserves sub... |
| zct 45999 | The set of integer numbers... |
| pwfin0 46000 | A finite set always belong... |
| uzct 46001 | An upper integer set is co... |
| iunxsnf 46002 | A singleton index picks ou... |
| fiiuncl 46003 | If a set is closed under t... |
| iunp1 46004 | The addition of the next s... |
| fiunicl 46005 | If a set is closed under t... |
| ixpeq2d 46006 | Equality theorem for infin... |
| disjxp1 46007 | The sets of a cartesian pr... |
| disjsnxp 46008 | The sets in the cartesian ... |
| eliind 46009 | Membership in indexed inte... |
| rspcef 46010 | Restricted existential spe... |
| ixpssmapc 46011 | An infinite Cartesian prod... |
| elintd 46012 | Membership in class inters... |
| ssdf 46013 | A sufficient condition for... |
| brneqtrd 46014 | Substitution of equal clas... |
| ssnct 46015 | A set containing an uncoun... |
| ssuniint 46016 | Sufficient condition for b... |
| elintdv 46017 | Membership in class inters... |
| ssd 46018 | A sufficient condition for... |
| ralimralim 46019 | Introducing any antecedent... |
| snelmap 46020 | Membership of the element ... |
| xrnmnfpnf 46021 | An extended real that is n... |
| iuneq1i 46022 | Equality theorem for index... |
| ssinc 46023 | Inclusion relation for a m... |
| ssdec 46024 | Inclusion relation for a m... |
| elixpconstg 46025 | Membership in an infinite ... |
| iineq1d 46026 | Equality theorem for index... |
| metpsmet 46027 | A metric is a pseudometric... |
| ixpssixp 46028 | Subclass theorem for infin... |
| ballss3 46029 | A sufficient condition for... |
| iunincfi 46030 | Given a sequence of increa... |
| nsstr 46031 | If it's not a subclass, it... |
| rexanuz3 46032 | Combine two different uppe... |
| cbvmpo2 46033 | Rule to change the second ... |
| cbvmpo1 46034 | Rule to change the first b... |
| eliuniin 46035 | Indexed union of indexed i... |
| ssabf 46036 | Subclass of a class abstra... |
| pssnssi 46037 | A proper subclass does not... |
| rabidim2 46038 | Membership in a restricted... |
| eluni2f 46039 | Membership in class union.... |
| eliin2f 46040 | Membership in indexed inte... |
| nssd 46041 | Negation of subclass relat... |
| iineq12dv 46042 | Equality deduction for ind... |
| supxrcld 46043 | The supremum of an arbitra... |
| elrestd 46044 | A sufficient condition for... |
| eliuniincex 46045 | Counterexample to show tha... |
| eliincex 46046 | Counterexample to show tha... |
| eliinid 46047 | Membership in an indexed i... |
| abssf 46048 | Class abstraction in a sub... |
| supxrubd 46049 | A member of a set of exten... |
| ssrabf 46050 | Subclass of a restricted c... |
| ssrabdf 46051 | Subclass of a restricted c... |
| eliin2 46052 | Membership in indexed inte... |
| ssrab2f 46053 | Subclass relation for a re... |
| restuni3 46054 | The underlying set of a su... |
| rabssf 46055 | Restricted class abstracti... |
| eliuniin2 46056 | Indexed union of indexed i... |
| restuni4 46057 | The underlying set of a su... |
| restuni6 46058 | The underlying set of a su... |
| restuni5 46059 | The underlying set of a su... |
| unirestss 46060 | The union of an elementwis... |
| iniin1 46061 | Indexed intersection of in... |
| iniin2 46062 | Indexed intersection of in... |
| cbvrabv2 46063 | A more general version of ... |
| cbvrabv2w 46064 | A more general version of ... |
| iinssiin 46065 | Subset implication for an ... |
| eliind2 46066 | Membership in indexed inte... |
| iinssd 46067 | Subset implication for an ... |
| rabbida2 46068 | Equivalent wff's yield equ... |
| iinexd 46069 | The existence of an indexe... |
| rabexf 46070 | Separation Scheme in terms... |
| rabbida3 46071 | Equivalent wff's yield equ... |
| r19.36vf 46072 | Restricted quantifier vers... |
| raleqd 46073 | Equality deduction for res... |
| iinssf 46074 | Subset implication for an ... |
| iinssdf 46075 | Subset implication for an ... |
| resabs2i 46076 | Absorption law for restric... |
| ssdf2 46077 | A sufficient condition for... |
| rabssd 46078 | Restricted class abstracti... |
| rexnegd 46079 | Minus a real number. (Con... |
| rexlimd3 46080 | * Inference from Theorem 1... |
| nel1nelini 46081 | Membership in an intersect... |
| nel2nelini 46082 | Membership in an intersect... |
| eliunid 46083 | Membership in indexed unio... |
| reximdd 46084 | Deduction from Theorem 19.... |
| inopnd 46085 | The intersection of two op... |
| ss2rabdf 46086 | Deduction of restricted ab... |
| restopn3 46087 | If ` A ` is open, then ` A... |
| restopnssd 46088 | A topology restricted to a... |
| restsubel 46089 | A subset belongs in the sp... |
| toprestsubel 46090 | A subset is open in the to... |
| rabidd 46091 | An "identity" law of concr... |
| iunssdf 46092 | Subset theorem for an inde... |
| iinss2d 46093 | Subset implication for an ... |
| r19.3rzf 46094 | Restricted quantification ... |
| r19.28zf 46095 | Restricted quantifier vers... |
| iindif2f 46096 | Indexed intersection of cl... |
| ralfal 46097 | Two ways of expressing emp... |
| archd 46098 | Archimedean property of re... |
| nimnbi 46099 | If an implication is false... |
| nimnbi2 46100 | If an implication is false... |
| notbicom 46101 | Commutative law for the ne... |
| rexeqif 46102 | Equality inference for res... |
| rspced 46103 | Restricted existential spe... |
| fnresdmss 46104 | A function does not change... |
| fmptsnxp 46105 | Maps-to notation and Carte... |
| fvmpt2bd 46106 | Value of a function given ... |
| rnmptfi 46107 | The range of a function wi... |
| fresin2 46108 | Restriction of a function ... |
| ffi 46109 | A function with finite dom... |
| suprnmpt 46110 | An explicit bound for the ... |
| rnffi 46111 | The range of a function wi... |
| mptelpm 46112 | A function in maps-to nota... |
| rnmptpr 46113 | Range of a function define... |
| resmpti 46114 | Restriction of the mapping... |
| founiiun 46115 | Union expressed as an inde... |
| rnresun 46116 | Distribution law for range... |
| elrnmptf 46117 | The range of a function in... |
| rnmptssrn 46118 | Inclusion relation for two... |
| disjf1 46119 | A 1 to 1 mapping built fro... |
| rnsnf 46120 | The range of a function wh... |
| wessf1ornlem 46121 | Given a function ` F ` on ... |
| wessf1orn 46122 | Given a function ` F ` on ... |
| nelrnres 46123 | If ` A ` is not in the ran... |
| disjrnmpt2 46124 | Disjointness of the range ... |
| elrnmpt1sf 46125 | Elementhood in an image se... |
| founiiun0 46126 | Union expressed as an inde... |
| disjf1o 46127 | A bijection built from dis... |
| disjinfi 46128 | Only a finite number of di... |
| fvovco 46129 | Value of the composition o... |
| ssnnf1octb 46130 | There exists a bijection b... |
| nnf1oxpnn 46131 | There is a bijection betwe... |
| projf1o 46132 | A biijection from a set to... |
| fvmap 46133 | Function value for a membe... |
| fvixp2 46134 | Projection of a factor of ... |
| choicefi 46135 | For a finite set, a choice... |
| mpct 46136 | The exponentiation of a co... |
| cnmetcoval 46137 | Value of the distance func... |
| fcomptss 46138 | Express composition of two... |
| elmapsnd 46139 | Membership in a set expone... |
| mapss2 46140 | Subset inheritance for set... |
| difmap 46141 | Difference of two sets exp... |
| unirnmap 46142 | Given a subset of a set ex... |
| inmap 46143 | Intersection of two sets e... |
| fcoss 46144 | Composition of two mapping... |
| fsneqrn 46145 | Equality condition for two... |
| difmapsn 46146 | Difference of two sets exp... |
| mapssbi 46147 | Subset inheritance for set... |
| unirnmapsn 46148 | Equality theorem for a sub... |
| iunmapss 46149 | The indexed union of set e... |
| ssmapsn 46150 | A subset ` C ` of a set ex... |
| iunmapsn 46151 | The indexed union of set e... |
| absfico 46152 | Mapping domain and codomai... |
| icof 46153 | The set of left-closed rig... |
| elpmrn 46154 | The range of a partial fun... |
| imaexi 46155 | The image of a set is a se... |
| axccdom 46156 | Relax the constraint on ax... |
| dmmptdff 46157 | The domain of the mapping ... |
| dmmptdf 46158 | The domain of the mapping ... |
| elpmi2 46159 | The domain of a partial fu... |
| dmrelrnrel 46160 | A relation preserving func... |
| elrnmpoid 46161 | Membership in the range of... |
| axccd 46162 | An alternative version of ... |
| axccd2 46163 | An alternative version of ... |
| feqresmptf 46164 | Express a restricted funct... |
| dmmptssf 46165 | The domain of a mapping is... |
| dmmptdf2 46166 | The domain of the mapping ... |
| dmuz 46167 | Domain of the upper intege... |
| fmptd2f 46168 | Domain and codomain of the... |
| mpteq1df 46169 | An equality theorem for th... |
| mptexf 46170 | If the domain of a functio... |
| fvmpt4 46171 | Value of a function given ... |
| fmptf 46172 | Functionality of the mappi... |
| resimass 46173 | The image of a restriction... |
| mptssid 46174 | The mapping operation expr... |
| mptfnd 46175 | The maps-to notation defin... |
| rnmptlb 46176 | Boundness below of the ran... |
| rnmptbddlem 46177 | Boundness of the range of ... |
| rnmptbdd 46178 | Boundness of the range of ... |
| funimaeq 46179 | Membership relation for th... |
| rnmptssf 46180 | The range of a function gi... |
| rnmptbd2lem 46181 | Boundness below of the ran... |
| rnmptbd2 46182 | Boundness below of the ran... |
| infnsuprnmpt 46183 | The indexed infimum of rea... |
| suprclrnmpt 46184 | Closure of the indexed sup... |
| suprubrnmpt2 46185 | A member of a nonempty ind... |
| suprubrnmpt 46186 | A member of a nonempty ind... |
| rnmptssdf 46187 | The range of a function gi... |
| rnmptbdlem 46188 | Boundness above of the ran... |
| rnmptbd 46189 | Boundness above of the ran... |
| rnmptss2 46190 | The range of a function gi... |
| elmptima 46191 | The image of a function in... |
| ralrnmpt3 46192 | A restricted quantifier ov... |
| rnmptssbi 46193 | The range of a function gi... |
| imass2d 46194 | Subset theorem for image. ... |
| imassmpt 46195 | Membership relation for th... |
| fpmd 46196 | A total function is a part... |
| fconst7 46197 | An alternative way to expr... |
| fnmptif 46198 | Functionality and domain o... |
| dmmptif 46199 | Domain of the mapping oper... |
| mpteq2dfa 46200 | Slightly more general equa... |
| dmmpt1 46201 | The domain of the mapping ... |
| fmptff 46202 | Functionality of the mappi... |
| fvmptelcdmf 46203 | The value of a function at... |
| fmptdff 46204 | A version of ~ fmptd using... |
| fvmpt2df 46205 | Deduction version of ~ fvm... |
| rn1st 46206 | The range of a function wi... |
| rnmptssff 46207 | The range of a function gi... |
| rnmptssdff 46208 | The range of a function gi... |
| fvmpt4d 46209 | Value of a function given ... |
| sub2times 46210 | Subtracting from a number,... |
| nnxrd 46211 | A natural number is an ext... |
| nnxr 46212 | A natural number is an ext... |
| abssubrp 46213 | The distance of two distin... |
| elfzfzo 46214 | Relationship between membe... |
| oddfl 46215 | Odd number representation ... |
| abscosbd 46216 | Bound for the absolute val... |
| mul13d 46217 | Commutative/associative la... |
| negpilt0 46218 | Negative ` _pi ` is negati... |
| dstregt0 46219 | A complex number ` A ` tha... |
| subadd4b 46220 | Rearrangement of 4 terms i... |
| xrlttri5d 46221 | Not equal and not larger i... |
| zltlesub 46222 | If an integer ` N ` is les... |
| divlt0gt0d 46223 | The ratio of a negative nu... |
| subsub23d 46224 | Swap subtrahend and result... |
| 2timesgt 46225 | Double of a positive real ... |
| reopn 46226 | The reals are open with re... |
| sub31 46227 | Swap the first and third t... |
| nnne1ge2 46228 | A positive integer which i... |
| lefldiveq 46229 | A closed enough, smaller r... |
| negsubdi3d 46230 | Distribution of negative o... |
| ltdiv2dd 46231 | Division of a positive num... |
| abssinbd 46232 | Bound for the absolute val... |
| halffl 46233 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 46234 | Ordering relation for a st... |
| hashssle 46235 | The size of a subset of a ... |
| lttri5d 46236 | Not equal and not larger i... |
| fzisoeu 46237 | A finite ordered set has a... |
| lt3addmuld 46238 | If three real numbers are ... |
| absnpncan2d 46239 | Triangular inequality, com... |
| fperiodmullem 46240 | A function with period ` T... |
| fperiodmul 46241 | A function with period T i... |
| upbdrech 46242 | Choice of an upper bound f... |
| lt4addmuld 46243 | If four real numbers are l... |
| absnpncan3d 46244 | Triangular inequality, com... |
| upbdrech2 46245 | Choice of an upper bound f... |
| ssfiunibd 46246 | A finite union of bounded ... |
| fzdifsuc2 46247 | Remove a successor from th... |
| fzsscn 46248 | A finite sequence of integ... |
| divcan8d 46249 | A cancellation law for div... |
| dmmcand 46250 | Cancellation law for divis... |
| fzssre 46251 | A finite sequence of integ... |
| bccld 46252 | A binomial coefficient, in... |
| fzssnn0 46253 | A finite set of sequential... |
| xreqle 46254 | Equality implies 'less tha... |
| xaddlidd 46255 | ` 0 ` is a left identity f... |
| xadd0ge 46256 | A number is less than or e... |
| xrleneltd 46257 | 'Less than or equal to' an... |
| xaddcomd 46258 | The extended real addition... |
| supxrre3 46259 | The supremum of a nonempty... |
| uzfissfz 46260 | For any finite subset of t... |
| xleadd2d 46261 | Addition of extended reals... |
| suprltrp 46262 | The supremum of a nonempty... |
| xleadd1d 46263 | Addition of extended reals... |
| xreqled 46264 | Equality implies 'less tha... |
| xrgepnfd 46265 | An extended real greater t... |
| xrge0nemnfd 46266 | A nonnegative extended rea... |
| supxrgere 46267 | If a real number can be ap... |
| iuneqfzuzlem 46268 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 46269 | If two unions indexed by u... |
| xle2addd 46270 | Adding both side of two in... |
| supxrgelem 46271 | If an extended real number... |
| supxrge 46272 | If an extended real number... |
| suplesup 46273 | If any element of ` A ` ca... |
| infxrglb 46274 | The infimum of a set of ex... |
| xadd0ge2 46275 | A number is less than or e... |
| nepnfltpnf 46276 | An extended real that is n... |
| ltadd12dd 46277 | Addition to both sides of ... |
| nemnftgtmnft 46278 | An extended real that is n... |
| xrgtso 46279 | 'Greater than' is a strict... |
| rpex 46280 | The positive reals form a ... |
| xrge0ge0 46281 | A nonnegative extended rea... |
| xrssre 46282 | A subset of extended reals... |
| ssuzfz 46283 | A finite subset of the upp... |
| absfun 46284 | The absolute value is a fu... |
| infrpge 46285 | The infimum of a nonempty,... |
| xrlexaddrp 46286 | If an extended real number... |
| supsubc 46287 | The supremum function dist... |
| xralrple2 46288 | Show that ` A ` is less th... |
| nnuzdisj 46289 | The first ` N ` elements o... |
| ltdivgt1 46290 | Divsion by a number greate... |
| xrltned 46291 | 'Less than' implies not eq... |
| nnsplit 46292 | Express the set of positiv... |
| divdiv3d 46293 | Division into a fraction. ... |
| abslt2sqd 46294 | Comparison of the square o... |
| qenom 46295 | The set of rational number... |
| qct 46296 | The set of rational number... |
| lenlteq 46297 | 'less than or equal to' bu... |
| xrred 46298 | An extended real that is n... |
| rr2sscn2 46299 | The cartesian square of ` ... |
| infxr 46300 | The infimum of a set of ex... |
| infxrunb2 46301 | The infimum of an unbounde... |
| infxrbnd2 46302 | The infimum of a bounded-b... |
| infleinflem1 46303 | Lemma for ~ infleinf , cas... |
| infleinflem2 46304 | Lemma for ~ infleinf , whe... |
| infleinf 46305 | If any element of ` B ` ca... |
| xralrple4 46306 | Show that ` A ` is less th... |
| xralrple3 46307 | Show that ` A ` is less th... |
| eluzelzd 46308 | A member of an upper set o... |
| suplesup2 46309 | If any element of ` A ` is... |
| recnnltrp 46310 | ` N ` is a natural number ... |
| nnn0 46311 | The set of positive intege... |
| fzct 46312 | A finite set of sequential... |
| rpgtrecnn 46313 | Any positive real number i... |
| fzossuz 46314 | A half-open integer interv... |
| infxrrefi 46315 | The real and extended real... |
| xrralrecnnle 46316 | Show that ` A ` is less th... |
| fzoct 46317 | A finite set of sequential... |
| frexr 46318 | A function taking real val... |
| nnrecrp 46319 | The reciprocal of a positi... |
| reclt0d 46320 | The reciprocal of a negati... |
| lt0neg1dd 46321 | If a number is negative, i... |
| infxrcld 46322 | The infimum of an arbitrar... |
| xrralrecnnge 46323 | Show that ` A ` is less th... |
| reclt0 46324 | The reciprocal of a negati... |
| ltmulneg 46325 | Multiplying by a negative ... |
| allbutfi 46326 | For all but finitely many.... |
| ltdiv23neg 46327 | Swap denominator with othe... |
| xreqnltd 46328 | A consequence of trichotom... |
| mnfnre2 46329 | Minus infinity is not a re... |
| zssxr 46330 | The integers are a subset ... |
| fisupclrnmpt 46331 | A nonempty finite indexed ... |
| supxrunb3 46332 | The supremum of an unbound... |
| fimaxre4 46333 | A nonempty finite set of r... |
| ren0 46334 | The set of reals is nonemp... |
| eluzelz2 46335 | A member of an upper set o... |
| resabs2d 46336 | Absorption law for restric... |
| uzid2 46337 | Membership of the least me... |
| supxrleubrnmpt 46338 | The supremum of a nonempty... |
| uzssre2 46339 | An upper set of integers i... |
| uzssd 46340 | Subset relationship for tw... |
| eluzd 46341 | Membership in an upper set... |
| infxrlbrnmpt2 46342 | A member of a nonempty ind... |
| xrre4 46343 | An extended real is real i... |
| uz0 46344 | The upper integers functio... |
| eluzelz2d 46345 | A member of an upper set o... |
| infleinf2 46346 | If any element in ` B ` is... |
| unb2ltle 46347 | "Unbounded below" expresse... |
| uzidd2 46348 | Membership of the least me... |
| uzssd2 46349 | Subset relationship for tw... |
| rexabslelem 46350 | An indexed set of absolute... |
| rexabsle 46351 | An indexed set of absolute... |
| allbutfiinf 46352 | Given a "for all but finit... |
| supxrrernmpt 46353 | The real and extended real... |
| suprleubrnmpt 46354 | The supremum of a nonempty... |
| infrnmptle 46355 | An indexed infimum of exte... |
| infxrunb3 46356 | The infimum of an unbounde... |
| uzn0d 46357 | The upper integers are all... |
| uzssd3 46358 | Subset relationship for tw... |
| rexabsle2 46359 | An indexed set of absolute... |
| infxrunb3rnmpt 46360 | The infimum of an unbounde... |
| supxrre3rnmpt 46361 | The indexed supremum of a ... |
| uzublem 46362 | A set of reals, indexed by... |
| uzub 46363 | A set of reals, indexed by... |
| ssrexr 46364 | A subset of the reals is a... |
| supxrmnf2 46365 | Removing minus infinity fr... |
| supxrcli 46366 | The supremum of an arbitra... |
| uzid3 46367 | Membership of the least me... |
| infxrlesupxr 46368 | The supremum of a nonempty... |
| xnegeqd 46369 | Equality of two extended n... |
| xnegrecl 46370 | The extended real negative... |
| xnegnegi 46371 | Extended real version of ~... |
| xnegeqi 46372 | Equality of two extended n... |
| nfxnegd 46373 | Deduction version of ~ nfx... |
| xnegnegd 46374 | Extended real version of ~... |
| uzred 46375 | An upper integer is a real... |
| xnegcli 46376 | Closure of extended real n... |
| supminfrnmpt 46377 | The indexed supremum of a ... |
| infxrpnf 46378 | Adding plus infinity to a ... |
| infxrrnmptcl 46379 | The infimum of an arbitrar... |
| leneg2d 46380 | Negative of one side of 'l... |
| supxrltinfxr 46381 | The supremum of the empty ... |
| max1d 46382 | A number is less than or e... |
| supxrleubrnmptf 46383 | The supremum of a nonempty... |
| nleltd 46384 | 'Not less than or equal to... |
| zxrd 46385 | An integer is an extended ... |
| infxrgelbrnmpt 46386 | The infimum of an indexed ... |
| rphalfltd 46387 | Half of a positive real is... |
| uzssz2 46388 | An upper set of integers i... |
| leneg3d 46389 | Negative of one side of 'l... |
| max2d 46390 | A number is less than or e... |
| uzn0bi 46391 | The upper integers functio... |
| xnegrecl2 46392 | If the extended real negat... |
| nfxneg 46393 | Bound-variable hypothesis ... |
| uzxrd 46394 | An upper integer is an ext... |
| infxrpnf2 46395 | Removing plus infinity fro... |
| supminfxr 46396 | The extended real suprema ... |
| infrpgernmpt 46397 | The infimum of a nonempty,... |
| xnegre 46398 | An extended real is real i... |
| xnegrecl2d 46399 | If the extended real negat... |
| uzxr 46400 | An upper integer is an ext... |
| supminfxr2 46401 | The extended real suprema ... |
| xnegred 46402 | An extended real is real i... |
| supminfxrrnmpt 46403 | The indexed supremum of a ... |
| min1d 46404 | The minimum of two numbers... |
| min2d 46405 | The minimum of two numbers... |
| xrnpnfmnf 46406 | An extended real that is n... |
| uzsscn 46407 | An upper set of integers i... |
| absimnre 46408 | The absolute value of the ... |
| uzsscn2 46409 | An upper set of integers i... |
| xrtgcntopre 46410 | The standard topologies on... |
| absimlere 46411 | The absolute value of the ... |
| rpssxr 46412 | The positive reals are a s... |
| monoordxrv 46413 | Ordering relation for a mo... |
| monoordxr 46414 | Ordering relation for a mo... |
| monoord2xrv 46415 | Ordering relation for a mo... |
| monoord2xr 46416 | Ordering relation for a mo... |
| xrpnf 46417 | An extended real is plus i... |
| xlenegcon1 46418 | Extended real version of ~... |
| xlenegcon2 46419 | Extended real version of ~... |
| pimxrneun 46420 | The preimage of a set of e... |
| caucvgbf 46421 | A function is convergent i... |
| cvgcau 46422 | A convergent function is C... |
| cvgcaule 46423 | A convergent function is C... |
| rexanuz2nf 46424 | A simple counterexample re... |
| gtnelioc 46425 | A real number larger than ... |
| ioossioc 46426 | An open interval is a subs... |
| ioondisj2 46427 | A condition for two open i... |
| ioondisj1 46428 | A condition for two open i... |
| ioogtlb 46429 | An element of a closed int... |
| evthiccabs 46430 | Extreme Value Theorem on y... |
| ltnelicc 46431 | A real number smaller than... |
| eliood 46432 | Membership in an open real... |
| iooabslt 46433 | An upper bound for the dis... |
| gtnelicc 46434 | A real number greater than... |
| iooinlbub 46435 | An open interval has empty... |
| iocgtlb 46436 | An element of a left-open ... |
| iocleub 46437 | An element of a left-open ... |
| eliccd 46438 | Membership in a closed rea... |
| eliccre 46439 | A member of a closed inter... |
| eliooshift 46440 | Element of an open interva... |
| eliocd 46441 | Membership in a left-open ... |
| icoltub 46442 | An element of a left-close... |
| eliocre 46443 | A member of a left-open ri... |
| iooltub 46444 | An element of an open inte... |
| ioontr 46445 | The interior of an interva... |
| snunioo1 46446 | The closure of one end of ... |
| lbioc 46447 | A left-open right-closed i... |
| ioomidp 46448 | The midpoint is an element... |
| iccdifioo 46449 | If the open inverval is re... |
| iccdifprioo 46450 | An open interval is the cl... |
| ioossioobi 46451 | Biconditional form of ~ io... |
| iccshift 46452 | A closed interval shifted ... |
| iccsuble 46453 | An upper bound to the dist... |
| iocopn 46454 | A left-open right-closed i... |
| eliccelioc 46455 | Membership in a closed int... |
| iooshift 46456 | An open interval shifted b... |
| iccintsng 46457 | Intersection of two adiace... |
| icoiccdif 46458 | Left-closed right-open int... |
| icoopn 46459 | A left-closed right-open i... |
| icoub 46460 | A left-closed, right-open ... |
| eliccxrd 46461 | Membership in a closed rea... |
| pnfel0pnf 46462 | ` +oo ` is a nonnegative e... |
| eliccnelico 46463 | An element of a closed int... |
| eliccelicod 46464 | A member of a closed inter... |
| ge0xrre 46465 | A nonnegative extended rea... |
| ge0lere 46466 | A nonnegative extended Rea... |
| elicores 46467 | Membership in a left-close... |
| inficc 46468 | The infimum of a nonempty ... |
| qinioo 46469 | The rational numbers are d... |
| lenelioc 46470 | A real number smaller than... |
| ioonct 46471 | A nonempty open interval i... |
| xrgtnelicc 46472 | A real number greater than... |
| iccdificc 46473 | The difference of two clos... |
| iocnct 46474 | A nonempty left-open, righ... |
| iccnct 46475 | A closed interval, with mo... |
| iooiinicc 46476 | A closed interval expresse... |
| iccgelbd 46477 | An element of a closed int... |
| iooltubd 46478 | An element of an open inte... |
| icoltubd 46479 | An element of a left-close... |
| qelioo 46480 | The rational numbers are d... |
| tgqioo2 46481 | Every open set of reals is... |
| iccleubd 46482 | An element of a closed int... |
| elioored 46483 | A member of an open interv... |
| ioogtlbd 46484 | An element of a closed int... |
| ioofun 46485 | ` (,) ` is a function. (C... |
| icomnfinre 46486 | A left-closed, right-open,... |
| sqrlearg 46487 | The square compared with i... |
| ressiocsup 46488 | If the supremum belongs to... |
| ressioosup 46489 | If the supremum does not b... |
| iooiinioc 46490 | A left-open, right-closed ... |
| ressiooinf 46491 | If the infimum does not be... |
| iocleubd 46492 | An element of a left-open ... |
| uzinico 46493 | An upper interval of integ... |
| preimaiocmnf 46494 | Preimage of a right-closed... |
| uzinico2 46495 | An upper interval of integ... |
| uzinico3 46496 | An upper interval of integ... |
| dmico 46497 | The domain of the closed-b... |
| ndmico 46498 | The closed-below, open-abo... |
| uzubioo 46499 | The upper integers are unb... |
| uzubico 46500 | The upper integers are unb... |
| uzubioo2 46501 | The upper integers are unb... |
| uzubico2 46502 | The upper integers are unb... |
| iocgtlbd 46503 | An element of a left-open ... |
| xrtgioo2 46504 | The topology on the extend... |
| fsummulc1f 46505 | Closure of a finite sum of... |
| fsumnncl 46506 | Closure of a nonempty, fin... |
| fsumge0cl 46507 | The finite sum of nonnegat... |
| fsumf1of 46508 | Re-index a finite sum usin... |
| fsumiunss 46509 | Sum over a disjoint indexe... |
| fsumreclf 46510 | Closure of a finite sum of... |
| fsumlessf 46511 | A shorter sum of nonnegati... |
| fsumsupp0 46512 | Finite sum of function val... |
| fsumsermpt 46513 | A finite sum expressed in ... |
| fmul01 46514 | Multiplying a finite numbe... |
| fmulcl 46515 | If ' Y ' is closed under t... |
| fmuldfeqlem1 46516 | induction step for the pro... |
| fmuldfeq 46517 | X and Z are two equivalent... |
| fmul01lt1lem1 46518 | Given a finite multiplicat... |
| fmul01lt1lem2 46519 | Given a finite multiplicat... |
| fmul01lt1 46520 | Given a finite multiplicat... |
| cncfmptss 46521 | A continuous complex funct... |
| rrpsscn 46522 | The positive reals are a s... |
| mulc1cncfg 46523 | A version of ~ mulc1cncf u... |
| infrglb 46524 | The infimum of a nonempty ... |
| expcnfg 46525 | If ` F ` is a complex cont... |
| prodeq2ad 46526 | Equality deduction for pro... |
| fprodsplit1 46527 | Separate out a term in a f... |
| fprodexp 46528 | Positive integer exponenti... |
| fprodabs2 46529 | The absolute value of a fi... |
| fprod0 46530 | A finite product with a ze... |
| mccllem 46531 | * Induction step for ~ mcc... |
| mccl 46532 | A multinomial coefficient,... |
| fprodcnlem 46533 | A finite product of functi... |
| fprodcn 46534 | A finite product of functi... |
| clim1fr1 46535 | A class of sequences of fr... |
| isumneg 46536 | Negation of a converging s... |
| climrec 46537 | Limit of the reciprocal of... |
| climmulf 46538 | A version of ~ climmul usi... |
| climexp 46539 | The limit of natural power... |
| climinf 46540 | A bounded monotonic noninc... |
| climsuselem1 46541 | The subsequence index ` I ... |
| climsuse 46542 | A subsequence ` G ` of a c... |
| climrecf 46543 | A version of ~ climrec usi... |
| climneg 46544 | Complex limit of the negat... |
| climinff 46545 | A version of ~ climinf usi... |
| climdivf 46546 | Limit of the ratio of two ... |
| climreeq 46547 | If ` F ` is a real functio... |
| ellimciota 46548 | An explicit value for the ... |
| climaddf 46549 | A version of ~ climadd usi... |
| mullimc 46550 | Limit of the product of tw... |
| ellimcabssub0 46551 | An equivalent condition fo... |
| limcdm0 46552 | If a function has empty do... |
| islptre 46553 | An equivalence condition f... |
| limccog 46554 | Limit of the composition o... |
| limciccioolb 46555 | The limit of a function at... |
| climf 46556 | Express the predicate: Th... |
| mullimcf 46557 | Limit of the multiplicatio... |
| constlimc 46558 | Limit of constant function... |
| rexlim2d 46559 | Inference removing two res... |
| idlimc 46560 | Limit of the identity func... |
| divcnvg 46561 | The sequence of reciprocal... |
| limcperiod 46562 | If ` F ` is a periodic fun... |
| limcrecl 46563 | If ` F ` is a real-valued ... |
| sumnnodd 46564 | A series indexed by ` NN `... |
| lptioo2 46565 | The upper bound of an open... |
| lptioo1 46566 | The lower bound of an open... |
| limcmptdm 46567 | The domain of a maps-to fu... |
| clim2f 46568 | Express the predicate: Th... |
| limcicciooub 46569 | The limit of a function at... |
| ltmod 46570 | A sufficient condition for... |
| islpcn 46571 | A characterization for a l... |
| lptre2pt 46572 | If a set in the real line ... |
| limsupre 46573 | If a sequence is bounded, ... |
| limcresiooub 46574 | The left limit doesn't cha... |
| limcresioolb 46575 | The right limit doesn't ch... |
| limcleqr 46576 | If the left and the right ... |
| lptioo2cn 46577 | The upper bound of an open... |
| lptioo1cn 46578 | The lower bound of an open... |
| neglimc 46579 | Limit of the negative func... |
| addlimc 46580 | Sum of two limits. (Contr... |
| 0ellimcdiv 46581 | If the numerator converges... |
| clim2cf 46582 | Express the predicate ` F ... |
| limclner 46583 | For a limit point, both fr... |
| sublimc 46584 | Subtraction of two limits.... |
| reclimc 46585 | Limit of the reciprocal of... |
| clim0cf 46586 | Express the predicate ` F ... |
| limclr 46587 | For a limit point, both fr... |
| divlimc 46588 | Limit of the quotient of t... |
| expfac 46589 | Factorial grows faster tha... |
| climconstmpt 46590 | A constant sequence conver... |
| climresmpt 46591 | A function restricted to u... |
| climsubmpt 46592 | Limit of the difference of... |
| climsubc2mpt 46593 | Limit of the difference of... |
| climsubc1mpt 46594 | Limit of the difference of... |
| fnlimfv 46595 | The value of the limit fun... |
| climreclf 46596 | The limit of a convergent ... |
| climeldmeq 46597 | Two functions that are eve... |
| climf2 46598 | Express the predicate: Th... |
| fnlimcnv 46599 | The sequence of function v... |
| climeldmeqmpt 46600 | Two functions that are eve... |
| climfveq 46601 | Two functions that are eve... |
| clim2f2 46602 | Express the predicate: Th... |
| climfveqmpt 46603 | Two functions that are eve... |
| climd 46604 | Express the predicate: Th... |
| clim2d 46605 | The limit of complex numbe... |
| fnlimfvre 46606 | The limit function of real... |
| allbutfifvre 46607 | Given a sequence of real-v... |
| climleltrp 46608 | The limit of complex numbe... |
| fnlimfvre2 46609 | The limit function of real... |
| fnlimf 46610 | The limit function of real... |
| fnlimabslt 46611 | A sequence of function val... |
| climfveqf 46612 | Two functions that are eve... |
| climmptf 46613 | Exhibit a function ` G ` w... |
| climfveqmpt3 46614 | Two functions that are eve... |
| climeldmeqf 46615 | Two functions that are eve... |
| climreclmpt 46616 | The limit of B convergent ... |
| limsupref 46617 | If a sequence is bounded, ... |
| limsupbnd1f 46618 | If a sequence is eventuall... |
| climbddf 46619 | A converging sequence of c... |
| climeqf 46620 | Two functions that are eve... |
| climeldmeqmpt3 46621 | Two functions that are eve... |
| limsupcld 46622 | Closure of the superior li... |
| climfv 46623 | The limit of a convergent ... |
| limsupval3 46624 | The superior limit of an i... |
| climfveqmpt2 46625 | Two functions that are eve... |
| limsup0 46626 | The superior limit of the ... |
| climeldmeqmpt2 46627 | Two functions that are eve... |
| limsupresre 46628 | The supremum limit of a fu... |
| climeqmpt 46629 | Two functions that are eve... |
| climfvd 46630 | The limit of a convergent ... |
| limsuplesup 46631 | An upper bound for the sup... |
| limsupresico 46632 | The superior limit doesn't... |
| limsuppnfdlem 46633 | If the restriction of a fu... |
| limsuppnfd 46634 | If the restriction of a fu... |
| limsupresuz 46635 | If the real part of the do... |
| limsupub 46636 | If the limsup is not ` +oo... |
| limsupres 46637 | The superior limit of a re... |
| climinf2lem 46638 | A convergent, nonincreasin... |
| climinf2 46639 | A convergent, nonincreasin... |
| limsupvaluz 46640 | The superior limit, when t... |
| limsupresuz2 46641 | If the domain of a functio... |
| limsuppnflem 46642 | If the restriction of a fu... |
| limsuppnf 46643 | If the restriction of a fu... |
| limsupubuzlem 46644 | If the limsup is not ` +oo... |
| limsupubuz 46645 | For a real-valued function... |
| climinf2mpt 46646 | A bounded below, monotonic... |
| climinfmpt 46647 | A bounded below, monotonic... |
| climinf3 46648 | A convergent, nonincreasin... |
| limsupvaluzmpt 46649 | The superior limit, when t... |
| limsupequzmpt2 46650 | Two functions that are eve... |
| limsupubuzmpt 46651 | If the limsup is not ` +oo... |
| limsupmnflem 46652 | The superior limit of a fu... |
| limsupmnf 46653 | The superior limit of a fu... |
| limsupequzlem 46654 | Two functions that are eve... |
| limsupequz 46655 | Two functions that are eve... |
| limsupre2lem 46656 | Given a function on the ex... |
| limsupre2 46657 | Given a function on the ex... |
| limsupmnfuzlem 46658 | The superior limit of a fu... |
| limsupmnfuz 46659 | The superior limit of a fu... |
| limsupequzmptlem 46660 | Two functions that are eve... |
| limsupequzmpt 46661 | Two functions that are eve... |
| limsupre2mpt 46662 | Given a function on the ex... |
| limsupequzmptf 46663 | Two functions that are eve... |
| limsupre3lem 46664 | Given a function on the ex... |
| limsupre3 46665 | Given a function on the ex... |
| limsupre3mpt 46666 | Given a function on the ex... |
| limsupre3uzlem 46667 | Given a function on the ex... |
| limsupre3uz 46668 | Given a function on the ex... |
| limsupreuz 46669 | Given a function on the re... |
| limsupvaluz2 46670 | The superior limit, when t... |
| limsupreuzmpt 46671 | Given a function on the re... |
| supcnvlimsup 46672 | If a function on a set of ... |
| supcnvlimsupmpt 46673 | If a function on a set of ... |
| 0cnv 46674 | If ` (/) ` is a complex nu... |
| climuzlem 46675 | Express the predicate: Th... |
| climuz 46676 | Express the predicate: Th... |
| lmbr3v 46677 | Express the binary relatio... |
| climisp 46678 | If a sequence converges to... |
| lmbr3 46679 | Express the binary relatio... |
| climrescn 46680 | A sequence converging w.r.... |
| climxrrelem 46681 | If a sequence ranging over... |
| climxrre 46682 | If a sequence ranging over... |
| limsuplt2 46685 | The defining property of t... |
| liminfgord 46686 | Ordering property of the i... |
| limsupvald 46687 | The superior limit of a se... |
| limsupresicompt 46688 | The superior limit doesn't... |
| limsupcli 46689 | Closure of the superior li... |
| liminfgf 46690 | Closure of the inferior li... |
| liminfval 46691 | The inferior limit of a se... |
| climlimsup 46692 | A sequence of real numbers... |
| limsupge 46693 | The defining property of t... |
| liminfgval 46694 | Value of the inferior limi... |
| liminfcl 46695 | Closure of the inferior li... |
| liminfvald 46696 | The inferior limit of a se... |
| liminfval5 46697 | The inferior limit of an i... |
| limsupresxr 46698 | The superior limit of a fu... |
| liminfresxr 46699 | The inferior limit of a fu... |
| liminfval2 46700 | The superior limit, relati... |
| climlimsupcex 46701 | Counterexample for ~ climl... |
| liminfcld 46702 | Closure of the inferior li... |
| liminfresico 46703 | The inferior limit doesn't... |
| limsup10exlem 46704 | The range of the given fun... |
| limsup10ex 46705 | The superior limit of a fu... |
| liminf10ex 46706 | The inferior limit of a fu... |
| liminflelimsuplem 46707 | The superior limit is grea... |
| liminflelimsup 46708 | The superior limit is grea... |
| limsupgtlem 46709 | For any positive real, the... |
| limsupgt 46710 | Given a sequence of real n... |
| liminfresre 46711 | The inferior limit of a fu... |
| liminfresicompt 46712 | The inferior limit doesn't... |
| liminfltlimsupex 46713 | An example where the ` lim... |
| liminfgelimsup 46714 | The inferior limit is grea... |
| liminfvalxr 46715 | Alternate definition of ` ... |
| liminfresuz 46716 | If the real part of the do... |
| liminflelimsupuz 46717 | The superior limit is grea... |
| liminfvalxrmpt 46718 | Alternate definition of ` ... |
| liminfresuz2 46719 | If the domain of a functio... |
| liminfgelimsupuz 46720 | The inferior limit is grea... |
| liminfval4 46721 | Alternate definition of ` ... |
| liminfval3 46722 | Alternate definition of ` ... |
| liminfequzmpt2 46723 | Two functions that are eve... |
| liminfvaluz 46724 | Alternate definition of ` ... |
| liminf0 46725 | The inferior limit of the ... |
| limsupval4 46726 | Alternate definition of ` ... |
| liminfvaluz2 46727 | Alternate definition of ` ... |
| liminfvaluz3 46728 | Alternate definition of ` ... |
| liminflelimsupcex 46729 | A counterexample for ~ lim... |
| limsupvaluz3 46730 | Alternate definition of ` ... |
| liminfvaluz4 46731 | Alternate definition of ` ... |
| limsupvaluz4 46732 | Alternate definition of ` ... |
| climliminflimsupd 46733 | If a sequence of real numb... |
| liminfreuzlem 46734 | Given a function on the re... |
| liminfreuz 46735 | Given a function on the re... |
| liminfltlem 46736 | Given a sequence of real n... |
| liminflt 46737 | Given a sequence of real n... |
| climliminf 46738 | A sequence of real numbers... |
| liminflimsupclim 46739 | A sequence of real numbers... |
| climliminflimsup 46740 | A sequence of real numbers... |
| climliminflimsup2 46741 | A sequence of real numbers... |
| climliminflimsup3 46742 | A sequence of real numbers... |
| climliminflimsup4 46743 | A sequence of real numbers... |
| limsupub2 46744 | A extended real valued fun... |
| limsupubuz2 46745 | A sequence with values in ... |
| xlimpnfxnegmnf 46746 | A sequence converges to ` ... |
| liminflbuz2 46747 | A sequence with values in ... |
| liminfpnfuz 46748 | The inferior limit of a fu... |
| liminflimsupxrre 46749 | A sequence with values in ... |
| xlimrel 46752 | The limit on extended real... |
| xlimres 46753 | A function converges iff i... |
| xlimcl 46754 | The limit of a sequence of... |
| rexlimddv2 46755 | Restricted existential eli... |
| xlimclim 46756 | Given a sequence of reals,... |
| xlimconst 46757 | A constant sequence conver... |
| climxlim 46758 | A converging sequence in t... |
| xlimbr 46759 | Express the binary relatio... |
| fuzxrpmcn 46760 | A function mapping from an... |
| cnrefiisplem 46761 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 46762 | A non-real, complex number... |
| xlimxrre 46763 | If a sequence ranging over... |
| xlimmnfvlem1 46764 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 46765 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 46766 | A function converges to mi... |
| xlimconst2 46767 | A sequence that eventually... |
| xlimpnfvlem1 46768 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 46769 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 46770 | A function converges to pl... |
| xlimclim2lem 46771 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 46772 | Given a sequence of extend... |
| xlimmnf 46773 | A function converges to mi... |
| xlimpnf 46774 | A function converges to pl... |
| xlimmnfmpt 46775 | A function converges to pl... |
| xlimpnfmpt 46776 | A function converges to pl... |
| climxlim2lem 46777 | In this lemma for ~ climxl... |
| climxlim2 46778 | A sequence of extended rea... |
| dfxlim2v 46779 | An alternative definition ... |
| dfxlim2 46780 | An alternative definition ... |
| climresd 46781 | A function restricted to u... |
| climresdm 46782 | A real function converges ... |
| dmclimxlim 46783 | A real valued sequence tha... |
| xlimmnflimsup2 46784 | A sequence of extended rea... |
| xlimuni 46785 | An infinite sequence conve... |
| xlimclimdm 46786 | A sequence of extended rea... |
| xlimfun 46787 | The convergence relation o... |
| xlimmnflimsup 46788 | If a sequence of extended ... |
| xlimdm 46789 | Two ways to express that a... |
| xlimpnfxnegmnf2 46790 | A sequence converges to ` ... |
| xlimresdm 46791 | A function converges in th... |
| xlimpnfliminf 46792 | If a sequence of extended ... |
| xlimpnfliminf2 46793 | A sequence of extended rea... |
| xlimliminflimsup 46794 | A sequence of extended rea... |
| xlimlimsupleliminf 46795 | A sequence of extended rea... |
| coseq0 46796 | A complex number whose cos... |
| sinmulcos 46797 | Multiplication formula for... |
| coskpi2 46798 | The cosine of an integer m... |
| cosnegpi 46799 | The cosine of negative ` _... |
| sinaover2ne0 46800 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 46801 | The cosine of an integer m... |
| mulcncff 46802 | The multiplication of two ... |
| cncfmptssg 46803 | A continuous complex funct... |
| constcncfg 46804 | A constant function is a c... |
| idcncfg 46805 | The identity function is a... |
| cncfshift 46806 | A periodic continuous func... |
| resincncf 46807 | ` sin ` restricted to real... |
| addccncf2 46808 | Adding a constant is a con... |
| 0cnf 46809 | The empty set is a continu... |
| fsumcncf 46810 | The finite sum of continuo... |
| cncfperiod 46811 | A periodic continuous func... |
| subcncff 46812 | The subtraction of two con... |
| negcncfg 46813 | The opposite of a continuo... |
| cnfdmsn 46814 | A function with a singleto... |
| cncfcompt 46815 | Composition of continuous ... |
| addcncff 46816 | The sum of two continuous ... |
| ioccncflimc 46817 | Limit at the upper bound o... |
| cncfuni 46818 | A complex function on a su... |
| icccncfext 46819 | A continuous function on a... |
| cncficcgt0 46820 | A the absolute value of a ... |
| icocncflimc 46821 | Limit at the lower bound, ... |
| cncfdmsn 46822 | A complex function with a ... |
| divcncff 46823 | The quotient of two contin... |
| cncfshiftioo 46824 | A periodic continuous func... |
| cncfiooicclem1 46825 | A continuous function ` F ... |
| cncfiooicc 46826 | A continuous function ` F ... |
| cncfiooiccre 46827 | A continuous function ` F ... |
| cncfioobdlem 46828 | ` G ` actually extends ` F... |
| cncfioobd 46829 | A continuous function ` F ... |
| jumpncnp 46830 | Jump discontinuity or disc... |
| cxpcncf2 46831 | The complex power function... |
| fprodcncf 46832 | The finite product of cont... |
| add1cncf 46833 | Addition to a constant is ... |
| add2cncf 46834 | Addition to a constant is ... |
| sub1cncfd 46835 | Subtracting a constant is ... |
| sub2cncfd 46836 | Subtraction from a constan... |
| fprodsub2cncf 46837 | ` F ` is continuous. (Con... |
| fprodadd2cncf 46838 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 46839 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 46840 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 46841 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 46842 | The sequence ` S ` of fini... |
| dvsinexp 46843 | The derivative of sin^N . ... |
| dvcosre 46844 | The real derivative of the... |
| dvsinax 46845 | Derivative exercise: the d... |
| dvsubf 46846 | The subtraction rule for e... |
| dvmptconst 46847 | Function-builder for deriv... |
| dvcnre 46848 | From complex differentiati... |
| dvmptidg 46849 | Function-builder for deriv... |
| dvresntr 46850 | Function-builder for deriv... |
| fperdvper 46851 | The derivative of a period... |
| dvasinbx 46852 | Derivative exercise: the d... |
| dvresioo 46853 | Restriction of a derivativ... |
| dvdivf 46854 | The quotient rule for ever... |
| dvdivbd 46855 | A sufficient condition for... |
| dvsubcncf 46856 | A sufficient condition for... |
| dvmulcncf 46857 | A sufficient condition for... |
| dvcosax 46858 | Derivative exercise: the d... |
| dvdivcncf 46859 | A sufficient condition for... |
| dvbdfbdioolem1 46860 | Given a function with boun... |
| dvbdfbdioolem2 46861 | A function on an open inte... |
| dvbdfbdioo 46862 | A function on an open inte... |
| ioodvbdlimc1lem1 46863 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 46864 | Limit at the lower bound o... |
| ioodvbdlimc1 46865 | A real function with bound... |
| ioodvbdlimc2lem 46866 | Limit at the upper bound o... |
| ioodvbdlimc2 46867 | A real function with bound... |
| dvdmsscn 46868 | ` X ` is a subset of ` CC ... |
| dvmptmulf 46869 | Function-builder for deriv... |
| dvnmptdivc 46870 | Function-builder for itera... |
| dvdsn1add 46871 | If ` K ` divides ` N ` but... |
| dvxpaek 46872 | Derivative of the polynomi... |
| dvnmptconst 46873 | The ` N ` -th derivative o... |
| dvnxpaek 46874 | The ` n ` -th derivative o... |
| dvnmul 46875 | Function-builder for the `... |
| dvmptfprodlem 46876 | Induction step for ~ dvmpt... |
| dvmptfprod 46877 | Function-builder for deriv... |
| dvnprodlem1 46878 | ` D ` is bijective. (Cont... |
| dvnprodlem2 46879 | Induction step for ~ dvnpr... |
| dvnprodlem3 46880 | The multinomial formula fo... |
| dvnprod 46881 | The multinomial formula fo... |
| itgsin0pilem1 46882 | Calculation of the integra... |
| ibliccsinexp 46883 | sin^n on a closed interval... |
| itgsin0pi 46884 | Calculation of the integra... |
| iblioosinexp 46885 | sin^n on an open integral ... |
| itgsinexplem1 46886 | Integration by parts is ap... |
| itgsinexp 46887 | A recursive formula for th... |
| iblconstmpt 46888 | A constant function is int... |
| itgeq1d 46889 | Equality theorem for an in... |
| mbfres2cn 46890 | Measurability of a piecewi... |
| vol0 46891 | The measure of the empty s... |
| ditgeqiooicc 46892 | A function ` F ` on an ope... |
| volge0 46893 | The volume of a set is alw... |
| cnbdibl 46894 | A continuous bounded funct... |
| snmbl 46895 | A singleton is measurable.... |
| ditgeq3d 46896 | Equality theorem for the d... |
| iblempty 46897 | The empty function is inte... |
| iblsplit 46898 | The union of two integrabl... |
| volsn 46899 | A singleton has 0 Lebesgue... |
| itgvol0 46900 | If the domani is negligibl... |
| itgcoscmulx 46901 | Exercise: the integral of ... |
| iblsplitf 46902 | A version of ~ iblsplit us... |
| ibliooicc 46903 | If a function is integrabl... |
| volioc 46904 | The measure of a left-open... |
| iblspltprt 46905 | If a function is integrabl... |
| itgsincmulx 46906 | Exercise: the integral of ... |
| itgsubsticclem 46907 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 46908 | Integration by u-substitut... |
| itgioocnicc 46909 | The integral of a piecewis... |
| iblcncfioo 46910 | A continuous function ` F ... |
| itgspltprt 46911 | The ` S. ` integral splits... |
| itgiccshift 46912 | The integral of a function... |
| itgperiod 46913 | The integral of a periodic... |
| itgsbtaddcnst 46914 | Integral substitution, add... |
| volico 46915 | The measure of left-closed... |
| sublevolico 46916 | The Lebesgue measure of a ... |
| dmvolss 46917 | Lebesgue measurable sets a... |
| ismbl3 46918 | The predicate " ` A ` is L... |
| volioof 46919 | The function that assigns ... |
| ovolsplit 46920 | The Lebesgue outer measure... |
| fvvolioof 46921 | The function value of the ... |
| volioore 46922 | The measure of an open int... |
| fvvolicof 46923 | The function value of the ... |
| voliooico 46924 | An open interval and a lef... |
| ismbl4 46925 | The predicate " ` A ` is L... |
| volioofmpt 46926 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 46927 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 46928 | The Lebesgue measure of op... |
| volicofmpt 46929 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 46930 | The Lebesgue measure of a ... |
| voliccico 46931 | A closed interval and a le... |
| mbfdmssre 46932 | The domain of a measurable... |
| stoweidlem1 46933 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 46934 | lemma for ~ stoweid : here... |
| stoweidlem3 46935 | Lemma for ~ stoweid : if `... |
| stoweidlem4 46936 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 46937 | There exists a δ as ... |
| stoweidlem6 46938 | Lemma for ~ stoweid : two ... |
| stoweidlem7 46939 | This lemma is used to prov... |
| stoweidlem8 46940 | Lemma for ~ stoweid : two ... |
| stoweidlem9 46941 | Lemma for ~ stoweid : here... |
| stoweidlem10 46942 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 46943 | This lemma is used to prov... |
| stoweidlem12 46944 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 46945 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 46946 | There exists a ` k ` as in... |
| stoweidlem15 46947 | This lemma is used to prov... |
| stoweidlem16 46948 | Lemma for ~ stoweid . The... |
| stoweidlem17 46949 | This lemma proves that the... |
| stoweidlem18 46950 | This theorem proves Lemma ... |
| stoweidlem19 46951 | If a set of real functions... |
| stoweidlem20 46952 | If a set A of real functio... |
| stoweidlem21 46953 | Once the Stone Weierstrass... |
| stoweidlem22 46954 | If a set of real functions... |
| stoweidlem23 46955 | This lemma is used to prov... |
| stoweidlem24 46956 | This lemma proves that for... |
| stoweidlem25 46957 | This lemma proves that for... |
| stoweidlem26 46958 | This lemma is used to prov... |
| stoweidlem27 46959 | This lemma is used to prov... |
| stoweidlem28 46960 | There exists a δ as ... |
| stoweidlem29 46961 | When the hypothesis for th... |
| stoweidlem30 46962 | This lemma is used to prov... |
| stoweidlem31 46963 | This lemma is used to prov... |
| stoweidlem32 46964 | If a set A of real functio... |
| stoweidlem33 46965 | If a set of real functions... |
| stoweidlem34 46966 | This lemma proves that for... |
| stoweidlem35 46967 | This lemma is used to prov... |
| stoweidlem36 46968 | This lemma is used to prov... |
| stoweidlem37 46969 | This lemma is used to prov... |
| stoweidlem38 46970 | This lemma is used to prov... |
| stoweidlem39 46971 | This lemma is used to prov... |
| stoweidlem40 46972 | This lemma proves that q_n... |
| stoweidlem41 46973 | This lemma is used to prov... |
| stoweidlem42 46974 | This lemma is used to prov... |
| stoweidlem43 46975 | This lemma is used to prov... |
| stoweidlem44 46976 | This lemma is used to prov... |
| stoweidlem45 46977 | This lemma proves that, gi... |
| stoweidlem46 46978 | This lemma proves that set... |
| stoweidlem47 46979 | Subtracting a constant fro... |
| stoweidlem48 46980 | This lemma is used to prov... |
| stoweidlem49 46981 | There exists a function q_... |
| stoweidlem50 46982 | This lemma proves that set... |
| stoweidlem51 46983 | There exists a function x ... |
| stoweidlem52 46984 | There exists a neighborhoo... |
| stoweidlem53 46985 | This lemma is used to prov... |
| stoweidlem54 46986 | There exists a function ` ... |
| stoweidlem55 46987 | This lemma proves the exis... |
| stoweidlem56 46988 | This theorem proves Lemma ... |
| stoweidlem57 46989 | There exists a function x ... |
| stoweidlem58 46990 | This theorem proves Lemma ... |
| stoweidlem59 46991 | This lemma proves that the... |
| stoweidlem60 46992 | This lemma proves that the... |
| stoweidlem61 46993 | This lemma proves that the... |
| stoweidlem62 46994 | This theorem proves the St... |
| stoweid 46995 | This theorem proves the St... |
| stowei 46996 | This theorem proves the St... |
| wallispilem1 46997 | ` I ` is monotone: increas... |
| wallispilem2 46998 | A first set of properties ... |
| wallispilem3 46999 | I maps to real values. (C... |
| wallispilem4 47000 | ` F ` maps to explicit exp... |
| wallispilem5 47001 | The sequence ` H ` converg... |
| wallispi 47002 | Wallis' formula for π :... |
| wallispi2lem1 47003 | An intermediate step betwe... |
| wallispi2lem2 47004 | Two expressions are proven... |
| wallispi2 47005 | An alternative version of ... |
| stirlinglem1 47006 | A simple limit of fraction... |
| stirlinglem2 47007 | ` A ` maps to positive rea... |
| stirlinglem3 47008 | Long but simple algebraic ... |
| stirlinglem4 47009 | Algebraic manipulation of ... |
| stirlinglem5 47010 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 47011 | A series that converges to... |
| stirlinglem7 47012 | Algebraic manipulation of ... |
| stirlinglem8 47013 | If ` A ` converges to ` C ... |
| stirlinglem9 47014 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 47015 | A bound for any B(N)-B(N +... |
| stirlinglem11 47016 | ` B ` is decreasing. (Con... |
| stirlinglem12 47017 | The sequence ` B ` is boun... |
| stirlinglem13 47018 | ` B ` is decreasing and ha... |
| stirlinglem14 47019 | The sequence ` A ` converg... |
| stirlinglem15 47020 | The Stirling's formula is ... |
| stirling 47021 | Stirling's approximation f... |
| stirlingr 47022 | Stirling's approximation f... |
| dirkerval 47023 | The N_th Dirichlet kernel.... |
| dirker2re 47024 | The Dirichlet kernel value... |
| dirkerdenne0 47025 | The Dirichlet kernel denom... |
| dirkerval2 47026 | The N_th Dirichlet kernel ... |
| dirkerre 47027 | The Dirichlet kernel at an... |
| dirkerper 47028 | the Dirichlet kernel has p... |
| dirkerf 47029 | For any natural number ` N... |
| dirkertrigeqlem1 47030 | Sum of an even number of a... |
| dirkertrigeqlem2 47031 | Trigonometric equality lem... |
| dirkertrigeqlem3 47032 | Trigonometric equality lem... |
| dirkertrigeq 47033 | Trigonometric equality for... |
| dirkeritg 47034 | The definite integral of t... |
| dirkercncflem1 47035 | If ` Y ` is a multiple of ... |
| dirkercncflem2 47036 | Lemma used to prove that t... |
| dirkercncflem3 47037 | The Dirichlet kernel is co... |
| dirkercncflem4 47038 | The Dirichlet kernel is co... |
| dirkercncf 47039 | For any natural number ` N... |
| fourierdlem1 47040 | A partition interval is a ... |
| fourierdlem2 47041 | Membership in a partition.... |
| fourierdlem3 47042 | Membership in a partition.... |
| fourierdlem4 47043 | ` E ` is a function that m... |
| fourierdlem5 47044 | ` S ` is a function. (Con... |
| fourierdlem6 47045 | ` X ` is in the periodic p... |
| fourierdlem7 47046 | The difference between the... |
| fourierdlem8 47047 | A partition interval is a ... |
| fourierdlem9 47048 | ` H ` is a complex functio... |
| fourierdlem10 47049 | Condition on the bounds of... |
| fourierdlem11 47050 | If there is a partition, t... |
| fourierdlem12 47051 | A point of a partition is ... |
| fourierdlem13 47052 | Value of ` V ` in terms of... |
| fourierdlem14 47053 | Given the partition ` V ` ... |
| fourierdlem15 47054 | The range of the partition... |
| fourierdlem16 47055 | The coefficients of the fo... |
| fourierdlem17 47056 | The defined ` L ` is actua... |
| fourierdlem18 47057 | The function ` S ` is cont... |
| fourierdlem19 47058 | If two elements of ` D ` h... |
| fourierdlem20 47059 | Every interval in the part... |
| fourierdlem21 47060 | The coefficients of the fo... |
| fourierdlem22 47061 | The coefficients of the fo... |
| fourierdlem23 47062 | If ` F ` is continuous and... |
| fourierdlem24 47063 | A sufficient condition for... |
| fourierdlem25 47064 | If ` C ` is not in the ran... |
| fourierdlem26 47065 | Periodic image of a point ... |
| fourierdlem27 47066 | A partition open interval ... |
| fourierdlem28 47067 | Derivative of ` ( F `` ( X... |
| fourierdlem29 47068 | Explicit function value fo... |
| fourierdlem30 47069 | Sum of three small pieces ... |
| fourierdlem31 47070 | If ` A ` is finite and for... |
| fourierdlem32 47071 | Limit of a continuous func... |
| fourierdlem33 47072 | Limit of a continuous func... |
| fourierdlem34 47073 | A partition is one to one.... |
| fourierdlem35 47074 | There is a single point in... |
| fourierdlem36 47075 | ` F ` is an isomorphism. ... |
| fourierdlem37 47076 | ` I ` is a function that m... |
| fourierdlem38 47077 | The function ` F ` is cont... |
| fourierdlem39 47078 | Integration by parts of ... |
| fourierdlem40 47079 | ` H ` is a continuous func... |
| fourierdlem41 47080 | Lemma used to prove that e... |
| fourierdlem42 47081 | The set of points in a mov... |
| fourierdlem43 47082 | ` K ` is a real function. ... |
| fourierdlem44 47083 | A condition for having ` (... |
| fourierdlem46 47084 | The function ` F ` has a l... |
| fourierdlem47 47085 | For ` r ` large enough, th... |
| fourierdlem48 47086 | The given periodic functio... |
| fourierdlem49 47087 | The given periodic functio... |
| fourierdlem50 47088 | Continuity of ` O ` and it... |
| fourierdlem51 47089 | ` X ` is in the periodic p... |
| fourierdlem52 47090 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 47091 | The limit of ` F ( s ) ` a... |
| fourierdlem54 47092 | Given a partition ` Q ` an... |
| fourierdlem55 47093 | ` U ` is a real function. ... |
| fourierdlem56 47094 | Derivative of the ` K ` fu... |
| fourierdlem57 47095 | The derivative of ` O ` . ... |
| fourierdlem58 47096 | The derivative of ` K ` is... |
| fourierdlem59 47097 | The derivative of ` H ` is... |
| fourierdlem60 47098 | Given a differentiable fun... |
| fourierdlem61 47099 | Given a differentiable fun... |
| fourierdlem62 47100 | The function ` K ` is cont... |
| fourierdlem63 47101 | The upper bound of interva... |
| fourierdlem64 47102 | The partition ` V ` is fin... |
| fourierdlem65 47103 | The distance of two adjace... |
| fourierdlem66 47104 | Value of the ` G ` functio... |
| fourierdlem67 47105 | ` G ` is a function. (Con... |
| fourierdlem68 47106 | The derivative of ` O ` is... |
| fourierdlem69 47107 | A piecewise continuous fun... |
| fourierdlem70 47108 | A piecewise continuous fun... |
| fourierdlem71 47109 | A periodic piecewise conti... |
| fourierdlem72 47110 | The derivative of ` O ` is... |
| fourierdlem73 47111 | A version of the Riemann L... |
| fourierdlem74 47112 | Given a piecewise smooth f... |
| fourierdlem75 47113 | Given a piecewise smooth f... |
| fourierdlem76 47114 | Continuity of ` O ` and it... |
| fourierdlem77 47115 | If ` H ` is bounded, then ... |
| fourierdlem78 47116 | ` G ` is continuous when r... |
| fourierdlem79 47117 | ` E ` projects every inter... |
| fourierdlem80 47118 | The derivative of ` O ` is... |
| fourierdlem81 47119 | The integral of a piecewis... |
| fourierdlem82 47120 | Integral by substitution, ... |
| fourierdlem83 47121 | The fourier partial sum fo... |
| fourierdlem84 47122 | If ` F ` is piecewise cont... |
| fourierdlem85 47123 | Limit of the function ` G ... |
| fourierdlem86 47124 | Continuity of ` O ` and it... |
| fourierdlem87 47125 | The integral of ` G ` goes... |
| fourierdlem88 47126 | Given a piecewise continuo... |
| fourierdlem89 47127 | Given a piecewise continuo... |
| fourierdlem90 47128 | Given a piecewise continuo... |
| fourierdlem91 47129 | Given a piecewise continuo... |
| fourierdlem92 47130 | The integral of a piecewis... |
| fourierdlem93 47131 | Integral by substitution (... |
| fourierdlem94 47132 | For a piecewise smooth fun... |
| fourierdlem95 47133 | Algebraic manipulation of ... |
| fourierdlem96 47134 | limit for ` F ` at the low... |
| fourierdlem97 47135 | ` F ` is continuous on the... |
| fourierdlem98 47136 | ` F ` is continuous on the... |
| fourierdlem99 47137 | limit for ` F ` at the upp... |
| fourierdlem100 47138 | A piecewise continuous fun... |
| fourierdlem101 47139 | Integral by substitution f... |
| fourierdlem102 47140 | For a piecewise smooth fun... |
| fourierdlem103 47141 | The half lower part of the... |
| fourierdlem104 47142 | The half upper part of the... |
| fourierdlem105 47143 | A piecewise continuous fun... |
| fourierdlem106 47144 | For a piecewise smooth fun... |
| fourierdlem107 47145 | The integral of a piecewis... |
| fourierdlem108 47146 | The integral of a piecewis... |
| fourierdlem109 47147 | The integral of a piecewis... |
| fourierdlem110 47148 | The integral of a piecewis... |
| fourierdlem111 47149 | The fourier partial sum fo... |
| fourierdlem112 47150 | Here abbreviations (local ... |
| fourierdlem113 47151 | Fourier series convergence... |
| fourierdlem114 47152 | Fourier series convergence... |
| fourierdlem115 47153 | Fourier serier convergence... |
| fourierd 47154 | Fourier series convergence... |
| fourierclimd 47155 | Fourier series convergence... |
| fourierclim 47156 | Fourier series convergence... |
| fourier 47157 | Fourier series convergence... |
| fouriercnp 47158 | If ` F ` is continuous at ... |
| fourier2 47159 | Fourier series convergence... |
| sqwvfoura 47160 | Fourier coefficients for t... |
| sqwvfourb 47161 | Fourier series ` B ` coeff... |
| fourierswlem 47162 | The Fourier series for the... |
| fouriersw 47163 | Fourier series convergence... |
| fouriercn 47164 | If the derivative of ` F `... |
| elaa2lem 47165 | Elementhood in the set of ... |
| elaa2 47166 | Elementhood in the set of ... |
| etransclem1 47167 | ` H ` is a function. (Con... |
| etransclem2 47168 | Derivative of ` G ` . (Co... |
| etransclem3 47169 | The given ` if ` term is a... |
| etransclem4 47170 | ` F ` expressed as a finit... |
| etransclem5 47171 | A change of bound variable... |
| etransclem6 47172 | A change of bound variable... |
| etransclem7 47173 | The given product is an in... |
| etransclem8 47174 | ` F ` is a function. (Con... |
| etransclem9 47175 | If ` K ` divides ` N ` but... |
| etransclem10 47176 | The given ` if ` term is a... |
| etransclem11 47177 | A change of bound variable... |
| etransclem12 47178 | ` C ` applied to ` N ` . ... |
| etransclem13 47179 | ` F ` applied to ` Y ` . ... |
| etransclem14 47180 | Value of the term ` T ` , ... |
| etransclem15 47181 | Value of the term ` T ` , ... |
| etransclem16 47182 | Every element in the range... |
| etransclem17 47183 | The ` N ` -th derivative o... |
| etransclem18 47184 | The given function is inte... |
| etransclem19 47185 | The ` N ` -th derivative o... |
| etransclem20 47186 | ` H ` is smooth. (Contrib... |
| etransclem21 47187 | The ` N ` -th derivative o... |
| etransclem22 47188 | The ` N ` -th derivative o... |
| etransclem23 47189 | This is the claim proof in... |
| etransclem24 47190 | ` P ` divides the I -th de... |
| etransclem25 47191 | ` P ` factorial divides th... |
| etransclem26 47192 | Every term in the sum of t... |
| etransclem27 47193 | The ` N ` -th derivative o... |
| etransclem28 47194 | ` ( P - 1 ) ` factorial di... |
| etransclem29 47195 | The ` N ` -th derivative o... |
| etransclem30 47196 | The ` N ` -th derivative o... |
| etransclem31 47197 | The ` N ` -th derivative o... |
| etransclem32 47198 | This is the proof for the ... |
| etransclem33 47199 | ` F ` is smooth. (Contrib... |
| etransclem34 47200 | The ` N ` -th derivative o... |
| etransclem35 47201 | ` P ` does not divide the ... |
| etransclem36 47202 | The ` N ` -th derivative o... |
| etransclem37 47203 | ` ( P - 1 ) ` factorial di... |
| etransclem38 47204 | ` P ` divides the I -th de... |
| etransclem39 47205 | ` G ` is a function. (Con... |
| etransclem40 47206 | The ` N ` -th derivative o... |
| etransclem41 47207 | ` P ` does not divide the ... |
| etransclem42 47208 | The ` N ` -th derivative o... |
| etransclem43 47209 | ` G ` is a continuous func... |
| etransclem44 47210 | The given finite sum is no... |
| etransclem45 47211 | ` K ` is an integer. (Con... |
| etransclem46 47212 | This is the proof for equa... |
| etransclem47 47213 | ` _e ` is transcendental. ... |
| etransclem48 47214 | ` _e ` is transcendental. ... |
| etransc 47215 | ` _e ` is transcendental. ... |
| rrxtopn 47216 | The topology of the genera... |
| rrxngp 47217 | Generalized Euclidean real... |
| rrxtps 47218 | Generalized Euclidean real... |
| rrxtopnfi 47219 | The topology of the n-dime... |
| rrxtopon 47220 | The topology on generalize... |
| rrxtop 47221 | The topology on generalize... |
| rrndistlt 47222 | Given two points in the sp... |
| rrxtoponfi 47223 | The topology on n-dimensio... |
| rrxunitopnfi 47224 | The base set of the standa... |
| rrxtopn0 47225 | The topology of the zero-d... |
| qndenserrnbllem 47226 | n-dimensional rational num... |
| qndenserrnbl 47227 | n-dimensional rational num... |
| rrxtopn0b 47228 | The topology of the zero-d... |
| qndenserrnopnlem 47229 | n-dimensional rational num... |
| qndenserrnopn 47230 | n-dimensional rational num... |
| qndenserrn 47231 | n-dimensional rational num... |
| rrxsnicc 47232 | A multidimensional singlet... |
| rrnprjdstle 47233 | The distance between two p... |
| rrndsmet 47234 | ` D ` is a metric for the ... |
| rrndsxmet 47235 | ` D ` is an extended metri... |
| ioorrnopnlem 47236 | The a point in an indexed ... |
| ioorrnopn 47237 | The indexed product of ope... |
| ioorrnopnxrlem 47238 | Given a point ` F ` that b... |
| ioorrnopnxr 47239 | The indexed product of ope... |
| issal 47246 | Express the predicate " ` ... |
| pwsal 47247 | The power set of a given s... |
| salunicl 47248 | SAlg sigma-algebra is clos... |
| saluncl 47249 | The union of two sets in a... |
| prsal 47250 | The pair of the empty set ... |
| saldifcl 47251 | The complement of an eleme... |
| 0sal 47252 | The empty set belongs to e... |
| salgenval 47253 | The sigma-algebra generate... |
| saliunclf 47254 | SAlg sigma-algebra is clos... |
| saliuncl 47255 | SAlg sigma-algebra is clos... |
| salincl 47256 | The intersection of two se... |
| saluni 47257 | A set is an element of any... |
| saliinclf 47258 | SAlg sigma-algebra is clos... |
| saliincl 47259 | SAlg sigma-algebra is clos... |
| saldifcl2 47260 | The difference of two elem... |
| intsaluni 47261 | The union of an arbitrary ... |
| intsal 47262 | The arbitrary intersection... |
| salgenn0 47263 | The set used in the defini... |
| salgencl 47264 | ` SalGen ` actually genera... |
| issald 47265 | Sufficient condition to pr... |
| salexct 47266 | An example of nontrivial s... |
| sssalgen 47267 | A set is a subset of the s... |
| salgenss 47268 | The sigma-algebra generate... |
| salgenuni 47269 | The base set of the sigma-... |
| issalgend 47270 | One side of ~ dfsalgen2 . ... |
| salexct2 47271 | An example of a subset tha... |
| unisalgen 47272 | The union of a set belongs... |
| dfsalgen2 47273 | Alternate characterization... |
| salexct3 47274 | An example of a sigma-alge... |
| salgencntex 47275 | This counterexample shows ... |
| salgensscntex 47276 | This counterexample shows ... |
| issalnnd 47277 | Sufficient condition to pr... |
| dmvolsal 47278 | Lebesgue measurable sets f... |
| saldifcld 47279 | The complement of an eleme... |
| saluncld 47280 | The union of two sets in a... |
| salgencld 47281 | ` SalGen ` actually genera... |
| 0sald 47282 | The empty set belongs to e... |
| iooborel 47283 | An open interval is a Bore... |
| salincld 47284 | The intersection of two se... |
| salunid 47285 | A set is an element of any... |
| unisalgen2 47286 | The union of a set belongs... |
| bor1sal 47287 | The Borel sigma-algebra on... |
| iocborel 47288 | A left-open, right-closed ... |
| subsaliuncllem 47289 | A subspace sigma-algebra i... |
| subsaliuncl 47290 | A subspace sigma-algebra i... |
| subsalsal 47291 | A subspace sigma-algebra i... |
| subsaluni 47292 | A set belongs to the subsp... |
| salrestss 47293 | A sigma-algebra restricted... |
| sge0rnre 47296 | When ` sum^ ` is applied t... |
| fge0icoicc 47297 | If ` F ` maps to nonnegati... |
| sge0val 47298 | The value of the sum of no... |
| fge0npnf 47299 | If ` F ` maps to nonnegati... |
| sge0rnn0 47300 | The range used in the defi... |
| sge0vald 47301 | The value of the sum of no... |
| fge0iccico 47302 | A range of nonnegative ext... |
| gsumge0cl 47303 | Closure of group sum, for ... |
| sge0reval 47304 | Value of the sum of nonneg... |
| sge0pnfval 47305 | If a term in the sum of no... |
| fge0iccre 47306 | A range of nonnegative ext... |
| sge0z 47307 | Any nonnegative extended s... |
| sge00 47308 | The sum of nonnegative ext... |
| fsumlesge0 47309 | Every finite subsum of non... |
| sge0revalmpt 47310 | Value of the sum of nonneg... |
| sge0sn 47311 | A sum of a nonnegative ext... |
| sge0tsms 47312 | ` sum^ ` applied to a nonn... |
| sge0cl 47313 | The arbitrary sum of nonne... |
| sge0f1o 47314 | Re-index a nonnegative ext... |
| sge0snmpt 47315 | A sum of a nonnegative ext... |
| sge0ge0 47316 | The sum of nonnegative ext... |
| sge0xrcl 47317 | The arbitrary sum of nonne... |
| sge0repnf 47318 | The of nonnegative extende... |
| sge0fsum 47319 | The arbitrary sum of a fin... |
| sge0rern 47320 | If the sum of nonnegative ... |
| sge0supre 47321 | If the arbitrary sum of no... |
| sge0fsummpt 47322 | The arbitrary sum of a fin... |
| sge0sup 47323 | The arbitrary sum of nonne... |
| sge0less 47324 | A shorter sum of nonnegati... |
| sge0rnbnd 47325 | The range used in the defi... |
| sge0pr 47326 | Sum of a pair of nonnegati... |
| sge0gerp 47327 | The arbitrary sum of nonne... |
| sge0pnffigt 47328 | If the sum of nonnegative ... |
| sge0ssre 47329 | If a sum of nonnegative ex... |
| sge0lefi 47330 | A sum of nonnegative exten... |
| sge0lessmpt 47331 | A shorter sum of nonnegati... |
| sge0ltfirp 47332 | If the sum of nonnegative ... |
| sge0prle 47333 | The sum of a pair of nonne... |
| sge0gerpmpt 47334 | The arbitrary sum of nonne... |
| sge0resrnlem 47335 | The sum of nonnegative ext... |
| sge0resrn 47336 | The sum of nonnegative ext... |
| sge0ssrempt 47337 | If a sum of nonnegative ex... |
| sge0resplit 47338 | ` sum^ ` splits into two p... |
| sge0le 47339 | If all of the terms of sum... |
| sge0ltfirpmpt 47340 | If the extended sum of non... |
| sge0split 47341 | Split a sum of nonnegative... |
| sge0lempt 47342 | If all of the terms of sum... |
| sge0splitmpt 47343 | Split a sum of nonnegative... |
| sge0ss 47344 | Change the index set to a ... |
| sge0iunmptlemfi 47345 | Sum of nonnegative extende... |
| sge0p1 47346 | The addition of the next t... |
| sge0iunmptlemre 47347 | Sum of nonnegative extende... |
| sge0fodjrnlem 47348 | Re-index a nonnegative ext... |
| sge0fodjrn 47349 | Re-index a nonnegative ext... |
| sge0iunmpt 47350 | Sum of nonnegative extende... |
| sge0iun 47351 | Sum of nonnegative extende... |
| sge0nemnf 47352 | The generalized sum of non... |
| sge0rpcpnf 47353 | The sum of an infinite num... |
| sge0rernmpt 47354 | If the sum of nonnegative ... |
| sge0lefimpt 47355 | A sum of nonnegative exten... |
| nn0ssge0 47356 | Nonnegative integers are n... |
| sge0clmpt 47357 | The generalized sum of non... |
| sge0ltfirpmpt2 47358 | If the extended sum of non... |
| sge0isum 47359 | If a series of nonnegative... |
| sge0xrclmpt 47360 | The generalized sum of non... |
| sge0xp 47361 | Combine two generalized su... |
| sge0isummpt 47362 | If a series of nonnegative... |
| sge0ad2en 47363 | The value of the infinite ... |
| sge0isummpt2 47364 | If a series of nonnegative... |
| sge0xaddlem1 47365 | The extended addition of t... |
| sge0xaddlem2 47366 | The extended addition of t... |
| sge0xadd 47367 | The extended addition of t... |
| sge0fsummptf 47368 | The generalized sum of a f... |
| sge0snmptf 47369 | A sum of a nonnegative ext... |
| sge0ge0mpt 47370 | The sum of nonnegative ext... |
| sge0repnfmpt 47371 | The of nonnegative extende... |
| sge0pnffigtmpt 47372 | If the generalized sum of ... |
| sge0splitsn 47373 | Separate out a term in a g... |
| sge0pnffsumgt 47374 | If the sum of nonnegative ... |
| sge0gtfsumgt 47375 | If the generalized sum of ... |
| sge0uzfsumgt 47376 | If a real number is smalle... |
| sge0pnfmpt 47377 | If a term in the sum of no... |
| sge0seq 47378 | A series of nonnegative re... |
| sge0reuz 47379 | Value of the generalized s... |
| sge0reuzb 47380 | Value of the generalized s... |
| ismea 47383 | Express the predicate " ` ... |
| dmmeasal 47384 | The domain of a measure is... |
| meaf 47385 | A measure is a function th... |
| mea0 47386 | The measure of the empty s... |
| nnfoctbdjlem 47387 | There exists a mapping fro... |
| nnfoctbdj 47388 | There exists a mapping fro... |
| meadjuni 47389 | The measure of the disjoin... |
| meacl 47390 | The measure of a set is a ... |
| iundjiunlem 47391 | The sets in the sequence `... |
| iundjiun 47392 | Given a sequence ` E ` of ... |
| meaxrcl 47393 | The measure of a set is an... |
| meadjun 47394 | The measure of the union o... |
| meassle 47395 | The measure of a set is gr... |
| meaunle 47396 | The measure of the union o... |
| meadjiunlem 47397 | The sum of nonnegative ext... |
| meadjiun 47398 | The measure of the disjoin... |
| ismeannd 47399 | Sufficient condition to pr... |
| meaiunlelem 47400 | The measure of the union o... |
| meaiunle 47401 | The measure of the union o... |
| psmeasurelem 47402 | ` M ` applied to a disjoin... |
| psmeasure 47403 | Point supported measure, R... |
| voliunsge0lem 47404 | The Lebesgue measure funct... |
| voliunsge0 47405 | The Lebesgue measure funct... |
| volmea 47406 | The Lebesgue measure on th... |
| meage0 47407 | If the measure of a measur... |
| meadjunre 47408 | The measure of the union o... |
| meassre 47409 | If the measure of a measur... |
| meale0eq0 47410 | A measure that is less tha... |
| meadif 47411 | The measure of the differe... |
| meaiuninclem 47412 | Measures are continuous fr... |
| meaiuninc 47413 | Measures are continuous fr... |
| meaiuninc2 47414 | Measures are continuous fr... |
| meaiunincf 47415 | Measures are continuous fr... |
| meaiuninc3v 47416 | Measures are continuous fr... |
| meaiuninc3 47417 | Measures are continuous fr... |
| meaiininclem 47418 | Measures are continuous fr... |
| meaiininc 47419 | Measures are continuous fr... |
| meaiininc2 47420 | Measures are continuous fr... |
| caragenval 47425 | The sigma-algebra generate... |
| isome 47426 | Express the predicate " ` ... |
| caragenel 47427 | Membership in the Caratheo... |
| omef 47428 | An outer measure is a func... |
| ome0 47429 | The outer measure of the e... |
| omessle 47430 | The outer measure of a set... |
| omedm 47431 | The domain of an outer mea... |
| caragensplit 47432 | If ` E ` is in the set gen... |
| caragenelss 47433 | An element of the Caratheo... |
| carageneld 47434 | Membership in the Caratheo... |
| omecl 47435 | The outer measure of a set... |
| caragenss 47436 | The sigma-algebra generate... |
| omeunile 47437 | The outer measure of the u... |
| caragen0 47438 | The empty set belongs to a... |
| omexrcl 47439 | The outer measure of a set... |
| caragenunidm 47440 | The base set of an outer m... |
| caragensspw 47441 | The sigma-algebra generate... |
| omessre 47442 | If the outer measure of a ... |
| caragenuni 47443 | The base set of the sigma-... |
| caragenuncllem 47444 | The Caratheodory's constru... |
| caragenuncl 47445 | The Caratheodory's constru... |
| caragendifcl 47446 | The Caratheodory's constru... |
| caragenfiiuncl 47447 | The Caratheodory's constru... |
| omeunle 47448 | The outer measure of the u... |
| omeiunle 47449 | The outer measure of the i... |
| omelesplit 47450 | The outer measure of a set... |
| omeiunltfirp 47451 | If the outer measure of a ... |
| omeiunlempt 47452 | The outer measure of the i... |
| carageniuncllem1 47453 | The outer measure of ` A i... |
| carageniuncllem2 47454 | The Caratheodory's constru... |
| carageniuncl 47455 | The Caratheodory's constru... |
| caragenunicl 47456 | The Caratheodory's constru... |
| caragensal 47457 | Caratheodory's method gene... |
| caratheodorylem1 47458 | Lemma used to prove that C... |
| caratheodorylem2 47459 | Caratheodory's constructio... |
| caratheodory 47460 | Caratheodory's constructio... |
| 0ome 47461 | The map that assigns 0 to ... |
| isomenndlem 47462 | ` O ` is sub-additive w.r.... |
| isomennd 47463 | Sufficient condition to pr... |
| caragenel2d 47464 | Membership in the Caratheo... |
| omege0 47465 | If the outer measure of a ... |
| omess0 47466 | If the outer measure of a ... |
| caragencmpl 47467 | A measure built with the C... |
| vonval 47472 | Value of the Lebesgue meas... |
| ovnval 47473 | Value of the Lebesgue oute... |
| elhoi 47474 | Membership in a multidimen... |
| icoresmbl 47475 | A closed-below, open-above... |
| hoissre 47476 | The projection of a half-o... |
| ovnval2 47477 | Value of the Lebesgue oute... |
| volicorecl 47478 | The Lebesgue measure of a ... |
| hoiprodcl 47479 | The pre-measure of half-op... |
| hoicvr 47480 | ` I ` is a countable set o... |
| hoissrrn 47481 | A half-open interval is a ... |
| ovn0val 47482 | The Lebesgue outer measure... |
| ovnn0val 47483 | The value of a (multidimen... |
| ovnval2b 47484 | Value of the Lebesgue oute... |
| volicorescl 47485 | The Lebesgue measure of a ... |
| ovnprodcl 47486 | The product used in the de... |
| hoiprodcl2 47487 | The pre-measure of half-op... |
| hoicvrrex 47488 | Any subset of the multidim... |
| ovnsupge0 47489 | The set used in the defini... |
| ovnlecvr 47490 | Given a subset of multidim... |
| ovnpnfelsup 47491 | ` +oo ` is an element of t... |
| ovnsslelem 47492 | The (multidimensional, non... |
| ovnssle 47493 | The (multidimensional) Leb... |
| ovnlerp 47494 | The Lebesgue outer measure... |
| ovnf 47495 | The Lebesgue outer measure... |
| ovncvrrp 47496 | The Lebesgue outer measure... |
| ovn0lem 47497 | For any finite dimension, ... |
| ovn0 47498 | For any finite dimension, ... |
| ovncl 47499 | The Lebesgue outer measure... |
| ovn02 47500 | For the zero-dimensional s... |
| ovnxrcl 47501 | The Lebesgue outer measure... |
| ovnsubaddlem1 47502 | The Lebesgue outer measure... |
| ovnsubaddlem2 47503 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 47504 | ` ( voln* `` X ) ` is suba... |
| ovnome 47505 | ` ( voln* `` X ) ` is an o... |
| vonmea 47506 | ` ( voln `` X ) ` is a mea... |
| volicon0 47507 | The measure of a nonempty ... |
| hsphoif 47508 | ` H ` is a function (that ... |
| hoidmvval 47509 | The dimensional volume of ... |
| hoissrrn2 47510 | A half-open interval is a ... |
| hsphoival 47511 | ` H ` is a function (that ... |
| hoiprodcl3 47512 | The pre-measure of half-op... |
| volicore 47513 | The Lebesgue measure of a ... |
| hoidmvcl 47514 | The dimensional volume of ... |
| hoidmv0val 47515 | The dimensional volume of ... |
| hoidmvn0val 47516 | The dimensional volume of ... |
| hsphoidmvle2 47517 | The dimensional volume of ... |
| hsphoidmvle 47518 | The dimensional volume of ... |
| hoidmvval0 47519 | The dimensional volume of ... |
| hoiprodp1 47520 | The dimensional volume of ... |
| sge0hsphoire 47521 | If the generalized sum of ... |
| hoidmvval0b 47522 | The dimensional volume of ... |
| hoidmv1lelem1 47523 | The supremum of ` U ` belo... |
| hoidmv1lelem2 47524 | This is the contradiction ... |
| hoidmv1lelem3 47525 | The dimensional volume of ... |
| hoidmv1le 47526 | The dimensional volume of ... |
| hoidmvlelem1 47527 | The supremum of ` U ` belo... |
| hoidmvlelem2 47528 | This is the contradiction ... |
| hoidmvlelem3 47529 | This is the contradiction ... |
| hoidmvlelem4 47530 | The dimensional volume of ... |
| hoidmvlelem5 47531 | The dimensional volume of ... |
| hoidmvle 47532 | The dimensional volume of ... |
| ovnhoilem1 47533 | The Lebesgue outer measure... |
| ovnhoilem2 47534 | The Lebesgue outer measure... |
| ovnhoi 47535 | The Lebesgue outer measure... |
| dmovn 47536 | The domain of the Lebesgue... |
| hoicoto2 47537 | The half-open interval exp... |
| dmvon 47538 | Lebesgue measurable n-dime... |
| hoi2toco 47539 | The half-open interval exp... |
| hoidifhspval 47540 | ` D ` is a function that r... |
| hspval 47541 | The value of the half-spac... |
| ovnlecvr2 47542 | Given a subset of multidim... |
| ovncvr2 47543 | ` B ` and ` T ` are the le... |
| dmovnsal 47544 | The domain of the Lebesgue... |
| unidmovn 47545 | Base set of the n-dimensio... |
| rrnmbl 47546 | The set of n-dimensional R... |
| hoidifhspval2 47547 | ` D ` is a function that r... |
| hspdifhsp 47548 | A n-dimensional half-open ... |
| unidmvon 47549 | Base set of the n-dimensio... |
| hoidifhspf 47550 | ` D ` is a function that r... |
| hoidifhspval3 47551 | ` D ` is a function that r... |
| hoidifhspdmvle 47552 | The dimensional volume of ... |
| voncmpl 47553 | The Lebesgue measure is co... |
| hoiqssbllem1 47554 | The center of the n-dimens... |
| hoiqssbllem2 47555 | The center of the n-dimens... |
| hoiqssbllem3 47556 | A n-dimensional ball conta... |
| hoiqssbl 47557 | A n-dimensional ball conta... |
| hspmbllem1 47558 | Any half-space of the n-di... |
| hspmbllem2 47559 | Any half-space of the n-di... |
| hspmbllem3 47560 | Any half-space of the n-di... |
| hspmbl 47561 | Any half-space of the n-di... |
| hoimbllem 47562 | Any n-dimensional half-ope... |
| hoimbl 47563 | Any n-dimensional half-ope... |
| opnvonmbllem1 47564 | The half-open interval exp... |
| opnvonmbllem2 47565 | An open subset of the n-di... |
| opnvonmbl 47566 | An open subset of the n-di... |
| opnssborel 47567 | Open sets of a generalized... |
| borelmbl 47568 | All Borel subsets of the n... |
| volicorege0 47569 | The Lebesgue measure of a ... |
| isvonmbl 47570 | The predicate " ` A ` is m... |
| mblvon 47571 | The n-dimensional Lebesgue... |
| vonmblss 47572 | n-dimensional Lebesgue mea... |
| volico2 47573 | The measure of left-closed... |
| vonmblss2 47574 | n-dimensional Lebesgue mea... |
| ovolval2lem 47575 | The value of the Lebesgue ... |
| ovolval2 47576 | The value of the Lebesgue ... |
| ovnsubadd2lem 47577 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 47578 | ` ( voln* `` X ) ` is suba... |
| ovolval3 47579 | The value of the Lebesgue ... |
| ovnsplit 47580 | The n-dimensional Lebesgue... |
| ovolval4lem1 47581 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 47582 | The value of the Lebesgue ... |
| ovolval4 47583 | The value of the Lebesgue ... |
| ovolval5lem1 47584 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 47585 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 47586 | The value of the Lebesgue ... |
| ovolval5 47587 | The value of the Lebesgue ... |
| ovnovollem1 47588 | if ` F ` is a cover of ` B... |
| ovnovollem2 47589 | if ` I ` is a cover of ` (... |
| ovnovollem3 47590 | The 1-dimensional Lebesgue... |
| ovnovol 47591 | The 1-dimensional Lebesgue... |
| vonvolmbllem 47592 | If a subset ` B ` of real ... |
| vonvolmbl 47593 | A subset of Real numbers i... |
| vonvol 47594 | The 1-dimensional Lebesgue... |
| vonvolmbl2 47595 | A subset ` X ` of the spac... |
| vonvol2 47596 | The 1-dimensional Lebesgue... |
| hoimbl2 47597 | Any n-dimensional half-ope... |
| voncl 47598 | The Lebesgue measure of a ... |
| vonhoi 47599 | The Lebesgue outer measure... |
| vonxrcl 47600 | The Lebesgue measure of a ... |
| ioosshoi 47601 | A n-dimensional open inter... |
| vonn0hoi 47602 | The Lebesgue outer measure... |
| von0val 47603 | The Lebesgue measure (for ... |
| vonhoire 47604 | The Lebesgue measure of a ... |
| iinhoiicclem 47605 | A n-dimensional closed int... |
| iinhoiicc 47606 | A n-dimensional closed int... |
| iunhoiioolem 47607 | A n-dimensional open inter... |
| iunhoiioo 47608 | A n-dimensional open inter... |
| ioovonmbl 47609 | Any n-dimensional open int... |
| iccvonmbllem 47610 | Any n-dimensional closed i... |
| iccvonmbl 47611 | Any n-dimensional closed i... |
| vonioolem1 47612 | The sequence of the measur... |
| vonioolem2 47613 | The n-dimensional Lebesgue... |
| vonioo 47614 | The n-dimensional Lebesgue... |
| vonicclem1 47615 | The sequence of the measur... |
| vonicclem2 47616 | The n-dimensional Lebesgue... |
| vonicc 47617 | The n-dimensional Lebesgue... |
| snvonmbl 47618 | A n-dimensional singleton ... |
| vonn0ioo 47619 | The n-dimensional Lebesgue... |
| vonn0icc 47620 | The n-dimensional Lebesgue... |
| ctvonmbl 47621 | Any n-dimensional countabl... |
| vonn0ioo2 47622 | The n-dimensional Lebesgue... |
| vonsn 47623 | The n-dimensional Lebesgue... |
| vonn0icc2 47624 | The n-dimensional Lebesgue... |
| vonct 47625 | The n-dimensional Lebesgue... |
| vitali2 47626 | There are non-measurable s... |
| pimltmnf2f 47629 | Given a real-valued functi... |
| pimltmnf2 47630 | Given a real-valued functi... |
| preimagelt 47631 | The preimage of a right-op... |
| preimalegt 47632 | The preimage of a left-ope... |
| pimconstlt0 47633 | Given a constant function,... |
| pimconstlt1 47634 | Given a constant function,... |
| pimltpnff 47635 | Given a real-valued functi... |
| pimltpnf 47636 | Given a real-valued functi... |
| pimgtpnf2f 47637 | Given a real-valued functi... |
| pimgtpnf2 47638 | Given a real-valued functi... |
| salpreimagelt 47639 | If all the preimages of le... |
| pimrecltpos 47640 | The preimage of an unbound... |
| salpreimalegt 47641 | If all the preimages of ri... |
| pimiooltgt 47642 | The preimage of an open in... |
| preimaicomnf 47643 | Preimage of an open interv... |
| pimltpnf2f 47644 | Given a real-valued functi... |
| pimltpnf2 47645 | Given a real-valued functi... |
| pimgtmnf2 47646 | Given a real-valued functi... |
| pimdecfgtioc 47647 | Given a nonincreasing func... |
| pimincfltioc 47648 | Given a nondecreasing func... |
| pimdecfgtioo 47649 | Given a nondecreasing func... |
| pimincfltioo 47650 | Given a nondecreasing func... |
| preimaioomnf 47651 | Preimage of an open interv... |
| preimageiingt 47652 | A preimage of a left-close... |
| preimaleiinlt 47653 | A preimage of a left-open,... |
| pimgtmnff 47654 | Given a real-valued functi... |
| pimgtmnf 47655 | Given a real-valued functi... |
| pimrecltneg 47656 | The preimage of an unbound... |
| salpreimagtge 47657 | If all the preimages of le... |
| salpreimaltle 47658 | If all the preimages of ri... |
| issmflem 47659 | The predicate " ` F ` is a... |
| issmf 47660 | The predicate " ` F ` is a... |
| salpreimalelt 47661 | If all the preimages of ri... |
| salpreimagtlt 47662 | If all the preimages of le... |
| smfpreimalt 47663 | Given a function measurabl... |
| smff 47664 | A function measurable w.r.... |
| smfdmss 47665 | The domain of a function m... |
| issmff 47666 | The predicate " ` F ` is a... |
| issmfd 47667 | A sufficient condition for... |
| smfpreimaltf 47668 | Given a function measurabl... |
| issmfdf 47669 | A sufficient condition for... |
| sssmf 47670 | The restriction of a sigma... |
| mbfresmf 47671 | A real-valued measurable f... |
| cnfsmf 47672 | A continuous function is m... |
| incsmflem 47673 | A nondecreasing function i... |
| incsmf 47674 | A real-valued, nondecreasi... |
| smfsssmf 47675 | If a function is measurabl... |
| issmflelem 47676 | The predicate " ` F ` is a... |
| issmfle 47677 | The predicate " ` F ` is a... |
| smfpimltmpt 47678 | Given a function measurabl... |
| smfpimltxr 47679 | Given a function measurabl... |
| issmfdmpt 47680 | A sufficient condition for... |
| smfconst 47681 | Given a sigma-algebra over... |
| sssmfmpt 47682 | The restriction of a sigma... |
| cnfrrnsmf 47683 | A function, continuous fro... |
| smfid 47684 | The identity function is B... |
| bormflebmf 47685 | A Borel measurable functio... |
| smfpreimale 47686 | Given a function measurabl... |
| issmfgtlem 47687 | The predicate " ` F ` is a... |
| issmfgt 47688 | The predicate " ` F ` is a... |
| issmfled 47689 | A sufficient condition for... |
| smfpimltxrmptf 47690 | Given a function measurabl... |
| smfpimltxrmpt 47691 | Given a function measurabl... |
| smfmbfcex 47692 | A constant function, with ... |
| issmfgtd 47693 | A sufficient condition for... |
| smfpreimagt 47694 | Given a function measurabl... |
| smfaddlem1 47695 | Given the sum of two funct... |
| smfaddlem2 47696 | The sum of two sigma-measu... |
| smfadd 47697 | The sum of two sigma-measu... |
| decsmflem 47698 | A nonincreasing function i... |
| decsmf 47699 | A real-valued, nonincreasi... |
| smfpreimagtf 47700 | Given a function measurabl... |
| issmfgelem 47701 | The predicate " ` F ` is a... |
| issmfge 47702 | The predicate " ` F ` is a... |
| smflimlem1 47703 | Lemma for the proof that t... |
| smflimlem2 47704 | Lemma for the proof that t... |
| smflimlem3 47705 | The limit of sigma-measura... |
| smflimlem4 47706 | Lemma for the proof that t... |
| smflimlem5 47707 | Lemma for the proof that t... |
| smflimlem6 47708 | Lemma for the proof that t... |
| smflim 47709 | The limit of sigma-measura... |
| nsssmfmbflem 47710 | The sigma-measurable funct... |
| nsssmfmbf 47711 | The sigma-measurable funct... |
| smfpimgtxr 47712 | Given a function measurabl... |
| smfpimgtmpt 47713 | Given a function measurabl... |
| smfpreimage 47714 | Given a function measurabl... |
| mbfpsssmf 47715 | Real-valued measurable fun... |
| smfpimgtxrmptf 47716 | Given a function measurabl... |
| smfpimgtxrmpt 47717 | Given a function measurabl... |
| smfpimioompt 47718 | Given a function measurabl... |
| smfpimioo 47719 | Given a function measurabl... |
| smfresal 47720 | Given a sigma-measurable f... |
| smfrec 47721 | The reciprocal of a sigma-... |
| smfres 47722 | The restriction of sigma-m... |
| smfmullem1 47723 | The multiplication of two ... |
| smfmullem2 47724 | The multiplication of two ... |
| smfmullem3 47725 | The multiplication of two ... |
| smfmullem4 47726 | The multiplication of two ... |
| smfmul 47727 | The multiplication of two ... |
| smfmulc1 47728 | A sigma-measurable functio... |
| smfdiv 47729 | The fraction of two sigma-... |
| smfpimbor1lem1 47730 | Every open set belongs to ... |
| smfpimbor1lem2 47731 | Given a sigma-measurable f... |
| smfpimbor1 47732 | Given a sigma-measurable f... |
| smf2id 47733 | Twice the identity functio... |
| smfco 47734 | The composition of a Borel... |
| smfneg 47735 | The negative of a sigma-me... |
| smffmptf 47736 | A function measurable w.r.... |
| smffmpt 47737 | A function measurable w.r.... |
| smflim2 47738 | The limit of a sequence of... |
| smfpimcclem 47739 | Lemma for ~ smfpimcc given... |
| smfpimcc 47740 | Given a countable set of s... |
| issmfle2d 47741 | A sufficient condition for... |
| smflimmpt 47742 | The limit of a sequence of... |
| smfsuplem1 47743 | The supremum of a countabl... |
| smfsuplem2 47744 | The supremum of a countabl... |
| smfsuplem3 47745 | The supremum of a countabl... |
| smfsup 47746 | The supremum of a countabl... |
| smfsupmpt 47747 | The supremum of a countabl... |
| smfsupxr 47748 | The supremum of a countabl... |
| smfinflem 47749 | The infimum of a countable... |
| smfinf 47750 | The infimum of a countable... |
| smfinfmpt 47751 | The infimum of a countable... |
| smflimsuplem1 47752 | If ` H ` converges, the ` ... |
| smflimsuplem2 47753 | The superior limit of a se... |
| smflimsuplem3 47754 | The limit of the ` ( H `` ... |
| smflimsuplem4 47755 | If ` H ` converges, the ` ... |
| smflimsuplem5 47756 | ` H ` converges to the sup... |
| smflimsuplem6 47757 | The superior limit of a se... |
| smflimsuplem7 47758 | The superior limit of a se... |
| smflimsuplem8 47759 | The superior limit of a se... |
| smflimsup 47760 | The superior limit of a se... |
| smflimsupmpt 47761 | The superior limit of a se... |
| smfliminflem 47762 | The inferior limit of a co... |
| smfliminf 47763 | The inferior limit of a co... |
| smfliminfmpt 47764 | The inferior limit of a co... |
| adddmmbl 47765 | If two functions have doma... |
| adddmmbl2 47766 | If two functions have doma... |
| muldmmbl 47767 | If two functions have doma... |
| muldmmbl2 47768 | If two functions have doma... |
| smfdmmblpimne 47769 | If a measurable function w... |
| smfdivdmmbl 47770 | If a functions and a sigma... |
| smfpimne 47771 | Given a function measurabl... |
| smfpimne2 47772 | Given a function measurabl... |
| smfdivdmmbl2 47773 | If a functions and a sigma... |
| fsupdm 47774 | The domain of the sup func... |
| fsupdm2 47775 | The domain of the sup func... |
| smfsupdmmbllem 47776 | If a countable set of sigm... |
| smfsupdmmbl 47777 | If a countable set of sigm... |
| finfdm 47778 | The domain of the inf func... |
| finfdm2 47779 | The domain of the inf func... |
| smfinfdmmbllem 47780 | If a countable set of sigm... |
| smfinfdmmbl 47781 | If a countable set of sigm... |
| sigarval 47782 | Define the signed area by ... |
| sigarim 47783 | Signed area takes value in... |
| sigarac 47784 | Signed area is anticommuta... |
| sigaraf 47785 | Signed area is additive by... |
| sigarmf 47786 | Signed area is additive (w... |
| sigaras 47787 | Signed area is additive by... |
| sigarms 47788 | Signed area is additive (w... |
| sigarls 47789 | Signed area is linear by t... |
| sigarid 47790 | Signed area of a flat para... |
| sigarexp 47791 | Expand the signed area for... |
| sigarperm 47792 | Signed area ` ( A - C ) G ... |
| sigardiv 47793 | If signed area between vec... |
| sigarimcd 47794 | Signed area takes value in... |
| sigariz 47795 | If signed area is zero, th... |
| sigarcol 47796 | Given three points ` A ` ,... |
| sharhght 47797 | Let ` A B C ` be a triangl... |
| sigaradd 47798 | Subtracting (double) area ... |
| cevathlem1 47799 | Ceva's theorem first lemma... |
| cevathlem2 47800 | Ceva's theorem second lemm... |
| cevath 47801 | Ceva's theorem. Let ` A B... |
| simpcntrab 47802 | The center of a simple gro... |
| et-ltneverrefl 47803 | Less-than class is never r... |
| et-equeucl 47804 | Alternative proof that equ... |
| et-sqrtnegnre 47805 | The square root of a negat... |
| quantgodel 47806 | There can be no formula as... |
| quantgodelALT 47807 | There can be no formula as... |
| ormklocald 47808 | If elements of a certain s... |
| ormkglobd 47809 | If all adjacent elements o... |
| chnsubseqword 47810 | A subsequence of a chain i... |
| chnsubseqwl 47811 | A subsequence of a chain h... |
| chnsubseq 47812 | An order-preserving subseq... |
| chnsuslle 47813 | Length of a subsequence is... |
| chnerlem1 47814 | In a chain constructed on ... |
| chnerlem2 47815 | Lemma for ~ chner where th... |
| chnerlem3 47816 | Lemma for ~ chner - tricho... |
| chner 47817 | Any two elements are equiv... |
| wrddin 47818 | A word in two alphabets is... |
| wrddrin 47819 | A word whose alphabet is i... |
| wrddin2 47820 | Distribution of word class... |
| wrddun 47821 | Words in either of two alp... |
| wrddun2 47822 | Superadditivity of word co... |
| chndin 47823 | A chain in two alphabets a... |
| chndrin 47824 | A chain whose alphabet is ... |
| chndin2 47825 | Distribution of chain clas... |
| chndun 47826 | Chains in either of two al... |
| chndun2 47827 | Superaddivity of chain con... |
| chnrin 47828 | Satisfying two chain relat... |
| chnrrin 47829 | A chain of elements satisf... |
| chnrin2 47830 | Distribution of chain clas... |
| chnrun 47831 | Satisfying either of two c... |
| chnrun2 47832 | Superadditivity of chain c... |
| evenwodadd 47833 | If an integer is multiplie... |
| squeezedltsq 47834 | If a real value is squeeze... |
| sqrtnnaa 47835 | Square root of a natural n... |
| sqrtnzqaa 47836 | Square root of a nonzero r... |
| sqrtqaa 47837 | Square root of a rational ... |
| numtowerdt 47838 | Certain number sets and fi... |
| sin3t 47839 | Triple-angle formula for s... |
| cos3t 47840 | Triple-angle formula for c... |
| sin5tlem1 47841 | Lemma 1 for quintupled ang... |
| sin5tlem2 47842 | Lemma 2 for quintupled ang... |
| sin5tlem3 47843 | Lemma 3 for quintupled ang... |
| sin5tlem4 47844 | Lemma 4 for quintupled ang... |
| sin5tlem5 47845 | Lemma 5 for quintupled ang... |
| sin5t 47846 | Five-times-angle formula f... |
| cos5t 47847 | Five-times-angle formula f... |
| cos5teq 47848 | Five-times-angle formula f... |
| goldpolyfactor 47849 | Factorization of a polynom... |
| goldrarr 47850 | The golden ratio is a real... |
| goldrasin 47851 | Alternative trigonometric ... |
| goldrapos 47852 | Golden ratio is positive. ... |
| goldrarp 47853 | The golden ratio is a posi... |
| goldracos5teq 47854 | Lemma 1 for determining th... |
| goldratmolem2 47855 | Lemma 2 for determining th... |
| goldratmolem3 47856 | Lemma 3 for determining th... |
| goldratmolem4 47857 | Lemma 4 for determining th... |
| goldratval 47858 | Value of the golden ratio.... |
| lambert0 47859 | A value of Lambert W (prod... |
| lamberte 47860 | A value of Lambert W (prod... |
| cjnpoly 47861 | Complex conjugation operat... |
| tannpoly 47862 | The tangent function is no... |
| sinnpoly 47863 | Sine function is not a pol... |
| sqrtrrnpoly 47864 | Real square root is not a ... |
| sqrtnpoly 47865 | Square root function is no... |
| tmachlem-extapes 47866 | The class of all tapes is ... |
| tmachlem-finscan 47867 | Execution on any tape only... |
| tmachlem-agreeself 47868 | Any tape belongs to its ow... |
| tmachlem-agreeprod 47869 | Agreement set can be writt... |
| tmachlem-tpcomp 47870 | Product (discrete) topolog... |
| tmachlem-tpbase 47871 | The base set of product to... |
| tmachlem-tpitem 47872 | Topology lemma. (Contribu... |
| tmachlem-tpopen 47873 | Agreement sets are open in... |
| tmachlem-tpopen2 47874 | Variable-renaming lemma co... |
| tmachlem-uassst 47875 | Union of all agreement set... |
| tmachlem-exlargecover 47876 | Product topology of tapes ... |
| tmachlem-extpcover 47877 | Product topology of tapes ... |
| tmachlem-exagreecover 47878 | Particular properties of t... |
| tmachlem-agreesn 47879 | Scans for all tapes of a s... |
| tmachlem-agreefin 47880 | Any agreement set has a fi... |
| tmachlem-franscan 47881 | There is a finite number o... |
| tmachlem-fssscan 47882 | Any scan set is finite. (... |
| tmachfullfin 47883 | Folk theorem. For any alg... |
| hirstL-ax3 47884 | The third axiom of a syste... |
| ax3h 47885 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 47886 | A closed form showing (a i... |
| aibandbiaiaiffb 47887 | A closed form showing (a i... |
| notatnand 47888 | Do not use. Use intnanr i... |
| aistia 47889 | Given a is equivalent to `... |
| aisfina 47890 | Given a is equivalent to `... |
| bothtbothsame 47891 | Given both a, b are equiva... |
| bothfbothsame 47892 | Given both a, b are equiva... |
| aiffbbtat 47893 | Given a is equivalent to b... |
| aisbbisfaisf 47894 | Given a is equivalent to b... |
| axorbtnotaiffb 47895 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 47896 | Given a is equivalent to (... |
| axorbciffatcxorb 47897 | Given a is equivalent to (... |
| aibnbna 47898 | Given a implies b, (not b)... |
| aibnbaif 47899 | Given a implies b, not b, ... |
| aiffbtbat 47900 | Given a is equivalent to b... |
| astbstanbst 47901 | Given a is equivalent to T... |
| aistbistaandb 47902 | Given a is equivalent to T... |
| aisbnaxb 47903 | Given a is equivalent to b... |
| atbiffatnnb 47904 | If a implies b, then a imp... |
| bisaiaisb 47905 | Application of bicom1 with... |
| atbiffatnnbalt 47906 | If a implies b, then a imp... |
| abnotbtaxb 47907 | Assuming a, not b, there e... |
| abnotataxb 47908 | Assuming not a, b, there e... |
| conimpf 47909 | Assuming a, not b, and a i... |
| conimpfalt 47910 | Assuming a, not b, and a i... |
| aistbisfiaxb 47911 | Given a is equivalent to T... |
| aisfbistiaxb 47912 | Given a is equivalent to F... |
| aifftbifffaibif 47913 | Given a is equivalent to T... |
| aifftbifffaibifff 47914 | Given a is equivalent to T... |
| atnaiana 47915 | Given a, it is not the cas... |
| ainaiaandna 47916 | Given a, a implies it is n... |
| abcdta 47917 | Given (((a and b) and c) a... |
| abcdtb 47918 | Given (((a and b) and c) a... |
| abcdtc 47919 | Given (((a and b) and c) a... |
| abcdtd 47920 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 47921 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 47922 | Operands in a biconditiona... |
| nabctnabc 47923 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 47924 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 47925 | From one negated implicati... |
| twonotinotbothi 47926 | From these two negated imp... |
| clifte 47927 | show d is the same as an i... |
| cliftet 47928 | show d is the same as an i... |
| clifteta 47929 | show d is the same as an i... |
| cliftetb 47930 | show d is the same as an i... |
| confun 47931 | Given the hypotheses there... |
| confun2 47932 | Confun simplified to two p... |
| confun3 47933 | Confun's more complex form... |
| confun4 47934 | An attempt at derivative. ... |
| confun5 47935 | An attempt at derivative. ... |
| plcofph 47936 | Given, a,b and a "definiti... |
| pldofph 47937 | Given, a,b c, d, "definiti... |
| plvcofph 47938 | Given, a,b,d, and "definit... |
| plvcofphax 47939 | Given, a,b,d, and "definit... |
| plvofpos 47940 | rh is derivable because ON... |
| mdandyv0 47941 | Given the equivalences set... |
| mdandyv1 47942 | Given the equivalences set... |
| mdandyv2 47943 | Given the equivalences set... |
| mdandyv3 47944 | Given the equivalences set... |
| mdandyv4 47945 | Given the equivalences set... |
| mdandyv5 47946 | Given the equivalences set... |
| mdandyv6 47947 | Given the equivalences set... |
| mdandyv7 47948 | Given the equivalences set... |
| mdandyv8 47949 | Given the equivalences set... |
| mdandyv9 47950 | Given the equivalences set... |
| mdandyv10 47951 | Given the equivalences set... |
| mdandyv11 47952 | Given the equivalences set... |
| mdandyv12 47953 | Given the equivalences set... |
| mdandyv13 47954 | Given the equivalences set... |
| mdandyv14 47955 | Given the equivalences set... |
| mdandyv15 47956 | Given the equivalences set... |
| mdandyvr0 47957 | Given the equivalences set... |
| mdandyvr1 47958 | Given the equivalences set... |
| mdandyvr2 47959 | Given the equivalences set... |
| mdandyvr3 47960 | Given the equivalences set... |
| mdandyvr4 47961 | Given the equivalences set... |
| mdandyvr5 47962 | Given the equivalences set... |
| mdandyvr6 47963 | Given the equivalences set... |
| mdandyvr7 47964 | Given the equivalences set... |
| mdandyvr8 47965 | Given the equivalences set... |
| mdandyvr9 47966 | Given the equivalences set... |
| mdandyvr10 47967 | Given the equivalences set... |
| mdandyvr11 47968 | Given the equivalences set... |
| mdandyvr12 47969 | Given the equivalences set... |
| mdandyvr13 47970 | Given the equivalences set... |
| mdandyvr14 47971 | Given the equivalences set... |
| mdandyvr15 47972 | Given the equivalences set... |
| mdandyvrx0 47973 | Given the exclusivities se... |
| mdandyvrx1 47974 | Given the exclusivities se... |
| mdandyvrx2 47975 | Given the exclusivities se... |
| mdandyvrx3 47976 | Given the exclusivities se... |
| mdandyvrx4 47977 | Given the exclusivities se... |
| mdandyvrx5 47978 | Given the exclusivities se... |
| mdandyvrx6 47979 | Given the exclusivities se... |
| mdandyvrx7 47980 | Given the exclusivities se... |
| mdandyvrx8 47981 | Given the exclusivities se... |
| mdandyvrx9 47982 | Given the exclusivities se... |
| mdandyvrx10 47983 | Given the exclusivities se... |
| mdandyvrx11 47984 | Given the exclusivities se... |
| mdandyvrx12 47985 | Given the exclusivities se... |
| mdandyvrx13 47986 | Given the exclusivities se... |
| mdandyvrx14 47987 | Given the exclusivities se... |
| mdandyvrx15 47988 | Given the exclusivities se... |
| H15NH16TH15IH16 47989 | Given 15 hypotheses and a ... |
| dandysum2p2e4 47990 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 47991 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 47992 | Replacement of a nested an... |
| adh-minim 47993 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 47994 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 47995 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 47996 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 47997 | Fourth lemma for the deriv... |
| adh-minim-ax1 47998 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 47999 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 48000 | Sixth lemma for the deriva... |
| adh-minim-ax2c 48001 | Derivation of a commuted f... |
| adh-minim-ax2 48002 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 48003 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 48004 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 48005 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 48006 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 48007 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 48008 | Third lemma for the deriva... |
| adh-minimp-sylsimp 48009 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 48010 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 48011 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 48012 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 48013 | Fourth lemma for the deriv... |
| adh-minimp-ax2 48014 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 48015 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 48016 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 48017 | If a class with one elemen... |
| eusnsn 48018 | There is a unique element ... |
| absnsb 48019 | If the class abstraction `... |
| euabsneu 48020 | Another way to express exi... |
| elprneb 48021 | An element of a proper uno... |
| oppr 48022 | Equality for ordered pairs... |
| opprb 48023 | Equality for unordered pai... |
| or2expropbilem1 48024 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 48025 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 48026 | If two classes are strictl... |
| eubrv 48027 | If there is a unique set w... |
| eubrdm 48028 | If there is a unique set w... |
| eldmressn 48029 | Element of the domain of a... |
| iota0def 48030 | Example for a defined iota... |
| iota0ndef 48031 | Example for an undefined i... |
| fveqvfvv 48032 | If a function's value at a... |
| fnresfnco 48033 | Composition of two functio... |
| funcoressn 48034 | A composition restricted t... |
| funressnfv 48035 | A restriction to a singlet... |
| funressndmfvrn 48036 | The value of a function ` ... |
| funressnvmo 48037 | A function restricted to a... |
| funressnmo 48038 | A function restricted to a... |
| funressneu 48039 | There is exactly one value... |
| fresfo 48040 | Conditions for a restricti... |
| fsetsniunop 48041 | The class of all functions... |
| fsetabsnop 48042 | The class of all functions... |
| fsetsnf 48043 | The mapping of an element ... |
| fsetsnf1 48044 | The mapping of an element ... |
| fsetsnfo 48045 | The mapping of an element ... |
| fsetsnf1o 48046 | The mapping of an element ... |
| fsetsnprcnex 48047 | The class of all functions... |
| cfsetssfset 48048 | The class of constant func... |
| cfsetsnfsetfv 48049 | The function value of the ... |
| cfsetsnfsetf 48050 | The mapping of the class o... |
| cfsetsnfsetf1 48051 | The mapping of the class o... |
| cfsetsnfsetfo 48052 | The mapping of the class o... |
| cfsetsnfsetf1o 48053 | The mapping of the class o... |
| fsetprcnexALT 48054 | First version of proof for... |
| fcoreslem1 48055 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 48056 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 48057 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 48058 | Lemma 4 for ~ fcores . (C... |
| fcores 48059 | Every composite function `... |
| fcoresf1lem 48060 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 48061 | If a composition is inject... |
| fcoresf1b 48062 | A composition is injective... |
| fcoresfo 48063 | If a composition is surjec... |
| fcoresfob 48064 | A composition is surjectiv... |
| fcoresf1ob 48065 | A composition is bijective... |
| f1cof1blem 48066 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 48067 | The composition of three b... |
| 3f1oss2 48068 | The composition of three b... |
| f1cof1b 48069 | If the range of ` F ` equa... |
| funfocofob 48070 | If the domain of a functio... |
| fnfocofob 48071 | If the domain of a functio... |
| focofob 48072 | If the domain of a functio... |
| f1ocof1ob 48073 | If the range of ` F ` equa... |
| f1ocof1ob2 48074 | If the range of ` F ` equa... |
| aiotajust 48076 | Soundness justification th... |
| dfaiota2 48078 | Alternate definition of th... |
| reuabaiotaiota 48079 | The iota and the alternate... |
| reuaiotaiota 48080 | The iota and the alternate... |
| aiotaexb 48081 | The alternate iota over a ... |
| aiotavb 48082 | The alternate iota over a ... |
| aiotaint 48083 | This is to ~ df-aiota what... |
| dfaiota3 48084 | Alternate definition of ` ... |
| iotan0aiotaex 48085 | If the iota over a wff ` p... |
| aiotaexaiotaiota 48086 | The alternate iota over a ... |
| aiotaval 48087 | Theorem 8.19 in [Quine] p.... |
| aiota0def 48088 | Example for a defined alte... |
| aiota0ndef 48089 | Example for an undefined a... |
| r19.32 48090 | Theorem 19.32 of [Margaris... |
| rexsb 48091 | An equivalent expression f... |
| rexrsb 48092 | An equivalent expression f... |
| 2rexsb 48093 | An equivalent expression f... |
| 2rexrsb 48094 | An equivalent expression f... |
| cbvral2 48095 | Change bound variables of ... |
| cbvrex2 48096 | Change bound variables of ... |
| ralndv1 48097 | Example for a theorem abou... |
| ralndv2 48098 | Second example for a theor... |
| reuf1odnf 48099 | There is exactly one eleme... |
| reuf1od 48100 | There is exactly one eleme... |
| euoreqb 48101 | There is a set which is eq... |
| 2reu3 48102 | Double restricted existent... |
| 2reu7 48103 | Two equivalent expressions... |
| 2reu8 48104 | Two equivalent expressions... |
| 2reu8i 48105 | Implication of a double re... |
| 2reuimp0 48106 | Implication of a double re... |
| 2reuimp 48107 | Implication of a double re... |
| ralbinrald 48114 | Elemination of a restricte... |
| nvelim 48115 | If a class is the universa... |
| alneu 48116 | If a statement holds for a... |
| eu2ndop1stv 48117 | If there is a unique secon... |
| dfateq12d 48118 | Equality deduction for "de... |
| nfdfat 48119 | Bound-variable hypothesis ... |
| dfdfat2 48120 | Alternate definition of th... |
| fundmdfat 48121 | A function is defined at a... |
| dfatprc 48122 | A function is not defined ... |
| dfatelrn 48123 | The value of a function ` ... |
| dfafv2 48124 | Alternative definition of ... |
| afveq12d 48125 | Equality deduction for fun... |
| afveq1 48126 | Equality theorem for funct... |
| afveq2 48127 | Equality theorem for funct... |
| nfafv 48128 | Bound-variable hypothesis ... |
| csbafv12g 48129 | Move class substitution in... |
| afvfundmfveq 48130 | If a class is a function r... |
| afvnfundmuv 48131 | If a set is not in the dom... |
| ndmafv 48132 | The value of a class outsi... |
| afvvdm 48133 | If the function value of a... |
| nfunsnafv 48134 | If the restriction of a cl... |
| afvvfunressn 48135 | If the function value of a... |
| afvprc 48136 | A function's value at a pr... |
| afvvv 48137 | If a function's value at a... |
| afvpcfv0 48138 | If the value of the altern... |
| afvnufveq 48139 | The value of the alternati... |
| afvvfveq 48140 | The value of the alternati... |
| afv0fv0 48141 | If the value of the altern... |
| afvfvn0fveq 48142 | If the function's value at... |
| afv0nbfvbi 48143 | The function's value at an... |
| afvfv0bi 48144 | The function's value at an... |
| afveu 48145 | The value of a function at... |
| fnbrafvb 48146 | Equivalence of function va... |
| fnopafvb 48147 | Equivalence of function va... |
| funbrafvb 48148 | Equivalence of function va... |
| funopafvb 48149 | Equivalence of function va... |
| funbrafv 48150 | The second argument of a b... |
| funbrafv2b 48151 | Function value in terms of... |
| dfafn5a 48152 | Representation of a functi... |
| dfafn5b 48153 | Representation of a functi... |
| fnrnafv 48154 | The range of a function ex... |
| afvelrnb 48155 | A member of a function's r... |
| afvelrnb0 48156 | A member of a function's r... |
| dfaimafn 48157 | Alternate definition of th... |
| dfaimafn2 48158 | Alternate definition of th... |
| afvelima 48159 | Function value in an image... |
| afvelrn 48160 | A function's value belongs... |
| fnafvelrn 48161 | A function's value belongs... |
| fafvelcdm 48162 | A function's value belongs... |
| ffnafv 48163 | A function maps to a class... |
| afvres 48164 | The value of a restricted ... |
| tz6.12-afv 48165 | Function value. Theorem 6... |
| tz6.12-1-afv 48166 | Function value (Theorem 6.... |
| dmfcoafv 48167 | Domains of a function comp... |
| afvco2 48168 | Value of a function compos... |
| rlimdmafv 48169 | Two ways to express that a... |
| aoveq123d 48170 | Equality deduction for ope... |
| nfaov 48171 | Bound-variable hypothesis ... |
| csbaovg 48172 | Move class substitution in... |
| aovfundmoveq 48173 | If a class is a function r... |
| aovnfundmuv 48174 | If an ordered pair is not ... |
| ndmaov 48175 | The value of an operation ... |
| ndmaovg 48176 | The value of an operation ... |
| aovvdm 48177 | If the operation value of ... |
| nfunsnaov 48178 | If the restriction of a cl... |
| aovvfunressn 48179 | If the operation value of ... |
| aovprc 48180 | The value of an operation ... |
| aovrcl 48181 | Reverse closure for an ope... |
| aovpcov0 48182 | If the alternative value o... |
| aovnuoveq 48183 | The alternative value of t... |
| aovvoveq 48184 | The alternative value of t... |
| aov0ov0 48185 | If the alternative value o... |
| aovovn0oveq 48186 | If the operation's value a... |
| aov0nbovbi 48187 | The operation's value on a... |
| aovov0bi 48188 | The operation's value on a... |
| rspceaov 48189 | A frequently used special ... |
| fnotaovb 48190 | Equivalence of operation v... |
| ffnaov 48191 | An operation maps to a cla... |
| faovcl 48192 | Closure law for an operati... |
| aovmpt4g 48193 | Value of a function given ... |
| aoprssdm 48194 | Domain of closure of an op... |
| ndmaovcl 48195 | The "closure" of an operat... |
| ndmaovrcl 48196 | Reverse closure law, in co... |
| ndmaovcom 48197 | Any operation is commutati... |
| ndmaovass 48198 | Any operation is associati... |
| ndmaovdistr 48199 | Any operation is distribut... |
| dfatafv2iota 48202 | If a function is defined a... |
| ndfatafv2 48203 | The alternate function val... |
| ndfatafv2undef 48204 | The alternate function val... |
| dfatafv2ex 48205 | The alternate function val... |
| afv2ex 48206 | The alternate function val... |
| afv2eq12d 48207 | Equality deduction for fun... |
| afv2eq1 48208 | Equality theorem for funct... |
| afv2eq2 48209 | Equality theorem for funct... |
| nfafv2 48210 | Bound-variable hypothesis ... |
| csbafv212g 48211 | Move class substitution in... |
| fexafv2ex 48212 | The alternate function val... |
| ndfatafv2nrn 48213 | The alternate function val... |
| ndmafv2nrn 48214 | The value of a class outsi... |
| funressndmafv2rn 48215 | The alternate function val... |
| afv2ndefb 48216 | Two ways to say that an al... |
| nfunsnafv2 48217 | If the restriction of a cl... |
| afv2prc 48218 | A function's value at a pr... |
| dfatafv2rnb 48219 | The alternate function val... |
| afv2orxorb 48220 | If a set is in the range o... |
| dmafv2rnb 48221 | The alternate function val... |
| fundmafv2rnb 48222 | The alternate function val... |
| afv2elrn 48223 | An alternate function valu... |
| afv20defat 48224 | If the alternate function ... |
| fnafv2elrn 48225 | An alternate function valu... |
| fafv2elcdm 48226 | An alternate function valu... |
| fafv2elrnb 48227 | An alternate function valu... |
| fcdmvafv2v 48228 | If the codomain of a funct... |
| tz6.12-2-afv2 48229 | Function value when ` F ` ... |
| afv2eu 48230 | The value of a function at... |
| afv2res 48231 | The value of a restricted ... |
| tz6.12-afv2 48232 | Function value (Theorem 6.... |
| tz6.12-1-afv2 48233 | Function value (Theorem 6.... |
| tz6.12c-afv2 48234 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 48235 | Corollary of Theorem 6.12(... |
| funressnbrafv2 48236 | The second argument of a b... |
| dfatbrafv2b 48237 | Equivalence of function va... |
| dfatopafv2b 48238 | Equivalence of function va... |
| funbrafv2 48239 | The second argument of a b... |
| fnbrafv2b 48240 | Equivalence of function va... |
| fnopafv2b 48241 | Equivalence of function va... |
| funbrafv22b 48242 | Equivalence of function va... |
| funopafv2b 48243 | Equivalence of function va... |
| dfatsnafv2 48244 | Singleton of function valu... |
| dfafv23 48245 | A definition of function v... |
| dfatdmfcoafv2 48246 | Domain of a function compo... |
| dfatcolem 48247 | Lemma for ~ dfatco . (Con... |
| dfatco 48248 | The predicate "defined at"... |
| afv2co2 48249 | Value of a function compos... |
| rlimdmafv2 48250 | Two ways to express that a... |
| dfafv22 48251 | Alternate definition of ` ... |
| afv2ndeffv0 48252 | If the alternate function ... |
| dfatafv2eqfv 48253 | If a function is defined a... |
| afv2rnfveq 48254 | If the alternate function ... |
| afv20fv0 48255 | If the alternate function ... |
| afv2fvn0fveq 48256 | If the function's value at... |
| afv2fv0 48257 | If the function's value at... |
| afv2fv0b 48258 | The function's value at an... |
| afv2fv0xorb 48259 | If a set is in the range o... |
| an4com24 48260 | Rearrangement of 4 conjunc... |
| 3an4ancom24 48261 | Commutative law for a conj... |
| 4an21 48262 | Rearrangement of 4 conjunc... |
| dfnelbr2 48265 | Alternate definition of th... |
| nelbr 48266 | The binary relation of a s... |
| nelbrim 48267 | If a set is related to ano... |
| nelbrnel 48268 | A set is related to anothe... |
| nelbrnelim 48269 | If a set is related to ano... |
| ralralimp 48270 | Selecting one of two alter... |
| otiunsndisjX 48271 | The union of singletons co... |
| fvifeq 48272 | Equality of function value... |
| rnfdmpr 48273 | The range of a one-to-one ... |
| imarnf1pr 48274 | The image of the range of ... |
| funop1 48275 | A function is an ordered p... |
| fun2dmnopgexmpl 48276 | A function with a domain c... |
| opabresex0d 48277 | A collection of ordered pa... |
| opabbrfex0d 48278 | A collection of ordered pa... |
| opabresexd 48279 | A collection of ordered pa... |
| opabbrfexd 48280 | A collection of ordered pa... |
| f1oresf1orab 48281 | Build a bijection by restr... |
| f1oresf1o 48282 | Build a bijection by restr... |
| f1oresf1o2 48283 | Build a bijection by restr... |
| fvmptrab 48284 | Value of a function mappin... |
| fvmptrabdm 48285 | Value of a function mappin... |
| cnambpcma 48286 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 48287 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 48288 | The sum of two complex num... |
| leaddsuble 48289 | Addition and subtraction o... |
| 2leaddle2 48290 | If two real numbers are le... |
| ltnltne 48291 | Variant of trichotomy law ... |
| p1lep2 48292 | A real number increasd by ... |
| ltsubsubaddltsub 48293 | If the result of subtracti... |
| zm1nn 48294 | An integer minus 1 is posi... |
| readdcnnred 48295 | The sum of a real number a... |
| resubcnnred 48296 | The difference of a real n... |
| recnmulnred 48297 | The product of a real numb... |
| cndivrenred 48298 | The quotient of an imagina... |
| sqrtnegnre 48299 | The square root of a negat... |
| nn0resubcl 48300 | Closure law for subtractio... |
| zgeltp1eq 48301 | If an integer is between a... |
| 1t10e1p1e11 48302 | 11 is 1 times 10 to the po... |
| deccarry 48303 | Add 1 to a 2 digit number ... |
| eluzge0nn0 48304 | If an integer is greater t... |
| nltle2tri 48305 | Negated extended trichotom... |
| ssfz12 48306 | Subset relationship for fi... |
| elfz2z 48307 | Membership of an integer i... |
| 2elfz3nn0 48308 | If there are two elements ... |
| fz0addcom 48309 | The addition of two member... |
| 2elfz2melfz 48310 | If the sum of two integers... |
| fz0addge0 48311 | The sum of two integers in... |
| elfzlble 48312 | Membership of an integer i... |
| elfzelfzlble 48313 | Membership of an element o... |
| elfz2nn 48314 | A member of a finite set o... |
| fzopred 48315 | Join a predecessor to the ... |
| fzopredsuc 48316 | Join a predecessor and a s... |
| 1fzopredsuc 48317 | Join 0 and a successor to ... |
| el1fzopredsuc 48318 | An element of an open inte... |
| subsubelfzo0 48319 | Subtracting a difference f... |
| 2ffzoeq 48320 | Two functions over a half-... |
| elfzo2nn 48321 | A member of a half-open ra... |
| nnmul2 48322 | If one factor of a product... |
| nnmul2b 48323 | A factor of a product of i... |
| 2ltceilhalf 48324 | The ceiling of half of an ... |
| ceilhalfgt1 48325 | The ceiling of half of an ... |
| ceilhalfelfzo1 48326 | A positive integer less th... |
| gpgedgvtx1lem 48327 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 48328 | Two times a positive integ... |
| ceilbi 48329 | A condition equivalent to ... |
| ceilhalf1 48330 | The ceiling of one half is... |
| rehalfge1 48331 | Half of a real number grea... |
| ceilhalfnn 48332 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 48333 | 1 is in the half-open inte... |
| nnge2recfl0 48334 | The floor of the reciproca... |
| flmrecm1 48335 | The floor of an integer mi... |
| fldivmod 48336 | Expressing the floor of a ... |
| ceildivmod 48337 | Expressing the ceiling of ... |
| ceil5half3 48338 | The ceiling of half of 5 i... |
| submodaddmod 48339 | Subtraction and addition m... |
| difltmodne 48340 | Two nonnegative integers a... |
| zplusmodne 48341 | A nonnegative integer is n... |
| addmodne 48342 | The sum of a nonnegative i... |
| plusmod5ne 48343 | A nonnegative integer is n... |
| zp1modne 48344 | An integer is not itself p... |
| p1modne 48345 | A nonnegative integer is n... |
| m1modne 48346 | A nonnegative integer is n... |
| minusmod5ne 48347 | A nonnegative integer is n... |
| submodlt 48348 | The difference of an eleme... |
| submodneaddmod 48349 | An integer minus ` B ` is ... |
| m1modnep2mod 48350 | A nonnegative integer minu... |
| minusmodnep2tmod 48351 | A nonnegative integer minu... |
| m1mod0mod1 48352 | An integer decreased by 1 ... |
| elmod2 48353 | An integer modulo 2 is eit... |
| mod0mul 48354 | If an integer is 0 modulo ... |
| modn0mul 48355 | If an integer is not 0 mod... |
| m1modmmod 48356 | An integer decreased by 1 ... |
| difmodm1lt 48357 | The difference between an ... |
| 8mod5e3 48358 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 48359 | If an integer minus a cons... |
| modmknepk 48360 | A nonnegative integer less... |
| modlt0b 48361 | An integer with an absolut... |
| mod2addne 48362 | The sums of a nonnegative ... |
| modm1nep1 48363 | A nonnegative integer less... |
| modm2nep1 48364 | A nonnegative integer less... |
| modp2nep1 48365 | A nonnegative integer less... |
| modm1nep2 48366 | A nonnegative integer less... |
| modm1nem2 48367 | A nonnegative integer less... |
| modm1p1ne 48368 | If an integer minus one eq... |
| smonoord 48369 | Ordering relation for a st... |
| 2timesltsq 48370 | Two times an integer great... |
| 2timesltsqm1 48371 | Two times an integer great... |
| fsummsndifre 48372 | A finite sum with one of i... |
| fsumsplitsndif 48373 | Separate out a term in a f... |
| fsummmodsndifre 48374 | A finite sum of summands m... |
| fsummmodsnunz 48375 | A finite sum of summands m... |
| nndivides2 48376 | Definition of the divides ... |
| facnn0dvdsfac 48377 | The factorial of a nonnega... |
| muldvdsfacgt 48378 | The product of two differe... |
| muldvdsfacm1 48379 | The product of two differe... |
| setsidel 48380 | The injected slot is an el... |
| setsnidel 48381 | The injected slot is an el... |
| setsv 48382 | The value of the structure... |
| preimafvsnel 48383 | The preimage of a function... |
| preimafvn0 48384 | The preimage of a function... |
| uniimafveqt 48385 | The union of the image of ... |
| uniimaprimaeqfv 48386 | The union of the image of ... |
| setpreimafvex 48387 | The class ` P ` of all pre... |
| elsetpreimafvb 48388 | The characterization of an... |
| elsetpreimafv 48389 | An element of the class ` ... |
| elsetpreimafvssdm 48390 | An element of the class ` ... |
| fvelsetpreimafv 48391 | There is an element in a p... |
| preimafvelsetpreimafv 48392 | The preimage of a function... |
| preimafvsspwdm 48393 | The class ` P ` of all pre... |
| 0nelsetpreimafv 48394 | The empty set is not an el... |
| elsetpreimafvbi 48395 | An element of the preimage... |
| elsetpreimafveqfv 48396 | The elements of the preima... |
| eqfvelsetpreimafv 48397 | If an element of the domai... |
| elsetpreimafvrab 48398 | An element of the preimage... |
| imaelsetpreimafv 48399 | The image of an element of... |
| uniimaelsetpreimafv 48400 | The union of the image of ... |
| elsetpreimafveq 48401 | If two preimages of functi... |
| fundcmpsurinjlem1 48402 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 48403 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 48404 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 48405 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 48406 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 48407 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 48408 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 48409 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 48410 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 48411 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 48412 | Every function ` F : A -->... |
| fundcmpsurinj 48413 | Every function ` F : A -->... |
| fundcmpsurbijinj 48414 | Every function ` F : A -->... |
| fundcmpsurinjimaid 48415 | Every function ` F : A -->... |
| fundcmpsurinjALT 48416 | Alternate proof of ~ fundc... |
| iccpval 48419 | Partition consisting of a ... |
| iccpart 48420 | A special partition. Corr... |
| iccpartimp 48421 | Implications for a class b... |
| iccpartres 48422 | The restriction of a parti... |
| iccpartxr 48423 | If there is a partition, t... |
| iccpartgtprec 48424 | If there is a partition, t... |
| iccpartipre 48425 | If there is a partition, t... |
| iccpartiltu 48426 | If there is a partition, t... |
| iccpartigtl 48427 | If there is a partition, t... |
| iccpartlt 48428 | If there is a partition, t... |
| iccpartltu 48429 | If there is a partition, t... |
| iccpartgtl 48430 | If there is a partition, t... |
| iccpartgt 48431 | If there is a partition, t... |
| iccpartleu 48432 | If there is a partition, t... |
| iccpartgel 48433 | If there is a partition, t... |
| iccpartrn 48434 | If there is a partition, t... |
| iccpartf 48435 | The range of the partition... |
| iccpartel 48436 | If there is a partition, t... |
| iccelpart 48437 | An element of any partitio... |
| iccpartiun 48438 | A half-open interval of ex... |
| icceuelpartlem 48439 | Lemma for ~ icceuelpart . ... |
| icceuelpart 48440 | An element of a partitione... |
| iccpartdisj 48441 | The segments of a partitio... |
| iccpartnel 48442 | A point of a partition is ... |
| fargshiftfv 48443 | If a class is a function, ... |
| fargshiftf 48444 | If a class is a function, ... |
| fargshiftf1 48445 | If a function is 1-1, then... |
| fargshiftfo 48446 | If a function is onto, the... |
| fargshiftfva 48447 | The values of a shifted fu... |
| lswn0 48448 | The last symbol of a nonem... |
| nfich1 48451 | The first interchangeable ... |
| nfich2 48452 | The second interchangeable... |
| ichv 48453 | Setvar variables are inter... |
| ichf 48454 | Setvar variables are inter... |
| ichid 48455 | A setvar variable is alway... |
| icht 48456 | A theorem is interchangeab... |
| ichbidv 48457 | Formula building rule for ... |
| ichcircshi 48458 | The setvar variables are i... |
| ichan 48459 | If two setvar variables ar... |
| ichn 48460 | Negation does not affect i... |
| ichim 48461 | Formula building rule for ... |
| dfich2 48462 | Alternate definition of th... |
| ichcom 48463 | The interchangeability of ... |
| ichbi12i 48464 | Equivalence for interchang... |
| icheqid 48465 | In an equality for the sam... |
| icheq 48466 | In an equality of setvar v... |
| ichnfimlem 48467 | Lemma for ~ ichnfim : A s... |
| ichnfim 48468 | If in an interchangeabilit... |
| ichnfb 48469 | If ` x ` and ` y ` are int... |
| ichal 48470 | Move a universal quantifie... |
| ich2al 48471 | Two setvar variables are a... |
| ich2ex 48472 | Two setvar variables are a... |
| ichexmpl1 48473 | Example for interchangeabl... |
| ichexmpl2 48474 | Example for interchangeabl... |
| ich2exprop 48475 | If the setvar variables ar... |
| ichnreuop 48476 | If the setvar variables ar... |
| ichreuopeq 48477 | If the setvar variables ar... |
| sprid 48478 | Two identical representati... |
| elsprel 48479 | An unordered pair is an el... |
| spr0nelg 48480 | The empty set is not an el... |
| sprval 48483 | The set of all unordered p... |
| sprvalpw 48484 | The set of all unordered p... |
| sprssspr 48485 | The set of all unordered p... |
| spr0el 48486 | The empty set is not an un... |
| sprvalpwn0 48487 | The set of all unordered p... |
| sprel 48488 | An element of the set of a... |
| prssspr 48489 | An element of a subset of ... |
| prelspr 48490 | An unordered pair of eleme... |
| prsprel 48491 | The elements of a pair fro... |
| prsssprel 48492 | The elements of a pair fro... |
| sprvalpwle2 48493 | The set of all unordered p... |
| sprsymrelfvlem 48494 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 48495 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 48496 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 48497 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 48498 | The value of the function ... |
| sprsymrelf 48499 | The mapping ` F ` is a fun... |
| sprsymrelf1 48500 | The mapping ` F ` is a one... |
| sprsymrelfo 48501 | The mapping ` F ` is a fun... |
| sprsymrelf1o 48502 | The mapping ` F ` is a bij... |
| sprbisymrel 48503 | There is a bijection betwe... |
| sprsymrelen 48504 | The class ` P ` of subsets... |
| prpair 48505 | Characterization of a prop... |
| prproropf1olem0 48506 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 48507 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 48508 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 48509 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 48510 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 48511 | There is a bijection betwe... |
| prproropen 48512 | The set of proper pairs an... |
| prproropreud 48513 | There is exactly one order... |
| pairreueq 48514 | Two equivalent representat... |
| paireqne 48515 | Two sets are not equal iff... |
| prprval 48518 | The set of all proper unor... |
| prprvalpw 48519 | The set of all proper unor... |
| prprelb 48520 | An element of the set of a... |
| prprelprb 48521 | A set is an element of the... |
| prprspr2 48522 | The set of all proper unor... |
| prprsprreu 48523 | There is a unique proper u... |
| prprreueq 48524 | There is a unique proper u... |
| sbcpr 48525 | The proper substitution of... |
| reupr 48526 | There is a unique unordere... |
| reuprpr 48527 | There is a unique proper u... |
| poprelb 48528 | Equality for unordered pai... |
| 2exopprim 48529 | The existence of an ordere... |
| reuopreuprim 48530 | There is a unique unordere... |
| nprmmul1 48531 | Special factorization of a... |
| nprmmul2 48532 | Special factorization of a... |
| nprmmul3 48533 | Special factorization of a... |
| fmtno 48536 | The ` N ` th Fermat number... |
| fmtnoge3 48537 | Each Fermat number is grea... |
| fmtnonn 48538 | Each Fermat number is a po... |
| fmtnom1nn 48539 | A Fermat number minus one ... |
| fmtnoodd 48540 | Each Fermat number is odd.... |
| fmtnorn 48541 | A Fermat number is a funct... |
| fmtnof1 48542 | The enumeration of the Fer... |
| fmtnoinf 48543 | The set of Fermat numbers ... |
| fmtnorec1 48544 | The first recurrence relat... |
| sqrtpwpw2p 48545 | The floor of the square ro... |
| fmtnosqrt 48546 | The floor of the square ro... |
| fmtno0 48547 | The ` 0 ` th Fermat number... |
| fmtno1 48548 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 48549 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 48550 | The second recurrence rela... |
| fmtnodvds 48551 | Any Fermat number divides ... |
| goldbachthlem1 48552 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 48553 | Lemma 2 for ~ goldbachth .... |
| goldbachth 48554 | Goldbach's theorem: Two d... |
| fmtnorec3 48555 | The third recurrence relat... |
| fmtnorec4 48556 | The fourth recurrence rela... |
| fmtno2 48557 | The ` 2 ` nd Fermat number... |
| fmtno3 48558 | The ` 3 ` rd Fermat number... |
| fmtno4 48559 | The ` 4 ` th Fermat number... |
| fmtno5lem1 48560 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 48561 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 48562 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 48563 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 48564 | The ` 5 ` th Fermat number... |
| fmtno0prm 48565 | The ` 0 ` th Fermat number... |
| fmtno1prm 48566 | The ` 1 ` st Fermat number... |
| fmtno2prm 48567 | The ` 2 ` nd Fermat number... |
| 257prm 48568 | 257 is a prime number (the... |
| fmtno3prm 48569 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 48570 | Any power of two is coprim... |
| fmtnoprmfac1lem 48571 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 48572 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 48573 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 48574 | Divisor of Fermat number (... |
| fmtnofac2lem 48575 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 48576 | Divisor of Fermat number (... |
| fmtnofac1 48577 | Divisor of Fermat number (... |
| fmtno4sqrt 48578 | The floor of the square ro... |
| fmtno4prmfac 48579 | If P was a (prime) factor ... |
| fmtno4prmfac193 48580 | If P was a (prime) factor ... |
| fmtno4nprmfac193 48581 | 193 is not a (prime) facto... |
| fmtno4prm 48582 | The ` 4 `-th Fermat number... |
| 65537prm 48583 | 65537 is a prime number (t... |
| fmtnofz04prm 48584 | The first five Fermat numb... |
| fmtnole4prm 48585 | The first five Fermat numb... |
| fmtno5faclem1 48586 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 48587 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 48588 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 48589 | The factorization of the `... |
| fmtno5nprm 48590 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 48591 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 48592 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 48593 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 48594 | The mapping of a Fermat nu... |
| prminf2 48595 | The set of prime numbers i... |
| 2pwp1prm 48596 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 48597 | Every prime number of the ... |
| m2prm 48598 | The second Mersenne number... |
| m3prm 48599 | The third Mersenne number ... |
| flsqrt 48600 | A condition equivalent to ... |
| flsqrt5 48601 | The floor of the square ro... |
| 3ndvds4 48602 | 3 does not divide 4. (Con... |
| 139prmALT 48603 | 139 is a prime number. In... |
| 31prm 48604 | 31 is a prime number. In ... |
| m5prm 48605 | The fifth Mersenne number ... |
| 127prm 48606 | 127 is a prime number. (C... |
| m7prm 48607 | The seventh Mersenne numbe... |
| m11nprm 48608 | The eleventh Mersenne numb... |
| mod42tp1mod8 48609 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 48610 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 48611 | If ` P ` is a Sophie Germa... |
| lighneallem1 48612 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 48613 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 48614 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 48615 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 48616 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 48617 | Lemma 3 for ~ lighneal . ... |
| lighneal 48618 | If a power of a prime ` P ... |
| modexp2m1d 48619 | The square of an integer w... |
| proththdlem 48620 | Lemma for ~ proththd . (C... |
| proththd 48621 | Proth's theorem (1878). I... |
| 5tcu2e40 48622 | 5 times the cube of 2 is 4... |
| 3exp4mod41 48623 | 3 to the fourth power is -... |
| 41prothprmlem1 48624 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 48625 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 48626 | 41 is a _Proth prime_. (C... |
| nprmdvdsfacm1lem1 48627 | Lemma 1 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem2 48628 | Lemma 2 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem3 48629 | Lemma 3 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem4 48630 | Lemma 4 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1 48631 | A non-prime integer greate... |
| ppivalnnprm 48632 | Value of a term of the pri... |
| ppivalnnnprmge6 48633 | Value of a term of the pri... |
| ppivalnn4 48634 | Value of the term of the p... |
| ppivalnnnprm 48635 | Value of a term of the pri... |
| indprm 48636 | An indicator function for ... |
| indprmfz 48637 | An indicator function for ... |
| ppi1sum 48638 | Value of the prime-countin... |
| ppivalnn 48639 | Value of the prime-countin... |
| quad1 48640 | A condition for a quadrati... |
| requad01 48641 | A condition for a quadrati... |
| requad1 48642 | A condition for a quadrati... |
| requad2 48643 | A condition for a quadrati... |
| iseven 48648 | The predicate "is an even ... |
| isodd 48649 | The predicate "is an odd n... |
| evenz 48650 | An even number is an integ... |
| oddz 48651 | An odd number is an intege... |
| evendiv2z 48652 | The result of dividing an ... |
| oddp1div2z 48653 | The result of dividing an ... |
| oddm1div2z 48654 | The result of dividing an ... |
| isodd2 48655 | The predicate "is an odd n... |
| dfodd2 48656 | Alternate definition for o... |
| dfodd6 48657 | Alternate definition for o... |
| dfeven4 48658 | Alternate definition for e... |
| evenm1odd 48659 | The predecessor of an even... |
| evenp1odd 48660 | The successor of an even n... |
| oddp1eveni 48661 | The successor of an odd nu... |
| oddm1eveni 48662 | The predecessor of an odd ... |
| evennodd 48663 | An even number is not an o... |
| oddneven 48664 | An odd number is not an ev... |
| enege 48665 | The negative of an even nu... |
| onego 48666 | The negative of an odd num... |
| m1expevenALTV 48667 | Exponentiation of -1 by an... |
| m1expoddALTV 48668 | Exponentiation of -1 by an... |
| dfeven2 48669 | Alternate definition for e... |
| dfodd3 48670 | Alternate definition for o... |
| iseven2 48671 | The predicate "is an even ... |
| isodd3 48672 | The predicate "is an odd n... |
| 2dvdseven 48673 | 2 divides an even number. ... |
| m2even 48674 | A multiple of 2 is an even... |
| 2ndvdsodd 48675 | 2 does not divide an odd n... |
| 2dvdsoddp1 48676 | 2 divides an odd number in... |
| 2dvdsoddm1 48677 | 2 divides an odd number de... |
| dfeven3 48678 | Alternate definition for e... |
| dfodd4 48679 | Alternate definition for o... |
| dfodd5 48680 | Alternate definition for o... |
| zefldiv2ALTV 48681 | The floor of an even numbe... |
| zofldiv2ALTV 48682 | The floor of an odd number... |
| oddflALTV 48683 | Odd number representation ... |
| iseven5 48684 | The predicate "is an even ... |
| isodd7 48685 | The predicate "is an odd n... |
| dfeven5 48686 | Alternate definition for e... |
| dfodd7 48687 | Alternate definition for o... |
| gcd2odd1 48688 | The greatest common diviso... |
| zneoALTV 48689 | No even integer equals an ... |
| zeoALTV 48690 | An integer is even or odd.... |
| zeo2ALTV 48691 | An integer is even or odd ... |
| nneoALTV 48692 | A positive integer is even... |
| nneoiALTV 48693 | A positive integer is even... |
| odd2np1ALTV 48694 | An integer is odd iff it i... |
| oddm1evenALTV 48695 | An integer is odd iff its ... |
| oddp1evenALTV 48696 | An integer is odd iff its ... |
| oexpnegALTV 48697 | The exponential of the neg... |
| oexpnegnz 48698 | The exponential of the neg... |
| bits0ALTV 48699 | Value of the zeroth bit. ... |
| bits0eALTV 48700 | The zeroth bit of an even ... |
| bits0oALTV 48701 | The zeroth bit of an odd n... |
| divgcdoddALTV 48702 | Either ` A / ( A gcd B ) `... |
| opoeALTV 48703 | The sum of two odds is eve... |
| opeoALTV 48704 | The sum of an odd and an e... |
| omoeALTV 48705 | The difference of two odds... |
| omeoALTV 48706 | The difference of an odd a... |
| oddprmALTV 48707 | A prime not equal to ` 2 `... |
| 0evenALTV 48708 | 0 is an even number. (Con... |
| 0noddALTV 48709 | 0 is not an odd number. (... |
| 1oddALTV 48710 | 1 is an odd number. (Cont... |
| 1nevenALTV 48711 | 1 is not an even number. ... |
| 2evenALTV 48712 | 2 is an even number. (Con... |
| 2noddALTV 48713 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 48714 | An odd nonnegative integer... |
| nnoALTV 48715 | An alternate characterizat... |
| nn0oALTV 48716 | An alternate characterizat... |
| nn0e 48717 | An alternate characterizat... |
| nneven 48718 | An alternate characterizat... |
| nn0onn0exALTV 48719 | For each odd nonnegative i... |
| nn0enn0exALTV 48720 | For each even nonnegative ... |
| nnennexALTV 48721 | For each even positive int... |
| nnpw2evenALTV 48722 | 2 to the power of a positi... |
| epoo 48723 | The sum of an even and an ... |
| emoo 48724 | The difference of an even ... |
| epee 48725 | The sum of two even number... |
| emee 48726 | The difference of two even... |
| evensumeven 48727 | If a summand is even, the ... |
| 3odd 48728 | 3 is an odd number. (Cont... |
| 4even 48729 | 4 is an even number. (Con... |
| 5odd 48730 | 5 is an odd number. (Cont... |
| 6even 48731 | 6 is an even number. (Con... |
| 7odd 48732 | 7 is an odd number. (Cont... |
| 8even 48733 | 8 is an even number. (Con... |
| evenprm2 48734 | A prime number is even iff... |
| oddprmne2 48735 | Every prime number not bei... |
| oddprmuzge3 48736 | A prime number which is od... |
| evenltle 48737 | If an even number is great... |
| odd2prm2 48738 | If an odd number is the su... |
| even3prm2 48739 | If an even number is the s... |
| mogoldbblem 48740 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 48741 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 48742 | Lemma for ~ perfectALTV . ... |
| perfectALTV 48743 | The Euclid-Euler theorem, ... |
| fppr 48746 | The set of Fermat pseudopr... |
| fpprmod 48747 | The set of Fermat pseudopr... |
| fpprel 48748 | A Fermat pseudoprime to th... |
| fpprbasnn 48749 | The base of a Fermat pseud... |
| fpprnn 48750 | A Fermat pseudoprime to th... |
| fppr2odd 48751 | A Fermat pseudoprime to th... |
| 11t31e341 48752 | 341 is the product of 11 a... |
| 2exp340mod341 48753 | Eight to the eighth power ... |
| 341fppr2 48754 | 341 is the (smallest) _Pou... |
| 4fppr1 48755 | 4 is the (smallest) Fermat... |
| 8exp8mod9 48756 | Eight to the eighth power ... |
| 9fppr8 48757 | 9 is the (smallest) Fermat... |
| dfwppr 48758 | Alternate definition of a ... |
| fpprwppr 48759 | A Fermat pseudoprime to th... |
| fpprwpprb 48760 | An integer ` X ` which is ... |
| fpprel2 48761 | An alternate definition fo... |
| nfermltl8rev 48762 | Fermat's little theorem wi... |
| nfermltl2rev 48763 | Fermat's little theorem wi... |
| nfermltlrev 48764 | Fermat's little theorem re... |
| isgbe 48771 | The predicate "is an even ... |
| isgbow 48772 | The predicate "is a weak o... |
| isgbo 48773 | The predicate "is an odd G... |
| gbeeven 48774 | An even Goldbach number is... |
| gbowodd 48775 | A weak odd Goldbach number... |
| gbogbow 48776 | A (strong) odd Goldbach nu... |
| gboodd 48777 | An odd Goldbach number is ... |
| gbepos 48778 | Any even Goldbach number i... |
| gbowpos 48779 | Any weak odd Goldbach numb... |
| gbopos 48780 | Any odd Goldbach number is... |
| gbegt5 48781 | Any even Goldbach number i... |
| gbowgt5 48782 | Any weak odd Goldbach numb... |
| gbowge7 48783 | Any weak odd Goldbach numb... |
| gboge9 48784 | Any odd Goldbach number is... |
| gbege6 48785 | Any even Goldbach number i... |
| gbpart6 48786 | The Goldbach partition of ... |
| gbpart7 48787 | The (weak) Goldbach partit... |
| gbpart8 48788 | The Goldbach partition of ... |
| gbpart9 48789 | The (strong) Goldbach part... |
| gbpart11 48790 | The (strong) Goldbach part... |
| 6gbe 48791 | 6 is an even Goldbach numb... |
| 7gbow 48792 | 7 is a weak odd Goldbach n... |
| 8gbe 48793 | 8 is an even Goldbach numb... |
| 9gbo 48794 | 9 is an odd Goldbach numbe... |
| 11gbo 48795 | 11 is an odd Goldbach numb... |
| stgoldbwt 48796 | If the strong ternary Gold... |
| sbgoldbwt 48797 | If the strong binary Goldb... |
| sbgoldbst 48798 | If the strong binary Goldb... |
| sbgoldbaltlem1 48799 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 48800 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 48801 | An alternate (related to t... |
| sbgoldbb 48802 | If the strong binary Goldb... |
| sgoldbeven3prm 48803 | If the binary Goldbach con... |
| sbgoldbm 48804 | If the strong binary Goldb... |
| mogoldbb 48805 | If the modern version of t... |
| sbgoldbmb 48806 | The strong binary Goldbach... |
| sbgoldbo 48807 | If the strong binary Goldb... |
| nnsum3primes4 48808 | 4 is the sum of at most 3 ... |
| nnsum4primes4 48809 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 48810 | Every prime is "the sum of... |
| nnsum4primesprm 48811 | Every prime is "the sum of... |
| nnsum3primesgbe 48812 | Any even Goldbach number i... |
| nnsum4primesgbe 48813 | Any even Goldbach number i... |
| nnsum3primesle9 48814 | Every integer greater than... |
| nnsum4primesle9 48815 | Every integer greater than... |
| nnsum4primesodd 48816 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 48817 | If the (strong) ternary Go... |
| evengpop3 48818 | If the (weak) ternary Gold... |
| evengpoap3 48819 | If the (strong) ternary Go... |
| nnsum4primeseven 48820 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 48821 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 48822 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 48823 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 48824 | If the binary Goldbach con... |
| bgoldbtbndlem1 48825 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 48826 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 48827 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 48828 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 48829 | If the binary Goldbach con... |
| tgoldbachgtALTV 48832 | Variant of Thierry Arnoux'... |
| bgoldbachlt 48833 | The binary Goldbach conjec... |
| tgblthelfgott 48835 | The ternary Goldbach conje... |
| tgoldbachlt 48836 | The ternary Goldbach conje... |
| tgoldbach 48837 | The ternary Goldbach conje... |
| clnbgrprc0 48840 | The closed neighborhood is... |
| clnbgrcl 48841 | If a class ` X ` has at le... |
| clnbgrval 48842 | The closed neighborhood of... |
| dfclnbgr2 48843 | Alternate definition of th... |
| dfclnbgr4 48844 | Alternate definition of th... |
| elclnbgrelnbgr 48845 | An element of the closed n... |
| dfclnbgr3 48846 | Alternate definition of th... |
| clnbgrnvtx0 48847 | If a class ` X ` is not a ... |
| clnbgrel 48848 | Characterization of a memb... |
| clnbgrvtxel 48849 | Every vertex ` K ` is a me... |
| clnbgrisvtx 48850 | Every member ` N ` of the ... |
| clnbgrssvtx 48851 | The closed neighborhood of... |
| clnbgrn0 48852 | The closed neighborhood of... |
| clnbupgr 48853 | The closed neighborhood of... |
| clnbupgrel 48854 | A member of the closed nei... |
| clnbupgreli 48855 | A member of the closed nei... |
| clnbgr0vtx 48856 | In a null graph (with no v... |
| clnbgr0edg 48857 | In an empty graph (with no... |
| clnbgrsym 48858 | In a graph, the closed nei... |
| predgclnbgrel 48859 | If a (not necessarily prop... |
| clnbgredg 48860 | A vertex connected by an e... |
| clnbgrssedg 48861 | The vertices connected by ... |
| edgusgrclnbfin 48862 | The size of the closed nei... |
| clnbusgrfi 48863 | The closed neighborhood of... |
| clnbfiusgrfi 48864 | The closed neighborhood of... |
| clnbgrlevtx 48865 | The size of the closed nei... |
| dfsclnbgr2 48866 | Alternate definition of th... |
| sclnbgrel 48867 | Characterization of a memb... |
| sclnbgrelself 48868 | A vertex ` N ` is a member... |
| sclnbgrisvtx 48869 | Every member ` X ` of the ... |
| dfclnbgr5 48870 | Alternate definition of th... |
| dfnbgr5 48871 | Alternate definition of th... |
| dfnbgrss 48872 | Subset chain for different... |
| dfvopnbgr2 48873 | Alternate definition of th... |
| vopnbgrel 48874 | Characterization of a memb... |
| vopnbgrelself 48875 | A vertex ` N ` is a member... |
| dfclnbgr6 48876 | Alternate definition of th... |
| dfnbgr6 48877 | Alternate definition of th... |
| dfsclnbgr6 48878 | Alternate definition of a ... |
| dfnbgrss2 48879 | Subset chain for different... |
| isisubgr 48882 | The subgraph induced by a ... |
| isubgriedg 48883 | The edges of an induced su... |
| isubgrvtxuhgr 48884 | The subgraph induced by th... |
| isubgredgss 48885 | The edges of an induced su... |
| isubgredg 48886 | An edge of an induced subg... |
| isubgrvtx 48887 | The vertices of an induced... |
| isubgruhgr 48888 | An induced subgraph of a h... |
| isubgrsubgr 48889 | An induced subgraph of a h... |
| isubgrupgr 48890 | An induced subgraph of a p... |
| isubgrumgr 48891 | An induced subgraph of a m... |
| isubgrusgr 48892 | An induced subgraph of a s... |
| isubgr0uhgr 48893 | The subgraph induced by an... |
| grimfn 48899 | The graph isomorphism func... |
| grimdmrel 48900 | The domain of the graph is... |
| isgrim 48902 | An isomorphism of graphs i... |
| grimprop 48903 | Properties of an isomorphi... |
| grimf1o 48904 | An isomorphism of graphs i... |
| grimidvtxedg 48905 | The identity relation rest... |
| grimid 48906 | The identity relation rest... |
| grimuhgr 48907 | If there is a graph isomor... |
| grimcnv 48908 | The converse of a graph is... |
| grimco 48909 | The composition of graph i... |
| uhgrimedgi 48910 | An isomorphism between gra... |
| uhgrimedg 48911 | An isomorphism between gra... |
| uhgrimprop 48912 | An isomorphism between hyp... |
| isuspgrim0lem 48913 | An isomorphism of simple p... |
| isuspgrim0 48914 | An isomorphism of simple p... |
| isuspgrimlem 48915 | Lemma for ~ isuspgrim . (... |
| isuspgrim 48916 | A class is an isomorphism ... |
| upgrimwlklem1 48917 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 48918 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 48919 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 48920 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 48921 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 48922 | Graph isomorphisms between... |
| upgrimwlklen 48923 | Graph isomorphisms between... |
| upgrimtrlslem1 48924 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 48925 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 48926 | Graph isomorphisms between... |
| upgrimpthslem1 48927 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 48928 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 48929 | Graph isomorphisms between... |
| upgrimspths 48930 | Graph isomorphisms between... |
| upgrimcycls 48931 | Graph isomorphisms between... |
| brgric 48932 | The relation "is isomorphi... |
| brgrici 48933 | Prove that two graphs are ... |
| gricrcl 48934 | Reverse closure of the "is... |
| dfgric2 48935 | Alternate, explicit defini... |
| gricbri 48936 | Implications of two graphs... |
| gricushgr 48937 | The "is isomorphic to" rel... |
| gricuspgr 48938 | The "is isomorphic to" rel... |
| gricrel 48939 | The "is isomorphic to" rel... |
| gricref 48940 | Graph isomorphism is refle... |
| gricsym 48941 | Graph isomorphism is symme... |
| gricsymb 48942 | Graph isomorphism is symme... |
| grictr 48943 | Graph isomorphism is trans... |
| gricer 48944 | Isomorphism is an equivale... |
| gricen 48945 | Isomorphic graphs have equ... |
| opstrgric 48946 | A graph represented as an ... |
| ushggricedg 48947 | A simple hypergraph (with ... |
| cycldlenngric 48948 | Two simple pseudographs ar... |
| isubgrgrim 48949 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 48950 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 48951 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 48952 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 48953 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 48954 | Graph isomorphisms between... |
| grimedg 48955 | For two isomorphic graphs,... |
| grimedgi 48956 | Graph isomorphisms map edg... |
| grtriproplem 48959 | Lemma for ~ grtriprop . (... |
| grtri 48960 | The triangles in a graph. ... |
| grtriprop 48961 | The properties of a triang... |
| grtrif1o 48962 | Any bijection onto a trian... |
| isgrtri 48963 | A triangle in a graph. (C... |
| grtrissvtx 48964 | A triangle is a subset of ... |
| grtriclwlk3 48965 | A triangle induces a close... |
| cycl3grtrilem 48966 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 48967 | The vertices of a cycle of... |
| grtrimap 48968 | Conditions for mapping tri... |
| grimgrtri 48969 | Graph isomorphisms map tri... |
| usgrgrtrirex 48970 | Conditions for a simple gr... |
| stgrfv 48973 | The star graph S_N. (Contr... |
| stgrvtx 48974 | The vertices of the star g... |
| stgriedg 48975 | The indexed edges of the s... |
| stgredg 48976 | The edges of the star grap... |
| stgredgel 48977 | An edge of the star graph ... |
| stgredgiun 48978 | The edges of the star grap... |
| stgrusgra 48979 | The star graph S_N is a si... |
| stgr0 48980 | The star graph S_0 consist... |
| stgr1 48981 | The star graph S_1 consist... |
| stgrvtx0 48982 | The center ("internal node... |
| stgrorder 48983 | The order of a star graph ... |
| stgrnbgr0 48984 | All vertices of a star gra... |
| stgrclnbgr0 48985 | All vertices of a star gra... |
| isubgr3stgrlem1 48986 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 48987 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 48988 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 48989 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 48990 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 48991 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 48992 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 48993 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 48994 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 48995 | If a vertex of a simple gr... |
| grlimfn 48999 | The graph local isomorphis... |
| grlimdmrel 49000 | The domain of the graph lo... |
| isgrlim 49002 | A local isomorphism of gra... |
| isgrlim2 49003 | A local isomorphism of gra... |
| grlimprop 49004 | Properties of a local isom... |
| grlimf1o 49005 | A local isomorphism of gra... |
| grlimprop2 49006 | Properties of a local isom... |
| uhgrimgrlim 49007 | An isomorphism of hypergra... |
| uspgrlimlem1 49008 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 49009 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 49010 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 49011 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 49012 | A local isomorphism of sim... |
| usgrlimprop 49013 | Properties of a local isom... |
| clnbgrvtxedg 49014 | An edge ` E ` containing a... |
| grlimedgclnbgr 49015 | For two locally isomorphic... |
| grlimprclnbgr 49016 | For two locally isomorphic... |
| grlimprclnbgredg 49017 | For two locally isomorphic... |
| grlimpredg 49018 | For two locally isomorphic... |
| grlimprclnbgrvtx 49019 | For two locally isomorphic... |
| grlimgredgex 49020 | Local isomorphisms between... |
| grlimgrtrilem1 49021 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 49022 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 49023 | If one of two locally isom... |
| brgrlic 49024 | The relation "is locally i... |
| brgrilci 49025 | Prove that two graphs are ... |
| grlicrel 49026 | The "is locally isomorphic... |
| grlicrcl 49027 | Reverse closure of the "is... |
| dfgrlic2 49028 | Alternate, explicit defini... |
| grilcbri 49029 | Implications of two graphs... |
| dfgrlic3 49030 | Alternate, explicit defini... |
| grilcbri2 49031 | Implications of two graphs... |
| grlicref 49032 | Graph local isomorphism is... |
| grlicsym 49033 | Graph local isomorphism is... |
| grlicsymb 49034 | Graph local isomorphism is... |
| grlictr 49035 | Graph local isomorphism is... |
| grlicer 49036 | Local isomorphism is an eq... |
| grlicen 49037 | Locally isomorphic graphs ... |
| gricgrlic 49038 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 49039 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 49040 | If all (closed) neighborho... |
| usgrexmpl1lem 49041 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 49042 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 49043 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 49044 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 49045 | ` G ` contains a triangle ... |
| usgrexmpl2lem 49046 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 49047 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 49048 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 49049 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 49050 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 49051 | The neighborhood of the fi... |
| usgrexmpl2nb1 49052 | The neighborhood of the se... |
| usgrexmpl2nb2 49053 | The neighborhood of the th... |
| usgrexmpl2nb3 49054 | The neighborhood of the fo... |
| usgrexmpl2nb4 49055 | The neighborhood of the fi... |
| usgrexmpl2nb5 49056 | The neighborhood of the si... |
| usgrexmpl2trifr 49057 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 49058 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 49059 | The graphs ` H ` and ` G `... |
| gpgov 49062 | The generalized Petersen g... |
| gpgvtx 49063 | The vertices of the genera... |
| gpgiedg 49064 | The indexed edges of the g... |
| gpgedg 49065 | The edges of the generaliz... |
| gpgiedgdmellem 49066 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 49067 | A vertex in a generalized ... |
| gpgvtxel2 49068 | The second component of a ... |
| gpgiedgdmel 49069 | An index of edges of the g... |
| gpgedgel 49070 | An edge in a generalized P... |
| gpgprismgriedgdmel 49071 | An index of edges of the g... |
| gpgprismgriedgdmss 49072 | A subset of the index of e... |
| gpgvtx0 49073 | The outside vertices in a ... |
| gpgvtx1 49074 | The inside vertices in a g... |
| opgpgvtx 49075 | A vertex in a generalized ... |
| gpgusgralem 49076 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 49077 | The generalized Petersen g... |
| gpgprismgrusgra 49078 | The generalized Petersen g... |
| gpgorder 49079 | The order of the generaliz... |
| gpg5order 49080 | The order of a generalized... |
| gpgedgvtx0 49081 | The edges starting at an o... |
| gpgedgvtx1 49082 | The edges starting at an i... |
| gpgvtxedg0 49083 | The edges starting at an o... |
| gpgvtxedg1 49084 | The edges starting at an i... |
| gpgedgiov 49085 | The edges of the generaliz... |
| gpgedg2ov 49086 | The edges of the generaliz... |
| gpgedg2iv 49087 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 49088 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 49089 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 49090 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 49091 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 49092 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 49093 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 49094 | The (open) neighborhood of... |
| gpgnbgrvtx1 49095 | The (open) neighborhood of... |
| gpg3nbgrvtx0 49096 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 49097 | In a generalized Petersen ... |
| gpg3nbgrvtx1 49098 | In a generalized Petersen ... |
| gpgcubic 49099 | Every generalized Petersen... |
| gpg5nbgrvtx03star 49100 | In a generalized Petersen ... |
| gpg5nbgr3star 49101 | In a generalized Petersen ... |
| gpgvtxdg3 49102 | Every vertex in a generali... |
| gpg3kgrtriexlem1 49103 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 49104 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 49105 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 49106 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 49107 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 49108 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 49109 | All generalized Petersen g... |
| gpg5gricstgr3 49110 | Each closed neighborhood i... |
| pglem 49111 | Lemma for theorems about P... |
| pgjsgr 49112 | A Petersen graph is a simp... |
| gpg5grlim 49113 | A local isomorphism betwee... |
| gpg5grlic 49114 | The two generalized Peters... |
| gpgprismgr4cycllem1 49115 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 49116 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 49117 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 49118 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 49119 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 49120 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 49121 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 49122 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 49123 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 49124 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 49125 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 49126 | The generalized Petersen g... |
| gpgprismgr4cyclex 49127 | The generalized Petersen g... |
| pgnioedg1 49128 | An inside and an outside v... |
| pgnioedg2 49129 | An inside and an outside v... |
| pgnioedg3 49130 | An inside and an outside v... |
| pgnioedg4 49131 | An inside and an outside v... |
| pgnioedg5 49132 | An inside and an outside v... |
| pgnbgreunbgrlem1 49133 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 49134 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 49135 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 49136 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 49137 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 49138 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 49139 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 49140 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 49141 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 49142 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 49143 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 49144 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 49145 | In a Petersen graph, two d... |
| pgn4cyclex 49146 | A cycle in a Petersen grap... |
| pg4cyclnex 49147 | In the Petersen graph G(5,... |
| gpg5ngric 49148 | The two generalized Peters... |
| lgricngricex 49149 | There are two different lo... |
| gpg5edgnedg 49150 | Two consecutive (according... |
| grlimedgnedg 49151 | In general, the image of a... |
| 1hegrlfgr 49152 | A graph ` G ` with one hyp... |
| upwlksfval 49155 | The set of simple walks (i... |
| isupwlk 49156 | Properties of a pair of fu... |
| isupwlkg 49157 | Generalization of ~ isupwl... |
| upwlkbprop 49158 | Basic properties of a simp... |
| upwlkwlk 49159 | A simple walk is a walk. ... |
| upgrwlkupwlk 49160 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 49161 | In a pseudograph, the defi... |
| upgrisupwlkALT 49162 | Alternate proof of ~ upgri... |
| upgredgssspr 49163 | The set of edges of a pseu... |
| uspgropssxp 49164 | The set ` G ` of "simple p... |
| uspgrsprfv 49165 | The value of the function ... |
| uspgrsprf 49166 | The mapping ` F ` is a fun... |
| uspgrsprf1 49167 | The mapping ` F ` is a one... |
| uspgrsprfo 49168 | The mapping ` F ` is a fun... |
| uspgrsprf1o 49169 | The mapping ` F ` is a bij... |
| uspgrex 49170 | The class ` G ` of all "si... |
| uspgrbispr 49171 | There is a bijection betwe... |
| uspgrspren 49172 | The set ` G ` of the "simp... |
| uspgrymrelen 49173 | The set ` G ` of the "simp... |
| uspgrbisymrel 49174 | There is a bijection betwe... |
| uspgrbisymrelALT 49175 | Alternate proof of ~ uspgr... |
| ovn0dmfun 49176 | If a class operation value... |
| xpsnopab 49177 | A Cartesian product with a... |
| xpiun 49178 | A Cartesian product expres... |
| fnxpdmdm 49179 | The domain of the domain o... |
| cnfldsrngbas 49180 | The base set of a subring ... |
| cnfldsrngadd 49181 | The group addition operati... |
| cnfldsrngmul 49182 | The ring multiplication op... |
| plusfreseq 49183 | If the empty set is not co... |
| mgmplusfreseq 49184 | If the empty set is not co... |
| 0mgm 49185 | A set with an empty base s... |
| opmpoismgm 49186 | A structure with a group a... |
| copissgrp 49187 | A structure with a constan... |
| copisnmnd 49188 | A structure with a constan... |
| 0nodd 49189 | 0 is not an odd integer. ... |
| 1odd 49190 | 1 is an odd integer. (Con... |
| 2nodd 49191 | 2 is not an odd integer. ... |
| oddibas 49192 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 49193 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 49194 | The structure of all odd i... |
| nnsgrpmgm 49195 | The structure of positive ... |
| nnsgrp 49196 | The structure of positive ... |
| nnsgrpnmnd 49197 | The structure of positive ... |
| nn0mnd 49198 | The set of nonnegative int... |
| gsumsplit2f 49199 | Split a group sum into two... |
| gsumdifsndf 49200 | Extract a summand from a f... |
| gsumfsupp 49201 | A group sum of a family ca... |
| iscllaw 49208 | The predicate "is a closed... |
| iscomlaw 49209 | The predicate "is a commut... |
| clcllaw 49210 | Closure of a closed operat... |
| isasslaw 49211 | The predicate "is an assoc... |
| asslawass 49212 | Associativity of an associ... |
| mgmplusgiopALT 49213 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 49214 | Slot 2 (group operation) o... |
| intopval 49221 | The internal (binary) oper... |
| intop 49222 | An internal (binary) opera... |
| clintopval 49223 | The closed (internal binar... |
| assintopval 49224 | The associative (closed in... |
| assintopmap 49225 | The associative (closed in... |
| isclintop 49226 | The predicate "is a closed... |
| clintop 49227 | A closed (internal binary)... |
| assintop 49228 | An associative (closed int... |
| isassintop 49229 | The predicate "is an assoc... |
| clintopcllaw 49230 | The closure law holds for ... |
| assintopcllaw 49231 | The closure low holds for ... |
| assintopasslaw 49232 | The associative low holds ... |
| assintopass 49233 | An associative (closed int... |
| ismgmALT 49242 | The predicate "is a magma"... |
| iscmgmALT 49243 | The predicate "is a commut... |
| issgrpALT 49244 | The predicate "is a semigr... |
| iscsgrpALT 49245 | The predicate "is a commut... |
| mgm2mgm 49246 | Equivalence of the two def... |
| sgrp2sgrp 49247 | Equivalence of the two def... |
| lmod0rng 49248 | If the scalar ring of a mo... |
| nzrneg1ne0 49249 | The additive inverse of th... |
| lidldomn1 49250 | If a (left) ideal (which i... |
| lidlabl 49251 | A (left) ideal of a ring i... |
| lidlrng 49252 | A (left) ideal of a ring i... |
| zlidlring 49253 | The zero (left) ideal of a... |
| uzlidlring 49254 | Only the zero (left) ideal... |
| lidldomnnring 49255 | A (left) ideal of a domain... |
| 0even 49256 | 0 is an even integer. (Co... |
| 1neven 49257 | 1 is not an even integer. ... |
| 2even 49258 | 2 is an even integer. (Co... |
| 2zlidl 49259 | The even integers are a (l... |
| 2zrng 49260 | The ring of integers restr... |
| 2zrngbas 49261 | The base set of R is the s... |
| 2zrngadd 49262 | The group addition operati... |
| 2zrng0 49263 | The additive identity of R... |
| 2zrngamgm 49264 | R is an (additive) magma. ... |
| 2zrngasgrp 49265 | R is an (additive) semigro... |
| 2zrngamnd 49266 | R is an (additive) monoid.... |
| 2zrngacmnd 49267 | R is a commutative (additi... |
| 2zrngagrp 49268 | R is an (additive) group. ... |
| 2zrngaabl 49269 | R is an (additive) abelian... |
| 2zrngmul 49270 | The ring multiplication op... |
| 2zrngmmgm 49271 | R is a (multiplicative) ma... |
| 2zrngmsgrp 49272 | R is a (multiplicative) se... |
| 2zrngALT 49273 | The ring of integers restr... |
| 2zrngnmlid 49274 | R has no multiplicative (l... |
| 2zrngnmrid 49275 | R has no multiplicative (r... |
| 2zrngnmlid2 49276 | R has no multiplicative (l... |
| 2zrngnring 49277 | R is not a unital ring. (... |
| cznrnglem 49278 | Lemma for ~ cznrng : The ... |
| cznabel 49279 | The ring constructed from ... |
| cznrng 49280 | The ring constructed from ... |
| cznnring 49281 | The ring constructed from ... |
| rngcvalALTV 49284 | Value of the category of n... |
| rngcbasALTV 49285 | Set of objects of the cate... |
| rngchomfvalALTV 49286 | Set of arrows of the categ... |
| rngchomALTV 49287 | Set of arrows of the categ... |
| elrngchomALTV 49288 | A morphism of non-unital r... |
| rngccofvalALTV 49289 | Composition in the categor... |
| rngccoALTV 49290 | Composition in the categor... |
| rngccatidALTV 49291 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 49292 | The category of non-unital... |
| rngcidALTV 49293 | The identity arrow in the ... |
| rngcsectALTV 49294 | A section in the category ... |
| rngcinvALTV 49295 | An inverse in the category... |
| rngcisoALTV 49296 | An isomorphism in the cate... |
| rngchomffvalALTV 49297 | The value of the functiona... |
| rngchomrnghmresALTV 49298 | The value of the functiona... |
| rngcrescrhmALTV 49299 | The category of non-unital... |
| rhmsubcALTVlem1 49300 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 49301 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 49302 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 49303 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 49304 | According to ~ df-subc , t... |
| rhmsubcALTVcat 49305 | The restriction of the cat... |
| ringcvalALTV 49308 | Value of the category of r... |
| funcringcsetcALTV2lem1 49309 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 49310 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 49311 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 49312 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 49313 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 49314 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 49315 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 49316 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 49317 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 49318 | The "natural forgetful fun... |
| ringcbasALTV 49319 | Set of objects of the cate... |
| ringchomfvalALTV 49320 | Set of arrows of the categ... |
| ringchomALTV 49321 | Set of arrows of the categ... |
| elringchomALTV 49322 | A morphism of rings is a f... |
| ringccofvalALTV 49323 | Composition in the categor... |
| ringccoALTV 49324 | Composition in the categor... |
| ringccatidALTV 49325 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 49326 | The category of rings is a... |
| ringcidALTV 49327 | The identity arrow in the ... |
| ringcsectALTV 49328 | A section in the category ... |
| ringcinvALTV 49329 | An inverse in the category... |
| ringcisoALTV 49330 | An isomorphism in the cate... |
| ringcbasbasALTV 49331 | An element of the base set... |
| funcringcsetclem1ALTV 49332 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 49333 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 49334 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 49335 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 49336 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 49337 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 49338 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 49339 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 49340 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 49341 | The "natural forgetful fun... |
| srhmsubcALTVlem1 49342 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 49343 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 49344 | According to ~ df-subc , t... |
| sringcatALTV 49345 | The restriction of the cat... |
| crhmsubcALTV 49346 | According to ~ df-subc , t... |
| cringcatALTV 49347 | The restriction of the cat... |
| drhmsubcALTV 49348 | According to ~ df-subc , t... |
| drngcatALTV 49349 | The restriction of the cat... |
| fldcatALTV 49350 | The restriction of the cat... |
| fldcALTV 49351 | The restriction of the cat... |
| fldhmsubcALTV 49352 | According to ~ df-subc , t... |
| isprmrng 49355 | The predicate "is a prime ... |
| prmringnzring 49356 | A prime ring is a nonzero ... |
| prmrngring 49357 | A prime ring is a ring. (... |
| smprngprmrng 49358 | A simple ring (a nonzero r... |
| drngprmrng 49359 | A division ring is a prime... |
| crngprmringidom 49360 | A commutative ring is a pr... |
| crngprmringdom 49361 | A commutative ring is a pr... |
| dfidom2 49362 | Alternate definition of th... |
| isidom2 49363 | The predicate "is an integ... |
| isidom3 49364 | The predicate "is a domain... |
| idomnzd 49365 | A domain has no zero-divis... |
| idomcanl 49366 | Cancellation law for domai... |
| idomcanr 49367 | Cancellation law for domai... |
| eliunxp2 49368 | Membership in a union of C... |
| mpomptx2 49369 | Express a two-argument fun... |
| cbvmpox2 49370 | Rule to change the bound v... |
| dmmpossx2 49371 | The domain of a mapping is... |
| mpoexxg2 49372 | Existence of an operation ... |
| ovmpordxf 49373 | Value of an operation give... |
| ovmpordx 49374 | Value of an operation give... |
| ovmpox2 49375 | The value of an operation ... |
| fdmdifeqresdif 49376 | The restriction of a condi... |
| ofaddmndmap 49377 | The function operation app... |
| mapsnop 49378 | A singleton of an ordered ... |
| fprmappr 49379 | A function with a domain o... |
| mapprop 49380 | An unordered pair containi... |
| ztprmneprm 49381 | A prime is not an integer ... |
| 2t6m3t4e0 49382 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 49383 | For any finite subset of `... |
| nn0sumltlt 49384 | If the sum of two nonnegat... |
| bcpascm1 49385 | Pascal's rule for the bino... |
| altgsumbc 49386 | The sum of binomial coeffi... |
| altgsumbcALT 49387 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 49388 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 49389 | An element of the (base se... |
| zlmodzxz0 49390 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 49391 | The scalar multiplication ... |
| zlmodzxzadd 49392 | The addition of the ` ZZ `... |
| zlmodzxzsubm 49393 | The subtraction of the ` Z... |
| zlmodzxzsub 49394 | The subtraction of the ` Z... |
| mgpsumunsn 49395 | Extract a summand/factor f... |
| mgpsumz 49396 | If the group sum for the m... |
| mgpsumn 49397 | If the group sum for the m... |
| exple2lt6 49398 | A nonnegative integer to t... |
| pgrple2abl 49399 | Every symmetric group on a... |
| pgrpgt2nabl 49400 | Every symmetric group on a... |
| invginvrid 49401 | Identity for a multiplicat... |
| rmsupp0 49402 | The support of a mapping o... |
| domnmsuppn0 49403 | The support of a mapping o... |
| rmsuppss 49404 | The support of a mapping o... |
| scmsuppss 49405 | The support of a mapping o... |
| rmsuppfi 49406 | The support of a mapping o... |
| rmfsupp 49407 | A mapping of a multiplicat... |
| scmsuppfi 49408 | The support of a mapping o... |
| scmfsupp 49409 | A mapping of a scalar mult... |
| suppmptcfin 49410 | The support of a mapping w... |
| mptcfsupp 49411 | A mapping with value 0 exc... |
| fsuppmptdmf 49412 | A mapping with a finite do... |
| lmodvsmdi 49413 | Multiple distributive law ... |
| gsumlsscl 49414 | Closure of a group sum in ... |
| assaascl0 49415 | The scalar 0 embedded into... |
| assaascl1 49416 | The scalar 1 embedded into... |
| ply1vr1smo 49417 | The variable in a polynomi... |
| ply1sclrmsm 49418 | The ring multiplication of... |
| coe1sclmulval 49419 | The value of the coefficie... |
| ply1mulgsumlem1 49420 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 49421 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 49422 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 49423 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 49424 | The product of two polynom... |
| evl1at0 49425 | Polynomial evaluation for ... |
| evl1at1 49426 | Polynomial evaluation for ... |
| linply1 49427 | A term of the form ` x - C... |
| lineval 49428 | A term of the form ` x - C... |
| linevalexample 49429 | The polynomial ` x - 3 ` o... |
| dmatALTval 49434 | The algebra of ` N ` x ` N... |
| dmatALTbas 49435 | The base set of the algebr... |
| dmatALTbasel 49436 | An element of the base set... |
| dmatbas 49437 | The set of all ` N ` x ` N... |
| lincop 49442 | A linear combination as op... |
| lincval 49443 | The value of a linear comb... |
| dflinc2 49444 | Alternative definition of ... |
| lcoop 49445 | A linear combination as op... |
| lcoval 49446 | The value of a linear comb... |
| lincfsuppcl 49447 | A linear combination of ve... |
| linccl 49448 | A linear combination of ve... |
| lincval0 49449 | The value of an empty line... |
| lincvalsng 49450 | The linear combination ove... |
| lincvalsn 49451 | The linear combination ove... |
| lincvalpr 49452 | The linear combination ove... |
| lincval1 49453 | The linear combination ove... |
| lcosn0 49454 | Properties of a linear com... |
| lincvalsc0 49455 | The linear combination whe... |
| lcoc0 49456 | Properties of a linear com... |
| linc0scn0 49457 | If a set contains the zero... |
| lincdifsn 49458 | A vector is a linear combi... |
| linc1 49459 | A vector is a linear combi... |
| lincellss 49460 | A linear combination of a ... |
| lco0 49461 | The set of empty linear co... |
| lcoel0 49462 | The zero vector is always ... |
| lincsum 49463 | The sum of two linear comb... |
| lincscm 49464 | A linear combinations mult... |
| lincsumcl 49465 | The sum of two linear comb... |
| lincscmcl 49466 | The multiplication of a li... |
| lincsumscmcl 49467 | The sum of a linear combin... |
| lincolss 49468 | According to the statement... |
| ellcoellss 49469 | Every linear combination o... |
| lcoss 49470 | A set of vectors of a modu... |
| lspsslco 49471 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 49472 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 49473 | Equivalence of a _span_ of... |
| rellininds 49477 | The class defining the rel... |
| linindsv 49479 | The classes of the module ... |
| islininds 49480 | The property of being a li... |
| linindsi 49481 | The implications of being ... |
| linindslinci 49482 | The implications of being ... |
| islinindfis 49483 | The property of being a li... |
| islinindfiss 49484 | The property of being a li... |
| linindscl 49485 | A linearly independent set... |
| lindepsnlininds 49486 | A linearly dependent subse... |
| islindeps 49487 | The property of being a li... |
| lincext1 49488 | Property 1 of an extension... |
| lincext2 49489 | Property 2 of an extension... |
| lincext3 49490 | Property 3 of an extension... |
| lindslinindsimp1 49491 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 49492 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 49493 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 49494 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 49495 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 49496 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 49497 | Implication 2 for ~ lindsl... |
| lindslininds 49498 | Equivalence of definitions... |
| linds0 49499 | The empty set is always a ... |
| el0ldep 49500 | A set containing the zero ... |
| el0ldepsnzr 49501 | A set containing the zero ... |
| lindsrng01 49502 | Any subset of a module is ... |
| lindszr 49503 | Any subset of a module ove... |
| snlindsntorlem 49504 | Lemma for ~ snlindsntor . ... |
| snlindsntor 49505 | A singleton is linearly in... |
| ldepsprlem 49506 | Lemma for ~ ldepspr . (Co... |
| ldepspr 49507 | If a vector is a scalar mu... |
| lincresunit3lem3 49508 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 49509 | Lemma 1 for properties of ... |
| lincresunitlem2 49510 | Lemma for properties of a ... |
| lincresunit1 49511 | Property 1 of a specially ... |
| lincresunit2 49512 | Property 2 of a specially ... |
| lincresunit3lem1 49513 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 49514 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 49515 | Property 3 of a specially ... |
| lincreslvec3 49516 | Property 3 of a specially ... |
| islindeps2 49517 | Conditions for being a lin... |
| islininds2 49518 | Implication of being a lin... |
| isldepslvec2 49519 | Alternative definition of ... |
| lindssnlvec 49520 | A singleton not containing... |
| lmod1lem1 49521 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 49522 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 49523 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 49524 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 49525 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 49526 | The (smallest) structure r... |
| lmod1zr 49527 | The (smallest) structure r... |
| lmod1zrnlvec 49528 | There is a (left) module (... |
| lmodn0 49529 | Left modules exist. (Cont... |
| zlmodzxzequa 49530 | Example of an equation wit... |
| zlmodzxznm 49531 | Example of a linearly depe... |
| zlmodzxzldeplem 49532 | A and B are not equal. (C... |
| zlmodzxzequap 49533 | Example of an equation wit... |
| zlmodzxzldeplem1 49534 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 49535 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 49536 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 49537 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 49538 | { A , B } is a linearly de... |
| ldepsnlinclem1 49539 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 49540 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 49541 | The class of all (left) ve... |
| ldepsnlinc 49542 | The reverse implication of... |
| ldepslinc 49543 | For (left) vector spaces, ... |
| suppdm 49544 | If the range of a function... |
| eluz2cnn0n1 49545 | An integer greater than 1 ... |
| divge1b 49546 | The ratio of a real number... |
| divgt1b 49547 | The ratio of a real number... |
| ltsubaddb 49548 | Equivalence for the "less ... |
| ltsubsubb 49549 | Equivalence for the "less ... |
| ltsubadd2b 49550 | Equivalence for the "less ... |
| divsub1dir 49551 | Distribution of division o... |
| expnegico01 49552 | An integer greater than 1 ... |
| elfzolborelfzop1 49553 | An element of a half-open ... |
| pw2m1lepw2m1 49554 | 2 to the power of a positi... |
| zgtp1leeq 49555 | If an integer is between a... |
| flsubz 49556 | An integer can be moved in... |
| nn0onn0ex 49557 | For each odd nonnegative i... |
| nn0enn0ex 49558 | For each even nonnegative ... |
| nnennex 49559 | For each even positive int... |
| nneop 49560 | A positive integer is even... |
| nneom 49561 | A positive integer is even... |
| nn0eo 49562 | A nonnegative integer is e... |
| nnpw2even 49563 | 2 to the power of a positi... |
| zefldiv2 49564 | The floor of an even integ... |
| zofldiv2 49565 | The floor of an odd intege... |
| nn0ofldiv2 49566 | The floor of an odd nonneg... |
| flnn0div2ge 49567 | The floor of a positive in... |
| flnn0ohalf 49568 | The floor of the half of a... |
| logcxp0 49569 | Logarithm of a complex pow... |
| regt1loggt0 49570 | The natural logarithm for ... |
| fdivval 49573 | The quotient of two functi... |
| fdivmpt 49574 | The quotient of two functi... |
| fdivmptf 49575 | The quotient of two functi... |
| refdivmptf 49576 | The quotient of two functi... |
| fdivpm 49577 | The quotient of two functi... |
| refdivpm 49578 | The quotient of two functi... |
| fdivmptfv 49579 | The function value of a qu... |
| refdivmptfv 49580 | The function value of a qu... |
| bigoval 49583 | Set of functions of order ... |
| elbigofrcl 49584 | Reverse closure of the "bi... |
| elbigo 49585 | Properties of a function o... |
| elbigo2 49586 | Properties of a function o... |
| elbigo2r 49587 | Sufficient condition for a... |
| elbigof 49588 | A function of order G(x) i... |
| elbigodm 49589 | The domain of a function o... |
| elbigoimp 49590 | The defining property of a... |
| elbigolo1 49591 | A function (into the posit... |
| rege1logbrege0 49592 | The general logarithm, wit... |
| rege1logbzge0 49593 | The general logarithm, wit... |
| fllogbd 49594 | A real number is between t... |
| relogbmulbexp 49595 | The logarithm of the produ... |
| relogbdivb 49596 | The logarithm of the quoti... |
| logbge0b 49597 | The logarithm of a number ... |
| logblt1b 49598 | The logarithm of a number ... |
| fldivexpfllog2 49599 | The floor of a positive re... |
| nnlog2ge0lt1 49600 | A positive integer is 1 if... |
| logbpw2m1 49601 | The floor of the binary lo... |
| fllog2 49602 | The floor of the binary lo... |
| blenval 49605 | The binary length of an in... |
| blen0 49606 | The binary length of 0. (... |
| blenn0 49607 | The binary length of a "nu... |
| blenre 49608 | The binary length of a pos... |
| blennn 49609 | The binary length of a pos... |
| blennnelnn 49610 | The binary length of a pos... |
| blennn0elnn 49611 | The binary length of a non... |
| blenpw2 49612 | The binary length of a pow... |
| blenpw2m1 49613 | The binary length of a pow... |
| nnpw2blen 49614 | A positive integer is betw... |
| nnpw2blenfzo 49615 | A positive integer is betw... |
| nnpw2blenfzo2 49616 | A positive integer is eith... |
| nnpw2pmod 49617 | Every positive integer can... |
| blen1 49618 | The binary length of 1. (... |
| blen2 49619 | The binary length of 2. (... |
| nnpw2p 49620 | Every positive integer can... |
| nnpw2pb 49621 | A number is a positive int... |
| blen1b 49622 | The binary length of a non... |
| blennnt2 49623 | The binary length of a pos... |
| nnolog2flm1 49624 | The floor of the binary lo... |
| blennn0em1 49625 | The binary length of the h... |
| blennngt2o2 49626 | The binary length of an od... |
| blengt1fldiv2p1 49627 | The binary length of an in... |
| blennn0e2 49628 | The binary length of an ev... |
| digfval 49631 | Operation to obtain the ` ... |
| digval 49632 | The ` K ` th digit of a no... |
| digvalnn0 49633 | The ` K ` th digit of a no... |
| nn0digval 49634 | The ` K ` th digit of a no... |
| dignn0fr 49635 | The digits of the fraction... |
| dignn0ldlem 49636 | Lemma for ~ dignnld . (Co... |
| dignnld 49637 | The leading digits of a po... |
| dig2nn0ld 49638 | The leading digits of a po... |
| dig2nn1st 49639 | The first (relevant) digit... |
| dig0 49640 | All digits of 0 are 0. (C... |
| digexp 49641 | The ` K ` th digit of a po... |
| dig1 49642 | All but one digits of 1 ar... |
| 0dig1 49643 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 49644 | The integers 0 and 1 corre... |
| dig2nn0 49645 | A digit of a nonnegative i... |
| 0dig2nn0e 49646 | The last bit of an even in... |
| 0dig2nn0o 49647 | The last bit of an odd int... |
| dig2bits 49648 | The ` K ` th digit of a no... |
| dignn0flhalflem1 49649 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 49650 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 49651 | The digits of the half of ... |
| dignn0flhalf 49652 | The digits of the rounded ... |
| nn0sumshdiglemA 49653 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 49654 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 49655 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 49656 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 49657 | A nonnegative integer can ... |
| nn0mulfsum 49658 | Trivial algorithm to calcu... |
| nn0mullong 49659 | Standard algorithm (also k... |
| naryfval 49662 | The set of the n-ary (endo... |
| naryfvalixp 49663 | The set of the n-ary (endo... |
| naryfvalel 49664 | An n-ary (endo)function on... |
| naryrcl 49665 | Reverse closure for n-ary ... |
| naryfvalelfv 49666 | The value of an n-ary (end... |
| naryfvalelwrdf 49667 | An n-ary (endo)function on... |
| 0aryfvalel 49668 | A nullary (endo)function o... |
| 0aryfvalelfv 49669 | The value of a nullary (en... |
| 1aryfvalel 49670 | A unary (endo)function on ... |
| fv1arycl 49671 | Closure of a unary (endo)f... |
| 1arympt1 49672 | A unary (endo)function in ... |
| 1arympt1fv 49673 | The value of a unary (endo... |
| 1arymaptfv 49674 | The value of the mapping o... |
| 1arymaptf 49675 | The mapping of unary (endo... |
| 1arymaptf1 49676 | The mapping of unary (endo... |
| 1arymaptfo 49677 | The mapping of unary (endo... |
| 1arymaptf1o 49678 | The mapping of unary (endo... |
| 1aryenef 49679 | The set of unary (endo)fun... |
| 1aryenefmnd 49680 | The set of unary (endo)fun... |
| 2aryfvalel 49681 | A binary (endo)function on... |
| fv2arycl 49682 | Closure of a binary (endo)... |
| 2arympt 49683 | A binary (endo)function in... |
| 2arymptfv 49684 | The value of a binary (end... |
| 2arymaptfv 49685 | The value of the mapping o... |
| 2arymaptf 49686 | The mapping of binary (end... |
| 2arymaptf1 49687 | The mapping of binary (end... |
| 2arymaptfo 49688 | The mapping of binary (end... |
| 2arymaptf1o 49689 | The mapping of binary (end... |
| 2aryenef 49690 | The set of binary (endo)fu... |
| itcoval 49695 | The value of the function ... |
| itcoval0 49696 | A function iterated zero t... |
| itcoval1 49697 | A function iterated once. ... |
| itcoval2 49698 | A function iterated twice.... |
| itcoval3 49699 | A function iterated three ... |
| itcoval0mpt 49700 | A mapping iterated zero ti... |
| itcovalsuc 49701 | The value of the function ... |
| itcovalsucov 49702 | The value of the function ... |
| itcovalendof 49703 | The n-th iterate of an end... |
| itcovalpclem1 49704 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 49705 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 49706 | The value of the function ... |
| itcovalt2lem2lem1 49707 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 49708 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 49709 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 49710 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 49711 | The value of the function ... |
| ackvalsuc1mpt 49712 | The Ackermann function at ... |
| ackvalsuc1 49713 | The Ackermann function at ... |
| ackval0 49714 | The Ackermann function at ... |
| ackval1 49715 | The Ackermann function at ... |
| ackval2 49716 | The Ackermann function at ... |
| ackval3 49717 | The Ackermann function at ... |
| ackendofnn0 49718 | The Ackermann function at ... |
| ackfnnn0 49719 | The Ackermann function at ... |
| ackval0val 49720 | The Ackermann function at ... |
| ackvalsuc0val 49721 | The Ackermann function at ... |
| ackvalsucsucval 49722 | The Ackermann function at ... |
| ackval0012 49723 | The Ackermann function at ... |
| ackval1012 49724 | The Ackermann function at ... |
| ackval2012 49725 | The Ackermann function at ... |
| ackval3012 49726 | The Ackermann function at ... |
| ackval40 49727 | The Ackermann function at ... |
| ackval41a 49728 | The Ackermann function at ... |
| ackval41 49729 | The Ackermann function at ... |
| ackval42 49730 | The Ackermann function at ... |
| ackval42a 49731 | The Ackermann function at ... |
| ackval50 49732 | The Ackermann function at ... |
| fv1prop 49733 | The function value of unor... |
| fv2prop 49734 | The function value of unor... |
| submuladdmuld 49735 | Transformation of a sum of... |
| affinecomb1 49736 | Combination of two real af... |
| affinecomb2 49737 | Combination of two real af... |
| affineid 49738 | Identity of an affine comb... |
| 1subrec1sub 49739 | Subtract the reciprocal of... |
| resum2sqcl 49740 | The sum of two squares of ... |
| resum2sqgt0 49741 | The sum of the square of a... |
| resum2sqrp 49742 | The sum of the square of a... |
| resum2sqorgt0 49743 | The sum of the square of t... |
| reorelicc 49744 | Membership in and outside ... |
| rrx2pxel 49745 | The x-coordinate of a poin... |
| rrx2pyel 49746 | The y-coordinate of a poin... |
| prelrrx2 49747 | An unordered pair of order... |
| prelrrx2b 49748 | An unordered pair of order... |
| rrx2pnecoorneor 49749 | If two different points ` ... |
| rrx2pnedifcoorneor 49750 | If two different points ` ... |
| rrx2pnedifcoorneorr 49751 | If two different points ` ... |
| rrx2xpref1o 49752 | There is a bijection betwe... |
| rrx2xpreen 49753 | The set of points in the t... |
| rrx2plord 49754 | The lexicographical orderi... |
| rrx2plord1 49755 | The lexicographical orderi... |
| rrx2plord2 49756 | The lexicographical orderi... |
| rrx2plordisom 49757 | The set of points in the t... |
| rrx2plordso 49758 | The lexicographical orderi... |
| ehl2eudisval0 49759 | The Euclidean distance of ... |
| ehl2eudis0lt 49760 | An upper bound of the Eucl... |
| lines 49765 | The lines passing through ... |
| line 49766 | The line passing through t... |
| rrxlines 49767 | Definition of lines passin... |
| rrxline 49768 | The line passing through t... |
| rrxlinesc 49769 | Definition of lines passin... |
| rrxlinec 49770 | The line passing through t... |
| eenglngeehlnmlem1 49771 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 49772 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 49773 | The line definition in the... |
| rrx2line 49774 | The line passing through t... |
| rrx2vlinest 49775 | The vertical line passing ... |
| rrx2linest 49776 | The line passing through t... |
| rrx2linesl 49777 | The line passing through t... |
| rrx2linest2 49778 | The line passing through t... |
| elrrx2linest2 49779 | The line passing through t... |
| spheres 49780 | The spheres for given cent... |
| sphere 49781 | A sphere with center ` X `... |
| rrxsphere 49782 | The sphere with center ` M... |
| 2sphere 49783 | The sphere with center ` M... |
| 2sphere0 49784 | The sphere around the orig... |
| line2ylem 49785 | Lemma for ~ line2y . This... |
| line2 49786 | Example for a line ` G ` p... |
| line2xlem 49787 | Lemma for ~ line2x . This... |
| line2x 49788 | Example for a horizontal l... |
| line2y 49789 | Example for a vertical lin... |
| itsclc0lem1 49790 | Lemma for theorems about i... |
| itsclc0lem2 49791 | Lemma for theorems about i... |
| itsclc0lem3 49792 | Lemma for theorems about i... |
| itscnhlc0yqe 49793 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 49794 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 49795 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 49796 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 49797 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 49798 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 49799 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 49800 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 49801 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 49802 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 49803 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 49804 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 49805 | The intersection points of... |
| itsclc0b 49806 | The intersection points of... |
| itsclinecirc0 49807 | The intersection points of... |
| itsclinecirc0b 49808 | The intersection points of... |
| itsclinecirc0in 49809 | The intersection points of... |
| itsclquadb 49810 | Quadratic equation for the... |
| itsclquadeu 49811 | Quadratic equation for the... |
| 2itscplem1 49812 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 49813 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 49814 | Lemma D for ~ 2itscp . (C... |
| 2itscp 49815 | A condition for a quadrati... |
| itscnhlinecirc02plem1 49816 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 49817 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 49818 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 49819 | Intersection of a nonhoriz... |
| inlinecirc02plem 49820 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 49821 | Intersection of a line wit... |
| inlinecirc02preu 49822 | Intersection of a line wit... |
| imbi12d2 49823 | Distribution of implicatio... |
| imbi12d2a 49824 | Variant of ~ imbi12d2 . (... |
| imbi12d3 49825 | Variant of ~ imbi12d2 . (... |
| pm5.32rda 49826 | Distribution of implicatio... |
| pm5.32dar 49827 | Reverse distribution of im... |
| exp12bd 49828 | The import-export theorem ... |
| mpbiran3d 49829 | Equivalence with a conjunc... |
| mpbiran4d 49830 | Equivalence with a conjunc... |
| dtrucor3 49831 | An example of how ~ ax-5 w... |
| ralbidb 49832 | Formula-building rule for ... |
| ralbidc 49833 | Formula-building rule for ... |
| r19.41dv 49834 | A complex deduction form o... |
| rmotru 49835 | Two ways of expressing "at... |
| reutru 49836 | Two ways of expressing "ex... |
| reutruALT 49837 | Alternate proof of ~ reutr... |
| reueqbidva 49838 | Formula-building rule for ... |
| reuxfr1dd 49839 | Transfer existential uniqu... |
| ssdisjd 49840 | Subset preserves disjointn... |
| ssdisjdr 49841 | Subset preserves disjointn... |
| disjdifb 49842 | Relative complement is ant... |
| predisj 49843 | Preimages of disjoint sets... |
| vsn 49844 | The singleton of the unive... |
| mosn 49845 | "At most one" element in a... |
| mo0 49846 | "At most one" element in a... |
| mosssn 49847 | "At most one" element in a... |
| mo0sn 49848 | Two ways of expressing "at... |
| mosssn2 49849 | Two ways of expressing "at... |
| unilbss 49850 | Superclass of the greatest... |
| iuneq0 49851 | An indexed union is empty ... |
| iineq0 49852 | An indexed intersection is... |
| iunlub 49853 | The indexed union is the t... |
| iinglb 49854 | The indexed intersection i... |
| iuneqconst2 49855 | Indexed union of identical... |
| iineqconst2 49856 | Indexed intersection of id... |
| inpw 49857 | Two ways of expressing a c... |
| opth1neg 49858 | Two ordered pairs are not ... |
| opth2neg 49859 | Two ordered pairs are not ... |
| brab2dd 49860 | Expressing that two sets a... |
| brab2ddw 49861 | Expressing that two sets a... |
| brab2ddw2 49862 | Expressing that two sets a... |
| iinxp 49863 | Indexed intersection of Ca... |
| intxpd 49864 | Intersection of Cartesian ... |
| coxp 49865 | Composition with a Cartesi... |
| cosn 49866 | Composition with an ordere... |
| cosni 49867 | Composition with an ordere... |
| inisegn0a 49868 | The inverse image of a sin... |
| dmrnxp 49869 | A Cartesian product is the... |
| mof0 49870 | There is at most one funct... |
| mof02 49871 | A variant of ~ mof0 . (Co... |
| mof0ALT 49872 | Alternate proof of ~ mof0 ... |
| eufsnlem 49873 | There is exactly one funct... |
| eufsn 49874 | There is exactly one funct... |
| eufsn2 49875 | There is exactly one funct... |
| mofsn 49876 | There is at most one funct... |
| mofsn2 49877 | There is at most one funct... |
| mofsssn 49878 | There is at most one funct... |
| mofmo 49879 | There is at most one funct... |
| mofeu 49880 | The uniqueness of a functi... |
| elfvne0 49881 | If a function value has a ... |
| fdomne0 49882 | A function with non-empty ... |
| f1sn2g 49883 | A function that maps a sin... |
| f102g 49884 | A function that maps the e... |
| f1mo 49885 | A function that maps a set... |
| f002 49886 | A function with an empty c... |
| map0cor 49887 | A function exists iff an e... |
| ffvbr 49888 | Relation with function val... |
| xpco2 49889 | Composition of a Cartesian... |
| ovsng 49890 | The operation value of a s... |
| ovsng2 49891 | The operation value of a s... |
| ovsn 49892 | The operation value of a s... |
| ovsn2 49893 | The operation value of a s... |
| ovconstbrd 49894 | Two ways of expressing ` A... |
| ovconstbrn0d 49895 | Two ways of expressing ` A... |
| elovconstbrd 49896 | Two ways of expressing ` A... |
| ovmpt4d 49897 | Deduction version of ~ ovm... |
| eqfnovd 49898 | Deduction for equality of ... |
| fonex 49899 | The domain of a surjection... |
| eloprab1st2nd 49900 | Reconstruction of a nested... |
| resinsnlem 49901 | Lemma for ~ resinsnALT . ... |
| resinsn 49902 | Restriction to the interse... |
| resinsnALT 49903 | Restriction to the interse... |
| dftpos5 49904 | Alternate definition of ` ... |
| dftpos6 49905 | Alternate definition of ` ... |
| dmtposss 49906 | The domain of ` tpos F ` i... |
| tposres0 49907 | The transposition of a set... |
| tposresg 49908 | The transposition restrict... |
| tposrescnv 49909 | The transposition restrict... |
| tposres2 49910 | The transposition restrict... |
| tposres3 49911 | The transposition restrict... |
| tposres 49912 | The transposition restrict... |
| tposresxp 49913 | The transposition restrict... |
| tposf1o 49914 | Condition of a bijective t... |
| tposid 49915 | Swap an ordered pair. (Co... |
| tposidres 49916 | Swap an ordered pair. (Co... |
| tposidf1o 49917 | The swap function, or the ... |
| tposideq 49918 | Two ways of expressing the... |
| tposideq2 49919 | Two ways of expressing the... |
| ixpv 49920 | Infinite Cartesian product... |
| fvconst0ci 49921 | A constant function's valu... |
| fvconstdomi 49922 | A constant function's valu... |
| f1omo 49923 | There is at most one eleme... |
| f1omoOLD 49924 | Obsolete version of ~ f1om... |
| f1omoALT 49925 | There is at most one eleme... |
| iccin 49926 | Intersection of two closed... |
| iccdisj2 49927 | If the upper bound of one ... |
| iccdisj 49928 | If the upper bound of one ... |
| slotresfo 49929 | The condition of a structu... |
| mreuniss 49930 | The union of a collection ... |
| clduni 49931 | The union of closed sets i... |
| opncldbid 49932 | Conditions on open sets ar... |
| opndisj 49933 | Two ways of saying that tw... |
| clddisj 49934 | Two ways of saying that tw... |
| neircl 49935 | Reverse closure of the nei... |
| opnneilem 49936 | Lemma factoring out common... |
| opnneir 49937 | If something is true for a... |
| opnneirv 49938 | A variant of ~ opnneir wit... |
| opnneilv 49939 | The converse of ~ opnneir ... |
| opnneil 49940 | A variant of ~ opnneilv . ... |
| opnneibid 49941 | The equivalence between ne... |
| opnneibid2 49942 | The equivalence between ne... |
| restcls2lem 49943 | A closed set in a subspace... |
| restcls2 49944 | A closed set in a subspace... |
| restclsseplem 49945 | Lemma for ~ restclssep . ... |
| restclssep 49946 | Two disjoint closed sets i... |
| cnneiima 49947 | Given a continuous functio... |
| iooii 49948 | Open intervals are open se... |
| icccldii 49949 | Closed intervals are close... |
| i0oii 49950 | ` ( 0 [,) A ) ` is open in... |
| io1ii 49951 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 49952 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 49953 | Open neighborhood and neig... |
| sepnsepo 49954 | Open neighborhood and neig... |
| sepdisj 49955 | Separated sets are disjoin... |
| seposep 49956 | If two sets are separated ... |
| sepcsepo 49957 | If two sets are separated ... |
| sepfsepc 49958 | If two sets are separated ... |
| seppsepf 49959 | If two sets are precisely ... |
| seppcld 49960 | If two sets are precisely ... |
| isnrm4 49961 | A topological space is nor... |
| dfnrm2 49962 | A topological space is nor... |
| dfnrm3 49963 | A topological space is nor... |
| iscnrm3lem1 49964 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 49965 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 49966 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 49967 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 49968 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 49969 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 49970 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 49971 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 49972 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 49973 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 49974 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 49975 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 49976 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 49977 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 49978 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 49979 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 49980 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 49981 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 49982 | A completely normal topolo... |
| iscnrm3v 49983 | A topology is completely n... |
| iscnrm4 49984 | A completely normal topolo... |
| isprsd 49985 | Property of being a preord... |
| lubeldm2 49986 | Member of the domain of th... |
| glbeldm2 49987 | Member of the domain of th... |
| lubeldm2d 49988 | Member of the domain of th... |
| glbeldm2d 49989 | Member of the domain of th... |
| lubsscl 49990 | If a subset of ` S ` conta... |
| glbsscl 49991 | If a subset of ` S ` conta... |
| lubprlem 49992 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 49993 | The set of two comparable ... |
| lubpr 49994 | The LUB of the set of two ... |
| glbprlem 49995 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 49996 | The set of two comparable ... |
| glbpr 49997 | The GLB of the set of two ... |
| joindm2 49998 | The join of any two elemen... |
| joindm3 49999 | The join of any two elemen... |
| meetdm2 50000 | The meet of any two elemen... |
| meetdm3 50001 | The meet of any two elemen... |
| posjidm 50002 | Poset join is idempotent. ... |
| posmidm 50003 | Poset meet is idempotent. ... |
| resiposbas 50004 | Construct a poset ( ~ resi... |
| resipos 50005 | A set equipped with an ord... |
| exbaspos 50006 | There exists a poset for a... |
| exbasprs 50007 | There exists a preordered ... |
| basresposfo 50008 | The base function restrict... |
| basresprsfo 50009 | The base function restrict... |
| posnex 50010 | The class of posets is a p... |
| prsnex 50011 | The class of preordered se... |
| toslat 50012 | A toset is a lattice. (Co... |
| isclatd 50013 | The predicate "is a comple... |
| intubeu 50014 | Existential uniqueness of ... |
| unilbeu 50015 | Existential uniqueness of ... |
| ipolublem 50016 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 50017 | The domain of the LUB of t... |
| ipolub 50018 | The LUB of the inclusion p... |
| ipoglblem 50019 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 50020 | The domain of the GLB of t... |
| ipoglb 50021 | The GLB of the inclusion p... |
| ipolub0 50022 | The LUB of the empty set i... |
| ipolub00 50023 | The LUB of the empty set i... |
| ipoglb0 50024 | The GLB of the empty set i... |
| mrelatlubALT 50025 | Least upper bounds in a Mo... |
| mrelatglbALT 50026 | Greatest lower bounds in a... |
| mreclat 50027 | A Moore space is a complet... |
| topclat 50028 | A topology is a complete l... |
| toplatglb0 50029 | The empty intersection in ... |
| toplatlub 50030 | Least upper bounds in a to... |
| toplatglb 50031 | Greatest lower bounds in a... |
| toplatjoin 50032 | Joins in a topology are re... |
| toplatmeet 50033 | Meets in a topology are re... |
| topdlat 50034 | A topology is a distributi... |
| elmgpcntrd 50035 | The center of a ring. (Co... |
| asclelbasALT 50036 | Alternate proof of ~ ascle... |
| asclcntr 50037 | The algebra scalar lifting... |
| asclcom 50038 | Scalars are commutative af... |
| homf0 50039 | The base is empty iff the ... |
| catprslem 50040 | Lemma for ~ catprs . (Con... |
| catprs 50041 | A preorder can be extracte... |
| catprs2 50042 | A category equipped with t... |
| catprsc 50043 | A construction of the preo... |
| catprsc2 50044 | An alternate construction ... |
| endmndlem 50045 | A diagonal hom-set in a ca... |
| oppccatb 50046 | An opposite category is a ... |
| oppcmndclem 50047 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 50048 | The opposite category of a... |
| oppcmndc 50049 | The opposite category of a... |
| idmon 50050 | An identity arrow, or an i... |
| idepi 50051 | An identity arrow, or an i... |
| sectrcl 50052 | Reverse closure for sectio... |
| sectrcl2 50053 | Reverse closure for sectio... |
| invrcl 50054 | Reverse closure for invers... |
| invrcl2 50055 | Reverse closure for invers... |
| isinv2 50056 | The property " ` F ` is an... |
| isisod 50057 | The predicate "is an isomo... |
| upeu2lem 50058 | Lemma for ~ upeu2 . There... |
| sectfn 50059 | The function value of the ... |
| invfn 50060 | The function value of the ... |
| isofnALT 50061 | The function value of the ... |
| isofval2 50062 | Function value of the func... |
| isorcl 50063 | Reverse closure for isomor... |
| isorcl2 50064 | Reverse closure for isomor... |
| isoval2 50065 | The isomorphisms are the d... |
| sectpropdlem 50066 | Lemma for ~ sectpropd . (... |
| sectpropd 50067 | Two structures with the sa... |
| invpropdlem 50068 | Lemma for ~ invpropd . (C... |
| invpropd 50069 | Two structures with the sa... |
| isopropdlem 50070 | Lemma for ~ isopropd . (C... |
| isopropd 50071 | Two structures with the sa... |
| cicfn 50072 | ` ~=c ` is a function on `... |
| cicrcl2 50073 | Isomorphism implies the st... |
| oppccic 50074 | Isomorphic objects are iso... |
| relcic 50075 | The set of isomorphic obje... |
| cicerALT 50076 | Isomorphism is an equivale... |
| cic1st2nd 50077 | Reconstruction of a pair o... |
| cic1st2ndbr 50078 | Rewrite the predicate of i... |
| cicpropdlem 50079 | Lemma for ~ cicpropd . (C... |
| cicpropd 50080 | Two structures with the sa... |
| oppccicb 50081 | Isomorphic objects are iso... |
| oppcciceq 50082 | The opposite category has ... |
| dmdm 50083 | The double domain of a fun... |
| iinfssclem1 50084 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 50085 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 50086 | Lemma for ~ iinfssc . (Co... |
| iinfssc 50087 | Indexed intersection of su... |
| iinfsubc 50088 | Indexed intersection of su... |
| iinfprg 50089 | Indexed intersection of fu... |
| infsubc 50090 | The intersection of two su... |
| infsubc2 50091 | The intersection of two su... |
| infsubc2d 50092 | The intersection of two su... |
| discsubclem 50093 | Lemma for ~ discsubc . (C... |
| discsubc 50094 | A discrete category, whose... |
| iinfconstbaslem 50095 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 50096 | The discrete category is t... |
| nelsubclem 50097 | Lemma for ~ nelsubc . (Co... |
| nelsubc 50098 | An empty "hom-set" for non... |
| nelsubc2 50099 | An empty "hom-set" for non... |
| nelsubc3lem 50100 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 50101 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 50102 | A category ` C ` restricte... |
| resccatlem 50103 | Lemma for ~ resccat . (Co... |
| resccat 50104 | A class ` C ` restricted b... |
| reldmfunc 50105 | The domain of ` Func ` is ... |
| func1st2nd 50106 | Rewrite the functor predic... |
| func1st 50107 | Extract the first member o... |
| func2nd 50108 | Extract the second member ... |
| funcrcl2 50109 | Reverse closure for a func... |
| funcrcl3 50110 | Reverse closure for a func... |
| funcf2lem 50111 | A utility theorem for prov... |
| funcf2lem2 50112 | A utility theorem for prov... |
| 0funcglem 50113 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 50114 | The functor from the empty... |
| 0funcg 50115 | The functor from the empty... |
| 0funclem 50116 | Lemma for ~ 0funcALT . (C... |
| 0func 50117 | The functor from the empty... |
| 0funcALT 50118 | Alternate proof of ~ 0func... |
| func0g 50119 | The source category of a f... |
| func0g2 50120 | The source category of a f... |
| initc 50121 | Sets with empty base are t... |
| cofu1st2nd 50122 | Rewrite the functor compos... |
| rescofuf 50123 | The restriction of functor... |
| cofu1a 50124 | Value of the object part o... |
| cofu2a 50125 | Value of the morphism part... |
| cofucla 50126 | The composition of two fun... |
| funchomf 50127 | Source categories of a fun... |
| idfurcl 50128 | Reverse closure for an ide... |
| idfu1stf1o 50129 | The identity functor/inclu... |
| idfu1stalem 50130 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 50131 | Value of the object part o... |
| idfu1a 50132 | Value of the object part o... |
| idfu2nda 50133 | Value of the morphism part... |
| imasubclem1 50134 | Lemma for ~ imasubc . (Co... |
| imasubclem2 50135 | Lemma for ~ imasubc . (Co... |
| imasubclem3 50136 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 50137 | Lemma for ~ imaf1hom and o... |
| imaf1hom 50138 | The hom-set of an image of... |
| imaidfu2lem 50139 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 50140 | The image of the identity ... |
| imaidfu2 50141 | The image of the identity ... |
| cofid1a 50142 | Express the object part of... |
| cofid2a 50143 | Express the morphism part ... |
| cofid1 50144 | Express the object part of... |
| cofid2 50145 | Express the morphism part ... |
| cofidvala 50146 | The property " ` F ` is a ... |
| cofidf2a 50147 | If " ` F ` is a section of... |
| cofidf1a 50148 | If " ` F ` is a section of... |
| cofidval 50149 | The property " ` <. F , G ... |
| cofidf2 50150 | If " ` F ` is a section of... |
| cofidf1 50151 | If " ` <. F , G >. ` is a ... |
| oppffn 50154 | ` oppFunc ` is a function ... |
| reldmoppf 50155 | The domain of ` oppFunc ` ... |
| oppfvalg 50156 | Value of the opposite func... |
| oppfrcllem 50157 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 50158 | If an opposite functor of ... |
| oppfrcl2 50159 | If an opposite functor of ... |
| oppfrcl3 50160 | If an opposite functor of ... |
| oppf1st2nd 50161 | Rewrite the opposite funct... |
| 2oppf 50162 | The double opposite functo... |
| eloppf 50163 | The pre-image of a non-emp... |
| eloppf2 50164 | Both components of a pre-i... |
| oppfvallem 50165 | Lemma for ~ oppfval . (Co... |
| oppfval 50166 | Value of the opposite func... |
| oppfval2 50167 | Value of the opposite func... |
| oppfval3 50168 | Value of the opposite func... |
| oppf1 50169 | Value of the object part o... |
| oppf2 50170 | Value of the morphism part... |
| oppfoppc 50171 | The opposite functor is a ... |
| oppfoppc2 50172 | The opposite functor is a ... |
| funcoppc2 50173 | A functor on opposite cate... |
| funcoppc4 50174 | A functor on opposite cate... |
| funcoppc5 50175 | A functor on opposite cate... |
| 2oppffunc 50176 | The opposite functor of an... |
| funcoppc3 50177 | A functor on opposite cate... |
| oppff1 50178 | The operation generating o... |
| oppff1o 50179 | The operation generating o... |
| cofuoppf 50180 | Composition of opposite fu... |
| imasubc 50181 | An image of a full functor... |
| imasubc2 50182 | An image of a full functor... |
| imassc 50183 | An image of a functor sati... |
| imaid 50184 | An image of a functor pres... |
| imaf1co 50185 | An image of a functor whos... |
| imasubc3 50186 | An image of a functor inje... |
| fthcomf 50187 | Source categories of a fai... |
| idfth 50188 | The inclusion functor is a... |
| idemb 50189 | The inclusion functor is a... |
| idsubc 50190 | The source category of an ... |
| idfullsubc 50191 | The source category of an ... |
| cofidfth 50192 | If " ` F ` is a section of... |
| fulloppf 50193 | The opposite functor of a ... |
| fthoppf 50194 | The opposite functor of a ... |
| ffthoppf 50195 | The opposite functor of a ... |
| upciclem1 50196 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 50197 | Lemma for ~ upciclem3 and ... |
| upciclem3 50198 | Lemma for ~ upciclem4 . (... |
| upciclem4 50199 | Lemma for ~ upcic and ~ up... |
| upcic 50200 | A universal property defin... |
| upeu 50201 | A universal property defin... |
| upeu2 50202 | Generate new universal mor... |
| reldmup 50205 | The domain of ` UP ` is a ... |
| upfval 50206 | Function value of the clas... |
| upfval2 50207 | Function value of the clas... |
| upfval3 50208 | Function value of the clas... |
| isuplem 50209 | Lemma for ~ isup and other... |
| isup 50210 | The predicate "is a univer... |
| uppropd 50211 | If two categories have the... |
| reldmup2 50212 | The domain of ` ( D UP E )... |
| relup 50213 | The set of universal pairs... |
| uprcl 50214 | Reverse closure for the cl... |
| up1st2nd 50215 | Rewrite the universal prop... |
| up1st2ndr 50216 | Combine separated parts in... |
| up1st2ndb 50217 | Combine/separate parts in ... |
| up1st2nd2 50218 | Rewrite the universal prop... |
| uprcl2 50219 | Reverse closure for the cl... |
| uprcl3 50220 | Reverse closure for the cl... |
| uprcl4 50221 | Reverse closure for the cl... |
| uprcl5 50222 | Reverse closure for the cl... |
| uobrcl 50223 | Reverse closure for univer... |
| isup2 50224 | The universal property of ... |
| upeu3 50225 | The universal pair ` <. X ... |
| upeu4 50226 | Generate a new universal m... |
| uptposlem 50227 | Lemma for ~ uptpos . (Con... |
| uptpos 50228 | Rewrite the predicate of u... |
| oppcuprcl4 50229 | Reverse closure for the cl... |
| oppcuprcl3 50230 | Reverse closure for the cl... |
| oppcuprcl5 50231 | Reverse closure for the cl... |
| oppcuprcl2 50232 | Reverse closure for the cl... |
| uprcl2a 50233 | Reverse closure for the cl... |
| oppfuprcl 50234 | Reverse closure for the cl... |
| oppfuprcl2 50235 | Reverse closure for the cl... |
| oppcup3lem 50236 | Lemma for ~ oppcup3 . (Co... |
| oppcup 50237 | The universal pair ` <. X ... |
| oppcup2 50238 | The universal property for... |
| oppcup3 50239 | The universal property for... |
| uptrlem1 50240 | Lemma for ~ uptr . (Contr... |
| uptrlem2 50241 | Lemma for ~ uptr . (Contr... |
| uptrlem3 50242 | Lemma for ~ uptr . (Contr... |
| uptr 50243 | Universal property and ful... |
| uptri 50244 | Universal property and ful... |
| uptra 50245 | Universal property and ful... |
| uptrar 50246 | Universal property and ful... |
| uptrai 50247 | Universal property and ful... |
| uobffth 50248 | A fully faithful functor g... |
| uobeqw 50249 | If a full functor (in fact... |
| uobeq 50250 | If a full functor (in fact... |
| uptr2 50251 | Universal property and ful... |
| uptr2a 50252 | Universal property and ful... |
| isnatd 50253 | Property of being a natura... |
| natrcl2 50254 | Reverse closure for a natu... |
| natrcl3 50255 | Reverse closure for a natu... |
| catbas 50256 | The base of the category s... |
| cathomfval 50257 | The hom-sets of the catego... |
| catcofval 50258 | Composition of the categor... |
| natoppf 50259 | A natural transformation i... |
| natoppf2 50260 | A natural transformation i... |
| natoppfb 50261 | A natural transformation i... |
| initoo2 50262 | An initial object is an ob... |
| termoo2 50263 | A terminal object is an ob... |
| zeroo2 50264 | A zero object is an object... |
| oppcinito 50265 | Initial objects are termin... |
| oppctermo 50266 | Terminal objects are initi... |
| oppczeroo 50267 | Zero objects are zero in t... |
| termoeu2 50268 | Terminal objects are essen... |
| initopropdlemlem 50269 | Lemma for ~ initopropdlem ... |
| initopropdlem 50270 | Lemma for ~ initopropd . ... |
| termopropdlem 50271 | Lemma for ~ termopropd . ... |
| zeroopropdlem 50272 | Lemma for ~ zeroopropd . ... |
| initopropd 50273 | Two structures with the sa... |
| termopropd 50274 | Two structures with the sa... |
| zeroopropd 50275 | Two structures with the sa... |
| reldmxpc 50276 | The binary product of cate... |
| reldmxpcALT 50277 | Alternate proof of ~ reldm... |
| elxpcbasex1 50278 | A non-empty base set of th... |
| elxpcbasex1ALT 50279 | Alternate proof of ~ elxpc... |
| elxpcbasex2 50280 | A non-empty base set of th... |
| elxpcbasex2ALT 50281 | Alternate proof of ~ elxpc... |
| xpcfucbas 50282 | The base set of the produc... |
| xpcfuchomfval 50283 | Set of morphisms of the bi... |
| xpcfuchom 50284 | Set of morphisms of the bi... |
| xpcfuchom2 50285 | Value of the set of morphi... |
| xpcfucco2 50286 | Value of composition in th... |
| xpcfuccocl 50287 | The composition of two nat... |
| xpcfucco3 50288 | Value of composition in th... |
| dfswapf2 50291 | Alternate definition of ` ... |
| swapfval 50292 | Value of the swap functor.... |
| swapfelvv 50293 | A swap functor is an order... |
| swapf2fvala 50294 | The morphism part of the s... |
| swapf2fval 50295 | The morphism part of the s... |
| swapf1vala 50296 | The object part of the swa... |
| swapf1val 50297 | The object part of the swa... |
| swapf2fn 50298 | The morphism part of the s... |
| swapf1a 50299 | The object part of the swa... |
| swapf2vala 50300 | The morphism part of the s... |
| swapf2a 50301 | The morphism part of the s... |
| swapf1 50302 | The object part of the swa... |
| swapf2val 50303 | The morphism part of the s... |
| swapf2 50304 | The morphism part of the s... |
| swapf1f1o 50305 | The object part of the swa... |
| swapf2f1o 50306 | The morphism part of the s... |
| swapf2f1oa 50307 | The morphism part of the s... |
| swapf2f1oaALT 50308 | Alternate proof of ~ swapf... |
| swapfid 50309 | Each identity morphism in ... |
| swapfida 50310 | Each identity morphism in ... |
| swapfcoa 50311 | Composition in the source ... |
| swapffunc 50312 | The swap functor is a func... |
| swapfffth 50313 | The swap functor is a full... |
| swapffunca 50314 | The swap functor is a func... |
| swapfiso 50315 | The swap functor is an iso... |
| swapciso 50316 | The product category is ca... |
| oppc1stflem 50317 | A utility theorem for prov... |
| oppc1stf 50318 | The opposite functor of th... |
| oppc2ndf 50319 | The opposite functor of th... |
| 1stfpropd 50320 | If two categories have the... |
| 2ndfpropd 50321 | If two categories have the... |
| diagpropd 50322 | If two categories have the... |
| cofuswapfcl 50323 | The bifunctor pre-composed... |
| cofuswapf1 50324 | The object part of a bifun... |
| cofuswapf2 50325 | The morphism part of a bif... |
| tposcurf1cl 50326 | The partially evaluated tr... |
| tposcurf11 50327 | Value of the double evalua... |
| tposcurf12 50328 | The partially evaluated tr... |
| tposcurf1 50329 | Value of the object part o... |
| tposcurf2 50330 | Value of the transposed cu... |
| tposcurf2val 50331 | Value of a component of th... |
| tposcurf2cl 50332 | The transposed curry funct... |
| tposcurfcl 50333 | The transposed curry funct... |
| diag1 50334 | The constant functor of ` ... |
| diag1a 50335 | The constant functor of ` ... |
| diag1f1lem 50336 | The object part of the dia... |
| diag1f1 50337 | The object part of the dia... |
| diag2f1lem 50338 | Lemma for ~ diag2f1 . The... |
| diag2f1 50339 | If ` B ` is non-empty, the... |
| fucofulem1 50340 | Lemma for proving functor ... |
| fucofulem2 50341 | Lemma for proving functor ... |
| fuco2el 50342 | Equivalence of product fun... |
| fuco2eld 50343 | Equivalence of product fun... |
| fuco2eld2 50344 | Equivalence of product fun... |
| fuco2eld3 50345 | Equivalence of product fun... |
| fucofvalg 50348 | Value of the function givi... |
| fucofval 50349 | Value of the function givi... |
| fucoelvv 50350 | A functor composition bifu... |
| fuco1 50351 | The object part of the fun... |
| fucof1 50352 | The object part of the fun... |
| fuco2 50353 | The morphism part of the f... |
| fucofn2 50354 | The morphism part of the f... |
| fucofvalne 50355 | Value of the function givi... |
| fuco11 50356 | The object part of the fun... |
| fuco11cl 50357 | The object part of the fun... |
| fuco11a 50358 | The object part of the fun... |
| fuco112 50359 | The object part of the fun... |
| fuco111 50360 | The object part of the fun... |
| fuco111x 50361 | The object part of the fun... |
| fuco112x 50362 | The object part of the fun... |
| fuco112xa 50363 | The object part of the fun... |
| fuco11id 50364 | The identity morphism of t... |
| fuco11idx 50365 | The identity morphism of t... |
| fuco21 50366 | The morphism part of the f... |
| fuco11b 50367 | The object part of the fun... |
| fuco11bALT 50368 | Alternate proof of ~ fuco1... |
| fuco22 50369 | The morphism part of the f... |
| fucofn22 50370 | The morphism part of the f... |
| fuco23 50371 | The morphism part of the f... |
| fuco22natlem1 50372 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 50373 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 50374 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 50375 | The composed natural trans... |
| fuco22nat 50376 | The composed natural trans... |
| fucof21 50377 | The morphism part of the f... |
| fucoid 50378 | Each identity morphism in ... |
| fucoid2 50379 | Each identity morphism in ... |
| fuco22a 50380 | The morphism part of the f... |
| fuco23alem 50381 | The naturality property ( ... |
| fuco23a 50382 | The morphism part of the f... |
| fucocolem1 50383 | Lemma for ~ fucoco . Asso... |
| fucocolem2 50384 | Lemma for ~ fucoco . The ... |
| fucocolem3 50385 | Lemma for ~ fucoco . The ... |
| fucocolem4 50386 | Lemma for ~ fucoco . The ... |
| fucoco 50387 | Composition in the source ... |
| fucoco2 50388 | Composition in the source ... |
| fucofunc 50389 | The functor composition bi... |
| fucofunca 50390 | The functor composition bi... |
| fucolid 50391 | Post-compose a natural tra... |
| fucorid 50392 | Pre-composing a natural tr... |
| fucorid2 50393 | Pre-composing a natural tr... |
| postcofval 50394 | Value of the post-composit... |
| postcofcl 50395 | The post-composition funct... |
| precofvallem 50396 | Lemma for ~ precofval to e... |
| precofval 50397 | Value of the pre-compositi... |
| precofvalALT 50398 | Alternate proof of ~ preco... |
| precofval2 50399 | Value of the pre-compositi... |
| precofcl 50400 | The pre-composition functo... |
| precofval3 50401 | Value of the pre-compositi... |
| precoffunc 50402 | The pre-composition functo... |
| reldmprcof 50405 | The domain of ` -o.F ` is ... |
| prcofvalg 50406 | Value of the pre-compositi... |
| prcofvala 50407 | Value of the pre-compositi... |
| prcofval 50408 | Value of the pre-compositi... |
| prcofpropd 50409 | If the categories have the... |
| prcofelvv 50410 | The pre-composition functo... |
| reldmprcof1 50411 | The domain of the object p... |
| reldmprcof2 50412 | The domain of the morphism... |
| prcoftposcurfuco 50413 | The pre-composition functo... |
| prcoftposcurfucoa 50414 | The pre-composition functo... |
| prcoffunc 50415 | The pre-composition functo... |
| prcoffunca 50416 | The pre-composition functo... |
| prcoffunca2 50417 | The pre-composition functo... |
| prcof1 50418 | The object part of the pre... |
| prcof2a 50419 | The morphism part of the p... |
| prcof2 50420 | The morphism part of the p... |
| prcof21a 50421 | The morphism part of the p... |
| prcof22a 50422 | The morphism part of the p... |
| prcofdiag1 50423 | A constant functor pre-com... |
| prcofdiag 50424 | A diagonal functor post-co... |
| catcrcl 50425 | Reverse closure for the ca... |
| catcrcl2 50426 | Reverse closure for the ca... |
| elcatchom 50427 | A morphism of the category... |
| catcsect 50428 | The property " ` F ` is a ... |
| catcinv 50429 | The property " ` F ` is an... |
| catcisoi 50430 | A functor is an isomorphis... |
| uobeq2 50431 | If a full functor (in fact... |
| uobeq3 50432 | An isomorphism between cat... |
| opf11 50433 | The object part of the op ... |
| opf12 50434 | The object part of the op ... |
| opf2fval 50435 | The morphism part of the o... |
| opf2 50436 | The morphism part of the o... |
| fucoppclem 50437 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 50438 | The opposite category of f... |
| fucoppcco 50439 | The opposite category of f... |
| fucoppc 50440 | The isomorphism from the o... |
| fucoppcffth 50441 | A fully faithful functor f... |
| fucoppcfunc 50442 | A functor from the opposit... |
| fucoppccic 50443 | The opposite category of f... |
| oppfdiag1 50444 | A constant functor for opp... |
| oppfdiag1a 50445 | A constant functor for opp... |
| oppfdiag 50446 | A diagonal functor for opp... |
| isthinc 50449 | The predicate "is a thin c... |
| isthinc2 50450 | A thin category is a categ... |
| isthinc3 50451 | A thin category is a categ... |
| thinccat 50452 | A thin category is a categ... |
| thinccatd 50453 | A thin category is a categ... |
| thincssc 50454 | A thin category is a categ... |
| isthincd2lem1 50455 | Lemma for ~ isthincd2 and ... |
| thincmo2 50456 | Morphisms in the same hom-... |
| thinchom 50457 | A non-empty hom-set of a t... |
| thincmo 50458 | There is at most one morph... |
| thincmoALT 50459 | Alternate proof of ~ thinc... |
| thincmod 50460 | At most one morphism in ea... |
| thincn0eu 50461 | In a thin category, a hom-... |
| thincid 50462 | In a thin category, a morp... |
| thincmon 50463 | In a thin category, all mo... |
| thincepi 50464 | In a thin category, all mo... |
| isthincd2lem2 50465 | Lemma for ~ isthincd2 . (... |
| isthincd 50466 | The predicate "is a thin c... |
| isthincd2 50467 | The predicate " ` C ` is a... |
| oppcthin 50468 | The opposite category of a... |
| oppcthinco 50469 | If the opposite category o... |
| oppcthinendc 50470 | The opposite category of a... |
| oppcthinendcALT 50471 | Alternate proof of ~ oppct... |
| thincpropd 50472 | Two structures with the sa... |
| subthinc 50473 | A subcategory of a thin ca... |
| functhinclem1 50474 | Lemma for ~ functhinc . G... |
| functhinclem2 50475 | Lemma for ~ functhinc . (... |
| functhinclem3 50476 | Lemma for ~ functhinc . T... |
| functhinclem4 50477 | Lemma for ~ functhinc . O... |
| functhinc 50478 | A functor to a thin catego... |
| functhincfun 50479 | A functor to a thin catego... |
| fullthinc 50480 | A functor to a thin catego... |
| fullthinc2 50481 | A full functor to a thin c... |
| thincfth 50482 | A functor from a thin cate... |
| thincciso 50483 | Two thin categories are is... |
| thinccisod 50484 | Two thin categories are is... |
| thincciso2 50485 | Categories isomorphic to a... |
| thincciso3 50486 | Categories isomorphic to a... |
| thincciso4 50487 | Two isomorphic categories ... |
| 0thincg 50488 | Any structure with an empt... |
| 0thinc 50489 | The empty category (see ~ ... |
| indcthing 50490 | An indiscrete category, i.... |
| discthing 50491 | A discrete category, i.e.,... |
| indthinc 50492 | An indiscrete category in ... |
| indthincALT 50493 | An alternate proof of ~ in... |
| prsthinc 50494 | Preordered sets as categor... |
| setcthin 50495 | A category of sets all of ... |
| setc2othin 50496 | The category ` ( SetCat ``... |
| thincsect 50497 | In a thin category, one mo... |
| thincsect2 50498 | In a thin category, ` F ` ... |
| thincinv 50499 | In a thin category, ` F ` ... |
| thinciso 50500 | In a thin category, ` F : ... |
| thinccic 50501 | In a thin category, two ob... |
| istermc 50504 | The predicate "is a termin... |
| istermc2 50505 | The predicate "is a termin... |
| istermc3 50506 | The predicate "is a termin... |
| termcthin 50507 | A terminal category is a t... |
| termcthind 50508 | A terminal category is a t... |
| termccatd 50509 | A terminal category is a c... |
| termcbas 50510 | The base of a terminal cat... |
| termco 50511 | The object of a terminal c... |
| termcbas2 50512 | The base of a terminal cat... |
| termcbasmo 50513 | Two objects in a terminal ... |
| termchomn0 50514 | All hom-sets of a terminal... |
| termchommo 50515 | All morphisms of a termina... |
| termcid 50516 | The morphism of a terminal... |
| termcid2 50517 | The morphism of a terminal... |
| termchom 50518 | The hom-set of a terminal ... |
| termchom2 50519 | The hom-set of a terminal ... |
| setcsnterm 50520 | The category of one set, e... |
| setc1oterm 50521 | The category ` ( SetCat ``... |
| setc1obas 50522 | The base of the trivial ca... |
| setc1ohomfval 50523 | Set of morphisms of the tr... |
| setc1ocofval 50524 | Composition in the trivial... |
| setc1oid 50525 | The identity morphism of t... |
| funcsetc1ocl 50526 | The functor to the trivial... |
| funcsetc1o 50527 | Value of the functor to th... |
| isinito2lem 50528 | The predicate "is an initi... |
| isinito2 50529 | The predicate "is an initi... |
| isinito3 50530 | The predicate "is an initi... |
| dfinito4 50531 | An alternate definition of... |
| dftermo4 50532 | An alternate definition of... |
| termcpropd 50533 | Two structures with the sa... |
| oppctermhom 50534 | The opposite category of a... |
| oppctermco 50535 | The opposite category of a... |
| oppcterm 50536 | The opposite category of a... |
| functermclem 50537 | Lemma for ~ functermc . (... |
| functermc 50538 | Functor to a terminal cate... |
| functermc2 50539 | Functor to a terminal cate... |
| functermceu 50540 | There exists a unique func... |
| fulltermc 50541 | A functor to a terminal ca... |
| fulltermc2 50542 | Given a full functor to a ... |
| termcterm 50543 | A terminal category is a t... |
| termcterm2 50544 | A terminal object of the c... |
| termcterm3 50545 | In the category of small c... |
| termcciso 50546 | A category is isomorphic t... |
| termccisoeu 50547 | The isomorphism between te... |
| termc2 50548 | If there exists a unique f... |
| termc 50549 | Alternate definition of ` ... |
| dftermc2 50550 | Alternate definition of ` ... |
| eufunclem 50551 | If there exists a unique f... |
| eufunc 50552 | If there exists a unique f... |
| idfudiag1lem 50553 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 50554 | If the identity functor of... |
| idfudiag1 50555 | If the identity functor of... |
| euendfunc 50556 | If there exists a unique e... |
| euendfunc2 50557 | If there exists a unique e... |
| termcarweu 50558 | There exists a unique disj... |
| arweuthinc 50559 | If a structure has a uniqu... |
| arweutermc 50560 | If a structure has a uniqu... |
| dftermc3 50561 | Alternate definition of ` ... |
| termcfuncval 50562 | The value of a functor fro... |
| diag1f1olem 50563 | To any functor from a term... |
| diag1f1o 50564 | The object part of the dia... |
| termcnatval 50565 | Value of natural transform... |
| diag2f1olem 50566 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 50567 | If ` D ` is terminal, the ... |
| diagffth 50568 | The diagonal functor is a ... |
| diagciso 50569 | The diagonal functor is an... |
| diagcic 50570 | Any category ` C ` is isom... |
| funcsn 50571 | The category of one functo... |
| fucterm 50572 | The category of functors t... |
| 0fucterm 50573 | The category of functors f... |
| termfucterm 50574 | All functors between two t... |
| cofuterm 50575 | Post-compose with a functo... |
| uobeqterm 50576 | Universal objects and term... |
| isinito4 50577 | The predicate "is an initi... |
| isinito4a 50578 | The predicate "is an initi... |
| prstcval 50581 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 50582 | Lemma for ~ prstcnid and ~... |
| prstcnid 50583 | Components other than ` Ho... |
| prstcbas 50584 | The base set is unchanged.... |
| prstcleval 50585 | Value of the less-than-or-... |
| prstcle 50586 | Value of the less-than-or-... |
| prstcocval 50587 | Orthocomplementation is un... |
| prstcoc 50588 | Orthocomplementation is un... |
| prstchomval 50589 | Hom-sets of the constructe... |
| prstcprs 50590 | The category is a preorder... |
| prstcthin 50591 | The preordered set is equi... |
| prstchom 50592 | Hom-sets of the constructe... |
| prstchom2 50593 | Hom-sets of the constructe... |
| prstchom2ALT 50594 | Hom-sets of the constructe... |
| oduoppcbas 50595 | The dual of a preordered s... |
| oduoppcciso 50596 | The dual of a preordered s... |
| postcpos 50597 | The converted category is ... |
| postcposALT 50598 | Alternate proof of ~ postc... |
| postc 50599 | The converted category is ... |
| discsntermlem 50600 | A singlegon is an element ... |
| basrestermcfolem 50601 | An element of the class of... |
| discbas 50602 | A discrete category (a cat... |
| discthin 50603 | A discrete category (a cat... |
| discsnterm 50604 | A discrete category (a cat... |
| basrestermcfo 50605 | The base function restrict... |
| termcnex 50606 | The class of all terminal ... |
| mndtcval 50609 | Value of the category buil... |
| mndtcbasval 50610 | The base set of the catego... |
| mndtcbaseu 50611 | The category built from a ... |
| mndtcob 50612 | Lemma for ~ mndtchom and ~... |
| mndtcobeq 50613 | Two objects in a category ... |
| mndtchom 50614 | The only hom-set of the ca... |
| mndtcco 50615 | The composition of the cat... |
| mndtcco2 50616 | The composition of the cat... |
| mndtccatid 50617 | Lemma for ~ mndtccat and ~... |
| mndtccat 50618 | The function value is a ca... |
| mndtcid 50619 | The identity morphism, or ... |
| oppgoppchom 50620 | The converted opposite mon... |
| oppgoppcco 50621 | The converted opposite mon... |
| oppgoppcid 50622 | The converted opposite mon... |
| grptcmon 50623 | All morphisms in a categor... |
| grptcepi 50624 | All morphisms in a categor... |
| 2arwcatlem1 50625 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 50626 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 50627 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 50628 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 50629 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 50630 | The condition for a struct... |
| incat 50631 | Constructing a category wi... |
| setc1onsubc 50632 | Construct a category with ... |
| cnelsubclem 50633 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 50634 | Remark 4.2(2) of [Adamek] ... |
| lanfn 50639 | ` Lan ` is a function on `... |
| ranfn 50640 | ` Ran ` is a function on `... |
| reldmlan 50641 | The domain of ` Lan ` is a... |
| reldmran 50642 | The domain of ` Ran ` is a... |
| lanfval 50643 | Value of the function gene... |
| ranfval 50644 | Value of the function gene... |
| lanpropd 50645 | If the categories have the... |
| ranpropd 50646 | If the categories have the... |
| reldmlan2 50647 | The domain of ` ( P Lan E ... |
| reldmran2 50648 | The domain of ` ( P Ran E ... |
| lanval 50649 | Value of the set of left K... |
| ranval 50650 | Value of the set of right ... |
| lanrcl 50651 | Reverse closure for left K... |
| ranrcl 50652 | Reverse closure for right ... |
| rellan 50653 | The set of left Kan extens... |
| relran 50654 | The set of right Kan exten... |
| islan 50655 | A left Kan extension is a ... |
| islan2 50656 | A left Kan extension is a ... |
| lanval2 50657 | The set of left Kan extens... |
| isran 50658 | A right Kan extension is a... |
| isran2 50659 | A right Kan extension is a... |
| ranval2 50660 | The set of right Kan exten... |
| ranval3 50661 | The set of right Kan exten... |
| lanrcl2 50662 | Reverse closure for left K... |
| lanrcl3 50663 | Reverse closure for left K... |
| lanrcl4 50664 | The first component of a l... |
| lanrcl5 50665 | The second component of a ... |
| ranrcl2 50666 | Reverse closure for right ... |
| ranrcl3 50667 | Reverse closure for right ... |
| ranrcl4lem 50668 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 50669 | The first component of a r... |
| ranrcl5 50670 | The second component of a ... |
| lanup 50671 | The universal property of ... |
| ranup 50672 | The universal property of ... |
| reldmlmd 50677 | The domain of ` Limit ` is... |
| reldmcmd 50678 | The domain of ` Colimit ` ... |
| lmdfval 50679 | Function value of ` Limit ... |
| cmdfval 50680 | Function value of ` Colimi... |
| lmdrcl 50681 | Reverse closure for a limi... |
| cmdrcl 50682 | Reverse closure for a coli... |
| reldmlmd2 50683 | The domain of ` ( C Limit ... |
| reldmcmd2 50684 | The domain of ` ( C Colimi... |
| lmdfval2 50685 | The set of limits of a dia... |
| cmdfval2 50686 | The set of colimits of a d... |
| lmdpropd 50687 | If the categories have the... |
| cmdpropd 50688 | If the categories have the... |
| rellmd 50689 | The set of limits of a dia... |
| relcmd 50690 | The set of colimits of a d... |
| concl 50691 | A natural transformation f... |
| coccl 50692 | A natural transformation t... |
| concom 50693 | A cone to a diagram commut... |
| coccom 50694 | A co-cone to a diagram com... |
| islmd 50695 | The universal property of ... |
| iscmd 50696 | The universal property of ... |
| lmddu 50697 | The duality of limits and ... |
| cmddu 50698 | The duality of limits and ... |
| initocmd 50699 | Initial objects are the ob... |
| termolmd 50700 | Terminal objects are the o... |
| lmdran 50701 | To each limit of a diagram... |
| cmdlan 50702 | To each colimit of a diagr... |
| nfintd 50703 | Bound-variable hypothesis ... |
| nfiund 50704 | Bound-variable hypothesis ... |
| nfiundg 50705 | Bound-variable hypothesis ... |
| iunord 50706 | The indexed union of a col... |
| iunordi 50707 | The indexed union of a col... |
| spd 50708 | Specialization deduction, ... |
| tfis2d 50709 | Transfinite Induction Sche... |
| setrecseq 50710 | Equality theorem for set r... |
| nfsetrecs 50711 | Bound-variable hypothesis ... |
| setrec2mpt 50712 | Version of ~ setrec2 where... |
| setis 50713 | Version of ~ setrec2 expre... |
| elsetrecslem 50714 | Lemma for ~ elsetrecs . A... |
| elsetrecs 50715 | A set ` A ` is an element ... |
| setrecsss 50716 | The ` setrecs ` operator r... |
| setrecsres 50717 | A recursively generated cl... |
| vsetrec 50718 | Construct ` _V ` using set... |
| 0setrec 50719 | If a function sends the em... |
| onsetreclem1 50720 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 50721 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 50722 | Lemma for ~ onsetrec . (C... |
| onsetrec 50723 | Construct ` On ` using set... |
| elpglem1 50726 | Lemma for ~ elpg . (Contr... |
| elpglem2 50727 | Lemma for ~ elpg . (Contr... |
| elpglem3 50728 | Lemma for ~ elpg . (Contr... |
| elpg 50729 | Membership in the class of... |
| pgindlem 50730 | Lemma for ~ pgind . (Cont... |
| pgindnf 50731 | Version of ~ pgind with ex... |
| pgind 50732 | Induction on partizan game... |
| sbidd 50733 | An identity theorem for su... |
| sbidd-misc 50734 | An identity theorem for su... |
| gte-lte 50739 | Simple relationship betwee... |
| gt-lt 50740 | Simple relationship betwee... |
| gte-lteh 50741 | Relationship between ` <_ ... |
| gt-lth 50742 | Relationship between ` < `... |
| ex-gt 50743 | Simple example of ` > ` , ... |
| ex-gte 50744 | Simple example of ` >_ ` ,... |
| sinhval-named 50751 | Value of the named sinh fu... |
| coshval-named 50752 | Value of the named cosh fu... |
| tanhval-named 50753 | Value of the named tanh fu... |
| sinh-conventional 50754 | Conventional definition of... |
| sinhpcosh 50755 | Prove that ` ( sinh `` A )... |
| secval 50762 | Value of the secant functi... |
| cscval 50763 | Value of the cosecant func... |
| cotval 50764 | Value of the cotangent fun... |
| seccl 50765 | The closure of the secant ... |
| csccl 50766 | The closure of the cosecan... |
| cotcl 50767 | The closure of the cotange... |
| reseccl 50768 | The closure of the secant ... |
| recsccl 50769 | The closure of the cosecan... |
| recotcl 50770 | The closure of the cotange... |
| recsec 50771 | The reciprocal of secant i... |
| reccsc 50772 | The reciprocal of cosecant... |
| reccot 50773 | The reciprocal of cotangen... |
| rectan 50774 | The reciprocal of tangent ... |
| sec0 50775 | The value of the secant fu... |
| onetansqsecsq 50776 | Prove the tangent squared ... |
| cotsqcscsq 50777 | Prove the tangent squared ... |
| dvsec 50778 | Derivative of the secant f... |
| dvcsc 50779 | Derivative of the cosecant... |
| dvcot 50780 | Derivative of the cotangen... |
| ifnmfalse 50781 | If A is not a member of B,... |
| logb2aval 50782 | Define the value of the ` ... |
| mvlraddi 50789 | Move the right term in a s... |
| assraddsubi 50790 | Associate RHS addition-sub... |
| joinlmuladdmuli 50791 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 50792 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 50793 | Join AB-CB into (A-C) on L... |
| mvlrmuld 50794 | Move the right term in a p... |
| mvlrmuli 50795 | Move the right term in a p... |
| i2linesi 50796 | Solve for the intersection... |
| i2linesd 50797 | Solve for the intersection... |
| alimp-surprise 50798 | Demonstrate that when usin... |
| alimp-no-surprise 50799 | There is no "surprise" in ... |
| empty-surprise 50800 | Demonstrate that when usin... |
| empty-surprise2 50801 | "Prove" that false is true... |
| eximp-surprise 50802 | Show what implication insi... |
| eximp-surprise2 50803 | Show that "there exists" w... |
| dfrals2 50808 | The bounded "all some" for... |
| alsd 50809 | Introduction rule: "all s... |
| ralsd 50810 | Introduction rule for "all... |
| als1d 50811 | Deduction rule: Given "al... |
| als2d 50812 | Deduction rule: Given "al... |
| rals1d 50813 | Deduction rule: Given "al... |
| rals2d 50814 | Deduction rule: Given "al... |
| ralsn0d 50815 | Deduction rule: Given "al... |
| alsex 50816 | The consequent of an "all ... |
| ralsex 50817 | The consequent of an "all ... |
| alsbii 50818 | Congruence: equivalents ma... |
| ralsbii 50819 | Congruence for "all some" ... |
| alsbid 50820 | Deduction form of ~ alsbii... |
| nfals 50821 | Bound-variable hypothesis ... |
| nfrals 50822 | Bound-variable hypothesis ... |
| cbvals 50823 | Rule used to change bound ... |
| als-no-surprise 50824 | Demonstrate that there is ... |
| rals-no-surprise 50825 | Demonstrate that there is ... |
| ralrals 50826 | If the universal part of a... |
| rexrals 50827 | If a member of ` A ` satis... |
| alsanmo 50828 | An "all some" statement co... |
| ralsanmo 50829 | An "all some" statement re... |
| alsralrex 50830 | The general "all some" qua... |
| alsraln0 50831 | The general "all some" qua... |
| ralals 50832 | If ` ph ` holds for every ... |
| rexals 50833 | If some ` x ` in ` A ` sat... |
| n0als 50834 | If ` A ` is not empty, the... |
| 2alsraln0 50835 | Nested general "all some" ... |
| 2alsraln0id 50836 | Nested general "all some" ... |
| dfralseu2 50841 | The bounded "all some one"... |
| alseuals 50842 | "All some one" implies "al... |
| ralseurals 50843 | "All some one" restricted ... |
| alseud 50844 | Introduction rule: "all s... |
| ralseud 50845 | Introduction rule for "all... |
| alseu1d 50846 | Deduction rule: Given "al... |
| alseu2d 50847 | Deduction rule: Given "al... |
| ralseu1d 50848 | Deduction rule: Given "al... |
| ralseu2d 50849 | Deduction rule: Given "al... |
| alseubii 50850 | Congruence: equivalents ma... |
| ralseubii 50851 | Congruence for "all some o... |
| nfalseu 50852 | Bound-variable hypothesis ... |
| nfralseu 50853 | Bound-variable hypothesis ... |
| dfalseu2 50854 | An "all some one" statemen... |
| alseueu 50855 | "The ` ph ` is ` ps ` " im... |
| alseu-no-surprise 50856 | Demonstrate that there is ... |
| 5m4e1 50857 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 50858 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 50859 | Resolution rule. This is ... |
| testable 50860 | In classical logic all wff... |
| aacllem 50861 | Lemma for other theorems a... |
| wrdf1d 50862 | A one-to-one word maps its... |
| 1ne3 50863 | ` 1 ` is not equal to ` 3 ... |
| 2ne3 50864 | ` 2 ` is not equal to ` 3 ... |
| 1elfz13 50865 | Membership of 1 in the int... |
| 2elfz13 50866 | Membership of 2 in the int... |
| 3elfz13 50867 | Membership of 3 in the int... |
| rr3fvcl 50868 | The components of a 3-dime... |
| rr3fv1cld 50869 | First component of a 3-dim... |
| rr3fv2cld 50870 | Second component of a 3-di... |
| rr3fv3cld 50871 | Third component of a 3-dim... |
| crosspval 50876 | Value of the cross product... |
| crosspcle1d 50877 | Closure of the first compo... |
| crosspcle2d 50878 | Closure of the second comp... |
| crosspcle3d 50879 | Closure of the third compo... |
| crosspclem 50880 | Lemma for ~ crosspcld . C... |
| crosspcld 50881 | Closure of the cross produ... |
| crosspv1d 50882 | Value of the first compone... |
| crosspv2d 50883 | Value of the second compon... |
| crosspv3d 50884 | Value of the third compone... |
| crosspdot0lem 50885 | Lemma for ~ crosspdotd . ... |
| crosspdotsumlem 50886 | Lemma for ~ crosspdotd . ... |
| crosspdotd 50887 | Value of the scalar triple... |
| crosspaltd 50888 | Antisymmetry of the cross ... |
| crossp3d 50889 | The vector triple product ... |
| nellindf 50892 | A nonzero coefficient vect... |
| veronesevald 50893 | Value of the Veronese map ... |
| veronesefvcl 50894 | Every coordinate of the Ve... |
| veronesev1lem 50895 | Lemma for ~ veronesevrowd ... |
| veronesev2lem 50896 | Lemma for ~ veronesevrowd ... |
| veronesev3lem 50897 | Lemma for ~ veronesevrowd ... |
| veronesev4lem 50898 | Lemma for ~ veronesevrowd ... |
| veronesev5lem 50899 | Lemma for ~ veronesevrowd ... |
| veronesev6lem 50900 | Lemma for ~ veronesevrowd ... |
| veronesevrowd 50901 | The Veronese map at a poin... |
| veronesematbasd 50902 | The matrix whose ` i ` -th... |
| veronesematrowd 50903 | Currying the Veronese matr... |
| veronesematrowexpd 50904 | Currying the Veronese matr... |
| veroquadgsumlem 50905 | Lemma for ~ veroquadmodzer... |
| veroquadmodzerod 50906 | The columns of the Verones... |
| veroquadnolindfd 50907 | A nonzero homogeneous quad... |
| veroquaddetzerod 50908 | The Veronese matrix of six... |
| amgmwlem 50909 | Weighted version of ~ amgm... |
| amgmlemALT 50910 | Alternate proof of ~ amgml... |
| amgmw2d 50911 | Weighted arithmetic-geomet... |
| young2d 50912 | Young's inequality for ` n... |
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