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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... |
| 3orel3 1516 | Partial elimination of a t... |
| 3orel13 1517 | Elimination of two disjunc... |
| 3pm3.2ni 1518 | Triple negated disjunction... |
| an42ds 1519 | Inference exchanging the l... |
| nanan 1522 | Conjunction in terms of al... |
| dfnan2 1523 | Alternative denial in term... |
| nanor 1524 | Alternative denial in term... |
| nancom 1525 | Alternative denial is comm... |
| nannan 1526 | Nested alternative denials... |
| nanim 1527 | Implication in terms of al... |
| nannot 1528 | Negation in terms of alter... |
| nanbi 1529 | Biconditional in terms of ... |
| nanbi1 1530 | Introduce a right anti-con... |
| nanbi2 1531 | Introduce a left anti-conj... |
| nanbi12 1532 | Join two logical equivalen... |
| nanbi1i 1533 | Introduce a right anti-con... |
| nanbi2i 1534 | Introduce a left anti-conj... |
| nanbi12i 1535 | Join two logical equivalen... |
| nanbi1d 1536 | Introduce a right anti-con... |
| nanbi2d 1537 | Introduce a left anti-conj... |
| nanbi12d 1538 | Join two logical equivalen... |
| nanass 1539 | A characterization of when... |
| xnor 1542 | Two ways to write XNOR (ex... |
| xorcom 1543 | The connector ` \/_ ` is c... |
| xorass 1544 | The connector ` \/_ ` is a... |
| excxor 1545 | This tautology shows that ... |
| xor2 1546 | Two ways to express "exclu... |
| xoror 1547 | Exclusive disjunction impl... |
| xornan 1548 | Exclusive disjunction impl... |
| xornan2 1549 | XOR implies NAND (written ... |
| xorneg2 1550 | The connector ` \/_ ` is n... |
| xorneg1 1551 | The connector ` \/_ ` is n... |
| xorneg 1552 | The connector ` \/_ ` is u... |
| xorbi12i 1553 | Equality property for excl... |
| xorbi12d 1554 | Equality property for excl... |
| anxordi 1555 | Conjunction distributes ov... |
| xorexmid 1556 | Exclusive-or variant of th... |
| norcom 1559 | The connector ` -\/ ` is c... |
| nornot 1560 | ` -. ` is expressible via ... |
| noran 1561 | ` /\ ` is expressible via ... |
| noror 1562 | ` \/ ` is expressible via ... |
| norasslem1 1563 | This lemma shows the equiv... |
| norasslem2 1564 | This lemma specializes ~ b... |
| norasslem3 1565 | This lemma specializes ~ b... |
| norass 1566 | A characterization of when... |
| trujust 1571 | Soundness justification th... |
| tru 1573 | The truth value ` T. ` is ... |
| dftru2 1574 | An alternate definition of... |
| trut 1575 | A proposition is equivalen... |
| mptru 1576 | Eliminate ` T. ` as an ant... |
| tbtru 1577 | A proposition is equivalen... |
| bitru 1578 | A theorem is equivalent to... |
| trud 1579 | Anything implies ` T. ` . ... |
| truan 1580 | True can be removed from a... |
| fal 1583 | The truth value ` F. ` is ... |
| nbfal 1584 | The negation of a proposit... |
| bifal 1585 | A contradiction is equival... |
| falim 1586 | The truth value ` F. ` imp... |
| falimd 1587 | The truth value ` F. ` imp... |
| dfnot 1588 | Given falsum ` F. ` , we c... |
| inegd 1589 | Negation introduction rule... |
| efald 1590 | Deduction based on reducti... |
| pm2.21fal 1591 | If a wff and its negation ... |
| truimtru 1592 | A ` -> ` identity. (Contr... |
| truimfal 1593 | A ` -> ` identity. (Contr... |
| falimtru 1594 | A ` -> ` identity. (Contr... |
| falimfal 1595 | A ` -> ` identity. (Contr... |
| nottru 1596 | A ` -. ` identity. (Contr... |
| notfal 1597 | A ` -. ` identity. (Contr... |
| trubitru 1598 | A ` <-> ` identity. (Cont... |
| falbitru 1599 | A ` <-> ` identity. (Cont... |
| trubifal 1600 | A ` <-> ` identity. (Cont... |
| falbifal 1601 | A ` <-> ` identity. (Cont... |
| truantru 1602 | A ` /\ ` identity. (Contr... |
| truanfal 1603 | A ` /\ ` identity. (Contr... |
| falantru 1604 | A ` /\ ` identity. (Contr... |
| falanfal 1605 | A ` /\ ` identity. (Contr... |
| truortru 1606 | A ` \/ ` identity. (Contr... |
| truorfal 1607 | A ` \/ ` identity. (Contr... |
| falortru 1608 | A ` \/ ` identity. (Contr... |
| falorfal 1609 | A ` \/ ` identity. (Contr... |
| trunantru 1610 | A ` -/\ ` identity. (Cont... |
| trunanfal 1611 | A ` -/\ ` identity. (Cont... |
| falnantru 1612 | A ` -/\ ` identity. (Cont... |
| falnanfal 1613 | A ` -/\ ` identity. (Cont... |
| truxortru 1614 | A ` \/_ ` identity. (Cont... |
| truxorfal 1615 | A ` \/_ ` identity. (Cont... |
| falxortru 1616 | A ` \/_ ` identity. (Cont... |
| falxorfal 1617 | A ` \/_ ` identity. (Cont... |
| trunortru 1618 | A ` -\/ ` identity. (Cont... |
| trunorfal 1619 | A ` -\/ ` identity. (Cont... |
| falnortru 1620 | A ` -\/ ` identity. (Cont... |
| falnorfal 1621 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1624 | Equality theorem for the a... |
| hadbi123i 1625 | Equality theorem for the a... |
| hadass 1626 | Associative law for the ad... |
| hadbi 1627 | The adder sum is the same ... |
| hadcoma 1628 | Commutative law for the ad... |
| hadcomb 1629 | Commutative law for the ad... |
| hadrot 1630 | Rotation law for the adder... |
| hadnot 1631 | The adder sum distributes ... |
| had1 1632 | If the first input is true... |
| had0 1633 | If the first input is fals... |
| had1OLD 1634 | Obsolete version of ~ had1... |
| had0OLD 1635 | Obsolete version of ~ had0... |
| hadifp 1636 | The value of the adder sum... |
| hadifpOLD 1637 | Obsolete version of ~ hadi... |
| cador 1640 | The adder carry in disjunc... |
| cadan 1641 | The adder carry in conjunc... |
| cadbi123d 1642 | Equality theorem for the a... |
| cadbi123i 1643 | Equality theorem for the a... |
| cadcoma 1644 | Commutative law for the ad... |
| cadcomb 1645 | Commutative law for the ad... |
| cadrot 1646 | Rotation law for the adder... |
| cadnot 1647 | The adder carry distribute... |
| cad11 1648 | If (at least) two inputs a... |
| cad1 1649 | If one input is true, then... |
| cad0 1650 | If one input is false, the... |
| cadifp 1651 | The value of the carry is,... |
| cadtru 1652 | The adder carry is true as... |
| minimp 1653 | A single axiom for minimal... |
| minimp-syllsimp 1654 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1655 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1656 | Derivation of a commuted f... |
| minimp-ax2 1657 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1658 | Derivation of ~ pm2.43 (al... |
| impsingle 1659 | The shortest single axiom ... |
| impsingle-step4 1660 | Derivation of impsingle-st... |
| impsingle-step8 1661 | Derivation of impsingle-st... |
| impsingle-ax1 1662 | Derivation of impsingle-ax... |
| impsingle-step15 1663 | Derivation of impsingle-st... |
| impsingle-step18 1664 | Derivation of impsingle-st... |
| impsingle-step19 1665 | Derivation of impsingle-st... |
| impsingle-step20 1666 | Derivation of impsingle-st... |
| impsingle-step21 1667 | Derivation of impsingle-st... |
| impsingle-step22 1668 | Derivation of impsingle-st... |
| impsingle-step25 1669 | Derivation of impsingle-st... |
| impsingle-imim1 1670 | Derivation of impsingle-im... |
| impsingle-peirce 1671 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1672 | Derivation of ~ ax-2 from ... |
| meredith 1673 | Carew Meredith's sole axio... |
| merlem1 1674 | Step 3 of Meredith's proof... |
| merlem2 1675 | Step 4 of Meredith's proof... |
| merlem3 1676 | Step 7 of Meredith's proof... |
| merlem4 1677 | Step 8 of Meredith's proof... |
| merlem5 1678 | Step 11 of Meredith's proo... |
| merlem6 1679 | Step 12 of Meredith's proo... |
| merlem7 1680 | Between steps 14 and 15 of... |
| merlem8 1681 | Step 15 of Meredith's proo... |
| merlem9 1682 | Step 18 of Meredith's proo... |
| merlem10 1683 | Step 19 of Meredith's proo... |
| merlem11 1684 | Step 20 of Meredith's proo... |
| merlem12 1685 | Step 28 of Meredith's proo... |
| merlem13 1686 | Step 35 of Meredith's proo... |
| luk-1 1687 | 1 of 3 axioms for proposit... |
| luk-2 1688 | 2 of 3 axioms for proposit... |
| luk-3 1689 | 3 of 3 axioms for proposit... |
| luklem1 1690 | Used to rederive standard ... |
| luklem2 1691 | Used to rederive standard ... |
| luklem3 1692 | Used to rederive standard ... |
| luklem4 1693 | Used to rederive standard ... |
| luklem5 1694 | Used to rederive standard ... |
| luklem6 1695 | Used to rederive standard ... |
| luklem7 1696 | Used to rederive standard ... |
| luklem8 1697 | Used to rederive standard ... |
| ax1 1698 | Standard propositional axi... |
| ax2 1699 | Standard propositional axi... |
| ax3 1700 | Standard propositional axi... |
| nic-dfim 1701 | This theorem "defines" imp... |
| nic-dfneg 1702 | This theorem "defines" neg... |
| nic-mp 1703 | Derive Nicod's rule of mod... |
| nic-mpALT 1704 | A direct proof of ~ nic-mp... |
| nic-ax 1705 | Nicod's axiom derived from... |
| nic-axALT 1706 | A direct proof of ~ nic-ax... |
| nic-imp 1707 | Inference for ~ nic-mp usi... |
| nic-idlem1 1708 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1709 | Lemma for ~ nic-id . Infe... |
| nic-id 1710 | Theorem ~ id expressed wit... |
| nic-swap 1711 | The connector ` -/\ ` is s... |
| nic-isw1 1712 | Inference version of ~ nic... |
| nic-isw2 1713 | Inference for swapping nes... |
| nic-iimp1 1714 | Inference version of ~ nic... |
| nic-iimp2 1715 | Inference version of ~ nic... |
| nic-idel 1716 | Inference to remove the tr... |
| nic-ich 1717 | Chained inference. (Contr... |
| nic-idbl 1718 | Double the terms. Since d... |
| nic-bijust 1719 | Biconditional justificatio... |
| nic-bi1 1720 | Inference to extract one s... |
| nic-bi2 1721 | Inference to extract the o... |
| nic-stdmp 1722 | Derive the standard modus ... |
| nic-luk1 1723 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1724 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1725 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1726 | This alternative axiom for... |
| lukshefth1 1727 | Lemma for ~ renicax . (Co... |
| lukshefth2 1728 | Lemma for ~ renicax . (Co... |
| renicax 1729 | A rederivation of ~ nic-ax... |
| tbw-bijust 1730 | Justification for ~ tbw-ne... |
| tbw-negdf 1731 | The definition of negation... |
| tbw-ax1 1732 | The first of four axioms i... |
| tbw-ax2 1733 | The second of four axioms ... |
| tbw-ax3 1734 | The third of four axioms i... |
| tbw-ax4 1735 | The fourth of four axioms ... |
| tbwsyl 1736 | Used to rederive the Lukas... |
| tbwlem1 1737 | Used to rederive the Lukas... |
| tbwlem2 1738 | Used to rederive the Lukas... |
| tbwlem3 1739 | Used to rederive the Lukas... |
| tbwlem4 1740 | Used to rederive the Lukas... |
| tbwlem5 1741 | Used to rederive the Lukas... |
| re1luk1 1742 | ~ luk-1 derived from the T... |
| re1luk2 1743 | ~ luk-2 derived from the T... |
| re1luk3 1744 | ~ luk-3 derived from the T... |
| merco1 1745 | A single axiom for proposi... |
| merco1lem1 1746 | Used to rederive the Tarsk... |
| retbwax4 1747 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1748 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1749 | Used to rederive the Tarsk... |
| merco1lem3 1750 | Used to rederive the Tarsk... |
| merco1lem4 1751 | Used to rederive the Tarsk... |
| merco1lem5 1752 | Used to rederive the Tarsk... |
| merco1lem6 1753 | Used to rederive the Tarsk... |
| merco1lem7 1754 | Used to rederive the Tarsk... |
| retbwax3 1755 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1756 | Used to rederive the Tarsk... |
| merco1lem9 1757 | Used to rederive the Tarsk... |
| merco1lem10 1758 | Used to rederive the Tarsk... |
| merco1lem11 1759 | Used to rederive the Tarsk... |
| merco1lem12 1760 | Used to rederive the Tarsk... |
| merco1lem13 1761 | Used to rederive the Tarsk... |
| merco1lem14 1762 | Used to rederive the Tarsk... |
| merco1lem15 1763 | Used to rederive the Tarsk... |
| merco1lem16 1764 | Used to rederive the Tarsk... |
| merco1lem17 1765 | Used to rederive the Tarsk... |
| merco1lem18 1766 | Used to rederive the Tarsk... |
| retbwax1 1767 | ~ tbw-ax1 rederived from ~... |
| merco2 1768 | A single axiom for proposi... |
| mercolem1 1769 | Used to rederive the Tarsk... |
| mercolem2 1770 | Used to rederive the Tarsk... |
| mercolem3 1771 | Used to rederive the Tarsk... |
| mercolem4 1772 | Used to rederive the Tarsk... |
| mercolem5 1773 | Used to rederive the Tarsk... |
| mercolem6 1774 | Used to rederive the Tarsk... |
| mercolem7 1775 | Used to rederive the Tarsk... |
| mercolem8 1776 | Used to rederive the Tarsk... |
| re1tbw1 1777 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1778 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1779 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1780 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1781 | Justification for ~ rb-imd... |
| rb-imdf 1782 | The definition of implicat... |
| anmp 1783 | Modus ponens for ` { \/ , ... |
| rb-ax1 1784 | The first of four axioms i... |
| rb-ax2 1785 | The second of four axioms ... |
| rb-ax3 1786 | The third of four axioms i... |
| rb-ax4 1787 | The fourth of four axioms ... |
| rbsyl 1788 | Used to rederive the Lukas... |
| rblem1 1789 | Used to rederive the Lukas... |
| rblem2 1790 | Used to rederive the Lukas... |
| rblem3 1791 | Used to rederive the Lukas... |
| rblem4 1792 | Used to rederive the Lukas... |
| rblem5 1793 | Used to rederive the Lukas... |
| rblem6 1794 | Used to rederive the Lukas... |
| rblem7 1795 | Used to rederive the Lukas... |
| re1axmp 1796 | ~ ax-mp derived from Russe... |
| re2luk1 1797 | ~ luk-1 derived from Russe... |
| re2luk2 1798 | ~ luk-2 derived from Russe... |
| re2luk3 1799 | ~ luk-3 derived from Russe... |
| mptnan 1800 | Modus ponendo tollens 1, o... |
| mptxor 1801 | Modus ponendo tollens 2, o... |
| mtpor 1802 | Modus tollendo ponens (inc... |
| mtpxor 1803 | Modus tollendo ponens (ori... |
| stoic1a 1804 | Stoic logic Thema 1 (part ... |
| stoic1b 1805 | Stoic logic Thema 1 (part ... |
| stoic2a 1806 | Stoic logic Thema 2 versio... |
| stoic2b 1807 | Stoic logic Thema 2 versio... |
| stoic3 1808 | Stoic logic Thema 3. Stat... |
| stoic4a 1809 | Stoic logic Thema 4 versio... |
| stoic4b 1810 | Stoic logic Thema 4 versio... |
| alnex 1813 | Universal quantification o... |
| eximal 1814 | An equivalence between an ... |
| nf2 1817 | Alternate definition of no... |
| nf3 1818 | Alternate definition of no... |
| nf4 1819 | Alternate definition of no... |
| nfi 1820 | Deduce that ` x ` is not f... |
| nfri 1821 | Consequence of the definit... |
| nfd 1822 | Deduce that ` x ` is not f... |
| nfrd 1823 | Consequence of the definit... |
| nftht 1824 | Closed form of ~ nfth . (... |
| nfntht 1825 | Closed form of ~ nfnth . ... |
| nfntht2 1826 | Closed form of ~ nfnth . ... |
| gen2 1828 | Generalization applied twi... |
| mpg 1829 | Modus ponens combined with... |
| mpgbi 1830 | Modus ponens on biconditio... |
| mpgbir 1831 | Modus ponens on biconditio... |
| nex 1832 | Generalization rule for ne... |
| nfth 1833 | No variable is (effectivel... |
| nfnth 1834 | No variable is (effectivel... |
| hbth 1835 | No variable is (effectivel... |
| nftru 1836 | The true constant has no f... |
| nffal 1837 | The false constant has no ... |
| sptruw 1838 | Version of ~ sp when ` ph ... |
| altru 1839 | For all sets, ` T. ` is tr... |
| alfal 1840 | For all sets, ` -. F. ` is... |
| alim 1842 | Restatement of Axiom ~ ax-... |
| alimi 1843 | Inference quantifying both... |
| 2alimi 1844 | Inference doubly quantifyi... |
| ala1 1845 | Add an antecedent in a uni... |
| al2im 1846 | Closed form of ~ al2imi . ... |
| al2imi 1847 | Inference quantifying ante... |
| alanimi 1848 | Variant of ~ al2imi with c... |
| alimdh 1849 | Deduction form of Theorem ... |
| albi 1850 | Theorem 19.15 of [Margaris... |
| albii 1851 | Inference adding universal... |
| 2albii 1852 | Inference adding two unive... |
| 3albii 1853 | Inference adding three uni... |
| sylgt 1854 | Closed form of ~ sylg . (... |
| sylg 1855 | A syllogism combined with ... |
| alrimih 1856 | Inference form of Theorem ... |
| hbxfrbi 1857 | A utility lemma to transfe... |
| alex 1858 | Universal quantifier in te... |
| exnal 1859 | Existential quantification... |
| 2nalexn 1860 | Part of theorem *11.5 in [... |
| 2exnaln 1861 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1862 | Theorem *11.25 in [Whitehe... |
| alimex 1863 | An equivalence between an ... |
| aleximi 1864 | A variant of ~ al2imi : in... |
| alexbii 1865 | Biconditional form of ~ al... |
| exim 1866 | Theorem 19.22 of [Margaris... |
| eximi 1867 | Inference adding existenti... |
| 2eximi 1868 | Inference adding two exist... |
| eximii 1869 | Inference associated with ... |
| exa1 1870 | Add an antecedent in an ex... |
| 19.38 1871 | Theorem 19.38 of [Margaris... |
| 19.38a 1872 | Under a nonfreeness hypoth... |
| 19.38b 1873 | Under a nonfreeness hypoth... |
| imnang 1874 | Quantified implication in ... |
| alinexa 1875 | A transformation of quanti... |
| exnalimn 1876 | Existential quantification... |
| alexn 1877 | A relationship between two... |
| 2exnexn 1878 | Theorem *11.51 in [Whitehe... |
| exbi 1879 | Theorem 19.18 of [Margaris... |
| exbii 1880 | Inference adding existenti... |
| 2exbii 1881 | Inference adding two exist... |
| 3exbii 1882 | Inference adding three exi... |
| nfbiit 1883 | Equivalence theorem for th... |
| nfbii 1884 | Equality theorem for the n... |
| nfxfr 1885 | A utility lemma to transfe... |
| nfxfrd 1886 | A utility lemma to transfe... |
| nfnbi 1887 | A variable is nonfree in a... |
| nfnt 1888 | If a variable is nonfree i... |
| nfn 1889 | Inference associated with ... |
| nfnd 1890 | Deduction associated with ... |
| exanali 1891 | A transformation of quanti... |
| 2exanali 1892 | Theorem *11.521 in [Whiteh... |
| exancom 1893 | Commutation of conjunction... |
| exan 1894 | Place a conjunct in the sc... |
| alrimdh 1895 | Deduction form of Theorem ... |
| eximdh 1896 | Deduction from Theorem 19.... |
| nexdh 1897 | Deduction for generalizati... |
| albidh 1898 | Formula-building rule for ... |
| exbidh 1899 | Formula-building rule for ... |
| exsimpl 1900 | Simplification of an exist... |
| exsimpr 1901 | Simplification of an exist... |
| 19.26 1902 | Theorem 19.26 of [Margaris... |
| 19.26-2 1903 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1904 | Theorem ~ 19.26 with tripl... |
| 19.29 1905 | Theorem 19.29 of [Margaris... |
| 19.29r 1906 | Variation of ~ 19.29 . (C... |
| 19.29r2 1907 | Variation of ~ 19.29r with... |
| 19.29x 1908 | Variation of ~ 19.29 with ... |
| 19.35 1909 | Theorem 19.35 of [Margaris... |
| 19.35i 1910 | Inference associated with ... |
| 19.35ri 1911 | Inference associated with ... |
| 19.25 1912 | Theorem 19.25 of [Margaris... |
| 19.30 1913 | Theorem 19.30 of [Margaris... |
| 19.43 1914 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1915 | Obsolete proof of ~ 19.43 ... |
| 19.33 1916 | Theorem 19.33 of [Margaris... |
| 19.33b 1917 | The antecedent provides a ... |
| 19.40 1918 | Theorem 19.40 of [Margaris... |
| 19.40-2 1919 | Theorem *11.42 in [Whitehe... |
| 19.40b 1920 | The antecedent provides a ... |
| albiim 1921 | Split a biconditional and ... |
| 2albiim 1922 | Split a biconditional and ... |
| exintrbi 1923 | Add/remove a conjunct in t... |
| exintr 1924 | Introduce a conjunct in th... |
| alsyl 1925 | Universally quantified and... |
| nfimd 1926 | If in a context ` x ` is n... |
| nfimt 1927 | Closed form of ~ nfim and ... |
| nfim 1928 | If ` x ` is not free in ` ... |
| nfand 1929 | If in a context ` x ` is n... |
| nf3and 1930 | Deduction form of bound-va... |
| nfan 1931 | If ` x ` is not free in ` ... |
| nfnan 1932 | If ` x ` is not free in ` ... |
| nf3an 1933 | If ` x ` is not free in ` ... |
| nfbid 1934 | If in a context ` x ` is n... |
| nfbi 1935 | If ` x ` is not free in ` ... |
| nfor 1936 | If ` x ` is not free in ` ... |
| nf3or 1937 | If ` x ` is not free in ` ... |
| empty 1938 | Two characterizations of t... |
| emptyex 1939 | On the empty domain, any e... |
| emptyal 1940 | On the empty domain, any u... |
| emptynf 1941 | On the empty domain, any v... |
| ax5d 1943 | Version of ~ ax-5 with ant... |
| ax5e 1944 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1945 | If a formula holds for som... |
| nfv 1946 | If ` x ` is not present in... |
| nfvd 1947 | ~ nfv with antecedent. Us... |
| alimdv 1948 | Deduction form of Theorem ... |
| eximdv 1949 | Deduction form of Theorem ... |
| 2alimdv 1950 | Deduction form of Theorem ... |
| 2eximdv 1951 | Deduction form of Theorem ... |
| albidv 1952 | Formula-building rule for ... |
| exbidv 1953 | Formula-building rule for ... |
| nfbidv 1954 | An equality theorem for no... |
| 2albidv 1955 | Formula-building rule for ... |
| 2exbidv 1956 | Formula-building rule for ... |
| 3exbidv 1957 | Formula-building rule for ... |
| 4exbidv 1958 | Formula-building rule for ... |
| alrimiv 1959 | Inference form of Theorem ... |
| alrimivv 1960 | Inference form of Theorem ... |
| alrimdv 1961 | Deduction form of Theorem ... |
| exlimiv 1962 | Inference form of Theorem ... |
| exlimiiv 1963 | Inference (Rule C) associa... |
| exlimivv 1964 | Inference form of Theorem ... |
| exlimdv 1965 | Deduction form of Theorem ... |
| exlimdvv 1966 | Deduction form of Theorem ... |
| exlimddv 1967 | Existential elimination ru... |
| nexdv 1968 | Deduction for generalizati... |
| 2ax5 1969 | Quantification of two vari... |
| stdpc5v 1970 | Version of ~ stdpc5 with a... |
| 19.21v 1971 | Version of ~ 19.21 with a ... |
| 19.32v 1972 | Version of ~ 19.32 with a ... |
| 19.31v 1973 | Version of ~ 19.31 with a ... |
| 19.23v 1974 | Version of ~ 19.23 with a ... |
| 19.23vv 1975 | Theorem ~ 19.23v extended ... |
| pm11.53v 1976 | Version of ~ pm11.53 with ... |
| 19.36imv 1977 | One direction of ~ 19.36v ... |
| 19.36iv 1978 | Inference associated with ... |
| 19.37imv 1979 | One direction of ~ 19.37v ... |
| 19.37iv 1980 | Inference associated with ... |
| 19.41v 1981 | Version of ~ 19.41 with a ... |
| 19.41vv 1982 | Version of ~ 19.41 with tw... |
| 19.41vvv 1983 | Version of ~ 19.41 with th... |
| 19.41vvvv 1984 | Version of ~ 19.41 with fo... |
| 19.42v 1985 | Version of ~ 19.42 with a ... |
| exdistr 1986 | Distribution of existentia... |
| exdistrv 1987 | Distribute a pair of exist... |
| 4exdistrv 1988 | Distribute two pairs of ex... |
| 19.42vv 1989 | Version of ~ 19.42 with tw... |
| exdistr2 1990 | Distribution of existentia... |
| 19.42vvv 1991 | Version of ~ 19.42 with th... |
| 3exdistr 1992 | Distribution of existentia... |
| 4exdistr 1993 | Distribution of existentia... |
| weq 1994 | Extend wff definition to i... |
| speimfw 1995 | Specialization, with addit... |
| speimfwALT 1996 | Alternate proof of ~ speim... |
| spimfw 1997 | Specialization, with addit... |
| ax12i 1998 | Inference that has ~ ax-12... |
| ax6v 2000 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 2001 | At least one individual ex... |
| spimw 2002 | Specialization. Lemma 8 o... |
| spimew 2003 | Existential introduction, ... |
| speiv 2004 | Inference from existential... |
| speivw 2005 | Version of ~ spei with a d... |
| exgen 2006 | Rule of existential genera... |
| extru 2007 | There exists a variable su... |
| 19.2 2008 | Theorem 19.2 of [Margaris]... |
| 19.2d 2009 | Deduction associated with ... |
| 19.8w 2010 | Weak version of ~ 19.8a an... |
| spnfw 2011 | Weak version of ~ sp . Us... |
| spfalw 2012 | Version of ~ sp when ` ph ... |
| spvw 2013 | Version of ~ sp when ` x `... |
| 19.3v 2014 | Version of ~ 19.3 with a d... |
| 19.8v 2015 | Version of ~ 19.8a with a ... |
| 19.9v 2016 | Version of ~ 19.9 with a d... |
| spimevw 2017 | Existential introduction, ... |
| spimvw 2018 | A weak form of specializat... |
| spsv 2019 | Generalization of antecede... |
| spvv 2020 | Specialization, using impl... |
| chvarvv 2021 | Implicit substitution of `... |
| 19.39 2022 | Theorem 19.39 of [Margaris... |
| 19.24 2023 | Theorem 19.24 of [Margaris... |
| 19.34 2024 | Theorem 19.34 of [Margaris... |
| 19.36v 2025 | Version of ~ 19.36 with a ... |
| 19.12vvv 2026 | Version of ~ 19.12vv with ... |
| 19.27v 2027 | Version of ~ 19.27 with a ... |
| 19.28v 2028 | Version of ~ 19.28 with a ... |
| 19.37v 2029 | Version of ~ 19.37 with a ... |
| 19.44v 2030 | Version of ~ 19.44 with a ... |
| 19.45v 2031 | Version of ~ 19.45 with a ... |
| equs4v 2032 | Version of ~ equs4 with a ... |
| alequexv 2033 | Version of ~ equs4v with i... |
| exsbim 2034 | One direction of the equiv... |
| equsv 2035 | If a formula does not cont... |
| equsalvw 2036 | Version of ~ equsalv with ... |
| equsexvw 2037 | Version of ~ equsexv with ... |
| cbvaliw 2038 | Change bound variable. Us... |
| cbvalivw 2039 | Change bound variable. Us... |
| ax7v 2041 | Weakened version of ~ ax-7... |
| ax7v1 2042 | First of two weakened vers... |
| ax7v2 2043 | Second of two weakened ver... |
| equid 2044 | Identity law for equality.... |
| nfequid 2045 | Bound-variable hypothesis ... |
| equcomiv 2046 | Weaker form of ~ equcomi w... |
| ax6evr 2047 | A commuted form of ~ ax6ev... |
| ax7 2048 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2049 | Commutative law for equali... |
| equcom 2050 | Commutative law for equali... |
| equcomd 2051 | Deduction form of ~ equcom... |
| equcoms 2052 | An inference commuting equ... |
| equtr 2053 | A transitive law for equal... |
| equtrr 2054 | A transitive law for equal... |
| equeuclr 2055 | Commuted version of ~ eque... |
| equeucl 2056 | Equality is a left-Euclide... |
| equequ1 2057 | An equivalence law for equ... |
| equequ2 2058 | An equivalence law for equ... |
| equtr2 2059 | Equality is a left-Euclide... |
| stdpc6 2060 | One of the two equality ax... |
| equvinv 2061 | A variable introduction la... |
| equvinva 2062 | A modified version of the ... |
| equvelv 2063 | A biconditional form of ~ ... |
| ax13b 2064 | An equivalence between two... |
| spfw 2065 | Weak version of ~ sp . Us... |
| spw 2066 | Weak version of the specia... |
| cbvalw 2067 | Change bound variable. Us... |
| cbvalvw 2068 | Change bound variable. Us... |
| cbvexvw 2069 | Change bound variable. Us... |
| cbvaldvaw 2070 | Rule used to change the bo... |
| cbvexdvaw 2071 | Rule used to change the bo... |
| cbval2vw 2072 | Rule used to change bound ... |
| cbvex2vw 2073 | Rule used to change bound ... |
| cbvex4vw 2074 | Rule used to change bound ... |
| alcomimw 2075 | Weak version of ~ ax-11 . ... |
| excomimw 2076 | Weak version of ~ excomim ... |
| alcomw 2077 | Weak version of ~ alcom an... |
| excomw 2078 | Weak version of ~ excom an... |
| hbn1fw 2079 | Weak version of ~ ax-10 fr... |
| hbn1w 2080 | Weak version of ~ hbn1 . ... |
| hba1w 2081 | Weak version of ~ hba1 . ... |
| hbe1w 2082 | Weak version of ~ hbe1 . ... |
| hbalw 2083 | Weak version of ~ hbal . ... |
| 19.8aw 2084 | If a formula is true, then... |
| exexw 2085 | Existential quantification... |
| spaev 2086 | A special instance of ~ sp... |
| cbvaev 2087 | Change bound variable in a... |
| aevlem0 2088 | Lemma for ~ aevlem . Inst... |
| aevlem 2089 | Lemma for ~ aev and ~ axc1... |
| aeveq 2090 | The antecedent ` A. x x = ... |
| aev 2091 | A "distinctor elimination"... |
| aev2 2092 | A version of ~ aev with tw... |
| hbaev 2093 | All variables are effectiv... |
| naev 2094 | If some set variables can ... |
| naev2 2095 | Generalization of ~ hbnaev... |
| hbnaev 2096 | Any variable is free in ` ... |
| sbjust 2097 | Justification theorem for ... |
| dfsb 2100 | Simplify definition ~ df-s... |
| sbtlem 2101 | In the case of ~ sbt , the... |
| sbt 2102 | A substitution into a theo... |
| sbtru 2103 | The result of substituting... |
| stdpc4 2104 | The specialization axiom o... |
| sbtALT 2105 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2106 | A double specialization us... |
| sbi1 2107 | Distribute substitution ov... |
| spsbim 2108 | Distribute substitution ov... |
| spsbbi 2109 | Biconditional property for... |
| sbimi 2110 | Distribute substitution ov... |
| sb2imi 2111 | Distribute substitution ov... |
| sbbii 2112 | Infer substitution into bo... |
| 2sbbii 2113 | Infer double substitution ... |
| sbimdv 2114 | Deduction substituting bot... |
| sbbidv 2115 | Deduction substituting bot... |
| sban 2116 | Conjunction inside and out... |
| sb3an 2117 | Threefold conjunction insi... |
| spsbe 2118 | Existential generalization... |
| sbequ 2119 | Equality property for subs... |
| sbequiOLD 2120 | Obsolete version of ~ sbeq... |
| sb6 2121 | Alternate definition of su... |
| 2sb6 2122 | Equivalence for double sub... |
| sb1v 2123 | One direction of ~ sb5 , p... |
| sbv 2124 | Substitution for a variabl... |
| sbcom4 2125 | Commutativity law for subs... |
| pm11.07 2126 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2127 | Substitution in an implica... |
| sbrimvwOLD 2128 | Obsolete version of ~ sbri... |
| sbbiiev 2129 | An equivalence of substitu... |
| sbievw 2130 | Conversion of implicit sub... |
| sbiedvw 2131 | Conversion of implicit sub... |
| 2sbievw 2132 | Conversion of double impli... |
| sbcom3vv 2133 | Substituting ` y ` for ` x... |
| sbievw2 2134 | ~ sbievw applied twice, av... |
| sbco2vv 2135 | A composition law for subs... |
| cbvsbv 2136 | Change the bound variable ... |
| sbco4lem 2137 | Lemma for ~ sbco4 . It re... |
| sbco4 2138 | Two ways of exchanging two... |
| equsb3 2139 | Substitution in an equalit... |
| equsb3r 2140 | Substitution applied to th... |
| equsb1v 2141 | Substitution applied to an... |
| nsb 2142 | Any substitution in an alw... |
| sbn1 2143 | One direction of ~ sbn , u... |
| wel 2145 | Extend wff definition to i... |
| ax8v 2147 | Weakened version of ~ ax-8... |
| ax8v1 2148 | First of two weakened vers... |
| ax8v2 2149 | Second of two weakened ver... |
| ax8 2150 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2151 | An identity law for the no... |
| elsb1 2152 | Substitution for the first... |
| cleljust 2153 | When the class variables i... |
| ax9v 2155 | Weakened version of ~ ax-9... |
| ax9v1 2156 | First of two weakened vers... |
| ax9v2 2157 | Second of two weakened ver... |
| ax9 2158 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2159 | An identity law for the no... |
| elequ2g 2160 | A form of ~ elequ2 with a ... |
| elsb2 2161 | Substitution for the secon... |
| elequ12 2162 | An identity law for the no... |
| ru0 2163 | The FOL statement used in ... |
| ax6dgen 2164 | Tarski's system uses the w... |
| ax10w 2165 | Weak version of ~ ax-10 fr... |
| ax11w 2166 | Weak version of ~ ax-11 fr... |
| ax11dgen 2167 | Degenerate instance of ~ a... |
| ax12wlem 2168 | Lemma for weak version of ... |
| ax12w 2169 | Weak version of ~ ax-12 fr... |
| ax12dgen 2170 | Degenerate instance of ~ a... |
| ax12wdemo 2171 | Example of an application ... |
| ax13w 2172 | Weak version (principal in... |
| ax13dgen1 2173 | Degenerate instance of ~ a... |
| ax13dgen2 2174 | Degenerate instance of ~ a... |
| ax13dgen3 2175 | Degenerate instance of ~ a... |
| ax13dgen4 2176 | Degenerate instance of ~ a... |
| hbn1 2178 | Alias for ~ ax-10 to be us... |
| hbe1 2179 | The setvar ` x ` is not fr... |
| hbe1a 2180 | Dual statement of ~ hbe1 .... |
| nf5-1 2181 | One direction of ~ nf5 can... |
| nf5i 2182 | Deduce that ` x ` is not f... |
| nf5dh 2183 | Deduce that ` x ` is not f... |
| nf5dv 2184 | Apply the definition of no... |
| nfnaew 2185 | All variables are effectiv... |
| nfe1 2186 | The setvar ` x ` is not fr... |
| nfa1 2187 | The setvar ` x ` is not fr... |
| nfna1 2188 | A convenience theorem part... |
| nfia1 2189 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2190 | The setvar ` x ` is not fr... |
| modal5 2191 | The analogue in our predic... |
| nfs1v 2192 | The setvar ` x ` is not fr... |
| alcoms 2194 | Swap quantifiers in an ant... |
| alcom 2195 | Theorem 19.5 of [Margaris]... |
| alrot3 2196 | Theorem *11.21 in [Whitehe... |
| alrot4 2197 | Rotate four universal quan... |
| excom 2198 | Theorem 19.11 of [Margaris... |
| excomim 2199 | One direction of Theorem 1... |
| excom13 2200 | Swap 1st and 3rd existenti... |
| exrot3 2201 | Rotate existential quantif... |
| exrot4 2202 | Rotate existential quantif... |
| hbal 2203 | If ` x ` is not free in ` ... |
| hbald 2204 | Deduction form of bound-va... |
| sbal 2205 | Move universal quantifier ... |
| sbalv 2206 | Quantify with new variable... |
| hbsbw 2207 | If ` z ` is not free in ` ... |
| sbcom2 2208 | Commutativity law for subs... |
| nfa2 2209 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2210 | Version of ~ nfex with the... |
| nfexa2 2211 | An inner universal quantif... |
| ax12v 2213 | This is essentially Axiom ... |
| ax12v2 2214 | It is possible to remove a... |
| ax12ev2 2215 | Version of ~ ax12v2 rewrit... |
| ax12ev2c 2216 | A commuted form of ~ ax12e... |
| 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... |
| dfss3 3897 | Alternate definition of su... |
| dfss6 3898 | Alternate definition of su... |
| dfssf 3899 | Equivalence for subclass r... |
| dfss3f 3900 | Equivalence for subclass r... |
| nfss 3901 | If ` x ` is not free in ` ... |
| ssel 3902 | Membership relationships f... |
| ssel2 3903 | Membership relationships f... |
| sseli 3904 | Membership implication fro... |
| sselii 3905 | Membership inference from ... |
| sselid 3906 | Membership inference from ... |
| sseld 3907 | Membership deduction from ... |
| sselda 3908 | Membership deduction from ... |
| sseldd 3909 | Membership inference from ... |
| ssneld 3910 | If a class is not in anoth... |
| ssneldd 3911 | If an element is not in a ... |
| ssriv 3912 | Inference based on subclas... |
| ssrd 3913 | Deduction based on subclas... |
| ssrdv 3914 | Deduction based on subclas... |
| sstr2 3915 | Transitivity of subclass r... |
| sstr 3916 | Transitivity of subclass r... |
| sstri 3917 | Subclass transitivity infe... |
| sstrd 3918 | Subclass transitivity dedu... |
| sstrid 3919 | Subclass transitivity dedu... |
| sstrdi 3920 | Subclass transitivity dedu... |
| sylan9ss 3921 | A subclass transitivity de... |
| sylan9ssr 3922 | A subclass transitivity de... |
| eqss 3923 | The subclass relationship ... |
| eqssi 3924 | Infer equality from two su... |
| eqssd 3925 | Equality deduction from tw... |
| sssseq 3926 | If a class is a subclass o... |
| eqrd 3927 | Deduce equality of classes... |
| eqri 3928 | Infer equality of classes ... |
| eqelssd 3929 | Equality deduction from su... |
| ssid 3930 | Any class is a subclass of... |
| ssidd 3931 | Weakening of ~ ssid . (Co... |
| ssv 3932 | Any class is a subclass of... |
| sseq1 3933 | Equality theorem for subcl... |
| sseq2 3934 | Equality theorem for the s... |
| sseq12 3935 | Equality theorem for the s... |
| sseq1i 3936 | An equality inference for ... |
| sseq2i 3937 | An equality inference for ... |
| sseq12i 3938 | An equality inference for ... |
| sseq1d 3939 | An equality deduction for ... |
| sseq2d 3940 | An equality deduction for ... |
| sseq12d 3941 | An equality deduction for ... |
| eqsstrd 3942 | Substitution of equality i... |
| eqsstrrd 3943 | Substitution of equality i... |
| sseqtrd 3944 | Substitution of equality i... |
| sseqtrrd 3945 | Substitution of equality i... |
| eqsstrid 3946 | A chained subclass and equ... |
| eqsstrrid 3947 | A chained subclass and equ... |
| sseqtrdi 3948 | A chained subclass and equ... |
| sseqtrrdi 3949 | A chained subclass and equ... |
| sseqtrid 3950 | Subclass transitivity dedu... |
| sseqtrrid 3951 | Subclass transitivity dedu... |
| eqsstrdi 3952 | A chained subclass and equ... |
| eqsstrrdi 3953 | A chained subclass and equ... |
| eqsstri 3954 | Substitution of equality i... |
| eqsstrri 3955 | Substitution of equality i... |
| sseqtri 3956 | Substitution of equality i... |
| sseqtrri 3957 | Substitution of equality i... |
| 3sstr3i 3958 | Substitution of equality i... |
| 3sstr4i 3959 | Substitution of equality i... |
| 3sstr3g 3960 | Substitution of equality i... |
| 3sstr4g 3961 | Substitution of equality i... |
| 3sstr3d 3962 | Substitution of equality i... |
| 3sstr4d 3963 | Substitution of equality i... |
| eqimssd 3964 | Equality implies inclusion... |
| eqimsscd 3965 | Equality implies inclusion... |
| eqimss 3966 | Equality implies inclusion... |
| eqimss2 3967 | Equality implies inclusion... |
| eqimssi 3968 | Infer subclass relationshi... |
| eqimss2i 3969 | Infer subclass relationshi... |
| nssne1 3970 | Two classes are different ... |
| nssne2 3971 | Two classes are different ... |
| nss 3972 | Negation of subclass relat... |
| nssrex 3973 | Negation of subclass relat... |
| nelss 3974 | Demonstrate by witnesses t... |
| ssrexf 3975 | Restricted existential qua... |
| ssrmof 3976 | "At most one" existential ... |
| ssralv 3977 | Quantification restricted ... |
| ssrexv 3978 | Existential quantification... |
| ss2ralv 3979 | Two quantifications restri... |
| ss2rexv 3980 | Two existential quantifica... |
| ralss 3981 | Restricted universal quant... |
| rexss 3982 | Restricted existential qua... |
| ralssOLD 3983 | Obsolete version of ~ rals... |
| rexssOLD 3984 | Obsolete version of ~ rexs... |
| ss2abim 3985 | Class abstractions in a su... |
| ss2ab 3986 | Class abstractions in a su... |
| abss 3987 | Class abstraction in a sub... |
| ssab 3988 | Subclass of a class abstra... |
| ssabral 3989 | The relation for a subclas... |
| ss2abdv 3990 | Deduction of abstraction s... |
| ss2abi 3991 | Inference of abstraction s... |
| abssdv 3992 | Deduction of abstraction s... |
| abssi 3993 | Inference of abstraction s... |
| ss2rab 3994 | Restricted abstraction cla... |
| rabss 3995 | Restricted class abstracti... |
| ssrab 3996 | Subclass of a restricted c... |
| ss2rabd 3997 | Subclass of a restricted c... |
| ssrabdv 3998 | Subclass of a restricted c... |
| rabssdv 3999 | Subclass of a restricted c... |
| ss2rabdv 4000 | Deduction of restricted ab... |
| ss2rabi 4001 | Inference of restricted ab... |
| rabss2 4002 | Subclass law for restricte... |
| rabss2OLD 4003 | Obsolete version of ~ rabs... |
| ssab2 4004 | Subclass relation for the ... |
| ssrab2 4005 | Subclass relation for a re... |
| rabss3d 4006 | Subclass law for restricte... |
| ssrab3 4007 | Subclass relation for a re... |
| rabssrabd 4008 | Subclass of a restricted c... |
| ssrabeq 4009 | If the restricting class o... |
| rabssab 4010 | A restricted class is a su... |
| eqrrabd 4011 | Deduce equality with a res... |
| uniiunlem 4012 | A subset relationship usef... |
| dfpss2 4015 | Alternate definition of pr... |
| dfpss3 4016 | Alternate definition of pr... |
| psseq1 4017 | Equality theorem for prope... |
| psseq2 4018 | Equality theorem for prope... |
| psseq1i 4019 | An equality inference for ... |
| psseq2i 4020 | An equality inference for ... |
| psseq12i 4021 | An equality inference for ... |
| psseq1d 4022 | An equality deduction for ... |
| psseq2d 4023 | An equality deduction for ... |
| psseq12d 4024 | An equality deduction for ... |
| pssss 4025 | A proper subclass is a sub... |
| pssne 4026 | Two classes in a proper su... |
| pssssd 4027 | Deduce subclass from prope... |
| pssned 4028 | Proper subclasses are uneq... |
| sspss 4029 | Subclass in terms of prope... |
| pssirr 4030 | Proper subclass is irrefle... |
| pssirrOLD 4031 | Obsolete version of ~ pssi... |
| pssn2lp 4032 | Proper subclass has no 2-c... |
| sspsstri 4033 | Two ways of stating tricho... |
| ssnpss 4034 | Partial trichotomy law for... |
| psstr 4035 | Transitive law for proper ... |
| sspsstr 4036 | Transitive law for subclas... |
| psssstr 4037 | Transitive law for subclas... |
| psstrd 4038 | Proper subclass inclusion ... |
| sspsstrd 4039 | Transitivity involving sub... |
| psssstrd 4040 | Transitivity involving sub... |
| npss 4041 | A class is not a proper su... |
| ssnelpss 4042 | A subclass missing a membe... |
| ssnelpssd 4043 | Subclass inclusion with on... |
| ssexnelpss 4044 | If there is an element of ... |
| difjust 4046 | Soundness justification th... |
| dfdif2 4048 | Alternate definition of cl... |
| eldif 4049 | Expansion of membership in... |
| eldifd 4050 | If a class is in one class... |
| eldifad 4051 | If a class is in the diffe... |
| eldifbd 4052 | If a class is in the diffe... |
| elneeldif 4053 | The elements of a set diff... |
| velcomp 4054 | Characterization of setvar... |
| dfdif3 4055 | Alternate definition of cl... |
| difeq1 4056 | Equality theorem for class... |
| difeq2 4057 | Equality theorem for class... |
| difeq12 4058 | Equality theorem for class... |
| difeq1i 4059 | Inference adding differenc... |
| difeq2i 4060 | Inference adding differenc... |
| difeq12i 4061 | Equality inference for cla... |
| difeq1d 4062 | Deduction adding differenc... |
| difeq2d 4063 | Deduction adding differenc... |
| difeq12d 4064 | Equality deduction for cla... |
| difeqri 4065 | Inference from membership ... |
| nfdif 4066 | Bound-variable hypothesis ... |
| eldifi 4067 | Implication of membership ... |
| eldifn 4068 | Implication of membership ... |
| elndif 4069 | A set does not belong to a... |
| neldif 4070 | Implication of membership ... |
| difdif 4071 | Double class difference. ... |
| difss 4072 | Subclass relationship for ... |
| difssd 4073 | A difference of two classe... |
| difss2 4074 | If a class is contained in... |
| difss2d 4075 | If a class is contained in... |
| ssdifss 4076 | Preservation of a subclass... |
| ddif 4077 | Double complement under un... |
| ssconb 4078 | Contraposition law for sub... |
| sscon 4079 | Contraposition law for sub... |
| ssdif 4080 | Difference law for subsets... |
| ssdifd 4081 | If ` A ` is contained in `... |
| sscond 4082 | If ` A ` is contained in `... |
| ssdifssd 4083 | If ` A ` is contained in `... |
| ssdif2d 4084 | If ` A ` is contained in `... |
| raldifb 4085 | Restricted universal quant... |
| rexdifi 4086 | Restricted existential qua... |
| complss 4087 | Complementation reverses i... |
| compleq 4088 | Two classes are equal if a... |
| injust 4090 | Soundness justification th... |
| dfin5 4092 | Alternate definition for t... |
| elin 4093 | Expansion of membership in... |
| dfss2 4094 | Alternate definition of th... |
| dfss 4095 | Variant of subclass defini... |
| elini 4096 | Membership in an intersect... |
| elind 4097 | Deduce membership in an in... |
| elinel1 4098 | Membership in an intersect... |
| elinel2 4099 | Membership in an intersect... |
| elin2 4100 | Membership in a class defi... |
| elin1d 4101 | Elementhood in the first s... |
| elin2d 4102 | Elementhood in the first s... |
| elin3 4103 | Membership in a class defi... |
| nel1nelin 4104 | Membership in an intersect... |
| nel2nelin 4105 | Membership in an intersect... |
| incom 4106 | Commutative law for inters... |
| ineqcom 4107 | Two ways of expressing tha... |
| ineqcomi 4108 | Two ways of expressing tha... |
| ineqri 4109 | Inference from membership ... |
| ineq1 4110 | Equality theorem for inter... |
| ineq2 4111 | Equality theorem for inter... |
| ineq12 4112 | Equality theorem for inter... |
| ineq1i 4113 | Equality inference for int... |
| ineq2i 4114 | Equality inference for int... |
| ineq12i 4115 | Equality inference for int... |
| ineq1d 4116 | Equality deduction for int... |
| ineq2d 4117 | Equality deduction for int... |
| ineq12d 4118 | Equality deduction for int... |
| ineqan12d 4119 | Equality deduction for int... |
| sseqin2 4120 | A relationship between sub... |
| nfin 4121 | Bound-variable hypothesis ... |
| rabbi2dva 4122 | Deduction from a wff to a ... |
| inidm 4123 | Idempotent law for interse... |
| inass 4124 | Associative law for inters... |
| in12 4125 | A rearrangement of interse... |
| in32 4126 | A rearrangement of interse... |
| in13 4127 | A rearrangement of interse... |
| in31 4128 | A rearrangement of interse... |
| inrot 4129 | Rotate the intersection of... |
| in4 4130 | Rearrangement of intersect... |
| inindi 4131 | Intersection distributes o... |
| inindir 4132 | Intersection distributes o... |
| inss1 4133 | The intersection of two cl... |
| inss2 4134 | The intersection of two cl... |
| ssin 4135 | Subclass of intersection. ... |
| ssini 4136 | An inference showing that ... |
| ssind 4137 | A deduction showing that a... |
| ssrin 4138 | Add right intersection to ... |
| sslin 4139 | Add left intersection to s... |
| ssrind 4140 | Add right intersection to ... |
| ss2in 4141 | Intersection of subclasses... |
| ssinss1 4142 | Intersection preserves sub... |
| ssinss1OLD 4143 | Obsolete version of ~ ssin... |
| ssinss1d 4144 | Intersection preserves sub... |
| inss 4145 | Inclusion of an intersecti... |
| ralin 4146 | Restricted universal quant... |
| rexin 4147 | Restricted existential qua... |
| dfss7 4148 | Alternate definition of su... |
| unjust 4150 | Soundness justification th... |
| elun 4152 | Expansion of membership in... |
| elunnel1 4153 | A member of a union that i... |
| elunnel2 4154 | A member of a union that i... |
| uneqri 4155 | Inference from membership ... |
| unidm 4156 | Idempotent law for union o... |
| uncom 4157 | Commutative law for union ... |
| equncom 4158 | If a class equals the unio... |
| equncomi 4159 | Inference form of ~ equnco... |
| uneq1 4160 | Equality theorem for the u... |
| uneq2 4161 | Equality theorem for the u... |
| uneq12 4162 | Equality theorem for the u... |
| uneq1i 4163 | Inference adding union to ... |
| uneq2i 4164 | Inference adding union to ... |
| uneq12i 4165 | Equality inference for the... |
| uneq1d 4166 | Deduction adding union to ... |
| uneq2d 4167 | Deduction adding union to ... |
| uneq12d 4168 | Equality deduction for the... |
| nfun 4169 | Bound-variable hypothesis ... |
| unass 4170 | Associative law for union ... |
| un12 4171 | A rearrangement of union. ... |
| un23 4172 | A rearrangement of union. ... |
| un4 4173 | A rearrangement of the uni... |
| unundi 4174 | Union distributes over its... |
| unundir 4175 | Union distributes over its... |
| ssun1 4176 | Subclass relationship for ... |
| ssun2 4177 | Subclass relationship for ... |
| ssun3 4178 | Subclass law for union of ... |
| ssun4 4179 | Subclass law for union of ... |
| elun1 4180 | Membership law for union o... |
| elun2 4181 | Membership law for union o... |
| elunant 4182 | A statement is true for ev... |
| unss1 4183 | Subclass law for union of ... |
| ssequn1 4184 | A relationship between sub... |
| unss2 4185 | Subclass law for union of ... |
| unss12 4186 | Subclass law for union of ... |
| ssequn2 4187 | A relationship between sub... |
| unss 4188 | The union of two subclasse... |
| unssi 4189 | An inference showing the u... |
| unssd 4190 | A deduction showing the un... |
| unssad 4191 | If ` ( A u. B ) ` is conta... |
| unssbd 4192 | If ` ( A u. B ) ` is conta... |
| ssun 4193 | A condition that implies i... |
| rexun 4194 | Restricted existential qua... |
| ralunb 4195 | Restricted quantification ... |
| ralun 4196 | Restricted quantification ... |
| 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... |
| brsymdif 5158 | Characterization of the sy... |
| sbcbr123 5159 | Move substitution in and o... |
| sbcbr 5160 | Move substitution in and o... |
| sbcbr12g 5161 | Move substitution in and o... |
| sbcbr1g 5162 | Move substitution in and o... |
| sbcbr2g 5163 | Move substitution in and o... |
| 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... |
| eqvinot 5456 | A variable introduction la... |
| sbcop1 5457 | The proper substitution of... |
| sbcop 5458 | The proper substitution of... |
| copsexgw 5459 | Version of ~ copsexg with ... |
| copsexgwOLD 5460 | Obsolete version of ~ cops... |
| copsexg 5461 | Substitution of class ` A ... |
| cotsexgw 5462 | Substitution of class ` A ... |
| copsex2t 5463 | Closed theorem form of ~ c... |
| copsex2g 5464 | Implicit substitution infe... |
| copsex2dv 5465 | Implicit substitution dedu... |
| copsex4g 5466 | An implicit substitution i... |
| 0nelop 5467 | A property of ordered pair... |
| opwo0id 5468 | An ordered pair is equal t... |
| opeqex 5469 | Equivalence of existence i... |
| oteqex2 5470 | Equivalence of existence i... |
| oteqex 5471 | Equivalence of existence i... |
| opcom 5472 | An ordered pair commutes i... |
| moop2 5473 | "At most one" property of ... |
| opeqsng 5474 | Equivalence for an ordered... |
| opeqsn 5475 | Equivalence for an ordered... |
| opeqpr 5476 | Equivalence for an ordered... |
| snopeqop 5477 | Equivalence for an ordered... |
| propeqop 5478 | Equivalence for an ordered... |
| propssopi 5479 | If a pair of ordered pairs... |
| snopeqopsnid 5480 | Equivalence for an ordered... |
| mosubopt 5481 | "At most one" remains true... |
| mosubop 5482 | "At most one" remains true... |
| mosubott 5483 | "At most one" remains true... |
| euop2 5484 | Transfer existential uniqu... |
| euotd 5485 | Prove existential uniquene... |
| opthwiener 5486 | Justification theorem for ... |
| uniop 5487 | The union of an ordered pa... |
| uniopel 5488 | Ordered pair membership is... |
| opthhausdorff 5489 | Justification theorem for ... |
| opthhausdorff0 5490 | Justification theorem for ... |
| otsndisj 5491 | The singletons consisting ... |
| otiunsndisj 5492 | The union of singletons co... |
| iunopeqop 5493 | Implication of an ordered ... |
| iunopeqopOLD 5494 | Obsolete version of ~ iuno... |
| brsnop 5495 | Binary relation for an ord... |
| brtp 5496 | A necessary and sufficient... |
| opabidw 5497 | The law of concretion. Sp... |
| opabid 5498 | The law of concretion. Sp... |
| elopabw 5499 | Membership in a class abst... |
| elopab 5500 | Membership in a class abst... |
| rexopabb 5501 | Restricted existential qua... |
| vopelopabsb 5502 | The law of concretion in t... |
| opelopabsb 5503 | The law of concretion in t... |
| brabsb 5504 | The law of concretion in t... |
| opelopabt 5505 | Closed theorem form of ~ o... |
| opelopabga 5506 | The law of concretion. Th... |
| brabga 5507 | The law of concretion for ... |
| opelopab2a 5508 | Ordered pair membership in... |
| opelopaba 5509 | The law of concretion. Th... |
| braba 5510 | The law of concretion for ... |
| brab2d 5511 | Expressing that two sets a... |
| opelopabg 5512 | The law of concretion. Th... |
| brabg 5513 | The law of concretion for ... |
| opelopabgf 5514 | The law of concretion. Th... |
| opelopab2 5515 | Ordered pair membership in... |
| opelopab 5516 | The law of concretion. Th... |
| brab 5517 | The law of concretion for ... |
| opelopabaf 5518 | The law of concretion. Th... |
| opelopabf 5519 | The law of concretion. Th... |
| ssopab2 5520 | Equivalence of ordered pai... |
| ssopab2bw 5521 | Equivalence of ordered pai... |
| eqopab2bw 5522 | Equivalence of ordered pai... |
| ssopab2b 5523 | Equivalence of ordered pai... |
| ssopab2i 5524 | Inference of ordered pair ... |
| ssopab2dv 5525 | Inference of ordered pair ... |
| eqopab2b 5526 | Equivalence of ordered pai... |
| opabn0 5527 | Nonempty ordered pair clas... |
| opab0 5528 | Empty ordered pair class a... |
| csbopab 5529 | Move substitution into a c... |
| csbopabw 5530 | Move substitution into a c... |
| csbmpt12 5531 | Move substitution into a m... |
| csbmpt2 5532 | Move substitution into the... |
| iunopab 5533 | Move indexed union inside ... |
| elopabr 5534 | Membership in an ordered-p... |
| elopabran 5535 | Membership in an ordered-p... |
| rbropapd 5536 | Properties of a pair in an... |
| rbropap 5537 | Properties of a pair in a ... |
| 2rbropap 5538 | Properties of a pair in a ... |
| 0nelopab 5539 | The empty set is not an el... |
| brabv 5540 | If two classes are in a re... |
| pwin 5541 | The power class of the int... |
| pwssun 5542 | The power class of the uni... |
| pwun 5543 | The power class of the uni... |
| dfid4 5546 | The identity function expr... |
| dfid2 5547 | Alternate definition of th... |
| dfid3 5548 | A stronger version of ~ df... |
| epelg 5551 | The membership relation an... |
| epeli 5552 | The membership relation an... |
| epel 5553 | The membership relation an... |
| 0sn0ep 5554 | An example for the members... |
| epn0 5555 | The membership relation is... |
| poss 5560 | Subset theorem for the par... |
| poeq1 5561 | Equality theorem for parti... |
| poeq2 5562 | Equality theorem for parti... |
| poeq12d 5563 | Equality deduction for par... |
| nfpo 5564 | Bound-variable hypothesis ... |
| nfso 5565 | Bound-variable hypothesis ... |
| pocl 5566 | Characteristic properties ... |
| ispod 5567 | Sufficient conditions for ... |
| swopolem 5568 | Perform the substitutions ... |
| swopo 5569 | A strict weak order is a p... |
| poirr 5570 | A partial order is irrefle... |
| potr 5571 | A partial order is a trans... |
| po2nr 5572 | A partial order has no 2-c... |
| po3nr 5573 | A partial order has no 3-c... |
| po2ne 5574 | Two sets related by a part... |
| po0 5575 | Any relation is a partial ... |
| pofun 5576 | The inverse image of a par... |
| sopo 5577 | A strict linear order is a... |
| soss 5578 | Subset theorem for the str... |
| soeq1 5579 | Equality theorem for the s... |
| soeq2 5580 | Equality theorem for the s... |
| soeq12d 5581 | Equality deduction for tot... |
| sonr 5582 | A strict order relation is... |
| sotr 5583 | A strict order relation is... |
| sotrd 5584 | Transitivity law for stric... |
| solin 5585 | A strict order relation is... |
| so2nr 5586 | A strict order relation ha... |
| so3nr 5587 | A strict order relation ha... |
| sotric 5588 | A strict order relation sa... |
| sotrieq 5589 | Trichotomy law for strict ... |
| sotrieq2 5590 | Trichotomy law for strict ... |
| soasym 5591 | Asymmetry law for strict o... |
| sotr2 5592 | A transitivity relation. ... |
| issod 5593 | An irreflexive, transitive... |
| issoi 5594 | An irreflexive, transitive... |
| isso2i 5595 | Deduce strict ordering fro... |
| so0 5596 | Any relation is a strict o... |
| somo 5597 | A totally ordered set has ... |
| sotrine 5598 | Trichotomy law for strict ... |
| sotr3 5599 | Transitivity law for stric... |
| dffr6 5606 | Alternate definition of ~ ... |
| frd 5607 | A nonempty subset of an ` ... |
| fri 5608 | A nonempty subset of an ` ... |
| seex 5609 | The ` R ` -preimage of an ... |
| exse 5610 | Any relation on a set is s... |
| dffr2 5611 | Alternate definition of we... |
| dffr2ALT 5612 | Alternate proof of ~ dffr2... |
| frc 5613 | Property of well-founded r... |
| frss 5614 | Subset theorem for the wel... |
| sess1 5615 | Subset theorem for the set... |
| sess2 5616 | Subset theorem for the set... |
| freq1 5617 | Equality theorem for the w... |
| freq2 5618 | Equality theorem for the w... |
| freq12d 5619 | Equality deduction for wel... |
| seeq1 5620 | Equality theorem for the s... |
| seeq2 5621 | Equality theorem for the s... |
| seeq12d 5622 | Equality deduction for the... |
| nffr 5623 | Bound-variable hypothesis ... |
| nfse 5624 | Bound-variable hypothesis ... |
| nfwe 5625 | Bound-variable hypothesis ... |
| frirr 5626 | A well-founded relation is... |
| fr2nr 5627 | A well-founded relation ha... |
| fr0 5628 | Any relation is well-found... |
| frminex 5629 | If an element of a well-fo... |
| efrirr 5630 | A well-founded class does ... |
| efrn2lp 5631 | A well-founded class conta... |
| epse 5632 | The membership relation is... |
| tz7.2 5633 | Similar to Theorem 7.2 of ... |
| dfepfr 5634 | An alternate way of saying... |
| epfrc 5635 | A subset of a well-founded... |
| wess 5636 | Subset theorem for the wel... |
| weeq1 5637 | Equality theorem for the w... |
| weeq2 5638 | Equality theorem for the w... |
| weeq12d 5639 | Equality deduction for wel... |
| wefr 5640 | A well-ordering is well-fo... |
| weso 5641 | A well-ordering is a stric... |
| wecmpep 5642 | The elements of a class we... |
| wetrep 5643 | On a class well-ordered by... |
| wefrc 5644 | A nonempty subclass of a c... |
| we0 5645 | Any relation is a well-ord... |
| wereu 5646 | A nonempty subset of an ` ... |
| wereu2 5647 | A nonempty subclass of an ... |
| xpeq1 5664 | Equality theorem for Carte... |
| xpss12 5665 | Subset theorem for Cartesi... |
| xpss 5666 | A Cartesian product is inc... |
| inxpssres 5667 | Intersection with a Cartes... |
| relxp 5668 | A Cartesian product is a r... |
| xpss1 5669 | Subset relation for Cartes... |
| xpss2 5670 | Subset relation for Cartes... |
| xpeq2 5671 | Equality theorem for Carte... |
| elxpi 5672 | Membership in a Cartesian ... |
| elxp 5673 | Membership in a Cartesian ... |
| elxp2 5674 | Membership in a Cartesian ... |
| xpeq12 5675 | Equality theorem for Carte... |
| xpeq1i 5676 | Equality inference for Car... |
| xpeq2i 5677 | Equality inference for Car... |
| xpeq12i 5678 | Equality inference for Car... |
| xpeq1d 5679 | Equality deduction for Car... |
| xpeq2d 5680 | Equality deduction for Car... |
| xpeq12d 5681 | Equality deduction for Car... |
| sqxpeqd 5682 | Equality deduction for a C... |
| nfxp 5683 | Bound-variable hypothesis ... |
| 0nelxp 5684 | The empty set is not a mem... |
| 0nelelxp 5685 | A member of a Cartesian pr... |
| opelxp 5686 | Ordered pair membership in... |
| opelxpi 5687 | Ordered pair membership in... |
| opelxpii 5688 | Ordered pair membership in... |
| opelxpd 5689 | Ordered pair membership in... |
| opelvv 5690 | Ordered pair membership in... |
| opelvvg 5691 | Ordered pair membership in... |
| opelxp1 5692 | The first member of an ord... |
| opelxp2 5693 | The second member of an or... |
| otelxp 5694 | Ordered triple membership ... |
| otelxp1 5695 | The first member of an ord... |
| otel3xp 5696 | An ordered triple is an el... |
| opabssxpd 5697 | An ordered-pair class abst... |
| rabxp 5698 | Class abstraction restrict... |
| brxp 5699 | Binary relation on a Carte... |
| pwvrel 5700 | A set is a binary relation... |
| pwvabrel 5701 | The powerclass of the cart... |
| brrelex12 5702 | Two classes related by a b... |
| brrelex1 5703 | If two classes are related... |
| brrelex2 5704 | If two classes are related... |
| brrelex12i 5705 | Two classes that are relat... |
| brrelex1i 5706 | The first argument of a bi... |
| brrelex2i 5707 | The second argument of a b... |
| nprrel12 5708 | Proper classes are not rel... |
| nprrel 5709 | No proper class is related... |
| 0nelrel0 5710 | A binary relation does not... |
| 0nelrel 5711 | A binary relation does not... |
| fconstmpt 5712 | Representation of a consta... |
| vtoclr 5713 | Variable to class conversi... |
| opthprc 5714 | Justification theorem for ... |
| brel 5715 | Two things in a binary rel... |
| elxp3 5716 | Membership in a Cartesian ... |
| opeliunxp 5717 | Membership in a union of C... |
| opeliun2xp 5718 | Membership of an ordered p... |
| xpundi 5719 | Distributive law for Carte... |
| xpundir 5720 | Distributive law for Carte... |
| xpiundi 5721 | Distributive law for Carte... |
| xpiundir 5722 | Distributive law for Carte... |
| iunxpconst 5723 | Membership in a union of C... |
| xpun 5724 | The Cartesian product of t... |
| elvv 5725 | Membership in universal cl... |
| elvvv 5726 | Membership in universal cl... |
| elvvuni 5727 | An ordered pair contains i... |
| brinxp2 5728 | Intersection of binary rel... |
| brinxp 5729 | Intersection of binary rel... |
| opelinxp 5730 | Ordered pair element in an... |
| poinxp 5731 | Intersection of partial or... |
| soinxp 5732 | Intersection of total orde... |
| frinxp 5733 | Intersection of well-found... |
| seinxp 5734 | Intersection of set-like r... |
| weinxp 5735 | Intersection of well-order... |
| posn 5736 | Partial ordering of a sing... |
| sosn 5737 | Strict ordering on a singl... |
| frsn 5738 | Founded relation on a sing... |
| wesn 5739 | Well-ordering of a singlet... |
| elopaelxp 5740 | Membership in an ordered-p... |
| bropaex12 5741 | Two classes related by an ... |
| opabssxp 5742 | An abstraction relation is... |
| brab2a 5743 | The law of concretion for ... |
| optocl 5744 | Implicit substitution of c... |
| optoclOLD 5745 | Obsolete version of ~ opto... |
| 2optocl 5746 | Implicit substitution of c... |
| 3optocl 5747 | Implicit substitution of c... |
| opbrop 5748 | Ordered pair membership in... |
| 0xp 5749 | The Cartesian product with... |
| xp0 5750 | The Cartesian product with... |
| csbxp 5751 | Distribute proper substitu... |
| releq 5752 | Equality theorem for the r... |
| releqi 5753 | Equality inference for the... |
| releqd 5754 | Equality deduction for the... |
| nfrel 5755 | Bound-variable hypothesis ... |
| sbcrel 5756 | Distribute proper substitu... |
| relss 5757 | Subclass theorem for relat... |
| ssrel 5758 | A subclass relationship de... |
| eqrel 5759 | Extensionality principle f... |
| ssrel2 5760 | A subclass relationship de... |
| ssrel3 5761 | Subclass relation in anoth... |
| relssi 5762 | Inference from subclass pr... |
| relssdv 5763 | Deduction from subclass pr... |
| eqrelriv 5764 | Inference from extensional... |
| eqrelriiv 5765 | Inference from extensional... |
| eqbrriv 5766 | Inference from extensional... |
| eqrelrdv 5767 | Deduce equality of relatio... |
| eqbrrdv 5768 | Deduction from extensional... |
| eqbrrdiv 5769 | Deduction from extensional... |
| eqrelrdv2 5770 | A version of ~ eqrelrdv . ... |
| ssrelrel 5771 | A subclass relationship de... |
| eqrelrel 5772 | Extensionality principle f... |
| elrel 5773 | A member of a relation is ... |
| elrelb 5774 | A member of a relation exp... |
| rel0 5775 | The empty set is a relatio... |
| nrelv 5776 | The universal class is not... |
| nrelvOLD 5777 | Obsolete version of ~ nrel... |
| relsng 5778 | A singleton is a relation ... |
| relsnb 5779 | An at-most-singleton is a ... |
| relsnopg 5780 | A singleton of an ordered ... |
| relsn 5781 | A singleton is a relation ... |
| relsnop 5782 | A singleton of an ordered ... |
| copsex2gb 5783 | Implicit substitution infe... |
| copsex2ga 5784 | Implicit substitution infe... |
| elopaba 5785 | Membership in an ordered-p... |
| xpsspw 5786 | A Cartesian product is inc... |
| unixpss 5787 | The double class union of ... |
| relun 5788 | The union of two relations... |
| relin1 5789 | The intersection with a re... |
| relin2 5790 | The intersection with a re... |
| relinxp 5791 | Intersection with a Cartes... |
| reldif 5792 | A difference cutting down ... |
| reliun 5793 | An indexed union is a rela... |
| reliin 5794 | An indexed intersection is... |
| reluni 5795 | The union of a class is a ... |
| relint 5796 | The intersection of a clas... |
| relopabiv 5797 | A class of ordered pairs i... |
| relopabv 5798 | A class of ordered pairs i... |
| relopabi 5799 | A class of ordered pairs i... |
| relopabiALT 5800 | Alternate proof of ~ relop... |
| relopab 5801 | A class of ordered pairs i... |
| mptrel 5802 | The maps-to notation alway... |
| reli 5803 | The identity relation is a... |
| rele 5804 | The membership relation is... |
| opabid2 5805 | A relation expressed as an... |
| inopab 5806 | Intersection of two ordere... |
| difopab 5807 | Difference of two ordered-... |
| inxp 5808 | Intersection of two Cartes... |
| xpindi 5809 | Distributive law for Carte... |
| xpindir 5810 | Distributive law for Carte... |
| xpiindi 5811 | Distributive law for Carte... |
| xpriindi 5812 | Distributive law for Carte... |
| eliunxp 5813 | Membership in a union of C... |
| opeliunxp2 5814 | Membership in a union of C... |
| raliunxp 5815 | Write a double restricted ... |
| rexiunxp 5816 | Write a double restricted ... |
| ralxp 5817 | Universal quantification r... |
| rexxp 5818 | Existential quantification... |
| el2xptp 5819 | A member of a nested Carte... |
| exopxfr 5820 | Transfer ordered-pair exis... |
| exopxfr2 5821 | Transfer ordered-pair exis... |
| djussxp 5822 | Disjoint union is a subset... |
| ralxpf 5823 | Version of ~ ralxp with bo... |
| rexxpf 5824 | Version of ~ rexxp with bo... |
| iunxpf 5825 | Indexed union on a Cartesi... |
| opabbi2dv 5826 | Deduce equality of a relat... |
| relop 5827 | A necessary and sufficient... |
| ideqg 5828 | For sets, the identity rel... |
| ideq 5829 | For sets, the identity rel... |
| ididg 5830 | A set is identical to itse... |
| issetid 5831 | Two ways of expressing set... |
| coss1 5832 | Subclass theorem for compo... |
| coss2 5833 | Subclass theorem for compo... |
| coeq1 5834 | Equality theorem for compo... |
| coeq2 5835 | Equality theorem for compo... |
| coeq1i 5836 | Equality inference for com... |
| coeq2i 5837 | Equality inference for com... |
| coeq1d 5838 | Equality deduction for com... |
| coeq2d 5839 | Equality deduction for com... |
| coeq12i 5840 | Equality inference for com... |
| coeq12d 5841 | Equality deduction for com... |
| nfco 5842 | Bound-variable hypothesis ... |
| brcog 5843 | Ordered pair membership in... |
| opelco2g 5844 | Ordered pair membership in... |
| brcogw 5845 | Ordered pair membership in... |
| eqbrrdva 5846 | Deduction from extensional... |
| brco 5847 | Binary relation on a compo... |
| opelco 5848 | Ordered pair membership in... |
| cnvss 5849 | Subset theorem for convers... |
| cnveq 5850 | Equality theorem for conve... |
| cnveqi 5851 | Equality inference for con... |
| cnveqd 5852 | Equality deduction for con... |
| elcnv 5853 | Membership in a converse r... |
| elcnv2 5854 | Membership in a converse r... |
| nfcnv 5855 | Bound-variable hypothesis ... |
| brcnvg 5856 | The converse of a binary r... |
| opelcnvg 5857 | Ordered-pair membership in... |
| opelcnv 5858 | Ordered-pair membership in... |
| brcnv 5859 | The converse of a binary r... |
| cnv0 5860 | The converse of the empty ... |
| cnv0OLD 5861 | Obsolete version of ~ cnv0... |
| cnvi 5862 | The converse of the identi... |
| csbcnv 5863 | Move class substitution in... |
| csbcnvOLD 5864 | Obsolete version of ~ csbc... |
| csbcnvgALTOLD 5865 | Obsolete version of ~ csbc... |
| cnvco 5866 | Distributive law of conver... |
| cnvuni 5867 | The converse of a class un... |
| dfdm3 5868 | Alternate definition of do... |
| dfrn2 5869 | Alternate definition of ra... |
| dfrn3 5870 | Alternate definition of ra... |
| elrn2g 5871 | Membership in a range. (C... |
| elrng 5872 | Membership in a range. (C... |
| elrn2 5873 | Membership in a range. (C... |
| elrn 5874 | Membership in a range. (C... |
| ssrelrn 5875 | If a relation is a subset ... |
| dfdm4 5876 | Alternate definition of do... |
| dfdmf 5877 | Definition of domain, usin... |
| csbdm 5878 | Distribute proper substitu... |
| eldmg 5879 | Domain membership. Theore... |
| eldm2g 5880 | Domain membership. Theore... |
| eldm 5881 | Membership in a domain. T... |
| eldm2 5882 | Membership in a domain. T... |
| dmss 5883 | Subset theorem for domain.... |
| dmeq 5884 | Equality theorem for domai... |
| dmeqi 5885 | Equality inference for dom... |
| dmeqd 5886 | Equality deduction for dom... |
| opeldmd 5887 | Membership of first of an ... |
| opeldm 5888 | Membership of first of an ... |
| breldm 5889 | Membership of first of a b... |
| breldmg 5890 | Membership of first of a b... |
| dmun 5891 | The domain of a union is t... |
| dmin 5892 | The domain of an intersect... |
| breldmd 5893 | Membership of first of a b... |
| dmiun 5894 | The domain of an indexed u... |
| dmuni 5895 | The domain of a union. Pa... |
| dmopab 5896 | The domain of a class of o... |
| dmopabelb 5897 | A set is an element of the... |
| dmopab2rex 5898 | The domain of an ordered p... |
| dmopabss 5899 | Upper bound for the domain... |
| dmopab3 5900 | The domain of a restricted... |
| dm0 5901 | The domain of the empty se... |
| dmi 5902 | The domain of the identity... |
| dmv 5903 | The domain of the universe... |
| dmep 5904 | The domain of the membersh... |
| dm0rn0 5905 | An empty domain is equival... |
| dm0rn0OLD 5906 | Obsolete version of ~ dm0r... |
| rn0 5907 | The range of the empty set... |
| rnep 5908 | The range of the membershi... |
| reldm0 5909 | A relation is empty iff it... |
| dmxp 5910 | The domain of a Cartesian ... |
| dmxpid 5911 | The domain of a Cartesian ... |
| dmxpin 5912 | The domain of the intersec... |
| xpid11 5913 | The Cartesian square is a ... |
| dmcnvcnv 5914 | The domain of the double c... |
| rncnvcnv 5915 | The range of the double co... |
| elreldm 5916 | The first member of an ord... |
| rneq 5917 | Equality theorem for range... |
| rneqi 5918 | Equality inference for ran... |
| rneqd 5919 | Equality deduction for ran... |
| rnss 5920 | Subset theorem for range. ... |
| rnssi 5921 | Subclass inference for ran... |
| brelrng 5922 | The second argument of a b... |
| brelrn 5923 | The second argument of a b... |
| opelrn 5924 | Membership of second membe... |
| releldm 5925 | The first argument of a bi... |
| relelrn 5926 | The second argument of a b... |
| releldmb 5927 | Membership in a domain. (... |
| relelrnb 5928 | Membership in a range. (C... |
| releldmi 5929 | The first argument of a bi... |
| relelrni 5930 | The second argument of a b... |
| dfrnf 5931 | Definition of range, using... |
| nfdm 5932 | Bound-variable hypothesis ... |
| nfrn 5933 | Bound-variable hypothesis ... |
| dmiin 5934 | Domain of an intersection.... |
| rnopab 5935 | The range of a class of or... |
| rnopabss 5936 | Upper bound for the range ... |
| rnopab3 5937 | The range of a restricted ... |
| rnmpt 5938 | The range of a function in... |
| elrnmpt 5939 | The range of a function in... |
| elrnmpt1s 5940 | Elementhood in an image se... |
| elrnmpt1 5941 | Elementhood in an image se... |
| elrnmptg 5942 | Membership in the range of... |
| elrnmpti 5943 | Membership in the range of... |
| elrnmptd 5944 | The range of a function in... |
| elrnmpt1d 5945 | Elementhood in an image se... |
| elrnmptdv 5946 | Elementhood in the range o... |
| elrnmpt2d 5947 | Elementhood in the range o... |
| nelrnmpt 5948 | Non-membership in the rang... |
| dfiun3g 5949 | Alternate definition of in... |
| dfiin3g 5950 | Alternate definition of in... |
| dfiun3 5951 | Alternate definition of in... |
| dfiin3 5952 | Alternate definition of in... |
| riinint 5953 | Express a relative indexed... |
| relrn0 5954 | A relation is empty iff it... |
| dmrnssfld 5955 | The domain and range of a ... |
| dmcoss 5956 | Domain of a composition. ... |
| dmcossOLD 5957 | Obsolete version of ~ dmco... |
| rncoss 5958 | Range of a composition. (... |
| dmcosseq 5959 | Domain of a composition. ... |
| dmcosseqOLD 5960 | Obsolete version of ~ dmco... |
| dmcoeq 5961 | Domain of a composition. ... |
| rncoeq 5962 | Range of a composition. (... |
| reseq1 5963 | Equality theorem for restr... |
| reseq2 5964 | Equality theorem for restr... |
| reseq1i 5965 | Equality inference for res... |
| reseq2i 5966 | Equality inference for res... |
| reseq12i 5967 | Equality inference for res... |
| reseq1d 5968 | Equality deduction for res... |
| reseq2d 5969 | Equality deduction for res... |
| reseq12d 5970 | Equality deduction for res... |
| nfres 5971 | Bound-variable hypothesis ... |
| csbres 5972 | Distribute proper substitu... |
| res0 5973 | A restriction to the empty... |
| dfres3 5974 | Alternate definition of re... |
| opelres 5975 | Ordered pair elementhood i... |
| brres 5976 | Binary relation on a restr... |
| opelresi 5977 | Ordered pair membership in... |
| brresi 5978 | Binary relation on a restr... |
| opres 5979 | Ordered pair membership in... |
| resieq 5980 | A restricted identity rela... |
| opelidres 5981 | ` <. A , A >. ` belongs to... |
| resres 5982 | The restriction of a restr... |
| resundi 5983 | Distributive law for restr... |
| resundir 5984 | Distributive law for restr... |
| resindi 5985 | Class restriction distribu... |
| resindir 5986 | Class restriction distribu... |
| inres 5987 | Move intersection into cla... |
| resdifcom 5988 | Commutative law for restri... |
| resiun1 5989 | Distribution of restrictio... |
| resiun2 5990 | Distribution of restrictio... |
| resss 5991 | A class includes its restr... |
| rescom 5992 | Commutative law for restri... |
| ssres 5993 | Subclass theorem for restr... |
| ssres2 5994 | Subclass theorem for restr... |
| relres 5995 | A restriction is a relatio... |
| resabs1 5996 | Absorption law for restric... |
| resabs1i 5997 | Absorption law for restric... |
| resabs1d 5998 | Absorption law for restric... |
| resabs2 5999 | Absorption law for restric... |
| residm 6000 | Idempotent law for restric... |
| dmresss 6001 | The domain of a restrictio... |
| dmres 6002 | The domain of a restrictio... |
| ssdmres 6003 | A domain restricted to a s... |
| dmresexg 6004 | The domain of a restrictio... |
| rnresss 6005 | The range of a restriction... |
| xpssres 6006 | Restriction of a constant ... |
| elinxp 6007 | Membership in an intersect... |
| elres 6008 | Membership in a restrictio... |
| elsnres 6009 | Membership in restriction ... |
| relssres 6010 | Simplification law for res... |
| dmressnsn 6011 | The domain of a restrictio... |
| eldmressnsn 6012 | The element of the domain ... |
| eldmeldmressn 6013 | An element of the domain (... |
| resdm 6014 | A relation restricted to i... |
| resexg 6015 | The restriction of a set i... |
| resexd 6016 | The restriction of a set i... |
| resex 6017 | The restriction of a set i... |
| resindm 6018 | When restricting a class, ... |
| resindmOLD 6019 | Obsolete version of ~ resi... |
| resdmdfsn 6020 | Restricting a class to its... |
| resdmdfsnOLD 6021 | Obsolete version of ~ resd... |
| reldmun 6022 | Split a relation into two ... |
| reldisjunOLD 6023 | Obsolete version of ~ reld... |
| relresdm1 6024 | Restriction of a disjoint ... |
| resopab 6025 | Restriction of a class abs... |
| iss 6026 | A subclass of the identity... |
| resopab2 6027 | Restriction of a class abs... |
| resmpt 6028 | Restriction of the mapping... |
| resmpt3 6029 | Unconditional restriction ... |
| resmptf 6030 | Restriction of the mapping... |
| resmptd 6031 | Restriction of the mapping... |
| dfres2 6032 | Alternate definition of th... |
| mptss 6033 | Sufficient condition for i... |
| elimampt 6034 | Membership in the image of... |
| elidinxp 6035 | Characterization of the el... |
| elidinxpid 6036 | Characterization of the el... |
| elrid 6037 | Characterization of the el... |
| idinxpres 6038 | The intersection of the id... |
| idinxpresid 6039 | The intersection of the id... |
| idssxp 6040 | A diagonal set as a subset... |
| opabresid 6041 | The restricted identity re... |
| mptresid 6042 | The restricted identity re... |
| dmresi 6043 | The domain of a restricted... |
| restidsing 6044 | Restriction of the identit... |
| iresn0n0 6045 | The identity function rest... |
| imaeq1 6046 | Equality theorem for image... |
| imaeq2 6047 | Equality theorem for image... |
| imaeq1i 6048 | Equality theorem for image... |
| imaeq2i 6049 | Equality theorem for image... |
| imaeq1d 6050 | Equality theorem for image... |
| imaeq2d 6051 | Equality theorem for image... |
| imaeq12d 6052 | Equality theorem for image... |
| imass1 6053 | Subset theorem for image. ... |
| imass2 6054 | Subset theorem for image. ... |
| resima 6055 | A restriction to an image.... |
| resima2 6056 | Image under a restricted c... |
| dfima2 6057 | Alternate definition of im... |
| dfima3 6058 | Alternate definition of im... |
| elimag 6059 | Membership in an image. T... |
| elima 6060 | Membership in an image. T... |
| elima2 6061 | Membership in an image. T... |
| elima3 6062 | Membership in an image. T... |
| nfima 6063 | Bound-variable hypothesis ... |
| nfimad 6064 | Deduction version of bound... |
| dfrn7 6065 | The range is equal to the ... |
| imadmrn 6066 | The image of the domain of... |
| imadmrnOLD 6067 | Obsolete version of ~ imad... |
| imassrnOLD 6068 | Obsolete version of ~ imas... |
| mptima 6069 | Image by a function in map... |
| mptimass 6070 | Image by a function in map... |
| imai 6071 | Image by the identity rela... |
| rnresi 6072 | The range of the restricte... |
| resiima 6073 | The image of a restriction... |
| ima0 6074 | Image of the empty set. T... |
| csbima12 6075 | Move class substitution in... |
| imadisj 6076 | A class whose image under ... |
| imadisjlnd 6077 | Deduction form of one nega... |
| cnvimassrndm 6078 | The preimage of a superset... |
| cnvimarndm 6079 | The preimage of the range ... |
| cnvimarndmOLD 6080 | Obsolete version of ~ cnvi... |
| imasng 6081 | The image of a singleton. ... |
| relimasn 6082 | The image of a singleton. ... |
| elrelimasn 6083 | Elementhood in the image o... |
| elimasng1 6084 | Membership in an image of ... |
| elimasn1 6085 | Membership in an image of ... |
| elimasng 6086 | Membership in an image of ... |
| elimasn 6087 | Membership in an image of ... |
| elimasni 6088 | Membership in an image of ... |
| args 6089 | Two ways to express the cl... |
| elinisegg 6090 | Membership in the inverse ... |
| eliniseg 6091 | Membership in the inverse ... |
| epin 6092 | Any set is equal to its pr... |
| epini 6093 | Any set is equal to its pr... |
| iniseg 6094 | An idiom that signifies an... |
| inisegn0 6095 | Nonemptiness of an initial... |
| dffr3 6096 | Alternate definition of we... |
| dfse2 6097 | Alternate definition of se... |
| ndmima 6098 | The image of a singleton o... |
| relcnv 6099 | A converse is a relation. ... |
| relbrcnvg 6100 | When ` R ` is a relation, ... |
| eliniseg2 6101 | Eliminate the class existe... |
| relbrcnv 6102 | When ` R ` is a relation, ... |
| relco 6103 | A composition is a relatio... |
| cotrg 6104 | Two ways of saying that th... |
| cotr 6105 | Two ways of saying a relat... |
| idrefALT 6106 | Alternate proof of ~ idref... |
| cnvsym 6107 | Two ways of saying a relat... |
| intasym 6108 | Two ways of saying a relat... |
| asymref 6109 | Two ways of saying a relat... |
| asymref2 6110 | Two ways of saying a relat... |
| intirr 6111 | Two ways of saying a relat... |
| brcodir 6112 | Two ways of saying that tw... |
| codir 6113 | Two ways of saying a relat... |
| qfto 6114 | A quantifier-free way of e... |
| xpidtr 6115 | A Cartesian square is a tr... |
| trin2 6116 | The intersection of two tr... |
| poirr2 6117 | A partial order is irrefle... |
| trinxp 6118 | The relation induced by a ... |
| soirri 6119 | A strict order relation is... |
| sotri 6120 | A strict order relation is... |
| son2lpi 6121 | A strict order relation ha... |
| sotri2 6122 | A transitivity relation. ... |
| sotri3 6123 | A transitivity relation. ... |
| poleloe 6124 | Express "less than or equa... |
| poltletr 6125 | Transitive law for general... |
| somin1 6126 | Property of a minimum in a... |
| somincom 6127 | Commutativity of minimum i... |
| somin2 6128 | Property of a minimum in a... |
| soltmin 6129 | Being less than a minimum,... |
| cnvopab 6130 | The converse of a class ab... |
| mptcnv 6131 | The converse of a mapping ... |
| cnvun 6132 | The converse of a union is... |
| cnvdif 6133 | Distributive law for conve... |
| cnvin 6134 | Distributive law for conve... |
| rnun 6135 | Distributive law for range... |
| rnin 6136 | The range of an intersecti... |
| rninOLD 6137 | Obsolete version of ~ rnin... |
| rniun 6138 | The range of an indexed un... |
| rnuni 6139 | The range of a union. Par... |
| imaundi 6140 | Distributive law for image... |
| imaundir 6141 | The image of a union. (Co... |
| dminss 6142 | An upper bound for interse... |
| imainss 6143 | An upper bound for interse... |
| inimass 6144 | The image of an intersecti... |
| inimasn 6145 | The intersection of the im... |
| cnvxp 6146 | The converse of a Cartesia... |
| cnvxpOLD 6147 | Obsolete version of ~ cnvx... |
| xp0OLD 6148 | Obsolete version of ~ xp0 ... |
| xpnz 6149 | The Cartesian product of n... |
| xpeq0 6150 | At least one member of an ... |
| xpdisj1 6151 | Cartesian products with di... |
| xpdisj2 6152 | Cartesian products with di... |
| xpsndisj 6153 | Cartesian products with tw... |
| difxp 6154 | Difference of Cartesian pr... |
| difxp1 6155 | Difference law for Cartesi... |
| difxp2 6156 | Difference law for Cartesi... |
| djudisj 6157 | Disjoint unions with disjo... |
| xpdifid 6158 | The set of distinct couple... |
| xpdifcnvepel 6159 | The set of couples in a Ca... |
| resdisj 6160 | A double restriction to di... |
| rnxp 6161 | The range of a Cartesian p... |
| dmxpss 6162 | The domain of a Cartesian ... |
| rnxpss 6163 | The range of a Cartesian p... |
| rnxpid 6164 | The range of a Cartesian s... |
| ssxpb 6165 | A Cartesian product subcla... |
| xp11 6166 | The Cartesian product of n... |
| xpcan 6167 | Cancellation law for Carte... |
| xpcan2 6168 | Cancellation law for Carte... |
| ssrnres 6169 | Two ways to express surjec... |
| rninxp 6170 | Two ways to express surjec... |
| dminxp 6171 | Two ways to express totali... |
| imainrect 6172 | Image by a restricted and ... |
| xpima 6173 | Direct image by a Cartesia... |
| xpima1 6174 | Direct image by a Cartesia... |
| xpima2 6175 | Direct image by a Cartesia... |
| xpimasn 6176 | Direct image of a singleto... |
| sossfld 6177 | The base set of a strict o... |
| sofld 6178 | The base set of a nonempty... |
| cnvcnv3 6179 | The set of all ordered pai... |
| dfrel2 6180 | Alternate definition of re... |
| dfrel4v 6181 | A relation can be expresse... |
| dfrel4 6182 | A relation can be expresse... |
| cnvcnv 6183 | The double converse of a c... |
| cnvcnv2 6184 | The double converse of a c... |
| cnvcnvss 6185 | The double converse of a c... |
| cnvcnvssOLD 6186 | Obsolete version of ~ cnvc... |
| cnvrescnv 6187 | Two ways to express the co... |
| cnvssb 6188 | Subclass theorem for conve... |
| cnveqb 6189 | Equality theorem for conve... |
| cnveq0 6190 | A relation is empty iff it... |
| cnveq0b 6191 | A class is empty iff it is... |
| dfrel3 6192 | Alternate definition of re... |
| elid 6193 | Characterization of the el... |
| dmresv 6194 | The domain of a universal ... |
| rnresv 6195 | The range of a universal r... |
| dfrn4 6196 | Range defined in terms of ... |
| imassrn 6197 | Any image by a class is in... |
| cnvimass 6198 | Any preimage by a class is... |
| cnvimassrndmOLD 6199 | Obsolete version of ~ cnvi... |
| 0ima 6200 | Image under the empty rela... |
| imadifssrn 6201 | Condition for the range of... |
| imadifssranOLD 6202 | Obsolete version of ~ imad... |
| imadifssranOLDOLD 6203 | Obsolete version of ~ imad... |
| csbrn 6204 | Distribute proper substitu... |
| rescnvcnv 6205 | The restriction of the dou... |
| cnvcnvres 6206 | The double converse of the... |
| imacnvcnv 6207 | The image of the double co... |
| dmsnn0 6208 | The domain of a singleton ... |
| rnsnn0 6209 | The range of a singleton i... |
| dmsn0 6210 | The domain of the singleto... |
| cnvsn0 6211 | The converse of the single... |
| dmsn0el 6212 | The domain of a singleton ... |
| relsn2 6213 | A singleton is a relation ... |
| dmsnopg 6214 | The domain of a singleton ... |
| dmsnopss 6215 | The domain of a singleton ... |
| dmpropg 6216 | The domain of an unordered... |
| dmsnop 6217 | The domain of a singleton ... |
| dmprop 6218 | The domain of an unordered... |
| dmtpop 6219 | The domain of an unordered... |
| cnvcnvsn 6220 | Double converse of a singl... |
| dmsnsnsn 6221 | The domain of the singleto... |
| rnsnopg 6222 | The range of a singleton o... |
| rnpropg 6223 | The range of a pair of ord... |
| cnvsng 6224 | Converse of a singleton of... |
| rnsnop 6225 | The range of a singleton o... |
| op1sta 6226 | Extract the first member o... |
| cnvsn 6227 | Converse of a singleton of... |
| op2ndb 6228 | Extract the second member ... |
| op2nda 6229 | Extract the second member ... |
| opswap 6230 | Swap the members of an ord... |
| cnvresima 6231 | An image under the convers... |
| resdm2 6232 | A class restricted to its ... |
| resdmres 6233 | Restriction to the domain ... |
| resresdm 6234 | A restriction by an arbitr... |
| imadmres 6235 | The image of the domain of... |
| resdmss 6236 | Subset relationship for th... |
| resdifdi 6237 | Distributive law for restr... |
| resdifdir 6238 | Distributive law for restr... |
| mptpreima 6239 | The preimage of a function... |
| mptiniseg 6240 | Converse singleton image o... |
| dmmpt 6241 | The domain of the mapping ... |
| dmmptss 6242 | The domain of a mapping is... |
| dmmptg 6243 | The domain of the mapping ... |
| rnmpt0f 6244 | The range of a function in... |
| rnmptn0 6245 | The range of a function in... |
| dfco2 6246 | Alternate definition of a ... |
| dfco2a 6247 | Generalization of ~ dfco2 ... |
| coundi 6248 | Class composition distribu... |
| coundir 6249 | Class composition distribu... |
| cores 6250 | Restricted first member of... |
| resco 6251 | Associative law for the re... |
| imaco 6252 | Image of the composition o... |
| rnco 6253 | The range of the compositi... |
| rncoOLD 6254 | Obsolete version of ~ rnco... |
| rnco2 6255 | The range of the compositi... |
| dmco 6256 | The domain of a compositio... |
| coeq0 6257 | A composition of two relat... |
| coiun 6258 | Composition with an indexe... |
| cocnvcnv1 6259 | A composition is not affec... |
| cocnvcnv2 6260 | A composition is not affec... |
| cores2 6261 | Absorption of a reverse (p... |
| co02 6262 | Composition with the empty... |
| co01 6263 | Composition with the empty... |
| coi1 6264 | Composition with the ident... |
| coi2 6265 | Composition with the ident... |
| coires1 6266 | Composition with a restric... |
| coass 6267 | Associative law for class ... |
| relcnvtrg 6268 | Subclass law for converse ... |
| relcnvtrgOLD 6269 | Obsolete form of ~ relcnvt... |
| relcnvtrOLD 6270 | Obsolete form of ~ relcnvt... |
| relssdmrn 6271 | A relation is included in ... |
| resssxp 6272 | If the ` R ` -image of a c... |
| cnvssrndm 6273 | The converse is a subset o... |
| cossxp 6274 | Composition as a subset of... |
| relrelss 6275 | Two ways to describe the s... |
| unielrel 6276 | The membership relation fo... |
| relfld 6277 | The double union of a rela... |
| relresfld 6278 | Restriction of a relation ... |
| relresfldOLD 6279 | Obsolete version of ~ relr... |
| relcoi2 6280 | Composition with the ident... |
| relcoi1 6281 | Composition with the ident... |
| unidmrn 6282 | The double union of the co... |
| relcnvfld 6283 | if ` R ` is a relation, it... |
| dfdm2 6284 | Alternate definition of do... |
| unixp 6285 | The double class union of ... |
| unixp0 6286 | A Cartesian product is emp... |
| unixpid 6287 | Field of a Cartesian squar... |
| ressn 6288 | Restriction of a class to ... |
| cnviin 6289 | The converse of an interse... |
| cnvpo 6290 | The converse of a partial ... |
| cnvso 6291 | The converse of a strict o... |
| xpco 6292 | Composition of two Cartesi... |
| xpcoid 6293 | Composition of two Cartesi... |
| elsnxp 6294 | Membership in a Cartesian ... |
| reu3op 6295 | There is a unique ordered ... |
| reuop 6296 | There is a unique ordered ... |
| opreu2reurex 6297 | There is a unique ordered ... |
| opreu2reu 6298 | If there is a unique order... |
| dfpo2 6299 | Quantifier-free definition... |
| csbcog 6300 | Distribute proper substitu... |
| snres0 6301 | Condition for restriction ... |
| imaindm 6302 | The image is unaffected by... |
| predeq123 6305 | Equality theorem for the p... |
| predeq1 6306 | Equality theorem for the p... |
| predeq2 6307 | Equality theorem for the p... |
| predeq3 6308 | Equality theorem for the p... |
| nfpred 6309 | Bound-variable hypothesis ... |
| csbpredg 6310 | Move class substitution in... |
| predpredss 6311 | If ` A ` is a subset of ` ... |
| predss 6312 | The predecessor class of `... |
| sspred 6313 | Another subset/predecessor... |
| dfpred2 6314 | An alternate definition of... |
| dfpred3 6315 | An alternate definition of... |
| dfpred3g 6316 | An alternate definition of... |
| elpredgg 6317 | Membership in a predecesso... |
| elpredg 6318 | Membership in a predecesso... |
| elpredimg 6319 | Membership in a predecesso... |
| elpredim 6320 | Membership in a predecesso... |
| elpred 6321 | Membership in a predecesso... |
| predexg 6322 | The predecessor class exis... |
| dffr4 6323 | Alternate definition of we... |
| predel 6324 | Membership in the predeces... |
| predtrss 6325 | If ` R ` is transitive ove... |
| predpo 6326 | Property of the predecesso... |
| predso 6327 | Property of the predecesso... |
| setlikespec 6328 | If ` R ` is set-like in ` ... |
| predidm 6329 | Idempotent law for the pre... |
| predin 6330 | Intersection law for prede... |
| predun 6331 | Union law for predecessor ... |
| preddif 6332 | Difference law for predece... |
| predep 6333 | The predecessor under the ... |
| trpred 6334 | The class of predecessors ... |
| preddowncl 6335 | A property of classes that... |
| predpoirr 6336 | Given a partial ordering, ... |
| predfrirr 6337 | Given a well-founded relat... |
| pred0 6338 | The predecessor class over... |
| dfse3 6339 | Alternate definition of se... |
| predrelss 6340 | Subset carries from relati... |
| predprc 6341 | The predecessor of a prope... |
| predres 6342 | Predecessor class is unaff... |
| frpomin 6343 | Every nonempty (possibly p... |
| frpomin2 6344 | Every nonempty (possibly p... |
| frpoind 6345 | The principle of well-foun... |
| frpoinsg 6346 | Well-Founded Induction Sch... |
| frpoins2fg 6347 | Well-Founded Induction sch... |
| frpoins2g 6348 | Well-Founded Induction sch... |
| frpoins3g 6349 | Well-Founded Induction sch... |
| tz6.26 6350 | All nonempty subclasses of... |
| tz6.26i 6351 | All nonempty subclasses of... |
| wfi 6352 | The Principle of Well-Orde... |
| wfii 6353 | The Principle of Well-Orde... |
| wfisg 6354 | Well-Ordered Induction Sch... |
| wfis 6355 | Well-Ordered Induction Sch... |
| wfis2fg 6356 | Well-Ordered Induction Sch... |
| wfis2f 6357 | Well-Ordered Induction sch... |
| wfis2g 6358 | Well-Ordered Induction Sch... |
| wfis2 6359 | Well-Ordered Induction sch... |
| wfis3 6360 | Well-Ordered Induction sch... |
| ordeq 6369 | Equality theorem for the o... |
| elong 6370 | An ordinal number is an or... |
| elon 6371 | An ordinal number is an or... |
| eloni 6372 | An ordinal number has the ... |
| elon2 6373 | An ordinal number is an or... |
| limeq 6374 | Equality theorem for the l... |
| ordwe 6375 | Membership well-orders eve... |
| ordtr 6376 | An ordinal class is transi... |
| ordfr 6377 | Membership is well-founded... |
| ordelss 6378 | An element of an ordinal c... |
| trssord 6379 | A transitive subclass of a... |
| ordirr 6380 | No ordinal class is a memb... |
| nordeq 6381 | A member of an ordinal cla... |
| ordn2lp 6382 | An ordinal class cannot be... |
| tz7.5 6383 | A nonempty subclass of an ... |
| ordelord 6384 | An element of an ordinal c... |
| tron 6385 | The class of all ordinal n... |
| ordelon 6386 | An element of an ordinal c... |
| onelon 6387 | An element of an ordinal n... |
| tz7.7 6388 | A transitive class belongs... |
| ordelssne 6389 | For ordinal classes, membe... |
| ordelpss 6390 | For ordinal classes, membe... |
| ordpss 6391 | ~ ordelpss with an anteced... |
| ordsseleq 6392 | For ordinal classes, inclu... |
| ordin 6393 | The intersection of two or... |
| onin 6394 | The intersection of two or... |
| ordtri3or 6395 | A trichotomy law for ordin... |
| ordtri1 6396 | A trichotomy law for ordin... |
| ontri1 6397 | A trichotomy law for ordin... |
| ordtri2 6398 | A trichotomy law for ordin... |
| ordtri3 6399 | A trichotomy law for ordin... |
| ordtri4 6400 | A trichotomy law for ordin... |
| orddisj 6401 | An ordinal class and its s... |
| onfr 6402 | The ordinal class is well-... |
| onelpss 6403 | Relationship between membe... |
| onsseleq 6404 | Relationship between subse... |
| onelss 6405 | An element of an ordinal n... |
| oneltri 6406 | The elementhood relation o... |
| ordtr1 6407 | Transitive law for ordinal... |
| ordtr2 6408 | Transitive law for ordinal... |
| ordtr3 6409 | Transitive law for ordinal... |
| ontr1 6410 | Transitive law for ordinal... |
| ontr2 6411 | Transitive law for ordinal... |
| onelssex 6412 | Ordinal less than is equiv... |
| ordunidif 6413 | The union of an ordinal st... |
| ordintdif 6414 | If ` B ` is smaller than `... |
| onintss 6415 | If a property is true for ... |
| oneqmini 6416 | A way to show that an ordi... |
| ord0 6417 | The empty set is an ordina... |
| 0elon 6418 | The empty set is an ordina... |
| ord0eln0 6419 | A nonempty ordinal contain... |
| on0eln0 6420 | An ordinal number contains... |
| dflim2 6421 | An alternate definition of... |
| inton 6422 | The intersection of the cl... |
| nlim0 6423 | The empty set is not a lim... |
| limord 6424 | A limit ordinal is ordinal... |
| limuni 6425 | A limit ordinal is its own... |
| limuni2 6426 | The union of a limit ordin... |
| 0ellim 6427 | A limit ordinal contains t... |
| limelon 6428 | A limit ordinal class that... |
| onn0 6429 | The class of all ordinal n... |
| suceqd 6430 | Deduction associated with ... |
| suceq 6431 | Equality of successors. (... |
| elsuci 6432 | Membership in a successor.... |
| elsucg 6433 | Membership in a successor.... |
| elsuc2g 6434 | Variant of membership in a... |
| elsuc 6435 | Membership in a successor.... |
| elsuc2 6436 | Membership in a successor.... |
| nfsuc 6437 | Bound-variable hypothesis ... |
| elelsuc 6438 | Membership in a successor.... |
| sucel 6439 | Membership of a successor ... |
| suc0 6440 | The successor of the empty... |
| sucprc 6441 | A proper class is its own ... |
| unisucs 6442 | The union of the successor... |
| unisucg 6443 | A transitive class is equa... |
| unisuc 6444 | A transitive class is equa... |
| sssucid 6445 | A class is included in its... |
| sucidg 6446 | Part of Proposition 7.23 o... |
| sucid 6447 | A set belongs to its succe... |
| nsuceq0 6448 | No successor is empty. (C... |
| eqelsuc 6449 | A set belongs to the succe... |
| iunsuc 6450 | Inductive definition for t... |
| suctr 6451 | The successor of a transit... |
| trsuc 6452 | A set whose successor belo... |
| trsucss 6453 | A member of the successor ... |
| ordsssuc 6454 | An ordinal is a subset of ... |
| onsssuc 6455 | A subset of an ordinal num... |
| ordsssuc2 6456 | An ordinal subset of an or... |
| onmindif 6457 | When its successor is subt... |
| ordnbtwn 6458 | There is no set between an... |
| onnbtwn 6459 | There is no set between an... |
| sucssel 6460 | A set whose successor is a... |
| orddif 6461 | Ordinal derived from its s... |
| orduniss 6462 | An ordinal class includes ... |
| ordtri2or 6463 | A trichotomy law for ordin... |
| ordtri2or2 6464 | A trichotomy law for ordin... |
| ordtri2or3 6465 | A consequence of total ord... |
| ordelinel 6466 | The intersection of two or... |
| ordssun 6467 | Property of a subclass of ... |
| ordequn 6468 | The maximum (i.e. union) o... |
| ordun 6469 | The maximum (i.e., union) ... |
| onunel 6470 | The union of two ordinals ... |
| ordunisssuc 6471 | A subclass relationship fo... |
| suc11 6472 | The successor operation be... |
| onun2 6473 | The union of two ordinals ... |
| ontr 6474 | An ordinal number is a tra... |
| onunisuc 6475 | An ordinal number is equal... |
| onordi 6476 | An ordinal number is an or... |
| onirri 6477 | An ordinal number is not a... |
| oneli 6478 | A member of an ordinal num... |
| onelssi 6479 | A member of an ordinal num... |
| onssneli 6480 | An ordering law for ordina... |
| onssnel2i 6481 | An ordering law for ordina... |
| onelini 6482 | An element of an ordinal n... |
| oneluni 6483 | An ordinal number equals i... |
| onunisuci 6484 | An ordinal number is equal... |
| onsseli 6485 | Subset is equivalent to me... |
| onun2i 6486 | The union of two ordinal n... |
| unizlim 6487 | An ordinal equal to its ow... |
| on0eqel 6488 | An ordinal number either e... |
| snsn0non 6489 | The singleton of the singl... |
| onxpdisj 6490 | Ordinal numbers and ordere... |
| onnev 6491 | The class of ordinal numbe... |
| iotajust 6493 | Soundness justification th... |
| dfiota2 6495 | Alternate definition for d... |
| nfiota1 6496 | Bound-variable hypothesis ... |
| nfiotadw 6497 | Deduction version of ~ nfi... |
| nfiotaw 6498 | Bound-variable hypothesis ... |
| nfiotad 6499 | Deduction version of ~ nfi... |
| nfiota 6500 | Bound-variable hypothesis ... |
| cbviotaw 6501 | Change bound variables in ... |
| cbviotavw 6502 | Change bound variables in ... |
| cbviota 6503 | Change bound variables in ... |
| cbviotav 6504 | Change bound variables in ... |
| sb8iota 6505 | Variable substitution in d... |
| iotaeq 6506 | Equality theorem for descr... |
| iotabi 6507 | Equivalence theorem for de... |
| uniabio 6508 | Part of Theorem 8.17 in [Q... |
| iotaval2 6509 | Version of ~ iotaval using... |
| iotauni2 6510 | Version of ~ iotauni using... |
| iotanul2 6511 | Version of ~ iotanul using... |
| iotaval 6512 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6513 | The ` iota ` class is a su... |
| iotaex 6514 | Theorem 8.23 in [Quine] p.... |
| iotauni 6515 | Equivalence between two di... |
| iotaint 6516 | Equivalence between two di... |
| iota1 6517 | Property of iota. (Contri... |
| iotanul 6518 | Theorem 8.22 in [Quine] p.... |
| iota4 6519 | Theorem *14.22 in [Whitehe... |
| iota4an 6520 | Theorem *14.23 in [Whitehe... |
| iota5 6521 | A method for computing iot... |
| iotabidv 6522 | Formula-building deduction... |
| iotabii 6523 | Formula-building deduction... |
| iotacl 6524 | Membership law for descrip... |
| iota2df 6525 | A condition that allows to... |
| iota2d 6526 | A condition that allows to... |
| iota2 6527 | The unique element such th... |
| iotan0 6528 | Representation of "the uni... |
| sniota 6529 | A class abstraction with a... |
| dfiota4 6530 | The ` iota ` operation usi... |
| csbiota 6531 | Class substitution within ... |
| dffun2 6548 | Alternate definition of a ... |
| dffun6 6549 | Alternate definition of a ... |
| dffun3 6550 | Alternate definition of fu... |
| dffun4 6551 | Alternate definition of a ... |
| dffun5 6552 | Alternate definition of fu... |
| dffun6f 6553 | Definition of function, us... |
| funmo 6554 | A function has at most one... |
| fununiq 6555 | Functions are single-value... |
| funrel 6556 | A function is a relation. ... |
| 0nelfun 6557 | A function does not contai... |
| funss 6558 | Subclass theorem for funct... |
| funeq 6559 | Equality theorem for funct... |
| funeqi 6560 | Equality inference for the... |
| funeqd 6561 | Equality deduction for the... |
| nffun 6562 | Bound-variable hypothesis ... |
| sbcfung 6563 | Distribute proper substitu... |
| sbcfungOLD 6564 | Obsolete version of ~ sbcf... |
| funeu 6565 | There is exactly one value... |
| funeu2 6566 | There is exactly one value... |
| dffun7 6567 | Alternate definition of a ... |
| dffun8 6568 | Alternate definition of a ... |
| dffun9 6569 | Alternate definition of a ... |
| funfn 6570 | A class is a function if a... |
| funfnd 6571 | A function is a function o... |
| funi 6572 | The identity relation is a... |
| nfunv 6573 | The universal class is not... |
| funopg 6574 | A Kuratowski ordered pair ... |
| funopab 6575 | A class of ordered pairs i... |
| funopabeq 6576 | A class of ordered pairs o... |
| funopab4 6577 | A class of ordered pairs o... |
| funmpt 6578 | A function in maps-to nota... |
| funmpt2 6579 | Functionality of a class g... |
| funco 6580 | The composition of two fun... |
| funresfunco 6581 | Composition of two functio... |
| funres 6582 | A restriction of a functio... |
| funresd 6583 | A restriction of a functio... |
| funssres 6584 | The restriction of a funct... |
| fun2ssres 6585 | Equality of restrictions o... |
| funun 6586 | The union of functions wit... |
| fununmo 6587 | If the union of classes is... |
| fununfun 6588 | If the union of classes is... |
| fundif 6589 | A function with removed el... |
| funcnvsn 6590 | The converse singleton of ... |
| funsng 6591 | A singleton of an ordered ... |
| fnsng 6592 | Functionality and domain o... |
| funsn 6593 | A singleton of an ordered ... |
| funprg 6594 | A set of two pairs is a fu... |
| funtpg 6595 | A set of three pairs is a ... |
| funpr 6596 | A function with a domain o... |
| funtp 6597 | A function with a domain o... |
| fnsn 6598 | Functionality and domain o... |
| fnprg 6599 | Function with a domain of ... |
| fntpg 6600 | Function with a domain of ... |
| fntp 6601 | A function with a domain o... |
| funcnvpr 6602 | The converse pair of order... |
| funcnvtp 6603 | The converse triple of ord... |
| funcnvqp 6604 | The converse quadruple of ... |
| fun0 6605 | The empty set is a functio... |
| funcnv0 6606 | The converse of the empty ... |
| funcnvcnv 6607 | The double converse of a f... |
| funcnv2 6608 | A simpler equivalence for ... |
| funcnv 6609 | The converse of a class is... |
| funcnv3 6610 | A condition showing a clas... |
| fun2cnv 6611 | The double converse of a c... |
| svrelfun 6612 | A single-valued relation i... |
| fncnv 6613 | Single-rootedness (see ~ f... |
| fun11 6614 | Two ways of stating that `... |
| fununi 6615 | The union of a chain (with... |
| funin 6616 | The intersection with a fu... |
| funres11 6617 | The restriction of a one-t... |
| funcnvres 6618 | The converse of a restrict... |
| cnvresid 6619 | Converse of a restricted i... |
| funcnvres2 6620 | The converse of a restrict... |
| funimacnv 6621 | The image of the preimage ... |
| funimass1 6622 | A kind of contraposition l... |
| funimass2 6623 | A kind of contraposition l... |
| imadif 6624 | The image of a difference ... |
| imain 6625 | The image of an intersecti... |
| funimaexg 6626 | Axiom of Replacement using... |
| funimaex 6627 | The image of a set under a... |
| isarep1 6628 | Part of a study of the Axi... |
| isarep2 6629 | Part of a study of the Axi... |
| fneq1 6630 | Equality theorem for funct... |
| fneq2 6631 | Equality theorem for funct... |
| fneq1d 6632 | Equality deduction for fun... |
| fneq2d 6633 | Equality deduction for fun... |
| fneq12d 6634 | Equality deduction for fun... |
| fneq12 6635 | Equality theorem for funct... |
| fneq1i 6636 | Equality inference for fun... |
| fneq2i 6637 | Equality inference for fun... |
| nffn 6638 | Bound-variable hypothesis ... |
| fnfun 6639 | A function with domain is ... |
| fnfund 6640 | A function with domain is ... |
| fnrel 6641 | A function with domain is ... |
| fndm 6642 | The domain of a function. ... |
| fndmi 6643 | The domain of a function. ... |
| fndmd 6644 | The domain of a function. ... |
| funfni 6645 | Inference to convert a fun... |
| fndmu 6646 | A function has a unique do... |
| fnbr 6647 | The first argument of bina... |
| fnop 6648 | The first argument of an o... |
| fneu 6649 | There is exactly one value... |
| fneu2 6650 | There is exactly one value... |
| fnunres1 6651 | Restriction of a disjoint ... |
| fnunres2 6652 | Restriction of a disjoint ... |
| fnun 6653 | The union of two functions... |
| fnund 6654 | The union of two functions... |
| fnunop 6655 | Extension of a function wi... |
| fncofn 6656 | Composition of a function ... |
| fnco 6657 | Composition of two functio... |
| fnresdm 6658 | A function does not change... |
| fnresdisj 6659 | A function restricted to a... |
| 2elresin 6660 | Membership in two function... |
| fnssresb 6661 | Restriction of a function ... |
| fnssres 6662 | Restriction of a function ... |
| fnssresd 6663 | Restriction of a function ... |
| fnresin1 6664 | Restriction of a function'... |
| fnresin2 6665 | Restriction of a function'... |
| fnres 6666 | An equivalence for functio... |
| idfn 6667 | The identity relation is a... |
| fnresi 6668 | The restricted identity re... |
| fnima 6669 | The image of a function's ... |
| fn0 6670 | A function with empty doma... |
| fnimadisj 6671 | A class that is disjoint w... |
| fnimaeq0 6672 | Images under a function ne... |
| dfmpt3 6673 | Alternate definition for t... |
| mptfnf 6674 | The maps-to notation defin... |
| fnmptf 6675 | The maps-to notation defin... |
| fnopabg 6676 | Functionality and domain o... |
| fnopab 6677 | Functionality and domain o... |
| mptfng 6678 | The maps-to notation defin... |
| fnmpt 6679 | The maps-to notation defin... |
| fnmptd 6680 | The maps-to notation defin... |
| mpt0 6681 | A mapping operation with e... |
| fnmpti 6682 | Functionality and domain o... |
| dmmpti 6683 | Domain of the mapping oper... |
| dmmptd 6684 | The domain of the mapping ... |
| mptun 6685 | Union of mappings which ar... |
| partfun 6686 | Rewrite a function defined... |
| feq1 6687 | Equality theorem for funct... |
| feq2 6688 | Equality theorem for funct... |
| feq3 6689 | Equality theorem for funct... |
| feq23 6690 | Equality theorem for funct... |
| feq1d 6691 | Equality deduction for fun... |
| feq1dd 6692 | Equality deduction for fun... |
| feq2d 6693 | Equality deduction for fun... |
| feq3d 6694 | Equality deduction for fun... |
| feq2dd 6695 | Equality deduction for fun... |
| feq3dd 6696 | Equality deduction for fun... |
| feq12d 6697 | Equality deduction for fun... |
| feq123d 6698 | Equality deduction for fun... |
| feq123 6699 | Equality theorem for funct... |
| feq1i 6700 | Equality inference for fun... |
| feq2i 6701 | Equality inference for fun... |
| feq12i 6702 | Equality inference for fun... |
| feq23i 6703 | Equality inference for fun... |
| feq23d 6704 | Equality deduction for fun... |
| nff 6705 | Bound-variable hypothesis ... |
| sbcfng 6706 | Distribute proper substitu... |
| sbcfg 6707 | Distribute proper substitu... |
| elimf 6708 | Eliminate a mapping hypoth... |
| ffn 6709 | A mapping is a function wi... |
| ffnd 6710 | A mapping is a function wi... |
| dffn2 6711 | Any function is a mapping ... |
| ffun 6712 | A mapping is a function. ... |
| ffunOLD 6713 | Obsolete version of ~ ffun... |
| ffund 6714 | A mapping is a function, d... |
| frel 6715 | A mapping is a relation. ... |
| freld 6716 | A mapping is a relation. ... |
| frn 6717 | The range of a mapping. (... |
| frnd 6718 | Deduction form of ~ frn . ... |
| fdm 6719 | The domain of a mapping. ... |
| fdmd 6720 | Deduction form of ~ fdm . ... |
| fdmi 6721 | Inference associated with ... |
| dffn3 6722 | A function maps to its ran... |
| ffrn 6723 | A function maps to its ran... |
| ffrnb 6724 | Characterization of a func... |
| ffrnbd 6725 | A function maps to its ran... |
| fss 6726 | Expanding the codomain of ... |
| fssd 6727 | Expanding the codomain of ... |
| fssdmd 6728 | Expressing that a class is... |
| fssdm 6729 | Expressing that a class is... |
| fimass 6730 | The image of a class under... |
| fimassd 6731 | The image of a class is a ... |
| fimacnv 6732 | The preimage of the codoma... |
| fcof 6733 | Composition of a function ... |
| fco 6734 | Composition of two functio... |
| fcod 6735 | Composition of two mapping... |
| fco2 6736 | Functionality of a composi... |
| fssxp 6737 | A mapping is a class of or... |
| funssxp 6738 | Two ways of specifying a p... |
| ffdm 6739 | A mapping is a partial fun... |
| ffdmd 6740 | The domain of a function. ... |
| fdmrn 6741 | A different way to write `... |
| funcofd 6742 | Composition of two functio... |
| opelf 6743 | The members of an ordered ... |
| fun 6744 | The union of two functions... |
| fun2 6745 | The union of two functions... |
| fun2d 6746 | The union of functions wit... |
| fnfco 6747 | Composition of two functio... |
| fssres 6748 | Restriction of a function ... |
| fssresd 6749 | Restriction of a function ... |
| fssres2 6750 | Restriction of a restricte... |
| fresin 6751 | An identity for the mappin... |
| resasplit 6752 | If two functions agree on ... |
| fresaun 6753 | The union of two functions... |
| fresaunres2 6754 | From the union of two func... |
| fresaunres1 6755 | From the union of two func... |
| fcoi1 6756 | Composition of a mapping a... |
| fcoi2 6757 | Composition of restricted ... |
| feu 6758 | There is exactly one value... |
| fcnvres 6759 | The converse of a restrict... |
| fimacnvdisj 6760 | The preimage of a class di... |
| fint 6761 | Function into an intersect... |
| fin 6762 | Mapping into an intersecti... |
| f0 6763 | The empty function. (Cont... |
| f00 6764 | A class is a function with... |
| f0bi 6765 | A function with empty doma... |
| f0dom0 6766 | A function is empty iff it... |
| f0rn0 6767 | If there is no element in ... |
| fconst 6768 | A Cartesian product with a... |
| fconstg 6769 | A Cartesian product with a... |
| fnconstg 6770 | A Cartesian product with a... |
| fconst6g 6771 | Constant function with loo... |
| fconst6 6772 | A constant function as a m... |
| f1eq1 6773 | Equality theorem for one-t... |
| f1eq2 6774 | Equality theorem for one-t... |
| f1eq3 6775 | Equality theorem for one-t... |
| nff1 6776 | Bound-variable hypothesis ... |
| dff12 6777 | Alternate definition of a ... |
| f1f 6778 | A one-to-one mapping is a ... |
| f1fn 6779 | A one-to-one mapping is a ... |
| f1fun 6780 | A one-to-one mapping is a ... |
| f1funOLD 6781 | Obsolete version of ~ f1fu... |
| f1rel 6782 | A one-to-one onto mapping ... |
| f1relOLD 6783 | Obsolete version of ~ f1re... |
| f1dm 6784 | The domain of a one-to-one... |
| f1ss 6785 | A function that is one-to-... |
| f1ssr 6786 | A function that is one-to-... |
| f1ssres 6787 | A function that is one-to-... |
| f1resf1 6788 | The restriction of an inje... |
| f1cnvcnv 6789 | Two ways to express that a... |
| f1cof1 6790 | Composition of two one-to-... |
| f1co 6791 | Composition of one-to-one ... |
| foeq1 6792 | Equality theorem for onto ... |
| foeq2 6793 | Equality theorem for onto ... |
| foeq3 6794 | Equality theorem for onto ... |
| nffo 6795 | Bound-variable hypothesis ... |
| fof 6796 | An onto mapping is a mappi... |
| fofun 6797 | An onto mapping is a funct... |
| fofn 6798 | An onto mapping is a funct... |
| forn 6799 | The codomain of an onto fu... |
| dffo2 6800 | Alternate definition of an... |
| foima 6801 | The image of the domain of... |
| dffn4 6802 | A function maps onto its r... |
| funforn 6803 | A function maps its domain... |
| fodmrnu 6804 | An onto function has uniqu... |
| fimadmfo 6805 | A function is a function o... |
| fores 6806 | Restriction of an onto fun... |
| fimadmfoALT 6807 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6808 | The preimage of the codoma... |
| focofo 6809 | Composition of onto functi... |
| foco 6810 | Composition of onto functi... |
| foconst 6811 | A nonzero constant functio... |
| f1oeq1 6812 | Equality theorem for one-t... |
| f1oeq2 6813 | Equality theorem for one-t... |
| f1oeq3 6814 | Equality theorem for one-t... |
| f1oeq23 6815 | Equality theorem for one-t... |
| f1eq123d 6816 | Equality deduction for one... |
| foeq123d 6817 | Equality deduction for ont... |
| f1oeq123d 6818 | Equality deduction for one... |
| f1oeq1d 6819 | Equality deduction for one... |
| f1oeq2d 6820 | Equality deduction for one... |
| f1oeq3d 6821 | Equality deduction for one... |
| nff1o 6822 | Bound-variable hypothesis ... |
| f1of1 6823 | A one-to-one onto mapping ... |
| f1of 6824 | A one-to-one onto mapping ... |
| f1ofn 6825 | A one-to-one onto mapping ... |
| f1ofun 6826 | A one-to-one onto mapping ... |
| f1orel 6827 | A one-to-one onto mapping ... |
| f1odm 6828 | The domain of a one-to-one... |
| f1odmOLD 6829 | Obsolete version of ~ f1od... |
| dff1o2 6830 | Alternate definition of on... |
| dff1o3 6831 | Alternate definition of on... |
| f1ofo 6832 | A one-to-one onto function... |
| dff1o4 6833 | Alternate definition of on... |
| dff1o5 6834 | Alternate definition of on... |
| f1orn 6835 | A one-to-one function maps... |
| f1f1orn 6836 | A one-to-one function maps... |
| f1ocnv 6837 | The converse of a one-to-o... |
| f1ocnvb 6838 | A relation is a one-to-one... |
| f1ores 6839 | The restriction of a one-t... |
| f1orescnv 6840 | The converse of a one-to-o... |
| f1imacnv 6841 | Preimage of an image. (Co... |
| foimacnv 6842 | A reverse version of ~ f1i... |
| foun 6843 | The union of two onto func... |
| f1oun 6844 | The union of two one-to-on... |
| f1un 6845 | The union of two one-to-on... |
| resdif 6846 | The restriction of a one-t... |
| resin 6847 | The restriction of a one-t... |
| f1oco 6848 | Composition of one-to-one ... |
| f1cnv 6849 | The converse of an injecti... |
| funcocnv2 6850 | Composition with the conve... |
| fococnv2 6851 | The composition of an onto... |
| f1ococnv2 6852 | The composition of a one-t... |
| f1cocnv2 6853 | Composition of an injectiv... |
| f1ococnv1 6854 | The composition of a one-t... |
| f1cocnv1 6855 | Composition of an injectiv... |
| funcoeqres 6856 | Express a constraint on a ... |
| f1ssf1 6857 | A subset of an injective f... |
| f10 6858 | The empty set maps one-to-... |
| f10d 6859 | The empty set maps one-to-... |
| f1o00 6860 | One-to-one onto mapping of... |
| fo00 6861 | Onto mapping of the empty ... |
| f1o0 6862 | One-to-one onto mapping of... |
| f1oi 6863 | A restriction of the ident... |
| f1oiOLD 6864 | Obsolete version of ~ f1oi... |
| f1ovi 6865 | The identity relation is a... |
| f1osn 6866 | A singleton of an ordered ... |
| f1osng 6867 | A singleton of an ordered ... |
| f1sng 6868 | A singleton of an ordered ... |
| fsnd 6869 | A singleton of an ordered ... |
| f1oprswap 6870 | A two-element swap is a bi... |
| f1oprg 6871 | An unordered pair of order... |
| tz6.12-2 6872 | Function value when ` F ` ... |
| tz6.12-2OLD 6873 | Obsolete version of ~ tz6.... |
| fveu 6874 | The value of a function at... |
| brprcneu 6875 | If ` A ` is a proper class... |
| brprcneuALT 6876 | Alternate proof of ~ brprc... |
| fvprc 6877 | A function's value at a pr... |
| fvprcALT 6878 | Alternate proof of ~ fvprc... |
| rnfvprc 6879 | The range of a function va... |
| fv2 6880 | Alternate definition of fu... |
| dffv3 6881 | A definition of function v... |
| dffv4 6882 | The previous definition of... |
| elfv 6883 | Membership in a function v... |
| fveq1 6884 | Equality theorem for funct... |
| fveq2 6885 | Equality theorem for funct... |
| fveq1i 6886 | Equality inference for fun... |
| fveq1d 6887 | Equality deduction for fun... |
| fveq2i 6888 | Equality inference for fun... |
| fveq2d 6889 | Equality deduction for fun... |
| 2fveq3 6890 | Equality theorem for neste... |
| fveq12i 6891 | Equality deduction for fun... |
| fveq12d 6892 | Equality deduction for fun... |
| fveqeq2d 6893 | Equality deduction for fun... |
| fveqeq2 6894 | Equality deduction for fun... |
| nffv 6895 | Bound-variable hypothesis ... |
| nffvmpt1 6896 | Bound-variable hypothesis ... |
| nffvd 6897 | Deduction version of bound... |
| fvex 6898 | The value of a class exist... |
| fvexi 6899 | The value of a class exist... |
| fvexd 6900 | The value of a class exist... |
| fvif 6901 | Move a conditional outside... |
| iffv 6902 | Move a conditional outside... |
| fv3 6903 | Alternate definition of th... |
| fvres 6904 | The value of a restricted ... |
| fvresd 6905 | The value of a restricted ... |
| funssfv 6906 | The value of a member of t... |
| tz6.12c 6907 | Corollary of Theorem 6.12(... |
| tz6.12-1 6908 | Function value. Theorem 6... |
| tz6.12 6909 | Function value. Theorem 6... |
| tz6.12f 6910 | Function value, using boun... |
| tz6.12i 6911 | Corollary of Theorem 6.12(... |
| fvbr0 6912 | Two possibilities for the ... |
| fvrn0 6913 | A function value is a memb... |
| fvn0fvelrn 6914 | If the value of a function... |
| elfvunirn 6915 | A function value is a subs... |
| fvssunirn 6916 | The result of a function v... |
| ndmfv 6917 | The value of a class outsi... |
| ndmfvrcl 6918 | Reverse closure law for fu... |
| elfvdm 6919 | If a function value has a ... |
| elfvex 6920 | If a function value has a ... |
| elfvexd 6921 | If a function value has a ... |
| eliman0 6922 | A nonempty function value ... |
| nfvres 6923 | The value of a non-member ... |
| nfunsn 6924 | If the restriction of a cl... |
| fvfundmfvn0 6925 | If the "value of a class" ... |
| 0fv 6926 | Function value of the empt... |
| fv2prc 6927 | A function value of a func... |
| elfv2ex 6928 | If a function value of a f... |
| fveqres 6929 | Equal values imply equal v... |
| csbfv12 6930 | Move class substitution in... |
| csbfv2g 6931 | Move class substitution in... |
| csbfv 6932 | Substitution for a functio... |
| funbrfv 6933 | The second argument of a b... |
| funopfv 6934 | The second element in an o... |
| fnbrfvb 6935 | Equivalence of function va... |
| fnopfvb 6936 | Equivalence of function va... |
| fvelima2 6937 | Function value in an image... |
| funbrfvb 6938 | Equivalence of function va... |
| funopfvb 6939 | Equivalence of function va... |
| fnbrfvb2 6940 | Version of ~ fnbrfvb for f... |
| fdmeu 6941 | There is exactly one codom... |
| funbrfv2b 6942 | Function value in terms of... |
| dffn5 6943 | Representation of a functi... |
| fnrnfv 6944 | The range of a function ex... |
| fvelrnb 6945 | A member of a function's r... |
| foelcdmi 6946 | A member of a surjective f... |
| dfimafn 6947 | Alternate definition of th... |
| dfimafn2 6948 | Alternate definition of th... |
| funimass4 6949 | Membership relation for th... |
| fvelima 6950 | Function value in an image... |
| funimassd 6951 | Sufficient condition for t... |
| fvelimad 6952 | Function value in an image... |
| feqmptd 6953 | Deduction form of ~ dffn5 ... |
| feqresmpt 6954 | Express a restricted funct... |
| feqmptdf 6955 | Deduction form of ~ dffn5f... |
| dffn5f 6956 | Representation of a functi... |
| fvelimab 6957 | Function value in an image... |
| fvelimabd 6958 | Deduction form of ~ fvelim... |
| fimarab 6959 | Expressing the image of a ... |
| unima 6960 | Image of a union. (Contri... |
| fvi 6961 | The value of the identity ... |
| fviss 6962 | The value of the identity ... |
| fniinfv 6963 | The indexed intersection o... |
| fnsnfv 6964 | Singleton of function valu... |
| opabiotafun 6965 | Define a function whose va... |
| opabiotadm 6966 | Define a function whose va... |
| opabiota 6967 | Define a function whose va... |
| fnimapr 6968 | The image of a pair under ... |
| fnimatpd 6969 | The image of an unordered ... |
| ssimaex 6970 | The existence of a subimag... |
| ssimaexg 6971 | The existence of a subimag... |
| funfv 6972 | A simplified expression fo... |
| funfv2 6973 | The value of a function. ... |
| funfv2f 6974 | The value of a function. ... |
| fvun 6975 | Value of the union of two ... |
| fvun1 6976 | The value of a union when ... |
| fvun2 6977 | The value of a union when ... |
| fvun1d 6978 | The value of a union when ... |
| fvun2d 6979 | The value of a union when ... |
| dffv2 6980 | Alternate definition of fu... |
| dmfco 6981 | Domains of a function comp... |
| fvco2 6982 | Value of a function compos... |
| fvco 6983 | Value of a function compos... |
| fvcod 6984 | Value of a function compos... |
| fvco3 6985 | Value of a function compos... |
| fvco3d 6986 | Value of a function compos... |
| fvco4i 6987 | Conditions for a compositi... |
| fvopab3g 6988 | Value of a function given ... |
| fvopab3ig 6989 | Value of a function given ... |
| brfvopabrbr 6990 | The binary relation of a f... |
| fvmptg 6991 | Value of a function given ... |
| fvmpti 6992 | Value of a function given ... |
| fvmpt 6993 | Value of a function given ... |
| fvmpt2f 6994 | Value of a function given ... |
| funcnvmpt 6995 | Condition for a function i... |
| fvtresfn 6996 | Functionality of a tuple-r... |
| fvmpts 6997 | Value of a function given ... |
| fvmpt3 6998 | Value of a function given ... |
| fvmpt3i 6999 | Value of a function given ... |
| fvmptdf 7000 | Deduction version of ~ fvm... |
| fvmptd 7001 | Deduction version of ~ fvm... |
| fvmptd2 7002 | Deduction version of ~ fvm... |
| mptrcl 7003 | Reverse closure for a mapp... |
| fvmpt2i 7004 | Value of a function given ... |
| fvmpt2 7005 | Value of a function given ... |
| fvmptss 7006 | If all the values of the m... |
| fvmpt2d 7007 | Deduction version of ~ fvm... |
| fvmptex 7008 | Express a function ` F ` w... |
| fvmptd3f 7009 | Alternate deduction versio... |
| fvmptd2f 7010 | Alternate deduction versio... |
| fvmptdv 7011 | Alternate deduction versio... |
| fvmptdv2 7012 | Alternate deduction versio... |
| mpteqb 7013 | Bidirectional equality the... |
| fvmptt 7014 | Closed theorem form of ~ f... |
| fvmptf 7015 | Value of a function given ... |
| fvmptnf 7016 | The value of a function gi... |
| fvmptd3 7017 | Deduction version of ~ fvm... |
| fvmptd4 7018 | Deduction version of ~ fvm... |
| fvmptn 7019 | This somewhat non-intuitiv... |
| fvmptss2 7020 | A mapping always evaluates... |
| elfvmptrab1w 7021 | Implications for the value... |
| elfvmptrab1 7022 | Implications for the value... |
| elfvmptrab 7023 | Implications for the value... |
| fvopab4ndm 7024 | Value of a function given ... |
| fvmptndm 7025 | Value of a function given ... |
| fvmptrabfv 7026 | Value of a function mappin... |
| fvopab5 7027 | The value of a function th... |
| fvopab6 7028 | Value of a function given ... |
| eqfnfv 7029 | Equality of functions is d... |
| eqfnfv2 7030 | Equality of functions is d... |
| eqfnfv3 7031 | Derive equality of functio... |
| eqfnfvd 7032 | Deduction for equality of ... |
| eqfnfv2f 7033 | Equality of functions is d... |
| fsneq 7034 | Equality condition for two... |
| eqfunfv 7035 | Equality of functions is d... |
| eqfnun 7036 | Two functions on ` A u. B ... |
| fvreseq0 7037 | Equality of restricted fun... |
| fvreseq1 7038 | Equality of a function res... |
| fvreseq 7039 | Equality of restricted fun... |
| fnmptfvd 7040 | A function with a given do... |
| fndmdif 7041 | Two ways to express the lo... |
| fndmdifcom 7042 | The difference set between... |
| fndmdifeq0 7043 | The difference set of two ... |
| fndmin 7044 | Two ways to express the lo... |
| fneqeql 7045 | Two functions are equal if... |
| fneqeql2 7046 | Two functions are equal if... |
| fnreseql 7047 | Two functions are equal on... |
| chfnrn 7048 | The range of a choice func... |
| funfvop 7049 | Ordered pair with function... |
| funfvbrb 7050 | Two ways to say that ` A `... |
| fvimacnvi 7051 | A member of a preimage is ... |
| fvimacnv 7052 | The argument of a function... |
| funimass3 7053 | A kind of contraposition l... |
| funimass5 7054 | A subclass of a preimage i... |
| funconstss 7055 | Two ways of specifying tha... |
| fvimacnvALT 7056 | Alternate proof of ~ fvima... |
| elpreima 7057 | Membership in the preimage... |
| elpreimad 7058 | Membership in the preimage... |
| fniniseg 7059 | Membership in the preimage... |
| fncnvima2 7060 | Inverse images under funct... |
| fniniseg2 7061 | Inverse point images under... |
| unpreima 7062 | Preimage of a union. (Con... |
| inpreima 7063 | Preimage of an intersectio... |
| difpreima 7064 | Preimage of a difference. ... |
| respreima 7065 | The preimage of a restrict... |
| cnvimainrn 7066 | The preimage of the inters... |
| sspreima 7067 | The preimage of a subset i... |
| iunpreima 7068 | Preimage of an indexed uni... |
| iinpreima 7069 | Preimage of an intersectio... |
| intpreima 7070 | Preimage of an intersectio... |
| fimacnvinrn 7071 | Taking the converse image ... |
| fimacnvinrn2 7072 | Taking the converse image ... |
| rescnvimafod 7073 | The restriction of a funct... |
| fvn0ssdmfun 7074 | If a class' function value... |
| fnopfv 7075 | Ordered pair with function... |
| fvelrn 7076 | A function's value belongs... |
| nelrnfvne 7077 | A function value cannot be... |
| fveqdmss 7078 | If the empty set is not co... |
| fveqressseq 7079 | If the empty set is not co... |
| fnfvelrn 7080 | A function's value belongs... |
| ffvelcdm 7081 | A function's value belongs... |
| fnfvelrnd 7082 | A function's value belongs... |
| ffvelcdmi 7083 | A function's value belongs... |
| ffvelcdmda 7084 | A function's value belongs... |
| ffvelcdmd 7085 | A function's value belongs... |
| feldmfvelcdm 7086 | A class is an element of t... |
| rexrn 7087 | Restricted existential qua... |
| ralrn 7088 | Restricted universal quant... |
| elrnrexdm 7089 | For any element in the ran... |
| elrnrexdmb 7090 | For any element in the ran... |
| eldmrexrn 7091 | For any element in the dom... |
| eldmrexrnb 7092 | For any element in the dom... |
| fvcofneq 7093 | The values of two function... |
| ralrnmptw 7094 | A restricted quantifier ov... |
| rexrnmptw 7095 | A restricted quantifier ov... |
| ralrnmpt 7096 | A restricted quantifier ov... |
| rexrnmpt 7097 | A restricted quantifier ov... |
| f0cli 7098 | Unconditional closure of a... |
| dff2 7099 | Alternate definition of a ... |
| dff3 7100 | Alternate definition of a ... |
| dff4 7101 | Alternate definition of a ... |
| dffo3 7102 | An onto mapping expressed ... |
| dffo4 7103 | Alternate definition of an... |
| dffo5 7104 | Alternate definition of an... |
| exfo 7105 | A relation equivalent to t... |
| dffo3f 7106 | An onto mapping expressed ... |
| foelrn 7107 | Property of a surjective f... |
| foelrnf 7108 | Property of a surjective f... |
| foco2 7109 | If a composition of two fu... |
| fmpt 7110 | Functionality of the mappi... |
| f1ompt 7111 | Express bijection for a ma... |
| fmpti 7112 | Functionality of the mappi... |
| fvmptelcdm 7113 | The value of a function at... |
| fmptd 7114 | Domain and codomain of the... |
| fmpttd 7115 | Version of ~ fmptd with in... |
| fmpt3d 7116 | Domain and codomain of the... |
| fmptdf 7117 | A version of ~ fmptd using... |
| fompt 7118 | Express being onto for a m... |
| ffnfv 7119 | A function maps to a class... |
| ffnfvf 7120 | A function maps to a class... |
| fnfvrnss 7121 | An upper bound for range d... |
| fcdmssb 7122 | A function is a function i... |
| rnmptss 7123 | The range of an operation ... |
| rnmptssd 7124 | The range of a function gi... |
| fmpt2d 7125 | Domain and codomain of the... |
| ffvresb 7126 | A necessary and sufficient... |
| fssrescdmd 7127 | Restriction of a function ... |
| f1oresrab 7128 | Build a bijection between ... |
| f1ossf1o 7129 | Restricting a bijection, w... |
| fmptco 7130 | Composition of two functio... |
| fmptcof 7131 | Version of ~ fmptco where ... |
| fmptcos 7132 | Composition of two functio... |
| cofmpt 7133 | Express composition of a m... |
| fcompt 7134 | Express composition of two... |
| fcoconst 7135 | Composition with a constan... |
| fsn 7136 | A function maps a singleto... |
| fsn2 7137 | A function that maps a sin... |
| fsng 7138 | A function maps a singleto... |
| fsn2g 7139 | A function that maps a sin... |
| xpsng 7140 | The Cartesian product of t... |
| xpsn 7141 | The Cartesian product of t... |
| xpprsng 7142 | The Cartesian product of a... |
| xpsnprg 7143 | The Cartesian product of a... |
| xpsntpg 7144 | The Cartesian product of a... |
| f1o2sn 7145 | A singleton consisting in ... |
| residpr 7146 | Restriction of the identit... |
| dfmpt 7147 | Alternate definition for t... |
| fnasrn 7148 | A function expressed as th... |
| idref 7149 | Two ways to state that a r... |
| funiun 7150 | A function is a union of s... |
| funopsn 7151 | If a function is an ordere... |
| funopsnOLD 7152 | Obsolete version of ~ funo... |
| funop 7153 | An ordered pair is a funct... |
| funopdmsn 7154 | The domain of a function w... |
| funsndifnop 7155 | A singleton of an ordered ... |
| funsneqopb 7156 | A singleton of an ordered ... |
| ressnop0 7157 | If ` A ` is not in ` C ` ,... |
| fpr 7158 | A function with a domain o... |
| fprg 7159 | A function with a domain o... |
| ftpg 7160 | A function with a domain o... |
| ftp 7161 | A function with a domain o... |
| fnressn 7162 | A function restricted to a... |
| funressn 7163 | A function restricted to a... |
| fressnfv 7164 | The value of a function re... |
| fvrnressn 7165 | If the value of a function... |
| fvressn 7166 | The value of a function re... |
| fvconst 7167 | The value of a constant fu... |
| fnsnr 7168 | If a class belongs to a fu... |
| fnsnbg 7169 | A function's domain is a s... |
| fnsnb 7170 | A function whose domain is... |
| fnsnbOLD 7171 | Obsolete version of ~ fnsn... |
| fmptsn 7172 | Express a singleton functi... |
| fmptsng 7173 | Express a singleton functi... |
| fmptsnd 7174 | Express a singleton functi... |
| fmptap 7175 | Append an additional value... |
| fmptapd 7176 | Append an additional value... |
| fmptpr 7177 | Express a pair function in... |
| fvresi 7178 | The value of a restricted ... |
| fninfp 7179 | Express the class of fixed... |
| fnelfp 7180 | Property of a fixed point ... |
| fndifnfp 7181 | Express the class of non-f... |
| fnelnfp 7182 | Property of a non-fixed po... |
| fnnfpeq0 7183 | A function is the identity... |
| fvunsn 7184 | Remove an ordered pair not... |
| fvsng 7185 | The value of a singleton o... |
| fvsn 7186 | The value of a singleton o... |
| fvsnun1 7187 | The value of a function wi... |
| fvsnun2 7188 | The value of a function wi... |
| fnsnsplit 7189 | Split a function into a si... |
| fsnunf 7190 | Adjoining a point to a fun... |
| fsnunf2 7191 | Adjoining a point to a pun... |
| fsnunfv 7192 | Recover the added point fr... |
| fsnunres 7193 | Recover the original funct... |
| funresdfunsn 7194 | Restricting a function to ... |
| fvpr1g 7195 | The value of a function wi... |
| fvpr2g 7196 | The value of a function wi... |
| fvpr1 7197 | The value of a function wi... |
| fvpr2 7198 | The value of a function wi... |
| fprb 7199 | A condition for functionho... |
| fvtp1 7200 | The first value of a funct... |
| fvtp2 7201 | The second value of a func... |
| fvtp3 7202 | The third value of a funct... |
| fvtp1g 7203 | The value of a function wi... |
| fvtp2g 7204 | The value of a function wi... |
| fvtp3g 7205 | The value of a function wi... |
| fvtp0 7206 | The undefined value of a f... |
| tpres 7207 | An unordered triple of ord... |
| fvconst2g 7208 | The value of a constant fu... |
| fconst2g 7209 | A constant function expres... |
| fvconst2 7210 | The value of a constant fu... |
| fconst2 7211 | A constant function expres... |
| fconst5 7212 | Two ways to express that a... |
| rnmptc 7213 | Range of a constant functi... |
| fnprb 7214 | A function whose domain ha... |
| fntpb 7215 | A function whose domain ha... |
| fnpr2g 7216 | A function whose domain ha... |
| fpr2g 7217 | A function that maps a pai... |
| fconstfv 7218 | A constant function expres... |
| fconst3 7219 | Two ways to express a cons... |
| fconst4 7220 | Two ways to express a cons... |
| resfunexg 7221 | The restriction of a funct... |
| resiexd 7222 | The restriction of the ide... |
| fnex 7223 | If the domain of a functio... |
| fnexd 7224 | If the domain of a functio... |
| funex 7225 | If the domain of a functio... |
| opabex 7226 | Existence of a function ex... |
| mptexg 7227 | If the domain of a functio... |
| mptexgf 7228 | If the domain of a functio... |
| mptex 7229 | If the domain of a functio... |
| mptexd 7230 | If the domain of a functio... |
| mptrabex 7231 | If the domain of a functio... |
| fex 7232 | If the domain of a mapping... |
| fexd 7233 | If the domain of a mapping... |
| mptfvmpt 7234 | A function in maps-to nota... |
| eufnfv 7235 | A function is uniquely det... |
| funfvima 7236 | A function's value in a pr... |
| funfvima2 7237 | A function's value in an i... |
| funfvima2d 7238 | A function's value in a pr... |
| fnfvima 7239 | The function value of an o... |
| fnfvimad 7240 | A function's value belongs... |
| resfvresima 7241 | The value of the function ... |
| funfvima3 7242 | A class including a functi... |
| ralima 7243 | Universal quantification u... |
| rexima 7244 | Existential quantification... |
| fvclss 7245 | Upper bound for the class ... |
| elabrex 7246 | Elementhood in an image se... |
| elabrexg 7247 | Elementhood in an image se... |
| abrexco 7248 | Composition of two image m... |
| imaiun 7249 | The image of an indexed un... |
| imauni 7250 | The image of a union is th... |
| fniunfv 7251 | The indexed union of a fun... |
| funiunfv 7252 | The indexed union of a fun... |
| funiunfvf 7253 | The indexed union of a fun... |
| eluniima 7254 | Membership in the union of... |
| elunirn 7255 | Membership in the union of... |
| elunirnALT 7256 | Alternate proof of ~ eluni... |
| fnunirn 7257 | Membership in a union of s... |
| dff13 7258 | A one-to-one function in t... |
| dff13f 7259 | A one-to-one function in t... |
| f1veqaeq 7260 | If the values of a one-to-... |
| f1cofveqaeq 7261 | If the values of a composi... |
| f1cofveqaeqALT 7262 | Alternate proof of ~ f1cof... |
| dff14i 7263 | A one-to-one function maps... |
| 2f1fvneq 7264 | If two one-to-one function... |
| f1mpt 7265 | Express injection for a ma... |
| f1fveq 7266 | Equality of function value... |
| f1elima 7267 | Membership in the image of... |
| f1imass 7268 | Taking images under a one-... |
| f1imaeq 7269 | Taking images under a one-... |
| f1imapss 7270 | Taking images under a one-... |
| fpropnf1 7271 | A function, given by an un... |
| f1dom3fv3dif 7272 | The function values for a ... |
| f1dom3el3dif 7273 | The codomain of a 1-1 func... |
| dff14a 7274 | A one-to-one function in t... |
| dff14b 7275 | A one-to-one function in t... |
| dff15 7276 | A one-to-one function in t... |
| f1resveqaeq 7277 | If a function restricted t... |
| f1resrcmplf1dlem 7278 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 7279 | If a function's restrictio... |
| f1ounsn 7280 | Extension of a bijection b... |
| f12dfv 7281 | A one-to-one function with... |
| f13dfv 7282 | A one-to-one function with... |
| dff1o6 7283 | A one-to-one onto function... |
| f1ocnvfv1 7284 | The converse value of the ... |
| f1ocnvfv2 7285 | The value of the converse ... |
| f1ocnvfv 7286 | Relationship between the v... |
| f1ocnvfvb 7287 | Relationship between the v... |
| nvof1o 7288 | An involution is a bijecti... |
| nvocnv 7289 | The converse of an involut... |
| f1cdmsn 7290 | If a one-to-one function w... |
| fsnex 7291 | Relate a function with a s... |
| f1prex 7292 | Relate a one-to-one functi... |
| f1ocnvdm 7293 | The value of the converse ... |
| f1ocnvfvrneq 7294 | If the values of a one-to-... |
| fcof1 7295 | An application is injectiv... |
| fcofo 7296 | An application is surjecti... |
| cbvfo 7297 | Change bound variable betw... |
| cbvexfo 7298 | Change bound variable betw... |
| cocan1 7299 | An injection is left-cance... |
| cocan2 7300 | A surjection is right-canc... |
| fcof1oinvd 7301 | Show that a function is th... |
| fcof1od 7302 | A function is bijective if... |
| 2fcoidinvd 7303 | Show that a function is th... |
| fcof1o 7304 | Show that two functions ar... |
| 2fvcoidd 7305 | Show that the composition ... |
| 2fvidf1od 7306 | A function is bijective if... |
| 2fvidinvd 7307 | Show that two functions ar... |
| foeqcnvco 7308 | Condition for function equ... |
| f1eqcocnv 7309 | Condition for function equ... |
| fveqf1o 7310 | Given a bijection ` F ` , ... |
| f1ocoima 7311 | The composition of two bij... |
| nf1const 7312 | A constant function from a... |
| nf1oconst 7313 | A constant function from a... |
| f1ofvswap 7314 | Swapping two values in a b... |
| fvf1pr 7315 | Values of a one-to-one fun... |
| fliftrel 7316 | ` F ` , a function lift, i... |
| fliftel 7317 | Elementhood in the relatio... |
| fliftel1 7318 | Elementhood in the relatio... |
| fliftcnv 7319 | Converse of the relation `... |
| fliftfun 7320 | The function ` F ` is the ... |
| fliftfund 7321 | The function ` F ` is the ... |
| fliftfuns 7322 | The function ` F ` is the ... |
| fliftf 7323 | The domain and range of th... |
| fliftval 7324 | The value of the function ... |
| isoeq1 7325 | Equality theorem for isomo... |
| isoeq2 7326 | Equality theorem for isomo... |
| isoeq3 7327 | Equality theorem for isomo... |
| isoeq4 7328 | Equality theorem for isomo... |
| isoeq5 7329 | Equality theorem for isomo... |
| nfiso 7330 | Bound-variable hypothesis ... |
| isof1o 7331 | An isomorphism is a one-to... |
| isof1oidb 7332 | A function is a bijection ... |
| isof1oopb 7333 | A function is a bijection ... |
| isorel 7334 | An isomorphism connects bi... |
| soisores 7335 | Express the condition of i... |
| soisoi 7336 | Infer isomorphism from one... |
| isoid 7337 | Identity law for isomorphi... |
| isocnv 7338 | Converse law for isomorphi... |
| isocnv2 7339 | Converse law for isomorphi... |
| isocnv3 7340 | Complementation law for is... |
| isores2 7341 | An isomorphism from one we... |
| isores1 7342 | An isomorphism from one we... |
| isores3 7343 | Induced isomorphism on a s... |
| isotr 7344 | Composition (transitive) l... |
| isomin 7345 | Isomorphisms preserve mini... |
| isoini 7346 | Isomorphisms preserve init... |
| isoini2 7347 | Isomorphisms are isomorphi... |
| isofrlem 7348 | Lemma for ~ isofr . (Cont... |
| isoselem 7349 | Lemma for ~ isose . (Cont... |
| isofr 7350 | An isomorphism preserves w... |
| isose 7351 | An isomorphism preserves s... |
| isofr2 7352 | A weak form of ~ isofr tha... |
| isopolem 7353 | Lemma for ~ isopo . (Cont... |
| isopo 7354 | An isomorphism preserves t... |
| isosolem 7355 | Lemma for ~ isoso . (Cont... |
| isoso 7356 | An isomorphism preserves t... |
| isowe 7357 | An isomorphism preserves t... |
| isowe2 7358 | A weak form of ~ isowe tha... |
| f1oiso 7359 | Any one-to-one onto functi... |
| f1oiso2 7360 | Any one-to-one onto functi... |
| f1owe 7361 | Well-ordering of isomorphi... |
| f1oweOLD 7362 | Obsolete version of f1owe ... |
| f1we 7363 | Pull back a well ordering ... |
| weniso 7364 | A set-like well-ordering h... |
| weisoeq 7365 | Thus, there is at most one... |
| weisoeq2 7366 | Thus, there is at most one... |
| knatar 7367 | The Knaster-Tarski theorem... |
| fvresval 7368 | The value of a restricted ... |
| funeldmb 7369 | If ` (/) ` is not part of ... |
| eqfunresadj 7370 | Law for adjoining an eleme... |
| eqfunressuc 7371 | Law for equality of restri... |
| fnssintima 7372 | Condition for subset of an... |
| fnimasnd 7373 | The image of a function by... |
| canth 7374 | No set ` A ` is equinumero... |
| ncanth 7375 | Cantor's theorem fails for... |
| riotaeqdv 7378 | Formula-building deduction... |
| riotabidv 7379 | Formula-building deduction... |
| riotaeqbidv 7380 | Equality deduction for res... |
| riotaex 7381 | Restricted iota is a set. ... |
| riotav 7382 | An iota restricted to the ... |
| riotauni 7383 | Restricted iota in terms o... |
| nfriota1 7384 | The abstraction variable i... |
| nfriotadw 7385 | Deduction version of ~ nfr... |
| cbvriotaw 7386 | Change bound variable in a... |
| cbvriotavw 7387 | Change bound variable in a... |
| nfriotad 7388 | Deduction version of ~ nfr... |
| nfriota 7389 | A variable not free in a w... |
| cbvriota 7390 | Change bound variable in a... |
| cbvriotav 7391 | Change bound variable in a... |
| csbriota 7392 | Interchange class substitu... |
| riotacl2 7393 | Membership law for "the un... |
| riotacl 7394 | Closure of restricted iota... |
| riotasbc 7395 | Substitution law for descr... |
| riotabidva 7396 | Equivalent wff's yield equ... |
| riotabiia 7397 | Equivalent wff's yield equ... |
| riota1 7398 | Property of restricted iot... |
| riota1a 7399 | Property of iota. (Contri... |
| riota2df 7400 | A deduction version of ~ r... |
| riota2f 7401 | This theorem shows a condi... |
| riota2 7402 | This theorem shows a condi... |
| riotaeqimp 7403 | If two restricted iota des... |
| riotaprop 7404 | Properties of a restricted... |
| riota5f 7405 | A method for computing res... |
| riota5 7406 | A method for computing res... |
| riotass2 7407 | Restriction of a unique el... |
| riotass 7408 | Restriction of a unique el... |
| moriotass 7409 | Restriction of a unique el... |
| snriota 7410 | A restricted class abstrac... |
| riotaxfrd 7411 | Change the variable ` x ` ... |
| eusvobj2 7412 | Specify the same property ... |
| eusvobj1 7413 | Specify the same object in... |
| f1ofveu 7414 | There is one domain elemen... |
| f1ocnvfv3 7415 | Value of the converse of a... |
| riotaund 7416 | Restricted iota equals the... |
| riotassuni 7417 | The restricted iota class ... |
| riotaclb 7418 | Bidirectional closure of r... |
| riotarab 7419 | Restricted iota of a restr... |
| oveq 7426 | Equality theorem for opera... |
| oveq1 7427 | Equality theorem for opera... |
| oveq2 7428 | Equality theorem for opera... |
| oveq12 7429 | Equality theorem for opera... |
| oveq1i 7430 | Equality inference for ope... |
| oveq2i 7431 | Equality inference for ope... |
| oveq12i 7432 | Equality inference for ope... |
| oveqi 7433 | Equality inference for ope... |
| oveq123i 7434 | Equality inference for ope... |
| oveq1d 7435 | Equality deduction for ope... |
| oveq2d 7436 | Equality deduction for ope... |
| oveqd 7437 | Equality deduction for ope... |
| oveq12d 7438 | Equality deduction for ope... |
| oveqan12d 7439 | Equality deduction for ope... |
| oveqan12rd 7440 | Equality deduction for ope... |
| oveq123d 7441 | Equality deduction for ope... |
| fvoveq1d 7442 | Equality deduction for nes... |
| fvoveq1 7443 | Equality theorem for neste... |
| ovanraleqv 7444 | Equality theorem for a con... |
| imbrov2fvoveq 7445 | Equality theorem for neste... |
| ovrspc2v 7446 | If an operation value is a... |
| oveqrspc2v 7447 | Restricted specialization ... |
| oveqdr 7448 | Equality of two operations... |
| nfovd 7449 | Deduction version of bound... |
| nfov 7450 | Bound-variable hypothesis ... |
| oprabidw 7451 | The law of concretion. Sp... |
| oprabid 7452 | The law of concretion. Sp... |
| ovex 7453 | The result of an operation... |
| ovexi 7454 | The result of an operation... |
| ovexd 7455 | The result of an operation... |
| ovssunirn 7456 | The result of an operation... |
| 0ov 7457 | Operation value of the emp... |
| ovprc 7458 | The value of an operation ... |
| ovprc1 7459 | The value of an operation ... |
| ovprc2 7460 | The value of an operation ... |
| ovrcl 7461 | Reverse closure for an ope... |
| elfvov1 7462 | Utility theorem: reverse c... |
| elfvov2 7463 | Utility theorem: reverse c... |
| csbov123 7464 | Move class substitution in... |
| csbov 7465 | Move class substitution in... |
| csbov12g 7466 | Move class substitution in... |
| csbov1g 7467 | Move class substitution in... |
| csbov2g 7468 | Move class substitution in... |
| rspceov 7469 | A frequently used special ... |
| elovimad 7470 | Elementhood of the image s... |
| fnbrovb 7471 | Value of a binary operatio... |
| fnotovb 7472 | Equivalence of operation v... |
| opabbrex 7473 | A collection of ordered pa... |
| opabresex2 7474 | Restrictions of a collecti... |
| fvmptopab 7475 | The function value of a ma... |
| f1opr 7476 | Condition for an operation... |
| brfvopab 7477 | The classes involved in a ... |
| dfoprab2 7478 | Class abstraction for oper... |
| reloprab 7479 | An operation class abstrac... |
| oprabv 7480 | If a pair and a class are ... |
| nfoprab1 7481 | The abstraction variables ... |
| nfoprab2 7482 | The abstraction variables ... |
| nfoprab3 7483 | The abstraction variables ... |
| nfoprab 7484 | Bound-variable hypothesis ... |
| oprabbid 7485 | Equivalent wff's yield equ... |
| oprabbidv 7486 | Equivalent wff's yield equ... |
| oprabbii 7487 | Equivalent wff's yield equ... |
| ssoprab2 7488 | Equivalence of ordered pai... |
| ssoprab2b 7489 | Equivalence of ordered pai... |
| eqoprab2bw 7490 | Equivalence of ordered pai... |
| eqoprab2b 7491 | Equivalence of ordered pai... |
| mpoeq123 7492 | An equality theorem for th... |
| mpoeq12 7493 | An equality theorem for th... |
| mpoeq123dva 7494 | An equality deduction for ... |
| mpoeq123dv 7495 | An equality deduction for ... |
| mpoeq123i 7496 | An equality inference for ... |
| mpoeq3dva 7497 | Slightly more general equa... |
| mpoeq3ia 7498 | An equality inference for ... |
| mpoeq3dv 7499 | An equality deduction for ... |
| nfmpo1 7500 | Bound-variable hypothesis ... |
| nfmpo2 7501 | Bound-variable hypothesis ... |
| nfmpo 7502 | Bound-variable hypothesis ... |
| 0mpo0 7503 | A mapping operation with e... |
| mpo0v 7504 | A mapping operation with e... |
| mpo0 7505 | A mapping operation with e... |
| oprab4 7506 | Two ways to state the doma... |
| cbvoprab1 7507 | Rule used to change first ... |
| cbvoprab2 7508 | Change the second bound va... |
| cbvoprab12 7509 | Rule used to change first ... |
| cbvoprab12v 7510 | Rule used to change first ... |
| cbvoprab3 7511 | Rule used to change the th... |
| cbvoprab3v 7512 | Rule used to change the th... |
| cbvmpox 7513 | Rule to change the bound v... |
| cbvmpo 7514 | Rule to change the bound v... |
| cbvmpov 7515 | Rule to change the bound v... |
| elimdelov 7516 | Eliminate a hypothesis whi... |
| brif1 7517 | Move a relation inside and... |
| ovif 7518 | Move a conditional outside... |
| ovif2 7519 | Move a conditional outside... |
| ovif12 7520 | Move a conditional outside... |
| ifov 7521 | Move a conditional outside... |
| ifmpt2v 7522 | Move a conditional inside ... |
| dmoprab 7523 | The domain of an operation... |
| dmoprabss 7524 | The domain of an operation... |
| rnoprab 7525 | The range of an operation ... |
| rnoprab2 7526 | The range of a restricted ... |
| reldmoprab 7527 | The domain of an operation... |
| oprabss 7528 | Structure of an operation ... |
| eloprabga 7529 | The law of concretion for ... |
| eloprabg 7530 | The law of concretion for ... |
| ssoprab2i 7531 | Inference of operation cla... |
| mpov 7532 | Operation with universal d... |
| mpomptx 7533 | Express a two-argument fun... |
| mpompt 7534 | Express a two-argument fun... |
| mpodifsnif 7535 | A mapping with two argumen... |
| mposnif 7536 | A mapping with two argumen... |
| fconstmpo 7537 | Representation of a consta... |
| resoprab 7538 | Restriction of an operatio... |
| resoprab2 7539 | Restriction of an operator... |
| resmpo 7540 | Restriction of the mapping... |
| funoprabg 7541 | "At most one" is a suffici... |
| funoprab 7542 | "At most one" is a suffici... |
| fnoprabg 7543 | Functionality and domain o... |
| mpofun 7544 | The maps-to notation for a... |
| fnoprab 7545 | Functionality and domain o... |
| ffnov 7546 | An operation maps to a cla... |
| fovcld 7547 | Closure law for an operati... |
| fovcl 7548 | Closure law for an operati... |
| eqfnov 7549 | Equality of two operations... |
| eqfnov2 7550 | Two operators with the sam... |
| fnov 7551 | Representation of a functi... |
| mpo2eqb 7552 | Bidirectional equality the... |
| rnmpo 7553 | The range of an operation ... |
| reldmmpo 7554 | The domain of an operation... |
| elrnmpog 7555 | Membership in the range of... |
| elrnmpo 7556 | Membership in the range of... |
| elimampo 7557 | Membership in the image of... |
| elrnmpores 7558 | Membership in the range of... |
| ralrnmpo 7559 | A restricted quantifier ov... |
| rexrnmpo 7560 | A restricted quantifier ov... |
| ovid 7561 | The value of an operation ... |
| ovidig 7562 | The value of an operation ... |
| ovidi 7563 | The value of an operation ... |
| ov 7564 | The value of an operation ... |
| ovigg 7565 | The value of an operation ... |
| ovig 7566 | The value of an operation ... |
| ovmpt4g 7567 | Value of a function given ... |
| ovmpos 7568 | Value of a function given ... |
| ov2gf 7569 | The value of an operation ... |
| ovmpodxf 7570 | Value of an operation give... |
| ovmpodx 7571 | Value of an operation give... |
| ovmpod 7572 | Value of an operation give... |
| ovmpox 7573 | The value of an operation ... |
| ovmpoga 7574 | Value of an operation give... |
| ovmpoa 7575 | Value of an operation give... |
| ovmpodf 7576 | Alternate deduction versio... |
| ovmpodv 7577 | Alternate deduction versio... |
| ovmpodv2 7578 | Alternate deduction versio... |
| ovmpog 7579 | Value of an operation give... |
| ovmpo 7580 | Value of an operation give... |
| ovmpot 7581 | The value of an operation ... |
| fvmpopr2d 7582 | Value of an operation give... |
| ov3 7583 | The value of an operation ... |
| ov6g 7584 | The value of an operation ... |
| ovg 7585 | The value of an operation ... |
| ovres 7586 | The value of a restricted ... |
| ovresd 7587 | Lemma for converting metri... |
| oprres 7588 | The restriction of an oper... |
| ovn0ssdmfun 7589 | If a class' operation valu... |
| oprssov 7590 | The value of a member of t... |
| fovcdm 7591 | An operation's value belon... |
| fovcdmda 7592 | An operation's value belon... |
| fovcdmd 7593 | An operation's value belon... |
| fnrnov 7594 | The range of an operation ... |
| foov 7595 | An onto mapping of an oper... |
| fnovrn 7596 | An operation's value belon... |
| ovelrn 7597 | A member of an operation's... |
| funimassov 7598 | Membership relation for th... |
| ovelimab 7599 | Operation value in an imag... |
| ovima0 7600 | An operation value is a me... |
| ovconst2 7601 | The value of a constant op... |
| oprssdm 7602 | Domain of closure of an op... |
| nssdmovg 7603 | The value of an operation ... |
| ndmovg 7604 | The value of an operation ... |
| ndmov 7605 | The value of an operation ... |
| ndmovcl 7606 | The closure of an operatio... |
| ndmovrcl 7607 | Reverse closure law, when ... |
| ndmovcom 7608 | Any operation is commutati... |
| ndmovass 7609 | Any operation is associati... |
| ndmovdistr 7610 | Any operation is distribut... |
| ndmovord 7611 | Elimination of redundant a... |
| ndmovordi 7612 | Elimination of redundant a... |
| caovclg 7613 | Convert an operation closu... |
| caovcld 7614 | Convert an operation closu... |
| caovcl 7615 | Convert an operation closu... |
| caovcomg 7616 | Convert an operation commu... |
| caovcomd 7617 | Convert an operation commu... |
| caovcom 7618 | Convert an operation commu... |
| caovassg 7619 | Convert an operation assoc... |
| caovassd 7620 | Convert an operation assoc... |
| caovass 7621 | Convert an operation assoc... |
| caovcang 7622 | Convert an operation cance... |
| caovcand 7623 | Convert an operation cance... |
| caovcanrd 7624 | Commute the arguments of a... |
| caovcan 7625 | Convert an operation cance... |
| caovordig 7626 | Convert an operation order... |
| caovordid 7627 | Convert an operation order... |
| caovordg 7628 | Convert an operation order... |
| caovordd 7629 | Convert an operation order... |
| caovord2d 7630 | Operation ordering law wit... |
| caovord3d 7631 | Ordering law. (Contribute... |
| caovord 7632 | Convert an operation order... |
| caovord2 7633 | Operation ordering law wit... |
| caovord3 7634 | Ordering law. (Contribute... |
| caovdig 7635 | Convert an operation distr... |
| caovdid 7636 | Convert an operation distr... |
| caovdir2d 7637 | Convert an operation distr... |
| caovdirg 7638 | Convert an operation rever... |
| caovdird 7639 | Convert an operation distr... |
| caovdi 7640 | Convert an operation distr... |
| caov32d 7641 | Rearrange arguments in a c... |
| caov12d 7642 | Rearrange arguments in a c... |
| caov31d 7643 | Rearrange arguments in a c... |
| caov13d 7644 | Rearrange arguments in a c... |
| caov4d 7645 | Rearrange arguments in a c... |
| caov411d 7646 | Rearrange arguments in a c... |
| caov42d 7647 | Rearrange arguments in a c... |
| caov32 7648 | Rearrange arguments in a c... |
| caov12 7649 | Rearrange arguments in a c... |
| caov31 7650 | Rearrange arguments in a c... |
| caov13 7651 | Rearrange arguments in a c... |
| caov4 7652 | Rearrange arguments in a c... |
| caov411 7653 | Rearrange arguments in a c... |
| caov42 7654 | Rearrange arguments in a c... |
| caovdir 7655 | Reverse distributive law. ... |
| caovdilem 7656 | Lemma used by real number ... |
| caovlem2 7657 | Lemma used in real number ... |
| caovmo 7658 | Uniqueness of inverse elem... |
| imaeqexov 7659 | Substitute an operation va... |
| imaeqalov 7660 | Substitute an operation va... |
| mpondm0 7661 | The value of an operation ... |
| elmpocl 7662 | If a two-parameter class i... |
| elmpocl1 7663 | If a two-parameter class i... |
| elmpocl2 7664 | If a two-parameter class i... |
| elovmpod 7665 | Utility lemma for two-para... |
| elovmpo 7666 | Utility lemma for two-para... |
| elovmporab 7667 | Implications for the value... |
| elovmporab1w 7668 | Implications for the value... |
| elovmporab1 7669 | Implications for the value... |
| 2mpo0 7670 | If the operation value of ... |
| relmptopab 7671 | Any function to sets of or... |
| f1ocnvd 7672 | Describe an implicit one-t... |
| f1od 7673 | Describe an implicit one-t... |
| f1ocnv2d 7674 | Describe an implicit one-t... |
| f1o2d 7675 | Describe an implicit one-t... |
| f1opw2 7676 | A one-to-one mapping induc... |
| f1opw 7677 | A one-to-one mapping induc... |
| elovmpt3imp 7678 | If the value of a function... |
| ovmpt3rab1 7679 | The value of an operation ... |
| ovmpt3rabdm 7680 | If the value of a function... |
| elovmpt3rab1 7681 | Implications for the value... |
| elovmpt3rab 7682 | Implications for the value... |
| mpt3mpt 7685 | Express a three-argument f... |
| mpt3eqdv 7686 | An equality deduction for ... |
| funmpt3 7687 | A function in maps-to nota... |
| mpt3fvd 7688 | Value of a three-argument ... |
| mpt3fvotd 7689 | Value of a three-argument ... |
| mpt3fvot2d 7690 | Value of a three-argument ... |
| ofeqd 7695 | Equality theorem for funct... |
| ofeq 7696 | Equality theorem for funct... |
| ofreq 7697 | Equality theorem for funct... |
| ofexg 7698 | A function operation restr... |
| nfof 7699 | Hypothesis builder for fun... |
| nfofr 7700 | Hypothesis builder for fun... |
| ofrfvalg 7701 | Value of a relation applie... |
| offval 7702 | Value of an operation appl... |
| ofrfval 7703 | Value of a relation applie... |
| ofval 7704 | Evaluate a function operat... |
| ofrval 7705 | Exhibit a function relatio... |
| offn 7706 | The function operation pro... |
| offun 7707 | The function operation pro... |
| offval2f 7708 | The function operation exp... |
| ofmresval 7709 | Value of a restriction of ... |
| fnfvof 7710 | Function value of a pointw... |
| off 7711 | The function operation pro... |
| ofres 7712 | Restrict the operands of a... |
| offval2 7713 | The function operation exp... |
| ofrfval2 7714 | The function relation acti... |
| offvalfv 7715 | The function operation exp... |
| ofmpteq 7716 | Value of a pointwise opera... |
| coof 7717 | The composition of a _homo... |
| ofco 7718 | The composition of a funct... |
| offveq 7719 | Convert an identity of the... |
| offveqb 7720 | Equivalent expressions for... |
| ofc1 7721 | Left operation by a consta... |
| ofc2 7722 | Right operation by a const... |
| ofc12 7723 | Function operation on two ... |
| caofref 7724 | Transfer a reflexive law t... |
| caofinvl 7725 | Transfer a left inverse la... |
| caofid0l 7726 | Transfer a left identity l... |
| caofid0r 7727 | Transfer a right identity ... |
| caofid1 7728 | Transfer a right absorptio... |
| caofid2 7729 | Transfer a right absorptio... |
| caofcom 7730 | Transfer a commutative law... |
| caofidlcan 7731 | Transfer a cancellation/id... |
| caofrss 7732 | Transfer a relation subset... |
| caofass 7733 | Transfer an associative la... |
| caoftrn 7734 | Transfer a transitivity la... |
| caofdi 7735 | Transfer a distributive la... |
| caofdir 7736 | Transfer a reverse distrib... |
| caonncan 7737 | Transfer ~ nncan -shaped l... |
| relrpss 7740 | The proper subset relation... |
| brrpssg 7741 | The proper subset relation... |
| brrpss 7742 | The proper subset relation... |
| porpss 7743 | Every class is partially o... |
| sorpss 7744 | Express strict ordering un... |
| sorpssi 7745 | Property of a chain of set... |
| sorpssun 7746 | A chain of sets is closed ... |
| sorpssin 7747 | A chain of sets is closed ... |
| sorpssuni 7748 | In a chain of sets, a maxi... |
| sorpssint 7749 | In a chain of sets, a mini... |
| sorpsscmpl 7750 | The componentwise compleme... |
| zfun 7752 | Axiom of Union expressed w... |
| axun2 7753 | A variant of the Axiom of ... |
| uniex2 7754 | The Axiom of Union using t... |
| uniex2OLD 7755 | Obsolete version of ~ unie... |
| vuniex 7756 | The union of a setvar is a... |
| uniexg 7757 | The ZF Axiom of Union in c... |
| uniex 7758 | The Axiom of Union in clas... |
| uniexd 7759 | Deduction version of the Z... |
| unexg 7760 | The union of two sets is a... |
| unex 7761 | The union of two sets is a... |
| tpex 7762 | An unordered triple of cla... |
| unexb 7763 | Existence of union is equi... |
| xpexg 7764 | The Cartesian product of t... |
| xpexd 7765 | The Cartesian product of t... |
| 3xpexg 7766 | The Cartesian product of t... |
| xpex 7767 | The Cartesian product of t... |
| unexd 7768 | The union of two sets is a... |
| sqxpexg 7769 | The Cartesian square of a ... |
| abnexg 7770 | Sufficient condition for a... |
| abnex 7771 | Sufficient condition for a... |
| snnex 7772 | The class of all singleton... |
| pwnex 7773 | The class of all power set... |
| difex2 7774 | If the subtrahend of a cla... |
| difsnexi 7775 | If the difference of a cla... |
| uniuni 7776 | Expression for double unio... |
| uniexr 7777 | Converse of the Axiom of U... |
| uniexb 7778 | The Axiom of Union and its... |
| pwexr 7779 | Converse of the Axiom of P... |
| pwexb 7780 | The Axiom of Power Sets an... |
| elpwpwel 7781 | A class belongs to a doubl... |
| eldifpw 7782 | Membership in a power clas... |
| elpwun 7783 | Membership in the power cl... |
| pwuncl 7784 | Power classes are closed u... |
| iunpw 7785 | An indexed union of a powe... |
| fr3nr 7786 | A well-founded relation ha... |
| epne3 7787 | A well-founded class conta... |
| dfwe2 7788 | Alternate definition of we... |
| epweon 7789 | The membership relation we... |
| epweonALT 7790 | Alternate proof of ~ epweo... |
| ordon 7791 | The class of all ordinal n... |
| onprc 7792 | No set contains all ordina... |
| ssorduni 7793 | The union of a class of or... |
| ssonuni 7794 | The union of a set of ordi... |
| ssonunii 7795 | The union of a set of ordi... |
| ordeleqon 7796 | A way to express the ordin... |
| ordsson 7797 | Any ordinal class is a sub... |
| dford5 7798 | A class is ordinal iff it ... |
| onss 7799 | An ordinal number is a sub... |
| predon 7800 | The predecessor of an ordi... |
| ssonprc 7801 | Two ways of saying a class... |
| onuni 7802 | The union of an ordinal nu... |
| orduni 7803 | The union of an ordinal cl... |
| onint 7804 | The intersection (infimum)... |
| onint0 7805 | The intersection of a clas... |
| onssmin 7806 | A nonempty class of ordina... |
| onminesb 7807 | If a property is true for ... |
| onminsb 7808 | If a property is true for ... |
| oninton 7809 | The intersection of a none... |
| onintrab 7810 | The intersection of a clas... |
| onintrab2 7811 | An existence condition equ... |
| onnmin 7812 | No member of a set of ordi... |
| onnminsb 7813 | An ordinal number smaller ... |
| oneqmin 7814 | A way to show that an ordi... |
| uniordint 7815 | The union of a set of ordi... |
| onminex 7816 | If a wff is true for an or... |
| sucon 7817 | The class of all ordinal n... |
| sucexb 7818 | A successor exists iff its... |
| sucexg 7819 | The successor of a set is ... |
| sucex 7820 | The successor of a set is ... |
| onmindif2 7821 | The minimum of a class of ... |
| ordsuci 7822 | The successor of an ordina... |
| sucexeloni 7823 | If the successor of an ord... |
| onsuc 7824 | The successor of an ordina... |
| ordsuc 7825 | A class is ordinal if and ... |
| ordpwsuc 7826 | The collection of ordinals... |
| onpwsuc 7827 | The collection of ordinal ... |
| onsucb 7828 | A class is an ordinal numb... |
| ordsucss 7829 | The successor of an elemen... |
| onpsssuc 7830 | An ordinal number is a pro... |
| ordelsuc 7831 | A set belongs to an ordina... |
| onsucmin 7832 | The successor of an ordina... |
| ordsucelsuc 7833 | Membership is inherited by... |
| ordsucsssuc 7834 | The subclass relationship ... |
| ordsucuniel 7835 | Given an element ` A ` of ... |
| ordsucun 7836 | The successor of the maxim... |
| ordunpr 7837 | The maximum of two ordinal... |
| ordunel 7838 | The maximum of two ordinal... |
| onsucuni 7839 | A class of ordinal numbers... |
| ordsucuni 7840 | An ordinal class is a subc... |
| orduniorsuc 7841 | An ordinal class is either... |
| unon 7842 | The class of all ordinal n... |
| ordunisuc 7843 | An ordinal class is equal ... |
| orduniss2 7844 | The union of the ordinal s... |
| onsucuni2 7845 | A successor ordinal is the... |
| 0elsuc 7846 | The successor of an ordina... |
| limon 7847 | The class of ordinal numbe... |
| onuniorsuc 7848 | An ordinal number is eithe... |
| onssi 7849 | An ordinal number is a sub... |
| onsuci 7850 | The successor of an ordina... |
| onuninsuci 7851 | An ordinal is equal to its... |
| onsucssi 7852 | A set belongs to an ordina... |
| nlimsucg 7853 | A successor is not a limit... |
| orduninsuc 7854 | An ordinal class is equal ... |
| ordunisuc2 7855 | An ordinal equal to its un... |
| ordzsl 7856 | An ordinal is zero, a succ... |
| onzsl 7857 | An ordinal number is zero,... |
| dflim3 7858 | An alternate definition of... |
| dflim4 7859 | An alternate definition of... |
| limsuc 7860 | The successor of a member ... |
| limsssuc 7861 | A class includes a limit o... |
| nlimon 7862 | Two ways to express the cl... |
| limuni3 7863 | The union of a nonempty cl... |
| tfi 7864 | The Principle of Transfini... |
| tfisg 7865 | A closed form of ~ tfis . ... |
| tfis 7866 | Transfinite Induction Sche... |
| tfis2f 7867 | Transfinite Induction Sche... |
| tfis2 7868 | Transfinite Induction Sche... |
| tfis3 7869 | Transfinite Induction Sche... |
| tfisi 7870 | A transfinite induction sc... |
| tfinds 7871 | Principle of Transfinite I... |
| tfindsg 7872 | Transfinite Induction (inf... |
| tfindsg2 7873 | Transfinite Induction (inf... |
| tfindes 7874 | Transfinite Induction with... |
| tfinds2 7875 | Transfinite Induction (inf... |
| tfinds3 7876 | Principle of Transfinite I... |
| dfom2 7879 | An alternate definition of... |
| elom 7880 | Membership in omega. The ... |
| omsson 7881 | Omega is a subset of ` On ... |
| limomss 7882 | The class of natural numbe... |
| nnon 7883 | A natural number is an ord... |
| nnoni 7884 | A natural number is an ord... |
| nnord 7885 | A natural number is ordina... |
| trom 7886 | The class of finite ordina... |
| ordom 7887 | The class of finite ordina... |
| elnn 7888 | A member of a natural numb... |
| omon 7889 | The class of natural numbe... |
| omelon2 7890 | Omega is an ordinal number... |
| nnlim 7891 | A natural number is not a ... |
| omssnlim 7892 | The class of natural numbe... |
| limom 7893 | Omega is a limit ordinal. ... |
| peano2b 7894 | A class belongs to omega i... |
| nnsuc 7895 | A nonzero natural number i... |
| omsucne 7896 | A natural number is not th... |
| ssnlim 7897 | An ordinal subclass of non... |
| omsinds 7898 | Strong (or "total") induct... |
| omun 7899 | The union of two finite or... |
| peano1 7900 | Zero is a natural number. ... |
| peano2 7901 | The successor of any natur... |
| peano3 7902 | The successor of any natur... |
| peano3OLD 7903 | Obsolete version of ~ pean... |
| peano4 7904 | Two natural numbers are eq... |
| peano5 7905 | The induction postulate: a... |
| nn0suc 7906 | A natural number is either... |
| find 7907 | The Principle of Finite In... |
| finds 7908 | Principle of Finite Induct... |
| findsg 7909 | Principle of Finite Induct... |
| finds2 7910 | Principle of Finite Induct... |
| finds1 7911 | Principle of Finite Induct... |
| findes 7912 | Finite induction with expl... |
| dmexg 7913 | The domain of a set is a s... |
| rnexg 7914 | The range of a set is a se... |
| dmexd 7915 | The domain of a set is a s... |
| fndmexd 7916 | If a function is a set, it... |
| dmfex 7917 | If a mapping is a set, its... |
| fndmexb 7918 | The domain of a function i... |
| fdmexb 7919 | The domain of a function i... |
| dmfexALT 7920 | Alternate proof of ~ dmfex... |
| dmex 7921 | The domain of a set is a s... |
| rnex 7922 | The range of a set is a se... |
| iprc 7923 | The identity function is a... |
| resiexg 7924 | The existence of a restric... |
| imaexg 7925 | The image of a set is a se... |
| imaex 7926 | The image of a set is a se... |
| rnexd 7927 | The range of a set is a se... |
| imaexd 7928 | The image of a set is a se... |
| exse2 7929 | Any set relation is set-li... |
| xpexr 7930 | If a Cartesian product is ... |
| xpexr2 7931 | If a nonempty Cartesian pr... |
| xpexcnv 7932 | A condition where the conv... |
| soex 7933 | If the relation in a stric... |
| elxp4 7934 | Membership in a Cartesian ... |
| elxp5 7935 | Membership in a Cartesian ... |
| cnvexg 7936 | The converse of a set is a... |
| cnvex 7937 | The converse of a set is a... |
| relcnvexb 7938 | A relation is a set iff it... |
| f1oexrnex 7939 | If the range of a 1-1 onto... |
| f1oexbi 7940 | There is a one-to-one onto... |
| coexg 7941 | The composition of two set... |
| coex 7942 | The composition of two set... |
| coexd 7943 | The composition of two set... |
| funcnvuni 7944 | The union of a chain (with... |
| fun11uni 7945 | The union of a chain (with... |
| resf1extb 7946 | Extension of an injection ... |
| resf1ext2b 7947 | Extension of an injection ... |
| fex2 7948 | A function with bounded do... |
| fabexd 7949 | Existence of a set of func... |
| fabexg 7950 | Existence of a set of func... |
| fabex 7951 | Existence of a set of func... |
| mapex 7952 | The class of all functions... |
| f1oabexg 7953 | The class of all 1-1-onto ... |
| fiunlem 7954 | Lemma for ~ fiun and ~ f1i... |
| fiun 7955 | The union of a chain (with... |
| f1iun 7956 | The union of a chain (with... |
| fviunfun 7957 | The function value of an i... |
| ffoss 7958 | Relationship between a map... |
| f11o 7959 | Relationship between one-t... |
| resfunexgALT 7960 | Alternate proof of ~ resfu... |
| cofunexg 7961 | Existence of a composition... |
| cofunex2g 7962 | Existence of a composition... |
| fnexALT 7963 | Alternate proof of ~ fnex ... |
| funexw 7964 | Weak version of ~ funex th... |
| mptexw 7965 | Weak version of ~ mptex th... |
| funrnex 7966 | If the domain of a functio... |
| zfrep6OLD 7967 | Obsolete proof of ~ zfrep6... |
| focdmex 7968 | If the domain of an onto f... |
| f1dmex 7969 | If the codomain of a one-t... |
| f1ovv 7970 | The codomain/range of a 1-... |
| fvclex 7971 | Existence of the class of ... |
| fvresex 7972 | Existence of the class of ... |
| abrexexg 7973 | Existence of a class abstr... |
| abrexex 7974 | Existence of a class abstr... |
| iunexg 7975 | The existence of an indexe... |
| abrexex2g 7976 | Existence of an existentia... |
| opabex3d 7977 | Existence of an ordered pa... |
| opabex3rd 7978 | Existence of an ordered pa... |
| opabex3 7979 | Existence of an ordered pa... |
| iunex 7980 | The existence of an indexe... |
| abrexex2 7981 | Existence of an existentia... |
| abexssex 7982 | Existence of a class abstr... |
| abexex 7983 | A condition where a class ... |
| f1oweALT 7984 | Alternate proof of one dir... |
| wemoiso 7985 | Thus, there is at most one... |
| wemoiso2 7986 | Thus, there is at most one... |
| oprabexd 7987 | Existence of an operator a... |
| oprabex 7988 | Existence of an operation ... |
| oprabex3 7989 | Existence of an operation ... |
| oprabrexex2 7990 | Existence of an existentia... |
| ab2rexex 7991 | Existence of a class abstr... |
| ab2rexex2 7992 | Existence of an existentia... |
| xpexgALT 7993 | Alternate proof of ~ xpexg... |
| offval3 7994 | General value of ` ( F oF ... |
| offres 7995 | Pointwise combination comm... |
| ofmres 7996 | Equivalent expressions for... |
| ofmresex 7997 | Existence of a restriction... |
| mptcnfimad 7998 | The converse of a mapping ... |
| 1stval 8003 | The value of the function ... |
| 2ndval 8004 | The value of the function ... |
| 1stnpr 8005 | Value of the first-member ... |
| 2ndnpr 8006 | Value of the second-member... |
| 1st0 8007 | The value of the first-mem... |
| 2nd0 8008 | The value of the second-me... |
| op1st 8009 | Extract the first member o... |
| op2nd 8010 | Extract the second member ... |
| op1std 8011 | Extract the first member o... |
| op2ndd 8012 | Extract the second member ... |
| op1stg 8013 | Extract the first member o... |
| op2ndg 8014 | Extract the second member ... |
| ot1stg 8015 | Extract the first member o... |
| ot2ndg 8016 | Extract the second member ... |
| ot3rdg 8017 | Extract the third member o... |
| 1stval2 8018 | Alternate value of the fun... |
| 2ndval2 8019 | Alternate value of the fun... |
| oteqimp 8020 | The components of an order... |
| fo1st 8021 | The ` 1st ` function maps ... |
| fo2nd 8022 | The ` 2nd ` function maps ... |
| br1steqg 8023 | Uniqueness condition for t... |
| br2ndeqg 8024 | Uniqueness condition for t... |
| f1stres 8025 | Mapping of a restriction o... |
| f2ndres 8026 | Mapping of a restriction o... |
| fo1stres 8027 | Onto mapping of a restrict... |
| fo2ndres 8028 | Onto mapping of a restrict... |
| 1st2val 8029 | Value of an alternate defi... |
| 2nd2val 8030 | Value of an alternate defi... |
| 1stcof 8031 | Composition of the first m... |
| 2ndcof 8032 | Composition of the second ... |
| xp1st 8033 | Location of the first elem... |
| xp2nd 8034 | Location of the second ele... |
| elxp6 8035 | Membership in a Cartesian ... |
| elxp7 8036 | Membership in a Cartesian ... |
| eqopi 8037 | Equality with an ordered p... |
| xp2 8038 | Representation of Cartesia... |
| unielxp 8039 | The membership relation fo... |
| 1st2nd2 8040 | Reconstruction of a member... |
| 1st2ndb 8041 | Reconstruction of an order... |
| xpopth 8042 | An ordered pair theorem fo... |
| eqop 8043 | Two ways to express equali... |
| eqop2 8044 | Two ways to express equali... |
| op1steq 8045 | Two ways of expressing tha... |
| opreuopreu 8046 | There is a unique ordered ... |
| el2xptp0 8047 | A member of a nested Carte... |
| el2xpss 8048 | Version of ~ elrel for tri... |
| 2nd1st 8049 | Swap the members of an ord... |
| 1st2nd 8050 | Reconstruction of a member... |
| 1stdm 8051 | The first ordered pair com... |
| 2ndrn 8052 | The second ordered pair co... |
| 1st2ndbr 8053 | Express an element of a re... |
| releldm2 8054 | Two ways of expressing mem... |
| reldm 8055 | An expression for the doma... |
| releldmdifi 8056 | One way of expressing memb... |
| funfv1st2nd 8057 | The function value for the... |
| funelss 8058 | If the first component of ... |
| funeldmdif 8059 | Two ways of expressing mem... |
| sbcopeq1a 8060 | Equality theorem for subst... |
| csbopeq1a 8061 | Equality theorem for subst... |
| sbcoteq1a 8062 | Equality theorem for subst... |
| dfopab2 8063 | A way to define an ordered... |
| dfoprab3s 8064 | A way to define an operati... |
| dfoprab3 8065 | Operation class abstractio... |
| dfoprab4 8066 | Operation class abstractio... |
| dfoprab4f 8067 | Operation class abstractio... |
| opabex2 8068 | Condition for an operation... |
| opabn1stprc 8069 | An ordered-pair class abst... |
| opiota 8070 | The property of a uniquely... |
| cnvoprab 8071 | The converse of a class ab... |
| dfxp3 8072 | Define the Cartesian produ... |
| elopabi 8073 | A consequence of membershi... |
| eloprabi 8074 | A consequence of membershi... |
| mpomptsx 8075 | Express a two-argument fun... |
| mpompts 8076 | Express a two-argument fun... |
| dmmpossx 8077 | The domain of a mapping is... |
| fmpox 8078 | Functionality, domain and ... |
| fmpo 8079 | Functionality, domain and ... |
| fnmpo 8080 | Functionality and domain o... |
| fnmpoi 8081 | Functionality and domain o... |
| dmmpo 8082 | Domain of a class given by... |
| fmpodg 8083 | Domain and codomain of the... |
| fmpod 8084 | Domain and codomain of the... |
| ovmpoelrn 8085 | An operation's value belon... |
| dmmpoga 8086 | Domain of an operation giv... |
| dmmpog 8087 | Domain of an operation giv... |
| mpoexxg 8088 | Existence of an operation ... |
| mpoexg 8089 | Existence of an operation ... |
| mpoexga 8090 | If the domain of an operat... |
| mpoexw 8091 | Weak version of ~ mpoex th... |
| mpoex 8092 | If the domain of an operat... |
| mpoexd 8093 | Existence of an operation ... |
| mptmpoopabbrd 8094 | The operation value of a f... |
| mptmpoopabovd 8095 | The operation value of a f... |
| el2mpocsbcl 8096 | If the operation value of ... |
| el2mpocl 8097 | If the operation value of ... |
| fnmpoovd 8098 | A function with a Cartesia... |
| offval22 8099 | The function operation exp... |
| brovpreldm 8100 | If a binary relation holds... |
| bropopvvv 8101 | If a binary relation holds... |
| bropfvvvvlem 8102 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8103 | If a binary relation holds... |
| ovmptss 8104 | If all the values of the m... |
| relmpoopab 8105 | Any function to sets of or... |
| fmpoco 8106 | Composition of two functio... |
| oprabco 8107 | Composition of a function ... |
| oprab2co 8108 | Composition of operator ab... |
| df1st2 8109 | An alternate possible defi... |
| df2nd2 8110 | An alternate possible defi... |
| 1stconst 8111 | The mapping of a restricti... |
| 2ndconst 8112 | The mapping of a restricti... |
| dfmpo 8113 | Alternate definition for t... |
| mposn 8114 | An operation (in maps-to n... |
| curry1 8115 | Composition with ` ``' ( 2... |
| curry1val 8116 | The value of a curried fun... |
| curry1f 8117 | Functionality of a curried... |
| curry2 8118 | Composition with ` ``' ( 1... |
| curry2f 8119 | Functionality of a curried... |
| curry2val 8120 | The value of a curried fun... |
| cnvf1olem 8121 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8122 | Describe a function that m... |
| fparlem1 8123 | Lemma for ~ fpar . (Contr... |
| fparlem2 8124 | Lemma for ~ fpar . (Contr... |
| fparlem3 8125 | Lemma for ~ fpar . (Contr... |
| fparlem4 8126 | Lemma for ~ fpar . (Contr... |
| fpar 8127 | Merge two functions in par... |
| fsplit 8128 | A function that can be use... |
| fsplitfpar 8129 | Merge two functions with a... |
| offsplitfpar 8130 | Express the function opera... |
| f2ndf 8131 | The ` 2nd ` (second compon... |
| fo2ndf 8132 | The ` 2nd ` (second compon... |
| f1o2ndf1 8133 | The ` 2nd ` (second compon... |
| opco1 8134 | Value of an operation prec... |
| opco2 8135 | Value of an operation prec... |
| opco1i 8136 | Inference form of ~ opco1 ... |
| mpof1o2d 8137 | Sufficient condition for a... |
| frxp 8138 | A lexicographical ordering... |
| xporderlem 8139 | Lemma for lexicographical ... |
| poxp 8140 | A lexicographical ordering... |
| soxp 8141 | A lexicographical ordering... |
| wexp 8142 | A lexicographical ordering... |
| fnwelem 8143 | Lemma for ~ fnwe . (Contr... |
| fnwe 8144 | A variant on lexicographic... |
| fnwe2lem1 8145 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 8146 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem3 8147 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem4 8148 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 8149 | A well-ordering can be con... |
| fnse 8150 | Condition for the well-ord... |
| fvproj 8151 | Value of a function on ord... |
| fimaproj 8152 | Image of a cartesian produ... |
| ralxpes 8153 | A version of ~ ralxp with ... |
| ralxp3f 8154 | Restricted for all over a ... |
| ralxp3 8155 | Restricted for all over a ... |
| ralxp3es 8156 | Restricted for-all over a ... |
| frpoins3xpg 8157 | Special case of founded pa... |
| frpoins3xp3g 8158 | Special case of founded pa... |
| xpord2lem 8159 | Lemma for Cartesian produc... |
| poxp2 8160 | Another way of partially o... |
| frxp2 8161 | Another way of giving a we... |
| xpord2pred 8162 | Calculate the predecessor ... |
| sexp2 8163 | Condition for the relation... |
| xpord2indlem 8164 | Induction over the Cartesi... |
| xpord2ind 8165 | Induction over the Cartesi... |
| xpord3lem 8166 | Lemma for triple ordering.... |
| poxp3 8167 | Triple Cartesian product p... |
| frxp3 8168 | Give well-foundedness over... |
| xpord3pred 8169 | Calculate the predecsessor... |
| sexp3 8170 | Show that the triple order... |
| xpord3inddlem 8171 | Induction over the triple ... |
| xpord3indd 8172 | Induction over the triple ... |
| xpord3ind 8173 | Induction over the triple ... |
| orderseqlem 8174 | Lemma for ~ poseq and ~ so... |
| poseq 8175 | A partial ordering of ordi... |
| soseq 8176 | A linear ordering of ordin... |
| suppval 8179 | The value of the operation... |
| supp0prc 8180 | The support of a class is ... |
| suppvalbr 8181 | The value of the operation... |
| supp0 8182 | The support of the empty s... |
| suppval1 8183 | The value of the operation... |
| suppvalfng 8184 | The value of the operation... |
| suppvalfn 8185 | The value of the operation... |
| elsuppfng 8186 | An element of the support ... |
| elsuppfn 8187 | An element of the support ... |
| fvdifsupp 8188 | Function value is zero out... |
| cnvimadfsn 8189 | The support of functions "... |
| suppimacnvss 8190 | The support of functions "... |
| suppimacnv 8191 | Support sets of functions ... |
| fsuppeq 8192 | Two ways of writing the su... |
| fsuppeqg 8193 | Version of ~ fsuppeq avoid... |
| suppssdm 8194 | The support of a function ... |
| suppsnop 8195 | The support of a singleton... |
| snopsuppss 8196 | The support of a singleton... |
| fvn0elsupp 8197 | If the function value for ... |
| fvn0elsuppb 8198 | The function value for a g... |
| rexsupp 8199 | Existential quantification... |
| ressuppss 8200 | The support of the restric... |
| suppun 8201 | The support of a class/fun... |
| ressuppssdif 8202 | The support of the restric... |
| mptsuppdifd 8203 | The support of a function ... |
| mptsuppd 8204 | The support of a function ... |
| extmptsuppeq 8205 | The support of an extended... |
| suppfnss 8206 | The support of a function ... |
| funsssuppss 8207 | The support of a function ... |
| fnsuppres 8208 | Two ways to express restri... |
| fnsuppeq0 8209 | The support of a function ... |
| fczsupp0 8210 | The support of a constant ... |
| suppss 8211 | Show that the support of a... |
| suppssr 8212 | A function is zero outside... |
| suppssrg 8213 | A function is zero outside... |
| suppssov1 8214 | Formula building theorem f... |
| suppssov2 8215 | Formula building theorem f... |
| suppssof1 8216 | Formula building theorem f... |
| suppss2 8217 | Show that the support of a... |
| suppsssn 8218 | Show that the support of a... |
| suppssfv 8219 | Formula building theorem f... |
| suppofssd 8220 | Condition for the support ... |
| suppofss1d 8221 | Condition for the support ... |
| suppofss2d 8222 | Condition for the support ... |
| suppco 8223 | The support of the composi... |
| suppcoss 8224 | The support of the composi... |
| supp0cosupp0 8225 | The support of the composi... |
| imacosupp 8226 | The image of the support o... |
| opeliunxp2f 8227 | Membership in a union of C... |
| mpoxeldm 8228 | If there is an element of ... |
| mpoxneldm 8229 | If the first argument of a... |
| mpoxopn0yelv 8230 | If there is an element of ... |
| mpoxopynvov0g 8231 | If the second argument of ... |
| mpoxopxnop0 8232 | If the first argument of a... |
| mpoxopx0ov0 8233 | If the first argument of a... |
| mpoxopxprcov0 8234 | If the components of the f... |
| mpoxopynvov0 8235 | If the second argument of ... |
| mpoxopoveq 8236 | Value of an operation give... |
| mpoxopovel 8237 | Element of the value of an... |
| mpoxopoveqd 8238 | Value of an operation give... |
| brovex 8239 | A binary relation of the v... |
| brovmpoex 8240 | A binary relation of the v... |
| sprmpod 8241 | The extension of a binary ... |
| tposss 8244 | Subset theorem for transpo... |
| tposeq 8245 | Equality theorem for trans... |
| tposeqd 8246 | Equality theorem for trans... |
| tposssxp 8247 | The transposition is a sub... |
| reltpos 8248 | The transposition is a rel... |
| brtpos2 8249 | Value of the transposition... |
| brtpos0 8250 | The behavior of ` tpos ` w... |
| reldmtpos 8251 | Necessary and sufficient c... |
| brtpos 8252 | The transposition swaps ar... |
| ottpos 8253 | The transposition swaps th... |
| relbrtpos 8254 | The transposition swaps ar... |
| dmtpos 8255 | The domain of ` tpos F ` w... |
| rntpos 8256 | The range of ` tpos F ` wh... |
| tposexg 8257 | The transposition of a set... |
| ovtpos 8258 | The transposition swaps th... |
| tposfun 8259 | The transposition of a fun... |
| dftpos2 8260 | Alternate definition of ` ... |
| dftpos3 8261 | Alternate definition of ` ... |
| dftpos4 8262 | Alternate definition of ` ... |
| tpostpos 8263 | Value of the double transp... |
| tpostpos2 8264 | Value of the double transp... |
| tposfn2 8265 | The domain of a transposit... |
| tposfo2 8266 | Condition for a surjective... |
| tposf2 8267 | The domain and codomain of... |
| tposf12 8268 | Condition for an injective... |
| tposf1o2 8269 | Condition of a bijective t... |
| tposfo 8270 | The domain and codomain/ra... |
| tposf 8271 | The domain and codomain of... |
| tposfn 8272 | Functionality of a transpo... |
| tpos0 8273 | Transposition of the empty... |
| tposco 8274 | Transposition of a composi... |
| tpossym 8275 | Two ways to say a function... |
| tposeqi 8276 | Equality theorem for trans... |
| tposex 8277 | A transposition is a set. ... |
| nftpos 8278 | Hypothesis builder for tra... |
| tposoprab 8279 | Transposition of a class o... |
| tposmpo 8280 | Transposition of a two-arg... |
| tposconst 8281 | The transposition of a con... |
| mpocurryd 8286 | The currying of an operati... |
| mpocurryvald 8287 | The value of a curried ope... |
| fvmpocurryd 8288 | The value of the value of ... |
| pwuninel2 8291 | Proof of ~ pwuninel under ... |
| pwuninel 8292 | The powerclass of the unio... |
| pwuninelOLD 8293 | Obsolete version of ~ pwun... |
| undefval 8294 | Value of the undefined val... |
| undefnel2 8295 | The undefined value genera... |
| undefnel 8296 | The undefined value genera... |
| undefne0 8297 | The undefined value genera... |
| frecseq123 8300 | Equality theorem for the w... |
| nffrecs 8301 | Bound-variable hypothesis ... |
| csbfrecsg 8302 | Move class substitution in... |
| fpr3g 8303 | Functions defined by well-... |
| frrlem1 8304 | Lemma for well-founded rec... |
| frrlem2 8305 | Lemma for well-founded rec... |
| frrlem3 8306 | Lemma for well-founded rec... |
| frrlem4 8307 | Lemma for well-founded rec... |
| frrlem5 8308 | Lemma for well-founded rec... |
| frrlem6 8309 | Lemma for well-founded rec... |
| frrlem7 8310 | Lemma for well-founded rec... |
| frrlem8 8311 | Lemma for well-founded rec... |
| frrlem9 8312 | Lemma for well-founded rec... |
| frrlem10 8313 | Lemma for well-founded rec... |
| frrlem11 8314 | Lemma for well-founded rec... |
| frrlem12 8315 | Lemma for well-founded rec... |
| frrlem13 8316 | Lemma for well-founded rec... |
| frrlem14 8317 | Lemma for well-founded rec... |
| fprlem1 8318 | Lemma for well-founded rec... |
| fprlem2 8319 | Lemma for well-founded rec... |
| fpr2a 8320 | Weak version of ~ fpr2 whi... |
| fpr1 8321 | Law of well-founded recurs... |
| fpr2 8322 | Law of well-founded recurs... |
| fpr3 8323 | Law of well-founded recurs... |
| frrrel 8324 | Show without using the axi... |
| frrdmss 8325 | Show without using the axi... |
| frrdmcl 8326 | Show without using the axi... |
| fprfung 8327 | A "function" defined by we... |
| fprresex 8328 | The restriction of a funct... |
| wrecseq123 8331 | General equality theorem f... |
| nfwrecs 8332 | Bound-variable hypothesis ... |
| wrecseq1 8333 | Equality theorem for the w... |
| wrecseq2 8334 | Equality theorem for the w... |
| wrecseq3 8335 | Equality theorem for the w... |
| csbwrecsg 8336 | Move class substitution in... |
| wfr3g 8337 | Functions defined by well-... |
| wfrrel 8338 | The well-ordered recursion... |
| wfrdmss 8339 | The domain of the well-ord... |
| wfrdmcl 8340 | The predecessor class of a... |
| wfrfun 8341 | The "function" generated b... |
| wfrresex 8342 | Show without using the axi... |
| wfr2a 8343 | A weak version of ~ wfr2 w... |
| wfr1 8344 | The Principle of Well-Orde... |
| wfr2 8345 | The Principle of Well-Orde... |
| wfr3 8346 | The principle of Well-Orde... |
| iunon 8347 | The indexed union of a set... |
| iinon 8348 | The nonempty indexed inter... |
| onfununi 8349 | A property of functions on... |
| onovuni 8350 | A variant of ~ onfununi fo... |
| onoviun 8351 | A variant of ~ onovuni wit... |
| onnseq 8352 | There are no length ` _om ... |
| dfsmo2 8355 | Alternate definition of a ... |
| issmo 8356 | Conditions for which ` A `... |
| issmo2 8357 | Alternate definition of a ... |
| smoeq 8358 | Equality theorem for stric... |
| smodm 8359 | The domain of a strictly m... |
| smores 8360 | A strictly monotone functi... |
| smores3 8361 | A strictly monotone functi... |
| smores2 8362 | A strictly monotone ordina... |
| smodm2 8363 | The domain of a strictly m... |
| smofvon2 8364 | The function values of a s... |
| iordsmo 8365 | The identity relation rest... |
| smo0 8366 | The empty set is a strictl... |
| smofvon 8367 | If ` B ` is a strictly mon... |
| smoel 8368 | If ` x ` is less than ` y ... |
| smoiun 8369 | The value of a strictly mo... |
| smoiso 8370 | If ` F ` is an isomorphism... |
| smoel2 8371 | A strictly monotone ordina... |
| smo11 8372 | A strictly monotone ordina... |
| smoord 8373 | A strictly monotone ordina... |
| smoword 8374 | A strictly monotone ordina... |
| smogt 8375 | A strictly monotone ordina... |
| smocdmdom 8376 | The codomain of a strictly... |
| smoiso2 8377 | The strictly monotone ordi... |
| dfrecs3 8380 | The old definition of tran... |
| recseq 8381 | Equality theorem for ` rec... |
| nfrecs 8382 | Bound-variable hypothesis ... |
| tfrlem1 8383 | A technical lemma for tran... |
| tfrlem3a 8384 | Lemma for transfinite recu... |
| tfrlem3 8385 | Lemma for transfinite recu... |
| tfrlem4 8386 | Lemma for transfinite recu... |
| tfrlem5 8387 | Lemma for transfinite recu... |
| recsfval 8388 | Lemma for transfinite recu... |
| tfrlem6 8389 | Lemma for transfinite recu... |
| tfrlem6OLD 8390 | Obsolete version of ~ tfrl... |
| tfrlem7 8391 | Lemma for transfinite recu... |
| tfrlem8 8392 | Lemma for transfinite recu... |
| tfrlem9 8393 | Lemma for transfinite recu... |
| tfrlem9a 8394 | Lemma for transfinite recu... |
| tfrlem10 8395 | Lemma for transfinite recu... |
| tfrlem11 8396 | Lemma for transfinite recu... |
| tfrlem12 8397 | Lemma for transfinite recu... |
| tfrlem13 8398 | Lemma for transfinite recu... |
| tfrlem14 8399 | Lemma for transfinite recu... |
| tfrlem15 8400 | Lemma for transfinite recu... |
| tfrlem16 8401 | Lemma for finite recursion... |
| tfr1a 8402 | A weak version of ~ tfr1 w... |
| tfr2a 8403 | A weak version of ~ tfr2 w... |
| tfr2b 8404 | Without assuming ~ ax-rep ... |
| tfr1 8405 | Principle of Transfinite R... |
| tfr2 8406 | Principle of Transfinite R... |
| tfr3 8407 | Principle of Transfinite R... |
| tfr1ALT 8408 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8409 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8410 | Alternate proof of ~ tfr3 ... |
| recsfnon 8411 | Strong transfinite recursi... |
| recsval 8412 | Strong transfinite recursi... |
| tz7.44lem1 8413 | The ordered pair abstracti... |
| tz7.44-1 8414 | The value of ` F ` at ` (/... |
| tz7.44-2 8415 | The value of ` F ` at a su... |
| tz7.44-3 8416 | The value of ` F ` at a li... |
| rdgeq1 8419 | Equality theorem for the r... |
| rdgeq2 8420 | Equality theorem for the r... |
| rdgeq12 8421 | Equality theorem for the r... |
| nfrdg 8422 | Bound-variable hypothesis ... |
| rdglem1 8423 | Lemma used with the recurs... |
| rdgfun 8424 | The recursive definition g... |
| rdgdmlim 8425 | The domain of the recursiv... |
| rdgfnon 8426 | The recursive definition g... |
| rdgvalg 8427 | Value of the recursive def... |
| rdgval 8428 | Value of the recursive def... |
| rdg0 8429 | The initial value of the r... |
| rdgseg 8430 | The initial segments of th... |
| rdgsucg 8431 | The value of the recursive... |
| rdgsuc 8432 | The value of the recursive... |
| rdglimg 8433 | The value of the recursive... |
| rdglim 8434 | The value of the recursive... |
| rdg0g 8435 | The initial value of the r... |
| rdgsucmptf 8436 | The value of the recursive... |
| rdgsucmptnf 8437 | The value of the recursive... |
| rdgsucmpt2 8438 | This version of ~ rdgsucmp... |
| rdgsucmpt 8439 | The value of the recursive... |
| rdglim2 8440 | The value of the recursive... |
| rdglim2a 8441 | The value of the recursive... |
| rdg0n 8442 | If ` A ` is a proper class... |
| frfnom 8443 | The function generated by ... |
| fr0g 8444 | The initial value resultin... |
| frsuc 8445 | The successor value result... |
| frsucmpt 8446 | The successor value result... |
| frsucmptn 8447 | The value of the finite re... |
| frsucmpt2 8448 | The successor value result... |
| onelfvnef1 8449 | A sufficient condition for... |
| tz7.48lem 8450 | A way of showing an ordina... |
| tz7.48lemOLD 8451 | Obsolete version of ~ tz7.... |
| tz7.48-2 8452 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8453 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8454 | Proposition 7.48(3) of [Ta... |
| tz7.49 8455 | Proposition 7.49 of [Takeu... |
| tz7.49c 8456 | Corollary of Proposition 7... |
| seqomlem0 8459 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8460 | Lemma for ` seqom ` . The... |
| seqomlem2 8461 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8462 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8463 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8464 | Equality theorem for ` seq... |
| fnseqom 8465 | An index-aware recursive d... |
| seqom0g 8466 | Value of an index-aware re... |
| seqomsuc 8467 | Value of an index-aware re... |
| omsucelsucb 8468 | Membership is inherited by... |
| df1o2 8483 | Expanded value of the ordi... |
| df2o3 8484 | Expanded value of the ordi... |
| df2o2 8485 | Expanded value of the ordi... |
| 1oex 8486 | Ordinal 1 is a set. (Cont... |
| 1oelpr 8487 | ` 1o ` is an element of ` ... |
| 2oex 8488 | ` 2o ` is a set. (Contrib... |
| 1on 8489 | Ordinal 1 is an ordinal nu... |
| 2on 8490 | Ordinal 2 is an ordinal nu... |
| 2on0 8491 | Ordinal two is not zero. ... |
| ord3 8492 | Ordinal 3 is an ordinal cl... |
| 3on 8493 | Ordinal 3 is an ordinal nu... |
| 4on 8494 | Ordinal 4 is an ordinal nu... |
| 1n0 8495 | Ordinal one is not equal t... |
| 1n0OLD 8496 | Obsolete version of ~ 1n0 ... |
| nlim1 8497 | 1 is not a limit ordinal. ... |
| nlim2 8498 | 2 is not a limit ordinal. ... |
| xp01disj 8499 | Cartesian products with th... |
| xp01disjl 8500 | Cartesian products with th... |
| ordgt0ge1 8501 | Two ways to express that a... |
| ordge1n0 8502 | An ordinal greater than or... |
| el1o 8503 | Membership in ordinal one.... |
| ord1eln01 8504 | An ordinal that is not 0 o... |
| ord2eln012 8505 | An ordinal that is not 0, ... |
| 1ellim 8506 | A limit ordinal contains 1... |
| 2ellim 8507 | A limit ordinal contains 2... |
| dif1o 8508 | Two ways to say that ` A `... |
| ondif1 8509 | Two ways to say that ` A `... |
| ondif2 8510 | Two ways to say that ` A `... |
| 2oconcl 8511 | Closure of the pair swappi... |
| 0lt1o 8512 | Ordinal zero is less than ... |
| dif20el 8513 | An ordinal greater than on... |
| 0we1 8514 | The empty set is a well-or... |
| brwitnlem 8515 | Lemma for relations which ... |
| fnoa 8516 | Functionality and domain o... |
| fnom 8517 | Functionality and domain o... |
| fnoe 8518 | Functionality and domain o... |
| oav 8519 | Value of ordinal addition.... |
| omv 8520 | Value of ordinal multiplic... |
| oe0lem 8521 | A helper lemma for ~ oe0 a... |
| oev 8522 | Value of ordinal exponenti... |
| oevn0 8523 | Value of ordinal exponenti... |
| oa0 8524 | Addition with zero. Propo... |
| om0 8525 | Ordinal multiplication wit... |
| oe0m 8526 | Value of zero raised to an... |
| om0x 8527 | Ordinal multiplication wit... |
| oe0m0 8528 | Ordinal exponentiation wit... |
| oe0m1 8529 | Ordinal exponentiation wit... |
| oe0 8530 | Ordinal exponentiation wit... |
| oev2 8531 | Alternate value of ordinal... |
| oasuc 8532 | Addition with successor. ... |
| oesuclem 8533 | Lemma for ~ oesuc . (Cont... |
| omsuc 8534 | Multiplication with succes... |
| oesuc 8535 | Ordinal exponentiation wit... |
| onasuc 8536 | Addition with successor. ... |
| onmsuc 8537 | Multiplication with succes... |
| onesuc 8538 | Exponentiation with a succ... |
| oa1suc 8539 | Addition with 1 is same as... |
| oalim 8540 | Ordinal addition with a li... |
| omlim 8541 | Ordinal multiplication wit... |
| oelim 8542 | Ordinal exponentiation wit... |
| oacl 8543 | Closure law for ordinal ad... |
| omcl 8544 | Closure law for ordinal mu... |
| oecl 8545 | Closure law for ordinal ex... |
| oa0r 8546 | Ordinal addition with zero... |
| om0r 8547 | Ordinal multiplication wit... |
| o1p1e2 8548 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8549 | 2 + 2 = 4 for ordinal numb... |
| om1 8550 | Ordinal multiplication wit... |
| om1r 8551 | Ordinal multiplication wit... |
| oe1 8552 | Ordinal exponentiation wit... |
| oe1m 8553 | Ordinal exponentiation wit... |
| oaordi 8554 | Ordering property of ordin... |
| oaord 8555 | Ordering property of ordin... |
| oacan 8556 | Left cancellation law for ... |
| oaword 8557 | Weak ordering property of ... |
| oawordri 8558 | Weak ordering property of ... |
| oaord1 8559 | An ordinal is less than it... |
| oaword1 8560 | An ordinal is less than or... |
| oaword2 8561 | An ordinal is less than or... |
| oawordeulem 8562 | Lemma for ~ oawordex . (C... |
| oawordeu 8563 | Existence theorem for weak... |
| oawordexr 8564 | Existence theorem for weak... |
| oawordex 8565 | Existence theorem for weak... |
| oaordex 8566 | Existence theorem for orde... |
| oa00 8567 | An ordinal sum is zero iff... |
| oalimcl 8568 | The ordinal sum with a lim... |
| oaass 8569 | Ordinal addition is associ... |
| oarec 8570 | Recursive definition of or... |
| oaf1o 8571 | Left addition by a constan... |
| oacomf1olem 8572 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8573 | Define a bijection from ` ... |
| omordi 8574 | Ordering property of ordin... |
| omord2 8575 | Ordering property of ordin... |
| omord 8576 | Ordering property of ordin... |
| omcan 8577 | Left cancellation law for ... |
| omword 8578 | Weak ordering property of ... |
| omwordi 8579 | Weak ordering property of ... |
| omwordri 8580 | Weak ordering property of ... |
| omword1 8581 | An ordinal is less than or... |
| omword2 8582 | An ordinal is less than or... |
| om00 8583 | The product of two ordinal... |
| om00el 8584 | The product of two nonzero... |
| omordlim 8585 | Ordering involving the pro... |
| omlimcl 8586 | The product of any nonzero... |
| odi 8587 | Distributive law for ordin... |
| omass 8588 | Multiplication of ordinal ... |
| oneo 8589 | If an ordinal number is ev... |
| omeulem1 8590 | Lemma for ~ omeu : existen... |
| omeulem2 8591 | Lemma for ~ omeu : uniquen... |
| omopth2 8592 | An ordered pair-like theor... |
| omeu 8593 | The division algorithm for... |
| om2 8594 | Two ways to double an ordi... |
| oen0 8595 | Ordinal exponentiation wit... |
| oeordi 8596 | Ordering law for ordinal e... |
| oeord 8597 | Ordering property of ordin... |
| oecan 8598 | Left cancellation law for ... |
| oeword 8599 | Weak ordering property of ... |
| oewordi 8600 | Weak ordering property of ... |
| oewordri 8601 | Weak ordering property of ... |
| oeworde 8602 | Ordinal exponentiation com... |
| oeordsuc 8603 | Ordering property of ordin... |
| oelim2 8604 | Ordinal exponentiation wit... |
| oeoalem 8605 | Lemma for ~ oeoa . (Contr... |
| oeoa 8606 | Sum of exponents law for o... |
| oeoelem 8607 | Lemma for ~ oeoe . (Contr... |
| oeoe 8608 | Product of exponents law f... |
| oelimcl 8609 | The ordinal exponential wi... |
| oeeulem 8610 | Lemma for ~ oeeu . (Contr... |
| oeeui 8611 | The division algorithm for... |
| oeeu 8612 | The division algorithm for... |
| nna0 8613 | Addition with zero. Theor... |
| nnm0 8614 | Multiplication with zero. ... |
| nnasuc 8615 | Addition with successor. ... |
| nnmsuc 8616 | Multiplication with succes... |
| nnesuc 8617 | Exponentiation with a succ... |
| nna0r 8618 | Addition to zero. Remark ... |
| nnm0r 8619 | Multiplication with zero. ... |
| nnacl 8620 | Closure of addition of nat... |
| nnmcl 8621 | Closure of multiplication ... |
| nnecl 8622 | Closure of exponentiation ... |
| nnacli 8623 | ` _om ` is closed under ad... |
| nnmcli 8624 | ` _om ` is closed under mu... |
| nnarcl 8625 | Reverse closure law for ad... |
| nnacom 8626 | Addition of natural number... |
| nnaordi 8627 | Ordering property of addit... |
| nnaord 8628 | Ordering property of addit... |
| nnaordr 8629 | Ordering property of addit... |
| nnawordi 8630 | Adding to both sides of an... |
| nnaass 8631 | Addition of natural number... |
| nndi 8632 | Distributive law for natur... |
| nnmass 8633 | Multiplication of natural ... |
| nnmsucr 8634 | Multiplication with succes... |
| nnmcom 8635 | Multiplication of natural ... |
| nnaword 8636 | Weak ordering property of ... |
| nnacan 8637 | Cancellation law for addit... |
| nnaword1 8638 | Weak ordering property of ... |
| nnaword2 8639 | Weak ordering property of ... |
| nnmordi 8640 | Ordering property of multi... |
| nnmord 8641 | Ordering property of multi... |
| nnmword 8642 | Weak ordering property of ... |
| nnmcan 8643 | Cancellation law for multi... |
| nnmwordi 8644 | Weak ordering property of ... |
| nnmwordri 8645 | Weak ordering property of ... |
| nnawordex 8646 | Equivalence for weak order... |
| nnaordex 8647 | Equivalence for ordering. ... |
| nnaordex2 8648 | Equivalence for ordering. ... |
| 1onn 8649 | The ordinal 1 is a natural... |
| 1onnALT 8650 | Shorter proof of ~ 1onn us... |
| 2onn 8651 | The ordinal 2 is a natural... |
| 2onnALT 8652 | Shorter proof of ~ 2onn us... |
| 3onn 8653 | The ordinal 3 is a natural... |
| 4onn 8654 | The ordinal 4 is a natural... |
| 1one2o 8655 | Ordinal one is not ordinal... |
| oaabslem 8656 | Lemma for ~ oaabs . (Cont... |
| oaabs 8657 | Ordinal addition absorbs a... |
| oaabs2 8658 | The absorption law ~ oaabs... |
| omabslem 8659 | Lemma for ~ omabs . (Cont... |
| omabs 8660 | Ordinal multiplication is ... |
| nnm1 8661 | Multiply an element of ` _... |
| nnm2 8662 | Multiply an element of ` _... |
| nn2m 8663 | Multiply an element of ` _... |
| nnneo 8664 | If a natural number is eve... |
| nneob 8665 | A natural number is even i... |
| omsmolem 8666 | Lemma for ~ omsmo . (Cont... |
| omsmo 8667 | A strictly monotonic ordin... |
| omopthlem1 8668 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8669 | Lemma for ~ omopthi . (Co... |
| omopthi 8670 | An ordered pair theorem fo... |
| omopth 8671 | An ordered pair theorem fo... |
| nnasmo 8672 | There is at most one left ... |
| eldifsucnn 8673 | Condition for membership i... |
| on2recsfn 8676 | Show that double recursion... |
| on2recsov 8677 | Calculate the value of the... |
| on2ind 8678 | Double induction over ordi... |
| on3ind 8679 | Triple induction over ordi... |
| coflton 8680 | Cofinality theorem for ord... |
| cofon1 8681 | Cofinality theorem for ord... |
| cofon2 8682 | Cofinality theorem for ord... |
| cofonr 8683 | Inverse cofinality law for... |
| naddfn 8684 | Natural addition is a func... |
| naddcllem 8685 | Lemma for ordinal addition... |
| naddcl 8686 | Closure law for natural ad... |
| naddov 8687 | The value of natural addit... |
| naddov2 8688 | Alternate expression for n... |
| naddcld 8689 | Closure law for natural ad... |
| naddov3 8690 | Alternate expression for n... |
| naddf 8691 | Function statement for nat... |
| naddcom 8692 | Natural addition is commut... |
| naddrid 8693 | Ordinal zero is the additi... |
| naddlid 8694 | Ordinal zero is the additi... |
| naddssim 8695 | Ordinal less-than-or-equal... |
| naddelim 8696 | Ordinal less-than is prese... |
| naddel1 8697 | Ordinal less-than is not a... |
| naddel2 8698 | Ordinal less-than is not a... |
| naddss1 8699 | Ordinal less-than-or-equal... |
| naddss2 8700 | Ordinal less-than-or-equal... |
| naddword1 8701 | Weak-ordering principle fo... |
| naddword2 8702 | Weak-ordering principle fo... |
| naddunif 8703 | Uniformity theorem for nat... |
| naddasslem1 8704 | Lemma for ~ naddass . Exp... |
| naddasslem2 8705 | Lemma for ~ naddass . Exp... |
| naddass 8706 | Natural ordinal addition i... |
| nadd32 8707 | Commutative/associative la... |
| nadd4 8708 | Rearragement of terms in a... |
| nadd42 8709 | Rearragement of terms in a... |
| naddel12 8710 | Natural addition to both s... |
| naddsuc2 8711 | Natural addition with succ... |
| naddoa 8712 | Natural addition of a natu... |
| omnaddcl 8713 | The naturals are closed un... |
| dfer2 8718 | Alternate definition of eq... |
| dfec2 8720 | Alternate definition of ` ... |
| ecexg 8721 | An equivalence class modul... |
| ecexr 8722 | A nonempty equivalence cla... |
| dfqs2 8724 | Alternate definition of qu... |
| ereq1 8725 | Equality theorem for equiv... |
| ereq2 8726 | Equality theorem for equiv... |
| errel 8727 | An equivalence relation is... |
| erdm 8728 | The domain of an equivalen... |
| ercl 8729 | Elementhood in the field o... |
| ersym 8730 | An equivalence relation is... |
| ercl2 8731 | Elementhood in the field o... |
| ersymb 8732 | An equivalence relation is... |
| ertr 8733 | An equivalence relation is... |
| ertrd 8734 | A transitivity relation fo... |
| ertr2d 8735 | A transitivity relation fo... |
| ertr3d 8736 | A transitivity relation fo... |
| ertr4d 8737 | A transitivity relation fo... |
| erref 8738 | An equivalence relation is... |
| ercnv 8739 | The converse of an equival... |
| errn 8740 | The range and domain of an... |
| erssxp 8741 | An equivalence relation is... |
| erex 8742 | An equivalence relation is... |
| erexb 8743 | An equivalence relation is... |
| iserd 8744 | A reflexive, symmetric, tr... |
| iseri 8745 | A reflexive, symmetric, tr... |
| iseriALT 8746 | Alternate proof of ~ iseri... |
| brinxper 8747 | Conditions for a reflexive... |
| brdifun 8748 | Evaluate the incomparabili... |
| swoer 8749 | Incomparability under a st... |
| swoord1 8750 | The incomparability equiva... |
| swoord2 8751 | The incomparability equiva... |
| swoso 8752 | If the incomparability rel... |
| eqerlem 8753 | Lemma for ~ eqer . (Contr... |
| eqer 8754 | Equivalence relation invol... |
| ider 8755 | The identity relation is a... |
| 0er 8756 | The empty set is an equiva... |
| eceq1 8757 | Equality theorem for equiv... |
| eceq1d 8758 | Equality theorem for equiv... |
| eceq2 8759 | Equality theorem for equiv... |
| eceq2i 8760 | Equality theorem for the `... |
| eceq2d 8761 | Equality theorem for the `... |
| elecg 8762 | Membership in an equivalen... |
| ecref 8763 | All elements are in their ... |
| elec 8764 | Membership in an equivalen... |
| relelec 8765 | Membership in an equivalen... |
| elecres 8766 | Elementhood in the restric... |
| elecreseq 8767 | The restricted coset of ` ... |
| elecex 8768 | Condition for a coset to b... |
| ecss 8769 | An equivalence class is a ... |
| ecdmn0 8770 | A representative of a none... |
| ereldm 8771 | Equality of equivalence cl... |
| erth 8772 | Basic property of equivale... |
| erth2 8773 | Basic property of equivale... |
| erthi 8774 | Basic property of equivale... |
| erdisj 8775 | Equivalence classes do not... |
| ecidsn 8776 | An equivalence class modul... |
| qseq1 8777 | Equality theorem for quoti... |
| qseq2 8778 | Equality theorem for quoti... |
| qseq2i 8779 | Equality theorem for quoti... |
| qseq1d 8780 | Equality theorem for quoti... |
| qseq2d 8781 | Equality theorem for quoti... |
| qseq12 8782 | Equality theorem for quoti... |
| 0qs 8783 | Quotient set with the empt... |
| elqsg 8784 | Closed form of ~ elqs . (... |
| elqs 8785 | Membership in a quotient s... |
| elqsi 8786 | Membership in a quotient s... |
| elqsecl 8787 | Membership in a quotient s... |
| ecelqs 8788 | Membership of an equivalen... |
| ecelqsw 8789 | Membership of an equivalen... |
| ecelqsi 8790 | Membership of an equivalen... |
| ecopqsi 8791 | "Closure" law for equivale... |
| qsexg 8792 | A quotient set exists. (C... |
| qsex 8793 | A quotient set exists. (C... |
| uniqs 8794 | The union of a quotient se... |
| uniqsw 8795 | The union of a quotient se... |
| qsss 8796 | A quotient set is a set of... |
| uniqs2 8797 | The union of a quotient se... |
| snecg 8798 | The singleton of a coset i... |
| snec 8799 | The singleton of an equiva... |
| ecqs 8800 | Equivalence class in terms... |
| ecid 8801 | A set is equal to its cose... |
| qsid 8802 | A set is equal to its quot... |
| ectocld 8803 | Implicit substitution of c... |
| ectocl 8804 | Implicit substitution of c... |
| elqsn0 8805 | A quotient set does not co... |
| ecelqsdm 8806 | Membership of an equivalen... |
| ecelqsdmb 8807 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8808 | ` R ` -coset in its domain... |
| xpider 8809 | A Cartesian square is an e... |
| iiner 8810 | The intersection of a none... |
| riiner 8811 | The relative intersection ... |
| erinxp 8812 | A restricted equivalence r... |
| ecinxp 8813 | Restrict the relation in a... |
| qsinxp 8814 | Restrict the equivalence r... |
| qsdisj 8815 | Members of a quotient set ... |
| qsdisj2 8816 | A quotient set is a disjoi... |
| qsel 8817 | If an element of a quotien... |
| uniinqs 8818 | Class union distributes ov... |
| qliftlem 8819 | Lemma for theorems about a... |
| qliftrel 8820 | ` F ` , a function lift, i... |
| qliftel 8821 | Elementhood in the relatio... |
| qliftel1 8822 | Elementhood in the relatio... |
| qliftfun 8823 | The function ` F ` is the ... |
| qliftfund 8824 | The function ` F ` is the ... |
| qliftfuns 8825 | The function ` F ` is the ... |
| qliftf 8826 | The domain and codomain of... |
| qliftval 8827 | The value of the function ... |
| ecoptocl 8828 | Implicit substitution of c... |
| 2ecoptocl 8829 | Implicit substitution of c... |
| 3ecoptocl 8830 | Implicit substitution of c... |
| brecop 8831 | Binary relation on a quoti... |
| brecop2 8832 | Binary relation on a quoti... |
| eroveu 8833 | Lemma for ~ erov and ~ ero... |
| erovlem 8834 | Lemma for ~ erov and ~ ero... |
| erov 8835 | The value of an operation ... |
| eroprf 8836 | Functionality of an operat... |
| erov2 8837 | The value of an operation ... |
| eroprf2 8838 | Functionality of an operat... |
| ecopoveq 8839 | This is the first of sever... |
| ecopovsym 8840 | Assuming the operation ` F... |
| ecopovtrn 8841 | Assuming that operation ` ... |
| ecopover 8842 | Assuming that operation ` ... |
| eceqoveq 8843 | Equality of equivalence re... |
| ecovcom 8844 | Lemma used to transfer a c... |
| ecovass 8845 | Lemma used to transfer an ... |
| ecovdi 8846 | Lemma used to transfer a d... |
| mapprc 8851 | When ` A ` is a proper cla... |
| pmex 8852 | The class of all partial f... |
| fnmap 8853 | Set exponentiation has a u... |
| fnpm 8854 | Partial function exponenti... |
| reldmmap 8855 | Set exponentiation is a we... |
| mapvalg 8856 | The value of set exponenti... |
| pmvalg 8857 | The value of the partial m... |
| mapval 8858 | The value of set exponenti... |
| elmapg 8859 | Membership relation for se... |
| elmapd 8860 | Deduction form of ~ elmapg... |
| elmapdd 8861 | Deduction associated with ... |
| mapdm0 8862 | The empty set is the only ... |
| elpmg 8863 | The predicate "is a partia... |
| elpm2g 8864 | The predicate "is a partia... |
| elpm2r 8865 | Sufficient condition for b... |
| elpmi 8866 | A partial function is a fu... |
| pmfun 8867 | A partial function is a fu... |
| elmapex 8868 | Eliminate antecedent for m... |
| elmapi 8869 | A mapping is a function, f... |
| elmaprd 8870 | Deduction associated with ... |
| elmaprdOLD 8871 | Obsolete version of ~ elma... |
| mapfset 8872 | If ` B ` is a set, the val... |
| mapssfset 8873 | The value of the set expon... |
| mapfoss 8874 | The value of the set expon... |
| fsetsspwxp 8875 | The class of all functions... |
| fset0 8876 | The set of functions from ... |
| fsetdmprc0 8877 | The set of functions with ... |
| fsetex 8878 | The set of functions betwe... |
| f1setex 8879 | The set of injections betw... |
| fosetex 8880 | The set of surjections bet... |
| f1osetex 8881 | The set of bijections betw... |
| fsetfcdm 8882 | The class of functions wit... |
| fsetfocdm 8883 | The class of functions wit... |
| fsetprcnex 8884 | The class of all functions... |
| fsetcdmex 8885 | The class of all functions... |
| fsetexb 8886 | The class of all functions... |
| elmapfn 8887 | A mapping is a function wi... |
| elmapfun 8888 | A mapping is always a func... |
| cureq 8889 | Equality theorem for curry... |
| curf 8890 | Functional property of cur... |
| uncf 8891 | Functional property of unc... |
| curfv 8892 | Value of currying. (Contr... |
| uncov 8893 | Value of uncurrying. (Con... |
| elmapssres 8894 | A restricted mapping is a ... |
| elmapssresd 8895 | A restricted mapping is a ... |
| fpmg 8896 | A total function is a part... |
| pmss12g 8897 | Subset relation for the se... |
| pmresg 8898 | Elementhood of a restricte... |
| elmap 8899 | Membership relation for se... |
| mapval2 8900 | Alternate expression for t... |
| elpm 8901 | The predicate "is a partia... |
| elpm2 8902 | The predicate "is a partia... |
| fpm 8903 | A total function is a part... |
| mapsspm 8904 | Set exponentiation is a su... |
| pmsspw 8905 | Partial maps are a subset ... |
| mapsspw 8906 | Set exponentiation is a su... |
| mapfvd 8907 | The value of a function th... |
| elmapresaun 8908 | ~ fresaun transposed to ma... |
| fvmptmap 8909 | Special case of ~ fvmpt fo... |
| map0e 8910 | Set exponentiation with an... |
| map0b 8911 | Set exponentiation with an... |
| map0g 8912 | Set exponentiation is empt... |
| 0map0sn0 8913 | The set of mappings of the... |
| mapsnd 8914 | The value of set exponenti... |
| map0 8915 | Set exponentiation is empt... |
| mapsn 8916 | The value of set exponenti... |
| mapss 8917 | Subset inheritance for set... |
| fdiagfn 8918 | Functionality of the diago... |
| fvdiagfn 8919 | Functionality of the diago... |
| mapsnconst 8920 | Every singleton map is a c... |
| mapsncnv 8921 | Expression for the inverse... |
| mapsnf1o2 8922 | Explicit bijection between... |
| mapsnf1o3 8923 | Explicit bijection in the ... |
| ralxpmap 8924 | Quantification over functi... |
| dfixp 8927 | Eliminate the expression `... |
| ixpsnval 8928 | The value of an infinite C... |
| elixp2 8929 | Membership in an infinite ... |
| fvixp 8930 | Projection of a factor of ... |
| ixpfn 8931 | A nuple is a function. (C... |
| elixp 8932 | Membership in an infinite ... |
| elixpconst 8933 | Membership in an infinite ... |
| ixpconstg 8934 | Infinite Cartesian product... |
| ixpconst 8935 | Infinite Cartesian product... |
| ixpeq1 8936 | Equality theorem for infin... |
| ixpeq1d 8937 | Equality theorem for infin... |
| ss2ixp 8938 | Subclass theorem for infin... |
| ixpeq2 8939 | Equality theorem for infin... |
| ixpeq2dva 8940 | Equality theorem for infin... |
| ixpeq2dv 8941 | Equality theorem for infin... |
| cbvixp 8942 | Change bound variable in a... |
| cbvixpv 8943 | Change bound variable in a... |
| nfixpw 8944 | Bound-variable hypothesis ... |
| nfixp 8945 | Bound-variable hypothesis ... |
| nfixp1 8946 | The index variable in an i... |
| ixpprc 8947 | A cartesian product of pro... |
| ixpf 8948 | A member of an infinite Ca... |
| uniixp 8949 | The union of an infinite C... |
| ixpexg 8950 | The existence of an infini... |
| ixpin 8951 | The intersection of two in... |
| ixpiin 8952 | The indexed intersection o... |
| ixpint 8953 | The intersection of a coll... |
| ixp0x 8954 | An infinite Cartesian prod... |
| ixpssmap2g 8955 | An infinite Cartesian prod... |
| ixpssmapg 8956 | An infinite Cartesian prod... |
| 0elixp 8957 | Membership of the empty se... |
| ixpn0 8958 | The infinite Cartesian pro... |
| ixp0 8959 | The infinite Cartesian pro... |
| ixpssmap 8960 | An infinite Cartesian prod... |
| resixp 8961 | Restriction of an element ... |
| undifixp 8962 | Union of two projections o... |
| mptelixpg 8963 | Condition for an explicit ... |
| resixpfo 8964 | Restriction of elements of... |
| elixpsn 8965 | Membership in a class of s... |
| ixpsnf1o 8966 | A bijection between a clas... |
| mapsnf1o 8967 | A bijection between a set ... |
| boxriin 8968 | A rectangular subset of a ... |
| boxcutc 8969 | The relative complement of... |
| relen 8978 | Equinumerosity is a relati... |
| reldom 8979 | Dominance is a relation. ... |
| relsdom 8980 | Strict dominance is a rela... |
| encv 8981 | If two classes are equinum... |
| breng 8982 | Equinumerosity relation. ... |
| bren 8983 | Equinumerosity relation. ... |
| brdom2g 8984 | Dominance relation. This ... |
| brdomg 8985 | Dominance relation. (Cont... |
| brdomi 8986 | Dominance relation. (Cont... |
| brdom 8987 | Dominance relation. (Cont... |
| domen 8988 | Dominance in terms of equi... |
| domeng 8989 | Dominance in terms of equi... |
| ctex 8990 | A countable set is a set. ... |
| f1oen4g 8991 | The domain and range of a ... |
| f1dom4g 8992 | The domain of a one-to-one... |
| f1oen3g 8993 | The domain and range of a ... |
| f1dom3g 8994 | The domain of a one-to-one... |
| f1oen2g 8995 | The domain and range of a ... |
| f1dom2g 8996 | The domain of a one-to-one... |
| f1oeng 8997 | The domain and range of a ... |
| f1domg 8998 | The domain of a one-to-one... |
| f1oen 8999 | The domain and range of a ... |
| f1dom 9000 | The domain of a one-to-one... |
| brsdom 9001 | Strict dominance relation,... |
| isfi 9002 | Express " ` A ` is finite"... |
| enssdom 9003 | Equinumerosity implies dom... |
| enssdomOLD 9004 | Obsolete version of ~ enss... |
| dfdom2 9005 | Alternate definition of do... |
| endom 9006 | Equinumerosity implies dom... |
| sdomdom 9007 | Strict dominance implies d... |
| sdomnen 9008 | Strict dominance implies n... |
| brdom2 9009 | Dominance in terms of stri... |
| bren2 9010 | Equinumerosity expressed i... |
| enrefg 9011 | Equinumerosity is reflexiv... |
| enref 9012 | Equinumerosity is reflexiv... |
| eqeng 9013 | Equality implies equinumer... |
| domrefg 9014 | Dominance is reflexive. (... |
| en2d 9015 | Equinumerosity inference f... |
| en3d 9016 | Equinumerosity inference f... |
| en2i 9017 | Equinumerosity inference f... |
| en3i 9018 | Equinumerosity inference f... |
| dom2lem 9019 | A mapping (first hypothesi... |
| dom2d 9020 | A mapping (first hypothesi... |
| dom3d 9021 | A mapping (first hypothesi... |
| dom2 9022 | A mapping (first hypothesi... |
| dom3 9023 | A mapping (first hypothesi... |
| idssen 9024 | Equality implies equinumer... |
| domssl 9025 | If ` A ` is a subset of ` ... |
| domssr 9026 | If ` C ` is a superset of ... |
| ssdomg 9027 | A set dominates its subset... |
| ener 9028 | Equinumerosity is an equiv... |
| ensymb 9029 | Symmetry of equinumerosity... |
| ensym 9030 | Symmetry of equinumerosity... |
| ensymi 9031 | Symmetry of equinumerosity... |
| ensymd 9032 | Symmetry of equinumerosity... |
| entr 9033 | Transitivity of equinumero... |
| domtr 9034 | Transitivity of dominance ... |
| entri 9035 | A chained equinumerosity i... |
| entr2i 9036 | A chained equinumerosity i... |
| entr3i 9037 | A chained equinumerosity i... |
| entr4i 9038 | A chained equinumerosity i... |
| endomtr 9039 | Transitivity of equinumero... |
| domentr 9040 | Transitivity of dominance ... |
| f1imaeng 9041 | If a function is one-to-on... |
| f1imaen2g 9042 | If a function is one-to-on... |
| f1imaen3g 9043 | If a set function is one-t... |
| f1imaen 9044 | If a function is one-to-on... |
| en0 9045 | The empty set is equinumer... |
| en0ALT 9046 | Shorter proof of ~ en0 , d... |
| en0r 9047 | The empty set is equinumer... |
| ensn1 9048 | A singleton is equinumerou... |
| ensn1g 9049 | A singleton is equinumerou... |
| enpr1g 9050 | ` { A , A } ` has only one... |
| en1 9051 | A set is equinumerous to o... |
| en1b 9052 | A set is equinumerous to o... |
| reuen1 9053 | Two ways to express "exact... |
| euen1 9054 | Two ways to express "exact... |
| euen1b 9055 | Two ways to express " ` A ... |
| funen1cnv 9056 | If a function is equinumer... |
| en1uniel 9057 | A singleton contains its s... |
| 2dom 9058 | A set that dominates ordin... |
| fundmen 9059 | A function is equinumerous... |
| fundmeng 9060 | A function is equinumerous... |
| cnven 9061 | A relational set is equinu... |
| cnvct 9062 | If a set is countable, so ... |
| fndmeng 9063 | A function is equinumerate... |
| mapsnend 9064 | Set exponentiation to a si... |
| mapsnen 9065 | Set exponentiation to a si... |
| snmapen 9066 | Set exponentiation: a sing... |
| snmapen1 9067 | Set exponentiation: a sing... |
| map1 9068 | Set exponentiation: ordina... |
| en2sn 9069 | Two singletons are equinum... |
| 0fi 9070 | The empty set is finite. ... |
| snfi 9071 | A singleton is finite. (C... |
| fiprc 9072 | The class of finite sets i... |
| unen 9073 | Equinumerosity of union of... |
| enrefnn 9074 | Equinumerosity is reflexiv... |
| en2prd 9075 | Two proper unordered pairs... |
| enpr2d 9076 | A pair with distinct eleme... |
| ssct 9077 | Any subset of a countable ... |
| difsnen 9078 | All decrements of a set ar... |
| domdifsn 9079 | Dominance over a set with ... |
| xpsnen 9080 | A set is equinumerous to i... |
| xpsneng 9081 | A set is equinumerous to i... |
| xp1en 9082 | One times a cardinal numbe... |
| endisj 9083 | Any two sets are equinumer... |
| undom 9084 | Dominance law for union. ... |
| xpcomf1o 9085 | The canonical bijection fr... |
| xpcomco 9086 | Composition with the bijec... |
| xpcomen 9087 | Commutative law for equinu... |
| xpcomeng 9088 | Commutative law for equinu... |
| xpsnen2g 9089 | A set is equinumerous to i... |
| xpassen 9090 | Associative law for equinu... |
| xpdom2 9091 | Dominance law for Cartesia... |
| xpdom2g 9092 | Dominance law for Cartesia... |
| xpdom1g 9093 | Dominance law for Cartesia... |
| xpdom3 9094 | A set is dominated by its ... |
| xpdom1 9095 | Dominance law for Cartesia... |
| domunsncan 9096 | A singleton cancellation l... |
| omxpenlem 9097 | Lemma for ~ omxpen . (Con... |
| omxpen 9098 | The cardinal and ordinal p... |
| omf1o 9099 | Construct an explicit bije... |
| pw2f1olem 9100 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9101 | The power set of a set is ... |
| pw2eng 9102 | The power set of a set is ... |
| pw2en 9103 | The power set of a set is ... |
| fopwdom 9104 | Covering implies injection... |
| enfixsn 9105 | Given two equipollent sets... |
| sbthlem1 9106 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9107 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9108 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9109 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9110 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9111 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9112 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9113 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9114 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9115 | Lemma for ~ sbth . (Contr... |
| sbth 9116 | Schroeder-Bernstein Theore... |
| sbthb 9117 | Schroeder-Bernstein Theore... |
| sbthcl 9118 | Schroeder-Bernstein Theore... |
| dfsdom2 9119 | Alternate definition of st... |
| brsdom2 9120 | Alternate definition of st... |
| sdomnsym 9121 | Strict dominance is asymme... |
| domnsym 9122 | Theorem 22(i) of [Suppes] ... |
| 0domg 9123 | Any set dominates the empt... |
| dom0 9124 | A set dominated by the emp... |
| 0sdomg 9125 | A set strictly dominates t... |
| 0dom 9126 | Any set dominates the empt... |
| 0sdom 9127 | A set strictly dominates t... |
| sdom0 9128 | The empty set does not str... |
| sdomdomtr 9129 | Transitivity of strict dom... |
| sdomentr 9130 | Transitivity of strict dom... |
| domsdomtr 9131 | Transitivity of dominance ... |
| ensdomtr 9132 | Transitivity of equinumero... |
| sdomirr 9133 | Strict dominance is irrefl... |
| sdomtr 9134 | Strict dominance is transi... |
| sdomn2lp 9135 | Strict dominance has no 2-... |
| enen1 9136 | Equality-like theorem for ... |
| enen2 9137 | Equality-like theorem for ... |
| domen1 9138 | Equality-like theorem for ... |
| domen2 9139 | Equality-like theorem for ... |
| sdomen1 9140 | Equality-like theorem for ... |
| sdomen2 9141 | Equality-like theorem for ... |
| domtriord 9142 | Dominance is trichotomous ... |
| sdomel 9143 | For ordinals, strict domin... |
| sdomdif 9144 | The difference of a set fr... |
| onsdominel 9145 | An ordinal with more eleme... |
| domunsn 9146 | Dominance over a set with ... |
| fodomr 9147 | There exists a mapping fro... |
| pwdom 9148 | Injection of sets implies ... |
| canth2 9149 | Cantor's Theorem. No set ... |
| canth2g 9150 | Cantor's theorem with the ... |
| 2pwuninel 9151 | The power set of the power... |
| 2pwne 9152 | No set equals the power se... |
| disjen 9153 | A stronger form of ~ pwuni... |
| disjenex 9154 | Existence version of ~ dis... |
| domss2 9155 | A corollary of ~ disjenex ... |
| domssex2 9156 | A corollary of ~ disjenex ... |
| domssex 9157 | Weakening of ~ domssex2 to... |
| xpf1o 9158 | Construct a bijection on a... |
| xpen 9159 | Equinumerosity law for Car... |
| mapen 9160 | Two set exponentiations ar... |
| mapdom1 9161 | Order-preserving property ... |
| mapxpen 9162 | Equinumerosity law for dou... |
| xpmapenlem 9163 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9164 | Equinumerosity law for set... |
| mapunen 9165 | Equinumerosity law for set... |
| map2xp 9166 | A cardinal power with expo... |
| mapdom2 9167 | Order-preserving property ... |
| mapdom3 9168 | Set exponentiation dominat... |
| pwen 9169 | If two sets are equinumero... |
| ssenen 9170 | Equinumerosity of equinume... |
| limenpsi 9171 | A limit ordinal is equinum... |
| limensuci 9172 | A limit ordinal is equinum... |
| limensuc 9173 | A limit ordinal is equinum... |
| infensuc 9174 | Any infinite ordinal is eq... |
| dif1enlem 9175 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9176 | If a set is equinumerous t... |
| dif1en 9177 | If a set ` A ` is equinume... |
| dif1ennn 9178 | If a set ` A ` is equinume... |
| findcard 9179 | Schema for induction on th... |
| findcard2 9180 | Schema for induction on th... |
| findcard2s 9181 | Variation of ~ findcard2 r... |
| findcard2d 9182 | Deduction version of ~ fin... |
| nnfi 9183 | Natural numbers are finite... |
| pssnn 9184 | A proper subset of a natur... |
| ssnnfi 9185 | A subset of a natural numb... |
| unfi 9186 | The union of two finite se... |
| unfid 9187 | The union of two finite se... |
| ssfi 9188 | A subset of a finite set i... |
| ssfiALT 9189 | Shorter proof of ~ ssfi us... |
| diffi 9190 | If ` A ` is finite, ` ( A ... |
| cnvfi 9191 | If a set is finite, its co... |
| pwssfi 9192 | Every element of the power... |
| fnfi 9193 | A version of ~ fnex for fi... |
| f1oenfi 9194 | If the domain of a one-to-... |
| f1oenfirn 9195 | If the range of a one-to-o... |
| f1domfi 9196 | If the codomain of a one-t... |
| f1domfi2 9197 | If the domain of a one-to-... |
| enreffi 9198 | Equinumerosity is reflexiv... |
| ensymfib 9199 | Symmetry of equinumerosity... |
| entrfil 9200 | Transitivity of equinumero... |
| enfii 9201 | A set equinumerous to a fi... |
| enfi 9202 | Equinumerous sets have the... |
| enfiALT 9203 | Shorter proof of ~ enfi us... |
| domfi 9204 | A set dominated by a finit... |
| entrfi 9205 | Transitivity of equinumero... |
| entrfir 9206 | Transitivity of equinumero... |
| domtrfil 9207 | Transitivity of dominance ... |
| domtrfi 9208 | Transitivity of dominance ... |
| domtrfir 9209 | Transitivity of dominance ... |
| f1imaenfi 9210 | If a function is one-to-on... |
| ssdomfi 9211 | A finite set dominates its... |
| ssdomfi2 9212 | A set dominates its finite... |
| sbthfilem 9213 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9214 | Schroeder-Bernstein Theore... |
| domnsymfi 9215 | If a set dominates a finit... |
| sdomdomtrfi 9216 | Transitivity of strict dom... |
| domsdomtrfi 9217 | Transitivity of dominance ... |
| sucdom2 9218 | Strict dominance of a set ... |
| phplem1 9219 | Lemma for Pigeonhole Princ... |
| phplem2 9220 | Lemma for Pigeonhole Princ... |
| nneneq 9221 | Two equinumerous natural n... |
| php 9222 | Pigeonhole Principle. A n... |
| php2 9223 | Corollary of Pigeonhole Pr... |
| php3 9224 | Corollary of Pigeonhole Pr... |
| php4 9225 | Corollary of the Pigeonhol... |
| php5 9226 | Corollary of the Pigeonhol... |
| phpeqd 9227 | Corollary of the Pigeonhol... |
| nndomog 9228 | Cardinal ordering agrees w... |
| onomeneq 9229 | An ordinal number equinume... |
| onfin 9230 | An ordinal number is finit... |
| ordfin 9231 | A generalization of ~ onfi... |
| onfin2 9232 | A set is a natural number ... |
| nndomo 9233 | Cardinal ordering agrees w... |
| nnsdomo 9234 | Cardinal ordering agrees w... |
| sucdom 9235 | Strict dominance of a set ... |
| snnen2o 9236 | A singleton ` { A } ` is n... |
| 0sdom1dom 9237 | Strict dominance over 0 is... |
| 0sdom1domALT 9238 | Alternate proof of ~ 0sdom... |
| 1sdom2 9239 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9240 | Alternate proof of ~ 1sdom... |
| sdom1 9241 | A set has less than one me... |
| modom 9242 | Two ways to express "at mo... |
| modom2 9243 | Two ways to express "at mo... |
| rex2dom 9244 | A set that has at least 2 ... |
| 1sdom2dom 9245 | Strict dominance over 1 is... |
| 1sdom 9246 | A set that strictly domina... |
| unxpdomlem1 9247 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9248 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9249 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9250 | Cartesian product dominate... |
| unxpdom2 9251 | Corollary of ~ unxpdom . ... |
| sucxpdom 9252 | Cartesian product dominate... |
| pssinf 9253 | A set equinumerous to a pr... |
| fisseneq 9254 | A finite set is equal to i... |
| ominf 9255 | The set of natural numbers... |
| isinf 9256 | Any set that is not finite... |
| fineqvlem 9257 | Lemma for ~ fineqv . (Con... |
| fineqv 9258 | If the Axiom of Infinity i... |
| xpfir 9259 | The components of a nonemp... |
| ssfid 9260 | A subset of a finite set i... |
| infi 9261 | The intersection of two se... |
| rabfi 9262 | A restricted class built f... |
| finresfin 9263 | The restriction of a finit... |
| f1finf1o 9264 | Any injection from one fin... |
| nfielex 9265 | If a class is not finite, ... |
| en1eqsn 9266 | A set with one element is ... |
| en1eqsnbi 9267 | A set containing an elemen... |
| dif1ennnALT 9268 | Alternate proof of ~ dif1e... |
| enp1ilem 9269 | Lemma for uses of ~ enp1i ... |
| enp1i 9270 | Proof induction for ~ en2 ... |
| en2 9271 | A set equinumerous to ordi... |
| en3 9272 | A set equinumerous to ordi... |
| en4 9273 | A set equinumerous to ordi... |
| findcard3 9274 | Schema for strong inductio... |
| ac6sfi 9275 | A version of ~ ac6s for fi... |
| frfi 9276 | A partial order is well-fo... |
| fimax2g 9277 | A finite set has a maximum... |
| fimaxg 9278 | A finite set has a maximum... |
| fisupg 9279 | Lemma showing existence an... |
| wofi 9280 | A total order on a finite ... |
| ordunifi 9281 | The maximum of a finite co... |
| fissorduni 9282 | The union (supremum) of a ... |
| nnunifi 9283 | The union (supremum) of a ... |
| unblem1 9284 | Lemma for ~ unbnn . After... |
| unblem2 9285 | Lemma for ~ unbnn . The v... |
| unblem3 9286 | Lemma for ~ unbnn . The v... |
| unblem4 9287 | Lemma for ~ unbnn . The f... |
| unbnn 9288 | Any unbounded subset of na... |
| unbnn2 9289 | Version of ~ unbnn that do... |
| isfinite2 9290 | Any set strictly dominated... |
| nnsdomg 9291 | Omega strictly dominates a... |
| isfiniteg 9292 | A set is finite iff it is ... |
| infsdomnn 9293 | An infinite set strictly d... |
| infn0 9294 | An infinite set is not emp... |
| infn0ALT 9295 | Shorter proof of ~ infn0 u... |
| fin2inf 9296 | This (useless) theorem, wh... |
| unfilem1 9297 | Lemma for proving that the... |
| unfilem2 9298 | Lemma for proving that the... |
| unfilem3 9299 | Lemma for proving that the... |
| unfir 9300 | If a union is finite, the ... |
| unfib 9301 | A union is finite if and o... |
| unfi2 9302 | The union of two finite se... |
| difinf 9303 | An infinite set ` A ` minu... |
| fodomfi 9304 | An onto function implies d... |
| fofi 9305 | If an onto function has a ... |
| f1fi 9306 | If a 1-to-1 function has a... |
| imafi 9307 | Images of finite sets are ... |
| pwfir 9308 | If the power set of a set ... |
| pwfilem 9309 | Lemma for ~ pwfi . (Contr... |
| pwfi 9310 | The power set of a finite ... |
| xpfi 9311 | The Cartesian product of t... |
| 3xpfi 9312 | The Cartesian product of t... |
| domunfican 9313 | A finite set union cancell... |
| infcntss 9314 | Every infinite set has a d... |
| prfi 9315 | An unordered pair is finit... |
| prfiALT 9316 | Shorter proof of ~ prfi us... |
| tpfi 9317 | An unordered triple is fin... |
| fiint 9318 | Equivalent ways of stating... |
| fodomfir 9319 | There exists a mapping fro... |
| fodomfib 9320 | Equivalence of an onto map... |
| fofinf1o 9321 | Any surjection from one fi... |
| rneqdmfinf1o 9322 | Any function from a finite... |
| fidomdm 9323 | Any finite set dominates i... |
| dmfi 9324 | The domain of a finite set... |
| fundmfibi 9325 | A function is finite if an... |
| resfnfinfin 9326 | The restriction of a funct... |
| residfi 9327 | A restricted identity func... |
| cnvfiALT 9328 | Shorter proof of ~ cnvfi u... |
| rnfi 9329 | The range of a finite set ... |
| f1dmvrnfibi 9330 | A one-to-one function whos... |
| f1vrnfibi 9331 | A one-to-one function whic... |
| iunfi 9332 | The finite union of finite... |
| unifi 9333 | The finite union of finite... |
| unifi2 9334 | The finite union of finite... |
| infssuni 9335 | If an infinite set ` A ` i... |
| unirnffid 9336 | The union of the range of ... |
| mapfi 9337 | Set exponentiation of fini... |
| ixpfi 9338 | A Cartesian product of fin... |
| ixpfi2 9339 | A Cartesian product of fin... |
| mptfi 9340 | A finite mapping set is fi... |
| abrexfi 9341 | An image set from a finite... |
| cnvimamptfin 9342 | Any preimage by a mapping ... |
| elfpw 9343 | Membership in a class of f... |
| unifpw 9344 | A set is the union of its ... |
| f1opwfi 9345 | A one-to-one mapping induc... |
| fissuni 9346 | A finite subset of a union... |
| fipreima 9347 | Given a finite subset ` A ... |
| finsschain 9348 | A finite subset of the uni... |
| indexfi 9349 | If for every element of a ... |
| imafi2 9350 | The image by a finite set ... |
| unifi3 9351 | If a union is finite, then... |
| tfsnfin2 9352 | A transfinite sequence is ... |
| relfsupp 9355 | The property of a function... |
| relprcnfsupp 9356 | A proper class is never fi... |
| isfsupp 9357 | The property of a class to... |
| isfsuppd 9358 | Deduction form of ~ isfsup... |
| funisfsupp 9359 | The property of a function... |
| fsuppimp 9360 | Implications of a class be... |
| fsuppimpd 9361 | A finitely supported funct... |
| fsuppfund 9362 | A finitely supported funct... |
| fisuppfi 9363 | A function on a finite set... |
| fidmfisupp 9364 | A function with a finite d... |
| finnzfsuppd 9365 | If a function is zero outs... |
| fdmfisuppfi 9366 | The support of a function ... |
| fdmfifsupp 9367 | A function with a finite d... |
| fsuppmptdm 9368 | A mapping with a finite do... |
| fndmfisuppfi 9369 | The support of a function ... |
| fndmfifsupp 9370 | A function with a finite d... |
| suppeqfsuppbi 9371 | If two functions have the ... |
| suppssfifsupp 9372 | If the support of a functi... |
| fsuppsssupp 9373 | If the support of a functi... |
| fsuppsssuppgd 9374 | If the support of a functi... |
| fsuppss 9375 | A subset of a finitely sup... |
| fsuppssov1 9376 | Formula building theorem f... |
| fsuppxpfi 9377 | The cartesian product of t... |
| fczfsuppd 9378 | A constant function with v... |
| fsuppun 9379 | The union of two finitely ... |
| fsuppunfi 9380 | The union of the support o... |
| fsuppunbi 9381 | If the union of two classe... |
| 0fsupp 9382 | The empty set is a finitel... |
| snopfsupp 9383 | A singleton containing an ... |
| funsnfsupp 9384 | Finite support for a funct... |
| fsuppres 9385 | The restriction of a finit... |
| fmptssfisupp 9386 | The restriction of a mappi... |
| ressuppfi 9387 | If the support of the rest... |
| resfsupp 9388 | If the restriction of a fu... |
| resfifsupp 9389 | The restriction of a funct... |
| ffsuppbi 9390 | Two ways of saying that a ... |
| fsuppmptif 9391 | A function mapping an argu... |
| sniffsupp 9392 | A function mapping all but... |
| fsuppcolem 9393 | Lemma for ~ fsuppco . For... |
| fsuppco 9394 | The composition of a 1-1 f... |
| fsuppco2 9395 | The composition of a funct... |
| fsuppcor 9396 | The composition of a funct... |
| mapfienlem1 9397 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9398 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9399 | Lemma 3 for ~ mapfien . (... |
| mapfien 9400 | A bijection of the base se... |
| mapfien2 9401 | Equinumerousity relation f... |
| fival 9404 | The set of all the finite ... |
| elfi 9405 | Specific properties of an ... |
| elfi2 9406 | The empty intersection nee... |
| elfir 9407 | Sufficient condition for a... |
| intrnfi 9408 | Sufficient condition for t... |
| iinfi 9409 | An indexed intersection of... |
| inelfi 9410 | The intersection of two se... |
| ssfii 9411 | Any element of a set ` A `... |
| fi0 9412 | The set of finite intersec... |
| fieq0 9413 | A set is empty iff the cla... |
| fiin 9414 | The elements of ` ( fi `` ... |
| dffi2 9415 | The set of finite intersec... |
| fiss 9416 | Subset relationship for fu... |
| inficl 9417 | A set which is closed unde... |
| fipwuni 9418 | The set of finite intersec... |
| fisn 9419 | A singleton is closed unde... |
| fiuni 9420 | The union of the finite in... |
| fipwss 9421 | If a set is a family of su... |
| elfiun 9422 | A finite intersection of e... |
| dffi3 9423 | The set of finite intersec... |
| fifo 9424 | Describe a surjection from... |
| marypha1lem 9425 | Core induction for Philip ... |
| marypha1 9426 | (Philip) Hall's marriage t... |
| marypha2lem1 9427 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9428 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9429 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9430 | Lemma for ~ marypha2 . Pr... |
| marypha2 9431 | Version of ~ marypha1 usin... |
| dfsup2 9436 | Quantifier-free definition... |
| supeq1 9437 | Equality theorem for supre... |
| supeq1d 9438 | Equality deduction for sup... |
| supeq1i 9439 | Equality inference for sup... |
| supeq2 9440 | Equality theorem for supre... |
| supeq3 9441 | Equality theorem for supre... |
| supeq123d 9442 | Equality deduction for sup... |
| nfsup 9443 | Hypothesis builder for sup... |
| supmo 9444 | Any class ` B ` has at mos... |
| supexd 9445 | A supremum is a set. (Con... |
| supeu 9446 | A supremum is unique. Sim... |
| supval2 9447 | Alternate expression for t... |
| eqsup 9448 | Sufficient condition for a... |
| eqsupd 9449 | Sufficient condition for a... |
| supcl 9450 | A supremum belongs to its ... |
| supub 9451 | A supremum is an upper bou... |
| suplub 9452 | A supremum is the least up... |
| suplub2 9453 | Bidirectional form of ~ su... |
| supnub 9454 | An upper bound is not less... |
| supssd 9455 | Inequality deduction for s... |
| supex 9456 | A supremum is a set. (Con... |
| sup00 9457 | The supremum under an empt... |
| sup0riota 9458 | The supremum of an empty s... |
| sup0 9459 | The supremum of an empty s... |
| supmax 9460 | The greatest element of a ... |
| fisup2g 9461 | A finite set satisfies the... |
| fisupcl 9462 | A nonempty finite set cont... |
| supgtoreq 9463 | The supremum of a finite s... |
| suppr 9464 | The supremum of a pair. (... |
| supsn 9465 | The supremum of a singleto... |
| supisolem 9466 | Lemma for ~ supiso . (Con... |
| supisoex 9467 | Lemma for ~ supiso . (Con... |
| supiso 9468 | Image of a supremum under ... |
| infeq1 9469 | Equality theorem for infim... |
| infeq1d 9470 | Equality deduction for inf... |
| infeq1i 9471 | Equality inference for inf... |
| infeq2 9472 | Equality theorem for infim... |
| infeq3 9473 | Equality theorem for infim... |
| infeq123d 9474 | Equality deduction for inf... |
| nfinf 9475 | Hypothesis builder for inf... |
| infexd 9476 | An infimum is a set. (Con... |
| eqinf 9477 | Sufficient condition for a... |
| eqinfd 9478 | Sufficient condition for a... |
| infval 9479 | Alternate expression for t... |
| infcllem 9480 | Lemma for ~ infcl , ~ infl... |
| infcl 9481 | An infimum belongs to its ... |
| inflb 9482 | An infimum is a lower boun... |
| infglb 9483 | An infimum is the greatest... |
| infglbb 9484 | Bidirectional form of ~ in... |
| infnlb 9485 | A lower bound is not great... |
| infssd 9486 | Inequality deduction for i... |
| infex 9487 | An infimum is a set. (Con... |
| infmin 9488 | The smallest element of a ... |
| infmo 9489 | Any class ` B ` has at mos... |
| infeu 9490 | An infimum is unique. (Co... |
| fimin2g 9491 | A finite set has a minimum... |
| fiming 9492 | A finite set has a minimum... |
| fiinfg 9493 | Lemma showing existence an... |
| fiinf2g 9494 | A finite set satisfies the... |
| fiinfcl 9495 | A nonempty finite set cont... |
| infltoreq 9496 | The infimum of a finite se... |
| infpr 9497 | The infimum of a pair. (C... |
| infsupprpr 9498 | The infimum of a proper pa... |
| infsn 9499 | The infimum of a singleton... |
| inf00 9500 | The infimum regarding an e... |
| infempty 9501 | The infimum of an empty se... |
| infiso 9502 | Image of an infimum under ... |
| dfoi 9505 | Rewrite ~ df-oi with abbre... |
| oieq1 9506 | Equality theorem for ordin... |
| oieq2 9507 | Equality theorem for ordin... |
| nfoi 9508 | Hypothesis builder for ord... |
| ordiso2 9509 | Generalize ~ ordiso to pro... |
| ordiso 9510 | Order-isomorphic ordinal n... |
| ordtypecbv 9511 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9512 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9513 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9514 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9515 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9516 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9517 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9518 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9519 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9520 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9521 | Lemma for ~ ordtype . Usi... |
| oi0 9522 | Definition of the ordinal ... |
| oicl 9523 | The order type of the well... |
| oif 9524 | The order isomorphism of t... |
| oiiso2 9525 | The order isomorphism of t... |
| ordtype 9526 | For any set-like well-orde... |
| oiiniseg 9527 | ` ran F ` is an initial se... |
| ordtype2 9528 | For any set-like well-orde... |
| oiexg 9529 | The order isomorphism on a... |
| oion 9530 | The order type of the well... |
| oiiso 9531 | The order isomorphism of t... |
| oien 9532 | The order type of a well-o... |
| oieu 9533 | Uniqueness of the unique o... |
| oismo 9534 | When ` A ` is a subclass o... |
| oiid 9535 | The order type of an ordin... |
| hartogslem1 9536 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9537 | Lemma for ~ hartogs . (Co... |
| hartogs 9538 | The class of ordinals domi... |
| wofib 9539 | The only sets which are we... |
| wemaplem1 9540 | Value of the lexicographic... |
| wemaplem2 9541 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9542 | Lemma for ~ wemapso . Tra... |
| wemappo 9543 | Construct lexicographic or... |
| wemapsolem 9544 | Lemma for ~ wemapso . (Co... |
| wemapso 9545 | Construct lexicographic or... |
| wemapso2lem 9546 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9547 | An alternative to having a... |
| card2on 9548 | The alternate definition o... |
| card2inf 9549 | The alternate definition o... |
| harf 9552 | Functionality of the Harto... |
| harcl 9553 | Values of the Hartogs func... |
| harval 9554 | Function value of the Hart... |
| elharval 9555 | The Hartogs number of a se... |
| harndom 9556 | The Hartogs number of a se... |
| harword 9557 | Weak ordering property of ... |
| relwdom 9560 | Weak dominance is a relati... |
| brwdom 9561 | Property of weak dominance... |
| brwdomi 9562 | Property of weak dominance... |
| brwdomn0 9563 | Weak dominance over nonemp... |
| 0wdom 9564 | Any set weakly dominates t... |
| fowdom 9565 | An onto function implies w... |
| wdomref 9566 | Reflexivity of weak domina... |
| brwdom2 9567 | Alternate characterization... |
| domwdom 9568 | Weak dominance is implied ... |
| wdomtr 9569 | Transitivity of weak domin... |
| wdomen1 9570 | Equality-like theorem for ... |
| wdomen2 9571 | Equality-like theorem for ... |
| wdompwdom 9572 | Weak dominance strengthens... |
| canthwdom 9573 | Cantor's Theorem, stated u... |
| wdom2d 9574 | Deduce weak dominance from... |
| wdomd 9575 | Deduce weak dominance from... |
| brwdom3 9576 | Condition for weak dominan... |
| brwdom3i 9577 | Weak dominance implies exi... |
| unwdomg 9578 | Weak dominance of a (disjo... |
| xpwdomg 9579 | Weak dominance of a Cartes... |
| wdomima2g 9580 | A set is weakly dominant o... |
| wdomimag 9581 | A set is weakly dominant o... |
| unxpwdom2 9582 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9583 | If a Cartesian product is ... |
| ixpiunwdom 9584 | Describe an onto function ... |
| harwdom 9585 | The value of the Hartogs f... |
| axreg2 9587 | Axiom of Regularity expres... |
| zfregcl 9588 | The Axiom of Regularity wi... |
| zfregclOLD 9589 | Obsolete version of ~ zfre... |
| zfreg 9590 | The Axiom of Regularity us... |
| elirrv 9591 | The membership relation is... |
| elirrvOLD 9592 | Obsolete version of ~ elir... |
| elirrvOLDOLD 9593 | Obsolete version of ~ elir... |
| elirr 9594 | No class is a member of it... |
| elneq 9595 | A class is not equal to an... |
| nelaneq 9596 | A class is not an element ... |
| nelaneqOLD 9597 | Obsolete version of ~ nela... |
| nelaneqOLDOLD 9598 | Obsolete version of ~ nela... |
| epinid0 9599 | The membership relation an... |
| sucprcreg 9600 | A class is equal to its su... |
| sucprcregOLD 9601 | Obsolete version of ~ sucp... |
| ruv 9602 | The Russell class is equal... |
| ruALT 9603 | Alternate proof of ~ ru , ... |
| disjcsn 9604 | A class is disjoint from i... |
| zfregfr 9605 | The membership relation is... |
| elirrvALT 9606 | Alternate proof of ~ elirr... |
| en2lp 9607 | No class has 2-cycle membe... |
| elnanel 9608 | Two classes are not elemen... |
| cnvepnep 9609 | The membership (epsilon) r... |
| epnsym 9610 | The membership (epsilon) r... |
| elnotel 9611 | A class cannot be an eleme... |
| elnel 9612 | A class cannot be an eleme... |
| en3lplem1 9613 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9614 | Lemma for ~ en3lp . (Cont... |
| en3lp 9615 | No class has 3-cycle membe... |
| preleqg 9616 | Equality of two unordered ... |
| preleq 9617 | Equality of two unordered ... |
| preleqALT 9618 | Alternate proof of ~ prele... |
| opthreg 9619 | Theorem for alternate repr... |
| suc11reg 9620 | The successor operation be... |
| dford2 9621 | Assuming ~ ax-reg , an ord... |
| inf0 9622 | Existence of ` _om ` impli... |
| inf1 9623 | Variation of Axiom of Infi... |
| inf2 9624 | Variation of Axiom of Infi... |
| inf3lema 9625 | Lemma for our Axiom of Inf... |
| inf3lemb 9626 | Lemma for our Axiom of Inf... |
| inf3lemc 9627 | Lemma for our Axiom of Inf... |
| inf3lemd 9628 | Lemma for our Axiom of Inf... |
| inf3lem1 9629 | Lemma for our Axiom of Inf... |
| inf3lem2 9630 | Lemma for our Axiom of Inf... |
| inf3lem3 9631 | Lemma for our Axiom of Inf... |
| inf3lem4 9632 | Lemma for our Axiom of Inf... |
| inf3lem5 9633 | Lemma for our Axiom of Inf... |
| inf3lem6 9634 | Lemma for our Axiom of Inf... |
| inf3lem7 9635 | Lemma for our Axiom of Inf... |
| inf3 9636 | Our Axiom of Infinity ~ ax... |
| infeq5i 9637 | Half of ~ infeq5 . (Contr... |
| infeq5 9638 | The statement "there exist... |
| zfinf 9640 | Axiom of Infinity expresse... |
| axinf2 9641 | A standard version of Axio... |
| zfinf2 9643 | A standard version of the ... |
| omex 9644 | The existence of omega (th... |
| axinf 9645 | The first version of the A... |
| inf5 9646 | The statement "there exist... |
| omelon 9647 | Omega is an ordinal number... |
| dfom3 9648 | The class of natural numbe... |
| elom3 9649 | A simplification of ~ elom... |
| dfom4 9650 | A simplification of ~ df-o... |
| dfom5 9651 | ` _om ` is the smallest li... |
| oancom 9652 | Ordinal addition is not co... |
| isfinite 9653 | A set is finite iff it is ... |
| fict 9654 | A finite set is countable ... |
| nnsdom 9655 | A natural number is strict... |
| omenps 9656 | Omega is equinumerous to a... |
| omensuc 9657 | The set of natural numbers... |
| infdifsn 9658 | Removing a singleton from ... |
| infdiffi 9659 | Removing a finite set from... |
| unbnn3 9660 | Any unbounded subset of na... |
| noinfep 9661 | Using the Axiom of Regular... |
| cantnffval 9664 | The value of the Cantor no... |
| cantnfdm 9665 | The domain of the Cantor n... |
| cantnfvalf 9666 | Lemma for ~ cantnf . The ... |
| cantnfs 9667 | Elementhood in the set of ... |
| cantnfcl 9668 | Basic properties of the or... |
| cantnfval 9669 | The value of the Cantor no... |
| cantnfval2 9670 | Alternate expression for t... |
| cantnfsuc 9671 | The value of the recursive... |
| cantnfle 9672 | A lower bound on the ` CNF... |
| cantnflt 9673 | An upper bound on the part... |
| cantnflt2 9674 | An upper bound on the ` CN... |
| cantnff 9675 | The ` CNF ` function is a ... |
| cantnf0 9676 | The value of the zero func... |
| cantnfrescl 9677 | A function is finitely sup... |
| cantnfres 9678 | The ` CNF ` function respe... |
| cantnfp1lem1 9679 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9680 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9681 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9682 | If ` F ` is created by add... |
| oemapso 9683 | The relation ` T ` is a st... |
| oemapval 9684 | Value of the relation ` T ... |
| oemapvali 9685 | If ` F < G ` , then there ... |
| cantnflem1a 9686 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9687 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9688 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9689 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9690 | Lemma for ~ cantnf . This... |
| cantnflem2 9691 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9692 | Lemma for ~ cantnf . Here... |
| cantnflem4 9693 | Lemma for ~ cantnf . Comp... |
| cantnf 9694 | The Cantor Normal Form the... |
| oemapwe 9695 | The lexicographic order on... |
| cantnffval2 9696 | An alternate definition of... |
| cantnff1o 9697 | Simplify the isomorphism o... |
| wemapwe 9698 | Construct lexicographic or... |
| oef1o 9699 | A bijection of the base se... |
| cnfcomlem 9700 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9701 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9702 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9703 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9704 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9705 | Any infinite ordinal ` B `... |
| cnfcom3clem 9706 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9707 | Wrap the construction of ~... |
| ttrcleq 9710 | Equality theorem for trans... |
| nfttrcld 9711 | Bound variable hypothesis ... |
| nfttrcl 9712 | Bound variable hypothesis ... |
| relttrcl 9713 | The transitive closure of ... |
| brttrcl 9714 | Characterization of elemen... |
| brttrcl2 9715 | Characterization of elemen... |
| ssttrcl 9716 | If ` R ` is a relation, th... |
| ttrcltr 9717 | The transitive closure of ... |
| ttrclresv 9718 | The transitive closure of ... |
| ttrclco 9719 | Composition law for the tr... |
| cottrcl 9720 | Composition law for the tr... |
| ttrclss 9721 | If ` R ` is a subclass of ... |
| dmttrcl 9722 | The domain of a transitive... |
| rnttrcl 9723 | The range of a transitive ... |
| ttrclexg 9724 | If ` R ` is a set, then so... |
| dfttrcl2 9725 | When ` R ` is a set and a ... |
| ttrclselem1 9726 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9727 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9728 | If ` R ` is set-like over ... |
| trcl 9729 | For any set ` A ` , show t... |
| tz9.1 9730 | Every set has a transitive... |
| tz9.1c 9731 | Alternate expression for t... |
| epfrs 9732 | The strong form of the Axi... |
| zfregs 9733 | The strong form of the Axi... |
| zfregs2 9734 | Alternate strong form of t... |
| tcvalg 9737 | Value of the transitive cl... |
| tcid 9738 | Defining property of the t... |
| tctr 9739 | Defining property of the t... |
| tcmin 9740 | Defining property of the t... |
| tc2 9741 | A variant of the definitio... |
| tcsni 9742 | The transitive closure of ... |
| tcss 9743 | The transitive closure fun... |
| tcel 9744 | The transitive closure fun... |
| tcidm 9745 | The transitive closure fun... |
| tc0 9746 | The transitive closure of ... |
| tc00 9747 | The transitive closure is ... |
| setind 9748 | Set (epsilon) induction. ... |
| setind2 9749 | Set (epsilon) induction, s... |
| setinds 9750 | Principle of set induction... |
| setinds2f 9751 | ` _E ` induction schema, u... |
| setinds2 9752 | ` _E ` induction schema, u... |
| frmin 9753 | Every (possibly proper) su... |
| frind 9754 | A subclass of a well-found... |
| frinsg 9755 | Well-Founded Induction Sch... |
| frins 9756 | Well-Founded Induction Sch... |
| frins2f 9757 | Well-Founded Induction sch... |
| frins2 9758 | Well-Founded Induction sch... |
| frins3 9759 | Well-Founded Induction sch... |
| frr3g 9760 | Functions defined by well-... |
| frrlem15 9761 | Lemma for general well-fou... |
| frrlem16 9762 | Lemma for general well-fou... |
| frr1 9763 | Law of general well-founde... |
| frr2 9764 | Law of general well-founde... |
| frr3 9765 | Law of general well-founde... |
| r1funlimOLD 9770 | The cumulative hierarchy o... |
| r1fun 9771 | The cumulative hierarchy o... |
| r1dmlim 9772 | The domain of the cumulati... |
| r1fnon 9773 | The cumulative hierarchy o... |
| rnr1 9774 | The range of the cumulativ... |
| r10 9775 | Value of the cumulative hi... |
| r1sucg 9776 | Value of the cumulative hi... |
| r1suc 9777 | Value of the cumulative hi... |
| r1limg 9778 | Value of the cumulative hi... |
| r1lim 9779 | Value of the cumulative hi... |
| r1fin 9780 | The first ` _om ` stages o... |
| r1sdom 9781 | Each stage of the cumulati... |
| r111 9782 | The cumulative hierarchy o... |
| r1tr 9783 | Each stage of the cumulati... |
| r1tr2 9784 | Each stage of the cumulati... |
| r1ordg 9785 | Ordering relation for the ... |
| r1ord3g 9786 | Ordering relation for the ... |
| r1ord 9787 | Ordering relation for the ... |
| r1ord2 9788 | Ordering relation for the ... |
| r1ord3 9789 | Ordering relation for the ... |
| r1sssuc 9790 | A stage of the cumulative ... |
| r1pwss 9791 | Each stage of the cumulati... |
| r1sscl 9792 | Each stage of the cumulati... |
| r1val1 9793 | The value of the cumulativ... |
| tz9.12lem1 9794 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9795 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9796 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9797 | A set is well-founded if a... |
| tz9.13 9798 | Every set is well-founded,... |
| tz9.13g 9799 | Every set is well-founded,... |
| rankwflemb 9800 | Two ways of expressing tha... |
| rankwflembOLD 9801 | Obsolete version of ~ rank... |
| rankf 9802 | The domain and codomain of... |
| rankon 9803 | The rank of a set is an or... |
| r1elwf 9804 | Any element of (any stage ... |
| rankvalb 9805 | Value of the rank function... |
| rankr1ai 9806 | One direction of ~ rankr1a... |
| rankvaln 9807 | Value of the rank function... |
| rankidb 9808 | Identity law for the rank ... |
| rankdmr1 9809 | A rank is a member of the ... |
| rankr1ag 9810 | A version of ~ rankr1a tha... |
| rankr1bg 9811 | A relationship between ran... |
| r1rankidb 9812 | Any set is a subset of the... |
| r1elssi 9813 | The range of the ` R1 ` fu... |
| r1elss 9814 | The range of the ` R1 ` fu... |
| pwwf 9815 | A power set is well-founde... |
| sswf 9816 | A subset of a well-founded... |
| snwf 9817 | A singleton is well-founde... |
| unwf 9818 | A binary union is well-fou... |
| prwf 9819 | An unordered pair is well-... |
| opwf 9820 | An ordered pair is well-fo... |
| unir1 9821 | The cumulative hierarchy o... |
| jech9.3 9822 | Every set belongs to some ... |
| jech9.3OLD 9823 | Obsolete version of ~ jech... |
| rankwflem 9824 | Every set is well-founded,... |
| rankval 9825 | Value of the rank function... |
| rankvalg 9826 | Value of the rank function... |
| rankval2 9827 | Value of an alternate defi... |
| uniwf 9828 | A union is well-founded if... |
| rankr1clem 9829 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9830 | A relationship between the... |
| rankidn 9831 | A relationship between the... |
| rankpwi 9832 | The rank of a power set. ... |
| rankelb 9833 | The membership relation is... |
| wfelirr 9834 | A well-founded set is not ... |
| rankval2b 9835 | Value of an alternate defi... |
| rankval3b 9836 | The value of the rank func... |
| ranksnb 9837 | The rank of a singleton. ... |
| rankonidlem 9838 | Lemma for ~ rankonid . (C... |
| rankonid 9839 | The rank of an ordinal num... |
| onwf 9840 | The ordinals are all well-... |
| r1wf 9841 | Each stage in the cumulati... |
| elwf 9842 | An element of a well-found... |
| onssr1 9843 | Initial segments of the or... |
| rankr1g 9844 | A relationship between the... |
| rankid 9845 | Identity law for the rank ... |
| rankr1 9846 | A relationship between the... |
| ssrankr1 9847 | A relationship between an ... |
| rankr1a 9848 | A relationship between ran... |
| r1val2 9849 | The value of the cumulativ... |
| r1val3 9850 | The value of the cumulativ... |
| rankel 9851 | The membership relation is... |
| rankelg 9852 | The membership relation is... |
| rankval3 9853 | The value of the rank func... |
| bndrank 9854 | Any class whose elements h... |
| unbndrank 9855 | The elements of a proper c... |
| rankpw 9856 | The rank of the powerset i... |
| rankpwg 9857 | The rank of the powerset i... |
| ranklim 9858 | The rank of a set belongs ... |
| r1pw 9859 | A set is in a given stage ... |
| r1pwALT 9860 | Alternate shorter proof of... |
| r1pwcl 9861 | The cumulative hierarchy o... |
| rankssb 9862 | The subset relation is inh... |
| rankss 9863 | The subset relation is inh... |
| rankunb 9864 | The rank of the union of t... |
| rankprb 9865 | The rank of an unordered p... |
| rankopb 9866 | The rank of an ordered pai... |
| rankuni2b 9867 | The value of the rank func... |
| ranksn 9868 | The rank of a singleton. ... |
| rankuni2 9869 | The rank of a union. Part... |
| rankun 9870 | The rank of the union of t... |
| rankpr 9871 | The rank of an unordered p... |
| rankop 9872 | The rank of an ordered pai... |
| rankung 9873 | The rank of the union of t... |
| ranksng 9874 | The rank of a singleton. ... |
| r1rankid 9875 | Any set is a subset of the... |
| rankeq0b 9876 | A set is empty iff its ran... |
| rankeq0 9877 | A set is empty iff its ran... |
| rankr1id 9878 | The rank of the hierarchy ... |
| rankuni 9879 | The rank of a union. Part... |
| rankval4b 9880 | The rank of a set is the s... |
| rankr1b 9881 | A relationship between ran... |
| ranksuc 9882 | The rank of a successor. ... |
| rankuniss 9883 | Upper bound of the rank of... |
| rankval4 9884 | The rank of a set is the s... |
| rankbnd 9885 | The rank of a set is bound... |
| rankbnd2 9886 | The rank of a set is bound... |
| rankc1 9887 | A relationship that can be... |
| rankc2 9888 | A relationship that can be... |
| rankelun 9889 | Rank membership is inherit... |
| rankelpr 9890 | Rank membership is inherit... |
| rankelop 9891 | Rank membership is inherit... |
| rankxpl 9892 | A lower bound on the rank ... |
| rankxpu 9893 | An upper bound on the rank... |
| rankfu 9894 | An upper bound on the rank... |
| rankmapu 9895 | An upper bound on the rank... |
| rankxplim 9896 | The rank of a Cartesian pr... |
| rankxplim2 9897 | If the rank of a Cartesian... |
| rankxplim3 9898 | The rank of a Cartesian pr... |
| rankxpsuc 9899 | The rank of a Cartesian pr... |
| tcwf 9900 | The transitive closure fun... |
| tcrank 9901 | This theorem expresses two... |
| rankfilimbi 9902 | If all elements of a finit... |
| r1filimi 9903 | If all elements of a finit... |
| dfhf2 9906 | Alternate definition of th... |
| elhf 9907 | Membership in the heredita... |
| elhfOLD 9908 | Obsolete version of ~ elhf... |
| hffi 9909 | Hereditarily finite sets a... |
| elhf2 9910 | Alternate form of membersh... |
| elhf2g 9911 | Hereditarily finiteness vi... |
| elhf4 9912 | A set is hereditarily fini... |
| elhf3 9913 | A set is hereditarily fini... |
| hfelhf 9914 | Any member of a hereditari... |
| hfsshf 9915 | Any subset of a hereditari... |
| hfelhfOLD 9916 | Obsolete version of ~ elhf... |
| 0hf 9917 | The empty set is a heredit... |
| hfun 9918 | The union of two hereditar... |
| hfunOLD 9919 | Obsolete version of ~ hfun... |
| hfsn 9920 | The singleton of a heredit... |
| hfsnOLD 9921 | Obsolete version of ~ hfsn... |
| hfadj 9922 | Adjoining one hereditarily... |
| elhf3OLD 9923 | Obsolete version of ~ elhf... |
| hfuni 9924 | The union of a hereditaril... |
| hfuniOLD 9925 | Obsolete version of ~ hfun... |
| hfpw 9926 | The power class of a hered... |
| hfpwOLD 9927 | Obsolete version of ~ hfpw... |
| scotteqd 9930 | Equality theorem for the S... |
| scotteq 9931 | Closed form of ~ scotteqd ... |
| nfscott 9932 | Bound-variable hypothesis ... |
| scottex 9933 | Scott's trick produces a s... |
| scottexOLD 9934 | Obsolete version of ~ scot... |
| scottss 9935 | Scott's trick produces a s... |
| scott0 9936 | Applying Scott's trick to ... |
| scott0b 9937 | Applying Scott's trick yie... |
| scott0OLD 9938 | Obsolete version of ~ scot... |
| scottabf 9939 | Value of the Scott operati... |
| scottab 9940 | Value of the Scott operati... |
| scottabes 9941 | Value of the Scott operati... |
| elscottab 9942 | An element of the output o... |
| scottexsOLD 9943 | Obsolete theorem as of 27-... |
| scott0bs 9944 | Theorem scheme version of ... |
| scott0bsOLD 9945 | Obsolete version of ~ scot... |
| scottex2OLD 9946 | Obsolete version of ~ scot... |
| scotteld 9947 | The Scott operation sends ... |
| scottelrankd 9948 | Property of a Scott's tric... |
| scottrankd 9949 | Rank of a nonempty Scott's... |
| cplem1 9950 | Lemma for the Collection P... |
| cplem1OLD 9951 | Obsolete version of ~ cple... |
| cplem2 9952 | Lemma for the Collection P... |
| cplem2OLD 9953 | Obsolete version of ~ cple... |
| cp 9954 | Collection Principle. Thi... |
| bnd 9955 | A very strong generalizati... |
| bnd2 9956 | A variant of the Boundedne... |
| kardex 9957 | The collection of all sets... |
| kardexOLD 9958 | Obsolete version of ~ kard... |
| karden 9959 | If we allow the Axiom of R... |
| kardenOLD 9960 | Obsolete version of ~ kard... |
| htalem 9961 | Lemma for defining an emul... |
| hta 9962 | A ZFC emulation of Hilbert... |
| htaOLD 9963 | Obsolete version of ~ hta ... |
| setrec1lem1 9966 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 9967 | Lemma for ~ setrec1 . If ... |
| bnd2d 9968 | Deduction form of ~ bnd2 .... |
| setrec1lem3 9969 | Lemma for ~ setrec1 . If ... |
| spcdvw 9970 | A version of ~ spcdv where... |
| setrec1lem4 9971 | Lemma for ~ setrec1 . If ... |
| setrec1 9972 | This is the first of two f... |
| setrec2fun 9973 | This is the second of two ... |
| setrec2lem1 9974 | Lemma for ~ setrec2 . The... |
| dffun3f 9975 | Alternate definition of fu... |
| setrec2lem2 9976 | Lemma for ~ setrec2 . The... |
| setrec2 9977 | This is the second of two ... |
| setrec2v 9978 | Version of ~ setrec2 with ... |
| djueq12 9985 | Equality theorem for disjo... |
| djueq1 9986 | Equality theorem for disjo... |
| djueq2 9987 | Equality theorem for disjo... |
| nfdju 9988 | Bound-variable hypothesis ... |
| djuex 9989 | The disjoint union of sets... |
| djuexb 9990 | The disjoint union of two ... |
| djulcl 9991 | Left closure of disjoint u... |
| djurcl 9992 | Right closure of disjoint ... |
| djulf1o 9993 | The left injection functio... |
| djurf1o 9994 | The right injection functi... |
| inlresf 9995 | The left injection restric... |
| inlresf1 9996 | The left injection restric... |
| inrresf 9997 | The right injection restri... |
| inrresf1 9998 | The right injection restri... |
| djuin 9999 | The images of any classes ... |
| djur 10000 | A member of a disjoint uni... |
| djuss 10001 | A disjoint union is a subc... |
| djuunxp 10002 | The union of a disjoint un... |
| djuexALT 10003 | Alternate proof of ~ djuex... |
| eldju1st 10004 | The first component of an ... |
| eldju2ndl 10005 | The second component of an... |
| eldju2ndr 10006 | The second component of an... |
| djuun 10007 | The disjoint union of two ... |
| 1stinl 10008 | The first component of the... |
| 2ndinl 10009 | The second component of th... |
| 1stinr 10010 | The first component of the... |
| 2ndinr 10011 | The second component of th... |
| updjudhf 10012 | The mapping of an element ... |
| updjudhcoinlf 10013 | The composition of the map... |
| updjudhcoinrg 10014 | The composition of the map... |
| updjud 10015 | Universal property of the ... |
| cardf2 10024 | The cardinality function i... |
| cardon 10025 | The cardinal number of a s... |
| isnum2 10026 | A way to express well-orde... |
| isnumi 10027 | A set equinumerous to an o... |
| ennum 10028 | Equinumerous sets are equi... |
| finnum 10029 | Every finite set is numera... |
| onenon 10030 | Every ordinal number is nu... |
| tskwe 10031 | A Tarski set is well-order... |
| xpnum 10032 | The cartesian product of n... |
| cardval3 10033 | An alternate definition of... |
| cardid2 10034 | Any numerable set is equin... |
| isnum3 10035 | A set is numerable iff it ... |
| oncardval 10036 | The value of the cardinal ... |
| oncardid 10037 | Any ordinal number is equi... |
| cardonle 10038 | The cardinal of an ordinal... |
| card0 10039 | The cardinality of the emp... |
| cardidm 10040 | The cardinality function i... |
| oncard 10041 | A set is a cardinal number... |
| ficardom 10042 | The cardinal number of a f... |
| ficardid 10043 | A finite set is equinumero... |
| cardnn 10044 | The cardinality of a natur... |
| cardnueq0 10045 | The empty set is the only ... |
| cardne 10046 | No member of a cardinal nu... |
| carden2a 10047 | If two sets have equal non... |
| carden2b 10048 | If two sets are equinumero... |
| card1 10049 | A set has cardinality one ... |
| cardsn 10050 | A singleton has cardinalit... |
| carddomi2 10051 | Two sets have the dominanc... |
| sdomsdomcardi 10052 | A set strictly dominates i... |
| cardlim 10053 | An infinite cardinal is a ... |
| cardsdomelir 10054 | A cardinal strictly domina... |
| cardsdomel 10055 | A cardinal strictly domina... |
| iscard 10056 | Two ways to express the pr... |
| iscard2 10057 | Two ways to express the pr... |
| carddom2 10058 | Two numerable sets have th... |
| harcard 10059 | The class of ordinal numbe... |
| cardprclem 10060 | Lemma for ~ cardprc . (Co... |
| cardprc 10061 | The class of all cardinal ... |
| carduni 10062 | The union of a set of card... |
| cardiun 10063 | The indexed union of a set... |
| cardennn 10064 | If ` A ` is equinumerous t... |
| cardsucinf 10065 | The cardinality of the suc... |
| cardsucnn 10066 | The cardinality of the suc... |
| cardom 10067 | The set of natural numbers... |
| carden2 10068 | Two numerable sets are equ... |
| cardsdom2 10069 | A numerable set is strictl... |
| domtri2 10070 | Trichotomy of dominance fo... |
| nnsdomel 10071 | Strict dominance and eleme... |
| cardval2 10072 | An alternate version of th... |
| isinffi 10073 | An infinite set contains s... |
| fidomtri 10074 | Trichotomy of dominance wi... |
| fidomtri2 10075 | Trichotomy of dominance wi... |
| harsdom 10076 | The Hartogs number of a we... |
| onsdom 10077 | Any well-orderable set is ... |
| harval2 10078 | An alternate expression fo... |
| harsucnn 10079 | The next cardinal after a ... |
| cardmin2 10080 | The smallest ordinal that ... |
| pm54.43lem 10081 | In Theorem *54.43 of [Whit... |
| pm54.43 10082 | Theorem *54.43 of [Whitehe... |
| enpr2 10083 | An unordered pair with dis... |
| pr2ne 10084 | If an unordered pair has t... |
| prdom2 10085 | An unordered pair has at m... |
| en2eqpr 10086 | Building a set with two el... |
| en2eleq 10087 | Express a set of pair card... |
| en2other2 10088 | Taking the other element t... |
| dif1card 10089 | The cardinality of a nonem... |
| leweon 10090 | Lexicographical order is a... |
| r0weon 10091 | A set-like well-ordering o... |
| infxpenlem 10092 | Lemma for ~ infxpen . (Co... |
| infxpen 10093 | Every infinite ordinal is ... |
| xpomen 10094 | The Cartesian product of o... |
| xpct 10095 | The cartesian product of t... |
| infxpidm2 10096 | Every infinite well-ordera... |
| infxpenc 10097 | A canonical version of ~ i... |
| infxpenc2lem1 10098 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10099 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10100 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10101 | Existence form of ~ infxpe... |
| iunmapdisj 10102 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10103 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10104 | Lemma for ~ fseqen . (Con... |
| fseqdom 10105 | One half of ~ fseqen . (C... |
| fseqen 10106 | A set that is equinumerous... |
| infpwfidom 10107 | The collection of finite s... |
| dfac8alem 10108 | Lemma for ~ dfac8a . If t... |
| dfac8a 10109 | Numeration theorem: every ... |
| dfac8b 10110 | The well-ordering theorem:... |
| dfac8clem 10111 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10112 | If the union of a set is w... |
| ac10ct 10113 | A proof of the well-orderi... |
| ween 10114 | A set is numerable iff it ... |
| ac5num 10115 | A version of ~ ac5b with t... |
| ondomen 10116 | If a set is dominated by a... |
| numdom 10117 | A set dominated by a numer... |
| ssnum 10118 | A subset of a numerable se... |
| onssnum 10119 | All subsets of the ordinal... |
| indcardi 10120 | Indirect strong induction ... |
| acnrcl 10121 | Reverse closure for the ch... |
| acneq 10122 | Equality theorem for the c... |
| isacn 10123 | The property of being a ch... |
| acni 10124 | The property of being a ch... |
| acni2 10125 | The property of being a ch... |
| acni3 10126 | The property of being a ch... |
| acnlem 10127 | Construct a mapping satisf... |
| numacn 10128 | A well-orderable set has c... |
| finacn 10129 | Every set has finite choic... |
| acndom 10130 | A set with long choice seq... |
| acnnum 10131 | A set ` X ` which has choi... |
| acnen 10132 | The class of choice sets o... |
| acndom2 10133 | A set smaller than one wit... |
| acnen2 10134 | The class of sets with cho... |
| fodomacn 10135 | A version of ~ fodom that ... |
| fodomnum 10136 | A version of ~ fodom that ... |
| fonum 10137 | A surjection maps numerabl... |
| numwdom 10138 | A surjection maps numerabl... |
| fodomfi2 10139 | Onto functions define domi... |
| wdomfil 10140 | Weak dominance agrees with... |
| infpwfien 10141 | Any infinite well-orderabl... |
| inffien 10142 | The set of finite intersec... |
| wdomnumr 10143 | Weak dominance agrees with... |
| alephfnon 10144 | The aleph function is a fu... |
| aleph0 10145 | The first infinite cardina... |
| alephlim 10146 | Value of the aleph functio... |
| alephsuc 10147 | Value of the aleph functio... |
| alephon 10148 | An aleph is an ordinal num... |
| alephcard 10149 | Every aleph is a cardinal ... |
| alephnbtwn 10150 | No cardinal can be sandwic... |
| alephnbtwn2 10151 | No set has equinumerosity ... |
| alephordilem1 10152 | Lemma for ~ alephordi . (... |
| alephordi 10153 | Strict ordering property o... |
| alephord 10154 | Ordering property of the a... |
| alephord2 10155 | Ordering property of the a... |
| alephord2i 10156 | Ordering property of the a... |
| alephord3 10157 | Ordering property of the a... |
| alephsucdom 10158 | A set dominated by an alep... |
| alephsuc2 10159 | An alternate representatio... |
| alephdom 10160 | Relationship between inclu... |
| alephgeom 10161 | Every aleph is greater tha... |
| alephislim 10162 | Every aleph is a limit ord... |
| aleph11 10163 | The aleph function is one-... |
| alephf1 10164 | The aleph function is a on... |
| alephsdom 10165 | If an ordinal is smaller t... |
| alephdom2 10166 | A dominated initial ordina... |
| alephle 10167 | The argument of the aleph ... |
| cardaleph 10168 | Given any transfinite card... |
| cardalephex 10169 | Every transfinite cardinal... |
| infenaleph 10170 | An infinite numerable set ... |
| isinfcard 10171 | Two ways to express the pr... |
| iscard3 10172 | Two ways to express the pr... |
| cardnum 10173 | Two ways to express the cl... |
| alephinit 10174 | An infinite initial ordina... |
| carduniima 10175 | The union of the image of ... |
| cardinfima 10176 | If a mapping to cardinals ... |
| alephiso 10177 | Aleph is an order isomorph... |
| alephprc 10178 | The class of all transfini... |
| alephsson 10179 | The class of transfinite c... |
| unialeph 10180 | The union of the class of ... |
| alephsmo 10181 | The aleph function is stri... |
| alephf1ALT 10182 | Alternate proof of ~ aleph... |
| alephfplem1 10183 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10184 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10185 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10186 | Lemma for ~ alephfp . (Co... |
| alephfp 10187 | The aleph function has a f... |
| alephfp2 10188 | The aleph function has at ... |
| alephval3 10189 | An alternate way to expres... |
| alephsucpw2 10190 | The power set of an aleph ... |
| mappwen 10191 | Power rule for cardinal ar... |
| finnisoeu 10192 | A finite totally ordered s... |
| iunfictbso 10193 | Countability of a countabl... |
| aceq1 10196 | Equivalence of two version... |
| aceq0 10197 | Equivalence of two version... |
| aceq2 10198 | Equivalence of two version... |
| aceq3lem 10199 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10200 | Equivalence of two version... |
| dfac4 10201 | Equivalence of two version... |
| dfac5lem1 10202 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10203 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10204 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10205 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10206 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10207 | Equivalence of two version... |
| dfac2a 10208 | Our Axiom of Choice (in th... |
| dfac2b 10209 | Axiom of Choice (first for... |
| dfac2 10210 | Axiom of Choice (first for... |
| dfac7 10211 | Equivalence of the Axiom o... |
| dfac0 10212 | Equivalence of two version... |
| dfac1 10213 | Equivalence of two version... |
| dfac8 10214 | A proof of the equivalency... |
| dfac9 10215 | Equivalence of the axiom o... |
| dfac10 10216 | Axiom of Choice equivalent... |
| dfac10c 10217 | Axiom of Choice equivalent... |
| dfac10b 10218 | Axiom of Choice equivalent... |
| acacni 10219 | A choice equivalent: every... |
| dfacacn 10220 | A choice equivalent: every... |
| dfac13 10221 | The axiom of choice holds ... |
| dfac12lem1 10222 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10223 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10224 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10225 | The axiom of choice holds ... |
| dfac12k 10226 | Equivalence of ~ dfac12 an... |
| dfac12a 10227 | The axiom of choice holds ... |
| dfac12 10228 | The axiom of choice holds ... |
| kmlem1 10229 | Lemma for 5-quantifier AC ... |
| kmlem2 10230 | Lemma for 5-quantifier AC ... |
| kmlem3 10231 | Lemma for 5-quantifier AC ... |
| kmlem4 10232 | Lemma for 5-quantifier AC ... |
| kmlem5 10233 | Lemma for 5-quantifier AC ... |
| kmlem6 10234 | Lemma for 5-quantifier AC ... |
| kmlem7 10235 | Lemma for 5-quantifier AC ... |
| kmlem8 10236 | Lemma for 5-quantifier AC ... |
| kmlem9 10237 | Lemma for 5-quantifier AC ... |
| kmlem10 10238 | Lemma for 5-quantifier AC ... |
| kmlem11 10239 | Lemma for 5-quantifier AC ... |
| kmlem12 10240 | Lemma for 5-quantifier AC ... |
| kmlem13 10241 | Lemma for 5-quantifier AC ... |
| kmlem14 10242 | Lemma for 5-quantifier AC ... |
| kmlem15 10243 | Lemma for 5-quantifier AC ... |
| kmlem16 10244 | Lemma for 5-quantifier AC ... |
| dfackm 10245 | Equivalence of the Axiom o... |
| undjudom 10246 | Cardinal addition dominate... |
| endjudisj 10247 | Equinumerosity of a disjoi... |
| djuen 10248 | Disjoint unions of equinum... |
| djuenun 10249 | Disjoint union is equinume... |
| dju1en 10250 | Cardinal addition with car... |
| dju1dif 10251 | Adding and subtracting one... |
| dju1p1e2 10252 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10253 | Alternate proof of ~ dju1p... |
| dju0en 10254 | Cardinal addition with car... |
| xp2dju 10255 | Two times a cardinal numbe... |
| djucomen 10256 | Commutative law for cardin... |
| djuassen 10257 | Associative law for cardin... |
| xpdjuen 10258 | Cardinal multiplication di... |
| mapdjuen 10259 | Sum of exponents law for c... |
| pwdjuen 10260 | Sum of exponents law for c... |
| djudom1 10261 | Ordering law for cardinal ... |
| djudom2 10262 | Ordering law for cardinal ... |
| djudoml 10263 | A set is dominated by its ... |
| djuxpdom 10264 | Cartesian product dominate... |
| djufi 10265 | The disjoint union of two ... |
| cdainflem 10266 | Any partition of omega int... |
| djuinf 10267 | A set is infinite iff the ... |
| infdju1 10268 | An infinite set is equinum... |
| pwdju1 10269 | The sum of a powerset with... |
| pwdjuidm 10270 | If the natural numbers inj... |
| djulepw 10271 | If ` A ` is idempotent und... |
| onadju 10272 | The cardinal and ordinal s... |
| cardadju 10273 | The cardinal sum is equinu... |
| djunum 10274 | The disjoint union of two ... |
| unnum 10275 | The union of two numerable... |
| nnadju 10276 | The cardinal and ordinal s... |
| nnadjuALT 10277 | Shorter proof of ~ nnadju ... |
| ficardadju 10278 | The disjoint union of fini... |
| ficardun 10279 | The cardinality of the uni... |
| ficardun2 10280 | The cardinality of the uni... |
| pwsdompw 10281 | Lemma for ~ domtriom . Th... |
| unctb 10282 | The union of two countable... |
| infdjuabs 10283 | Absorption law for additio... |
| infunabs 10284 | An infinite set is equinum... |
| infdju 10285 | The sum of two cardinal nu... |
| infdif 10286 | The cardinality of an infi... |
| infdif2 10287 | Cardinality ordering for a... |
| infxpdom 10288 | Dominance law for multipli... |
| infxpabs 10289 | Absorption law for multipl... |
| infunsdom1 10290 | The union of two sets that... |
| infunsdom 10291 | The union of two sets that... |
| infxp 10292 | Absorption law for multipl... |
| pwdjudom 10293 | A property of dominance ov... |
| infpss 10294 | Every infinite set has an ... |
| infmap2 10295 | An exponentiation law for ... |
| ackbij2lem1 10296 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10297 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10298 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10299 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10300 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10301 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10302 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10303 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10304 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10305 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10306 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10307 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10308 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10309 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10310 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10311 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10312 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10313 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10314 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10315 | The Ackermann bijection, p... |
| ackbij1b 10316 | The Ackermann bijection, p... |
| ackbij2lem2 10317 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10318 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10319 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10320 | The Ackermann bijection, p... |
| hfom 10321 | The set of hereditarily fi... |
| fictb 10322 | A set is countable iff its... |
| cflem 10323 | A lemma used to simplify c... |
| cfval 10324 | Value of the cofinality fu... |
| cff 10325 | Cofinality is a function o... |
| cfub 10326 | An upper bound on cofinali... |
| cflm 10327 | Value of the cofinality fu... |
| cf0 10328 | Value of the cofinality fu... |
| cardcf 10329 | Cofinality is a cardinal n... |
| cflecard 10330 | Cofinality is bounded by t... |
| cfle 10331 | Cofinality is bounded by i... |
| cfon 10332 | The cofinality of any set ... |
| cfonOLD 10333 | Obsolete version of ~ cfon... |
| cfeq0 10334 | Only the ordinal zero has ... |
| cfsuc 10335 | Value of the cofinality fu... |
| cff1 10336 | There is always a map from... |
| cfflb 10337 | If there is a cofinal map ... |
| cfval2 10338 | Another expression for the... |
| coflim 10339 | A simpler expression for t... |
| cflim3 10340 | Another expression for the... |
| cflim2 10341 | The cofinality function is... |
| cfom 10342 | Value of the cofinality fu... |
| cfss 10343 | There is a cofinal subset ... |
| cfslb 10344 | Any cofinal subset of ` A ... |
| cfslbn 10345 | Any subset of ` A ` smalle... |
| cfslb2n 10346 | Any small collection of sm... |
| cofsmo 10347 | Any cofinal map implies th... |
| cfsmolem 10348 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10349 | The map in ~ cff1 can be a... |
| cfcoflem 10350 | Lemma for ~ cfcof , showin... |
| coftr 10351 | If there is a cofinal map ... |
| cfcof 10352 | If there is a cofinal map ... |
| cfidm 10353 | The cofinality function is... |
| alephsing 10354 | The cofinality of a limit ... |
| sornom 10355 | The range of a single-step... |
| isfin1a 10370 | Definition of a Ia-finite ... |
| fin1ai 10371 | Property of a Ia-finite se... |
| isfin2 10372 | Definition of a II-finite ... |
| fin2i 10373 | Property of a II-finite se... |
| isfin3 10374 | Definition of a III-finite... |
| isfin4 10375 | Definition of a IV-finite ... |
| fin4i 10376 | Infer that a set is IV-inf... |
| isfin5 10377 | Definition of a V-finite s... |
| isfin6 10378 | Definition of a VI-finite ... |
| isfin7 10379 | Definition of a VII-finite... |
| sdom2en01 10380 | A set with less than two e... |
| infpssrlem1 10381 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10382 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10383 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10384 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10385 | Lemma for ~ infpssr . (Co... |
| infpssr 10386 | Dedekind infinity implies ... |
| fin4en1 10387 | Dedekind finite is a cardi... |
| ssfin4 10388 | Dedekind finite sets have ... |
| domfin4 10389 | A set dominated by a Dedek... |
| ominf4 10390 | ` _om ` is Dedekind infini... |
| infpssALT 10391 | Alternate proof of ~ infps... |
| isfin4-2 10392 | Alternate definition of IV... |
| isfin4p1 10393 | Alternate definition of IV... |
| fin23lem7 10394 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10395 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10396 | A II-finite set contains m... |
| isfin2-2 10397 | ` Fin2 ` expressed in term... |
| ssfin2 10398 | A subset of a II-finite se... |
| enfin2i 10399 | II-finiteness is a cardina... |
| fin23lem24 10400 | Lemma for ~ fin23 . In a ... |
| fincssdom 10401 | In a chain of finite sets,... |
| fin23lem25 10402 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10403 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10404 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10405 | Lemma for ~ fin23 but coul... |
| fin23lem27 10406 | The mapping constructed in... |
| isfin3ds 10407 | Property of a III-finite s... |
| ssfin3ds 10408 | A subset of a III-finite s... |
| fin23lem12 10409 | The beginning of the proof... |
| fin23lem13 10410 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10411 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10412 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10413 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10414 | Lemma for ~ fin23 . The f... |
| fin23lem20 10415 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10416 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10417 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10418 | Lemma for ~ fin23 . The r... |
| fin23lem29 10419 | Lemma for ~ fin23 . The r... |
| fin23lem30 10420 | Lemma for ~ fin23 . The r... |
| fin23lem31 10421 | Lemma for ~ fin23 . The r... |
| fin23lem32 10422 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10423 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10424 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10425 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10426 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10427 | Lemma for ~ fin23 . The c... |
| fin23lem39 10428 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10429 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10430 | Lemma for ~ fin23 . A set... |
| isf32lem1 10431 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10432 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10433 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10434 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10435 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10436 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10437 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10438 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10439 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10440 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10441 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10442 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10443 | One half of ~ isfin3-2 . ... |
| isf33lem 10444 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10445 | Weakly Dedekind-infinite s... |
| isfin3-3 10446 | Weakly Dedekind-infinite s... |
| fin33i 10447 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10448 | Lemma for ~ compsscnv . (... |
| compsscnv 10449 | Complementation on a power... |
| isf34lem1 10450 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10451 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10452 | Complementation is an anti... |
| isf34lem3 10453 | Lemma for ~ isfin3-4 . (C... |
| compss 10454 | Express image under of the... |
| isf34lem4 10455 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10456 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10457 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10458 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10459 | Inference from ~ isfin3-4 ... |
| isfin3-4 10460 | Weakly Dedekind-infinite s... |
| fin11a 10461 | Every I-finite set is Ia-f... |
| enfin1ai 10462 | Ia-finiteness is a cardina... |
| isfin1-2 10463 | A set is finite in the usu... |
| isfin1-3 10464 | A set is I-finite iff ever... |
| isfin1-4 10465 | A set is I-finite iff ever... |
| dffin1-5 10466 | Compact quantifier-free ve... |
| fin23 10467 | Every II-finite set (every... |
| fin34 10468 | Every III-finite set is IV... |
| isfin5-2 10469 | Alternate definition of V-... |
| fin45 10470 | Every IV-finite set is V-f... |
| fin56 10471 | Every V-finite set is VI-f... |
| fin17 10472 | Every I-finite set is VII-... |
| fin67 10473 | Every VI-finite set is VII... |
| isfin7-2 10474 | A set is VII-finite iff it... |
| fin71num 10475 | A well-orderable set is VI... |
| dffin7-2 10476 | Class form of ~ isfin7-2 .... |
| dfacfin7 10477 | Axiom of Choice equivalent... |
| fin1a2lem1 10478 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10479 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10480 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10481 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10482 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10483 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10484 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10485 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10486 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10487 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10488 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10489 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10490 | Lemma for ~ fin1a2 . (Con... |
| fin12 10491 | Weak theorem which skips I... |
| fin1a2s 10492 | An II-infinite set can hav... |
| fin1a2 10493 | Every Ia-finite set is II-... |
| itunifval 10494 | Function value of iterated... |
| itunifn 10495 | Functionality of the itera... |
| ituni0 10496 | A zero-fold iterated union... |
| itunisuc 10497 | Successor iterated union. ... |
| itunitc1 10498 | Each union iterate is a me... |
| itunitc 10499 | The union of all union ite... |
| ituniiun 10500 | Unwrap an iterated union f... |
| hsmexlem7 10501 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10502 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10503 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10504 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10505 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10506 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10507 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10508 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10509 | Lemma for ~ hsmex . (Cont... |
| hsmex 10510 | The collection of heredita... |
| hsmex2 10511 | The set of hereditary size... |
| hsmex3 10512 | The set of hereditary size... |
| axcc2lem 10514 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10515 | A possibly more useful ver... |
| axcc3 10516 | A possibly more useful ver... |
| axcc4 10517 | A version of ~ axcc3 that ... |
| acncc 10518 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10519 | Relax the constraint on ~ ... |
| domtriomlem 10520 | Lemma for ~ domtriom . (C... |
| domtriom 10521 | Trichotomy of equinumerosi... |
| fin41 10522 | Under countable choice, th... |
| dominf 10523 | A nonempty set that is a s... |
| dcomex 10525 | The Axiom of Dependent Cho... |
| axdc2lem 10526 | Lemma for ~ axdc2 . We co... |
| axdc2 10527 | An apparent strengthening ... |
| axdc3lem 10528 | The class ` S ` of finite ... |
| axdc3lem2 10529 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10530 | Simple substitution lemma ... |
| axdc3lem4 10531 | Lemma for ~ axdc3 . We ha... |
| axdc3 10532 | Dependent Choice. Axiom D... |
| axdc4lem 10533 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10534 | A more general version of ... |
| axcclem 10535 | Lemma for ~ axcc . (Contr... |
| axcc 10536 | Although CC can be proven ... |
| zfac 10538 | Axiom of Choice expressed ... |
| ac2 10539 | Axiom of Choice equivalent... |
| ac3 10540 | Axiom of Choice using abbr... |
| axac3 10542 | This theorem asserts that ... |
| ackm 10543 | A remarkable equivalent to... |
| axac2 10544 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10545 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10546 | Apply a choice equivalent.... |
| cardeqv 10547 | All sets are well-orderabl... |
| numth3 10548 | All sets are well-orderabl... |
| numth2 10549 | Numeration theorem: any se... |
| numth 10550 | Numeration theorem: every ... |
| ac7 10551 | An Axiom of Choice equival... |
| ac7g 10552 | An Axiom of Choice equival... |
| ac4 10553 | Equivalent of Axiom of Cho... |
| ac4c 10554 | Equivalent of Axiom of Cho... |
| ac5 10555 | An Axiom of Choice equival... |
| ac5b 10556 | Equivalent of Axiom of Cho... |
| ac6num 10557 | A version of ~ ac6 which t... |
| ac6 10558 | Equivalent of Axiom of Cho... |
| ac6c4 10559 | Equivalent of Axiom of Cho... |
| ac6c5 10560 | Equivalent of Axiom of Cho... |
| ac9 10561 | An Axiom of Choice equival... |
| ac6s 10562 | Equivalent of Axiom of Cho... |
| ac6n 10563 | Equivalent of Axiom of Cho... |
| ac6s2 10564 | Generalization of the Axio... |
| ac6s3 10565 | Generalization of the Axio... |
| ac6sg 10566 | ~ ac6s with sethood as ant... |
| ac6sf 10567 | Version of ~ ac6 with boun... |
| ac6s4 10568 | Generalization of the Axio... |
| ac6s5 10569 | Generalization of the Axio... |
| ac8 10570 | An Axiom of Choice equival... |
| ac9s 10571 | An Axiom of Choice equival... |
| numthcor 10572 | Any set is strictly domina... |
| weth 10573 | Well-ordering theorem: any... |
| zorn2lem1 10574 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10575 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10576 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10577 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10578 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10579 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10580 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10581 | Zorn's Lemma of [Monk1] p.... |
| zorng 10582 | Zorn's Lemma. If the unio... |
| zornn0g 10583 | Variant of Zorn's lemma ~ ... |
| zorn2 10584 | Zorn's Lemma of [Monk1] p.... |
| zorn 10585 | Zorn's Lemma. If the unio... |
| zornn0 10586 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10587 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10588 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10589 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10590 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10591 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10592 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10593 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10594 | The Teichmüller-Tukey... |
| ttukeyg 10595 | The Teichmüller-Tukey... |
| ttukey 10596 | The Teichmüller-Tukey... |
| axdclem 10597 | Lemma for ~ axdc . (Contr... |
| axdclem2 10598 | Lemma for ~ axdc . Using ... |
| axdc 10599 | This theorem derives ~ ax-... |
| fodomg 10600 | An onto function implies d... |
| fodom 10601 | An onto function implies d... |
| dmct 10602 | The domain of a countable ... |
| dmctOLD 10603 | Obsolete version of ~ dmct... |
| rnct 10604 | The range of a countable s... |
| fodomb 10605 | Equivalence of an onto map... |
| wdomac 10606 | When assuming AC, weak and... |
| brdom3 10607 | Equivalence to a dominance... |
| brdom5 10608 | An equivalence to a domina... |
| brdom4 10609 | An equivalence to a domina... |
| brdom7disj 10610 | An equivalence to a domina... |
| brdom6disj 10611 | An equivalence to a domina... |
| fin71ac 10612 | Once we allow AC, the "str... |
| imadomg 10613 | An image of a function und... |
| imadomnum 10614 | A version of ~ imadomg tha... |
| fimact 10615 | The image by a function of... |
| fimactOLD 10616 | Obsolete version of ~ fima... |
| fnrndomnum 10617 | A version of ~ fnrndomg th... |
| fnrndomg 10618 | The range of a function is... |
| fnrndomgOLD 10619 | Obsolete version of ~ fnrn... |
| fnct 10620 | If the domain of a functio... |
| fnctOLD 10621 | Obsolete version of ~ fnct... |
| mptct 10622 | A countable mapping set is... |
| iunfo 10623 | Existence of an onto funct... |
| iundom2g 10624 | An upper bound for the car... |
| iundomg 10625 | An upper bound for the car... |
| iundom 10626 | An upper bound for the car... |
| unidom 10627 | An upper bound for the car... |
| uniimadom 10628 | An upper bound for the car... |
| uniimadomf 10629 | An upper bound for the car... |
| cardval 10630 | The value of the cardinal ... |
| cardid 10631 | Any set is equinumerous to... |
| cardidg 10632 | Any set is equinumerous to... |
| cardidd 10633 | Any set is equinumerous to... |
| cardf 10634 | The cardinality function i... |
| carden 10635 | Two sets are equinumerous ... |
| cardeq0 10636 | Only the empty set has car... |
| unsnen 10637 | Equinumerosity of a set wi... |
| carddom 10638 | Two sets have the dominanc... |
| cardsdom 10639 | Two sets have the strict d... |
| domtri 10640 | Trichotomy law for dominan... |
| entric 10641 | Trichotomy of equinumerosi... |
| entri2 10642 | Trichotomy of dominance an... |
| entri3 10643 | Trichotomy of dominance. ... |
| sdomsdomcard 10644 | A set strictly dominates i... |
| canth3 10645 | Cantor's theorem in terms ... |
| infxpidm 10646 | Every infinite class is eq... |
| ondomon 10647 | The class of ordinals domi... |
| cardmin 10648 | The smallest ordinal that ... |
| ficard 10649 | A set is finite iff its ca... |
| infinfg 10650 | Equivalence between two in... |
| infinf 10651 | Equivalence between two in... |
| unirnfdomd 10652 | The union of the range of ... |
| konigthlem 10653 | Lemma for ~ konigth . (Co... |
| konigth 10654 | Konig's Theorem. If ` m (... |
| alephsucpw 10655 | The power set of an aleph ... |
| aleph1 10656 | The set exponentiation of ... |
| alephval2 10657 | An alternate way to expres... |
| dominfac 10658 | A nonempty set that is a s... |
| iunctb 10659 | The countable union of cou... |
| unictb 10660 | The countable union of cou... |
| infmap 10661 | An exponentiation law for ... |
| alephadd 10662 | The sum of two alephs is t... |
| alephmul 10663 | The product of two alephs ... |
| alephexp1 10664 | An exponentiation law for ... |
| alephsuc3 10665 | An alternate representatio... |
| alephexp2 10666 | An expression equinumerous... |
| alephreg 10667 | A successor aleph is regul... |
| pwcfsdom 10668 | A corollary of Konig's The... |
| cfpwsdom 10669 | A corollary of Konig's The... |
| alephom 10670 | From ~ canth2 , we know th... |
| smobeth 10671 | The beth function is stric... |
| nd1 10672 | A lemma for proving condit... |
| nd2 10673 | A lemma for proving condit... |
| nd3 10674 | A lemma for proving condit... |
| nd4 10675 | A lemma for proving condit... |
| axextnd 10676 | A version of the Axiom of ... |
| axrepndlem1 10677 | Lemma for the Axiom of Rep... |
| axrepndlem2 10678 | Lemma for the Axiom of Rep... |
| axrepnd 10679 | A version of the Axiom of ... |
| axunndlem1 10680 | Lemma for the Axiom of Uni... |
| axunnd 10681 | A version of the Axiom of ... |
| axpowndlem1 10682 | Lemma for the Axiom of Pow... |
| axpowndlem2 10683 | Lemma for the Axiom of Pow... |
| axpowndlem3 10684 | Lemma for the Axiom of Pow... |
| axpowndlem4 10685 | Lemma for the Axiom of Pow... |
| axpownd 10686 | A version of the Axiom of ... |
| axregndlem1 10687 | Lemma for the Axiom of Reg... |
| axregndlem2 10688 | Lemma for the Axiom of Reg... |
| axregnd 10689 | A version of the Axiom of ... |
| axinfndlem1 10690 | Lemma for the Axiom of Inf... |
| axinfnd 10691 | A version of the Axiom of ... |
| axacndlem1 10692 | Lemma for the Axiom of Cho... |
| axacndlem2 10693 | Lemma for the Axiom of Cho... |
| axacndlem3 10694 | Lemma for the Axiom of Cho... |
| axacndlem4 10695 | Lemma for the Axiom of Cho... |
| axacndlem5 10696 | Lemma for the Axiom of Cho... |
| axacnd 10697 | A version of the Axiom of ... |
| zfcndext 10698 | Axiom of Extensionality ~ ... |
| zfcndrep 10699 | Axiom of Replacement ~ ax-... |
| zfcndun 10700 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10701 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10702 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10703 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10704 | Axiom of Choice ~ ax-ac , ... |
| elgch 10707 | Elementhood in the collect... |
| fingch 10708 | A finite set is a GCH-set.... |
| gchi 10709 | The only GCH-sets which ha... |
| gchen1 10710 | If ` A <_ B < ~P A ` , and... |
| gchen2 10711 | If ` A < B <_ ~P A ` , and... |
| gchor 10712 | If ` A <_ B <_ ~P A ` , an... |
| engch 10713 | The property of being a GC... |
| gchdomtri 10714 | Under certain conditions, ... |
| fpwwe2cbv 10715 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10716 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10717 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10718 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10719 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10720 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10721 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10722 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10723 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10724 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10725 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10726 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10727 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10728 | Given any function ` F ` f... |
| fpwwecbv 10729 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10730 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10731 | Given any function ` F ` f... |
| canth4 10732 | An "effective" form of Can... |
| canthnumlem 10733 | Lemma for ~ canthnum . (C... |
| canthnum 10734 | The set of well-orderable ... |
| canthwelem 10735 | Lemma for ~ canthwe . (Co... |
| canthwe 10736 | The set of well-orders of ... |
| canthp1lem1 10737 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10738 | Lemma for ~ canthp1 . (Co... |
| canthp1 10739 | A slightly stronger form o... |
| finngch 10740 | The exclusion of finite se... |
| gchdju1 10741 | An infinite GCH-set is ide... |
| gchinf 10742 | An infinite GCH-set is Ded... |
| pwfseqlem1 10743 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10744 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10745 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10746 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10747 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10748 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10749 | The powerset of a Dedekind... |
| pwxpndom2 10750 | The powerset of a Dedekind... |
| pwxpndom 10751 | The powerset of a Dedekind... |
| pwdjundom 10752 | The powerset of a Dedekind... |
| gchdjuidm 10753 | An infinite GCH-set is ide... |
| gchxpidm 10754 | An infinite GCH-set is ide... |
| gchpwdom 10755 | A relationship between dom... |
| gchaleph 10756 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10757 | If ` ( aleph `` A ) ` and ... |
| hargch 10758 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10759 | If ` ( aleph `` suc A ) ` ... |
| gch2 10760 | It is sufficient to requir... |
| gch3 10761 | An equivalent formulation ... |
| gch-kn 10762 | The equivalence of two ver... |
| gchaclem 10763 | Lemma for ~ gchac (obsolet... |
| gchhar 10764 | A "local" form of ~ gchac ... |
| gchacg 10765 | A "local" form of ~ gchac ... |
| gchac 10766 | The Generalized Continuum ... |
| elwina 10771 | Conditions of weak inacces... |
| elina 10772 | Conditions of strong inacc... |
| winaon 10773 | A weakly inaccessible card... |
| inawinalem 10774 | Lemma for ~ inawina . (Co... |
| inawina 10775 | Every strongly inaccessibl... |
| omina 10776 | ` _om ` is a strongly inac... |
| winacard 10777 | A weakly inaccessible card... |
| winainflem 10778 | A weakly inaccessible card... |
| winainf 10779 | A weakly inaccessible card... |
| winalim 10780 | A weakly inaccessible card... |
| winalim2 10781 | A nontrivial weakly inacce... |
| winafp 10782 | A nontrivial weakly inacce... |
| winafpi 10783 | This theorem, which states... |
| gchina 10784 | Assuming the GCH, weakly a... |
| iswun 10789 | Properties of a weak unive... |
| wuntr 10790 | A weak universe is transit... |
| wununi 10791 | A weak universe is closed ... |
| wunpw 10792 | A weak universe is closed ... |
| wunelss 10793 | The elements of a weak uni... |
| wunpr 10794 | A weak universe is closed ... |
| wunun 10795 | A weak universe is closed ... |
| wuntp 10796 | A weak universe is closed ... |
| wunss 10797 | A weak universe is closed ... |
| wunin 10798 | A weak universe is closed ... |
| wundif 10799 | A weak universe is closed ... |
| wunint 10800 | A weak universe is closed ... |
| wunsn 10801 | A weak universe is closed ... |
| wunsuc 10802 | A weak universe is closed ... |
| wun0 10803 | A weak universe contains t... |
| wunr1om 10804 | A weak universe is infinit... |
| wunom 10805 | A weak universe contains a... |
| wunfi 10806 | A weak universe contains a... |
| wunop 10807 | A weak universe is closed ... |
| wunot 10808 | A weak universe is closed ... |
| wunxp 10809 | A weak universe is closed ... |
| wunpm 10810 | A weak universe is closed ... |
| wunmap 10811 | A weak universe is closed ... |
| wunf 10812 | A weak universe is closed ... |
| wundm 10813 | A weak universe is closed ... |
| wunrn 10814 | A weak universe is closed ... |
| wuncnv 10815 | A weak universe is closed ... |
| wunres 10816 | A weak universe is closed ... |
| wunfv 10817 | A weak universe is closed ... |
| wunco 10818 | A weak universe is closed ... |
| wuntpos 10819 | A weak universe is closed ... |
| intwun 10820 | The intersection of a coll... |
| r1limwun 10821 | Each limit stage in the cu... |
| r1wunlim 10822 | The stages of the cumulati... |
| wunex2 10823 | Construct a weak universe ... |
| wunex 10824 | Construct a weak universe ... |
| uniwun 10825 | Every set is contained in ... |
| wunex3 10826 | Construct a weak universe ... |
| wuncval 10827 | Value of the weak universe... |
| wuncid 10828 | The weak universe closure ... |
| wunccl 10829 | The weak universe closure ... |
| wuncss 10830 | The weak universe closure ... |
| wuncidm 10831 | The weak universe closure ... |
| wuncval2 10832 | Our earlier expression for... |
| eltskg 10835 | Properties of a Tarski cla... |
| eltsk2g 10836 | Properties of a Tarski cla... |
| tskpwss 10837 | First axiom of a Tarski cl... |
| tskpw 10838 | Second axiom of a Tarski c... |
| tsken 10839 | Third axiom of a Tarski cl... |
| 0tsk 10840 | The empty set is a (transi... |
| tsksdom 10841 | An element of a Tarski cla... |
| tskssel 10842 | A part of a Tarski class s... |
| tskss 10843 | The subsets of an element ... |
| tskin 10844 | The intersection of two el... |
| tsksn 10845 | A singleton of an element ... |
| tsktrss 10846 | A transitive element of a ... |
| tsksuc 10847 | If an element of a Tarski ... |
| tsk0 10848 | A nonempty Tarski class co... |
| tsk1 10849 | One is an element of a non... |
| tsk2 10850 | Two is an element of a non... |
| 2domtsk 10851 | If a Tarski class is not e... |
| tskr1om 10852 | A nonempty Tarski class is... |
| tskhf 10853 | A nonempty Tarski class co... |
| tskinf 10854 | A nonempty Tarski class is... |
| tskpr 10855 | If ` A ` and ` B ` are mem... |
| tskop 10856 | If ` A ` and ` B ` are mem... |
| tskxpss 10857 | A Cartesian product of two... |
| tskwe2 10858 | A Tarski class is well-ord... |
| inttsk 10859 | The intersection of a coll... |
| inar1 10860 | ` ( R1 `` A ) ` for ` A ` ... |
| hfomALT 10861 | Alternate proof of ~ hfom ... |
| rankcf 10862 | Any set must be at least a... |
| inatsk 10863 | ` ( R1 `` A ) ` for ` A ` ... |
| hftsk 10864 | The set of hereditarily fi... |
| tskord 10865 | A Tarski class contains al... |
| tskcard 10866 | An even more direct relati... |
| r1tskina 10867 | There is a direct relation... |
| tskuni 10868 | The union of an element of... |
| tskwun 10869 | A nonempty transitive Tars... |
| tskint 10870 | The intersection of an ele... |
| tskun 10871 | The union of two elements ... |
| tskxp 10872 | The Cartesian product of t... |
| tskmap 10873 | Set exponentiation is an e... |
| tskurn 10874 | A transitive Tarski class ... |
| elgrug 10877 | Properties of a Grothendie... |
| grutr 10878 | A Grothendieck universe is... |
| gruelss 10879 | A Grothendieck universe is... |
| grupw 10880 | A Grothendieck universe co... |
| gruss 10881 | Any subset of an element o... |
| grupr 10882 | A Grothendieck universe co... |
| gruurn 10883 | A Grothendieck universe co... |
| gruiun 10884 | If ` B ( x ) ` is a family... |
| gruuni 10885 | A Grothendieck universe co... |
| grurn 10886 | A Grothendieck universe co... |
| gruima 10887 | A Grothendieck universe co... |
| gruel 10888 | Any element of an element ... |
| grusn 10889 | A Grothendieck universe co... |
| gruop 10890 | A Grothendieck universe co... |
| gruun 10891 | A Grothendieck universe co... |
| gruxp 10892 | A Grothendieck universe co... |
| grumap 10893 | A Grothendieck universe co... |
| gruixp 10894 | A Grothendieck universe co... |
| gruiin 10895 | A Grothendieck universe co... |
| gruf 10896 | A Grothendieck universe co... |
| gruen 10897 | A Grothendieck universe co... |
| gruwun 10898 | A nonempty Grothendieck un... |
| intgru 10899 | The intersection of a fami... |
| ingru 10900 | The intersection of a univ... |
| wfgru 10901 | The wellfounded part of a ... |
| grudomon 10902 | Each ordinal that is compa... |
| gruina 10903 | If a Grothendieck universe... |
| grur1a 10904 | A characterization of Grot... |
| grur1 10905 | A characterization of Grot... |
| grutsk1 10906 | Grothendieck universes are... |
| grutsk 10907 | Grothendieck universes are... |
| axgroth5 10909 | The Tarski-Grothendieck ax... |
| axgroth2 10910 | Alternate version of the T... |
| grothpw 10911 | Derive the Axiom of Power ... |
| grothpwex 10912 | Derive the Axiom of Power ... |
| axgroth6 10913 | The Tarski-Grothendieck ax... |
| grothomex 10914 | The Tarski-Grothendieck Ax... |
| grothac 10915 | The Tarski-Grothendieck Ax... |
| axgroth3 10916 | Alternate version of the T... |
| axgroth4 10917 | Alternate version of the T... |
| grothprimlem 10918 | Lemma for ~ grothprim . E... |
| grothprim 10919 | The Tarski-Grothendieck Ax... |
| grothtsk 10920 | The Tarski-Grothendieck Ax... |
| inaprc 10921 | An equivalent to the Tarsk... |
| tskmval 10924 | Value of our tarski map. ... |
| tskmid 10925 | The set ` A ` is an elemen... |
| tskmcl 10926 | A Tarski class that contai... |
| sstskm 10927 | Being a part of ` ( tarski... |
| eltskm 10928 | Belonging to ` ( tarskiMap... |
| elni 10961 | Membership in the class of... |
| elni2 10962 | Membership in the class of... |
| pinn 10963 | A positive integer is a na... |
| pion 10964 | A positive integer is an o... |
| piord 10965 | A positive integer is ordi... |
| niex 10966 | The class of positive inte... |
| 0npi 10967 | The empty set is not a pos... |
| 1pi 10968 | Ordinal 'one' is a positiv... |
| addpiord 10969 | Positive integer addition ... |
| mulpiord 10970 | Positive integer multiplic... |
| mulidpi 10971 | 1 is an identity element f... |
| ltpiord 10972 | Positive integer 'less tha... |
| ltsopi 10973 | Positive integer 'less tha... |
| ltrelpi 10974 | Positive integer 'less tha... |
| dmaddpi 10975 | Domain of addition on posi... |
| dmmulpi 10976 | Domain of multiplication o... |
| addclpi 10977 | Closure of addition of pos... |
| mulclpi 10978 | Closure of multiplication ... |
| addcompi 10979 | Addition of positive integ... |
| addasspi 10980 | Addition of positive integ... |
| mulcompi 10981 | Multiplication of positive... |
| mulasspi 10982 | Multiplication of positive... |
| distrpi 10983 | Multiplication of positive... |
| addcanpi 10984 | Addition cancellation law ... |
| mulcanpi 10985 | Multiplication cancellatio... |
| addnidpi 10986 | There is no identity eleme... |
| ltexpi 10987 | Ordering on positive integ... |
| ltapi 10988 | Ordering property of addit... |
| ltmpi 10989 | Ordering property of multi... |
| 1lt2pi 10990 | One is less than two (one ... |
| nlt1pi 10991 | No positive integer is les... |
| indpi 10992 | Principle of Finite Induct... |
| enqbreq 11004 | Equivalence relation for p... |
| enqbreq2 11005 | Equivalence relation for p... |
| enqer 11006 | The equivalence relation f... |
| enqex 11007 | The equivalence relation f... |
| nqex 11008 | The class of positive frac... |
| 0nnq 11009 | The empty set is not a pos... |
| elpqn 11010 | Each positive fraction is ... |
| ltrelnq 11011 | Positive fraction 'less th... |
| pinq 11012 | The representatives of pos... |
| 1nq 11013 | The positive fraction 'one... |
| nqereu 11014 | There is a unique element ... |
| nqerf 11015 | Corollary of ~ nqereu : th... |
| nqercl 11016 | Corollary of ~ nqereu : cl... |
| nqerrel 11017 | Any member of ` ( N. X. N.... |
| nqerid 11018 | Corollary of ~ nqereu : th... |
| enqeq 11019 | Corollary of ~ nqereu : if... |
| nqereq 11020 | The function ` /Q ` acts a... |
| addpipq2 11021 | Addition of positive fract... |
| addpipq 11022 | Addition of positive fract... |
| addpqnq 11023 | Addition of positive fract... |
| mulpipq2 11024 | Multiplication of positive... |
| mulpipq 11025 | Multiplication of positive... |
| mulpqnq 11026 | Multiplication of positive... |
| ordpipq 11027 | Ordering of positive fract... |
| ordpinq 11028 | Ordering of positive fract... |
| addpqf 11029 | Closure of addition on pos... |
| addclnq 11030 | Closure of addition on pos... |
| mulpqf 11031 | Closure of multiplication ... |
| mulclnq 11032 | Closure of multiplication ... |
| addnqf 11033 | Domain of addition on posi... |
| mulnqf 11034 | Domain of multiplication o... |
| addcompq 11035 | Addition of positive fract... |
| addcomnq 11036 | Addition of positive fract... |
| mulcompq 11037 | Multiplication of positive... |
| mulcomnq 11038 | Multiplication of positive... |
| adderpqlem 11039 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 11040 | Lemma for ~ mulerpq . (Co... |
| adderpq 11041 | Addition is compatible wit... |
| mulerpq 11042 | Multiplication is compatib... |
| addassnq 11043 | Addition of positive fract... |
| mulassnq 11044 | Multiplication of positive... |
| mulcanenq 11045 | Lemma for distributive law... |
| distrnq 11046 | Multiplication of positive... |
| 1nqenq 11047 | The equivalence class of r... |
| mulidnq 11048 | Multiplication identity el... |
| recmulnq 11049 | Relationship between recip... |
| recidnq 11050 | A positive fraction times ... |
| recclnq 11051 | Closure law for positive f... |
| recrecnq 11052 | Reciprocal of reciprocal o... |
| dmrecnq 11053 | Domain of reciprocal on po... |
| ltsonq 11054 | 'Less than' is a strict or... |
| lterpq 11055 | Compatibility of ordering ... |
| ltanq 11056 | Ordering property of addit... |
| ltmnq 11057 | Ordering property of multi... |
| 1lt2nq 11058 | One is less than two (one ... |
| ltaddnq 11059 | The sum of two fractions i... |
| ltexnq 11060 | Ordering on positive fract... |
| halfnq 11061 | One-half of any positive f... |
| nsmallnq 11062 | The is no smallest positiv... |
| ltbtwnnq 11063 | There exists a number betw... |
| ltrnq 11064 | Ordering property of recip... |
| archnq 11065 | For any fraction, there is... |
| npex 11071 | The class of positive real... |
| elnp 11072 | Membership in positive rea... |
| elnpi 11073 | Membership in positive rea... |
| prn0 11074 | A positive real is not emp... |
| prpssnq 11075 | A positive real is a subse... |
| elprnq 11076 | A positive real is a set o... |
| 0npr 11077 | The empty set is not a pos... |
| prcdnq 11078 | A positive real is closed ... |
| prub 11079 | A positive fraction not in... |
| prnmax 11080 | A positive real has no lar... |
| npomex 11081 | A simplifying observation,... |
| prnmadd 11082 | A positive real has no lar... |
| ltrelpr 11083 | Positive real 'less than' ... |
| genpv 11084 | Value of general operation... |
| genpelv 11085 | Membership in value of gen... |
| genpprecl 11086 | Pre-closure law for genera... |
| genpdm 11087 | Domain of general operatio... |
| genpn0 11088 | The result of an operation... |
| genpss 11089 | The result of an operation... |
| genpnnp 11090 | The result of an operation... |
| genpcd 11091 | Downward closure of an ope... |
| genpnmax 11092 | An operation on positive r... |
| genpcl 11093 | Closure of an operation on... |
| genpass 11094 | Associativity of an operat... |
| plpv 11095 | Value of addition on posit... |
| mpv 11096 | Value of multiplication on... |
| dmplp 11097 | Domain of addition on posi... |
| dmmp 11098 | Domain of multiplication o... |
| nqpr 11099 | The canonical embedding of... |
| 1pr 11100 | The positive real number '... |
| addclprlem1 11101 | Lemma to prove downward cl... |
| addclprlem2 11102 | Lemma to prove downward cl... |
| addclpr 11103 | Closure of addition on pos... |
| mulclprlem 11104 | Lemma to prove downward cl... |
| mulclpr 11105 | Closure of multiplication ... |
| addcompr 11106 | Addition of positive reals... |
| addasspr 11107 | Addition of positive reals... |
| mulcompr 11108 | Multiplication of positive... |
| mulasspr 11109 | Multiplication of positive... |
| distrlem1pr 11110 | Lemma for distributive law... |
| distrlem4pr 11111 | Lemma for distributive law... |
| distrlem5pr 11112 | Lemma for distributive law... |
| distrpr 11113 | Multiplication of positive... |
| 1idpr 11114 | 1 is an identity element f... |
| ltprord 11115 | Positive real 'less than' ... |
| psslinpr 11116 | Proper subset is a linear ... |
| ltsopr 11117 | Positive real 'less than' ... |
| prlem934 11118 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11119 | The sum of two positive re... |
| ltaddpr2 11120 | The sum of two positive re... |
| ltexprlem1 11121 | Lemma for Proposition 9-3.... |
| ltexprlem2 11122 | Lemma for Proposition 9-3.... |
| ltexprlem3 11123 | Lemma for Proposition 9-3.... |
| ltexprlem4 11124 | Lemma for Proposition 9-3.... |
| ltexprlem5 11125 | Lemma for Proposition 9-3.... |
| ltexprlem6 11126 | Lemma for Proposition 9-3.... |
| ltexprlem7 11127 | Lemma for Proposition 9-3.... |
| ltexpri 11128 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11129 | Lemma for Proposition 9-3.... |
| ltapr 11130 | Ordering property of addit... |
| addcanpr 11131 | Addition cancellation law ... |
| prlem936 11132 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11133 | Lemma for Proposition 9-3.... |
| reclem3pr 11134 | Lemma for Proposition 9-3.... |
| reclem4pr 11135 | Lemma for Proposition 9-3.... |
| recexpr 11136 | The reciprocal of a positi... |
| suplem1pr 11137 | The union of a nonempty, b... |
| suplem2pr 11138 | The union of a set of posi... |
| supexpr 11139 | The union of a nonempty, b... |
| enrer 11148 | The equivalence relation f... |
| nrex1 11149 | The class of signed reals ... |
| enrbreq 11150 | Equivalence relation for s... |
| enreceq 11151 | Equivalence class equality... |
| enrex 11152 | The equivalence relation f... |
| ltrelsr 11153 | Signed real 'less than' is... |
| addcmpblnr 11154 | Lemma showing compatibilit... |
| mulcmpblnrlem 11155 | Lemma used in lemma showin... |
| mulcmpblnr 11156 | Lemma showing compatibilit... |
| prsrlem1 11157 | Decomposing signed reals i... |
| addsrmo 11158 | There is at most one resul... |
| mulsrmo 11159 | There is at most one resul... |
| addsrpr 11160 | Addition of signed reals i... |
| mulsrpr 11161 | Multiplication of signed r... |
| ltsrpr 11162 | Ordering of signed reals i... |
| gt0srpr 11163 | Greater than zero in terms... |
| 0nsr 11164 | The empty set is not a sig... |
| 0r 11165 | The constant ` 0R ` is a s... |
| 1sr 11166 | The constant ` 1R ` is a s... |
| m1r 11167 | The constant ` -1R ` is a ... |
| addclsr 11168 | Closure of addition on sig... |
| mulclsr 11169 | Closure of multiplication ... |
| dmaddsr 11170 | Domain of addition on sign... |
| dmmulsr 11171 | Domain of multiplication o... |
| addcomsr 11172 | Addition of signed reals i... |
| addasssr 11173 | Addition of signed reals i... |
| mulcomsr 11174 | Multiplication of signed r... |
| mulasssr 11175 | Multiplication of signed r... |
| distrsr 11176 | Multiplication of signed r... |
| m1p1sr 11177 | Minus one plus one is zero... |
| m1m1sr 11178 | Minus one times minus one ... |
| ltsosr 11179 | Signed real 'less than' is... |
| 0lt1sr 11180 | 0 is less than 1 for signe... |
| 1ne0sr 11181 | 1 and 0 are distinct for s... |
| 0idsr 11182 | The signed real number 0 i... |
| 1idsr 11183 | 1 is an identity element f... |
| 00sr 11184 | A signed real times 0 is 0... |
| ltasr 11185 | Ordering property of addit... |
| pn0sr 11186 | A signed real plus its neg... |
| negexsr 11187 | Existence of negative sign... |
| recexsrlem 11188 | The reciprocal of a positi... |
| addgt0sr 11189 | The sum of two positive si... |
| mulgt0sr 11190 | The product of two positiv... |
| sqgt0sr 11191 | The square of a nonzero si... |
| recexsr 11192 | The reciprocal of a nonzer... |
| mappsrpr 11193 | Mapping from positive sign... |
| ltpsrpr 11194 | Mapping of order from posi... |
| map2psrpr 11195 | Equivalence for positive s... |
| supsrlem 11196 | Lemma for supremum theorem... |
| supsr 11197 | A nonempty, bounded set of... |
| opelcn 11214 | Ordered pair membership in... |
| opelreal 11215 | Ordered pair membership in... |
| elreal 11216 | Membership in class of rea... |
| elreal2 11217 | Ordered pair membership in... |
| 0ncn 11218 | The empty set is not a com... |
| ltrelre 11219 | 'Less than' is a relation ... |
| addcnsr 11220 | Addition of complex number... |
| mulcnsr 11221 | Multiplication of complex ... |
| eqresr 11222 | Equality of real numbers i... |
| addresr 11223 | Addition of real numbers i... |
| mulresr 11224 | Multiplication of real num... |
| ltresr 11225 | Ordering of real subset of... |
| ltresr2 11226 | Ordering of real subset of... |
| dfcnqs 11227 | Technical trick to permit ... |
| addcnsrec 11228 | Technical trick to permit ... |
| mulcnsrec 11229 | Technical trick to permit ... |
| axaddf 11230 | Addition is an operation o... |
| axmulf 11231 | Multiplication is an opera... |
| axcnex 11232 | The complex numbers form a... |
| axresscn 11233 | The real numbers are a sub... |
| ax1cn 11234 | 1 is a complex number. Ax... |
| axicn 11235 | ` _i ` is a complex number... |
| axaddcl 11236 | Closure law for addition o... |
| axaddrcl 11237 | Closure law for addition i... |
| axmulcl 11238 | Closure law for multiplica... |
| axmulrcl 11239 | Closure law for multiplica... |
| axmulcom 11240 | Multiplication of complex ... |
| axaddass 11241 | Addition of complex number... |
| axmulass 11242 | Multiplication of complex ... |
| axdistr 11243 | Distributive law for compl... |
| axi2m1 11244 | i-squared equals -1 (expre... |
| ax1ne0 11245 | 1 and 0 are distinct. Axi... |
| ax1rid 11246 | ` 1 ` is an identity eleme... |
| axrnegex 11247 | Existence of negative of r... |
| axrrecex 11248 | Existence of reciprocal of... |
| axcnre 11249 | A complex number can be ex... |
| axpre-lttri 11250 | Ordering on reals satisfie... |
| axpre-lttrn 11251 | Ordering on reals is trans... |
| axpre-ltadd 11252 | Ordering property of addit... |
| axpre-mulgt0 11253 | The product of two positiv... |
| axpre-sup 11254 | A nonempty, bounded-above ... |
| wuncn 11255 | A weak universe containing... |
| cnex 11281 | Alias for ~ ax-cnex . See... |
| addcl 11282 | Alias for ~ ax-addcl , for... |
| readdcl 11283 | Alias for ~ ax-addrcl , fo... |
| mulcl 11284 | Alias for ~ ax-mulcl , for... |
| remulcl 11285 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11286 | Alias for ~ ax-mulcom , fo... |
| addass 11287 | Alias for ~ ax-addass , fo... |
| mulass 11288 | Alias for ~ ax-mulass , fo... |
| adddi 11289 | Alias for ~ ax-distr , for... |
| recn 11290 | A real number is a complex... |
| reex 11291 | The real numbers form a se... |
| reelprrecn 11292 | Reals are a subset of the ... |
| cnelprrecn 11293 | Complex numbers are a subs... |
| mpoaddf 11294 | Addition is an operation o... |
| mpomulf 11295 | Multiplication is an opera... |
| elimne0 11296 | Hypothesis for weak deduct... |
| adddir 11297 | Distributive law for compl... |
| 0cn 11298 | Zero is a complex number. ... |
| 0cnd 11299 | Zero is a complex number, ... |
| c0ex 11300 | Zero is a set. (Contribut... |
| 0elpr01 11301 | 0 is an element of ` { 0 ,... |
| 1cnd 11302 | One is a complex number, d... |
| 1ex 11303 | One is a set. (Contribute... |
| 1elpr01 11304 | 1 is an element of ` { 0 ,... |
| cnre 11305 | Alias for ~ ax-cnre , for ... |
| mulrid 11306 | The number 1 is an identit... |
| mullid 11307 | Identity law for multiplic... |
| 1re 11308 | The number 1 is real. Thi... |
| 1red 11309 | The number 1 is real, dedu... |
| 0re 11310 | The number 0 is real. Rem... |
| 0red 11311 | The number 0 is real, dedu... |
| pr01ssre 11312 | The pair ` { 0 , 1 } ` is ... |
| mulridi 11313 | Identity law for multiplic... |
| mullidi 11314 | Identity law for multiplic... |
| addcli 11315 | Closure law for addition. ... |
| mulcli 11316 | Closure law for multiplica... |
| mulcomi 11317 | Commutative law for multip... |
| mulcomli 11318 | Commutative law for multip... |
| addassi 11319 | Associative law for additi... |
| mulassi 11320 | Associative law for multip... |
| adddii 11321 | Distributive law (left-dis... |
| adddiri 11322 | Distributive law (right-di... |
| recni 11323 | A real number is a complex... |
| readdcli 11324 | Closure law for addition o... |
| remulcli 11325 | Closure law for multiplica... |
| mulridd 11326 | Identity law for multiplic... |
| mullidd 11327 | Identity law for multiplic... |
| addcld 11328 | Closure law for addition. ... |
| mulcld 11329 | Closure law for multiplica... |
| mulcomd 11330 | Commutative law for multip... |
| addassd 11331 | Associative law for additi... |
| mulassd 11332 | Associative law for multip... |
| adddid 11333 | Distributive law (left-dis... |
| adddird 11334 | Distributive law (right-di... |
| adddirp1d 11335 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11336 | Join AB+CB into (A+C) on L... |
| recnd 11337 | Deduction from real number... |
| readdcld 11338 | Closure law for addition o... |
| remulcld 11339 | Closure law for multiplica... |
| pnfnre 11350 | Plus infinity is not a rea... |
| pnfnre2 11351 | Plus infinity is not a rea... |
| mnfnre 11352 | Minus infinity is not a re... |
| ressxr 11353 | The standard reals are a s... |
| rexpssxrxp 11354 | The Cartesian product of s... |
| rexr 11355 | A standard real is an exte... |
| 0xr 11356 | Zero is an extended real. ... |
| renepnf 11357 | No (finite) real equals pl... |
| renemnf 11358 | No real equals minus infin... |
| rexrd 11359 | A standard real is an exte... |
| renepnfd 11360 | No (finite) real equals pl... |
| renemnfd 11361 | No real equals minus infin... |
| pnfex 11362 | Plus infinity exists. (Co... |
| pnfxr 11363 | Plus infinity belongs to t... |
| pnfnemnf 11364 | Plus and minus infinity ar... |
| mnfnepnf 11365 | Minus and plus infinity ar... |
| mnfxr 11366 | Minus infinity belongs to ... |
| rexri 11367 | A standard real is an exte... |
| 1xr 11368 | ` 1 ` is an extended real ... |
| renfdisj 11369 | The reals and the infiniti... |
| ltrelxr 11370 | "Less than" is a relation ... |
| ltrel 11371 | "Less than" is a relation.... |
| lerelxr 11372 | "Less than or equal to" is... |
| lerel 11373 | "Less than or equal to" is... |
| xrlenlt 11374 | "Less than or equal to" ex... |
| xrlenltd 11375 | "Less than or equal to" ex... |
| xrltnle 11376 | "Less than" expressed in t... |
| xrltnled 11377 | 'Less than' in terms of 'l... |
| xrnltled 11378 | "Not less than" implies "l... |
| ssxr 11379 | The three (non-exclusive) ... |
| ltxrlt 11380 | The standard less-than ` <... |
| axlttri 11381 | Ordering on reals satisfie... |
| axlttrn 11382 | Ordering on reals is trans... |
| axltadd 11383 | Ordering property of addit... |
| axmulgt0 11384 | The product of two positiv... |
| axsup 11385 | A nonempty, bounded-above ... |
| lttr 11386 | Alias for ~ axlttrn , for ... |
| mulgt0 11387 | The product of two positiv... |
| lenlt 11388 | 'Less than or equal to' ex... |
| ltnle 11389 | 'Less than' expressed in t... |
| ltso 11390 | 'Less than' is a strict or... |
| gtso 11391 | 'Greater than' is a strict... |
| lttri2 11392 | Consequence of trichotomy.... |
| lttri3 11393 | Trichotomy law for 'less t... |
| lttri4 11394 | Trichotomy law for 'less t... |
| letri3 11395 | Trichotomy law. (Contribu... |
| leloe 11396 | 'Less than or equal to' ex... |
| eqlelt 11397 | Equality in terms of 'less... |
| ltle 11398 | 'Less than' implies 'less ... |
| leltne 11399 | 'Less than or equal to' im... |
| lelttr 11400 | Transitive law. (Contribu... |
| leltletr 11401 | Transitive law, weaker for... |
| ltletr 11402 | Transitive law. (Contribu... |
| ltleletr 11403 | Transitive law, weaker for... |
| letr 11404 | Transitive law. (Contribu... |
| ltnr 11405 | 'Less than' is irreflexive... |
| leid 11406 | 'Less than or equal to' is... |
| ltne 11407 | 'Less than' implies not eq... |
| ltnsym 11408 | 'Less than' is not symmetr... |
| ltnsym2 11409 | 'Less than' is antisymmetr... |
| letric 11410 | Trichotomy law. (Contribu... |
| ltlen 11411 | 'Less than' expressed in t... |
| eqle 11412 | Equality implies 'less tha... |
| eqled 11413 | Equality implies 'less tha... |
| ltadd2 11414 | Addition to both sides of ... |
| ne0gt0 11415 | A nonzero nonnegative numb... |
| lecasei 11416 | Ordering elimination by ca... |
| lelttric 11417 | Trichotomy law. (Contribu... |
| ltlecasei 11418 | Ordering elimination by ca... |
| ltnri 11419 | 'Less than' is irreflexive... |
| eqlei 11420 | Equality implies 'less tha... |
| eqlei2 11421 | Equality implies 'less tha... |
| gtneii 11422 | 'Less than' implies not eq... |
| ltneii 11423 | 'Greater than' implies not... |
| lttri2i 11424 | Consequence of trichotomy.... |
| lttri3i 11425 | Consequence of trichotomy.... |
| letri3i 11426 | Consequence of trichotomy.... |
| leloei 11427 | 'Less than or equal to' in... |
| ltleni 11428 | 'Less than' expressed in t... |
| ltnsymi 11429 | 'Less than' is not symmetr... |
| lenlti 11430 | 'Less than or equal to' in... |
| ltnlei 11431 | 'Less than' in terms of 'l... |
| ltlei 11432 | 'Less than' implies 'less ... |
| ltleii 11433 | 'Less than' implies 'less ... |
| ltnei 11434 | 'Less than' implies not eq... |
| letrii 11435 | Trichotomy law for 'less t... |
| lttri 11436 | 'Less than' is transitive.... |
| lelttri 11437 | 'Less than or equal to', '... |
| ltletri 11438 | 'Less than', 'less than or... |
| letri 11439 | 'Less than or equal to' is... |
| le2tri3i 11440 | Extended trichotomy law fo... |
| ltadd2i 11441 | Addition to both sides of ... |
| mulgt0i 11442 | The product of two positiv... |
| mulgt0ii 11443 | The product of two positiv... |
| ltnrd 11444 | 'Less than' is irreflexive... |
| gtned 11445 | 'Less than' implies not eq... |
| ltned 11446 | 'Greater than' implies not... |
| ne0gt0d 11447 | A nonzero nonnegative numb... |
| lttrid 11448 | Ordering on reals satisfie... |
| lttri2d 11449 | Consequence of trichotomy.... |
| lttri3d 11450 | Consequence of trichotomy.... |
| lttri4d 11451 | Trichotomy law for 'less t... |
| letri3d 11452 | Consequence of trichotomy.... |
| leloed 11453 | 'Less than or equal to' in... |
| eqleltd 11454 | Equality in terms of 'less... |
| ltlend 11455 | 'Less than' expressed in t... |
| lenltd 11456 | 'Less than or equal to' in... |
| ltnled 11457 | 'Less than' in terms of 'l... |
| ltled 11458 | 'Less than' implies 'less ... |
| ltnsymd 11459 | 'Less than' implies 'less ... |
| nltled 11460 | 'Not less than ' implies '... |
| lensymd 11461 | 'Less than or equal to' im... |
| letrid 11462 | Trichotomy law for 'less t... |
| leltned 11463 | 'Less than or equal to' im... |
| leneltd 11464 | 'Less than or equal to' an... |
| mulgt0d 11465 | The product of two positiv... |
| ltadd2d 11466 | Addition to both sides of ... |
| letrd 11467 | Transitive law deduction f... |
| lelttrd 11468 | Transitive law deduction f... |
| ltadd2dd 11469 | Addition to both sides of ... |
| ltletrd 11470 | Transitive law deduction f... |
| lttrd 11471 | Transitive law deduction f... |
| lelttrdi 11472 | If a number is less than a... |
| dedekind 11473 | The Dedekind cut theorem. ... |
| dedekindle 11474 | The Dedekind cut theorem, ... |
| mul12 11475 | Commutative/associative la... |
| mul32 11476 | Commutative/associative la... |
| mul31 11477 | Commutative/associative la... |
| mul4 11478 | Rearrangement of 4 factors... |
| mul4r 11479 | Rearrangement of 4 factors... |
| muladd11 11480 | A simple product of sums e... |
| 1p1times 11481 | Two times a number. (Cont... |
| peano2cn 11482 | A theorem for complex numb... |
| peano2re 11483 | A theorem for reals analog... |
| readdcan 11484 | Cancellation law for addit... |
| 00id 11485 | ` 0 ` is its own additive ... |
| mul02lem1 11486 | Lemma for ~ mul02 . If an... |
| mul02lem2 11487 | Lemma for ~ mul02 . Zero ... |
| mul02 11488 | Multiplication by ` 0 ` . ... |
| mul01 11489 | Multiplication by ` 0 ` . ... |
| addrid 11490 | ` 0 ` is an additive ident... |
| cnegex 11491 | Existence of the negative ... |
| cnegex2 11492 | Existence of a left invers... |
| addlid 11493 | ` 0 ` is a left identity f... |
| addcan 11494 | Cancellation law for addit... |
| addcan2 11495 | Cancellation law for addit... |
| addcom 11496 | Addition is commutative. ... |
| addridi 11497 | ` 0 ` is an additive ident... |
| addlidi 11498 | ` 0 ` is a left identity f... |
| mul02i 11499 | Multiplication by 0. Theo... |
| mul01i 11500 | Multiplication by ` 0 ` . ... |
| addcomi 11501 | Addition is commutative. ... |
| addcomli 11502 | Addition is commutative. ... |
| addcani 11503 | Cancellation law for addit... |
| addcan2i 11504 | Cancellation law for addit... |
| mul12i 11505 | Commutative/associative la... |
| mul32i 11506 | Commutative/associative la... |
| mul4i 11507 | Rearrangement of 4 factors... |
| mul02d 11508 | Multiplication by 0. Theo... |
| mul01d 11509 | Multiplication by ` 0 ` . ... |
| addridd 11510 | ` 0 ` is an additive ident... |
| addlidd 11511 | ` 0 ` is a left identity f... |
| addcomd 11512 | Addition is commutative. ... |
| addcand 11513 | Cancellation law for addit... |
| addcan2d 11514 | Cancellation law for addit... |
| addcanad 11515 | Cancelling a term on the l... |
| addcan2ad 11516 | Cancelling a term on the r... |
| addneintrd 11517 | Introducing a term on the ... |
| addneintr2d 11518 | Introducing a term on the ... |
| mul12d 11519 | Commutative/associative la... |
| mul32d 11520 | Commutative/associative la... |
| mul31d 11521 | Commutative/associative la... |
| mul4d 11522 | Rearrangement of 4 factors... |
| muladd11r 11523 | A simple product of sums e... |
| comraddd 11524 | Commute RHS addition, in d... |
| comraddi 11525 | Commute RHS addition. See... |
| ltaddneg 11526 | Adding a negative number t... |
| ltaddnegr 11527 | Adding a negative number t... |
| add12 11528 | Commutative/associative la... |
| add32 11529 | Commutative/associative la... |
| add32r 11530 | Commutative/associative la... |
| add4 11531 | Rearrangement of 4 terms i... |
| add42 11532 | Rearrangement of 4 terms i... |
| add12i 11533 | Commutative/associative la... |
| add32i 11534 | Commutative/associative la... |
| add4i 11535 | Rearrangement of 4 terms i... |
| add42i 11536 | Rearrangement of 4 terms i... |
| add12d 11537 | Commutative/associative la... |
| add32d 11538 | Commutative/associative la... |
| add4d 11539 | Rearrangement of 4 terms i... |
| add42d 11540 | Rearrangement of 4 terms i... |
| 0cnALT 11545 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11546 | Alternate proof of ~ 0cnAL... |
| negeu 11547 | Existential uniqueness of ... |
| subval 11548 | Value of subtraction, whic... |
| negeq 11549 | Equality theorem for negat... |
| negeqi 11550 | Equality inference for neg... |
| negeqd 11551 | Equality deduction for neg... |
| nfnegd 11552 | Deduction version of ~ nfn... |
| nfneg 11553 | Bound-variable hypothesis ... |
| csbnegg 11554 | Move class substitution in... |
| negex 11555 | A negative is a set. (Con... |
| subcl 11556 | Closure law for subtractio... |
| negcl 11557 | Closure law for negative. ... |
| negicn 11558 | ` -u _i ` is a complex num... |
| subf 11559 | Subtraction is an operatio... |
| subadd 11560 | Relationship between subtr... |
| subadd2 11561 | Relationship between subtr... |
| subsub23 11562 | Swap subtrahend and result... |
| pncan 11563 | Cancellation law for subtr... |
| pncan2 11564 | Cancellation law for subtr... |
| pncan3 11565 | Subtraction and addition o... |
| npcan 11566 | Cancellation law for subtr... |
| addsubass 11567 | Associative-type law for a... |
| addsub 11568 | Law for addition and subtr... |
| subadd23 11569 | Commutative/associative la... |
| addsub12 11570 | Commutative/associative la... |
| 2addsub 11571 | Law for subtraction and ad... |
| addsubeq4 11572 | Relation between sums and ... |
| pncan3oi 11573 | Subtraction and addition o... |
| mvrraddi 11574 | Move the right term in a s... |
| mvrladdi 11575 | Move the left term in a su... |
| mvlladdi 11576 | Move the left term in a su... |
| subid 11577 | Subtraction of a number fr... |
| subid1 11578 | Identity law for subtracti... |
| npncan 11579 | Cancellation law for subtr... |
| nppcan 11580 | Cancellation law for subtr... |
| nnpcan 11581 | Cancellation law for subtr... |
| nppcan3 11582 | Cancellation law for subtr... |
| subcan2 11583 | Cancellation law for subtr... |
| subeq0 11584 | If the difference between ... |
| npncan2 11585 | Cancellation law for subtr... |
| subsub2 11586 | Law for double subtraction... |
| nncan 11587 | Cancellation law for subtr... |
| subsub 11588 | Law for double subtraction... |
| nppcan2 11589 | Cancellation law for subtr... |
| subsub3 11590 | Law for double subtraction... |
| subsub4 11591 | Law for double subtraction... |
| sub32 11592 | Swap the second and third ... |
| nnncan 11593 | Cancellation law for subtr... |
| nnncan1 11594 | Cancellation law for subtr... |
| nnncan2 11595 | Cancellation law for subtr... |
| npncan3 11596 | Cancellation law for subtr... |
| pnpcan 11597 | Cancellation law for mixed... |
| pnpcan2 11598 | Cancellation law for mixed... |
| pnncan 11599 | Cancellation law for mixed... |
| ppncan 11600 | Cancellation law for mixed... |
| addsub4 11601 | Rearrangement of 4 terms i... |
| subadd4 11602 | Rearrangement of 4 terms i... |
| sub4 11603 | Rearrangement of 4 terms i... |
| neg0 11604 | Minus 0 equals 0. (Contri... |
| negid 11605 | Addition of a number and i... |
| negsub 11606 | Relationship between subtr... |
| subneg 11607 | Relationship between subtr... |
| negneg 11608 | A number is equal to the n... |
| neg11 11609 | Negative is one-to-one. (... |
| negcon1 11610 | Negative contraposition la... |
| negcon2 11611 | Negative contraposition la... |
| negeq0 11612 | A number is zero iff its n... |
| subcan 11613 | Cancellation law for subtr... |
| negsubdi 11614 | Distribution of negative o... |
| negdi 11615 | Distribution of negative o... |
| negdi2 11616 | Distribution of negative o... |
| negsubdi2 11617 | Distribution of negative o... |
| neg2sub 11618 | Relationship between subtr... |
| renegcli 11619 | Closure law for negative o... |
| resubcli 11620 | Closure law for subtractio... |
| renegcl 11621 | Closure law for negative o... |
| resubcl 11622 | Closure law for subtractio... |
| negreb 11623 | The negative of a real is ... |
| peano2cnm 11624 | "Reverse" second Peano pos... |
| peano2rem 11625 | "Reverse" second Peano pos... |
| negcli 11626 | Closure law for negative. ... |
| negidi 11627 | Addition of a number and i... |
| negnegi 11628 | A number is equal to the n... |
| subidi 11629 | Subtraction of a number fr... |
| subid1i 11630 | Identity law for subtracti... |
| negne0bi 11631 | A number is nonzero iff it... |
| negrebi 11632 | The negative of a real is ... |
| negne0i 11633 | The negative of a nonzero ... |
| subcli 11634 | Closure law for subtractio... |
| pncan3i 11635 | Subtraction and addition o... |
| negsubi 11636 | Relationship between subtr... |
| subnegi 11637 | Relationship between subtr... |
| subeq0i 11638 | If the difference between ... |
| neg11i 11639 | Negative is one-to-one. (... |
| negcon1i 11640 | Negative contraposition la... |
| negcon2i 11641 | Negative contraposition la... |
| negdii 11642 | Distribution of negative o... |
| negsubdii 11643 | Distribution of negative o... |
| negsubdi2i 11644 | Distribution of negative o... |
| subaddi 11645 | Relationship between subtr... |
| subadd2i 11646 | Relationship between subtr... |
| subaddrii 11647 | Relationship between subtr... |
| subsub23i 11648 | Swap subtrahend and result... |
| addsubassi 11649 | Associative-type law for s... |
| addsubi 11650 | Law for subtraction and ad... |
| subcani 11651 | Cancellation law for subtr... |
| subcan2i 11652 | Cancellation law for subtr... |
| pnncani 11653 | Cancellation law for mixed... |
| addsub4i 11654 | Rearrangement of 4 terms i... |
| 0reALT 11655 | Alternate proof of ~ 0re .... |
| negcld 11656 | Closure law for negative. ... |
| subidd 11657 | Subtraction of a number fr... |
| subid1d 11658 | Identity law for subtracti... |
| negidd 11659 | Addition of a number and i... |
| negnegd 11660 | A number is equal to the n... |
| negeq0d 11661 | A number is zero iff its n... |
| negne0bd 11662 | A number is nonzero iff it... |
| negcon1d 11663 | Contraposition law for una... |
| negcon1ad 11664 | Contraposition law for una... |
| neg11ad 11665 | The negatives of two compl... |
| negned 11666 | If two complex numbers are... |
| negne0d 11667 | The negative of a nonzero ... |
| negrebd 11668 | The negative of a real is ... |
| subcld 11669 | Closure law for subtractio... |
| pncand 11670 | Cancellation law for subtr... |
| pncan2d 11671 | Cancellation law for subtr... |
| pncan3d 11672 | Subtraction and addition o... |
| npcand 11673 | Cancellation law for subtr... |
| nncand 11674 | Cancellation law for subtr... |
| negsubd 11675 | Relationship between subtr... |
| subnegd 11676 | Relationship between subtr... |
| subeq0ad 11677 | The difference of two comp... |
| subeq0d 11678 | If the difference between ... |
| subne0d 11679 | Two unequal numbers have n... |
| subne0ad 11680 | If the difference of two c... |
| neg11d 11681 | If the difference between ... |
| negdid 11682 | Distribution of negative o... |
| negdi2d 11683 | Distribution of negative o... |
| negsubdid 11684 | Distribution of negative o... |
| negsubdi2d 11685 | Distribution of negative o... |
| neg2subd 11686 | Relationship between subtr... |
| subaddd 11687 | Relationship between subtr... |
| subadd2d 11688 | Relationship between subtr... |
| addsubassd 11689 | Associative-type law for s... |
| addsubd 11690 | Law for subtraction and ad... |
| subadd23d 11691 | Commutative/associative la... |
| addsub12d 11692 | Commutative/associative la... |
| npncand 11693 | Cancellation law for subtr... |
| nppcand 11694 | Cancellation law for subtr... |
| nppcan2d 11695 | Cancellation law for subtr... |
| nppcan3d 11696 | Cancellation law for subtr... |
| subsubd 11697 | Law for double subtraction... |
| subsub2d 11698 | Law for double subtraction... |
| subsub3d 11699 | Law for double subtraction... |
| subsub4d 11700 | Law for double subtraction... |
| sub32d 11701 | Swap the second and third ... |
| nnncand 11702 | Cancellation law for subtr... |
| nnncan1d 11703 | Cancellation law for subtr... |
| nnncan2d 11704 | Cancellation law for subtr... |
| npncan3d 11705 | Cancellation law for subtr... |
| pnpcand 11706 | Cancellation law for mixed... |
| pnpcan2d 11707 | Cancellation law for mixed... |
| pnncand 11708 | Cancellation law for mixed... |
| ppncand 11709 | Cancellation law for mixed... |
| subcand 11710 | Cancellation law for subtr... |
| subcan2d 11711 | Cancellation law for subtr... |
| subcanad 11712 | Cancellation law for subtr... |
| subneintrd 11713 | Introducing subtraction on... |
| subcan2ad 11714 | Cancellation law for subtr... |
| subneintr2d 11715 | Introducing subtraction on... |
| addsub4d 11716 | Rearrangement of 4 terms i... |
| subadd4d 11717 | Rearrangement of 4 terms i... |
| sub4d 11718 | Rearrangement of 4 terms i... |
| 2addsubd 11719 | Law for subtraction and ad... |
| addsubeq4d 11720 | Relation between sums and ... |
| subsubadd23 11721 | Swap the second and the th... |
| addsubsub23 11722 | Swap the second and the th... |
| subeqxfrd 11723 | Transfer two terms of a su... |
| mvlraddd 11724 | Move the right term in a s... |
| mvlladdd 11725 | Move the left term in a su... |
| mvlladdcd 11726 | Rotate the variables right... |
| mvrraddd 11727 | Move the right term in a s... |
| mvrladdd 11728 | Move the left term in a su... |
| mvrrsubd 11729 | Move a subtraction in the ... |
| assraddsubd 11730 | Associate RHS addition-sub... |
| subaddeqd 11731 | Transfer two terms of a su... |
| addlsub 11732 | Left-subtraction: Subtrac... |
| addrsub 11733 | Right-subtraction: Subtra... |
| subexsub 11734 | A subtraction law: Exchan... |
| addid0 11735 | If adding a number to a an... |
| addn0nid 11736 | Adding a nonzero number to... |
| pnpncand 11737 | Addition/subtraction cance... |
| subeqrev 11738 | Reverse the order of subtr... |
| addeq0 11739 | Two complex numbers add up... |
| pncan1 11740 | Cancellation law for addit... |
| npcan1 11741 | Cancellation law for subtr... |
| subeq0bd 11742 | If two complex numbers are... |
| renegcld 11743 | Closure law for negative o... |
| resubcld 11744 | Closure law for subtractio... |
| negn0 11745 | The image under negation o... |
| negf1o 11746 | Negation is an isomorphism... |
| kcnktkm1cn 11747 | k times k minus 1 is a com... |
| muladd 11748 | Product of two sums. (Con... |
| subdi 11749 | Distribution of multiplica... |
| subdir 11750 | Distribution of multiplica... |
| ine0 11751 | The imaginary unit ` _i ` ... |
| mulneg1 11752 | Product with negative is n... |
| mulneg2 11753 | The product with a negativ... |
| mulneg12 11754 | Swap the negative sign in ... |
| mul2neg 11755 | Product of two negatives. ... |
| submul2 11756 | Convert a subtraction to a... |
| mulm1 11757 | Product with minus one is ... |
| addneg1mul 11758 | Addition with product with... |
| mulsub 11759 | Product of two differences... |
| mulsub2 11760 | Swap the order of subtract... |
| mulm1i 11761 | Product with minus one is ... |
| mulneg1i 11762 | Product with negative is n... |
| mulneg2i 11763 | Product with negative is n... |
| mul2negi 11764 | Product of two negatives. ... |
| subdii 11765 | Distribution of multiplica... |
| subdiri 11766 | Distribution of multiplica... |
| muladdi 11767 | Product of two sums. (Con... |
| mulm1d 11768 | Product with minus one is ... |
| mulneg1d 11769 | Product with negative is n... |
| mulneg2d 11770 | Product with negative is n... |
| mul2negd 11771 | Product of two negatives. ... |
| subdid 11772 | Distribution of multiplica... |
| subdird 11773 | Distribution of multiplica... |
| muladdd 11774 | Product of two sums. (Con... |
| mulsubd 11775 | Product of two differences... |
| muls1d 11776 | Multiplication by one minu... |
| mulsubfacd 11777 | Multiplication followed by... |
| addmulsub 11778 | The product of a sum and a... |
| subaddmulsub 11779 | The difference with a prod... |
| mulsubaddmulsub 11780 | A special difference of a ... |
| gt0ne0 11781 | Positive implies nonzero. ... |
| lt0ne0 11782 | A number which is less tha... |
| ltadd1 11783 | Addition to both sides of ... |
| leadd1 11784 | Addition to both sides of ... |
| leadd2 11785 | Addition to both sides of ... |
| ltsubadd 11786 | 'Less than' relationship b... |
| ltsubadd2 11787 | 'Less than' relationship b... |
| lesubadd 11788 | 'Less than or equal to' re... |
| lesubadd2 11789 | 'Less than or equal to' re... |
| ltaddsub 11790 | 'Less than' relationship b... |
| ltaddsub2 11791 | 'Less than' relationship b... |
| leaddsub 11792 | 'Less than or equal to' re... |
| leaddsub2 11793 | 'Less than or equal to' re... |
| suble 11794 | Swap subtrahends in an ine... |
| lesub 11795 | Swap subtrahends in an ine... |
| ltsub23 11796 | 'Less than' relationship b... |
| ltsub13 11797 | 'Less than' relationship b... |
| le2add 11798 | Adding both sides of two '... |
| ltleadd 11799 | Adding both sides of two o... |
| leltadd 11800 | Adding both sides of two o... |
| lt2add 11801 | Adding both sides of two '... |
| addgt0 11802 | The sum of 2 positive numb... |
| addgegt0 11803 | The sum of nonnegative and... |
| addgtge0 11804 | The sum of nonnegative and... |
| addge0 11805 | The sum of 2 nonnegative n... |
| ltaddpos 11806 | Adding a positive number t... |
| ltaddpos2 11807 | Adding a positive number t... |
| ltsubpos 11808 | Subtracting a positive num... |
| posdif 11809 | Comparison of two numbers ... |
| lesub1 11810 | Subtraction from both side... |
| lesub2 11811 | Subtraction of both sides ... |
| ltsub1 11812 | Subtraction from both side... |
| ltsub2 11813 | Subtraction of both sides ... |
| lt2sub 11814 | Subtracting both sides of ... |
| le2sub 11815 | Subtracting both sides of ... |
| ltneg 11816 | Negative of both sides of ... |
| ltnegcon1 11817 | Contraposition of negative... |
| ltnegcon2 11818 | Contraposition of negative... |
| leneg 11819 | Negative of both sides of ... |
| lenegcon1 11820 | Contraposition of negative... |
| lenegcon2 11821 | Contraposition of negative... |
| lt0neg1 11822 | Comparison of a number and... |
| lt0neg2 11823 | Comparison of a number and... |
| le0neg1 11824 | Comparison of a number and... |
| le0neg2 11825 | Comparison of a number and... |
| addge01 11826 | A number is less than or e... |
| addge02 11827 | A number is less than or e... |
| add20 11828 | Two nonnegative numbers ar... |
| subge0 11829 | Nonnegative subtraction. ... |
| suble0 11830 | Nonpositive subtraction. ... |
| leaddle0 11831 | The sum of a real number a... |
| subge02 11832 | Nonnegative subtraction. ... |
| lesub0 11833 | Lemma to show a nonnegativ... |
| mulge0 11834 | The product of two nonnega... |
| mullt0 11835 | The product of two negativ... |
| msqgt0 11836 | A nonzero square is positi... |
| msqge0 11837 | A square is nonnegative. ... |
| 0lt1 11838 | 0 is less than 1. Theorem... |
| 0le1 11839 | 0 is less than or equal to... |
| relin01 11840 | An interval law for less t... |
| ltordlem 11841 | Lemma for ~ ltord1 . (Con... |
| ltord1 11842 | Infer an ordering relation... |
| leord1 11843 | Infer an ordering relation... |
| eqord1 11844 | A strictly increasing real... |
| ltord2 11845 | Infer an ordering relation... |
| leord2 11846 | Infer an ordering relation... |
| eqord2 11847 | A strictly decreasing real... |
| wloglei 11848 | Form of ~ wlogle where bot... |
| wlogle 11849 | If the predicate ` ch ( x ... |
| leidi 11850 | 'Less than or equal to' is... |
| gt0ne0i 11851 | Positive means nonzero (us... |
| gt0ne0ii 11852 | Positive implies nonzero. ... |
| msqgt0i 11853 | A nonzero square is positi... |
| msqge0i 11854 | A square is nonnegative. ... |
| addgt0i 11855 | Addition of 2 positive num... |
| addge0i 11856 | Addition of 2 nonnegative ... |
| addgegt0i 11857 | Addition of nonnegative an... |
| addgt0ii 11858 | Addition of 2 positive num... |
| add20i 11859 | Two nonnegative numbers ar... |
| ltnegi 11860 | Negative of both sides of ... |
| lenegi 11861 | Negative of both sides of ... |
| ltnegcon2i 11862 | Contraposition of negative... |
| mulge0i 11863 | The product of two nonnega... |
| lesub0i 11864 | Lemma to show a nonnegativ... |
| ltaddposi 11865 | Adding a positive number t... |
| posdifi 11866 | Comparison of two numbers ... |
| ltnegcon1i 11867 | Contraposition of negative... |
| lenegcon1i 11868 | Contraposition of negative... |
| subge0i 11869 | Nonnegative subtraction. ... |
| ltadd1i 11870 | Addition to both sides of ... |
| leadd1i 11871 | Addition to both sides of ... |
| leadd2i 11872 | Addition to both sides of ... |
| ltsubaddi 11873 | 'Less than' relationship b... |
| lesubaddi 11874 | 'Less than or equal to' re... |
| ltsubadd2i 11875 | 'Less than' relationship b... |
| lesubadd2i 11876 | 'Less than or equal to' re... |
| ltaddsubi 11877 | 'Less than' relationship b... |
| lt2addi 11878 | Adding both side of two in... |
| le2addi 11879 | Adding both side of two in... |
| gt0ne0d 11880 | Positive implies nonzero. ... |
| lt0ne0d 11881 | Something less than zero i... |
| leidd 11882 | 'Less than or equal to' is... |
| msqgt0d 11883 | A nonzero square is positi... |
| msqge0d 11884 | A square is nonnegative. ... |
| lt0neg1d 11885 | Comparison of a number and... |
| lt0neg2d 11886 | Comparison of a number and... |
| le0neg1d 11887 | Comparison of a number and... |
| le0neg2d 11888 | Comparison of a number and... |
| addgegt0d 11889 | Addition of nonnegative an... |
| addgtge0d 11890 | Addition of positive and n... |
| addgt0d 11891 | Addition of 2 positive num... |
| addge0d 11892 | Addition of 2 nonnegative ... |
| mulge0d 11893 | The product of two nonnega... |
| ltnegd 11894 | Negative of both sides of ... |
| lenegd 11895 | Negative of both sides of ... |
| ltnegcon1d 11896 | Contraposition of negative... |
| ltnegcon2d 11897 | Contraposition of negative... |
| lenegcon1d 11898 | Contraposition of negative... |
| lenegcon2d 11899 | Contraposition of negative... |
| ltaddposd 11900 | Adding a positive number t... |
| ltaddpos2d 11901 | Adding a positive number t... |
| ltsubposd 11902 | Subtracting a positive num... |
| posdifd 11903 | Comparison of two numbers ... |
| addge01d 11904 | A number is less than or e... |
| addge02d 11905 | A number is less than or e... |
| subge0d 11906 | Nonnegative subtraction. ... |
| suble0d 11907 | Nonpositive subtraction. ... |
| subge02d 11908 | Nonnegative subtraction. ... |
| ltadd1d 11909 | Addition to both sides of ... |
| leadd1d 11910 | Addition to both sides of ... |
| leadd2d 11911 | Addition to both sides of ... |
| ltsubaddd 11912 | 'Less than' relationship b... |
| lesubaddd 11913 | 'Less than or equal to' re... |
| ltsubadd2d 11914 | 'Less than' relationship b... |
| lesubadd2d 11915 | 'Less than or equal to' re... |
| ltaddsubd 11916 | 'Less than' relationship b... |
| ltaddsub2d 11917 | 'Less than' relationship b... |
| leaddsub2d 11918 | 'Less than or equal to' re... |
| subled 11919 | Swap subtrahends in an ine... |
| lesubd 11920 | Swap subtrahends in an ine... |
| ltsub23d 11921 | 'Less than' relationship b... |
| ltsub13d 11922 | 'Less than' relationship b... |
| lesub1d 11923 | Subtraction from both side... |
| lesub2d 11924 | Subtraction of both sides ... |
| ltsub1d 11925 | Subtraction from both side... |
| ltsub2d 11926 | Subtraction of both sides ... |
| ltadd1dd 11927 | Addition to both sides of ... |
| ltsub1dd 11928 | Subtraction from both side... |
| ltsub2dd 11929 | Subtraction of both sides ... |
| leadd1dd 11930 | Addition to both sides of ... |
| leadd2dd 11931 | Addition to both sides of ... |
| lesub1dd 11932 | Subtraction from both side... |
| lesub2dd 11933 | Subtraction of both sides ... |
| lesub3d 11934 | The result of subtracting ... |
| le2addd 11935 | Adding both side of two in... |
| le2subd 11936 | Subtracting both sides of ... |
| ltleaddd 11937 | Adding both sides of two o... |
| leltaddd 11938 | Adding both sides of two o... |
| lt2addd 11939 | Adding both side of two in... |
| lt2subd 11940 | Subtracting both sides of ... |
| possumd 11941 | Condition for a positive s... |
| sublt0d 11942 | When a subtraction gives a... |
| ltaddsublt 11943 | Addition and subtraction o... |
| 1le1 11944 | One is less than or equal ... |
| ixi 11945 | ` _i ` times itself is min... |
| recextlem1 11946 | Lemma for ~ recex . (Cont... |
| recextlem2 11947 | Lemma for ~ recex . (Cont... |
| recex 11948 | Existence of reciprocal of... |
| mulcand 11949 | Cancellation law for multi... |
| mulcan2d 11950 | Cancellation law for multi... |
| mulcanad 11951 | Cancellation of a nonzero ... |
| mulcan2ad 11952 | Cancellation of a nonzero ... |
| mulcan 11953 | Cancellation law for multi... |
| mulcan2 11954 | Cancellation law for multi... |
| mulcani 11955 | Cancellation law for multi... |
| mul0or 11956 | If a product is zero, one ... |
| mulne0b 11957 | The product of two nonzero... |
| mulne0 11958 | The product of two nonzero... |
| mulne0i 11959 | The product of two nonzero... |
| muleqadd 11960 | Property of numbers whose ... |
| receu 11961 | Existential uniqueness of ... |
| mulnzcnf 11962 | Multiplication maps nonzer... |
| mul0ori 11963 | If a product is zero, one ... |
| mul0ord 11964 | If a product is zero, one ... |
| msq0i 11965 | A number is zero iff its s... |
| msq0d 11966 | A number is zero iff its s... |
| mulne0bd 11967 | The product of two nonzero... |
| mulne0d 11968 | The product of two nonzero... |
| mulcan1g 11969 | A generalized form of the ... |
| mulcan2g 11970 | A generalized form of the ... |
| mulne0bad 11971 | A factor of a nonzero comp... |
| mulne0bbd 11972 | A factor of a nonzero comp... |
| 1div0 11975 | You can't divide by zero, ... |
| divval 11976 | Value of division: if ` A ... |
| divmul 11977 | Relationship between divis... |
| divmul2 11978 | Relationship between divis... |
| divmul3 11979 | Relationship between divis... |
| divcl 11980 | Closure law for division. ... |
| reccl 11981 | Closure law for reciprocal... |
| divcan2 11982 | A cancellation law for div... |
| divcan1 11983 | A cancellation law for div... |
| diveq0 11984 | A ratio is zero iff the nu... |
| divne0b 11985 | The ratio of nonzero numbe... |
| divne0 11986 | The ratio of nonzero numbe... |
| recne0 11987 | The reciprocal of a nonzer... |
| recid 11988 | Multiplication of a number... |
| recid2 11989 | Multiplication of a number... |
| divrec 11990 | Relationship between divis... |
| divrec2 11991 | Relationship between divis... |
| divass 11992 | An associative law for div... |
| div23 11993 | A commutative/associative ... |
| div32 11994 | A commutative/associative ... |
| div13 11995 | A commutative/associative ... |
| div12 11996 | A commutative/associative ... |
| divmulass 11997 | An associative law for div... |
| divmulasscom 11998 | An associative/commutative... |
| divdir 11999 | Distribution of division o... |
| divcan3 12000 | A cancellation law for div... |
| divcan4 12001 | A cancellation law for div... |
| div11 12002 | One-to-one relationship fo... |
| diveq1 12003 | Equality in terms of unit ... |
| divid 12004 | A number divided by itself... |
| div0 12005 | Division into zero is zero... |
| div1 12006 | A number divided by 1 is i... |
| 1div1e1 12007 | 1 divided by 1 is 1. (Con... |
| divneg 12008 | Move negative sign inside ... |
| muldivdir 12009 | Distribution of division o... |
| divsubdir 12010 | Distribution of division o... |
| muldivdid 12011 | Distribution of division o... |
| subdivcomb1 12012 | Bring a term in a subtract... |
| subdivcomb2 12013 | Bring a term in a subtract... |
| recrec 12014 | A number is equal to the r... |
| rec11 12015 | Reciprocal is one-to-one. ... |
| rec11r 12016 | Mutual reciprocals. (Cont... |
| divmuldiv 12017 | Multiplication of two rati... |
| divdivdiv 12018 | Division of two ratios. T... |
| divcan5 12019 | Cancellation of common fac... |
| divmul13 12020 | Swap the denominators in t... |
| divmul24 12021 | Swap the numerators in the... |
| divmuleq 12022 | Cross-multiply in an equal... |
| recdiv 12023 | The reciprocal of a ratio.... |
| divcan6 12024 | Cancellation of inverted f... |
| divdiv32 12025 | Swap denominators in a div... |
| divcan7 12026 | Cancel equal divisors in a... |
| dmdcan 12027 | Cancellation law for divis... |
| divdiv1 12028 | Division into a fraction. ... |
| divdiv2 12029 | Division by a fraction. (... |
| recdiv2 12030 | Division into a reciprocal... |
| ddcan 12031 | Cancellation in a double d... |
| divadddiv 12032 | Addition of two ratios. T... |
| divsubdiv 12033 | Subtraction of two ratios.... |
| conjmul 12034 | Two numbers whose reciproc... |
| rereccl 12035 | Closure law for reciprocal... |
| redivcl 12036 | Closure law for division o... |
| eqneg 12037 | A number equal to its nega... |
| eqnegd 12038 | A complex number equals it... |
| eqnegad 12039 | If a complex number equals... |
| div2neg 12040 | Quotient of two negatives.... |
| divneg2 12041 | Move negative sign inside ... |
| recclzi 12042 | Closure law for reciprocal... |
| recne0zi 12043 | The reciprocal of a nonzer... |
| recidzi 12044 | Multiplication of a number... |
| div1i 12045 | A number divided by 1 is i... |
| eqnegi 12046 | A number equal to its nega... |
| reccli 12047 | Closure law for reciprocal... |
| recidi 12048 | Multiplication of a number... |
| recreci 12049 | A number is equal to the r... |
| dividi 12050 | A number divided by itself... |
| div0i 12051 | Division into zero is zero... |
| divclzi 12052 | Closure law for division. ... |
| divcan1zi 12053 | A cancellation law for div... |
| divcan2zi 12054 | A cancellation law for div... |
| divreczi 12055 | Relationship between divis... |
| divcan3zi 12056 | A cancellation law for div... |
| divcan4zi 12057 | A cancellation law for div... |
| rec11i 12058 | Reciprocal is one-to-one. ... |
| divcli 12059 | Closure law for division. ... |
| divcan2i 12060 | A cancellation law for div... |
| divcan1i 12061 | A cancellation law for div... |
| divreci 12062 | Relationship between divis... |
| divcan3i 12063 | A cancellation law for div... |
| divcan4i 12064 | A cancellation law for div... |
| divne0i 12065 | The ratio of nonzero numbe... |
| rec11ii 12066 | Reciprocal is one-to-one. ... |
| divasszi 12067 | An associative law for div... |
| divmulzi 12068 | Relationship between divis... |
| divdirzi 12069 | Distribution of division o... |
| divdiv23zi 12070 | Swap denominators in a div... |
| divmuli 12071 | Relationship between divis... |
| divdiv32i 12072 | Swap denominators in a div... |
| divassi 12073 | An associative law for div... |
| divdiri 12074 | Distribution of division o... |
| div23i 12075 | A commutative/associative ... |
| div11i 12076 | One-to-one relationship fo... |
| divmuldivi 12077 | Multiplication of two rati... |
| divmul13i 12078 | Swap denominators of two r... |
| divadddivi 12079 | Addition of two ratios. T... |
| divdivdivi 12080 | Division of two ratios. T... |
| rerecclzi 12081 | Closure law for reciprocal... |
| rereccli 12082 | Closure law for reciprocal... |
| redivclzi 12083 | Closure law for division o... |
| redivcli 12084 | Closure law for division o... |
| div1d 12085 | A number divided by 1 is i... |
| reccld 12086 | Closure law for reciprocal... |
| recne0d 12087 | The reciprocal of a nonzer... |
| recidd 12088 | Multiplication of a number... |
| recid2d 12089 | Multiplication of a number... |
| recrecd 12090 | A number is equal to the r... |
| dividd 12091 | A number divided by itself... |
| div0d 12092 | Division into zero is zero... |
| divcld 12093 | Closure law for division. ... |
| divcan1d 12094 | A cancellation law for div... |
| divcan2d 12095 | A cancellation law for div... |
| divrecd 12096 | Relationship between divis... |
| divrec2d 12097 | Relationship between divis... |
| divcan3d 12098 | A cancellation law for div... |
| divcan4d 12099 | A cancellation law for div... |
| diveq0d 12100 | A ratio is zero iff the nu... |
| diveq1d 12101 | Equality in terms of unit ... |
| diveq1ad 12102 | The quotient of two comple... |
| diveq0ad 12103 | A fraction of complex numb... |
| divne1d 12104 | If two complex numbers are... |
| divne0bd 12105 | A ratio is zero iff the nu... |
| divnegd 12106 | Move negative sign inside ... |
| divneg2d 12107 | Move negative sign inside ... |
| div2negd 12108 | Quotient of two negatives.... |
| divne0d 12109 | The ratio of nonzero numbe... |
| recdivd 12110 | The reciprocal of a ratio.... |
| recdiv2d 12111 | Division into a reciprocal... |
| divcan6d 12112 | Cancellation of inverted f... |
| ddcand 12113 | Cancellation in a double d... |
| rec11d 12114 | Reciprocal is one-to-one. ... |
| divmuld 12115 | Relationship between divis... |
| div32d 12116 | A commutative/associative ... |
| div13d 12117 | A commutative/associative ... |
| divdiv32d 12118 | Swap denominators in a div... |
| divcan5d 12119 | Cancellation of common fac... |
| divcan5rd 12120 | Cancellation of common fac... |
| divcan7d 12121 | Cancel equal divisors in a... |
| dmdcand 12122 | Cancellation law for divis... |
| dmdcan2d 12123 | Cancellation law for divis... |
| divdiv1d 12124 | Division into a fraction. ... |
| divdiv2d 12125 | Division by a fraction. (... |
| divmul2d 12126 | Relationship between divis... |
| divmul3d 12127 | Relationship between divis... |
| divassd 12128 | An associative law for div... |
| div12d 12129 | A commutative/associative ... |
| div23d 12130 | A commutative/associative ... |
| divdird 12131 | Distribution of division o... |
| divsubdird 12132 | Distribution of division o... |
| div11d 12133 | One-to-one relationship fo... |
| divmuldivd 12134 | Multiplication of two rati... |
| divmul13d 12135 | Swap denominators of two r... |
| divmul24d 12136 | Swap the numerators in the... |
| divadddivd 12137 | Addition of two ratios. T... |
| divsubdivd 12138 | Subtraction of two ratios.... |
| divmuleqd 12139 | Cross-multiply in an equal... |
| divdivdivd 12140 | Division of two ratios. T... |
| diveq1bd 12141 | If two complex numbers are... |
| div2sub 12142 | Swap the order of subtract... |
| div2subd 12143 | Swap subtrahend and minuen... |
| rereccld 12144 | Closure law for reciprocal... |
| redivcld 12145 | Closure law for division o... |
| subrecd 12146 | Subtraction of reciprocals... |
| subrec 12147 | Subtraction of reciprocals... |
| subreci 12148 | Subtraction of reciprocals... |
| mvllmuld 12149 | Move the left term in a pr... |
| mvllmuli 12150 | Move the left term in a pr... |
| ldiv 12151 | Left-division. (Contribut... |
| rdiv 12152 | Right-division. (Contribu... |
| mdiv 12153 | A division law. (Contribu... |
| lineq 12154 | Solution of a (scalar) lin... |
| elimgt0 12155 | Hypothesis for weak deduct... |
| elimge0 12156 | Hypothesis for weak deduct... |
| ltp1 12157 | A number is less than itse... |
| lep1 12158 | A number is less than or e... |
| ltm1 12159 | A number minus 1 is less t... |
| lem1 12160 | A number minus 1 is less t... |
| letrp1 12161 | A transitive property of '... |
| p1le 12162 | A transitive property of p... |
| recgt0 12163 | The reciprocal of a positi... |
| prodgt0 12164 | Infer that a multiplicand ... |
| prodgt02 12165 | Infer that a multiplier is... |
| ltmul1a 12166 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12167 | Multiplication of both sid... |
| ltmul2 12168 | Multiplication of both sid... |
| lemul1 12169 | Multiplication of both sid... |
| lemul2 12170 | Multiplication of both sid... |
| lemul1a 12171 | Multiplication of both sid... |
| lemul2a 12172 | Multiplication of both sid... |
| ltmul12a 12173 | Comparison of product of t... |
| lemul12b 12174 | Comparison of product of t... |
| lemul12a 12175 | Comparison of product of t... |
| ltmulgt11 12176 | Multiplication by a number... |
| ltmulgt12 12177 | Multiplication by a number... |
| mulgt1 12178 | The product of two numbers... |
| lemulge11 12179 | Multiplication by a number... |
| lemulge12 12180 | Multiplication by a number... |
| ltdiv1 12181 | Division of both sides of ... |
| lediv1 12182 | Division of both sides of ... |
| gt0div 12183 | Division of a positive num... |
| ge0div 12184 | Division of a nonnegative ... |
| divgt0 12185 | The ratio of two positive ... |
| divge0 12186 | The ratio of nonnegative a... |
| mulge0b 12187 | A condition for multiplica... |
| mulle0b 12188 | A condition for multiplica... |
| mulsuble0b 12189 | A condition for multiplica... |
| ltmuldiv 12190 | 'Less than' relationship b... |
| ltmuldiv2 12191 | 'Less than' relationship b... |
| ltdivmul 12192 | 'Less than' relationship b... |
| ledivmul 12193 | 'Less than or equal to' re... |
| ltdivmul2 12194 | 'Less than' relationship b... |
| lt2mul2div 12195 | 'Less than' relationship b... |
| ledivmul2 12196 | 'Less than or equal to' re... |
| lemuldiv 12197 | 'Less than or equal' relat... |
| lemuldiv2 12198 | 'Less than or equal' relat... |
| ltrec 12199 | The reciprocal of both sid... |
| lerec 12200 | The reciprocal of both sid... |
| lt2msq1 12201 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12202 | Two nonnegative numbers co... |
| ltdiv2 12203 | Division of a positive num... |
| ltrec1 12204 | Reciprocal swap in a 'less... |
| lerec2 12205 | Reciprocal swap in a 'less... |
| ledivdiv 12206 | Invert ratios of positive ... |
| lediv2 12207 | Division of a positive num... |
| ltdiv23 12208 | Swap denominator with othe... |
| lediv23 12209 | Swap denominator with othe... |
| lediv12a 12210 | Comparison of ratio of two... |
| lediv2a 12211 | Division of both sides of ... |
| reclt1 12212 | The reciprocal of a positi... |
| recgt1 12213 | The reciprocal of a positi... |
| recgt1i 12214 | The reciprocal of a number... |
| recp1lt1 12215 | Construct a number less th... |
| recreclt 12216 | Given a positive number ` ... |
| le2msq 12217 | The square function on non... |
| msq11 12218 | The square of a nonnegativ... |
| ledivp1 12219 | "Less than or equal to" an... |
| squeeze0 12220 | If a nonnegative number is... |
| ltp1i 12221 | A number is less than itse... |
| recgt0i 12222 | The reciprocal of a positi... |
| recgt0ii 12223 | The reciprocal of a positi... |
| prodgt0i 12224 | Infer that a multiplicand ... |
| divgt0i 12225 | The ratio of two positive ... |
| divge0i 12226 | The ratio of nonnegative a... |
| ltreci 12227 | The reciprocal of both sid... |
| lereci 12228 | The reciprocal of both sid... |
| lt2msqi 12229 | The square function on non... |
| le2msqi 12230 | The square function on non... |
| msq11i 12231 | The square of a nonnegativ... |
| divgt0i2i 12232 | The ratio of two positive ... |
| ltrecii 12233 | The reciprocal of both sid... |
| divgt0ii 12234 | The ratio of two positive ... |
| ltmul1i 12235 | Multiplication of both sid... |
| ltdiv1i 12236 | Division of both sides of ... |
| ltmuldivi 12237 | 'Less than' relationship b... |
| ltmul2i 12238 | Multiplication of both sid... |
| lemul1i 12239 | Multiplication of both sid... |
| lemul2i 12240 | Multiplication of both sid... |
| ltdiv23i 12241 | Swap denominator with othe... |
| ledivp1i 12242 | "Less than or equal to" an... |
| ltdivp1i 12243 | Less-than and division rel... |
| ltdiv23ii 12244 | Swap denominator with othe... |
| ltmul1ii 12245 | Multiplication of both sid... |
| ltdiv1ii 12246 | Division of both sides of ... |
| ltp1d 12247 | A number is less than itse... |
| lep1d 12248 | A number is less than or e... |
| ltm1d 12249 | A number minus 1 is less t... |
| lem1d 12250 | A number minus 1 is less t... |
| recgt0d 12251 | The reciprocal of a positi... |
| divgt0d 12252 | The ratio of two positive ... |
| mulgt1d 12253 | The product of two numbers... |
| lemulge11d 12254 | Multiplication by a number... |
| lemulge12d 12255 | Multiplication by a number... |
| lemul1ad 12256 | Multiplication of both sid... |
| lemul2ad 12257 | Multiplication of both sid... |
| ltmul12ad 12258 | Comparison of product of t... |
| lemul12ad 12259 | Comparison of product of t... |
| lemul12bd 12260 | Comparison of product of t... |
| fimaxre 12261 | A finite set of real numbe... |
| fimaxre2 12262 | A nonempty finite set of r... |
| fimaxre3 12263 | A nonempty finite set of r... |
| fiminre 12264 | A nonempty finite set of r... |
| fiminre2 12265 | A nonempty finite set of r... |
| negfi 12266 | The negation of a finite s... |
| lbreu 12267 | If a set of reals contains... |
| lbcl 12268 | If a set of reals contains... |
| lble 12269 | If a set of reals contains... |
| lbinf 12270 | If a set of reals contains... |
| lbinfcl 12271 | If a set of reals contains... |
| lbinfle 12272 | If a set of reals contains... |
| sup2 12273 | A nonempty, bounded-above ... |
| sup3 12274 | A version of the completen... |
| infm3lem 12275 | Lemma for ~ infm3 . (Cont... |
| infm3 12276 | The completeness axiom for... |
| suprcl 12277 | Closure of supremum of a n... |
| suprub 12278 | A member of a nonempty bou... |
| suprubd 12279 | Natural deduction form of ... |
| suprcld 12280 | Natural deduction form of ... |
| suprlub 12281 | The supremum of a nonempty... |
| suprnub 12282 | An upper bound is not less... |
| suprleub 12283 | The supremum of a nonempty... |
| supaddc 12284 | The supremum function dist... |
| supadd 12285 | The supremum function dist... |
| supmul1 12286 | The supremum function dist... |
| supmullem1 12287 | Lemma for ~ supmul . (Con... |
| supmullem2 12288 | Lemma for ~ supmul . (Con... |
| supmul 12289 | The supremum function dist... |
| sup3ii 12290 | A version of the completen... |
| suprclii 12291 | Closure of supremum of a n... |
| suprubii 12292 | A member of a nonempty bou... |
| suprlubii 12293 | The supremum of a nonempty... |
| suprnubii 12294 | An upper bound is not less... |
| suprleubii 12295 | The supremum of a nonempty... |
| riotaneg 12296 | The negative of the unique... |
| negiso 12297 | Negation is an order anti-... |
| dfinfre 12298 | The infimum of a set of re... |
| infrecl 12299 | Closure of infimum of a no... |
| infrenegsup 12300 | The infimum of a set of re... |
| infregelb 12301 | Any lower bound of a nonem... |
| infrelb 12302 | If a nonempty set of real ... |
| infrefilb 12303 | The infimum of a finite se... |
| supfirege 12304 | The supremum of a finite s... |
| neg1cn 12305 | -1 is a complex number. (... |
| neg1rr 12306 | -1 is a real number. (Con... |
| neg1ne0 12307 | -1 is nonzero. (Contribut... |
| neg1lt0 12308 | -1 is less than 0. (Contr... |
| negneg1e1 12309 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12310 | The imaginary unit ` _i ` ... |
| rimul 12311 | A real number times the im... |
| cru 12312 | The representation of comp... |
| crne0 12313 | The real representation of... |
| creur 12314 | The real part of a complex... |
| creui 12315 | The imaginary part of a co... |
| cju 12316 | The complex conjugate of a... |
| ofsubeq0 12317 | Function analogue of ~ sub... |
| ofnegsub 12318 | Function analogue of ~ neg... |
| ofsubge0 12319 | Function analogue of ~ sub... |
| indv 12322 | Value of the indicator fun... |
| indval 12323 | Value of the indicator fun... |
| indval0 12324 | The indicator function gen... |
| indval2 12325 | Alternate value of the ind... |
| indf 12326 | An indicator function as a... |
| indfval 12327 | Value of the indicator fun... |
| fvindre 12328 | The range of the indicator... |
| ind1 12329 | Value of the indicator fun... |
| ind0 12330 | Value of the indicator fun... |
| ind1a 12331 | Value of the indicator fun... |
| indconst0 12332 | Indicator of the empty set... |
| indconst1 12333 | Indicator of the whole set... |
| indpi1 12334 | Preimage of the singleton ... |
| nnexALT 12337 | Alternate proof of ~ nnex ... |
| peano5nni 12338 | Peano's inductive postulat... |
| nnssre 12339 | The positive integers are ... |
| nnsscn 12340 | The positive integers are ... |
| nnex 12341 | The set of positive intege... |
| nnre 12342 | A positive integer is a re... |
| nncn 12343 | A positive integer is a co... |
| nnrei 12344 | A positive integer is a re... |
| nncni 12345 | A positive integer is a co... |
| 1nn 12346 | Peano postulate: 1 is a po... |
| peano2nn 12347 | Peano postulate: a success... |
| dfnn2 12348 | Alternate definition of th... |
| dfnn3 12349 | Alternate definition of th... |
| nnred 12350 | A positive integer is a re... |
| nncnd 12351 | A positive integer is a co... |
| peano2nnd 12352 | Peano postulate: a success... |
| nnind 12353 | Principle of Mathematical ... |
| nnindALT 12354 | Principle of Mathematical ... |
| nnindd 12355 | Principle of Mathematical ... |
| nn1m1nn 12356 | Every positive integer is ... |
| nn1suc 12357 | If a statement holds for 1... |
| nnaddcl 12358 | Closure of addition of pos... |
| nnmulcl 12359 | Closure of multiplication ... |
| nnmulcli 12360 | Closure of multiplication ... |
| nnadd1com 12361 | Addition with 1 is commuta... |
| nnaddcom 12362 | Addition is commutative fo... |
| nnaddcomli 12363 | Version of ~ addcomli for ... |
| nnmtmip 12364 | "Minus times minus is plus... |
| nn2ge 12365 | There exists a positive in... |
| nnge1 12366 | A positive integer is one ... |
| nngt1ne1 12367 | A positive integer is grea... |
| nnle1eq1 12368 | A positive integer is less... |
| nngt0 12369 | A positive integer is posi... |
| nnnlt1 12370 | A positive integer is not ... |
| nnnle0 12371 | A positive integer is not ... |
| nnne0 12372 | A positive integer is nonz... |
| nnneneg 12373 | No positive integer is equ... |
| 0nnn 12374 | Zero is not a positive int... |
| 0nnnALT 12375 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12376 | Alternate version of ~ nnn... |
| nngt0i 12377 | A positive integer is posi... |
| nnne0i 12378 | A positive integer is nonz... |
| nndivre 12379 | The quotient of a real and... |
| nnrecre 12380 | The reciprocal of a positi... |
| nnrecgt0 12381 | The reciprocal of a positi... |
| nnsub 12382 | Subtraction of positive in... |
| nnsubi 12383 | Subtraction of positive in... |
| nndiv 12384 | Two ways to express " ` A ... |
| nndivtr 12385 | Transitive property of div... |
| nnge1d 12386 | A positive integer is one ... |
| nngt0d 12387 | A positive integer is posi... |
| nnne0d 12388 | A positive integer is nonz... |
| nnrecred 12389 | The reciprocal of a positi... |
| nnaddcld 12390 | Closure of addition of pos... |
| nnmulcld 12391 | Closure of multiplication ... |
| nndivred 12392 | A positive integer is one ... |
| 1t1e1ALT 12393 | Alternate proof of ~ 1t1e1... |
| nnadddir 12394 | Right-distributivity for n... |
| nnmul1com 12395 | Multiplication with 1 is c... |
| nnmulcom 12396 | Multiplication is commutat... |
| 1eltp012 12413 | 1 is an element of ` { 0 ,... |
| 0ne1 12414 | Zero is different from one... |
| 1m1e0 12415 | One minus one equals zero.... |
| 2nn 12416 | 2 is a positive integer. ... |
| 2re 12417 | The number 2 is real. (Co... |
| 2cn 12418 | The number 2 is a complex ... |
| 2cnALT 12419 | Alternate proof of ~ 2cn .... |
| 2ex 12420 | The number 2 is a set. (C... |
| 2cnd 12421 | The number 2 is a complex ... |
| 3nn 12422 | 3 is a positive integer. ... |
| 3re 12423 | The number 3 is real. (Co... |
| 3cn 12424 | The number 3 is a complex ... |
| 3ex 12425 | The number 3 is a set. (C... |
| 4nn 12426 | 4 is a positive integer. ... |
| 4re 12427 | The number 4 is real. (Co... |
| 4cn 12428 | The number 4 is a complex ... |
| 5nn 12429 | 5 is a positive integer. ... |
| 5re 12430 | The number 5 is real. (Co... |
| 5cn 12431 | The number 5 is a complex ... |
| 6nn 12432 | 6 is a positive integer. ... |
| 6re 12433 | The number 6 is real. (Co... |
| 6cn 12434 | The number 6 is a complex ... |
| 7nn 12435 | 7 is a positive integer. ... |
| 7re 12436 | The number 7 is real. (Co... |
| 7cn 12437 | The number 7 is a complex ... |
| 8nn 12438 | 8 is a positive integer. ... |
| 8re 12439 | The number 8 is real. (Co... |
| 8cn 12440 | The number 8 is a complex ... |
| 9nn 12441 | 9 is a positive integer. ... |
| 9re 12442 | The number 9 is real. (Co... |
| 9cn 12443 | The number 9 is a complex ... |
| 0le0 12444 | Zero is nonnegative. (Con... |
| 0le2 12445 | The number 0 is less than ... |
| 0le2OLD 12446 | Obsolete version of ~ 0le2... |
| 2pos 12447 | The number 2 is positive. ... |
| 2posOLD 12448 | Obsolete version of ~ 2pos... |
| 2ne0 12449 | The number 2 is nonzero. ... |
| 2thalfe1 12450 | 2 times one half equals 1.... |
| 3pos 12451 | The number 3 is positive. ... |
| 3ne0 12452 | The number 3 is nonzero. ... |
| 4pos 12453 | The number 4 is positive. ... |
| 4ne0 12454 | The number 4 is nonzero. ... |
| 5pos 12455 | The number 5 is positive. ... |
| 6pos 12456 | The number 6 is positive. ... |
| 7pos 12457 | The number 7 is positive. ... |
| 8pos 12458 | The number 8 is positive. ... |
| 9pos 12459 | The number 9 is positive. ... |
| 1pneg1e0 12460 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12461 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12462 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12463 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12464 | Function value at ` N + 1 ... |
| 1p0e1 12465 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12466 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12467 | 2 - 1 = 1. The result is ... |
| 2m1e1OLD 12468 | Obsolete version of ~ 2m1e... |
| 1e2m1 12469 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12470 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12471 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12472 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12473 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12474 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12475 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12476 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12477 | Two plus two equals four. ... |
| 2times 12478 | Two times a number. (Cont... |
| times2 12479 | A number times 2. (Contri... |
| 2timesi 12480 | Two times a number. (Cont... |
| times2i 12481 | A number times 2. (Contri... |
| 2txmxeqx 12482 | Two times a complex number... |
| 2div2e1 12483 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12484 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12485 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12486 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12487 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12488 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12489 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12490 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12491 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12492 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12493 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12494 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12495 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12496 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12497 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12498 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12499 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12500 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12501 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12502 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12503 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12504 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12505 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12506 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12507 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12508 | 3 times 2 equals 6. (Cont... |
| 2t3e6 12509 | 2 times 3 equals 6. (Cont... |
| 3t3e9 12510 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12511 | 4 times 2 equals 8. (Cont... |
| 2t4e8 12512 | 2 times 4 equals 8. (Cont... |
| 2t0e0 12513 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12514 | One half of four is two. ... |
| 1lt2 12515 | 1 is less than 2. (Contri... |
| 2lt3 12516 | 2 is less than 3. (Contri... |
| 2le3 12517 | 2 is less than or equal to... |
| 1lt3 12518 | 1 is less than 3. (Contri... |
| 3lt4 12519 | 3 is less than 4. (Contri... |
| 2lt4 12520 | 2 is less than 4. (Contri... |
| 1lt4 12521 | 1 is less than 4. (Contri... |
| 4lt5 12522 | 4 is less than 5. (Contri... |
| 3lt5 12523 | 3 is less than 5. (Contri... |
| 2lt5 12524 | 2 is less than 5. (Contri... |
| 1lt5 12525 | 1 is less than 5. (Contri... |
| 5lt6 12526 | 5 is less than 6. (Contri... |
| 4lt6 12527 | 4 is less than 6. (Contri... |
| 3lt6 12528 | 3 is less than 6. (Contri... |
| 2lt6 12529 | 2 is less than 6. (Contri... |
| 1lt6 12530 | 1 is less than 6. (Contri... |
| 6lt7 12531 | 6 is less than 7. (Contri... |
| 5lt7 12532 | 5 is less than 7. (Contri... |
| 4lt7 12533 | 4 is less than 7. (Contri... |
| 3lt7 12534 | 3 is less than 7. (Contri... |
| 2lt7 12535 | 2 is less than 7. (Contri... |
| 1lt7 12536 | 1 is less than 7. (Contri... |
| 7lt8 12537 | 7 is less than 8. (Contri... |
| 6lt8 12538 | 6 is less than 8. (Contri... |
| 5lt8 12539 | 5 is less than 8. (Contri... |
| 4lt8 12540 | 4 is less than 8. (Contri... |
| 3lt8 12541 | 3 is less than 8. (Contri... |
| 2lt8 12542 | 2 is less than 8. (Contri... |
| 1lt8 12543 | 1 is less than 8. (Contri... |
| 8lt9 12544 | 8 is less than 9. (Contri... |
| 7lt9 12545 | 7 is less than 9. (Contri... |
| 6lt9 12546 | 6 is less than 9. (Contri... |
| 5lt9 12547 | 5 is less than 9. (Contri... |
| 4lt9 12548 | 4 is less than 9. (Contri... |
| 3lt9 12549 | 3 is less than 9. (Contri... |
| 2lt9 12550 | 2 is less than 9. (Contri... |
| 1lt9 12551 | 1 is less than 9. (Contri... |
| 0ne2 12552 | 0 is not equal to 2. (Con... |
| 1ne2 12553 | 1 is not equal to 2. (Con... |
| 1le2 12554 | 1 is less than or equal to... |
| 2cnne0 12555 | 2 is a nonzero complex num... |
| 2rene0 12556 | 2 is a nonzero real number... |
| 1le3 12557 | 1 is less than or equal to... |
| neg1mulneg1e1 12558 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12559 | One-half is real. (Contri... |
| halfcn 12560 | One-half is a complex numb... |
| halfgt0 12561 | One-half is greater than z... |
| halfge0 12562 | One-half is not negative. ... |
| halflt1 12563 | One-half is less than one.... |
| 2halves 12564 | Two halves make a whole. ... |
| 1mhlfehlf 12565 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12566 | An eighth of four thirds i... |
| halfthird 12567 | Half minus a third. (Cont... |
| halfpm6th 12568 | One half plus or minus one... |
| it0e0 12569 | i times 0 equals 0. (Cont... |
| 2mulicn 12570 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12571 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12572 | Closure of half of a numbe... |
| rehalfcl 12573 | Real closure of half. (Co... |
| half0 12574 | Half of a number is zero i... |
| halfpos2 12575 | A number is positive iff i... |
| halfpos 12576 | A positive number is great... |
| halfnneg2 12577 | A number is nonnegative if... |
| halfaddsubcl 12578 | Closure of half-sum and ha... |
| halfaddsub 12579 | Sum and difference of half... |
| subhalfhalf 12580 | Subtracting the half of a ... |
| lt2halves 12581 | A sum is less than the who... |
| addltmul 12582 | Sum is less than product f... |
| nominpos 12583 | There is no smallest posit... |
| avglt1 12584 | Ordering property for aver... |
| avglt2 12585 | Ordering property for aver... |
| avgle1 12586 | Ordering property for aver... |
| avgle2 12587 | Ordering property for aver... |
| avgle 12588 | The average of two numbers... |
| 2timesd 12589 | Two times a number. (Cont... |
| times2d 12590 | A number times 2. (Contri... |
| halfcld 12591 | Closure of half of a numbe... |
| 2halvesd 12592 | Two halves make a whole. ... |
| rehalfcld 12593 | Real closure of half. (Co... |
| lt2halvesd 12594 | A sum is less than the who... |
| rehalfcli 12595 | Half a real number is real... |
| lt2addmuld 12596 | If two real numbers are le... |
| add1p1 12597 | Adding two times 1 to a nu... |
| sub1m1 12598 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12599 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12600 | A complex number increased... |
| div4p1lem1div2 12601 | An integer greater than 5,... |
| nnunb 12602 | The set of positive intege... |
| arch 12603 | Archimedean property of re... |
| nnrecl 12604 | There exists a positive in... |
| bndndx 12605 | A bounded real sequence ` ... |
| elnn0 12608 | Nonnegative integers expre... |
| nnssnn0 12609 | Positive naturals are a su... |
| nn0ssre 12610 | Nonnegative integers are a... |
| nn0sscn 12611 | Nonnegative integers are a... |
| nn0ex 12612 | The set of nonnegative int... |
| nnnn0 12613 | A positive integer is a no... |
| nnnn0i 12614 | A positive integer is a no... |
| nn0re 12615 | A nonnegative integer is a... |
| nn0cn 12616 | A nonnegative integer is a... |
| nn0rei 12617 | A nonnegative integer is a... |
| nn0cni 12618 | A nonnegative integer is a... |
| dfn2 12619 | The set of positive intege... |
| elnnne0 12620 | The positive integer prope... |
| 0nn0 12621 | 0 is a nonnegative integer... |
| 1nn0 12622 | 1 is a nonnegative integer... |
| 2nn0 12623 | 2 is a nonnegative integer... |
| 3nn0 12624 | 3 is a nonnegative integer... |
| 4nn0 12625 | 4 is a nonnegative integer... |
| 5nn0 12626 | 5 is a nonnegative integer... |
| 6nn0 12627 | 6 is a nonnegative integer... |
| 7nn0 12628 | 7 is a nonnegative integer... |
| 8nn0 12629 | 8 is a nonnegative integer... |
| 9nn0 12630 | 9 is a nonnegative integer... |
| nn0ge0 12631 | A nonnegative integer is g... |
| nn0nlt0 12632 | A nonnegative integer is n... |
| nn0ge0i 12633 | Nonnegative integers are n... |
| nn0le0eq0 12634 | A nonnegative integer is l... |
| nn0p1gt0 12635 | A nonnegative integer incr... |
| nnnn0addcl 12636 | A positive integer plus a ... |
| nn0nnaddcl 12637 | A nonnegative integer plus... |
| 0mnnnnn0 12638 | The result of subtracting ... |
| un0addcl 12639 | If ` S ` is closed under a... |
| un0mulcl 12640 | If ` S ` is closed under m... |
| nn0addcl 12641 | Closure of addition of non... |
| nn0mulcl 12642 | Closure of multiplication ... |
| nn0addcli 12643 | Closure of addition of non... |
| nn0mulcli 12644 | Closure of multiplication ... |
| nn0p1nn 12645 | A nonnegative integer plus... |
| peano2nn0 12646 | Second Peano postulate for... |
| nnm1nn0 12647 | A positive integer minus 1... |
| elnn0nn 12648 | The nonnegative integer pr... |
| elnnnn0 12649 | The positive integer prope... |
| elnnnn0b 12650 | The positive integer prope... |
| elnnnn0c 12651 | The positive integer prope... |
| nn0addge1 12652 | A number is less than or e... |
| nn0addge2 12653 | A number is less than or e... |
| nn0addge1i 12654 | A number is less than or e... |
| nn0addge2i 12655 | A number is less than or e... |
| nn0sub 12656 | Subtraction of nonnegative... |
| ltsubnn0 12657 | Subtracting a nonnegative ... |
| nn0negleid 12658 | A nonnegative integer is g... |
| difgtsumgt 12659 | If the difference of a rea... |
| nn0le2x 12660 | A nonnegative integer is l... |
| nn0le2xi 12661 | A nonnegative integer is l... |
| nn0lele2xi 12662 | 'Less than or equal to' im... |
| fcdmnn0supp 12663 | Two ways to write the supp... |
| fcdmnn0fsupp 12664 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12665 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12666 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12667 | A positive integer is a no... |
| nn0red 12668 | A nonnegative integer is a... |
| nn0cnd 12669 | A nonnegative integer is a... |
| nn0ge0d 12670 | A nonnegative integer is g... |
| nn0addcld 12671 | Closure of addition of non... |
| nn0mulcld 12672 | Closure of multiplication ... |
| nn0readdcl 12673 | Closure law for addition o... |
| nn0n0n1ge2 12674 | A nonnegative integer whic... |
| nn0n0n1ge2b 12675 | A nonnegative integer is n... |
| nn0ge2m1nn 12676 | If a nonnegative integer i... |
| nn0ge2m1nn0 12677 | If a nonnegative integer i... |
| nn0nndivcl 12678 | Closure law for dividing o... |
| elxnn0 12681 | An extended nonnegative in... |
| nn0ssxnn0 12682 | The standard nonnegative i... |
| nn0xnn0 12683 | A standard nonnegative int... |
| xnn0xr 12684 | An extended nonnegative in... |
| 0xnn0 12685 | Zero is an extended nonneg... |
| pnf0xnn0 12686 | Positive infinity is an ex... |
| nn0nepnf 12687 | No standard nonnegative in... |
| nn0xnn0d 12688 | A standard nonnegative int... |
| nn0nepnfd 12689 | No standard nonnegative in... |
| xnn0nemnf 12690 | No extended nonnegative in... |
| xnn0xrnemnf 12691 | The extended nonnegative i... |
| xnn0nnn0pnf 12692 | An extended nonnegative in... |
| elz 12695 | Membership in the set of i... |
| nnnegz 12696 | The negative of a positive... |
| zre 12697 | An integer is a real. (Co... |
| zcn 12698 | An integer is a complex nu... |
| zrei 12699 | An integer is a real numbe... |
| zssre 12700 | The integers are a subset ... |
| zsscn 12701 | The integers are a subset ... |
| zex 12702 | The set of integers exists... |
| elnnz 12703 | Positive integer property ... |
| 0z 12704 | Zero is an integer. (Cont... |
| 0zd 12705 | Zero is an integer, deduct... |
| elnn0z 12706 | Nonnegative integer proper... |
| elznn0nn 12707 | Integer property expressed... |
| elznn0 12708 | Integer property expressed... |
| elznn 12709 | Integer property expressed... |
| zle0orge1 12710 | There is no integer in the... |
| elz2 12711 | Membership in the set of i... |
| dfz2 12712 | Alternative definition of ... |
| zexALT 12713 | Alternate proof of ~ zex .... |
| nnz 12714 | A positive integer is an i... |
| nnssz 12715 | Positive integers are a su... |
| nn0ssz 12716 | Nonnegative integers are a... |
| nn0z 12717 | A nonnegative integer is a... |
| nn0zd 12718 | A nonnegative integer is a... |
| nnzd 12719 | A positive integer is an i... |
| nnzi 12720 | A positive integer is an i... |
| nn0zi 12721 | A nonnegative integer is a... |
| elnnz1 12722 | Positive integer property ... |
| znnnlt1 12723 | An integer is not a positi... |
| nnzrab 12724 | Positive integers expresse... |
| nn0zrab 12725 | Nonnegative integers expre... |
| 1z 12726 | One is an integer. (Contr... |
| 1zzd 12727 | One is an integer, deducti... |
| 2z 12728 | 2 is an integer. (Contrib... |
| 3z 12729 | 3 is an integer. (Contrib... |
| 4z 12730 | 4 is an integer. (Contrib... |
| znegcl 12731 | Closure law for negative i... |
| neg1z 12732 | -1 is an integer. (Contri... |
| znegclb 12733 | A complex number is an int... |
| nn0negz 12734 | The negative of a nonnegat... |
| nn0negzi 12735 | The negative of a nonnegat... |
| zaddcl 12736 | Closure of addition of int... |
| peano2z 12737 | Second Peano postulate gen... |
| zsubcl 12738 | Closure of subtraction of ... |
| peano2zm 12739 | "Reverse" second Peano pos... |
| zletr 12740 | Transitive law of ordering... |
| zrevaddcl 12741 | Reverse closure law for ad... |
| znnsub 12742 | The positive difference of... |
| znn0sub 12743 | The nonnegative difference... |
| nzadd 12744 | The sum of a real number n... |
| zmulcl 12745 | Closure of multiplication ... |
| zltp1le 12746 | Integer ordering relation.... |
| zleltp1 12747 | Integer ordering relation.... |
| zlem1lt 12748 | Integer ordering relation.... |
| zltlem1 12749 | Integer ordering relation.... |
| zltlem1d 12750 | Integer ordering relation,... |
| zltp1led 12751 | Integer ordering relation,... |
| zgt0ge1 12752 | An integer greater than ` ... |
| 0nn0m1nnn0 12753 | A number is zero if and on... |
| nnleltp1 12754 | Positive integer ordering ... |
| nnltp1le 12755 | Positive integer ordering ... |
| nnaddm1cl 12756 | Closure of addition of pos... |
| nn0ltp1le 12757 | Nonnegative integer orderi... |
| nn0leltp1 12758 | Nonnegative integer orderi... |
| nn0ltlem1 12759 | Nonnegative integer orderi... |
| nn0sub2 12760 | Subtraction of nonnegative... |
| nn0lt10b 12761 | A nonnegative integer less... |
| nn0lt2 12762 | A nonnegative integer less... |
| nn0le2is012 12763 | A nonnegative integer whic... |
| nn0lem1lt 12764 | Nonnegative integer orderi... |
| nnlem1lt 12765 | Positive integer ordering ... |
| nnltlem1 12766 | Positive integer ordering ... |
| nnm1ge0 12767 | A positive integer decreas... |
| nn0ge0div 12768 | Division of a nonnegative ... |
| zdiv 12769 | Two ways to express " ` M ... |
| zdivadd 12770 | Property of divisibility: ... |
| zdivmul 12771 | Property of divisibility: ... |
| zextle 12772 | An extensionality-like pro... |
| zextlt 12773 | An extensionality-like pro... |
| recnz 12774 | The reciprocal of a number... |
| btwnnz 12775 | A number between an intege... |
| gtndiv 12776 | A larger number does not d... |
| halfnz 12777 | One-half is not an integer... |
| 3halfnz 12778 | Three halves is not an int... |
| suprzcl 12779 | The supremum of a bounded-... |
| prime 12780 | Two ways to express " ` A ... |
| msqznn 12781 | The square of a nonzero in... |
| zneo 12782 | No even integer equals an ... |
| nneo 12783 | A positive integer is even... |
| nneoi 12784 | A positive integer is even... |
| zeo 12785 | An integer is even or odd.... |
| zeo2 12786 | An integer is even or odd ... |
| peano2uz2 12787 | Second Peano postulate for... |
| peano5uzi 12788 | Peano's inductive postulat... |
| peano5uzti 12789 | Peano's inductive postulat... |
| dfuzi 12790 | An expression for the uppe... |
| uzind 12791 | Induction on the upper int... |
| uzind2 12792 | Induction on the upper int... |
| uzind3 12793 | Induction on the upper int... |
| nn0ind 12794 | Principle of Mathematical ... |
| nn0indALT 12795 | Principle of Mathematical ... |
| nn0indd 12796 | Principle of Mathematical ... |
| fzind 12797 | Induction on the integers ... |
| fnn0ind 12798 | Induction on the integers ... |
| nn0ind-raph 12799 | Principle of Mathematical ... |
| zindd 12800 | Principle of Mathematical ... |
| fzindd 12801 | Induction on the integers ... |
| btwnz 12802 | Any real number can be san... |
| zred 12803 | An integer is a real numbe... |
| zcnd 12804 | An integer is a complex nu... |
| znegcld 12805 | Closure law for negative i... |
| peano2zd 12806 | Deduction from second Pean... |
| zaddcld 12807 | Closure of addition of int... |
| zsubcld 12808 | Closure of subtraction of ... |
| zmulcld 12809 | Closure of multiplication ... |
| znnn0nn 12810 | The negative of a negative... |
| zadd2cl 12811 | Increasing an integer by 2... |
| zriotaneg 12812 | The negative of the unique... |
| suprfinzcl 12813 | The supremum of a nonempty... |
| 9p1e10 12816 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12817 | Version of the definition ... |
| decex 12818 | A decimal number is a set.... |
| deceq1 12819 | Equality theorem for the d... |
| deceq2 12820 | Equality theorem for the d... |
| deceq1i 12821 | Equality theorem for the d... |
| deceq2i 12822 | Equality theorem for the d... |
| deceq12i 12823 | Equality theorem for the d... |
| numnncl 12824 | Closure for a numeral (wit... |
| num0u 12825 | Add a zero in the units pl... |
| num0h 12826 | Add a zero in the higher p... |
| numcl 12827 | Closure for a decimal inte... |
| numsuc 12828 | The successor of a decimal... |
| deccl 12829 | Closure for a numeral. (C... |
| 11nn0 12830 | 11 is a nonnegative intege... |
| 12nn0 12831 | 12 is a nonnegative intege... |
| 16nn0 12832 | 16 is a nonnegative intege... |
| 25nn0 12833 | 25 is a nonnegative intege... |
| 10nn 12834 | 10 is a positive integer. ... |
| 10pos 12835 | The number 10 is positive.... |
| 10nn0 12836 | 10 is a nonnegative intege... |
| 10re 12837 | The number 10 is real. (C... |
| decnncl 12838 | Closure for a numeral. (C... |
| 11nn 12839 | 11 is a positive integer. ... |
| dec0u 12840 | Add a zero in the units pl... |
| dec0h 12841 | Add a zero in the higher p... |
| numnncl2 12842 | Closure for a decimal inte... |
| decnncl2 12843 | Closure for a decimal inte... |
| numlt 12844 | Comparing two decimal inte... |
| numltc 12845 | Comparing two decimal inte... |
| le9lt10 12846 | A "decimal digit" (i.e. a ... |
| declt 12847 | Comparing two decimal inte... |
| decltc 12848 | Comparing two decimal inte... |
| declth 12849 | Comparing two decimal inte... |
| decsuc 12850 | The successor of a decimal... |
| 3declth 12851 | Comparing two decimal inte... |
| 3decltc 12852 | Comparing two decimal inte... |
| decle 12853 | Comparing two decimal inte... |
| decleh 12854 | Comparing two decimal inte... |
| declei 12855 | Comparing a digit to a dec... |
| numlti 12856 | Comparing a digit to a dec... |
| declti 12857 | Comparing a digit to a dec... |
| decltdi 12858 | Comparing a digit to a dec... |
| numsucc 12859 | The successor of a decimal... |
| decsucc 12860 | The successor of a decimal... |
| 1e0p1 12861 | The successor of zero. (C... |
| dec10p 12862 | Ten plus an integer. (Con... |
| numma 12863 | Perform a multiply-add of ... |
| nummac 12864 | Perform a multiply-add of ... |
| numma2c 12865 | Perform a multiply-add of ... |
| numadd 12866 | Add two decimal integers `... |
| numaddc 12867 | Add two decimal integers `... |
| nummul1c 12868 | The product of a decimal i... |
| nummul2c 12869 | The product of a decimal i... |
| decma 12870 | Perform a multiply-add of ... |
| decmac 12871 | Perform a multiply-add of ... |
| decma2c 12872 | Perform a multiply-add of ... |
| decadd 12873 | Add two numerals ` M ` and... |
| decaddc 12874 | Add two numerals ` M ` and... |
| decaddc2 12875 | Add two numerals ` M ` and... |
| decrmanc 12876 | Perform a multiply-add of ... |
| decrmac 12877 | Perform a multiply-add of ... |
| decaddm10 12878 | The sum of two multiples o... |
| decaddi 12879 | Add two numerals ` M ` and... |
| decaddci 12880 | Add two numerals ` M ` and... |
| decaddci2 12881 | Add two numerals ` M ` and... |
| decsubi 12882 | Difference between a numer... |
| decmul1 12883 | The product of a numeral w... |
| decmul1c 12884 | The product of a numeral w... |
| decmul2c 12885 | The product of a numeral w... |
| decmulnc 12886 | The product of a numeral w... |
| 11multnc 12887 | The product of 11 (as nume... |
| decmul10add 12888 | A multiplication of a numb... |
| 6p5lem 12889 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12890 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12891 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12892 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12893 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12894 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12895 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12896 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12897 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12898 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12899 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12900 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12901 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12902 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12903 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12904 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12905 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12906 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12907 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12908 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12909 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12910 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12911 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12912 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12913 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12914 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12915 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12916 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12917 | 4 times 3 equals 12. (Con... |
| 4t4e16 12918 | 4 times 4 equals 16. (Con... |
| 5t2e10 12919 | 5 times 2 equals 10. (Con... |
| 5t3e15 12920 | 5 times 3 equals 15. (Con... |
| 5t4e20 12921 | 5 times 4 equals 20. (Con... |
| 5t5e25 12922 | 5 times 5 equals 25. (Con... |
| 6t2e12 12923 | 6 times 2 equals 12. (Con... |
| 6t3e18 12924 | 6 times 3 equals 18. (Con... |
| 6t4e24 12925 | 6 times 4 equals 24. (Con... |
| 6t5e30 12926 | 6 times 5 equals 30. (Con... |
| 6t6e36 12927 | 6 times 6 equals 36. (Con... |
| 7t2e14 12928 | 7 times 2 equals 14. (Con... |
| 7t3e21 12929 | 7 times 3 equals 21. (Con... |
| 7t4e28 12930 | 7 times 4 equals 28. (Con... |
| 7t5e35 12931 | 7 times 5 equals 35. (Con... |
| 7t6e42 12932 | 7 times 6 equals 42. (Con... |
| 7t7e49 12933 | 7 times 7 equals 49. (Con... |
| 8t2e16 12934 | 8 times 2 equals 16. (Con... |
| 8t3e24 12935 | 8 times 3 equals 24. (Con... |
| 8t4e32 12936 | 8 times 4 equals 32. (Con... |
| 8t5e40 12937 | 8 times 5 equals 40. (Con... |
| 8t6e48 12938 | 8 times 6 equals 48. (Con... |
| 8t7e56 12939 | 8 times 7 equals 56. (Con... |
| 8t8e64 12940 | 8 times 8 equals 64. (Con... |
| 9t2e18 12941 | 9 times 2 equals 18. (Con... |
| 9t3e27 12942 | 9 times 3 equals 27. (Con... |
| 9t4e36 12943 | 9 times 4 equals 36. (Con... |
| 9t5e45 12944 | 9 times 5 equals 45. (Con... |
| 9t6e54 12945 | 9 times 6 equals 54. (Con... |
| 9t7e63 12946 | 9 times 7 equals 63. (Con... |
| 9t8e72 12947 | 9 times 8 equals 72. (Con... |
| 9t9e81 12948 | 9 times 9 equals 81. (Con... |
| 9t11e99 12949 | 9 times 11 equals 99. (Co... |
| 9t11e99OLD 12950 | Obsolete version of ~ 9t11... |
| 9lt10 12951 | 9 is less than 10. (Contr... |
| 8lt10 12952 | 8 is less than 10. (Contr... |
| 7lt10 12953 | 7 is less than 10. (Contr... |
| 6lt10 12954 | 6 is less than 10. (Contr... |
| 5lt10 12955 | 5 is less than 10. (Contr... |
| 4lt10 12956 | 4 is less than 10. (Contr... |
| 3lt10 12957 | 3 is less than 10. (Contr... |
| 2lt10 12958 | 2 is less than 10. (Contr... |
| 1lt10 12959 | 1 is less than 10. (Contr... |
| 1lt10OLD 12960 | Obsolete version of ~ 1lt1... |
| decbin0 12961 | Decompose base 4 into base... |
| decbin2 12962 | Decompose base 4 into base... |
| decbin3 12963 | Decompose base 4 into base... |
| 5recm6rec 12964 | One fifth minus one sixth.... |
| uzval 12967 | The value of the upper int... |
| uzf 12968 | The domain and codomain of... |
| eluz1 12969 | Membership in the upper se... |
| eluzel2 12970 | Implication of membership ... |
| eluz2 12971 | Membership in an upper set... |
| eluzmn 12972 | Membership in an earlier u... |
| eluz1i 12973 | Membership in an upper set... |
| eluzuzle 12974 | An integer in an upper set... |
| eluzelz 12975 | A member of an upper set o... |
| eluzelre 12976 | A member of an upper set o... |
| eluzelcn 12977 | A member of an upper set o... |
| eluzle 12978 | Implication of membership ... |
| eluz 12979 | Membership in an upper set... |
| uzid 12980 | Membership of the least me... |
| uzidd 12981 | Membership of the least me... |
| uzn0 12982 | The upper integers are all... |
| uztrn 12983 | Transitive law for sets of... |
| uztrn2 12984 | Transitive law for sets of... |
| uzneg 12985 | Contraposition law for upp... |
| uzssz 12986 | An upper set of integers i... |
| uzssre 12987 | An upper set of integers i... |
| uzss 12988 | Subset relationship for tw... |
| uztric 12989 | Totality of the ordering r... |
| uz11 12990 | The upper integers functio... |
| eluzp1m1 12991 | Membership in the next upp... |
| eluzp1l 12992 | Strict ordering implied by... |
| eluzp1p1 12993 | Membership in the next upp... |
| eluzadd 12994 | Membership in a later uppe... |
| eluzsub 12995 | Membership in an earlier u... |
| eluzaddi 12996 | Membership in a later uppe... |
| eluzsubi 12997 | Membership in an earlier u... |
| subeluzsub 12998 | Membership of a difference... |
| uzm1 12999 | Choices for an element of ... |
| uznn0sub 13000 | The nonnegative difference... |
| uzin 13001 | Intersection of two upper ... |
| uzp1 13002 | Choices for an element of ... |
| nn0uz 13003 | Nonnegative integers expre... |
| nnuz 13004 | Positive integers expresse... |
| elnnuz 13005 | A positive integer express... |
| elnn0uz 13006 | A nonnegative integer expr... |
| 1eluzge0 13007 | 1 is an integer greater th... |
| 2eluzge0 13008 | 2 is an integer greater th... |
| 2eluzge1 13009 | 2 is an integer greater th... |
| 5eluz3 13010 | 5 is an integer greater th... |
| uzuzle23 13011 | An integer greater than or... |
| uzuzle24 13012 | An integer greater than or... |
| uzuzle34 13013 | An integer greater than or... |
| uzuzle35 13014 | An integer greater than or... |
| eluz2nn 13015 | An integer greater than or... |
| eluz3nn 13016 | An integer greater than or... |
| eluz4nn 13017 | An integer greater than or... |
| eluz5nn 13018 | An integer greater than or... |
| eluzge2nn0 13019 | If an integer is greater t... |
| eluz2n0 13020 | An integer greater than or... |
| uz3m2nn 13021 | An integer greater than or... |
| uznnssnn 13022 | The upper integers startin... |
| raluz 13023 | Restricted universal quant... |
| raluz2 13024 | Restricted universal quant... |
| rexuz 13025 | Restricted existential qua... |
| rexuz2 13026 | Restricted existential qua... |
| 2rexuz 13027 | Double existential quantif... |
| peano2uz 13028 | Second Peano postulate for... |
| peano2uzs 13029 | Second Peano postulate for... |
| peano2uzr 13030 | Reversed second Peano axio... |
| uzaddcl 13031 | Addition closure law for a... |
| nn0pzuz 13032 | The sum of a nonnegative i... |
| uzind4 13033 | Induction on the upper set... |
| uzind4ALT 13034 | Induction on the upper set... |
| uzind4s 13035 | Induction on the upper set... |
| uzind4s2 13036 | Induction on the upper set... |
| uzind4i 13037 | Induction on the upper int... |
| uzwo 13038 | Well-ordering principle: a... |
| uzwo2 13039 | Well-ordering principle: a... |
| nnwo 13040 | Well-ordering principle: a... |
| nnwof 13041 | Well-ordering principle: a... |
| nnwos 13042 | Well-ordering principle: a... |
| indstr 13043 | Strong Mathematical Induct... |
| eluznn0 13044 | Membership in a nonnegativ... |
| eluznn 13045 | Membership in a positive u... |
| eluz2b1 13046 | Two ways to say "an intege... |
| eluz2gt1 13047 | An integer greater than or... |
| eluz2b2 13048 | Two ways to say "an intege... |
| eluz2b3 13049 | Two ways to say "an intege... |
| uz2m1nn 13050 | One less than an integer g... |
| 1nuz2 13051 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 13052 | A positive integer is eith... |
| uz2mulcl 13053 | Closure of multiplication ... |
| indstr2 13054 | Strong Mathematical Induct... |
| uzinfi 13055 | Extract the lower bound of... |
| nninf 13056 | The infimum of the set of ... |
| nn0inf 13057 | The infimum of the set of ... |
| infssuzle 13058 | The infimum of a subset of... |
| infssuzcl 13059 | The infimum of a subset of... |
| ublbneg 13060 | The image under negation o... |
| eqreznegel 13061 | Two ways to express the im... |
| supminf 13062 | The supremum of a bounded-... |
| lbzbi 13063 | If a set of reals is bound... |
| zsupss 13064 | Any nonempty bounded subse... |
| suprzcl2 13065 | The supremum of a bounded-... |
| suprzub 13066 | The supremum of a bounded-... |
| uzsupss 13067 | Any bounded subset of an u... |
| nn01to3 13068 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 13069 | Alternate proof of ~ nn0ge... |
| uzwo3 13070 | Well-ordering principle: a... |
| zmin 13071 | There is a unique smallest... |
| zmax 13072 | There is a unique largest ... |
| zbtwnre 13073 | There is a unique integer ... |
| rebtwnz 13074 | There is a unique greatest... |
| elq 13077 | Membership in the set of r... |
| qmulz 13078 | If ` A ` is rational, then... |
| znq 13079 | The ratio of an integer an... |
| qre 13080 | A rational number is a rea... |
| zq 13081 | An integer is a rational n... |
| qred 13082 | A rational number is a rea... |
| zssq 13083 | The integers are a subset ... |
| nn0ssq 13084 | The nonnegative integers a... |
| nnssq 13085 | The positive integers are ... |
| qssre 13086 | The rationals are a subset... |
| qsscn 13087 | The rationals are a subset... |
| qex 13088 | The set of rational number... |
| nnq 13089 | A positive integer is rati... |
| qcn 13090 | A rational number is a com... |
| qexALT 13091 | Alternate proof of ~ qex .... |
| 1q 13092 | The number 1 is rational. ... |
| qaddcl 13093 | Closure of addition of rat... |
| qnegcl 13094 | Closure law for the negati... |
| qmulcl 13095 | Closure of multiplication ... |
| qsubcl 13096 | Closure of subtraction of ... |
| qreccl 13097 | Closure of reciprocal of r... |
| qdivcl 13098 | Closure of division of rat... |
| qrevaddcl 13099 | Reverse closure law for ad... |
| nnrecq 13100 | The reciprocal of a positi... |
| irradd 13101 | The sum of an irrational n... |
| irrmul 13102 | The product of an irration... |
| elpq 13103 | A positive rational is the... |
| elpqb 13104 | A class is a positive rati... |
| rpnnen1lem2 13105 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 13106 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 13107 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 13108 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 13109 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 13110 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 13111 | One half of ~ rpnnen , whe... |
| reexALT 13112 | Alternate proof of ~ reex ... |
| cnref1o 13113 | There is a natural one-to-... |
| cnexALT 13114 | The set of complex numbers... |
| xrex 13115 | The set of extended reals ... |
| mpoaddex 13116 | The addition operation is ... |
| addex 13117 | The addition operation is ... |
| mpomulex 13118 | The multiplication operati... |
| mulex 13119 | The multiplication operati... |
| elrp 13122 | Membership in the set of p... |
| elrpii 13123 | Membership in the set of p... |
| 1rp 13124 | 1 is a positive real. (Co... |
| 2rp 13125 | 2 is a positive real. (Co... |
| 3rp 13126 | 3 is a positive real. (Co... |
| 5rp 13127 | 5 is a positive real. (Co... |
| rpssre 13128 | The positive reals are a s... |
| rpre 13129 | A positive real is a real.... |
| rpxr 13130 | A positive real is an exte... |
| rpcn 13131 | A positive real is a compl... |
| nnrp 13132 | A positive integer is a po... |
| rpgt0 13133 | A positive real is greater... |
| rpge0 13134 | A positive real is greater... |
| rpregt0 13135 | A positive real is a posit... |
| rprege0 13136 | A positive real is a nonne... |
| rpne0 13137 | A positive real is nonzero... |
| rprene0 13138 | A positive real is a nonze... |
| rpcnne0 13139 | A positive real is a nonze... |
| neglt 13140 | The negative of a positive... |
| rpcndif0 13141 | A positive real number is ... |
| ralrp 13142 | Quantification over positi... |
| rexrp 13143 | Quantification over positi... |
| rpaddcl 13144 | Closure law for addition o... |
| rpmulcl 13145 | Closure law for multiplica... |
| rpmtmip 13146 | "Minus times minus is plus... |
| rpdivcl 13147 | Closure law for division o... |
| rpreccl 13148 | Closure law for reciprocat... |
| rphalfcl 13149 | Closure law for half of a ... |
| rpgecl 13150 | A number greater than or e... |
| rphalflt 13151 | Half of a positive real is... |
| rerpdivcl 13152 | Closure law for division o... |
| ge0p1rp 13153 | A nonnegative number plus ... |
| rpneg 13154 | Either a nonzero real or i... |
| negelrp 13155 | Elementhood of a negation ... |
| negelrpd 13156 | The negation of a negative... |
| 0nrp 13157 | Zero is not a positive rea... |
| ltsubrp 13158 | Subtracting a positive rea... |
| ltaddrp 13159 | Adding a positive number t... |
| difrp 13160 | Two ways to say one number... |
| elrpd 13161 | Membership in the set of p... |
| nnrpd 13162 | A positive integer is a po... |
| zgt1rpn0n1 13163 | An integer greater than 1 ... |
| rpred 13164 | A positive real is a real.... |
| rpxrd 13165 | A positive real is an exte... |
| rpcnd 13166 | A positive real is a compl... |
| rpgt0d 13167 | A positive real is greater... |
| rpge0d 13168 | A positive real is greater... |
| rpne0d 13169 | A positive real is nonzero... |
| rpregt0d 13170 | A positive real is real an... |
| rprege0d 13171 | A positive real is real an... |
| rprene0d 13172 | A positive real is a nonze... |
| rpcnne0d 13173 | A positive real is a nonze... |
| rpreccld 13174 | Closure law for reciprocat... |
| rprecred 13175 | Closure law for reciprocat... |
| rphalfcld 13176 | Closure law for half of a ... |
| reclt1d 13177 | The reciprocal of a positi... |
| recgt1d 13178 | The reciprocal of a positi... |
| rpaddcld 13179 | Closure law for addition o... |
| rpmulcld 13180 | Closure law for multiplica... |
| rpdivcld 13181 | Closure law for division o... |
| ltrecd 13182 | The reciprocal of both sid... |
| lerecd 13183 | The reciprocal of both sid... |
| ltrec1d 13184 | Reciprocal swap in a 'less... |
| lerec2d 13185 | Reciprocal swap in a 'less... |
| lediv2ad 13186 | Division of both sides of ... |
| ltdiv2d 13187 | Division of a positive num... |
| lediv2d 13188 | Division of a positive num... |
| ledivdivd 13189 | Invert ratios of positive ... |
| divge1 13190 | The ratio of a number over... |
| divlt1lt 13191 | A real number divided by a... |
| divle1le 13192 | A real number divided by a... |
| ledivge1le 13193 | If a number is less than o... |
| ge0p1rpd 13194 | A nonnegative number plus ... |
| rerpdivcld 13195 | Closure law for division o... |
| ltsubrpd 13196 | Subtracting a positive rea... |
| ltaddrpd 13197 | Adding a positive number t... |
| ltaddrp2d 13198 | Adding a positive number t... |
| ltmulgt11d 13199 | Multiplication by a number... |
| ltmulgt12d 13200 | Multiplication by a number... |
| gt0divd 13201 | Division of a positive num... |
| ge0divd 13202 | Division of a nonnegative ... |
| rpgecld 13203 | A number greater than or e... |
| divge0d 13204 | The ratio of nonnegative a... |
| ltmul1d 13205 | The ratio of nonnegative a... |
| ltmul2d 13206 | Multiplication of both sid... |
| lemul1d 13207 | Multiplication of both sid... |
| lemul2d 13208 | Multiplication of both sid... |
| ltdiv1d 13209 | Division of both sides of ... |
| lediv1d 13210 | Division of both sides of ... |
| ltmuldivd 13211 | 'Less than' relationship b... |
| ltmuldiv2d 13212 | 'Less than' relationship b... |
| lemuldivd 13213 | 'Less than or equal to' re... |
| lemuldiv2d 13214 | 'Less than or equal to' re... |
| ltdivmuld 13215 | 'Less than' relationship b... |
| ltdivmul2d 13216 | 'Less than' relationship b... |
| ledivmuld 13217 | 'Less than or equal to' re... |
| ledivmul2d 13218 | 'Less than or equal to' re... |
| ltmul1dd 13219 | The ratio of nonnegative a... |
| ltmul2dd 13220 | Multiplication of both sid... |
| ltdiv1dd 13221 | Division of both sides of ... |
| lediv1dd 13222 | Division of both sides of ... |
| lediv12ad 13223 | Comparison of ratio of two... |
| mul2lt0rlt0 13224 | If the result of a multipl... |
| mul2lt0rgt0 13225 | If the result of a multipl... |
| mul2lt0llt0 13226 | If the result of a multipl... |
| mul2lt0lgt0 13227 | If the result of a multipl... |
| mul2lt0bi 13228 | If the result of a multipl... |
| prodge0rd 13229 | Infer that a multiplicand ... |
| prodge0ld 13230 | Infer that a multiplier is... |
| ltdiv23d 13231 | Swap denominator with othe... |
| lediv23d 13232 | Swap denominator with othe... |
| lt2mul2divd 13233 | The ratio of nonnegative a... |
| nnledivrp 13234 | Division of a positive int... |
| nn0ledivnn 13235 | Division of a nonnegative ... |
| addlelt 13236 | If the sum of a real numbe... |
| ge2halflem1 13237 | Half of an integer greater... |
| ltxr 13244 | The 'less than' binary rel... |
| elxr 13245 | Membership in the set of e... |
| xrnemnf 13246 | An extended real other tha... |
| xrnepnf 13247 | An extended real other tha... |
| xrltnr 13248 | The extended real 'less th... |
| ltpnf 13249 | Any (finite) real is less ... |
| ltpnfd 13250 | Any (finite) real is less ... |
| 0ltpnf 13251 | Zero is less than plus inf... |
| mnflt 13252 | Minus infinity is less tha... |
| mnfltd 13253 | Minus infinity is less tha... |
| mnflt0 13254 | Minus infinity is less tha... |
| mnfltpnf 13255 | Minus infinity is less tha... |
| mnfltxr 13256 | Minus infinity is less tha... |
| pnfnlt 13257 | No extended real is greate... |
| nltmnf 13258 | No extended real is less t... |
| pnfge 13259 | Plus infinity is an upper ... |
| pnfged 13260 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13261 | An extended nonnegative in... |
| 0lepnf 13262 | 0 less than or equal to po... |
| xnn0ge0 13263 | An extended nonnegative in... |
| mnfle 13264 | Minus infinity is less tha... |
| mnfled 13265 | Minus infinity is less tha... |
| xrltnsym 13266 | Ordering on the extended r... |
| xrltnsym2 13267 | 'Less than' is antisymmetr... |
| xrlttri 13268 | Ordering on the extended r... |
| xrlttr 13269 | Ordering on the extended r... |
| xrltso 13270 | 'Less than' is a strict or... |
| xrlttri2 13271 | Trichotomy law for 'less t... |
| xrlttri3 13272 | Trichotomy law for 'less t... |
| xrleloe 13273 | 'Less than or equal' expre... |
| xrleltne 13274 | 'Less than or equal to' im... |
| xrltlen 13275 | 'Less than' expressed in t... |
| dfle2 13276 | Alternative definition of ... |
| dflt2 13277 | Alternative definition of ... |
| xrltle 13278 | 'Less than' implies 'less ... |
| xrltled 13279 | 'Less than' implies 'less ... |
| xrleid 13280 | 'Less than or equal to' is... |
| xrleidd 13281 | 'Less than or equal to' is... |
| xrletri 13282 | Trichotomy law for extende... |
| xrletri3 13283 | Trichotomy law for extende... |
| xrletrid 13284 | Trichotomy law for extende... |
| xrlelttr 13285 | Transitive law for orderin... |
| xrltletr 13286 | Transitive law for orderin... |
| xrletr 13287 | Transitive law for orderin... |
| xrlttrd 13288 | Transitive law for orderin... |
| xrlelttrd 13289 | Transitive law for orderin... |
| xrltletrd 13290 | Transitive law for orderin... |
| xrletrd 13291 | Transitive law for orderin... |
| xrltne 13292 | 'Less than' implies not eq... |
| xrgtned 13293 | 'Greater than' implies not... |
| nltpnft 13294 | An extended real is not le... |
| xgepnf 13295 | An extended real which is ... |
| ngtmnft 13296 | An extended real is not gr... |
| xlemnf 13297 | An extended real which is ... |
| xrrebnd 13298 | An extended real is real i... |
| xrre 13299 | A way of proving that an e... |
| xrre2 13300 | An extended real between t... |
| xrre3 13301 | A way of proving that an e... |
| ge0gtmnf 13302 | A nonnegative extended rea... |
| ge0nemnf 13303 | A nonnegative extended rea... |
| xrrege0 13304 | A nonnegative extended rea... |
| xrmax1 13305 | An extended real is less t... |
| xrmax2 13306 | An extended real is less t... |
| xrmin1 13307 | The minimum of two extende... |
| xrmin2 13308 | The minimum of two extende... |
| xrmaxeq 13309 | The maximum of two extende... |
| xrmineq 13310 | The minimum of two extende... |
| xrmaxlt 13311 | Two ways of saying the max... |
| xrltmin 13312 | Two ways of saying an exte... |
| xrmaxle 13313 | Two ways of saying the max... |
| xrlemin 13314 | Two ways of saying a numbe... |
| max1 13315 | A number is less than or e... |
| max1ALT 13316 | A number is less than or e... |
| max2 13317 | A number is less than or e... |
| 2resupmax 13318 | The supremum of two real n... |
| min1 13319 | The minimum of two numbers... |
| min2 13320 | The minimum of two numbers... |
| maxle 13321 | Two ways of saying the max... |
| lemin 13322 | Two ways of saying a numbe... |
| maxlt 13323 | Two ways of saying the max... |
| ltmin 13324 | Two ways of saying a numbe... |
| lemaxle 13325 | A real number which is les... |
| max0sub 13326 | Decompose a real number in... |
| ifle 13327 | An if statement transforms... |
| z2ge 13328 | There exists an integer gr... |
| qbtwnre 13329 | The rational numbers are d... |
| qbtwnxr 13330 | The rational numbers are d... |
| qsqueeze 13331 | If a nonnegative real is l... |
| qextltlem 13332 | Lemma for ~ qextlt and qex... |
| qextlt 13333 | An extensionality-like pro... |
| qextle 13334 | An extensionality-like pro... |
| xralrple 13335 | Show that ` A ` is less th... |
| alrple 13336 | Show that ` A ` is less th... |
| xnegeq 13337 | Equality of two extended n... |
| xnegex 13338 | A negative extended real e... |
| xnegpnf 13339 | Minus ` +oo ` . Remark of... |
| xnegmnf 13340 | Minus ` -oo ` . Remark of... |
| rexneg 13341 | Minus a real number. Rema... |
| xneg0 13342 | The negative of zero. (Co... |
| xnegcl 13343 | Closure of extended real n... |
| xnegneg 13344 | Extended real version of ~... |
| xneg11 13345 | Extended real version of ~... |
| xltnegi 13346 | Forward direction of ~ xlt... |
| xltneg 13347 | Extended real version of ~... |
| xleneg 13348 | Extended real version of ~... |
| xlt0neg1 13349 | Extended real version of ~... |
| xlt0neg2 13350 | Extended real version of ~... |
| xle0neg1 13351 | Extended real version of ~... |
| xle0neg2 13352 | Extended real version of ~... |
| xaddval 13353 | Value of the extended real... |
| xaddf 13354 | The extended real addition... |
| xmulval 13355 | Value of the extended real... |
| xaddpnf1 13356 | Addition of positive infin... |
| xaddpnf2 13357 | Addition of positive infin... |
| xaddmnf1 13358 | Addition of negative infin... |
| xaddmnf2 13359 | Addition of negative infin... |
| pnfaddmnf 13360 | Addition of positive and n... |
| mnfaddpnf 13361 | Addition of negative and p... |
| rexadd 13362 | The extended real addition... |
| rexsub 13363 | Extended real subtraction ... |
| rexaddd 13364 | The extended real addition... |
| xnn0xaddcl 13365 | The extended nonnegative i... |
| xaddnemnf 13366 | Closure of extended real a... |
| xaddnepnf 13367 | Closure of extended real a... |
| xnegid 13368 | Extended real version of ~... |
| xaddcl 13369 | The extended real addition... |
| xaddcom 13370 | The extended real addition... |
| xaddrid 13371 | Extended real version of ~... |
| xaddlid 13372 | Extended real version of ~... |
| xaddridd 13373 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13374 | Extended nonnegative integ... |
| xnn0lenn0nn0 13375 | An extended nonnegative in... |
| xnn0le2is012 13376 | An extended nonnegative in... |
| xnn0xadd0 13377 | The sum of two extended no... |
| xnegdi 13378 | Extended real version of ~... |
| xaddass 13379 | Associativity of extended ... |
| xaddass2 13380 | Associativity of extended ... |
| xpncan 13381 | Extended real version of ~... |
| xnpcan 13382 | Extended real version of ~... |
| xleadd1a 13383 | Extended real version of ~... |
| xleadd2a 13384 | Commuted form of ~ xleadd1... |
| xleadd1 13385 | Weakened version of ~ xlea... |
| xltadd1 13386 | Extended real version of ~... |
| xltadd2 13387 | Extended real version of ~... |
| xaddge0 13388 | The sum of nonnegative ext... |
| xle2add 13389 | Extended real version of ~... |
| xlt2add 13390 | Extended real version of ~... |
| xsubge0 13391 | Extended real version of ~... |
| xposdif 13392 | Extended real version of ~... |
| xlesubadd 13393 | Under certain conditions, ... |
| xmullem 13394 | Lemma for ~ rexmul . (Con... |
| xmullem2 13395 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13396 | Extended real multiplicati... |
| xmul01 13397 | Extended real version of ~... |
| xmul02 13398 | Extended real version of ~... |
| xmulneg1 13399 | Extended real version of ~... |
| xmulneg2 13400 | Extended real version of ~... |
| rexmul 13401 | The extended real multipli... |
| xmulf 13402 | The extended real multipli... |
| xmulcl 13403 | Closure of extended real m... |
| xmulpnf1 13404 | Multiplication by plus inf... |
| xmulpnf2 13405 | Multiplication by plus inf... |
| xmulmnf1 13406 | Multiplication by minus in... |
| xmulmnf2 13407 | Multiplication by minus in... |
| xmulpnf1n 13408 | Multiplication by plus inf... |
| xmulrid 13409 | Extended real version of ~... |
| xmullid 13410 | Extended real version of ~... |
| xmulm1 13411 | Extended real version of ~... |
| xmulasslem2 13412 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13413 | Extended real version of ~... |
| xmulge0 13414 | Extended real version of ~... |
| xmulasslem 13415 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13416 | Lemma for ~ xmulass . (Co... |
| xmulass 13417 | Associativity of the exten... |
| xlemul1a 13418 | Extended real version of ~... |
| xlemul2a 13419 | Extended real version of ~... |
| xlemul1 13420 | Extended real version of ~... |
| xlemul2 13421 | Extended real version of ~... |
| xltmul1 13422 | Extended real version of ~... |
| xltmul2 13423 | Extended real version of ~... |
| xadddilem 13424 | Lemma for ~ xadddi . (Con... |
| xadddi 13425 | Distributive property for ... |
| xadddir 13426 | Commuted version of ~ xadd... |
| xadddi2 13427 | The assumption that the mu... |
| xadddi2r 13428 | Commuted version of ~ xadd... |
| x2times 13429 | Extended real version of ~... |
| xnegcld 13430 | Closure of extended real n... |
| xaddcld 13431 | The extended real addition... |
| xmulcld 13432 | Closure of extended real m... |
| xadd4d 13433 | Rearrangement of 4 terms i... |
| xnn0add4d 13434 | Rearrangement of 4 terms i... |
| xrsupexmnf 13435 | Adding minus infinity to a... |
| xrinfmexpnf 13436 | Adding plus infinity to a ... |
| xrsupsslem 13437 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13438 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13439 | Any subset of extended rea... |
| xrinfmss 13440 | Any subset of extended rea... |
| xrinfmss2 13441 | Any subset of extended rea... |
| xrub 13442 | By quantifying only over r... |
| supxr 13443 | The supremum of a set of e... |
| supxr2 13444 | The supremum of a set of e... |
| supxrcl 13445 | The supremum of an arbitra... |
| supxrun 13446 | The supremum of the union ... |
| supxrmnf 13447 | Adding minus infinity to a... |
| supxrpnf 13448 | The supremum of a set of e... |
| supxrunb1 13449 | The supremum of an unbound... |
| supxrunb2 13450 | The supremum of an unbound... |
| supxrbnd1 13451 | The supremum of a bounded-... |
| supxrbnd2 13452 | The supremum of a bounded-... |
| xrsup0 13453 | The supremum of an empty s... |
| supxrub 13454 | A member of a set of exten... |
| supxrlub 13455 | The supremum of a set of e... |
| supxrleub 13456 | The supremum of a set of e... |
| supxrre 13457 | The real and extended real... |
| supxrbnd 13458 | The supremum of a bounded-... |
| supxrgtmnf 13459 | The supremum of a nonempty... |
| supxrre1 13460 | The supremum of a nonempty... |
| supxrre2 13461 | The supremum of a nonempty... |
| supxrss 13462 | Smaller sets of extended r... |
| xrsupssd 13463 | Inequality deduction for s... |
| infxrcl 13464 | The infimum of an arbitrar... |
| infxrlb 13465 | A member of a set of exten... |
| infxrgelb 13466 | The infimum of a set of ex... |
| infxrre 13467 | The real and extended real... |
| infxrmnf 13468 | The infinimum of a set of ... |
| xrinf0 13469 | The infimum of the empty s... |
| infxrss 13470 | Larger sets of extended re... |
| reltre 13471 | For all real numbers there... |
| rpltrp 13472 | For all positive real numb... |
| reltxrnmnf 13473 | For all extended real numb... |
| infmremnf 13474 | The infimum of the reals i... |
| infmrp1 13475 | The infimum of the positiv... |
| ixxval 13484 | Value of the interval func... |
| elixx1 13485 | Membership in an interval ... |
| ixxf 13486 | The set of intervals of ex... |
| ixxex 13487 | The set of intervals of ex... |
| ixxssxr 13488 | The set of intervals of ex... |
| elixx3g 13489 | Membership in a set of ope... |
| ixxssixx 13490 | An interval is a subset of... |
| ixxdisj 13491 | Split an interval into dis... |
| ixxun 13492 | Split an interval into two... |
| ixxin 13493 | Intersection of two interv... |
| ixxss1 13494 | Subset relationship for in... |
| ixxss2 13495 | Subset relationship for in... |
| ixxss12 13496 | Subset relationship for in... |
| ixxub 13497 | Extract the upper bound of... |
| ixxlb 13498 | Extract the lower bound of... |
| iooex 13499 | The set of open intervals ... |
| iooval 13500 | Value of the open interval... |
| ioo0 13501 | An empty open interval of ... |
| ioon0 13502 | An open interval of extend... |
| ndmioo 13503 | The open interval function... |
| iooid 13504 | An open interval with iden... |
| elioo3g 13505 | Membership in a set of ope... |
| elioore 13506 | A member of an open interv... |
| lbioo 13507 | An open interval does not ... |
| ubioo 13508 | An open interval does not ... |
| iooval2 13509 | Value of the open interval... |
| iooin 13510 | Intersection of two open i... |
| iooss1 13511 | Subset relationship for op... |
| iooss2 13512 | Subset relationship for op... |
| iocval 13513 | Value of the open-below, c... |
| icoval 13514 | Value of the closed-below,... |
| iccval 13515 | Value of the closed interv... |
| elioo1 13516 | Membership in an open inte... |
| elioo2 13517 | Membership in an open inte... |
| elioc1 13518 | Membership in an open-belo... |
| elico1 13519 | Membership in a closed-bel... |
| elicc1 13520 | Membership in a closed int... |
| iccid 13521 | A closed interval with ide... |
| ico0 13522 | An empty open interval of ... |
| ioc0 13523 | An empty open interval of ... |
| icc0 13524 | An empty closed interval o... |
| dfrp2 13525 | Alternate definition of th... |
| elicod 13526 | Membership in a left-close... |
| icogelb 13527 | An element of a left-close... |
| icogelbd 13528 | An element of a left-close... |
| elicore 13529 | A member of a left-closed ... |
| ubioc1 13530 | The upper bound belongs to... |
| lbico1 13531 | The lower bound belongs to... |
| iccleub 13532 | An element of a closed int... |
| iccgelb 13533 | An element of a closed int... |
| elioo5 13534 | Membership in an open inte... |
| eliooxr 13535 | A nonempty open interval s... |
| eliooord 13536 | Ordering implied by a memb... |
| elioo4g 13537 | Membership in an open inte... |
| ioossre 13538 | An open interval is a set ... |
| ioosscn 13539 | An open interval is a set ... |
| elioc2 13540 | Membership in an open-belo... |
| elico2 13541 | Membership in a closed-bel... |
| elicc2 13542 | Membership in a closed rea... |
| elicc2i 13543 | Inference for membership i... |
| elicc4 13544 | Membership in a closed rea... |
| iccss 13545 | Condition for a closed int... |
| iccssioo 13546 | Condition for a closed int... |
| icossico 13547 | Condition for a closed-bel... |
| iccss2 13548 | Condition for a closed int... |
| iccssico 13549 | Condition for a closed int... |
| iccssioo2 13550 | Condition for a closed int... |
| iccssico2 13551 | Condition for a closed int... |
| icossico2d 13552 | Condition for a closed-bel... |
| ioomax 13553 | The open interval from min... |
| iccmax 13554 | The closed interval from m... |
| ioopos 13555 | The set of positive reals ... |
| ioorp 13556 | The set of positive reals ... |
| iooshf 13557 | Shift the arguments of the... |
| iocssre 13558 | A closed-above interval wi... |
| icossre 13559 | A closed-below interval wi... |
| iccssre 13560 | A closed real interval is ... |
| iccssxr 13561 | A closed interval is a set... |
| iocssxr 13562 | An open-below, closed-abov... |
| icossxr 13563 | A closed-below, open-above... |
| ioossicc 13564 | An open interval is a subs... |
| iccssred 13565 | A closed real interval is ... |
| eliccxr 13566 | A member of a closed inter... |
| icossicc 13567 | A closed-below, open-above... |
| iocssicc 13568 | A closed-above, open-below... |
| ioossico 13569 | An open interval is a subs... |
| iocssioo 13570 | Condition for a closed int... |
| icossioo 13571 | Condition for a closed int... |
| ioossioo 13572 | Condition for an open inte... |
| iccsupr 13573 | A nonempty subset of a clo... |
| elioopnf 13574 | Membership in an unbounded... |
| elioomnf 13575 | Membership in an unbounded... |
| elicopnf 13576 | Membership in a closed unb... |
| repos 13577 | Two ways of saying that a ... |
| ioof 13578 | The set of open intervals ... |
| iccf 13579 | The set of closed interval... |
| unirnioo 13580 | The union of the range of ... |
| dfioo2 13581 | Alternate definition of th... |
| ioorebas 13582 | Open intervals are element... |
| xrge0neqmnf 13583 | A nonnegative extended rea... |
| xrge0nre 13584 | An extended real which is ... |
| elrege0 13585 | The predicate "is a nonneg... |
| nn0rp0 13586 | A nonnegative integer is a... |
| rge0ssre 13587 | Nonnegative real numbers a... |
| elxrge0 13588 | Elementhood in the set of ... |
| 0e0icopnf 13589 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13590 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13591 | The nonnegative reals are ... |
| ge0mulcl 13592 | The nonnegative reals are ... |
| ge0xaddcl 13593 | The nonnegative reals are ... |
| ge0xmulcl 13594 | The nonnegative extended r... |
| lbicc2 13595 | The lower bound of a close... |
| ubicc2 13596 | The upper bound of a close... |
| elicc01 13597 | Membership in the closed r... |
| elunitrn 13598 | The closed unit interval i... |
| elunitcn 13599 | The closed unit interval i... |
| 0elunit 13600 | Zero is an element of the ... |
| 1elunit 13601 | One is an element of the c... |
| iooneg 13602 | Membership in a negated op... |
| iccneg 13603 | Membership in a negated cl... |
| icoshft 13604 | A shifted real is a member... |
| icoshftf1o 13605 | Shifting a closed-below, o... |
| icoun 13606 | The union of two adjacent ... |
| icodisj 13607 | Adjacent left-closed right... |
| ioounsn 13608 | The union of an open inter... |
| snunioo 13609 | The closure of one end of ... |
| snunico 13610 | The closure of the open en... |
| snunioc 13611 | The closure of the open en... |
| prunioo 13612 | The closure of an open rea... |
| ioodisj 13613 | If the upper bound of one ... |
| ioojoin 13614 | Join two open intervals to... |
| difreicc 13615 | The class difference of ` ... |
| iccsplit 13616 | Split a closed interval in... |
| iccshftr 13617 | Membership in a shifted in... |
| iccshftri 13618 | Membership in a shifted in... |
| iccshftl 13619 | Membership in a shifted in... |
| iccshftli 13620 | Membership in a shifted in... |
| iccdil 13621 | Membership in a dilated in... |
| iccdili 13622 | Membership in a dilated in... |
| icccntr 13623 | Membership in a contracted... |
| icccntri 13624 | Membership in a contracted... |
| divelunit 13625 | A condition for a ratio to... |
| lincmb01cmp 13626 | A linear combination of tw... |
| iccf1o 13627 | Describe a bijection from ... |
| iccen 13628 | Any nontrivial closed inte... |
| xov1plusxeqvd 13629 | A complex number ` X ` is ... |
| unitssre 13630 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13631 | The closed unit interval i... |
| supicc 13632 | Supremum of a bounded set ... |
| supiccub 13633 | The supremum of a bounded ... |
| supicclub 13634 | The supremum of a bounded ... |
| supicclub2 13635 | The supremum of a bounded ... |
| zltaddlt1le 13636 | The sum of an integer and ... |
| xnn0xrge0 13637 | An extended nonnegative in... |
| nnge2recico01 13638 | The reciprocal of an integ... |
| fzval 13641 | The value of a finite set ... |
| fzval2 13642 | An alternative way of expr... |
| fzf 13643 | Establish the domain and c... |
| elfz1 13644 | Membership in a finite set... |
| elfz 13645 | Membership in a finite set... |
| elfz2 13646 | Membership in a finite set... |
| elfzd 13647 | Membership in a finite set... |
| elfz5 13648 | Membership in a finite set... |
| elfz4 13649 | Membership in a finite set... |
| elfzuzb 13650 | Membership in a finite set... |
| eluzfz 13651 | Membership in a finite set... |
| elfzuz 13652 | A member of a finite set o... |
| elfzuz3 13653 | Membership in a finite set... |
| elfzel2 13654 | Membership in a finite set... |
| elfzel1 13655 | Membership in a finite set... |
| elfzelz 13656 | A member of a finite set o... |
| elfzelzd 13657 | A member of a finite set o... |
| fzssz 13658 | A finite sequence of integ... |
| fzssre 13659 | A finite sequence of integ... |
| elfzle1 13660 | A member of a finite set o... |
| elfzle2 13661 | A member of a finite set o... |
| elfzuz2 13662 | Implication of membership ... |
| elfzle3 13663 | Membership in a finite set... |
| eluzfz1 13664 | Membership in a finite set... |
| eluzfz2 13665 | Membership in a finite set... |
| eluzfz2b 13666 | Membership in a finite set... |
| elfz3 13667 | Membership in a finite set... |
| elfz1eq 13668 | Membership in a finite set... |
| elfzubelfz 13669 | If there is a member in a ... |
| peano2fzr 13670 | A Peano-postulate-like the... |
| fzn0 13671 | Properties of a finite int... |
| fz0 13672 | A finite set of sequential... |
| fzn 13673 | A finite set of sequential... |
| fzen 13674 | A shifted finite set of se... |
| fz1n 13675 | A 1-based finite set of se... |
| 0nelfz1 13676 | 0 is not an element of a f... |
| 0fz1 13677 | Two ways to say a finite 1... |
| fz10 13678 | There are no integers betw... |
| fz00m1 13679 | There are no integers betw... |
| uzsubsubfz 13680 | Membership of an integer g... |
| uzsubsubfz1 13681 | Membership of an integer g... |
| ige3m2fz 13682 | Membership of an integer g... |
| fzsplit2 13683 | Split a finite interval of... |
| fzsplit 13684 | Split a finite interval of... |
| fzdisj 13685 | Condition for two finite i... |
| fz01en 13686 | 0-based and 1-based finite... |
| elfznn 13687 | A member of a finite set o... |
| elfz1end 13688 | A nonempty finite range of... |
| fz1ssnn 13689 | A finite set of positive i... |
| fznn0sub 13690 | Subtraction closure for a ... |
| fzmmmeqm 13691 | Subtracting the difference... |
| fzaddel 13692 | Membership of a sum in a f... |
| fzadd2 13693 | Membership of a sum in a f... |
| fzsubel 13694 | Membership of a difference... |
| fzopth 13695 | A finite set of sequential... |
| fzass4 13696 | Two ways to express a nond... |
| fzss1 13697 | Subset relationship for fi... |
| fzss2 13698 | Subset relationship for fi... |
| fzssuz 13699 | A finite set of sequential... |
| fzsn 13700 | A finite interval of integ... |
| fzssp1 13701 | Subset relationship for fi... |
| fzssnn 13702 | Finite sets of sequential ... |
| ssfzunsnext 13703 | A subset of a finite seque... |
| ssfzunsn 13704 | A subset of a finite seque... |
| fzsuc 13705 | Join a successor to the en... |
| fzpred 13706 | Join a predecessor to the ... |
| fzpreddisj 13707 | A finite set of sequential... |
| elfzp1 13708 | Append an element to a fin... |
| fzp1ss 13709 | Subset relationship for fi... |
| fzelp1 13710 | Membership in a set of seq... |
| fzp1elp1 13711 | Add one to an element of a... |
| fznatpl1 13712 | Shift membership in a fini... |
| fzpr 13713 | A finite interval of integ... |
| fztp 13714 | A finite interval of integ... |
| fz12pr 13715 | An integer range between 1... |
| fzsuc2 13716 | Join a successor to the en... |
| fzp1disj 13717 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13718 | Remove a successor from th... |
| fzprval 13719 | Two ways of defining the f... |
| fztpval 13720 | Two ways of defining the f... |
| fzrev 13721 | Reversal of start and end ... |
| fzrev2 13722 | Reversal of start and end ... |
| fzrev2i 13723 | Reversal of start and end ... |
| fzrev3 13724 | The "complement" of a memb... |
| fzrev3i 13725 | The "complement" of a memb... |
| fznn 13726 | Finite set of sequential i... |
| elfz1b 13727 | Membership in a 1-based fi... |
| elfz1uz 13728 | Membership in a 1-based fi... |
| elfzm11 13729 | Membership in a finite set... |
| uzsplit 13730 | Express an upper integer s... |
| uzdisj 13731 | The first ` N ` elements o... |
| fseq1p1m1 13732 | Add/remove an item to/from... |
| fseq1m1p1 13733 | Add/remove an item to/from... |
| fz1sbc 13734 | Quantification over a one-... |
| elfzp1b 13735 | An integer is a member of ... |
| elfzm1b 13736 | An integer is a member of ... |
| elfzp12 13737 | Options for membership in ... |
| fzne1 13738 | Elementhood in a finite se... |
| fzdif1 13739 | Split the first element of... |
| fz0dif1 13740 | Split the first element of... |
| fzm1 13741 | Choices for an element of ... |
| fzneuz 13742 | No finite set of sequentia... |
| fznuz 13743 | Disjointness of the upper ... |
| uznfz 13744 | Disjointness of the upper ... |
| fzp1nel 13745 | One plus the upper bound o... |
| fzrevral 13746 | Reversal of scanning order... |
| fzrevral2 13747 | Reversal of scanning order... |
| fzrevral3 13748 | Reversal of scanning order... |
| fzshftral 13749 | Shift the scanning order i... |
| ige2m1fz1 13750 | Membership of an integer g... |
| ige2m1fz 13751 | Membership in a 0-based fi... |
| elfz2nn0 13752 | Membership in a finite set... |
| fznn0 13753 | Characterization of a fini... |
| elfznn0 13754 | A member of a finite set o... |
| elfz3nn0 13755 | The upper bound of a nonem... |
| fz0ssnn0 13756 | Finite sets of sequential ... |
| fz1ssfz0 13757 | Subset relationship for fi... |
| 0elfz 13758 | 0 is an element of a finit... |
| nn0fz0 13759 | A nonnegative integer is a... |
| elfz0add 13760 | An element of a finite set... |
| fz0sn 13761 | An integer range from 0 to... |
| fz0tp 13762 | An integer range from 0 to... |
| fz0to3un2pr 13763 | An integer range from 0 to... |
| fz0to4untppr 13764 | An integer range from 0 to... |
| fz0to5un2tp 13765 | An integer range from 0 to... |
| elfz0ubfz0 13766 | An element of a finite set... |
| elfz0fzfz0 13767 | A member of a finite set o... |
| fz0fzelfz0 13768 | If a member of a finite se... |
| fznn0sub2 13769 | Subtraction closure for a ... |
| uzsubfz0 13770 | Membership of an integer g... |
| fz0fzdiffz0 13771 | The difference of an integ... |
| elfzmlbm 13772 | Subtracting the lower boun... |
| elfzmlbp 13773 | Subtracting the lower boun... |
| fzctr 13774 | Lemma for theorems about t... |
| difelfzle 13775 | The difference of two inte... |
| difelfznle 13776 | The difference of two inte... |
| nn0split 13777 | Express the set of nonnega... |
| nn0disj 13778 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13779 | A finite set of sequential... |
| fvffz0 13780 | The function value of a fu... |
| 1fv 13781 | A function on a singleton.... |
| 4fvwrd4 13782 | The first four function va... |
| 2ffzeq 13783 | Two functions over 0-based... |
| preduz 13784 | The value of the predecess... |
| prednn 13785 | The value of the predecess... |
| prednn0 13786 | The value of the predecess... |
| predfz 13787 | Calculate the predecessor ... |
| fzof 13790 | Functionality of the half-... |
| elfzoel1 13791 | Reverse closure for half-o... |
| elfzoel2 13792 | Reverse closure for half-o... |
| elfzoelz 13793 | Reverse closure for half-o... |
| fzoval 13794 | Value of the half-open int... |
| elfzo 13795 | Membership in a half-open ... |
| elfzo2 13796 | Membership in a half-open ... |
| elfzod 13797 | Membership in a half-open ... |
| elfzouz 13798 | Membership in a half-open ... |
| nelfzo 13799 | An integer not being a mem... |
| fzolb 13800 | The left endpoint of a hal... |
| fzolb2 13801 | The left endpoint of a hal... |
| elfzole1 13802 | A member in a half-open in... |
| elfzolt2 13803 | A member in a half-open in... |
| elfzolt3 13804 | Membership in a half-open ... |
| elfzolt2b 13805 | A member in a half-open in... |
| elfzolt3b 13806 | Membership in a half-open ... |
| elfzop1le2 13807 | A member in a half-open in... |
| fzonel 13808 | A half-open range does not... |
| elfzouz2 13809 | The upper bound of a half-... |
| elfzofz 13810 | A half-open range is conta... |
| elfzo3 13811 | Express membership in a ha... |
| fzon0 13812 | A half-open integer interv... |
| fzossfz 13813 | A half-open range is conta... |
| fzossz 13814 | A half-open integer interv... |
| fzon 13815 | A half-open set of sequent... |
| fzo0n 13816 | A half-open range of nonne... |
| fzonlt0 13817 | A half-open integer range ... |
| fzo0 13818 | Half-open sets with equal ... |
| fzonnsub 13819 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13820 | If ` M < N ` then ` N - M ... |
| fzoss1 13821 | Subset relationship for ha... |
| fzoss2 13822 | Subset relationship for ha... |
| fzossrbm1 13823 | Subset of a half-open rang... |
| fzo0ss1 13824 | Subset relationship for ha... |
| fzossnn0 13825 | A half-open integer range ... |
| fzospliti 13826 | One direction of splitting... |
| fzosplit 13827 | Split a half-open integer ... |
| fzodisj 13828 | Abutting half-open integer... |
| fzouzsplit 13829 | Split an upper integer set... |
| fzouzdisj 13830 | A half-open integer range ... |
| fzoun 13831 | A half-open integer range ... |
| fzodisjsn 13832 | A half-open integer range ... |
| prinfzo0 13833 | The intersection of a half... |
| lbfzo0 13834 | An integer is strictly gre... |
| elfzo0 13835 | Membership in a half-open ... |
| elfzo0z 13836 | Membership in a half-open ... |
| nn0p1elfzo 13837 | A nonnegative integer incr... |
| elfzo0le 13838 | A member in a half-open ra... |
| elfzolem1 13839 | A member in a half-open in... |
| elfzo0subge1 13840 | The difference of the uppe... |
| elfzo0suble 13841 | The difference of the uppe... |
| elfzonn0 13842 | A member of a half-open ra... |
| fzonmapblen 13843 | The result of subtracting ... |
| fzofzim 13844 | If a nonnegative integer i... |
| fz1fzo0m1 13845 | Translation of one between... |
| fzossnn 13846 | Half-open integer ranges s... |
| elfzo1 13847 | Membership in a half-open ... |
| fzo1lb 13848 | 1 is the left endpoint of ... |
| 1elfzo1 13849 | 1 is in a half-open range ... |
| fzo1fzo0n0 13850 | An integer between 1 and a... |
| fzo0n0 13851 | A half-open integer range ... |
| fzoaddel 13852 | Translate membership in a ... |
| fzo0addel 13853 | Translate membership in a ... |
| fzo0addelr 13854 | Translate membership in a ... |
| fzoaddel2 13855 | Translate membership in a ... |
| elfzoextl 13856 | Membership of an integer i... |
| elfzoext 13857 | Membership of an integer i... |
| elincfzoext 13858 | Membership of an increased... |
| fzosubel 13859 | Translate membership in a ... |
| fzosubel2 13860 | Membership in a translated... |
| fzosubel3 13861 | Membership in a translated... |
| eluzgtdifelfzo 13862 | Membership of the differen... |
| ige2m2fzo 13863 | Membership of an integer g... |
| fzocatel 13864 | Translate membership in a ... |
| ubmelfzo 13865 | If an integer in a 1-based... |
| elfzodifsumelfzo 13866 | If an integer is in a half... |
| elfzom1elp1fzo 13867 | Membership of an integer i... |
| elfzom1elfzo 13868 | Membership in a half-open ... |
| fzval3 13869 | Expressing a closed intege... |
| fz0add1fz1 13870 | Translate membership in a ... |
| fzosn 13871 | Expressing a singleton as ... |
| elfzomin 13872 | Membership of an integer i... |
| zpnn0elfzo 13873 | Membership of an integer i... |
| zpnn0elfzo1 13874 | Membership of an integer i... |
| fzosplitsnm1 13875 | Removing a singleton from ... |
| elfzonlteqm1 13876 | If an element of a half-op... |
| fzonn0p1 13877 | A nonnegative integer is a... |
| fzossfzop1 13878 | A half-open range of nonne... |
| fzonn0p1p1 13879 | If a nonnegative integer i... |
| elfzom1p1elfzo 13880 | Increasing an element of a... |
| fzo0ssnn0 13881 | Half-open integer ranges s... |
| fzo01 13882 | Expressing the singleton o... |
| fzo12sn 13883 | A 1-based half-open intege... |
| fzo13pr 13884 | A 1-based half-open intege... |
| fzo0to2pr 13885 | A half-open integer range ... |
| fz01pr 13886 | An integer range between 0... |
| fzo0to3tp 13887 | A half-open integer range ... |
| fzo0to42pr 13888 | A half-open integer range ... |
| fzo1to4tp 13889 | A half-open integer range ... |
| fzo0sn0fzo1 13890 | A half-open range of nonne... |
| elfzo0l 13891 | A member of a half-open ra... |
| fzoend 13892 | The endpoint of a half-ope... |
| fzo0end 13893 | The endpoint of a zero-bas... |
| ssfzo12 13894 | Subset relationship for ha... |
| ssfzoulel 13895 | If a half-open integer ran... |
| ssfzo12bi 13896 | Subset relationship for ha... |
| fzoopth 13897 | A half-open integer range ... |
| ubmelm1fzo 13898 | The result of subtracting ... |
| fzofzp1 13899 | If a point is in a half-op... |
| fzofzp1b 13900 | If a point is in a half-op... |
| elfzom1b 13901 | An integer is a member of ... |
| elfzom1elp1fzo1 13902 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13903 | Membership of a positive i... |
| elfzonelfzo 13904 | If an element of a half-op... |
| elfzodif0 13905 | If an integer ` M ` is in ... |
| fzonfzoufzol 13906 | If an element of a half-op... |
| elfzomelpfzo 13907 | An integer increased by an... |
| elfznelfzo 13908 | A value in a finite set of... |
| elfznelfzob 13909 | A value in a finite set of... |
| peano2fzor 13910 | A Peano-postulate-like the... |
| fzosplitsn 13911 | Extending a half-open rang... |
| fzosplitpr 13912 | Extending a half-open inte... |
| fzosplitprm1 13913 | Extending a half-open inte... |
| fzosplitsni 13914 | Membership in a half-open ... |
| fzisfzounsn 13915 | A finite interval of integ... |
| elfzr 13916 | A member of a finite inter... |
| elfzlmr 13917 | A member of a finite inter... |
| elfz0lmr 13918 | A member of a finite inter... |
| fzone1 13919 | Elementhood in a half-open... |
| fzom1ne1 13920 | Elementhood in a half-open... |
| fzostep1 13921 | Two possibilities for a nu... |
| fzoshftral 13922 | Shift the scanning order i... |
| fzind2 13923 | Induction on the integers ... |
| fvinim0ffz 13924 | The function values for th... |
| injresinjlem 13925 | Lemma for ~ injresinj . (... |
| injresinj 13926 | A function whose restricti... |
| f1resfz0f1d 13927 | If a function with a seque... |
| subfzo0 13928 | The difference between two... |
| fvf1tp 13929 | Values of a one-to-one fun... |
| flval 13934 | Value of the floor (greate... |
| flcl 13935 | The floor (greatest intege... |
| reflcl 13936 | The floor (greatest intege... |
| fllelt 13937 | A basic property of the fl... |
| flcld 13938 | The floor (greatest intege... |
| flle 13939 | A basic property of the fl... |
| flltp1 13940 | A basic property of the fl... |
| fllep1 13941 | A basic property of the fl... |
| fraclt1 13942 | The fractional part of a r... |
| fracle1 13943 | The fractional part of a r... |
| fracge0 13944 | The fractional part of a r... |
| flge 13945 | The floor function value i... |
| fllt 13946 | The floor function value i... |
| flflp1 13947 | Move floor function betwee... |
| flid 13948 | An integer is its own floo... |
| flidm 13949 | The floor function is idem... |
| flidz 13950 | A real number equals its f... |
| flltnz 13951 | The floor of a non-integer... |
| flwordi 13952 | Ordering relation for the ... |
| flword2 13953 | Ordering relation for the ... |
| flval2 13954 | An alternate way to define... |
| flval3 13955 | An alternate way to define... |
| flbi 13956 | A condition equivalent to ... |
| flbi2 13957 | A condition equivalent to ... |
| adddivflid 13958 | The floor of a sum of an i... |
| ico01fl0 13959 | The floor of a real number... |
| flge0nn0 13960 | The floor of a number grea... |
| flge1nn 13961 | The floor of a number grea... |
| fldivnn0 13962 | The floor function of a di... |
| refldivcl 13963 | The floor function of a di... |
| divfl0 13964 | The floor of a fraction is... |
| fladdz 13965 | An integer can be moved in... |
| flzadd 13966 | An integer can be moved in... |
| flmulnn0 13967 | Move a nonnegative integer... |
| btwnzge0 13968 | A real bounded between an ... |
| 2tnp1ge0ge0 13969 | Two times an integer plus ... |
| flhalf 13970 | Ordering relation for the ... |
| fldivle 13971 | The floor function of a di... |
| fldivnn0le 13972 | The floor function of a di... |
| flltdivnn0lt 13973 | The floor function of a di... |
| ltdifltdiv 13974 | If the dividend of a divis... |
| fldiv4p1lem1div2 13975 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13976 | The floor of an integer gr... |
| fldiv4lem1div2 13977 | The floor of a positive in... |
| ceilval 13978 | The value of the ceiling f... |
| dfceil2 13979 | Alternative definition of ... |
| ceilval2 13980 | The value of the ceiling f... |
| ceicl 13981 | The ceiling function retur... |
| ceilcl 13982 | Closure of the ceiling fun... |
| ceilcld 13983 | Closure of the ceiling fun... |
| ceige 13984 | The ceiling of a real numb... |
| ceilge 13985 | The ceiling of a real numb... |
| ceilged 13986 | The ceiling of a real numb... |
| ceim1l 13987 | One less than the ceiling ... |
| ceilm1lt 13988 | One less than the ceiling ... |
| ceile 13989 | The ceiling of a real numb... |
| ceille 13990 | The ceiling of a real numb... |
| ceilid 13991 | An integer is its own ceil... |
| ceilidz 13992 | A real number equals its c... |
| flleceil 13993 | The floor of a real number... |
| fleqceilz 13994 | A real number is an intege... |
| quoremz 13995 | Quotient and remainder of ... |
| quoremnn0 13996 | Quotient and remainder of ... |
| quoremnn0ALT 13997 | Alternate proof of ~ quore... |
| intfrac2 13998 | Decompose a real into inte... |
| intfracq 13999 | Decompose a rational numbe... |
| fldiv 14000 | Cancellation of the embedd... |
| fldiv2 14001 | Cancellation of an embedde... |
| fznnfl 14002 | Finite set of sequential i... |
| uzsup 14003 | An upper set of integers i... |
| ioopnfsup 14004 | An upper set of reals is u... |
| icopnfsup 14005 | An upper set of reals is u... |
| rpsup 14006 | The positive reals are unb... |
| resup 14007 | The real numbers are unbou... |
| xrsup 14008 | The extended real numbers ... |
| modval 14011 | The value of the modulo op... |
| modvalr 14012 | The value of the modulo op... |
| modcl 14013 | Closure law for the modulo... |
| flpmodeq 14014 | Partition of a division in... |
| modcld 14015 | Closure law for the modulo... |
| mod0 14016 | ` A mod B ` is zero iff ` ... |
| mulmod0 14017 | The product of an integer ... |
| negmod0 14018 | ` A ` is divisible by ` B ... |
| modge0 14019 | The modulo operation is no... |
| modlt 14020 | The modulo operation is le... |
| modelico 14021 | Modular reduction produces... |
| moddiffl 14022 | Value of the modulo operat... |
| moddifz 14023 | The modulo operation diffe... |
| modfrac 14024 | The fractional part of a n... |
| flmod 14025 | The floor function express... |
| intfrac 14026 | Break a number into its in... |
| zmod10 14027 | An integer modulo 1 is 0. ... |
| zmod1congr 14028 | Two arbitrary integers are... |
| modmulnn 14029 | Move a positive integer in... |
| modvalp1 14030 | The value of the modulo op... |
| zmodcl 14031 | Closure law for the modulo... |
| zmodcld 14032 | Closure law for the modulo... |
| zmodfz 14033 | An integer mod ` B ` lies ... |
| zmodfzo 14034 | An integer mod ` B ` lies ... |
| zmodfzp1 14035 | An integer mod ` B ` lies ... |
| modid 14036 | Identity law for modulo. ... |
| modid0 14037 | A positive real number mod... |
| modid2 14038 | Identity law for modulo. ... |
| zmodid2 14039 | Identity law for modulo re... |
| zmodidfzo 14040 | Identity law for modulo re... |
| zmodidfzoimp 14041 | Identity law for modulo re... |
| 0mod 14042 | Special case: 0 modulo a p... |
| 1mod 14043 | Special case: 1 modulo a r... |
| modabs 14044 | Absorption law for modulo.... |
| modabs2 14045 | Absorption law for modulo.... |
| modcyc 14046 | The modulo operation is pe... |
| modcyc2 14047 | The modulo operation is pe... |
| modadd1 14048 | Addition property of the m... |
| modaddb 14049 | Addition property of the m... |
| modaddid 14050 | The sums of two nonnegativ... |
| modaddabs 14051 | Absorption law for modulo.... |
| modaddmod 14052 | The sum of a real number m... |
| muladdmodid 14053 | The sum of a positive real... |
| mulp1mod1 14054 | The product of an integer ... |
| muladdmod 14055 | A real number is the sum o... |
| modmuladd 14056 | Decomposition of an intege... |
| modmuladdim 14057 | Implication of a decomposi... |
| modmuladdnn0 14058 | Implication of a decomposi... |
| negmod 14059 | The negation of a number m... |
| m1modnnsub1 14060 | Minus one modulo a positiv... |
| m1modge3gt1 14061 | Minus one modulo an intege... |
| addmodid 14062 | The sum of a positive inte... |
| addmodidr 14063 | The sum of a positive inte... |
| modadd2mod 14064 | The sum of a real number m... |
| modm1p1mod0 14065 | If a real number modulo a ... |
| modltm1p1mod 14066 | If a real number modulo a ... |
| modmul1 14067 | Multiplication property of... |
| modmul12d 14068 | Multiplication property of... |
| modnegd 14069 | Negation property of the m... |
| modadd12d 14070 | Additive property of the m... |
| modsub12d 14071 | Subtraction property of th... |
| modsubmod 14072 | The difference of a real n... |
| modsubmodmod 14073 | The difference of a real n... |
| 2txmodxeq0 14074 | Two times a positive real ... |
| 2submod 14075 | If a real number is betwee... |
| modifeq2int 14076 | If a nonnegative integer i... |
| modaddmodup 14077 | The sum of an integer modu... |
| modaddmodlo 14078 | The sum of an integer modu... |
| modmulmod 14079 | The product of a real numb... |
| modmulmodr 14080 | The product of an integer ... |
| modaddmulmod 14081 | The sum of a real number a... |
| moddi 14082 | Distribute multiplication ... |
| modsubdir 14083 | Distribute the modulo oper... |
| modeqmodmin 14084 | A real number equals the d... |
| modirr 14085 | A number modulo an irratio... |
| modfzo0difsn 14086 | For a number within a half... |
| modsumfzodifsn 14087 | The sum of a number within... |
| modlteq 14088 | Two nonnegative integers l... |
| addmodlteq 14089 | Two nonnegative integers l... |
| om2uz0i 14090 | The mapping ` G ` is a one... |
| om2uzsuci 14091 | The value of ` G ` (see ~ ... |
| om2uzuzi 14092 | The value ` G ` (see ~ om2... |
| om2uzlti 14093 | Less-than relation for ` G... |
| om2uzlt2i 14094 | The mapping ` G ` (see ~ o... |
| om2uzrani 14095 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 14096 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 14097 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 14098 | An alternative definition ... |
| om2uzrdg 14099 | A helper lemma for the val... |
| uzrdglem 14100 | A helper lemma for the val... |
| uzrdgfni 14101 | The recursive definition g... |
| uzrdg0i 14102 | Initial value of a recursi... |
| uzrdgsuci 14103 | Successor value of a recur... |
| ltweuz 14104 | ` < ` is a well-founded re... |
| ltwenn 14105 | Less than well-orders the ... |
| ltwefz 14106 | Less than well-orders a se... |
| uzenom 14107 | An upper integer set is de... |
| uzinf 14108 | An upper integer set is in... |
| nnnfi 14109 | The set of positive intege... |
| uzrdgxfr 14110 | Transfer the value of the ... |
| fzennn 14111 | The cardinality of a finit... |
| fzen2 14112 | The cardinality of a finit... |
| cardfz 14113 | The cardinality of a finit... |
| hashgf1o 14114 | ` G ` maps ` _om ` one-to-... |
| fzfi 14115 | A finite interval of integ... |
| fzfid 14116 | Commonly used special case... |
| fzofi 14117 | Half-open integer sets are... |
| fsequb 14118 | The values of a finite rea... |
| fsequb2 14119 | The values of a finite rea... |
| fseqsupcl 14120 | The values of a finite rea... |
| fseqsupubi 14121 | The values of a finite rea... |
| nn0ennn 14122 | The nonnegative integers a... |
| nnenom 14123 | The set of positive intege... |
| nnct 14124 | ` NN ` is countable. (Con... |
| uzindi 14125 | Indirect strong induction ... |
| axdc4uzlem 14126 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 14127 | A version of ~ axdc4 that ... |
| ssnn0fi 14128 | A subset of the nonnegativ... |
| rabssnn0fi 14129 | A subset of the nonnegativ... |
| uzsinds 14130 | Strong (or "total") induct... |
| nnsinds 14131 | Strong (or "total") induct... |
| nn0sinds 14132 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 14133 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 14134 | If all functions of a fini... |
| fsuppmapnn0fiubex 14135 | If all functions of a fini... |
| fsuppmapnn0fiub0 14136 | If all functions of a fini... |
| suppssfz 14137 | Condition for a function o... |
| fsuppmapnn0ub 14138 | If a function over the non... |
| fsuppmapnn0fz 14139 | If a function over the non... |
| mptnn0fsupp 14140 | A mapping from the nonnega... |
| mptnn0fsuppd 14141 | A mapping from the nonnega... |
| mptnn0fsuppr 14142 | A finitely supported mappi... |
| f13idfv 14143 | A one-to-one function with... |
| seqex 14146 | Existence of the sequence ... |
| seqeq1 14147 | Equality theorem for the s... |
| seqeq2 14148 | Equality theorem for the s... |
| seqeq3 14149 | Equality theorem for the s... |
| seqeq1d 14150 | Equality deduction for the... |
| seqeq2d 14151 | Equality deduction for the... |
| seqeq3d 14152 | Equality deduction for the... |
| seqeq123d 14153 | Equality deduction for the... |
| nfseq 14154 | Hypothesis builder for the... |
| seqval 14155 | Value of the sequence buil... |
| seqfn 14156 | The sequence builder funct... |
| seq1 14157 | Value of the sequence buil... |
| seq1i 14158 | Value of the sequence buil... |
| seqp1 14159 | Value of the sequence buil... |
| seqexw 14160 | Weak version of ~ seqex th... |
| seqp1d 14161 | Value of the sequence buil... |
| seqm1 14162 | Value of the sequence buil... |
| seqcl2 14163 | Closure properties of the ... |
| seqf2 14164 | Range of the recursive seq... |
| seqcl 14165 | Closure properties of the ... |
| seqf 14166 | Range of the recursive seq... |
| seqfveq2 14167 | Equality of sequences. (C... |
| seqfeq2 14168 | Equality of sequences. (C... |
| seqfveq 14169 | Equality of sequences. (C... |
| seqfeq 14170 | Equality of sequences. (C... |
| seqshft2 14171 | Shifting the index set of ... |
| seqres 14172 | Restricting its characteri... |
| serf 14173 | An infinite series of comp... |
| serfre 14174 | An infinite series of real... |
| monoord 14175 | Ordering relation for a mo... |
| monoord2 14176 | Ordering relation for a mo... |
| sermono 14177 | The partial sums in an inf... |
| seqsplit 14178 | Split a sequence into two ... |
| seq1p 14179 | Removing the first term fr... |
| seqcaopr3 14180 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14181 | The sum of two infinite se... |
| seqcaopr 14182 | The sum of two infinite se... |
| seqf1olem2a 14183 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14184 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14185 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14186 | Rearrange a sum via an arb... |
| seradd 14187 | The sum of two infinite se... |
| sersub 14188 | The difference of two infi... |
| seqid3 14189 | A sequence that consists e... |
| seqid 14190 | Discarding the first few t... |
| seqid2 14191 | The last few partial sums ... |
| seqhomo 14192 | Apply a homomorphism to a ... |
| seqz 14193 | If the operation ` .+ ` ha... |
| seqfeq4 14194 | Equality of series under d... |
| seqfeq3 14195 | Equality of series under d... |
| seqdistr 14196 | The distributive property ... |
| ser0 14197 | The value of the partial s... |
| ser0f 14198 | A zero-valued infinite ser... |
| serge0 14199 | A finite sum of nonnegativ... |
| serle 14200 | Comparison of partial sums... |
| ser1const 14201 | Value of the partial serie... |
| seqof 14202 | Distribute function operat... |
| seqof2 14203 | Distribute function operat... |
| expval 14206 | Value of exponentiation to... |
| expnnval 14207 | Value of exponentiation to... |
| exp0 14208 | Value of a complex number ... |
| 0exp0e1 14209 | The zeroth power of zero e... |
| exp1 14210 | Value of a complex number ... |
| expp1 14211 | Value of a complex number ... |
| expneg 14212 | Value of a complex number ... |
| expneg2 14213 | Value of a complex number ... |
| expn1 14214 | A complex number raised to... |
| expcllem 14215 | Lemma for proving nonnegat... |
| expcl2lem 14216 | Lemma for proving integer ... |
| nnexpcl 14217 | Closure of exponentiation ... |
| nn0expcl 14218 | Closure of exponentiation ... |
| zexpcl 14219 | Closure of exponentiation ... |
| qexpcl 14220 | Closure of exponentiation ... |
| reexpcl 14221 | Closure of exponentiation ... |
| expcl 14222 | Closure law for nonnegativ... |
| rpexpcl 14223 | Closure law for integer ex... |
| qexpclz 14224 | Closure of integer exponen... |
| reexpclz 14225 | Closure of integer exponen... |
| expclzlem 14226 | Lemma for ~ expclz . (Con... |
| expclz 14227 | Closure law for integer ex... |
| m1expcl2 14228 | Closure of integer exponen... |
| m1expcl 14229 | Closure of exponentiation ... |
| zexpcld 14230 | Closure of exponentiation ... |
| nn0expcli 14231 | Closure of exponentiation ... |
| nn0sqcl 14232 | The square of a nonnegativ... |
| expm1t 14233 | Exponentiation in terms of... |
| 1exp 14234 | Value of 1 raised to an in... |
| expeq0 14235 | A positive integer power i... |
| expne0 14236 | A positive integer power i... |
| expne0i 14237 | An integer power is nonzer... |
| expgt0 14238 | A positive real raised to ... |
| expnegz 14239 | Value of a nonzero complex... |
| 0exp 14240 | Value of zero raised to a ... |
| expge0 14241 | A nonnegative real raised ... |
| expge1 14242 | A real greater than or equ... |
| expgt1 14243 | A real greater than 1 rais... |
| mulexp 14244 | Nonnegative integer expone... |
| mulexpz 14245 | Integer exponentiation of ... |
| exprec 14246 | Integer exponentiation of ... |
| expadd 14247 | Sum of exponents law for n... |
| expaddzlem 14248 | Lemma for ~ expaddz . (Co... |
| expaddz 14249 | Sum of exponents law for i... |
| expmul 14250 | Product of exponents law f... |
| expmulz 14251 | Product of exponents law f... |
| m1expeven 14252 | Exponentiation of negative... |
| expsub 14253 | Exponent subtraction law f... |
| expp1z 14254 | Value of a nonzero complex... |
| expm1 14255 | Value of a nonzero complex... |
| expdiv 14256 | Nonnegative integer expone... |
| sqval 14257 | Value of the square of a c... |
| sqneg 14258 | The square of the negative... |
| sqnegd 14259 | The square of the negative... |
| sqsubswap 14260 | Swap the order of subtract... |
| sqcl 14261 | Closure of square. (Contr... |
| sqmul 14262 | Distribution of squaring o... |
| sqeq0 14263 | A complex number is zero i... |
| sqdiv 14264 | Distribution of squaring o... |
| sqdivid 14265 | The square of a nonzero co... |
| sqne0 14266 | A complex number is nonzer... |
| resqcl 14267 | Closure of squaring in rea... |
| resqcld 14268 | Closure of squaring in rea... |
| sqgt0 14269 | The square of a nonzero re... |
| sqn0rp 14270 | The square of a nonzero re... |
| nnsqcl 14271 | The positive naturals are ... |
| zsqcl 14272 | Integers are closed under ... |
| qsqcl 14273 | The square of a rational i... |
| sq11 14274 | The square function is one... |
| nn0sq11 14275 | The square function is one... |
| lt2sq 14276 | The square function is inc... |
| le2sq 14277 | The square function is non... |
| le2sq2 14278 | The square function is non... |
| sqge0 14279 | The square of a real is no... |
| sqge0d 14280 | The square of a real is no... |
| zsqcl2 14281 | The square of an integer i... |
| 0expd 14282 | Value of zero raised to a ... |
| exp0d 14283 | Value of a complex number ... |
| exp1d 14284 | Value of a complex number ... |
| expeq0d 14285 | If a positive integer powe... |
| sqvald 14286 | Value of square. Inferenc... |
| sqcld 14287 | Closure of square. (Contr... |
| sqeq0d 14288 | A number is zero iff its s... |
| expcld 14289 | Closure law for nonnegativ... |
| expp1d 14290 | Value of a complex number ... |
| expaddd 14291 | Sum of exponents law for n... |
| expmuld 14292 | Product of exponents law f... |
| sqrecd 14293 | Square of reciprocal is re... |
| expclzd 14294 | Closure law for integer ex... |
| expne0d 14295 | A nonnegative integer powe... |
| expnegd 14296 | Value of a nonzero complex... |
| exprecd 14297 | An integer power of a reci... |
| expp1zd 14298 | Value of a nonzero complex... |
| expm1d 14299 | Value of a nonzero complex... |
| expsubd 14300 | Exponent subtraction law f... |
| sqmuld 14301 | Distribution of squaring o... |
| sqdivd 14302 | Distribution of squaring o... |
| expdivd 14303 | Nonnegative integer expone... |
| mulexpd 14304 | Nonnegative integer expone... |
| znsqcld 14305 | The square of a nonzero in... |
| reexpcld 14306 | Closure of exponentiation ... |
| expge0d 14307 | A nonnegative real raised ... |
| expge1d 14308 | A real greater than or equ... |
| ltexp2a 14309 | Exponent ordering relation... |
| expmordi 14310 | Base ordering relationship... |
| rpexpmord 14311 | Base ordering relationship... |
| expcan 14312 | Cancellation law for integ... |
| ltexp2 14313 | Strict ordering law for ex... |
| leexp2 14314 | Ordering law for exponenti... |
| leexp2a 14315 | Weak ordering relationship... |
| ltexp2r 14316 | The integer powers of a fi... |
| leexp2r 14317 | Weak ordering relationship... |
| leexp1a 14318 | Weak base ordering relatio... |
| leexp1ad 14319 | Weak base ordering relatio... |
| exple1 14320 | A real between 0 and 1 inc... |
| expubnd 14321 | An upper bound on ` A ^ N ... |
| sumsqeq0 14322 | The sum of two squres of r... |
| sqvali 14323 | Value of square. Inferenc... |
| sqcli 14324 | Closure of square. (Contr... |
| sqeq0i 14325 | A complex number is zero i... |
| sqrecii 14326 | The square of a reciprocal... |
| sqmuli 14327 | Distribution of squaring o... |
| sqdivi 14328 | Distribution of squaring o... |
| resqcli 14329 | Closure of square in reals... |
| sqgt0i 14330 | The square of a nonzero re... |
| sqge0i 14331 | The square of a real is no... |
| lt2sqi 14332 | The square function on non... |
| le2sqi 14333 | The square function on non... |
| sq11i 14334 | The square function is one... |
| sq0 14335 | The square of 0 is 0. (Co... |
| sq0i 14336 | If a number is zero, then ... |
| sq0id 14337 | If a number is zero, then ... |
| sq1 14338 | The square of 1 is 1. (Co... |
| neg1sqe1 14339 | The square of ` -u 1 ` is ... |
| sq2 14340 | The square of 2 is 4. (Co... |
| sq3 14341 | The square of 3 is 9. (Co... |
| sq4e2t8 14342 | The square of 4 is 2 times... |
| cu2 14343 | The cube of 2 is 8. (Cont... |
| exp4sqsq 14344 | Raising a number to the fo... |
| irec 14345 | The reciprocal of ` _i ` .... |
| i2 14346 | ` _i ` squared. (Contribu... |
| i3 14347 | ` _i ` cubed. (Contribute... |
| i4 14348 | ` _i ` to the fourth power... |
| nnlesq 14349 | A positive integer is less... |
| zzlesq 14350 | An integer is less than or... |
| iexpcyc 14351 | Taking ` _i ` to the ` K `... |
| expnass 14352 | A counterexample showing t... |
| sqlecan 14353 | Cancel one factor of a squ... |
| subsq 14354 | Factor the difference of t... |
| subsq2 14355 | Express the difference of ... |
| binom2i 14356 | The square of a binomial. ... |
| subsqi 14357 | Factor the difference of t... |
| sqeqori 14358 | The squares of two complex... |
| subsq0i 14359 | The two solutions to the d... |
| sqeqor 14360 | The squares of two complex... |
| binom2 14361 | The square of a binomial. ... |
| binom2d 14362 | Deduction form of ~ binom2... |
| binom21 14363 | Special case of ~ binom2 w... |
| binom2sub 14364 | Expand the square of a sub... |
| binom2sub1 14365 | Special case of ~ binom2su... |
| binom2subi 14366 | Expand the square of a sub... |
| mulbinom2 14367 | The square of a binomial w... |
| binom3 14368 | The cube of a binomial. (... |
| sq01 14369 | If a complex number equals... |
| zesq 14370 | An integer is even iff its... |
| nnesq 14371 | A positive integer is even... |
| crreczi 14372 | Reciprocal of a complex nu... |
| bernneq 14373 | Bernoulli's inequality, du... |
| bernneq2 14374 | Variation of Bernoulli's i... |
| bernneq3 14375 | A corollary of ~ bernneq .... |
| expnbnd 14376 | Exponentiation with a base... |
| expnlbnd 14377 | The reciprocal of exponent... |
| expnlbnd2 14378 | The reciprocal of exponent... |
| expmulnbnd 14379 | Exponentiation with a base... |
| digit2 14380 | Two ways to express the ` ... |
| digit1 14381 | Two ways to express the ` ... |
| modexp 14382 | Exponentiation property of... |
| discr1 14383 | A nonnegative quadratic fo... |
| discr 14384 | If a quadratic polynomial ... |
| expnngt1 14385 | If an integer power with a... |
| expnngt1b 14386 | An integer power with an i... |
| sqoddm1div8 14387 | A squared odd number minus... |
| nnsqcld 14388 | The naturals are closed un... |
| nnexpcld 14389 | Closure of exponentiation ... |
| nn0expcld 14390 | Closure of exponentiation ... |
| rpexpcld 14391 | Closure law for exponentia... |
| ltexp2rd 14392 | The power of a positive nu... |
| reexpclzd 14393 | Closure of exponentiation ... |
| sqgt0d 14394 | The square of a nonzero re... |
| ltexp2d 14395 | Ordering relationship for ... |
| leexp2d 14396 | Ordering law for exponenti... |
| expcand 14397 | Ordering relationship for ... |
| leexp2ad 14398 | Ordering relationship for ... |
| leexp2rd 14399 | Ordering relationship for ... |
| lt2sqd 14400 | The square function on non... |
| le2sqd 14401 | The square function on non... |
| sq11d 14402 | The square function is one... |
| ltexp1d 14403 | Elevating to a positive po... |
| ltexp1dd 14404 | Raising both sides of 'les... |
| exp11nnd 14405 | The function elevating non... |
| mulsubdivbinom2 14406 | The square of a binomial w... |
| muldivbinom2 14407 | The square of a binomial w... |
| sq10 14408 | The square of 10 is 100. ... |
| sq10e99m1 14409 | The square of 10 is 99 plu... |
| 3dec 14410 | A "decimal constructor" wh... |
| nn0le2msqi 14411 | The square function on non... |
| nn0opthlem1 14412 | A rather pretty lemma for ... |
| nn0opthlem2 14413 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14414 | An ordered pair theorem fo... |
| nn0opth2i 14415 | An ordered pair theorem fo... |
| nn0opth2 14416 | An ordered pair theorem fo... |
| facnn 14419 | Value of the factorial fun... |
| fac0 14420 | The factorial of 0. (Cont... |
| fac1 14421 | The factorial of 1. (Cont... |
| facp1 14422 | The factorial of a success... |
| fac2 14423 | The factorial of 2. (Cont... |
| fac3 14424 | The factorial of 3. (Cont... |
| fac4 14425 | The factorial of 4. (Cont... |
| facnn2 14426 | Value of the factorial fun... |
| faccl 14427 | Closure of the factorial f... |
| faccld 14428 | Closure of the factorial f... |
| facmapnn 14429 | The factorial function res... |
| facne0 14430 | The factorial function is ... |
| facdiv 14431 | A positive integer divides... |
| facndiv 14432 | No positive integer (great... |
| facwordi 14433 | Ordering property of facto... |
| faclbnd 14434 | A lower bound for the fact... |
| faclbnd2 14435 | A lower bound for the fact... |
| faclbnd3 14436 | A lower bound for the fact... |
| faclbnd4lem1 14437 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14438 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14439 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14440 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14441 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14442 | The factorial function gro... |
| faclbnd6 14443 | Geometric lower bound for ... |
| facubnd 14444 | An upper bound for the fac... |
| facavg 14445 | The product of two factori... |
| bcval 14448 | Value of the binomial coef... |
| bcval2 14449 | Value of the binomial coef... |
| bcval3 14450 | Value of the binomial coef... |
| bcval4 14451 | Value of the binomial coef... |
| bcrpcl 14452 | Closure of the binomial co... |
| bccmpl 14453 | "Complementing" its second... |
| bcn0 14454 | ` N ` choose 0 is 1. Rema... |
| bc0k 14455 | The binomial coefficient "... |
| bcnn 14456 | ` N ` choose ` N ` is 1. ... |
| bcn1 14457 | Binomial coefficient: ` N ... |
| bcnp1n 14458 | Binomial coefficient: ` N ... |
| bcm1k 14459 | The proportion of one bino... |
| bcp1n 14460 | The proportion of one bino... |
| bcp1nk 14461 | The proportion of one bino... |
| bcval5 14462 | Write out the top and bott... |
| bcn2 14463 | Binomial coefficient: ` N ... |
| bcp1m1 14464 | Compute the binomial coeff... |
| bcpasc 14465 | Pascal's rule for the bino... |
| bccl 14466 | A binomial coefficient, in... |
| bccl2 14467 | A binomial coefficient, in... |
| bcn2m1 14468 | Compute the binomial coeff... |
| bcn2p1 14469 | Compute the binomial coeff... |
| permnn 14470 | The number of permutations... |
| bcnm1 14471 | The binomial coefficient o... |
| 4bc3eq4 14472 | The value of four choose t... |
| 4bc2eq6 14473 | The value of four choose t... |
| hashkf 14476 | The finite part of the siz... |
| hashgval 14477 | The value of the ` # ` fun... |
| hashginv 14478 | The converse of ` G ` maps... |
| hashinf 14479 | The value of the ` # ` fun... |
| hashbnd 14480 | If ` A ` has size bounded ... |
| hashfxnn0 14481 | The size function is a fun... |
| hashf 14482 | The size function maps all... |
| hashxnn0 14483 | The value of the hash func... |
| hashresfn 14484 | Restriction of the domain ... |
| dmhashres 14485 | Restriction of the domain ... |
| hashnn0pnf 14486 | The value of the hash func... |
| hashnnn0genn0 14487 | If the size of a set is no... |
| hashnemnf 14488 | The size of a set is never... |
| hashv01gt1 14489 | The size of a set is eithe... |
| hashfz1 14490 | The set ` ( 1 ... N ) ` ha... |
| hashen 14491 | Two finite sets have the s... |
| hasheni 14492 | Equinumerous sets have the... |
| hasheqf1o 14493 | The size of two finite set... |
| fiinfnf1o 14494 | There is no bijection betw... |
| hasheqf1oi 14495 | The size of two sets is eq... |
| hashf1rn 14496 | The size of a finite set w... |
| hasheqf1od 14497 | The size of two sets is eq... |
| fz1eqb 14498 | Two possibly-empty 1-based... |
| hashcard 14499 | The size function of the c... |
| hashcl 14500 | Closure of the ` # ` funct... |
| hashxrcl 14501 | Extended real closure of t... |
| hashclb 14502 | Reverse closure of the ` #... |
| nfile 14503 | The size of any infinite s... |
| hashvnfin 14504 | A set of finite size is a ... |
| hashnfinnn0 14505 | The size of an infinite se... |
| isfinite4 14506 | A finite set is equinumero... |
| hasheq0 14507 | Two ways of saying a set i... |
| hashneq0 14508 | Two ways of saying a set i... |
| hashgt0n0 14509 | If the size of a set is gr... |
| hashnncl 14510 | Positive natural closure o... |
| hash0 14511 | The empty set has size zer... |
| hashelne0d 14512 | A set with an element has ... |
| hashsng 14513 | The size of a singleton. ... |
| hashen1 14514 | A set has size 1 if and on... |
| hash1elsn 14515 | A set of size 1 with a kno... |
| hashrabrsn 14516 | The size of a restricted c... |
| hashrabsn01 14517 | The size of a restricted c... |
| hashrabsn1 14518 | If the size of a restricte... |
| hashfn 14519 | A function is equinumerous... |
| fseq1hash 14520 | The value of the size func... |
| hashgadd 14521 | ` G ` maps ordinal additio... |
| hashgval2 14522 | A short expression for the... |
| hashdom 14523 | Dominance relation for the... |
| hashdomi 14524 | Non-strict order relation ... |
| hashsdom 14525 | Strict dominance relation ... |
| hashun 14526 | The size of the union of d... |
| hashun2 14527 | The size of the union of f... |
| hashun3 14528 | The size of the union of f... |
| hashinfxadd 14529 | The extended real addition... |
| hashunx 14530 | The size of the union of d... |
| hashge0 14531 | The cardinality of a set i... |
| hashgt0 14532 | The cardinality of a nonem... |
| hashge1 14533 | The cardinality of a nonem... |
| 1elfz0hash 14534 | 1 is an element of the fin... |
| hashnn0n0nn 14535 | If a nonnegative integer i... |
| hashunsng 14536 | The size of the union of a... |
| hashunsngx 14537 | The size of the union of a... |
| hashunsnggt 14538 | The size of a set is great... |
| hashprg 14539 | The size of an unordered p... |
| elprchashprn2 14540 | If one element of an unord... |
| hashprb 14541 | The size of an unordered p... |
| hashprdifel 14542 | The elements of an unorder... |
| prhash2ex 14543 | There is (at least) one se... |
| hashle00 14544 | If the size of a set is le... |
| hashgt0elex 14545 | If the size of a set is gr... |
| hashgt0elexb 14546 | The size of a set is great... |
| hashp1i 14547 | Size of a finite ordinal. ... |
| hash1 14548 | Size of a finite ordinal. ... |
| hash2 14549 | Size of a finite ordinal. ... |
| hash3 14550 | Size of a finite ordinal. ... |
| hash4 14551 | Size of a finite ordinal. ... |
| pr0hash2ex 14552 | There is (at least) one se... |
| hashss 14553 | The size of a subset is le... |
| hashpss 14554 | The size of a proper subse... |
| prsshashgt1 14555 | The size of a superset of ... |
| hashin 14556 | The size of the intersecti... |
| hashssdif 14557 | The size of the difference... |
| hashdif 14558 | The size of the difference... |
| hashdifsn 14559 | The size of the difference... |
| hashdifpr 14560 | The size of the difference... |
| hashsn01 14561 | The size of a singleton is... |
| hashsnle1 14562 | The size of a singleton is... |
| hashsnlei 14563 | Get an upper bound on a co... |
| hash1snb 14564 | The size of a set is 1 if ... |
| euhash1 14565 | The size of a set is 1 in ... |
| hash1n0 14566 | If the size of a set is 1 ... |
| hashgt12el 14567 | In a set with more than on... |
| hashgt12el2 14568 | In a set with more than on... |
| hashgt23el 14569 | A set with more than two e... |
| hashunlei 14570 | Get an upper bound on a co... |
| hashsslei 14571 | Get an upper bound on a co... |
| hashfz 14572 | Value of the numeric cardi... |
| fzsdom2 14573 | Condition for finite range... |
| hashfzo 14574 | Cardinality of a half-open... |
| hashfzo0 14575 | Cardinality of a half-open... |
| hashfzp1 14576 | Value of the numeric cardi... |
| hashfz0 14577 | Value of the numeric cardi... |
| hashxplem 14578 | Lemma for ~ hashxp . (Con... |
| hashxp 14579 | The size of the Cartesian ... |
| hashmap 14580 | The size of the set expone... |
| hashpw 14581 | The size of the power set ... |
| hashfun 14582 | A finite set is a function... |
| hashres 14583 | The number of elements of ... |
| hashreshashfun 14584 | The number of elements of ... |
| hashimarn 14585 | The size of the image of a... |
| hashimarni 14586 | If the size of the image o... |
| hashfundm 14587 | The size of a set function... |
| hashf1dmrn 14588 | The size of the domain of ... |
| hashf1dmcdm 14589 | The size of the domain of ... |
| resunimafz0 14590 | TODO-AV: Revise using ` F... |
| fnfz0hash 14591 | The size of a function on ... |
| ffz0hash 14592 | The size of a function on ... |
| fnfz0hashnn0 14593 | The size of a function on ... |
| ffzo0hash 14594 | The size of a function on ... |
| fnfzo0hash 14595 | The size of a function on ... |
| fnfzo0hashnn0 14596 | The value of the size func... |
| hashbclem 14597 | Lemma for ~ hashbc : induc... |
| hashbc 14598 | The binomial coefficient c... |
| hashfacen 14599 | The number of bijections b... |
| hashf1lem1 14600 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14601 | Lemma for ~ hashf1 . (Con... |
| hashf1 14602 | The permutation number ` |... |
| hashfac 14603 | A factorial counts the num... |
| leiso 14604 | Two ways to write a strict... |
| leisorel 14605 | Version of ~ isorel for st... |
| fz1isolem 14606 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14607 | Any finite ordered set has... |
| ishashinf 14608 | Any set that is not finite... |
| seqcoll 14609 | The function ` F ` contain... |
| seqcoll2 14610 | The function ` F ` contain... |
| phphashd 14611 | Corollary of the Pigeonhol... |
| phphashrd 14612 | Corollary of the Pigeonhol... |
| hashprlei 14613 | An unordered pair has at m... |
| hash2pr 14614 | A set of size two is an un... |
| hash2prde 14615 | A set of size two is an un... |
| hash2exprb 14616 | A set of size two is an un... |
| hash2prb 14617 | A set of size two is a pro... |
| prprrab 14618 | The set of proper pairs of... |
| nehash2 14619 | The cardinality of a set w... |
| hash2prd 14620 | A set of size two is an un... |
| hash2pwpr 14621 | If the size of a subset of... |
| hashle2pr 14622 | A nonempty set of size les... |
| hashle2prv 14623 | A nonempty subset of a pow... |
| pr2pwpr 14624 | The set of subsets of a pa... |
| hashge2el2dif 14625 | A set with size at least 2... |
| hashge2el2difr 14626 | A set with at least 2 diff... |
| hashge2el2difb 14627 | A set has size at least 2 ... |
| hashdmpropge2 14628 | The size of the domain of ... |
| hashtplei 14629 | An unordered triple has at... |
| hashtpg 14630 | The size of an unordered t... |
| hash7g 14631 | The size of an unordered s... |
| hashge3el3dif 14632 | A set with size at least 3... |
| elss2prb 14633 | An element of the set of s... |
| hash2sspr 14634 | A subset of size two is an... |
| exprelprel 14635 | If there is an element of ... |
| hash3tr 14636 | A set of size three is an ... |
| hash1to3 14637 | If the size of a set is be... |
| hash3tpde 14638 | A set of size three is an ... |
| hash3tpexb 14639 | A set of size three is an ... |
| hash3tpb 14640 | A set of size three is a p... |
| tpf1ofv0 14641 | The value of a one-to-one ... |
| tpf1ofv1 14642 | The value of a one-to-one ... |
| tpf1ofv2 14643 | The value of a one-to-one ... |
| tpf 14644 | A function into a (proper)... |
| tpfo 14645 | A function onto a (proper)... |
| tpf1o 14646 | A bijection onto a (proper... |
| fundmge2nop0 14647 | A function with a domain c... |
| fundmge2nop 14648 | A function with a domain c... |
| fun2dmnop0 14649 | A function with a domain c... |
| fun2dmnop 14650 | A function with a domain c... |
| hashdifsnp1 14651 | If the size of a set is a ... |
| fi1uzind 14652 | Properties of an ordered p... |
| brfi1uzind 14653 | Properties of a binary rel... |
| brfi1ind 14654 | Properties of a binary rel... |
| brfi1indALT 14655 | Alternate proof of ~ brfi1... |
| opfi1uzind 14656 | Properties of an ordered p... |
| opfi1ind 14657 | Properties of an ordered p... |
| iswrd 14660 | Property of being a word o... |
| wrdval 14661 | Value of the set of words ... |
| iswrdi 14662 | A zero-based sequence is a... |
| wrdf 14663 | A word is a zero-based seq... |
| wrdfd 14664 | A word is a zero-based seq... |
| iswrdb 14665 | A word over an alphabet is... |
| wrddm 14666 | The indices of a word (i.e... |
| sswrd 14667 | The set of words respects ... |
| snopiswrd 14668 | A singleton of an ordered ... |
| wrdexg 14669 | The set of words over a se... |
| wrdexb 14670 | The set of words over a se... |
| wrdexi 14671 | The set of words over a se... |
| wrdsymbcl 14672 | A symbol within a word ove... |
| wrdfn 14673 | A word is a function with ... |
| wrdv 14674 | A word over an alphabet is... |
| wrdlndm 14675 | The length of a word is no... |
| iswrdsymb 14676 | An arbitrary word is a wor... |
| wrdfin 14677 | A word is a finite set. (... |
| lencl 14678 | The length of a word is a ... |
| lennncl 14679 | The length of a nonempty w... |
| wrdffz 14680 | A word is a function from ... |
| wrdeq 14681 | Equality theorem for the s... |
| wrdeqi 14682 | Equality theorem for the s... |
| iswrddm0 14683 | A function with empty doma... |
| wrd0 14684 | The empty set is a word (t... |
| 0wrd0 14685 | The empty word is the only... |
| ffz0iswrd 14686 | A sequence with zero-based... |
| wrdsymb 14687 | A word is a word over the ... |
| nfwrd 14688 | Hypothesis builder for ` W... |
| csbwrdg 14689 | Class substitution for the... |
| wrdnval 14690 | Words of a fixed length ar... |
| wrdmap 14691 | Words as a mapping. (Cont... |
| hashwrdn 14692 | If there is only a finite ... |
| wrdnfi 14693 | If there is only a finite ... |
| wrdsymb0 14694 | A symbol at a position "ou... |
| wrdlenge1n0 14695 | A word with length at leas... |
| len0nnbi 14696 | The length of a word is a ... |
| wrdlenge2n0 14697 | A word with length at leas... |
| wrdsymb1 14698 | The first symbol of a none... |
| wrdlen1 14699 | A word of length 1 starts ... |
| fstwrdne 14700 | The first symbol of a none... |
| fstwrdne0 14701 | The first symbol of a none... |
| eqwrd 14702 | Two words are equal iff th... |
| elovmpowrd 14703 | Implications for the value... |
| elovmptnn0wrd 14704 | Implications for the value... |
| wrdred1 14705 | A word truncated by a symb... |
| wrdred1hash 14706 | The length of a word trunc... |
| lsw 14709 | Extract the last symbol of... |
| lsw0 14710 | The last symbol of an empt... |
| lsw0g 14711 | The last symbol of an empt... |
| lsw1 14712 | The last symbol of a word ... |
| lswcl 14713 | Closure of the last symbol... |
| lswlgt0cl 14714 | The last symbol of a nonem... |
| ccatfn 14717 | The concatenation operator... |
| ccatfval 14718 | Value of the concatenation... |
| ccatcl 14719 | The concatenation of two w... |
| ccatlen 14720 | The length of a concatenat... |
| ccat0 14721 | The concatenation of two w... |
| ccatval1 14722 | Value of a symbol in the l... |
| ccatval2 14723 | Value of a symbol in the r... |
| ccatval3 14724 | Value of a symbol in the r... |
| elfzelfzccat 14725 | An element of a finite set... |
| ccatvalfn 14726 | The concatenation of two w... |
| ccatdmss 14727 | The domain of a concatenat... |
| ccatsymb 14728 | The symbol at a given posi... |
| ccatfv0 14729 | The first symbol of a conc... |
| ccatval1lsw 14730 | The last symbol of the lef... |
| ccatval21sw 14731 | The first symbol of the ri... |
| ccatlid 14732 | Concatenation of a word by... |
| ccatrid 14733 | Concatenation of a word by... |
| ccatass 14734 | Associative law for concat... |
| ccatrn 14735 | The range of a concatenate... |
| ccatf1 14736 | Conditions for a concatena... |
| ccatidid 14737 | Concatenation of the empty... |
| lswccatn0lsw 14738 | The last symbol of a word ... |
| lswccat0lsw 14739 | The last symbol of a word ... |
| ccatalpha 14740 | A concatenation of two arb... |
| ccatrcl1 14741 | Reverse closure of a conca... |
| ids1 14744 | Identity function protecti... |
| s1val 14745 | Value of a singleton word.... |
| s1rn 14746 | The range of a singleton w... |
| s1eq 14747 | Equality theorem for a sin... |
| s1eqd 14748 | Equality theorem for a sin... |
| s1cl 14749 | A singleton word is a word... |
| s1cld 14750 | A singleton word is a word... |
| s1prc 14751 | Value of a singleton word ... |
| s1cli 14752 | A singleton word is a word... |
| s1len 14753 | Length of a singleton word... |
| s1nz 14754 | A singleton word is not th... |
| s1dm 14755 | The domain of a singleton ... |
| s1f1 14756 | Conditions for a length 1 ... |
| s1dmALT 14757 | Alternate version of ~ s1d... |
| s1fv 14758 | Sole symbol of a singleton... |
| lsws1 14759 | The last symbol of a singl... |
| eqs1 14760 | A word of length 1 is a si... |
| wrdl1exs1 14761 | A word of length 1 is a si... |
| wrdl1s1 14762 | A word of length 1 is a si... |
| s111 14763 | The singleton word functio... |
| ccatws1cl 14764 | The concatenation of a wor... |
| ccatws1clv 14765 | The concatenation of a wor... |
| ccat2s1cl 14766 | The concatenation of two s... |
| ccats1alpha 14767 | A concatenation of a word ... |
| ccatws1len 14768 | The length of the concaten... |
| ccatws1lenp1b 14769 | The length of a word is ` ... |
| wrdlenccats1lenm1 14770 | The length of a word is th... |
| ccat2s1len 14771 | The length of the concaten... |
| ccatw2s1cl 14772 | The concatenation of a wor... |
| ccatw2s1len 14773 | The length of the concaten... |
| ccats1val1 14774 | Value of a symbol in the l... |
| ccats1val2 14775 | Value of the symbol concat... |
| ccat1st1st 14776 | The first symbol of a word... |
| ccat2s1p1 14777 | Extract the first of two c... |
| ccat2s1p2 14778 | Extract the second of two ... |
| ccatw2s1ass 14779 | Associative law for a conc... |
| ccatws1n0 14780 | The concatenation of a wor... |
| ccatws1ls 14781 | The last symbol of the con... |
| lswccats1 14782 | The last symbol of a word ... |
| lswccats1fst 14783 | The last symbol of a nonem... |
| ccatw2s1p1 14784 | Extract the symbol of the ... |
| ccatw2s1p2 14785 | Extract the second of two ... |
| ccat2s1fvw 14786 | Extract a symbol of a word... |
| ccat2s1fst 14787 | The first symbol of the co... |
| swrdnznd 14790 | The value of a subword ope... |
| swrdval 14791 | Value of a subword. (Cont... |
| swrd00 14792 | A zero length substring. ... |
| swrdcl 14793 | Closure of the subword ext... |
| swrdval2 14794 | Value of the subword extra... |
| swrdlen 14795 | Length of an extracted sub... |
| swrdfv 14796 | A symbol in an extracted s... |
| swrdfv0 14797 | The first symbol in an ext... |
| swrdf 14798 | A subword of a word is a f... |
| swrdf1 14799 | Condition for a subword to... |
| swrdvalfn 14800 | Value of the subword extra... |
| swrdrn 14801 | The range of a subword of ... |
| swrdrn3 14802 | Express the range of a sub... |
| swrdlend 14803 | The value of the subword e... |
| swrdnd 14804 | The value of the subword e... |
| swrdnd2 14805 | Value of the subword extra... |
| swrdnnn0nd 14806 | The value of a subword ope... |
| swrdnd0 14807 | The value of a subword ope... |
| swrd0 14808 | A subword of an empty set ... |
| swrdrlen 14809 | Length of a right-anchored... |
| swrdlen2 14810 | Length of an extracted sub... |
| swrdfv2 14811 | A symbol in an extracted s... |
| swrdwrdsymb 14812 | A subword is a word over t... |
| swrdsb0eq 14813 | Two subwords with the same... |
| swrdsbslen 14814 | Two subwords with the same... |
| swrdspsleq 14815 | Two words have a common su... |
| swrds1 14816 | Extract a single symbol fr... |
| swrdlsw 14817 | Extract the last single sy... |
| ccatswrd 14818 | Joining two adjacent subwo... |
| swrdccat2 14819 | Recover the right half of ... |
| pfxnndmnd 14822 | The value of a prefix oper... |
| pfxval 14823 | Value of a prefix operatio... |
| pfx00 14824 | The zero length prefix is ... |
| pfx0 14825 | A prefix of an empty set i... |
| pfxval0 14826 | Value of a prefix operatio... |
| pfxcl 14827 | Closure of the prefix extr... |
| pfxmpt 14828 | Value of the prefix extrac... |
| pfxres 14829 | Value of the prefix extrac... |
| pfxf 14830 | A prefix of a word is a fu... |
| pfxfn 14831 | Value of the prefix extrac... |
| pfxfv 14832 | A symbol in a prefix of a ... |
| pfxlen 14833 | Length of a prefix. (Cont... |
| pfxid 14834 | A word is a prefix of itse... |
| pfxrn 14835 | The range of a prefix of a... |
| pfxn0 14836 | A prefix consisting of at ... |
| pfxnd 14837 | The value of a prefix oper... |
| pfxnd0 14838 | The value of a prefix oper... |
| pfxwrdsymb 14839 | A prefix of a word is a wo... |
| addlenpfx 14840 | The sum of the lengths of ... |
| pfxfv0 14841 | The first symbol of a pref... |
| pfxtrcfv 14842 | A symbol in a word truncat... |
| pfxtrcfv0 14843 | The first symbol in a word... |
| pfxfvlsw 14844 | The last symbol in a nonem... |
| pfxeq 14845 | The prefixes of two words ... |
| pfxtrcfvl 14846 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14847 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14848 | Two (nonempty) words are e... |
| disjwrdpfx 14849 | Sets of words are disjoint... |
| ccatpfx 14850 | Concatenating a prefix wit... |
| pfxccat1 14851 | Recover the left half of a... |
| pfx1 14852 | The prefix of length one o... |
| swrdswrdlem 14853 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14854 | A subword of a subword is ... |
| pfxswrd 14855 | A prefix of a subword is a... |
| swrdpfx 14856 | A subword of a prefix is a... |
| pfxpfx 14857 | A prefix of a prefix is a ... |
| pfxpfxid 14858 | A prefix of a prefix with ... |
| pfxcctswrd 14859 | The concatenation of the p... |
| lenpfxcctswrd 14860 | The length of the concaten... |
| lenrevpfxcctswrd 14861 | The length of the concaten... |
| pfxlswccat 14862 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14863 | The last symbol of a word ... |
| ccats1pfxeqrex 14864 | There exists a symbol such... |
| ccatopth 14865 | An ~ opth -like theorem fo... |
| ccatopth2 14866 | An ~ opth -like theorem fo... |
| ccatlcan 14867 | Concatenation of words is ... |
| ccatrcan 14868 | Concatenation of words is ... |
| wrdeqs1cat 14869 | Decompose a nonempty word ... |
| cats1un 14870 | Express a word with an ext... |
| wrdind 14871 | Perform induction over the... |
| wrd2ind 14872 | Perform induction over the... |
| swrdccatfn 14873 | The subword of a concatena... |
| swrdccatin1 14874 | The subword of a concatena... |
| pfxccatin12lem4 14875 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14876 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14877 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14878 | The subword of a concatena... |
| pfxccatin12lem2c 14879 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14880 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14881 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14882 | The subword of a concatena... |
| pfxccat3 14883 | The subword of a concatena... |
| swrdccat 14884 | The subword of a concatena... |
| pfxccatpfx1 14885 | A prefix of a concatenatio... |
| pfxccatpfx2 14886 | A prefix of a concatenatio... |
| pfxccat3a 14887 | A prefix of a concatenatio... |
| swrdccat3blem 14888 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14889 | A suffix of a concatenatio... |
| pfxccatid 14890 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14891 | A word is a prefix of a wo... |
| swrdccatin1d 14892 | The subword of a concatena... |
| swrdccatin2d 14893 | The subword of a concatena... |
| pfxccatin12d 14894 | The subword of a concatena... |
| reuccatpfxs1lem 14895 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14896 | There is a unique word hav... |
| reuccatpfxs1v 14897 | There is a unique word hav... |
| splval 14900 | Value of the substring rep... |
| splcl 14901 | Closure of the substring r... |
| splid 14902 | Splicing a subword for the... |
| spllen 14903 | The length of a splice. (... |
| splfv1 14904 | Symbols to the left of a s... |
| splfv2a 14905 | Symbols within the replace... |
| splval2 14906 | Value of a splice, assumin... |
| revval 14909 | Value of the word reversin... |
| revcl 14910 | The reverse of a word is a... |
| revlen 14911 | The reverse of a word has ... |
| revfv 14912 | Reverse of a word at a poi... |
| rev0 14913 | The empty word is its own ... |
| revs1 14914 | Singleton words are their ... |
| revccat 14915 | Antiautomorphic property o... |
| revrev 14916 | Reversal is an involution ... |
| revpfxsfxrev 14917 | The reverse of a prefix of... |
| swrdrevpfx 14918 | A subword expressed in ter... |
| reps 14921 | Construct a function mappi... |
| repsundef 14922 | A function mapping a half-... |
| repsconst 14923 | Construct a function mappi... |
| repsf 14924 | The constructed function m... |
| repswsymb 14925 | The symbols of a "repeated... |
| repsw 14926 | A function mapping a half-... |
| repswlen 14927 | The length of a "repeated ... |
| repsw0 14928 | The "repeated symbol word"... |
| repsdf2 14929 | Alternative definition of ... |
| repswsymball 14930 | All the symbols of a "repe... |
| repswsymballbi 14931 | A word is a "repeated symb... |
| repswfsts 14932 | The first symbol of a none... |
| repswlsw 14933 | The last symbol of a nonem... |
| repsw1 14934 | The "repeated symbol word"... |
| repswswrd 14935 | A subword of a "repeated s... |
| repswpfx 14936 | A prefix of a repeated sym... |
| repswccat 14937 | The concatenation of two "... |
| repswrevw 14938 | The reverse of a "repeated... |
| cshfn 14941 | Perform a cyclical shift f... |
| cshword 14942 | Perform a cyclical shift f... |
| cshnz 14943 | A cyclical shift is the em... |
| 0csh0 14944 | Cyclically shifting an emp... |
| cshw0 14945 | A word cyclically shifted ... |
| cshwmodn 14946 | Cyclically shifting a word... |
| cshwsublen 14947 | Cyclically shifting a word... |
| cshwn 14948 | A word cyclically shifted ... |
| cshwcl 14949 | A cyclically shifted word ... |
| cshwlen 14950 | The length of a cyclically... |
| cshwf 14951 | A cyclically shifted word ... |
| cshwfn 14952 | A cyclically shifted word ... |
| cshwrn 14953 | The range of a cyclically ... |
| cshwidxmod 14954 | The symbol at a given inde... |
| cshwidxmodr 14955 | The symbol at a given inde... |
| cshwidx0mod 14956 | The symbol at index 0 of a... |
| cshwidx0 14957 | The symbol at index 0 of a... |
| cshwidxm1 14958 | The symbol at index ((n-N)... |
| cshwidxm 14959 | The symbol at index (n-N) ... |
| cshwidxn 14960 | The symbol at index (n-1) ... |
| cshf1 14961 | Cyclically shifting a word... |
| cshinj 14962 | If a word is injectiv (reg... |
| repswcshw 14963 | A cyclically shifted "repe... |
| 2cshw 14964 | Cyclically shifting a word... |
| 2cshwid 14965 | Cyclically shifting a word... |
| lswcshw 14966 | The last symbol of a word ... |
| 2cshwcom 14967 | Cyclically shifting a word... |
| cshwleneq 14968 | If the results of cyclical... |
| 3cshw 14969 | Cyclically shifting a word... |
| cshweqdif2 14970 | If cyclically shifting two... |
| cshweqdifid 14971 | If cyclically shifting a w... |
| cshweqrep 14972 | If cyclically shifting a w... |
| cshw1 14973 | If cyclically shifting a w... |
| cshw1repsw 14974 | If cyclically shifting a w... |
| cshwsexa 14975 | The class of (different!) ... |
| 2cshwcshw 14976 | If a word is a cyclically ... |
| scshwfzeqfzo 14977 | For a nonempty word the se... |
| cshwcshid 14978 | A cyclically shifted word ... |
| cshwcsh2id 14979 | A cyclically shifted word ... |
| cshimadifsn 14980 | The image of a cyclically ... |
| cshimadifsn0 14981 | The image of a cyclically ... |
| wrdco 14982 | Mapping a word by a functi... |
| lenco 14983 | Length of a mapped word is... |
| s1co 14984 | Mapping of a singleton wor... |
| revco 14985 | Mapping of words (i.e., a ... |
| ccatco 14986 | Mapping of words commutes ... |
| cshco 14987 | Mapping of words commutes ... |
| swrdco 14988 | Mapping of words commutes ... |
| pfxco 14989 | Mapping of words commutes ... |
| lswco 14990 | Mapping of (nonempty) word... |
| repsco 14991 | Mapping of words commutes ... |
| cats1cld 15006 | Closure of concatenation w... |
| cats1co 15007 | Closure of concatenation w... |
| cats1cli 15008 | Closure of concatenation w... |
| cats1fvn 15009 | The last symbol of a conca... |
| cats1fv 15010 | A symbol other than the la... |
| cats1len 15011 | The length of concatenatio... |
| cats1cat 15012 | Closure of concatenation w... |
| cats2cat 15013 | Closure of concatenation o... |
| s2eqd 15014 | Equality theorem for a dou... |
| s3eqd 15015 | Equality theorem for a len... |
| s4eqd 15016 | Equality theorem for a len... |
| s5eqd 15017 | Equality theorem for a len... |
| s6eqd 15018 | Equality theorem for a len... |
| s7eqd 15019 | Equality theorem for a len... |
| s8eqd 15020 | Equality theorem for a len... |
| s3eq2 15021 | Equality theorem for a len... |
| s2cld 15022 | A doubleton word is a word... |
| s3cld 15023 | A length 3 string is a wor... |
| s4cld 15024 | A length 4 string is a wor... |
| s5cld 15025 | A length 5 string is a wor... |
| s6cld 15026 | A length 6 string is a wor... |
| s7cld 15027 | A length 7 string is a wor... |
| s8cld 15028 | A length 8 string is a wor... |
| s2cl 15029 | A doubleton word is a word... |
| s3cl 15030 | A length 3 string is a wor... |
| s2cli 15031 | A doubleton word is a word... |
| s3cli 15032 | A length 3 string is a wor... |
| s4cli 15033 | A length 4 string is a wor... |
| s5cli 15034 | A length 5 string is a wor... |
| s6cli 15035 | A length 6 string is a wor... |
| s7cli 15036 | A length 7 string is a wor... |
| s8cli 15037 | A length 8 string is a wor... |
| s2fv0 15038 | Extract the first symbol f... |
| s2fv1 15039 | Extract the second symbol ... |
| s2len 15040 | The length of a doubleton ... |
| s2dm 15041 | The domain of a doubleton ... |
| s3fv0 15042 | Extract the first symbol f... |
| s3fv1 15043 | Extract the second symbol ... |
| s3fv2 15044 | Extract the third symbol f... |
| s3len 15045 | The length of a length 3 s... |
| s4fv0 15046 | Extract the first symbol f... |
| s4fv1 15047 | Extract the second symbol ... |
| s4fv2 15048 | Extract the third symbol f... |
| s4fv3 15049 | Extract the fourth symbol ... |
| s4len 15050 | The length of a length 4 s... |
| s5len 15051 | The length of a length 5 s... |
| s6len 15052 | The length of a length 6 s... |
| s7len 15053 | The length of a length 7 s... |
| s8len 15054 | The length of a length 8 s... |
| lsws2 15055 | The last symbol of a doubl... |
| lsws3 15056 | The last symbol of a 3 let... |
| lsws4 15057 | The last symbol of a 4 let... |
| s2prop 15058 | A length 2 word is an unor... |
| s2dmALT 15059 | Alternate version of ~ s2d... |
| s3tpop 15060 | A length 3 word is an unor... |
| s4prop 15061 | A length 4 word is a union... |
| s3fn 15062 | A length 3 word is a funct... |
| funcnvs1 15063 | The converse of a singleto... |
| funcnvs2 15064 | The converse of a length 2... |
| funcnvs3 15065 | The converse of a length 3... |
| funcnvs4 15066 | The converse of a length 4... |
| s2f1o 15067 | A length 2 word with mutua... |
| f1oun2prg 15068 | A union of unordered pairs... |
| s4f1o 15069 | A length 4 word with mutua... |
| s4dom 15070 | The domain of a length 4 w... |
| s2co 15071 | Mapping a doubleton word b... |
| s3co 15072 | Mapping a length 3 string ... |
| s0s1 15073 | Concatenation of fixed len... |
| s1s2 15074 | Concatenation of fixed len... |
| s1s3 15075 | Concatenation of fixed len... |
| s1s4 15076 | Concatenation of fixed len... |
| s1s5 15077 | Concatenation of fixed len... |
| s1s6 15078 | Concatenation of fixed len... |
| s1s7 15079 | Concatenation of fixed len... |
| s2s2 15080 | Concatenation of fixed len... |
| s4s2 15081 | Concatenation of fixed len... |
| s4s3 15082 | Concatenation of fixed len... |
| s4s4 15083 | Concatenation of fixed len... |
| s3s4 15084 | Concatenation of fixed len... |
| s2s5 15085 | Concatenation of fixed len... |
| s5s2 15086 | Concatenation of fixed len... |
| s2eq2s1eq 15087 | Two length 2 words are equ... |
| s2eq2seq 15088 | Two length 2 words are equ... |
| s3eqs2s1eq 15089 | Two length 3 words are equ... |
| s3eq3seq 15090 | Two length 3 words are equ... |
| swrds2 15091 | Extract two adjacent symbo... |
| swrds2m 15092 | Extract two adjacent symbo... |
| wrdlen2i 15093 | Implications of a word of ... |
| wrd2pr2op 15094 | A word of length two repre... |
| wrdlen2 15095 | A word of length two. (Co... |
| wrdlen2s2 15096 | A word of length two as do... |
| wrdl2exs2 15097 | A word of length two is a ... |
| pfx2 15098 | A prefix of length two. (... |
| wrd3tpop 15099 | A word of length three rep... |
| wrdlen3s3 15100 | A word of length three as ... |
| s3rex 15101 | Membership in a family of ... |
| s3rexrd 15102 | Converse of ~ s3rex , dedu... |
| repsw2 15103 | The "repeated symbol word"... |
| repsw3 15104 | The "repeated symbol word"... |
| swrd2lsw 15105 | Extract the last two symbo... |
| 2swrd2eqwrdeq 15106 | Two words of length at lea... |
| ccatw2s1ccatws2 15107 | The concatenation of a wor... |
| ccat2s1fvwALT 15108 | Alternate proof of ~ ccat2... |
| wwlktovf 15109 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 15110 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 15111 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 15112 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 15113 | There is a bijection betwe... |
| eqwrds3 15114 | A word is equal with a len... |
| wrdl3s3 15115 | A word of length 3 is a le... |
| s2rn 15116 | Range of a length 2 string... |
| s3rn 15117 | Range of a length 3 string... |
| s7rn 15118 | Range of a length 7 string... |
| s7f1o 15119 | A length 7 word with mutua... |
| s3sndisj 15120 | The singletons consisting ... |
| s3iunsndisj 15121 | The union of singletons co... |
| ofccat 15122 | Letterwise operations on w... |
| ofs1 15123 | Letterwise operations on a... |
| ofs2 15124 | Letterwise operations on a... |
| coss12d 15125 | Subset deduction for compo... |
| trrelssd 15126 | The composition of subclas... |
| xpcogend 15127 | The most interesting case ... |
| xpcoidgend 15128 | If two classes are not dis... |
| cotr2g 15129 | Two ways of saying that th... |
| cotr2 15130 | Two ways of saying a relat... |
| cotr3 15131 | Two ways of saying a relat... |
| coemptyd 15132 | Deduction about compositio... |
| xptrrel 15133 | The cross product is alway... |
| 0trrel 15134 | The empty class is a trans... |
| cleq1lem 15135 | Equality implies bijection... |
| cleq1 15136 | Equality of relations impl... |
| clsslem 15137 | The closure of a subclass ... |
| trcleq1 15142 | Equality of relations impl... |
| trclsslem 15143 | The transitive closure (as... |
| trcleq2lem 15144 | Equality implies bijection... |
| cvbtrcl 15145 | Change of bound variable i... |
| trcleq12lem 15146 | Equality implies bijection... |
| trclexlem 15147 | Existence of relation impl... |
| trclublem 15148 | If a relation exists then ... |
| trclubi 15149 | The Cartesian product of t... |
| trclubgi 15150 | The union with the Cartesi... |
| trclub 15151 | The Cartesian product of t... |
| trclubg 15152 | The union with the Cartesi... |
| trclfv 15153 | The transitive closure of ... |
| brintclab 15154 | Two ways to express a bina... |
| brtrclfv 15155 | Two ways of expressing the... |
| brcnvtrclfv 15156 | Two ways of expressing the... |
| brtrclfvcnv 15157 | Two ways of expressing the... |
| brcnvtrclfvcnv 15158 | Two ways of expressing the... |
| trclfvss 15159 | The transitive closure (as... |
| trclfvub 15160 | The transitive closure of ... |
| trclfvlb 15161 | The transitive closure of ... |
| trclfvcotr 15162 | The transitive closure of ... |
| trclfvlb2 15163 | The transitive closure of ... |
| trclfvlb3 15164 | The transitive closure of ... |
| cotrtrclfv 15165 | The transitive closure of ... |
| trclidm 15166 | The transitive closure of ... |
| trclun 15167 | Transitive closure of a un... |
| trclfvg 15168 | The value of the transitiv... |
| trclfvcotrg 15169 | The value of the transitiv... |
| reltrclfv 15170 | The transitive closure of ... |
| dmtrclfv 15171 | The domain of the transiti... |
| reldmrelexp 15174 | The domain of the repeated... |
| relexp0g 15175 | A relation composed zero t... |
| relexp0 15176 | A relation composed zero t... |
| relexp0d 15177 | A relation composed zero t... |
| relexpsucnnr 15178 | A reduction for relation e... |
| relexp1g 15179 | A relation composed once i... |
| dfid5 15180 | Identity relation is equal... |
| dfid6 15181 | Identity relation expresse... |
| relexp1d 15182 | A relation composed once i... |
| relexpsucnnl 15183 | A reduction for relation e... |
| relexpsucl 15184 | A reduction for relation e... |
| relexpsucr 15185 | A reduction for relation e... |
| relexpsucrd 15186 | A reduction for relation e... |
| relexpsucld 15187 | A reduction for relation e... |
| relexpcnv 15188 | Commutation of converse an... |
| relexpcnvd 15189 | Commutation of converse an... |
| relexp0rel 15190 | The exponentiation of a cl... |
| relexprelg 15191 | The exponentiation of a cl... |
| relexprel 15192 | The exponentiation of a re... |
| relexpreld 15193 | The exponentiation of a re... |
| relexpnndm 15194 | The domain of an exponenti... |
| relexpdmg 15195 | The domain of an exponenti... |
| relexpdm 15196 | The domain of an exponenti... |
| relexpdmd 15197 | The domain of an exponenti... |
| relexpnnrn 15198 | The range of an exponentia... |
| relexprng 15199 | The range of an exponentia... |
| relexprn 15200 | The range of an exponentia... |
| relexprnd 15201 | The range of an exponentia... |
| relexpfld 15202 | The field of an exponentia... |
| relexpfldd 15203 | The field of an exponentia... |
| relexpaddnn 15204 | Relation composition becom... |
| relexpuzrel 15205 | The exponentiation of a cl... |
| relexpaddg 15206 | Relation composition becom... |
| relexpaddd 15207 | Relation composition becom... |
| rtrclreclem1 15210 | The reflexive, transitive ... |
| dfrtrclrec2 15211 | If two elements are connec... |
| rtrclreclem2 15212 | The reflexive, transitive ... |
| rtrclreclem3 15213 | The reflexive, transitive ... |
| rtrclreclem4 15214 | The reflexive, transitive ... |
| dfrtrcl2 15215 | The two definitions ` t* `... |
| relexpindlem 15216 | Principle of transitive in... |
| relexpind 15217 | Principle of transitive in... |
| rtrclind 15218 | Principle of transitive in... |
| shftlem 15221 | Two ways to write a shifte... |
| shftuz 15222 | A shift of the upper integ... |
| shftfval 15223 | The value of the sequence ... |
| shftdm 15224 | Domain of a relation shift... |
| shftfib 15225 | Value of a fiber of the re... |
| shftfn 15226 | Functionality and domain o... |
| shftval 15227 | Value of a sequence shifte... |
| shftval2 15228 | Value of a sequence shifte... |
| shftval3 15229 | Value of a sequence shifte... |
| shftval4 15230 | Value of a sequence shifte... |
| shftval5 15231 | Value of a shifted sequenc... |
| shftf 15232 | Functionality of a shifted... |
| 2shfti 15233 | Composite shift operations... |
| shftidt2 15234 | Identity law for the shift... |
| shftidt 15235 | Identity law for the shift... |
| shftcan1 15236 | Cancellation law for the s... |
| shftcan2 15237 | Cancellation law for the s... |
| seqshft 15238 | Shifting the index set of ... |
| sgnval 15241 | Value of the signum functi... |
| sgn0 15242 | The signum of 0 is 0. (Co... |
| sgnp 15243 | The signum of a positive e... |
| sgnrrp 15244 | The signum of a positive r... |
| sgn1 15245 | The signum of 1 is 1. (Co... |
| sgnpnf 15246 | The signum of ` +oo ` is 1... |
| sgnn 15247 | The signum of a negative e... |
| sgnmnf 15248 | The signum of ` -oo ` is -... |
| sgndm 15249 | The domain of the signum f... |
| sgncl 15250 | Closure of the signum: Th... |
| sgnrn 15251 | The range of the signum fu... |
| sgnfo 15252 | The signum function as ont... |
| sgnneg 15253 | Negation of the signum. (... |
| sgn3da 15254 | A conditional containing a... |
| sgnclre 15255 | Closure of the signum for ... |
| sgn0bi 15256 | Zero signum. (Contributed... |
| sgnnbi 15257 | Negative signum. (Contrib... |
| sgnpbi 15258 | Positive signum. (Contrib... |
| sgnsub 15259 | Signum of a difference wit... |
| sgnmul 15260 | Signum of a product. (Con... |
| sgnmulrp2 15261 | Multiplication by a positi... |
| sgnmulsgn 15262 | If two real numbers are of... |
| cjval 15269 | The value of the conjugate... |
| cjth 15270 | The defining property of t... |
| cjf 15271 | Domain and codomain of the... |
| cjcl 15272 | The conjugate of a complex... |
| reval 15273 | The value of the real part... |
| imval 15274 | The value of the imaginary... |
| imre 15275 | The imaginary part of a co... |
| reim 15276 | The real part of a complex... |
| recl 15277 | The real part of a complex... |
| imcl 15278 | The imaginary part of a co... |
| ref 15279 | Domain and codomain of the... |
| imf 15280 | Domain and codomain of the... |
| crre 15281 | The real part of a complex... |
| crim 15282 | The real part of a complex... |
| replim 15283 | Reconstruct a complex numb... |
| remim 15284 | Value of the conjugate of ... |
| reim0 15285 | The imaginary part of a re... |
| reim0b 15286 | A number is real iff its i... |
| rereb 15287 | A number is real iff it eq... |
| mulre 15288 | A product with a nonzero r... |
| rere 15289 | A real number equals its r... |
| cjreb 15290 | A number is real iff it eq... |
| recj 15291 | Real part of a complex con... |
| reneg 15292 | Real part of negative. (C... |
| readd 15293 | Real part distributes over... |
| resub 15294 | Real part distributes over... |
| remullem 15295 | Lemma for ~ remul , ~ immu... |
| remul 15296 | Real part of a product. (... |
| remul2 15297 | Real part of a product. (... |
| rediv 15298 | Real part of a division. ... |
| imcj 15299 | Imaginary part of a comple... |
| imneg 15300 | The imaginary part of a ne... |
| imadd 15301 | Imaginary part distributes... |
| imsub 15302 | Imaginary part distributes... |
| immul 15303 | Imaginary part of a produc... |
| immul2 15304 | Imaginary part of a produc... |
| imdiv 15305 | Imaginary part of a divisi... |
| cjre 15306 | A real number equals its c... |
| cjcj 15307 | The conjugate of the conju... |
| cjadd 15308 | Complex conjugate distribu... |
| cjmul 15309 | Complex conjugate distribu... |
| ipcnval 15310 | Standard inner product on ... |
| cjmulrcl 15311 | A complex number times its... |
| cjmulval 15312 | A complex number times its... |
| cjmulge0 15313 | A complex number times its... |
| cjneg 15314 | Complex conjugate of negat... |
| addcj 15315 | A number plus its conjugat... |
| cjsub 15316 | Complex conjugate distribu... |
| cjexp 15317 | Complex conjugate of posit... |
| imval2 15318 | The imaginary part of a nu... |
| re0 15319 | The real part of zero. (C... |
| im0 15320 | The imaginary part of zero... |
| re1 15321 | The real part of one. (Co... |
| im1 15322 | The imaginary part of one.... |
| rei 15323 | The real part of ` _i ` . ... |
| imi 15324 | The imaginary part of ` _i... |
| cj0 15325 | The conjugate of zero. (C... |
| cji 15326 | The complex conjugate of t... |
| cjreim 15327 | The conjugate of a represe... |
| cjreim2 15328 | The conjugate of the repre... |
| cj11 15329 | Complex conjugate is a one... |
| cjne0 15330 | A number is nonzero iff it... |
| cjdiv 15331 | Complex conjugate distribu... |
| cnrecnv 15332 | The inverse to the canonic... |
| sqeqd 15333 | A deduction for showing tw... |
| recli 15334 | The real part of a complex... |
| imcli 15335 | The imaginary part of a co... |
| cjcli 15336 | Closure law for complex co... |
| replimi 15337 | Construct a complex number... |
| cjcji 15338 | The conjugate of the conju... |
| reim0bi 15339 | A number is real iff its i... |
| rerebi 15340 | A real number equals its r... |
| cjrebi 15341 | A number is real iff it eq... |
| recji 15342 | Real part of a complex con... |
| imcji 15343 | Imaginary part of a comple... |
| cjmulrcli 15344 | A complex number times its... |
| cjmulvali 15345 | A complex number times its... |
| cjmulge0i 15346 | A complex number times its... |
| renegi 15347 | Real part of negative. (C... |
| imnegi 15348 | Imaginary part of negative... |
| cjnegi 15349 | Complex conjugate of negat... |
| addcji 15350 | A number plus its conjugat... |
| readdi 15351 | Real part distributes over... |
| imaddi 15352 | Imaginary part distributes... |
| remuli 15353 | Real part of a product. (... |
| immuli 15354 | Imaginary part of a produc... |
| cjaddi 15355 | Complex conjugate distribu... |
| cjmuli 15356 | Complex conjugate distribu... |
| ipcni 15357 | Standard inner product on ... |
| cjdivi 15358 | Complex conjugate distribu... |
| crrei 15359 | The real part of a complex... |
| crimi 15360 | The imaginary part of a co... |
| recld 15361 | The real part of a complex... |
| imcld 15362 | The imaginary part of a co... |
| cjcld 15363 | Closure law for complex co... |
| replimd 15364 | Construct a complex number... |
| remimd 15365 | Value of the conjugate of ... |
| cjcjd 15366 | The conjugate of the conju... |
| reim0bd 15367 | A number is real iff its i... |
| rerebd 15368 | A real number equals its r... |
| cjrebd 15369 | A number is real iff it eq... |
| cjne0d 15370 | A number is nonzero iff it... |
| recjd 15371 | Real part of a complex con... |
| imcjd 15372 | Imaginary part of a comple... |
| cjmulrcld 15373 | A complex number times its... |
| cjmulvald 15374 | A complex number times its... |
| cjmulge0d 15375 | A complex number times its... |
| renegd 15376 | Real part of negative. (C... |
| imnegd 15377 | Imaginary part of negative... |
| cjnegd 15378 | Complex conjugate of negat... |
| addcjd 15379 | A number plus its conjugat... |
| cjexpd 15380 | Complex conjugate of posit... |
| readdd 15381 | Real part distributes over... |
| imaddd 15382 | Imaginary part distributes... |
| resubd 15383 | Real part distributes over... |
| imsubd 15384 | Imaginary part distributes... |
| remuld 15385 | Real part of a product. (... |
| immuld 15386 | Imaginary part of a produc... |
| cjaddd 15387 | Complex conjugate distribu... |
| cjmuld 15388 | Complex conjugate distribu... |
| ipcnd 15389 | Standard inner product on ... |
| cjdivd 15390 | Complex conjugate distribu... |
| rered 15391 | A real number equals its r... |
| reim0d 15392 | The imaginary part of a re... |
| cjred 15393 | A real number equals its c... |
| remul2d 15394 | Real part of a product. (... |
| immul2d 15395 | Imaginary part of a produc... |
| redivd 15396 | Real part of a division. ... |
| imdivd 15397 | Imaginary part of a divisi... |
| crred 15398 | The real part of a complex... |
| crimd 15399 | The imaginary part of a co... |
| sqrtval 15404 | Value of square root funct... |
| absval 15405 | The absolute value (modulu... |
| rennim 15406 | A real number does not lie... |
| cnpart 15407 | The specification of restr... |
| sqrt0 15408 | The square root of zero is... |
| 01sqrexlem1 15409 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15410 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15411 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15412 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15413 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15414 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15415 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15416 | Existence of a square root... |
| resqrex 15417 | Existence of a square root... |
| sqrmo 15418 | Uniqueness for the square ... |
| resqreu 15419 | Existence and uniqueness f... |
| resqrtcl 15420 | Closure of the square root... |
| resqrtthlem 15421 | Lemma for ~ resqrtth . (C... |
| resqrtth 15422 | Square root theorem over t... |
| remsqsqrt 15423 | Square of square root. (C... |
| sqrtge0 15424 | The square root function i... |
| sqrtgt0 15425 | The square root function i... |
| sqrtmul 15426 | Square root distributes ov... |
| sqrtle 15427 | Square root is monotonic. ... |
| sqrtlt 15428 | Square root is strictly mo... |
| sqrt11 15429 | The square root function i... |
| sqrt00 15430 | A square root is zero iff ... |
| rpsqrtcl 15431 | The square root of a posit... |
| sqrtdiv 15432 | Square root distributes ov... |
| sqrtneglem 15433 | The square root of a negat... |
| sqrtneg 15434 | The square root of a negat... |
| sqrtsq2 15435 | Relationship between squar... |
| sqrtsq 15436 | Square root of square. (C... |
| sqrtmsq 15437 | Square root of square. (C... |
| sqrt1 15438 | The square root of 1 is 1.... |
| sqrt4 15439 | The square root of 4 is 2.... |
| sqrt9 15440 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15441 | The square root of 2 is bo... |
| sqrtm1 15442 | The imaginary unit is the ... |
| nn0sqeq1 15443 | A natural number with squa... |
| absneg 15444 | Absolute value of the nega... |
| abscl 15445 | Real closure of absolute v... |
| abscj 15446 | The absolute value of a nu... |
| absvalsq 15447 | Square of value of absolut... |
| absvalsq2 15448 | Square of value of absolut... |
| sqabsadd 15449 | Square of absolute value o... |
| sqabssub 15450 | Square of absolute value o... |
| absval2 15451 | Value of absolute value fu... |
| abs0 15452 | The absolute value of 0. ... |
| absi 15453 | The absolute value of the ... |
| absge0 15454 | Absolute value is nonnegat... |
| absrpcl 15455 | The absolute value of a no... |
| abs00 15456 | The absolute value of a nu... |
| abs00ad 15457 | A complex number is zero i... |
| abs00bd 15458 | If a complex number is zer... |
| absreimsq 15459 | Square of the absolute val... |
| absreim 15460 | Absolute value of a number... |
| absmul 15461 | Absolute value distributes... |
| absdiv 15462 | Absolute value distributes... |
| absid 15463 | A nonnegative number is it... |
| abs1 15464 | The absolute value of one ... |
| absnid 15465 | For a negative number, its... |
| leabs 15466 | A real number is less than... |
| absor 15467 | The absolute value of a re... |
| absre 15468 | Absolute value of a real n... |
| absresq 15469 | Square of the absolute val... |
| absmod0 15470 | ` A ` is divisible by ` B ... |
| absexp 15471 | Absolute value of positive... |
| absexpz 15472 | Absolute value of integer ... |
| abssq 15473 | Square can be moved in and... |
| sqabs 15474 | The squares of two reals a... |
| absrele 15475 | The absolute value of a co... |
| absimle 15476 | The absolute value of a co... |
| max0add 15477 | The sum of the positive an... |
| absz 15478 | A real number is an intege... |
| nn0abscl 15479 | The absolute value of an i... |
| zabscl 15480 | The absolute value of an i... |
| zabs0b 15481 | An integer has an absolute... |
| abslt 15482 | Absolute value and 'less t... |
| absle 15483 | Absolute value and 'less t... |
| abssubne0 15484 | If the absolute value of a... |
| absdiflt 15485 | The absolute value of a di... |
| absdifle 15486 | The absolute value of a di... |
| elicc4abs 15487 | Membership in a symmetric ... |
| lenegsq 15488 | Comparison to a nonnegativ... |
| releabs 15489 | The real part of a number ... |
| recval 15490 | Reciprocal expressed with ... |
| absidm 15491 | The absolute value functio... |
| absgt0 15492 | The absolute value of a no... |
| nnabscl 15493 | The absolute value of a no... |
| abssub 15494 | Swapping order of subtract... |
| abssubge0 15495 | Absolute value of a nonneg... |
| abssuble0 15496 | Absolute value of a nonpos... |
| absmax 15497 | The maximum of two numbers... |
| abstri 15498 | Triangle inequality for ab... |
| abs3dif 15499 | Absolute value of differen... |
| abs2dif 15500 | Difference of absolute val... |
| abs2dif2 15501 | Difference of absolute val... |
| abs2difabs 15502 | Absolute value of differen... |
| abs1m 15503 | For any complex number, th... |
| recan 15504 | Cancellation law involving... |
| absf 15505 | Mapping domain and codomai... |
| abs3lem 15506 | Lemma involving absolute v... |
| abslem2 15507 | Lemma involving absolute v... |
| rddif 15508 | The difference between a r... |
| absrdbnd 15509 | Bound on the absolute valu... |
| fzomaxdiflem 15510 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15511 | A bound on the separation ... |
| uzin2 15512 | The upper integers are clo... |
| rexanuz 15513 | Combine two different uppe... |
| rexanre 15514 | Combine two different uppe... |
| rexfiuz 15515 | Combine finitely many diff... |
| rexuz3 15516 | Restrict the base of the u... |
| rexanuz2 15517 | Combine two different uppe... |
| r19.29uz 15518 | A version of ~ 19.29 for u... |
| r19.2uz 15519 | A version of ~ r19.2z for ... |
| rexuzre 15520 | Convert an upper real quan... |
| rexico 15521 | Restrict the base of an up... |
| cau3lem 15522 | Lemma for ~ cau3 . (Contr... |
| cau3 15523 | Convert between three-quan... |
| cau4 15524 | Change the base of a Cauch... |
| caubnd2 15525 | A Cauchy sequence of compl... |
| caubnd 15526 | A Cauchy sequence of compl... |
| sqreulem 15527 | Lemma for ~ sqreu : write ... |
| sqreu 15528 | Existence and uniqueness f... |
| sqrtcl 15529 | Closure of the square root... |
| sqrtthlem 15530 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15531 | Mapping domain and codomai... |
| sqrtth 15532 | Square root theorem over t... |
| sqrtrege0 15533 | The square root function m... |
| eqsqrtor 15534 | Solve an equation containi... |
| eqsqrtd 15535 | A deduction for showing th... |
| eqsqrt2d 15536 | A deduction for showing th... |
| amgm2 15537 | Arithmetic-geometric mean ... |
| sqrtthi 15538 | Square root theorem. Theo... |
| sqrtcli 15539 | The square root of a nonne... |
| sqrtgt0i 15540 | The square root of a posit... |
| sqrtmsqi 15541 | Square root of square. (C... |
| sqrtsqi 15542 | Square root of square. (C... |
| sqsqrti 15543 | Square of square root. (C... |
| sqrtge0i 15544 | The square root of a nonne... |
| absidi 15545 | A nonnegative number is it... |
| absnidi 15546 | A negative number is the n... |
| leabsi 15547 | A real number is less than... |
| absori 15548 | The absolute value of a re... |
| absrei 15549 | Absolute value of a real n... |
| sqrtpclii 15550 | The square root of a posit... |
| sqrtgt0ii 15551 | The square root of a posit... |
| sqrt11i 15552 | The square root function i... |
| sqrtmuli 15553 | Square root distributes ov... |
| sqrtmulii 15554 | Square root distributes ov... |
| sqrtmsq2i 15555 | Relationship between squar... |
| sqrtlei 15556 | Square root is monotonic. ... |
| sqrtlti 15557 | Square root is strictly mo... |
| abslti 15558 | Absolute value and 'less t... |
| abslei 15559 | Absolute value and 'less t... |
| cnsqrt00 15560 | A square root of a complex... |
| absvalsqi 15561 | Square of value of absolut... |
| absvalsq2i 15562 | Square of value of absolut... |
| abscli 15563 | Real closure of absolute v... |
| absge0i 15564 | Absolute value is nonnegat... |
| absval2i 15565 | Value of absolute value fu... |
| abs00i 15566 | The absolute value of a nu... |
| absgt0i 15567 | The absolute value of a no... |
| absnegi 15568 | Absolute value of negative... |
| abscji 15569 | The absolute value of a nu... |
| releabsi 15570 | The real part of a number ... |
| abssubi 15571 | Swapping order of subtract... |
| absmuli 15572 | Absolute value distributes... |
| sqabsaddi 15573 | Square of absolute value o... |
| sqabssubi 15574 | Square of absolute value o... |
| absdivzi 15575 | Absolute value distributes... |
| abstrii 15576 | Triangle inequality for ab... |
| abs3difi 15577 | Absolute value of differen... |
| abs3lemi 15578 | Lemma involving absolute v... |
| rpsqrtcld 15579 | The square root of a posit... |
| sqrtgt0d 15580 | The square root of a posit... |
| absnidd 15581 | A negative number is the n... |
| leabsd 15582 | A real number is less than... |
| absord 15583 | The absolute value of a re... |
| absred 15584 | Absolute value of a real n... |
| resqrtcld 15585 | The square root of a nonne... |
| sqrtmsqd 15586 | Square root of square. (C... |
| sqrtsqd 15587 | Square root of square. (C... |
| sqrtge0d 15588 | The square root of a nonne... |
| sqrtnegd 15589 | The square root of a negat... |
| absidd 15590 | A nonnegative number is it... |
| sqrtdivd 15591 | Square root distributes ov... |
| sqrtmuld 15592 | Square root distributes ov... |
| sqrtsq2d 15593 | Relationship between squar... |
| sqrtled 15594 | Square root is monotonic. ... |
| sqrtltd 15595 | Square root is strictly mo... |
| sqr11d 15596 | The square root function i... |
| nn0absid 15597 | A nonnegative integer is i... |
| nn0absidi 15598 | A nonnegative integer is i... |
| absltd 15599 | Absolute value and 'less t... |
| absled 15600 | Absolute value and 'less t... |
| abssubge0d 15601 | Absolute value of a nonneg... |
| abssuble0d 15602 | Absolute value of a nonpos... |
| absdifltd 15603 | The absolute value of a di... |
| absdifled 15604 | The absolute value of a di... |
| icodiamlt 15605 | Two elements in a half-ope... |
| abscld 15606 | Real closure of absolute v... |
| sqrtcld 15607 | Closure of the square root... |
| sqrtrege0d 15608 | The real part of the squar... |
| sqsqrtd 15609 | Square root theorem. Theo... |
| msqsqrtd 15610 | Square root theorem. Theo... |
| sqr00d 15611 | A square root is zero iff ... |
| absvalsqd 15612 | Square of value of absolut... |
| absvalsq2d 15613 | Square of value of absolut... |
| absge0d 15614 | Absolute value is nonnegat... |
| absval2d 15615 | Value of absolute value fu... |
| abs00d 15616 | The absolute value of a nu... |
| absne0d 15617 | The absolute value of a nu... |
| absrpcld 15618 | The absolute value of a no... |
| absnegd 15619 | Absolute value of negative... |
| abscjd 15620 | The absolute value of a nu... |
| releabsd 15621 | The real part of a number ... |
| absexpd 15622 | Absolute value of positive... |
| abssubd 15623 | Swapping order of subtract... |
| absmuld 15624 | Absolute value distributes... |
| absdivd 15625 | Absolute value distributes... |
| abstrid 15626 | Triangle inequality for ab... |
| abs2difd 15627 | Difference of absolute val... |
| abs2dif2d 15628 | Difference of absolute val... |
| abs2difabsd 15629 | Absolute value of differen... |
| abs3difd 15630 | Absolute value of differen... |
| abs3lemd 15631 | Lemma involving absolute v... |
| reusq0 15632 | A complex number is the sq... |
| bhmafibid1cn 15633 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15634 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15635 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15636 | The Brahmagupta-Fibonacci ... |
| limsupgord 15639 | Ordering property of the s... |
| limsupcl 15640 | Closure of the superior li... |
| limsupval 15641 | The superior limit of an i... |
| limsupgf 15642 | Closure of the superior li... |
| limsupgval 15643 | Value of the superior limi... |
| limsupgle 15644 | The defining property of t... |
| limsuple 15645 | The defining property of t... |
| limsuplt 15646 | The defining property of t... |
| limsupval2 15647 | The superior limit, relati... |
| limsupgre 15648 | If a sequence of real numb... |
| limsupbnd1 15649 | If a sequence is eventuall... |
| limsupbnd2 15650 | If a sequence is eventuall... |
| climrel 15659 | The limit relation is a re... |
| rlimrel 15660 | The limit relation is a re... |
| clim 15661 | Express the predicate: Th... |
| rlim 15662 | Express the predicate: Th... |
| rlim2 15663 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15664 | Use strictly less-than in ... |
| rlim3 15665 | Restrict the range of the ... |
| climcl 15666 | Closure of the limit of a ... |
| rlimpm 15667 | Closure of a function with... |
| rlimf 15668 | Closure of a function with... |
| rlimss 15669 | Domain closure of a functi... |
| rlimcl 15670 | Closure of the limit of a ... |
| clim2 15671 | Express the predicate: Th... |
| clim2c 15672 | Express the predicate ` F ... |
| clim0 15673 | Express the predicate ` F ... |
| clim0c 15674 | Express the predicate ` F ... |
| rlim0 15675 | Express the predicate ` B ... |
| rlim0lt 15676 | Use strictly less-than in ... |
| climi 15677 | Convergence of a sequence ... |
| climi2 15678 | Convergence of a sequence ... |
| climi0 15679 | Convergence of a sequence ... |
| rlimi 15680 | Convergence at infinity of... |
| rlimi2 15681 | Convergence at infinity of... |
| ello1 15682 | Elementhood in the set of ... |
| ello12 15683 | Elementhood in the set of ... |
| ello12r 15684 | Sufficient condition for e... |
| lo1f 15685 | An eventually upper bounde... |
| lo1dm 15686 | An eventually upper bounde... |
| lo1bdd 15687 | The defining property of a... |
| ello1mpt 15688 | Elementhood in the set of ... |
| ello1mpt2 15689 | Elementhood in the set of ... |
| ello1d 15690 | Sufficient condition for e... |
| lo1bdd2 15691 | If an eventually bounded f... |
| lo1bddrp 15692 | Refine ~ o1bdd2 to give a ... |
| elo1 15693 | Elementhood in the set of ... |
| elo12 15694 | Elementhood in the set of ... |
| elo12r 15695 | Sufficient condition for e... |
| o1f 15696 | An eventually bounded func... |
| o1dm 15697 | An eventually bounded func... |
| o1bdd 15698 | The defining property of a... |
| lo1o1 15699 | A function is eventually b... |
| lo1o12 15700 | A function is eventually b... |
| elo1mpt 15701 | Elementhood in the set of ... |
| elo1mpt2 15702 | Elementhood in the set of ... |
| elo1d 15703 | Sufficient condition for e... |
| o1lo1 15704 | A real function is eventua... |
| o1lo12 15705 | A lower bounded real funct... |
| o1lo1d 15706 | A real eventually bounded ... |
| icco1 15707 | Derive eventual boundednes... |
| o1bdd2 15708 | If an eventually bounded f... |
| o1bddrp 15709 | Refine ~ o1bdd2 to give a ... |
| climconst 15710 | An (eventually) constant s... |
| rlimconst 15711 | A constant sequence conver... |
| rlimclim1 15712 | Forward direction of ~ rli... |
| rlimclim 15713 | A sequence on an upper int... |
| climrlim2 15714 | Produce a real limit from ... |
| climconst2 15715 | A constant sequence conver... |
| climz 15716 | The zero sequence converge... |
| rlimuni 15717 | A real function whose doma... |
| rlimdm 15718 | Two ways to express that a... |
| climuni 15719 | An infinite sequence of co... |
| fclim 15720 | The limit relation is func... |
| climdm 15721 | Two ways to express that a... |
| climeu 15722 | An infinite sequence of co... |
| climreu 15723 | An infinite sequence of co... |
| climmo 15724 | An infinite sequence of co... |
| rlimres 15725 | The restriction of a funct... |
| lo1res 15726 | The restriction of an even... |
| o1res 15727 | The restriction of an even... |
| rlimres2 15728 | The restriction of a funct... |
| lo1res2 15729 | The restriction of a funct... |
| o1res2 15730 | The restriction of a funct... |
| lo1resb 15731 | The restriction of a funct... |
| rlimresb 15732 | The restriction of a funct... |
| o1resb 15733 | The restriction of a funct... |
| climeq 15734 | Two functions that are eve... |
| lo1eq 15735 | Two functions that are eve... |
| rlimeq 15736 | Two functions that are eve... |
| o1eq 15737 | Two functions that are eve... |
| climmpt 15738 | Exhibit a function ` G ` w... |
| 2clim 15739 | If two sequences converge ... |
| climmpt2 15740 | Relate an integer limit on... |
| climshftlem 15741 | A shifted function converg... |
| climres 15742 | A function restricted to u... |
| climshft 15743 | A shifted function converg... |
| serclim0 15744 | The zero series converges ... |
| rlimcld2 15745 | If ` D ` is a closed set i... |
| rlimrege0 15746 | The limit of a sequence of... |
| rlimrecl 15747 | The limit of a real sequen... |
| rlimge0 15748 | The limit of a sequence of... |
| climshft2 15749 | A shifted function converg... |
| climrecl 15750 | The limit of a convergent ... |
| climge0 15751 | A nonnegative sequence con... |
| climabs0 15752 | Convergence to zero of the... |
| o1co 15753 | Sufficient condition for t... |
| o1compt 15754 | Sufficient condition for t... |
| rlimcn1 15755 | Image of a limit under a c... |
| rlimcn1b 15756 | Image of a limit under a c... |
| rlimcn3 15757 | Image of a limit under a c... |
| rlimcn2 15758 | Image of a limit under a c... |
| climcn1 15759 | Image of a limit under a c... |
| climcn2 15760 | Image of a limit under a c... |
| addcn2 15761 | Complex number addition is... |
| subcn2 15762 | Complex number subtraction... |
| mulcn2 15763 | Complex number multiplicat... |
| reccn2 15764 | The reciprocal function is... |
| cn1lem 15765 | A sufficient condition for... |
| abscn2 15766 | The absolute value functio... |
| cjcn2 15767 | The complex conjugate func... |
| recn2 15768 | The real part function is ... |
| imcn2 15769 | The imaginary part functio... |
| climcn1lem 15770 | The limit of a continuous ... |
| climabs 15771 | Limit of the absolute valu... |
| climcj 15772 | Limit of the complex conju... |
| climre 15773 | Limit of the real part of ... |
| climim 15774 | Limit of the imaginary par... |
| rlimmptrcl 15775 | Reverse closure for a real... |
| rlimabs 15776 | Limit of the absolute valu... |
| rlimcj 15777 | Limit of the complex conju... |
| rlimre 15778 | Limit of the real part of ... |
| rlimim 15779 | Limit of the imaginary par... |
| o1of2 15780 | Show that a binary operati... |
| o1add 15781 | The sum of two eventually ... |
| o1mul 15782 | The product of two eventua... |
| o1sub 15783 | The difference of two even... |
| rlimo1 15784 | Any function with a finite... |
| rlimdmo1 15785 | A convergent function is e... |
| o1rlimmul 15786 | The product of an eventual... |
| o1const 15787 | A constant function is eve... |
| lo1const 15788 | A constant function is eve... |
| lo1mptrcl 15789 | Reverse closure for an eve... |
| o1mptrcl 15790 | Reverse closure for an eve... |
| o1add2 15791 | The sum of two eventually ... |
| o1mul2 15792 | The product of two eventua... |
| o1sub2 15793 | The product of two eventua... |
| lo1add 15794 | The sum of two eventually ... |
| lo1mul 15795 | The product of an eventual... |
| lo1mul2 15796 | The product of an eventual... |
| o1dif 15797 | If the difference of two f... |
| lo1sub 15798 | The difference of an event... |
| climadd 15799 | Limit of the sum of two co... |
| climmul 15800 | Limit of the product of tw... |
| climsub 15801 | Limit of the difference of... |
| climaddc1 15802 | Limit of a constant ` C ` ... |
| climaddc2 15803 | Limit of a constant ` C ` ... |
| climmulc2 15804 | Limit of a sequence multip... |
| climsubc1 15805 | Limit of a constant ` C ` ... |
| climsubc2 15806 | Limit of a constant ` C ` ... |
| climle 15807 | Comparison of the limits o... |
| climsqz 15808 | Convergence of a sequence ... |
| climsqz2 15809 | Convergence of a sequence ... |
| rlimadd 15810 | Limit of the sum of two co... |
| rlimsub 15811 | Limit of the difference of... |
| rlimmul 15812 | Limit of the product of tw... |
| rlimdiv 15813 | Limit of the quotient of t... |
| rlimneg 15814 | Limit of the negative of a... |
| rlimle 15815 | Comparison of the limits o... |
| rlimsqzlem 15816 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15817 | Convergence of a sequence ... |
| rlimsqz2 15818 | Convergence of a sequence ... |
| lo1le 15819 | Transfer eventual upper bo... |
| o1le 15820 | Transfer eventual boundedn... |
| rlimno1 15821 | A function whose inverse c... |
| clim2ser 15822 | The limit of an infinite s... |
| clim2ser2 15823 | The limit of an infinite s... |
| iserex 15824 | An infinite series converg... |
| isermulc2 15825 | Multiplication of an infin... |
| climlec2 15826 | Comparison of a constant t... |
| iserle 15827 | Comparison of the limits o... |
| iserge0 15828 | The limit of an infinite s... |
| climub 15829 | The limit of a monotonic s... |
| climserle 15830 | The partial sums of a conv... |
| isershft 15831 | Index shift of the limit o... |
| isercolllem1 15832 | Lemma for ~ isercoll . (C... |
| isercolllem2 15833 | Lemma for ~ isercoll . (C... |
| isercolllem3 15834 | Lemma for ~ isercoll . (C... |
| isercoll 15835 | Rearrange an infinite seri... |
| isercoll2 15836 | Generalize ~ isercoll so t... |
| climsup 15837 | A bounded monotonic sequen... |
| climcau 15838 | A converging sequence of c... |
| climbdd 15839 | A converging sequence of c... |
| caucvgrlem 15840 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15841 | A Cauchy sequence of real ... |
| caucvgrlem2 15842 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15843 | A Cauchy sequence of compl... |
| caurcvg 15844 | A Cauchy sequence of real ... |
| caurcvg2 15845 | A Cauchy sequence of real ... |
| caucvg 15846 | A Cauchy sequence of compl... |
| caucvgb 15847 | A function is convergent i... |
| serf0 15848 | If an infinite series conv... |
| iseraltlem1 15849 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15850 | Lemma for ~ iseralt . The... |
| iseraltlem3 15851 | Lemma for ~ iseralt . Fro... |
| iseralt 15852 | The alternating series tes... |
| sumex 15855 | A sum is a set. (Contribu... |
| sumeq1 15856 | Equality theorem for a sum... |
| nfsum1 15857 | Bound-variable hypothesis ... |
| nfsum 15858 | Bound-variable hypothesis ... |
| sumeq2w 15859 | Equality theorem for sum, ... |
| sumeq2ii 15860 | Equality theorem for sum, ... |
| sumeq2 15861 | Equality theorem for sum. ... |
| cbvsum 15862 | Change bound variable in a... |
| cbvsumv 15863 | Change bound variable in a... |
| sumeq1i 15864 | Equality inference for sum... |
| sumeq2i 15865 | Equality inference for sum... |
| sumeq12i 15866 | Equality inference for sum... |
| sumeq1d 15867 | Equality deduction for sum... |
| sumeq2d 15868 | Equality deduction for sum... |
| sumeq2dv 15869 | Equality deduction for sum... |
| sumeq2sdv 15870 | Equality deduction for sum... |
| 2sumeq2dv 15871 | Equality deduction for dou... |
| sumeq12dv 15872 | Equality deduction for sum... |
| sumeq12rdv 15873 | Equality deduction for sum... |
| sum2id 15874 | The second class argument ... |
| sumfc 15875 | A lemma to facilitate conv... |
| fz1f1o 15876 | A lemma for working with f... |
| sumrblem 15877 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15878 | The sequence of partial su... |
| sumrb 15879 | Rebase the starting point ... |
| summolem3 15880 | Lemma for ~ summo . (Cont... |
| summolem2a 15881 | Lemma for ~ summo . (Cont... |
| summolem2 15882 | Lemma for ~ summo . (Cont... |
| summo 15883 | A sum has at most one limi... |
| zsum 15884 | Series sum with index set ... |
| isum 15885 | Series sum with an upper i... |
| fsum 15886 | The value of a sum over a ... |
| sum0 15887 | Any sum over the empty set... |
| sumz 15888 | Any sum of zero over a sum... |
| fsumf1o 15889 | Re-index a finite sum usin... |
| sumss 15890 | Change the index set to a ... |
| fsumss 15891 | Change the index set to a ... |
| sumss2 15892 | Change the index set of a ... |
| fsumcvg2 15893 | The sequence of partial su... |
| fsumsers 15894 | Special case of series sum... |
| fsumcvg3 15895 | A finite sum is convergent... |
| fsumser 15896 | A finite sum expressed in ... |
| fsumcl2lem 15897 | - Lemma for finite sum clo... |
| fsumcllem 15898 | - Lemma for finite sum clo... |
| fsumcl 15899 | Closure of a finite sum of... |
| fsumrecl 15900 | Closure of a finite sum of... |
| fsumzcl 15901 | Closure of a finite sum of... |
| fsumnn0cl 15902 | Closure of a finite sum of... |
| fsumrpcl 15903 | Closure of a finite sum of... |
| fsumclf 15904 | Closure of a finite sum of... |
| fsumzcl2 15905 | A finite sum with integer ... |
| fsumadd 15906 | The sum of two finite sums... |
| fsumsplit 15907 | Split a sum into two parts... |
| fsumsplitf 15908 | Split a sum into two parts... |
| sumsnf 15909 | A sum of a singleton is th... |
| fsumsplitsn 15910 | Separate out a term in a f... |
| fsumsplit1 15911 | Separate out a term in a f... |
| sumsn 15912 | A sum of a singleton is th... |
| fsum1 15913 | The finite sum of ` A ( k ... |
| sumpr 15914 | A sum over a pair is the s... |
| sumtp 15915 | A sum over a triple is the... |
| sumsns 15916 | A sum of a singleton is th... |
| fsumm1 15917 | Separate out the last term... |
| fzosump1 15918 | Separate out the last term... |
| fsum1p 15919 | Separate out the first ter... |
| fsummsnunz 15920 | A finite sum all of whose ... |
| fsumsplitsnun 15921 | Separate out a term in a f... |
| fsump1 15922 | The addition of the next t... |
| isumclim 15923 | An infinite sum equals the... |
| isumclim2 15924 | A converging series conver... |
| isumclim3 15925 | The sequence of partial fi... |
| sumnul 15926 | The sum of a non-convergen... |
| isumcl 15927 | The sum of a converging in... |
| isummulc2 15928 | An infinite sum multiplied... |
| isummulc1 15929 | An infinite sum multiplied... |
| isumdivc 15930 | An infinite sum divided by... |
| isumrecl 15931 | The sum of a converging in... |
| isumge0 15932 | An infinite sum of nonnega... |
| isumadd 15933 | Addition of infinite sums.... |
| sumsplit 15934 | Split a sum into two parts... |
| fsump1i 15935 | Optimized version of ~ fsu... |
| fsum2dlem 15936 | Lemma for ~ fsum2d - induc... |
| fsum2d 15937 | Write a double sum as a su... |
| fsumxp 15938 | Combine two sums into a si... |
| fsumcnv 15939 | Transform a region of summ... |
| fsumcom2 15940 | Interchange order of summa... |
| fsumcom 15941 | Interchange order of summa... |
| fsum0diaglem 15942 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15943 | Two ways to express "the s... |
| mptfzshft 15944 | 1-1 onto function in maps-... |
| fsumrev 15945 | Reversal of a finite sum. ... |
| fsumshft 15946 | Index shift of a finite su... |
| fsumshftm 15947 | Negative index shift of a ... |
| fsumrev2 15948 | Reversal of a finite sum. ... |
| fsum0diag2 15949 | Two ways to express "the s... |
| fsummulc2 15950 | A finite sum multiplied by... |
| fsummulc1 15951 | A finite sum multiplied by... |
| fsumdivc 15952 | A finite sum divided by a ... |
| fsumneg 15953 | Negation of a finite sum. ... |
| fsumsub 15954 | Split a finite sum over a ... |
| fsum2mul 15955 | Separate the nested sum of... |
| fsumconst 15956 | The sum of constant terms ... |
| fsumconst1 15957 | The sum of 1 over a finite... |
| fsumdifsnconst 15958 | The sum of constant terms ... |
| modfsummodslem1 15959 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15960 | Induction step for ~ modfs... |
| modfsummod 15961 | A finite sum modulo a posi... |
| fsumge0 15962 | If all of the terms of a f... |
| fsumless 15963 | A shorter sum of nonnegati... |
| fsumge1 15964 | A sum of nonnegative numbe... |
| fsum00 15965 | A sum of nonnegative numbe... |
| fsumle 15966 | If all of the terms of fin... |
| fsumlt 15967 | If every term in one finit... |
| fsumabs 15968 | Generalized triangle inequ... |
| telfsumo 15969 | Sum of a telescoping serie... |
| telfsumo2 15970 | Sum of a telescoping serie... |
| telfsum 15971 | Sum of a telescoping serie... |
| telfsum2 15972 | Sum of a telescoping serie... |
| fsumparts 15973 | Summation by parts. (Cont... |
| fsumrelem 15974 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15975 | The real part of a sum. (... |
| fsumim 15976 | The imaginary part of a su... |
| fsumcj 15977 | The complex conjugate of a... |
| fsumrlim 15978 | Limit of a finite sum of c... |
| fsumo1 15979 | The finite sum of eventual... |
| o1fsum 15980 | If ` A ( k ) ` is O(1), th... |
| seqabs 15981 | Generalized triangle inequ... |
| iserabs 15982 | Generalized triangle inequ... |
| cvgcmp 15983 | A comparison test for conv... |
| cvgcmpub 15984 | An upper bound for the lim... |
| cvgcmpce 15985 | A comparison test for conv... |
| abscvgcvg 15986 | An absolutely convergent s... |
| climfsum 15987 | Limit of a finite sum of c... |
| fsumiun 15988 | Sum over a disjoint indexe... |
| hashiun 15989 | The cardinality of a disjo... |
| hash2iun 15990 | The cardinality of a neste... |
| hash2iun1dif1 15991 | The cardinality of a neste... |
| hashrabrex 15992 | The number of elements in ... |
| hashuni 15993 | The cardinality of a disjo... |
| qshash 15994 | The cardinality of a set w... |
| indsum 15995 | Finite sum of a product wi... |
| indsumhash 15996 | The finite sum of the indi... |
| ackbijnn 15997 | Translate the Ackermann bi... |
| binomlem 15998 | Lemma for ~ binom (binomia... |
| binom 15999 | The binomial theorem: ` ( ... |
| binom1p 16000 | Special case of the binomi... |
| binom11 16001 | Special case of the binomi... |
| binom1dif 16002 | A summation for the differ... |
| bcxmaslem1 16003 | Lemma for ~ bcxmas . (Con... |
| bcxmas 16004 | Parallel summation (Christ... |
| incexclem 16005 | Lemma for ~ incexc . (Con... |
| incexc 16006 | The inclusion/exclusion pr... |
| incexc2 16007 | The inclusion/exclusion pr... |
| isumshft 16008 | Index shift of an infinite... |
| isumsplit 16009 | Split off the first ` N ` ... |
| isum1p 16010 | The infinite sum of a conv... |
| isumnn0nn 16011 | Sum from 0 to infinity in ... |
| isumrpcl 16012 | The infinite sum of positi... |
| isumle 16013 | Comparison of two infinite... |
| isumless 16014 | A finite sum of nonnegativ... |
| isumsup2 16015 | An infinite sum of nonnega... |
| isumsup 16016 | An infinite sum of nonnega... |
| isumltss 16017 | A partial sum of a series ... |
| climcndslem1 16018 | Lemma for ~ climcnds : bou... |
| climcndslem2 16019 | Lemma for ~ climcnds : bou... |
| climcnds 16020 | The Cauchy condensation te... |
| divrcnv 16021 | The sequence of reciprocal... |
| divcnv 16022 | The sequence of reciprocal... |
| flo1 16023 | The floor function satisfi... |
| divcnvshft 16024 | Limit of a ratio function.... |
| supcvg 16025 | Extract a sequence ` f ` i... |
| infcvgaux1i 16026 | Auxiliary theorem for appl... |
| infcvgaux2i 16027 | Auxiliary theorem for appl... |
| harmonic 16028 | The harmonic series ` H ` ... |
| arisum 16029 | Arithmetic series sum of t... |
| arisum2 16030 | Arithmetic series sum of t... |
| trireciplem 16031 | Lemma for ~ trirecip . Sh... |
| trirecip 16032 | The sum of the reciprocals... |
| expcnv 16033 | A sequence of powers of a ... |
| explecnv 16034 | A sequence of terms conver... |
| geoserg 16035 | The value of the finite ge... |
| geoser 16036 | The value of the finite ge... |
| pwdif 16037 | The difference of two numb... |
| pwm1geoser 16038 | The n-th power of a number... |
| geolim 16039 | The partial sums in the in... |
| geolim2 16040 | The partial sums in the ge... |
| georeclim 16041 | The limit of a geometric s... |
| geo2sum 16042 | The value of the finite ge... |
| geo2sum2 16043 | The value of the finite ge... |
| geo2lim 16044 | The value of the infinite ... |
| geomulcvg 16045 | The geometric series conve... |
| geoisum 16046 | The infinite sum of ` 1 + ... |
| geoisumr 16047 | The infinite sum of recipr... |
| geoisum1 16048 | The infinite sum of ` A ^ ... |
| geoisum1c 16049 | The infinite sum of ` A x.... |
| 0.999... 16050 | The recurring decimal 0.99... |
| geoihalfsum 16051 | Prove that the infinite ge... |
| cvgrat 16052 | Ratio test for convergence... |
| mertenslem1 16053 | Lemma for ~ mertens . (Co... |
| mertenslem2 16054 | Lemma for ~ mertens . (Co... |
| mertens 16055 | Mertens' theorem. If ` A ... |
| prodf 16056 | An infinite product of com... |
| clim2prod 16057 | The limit of an infinite p... |
| clim2div 16058 | The limit of an infinite p... |
| prodfmul 16059 | The product of two infinit... |
| prodf1 16060 | The value of the partial p... |
| prodf1f 16061 | A one-valued infinite prod... |
| prodfclim1 16062 | The constant one product c... |
| prodfn0 16063 | No term of a nonzero infin... |
| prodfrec 16064 | The reciprocal of an infin... |
| prodfdiv 16065 | The quotient of two infini... |
| ntrivcvg 16066 | A non-trivially converging... |
| ntrivcvgn0 16067 | A product that converges t... |
| ntrivcvgfvn0 16068 | Any value of a product seq... |
| ntrivcvgtail 16069 | A tail of a non-trivially ... |
| ntrivcvgmullem 16070 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 16071 | The product of two non-tri... |
| prodex 16074 | A product is a set. (Cont... |
| prodeq1f 16075 | Equality theorem for a pro... |
| prodeq1 16076 | Equality theorem for a pro... |
| nfcprod1 16077 | Bound-variable hypothesis ... |
| nfcprod 16078 | Bound-variable hypothesis ... |
| prodeq2w 16079 | Equality theorem for produ... |
| prodeq2ii 16080 | Equality theorem for produ... |
| prodeq2 16081 | Equality theorem for produ... |
| cbvprod 16082 | Change bound variable in a... |
| cbvprodv 16083 | Change bound variable in a... |
| cbvprodi 16084 | Change bound variable in a... |
| prodeq1i 16085 | Equality inference for pro... |
| prodeq2i 16086 | Equality inference for pro... |
| prodeq12i 16087 | Equality inference for pro... |
| prodeq1d 16088 | Equality deduction for pro... |
| prodeq2d 16089 | Equality deduction for pro... |
| prodeq2dv 16090 | Equality deduction for pro... |
| prodeq2sdv 16091 | Equality deduction for pro... |
| 2cprodeq2dv 16092 | Equality deduction for dou... |
| prodeq12dv 16093 | Equality deduction for pro... |
| prodeq12rdv 16094 | Equality deduction for pro... |
| prod2id 16095 | The second class argument ... |
| prodrblem 16096 | Lemma for ~ prodrb . (Con... |
| fprodcvg 16097 | The sequence of partial pr... |
| prodrblem2 16098 | Lemma for ~ prodrb . (Con... |
| prodrb 16099 | Rebase the starting point ... |
| prodmolem3 16100 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 16101 | Lemma for ~ prodmo . (Con... |
| prodmolem2 16102 | Lemma for ~ prodmo . (Con... |
| prodmo 16103 | A product has at most one ... |
| zprod 16104 | Series product with index ... |
| iprod 16105 | Series product with an upp... |
| zprodn0 16106 | Nonzero series product wit... |
| iprodn0 16107 | Nonzero series product wit... |
| fprod 16108 | The value of a product ove... |
| fprodntriv 16109 | A non-triviality lemma for... |
| prod0 16110 | A product over the empty s... |
| prod1 16111 | Any product of one over a ... |
| prodfc 16112 | A lemma to facilitate conv... |
| fprodf1o 16113 | Re-index a finite product ... |
| prodss 16114 | Change the index set to a ... |
| fprodss 16115 | Change the index set to a ... |
| fprodser 16116 | A finite product expressed... |
| fprodcl2lem 16117 | Finite product closure lem... |
| fprodcllem 16118 | Finite product closure lem... |
| fprodcl 16119 | Closure of a finite produc... |
| fprodrecl 16120 | Closure of a finite produc... |
| fprodzcl 16121 | Closure of a finite produc... |
| fprodnncl 16122 | Closure of a finite produc... |
| fprodrpcl 16123 | Closure of a finite produc... |
| fprodnn0cl 16124 | Closure of a finite produc... |
| fprodcllemf 16125 | Finite product closure lem... |
| fprodreclf 16126 | Closure of a finite produc... |
| fprodmul 16127 | The product of two finite ... |
| fproddiv 16128 | The quotient of two finite... |
| prodsn 16129 | A product of a singleton i... |
| fprod1 16130 | A finite product of only o... |
| prodsnf 16131 | A product of a singleton i... |
| climprod1 16132 | The limit of a product ove... |
| fprodsplit 16133 | Split a finite product int... |
| fprodm1 16134 | Separate out the last term... |
| fprod1p 16135 | Separate out the first ter... |
| fprodp1 16136 | Multiply in the last term ... |
| fprodm1s 16137 | Separate out the last term... |
| fprodp1s 16138 | Multiply in the last term ... |
| prodsns 16139 | A product of the singleton... |
| fprodfac 16140 | Factorial using product no... |
| fprodabs 16141 | The absolute value of a fi... |
| fprodeq0 16142 | Any finite product contain... |
| fprodshft 16143 | Shift the index of a finit... |
| fprodrev 16144 | Reversal of a finite produ... |
| fprodconst 16145 | The product of constant te... |
| fprodn0 16146 | A finite product of nonzer... |
| fprod2dlem 16147 | Lemma for ~ fprod2d - indu... |
| fprod2d 16148 | Write a double product as ... |
| fprodxp 16149 | Combine two products into ... |
| fprodcnv 16150 | Transform a product region... |
| fprodcom2 16151 | Interchange order of multi... |
| fprodcom 16152 | Interchange product order.... |
| fprod0diag 16153 | Two ways to express "the p... |
| fproddivf 16154 | The quotient of two finite... |
| fprodsplitf 16155 | Split a finite product int... |
| fprodsplitsn 16156 | Separate out a term in a f... |
| fprodsplit1f 16157 | Separate out a term in a f... |
| fprodn0f 16158 | A finite product of nonzer... |
| fprodclf 16159 | Closure of a finite produc... |
| fprodge0 16160 | If all the terms of a fini... |
| fprodeq0g 16161 | Any finite product contain... |
| fprodge1 16162 | If all of the terms of a f... |
| fprodle 16163 | If all the terms of two fi... |
| fprodmodd 16164 | If all factors of two fini... |
| iprodclim 16165 | An infinite product equals... |
| iprodclim2 16166 | A converging product conve... |
| iprodclim3 16167 | The sequence of partial fi... |
| iprodcl 16168 | The product of a non-trivi... |
| iprodrecl 16169 | The product of a non-trivi... |
| iprodmul 16170 | Multiplication of infinite... |
| risefacval 16175 | The value of the rising fa... |
| fallfacval 16176 | The value of the falling f... |
| risefacval2 16177 | One-based value of rising ... |
| fallfacval2 16178 | One-based value of falling... |
| fallfacval3 16179 | A product representation o... |
| risefaccllem 16180 | Lemma for rising factorial... |
| fallfaccllem 16181 | Lemma for falling factoria... |
| risefaccl 16182 | Closure law for rising fac... |
| fallfaccl 16183 | Closure law for falling fa... |
| rerisefaccl 16184 | Closure law for rising fac... |
| refallfaccl 16185 | Closure law for falling fa... |
| nnrisefaccl 16186 | Closure law for rising fac... |
| zrisefaccl 16187 | Closure law for rising fac... |
| zfallfaccl 16188 | Closure law for falling fa... |
| nn0risefaccl 16189 | Closure law for rising fac... |
| rprisefaccl 16190 | Closure law for rising fac... |
| risefallfac 16191 | A relationship between ris... |
| fallrisefac 16192 | A relationship between fal... |
| risefac0 16193 | The value of the rising fa... |
| fallfac0 16194 | The value of the falling f... |
| risefacp1 16195 | The value of the rising fa... |
| fallfacp1 16196 | The value of the falling f... |
| risefacp1d 16197 | The value of the rising fa... |
| fallfacp1d 16198 | The value of the falling f... |
| risefac1 16199 | The value of rising factor... |
| fallfac1 16200 | The value of falling facto... |
| risefacfac 16201 | Relate rising factorial to... |
| fallfacfwd 16202 | The forward difference of ... |
| 0fallfac 16203 | The value of the zero fall... |
| 0risefac 16204 | The value of the zero risi... |
| binomfallfaclem1 16205 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16206 | Lemma for ~ binomfallfac .... |
| binomfallfac 16207 | A version of the binomial ... |
| binomrisefac 16208 | A version of the binomial ... |
| fallfacval4 16209 | Represent the falling fact... |
| bcfallfac 16210 | Binomial coefficient in te... |
| fallfacfac 16211 | Relate falling factorial t... |
| bpolylem 16214 | Lemma for ~ bpolyval . (C... |
| bpolyval 16215 | The value of the Bernoulli... |
| bpoly0 16216 | The value of the Bernoulli... |
| bpoly1 16217 | The value of the Bernoulli... |
| bpolycl 16218 | Closure law for Bernoulli ... |
| bpolysum 16219 | A sum for Bernoulli polyno... |
| bpolydiflem 16220 | Lemma for ~ bpolydif . (C... |
| bpolydif 16221 | Calculate the difference b... |
| fsumkthpow 16222 | A closed-form expression f... |
| bpoly2 16223 | The Bernoulli polynomials ... |
| bpoly3 16224 | The Bernoulli polynomials ... |
| bpoly4 16225 | The Bernoulli polynomials ... |
| fsumcube 16226 | Express the sum of cubes i... |
| eftcl 16239 | Closure of a term in the s... |
| reeftcl 16240 | The terms of the series ex... |
| eftabs 16241 | The absolute value of a te... |
| eftval 16242 | The value of a term in the... |
| efcllem 16243 | Lemma for ~ efcl . The se... |
| ef0lem 16244 | The series defining the ex... |
| efval 16245 | Value of the exponential f... |
| esum 16246 | Value of Euler's constant ... |
| eff 16247 | Domain and codomain of the... |
| efcl 16248 | Closure law for the expone... |
| efcld 16249 | Closure law for the expone... |
| efval2 16250 | Value of the exponential f... |
| efcvg 16251 | The series that defines th... |
| efcvgfsum 16252 | Exponential function conve... |
| reefcl 16253 | The exponential function i... |
| reefcld 16254 | The exponential function i... |
| ere 16255 | Euler's constant ` _e ` = ... |
| ege2le3 16256 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16257 | Value of the exponential f... |
| efcj 16258 | The exponential of a compl... |
| efaddlem 16259 | Lemma for ~ efadd (exponen... |
| efadd 16260 | Sum of exponents law for e... |
| fprodefsum 16261 | Move the exponential funct... |
| efcan 16262 | Cancellation law for expon... |
| efne0d 16263 | The exponential of a compl... |
| efne0 16264 | The exponential of a compl... |
| efne0OLD 16265 | Obsolete version of ~ efne... |
| efneg 16266 | The exponential of the opp... |
| eff2 16267 | The exponential function m... |
| efsub 16268 | Difference of exponents la... |
| efexp 16269 | The exponential of an inte... |
| efzval 16270 | Value of the exponential f... |
| efgt0 16271 | The exponential of a real ... |
| rpefcl 16272 | The exponential of a real ... |
| rpefcld 16273 | The exponential of a real ... |
| eftlcvg 16274 | The tail series of the exp... |
| eftlcl 16275 | Closure of the sum of an i... |
| reeftlcl 16276 | Closure of the sum of an i... |
| eftlub 16277 | An upper bound on the abso... |
| efsep 16278 | Separate out the next term... |
| effsumlt 16279 | The partial sums of the se... |
| eft0val 16280 | The value of the first ter... |
| ef4p 16281 | Separate out the first fou... |
| efgt1p2 16282 | The exponential of a posit... |
| efgt1p 16283 | The exponential of a posit... |
| efgt1 16284 | The exponential of a posit... |
| eflt 16285 | The exponential function o... |
| efle 16286 | The exponential function o... |
| reef11 16287 | The exponential function o... |
| reeff1 16288 | The exponential function m... |
| eflegeo 16289 | The exponential function o... |
| sinval 16290 | Value of the sine function... |
| cosval 16291 | Value of the cosine functi... |
| sinf 16292 | Domain and codomain of the... |
| cosf 16293 | Domain and codomain of the... |
| sincl 16294 | Closure of the sine functi... |
| coscl 16295 | Closure of the cosine func... |
| tanval 16296 | Value of the tangent funct... |
| tancl 16297 | The closure of the tangent... |
| sincld 16298 | Closure of the sine functi... |
| coscld 16299 | Closure of the cosine func... |
| tancld 16300 | Closure of the tangent fun... |
| tanval2 16301 | Express the tangent functi... |
| tanval3 16302 | Express the tangent functi... |
| resinval 16303 | The sine of a real number ... |
| recosval 16304 | The cosine of a real numbe... |
| efi4p 16305 | Separate out the first fou... |
| resin4p 16306 | Separate out the first fou... |
| recos4p 16307 | Separate out the first fou... |
| resincl 16308 | The sine of a real number ... |
| recoscl 16309 | The cosine of a real numbe... |
| retancl 16310 | The closure of the tangent... |
| resincld 16311 | Closure of the sine functi... |
| recoscld 16312 | Closure of the cosine func... |
| retancld 16313 | Closure of the tangent fun... |
| sinneg 16314 | The sine of a negative is ... |
| cosneg 16315 | The cosines of a number an... |
| tanneg 16316 | The tangent of a negative ... |
| sin0 16317 | Value of the sine function... |
| cos0 16318 | Value of the cosine functi... |
| tan0 16319 | The value of the tangent f... |
| efival 16320 | The exponential function i... |
| efmival 16321 | The exponential function i... |
| sinhval 16322 | Value of the hyperbolic si... |
| coshval 16323 | Value of the hyperbolic co... |
| resinhcl 16324 | The hyperbolic sine of a r... |
| rpcoshcl 16325 | The hyperbolic cosine of a... |
| recoshcl 16326 | The hyperbolic cosine of a... |
| retanhcl 16327 | The hyperbolic tangent of ... |
| tanhlt1 16328 | The hyperbolic tangent of ... |
| tanhbnd 16329 | The hyperbolic tangent of ... |
| efeul 16330 | Eulerian representation of... |
| efieq 16331 | The exponentials of two im... |
| sinadd 16332 | Addition formula for sine.... |
| cosadd 16333 | Addition formula for cosin... |
| tanaddlem 16334 | A useful intermediate step... |
| tanadd 16335 | Addition formula for tange... |
| sinsub 16336 | Sine of difference. (Cont... |
| cossub 16337 | Cosine of difference. (Co... |
| addsin 16338 | Sum of sines. (Contribute... |
| subsin 16339 | Difference of sines. (Con... |
| sinmul 16340 | Product of sines can be re... |
| cosmul 16341 | Product of cosines can be ... |
| addcos 16342 | Sum of cosines. (Contribu... |
| subcos 16343 | Difference of cosines. (C... |
| sincossq 16344 | Sine squared plus cosine s... |
| sin2t 16345 | Double-angle formula for s... |
| cos2t 16346 | Double-angle formula for c... |
| cos2tsin 16347 | Double-angle formula for c... |
| sinbnd 16348 | The sine of a real number ... |
| cosbnd 16349 | The cosine of a real numbe... |
| sinbnd2 16350 | The sine of a real number ... |
| cosbnd2 16351 | The cosine of a real numbe... |
| ef01bndlem 16352 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16353 | Bounds on the sine of a po... |
| cos01bnd 16354 | Bounds on the cosine of a ... |
| cos1bnd 16355 | Bounds on the cosine of 1.... |
| cos2bnd 16356 | Bounds on the cosine of 2.... |
| sinltx 16357 | The sine of a positive rea... |
| sin01gt0 16358 | The sine of a positive rea... |
| cos01gt0 16359 | The cosine of a positive r... |
| sin02gt0 16360 | The sine of a positive rea... |
| sincos1sgn 16361 | The signs of the sine and ... |
| sincos2sgn 16362 | The signs of the sine and ... |
| sin4lt0 16363 | The sine of 4 is negative.... |
| absefi 16364 | The absolute value of the ... |
| absef 16365 | The absolute value of the ... |
| absefib 16366 | A complex number is real i... |
| efieq1re 16367 | A number whose imaginary e... |
| demoivre 16368 | De Moivre's Formula. Proo... |
| demoivreALT 16369 | Alternate proof of ~ demoi... |
| eirrlem 16372 | Lemma for ~ eirr . (Contr... |
| eirr 16373 | ` _e ` is irrational. (Co... |
| egt2lt3 16374 | Euler's constant ` _e ` = ... |
| epos 16375 | Euler's constant ` _e ` is... |
| epr 16376 | Euler's constant ` _e ` is... |
| ene0 16377 | ` _e ` is not 0. (Contrib... |
| ene1 16378 | ` _e ` is not 1. (Contrib... |
| xpnnen 16379 | The Cartesian product of t... |
| znnen 16380 | The set of integers and th... |
| qnnen 16381 | The rational numbers are c... |
| rpnnen2lem1 16382 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16383 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16384 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16385 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16386 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16387 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16388 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16389 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16390 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16391 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16392 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16393 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16394 | The other half of ~ rpnnen... |
| rpnnen 16395 | The cardinality of the con... |
| rexpen 16396 | The real numbers are equin... |
| cpnnen 16397 | The complex numbers are eq... |
| rucALT 16398 | Alternate proof of ~ ruc .... |
| ruclem1 16399 | Lemma for ~ ruc (the reals... |
| ruclem2 16400 | Lemma for ~ ruc . Orderin... |
| ruclem3 16401 | Lemma for ~ ruc . The con... |
| ruclem4 16402 | Lemma for ~ ruc . Initial... |
| ruclem6 16403 | Lemma for ~ ruc . Domain ... |
| ruclem7 16404 | Lemma for ~ ruc . Success... |
| ruclem8 16405 | Lemma for ~ ruc . The int... |
| ruclem9 16406 | Lemma for ~ ruc . The fir... |
| ruclem10 16407 | Lemma for ~ ruc . Every f... |
| ruclem11 16408 | Lemma for ~ ruc . Closure... |
| ruclem12 16409 | Lemma for ~ ruc . The sup... |
| ruclem13 16410 | Lemma for ~ ruc . There i... |
| ruc 16411 | The set of positive intege... |
| resdomq 16412 | The set of rationals is st... |
| aleph1re 16413 | There are at least aleph-o... |
| aleph1irr 16414 | There are at least aleph-o... |
| cnso 16415 | The complex numbers can be... |
| sqrt2irrlem 16416 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16417 | The square root of 2 is ir... |
| sqrt2re 16418 | The square root of 2 exist... |
| sqrt2irr0 16419 | The square root of 2 is an... |
| nthruc 16420 | The sequence ` NN ` , ` ZZ... |
| nthruz 16421 | The sequence ` NN ` , ` NN... |
| divides 16424 | Define the divides relatio... |
| dvdsval2 16425 | One nonzero integer divide... |
| dvdsval3 16426 | One nonzero integer divide... |
| dvdszrcl 16427 | Reverse closure for the di... |
| dvdsmod0 16428 | If a positive integer divi... |
| p1modz1 16429 | If a number greater than 1... |
| dvdsmodexp 16430 | If a positive integer divi... |
| nndivdvds 16431 | Strong form of ~ dvdsval2 ... |
| nndivides 16432 | Definition of the divides ... |
| moddvds 16433 | Two ways to say ` A == B `... |
| modm1div 16434 | An integer greater than on... |
| addmulmodb 16435 | An integer plus a product ... |
| dvds0lem 16436 | A lemma to assist theorems... |
| dvds1lem 16437 | A lemma to assist theorems... |
| dvds2lem 16438 | A lemma to assist theorems... |
| iddvds 16439 | An integer divides itself.... |
| 1dvds 16440 | 1 divides any integer. Th... |
| dvds0 16441 | Any integer divides 0. Th... |
| negdvdsb 16442 | An integer divides another... |
| dvdsnegb 16443 | An integer divides another... |
| absdvdsb 16444 | An integer divides another... |
| dvdsabsb 16445 | An integer divides another... |
| 0dvds 16446 | Only 0 is divisible by 0. ... |
| dvdsmul1 16447 | An integer divides a multi... |
| dvdsmul2 16448 | An integer divides a multi... |
| iddvdsexp 16449 | An integer divides a posit... |
| muldvds1 16450 | If a product divides an in... |
| muldvds2 16451 | If a product divides an in... |
| dvdscmul 16452 | Multiplication by a consta... |
| dvdsmulc 16453 | Multiplication by a consta... |
| dvdscmulr 16454 | Cancellation law for the d... |
| dvdsmulcr 16455 | Cancellation law for the d... |
| summodnegmod 16456 | The sum of two integers mo... |
| difmod0 16457 | The difference of two inte... |
| modmulconst 16458 | Constant multiplication in... |
| dvds2ln 16459 | If an integer divides each... |
| dvds2add 16460 | If an integer divides each... |
| dvds2sub 16461 | If an integer divides each... |
| dvds2addd 16462 | Deduction form of ~ dvds2a... |
| dvds2subd 16463 | Deduction form of ~ dvds2s... |
| dvdstr 16464 | The divides relation is tr... |
| dvdstrd 16465 | The divides relation is tr... |
| dvdsmultr1 16466 | If an integer divides anot... |
| dvdsmultr1d 16467 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16468 | If an integer divides anot... |
| dvdsmultr2d 16469 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16470 | If an integer divides eith... |
| dvdssub2 16471 | If an integer divides a di... |
| dvdsadd 16472 | An integer divides another... |
| dvdsaddr 16473 | An integer divides another... |
| dvdssub 16474 | An integer divides another... |
| dvdssubr 16475 | An integer divides another... |
| dvdsadd2b 16476 | Adding a multiple of the b... |
| dvdsaddre2b 16477 | Adding a multiple of the b... |
| fsumdvds 16478 | If every term in a sum is ... |
| dvdslelem 16479 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16480 | The divisors of a positive... |
| dvdsleabs 16481 | The divisors of a nonzero ... |
| dvdsleabs2 16482 | Transfer divisibility to a... |
| dvdsabseq 16483 | If two integers divide eac... |
| dvdseq 16484 | If two nonnegative integer... |
| divconjdvds 16485 | If a nonzero integer ` M `... |
| dvdsdivcl 16486 | The complement of a diviso... |
| dvdsflip 16487 | An involution of the divis... |
| dvdsssfz1 16488 | The set of divisors of a n... |
| dvds1 16489 | The only nonnegative integ... |
| alzdvds 16490 | Only 0 is divisible by all... |
| dvdsext 16491 | Poset extensionality for d... |
| fzm1ndvds 16492 | No number from ` 1 ` to ` ... |
| fzo0dvdseq 16493 | Zero is the only one of th... |
| fzocongeq 16494 | Two different elements of ... |
| addmodlteqALT 16495 | Two nonnegative integers l... |
| dvdsfac 16496 | A positive integer divides... |
| dvdsexp2im 16497 | If an integer divides anot... |
| dvdsexp 16498 | A power divides a power wi... |
| dvdsmod 16499 | Any number ` K ` whose mod... |
| mulmoddvds 16500 | If an integer is divisible... |
| 3dvds 16501 | A rule for divisibility by... |
| 3dvdsdec 16502 | A decimal number is divisi... |
| 3dvds2dec 16503 | A decimal number is divisi... |
| fprodfvdvdsd 16504 | A finite product of intege... |
| fproddvdsd 16505 | A finite product of intege... |
| evenelz 16506 | An even number is an integ... |
| zeo3 16507 | An integer is even or odd.... |
| zeo4 16508 | An integer is even or odd ... |
| zeneo 16509 | No even integer equals an ... |
| odd2np1lem 16510 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16511 | An integer is odd iff it i... |
| even2n 16512 | An integer is even iff it ... |
| oddm1even 16513 | An integer is odd iff its ... |
| oddp1even 16514 | An integer is odd iff its ... |
| oexpneg 16515 | The exponential of the neg... |
| mod2eq0even 16516 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16517 | An integer is 1 modulo 2 i... |
| oddnn02np1 16518 | A nonnegative integer is o... |
| oddge22np1 16519 | An integer greater than on... |
| evennn02n 16520 | A nonnegative integer is e... |
| evennn2n 16521 | A positive integer is even... |
| 2tp1odd 16522 | A number which is twice an... |
| mulsucdiv2z 16523 | An integer multiplied with... |
| sqoddm1div8z 16524 | A squared odd number minus... |
| 2teven 16525 | A number which is twice an... |
| zeo5 16526 | An integer is either even ... |
| evend2 16527 | An integer is even iff its... |
| oddp1d2 16528 | An integer is odd iff its ... |
| zob 16529 | Alternate characterization... |
| oddm1d2 16530 | An integer is odd iff its ... |
| ltoddhalfle 16531 | An integer is less than ha... |
| halfleoddlt 16532 | An integer is greater than... |
| opoe 16533 | The sum of two odds is eve... |
| omoe 16534 | The difference of two odds... |
| opeo 16535 | The sum of an odd and an e... |
| omeo 16536 | The difference of an odd a... |
| z0even 16537 | 2 divides 0. That means 0... |
| n2dvds1 16538 | 2 does not divide 1. That... |
| n2dvdsm1 16539 | 2 does not divide -1. Tha... |
| z2even 16540 | 2 divides 2. That means 2... |
| n2dvds3 16541 | 2 does not divide 3. That... |
| z4even 16542 | 2 divides 4. That means 4... |
| 4dvdseven 16543 | An integer which is divisi... |
| m1expe 16544 | Exponentiation of -1 by an... |
| m1expo 16545 | Exponentiation of -1 by an... |
| m1exp1 16546 | Exponentiation of negative... |
| nn0enne 16547 | A positive integer is an e... |
| nn0ehalf 16548 | The half of an even nonneg... |
| nnehalf 16549 | The half of an even positi... |
| nn0onn 16550 | An odd nonnegative integer... |
| nn0o1gt2 16551 | An odd nonnegative integer... |
| nno 16552 | An alternate characterizat... |
| nn0o 16553 | An alternate characterizat... |
| nn0ob 16554 | Alternate characterization... |
| nn0oddm1d2 16555 | A positive integer is odd ... |
| nnoddm1d2 16556 | A positive integer is odd ... |
| sumeven 16557 | If every term in a sum is ... |
| sumodd 16558 | If every term in a sum is ... |
| evensumodd 16559 | If every term in a sum wit... |
| oddsumodd 16560 | If every term in a sum wit... |
| pwp1fsum 16561 | The n-th power of a number... |
| oddpwp1fsum 16562 | An odd power of a number i... |
| divalglem0 16563 | Lemma for ~ divalg . (Con... |
| divalglem1 16564 | Lemma for ~ divalg . (Con... |
| divalglem2 16565 | Lemma for ~ divalg . (Con... |
| divalglem4 16566 | Lemma for ~ divalg . (Con... |
| divalglem5 16567 | Lemma for ~ divalg . (Con... |
| divalglem6 16568 | Lemma for ~ divalg . (Con... |
| divalglem7 16569 | Lemma for ~ divalg . (Con... |
| divalglem8 16570 | Lemma for ~ divalg . (Con... |
| divalglem9 16571 | Lemma for ~ divalg . (Con... |
| divalglem10 16572 | Lemma for ~ divalg . (Con... |
| divalg 16573 | The division algorithm (th... |
| divalgb 16574 | Express the division algor... |
| divalg2 16575 | The division algorithm (th... |
| divalgmod 16576 | The result of the ` mod ` ... |
| divalgmodcl 16577 | The result of the ` mod ` ... |
| modremain 16578 | The result of the modulo o... |
| ndvdssub 16579 | Corollary of the division ... |
| ndvdsadd 16580 | Corollary of the division ... |
| ndvdsp1 16581 | Special case of ~ ndvdsadd... |
| ndvdsi 16582 | A quick test for non-divis... |
| 5ndvds3 16583 | 5 does not divide 3. (Con... |
| 5ndvds6 16584 | 5 does not divide 6. (Con... |
| flodddiv4 16585 | The floor of an odd intege... |
| fldivndvdslt 16586 | The floor of an integer di... |
| flodddiv4lt 16587 | The floor of an odd number... |
| flodddiv4t2lthalf 16588 | The floor of an odd number... |
| bitsfval 16593 | Expand the definition of t... |
| bitsval 16594 | Expand the definition of t... |
| bitsval2 16595 | Expand the definition of t... |
| bitsss 16596 | The set of bits of an inte... |
| bitsf 16597 | The ` bits ` function is a... |
| bits0 16598 | Value of the zeroth bit. ... |
| bits0e 16599 | The zeroth bit of an even ... |
| bits0o 16600 | The zeroth bit of an odd n... |
| bitsp1 16601 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16602 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16603 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16604 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16605 | The bits of a number are a... |
| bitsmod 16606 | Truncating the bit sequenc... |
| bitsfi 16607 | Every number is associated... |
| bitscmp 16608 | The bit complement of ` N ... |
| 0bits 16609 | The bits of zero. (Contri... |
| m1bits 16610 | The bits of negative one. ... |
| bitsinv1lem 16611 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16612 | There is an explicit inver... |
| bitsinv2 16613 | There is an explicit inver... |
| bitsf1ocnv 16614 | The ` bits ` function rest... |
| bitsf1o 16615 | The ` bits ` function rest... |
| bitsf1 16616 | The ` bits ` function is a... |
| 2ebits 16617 | The bits of a power of two... |
| bitsinv 16618 | The inverse of the ` bits ... |
| bitsinvp1 16619 | Recursive definition of th... |
| sadadd2lem2 16620 | The core of the proof of ~... |
| sadfval 16622 | Define the addition of two... |
| sadcf 16623 | The carry sequence is a se... |
| sadc0 16624 | The initial element of the... |
| sadcp1 16625 | The carry sequence (which ... |
| sadval 16626 | The full adder sequence is... |
| sadcaddlem 16627 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16628 | Non-recursive definition o... |
| sadadd2lem 16629 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16630 | Sum of initial segments of... |
| sadadd3 16631 | Sum of initial segments of... |
| sadcl 16632 | The sum of two sequences i... |
| sadcom 16633 | The adder sequence functio... |
| saddisjlem 16634 | Lemma for ~ sadadd . (Con... |
| saddisj 16635 | The sum of disjoint sequen... |
| sadaddlem 16636 | Lemma for ~ sadadd . (Con... |
| sadadd 16637 | For sequences that corresp... |
| sadid1 16638 | The adder sequence functio... |
| sadid2 16639 | The adder sequence functio... |
| sadasslem 16640 | Lemma for ~ sadass . (Con... |
| sadass 16641 | Sequence addition is assoc... |
| sadeq 16642 | Any element of a sequence ... |
| bitsres 16643 | Restrict the bits of a num... |
| bitsuz 16644 | The bits of a number are a... |
| bitsshft 16645 | Shifting a bit sequence to... |
| smufval 16647 | The multiplication of two ... |
| smupf 16648 | The sequence of partial su... |
| smup0 16649 | The initial element of the... |
| smupp1 16650 | The initial element of the... |
| smuval 16651 | Define the addition of two... |
| smuval2 16652 | The partial sum sequence s... |
| smupvallem 16653 | If ` A ` only has elements... |
| smucl 16654 | The product of two sequenc... |
| smu01lem 16655 | Lemma for ~ smu01 and ~ sm... |
| smu01 16656 | Multiplication of a sequen... |
| smu02 16657 | Multiplication of a sequen... |
| smupval 16658 | Rewrite the elements of th... |
| smup1 16659 | Rewrite ~ smupp1 using onl... |
| smueqlem 16660 | Any element of a sequence ... |
| smueq 16661 | Any element of a sequence ... |
| smumullem 16662 | Lemma for ~ smumul . (Con... |
| smumul 16663 | For sequences that corresp... |
| gcdval 16666 | The value of the ` gcd ` o... |
| gcd0val 16667 | The value, by convention, ... |
| gcdn0val 16668 | The value of the ` gcd ` o... |
| gcdcllem1 16669 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16670 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16671 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16672 | Closure of the ` gcd ` ope... |
| gcddvds 16673 | The gcd of two integers di... |
| dvdslegcd 16674 | An integer which divides b... |
| nndvdslegcd 16675 | A positive integer which d... |
| gcdcl 16676 | Closure of the ` gcd ` ope... |
| gcdnncl 16677 | Closure of the ` gcd ` ope... |
| gcdcld 16678 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16679 | Closure of the ` gcd ` ope... |
| gcdle1d 16680 | The greatest common diviso... |
| gcdle2d 16681 | The greatest common diviso... |
| zeqzmulgcd 16682 | An integer is the product ... |
| divgcdz 16683 | An integer divided by the ... |
| gcdf 16684 | Domain and codomain of the... |
| gcdcom 16685 | The ` gcd ` operator is co... |
| gcdcomd 16686 | The ` gcd ` operator is co... |
| divgcdnn 16687 | A positive integer divided... |
| divgcdnnr 16688 | A positive integer divided... |
| gcdeq0 16689 | The gcd of two integers is... |
| gcdn0gt0 16690 | The gcd of two integers is... |
| gcd0id 16691 | The gcd of 0 and an intege... |
| gcdid0 16692 | The gcd of an integer and ... |
| nn0gcdid0 16693 | The gcd of a nonnegative i... |
| gcdneg 16694 | Negating one operand of th... |
| neggcd 16695 | Negating one operand of th... |
| gcdaddmlem 16696 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16697 | Adding a multiple of one o... |
| gcdadd 16698 | The gcd of two numbers is ... |
| gcdid 16699 | The gcd of a number and it... |
| gcdnn0id 16700 | The gcd of a nonnegative i... |
| gcd1 16701 | The gcd of a number with 1... |
| gcdabs1 16702 | The gcd of the absolute va... |
| gcdabs2 16703 | The gcd of the absolute va... |
| gcdabs 16704 | The gcd of two integers is... |
| modgcd 16705 | The gcd remains unchanged ... |
| 1gcd 16706 | The gcd of one and an inte... |
| gcdmultipled 16707 | The greatest common diviso... |
| gcdmultiplez 16708 | The gcd of a multiple of a... |
| gcdmultiple 16709 | The gcd of a multiple of a... |
| dvdsgcdidd 16710 | The greatest common diviso... |
| 6gcd4e2 16711 | The greatest common diviso... |
| bezoutlem1 16712 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16713 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16714 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16715 | Lemma for ~ bezout . (Con... |
| bezout 16716 | Bézout's identity: ... |
| dvdsgcd 16717 | An integer which divides e... |
| dvdsgcdb 16718 | Biconditional form of ~ dv... |
| dfgcd2 16719 | Alternate definition of th... |
| gcdass 16720 | Associative law for ` gcd ... |
| mulgcd 16721 | Distribute multiplication ... |
| absmulgcd 16722 | Distribute absolute value ... |
| mulgcdr 16723 | Reverse distribution law f... |
| gcddiv 16724 | Division law for the ` gcd... |
| gcdzeq 16725 | A positive integer ` A ` i... |
| gcdeq 16726 | ` A ` is equal to its gcd ... |
| dvdssqim 16727 | Unidirectional form of ~ d... |
| dvdsexpim 16728 | If two numbers are divisib... |
| dvdsexpad 16729 | Deduction associated with ... |
| dvdsmulgcd 16730 | A divisibility equivalent ... |
| rpmulgcd 16731 | If ` K ` and ` M ` are rel... |
| rplpwr 16732 | If ` A ` and ` B ` are rel... |
| rprpwr 16733 | If ` A ` and ` B ` are rel... |
| rppwr 16734 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16735 | If ` A ` and ` B ` are rel... |
| sqgcd 16736 | Square distributes over gc... |
| expgcd 16737 | Exponentiation distributes... |
| nn0expgcd 16738 | Exponentiation distributes... |
| zexpgcd 16739 | Exponentiation distributes... |
| dvdsexpnn 16740 | Two positive integers are ... |
| dvdssqnn 16741 | Two positive integers are ... |
| dvdssq 16742 | Two integers are divisible... |
| bezoutr 16743 | Partial converse to ~ bezo... |
| bezoutr1 16744 | Converse of ~ bezout for w... |
| nn0seqcvgd 16745 | A strictly-decreasing nonn... |
| seq1st 16746 | A sequence whose iteration... |
| algr0 16747 | The value of the algorithm... |
| algrf 16748 | An algorithm is a step fun... |
| algrp1 16749 | The value of the algorithm... |
| alginv 16750 | If ` I ` is an invariant o... |
| algcvg 16751 | One way to prove that an a... |
| algcvgblem 16752 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16753 | Two ways of expressing tha... |
| algcvga 16754 | The countdown function ` C... |
| algfx 16755 | If ` F ` reaches a fixed p... |
| eucalgval2 16756 | The value of the step func... |
| eucalgval 16757 | Euclid's Algorithm ~ eucal... |
| eucalgf 16758 | Domain and codomain of the... |
| eucalginv 16759 | The invariant of the step ... |
| eucalglt 16760 | The second member of the s... |
| eucalgcvga 16761 | Once Euclid's Algorithm ha... |
| eucalg 16762 | Euclid's Algorithm compute... |
| lcmval 16767 | Value of the ` lcm ` opera... |
| lcmcom 16768 | The ` lcm ` operator is co... |
| lcm0val 16769 | The value, by convention, ... |
| lcmn0val 16770 | The value of the ` lcm ` o... |
| lcmcllem 16771 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16772 | Closure of the ` lcm ` ope... |
| dvdslcm 16773 | The lcm of two integers is... |
| lcmledvds 16774 | A positive integer which b... |
| lcmeq0 16775 | The lcm of two integers is... |
| lcmcl 16776 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16777 | The greatest common diviso... |
| lcmneg 16778 | Negating one operand of th... |
| neglcm 16779 | Negating one operand of th... |
| lcmabs 16780 | The lcm of two integers is... |
| lcmgcdlem 16781 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16782 | The product of two numbers... |
| lcmdvds 16783 | The lcm of two integers di... |
| lcmid 16784 | The lcm of an integer and ... |
| lcm1 16785 | The lcm of an integer and ... |
| lcmgcdnn 16786 | The product of two positiv... |
| lcmgcdeq 16787 | Two integers' absolute val... |
| lcmdvdsb 16788 | Biconditional form of ~ lc... |
| lcmass 16789 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16790 | The least common multiple ... |
| 6lcm4e12 16791 | The least common multiple ... |
| absproddvds 16792 | The absolute value of the ... |
| absprodnn 16793 | The absolute value of the ... |
| fissn0dvds 16794 | For each finite subset of ... |
| fissn0dvdsn0 16795 | For each finite subset of ... |
| lcmfval 16796 | Value of the ` _lcm ` func... |
| lcmf0val 16797 | The value, by convention, ... |
| lcmfn0val 16798 | The value of the ` _lcm ` ... |
| lcmfnnval 16799 | The value of the ` _lcm ` ... |
| lcmfcllem 16800 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16801 | Closure of the ` _lcm ` fu... |
| lcmfpr 16802 | The value of the ` _lcm ` ... |
| lcmfcl 16803 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16804 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16805 | The least common multiple ... |
| dvdslcmf 16806 | The least common multiple ... |
| lcmfledvds 16807 | A positive integer which i... |
| lcmf 16808 | Characterization of the le... |
| lcmf0 16809 | The least common multiple ... |
| lcmfsn 16810 | The least common multiple ... |
| lcmftp 16811 | The least common multiple ... |
| lcmfunsnlem1 16812 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16813 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16814 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16815 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16816 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16817 | The least common multiple ... |
| lcmfdvdsb 16818 | Biconditional form of ~ lc... |
| lcmfunsn 16819 | The ` _lcm ` function for ... |
| lcmfun 16820 | The ` _lcm ` function for ... |
| lcmfass 16821 | Associative law for the ` ... |
| lcmf2a3a4e12 16822 | The least common multiple ... |
| lcmflefac 16823 | The least common multiple ... |
| coprmgcdb 16824 | Two positive integers are ... |
| ncoprmgcdne1b 16825 | Two positive integers are ... |
| ncoprmgcdgt1b 16826 | Two positive integers are ... |
| coprmdvds1 16827 | If two positive integers a... |
| coprmdvds 16828 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16829 | If an integer is divisible... |
| mulgcddvds 16830 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16831 | If ` M ` is relatively pri... |
| qredeq 16832 | Two equal reduced fraction... |
| qredeu 16833 | Every rational number has ... |
| rpmul 16834 | If ` K ` is relatively pri... |
| rpdvds 16835 | If ` K ` is relatively pri... |
| coprmprod 16836 | The product of the element... |
| coprmproddvdslem 16837 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16838 | If a positive integer is d... |
| congr 16839 | Definition of congruence b... |
| divgcdcoprm0 16840 | Integers divided by gcd ar... |
| divgcdcoprmex 16841 | Integers divided by gcd ar... |
| cncongr1 16842 | One direction of the bicon... |
| cncongr2 16843 | The other direction of the... |
| cncongr 16844 | Cancellability of Congruen... |
| cncongrcoprm 16845 | Corollary 1 of Cancellabil... |
| isprm 16848 | The predicate "is a prime ... |
| prmnn 16849 | A prime number is a positi... |
| prmz 16850 | A prime number is an integ... |
| prmssnn 16851 | The prime numbers are a su... |
| prmex 16852 | The set of prime numbers e... |
| 0nprm 16853 | 0 is not a prime number. ... |
| 1nprm 16854 | 1 is not a prime number. ... |
| 1idssfct 16855 | The positive divisors of a... |
| isprm2lem 16856 | Lemma for ~ isprm2 . (Con... |
| isprm2 16857 | The predicate "is a prime ... |
| isprm3 16858 | The predicate "is a prime ... |
| isprm4 16859 | The predicate "is a prime ... |
| prmind2 16860 | A variation on ~ prmind as... |
| prmind 16861 | Perform induction over the... |
| dvdsprime 16862 | If ` M ` divides a prime, ... |
| nprm 16863 | A product of two integers ... |
| nprmi 16864 | An inference for composite... |
| dvdsnprmd 16865 | If a number is divisible b... |
| prm2orodd 16866 | A prime number is either 2... |
| 2prm 16867 | 2 is a prime number. (Con... |
| 2mulprm 16868 | A multiple of two is prime... |
| 3prm 16869 | 3 is a prime number. (Con... |
| 4nprm 16870 | 4 is not a prime number. ... |
| prmuz2 16871 | A prime number is an integ... |
| prmssuz2 16872 | The primes are integers gr... |
| prmgt1 16873 | A prime number is an integ... |
| prmm2nn0 16874 | Subtracting 2 from a prime... |
| oddprmgt2 16875 | An odd prime is greater th... |
| oddprmge3 16876 | An odd prime is greater th... |
| ge2nprmge4 16877 | A composite integer greate... |
| sqnprm 16878 | A square is never prime. ... |
| dvdsprm 16879 | An integer greater than or... |
| exprmfct 16880 | Every integer greater than... |
| prmdvdsfz 16881 | Each integer greater than ... |
| nprmdvds1 16882 | No prime number divides 1.... |
| isprm5 16883 | One need only check prime ... |
| isprm7 16884 | One need only check prime ... |
| maxprmfct 16885 | The set of prime factors o... |
| divgcdodd 16886 | Either ` A / ( A gcd B ) `... |
| coprm 16887 | A prime number either divi... |
| prmrp 16888 | Unequal prime numbers are ... |
| euclemma 16889 | Euclid's lemma. A prime n... |
| isprm6 16890 | A number is prime iff it s... |
| prmdvdsexp 16891 | A prime divides a positive... |
| prmdvdsexpb 16892 | A prime divides a positive... |
| prmdvdsexpr 16893 | If a prime divides a nonne... |
| prmdvdssq 16894 | Condition for a prime divi... |
| prmexpb 16895 | Two positive prime powers ... |
| prmfac1 16896 | The factorial of a number ... |
| dvdszzq 16897 | Divisibility for an intege... |
| rpexp 16898 | If two numbers ` A ` and `... |
| rpexp1i 16899 | Relative primality passes ... |
| rpexp12i 16900 | Relative primality passes ... |
| prmndvdsfaclt 16901 | A prime number does not di... |
| prmdvdsbc 16902 | Condition for a prime numb... |
| prmdvdsncoprmbd 16903 | Two positive integers are ... |
| ncoprmlnprm 16904 | If two positive integers a... |
| cncongrprm 16905 | Corollary 2 of Cancellabil... |
| isevengcd2 16906 | The predicate "is an even ... |
| isoddgcd1 16907 | The predicate "is an odd n... |
| 3lcm2e6 16908 | The least common multiple ... |
| qnumval 16913 | Value of the canonical num... |
| qdenval 16914 | Value of the canonical den... |
| qnumdencl 16915 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16916 | The canonical numerator of... |
| qdencl 16917 | The canonical denominator ... |
| fnum 16918 | Canonical numerator define... |
| fden 16919 | Canonical denominator defi... |
| qnumdenbi 16920 | Two numbers are the canoni... |
| qnumdencoprm 16921 | The canonical representati... |
| qeqnumdivden 16922 | Recover a rational number ... |
| qmuldeneqnum 16923 | Multiplying a rational by ... |
| divnumden 16924 | Calculate the reduced form... |
| divdenle 16925 | Reducing a quotient never ... |
| qnumgt0 16926 | A rational is positive iff... |
| qgt0numnn 16927 | A rational is positive iff... |
| nn0gcdsq 16928 | Squaring commutes with GCD... |
| zgcdsq 16929 | ~ nn0gcdsq extended to int... |
| numdensq 16930 | Squaring a rational square... |
| numsq 16931 | Square commutes with canon... |
| densq 16932 | Square commutes with canon... |
| qden1elz 16933 | A rational is an integer i... |
| zsqrtelqelz 16934 | If an integer has a ration... |
| nonsq 16935 | Any integer strictly betwe... |
| posqsqznn 16936 | When a positive rational s... |
| numdenexp 16937 | Elevating a rational numbe... |
| numexp 16938 | Elevating to a nonnegative... |
| denexp 16939 | Elevating to a nonnegative... |
| phival 16944 | Value of the Euler ` phi `... |
| phicl2 16945 | Bounds and closure for the... |
| phicl 16946 | Closure for the value of t... |
| phibndlem 16947 | Lemma for ~ phibnd . (Con... |
| phibnd 16948 | A slightly tighter bound o... |
| phicld 16949 | Closure for the value of t... |
| phi1 16950 | Value of the Euler ` phi `... |
| dfphi2 16951 | Alternate definition of th... |
| hashdvds 16952 | The number of numbers in a... |
| phiprmpw 16953 | Value of the Euler ` phi `... |
| phiprm 16954 | Value of the Euler ` phi `... |
| crth 16955 | The Chinese Remainder Theo... |
| phimullem 16956 | Lemma for ~ phimul . (Con... |
| phimul 16957 | The Euler ` phi ` function... |
| eulerthlem1 16958 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16959 | Lemma for ~ eulerth . (Co... |
| eulerth 16960 | Euler's theorem, a general... |
| fermltl 16961 | Fermat's little theorem. ... |
| prmdiv 16962 | Show an explicit expressio... |
| prmdiveq 16963 | The modular inverse of ` A... |
| prmdivdiv 16964 | The (modular) inverse of t... |
| hashgcdlem 16965 | A correspondence between e... |
| dvdsfi 16966 | A natural number has finit... |
| hashgcdeq 16967 | Number of initial positive... |
| phisum 16968 | The divisor sum identity o... |
| odzval 16969 | Value of the order functio... |
| odzcllem 16970 | - Lemma for ~ odzcl , show... |
| odzcl 16971 | The order of a group eleme... |
| odzid 16972 | Any element raised to the ... |
| odzdvds 16973 | The only powers of ` A ` t... |
| odzphi 16974 | The order of any group ele... |
| modprm1div 16975 | A prime number divides an ... |
| m1dvdsndvds 16976 | If an integer minus 1 is d... |
| modprminv 16977 | Show an explicit expressio... |
| modprminveq 16978 | The modular inverse of ` A... |
| vfermltl 16979 | Variant of Fermat's little... |
| vfermltlALT 16980 | Alternate proof of ~ vferm... |
| powm2modprm 16981 | If an integer minus 1 is d... |
| reumodprminv 16982 | For any prime number and f... |
| modprm0 16983 | For two positive integers ... |
| nnnn0modprm0 16984 | For a positive integer and... |
| modprmn0modprm0 16985 | For an integer not being 0... |
| coprimeprodsq 16986 | If three numbers are copri... |
| coprimeprodsq2 16987 | If three numbers are copri... |
| oddprm 16988 | A prime not equal to ` 2 `... |
| nnoddn2prm 16989 | A prime not equal to ` 2 `... |
| oddn2prm 16990 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16991 | A number is a prime number... |
| prm23lt5 16992 | A prime less than 5 is eit... |
| prm23ge5 16993 | A prime is either 2 or 3 o... |
| pythagtriplem1 16994 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16995 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16996 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16997 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16998 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16999 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 17000 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 17001 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 17002 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 17003 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 17004 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 17005 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 17006 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 17007 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 17008 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 17009 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 17010 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 17011 | Lemma for ~ pythagtrip . ... |
| pythagtrip 17012 | Parameterize the Pythagore... |
| iserodd 17013 | Collect the odd terms in a... |
| pclem 17016 | - Lemma for the prime powe... |
| pcprecl 17017 | Closure of the prime power... |
| pcprendvds 17018 | Non-divisibility property ... |
| pcprendvds2 17019 | Non-divisibility property ... |
| pcpre1 17020 | Value of the prime power p... |
| pcpremul 17021 | Multiplicative property of... |
| pcval 17022 | The value of the prime pow... |
| pceulem 17023 | Lemma for ~ pceu . (Contr... |
| pceu 17024 | Uniqueness for the prime p... |
| pczpre 17025 | Connect the prime count pr... |
| pczcl 17026 | Closure of the prime power... |
| pccl 17027 | Closure of the prime power... |
| pccld 17028 | Closure of the prime power... |
| pcmul 17029 | Multiplication property of... |
| pcdiv 17030 | Division property of the p... |
| pcqmul 17031 | Multiplication property of... |
| pc0 17032 | The value of the prime pow... |
| pc1 17033 | Value of the prime count f... |
| pcqcl 17034 | Closure of the general pri... |
| pcqdiv 17035 | Division property of the p... |
| pcrec 17036 | Prime power of a reciproca... |
| pcexp 17037 | Prime power of an exponent... |
| pcxnn0cl 17038 | Extended nonnegative integ... |
| pcxcl 17039 | Extended real closure of t... |
| pcge0 17040 | The prime count of an inte... |
| pczdvds 17041 | Defining property of the p... |
| pcdvds 17042 | Defining property of the p... |
| pczndvds 17043 | Defining property of the p... |
| pcndvds 17044 | Defining property of the p... |
| pczndvds2 17045 | The remainder after dividi... |
| pcndvds2 17046 | The remainder after dividi... |
| pcdvdsb 17047 | ` P ^ A ` divides ` N ` if... |
| pcelnn 17048 | There are a positive numbe... |
| pceq0 17049 | There are zero powers of a... |
| pcidlem 17050 | The prime count of a prime... |
| pcid 17051 | The prime count of a prime... |
| pcneg 17052 | The prime count of a negat... |
| pcabs 17053 | The prime count of an abso... |
| pcdvdstr 17054 | The prime count increases ... |
| pcgcd1 17055 | The prime count of a GCD i... |
| pcgcd 17056 | The prime count of a GCD i... |
| pc2dvds 17057 | A characterization of divi... |
| pc11 17058 | The prime count function, ... |
| pcz 17059 | The prime count function c... |
| pcprmpw2 17060 | Self-referential expressio... |
| pcprmpw 17061 | Self-referential expressio... |
| dvdsprmpweq 17062 | If a positive integer divi... |
| dvdsprmpweqnn 17063 | If an integer greater than... |
| dvdsprmpweqle 17064 | If a positive integer divi... |
| difsqpwdvds 17065 | If the difference of two s... |
| pcaddlem 17066 | Lemma for ~ pcadd . The o... |
| pcadd 17067 | An inequality for the prim... |
| pcadd2 17068 | The inequality of ~ pcadd ... |
| pcmptcl 17069 | Closure for the prime powe... |
| pcmpt 17070 | Construct a function with ... |
| pcmpt2 17071 | Dividing two prime count m... |
| pcmptdvds 17072 | The partial products of th... |
| pcprod 17073 | The product of the primes ... |
| sumhash 17074 | The sum of 1 over a set is... |
| fldivp1 17075 | The difference between the... |
| pcfaclem 17076 | Lemma for ~ pcfac . (Cont... |
| pcfac 17077 | Calculate the prime count ... |
| pcbc 17078 | Calculate the prime count ... |
| qexpz 17079 | If a power of a rational n... |
| expnprm 17080 | A second or higher power o... |
| oddprmdvds 17081 | Every positive integer whi... |
| prmpwdvds 17082 | A relation involving divis... |
| pockthlem 17083 | Lemma for ~ pockthg . (Co... |
| pockthg 17084 | The generalized Pocklingto... |
| pockthi 17085 | Pocklington's theorem, whi... |
| unbenlem 17086 | Lemma for ~ unben . (Cont... |
| unben 17087 | An unbounded set of positi... |
| infpnlem1 17088 | Lemma for ~ infpn . The s... |
| infpnlem2 17089 | Lemma for ~ infpn . For a... |
| infpn 17090 | There exist infinitely man... |
| infpn2 17091 | There exist infinitely man... |
| prmunb 17092 | The primes are unbounded. ... |
| prminf 17093 | There are an infinite numb... |
| prmreclem1 17094 | Lemma for ~ prmrec . Prop... |
| prmreclem2 17095 | Lemma for ~ prmrec . Ther... |
| prmreclem3 17096 | Lemma for ~ prmrec . The ... |
| prmreclem4 17097 | Lemma for ~ prmrec . Show... |
| prmreclem5 17098 | Lemma for ~ prmrec . Here... |
| prmreclem6 17099 | Lemma for ~ prmrec . If t... |
| prmrec 17100 | The sum of the reciprocals... |
| 1arithlem1 17101 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 17102 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 17103 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 17104 | Lemma for ~ 1arith . (Con... |
| 1arith 17105 | Fundamental theorem of ari... |
| 1arith2 17106 | Fundamental theorem of ari... |
| elgz 17109 | Elementhood in the gaussia... |
| gzcn 17110 | A gaussian integer is a co... |
| zgz 17111 | An integer is a gaussian i... |
| igz 17112 | ` _i ` is a gaussian integ... |
| gznegcl 17113 | The gaussian integers are ... |
| gzcjcl 17114 | The gaussian integers are ... |
| gzaddcl 17115 | The gaussian integers are ... |
| gzmulcl 17116 | The gaussian integers are ... |
| gzreim 17117 | Construct a gaussian integ... |
| gzsubcl 17118 | The gaussian integers are ... |
| gzabssqcl 17119 | The squared norm of a gaus... |
| 4sqlem5 17120 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 17121 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 17122 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 17123 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 17124 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 17125 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 17126 | Lemma for ~ 4sq . The set... |
| 4sqlem2 17127 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 17128 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 17129 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 17130 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 17131 | Lemma for ~ mul4sq : algeb... |
| mul4sq 17132 | Euler's four-square identi... |
| 4sqlem11 17133 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 17134 | Lemma for ~ 4sq . For any... |
| 4sqlem13 17135 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 17136 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 17137 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 17138 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 17139 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 17140 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 17141 | Lemma for ~ 4sq . The pro... |
| 4sq 17142 | Lagrange's four-square the... |
| vdwapfval 17149 | Define the arithmetic prog... |
| vdwapf 17150 | The arithmetic progression... |
| vdwapval 17151 | Value of the arithmetic pr... |
| vdwapun 17152 | Remove the first element o... |
| vdwapid1 17153 | The first element of an ar... |
| vdwap0 17154 | Value of a length-1 arithm... |
| vdwap1 17155 | Value of a length-1 arithm... |
| vdwmc 17156 | The predicate " The ` <. R... |
| vdwmc2 17157 | Expand out the definition ... |
| vdwpc 17158 | The predicate " The colori... |
| vdwlem1 17159 | Lemma for ~ vdw . (Contri... |
| vdwlem2 17160 | Lemma for ~ vdw . (Contri... |
| vdwlem3 17161 | Lemma for ~ vdw . (Contri... |
| vdwlem4 17162 | Lemma for ~ vdw . (Contri... |
| vdwlem5 17163 | Lemma for ~ vdw . (Contri... |
| vdwlem6 17164 | Lemma for ~ vdw . (Contri... |
| vdwlem7 17165 | Lemma for ~ vdw . (Contri... |
| vdwlem8 17166 | Lemma for ~ vdw . (Contri... |
| vdwlem9 17167 | Lemma for ~ vdw . (Contri... |
| vdwlem10 17168 | Lemma for ~ vdw . Set up ... |
| vdwlem11 17169 | Lemma for ~ vdw . (Contri... |
| vdwlem12 17170 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 17171 | Lemma for ~ vdw . Main in... |
| vdw 17172 | Van der Waerden's theorem.... |
| vdwnnlem1 17173 | Corollary of ~ vdw , and l... |
| vdwnnlem2 17174 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 17175 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 17176 | Van der Waerden's theorem,... |
| ramtlecl 17178 | The set ` T ` of numbers w... |
| hashbcval 17180 | Value of the "binomial set... |
| hashbccl 17181 | The binomial set is a fini... |
| hashbcss 17182 | Subset relation for the bi... |
| hashbc0 17183 | The set of subsets of size... |
| hashbc2 17184 | The size of the binomial s... |
| 0hashbc 17185 | There are no subsets of th... |
| ramval 17186 | The value of the Ramsey nu... |
| ramcl2lem 17187 | Lemma for extended real cl... |
| ramtcl 17188 | The Ramsey number has the ... |
| ramtcl2 17189 | The Ramsey number is an in... |
| ramtub 17190 | The Ramsey number is a low... |
| ramub 17191 | The Ramsey number is a low... |
| ramub2 17192 | It is sufficient to check ... |
| rami 17193 | The defining property of a... |
| ramcl2 17194 | The Ramsey number is eithe... |
| ramxrcl 17195 | The Ramsey number is an ex... |
| ramubcl 17196 | If the Ramsey number is up... |
| ramlb 17197 | Establish a lower bound on... |
| 0ram 17198 | The Ramsey number when ` M... |
| 0ram2 17199 | The Ramsey number when ` M... |
| ram0 17200 | The Ramsey number when ` R... |
| 0ramcl 17201 | Lemma for ~ ramcl : Exist... |
| ramz2 17202 | The Ramsey number when ` F... |
| ramz 17203 | The Ramsey number when ` F... |
| ramub1lem1 17204 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 17205 | Lemma for ~ ramub1 . (Con... |
| ramub1 17206 | Inductive step for Ramsey'... |
| ramcl 17207 | Ramsey's theorem: the Rams... |
| ramsey 17208 | Ramsey's theorem with the ... |
| prmoval 17211 | Value of the primorial fun... |
| prmocl 17212 | Closure of the primorial f... |
| prmone0 17213 | The primorial function is ... |
| prmo0 17214 | The primorial of 0. (Cont... |
| prmo1 17215 | The primorial of 1. (Cont... |
| prmop1 17216 | The primorial of a success... |
| prmonn2 17217 | Value of the primorial fun... |
| prmo2 17218 | The primorial of 2. (Cont... |
| prmo3 17219 | The primorial of 3. (Cont... |
| prmdvdsprmo 17220 | The primorial of a number ... |
| prmdvdsprmop 17221 | The primorial of a number ... |
| fvprmselelfz 17222 | The value of the prime sel... |
| fvprmselgcd1 17223 | The greatest common diviso... |
| prmolefac 17224 | The primorial of a positiv... |
| prmodvdslcmf 17225 | The primorial of a nonnega... |
| prmolelcmf 17226 | The primorial of a positiv... |
| prmgaplem1 17227 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17228 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17229 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17230 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17231 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17232 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17233 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17234 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17235 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17236 | Lemma for ~ prmgap . (Con... |
| prmgap 17237 | The prime gap theorem: for... |
| prmgaplcm 17238 | Alternate proof of ~ prmga... |
| prmgapprmolem 17239 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17240 | Alternate proof of ~ prmga... |
| dec2dvds 17241 | Divisibility by two is obv... |
| dec5dvds 17242 | Divisibility by five is ob... |
| dec5dvds2 17243 | Divisibility by five is ob... |
| dec5nprm 17244 | A decimal number greater t... |
| dec2nprm 17245 | A decimal number greater t... |
| modxai 17246 | Add exponents in a power m... |
| mod2xi 17247 | Double exponents in a powe... |
| modxp1i 17248 | Add one to an exponent in ... |
| mod2xnegi 17249 | Version of ~ mod2xi where ... |
| modsubi 17250 | Subtract from within a mod... |
| gcdi 17251 | Calculate a GCD via Euclid... |
| gcdmodi 17252 | Calculate a GCD via Euclid... |
| numexp0 17253 | Calculate an integer power... |
| numexp1 17254 | Calculate an integer power... |
| numexpp1 17255 | Calculate an integer power... |
| numexp2x 17256 | Double an integer power. ... |
| decsplit0b 17257 | Split a decimal number int... |
| decsplit0 17258 | Split a decimal number int... |
| decsplit1 17259 | Split a decimal number int... |
| decsplit 17260 | Split a decimal number int... |
| karatsuba 17261 | The Karatsuba multiplicati... |
| 2exp4 17262 | Two to the fourth power is... |
| 2exp5 17263 | Two to the fifth power is ... |
| 2exp6 17264 | Two to the sixth power is ... |
| 2exp7 17265 | Two to the seventh power i... |
| 2exp8 17266 | Two to the eighth power is... |
| 2exp11 17267 | Two to the eleventh power ... |
| 2exp16 17268 | Two to the sixteenth power... |
| 3exp3 17269 | Three to the third power i... |
| 2expltfac 17270 | The factorial grows faster... |
| cshwsidrepsw 17271 | If cyclically shifting a w... |
| cshwsidrepswmod0 17272 | If cyclically shifting a w... |
| cshwshashlem1 17273 | If cyclically shifting a w... |
| cshwshashlem2 17274 | If cyclically shifting a w... |
| cshwshashlem3 17275 | If cyclically shifting a w... |
| cshwsdisj 17276 | The singletons resulting b... |
| cshwsiun 17277 | The set of (different!) wo... |
| cshwsex 17278 | The class of (different!) ... |
| cshws0 17279 | The size of the set of (di... |
| cshwrepswhash1 17280 | The size of the set of (di... |
| cshwshashnsame 17281 | If a word (not consisting ... |
| cshwshash 17282 | If a word has a length bei... |
| prmlem0 17283 | Lemma for ~ prmlem1a and ~... |
| prmlem1a 17284 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1 17285 | A quick proof skeleton to ... |
| 5prm 17286 | 5 is a prime number. (Con... |
| 6nprm 17287 | 6 is not a prime number. ... |
| 7prm 17288 | 7 is a prime number. (Con... |
| 8nprm 17289 | 8 is not a prime number. ... |
| 9nprm 17290 | 9 is not a prime number. ... |
| 10nprm 17291 | 10 is not a prime number. ... |
| 10nprmOLD 17292 | Obsolete version of ~ 10np... |
| 11prm 17293 | 11 is a prime number. (Co... |
| 13prm 17294 | 13 is a prime number. (Co... |
| 17prm 17295 | 17 is a prime number. (Co... |
| 19prm 17296 | 19 is a prime number. (Co... |
| 23prm 17297 | 23 is a prime number. (Co... |
| prmlem2 17298 | Our last proving session g... |
| 37prm 17299 | 37 is a prime number. (Co... |
| 43prm 17300 | 43 is a prime number. (Co... |
| 83prm 17301 | 83 is a prime number. (Co... |
| 139prm 17302 | 139 is a prime number. (C... |
| 163prm 17303 | 163 is a prime number. (C... |
| 317prm 17304 | 317 is a prime number. (C... |
| 631prm 17305 | 631 is a prime number. (C... |
| prmo4 17306 | The primorial of 4. (Cont... |
| prmo5 17307 | The primorial of 5. (Cont... |
| prmo6 17308 | The primorial of 6. (Cont... |
| 1259lem1 17309 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17310 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17311 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17312 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17313 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17314 | 1259 is a prime number. (... |
| 2503lem1 17315 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17316 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17317 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17318 | 2503 is a prime number. (... |
| 4001lem1 17319 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17320 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17321 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17322 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17323 | 4001 is a prime number. (... |
| brstruct 17326 | The structure relation is ... |
| isstruct2 17327 | The property of being a st... |
| structex 17328 | A structure is a set. (Co... |
| structn0fun 17329 | A structure without the em... |
| isstruct 17330 | The property of being a st... |
| structcnvcnv 17331 | Two ways to express the re... |
| structfung 17332 | The converse of the conver... |
| structfun 17333 | Convert between two kinds ... |
| structfn 17334 | Convert between two kinds ... |
| strleun 17335 | Combine two structures int... |
| strle1 17336 | Make a structure from a si... |
| strle2 17337 | Make a structure from a pa... |
| strle3 17338 | Make a structure from a tr... |
| sbcie2s 17339 | A special version of class... |
| sbcie3s 17340 | A special version of class... |
| reldmsets 17343 | The structure override ope... |
| setsvalg 17344 | Value of the structure rep... |
| setsval 17345 | Value of the structure rep... |
| fvsetsid 17346 | The value of the structure... |
| fsets 17347 | The structure replacement ... |
| setsdm 17348 | The domain of a structure ... |
| setsfun 17349 | A structure with replaceme... |
| setsfun0 17350 | A structure with replaceme... |
| setsn0fun 17351 | The value of the structure... |
| setsstruct2 17352 | An extensible structure wi... |
| setsexstruct2 17353 | An extensible structure wi... |
| setsstruct 17354 | An extensible structure wi... |
| wunsets 17355 | Closure of structure repla... |
| setsres 17356 | The structure replacement ... |
| setsabs 17357 | Replacing the same compone... |
| setscom 17358 | Different components can b... |
| sloteq 17361 | Equality theorem for the `... |
| slotfn 17362 | A slot is a function on se... |
| strfvnd 17363 | Deduction version of ~ str... |
| strfvn 17364 | Value of a structure compo... |
| strfvss 17365 | A structure component extr... |
| wunstr 17366 | Closure of a structure ind... |
| str0 17367 | All components of the empt... |
| strfvi 17368 | Structure slot extractors ... |
| fveqprc 17369 | Lemma for showing the equa... |
| oveqprc 17370 | Lemma for showing the equa... |
| wunndx 17373 | Closure of the index extra... |
| ndxarg 17374 | Get the numeric argument f... |
| ndxid 17375 | A structure component extr... |
| strndxid 17376 | The value of a structure c... |
| setsidvald 17377 | Value of the structure rep... |
| strfvd 17378 | Deduction version of ~ str... |
| strfv2d 17379 | Deduction version of ~ str... |
| strfv2 17380 | A variation on ~ strfv to ... |
| strfv 17381 | Extract a structure compon... |
| strfv3 17382 | Variant on ~ strfv for lar... |
| strssd 17383 | Deduction version of ~ str... |
| strss 17384 | Propagate component extrac... |
| setsid 17385 | Value of the structure rep... |
| setsnid 17386 | Value of the structure rep... |
| baseval 17389 | Value of the base set extr... |
| baseid 17390 | Utility theorem: index-ind... |
| basfn 17391 | The base set extractor is ... |
| base0 17392 | The base set of the empty ... |
| elbasfv 17393 | Utility theorem: reverse c... |
| elbasov 17394 | Utility theorem: reverse c... |
| strov2rcl 17395 | Partial reverse closure fo... |
| basendx 17396 | Index value of the base se... |
| basendxnn 17397 | The index value of the bas... |
| basndxelwund 17398 | The index of the base set ... |
| basprssdmsets 17399 | The pair of the base index... |
| opelstrbas 17400 | The base set of a structur... |
| 1strstr 17401 | A constructed one-slot str... |
| 1strbas 17402 | The base set of a construc... |
| 1strwunbndx 17403 | A constructed one-slot str... |
| 1strwun 17404 | A constructed one-slot str... |
| 2strstr 17405 | A constructed two-slot str... |
| 2strbas 17406 | The base set of a construc... |
| 2strop 17407 | The other slot of a constr... |
| reldmress 17410 | The structure restriction ... |
| ressval 17411 | Value of structure restric... |
| ressid2 17412 | General behavior of trivia... |
| ressval2 17413 | Value of nontrivial struct... |
| ressbas 17414 | Base set of a structure re... |
| ressbasssg 17415 | The base set of a restrict... |
| ressbas2 17416 | Base set of a structure re... |
| ressbasss 17417 | The base set of a restrict... |
| ressbasssOLD 17418 | Obsolete version of ~ ress... |
| ressbasss2 17419 | The base set of a restrict... |
| resseqnbas 17420 | The components of an exten... |
| ress0 17421 | All restrictions of the em... |
| ressid 17422 | Behavior of trivial restri... |
| ressinbas 17423 | Restriction only cares abo... |
| ressval3d 17424 | Value of structure restric... |
| ressress 17425 | Restriction composition la... |
| ressabs 17426 | Restriction absorption law... |
| wunress 17427 | Closure of structure restr... |
| plusgndx 17454 | Index value of the ~ df-pl... |
| plusgid 17455 | Utility theorem: index-ind... |
| plusgndxnn 17456 | The index of the slot for ... |
| basendxltplusgndx 17457 | The index of the slot for ... |
| basendxnplusgndx 17458 | The slot for the base set ... |
| grpstr 17459 | A constructed group is a s... |
| grpbase 17460 | The base set of a construc... |
| grpplusg 17461 | The operation of a constru... |
| ressplusg 17462 | ` +g ` is unaffected by re... |
| grpbasex 17463 | The base of an explicitly ... |
| grpplusgx 17464 | The operation of an explic... |
| mulrndx 17465 | Index value of the ~ df-mu... |
| mulridx 17466 | Utility theorem: index-ind... |
| basendxnmulrndx 17467 | The slot for the base set ... |
| plusgndxnmulrndx 17468 | The slot for the group (ad... |
| rngstr 17469 | A constructed ring is a st... |
| rngbase 17470 | The base set of a construc... |
| rngplusg 17471 | The additive operation of ... |
| rngmulr 17472 | The multiplicative operati... |
| starvndx 17473 | Index value of the ~ df-st... |
| starvid 17474 | Utility theorem: index-ind... |
| starvndxnbasendx 17475 | The slot for the involutio... |
| starvndxnplusgndx 17476 | The slot for the involutio... |
| starvndxnmulrndx 17477 | The slot for the involutio... |
| ressmulr 17478 | ` .r ` is unaffected by re... |
| ressstarv 17479 | ` *r ` is unaffected by re... |
| srngstr 17480 | A constructed star ring is... |
| srngbase 17481 | The base set of a construc... |
| srngplusg 17482 | The addition operation of ... |
| srngmulr 17483 | The multiplication operati... |
| srnginvl 17484 | The involution function of... |
| scandx 17485 | Index value of the ~ df-sc... |
| scaid 17486 | Utility theorem: index-ind... |
| scandxnbasendx 17487 | The slot for the scalar is... |
| scandxnplusgndx 17488 | The slot for the scalar fi... |
| scandxnmulrndx 17489 | The slot for the scalar fi... |
| vscandx 17490 | Index value of the ~ df-vs... |
| vscaid 17491 | Utility theorem: index-ind... |
| vscandxnbasendx 17492 | The slot for the scalar pr... |
| vscandxnplusgndx 17493 | The slot for the scalar pr... |
| vscandxnmulrndx 17494 | The slot for the scalar pr... |
| vscandxnscandx 17495 | The slot for the scalar pr... |
| lmodstr 17496 | A constructed left module ... |
| lmodbase 17497 | The base set of a construc... |
| lmodplusg 17498 | The additive operation of ... |
| lmodsca 17499 | The set of scalars of a co... |
| lmodvsca 17500 | The scalar product operati... |
| ipndx 17501 | Index value of the ~ df-ip... |
| ipid 17502 | Utility theorem: index-ind... |
| ipndxnbasendx 17503 | The slot for the inner pro... |
| ipndxnplusgndx 17504 | The slot for the inner pro... |
| ipndxnmulrndx 17505 | The slot for the inner pro... |
| slotsdifipndx 17506 | The slot for the scalar is... |
| ipsstr 17507 | Lemma to shorten proofs of... |
| ipsbase 17508 | The base set of a construc... |
| ipsaddg 17509 | The additive operation of ... |
| ipsmulr 17510 | The multiplicative operati... |
| ipssca 17511 | The set of scalars of a co... |
| ipsvsca 17512 | The scalar product operati... |
| ipsip 17513 | The multiplicative operati... |
| resssca 17514 | ` Scalar ` is unaffected b... |
| ressvsca 17515 | ` .s ` is unaffected by re... |
| ressip 17516 | The inner product is unaff... |
| phlstr 17517 | A constructed pre-Hilbert ... |
| phlbase 17518 | The base set of a construc... |
| phlplusg 17519 | The additive operation of ... |
| phlsca 17520 | The ring of scalars of a c... |
| phlvsca 17521 | The scalar product operati... |
| phlip 17522 | The inner product (Hermiti... |
| tsetndx 17523 | Index value of the ~ df-ts... |
| tsetid 17524 | Utility theorem: index-ind... |
| tsetndxnn 17525 | The index of the slot for ... |
| basendxlttsetndx 17526 | The index of the slot for ... |
| tsetndxnbasendx 17527 | The slot for the topology ... |
| tsetndxnplusgndx 17528 | The slot for the topology ... |
| tsetndxnmulrndx 17529 | The slot for the topology ... |
| tsetndxnstarvndx 17530 | The slot for the topology ... |
| slotstnscsi 17531 | The slots ` Scalar ` , ` .... |
| topgrpstr 17532 | A constructed topological ... |
| topgrpbas 17533 | The base set of a construc... |
| topgrpplusg 17534 | The additive operation of ... |
| topgrptset 17535 | The topology of a construc... |
| resstset 17536 | ` TopSet ` is unaffected b... |
| plendx 17537 | Index value of the ~ df-pl... |
| pleid 17538 | Utility theorem: self-refe... |
| plendxnn 17539 | The index value of the ord... |
| basendxltplendx 17540 | The index value of the ` B... |
| plendxnbasendx 17541 | The slot for the order is ... |
| plendxnplusgndx 17542 | The slot for the "less tha... |
| plendxnmulrndx 17543 | The slot for the "less tha... |
| plendxnscandx 17544 | The slot for the "less tha... |
| plendxnvscandx 17545 | The slot for the "less tha... |
| slotsdifplendx 17546 | The index of the slot for ... |
| otpsstr 17547 | Functionality of a topolog... |
| otpsbas 17548 | The base set of a topologi... |
| otpstset 17549 | The open sets of a topolog... |
| otpsle 17550 | The order of a topological... |
| ressle 17551 | ` le ` is unaffected by re... |
| ocndx 17552 | Index value of the ~ df-oc... |
| ocid 17553 | Utility theorem: index-ind... |
| basendxnocndx 17554 | The slot for the orthocomp... |
| plendxnocndx 17555 | The slot for the orthocomp... |
| dsndx 17556 | Index value of the ~ df-ds... |
| dsid 17557 | Utility theorem: index-ind... |
| dsndxnn 17558 | The index of the slot for ... |
| basendxltdsndx 17559 | The index of the slot for ... |
| dsndxnbasendx 17560 | The slot for the distance ... |
| dsndxnplusgndx 17561 | The slot for the distance ... |
| dsndxnmulrndx 17562 | The slot for the distance ... |
| slotsdnscsi 17563 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17564 | The slot for the distance ... |
| slotsdifdsndx 17565 | The index of the slot for ... |
| unifndx 17566 | Index value of the ~ df-un... |
| unifid 17567 | Utility theorem: index-ind... |
| unifndxnn 17568 | The index of the slot for ... |
| basendxltunifndx 17569 | The index of the slot for ... |
| unifndxnbasendx 17570 | The slot for the uniform s... |
| unifndxntsetndx 17571 | The slot for the uniform s... |
| slotsdifunifndx 17572 | The index of the slot for ... |
| ressunif 17573 | ` UnifSet ` is unaffected ... |
| odrngstr 17574 | Functionality of an ordere... |
| odrngbas 17575 | The base set of an ordered... |
| odrngplusg 17576 | The addition operation of ... |
| odrngmulr 17577 | The multiplication operati... |
| odrngtset 17578 | The open sets of an ordere... |
| odrngle 17579 | The order of an ordered me... |
| odrngds 17580 | The metric of an ordered m... |
| ressds 17581 | ` dist ` is unaffected by ... |
| homndx 17582 | Index value of the ~ df-ho... |
| homid 17583 | Utility theorem: index-ind... |
| ccondx 17584 | Index value of the ~ df-cc... |
| ccoid 17585 | Utility theorem: index-ind... |
| slotsbhcdif 17586 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17587 | The index of the slot for ... |
| slotsdifocndx 17588 | The index of the slot for ... |
| resshom 17589 | ` Hom ` is unaffected by r... |
| ressco 17590 | ` comp ` is unaffected by ... |
| restfn 17595 | The subspace topology oper... |
| topnfn 17596 | The topology extractor fun... |
| restval 17597 | The subspace topology indu... |
| elrest 17598 | The predicate "is an open ... |
| elrestr 17599 | Sufficient condition for b... |
| 0rest 17600 | Value of the structure res... |
| restid2 17601 | The subspace topology over... |
| restsspw 17602 | The subspace topology is a... |
| firest 17603 | The finite intersections o... |
| restid 17604 | The subspace topology of t... |
| topnval 17605 | Value of the topology extr... |
| topnid 17606 | Value of the topology extr... |
| topnpropd 17607 | The topology extractor fun... |
| reldmprds 17619 | The structure product is a... |
| prdsbasex 17621 | Lemma for structure produc... |
| imasvalstr 17622 | An image structure value i... |
| prdsvalstr 17623 | Structure product value is... |
| prdsbaslem 17624 | Lemma for ~ prdsbas and si... |
| prdsvallem 17625 | Lemma for ~ prdsval . (Co... |
| prdsval 17626 | Value of the structure pro... |
| prdssca 17627 | Scalar ring of a structure... |
| prdsbas 17628 | Base set of a structure pr... |
| prdsplusg 17629 | Addition in a structure pr... |
| prdsmulr 17630 | Multiplication in a struct... |
| prdsvsca 17631 | Scalar multiplication in a... |
| prdsip 17632 | Inner product in a structu... |
| prdsle 17633 | Structure product weak ord... |
| prdsless 17634 | Closure of the order relat... |
| prdsds 17635 | Structure product distance... |
| prdsdsfn 17636 | Structure product distance... |
| prdstset 17637 | Structure product topology... |
| prdshom 17638 | Structure product hom-sets... |
| prdsco 17639 | Structure product composit... |
| prdsbas2 17640 | The base set of a structur... |
| prdsbasmpt 17641 | A constructed tuple is a p... |
| prdsbasfn 17642 | Points in the structure pr... |
| prdsbasprj 17643 | Each point in a structure ... |
| prdsplusgval 17644 | Value of a componentwise s... |
| prdsplusgfval 17645 | Value of a structure produ... |
| prdsmulrval 17646 | Value of a componentwise r... |
| prdsmulrfval 17647 | Value of a structure produ... |
| prdsleval 17648 | Value of the product order... |
| prdsdsval 17649 | Value of the metric in a s... |
| prdsvscaval 17650 | Scalar multiplication in a... |
| prdsvscafval 17651 | Scalar multiplication of a... |
| prdsbas3 17652 | The base set of an indexed... |
| prdsbasmpt2 17653 | A constructed tuple is a p... |
| prdsbascl 17654 | An element of the base has... |
| prdsdsval2 17655 | Value of the metric in a s... |
| prdsdsval3 17656 | Value of the metric in a s... |
| pwsval 17657 | Value of a structure power... |
| pwsbas 17658 | Base set of a structure po... |
| pwselbasb 17659 | Membership in the base set... |
| pwselbas 17660 | An element of a structure ... |
| pwselbasr 17661 | The reverse direction of ~... |
| pwsplusgval 17662 | Value of addition in a str... |
| pwsmulrval 17663 | Value of multiplication in... |
| pwsle 17664 | Ordering in a structure po... |
| pwsleval 17665 | Ordering in a structure po... |
| pwsvscafval 17666 | Scalar multiplication in a... |
| pwsvscaval 17667 | Scalar multiplication of a... |
| pwssca 17668 | The ring of scalars of a s... |
| pwsdiagel 17669 | Membership of diagonal ele... |
| pwssnf1o 17670 | Triviality of singleton po... |
| imasval 17683 | Value of an image structur... |
| imasbas 17684 | The base set of an image s... |
| imasds 17685 | The distance function of a... |
| imasdsfn 17686 | The distance function is a... |
| imasdsval 17687 | The distance function of a... |
| imasdsval2 17688 | The distance function of a... |
| imasplusg 17689 | The group operation in an ... |
| imasmulr 17690 | The ring multiplication in... |
| imassca 17691 | The scalar field of an ima... |
| imasvsca 17692 | The scalar multiplication ... |
| imasip 17693 | The inner product of an im... |
| imastset 17694 | The topology of an image s... |
| imasle 17695 | The ordering of an image s... |
| f1ocpbllem 17696 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17697 | An injection is compatible... |
| f1ovscpbl 17698 | An injection is compatible... |
| f1olecpbl 17699 | An injection is compatible... |
| imasaddfnlem 17700 | The image structure operat... |
| imasaddvallem 17701 | The operation of an image ... |
| imasaddflem 17702 | The image set operations a... |
| imasaddfn 17703 | The image structure's grou... |
| imasaddval 17704 | The value of an image stru... |
| imasaddf 17705 | The image structure's grou... |
| imasmulfn 17706 | The image structure's ring... |
| imasmulval 17707 | The value of an image stru... |
| imasmulf 17708 | The image structure's ring... |
| imasvscafn 17709 | The image structure's scal... |
| imasvscaval 17710 | The value of an image stru... |
| imasvscaf 17711 | The image structure's scal... |
| imasless 17712 | The order relation defined... |
| imasleval 17713 | The value of the image str... |
| qusval 17714 | Value of a quotient struct... |
| quslem 17715 | The function in ~ qusval i... |
| qusin 17716 | Restrict the equivalence r... |
| qusbas 17717 | Base set of a quotient str... |
| quss 17718 | The scalar field of a quot... |
| divsfval 17719 | Value of the function in ~... |
| ercpbllem 17720 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17721 | Translate the function com... |
| erlecpbl 17722 | Translate the relation com... |
| qusaddvallem 17723 | Value of an operation defi... |
| qusaddflem 17724 | The operation of a quotien... |
| qusaddval 17725 | The addition in a quotient... |
| qusaddf 17726 | The addition in a quotient... |
| qusmulval 17727 | The multiplication in a qu... |
| qusmulf 17728 | The multiplication in a qu... |
| fnpr2o 17729 | Function with a domain of ... |
| fnpr2ob 17730 | Biconditional version of ~... |
| fvpr0o 17731 | The value of a function wi... |
| fvpr1o 17732 | The value of a function wi... |
| fvprif 17733 | The value of the pair func... |
| xpsfrnel 17734 | Elementhood in the target ... |
| xpsfeq 17735 | A function on ` 2o ` is de... |
| xpsfrnel2 17736 | Elementhood in the target ... |
| xpscf 17737 | Equivalent condition for t... |
| xpsfval 17738 | The value of the function ... |
| xpsff1o 17739 | The function appearing in ... |
| xpsfrn 17740 | A short expression for the... |
| xpsff1o2 17741 | The function appearing in ... |
| xpsval 17742 | Value of the binary struct... |
| xpsrnbas 17743 | The indexed structure prod... |
| xpsbas 17744 | The base set of the binary... |
| xpsaddlem 17745 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17746 | Value of the addition oper... |
| xpsmul 17747 | Value of the multiplicatio... |
| xpssca 17748 | Value of the scalar field ... |
| xpsvsca 17749 | Value of the scalar multip... |
| xpsless 17750 | Closure of the ordering in... |
| xpsle 17751 | Value of the ordering in a... |
| ismre 17760 | Property of being a Moore ... |
| fnmre 17761 | The Moore collection gener... |
| mresspw 17762 | A Moore collection is a su... |
| mress 17763 | A Moore-closed subset is a... |
| mre1cl 17764 | In any Moore collection th... |
| mreintcl 17765 | A nonempty collection of c... |
| mreiincl 17766 | A nonempty indexed interse... |
| mrerintcl 17767 | The relative intersection ... |
| mreriincl 17768 | The relative intersection ... |
| mreincl 17769 | Two closed sets have a clo... |
| mreuni 17770 | Since the entire base set ... |
| mreunirn 17771 | Two ways to express the no... |
| ismred 17772 | Properties that determine ... |
| ismred2 17773 | Properties that determine ... |
| mremre 17774 | The Moore collections of s... |
| submre 17775 | The subcollection of a clo... |
| xrsle 17776 | The ordering of the extend... |
| xrge0le 17777 | The "less than or equal to... |
| xrsbas 17778 | The base set of the extend... |
| xrge0base 17779 | The base of the extended n... |
| mrcflem 17780 | The domain and codomain of... |
| fnmrc 17781 | Moore-closure is a well-be... |
| mrcfval 17782 | Value of the function expr... |
| mrcf 17783 | The Moore closure is a fun... |
| mrcval 17784 | Evaluation of the Moore cl... |
| mrccl 17785 | The Moore closure of a set... |
| mrcsncl 17786 | The Moore closure of a sin... |
| mrcid 17787 | The closure of a closed se... |
| mrcssv 17788 | The closure of a set is a ... |
| mrcidb 17789 | A set is closed iff it is ... |
| mrcss 17790 | Closure preserves subset o... |
| mrcssid 17791 | The closure of a set is a ... |
| mrcidb2 17792 | A set is closed iff it con... |
| mrcidm 17793 | The closure operation is i... |
| mrcsscl 17794 | The closure is the minimal... |
| mrcuni 17795 | Idempotence of closure und... |
| mrcun 17796 | Idempotence of closure und... |
| mrcssvd 17797 | The Moore closure of a set... |
| mrcssd 17798 | Moore closure preserves su... |
| mrcssidd 17799 | A set is contained in its ... |
| mrcidmd 17800 | Moore closure is idempoten... |
| mressmrcd 17801 | In a Moore system, if a se... |
| submrc 17802 | In a closure system which ... |
| mrieqvlemd 17803 | In a Moore system, if ` Y ... |
| mrisval 17804 | Value of the set of indepe... |
| ismri 17805 | Criterion for a set to be ... |
| ismri2 17806 | Criterion for a subset of ... |
| ismri2d 17807 | Criterion for a subset of ... |
| ismri2dd 17808 | Definition of independence... |
| mriss 17809 | An independent set of a Mo... |
| mrissd 17810 | An independent set of a Mo... |
| ismri2dad 17811 | Consequence of a set in a ... |
| mrieqvd 17812 | In a Moore system, a set i... |
| mrieqv2d 17813 | In a Moore system, a set i... |
| mrissmrcd 17814 | In a Moore system, if an i... |
| mrissmrid 17815 | In a Moore system, subsets... |
| mreexd 17816 | In a Moore system, the clo... |
| mreexmrid 17817 | In a Moore system whose cl... |
| mreexexlemd 17818 | This lemma is used to gene... |
| mreexexlem2d 17819 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17820 | Base case of the induction... |
| mreexexlem4d 17821 | Induction step of the indu... |
| mreexexd 17822 | Exchange-type theorem. In... |
| mreexdomd 17823 | In a Moore system whose cl... |
| mreexfidimd 17824 | In a Moore system whose cl... |
| isacs 17825 | A set is an algebraic clos... |
| acsmre 17826 | Algebraic closure systems ... |
| isacs2 17827 | In the definition of an al... |
| acsfiel 17828 | A set is closed in an alge... |
| acsfiel2 17829 | A set is closed in an alge... |
| acsmred 17830 | An algebraic closure syste... |
| isacs1i 17831 | A closure system determine... |
| mreacs 17832 | Algebraicity is a composab... |
| acsfn 17833 | Algebraicity of a conditio... |
| acsfn0 17834 | Algebraicity of a point cl... |
| acsfn1 17835 | Algebraicity of a one-argu... |
| acsfn1c 17836 | Algebraicity of a one-argu... |
| acsfn2 17837 | Algebraicity of a two-argu... |
| iscat 17846 | The predicate "is a catego... |
| iscatd 17847 | Properties that determine ... |
| catidex 17848 | Each object in a category ... |
| catideu 17849 | Each object in a category ... |
| cidfval 17850 | Each object in a category ... |
| cidval 17851 | Each object in a category ... |
| cidffn 17852 | The identity arrow constru... |
| cidfn 17853 | The identity arrow operato... |
| catidd 17854 | Deduce the identity arrow ... |
| iscatd2 17855 | Version of ~ iscatd with a... |
| catidcl 17856 | Each object in a category ... |
| catlid 17857 | Left identity property of ... |
| catrid 17858 | Right identity property of... |
| catcocl 17859 | Closure of a composition a... |
| catass 17860 | Associativity of compositi... |
| catcone0 17861 | Composition of non-empty h... |
| 0catg 17862 | Any structure with an empt... |
| 0cat 17863 | The empty set is a categor... |
| homffval 17864 | Value of the functionalize... |
| fnhomeqhomf 17865 | If the Hom-set operation i... |
| homfval 17866 | Value of the functionalize... |
| homffn 17867 | The functionalized Hom-set... |
| homfeq 17868 | Condition for two categori... |
| homfeqd 17869 | If two structures have the... |
| homfeqbas 17870 | Deduce equality of base se... |
| homfeqval 17871 | Value of the functionalize... |
| comfffval 17872 | Value of the functionalize... |
| comffval 17873 | Value of the functionalize... |
| comfval 17874 | Value of the functionalize... |
| comfffval2 17875 | Value of the functionalize... |
| comffval2 17876 | Value of the functionalize... |
| comfval2 17877 | Value of the functionalize... |
| comfffn 17878 | The functionalized composi... |
| comffn 17879 | The functionalized composi... |
| comfeq 17880 | Condition for two categori... |
| comfeqd 17881 | Condition for two categori... |
| comfeqval 17882 | Equality of two compositio... |
| catpropd 17883 | Two structures with the sa... |
| cidpropd 17884 | Two structures with the sa... |
| oppcval 17887 | Value of the opposite cate... |
| oppchomfval 17888 | Hom-sets of the opposite c... |
| oppchom 17889 | Hom-sets of the opposite c... |
| oppccofval 17890 | Composition in the opposit... |
| oppcco 17891 | Composition in the opposit... |
| oppcbas 17892 | Base set of an opposite ca... |
| oppccatid 17893 | Lemma for ~ oppccat . (Co... |
| oppchomf 17894 | Hom-sets of the opposite c... |
| oppcid 17895 | Identity function of an op... |
| oppccat 17896 | An opposite category is a ... |
| 2oppcbas 17897 | The double opposite catego... |
| 2oppchomf 17898 | The double opposite catego... |
| 2oppccomf 17899 | The double opposite catego... |
| oppchomfpropd 17900 | If two categories have the... |
| oppccomfpropd 17901 | If two categories have the... |
| oppccatf 17902 | ` oppCat ` restricted to `... |
| monfval 17907 | Definition of a monomorphi... |
| ismon 17908 | Definition of a monomorphi... |
| ismon2 17909 | Write out the monomorphism... |
| monhom 17910 | A monomorphism is a morphi... |
| moni 17911 | Property of a monomorphism... |
| monpropd 17912 | If two categories have the... |
| oppcmon 17913 | A monomorphism in the oppo... |
| oppcepi 17914 | An epimorphism in the oppo... |
| isepi 17915 | Definition of an epimorphi... |
| isepi2 17916 | Write out the epimorphism ... |
| epihom 17917 | An epimorphism is a morphi... |
| epii 17918 | Property of an epimorphism... |
| sectffval 17925 | Value of the section opera... |
| sectfval 17926 | Value of the section relat... |
| sectss 17927 | The section relation is a ... |
| issect 17928 | The property " ` F ` is a ... |
| issect2 17929 | Property of being a sectio... |
| sectcan 17930 | If ` G ` is a section of `... |
| sectco 17931 | Composition of two section... |
| isofval 17932 | Function value of the func... |
| invffval 17933 | Value of the inverse relat... |
| invfval 17934 | Value of the inverse relat... |
| isinv 17935 | Value of the inverse relat... |
| invss 17936 | The inverse relation is a ... |
| invsym 17937 | The inverse relation is sy... |
| invsym2 17938 | The inverse relation is sy... |
| invfun 17939 | The inverse relation is a ... |
| isoval 17940 | The isomorphisms are the d... |
| inviso1 17941 | If ` G ` is an inverse to ... |
| inviso2 17942 | If ` G ` is an inverse to ... |
| invf 17943 | The inverse relation is a ... |
| invf1o 17944 | The inverse relation is a ... |
| invinv 17945 | The inverse of the inverse... |
| invco 17946 | The composition of two iso... |
| dfiso2 17947 | Alternate definition of an... |
| dfiso3 17948 | Alternate definition of an... |
| inveq 17949 | If there are two inverses ... |
| isofn 17950 | The function value of the ... |
| isohom 17951 | An isomorphism is a homomo... |
| isoco 17952 | The composition of two iso... |
| oppcsect 17953 | A section in the opposite ... |
| oppcsect2 17954 | A section in the opposite ... |
| oppcinv 17955 | An inverse in the opposite... |
| oppciso 17956 | An isomorphism in the oppo... |
| sectmon 17957 | If ` F ` is a section of `... |
| monsect 17958 | If ` F ` is a monomorphism... |
| sectepi 17959 | If ` F ` is a section of `... |
| episect 17960 | If ` F ` is an epimorphism... |
| sectid 17961 | The identity is a section ... |
| invid 17962 | The inverse of the identit... |
| idiso 17963 | The identity is an isomorp... |
| idinv 17964 | The inverse of the identit... |
| invisoinvl 17965 | The inverse of an isomorph... |
| invisoinvr 17966 | The inverse of an isomorph... |
| invcoisoid 17967 | The inverse of an isomorph... |
| isocoinvid 17968 | The inverse of an isomorph... |
| rcaninv 17969 | Right cancellation of an i... |
| cicfval 17972 | The set of isomorphic obje... |
| brcic 17973 | The relation "is isomorphi... |
| cic 17974 | Objects ` X ` and ` Y ` in... |
| brcici 17975 | Prove that two objects are... |
| cicref 17976 | Isomorphism is reflexive. ... |
| ciclcl 17977 | Isomorphism implies the le... |
| cicrcl 17978 | Isomorphism implies the ri... |
| cicsym 17979 | Isomorphism is symmetric. ... |
| cictr 17980 | Isomorphism is transitive.... |
| cicer 17981 | Isomorphism is an equivale... |
| sscrel 17988 | The subcategory subset rel... |
| brssc 17989 | The subcategory subset rel... |
| sscpwex 17990 | An analogue of ~ pwex for ... |
| subcrcl 17991 | Reverse closure for the su... |
| sscfn1 17992 | The subcategory subset rel... |
| sscfn2 17993 | The subcategory subset rel... |
| ssclem 17994 | Lemma for ~ ssc1 and simil... |
| isssc 17995 | Value of the subcategory s... |
| ssc1 17996 | Infer subset relation on o... |
| ssc2 17997 | Infer subset relation on m... |
| sscres 17998 | Any function restricted to... |
| sscid 17999 | The subcategory subset rel... |
| ssctr 18000 | The subcategory subset rel... |
| ssceq 18001 | The subcategory subset rel... |
| rescval 18002 | Value of the category rest... |
| rescval2 18003 | Value of the category rest... |
| rescbas 18004 | Base set of the category r... |
| reschom 18005 | Hom-sets of the category r... |
| reschomf 18006 | Hom-sets of the category r... |
| rescco 18007 | Composition in the categor... |
| rescabs 18008 | Restriction absorption law... |
| rescabs2 18009 | Restriction absorption law... |
| issubc 18010 | Elementhood in the set of ... |
| issubc2 18011 | Elementhood in the set of ... |
| 0ssc 18012 | For any category ` C ` , t... |
| 0subcat 18013 | For any category ` C ` , t... |
| catsubcat 18014 | For any category ` C ` , `... |
| subcssc 18015 | An element in the set of s... |
| subcfn 18016 | An element in the set of s... |
| subcss1 18017 | The objects of a subcatego... |
| subcss2 18018 | The morphisms of a subcate... |
| subcidcl 18019 | The identity of the origin... |
| subccocl 18020 | A subcategory is closed un... |
| subccatid 18021 | A subcategory is a categor... |
| subcid 18022 | The identity in a subcateg... |
| subccat 18023 | A subcategory is a categor... |
| issubc3 18024 | Alternate definition of a ... |
| fullsubc 18025 | The full subcategory gener... |
| fullresc 18026 | The category formed by str... |
| resscat 18027 | A category restricted to a... |
| subsubc 18028 | A subcategory of a subcate... |
| relfunc 18037 | The set of functors is a r... |
| funcrcl 18038 | Reverse closure for a func... |
| isfunc 18039 | Value of the set of functo... |
| isfuncd 18040 | Deduce that an operation i... |
| funcf1 18041 | The object part of a funct... |
| funcixp 18042 | The morphism part of a fun... |
| funcf2 18043 | The morphism part of a fun... |
| funcfn2 18044 | The morphism part of a fun... |
| funcid 18045 | A functor maps each identi... |
| funcco 18046 | A functor maps composition... |
| funcsect 18047 | The image of a section und... |
| funcinv 18048 | The image of an inverse un... |
| funciso 18049 | The image of an isomorphis... |
| funcoppc 18050 | A functor on categories yi... |
| idfuval 18051 | Value of the identity func... |
| idfu2nd 18052 | Value of the morphism part... |
| idfu2 18053 | Value of the morphism part... |
| idfu1st 18054 | Value of the object part o... |
| idfu1 18055 | Value of the object part o... |
| idfucl 18056 | The identity functor is a ... |
| cofuval 18057 | Value of the composition o... |
| cofu1st 18058 | Value of the object part o... |
| cofu1 18059 | Value of the object part o... |
| cofu2nd 18060 | Value of the morphism part... |
| cofu2 18061 | Value of the morphism part... |
| cofuval2 18062 | Value of the composition o... |
| cofucl 18063 | The composition of two fun... |
| cofuass 18064 | Functor composition is ass... |
| cofulid 18065 | The identity functor is a ... |
| cofurid 18066 | The identity functor is a ... |
| resfval 18067 | Value of the functor restr... |
| resfval2 18068 | Value of the functor restr... |
| resf1st 18069 | Value of the functor restr... |
| resf2nd 18070 | Value of the functor restr... |
| funcres 18071 | A functor restricted to a ... |
| funcres2b 18072 | Condition for a functor to... |
| funcres2 18073 | A functor into a restricte... |
| idfusubc0 18074 | The identity functor for a... |
| idfusubc 18075 | The identity functor for a... |
| wunfunc 18076 | A weak universe is closed ... |
| funcpropd 18077 | If two categories have the... |
| funcres2c 18078 | Condition for a functor to... |
| fullfunc 18083 | A full functor is a functo... |
| fthfunc 18084 | A faithful functor is a fu... |
| relfull 18085 | The set of full functors i... |
| relfth 18086 | The set of faithful functo... |
| isfull 18087 | Value of the set of full f... |
| isfull2 18088 | Equivalent condition for a... |
| fullfo 18089 | The morphism map of a full... |
| fulli 18090 | The morphism map of a full... |
| isfth 18091 | Value of the set of faithf... |
| isfth2 18092 | Equivalent condition for a... |
| isffth2 18093 | A fully faithful functor i... |
| fthf1 18094 | The morphism map of a fait... |
| fthi 18095 | The morphism map of a fait... |
| ffthf1o 18096 | The morphism map of a full... |
| fullpropd 18097 | If two categories have the... |
| fthpropd 18098 | If two categories have the... |
| fulloppc 18099 | The opposite functor of a ... |
| fthoppc 18100 | The opposite functor of a ... |
| ffthoppc 18101 | The opposite functor of a ... |
| fthsect 18102 | A faithful functor reflect... |
| fthinv 18103 | A faithful functor reflect... |
| fthmon 18104 | A faithful functor reflect... |
| fthepi 18105 | A faithful functor reflect... |
| ffthiso 18106 | A fully faithful functor r... |
| fthres2b 18107 | Condition for a faithful f... |
| fthres2c 18108 | Condition for a faithful f... |
| fthres2 18109 | A faithful functor into a ... |
| idffth 18110 | The identity functor is a ... |
| cofull 18111 | The composition of two ful... |
| cofth 18112 | The composition of two fai... |
| coffth 18113 | The composition of two ful... |
| rescfth 18114 | The inclusion functor from... |
| ressffth 18115 | The inclusion functor from... |
| fullres2c 18116 | Condition for a full funct... |
| ffthres2c 18117 | Condition for a fully fait... |
| inclfusubc 18118 | The "inclusion functor" fr... |
| fnfuc 18123 | The ` FuncCat ` operation ... |
| natfval 18124 | Value of the function givi... |
| isnat 18125 | Property of being a natura... |
| isnat2 18126 | Property of being a natura... |
| natffn 18127 | The natural transformation... |
| natrcl 18128 | Reverse closure for a natu... |
| nat1st2nd 18129 | Rewrite the natural transf... |
| natixp 18130 | A natural transformation i... |
| natcl 18131 | A component of a natural t... |
| natfn 18132 | A natural transformation i... |
| nati 18133 | Naturality property of a n... |
| wunnat 18134 | A weak universe is closed ... |
| catstr 18135 | A category structure is a ... |
| fucval 18136 | Value of the functor categ... |
| fuccofval 18137 | Value of the functor categ... |
| fucbas 18138 | The objects of the functor... |
| fuchom 18139 | The morphisms in the funct... |
| fucco 18140 | Value of the composition o... |
| fuccoval 18141 | Value of the functor categ... |
| fuccocl 18142 | The composition of two nat... |
| fucidcl 18143 | The identity natural trans... |
| fuclid 18144 | Left identity of natural t... |
| fucrid 18145 | Right identity of natural ... |
| fucass 18146 | Associativity of natural t... |
| fuccatid 18147 | The functor category is a ... |
| fuccat 18148 | The functor category is a ... |
| fucid 18149 | The identity morphism in t... |
| fucsect 18150 | Two natural transformation... |
| fucinv 18151 | Two natural transformation... |
| invfuc 18152 | If ` V ( x ) ` is an inver... |
| fuciso 18153 | A natural transformation i... |
| natpropd 18154 | If two categories have the... |
| fucpropd 18155 | If two categories have the... |
| initofn 18162 | ` InitO ` is a function on... |
| termofn 18163 | ` TermO ` is a function on... |
| zeroofn 18164 | ` ZeroO ` is a function on... |
| initorcl 18165 | Reverse closure for an ini... |
| termorcl 18166 | Reverse closure for a term... |
| zeroorcl 18167 | Reverse closure for a zero... |
| initoval 18168 | The value of the initial o... |
| termoval 18169 | The value of the terminal ... |
| zerooval 18170 | The value of the zero obje... |
| isinito 18171 | The predicate "is an initi... |
| istermo 18172 | The predicate "is a termin... |
| iszeroo 18173 | The predicate "is a zero o... |
| isinitoi 18174 | Implication of a class bei... |
| istermoi 18175 | Implication of a class bei... |
| initoid 18176 | For an initial object, the... |
| termoid 18177 | For a terminal object, the... |
| dfinito2 18178 | An initial object is a ter... |
| dftermo2 18179 | A terminal object is an in... |
| dfinito3 18180 | An alternate definition of... |
| dftermo3 18181 | An alternate definition of... |
| initoo 18182 | An initial object is an ob... |
| termoo 18183 | A terminal object is an ob... |
| iszeroi 18184 | Implication of a class bei... |
| 2initoinv 18185 | Morphisms between two init... |
| initoeu1 18186 | Initial objects are essent... |
| initoeu1w 18187 | Initial objects are essent... |
| initoeu2lem0 18188 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18189 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18190 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18191 | Initial objects are essent... |
| 2termoinv 18192 | Morphisms between two term... |
| termoeu1 18193 | Terminal objects are essen... |
| termoeu1w 18194 | Terminal objects are essen... |
| homarcl 18203 | Reverse closure for an arr... |
| homafval 18204 | Value of the disjointified... |
| homaf 18205 | Functionality of the disjo... |
| homaval 18206 | Value of the disjointified... |
| elhoma 18207 | Value of the disjointified... |
| elhomai 18208 | Produce an arrow from a mo... |
| elhomai2 18209 | Produce an arrow from a mo... |
| homarcl2 18210 | Reverse closure for the do... |
| homarel 18211 | An arrow is an ordered pai... |
| homa1 18212 | The first component of an ... |
| homahom2 18213 | The second component of an... |
| homahom 18214 | The second component of an... |
| homadm 18215 | The domain of an arrow wit... |
| homacd 18216 | The codomain of an arrow w... |
| homadmcd 18217 | Decompose an arrow into do... |
| arwval 18218 | The set of arrows is the u... |
| arwrcl 18219 | The first component of an ... |
| arwhoma 18220 | An arrow is contained in t... |
| homarw 18221 | A hom-set is a subset of t... |
| arwdm 18222 | The domain of an arrow is ... |
| arwcd 18223 | The codomain of an arrow i... |
| dmaf 18224 | The domain function is a f... |
| cdaf 18225 | The codomain function is a... |
| arwhom 18226 | The second component of an... |
| arwdmcd 18227 | Decompose an arrow into do... |
| idafval 18232 | Value of the identity arro... |
| idaval 18233 | Value of the identity arro... |
| ida2 18234 | Morphism part of the ident... |
| idahom 18235 | Domain and codomain of the... |
| idadm 18236 | Domain of the identity arr... |
| idacd 18237 | Codomain of the identity a... |
| idaf 18238 | The identity arrow functio... |
| coafval 18239 | The value of the compositi... |
| eldmcoa 18240 | A pair ` <. G , F >. ` is ... |
| dmcoass 18241 | The domain of composition ... |
| homdmcoa 18242 | If ` F : X --> Y ` and ` G... |
| coaval 18243 | Value of composition for c... |
| coa2 18244 | The morphism part of arrow... |
| coahom 18245 | The composition of two com... |
| coapm 18246 | Composition of arrows is a... |
| arwlid 18247 | Left identity of a categor... |
| arwrid 18248 | Right identity of a catego... |
| arwass 18249 | Associativity of compositi... |
| setcval 18252 | Value of the category of s... |
| setcbas 18253 | Set of objects of the cate... |
| setchomfval 18254 | Set of arrows of the categ... |
| setchom 18255 | Set of arrows of the categ... |
| elsetchom 18256 | A morphism of sets is a fu... |
| setccofval 18257 | Composition in the categor... |
| setcco 18258 | Composition in the categor... |
| setccatid 18259 | Lemma for ~ setccat . (Co... |
| setccat 18260 | The category of sets is a ... |
| setcid 18261 | The identity arrow in the ... |
| setcmon 18262 | A monomorphism of sets is ... |
| setcepi 18263 | An epimorphism of sets is ... |
| setcsect 18264 | A section in the category ... |
| setcinv 18265 | An inverse in the category... |
| setciso 18266 | An isomorphism in the cate... |
| resssetc 18267 | The restriction of the cat... |
| funcsetcres2 18268 | A functor into a smaller c... |
| setc2obas 18269 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18270 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18271 | The category of sets in a ... |
| cat1 18272 | The definition of category... |
| catcval 18275 | Value of the category of c... |
| catcbas 18276 | Set of objects of the cate... |
| catchomfval 18277 | Set of arrows of the categ... |
| catchom 18278 | Set of arrows of the categ... |
| catccofval 18279 | Composition in the categor... |
| catcco 18280 | Composition in the categor... |
| catccatid 18281 | Lemma for ~ catccat . (Co... |
| catcid 18282 | The identity arrow in the ... |
| catccat 18283 | The category of categories... |
| resscatc 18284 | The restriction of the cat... |
| catcisolem 18285 | Lemma for ~ catciso . (Co... |
| catciso 18286 | A functor is an isomorphis... |
| catcbascl 18287 | An element of the base set... |
| catcslotelcl 18288 | A slot entry of an element... |
| catcbaselcl 18289 | The base set of an element... |
| catchomcl 18290 | The Hom-set of an element ... |
| catcccocl 18291 | The composition operation ... |
| catcoppccl 18292 | The category of categories... |
| catcfuccl 18293 | The category of categories... |
| fncnvimaeqv 18294 | The inverse images of the ... |
| bascnvimaeqv 18295 | The inverse image of the u... |
| estrcval 18298 | Value of the category of e... |
| estrcbas 18299 | Set of objects of the cate... |
| estrchomfval 18300 | Set of morphisms ("arrows"... |
| estrchom 18301 | The morphisms between exte... |
| elestrchom 18302 | A morphism between extensi... |
| estrccofval 18303 | Composition in the categor... |
| estrcco 18304 | Composition in the categor... |
| estrcbasbas 18305 | An element of the base set... |
| estrccatid 18306 | Lemma for ~ estrccat . (C... |
| estrccat 18307 | The category of extensible... |
| estrcid 18308 | The identity arrow in the ... |
| estrchomfn 18309 | The Hom-set operation in t... |
| estrchomfeqhom 18310 | The functionalized Hom-set... |
| estrreslem1 18311 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18312 | Lemma 2 for ~ estrres . (... |
| estrres 18313 | Any restriction of a categ... |
| funcestrcsetclem1 18314 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18315 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18316 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18317 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18318 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18319 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18320 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18321 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18322 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18323 | The "natural forgetful fun... |
| fthestrcsetc 18324 | The "natural forgetful fun... |
| fullestrcsetc 18325 | The "natural forgetful fun... |
| equivestrcsetc 18326 | The "natural forgetful fun... |
| setc1strwun 18327 | A constructed one-slot str... |
| funcsetcestrclem1 18328 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18329 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18330 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18331 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18332 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18333 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18334 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18335 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18336 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18337 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18338 | The "embedding functor" fr... |
| fthsetcestrc 18339 | The "embedding functor" fr... |
| fullsetcestrc 18340 | The "embedding functor" fr... |
| embedsetcestrc 18341 | The "embedding functor" fr... |
| fnxpc 18350 | The binary product of cate... |
| xpcval 18351 | Value of the binary produc... |
| xpcbas 18352 | Set of objects of the bina... |
| xpchomfval 18353 | Set of morphisms of the bi... |
| xpchom 18354 | Set of morphisms of the bi... |
| relxpchom 18355 | A hom-set in the binary pr... |
| xpccofval 18356 | Value of composition in th... |
| xpcco 18357 | Value of composition in th... |
| xpcco1st 18358 | Value of composition in th... |
| xpcco2nd 18359 | Value of composition in th... |
| xpchom2 18360 | Value of the set of morphi... |
| xpcco2 18361 | Value of composition in th... |
| xpccatid 18362 | The product of two categor... |
| xpcid 18363 | The identity morphism in t... |
| xpccat 18364 | The product of two categor... |
| 1stfval 18365 | Value of the first project... |
| 1stf1 18366 | Value of the first project... |
| 1stf2 18367 | Value of the first project... |
| 2ndfval 18368 | Value of the first project... |
| 2ndf1 18369 | Value of the first project... |
| 2ndf2 18370 | Value of the first project... |
| 1stfcl 18371 | The first projection funct... |
| 2ndfcl 18372 | The second projection func... |
| prfval 18373 | Value of the pairing funct... |
| prf1 18374 | Value of the pairing funct... |
| prf2fval 18375 | Value of the pairing funct... |
| prf2 18376 | Value of the pairing funct... |
| prfcl 18377 | The pairing of functors ` ... |
| prf1st 18378 | Cancellation of pairing wi... |
| prf2nd 18379 | Cancellation of pairing wi... |
| 1st2ndprf 18380 | Break a functor into a pro... |
| catcxpccl 18381 | The category of categories... |
| xpcpropd 18382 | If two categories have the... |
| evlfval 18391 | Value of the evaluation fu... |
| evlf2 18392 | Value of the evaluation fu... |
| evlf2val 18393 | Value of the evaluation na... |
| evlf1 18394 | Value of the evaluation fu... |
| evlfcllem 18395 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18396 | The evaluation functor is ... |
| curfval 18397 | Value of the curry functor... |
| curf1fval 18398 | Value of the object part o... |
| curf1 18399 | Value of the object part o... |
| curf11 18400 | Value of the double evalua... |
| curf12 18401 | The partially evaluated cu... |
| curf1cl 18402 | The partially evaluated cu... |
| curf2 18403 | Value of the curry functor... |
| curf2val 18404 | Value of a component of th... |
| curf2cl 18405 | The curry functor at a mor... |
| curfcl 18406 | The curry functor of a fun... |
| curfpropd 18407 | If two categories have the... |
| uncfval 18408 | Value of the uncurry funct... |
| uncfcl 18409 | The uncurry operation take... |
| uncf1 18410 | Value of the uncurry funct... |
| uncf2 18411 | Value of the uncurry funct... |
| curfuncf 18412 | Cancellation of curry with... |
| uncfcurf 18413 | Cancellation of uncurry wi... |
| diagval 18414 | Define the diagonal functo... |
| diagcl 18415 | The diagonal functor is a ... |
| diag1cl 18416 | The constant functor of ` ... |
| diag11 18417 | Value of the constant func... |
| diag12 18418 | Value of the constant func... |
| diag2 18419 | Value of the diagonal func... |
| diag2cl 18420 | The diagonal functor at a ... |
| curf2ndf 18421 | As shown in ~ diagval , th... |
| hofval 18426 | Value of the Hom functor, ... |
| hof1fval 18427 | The object part of the Hom... |
| hof1 18428 | The object part of the Hom... |
| hof2fval 18429 | The morphism part of the H... |
| hof2val 18430 | The morphism part of the H... |
| hof2 18431 | The morphism part of the H... |
| hofcllem 18432 | Lemma for ~ hofcl . (Cont... |
| hofcl 18433 | Closure of the Hom functor... |
| oppchofcl 18434 | Closure of the opposite Ho... |
| yonval 18435 | Value of the Yoneda embedd... |
| yoncl 18436 | The Yoneda embedding is a ... |
| yon1cl 18437 | The Yoneda embedding at an... |
| yon11 18438 | Value of the Yoneda embedd... |
| yon12 18439 | Value of the Yoneda embedd... |
| yon2 18440 | Value of the Yoneda embedd... |
| hofpropd 18441 | If two categories have the... |
| yonpropd 18442 | If two categories have the... |
| oppcyon 18443 | Value of the opposite Yone... |
| oyoncl 18444 | The opposite Yoneda embedd... |
| oyon1cl 18445 | The opposite Yoneda embedd... |
| yonedalem1 18446 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18447 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18448 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18449 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18450 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18451 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18452 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18453 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18454 | Lemma for ~ yoneda . (Con... |
| yonedainv 18455 | The Yoneda Lemma with expl... |
| yonffthlem 18456 | Lemma for ~ yonffth . (Co... |
| yoneda 18457 | The Yoneda Lemma. There i... |
| yonffth 18458 | The Yoneda Lemma. The Yon... |
| yoniso 18459 | If the codomain is recover... |
| oduval 18462 | Value of an order dual str... |
| oduleval 18463 | Value of the less-equal re... |
| oduleg 18464 | Truth of the less-equal re... |
| odubas 18465 | Base set of an order dual ... |
| isprs 18470 | Property of being a preord... |
| prslem 18471 | Lemma for ~ prsref and ~ p... |
| prsref 18472 | "Less than or equal to" is... |
| prstr 18473 | "Less than or equal to" is... |
| oduprs 18474 | Being a proset is a self-d... |
| isdrs 18475 | Property of being a direct... |
| drsdir 18476 | Direction of a directed se... |
| drsprs 18477 | A directed set is a proset... |
| drsbn0 18478 | The base of a directed set... |
| drsdirfi 18479 | Any _finite_ number of ele... |
| isdrs2 18480 | Directed sets may be defin... |
| ispos 18488 | The predicate "is a poset"... |
| ispos2 18489 | A poset is an antisymmetri... |
| posprs 18490 | A poset is a proset. (Con... |
| posi 18491 | Lemma for poset properties... |
| posref 18492 | A poset ordering is reflex... |
| posasymb 18493 | A poset ordering is asymme... |
| postr 18494 | A poset ordering is transi... |
| 0pos 18495 | Technical lemma to simplif... |
| isposd 18496 | Properties that determine ... |
| isposi 18497 | Properties that determine ... |
| isposix 18498 | Properties that determine ... |
| pospropd 18499 | Posethood is determined on... |
| odupos 18500 | Being a poset is a self-du... |
| oduposb 18501 | Being a poset is a self-du... |
| pltfval 18503 | Value of the less-than rel... |
| pltval 18504 | Less-than relation. ( ~ d... |
| pltle 18505 | "Less than" implies "less ... |
| pltne 18506 | The "less than" relation i... |
| pltirr 18507 | The "less than" relation i... |
| pleval2i 18508 | One direction of ~ pleval2... |
| pleval2 18509 | "Less than or equal to" in... |
| pltnle 18510 | "Less than" implies not co... |
| pltval3 18511 | Alternate expression for t... |
| pltnlt 18512 | The less-than relation imp... |
| pltn2lp 18513 | The less-than relation has... |
| plttr 18514 | The less-than relation is ... |
| pltletr 18515 | Transitive law for chained... |
| plelttr 18516 | Transitive law for chained... |
| pospo 18517 | Write a poset structure in... |
| lubfval 18522 | Value of the least upper b... |
| lubdm 18523 | Domain of the least upper ... |
| lubfun 18524 | The LUB is a function. (C... |
| lubeldm 18525 | Member of the domain of th... |
| lubelss 18526 | A member of the domain of ... |
| lubeu 18527 | Unique existence proper of... |
| lubval 18528 | Value of the least upper b... |
| lubcl 18529 | The least upper bound func... |
| lubprop 18530 | Properties of greatest low... |
| luble 18531 | The greatest lower bound i... |
| lublecllem 18532 | Lemma for ~ lublecl and ~ ... |
| lublecl 18533 | The set of all elements le... |
| lubid 18534 | The LUB of elements less t... |
| glbfval 18535 | Value of the greatest lowe... |
| glbdm 18536 | Domain of the greatest low... |
| glbfun 18537 | The GLB is a function. (C... |
| glbeldm 18538 | Member of the domain of th... |
| glbelss 18539 | A member of the domain of ... |
| glbeu 18540 | Unique existence proper of... |
| glbval 18541 | Value of the greatest lowe... |
| glbcl 18542 | The least upper bound func... |
| glbprop 18543 | Properties of greatest low... |
| glble 18544 | The greatest lower bound i... |
| joinfval 18545 | Value of join function for... |
| joinfval2 18546 | Value of join function for... |
| joindm 18547 | Domain of join function fo... |
| joindef 18548 | Two ways to say that a joi... |
| joinval 18549 | Join value. Since both si... |
| joincl 18550 | Closure of join of element... |
| joindmss 18551 | Subset property of domain ... |
| joinval2lem 18552 | Lemma for ~ joinval2 and ~... |
| joinval2 18553 | Value of join for a poset ... |
| joineu 18554 | Uniqueness of join of elem... |
| joinlem 18555 | Lemma for join properties.... |
| lejoin1 18556 | A join's first argument is... |
| lejoin2 18557 | A join's second argument i... |
| joinle 18558 | A join is less than or equ... |
| meetfval 18559 | Value of meet function for... |
| meetfval2 18560 | Value of meet function for... |
| meetdm 18561 | Domain of meet function fo... |
| meetdef 18562 | Two ways to say that a mee... |
| meetval 18563 | Meet value. Since both si... |
| meetcl 18564 | Closure of meet of element... |
| meetdmss 18565 | Subset property of domain ... |
| meetval2lem 18566 | Lemma for ~ meetval2 and ~... |
| meetval2 18567 | Value of meet for a poset ... |
| meeteu 18568 | Uniqueness of meet of elem... |
| meetlem 18569 | Lemma for meet properties.... |
| lemeet1 18570 | A meet's first argument is... |
| lemeet2 18571 | A meet's second argument i... |
| meetle 18572 | A meet is less than or equ... |
| joincomALT 18573 | The join of a poset is com... |
| joincom 18574 | The join of a poset is com... |
| meetcomALT 18575 | The meet of a poset is com... |
| meetcom 18576 | The meet of a poset is com... |
| join0 18577 | Lemma for ~ odumeet . (Co... |
| meet0 18578 | Lemma for ~ odujoin . (Co... |
| odulub 18579 | Least upper bounds in a du... |
| odujoin 18580 | Joins in a dual order are ... |
| oduglb 18581 | Greatest lower bounds in a... |
| odumeet 18582 | Meets in a dual order are ... |
| poslubmo 18583 | Least upper bounds in a po... |
| posglbmo 18584 | Greatest lower bounds in a... |
| poslubd 18585 | Properties which determine... |
| poslubdg 18586 | Properties which determine... |
| posglbdg 18587 | Properties which determine... |
| istos 18590 | The predicate "is a toset"... |
| tosso 18591 | Write the totally ordered ... |
| tospos 18592 | A Toset is a Poset. (Cont... |
| tleile 18593 | In a Toset, any two elemen... |
| tltnle 18594 | In a Toset, "less than" is... |
| p0val 18599 | Value of poset zero. (Con... |
| p1val 18600 | Value of poset zero. (Con... |
| p0le 18601 | Any element is less than o... |
| ple1 18602 | Any element is less than o... |
| resspos 18603 | The restriction of a Poset... |
| resstos 18604 | The restriction of a Toset... |
| islat 18607 | The predicate "is a lattic... |
| odulatb 18608 | Being a lattice is self-du... |
| odulat 18609 | Being a lattice is self-du... |
| latcl2 18610 | The join and meet of any t... |
| latlem 18611 | Lemma for lattice properti... |
| latpos 18612 | A lattice is a poset. (Co... |
| latjcl 18613 | Closure of join operation ... |
| latmcl 18614 | Closure of meet operation ... |
| latref 18615 | A lattice ordering is refl... |
| latasymb 18616 | A lattice ordering is asym... |
| latasym 18617 | A lattice ordering is asym... |
| lattr 18618 | A lattice ordering is tran... |
| latasymd 18619 | Deduce equality from latti... |
| lattrd 18620 | A lattice ordering is tran... |
| latjcom 18621 | The join of a lattice comm... |
| latlej1 18622 | A join's first argument is... |
| latlej2 18623 | A join's second argument i... |
| latjle12 18624 | A join is less than or equ... |
| latleeqj1 18625 | "Less than or equal to" in... |
| latleeqj2 18626 | "Less than or equal to" in... |
| latjlej1 18627 | Add join to both sides of ... |
| latjlej2 18628 | Add join to both sides of ... |
| latjlej12 18629 | Add join to both sides of ... |
| latnlej 18630 | An idiom to express that a... |
| latnlej1l 18631 | An idiom to express that a... |
| latnlej1r 18632 | An idiom to express that a... |
| latnlej2 18633 | An idiom to express that a... |
| latnlej2l 18634 | An idiom to express that a... |
| latnlej2r 18635 | An idiom to express that a... |
| latjidm 18636 | Lattice join is idempotent... |
| latmcom 18637 | The join of a lattice comm... |
| latmle1 18638 | A meet is less than or equ... |
| latmle2 18639 | A meet is less than or equ... |
| latlem12 18640 | An element is less than or... |
| latleeqm1 18641 | "Less than or equal to" in... |
| latleeqm2 18642 | "Less than or equal to" in... |
| latmlem1 18643 | Add meet to both sides of ... |
| latmlem2 18644 | Add meet to both sides of ... |
| latmlem12 18645 | Add join to both sides of ... |
| latnlemlt 18646 | Negation of "less than or ... |
| latnle 18647 | Equivalent expressions for... |
| latmidm 18648 | Lattice meet is idempotent... |
| latabs1 18649 | Lattice absorption law. F... |
| latabs2 18650 | Lattice absorption law. F... |
| latledi 18651 | An ortholattice is distrib... |
| latmlej11 18652 | Ordering of a meet and joi... |
| latmlej12 18653 | Ordering of a meet and joi... |
| latmlej21 18654 | Ordering of a meet and joi... |
| latmlej22 18655 | Ordering of a meet and joi... |
| lubsn 18656 | The least upper bound of a... |
| latjass 18657 | Lattice join is associativ... |
| latj12 18658 | Swap 1st and 2nd members o... |
| latj32 18659 | Swap 2nd and 3rd members o... |
| latj13 18660 | Swap 1st and 3rd members o... |
| latj31 18661 | Swap 2nd and 3rd members o... |
| latjrot 18662 | Rotate lattice join of 3 c... |
| latj4 18663 | Rearrangement of lattice j... |
| latj4rot 18664 | Rotate lattice join of 4 c... |
| latjjdi 18665 | Lattice join distributes o... |
| latjjdir 18666 | Lattice join distributes o... |
| mod1ile 18667 | The weak direction of the ... |
| mod2ile 18668 | The weak direction of the ... |
| latmass 18669 | Lattice meet is associativ... |
| latdisdlem 18670 | Lemma for ~ latdisd . (Co... |
| latdisd 18671 | In a lattice, joins distri... |
| isclat 18674 | The predicate "is a comple... |
| clatpos 18675 | A complete lattice is a po... |
| clatlem 18676 | Lemma for properties of a ... |
| clatlubcl 18677 | Any subset of the base set... |
| clatlubcl2 18678 | Any subset of the base set... |
| clatglbcl 18679 | Any subset of the base set... |
| clatglbcl2 18680 | Any subset of the base set... |
| oduclatb 18681 | Being a complete lattice i... |
| clatl 18682 | A complete lattice is a la... |
| isglbd 18683 | Properties that determine ... |
| lublem 18684 | Lemma for the least upper ... |
| lubub 18685 | The LUB of a complete latt... |
| lubl 18686 | The LUB of a complete latt... |
| lubss 18687 | Subset law for least upper... |
| lubel 18688 | An element of a set is les... |
| lubun 18689 | The LUB of a union. (Cont... |
| clatglb 18690 | Properties of greatest low... |
| clatglble 18691 | The greatest lower bound i... |
| clatleglb 18692 | Two ways of expressing "le... |
| clatglbss 18693 | Subset law for greatest lo... |
| isdlat 18696 | Property of being a distri... |
| dlatmjdi 18697 | In a distributive lattice,... |
| dlatl 18698 | A distributive lattice is ... |
| odudlatb 18699 | The dual of a distributive... |
| dlatjmdi 18700 | In a distributive lattice,... |
| ipostr 18703 | The structure of ~ df-ipo ... |
| ipoval 18704 | Value of the inclusion pos... |
| ipobas 18705 | Base set of the inclusion ... |
| ipolerval 18706 | Relation of the inclusion ... |
| ipotset 18707 | Topology of the inclusion ... |
| ipole 18708 | Weak order condition of th... |
| ipolt 18709 | Strict order condition of ... |
| ipopos 18710 | The inclusion poset on a f... |
| isipodrs 18711 | Condition for a family of ... |
| ipodrscl 18712 | Direction by inclusion as ... |
| ipodrsfi 18713 | Finite upper bound propert... |
| fpwipodrs 18714 | The finite subsets of any ... |
| ipodrsima 18715 | The monotone image of a di... |
| isacs3lem 18716 | An algebraic closure syste... |
| acsdrsel 18717 | An algebraic closure syste... |
| isacs4lem 18718 | In a closure system in whi... |
| isacs5lem 18719 | If closure commutes with d... |
| acsdrscl 18720 | In an algebraic closure sy... |
| acsficl 18721 | A closure in an algebraic ... |
| isacs5 18722 | A closure system is algebr... |
| isacs4 18723 | A closure system is algebr... |
| isacs3 18724 | A closure system is algebr... |
| acsficld 18725 | In an algebraic closure sy... |
| acsficl2d 18726 | In an algebraic closure sy... |
| acsfiindd 18727 | In an algebraic closure sy... |
| acsmapd 18728 | In an algebraic closure sy... |
| acsmap2d 18729 | In an algebraic closure sy... |
| acsinfd 18730 | In an algebraic closure sy... |
| acsdomd 18731 | In an algebraic closure sy... |
| acsinfdimd 18732 | In an algebraic closure sy... |
| acsexdimd 18733 | In an algebraic closure sy... |
| mrelatglb 18734 | Greatest lower bounds in a... |
| mrelatglb0 18735 | The empty intersection in ... |
| mrelatlub 18736 | Least upper bounds in a Mo... |
| mreclatBAD 18737 | A Moore space is a complet... |
| isps 18742 | The predicate "is a poset"... |
| psrel 18743 | A poset is a relation. (C... |
| psref2 18744 | A poset is antisymmetric a... |
| pstr2 18745 | A poset is transitive. (C... |
| pslem 18746 | Lemma for ~ psref and othe... |
| psdmrn 18747 | The domain and range of a ... |
| psref 18748 | A poset is reflexive. (Co... |
| psrn 18749 | The range of a poset equal... |
| psasym 18750 | A poset is antisymmetric. ... |
| pstr 18751 | A poset is transitive. (C... |
| cnvps 18752 | The converse of a poset is... |
| cnvpsb 18753 | The converse of a poset is... |
| psss 18754 | Any subset of a partially ... |
| psssdm2 18755 | Field of a subposet. (Con... |
| psssdm 18756 | Field of a subposet. (Con... |
| istsr 18757 | The predicate is a toset. ... |
| istsr2 18758 | The predicate is a toset. ... |
| tsrlin 18759 | A toset is a linear order.... |
| tsrlemax 18760 | Two ways of saying a numbe... |
| tsrps 18761 | A toset is a poset. (Cont... |
| cnvtsr 18762 | The converse of a toset is... |
| tsrss 18763 | Any subset of a totally or... |
| ledm 18764 | The domain of ` <_ ` is ` ... |
| lern 18765 | The range of ` <_ ` is ` R... |
| lefld 18766 | The field of the 'less or ... |
| letsr 18767 | The "less than or equal to... |
| isdir 18772 | A condition for a relation... |
| reldir 18773 | A direction is a relation.... |
| dirdm 18774 | A direction's domain is eq... |
| dirref 18775 | A direction is reflexive. ... |
| dirtr 18776 | A direction is transitive.... |
| dirge 18777 | For any two elements of a ... |
| tsrdir 18778 | A totally ordered set is a... |
| ischn 18781 | Property of being a chain.... |
| chnwrd 18782 | A chain is an ordered sequ... |
| chnltm1 18783 | Basic property of a chain.... |
| pfxchn 18784 | A prefix of a chain is sti... |
| nfchnd 18785 | Bound-variable hypothesis ... |
| chneq1 18786 | Equality theorem for chain... |
| chneq2 18787 | Equality theorem for chain... |
| chneq12 18788 | Equality theorem for chain... |
| chnrss 18789 | Chains under a relation ar... |
| chndss 18790 | Chains with an alphabet ar... |
| chnrdss 18791 | Subset theorem for chains.... |
| chnexg 18792 | Chains with a set given fo... |
| nulchn 18793 | Empty set is an increasing... |
| s1chn 18794 | A singleton word is always... |
| chnind 18795 | Induction over a chain. S... |
| chnub 18796 | In a chain, the last eleme... |
| chnlt 18797 | Compare any two elements i... |
| chnso 18798 | A chain induces a total or... |
| chnccats1 18799 | Extend a chain with a sing... |
| chnccat 18800 | Concatenate two chains. (... |
| chnrev 18801 | Reverse of a chain is chai... |
| chnflenfi 18802 | There is a finite number o... |
| chnf 18803 | A chain is a zero-based fi... |
| chnpof1 18804 | A chain under relation whi... |
| chnpoadomd 18805 | A chain under relation whi... |
| chnpolleha 18806 | A chain under relation whi... |
| chnpolfz 18807 | Provided that chain's rela... |
| chnfi 18808 | There is a finite number o... |
| chninf 18809 | There is an infinite numbe... |
| chnfibg 18810 | Given a partial order, the... |
| ex-chn1 18811 | Example: a doubleton of tw... |
| ex-chn2 18812 | Example: sequence <" ZZ NN... |
| ismgm 18817 | The predicate "is a magma"... |
| ismgmn0 18818 | The predicate "is a magma"... |
| mgmcl 18819 | Closure of the operation o... |
| isnmgm 18820 | A condition for a structur... |
| mgmsscl 18821 | If the base set of a magma... |
| plusffval 18822 | The group addition operati... |
| plusfval 18823 | The group addition operati... |
| plusfeq 18824 | If the addition operation ... |
| plusffn 18825 | The group addition operati... |
| mgmplusf 18826 | The group addition functio... |
| mgmn0plusgf 18827 | The restriction of the gro... |
| mgmn0plusgplusf 18828 | The group addition functio... |
| mgmpropd 18829 | If two structures have the... |
| ismgmd 18830 | Deduce a magma from its pr... |
| issstrmgm 18831 | Characterize a substructur... |
| intopsn 18832 | The internal operation for... |
| mgmb1mgm1 18833 | The only magma with a base... |
| mgm0 18834 | Any set with an empty base... |
| mgm0b 18835 | The structure with an empt... |
| mgm1 18836 | The structure with one ele... |
| opifismgm 18837 | A structure with a group a... |
| mgmidmo 18838 | A two-sided identity eleme... |
| mgmideud 18839 | Uniqueness of the left and... |
| grpidval 18840 | The value of the group ide... |
| idvalriota 18841 | The unique value of the gr... |
| grpidpropd 18842 | If two structures have the... |
| fn0g 18843 | The group identity extract... |
| 0g0 18844 | The identity element funct... |
| ismgmid 18845 | Conditions for a class to ... |
| mgmidcl 18846 | The identity element of a ... |
| mgmlrid 18847 | The identity element of a ... |
| 0gisid 18848 | In a structure with an ide... |
| ismgmid2 18849 | Show that a given element ... |
| lidrideqd 18850 | If there is a left and rig... |
| lidrididd 18851 | If there is a left and rig... |
| grpidd 18852 | Deduce the identity elemen... |
| mgmidsssn0 18853 | Property of the set of ide... |
| grpinvalem 18854 | Lemma for ~ grpinva . (Co... |
| grpinva 18855 | Deduce right inverse from ... |
| grprida 18856 | Deduce right identity from... |
| mgmidpfod 18857 | The operation of a magma w... |
| mgmidprnd 18858 | Range of an operation with... |
| mgmfod 18859 | The operation of a magma w... |
| idressidex0 18860 | The restriction of a struc... |
| idressidex 18861 | The restriction of a struc... |
| idressid 18862 | The restriction of a struc... |
| imasmgm2 18863 | The image structure of a u... |
| qusmgm 18864 | Prove that a quotient stru... |
| gsumvalx 18865 | Expand out the substitutio... |
| gsumval 18866 | Expand out the substitutio... |
| gsumpropd 18867 | The group sum depends only... |
| gsumpropd2lem 18868 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18869 | A stronger version of ~ gs... |
| gsummgmpropd 18870 | A stronger version of ~ gs... |
| gsumress 18871 | The group sum in a substru... |
| gsumval1 18872 | Value of the group sum ope... |
| gsum0 18873 | Value of the empty group s... |
| gsumval2a 18874 | Value of the group sum ope... |
| gsumval2 18875 | Value of the group sum ope... |
| gsumsplit1r 18876 | Splitting off the rightmos... |
| gsumprval 18877 | Value of the group sum ope... |
| gsumpr12val 18878 | Value of the group sum ope... |
| mgmhmrcl 18883 | Reverse closure of a magma... |
| submgmrcl 18884 | Reverse closure for submag... |
| ismgmhm 18885 | Property of a magma homomo... |
| mgmhmf 18886 | A magma homomorphism is a ... |
| mgmhmpropd 18887 | Magma homomorphism depends... |
| mgmhmlin 18888 | A magma homomorphism prese... |
| mgmhmf1o 18889 | A magma homomorphism is bi... |
| idmgmhm 18890 | The identity homomorphism ... |
| issubmgm 18891 | Expand definition of a sub... |
| issubmgm2 18892 | Submagmas are subsets that... |
| rabsubmgmd 18893 | Deduction for proving that... |
| submgmss 18894 | Submagmas are subsets of t... |
| submgmid 18895 | Every magma is trivially a... |
| submgmcl 18896 | Submagmas are closed under... |
| submgmmgm 18897 | Submagmas are themselves m... |
| submgmbas 18898 | The base set of a submagma... |
| subsubmgm 18899 | A submagma of a submagma i... |
| resmgmhm 18900 | Restriction of a magma hom... |
| resmgmhm2 18901 | One direction of ~ resmgmh... |
| resmgmhm2b 18902 | Restriction of the codomai... |
| mgmhmco 18903 | The composition of magma h... |
| mgmhmima 18904 | The homomorphic image of a... |
| mgmhmeql 18905 | The equalizer of two magma... |
| submgmacs 18906 | Submagmas are an algebraic... |
| issgrp 18909 | The predicate "is a semigr... |
| issgrpv 18910 | The predicate "is a semigr... |
| issgrpn0 18911 | The predicate "is a semigr... |
| isnsgrp 18912 | A condition for a structur... |
| sgrpmgm 18913 | A semigroup is a magma. (... |
| sgrpass 18914 | A semigroup operation is a... |
| sgrpcl 18915 | Closure of the operation o... |
| sgrp0 18916 | Any set with an empty base... |
| sgrp0b 18917 | The structure with an empt... |
| sgrp1 18918 | The structure with one ele... |
| issgrpd 18919 | Deduce a semigroup from it... |
| sgrppropd 18920 | If two structures are sets... |
| prdsplusgsgrpcl 18921 | Structure product pointwis... |
| prdssgrpd 18922 | The product of a family of... |
| ismnddef 18925 | The predicate "is a monoid... |
| ismnd 18926 | The predicate "is a monoid... |
| isnmnd 18927 | A condition for a structur... |
| sgrpidmnd 18928 | A semigroup with an identi... |
| mndsgrp 18929 | A monoid is a semigroup. ... |
| mndmgm 18930 | A monoid is a magma. (Con... |
| mndcl 18931 | Closure of the operation o... |
| mndass 18932 | A monoid operation is asso... |
| mndid 18933 | A monoid has a two-sided i... |
| mndideu 18934 | The two-sided identity ele... |
| mndideuOLD 18935 | Obsolete version of ~ mndi... |
| mnd32g 18936 | Commutative/associative la... |
| mnd12g 18937 | Commutative/associative la... |
| mnd4g 18938 | Commutative/associative la... |
| mndidcl 18939 | The identity element of a ... |
| mndbn0 18940 | The base set of a monoid i... |
| hashfinmndnn 18941 | A finite monoid has positi... |
| mndplusf 18942 | The group addition operati... |
| mndlrid 18943 | A monoid's identity elemen... |
| mndlid 18944 | The identity element of a ... |
| mndrid 18945 | The identity element of a ... |
| ismndd 18946 | Deduce a monoid from its p... |
| mndpfo 18947 | The addition operation of ... |
| mndfo 18948 | The addition operation of ... |
| mndpfoOLD 18949 | Obsolete version of ~ mndp... |
| mndfoOLD 18950 | Obsolete version of ~ mndf... |
| mndpropd 18951 | If two structures have the... |
| mndprop 18952 | If two structures have the... |
| issubmnd 18953 | Characterize a submonoid b... |
| ress0g 18954 | ` 0g ` is unaffected by re... |
| ress0gOLD 18955 | Obsolete version of ~ ress... |
| submnd0 18956 | The zero of a submonoid is... |
| submnd0OLD 18957 | Obsolete version of ~ subm... |
| mndinvmod 18958 | Uniqueness of an inverse e... |
| mndpsuppss 18959 | The support of a mapping o... |
| mndpsuppfi 18960 | The support of a mapping o... |
| mndpfsupp 18961 | A mapping of a scalar mult... |
| prdsplusgcl 18962 | Structure product pointwis... |
| prdsidlem 18963 | Characterization of identi... |
| prdsmndd 18964 | The product of a family of... |
| prds0g 18965 | The identity in a product ... |
| pwsmnd 18966 | The structure power of a m... |
| pws0g 18967 | The identity in a structur... |
| imasmnd2 18968 | The image structure of a m... |
| imasmnd 18969 | The image structure of a m... |
| imasmndf1 18970 | The image of a monoid unde... |
| xpsmnd 18971 | The binary product of mono... |
| xpsmnd0 18972 | The identity element of a ... |
| mnd1 18973 | The (smallest) structure r... |
| mnd1id 18974 | The singleton element of a... |
| qusmnd 18975 | Prove that a quotient stru... |
| ismhm 18980 | Property of a monoid homom... |
| ismhmd 18981 | Deduction version of ~ ism... |
| mhmrcl1 18982 | Reverse closure of a monoi... |
| mhmrcl2 18983 | Reverse closure of a monoi... |
| mhmf 18984 | A monoid homomorphism is a... |
| ismhm0 18985 | Property of a monoid homom... |
| mhmismgmhm 18986 | Each monoid homomorphism i... |
| mhmpropd 18987 | Monoid homomorphism depend... |
| mhmlin 18988 | A monoid homomorphism comm... |
| mhm0 18989 | A monoid homomorphism pres... |
| idmhm 18990 | The identity homomorphism ... |
| mhmf1o 18991 | A monoid homomorphism is b... |
| mndvcl 18992 | Tuple-wise additive closur... |
| mndvass 18993 | Tuple-wise associativity i... |
| mndvlid 18994 | Tuple-wise left identity i... |
| mndvrid 18995 | Tuple-wise right identity ... |
| mhmvlin 18996 | Tuple extension of monoid ... |
| submrcl 18997 | Reverse closure for submon... |
| issubm 18998 | Expand definition of a sub... |
| issubm2 18999 | Submonoids are subsets tha... |
| issubmndb 19000 | The submonoid predicate. ... |
| issubmd 19001 | Deduction for proving a su... |
| mndissubm 19002 | If the base set of a monoi... |
| resmndismnd 19003 | If the base set of a monoi... |
| submss 19004 | Submonoids are subsets of ... |
| submid 19005 | Every monoid is trivially ... |
| subm0cl 19006 | Submonoids contain zero. ... |
| submcl 19007 | Submonoids are closed unde... |
| submcld 19008 | Submonoids are closed unde... |
| submmnd 19009 | Submonoids are themselves ... |
| submbas 19010 | The base set of a submonoi... |
| subm0 19011 | Submonoids have the same i... |
| subsubm 19012 | A submonoid of a submonoid... |
| 0subm 19013 | The zero submonoid of an a... |
| insubm 19014 | The intersection of two su... |
| 0mhm 19015 | The constant zero linear f... |
| resmhm 19016 | Restriction of a monoid ho... |
| resmhm2 19017 | One direction of ~ resmhm2... |
| resmhm2b 19018 | Restriction of the codomai... |
| mhmco 19019 | The composition of monoid ... |
| mhmimalem 19020 | Lemma for ~ mhmima and sim... |
| mhmima 19021 | The homomorphic image of a... |
| mhmeql 19022 | The equalizer of two monoi... |
| submacs 19023 | Submonoids are an algebrai... |
| mndind 19024 | Induction in a monoid. In... |
| prdspjmhm 19025 | A projection from a produc... |
| pwspjmhm 19026 | A projection from a struct... |
| pwsdiagmhm 19027 | Diagonal monoid homomorphi... |
| pwsco1mhm 19028 | Right composition with a f... |
| pwsco2mhm 19029 | Left composition with a mo... |
| gsumvallem2 19030 | Lemma for properties of th... |
| gsumsubm 19031 | Evaluate a group sum in a ... |
| gsumz 19032 | Value of a group sum over ... |
| gsumwsubmcl 19033 | Closure of the composite i... |
| gsumws1 19034 | A singleton composite reco... |
| gsumwcl 19035 | Closure of the composite o... |
| gsumsgrpccat 19036 | Homomorphic property of no... |
| gsumccat 19037 | Homomorphic property of co... |
| gsumws2 19038 | Valuation of a pair in a m... |
| gsumccatsn 19039 | Homomorphic property of co... |
| gsumspl 19040 | The primary purpose of the... |
| gsumwmhm 19041 | Behavior of homomorphisms ... |
| gsumwspan 19042 | The submonoid generated by... |
| frmdval 19047 | Value of the free monoid c... |
| frmdbas 19048 | The base set of a free mon... |
| frmdelbas 19049 | An element of the base set... |
| frmdplusg 19050 | The monoid operation of a ... |
| frmdadd 19051 | Value of the monoid operat... |
| vrmdfval 19052 | The canonical injection fr... |
| vrmdval 19053 | The value of the generatin... |
| vrmdf 19054 | The mapping from the index... |
| frmdmnd 19055 | A free monoid is a monoid.... |
| frmd0 19056 | The identity of the free m... |
| frmdsssubm 19057 | The set of words taking va... |
| frmdgsum 19058 | Any word in a free monoid ... |
| frmdss2 19059 | A subset of generators is ... |
| frmdup1 19060 | Any assignment of the gene... |
| frmdup2 19061 | The evaluation map has the... |
| frmdup3lem 19062 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 19063 | Universal property of the ... |
| efmnd 19066 | The monoid of endofunction... |
| efmndbas 19067 | The base set of the monoid... |
| efmndbasabf 19068 | The base set of the monoid... |
| elefmndbas 19069 | Two ways of saying a funct... |
| elefmndbas2 19070 | Two ways of saying a funct... |
| efmndbasf 19071 | Elements in the monoid of ... |
| efmndhash 19072 | The monoid of endofunction... |
| efmndbasfi 19073 | The monoid of endofunction... |
| efmndfv 19074 | The function value of an e... |
| efmndtset 19075 | The topology of the monoid... |
| efmndplusg 19076 | The group operation of a m... |
| efmndov 19077 | The value of the group ope... |
| efmndcl 19078 | The group operation of the... |
| efmndtopn 19079 | The topology of the monoid... |
| symggrplem 19080 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 19081 | The monoid of endofunction... |
| efmndsgrp 19082 | The monoid of endofunction... |
| ielefmnd 19083 | The identity function rest... |
| efmndid 19084 | The identity function rest... |
| efmndmnd 19085 | The monoid of endofunction... |
| efmnd0nmnd 19086 | Even the monoid of endofun... |
| efmndbas0 19087 | The base set of the monoid... |
| efmnd1hash 19088 | The monoid of endofunction... |
| efmnd1bas 19089 | The monoid of endofunction... |
| efmnd2hash 19090 | The monoid of endofunction... |
| submefmnd 19091 | If the base set of a monoi... |
| sursubmefmnd 19092 | The set of surjective endo... |
| injsubmefmnd 19093 | The set of injective endof... |
| idressubmefmnd 19094 | The singleton containing o... |
| idresefmnd 19095 | The structure with the sin... |
| smndex1ibas 19096 | The modulo function ` I ` ... |
| smndex1iidm 19097 | The modulo function ` I ` ... |
| smndex1gbas 19098 | The constant functions ` (... |
| smndex1gbasOLD 19099 | Obsolete version of ~ smnd... |
| smndex1gid 19100 | The composition of a const... |
| smndex1gidOLD 19101 | Obsolete version of ~ smnd... |
| smndex1igid 19102 | The composition of the mod... |
| smndex1igidOLD 19103 | Obsolete version of ~ smnd... |
| smndex1basss 19104 | The modulo function ` I ` ... |
| smndex1bas 19105 | The base set of the monoid... |
| smndex1mgm 19106 | The monoid of endofunction... |
| smndex1sgrp 19107 | The monoid of endofunction... |
| smndex1mndlem 19108 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 19109 | The monoid of endofunction... |
| smndex1id 19110 | The modulo function ` I ` ... |
| smndex1n0mnd 19111 | The identity of the monoid... |
| nsmndex1 19112 | The base set ` B ` of the ... |
| smndex2dbas 19113 | The doubling function ` D ... |
| smndex2dnrinv 19114 | The doubling function ` D ... |
| smndex2hbas 19115 | The halving functions ` H ... |
| smndex2dlinvh 19116 | The halving functions ` H ... |
| mgm2nsgrplem1 19117 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 19118 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 19119 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 19120 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 19121 | A small magma (with two el... |
| sgrp2nmndlem1 19122 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 19123 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 19124 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 19125 | A small semigroup (with tw... |
| sgrp2rid2ex 19126 | A small semigroup (with tw... |
| sgrp2nmndlem4 19127 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 19128 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 19129 | A small semigroup (with tw... |
| mgmnsgrpex 19130 | There is a magma which is ... |
| sgrpnmndex 19131 | There is a semigroup which... |
| sgrpssmgm 19132 | The class of all semigroup... |
| mndsssgrp 19133 | The class of all monoids i... |
| degenmgmopdm 19134 | The domain of the operatio... |
| degenmgmbas 19135 | The base set of a degenera... |
| degenmgmnfn 19136 | The operation of a degener... |
| degenmgm 19137 | A degenerate magma: althou... |
| degenmgm2opdm 19138 | The domain of the operatio... |
| degenmgm2nfun 19139 | The operation of a second ... |
| degenmgm2 19140 | A degenerate magma: althou... |
| pwmndgplus 19141 | The operation of the monoi... |
| pwmndid 19142 | The identity of the monoid... |
| pwmnd 19143 | The power set of a class `... |
| isgrp 19150 | The predicate "is a group"... |
| grpmnd 19151 | A group is a monoid. (Con... |
| grpcl 19152 | Closure of the operation o... |
| grpass 19153 | A group operation is assoc... |
| grpinvex 19154 | Every member of a group ha... |
| grpideu 19155 | The two-sided identity ele... |
| grpassd 19156 | A group operation is assoc... |
| grpmndd 19157 | A group is a monoid. (Con... |
| grpcld 19158 | Closure of the operation o... |
| grpplusf 19159 | The group addition operati... |
| grpplusfo 19160 | The group addition operati... |
| resgrpplusfrn 19161 | The underlying set of a gr... |
| grppropd 19162 | If two structures have the... |
| grpprop 19163 | If two structures have the... |
| grppropstr 19164 | Generalize a specific 2-el... |
| grpss 19165 | Show that a structure exte... |
| isgrpd2e 19166 | Deduce a group from its pr... |
| isgrpd2 19167 | Deduce a group from its pr... |
| isgrpde 19168 | Deduce a group from its pr... |
| isgrpd 19169 | Deduce a group from its pr... |
| isgrpi 19170 | Properties that determine ... |
| grpsgrp 19171 | A group is a semigroup. (... |
| grpmgmd 19172 | A group is a magma, deduct... |
| dfgrp2 19173 | Alternate definition of a ... |
| dfgrp2e 19174 | Alternate definition of a ... |
| isgrpix 19175 | Properties that determine ... |
| grpidcl 19176 | The identity element of a ... |
| grpbn0 19177 | The base set of a group is... |
| grplid 19178 | The identity element of a ... |
| grprid 19179 | The identity element of a ... |
| grplidd 19180 | The identity element of a ... |
| grpridd 19181 | The identity element of a ... |
| grpn0 19182 | A group is not empty. (Co... |
| hashfingrpnn 19183 | A finite group has positiv... |
| grprcan 19184 | Right cancellation law for... |
| grpinveu 19185 | The left inverse element o... |
| grpid 19186 | Two ways of saying that an... |
| isgrpid2 19187 | Properties showing that an... |
| grpidd2 19188 | Deduce the identity elemen... |
| grpinvfval 19189 | The inverse function of a ... |
| grpinvfvalALT 19190 | Shorter proof of ~ grpinvf... |
| grpinvval 19191 | The inverse of a group ele... |
| grpinvfn 19192 | Functionality of the group... |
| grpinvfvi 19193 | The group inverse function... |
| grpsubfval 19194 | Group subtraction (divisio... |
| grpsubfvalALT 19195 | Shorter proof of ~ grpsubf... |
| grpsubval 19196 | Group subtraction (divisio... |
| grpinvf 19197 | The group inversion operat... |
| grpinvcl 19198 | A group element's inverse ... |
| grpinvcld 19199 | A group element's inverse ... |
| grplinv 19200 | The left inverse of a grou... |
| grprinv 19201 | The right inverse of a gro... |
| grpinvid1 19202 | The inverse of a group ele... |
| grpinvid2 19203 | The inverse of a group ele... |
| isgrpinv 19204 | Properties showing that a ... |
| grplinvd 19205 | The left inverse of a grou... |
| grprinvd 19206 | The right inverse of a gro... |
| grplrinv 19207 | In a group, every member h... |
| grpidinv2 19208 | A group's properties using... |
| grpidinv 19209 | A group has a left and rig... |
| grpinvid 19210 | The inverse of the identit... |
| grplcan 19211 | Left cancellation law for ... |
| grpasscan1 19212 | An associative cancellatio... |
| grpasscan2 19213 | An associative cancellatio... |
| grpidrcan 19214 | If right adding an element... |
| grpidlcan 19215 | If left adding an element ... |
| grpinvinv 19216 | Double inverse law for gro... |
| grpinvcnv 19217 | The group inverse is its o... |
| grpinv11 19218 | The group inverse is one-t... |
| grpinvf1o 19219 | The group inverse is a one... |
| grpinvnz 19220 | The inverse of a nonzero g... |
| grpinvnzcl 19221 | The inverse of a nonzero g... |
| grpsubinv 19222 | Subtraction of an inverse.... |
| grplmulf1o 19223 | Left multiplication by a g... |
| grpraddf1o 19224 | Right addition by a group ... |
| grpinvpropd 19225 | If two structures have the... |
| grpidssd 19226 | If the base set of a group... |
| grpinvssd 19227 | If the base set of a group... |
| grpinvadd 19228 | The inverse of the group o... |
| grpsubf 19229 | Functionality of group sub... |
| grpsubcl 19230 | Closure of group subtracti... |
| grpsubrcan 19231 | Right cancellation law for... |
| grpinvsub 19232 | Inverse of a group subtrac... |
| grpinvval2 19233 | A ~ df-neg -like equation ... |
| grpsubid 19234 | Subtraction of a group ele... |
| grpsubid1 19235 | Subtraction of the identit... |
| grpsubeq0 19236 | If the difference between ... |
| grpsubadd0sub 19237 | Subtraction expressed as a... |
| grpsubadd 19238 | Relationship between group... |
| grpsubsub 19239 | Double group subtraction. ... |
| grpaddsubass 19240 | Associative-type law for g... |
| grppncan 19241 | Cancellation law for subtr... |
| grpnpcan 19242 | Cancellation law for subtr... |
| grpsubsub4 19243 | Double group subtraction (... |
| grppnpcan2 19244 | Cancellation law for mixed... |
| grpnpncan 19245 | Cancellation law for group... |
| grpnpncan0 19246 | Cancellation law for group... |
| grpnnncan2 19247 | Cancellation law for group... |
| dfgrp3lem 19248 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19249 | Alternate definition of a ... |
| dfgrp3e 19250 | Alternate definition of a ... |
| grplactfval 19251 | The left group action of e... |
| grplactval 19252 | The value of the left grou... |
| grplactcnv 19253 | The left group action of e... |
| grplactf1o 19254 | The left group action of e... |
| grpsubpropd 19255 | Weak property deduction fo... |
| grpsubpropd2 19256 | Strong property deduction ... |
| grp1 19257 | The (smallest) structure r... |
| grp1inv 19258 | The inverse function of th... |
| prdsinvlem 19259 | Characterization of invers... |
| prdsgrpd 19260 | The product of a family of... |
| prdsinvgd 19261 | Negation in a product of g... |
| pwsgrp 19262 | A structure power of a gro... |
| pwsinvg 19263 | Negation in a structure po... |
| pwssub 19264 | Subtraction in a structure... |
| imasgrp2 19265 | The image structure of a g... |
| imasgrp 19266 | The image structure of a g... |
| imasgrpf1 19267 | The image of a group under... |
| qusgrp2 19268 | Prove that a quotient stru... |
| xpsgrp 19269 | The binary product of grou... |
| xpsinv 19270 | Value of the negation oper... |
| xpsgrpsub 19271 | Value of the subtraction o... |
| mhmlem 19272 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19273 | A surjective monoid morphi... |
| mhmmnd 19274 | The image of a monoid ` G ... |
| mhmfmhm 19275 | The function fulfilling th... |
| ghmgrp 19276 | The image of a group ` G `... |
| mulgfval 19279 | Group multiple (exponentia... |
| mulgfvalALT 19280 | Shorter proof of ~ mulgfva... |
| mulgval 19281 | Value of the group multipl... |
| mulgfn 19282 | Functionality of the group... |
| mulgfvi 19283 | The group multiple operati... |
| mulg0 19284 | Group multiple (exponentia... |
| mulgnn 19285 | Group multiple (exponentia... |
| ressmulgnn 19286 | Values for the group multi... |
| ressmulgnn0 19287 | Values for the group multi... |
| ressmulgnnd 19288 | Values for the group multi... |
| mulgnngsum 19289 | Group multiple (exponentia... |
| mulgnn0gsum 19290 | Group multiple (exponentia... |
| mulg1 19291 | Group multiple (exponentia... |
| mulgnnp1 19292 | Group multiple (exponentia... |
| mulg2 19293 | Group multiple (exponentia... |
| mulgnegnn 19294 | Group multiple (exponentia... |
| mulgnn0p1 19295 | Group multiple (exponentia... |
| mulgnnsubcl 19296 | Closure of the group multi... |
| mulgnn0subcl 19297 | Closure of the group multi... |
| mulgsubcl 19298 | Closure of the group multi... |
| mulgnncl 19299 | Closure of the group multi... |
| mulgnn0cl 19300 | Closure of the group multi... |
| mulgcl 19301 | Closure of the group multi... |
| mulgneg 19302 | Group multiple (exponentia... |
| mulgnegneg 19303 | The inverse of a negative ... |
| mulgm1 19304 | Group multiple (exponentia... |
| mulgnn0cld 19305 | Closure of the group multi... |
| mulgcld 19306 | Deduction associated with ... |
| mulgaddcomlem 19307 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19308 | The group multiple operato... |
| mulginvcom 19309 | The group multiple operato... |
| mulginvinv 19310 | The group multiple operato... |
| mulgnn0z 19311 | A group multiple of the id... |
| mulgz 19312 | A group multiple of the id... |
| mulgnndir 19313 | Sum of group multiples, fo... |
| mulgnn0dir 19314 | Sum of group multiples, ge... |
| mulgdirlem 19315 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19316 | Sum of group multiples, ge... |
| mulgp1 19317 | Group multiple (exponentia... |
| mulgneg2 19318 | Group multiple (exponentia... |
| mulgnnass 19319 | Product of group multiples... |
| mulgnn0ass 19320 | Product of group multiples... |
| mulgass 19321 | Product of group multiples... |
| mulgassr 19322 | Reversed product of group ... |
| mulgmodid 19323 | Casting out multiples of t... |
| mulgsubdir 19324 | Distribution of group mult... |
| mhmmulg 19325 | A homomorphism of monoids ... |
| mulgpropd 19326 | Two structures with the sa... |
| submmulgcl 19327 | Closure of the group multi... |
| submmulg 19328 | A group multiple is the sa... |
| pwsmulg 19329 | Value of a group multiple ... |
| issubg 19336 | The subgroup predicate. (... |
| subgss 19337 | A subgroup is a subset. (... |
| subgid 19338 | A group is a subgroup of i... |
| subggrp 19339 | A subgroup is a group. (C... |
| subgbas 19340 | The base of the restricted... |
| subgrcl 19341 | Reverse closure for the su... |
| subg0 19342 | A subgroup of a group must... |
| subginv 19343 | The inverse of an element ... |
| subg0cl 19344 | The group identity is an e... |
| subginvcl 19345 | The inverse of an element ... |
| subgcl 19346 | A subgroup is closed under... |
| subgcld 19347 | A subgroup is closed under... |
| subgsubcl 19348 | A subgroup is closed under... |
| subgsub 19349 | The subtraction of element... |
| subgmulgcl 19350 | Closure of the group multi... |
| subgmulg 19351 | A group multiple is the sa... |
| issubg2 19352 | Characterize the subgroups... |
| issubgrpd2 19353 | Prove a subgroup by closur... |
| issubgrpd 19354 | Prove a subgroup by closur... |
| issubg3 19355 | A subgroup is a symmetric ... |
| issubg4 19356 | A subgroup is a nonempty s... |
| grpissubg 19357 | If the base set of a group... |
| resgrpisgrp 19358 | If the base set of a group... |
| subgsubm 19359 | A subgroup is a submonoid.... |
| subsubg 19360 | A subgroup of a subgroup i... |
| subgint 19361 | The intersection of a none... |
| 0subg 19362 | The zero subgroup of an ar... |
| trivsubgd 19363 | The only subgroup of a tri... |
| trivsubgsnd 19364 | The only subgroup of a tri... |
| isnsg 19365 | Property of being a normal... |
| isnsg2 19366 | Weaken the condition of ~ ... |
| nsgbi 19367 | Defining property of a nor... |
| nsgsubg 19368 | A normal subgroup is a sub... |
| nsgconj 19369 | The conjugation of an elem... |
| isnsg3 19370 | A subgroup is normal iff t... |
| subgacs 19371 | Subgroups are an algebraic... |
| nsgacs 19372 | Normal subgroups form an a... |
| elnmz 19373 | Elementhood in the normali... |
| nmzbi 19374 | Defining property of the n... |
| nmzsubg 19375 | The normalizer N_G(S) of a... |
| ssnmz 19376 | A subgroup is a subset of ... |
| isnsg4 19377 | A subgroup is normal iff i... |
| nmznsg 19378 | Any subgroup is a normal s... |
| 0nsg 19379 | The zero subgroup is norma... |
| nsgid 19380 | The whole group is a norma... |
| 0idnsgd 19381 | The whole group and the ze... |
| trivnsgd 19382 | The only normal subgroup o... |
| triv1nsgd 19383 | A trivial group has exactl... |
| 1nsgtrivd 19384 | A group with exactly one n... |
| releqg 19385 | The left coset equivalence... |
| ecxpid 19386 | The equivalence class of a... |
| qsxpid 19387 | The quotient set of a cart... |
| eqgfval 19388 | Value of the subgroup left... |
| eqgval 19389 | Value of the subgroup left... |
| eqger 19390 | The subgroup coset equival... |
| eqglact 19391 | A left coset can be expres... |
| eqgid 19392 | The left coset containing ... |
| eqgen 19393 | Each coset is equipotent t... |
| eqgcpbl 19394 | The subgroup coset equival... |
| qusxpid 19395 | The Group quotient equival... |
| qustriv 19396 | The quotient of a group ` ... |
| qustrivr 19397 | Converse of ~ qustriv . (... |
| eqg0el 19398 | Equivalence class of a quo... |
| quselbas 19399 | Membership in the base set... |
| quseccl0 19400 | Closure of the quotient ma... |
| qusgrp 19401 | If ` Y ` is a normal subgr... |
| quseccl 19402 | Closure of the quotient ma... |
| qusadd 19403 | Value of the group operati... |
| qus0 19404 | Value of the group identit... |
| qusinv 19405 | Value of the group inverse... |
| qussub 19406 | Value of the group subtrac... |
| ecqusaddd 19407 | Addition of equivalence cl... |
| ecqusaddcl 19408 | Closure of the addition in... |
| lagsubg2 19409 | Lagrange's theorem for fin... |
| lagsubg 19410 | Lagrange's theorem for Gro... |
| eqg0subg 19411 | The coset equivalence rela... |
| eqg0subgecsn 19412 | The equivalence classes mo... |
| qus0subgbas 19413 | The base set of a quotient... |
| qus0subgadd 19414 | The addition in a quotient... |
| cycsubmel 19415 | Characterization of an ele... |
| cycsubmcl 19416 | The set of nonnegative int... |
| cycsubm 19417 | The set of nonnegative int... |
| cyccom 19418 | Condition for an operation... |
| cycsubmcom 19419 | The operation of a monoid ... |
| cycsubggend 19420 | The cyclic subgroup genera... |
| cycsubgcl 19421 | The set of integer powers ... |
| cycsubgss 19422 | The cyclic subgroup genera... |
| cycsubg 19423 | The cyclic group generated... |
| cycsubgcld 19424 | The cyclic subgroup genera... |
| cycsubg2 19425 | The subgroup generated by ... |
| cycsubg2cl 19426 | Any multiple of an element... |
| reldmghm 19429 | Lemma for group homomorphi... |
| isghm 19430 | Property of being a homomo... |
| isghm3 19431 | Property of a group homomo... |
| ghmgrp1 19432 | A group homomorphism is on... |
| ghmgrp2 19433 | A group homomorphism is on... |
| ghmf 19434 | A group homomorphism is a ... |
| ghmlin 19435 | A homomorphism of groups i... |
| ghmid 19436 | A homomorphism of groups p... |
| ghminv 19437 | A homomorphism of groups p... |
| ghmsub 19438 | Linearity of subtraction t... |
| isghmd 19439 | Deduction for a group homo... |
| ghmmhm 19440 | A group homomorphism is a ... |
| ghmmhmb 19441 | Group homomorphisms and mo... |
| ghmmulg 19442 | A group homomorphism prese... |
| ghmrn 19443 | The range of a homomorphis... |
| 0ghm 19444 | The constant zero linear f... |
| idghm 19445 | The identity homomorphism ... |
| resghm 19446 | Restriction of a homomorph... |
| resghm2 19447 | One direction of ~ resghm2... |
| resghm2b 19448 | Restriction of the codomai... |
| ghmghmrn 19449 | A group homomorphism from ... |
| ghmco 19450 | The composition of group h... |
| ghmima 19451 | The image of a subgroup un... |
| ghmpreima 19452 | The inverse image of a sub... |
| ghmeql 19453 | The equalizer of two group... |
| ghmnsgima 19454 | The image of a normal subg... |
| ghmnsgpreima 19455 | The inverse image of a nor... |
| ghmker 19456 | The kernel of a homomorphi... |
| ghmeqker 19457 | Two source points map to t... |
| pwsdiagghm 19458 | Diagonal homomorphism into... |
| f1ghm0to0 19459 | If a group homomorphism ` ... |
| ghmf1 19460 | Two ways of saying a group... |
| kerf1ghm 19461 | A group homomorphism ` F `... |
| ghmf1o 19462 | A bijective group homomorp... |
| conjghm 19463 | Conjugation is an automorp... |
| conjsubg 19464 | A conjugated subgroup is a... |
| conjsubgen 19465 | A conjugated subgroup is e... |
| conjnmz 19466 | A subgroup is unchanged un... |
| conjnmzb 19467 | Alternative condition for ... |
| conjnsg 19468 | A normal subgroup is uncha... |
| qusghm 19469 | If ` Y ` is a normal subgr... |
| ghmpropd 19470 | Group homomorphism depends... |
| gimfn 19475 | The group isomorphism func... |
| isgim 19476 | An isomorphism of groups i... |
| gimf1o 19477 | An isomorphism of groups i... |
| gimghm 19478 | An isomorphism of groups i... |
| isgim2 19479 | A group isomorphism is a h... |
| subggim 19480 | Behavior of subgroups unde... |
| gimcnv 19481 | The converse of a group is... |
| gimco 19482 | The composition of group i... |
| gim0to0 19483 | A group isomorphism maps t... |
| brgic 19484 | The relation "is isomorphi... |
| brgici 19485 | Prove isomorphic by an exp... |
| gicref 19486 | Isomorphism is reflexive. ... |
| giclcl 19487 | Isomorphism implies the le... |
| gicrcl 19488 | Isomorphism implies the ri... |
| gicsym 19489 | Isomorphism is symmetric. ... |
| gictr 19490 | Isomorphism is transitive.... |
| gicer 19491 | Isomorphism is an equivale... |
| gicen 19492 | Isomorphic groups have equ... |
| gicsubgen 19493 | A less trivial example of ... |
| ghmqusnsglem1 19494 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19495 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19496 | The mapping ` H ` induced ... |
| ghmquskerlem1 19497 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19498 | In the case of theorem ~ g... |
| ghmquskerlem2 19499 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19500 | The mapping ` H ` induced ... |
| ghmqusker 19501 | A surjective group homomor... |
| gicqusker 19502 | The image ` H ` of a group... |
| isga 19505 | The predicate "is a (left)... |
| gagrp 19506 | The left argument of a gro... |
| gaset 19507 | The right argument of a gr... |
| gagrpid 19508 | The identity of the group ... |
| gaf 19509 | The mapping of the group a... |
| gafo 19510 | A group action is onto its... |
| gaass 19511 | An "associative" property ... |
| ga0 19512 | The action of a group on t... |
| gaid 19513 | The trivial action of a gr... |
| subgga 19514 | A subgroup acts on its par... |
| gass 19515 | A subset of a group action... |
| gasubg 19516 | The restriction of a group... |
| gaid2 19517 | A group operation is a lef... |
| galcan 19518 | The action of a particular... |
| gacan 19519 | Group inverses cancel in a... |
| gapm 19520 | The action of a particular... |
| gaorb 19521 | The orbit equivalence rela... |
| gaorber 19522 | The orbit equivalence rela... |
| gastacl 19523 | The stabilizer subgroup in... |
| gastacos 19524 | Write the coset relation f... |
| orbstafun 19525 | Existence and uniqueness f... |
| orbstaval 19526 | Value of the function at a... |
| orbsta 19527 | The Orbit-Stabilizer theor... |
| orbsta2 19528 | Relation between the size ... |
| cntrval 19533 | Substitute definition of t... |
| cntzfval 19534 | First level substitution f... |
| cntzval 19535 | Definition substitution fo... |
| elcntz 19536 | Elementhood in the central... |
| cntzel 19537 | Membership in a centralize... |
| cntzsnval 19538 | Special substitution for t... |
| elcntzsn 19539 | Value of the centralizer o... |
| sscntz 19540 | A centralizer expression f... |
| cntzrcl 19541 | Reverse closure for elemen... |
| cntzssv 19542 | The centralizer is uncondi... |
| cntzi 19543 | Membership in a centralize... |
| elcntr 19544 | Elementhood in the center ... |
| cntrss 19545 | The center is a subset of ... |
| cntri 19546 | Defining property of the c... |
| resscntz 19547 | Centralizer in a substruct... |
| cntzsgrpcl 19548 | Centralizers are closed un... |
| cntz2ss 19549 | Centralizers reverse the s... |
| cntzrec 19550 | Reciprocity relationship f... |
| cntziinsn 19551 | Express any centralizer as... |
| cntzsubm 19552 | Centralizers in a monoid a... |
| cntzsubg 19553 | Centralizers in a group ar... |
| cntzidss 19554 | If the elements of ` S ` c... |
| cntzmhm 19555 | Centralizers in a monoid a... |
| cntzmhm2 19556 | Centralizers in a monoid a... |
| cntrsubgnsg 19557 | A central subgroup is norm... |
| cntrnsg 19558 | The center of a group is a... |
| oppgval 19561 | Value of the opposite grou... |
| oppgplusfval 19562 | Value of the addition oper... |
| oppgplus 19563 | Value of the addition oper... |
| setsplusg 19564 | The other components of an... |
| oppgbas 19565 | Base set of an opposite gr... |
| oppgtset 19566 | Topology of an opposite gr... |
| oppgtopn 19567 | Topology of an opposite gr... |
| oppgmnd 19568 | The opposite of a monoid i... |
| oppgmndb 19569 | Bidirectional form of ~ op... |
| oppgid 19570 | Zero in a monoid is a symm... |
| oppggrp 19571 | The opposite of a group is... |
| oppggrpb 19572 | Bidirectional form of ~ op... |
| oppginv 19573 | Inverses in a group are a ... |
| invoppggim 19574 | The inverse is an antiauto... |
| oppggic 19575 | Every group is (naturally)... |
| oppgsubm 19576 | Being a submonoid is a sym... |
| oppgsubg 19577 | Being a subgroup is a symm... |
| oppgcntz 19578 | A centralizer in a group i... |
| oppgcntr 19579 | The center of a group is t... |
| gsumwrev 19580 | A sum in an opposite monoi... |
| oppgle 19581 | less-than relation of an o... |
| oppglt 19582 | less-than relation of an o... |
| symgval 19585 | The value of the symmetric... |
| symgbas 19586 | The base set of the symmet... |
| elsymgbas2 19587 | Two ways of saying a funct... |
| elsymgbas 19588 | Two ways of saying a funct... |
| symgbasf1o 19589 | Elements in the symmetric ... |
| symgbasf 19590 | A permutation (element of ... |
| symgbasmap 19591 | A permutation (element of ... |
| symghash 19592 | The symmetric group on ` n... |
| symgbasfi 19593 | The symmetric group on a f... |
| symgfv 19594 | The function value of a pe... |
| symgfvne 19595 | The function values of a p... |
| symgressbas 19596 | The symmetric group on ` A... |
| symgplusg 19597 | The group operation of a s... |
| symgov 19598 | The value of the group ope... |
| symgcl 19599 | The group operation of the... |
| idresperm 19600 | The identity function rest... |
| symgmov1 19601 | For a permutation of a set... |
| symgmov2 19602 | For a permutation of a set... |
| symgbas0 19603 | The base set of the symmet... |
| symg1hash 19604 | The symmetric group on a s... |
| symg1bas 19605 | The symmetric group on a s... |
| symg2hash 19606 | The symmetric group on a (... |
| symg2bas 19607 | The symmetric group on a p... |
| 0symgefmndeq 19608 | The symmetric group on the... |
| snsymgefmndeq 19609 | The symmetric group on a s... |
| symgpssefmnd 19610 | For a set ` A ` with more ... |
| symgvalstruct 19611 | The value of the symmetric... |
| symgsubmefmnd 19612 | The symmetric group on a s... |
| symgtset 19613 | The topology of the symmet... |
| symggrp 19614 | The symmetric group on a s... |
| symgid 19615 | The group identity element... |
| symginv 19616 | The group inverse in the s... |
| symgsubmefmndALT 19617 | The symmetric group on a s... |
| galactghm 19618 | The currying of a group ac... |
| lactghmga 19619 | The converse of ~ galactgh... |
| symgtopn 19620 | The topology of the symmet... |
| symgga 19621 | The symmetric group induce... |
| pgrpsubgsymgbi 19622 | Every permutation group is... |
| pgrpsubgsymg 19623 | Every permutation group is... |
| idressubgsymg 19624 | The singleton containing o... |
| idrespermg 19625 | The structure with the sin... |
| cayleylem1 19626 | Lemma for ~ cayley . (Con... |
| cayleylem2 19627 | Lemma for ~ cayley . (Con... |
| cayley 19628 | Cayley's Theorem (construc... |
| cayleyth 19629 | Cayley's Theorem (existenc... |
| symgfix2 19630 | If a permutation does not ... |
| symgextf 19631 | The extension of a permuta... |
| symgextfv 19632 | The function value of the ... |
| symgextfve 19633 | The function value of the ... |
| symgextf1lem 19634 | Lemma for ~ symgextf1 . (... |
| symgextf1 19635 | The extension of a permuta... |
| symgextfo 19636 | The extension of a permuta... |
| symgextf1o 19637 | The extension of a permuta... |
| symgextsymg 19638 | The extension of a permuta... |
| symgextres 19639 | The restriction of the ext... |
| gsumccatsymgsn 19640 | Homomorphic property of co... |
| gsmsymgrfixlem1 19641 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19642 | The composition of permuta... |
| fvcosymgeq 19643 | The values of two composit... |
| gsmsymgreqlem1 19644 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19645 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19646 | Two combination of permuta... |
| symgfixelq 19647 | A permutation of a set fix... |
| symgfixels 19648 | The restriction of a permu... |
| symgfixelsi 19649 | The restriction of a permu... |
| symgfixf 19650 | The mapping of a permutati... |
| symgfixf1 19651 | The mapping of a permutati... |
| symgfixfolem1 19652 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19653 | The mapping of a permutati... |
| symgfixf1o 19654 | The mapping of a permutati... |
| f1omvdmvd 19657 | A permutation of any class... |
| f1omvdcnv 19658 | A permutation and its inve... |
| mvdco 19659 | Composing two permutations... |
| f1omvdconj 19660 | Conjugation of a permutati... |
| f1otrspeq 19661 | A transposition is charact... |
| f1omvdco2 19662 | If exactly one of two perm... |
| f1omvdco3 19663 | If a point is moved by exa... |
| pmtrfval 19664 | The function generating tr... |
| pmtrval 19665 | A generated transposition,... |
| pmtrfv 19666 | General value of mapping a... |
| pmtrprfv 19667 | In a transposition of two ... |
| pmtrprfv3 19668 | In a transposition of two ... |
| pmtrf 19669 | Functionality of a transpo... |
| pmtrmvd 19670 | A transposition moves prec... |
| pmtrrn 19671 | Transposing two points giv... |
| pmtrfrn 19672 | A transposition (as a kind... |
| pmtrffv 19673 | Mapping of a point under a... |
| pmtrrn2 19674 | For any transposition ther... |
| pmtrfinv 19675 | A transposition function i... |
| pmtrfmvdn0 19676 | A transposition moves at l... |
| pmtrff1o 19677 | A transposition function i... |
| pmtrfcnv 19678 | A transposition function i... |
| pmtrfb 19679 | An intrinsic characterizat... |
| pmtrfconj 19680 | Any conjugate of a transpo... |
| symgsssg 19681 | The symmetric group has su... |
| symgfisg 19682 | The symmetric group has a ... |
| symgtrf 19683 | Transpositions are element... |
| symggen 19684 | The span of the transposit... |
| symggen2 19685 | A finite permutation group... |
| symgtrinv 19686 | To invert a permutation re... |
| pmtr3ncomlem1 19687 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19688 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19689 | Transpositions over sets w... |
| pmtrdifellem1 19690 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19691 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19692 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19693 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19694 | A transposition of element... |
| pmtrdifwrdellem1 19695 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19696 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19697 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19698 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19699 | A sequence of transpositio... |
| pmtrdifwrdel2 19700 | A sequence of transpositio... |
| pmtrprfval 19701 | The transpositions on a pa... |
| pmtrprfvalrn 19702 | The range of the transposi... |
| psgnunilem1 19707 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19708 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19709 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19710 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19711 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19712 | Addition and subtraction o... |
| psgnuni 19713 | If the same permutation ca... |
| psgnfval 19714 | Function definition of the... |
| psgnfn 19715 | Functionality and domain o... |
| psgndmsubg 19716 | The finitary permutations ... |
| psgneldm 19717 | Property of being a finita... |
| psgneldm2 19718 | The finitary permutations ... |
| psgneldm2i 19719 | A sequence of transpositio... |
| psgneu 19720 | A finitary permutation has... |
| psgnval 19721 | Value of the permutation s... |
| psgnvali 19722 | A finitary permutation has... |
| psgnvalii 19723 | Any representation of a pe... |
| psgnpmtr 19724 | All transpositions are odd... |
| psgn0fv0 19725 | The permutation sign funct... |
| sygbasnfpfi 19726 | The class of non-fixed poi... |
| psgnfvalfi 19727 | Function definition of the... |
| psgnvalfi 19728 | Value of the permutation s... |
| psgnran 19729 | The range of the permutati... |
| gsmtrcl 19730 | The group sum of transposi... |
| psgnfitr 19731 | A permutation of a finite ... |
| psgnfieu 19732 | A permutation of a finite ... |
| pmtrsn 19733 | The value of the transposi... |
| psgnsn 19734 | The permutation sign funct... |
| psgnprfval 19735 | The permutation sign funct... |
| psgnprfval1 19736 | The permutation sign of th... |
| psgnprfval2 19737 | The permutation sign of th... |
| odfval 19746 | Value of the order functio... |
| odfvalALT 19747 | Shorter proof of ~ odfval ... |
| odval 19748 | Second substitution for th... |
| odlem1 19749 | The group element order is... |
| odcl 19750 | The order of a group eleme... |
| odf 19751 | Functionality of the group... |
| odid 19752 | Any element to the power o... |
| odlem2 19753 | Any positive annihilator o... |
| odmodnn0 19754 | Reduce the argument of a g... |
| mndodconglem 19755 | Lemma for ~ mndodcong . (... |
| mndodcong 19756 | If two multipliers are con... |
| mndodcongi 19757 | If two multipliers are con... |
| oddvdsnn0 19758 | The only multiples of ` A ... |
| odnncl 19759 | If a nonzero multiple of a... |
| odmod 19760 | Reduce the argument of a g... |
| oddvds 19761 | The only multiples of ` A ... |
| oddvdsi 19762 | Any group element is annih... |
| odcong 19763 | If two multipliers are con... |
| odeq 19764 | The ~ oddvds property uniq... |
| odval2 19765 | A non-conditional definiti... |
| odcld 19766 | The order of a group eleme... |
| odm1inv 19767 | The (order-1)th multiple o... |
| odmulgid 19768 | A relationship between the... |
| odmulg2 19769 | The order of a multiple di... |
| odmulg 19770 | Relationship between the o... |
| odmulgeq 19771 | A multiple of a point of f... |
| odbezout 19772 | If ` N ` is coprime to the... |
| od1 19773 | The order of the group ide... |
| odeq1 19774 | The group identity is the ... |
| odinv 19775 | The order of the inverse o... |
| odf1 19776 | The multiples of an elemen... |
| odinf 19777 | The multiples of an elemen... |
| dfod2 19778 | An alternative definition ... |
| odcl2 19779 | The order of an element of... |
| oddvds2 19780 | The order of an element of... |
| finodsubmsubg 19781 | A submonoid whose elements... |
| 0subgALT 19782 | A shorter proof of ~ 0subg... |
| submod 19783 | The order of an element is... |
| subgod 19784 | The order of an element is... |
| odsubdvds 19785 | The order of an element of... |
| odf1o1 19786 | An element with zero order... |
| odf1o2 19787 | An element with nonzero or... |
| odhash 19788 | An element of zero order g... |
| odhash2 19789 | If an element has nonzero ... |
| odhash3 19790 | An element which generates... |
| odngen 19791 | A cyclic subgroup of size ... |
| gexval 19792 | Value of the exponent of a... |
| gexlem1 19793 | The group element order is... |
| gexcl 19794 | The exponent of a group is... |
| gexid 19795 | Any element to the power o... |
| gexlem2 19796 | Any positive annihilator o... |
| gexdvdsi 19797 | Any group element is annih... |
| gexdvds 19798 | The only ` N ` that annihi... |
| gexdvds2 19799 | An integer divides the gro... |
| gexod 19800 | Any group element is annih... |
| gexcl3 19801 | If the order of every grou... |
| gexnnod 19802 | Every group element has fi... |
| gexcl2 19803 | The exponent of a finite g... |
| gexdvds3 19804 | The exponent of a finite g... |
| gex1 19805 | A group or monoid has expo... |
| ispgp 19806 | A group is a ` P ` -group ... |
| pgpprm 19807 | Reverse closure for the fi... |
| pgpgrp 19808 | Reverse closure for the se... |
| pgpfi1 19809 | A finite group with order ... |
| pgp0 19810 | The identity subgroup is a... |
| subgpgp 19811 | A subgroup of a p-group is... |
| sylow1lem1 19812 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19813 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19814 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19815 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19816 | Lemma for ~ sylow1 . Usin... |
| sylow1 19817 | Sylow's first theorem. If... |
| odcau 19818 | Cauchy's theorem for the o... |
| pgpfi 19819 | The converse to ~ pgpfi1 .... |
| pgpfi2 19820 | Alternate version of ~ pgp... |
| pgphash 19821 | The order of a p-group. (... |
| isslw 19822 | The property of being a Sy... |
| slwprm 19823 | Reverse closure for the fi... |
| slwsubg 19824 | A Sylow ` P ` -subgroup is... |
| slwispgp 19825 | Defining property of a Syl... |
| slwpss 19826 | A proper superset of a Syl... |
| slwpgp 19827 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19828 | Every ` P ` -subgroup is c... |
| slwn0 19829 | Every finite group contain... |
| subgslw 19830 | A Sylow subgroup that is c... |
| sylow2alem1 19831 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19832 | Lemma for ~ sylow2a . All... |
| sylow2a 19833 | A named lemma of Sylow's s... |
| sylow2blem1 19834 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19835 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19836 | Sylow's second theorem. P... |
| sylow2b 19837 | Sylow's second theorem. A... |
| slwhash 19838 | A sylow subgroup has cardi... |
| fislw 19839 | The sylow subgroups of a f... |
| sylow2 19840 | Sylow's second theorem. S... |
| sylow3lem1 19841 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19842 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19843 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19844 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19845 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19846 | Lemma for ~ sylow3 , secon... |
| sylow3 19847 | Sylow's third theorem. Th... |
| lsmfval 19852 | The subgroup sum function ... |
| lsmvalx 19853 | Subspace sum value (for a ... |
| lsmelvalx 19854 | Subspace sum membership (f... |
| lsmelvalix 19855 | Subspace sum membership (f... |
| oppglsm 19856 | The subspace sum operation... |
| lsmssv 19857 | Subgroup sum is a subset o... |
| lsmless1x 19858 | Subset implies subgroup su... |
| lsmless2x 19859 | Subset implies subgroup su... |
| lsmub1x 19860 | Subgroup sum is an upper b... |
| lsmub2x 19861 | Subgroup sum is an upper b... |
| lsmval 19862 | Subgroup sum value (for a ... |
| lsmelval 19863 | Subgroup sum membership (f... |
| lsmelvali 19864 | Subgroup sum membership (f... |
| lsmelvalm 19865 | Subgroup sum membership an... |
| lsmelvalmi 19866 | Membership of vector subtr... |
| lsmsubm 19867 | The sum of two commuting s... |
| lsmsubg 19868 | The sum of two commuting s... |
| lsmcom2 19869 | Subgroup sum commutes. (C... |
| smndlsmidm 19870 | The direct product is idem... |
| lsmub1 19871 | Subgroup sum is an upper b... |
| lsmub2 19872 | Subgroup sum is an upper b... |
| lsmunss 19873 | Union of subgroups is a su... |
| lsmless1 19874 | Subset implies subgroup su... |
| lsmless2 19875 | Subset implies subgroup su... |
| lsmless12 19876 | Subset implies subgroup su... |
| lsmidm 19877 | Subgroup sum is idempotent... |
| lsmlub 19878 | The least upper bound prop... |
| lsmss1 19879 | Subgroup sum with a subset... |
| lsmss1b 19880 | Subgroup sum with a subset... |
| lsmss2 19881 | Subgroup sum with a subset... |
| lsmss2b 19882 | Subgroup sum with a subset... |
| lsmass 19883 | Subgroup sum is associativ... |
| mndlsmidm 19884 | Subgroup sum is idempotent... |
| lsm01 19885 | Subgroup sum with the zero... |
| lsm02 19886 | Subgroup sum with the zero... |
| subglsm 19887 | The subgroup sum evaluated... |
| lssnle 19888 | Equivalent expressions for... |
| lsmmod 19889 | The modular law holds for ... |
| lsmmod2 19890 | Modular law dual for subgr... |
| lsmpropd 19891 | If two structures have the... |
| cntzrecd 19892 | Commute the "subgroups com... |
| lsmcntz 19893 | The "subgroups commute" pr... |
| lsmcntzr 19894 | The "subgroups commute" pr... |
| lsmdisj 19895 | Disjointness from a subgro... |
| lsmdisj2 19896 | Association of the disjoin... |
| lsmdisj3 19897 | Association of the disjoin... |
| lsmdisjr 19898 | Disjointness from a subgro... |
| lsmdisj2r 19899 | Association of the disjoin... |
| lsmdisj3r 19900 | Association of the disjoin... |
| lsmdisj2a 19901 | Association of the disjoin... |
| lsmdisj2b 19902 | Association of the disjoin... |
| lsmdisj3a 19903 | Association of the disjoin... |
| lsmdisj3b 19904 | Association of the disjoin... |
| subgdisj1 19905 | Vectors belonging to disjo... |
| subgdisj2 19906 | Vectors belonging to disjo... |
| subgdisjb 19907 | Vectors belonging to disjo... |
| pj1fval 19908 | The left projection functi... |
| pj1val 19909 | The left projection functi... |
| pj1eu 19910 | Uniqueness of a left proje... |
| pj1f 19911 | The left projection functi... |
| pj2f 19912 | The right projection funct... |
| pj1id 19913 | Any element of a direct su... |
| pj1eq 19914 | Any element of a direct su... |
| pj1lid 19915 | The left projection functi... |
| pj1rid 19916 | The left projection functi... |
| pj1ghm 19917 | The left projection functi... |
| pj1ghm2 19918 | The left projection functi... |
| lsmhash 19919 | The order of the direct pr... |
| efgmval 19926 | Value of the formal invers... |
| efgmf 19927 | The formal inverse operati... |
| efgmnvl 19928 | The inversion function on ... |
| efgrcl 19929 | Lemma for ~ efgval . (Con... |
| efglem 19930 | Lemma for ~ efgval . (Con... |
| efgval 19931 | Value of the free group co... |
| efger 19932 | Value of the free group co... |
| efgi 19933 | Value of the free group co... |
| efgi0 19934 | Value of the free group co... |
| efgi1 19935 | Value of the free group co... |
| efgtf 19936 | Value of the free group co... |
| efgtval 19937 | Value of the extension fun... |
| efgval2 19938 | Value of the free group co... |
| efgi2 19939 | Value of the free group co... |
| efgtlen 19940 | Value of the free group co... |
| efginvrel2 19941 | The inverse of the reverse... |
| efginvrel1 19942 | The inverse of the reverse... |
| efgsf 19943 | Value of the auxiliary fun... |
| efgsdm 19944 | Elementhood in the domain ... |
| efgsval 19945 | Value of the auxiliary fun... |
| efgsdmi 19946 | Property of the last link ... |
| efgsval2 19947 | Value of the auxiliary fun... |
| efgsrel 19948 | The start and end of any e... |
| efgs1 19949 | A singleton of an irreduci... |
| efgs1b 19950 | Every extension sequence e... |
| efgsp1 19951 | If ` F ` is an extension s... |
| efgsres 19952 | An initial segment of an e... |
| efgsfo 19953 | For any word, there is a s... |
| efgredlema 19954 | The reduced word that form... |
| efgredlemf 19955 | Lemma for ~ efgredleme . ... |
| efgredlemg 19956 | Lemma for ~ efgred . (Con... |
| efgredleme 19957 | Lemma for ~ efgred . (Con... |
| efgredlemd 19958 | The reduced word that form... |
| efgredlemc 19959 | The reduced word that form... |
| efgredlemb 19960 | The reduced word that form... |
| efgredlem 19961 | The reduced word that form... |
| efgred 19962 | The reduced word that form... |
| efgrelexlema 19963 | If two words ` A , B ` are... |
| efgrelexlemb 19964 | If two words ` A , B ` are... |
| efgrelex 19965 | If two words ` A , B ` are... |
| efgredeu 19966 | There is a unique reduced ... |
| efgred2 19967 | Two extension sequences ha... |
| efgcpbllema 19968 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19969 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19970 | Two extension sequences ha... |
| efgcpbl2 19971 | Two extension sequences ha... |
| frgpval 19972 | Value of the free group co... |
| frgpcpbl 19973 | Compatibility of the group... |
| frgp0 19974 | The free group is a group.... |
| frgpeccl 19975 | Closure of the quotient ma... |
| frgpgrp 19976 | The free group is a group.... |
| frgpadd 19977 | Addition in the free group... |
| frgpinv 19978 | The inverse of an element ... |
| frgpmhm 19979 | The "natural map" from wor... |
| vrgpfval 19980 | The canonical injection fr... |
| vrgpval 19981 | The value of the generatin... |
| vrgpf 19982 | The mapping from the index... |
| vrgpinv 19983 | The inverse of a generatin... |
| frgpuptf 19984 | Any assignment of the gene... |
| frgpuptinv 19985 | Any assignment of the gene... |
| frgpuplem 19986 | Any assignment of the gene... |
| frgpupf 19987 | Any assignment of the gene... |
| frgpupval 19988 | Any assignment of the gene... |
| frgpup1 19989 | Any assignment of the gene... |
| frgpup2 19990 | The evaluation map has the... |
| frgpup3lem 19991 | The evaluation map has the... |
| frgpup3 19992 | Universal property of the ... |
| 0frgp 19993 | The free group on zero gen... |
| isabl 19998 | The predicate "is an Abeli... |
| ablgrp 19999 | An Abelian group is a grou... |
| ablgrpd 20000 | An Abelian group is a grou... |
| ablcmn 20001 | An Abelian group is a comm... |
| ablcmnd 20002 | An Abelian group is a comm... |
| iscmn 20003 | The predicate "is a commut... |
| isabl2 20004 | The predicate "is an Abeli... |
| cmnpropd 20005 | If two structures have the... |
| ablpropd 20006 | If two structures have the... |
| ablprop 20007 | If two structures have the... |
| iscmnd 20008 | Properties that determine ... |
| isabld 20009 | Properties that determine ... |
| isabli 20010 | Properties that determine ... |
| cmnmnd 20011 | A commutative monoid is a ... |
| cmncom 20012 | A commutative monoid is co... |
| ablcom 20013 | An Abelian group operation... |
| cmn32 20014 | Commutative/associative la... |
| cmn4 20015 | Commutative/associative la... |
| cmn12 20016 | Commutative/associative la... |
| abl32 20017 | Commutative/associative la... |
| cmnmndd 20018 | A commutative monoid is a ... |
| cmnbascntr 20019 | The base set of a commutat... |
| rinvmod 20020 | Uniqueness of a right inve... |
| ablinvadd 20021 | The inverse of an Abelian ... |
| ablsub2inv 20022 | Abelian group subtraction ... |
| ablsubadd 20023 | Relationship between Abeli... |
| ablsub4 20024 | Commutative/associative su... |
| abladdsub4 20025 | Abelian group addition/sub... |
| abladdsub 20026 | Associative-type law for g... |
| ablsubadd23 20027 | Commutative/associative la... |
| ablsubaddsub 20028 | Double subtraction and add... |
| ablpncan2 20029 | Cancellation law for subtr... |
| ablpncan3 20030 | A cancellation law for Abe... |
| ablsubsub 20031 | Law for double subtraction... |
| ablsubsub4 20032 | Law for double subtraction... |
| ablpnpcan 20033 | Cancellation law for mixed... |
| ablnncan 20034 | Cancellation law for group... |
| ablsub32 20035 | Swap the second and third ... |
| ablnnncan 20036 | Cancellation law for group... |
| ablnnncan1 20037 | Cancellation law for group... |
| ablsubsub23 20038 | Swap subtrahend and result... |
| mulgnn0di 20039 | Group multiple of a sum, f... |
| mulgdi 20040 | Group multiple of a sum. ... |
| mulgmhm 20041 | The map from ` x ` to ` n ... |
| mulgghm 20042 | The map from ` x ` to ` n ... |
| mulgsubdi 20043 | Group multiple of a differ... |
| ghmfghm 20044 | The function fulfilling th... |
| ghmcmn 20045 | The image of a commutative... |
| ghmabl 20046 | The image of an abelian gr... |
| invghm 20047 | The inversion map is a gro... |
| eqgabl 20048 | Value of the subgroup cose... |
| qusecsub 20049 | Two subgroup cosets are eq... |
| subgabl 20050 | A subgroup of an abelian g... |
| subcmn 20051 | A submonoid of a commutati... |
| submcmn 20052 | A submonoid of a commutati... |
| submcmn2 20053 | A submonoid is commutative... |
| cntzcmn 20054 | The centralizer of any sub... |
| cntzcmnss 20055 | Any subset in a commutativ... |
| cntrcmnd 20056 | The center of a monoid is ... |
| cntrabl 20057 | The center of a group is a... |
| cntzspan 20058 | If the generators commute,... |
| cntzcmnf 20059 | Discharge the centralizer ... |
| ghmplusg 20060 | The pointwise sum of two l... |
| ablnsg 20061 | Every subgroup of an abeli... |
| odadd1 20062 | The order of a product in ... |
| odadd2 20063 | The order of a product in ... |
| odadd 20064 | The order of a product is ... |
| gex2abl 20065 | A group with exponent 2 (o... |
| gexexlem 20066 | Lemma for ~ gexex . (Cont... |
| gexex 20067 | In an abelian group with f... |
| torsubg 20068 | The set of all elements of... |
| oddvdssubg 20069 | The set of all elements wh... |
| lsmcomx 20070 | Subgroup sum commutes (ext... |
| ablcntzd 20071 | All subgroups in an abelia... |
| lsmcom 20072 | Subgroup sum commutes. (C... |
| lsmsubg2 20073 | The sum of two subgroups i... |
| lsm4 20074 | Commutative/associative la... |
| prdscmnd 20075 | The product of a family of... |
| prdsabld 20076 | The product of a family of... |
| pwscmn 20077 | The structure power on a c... |
| pwsabl 20078 | The structure power on an ... |
| qusabl 20079 | If ` Y ` is a subgroup of ... |
| abl1 20080 | The (smallest) structure r... |
| abln0 20081 | Abelian groups (and theref... |
| cnaddablx 20082 | The complex numbers are an... |
| cnaddabl 20083 | The complex numbers are an... |
| cnaddid 20084 | The group identity element... |
| cnaddinv 20085 | Value of the group inverse... |
| zaddablx 20086 | The integers are an Abelia... |
| frgpnabllem1 20087 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 20088 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 20089 | The free group on two or m... |
| imasabl 20090 | The image structure of an ... |
| iscyg 20093 | Definition of a cyclic gro... |
| iscyggen 20094 | The property of being a cy... |
| iscyggen2 20095 | The property of being a cy... |
| iscyg2 20096 | A cyclic group is a group ... |
| cyggeninv 20097 | The inverse of a cyclic ge... |
| cyggenod 20098 | An element is the generato... |
| cyggenod2 20099 | In an infinite cyclic grou... |
| iscyg3 20100 | Definition of a cyclic gro... |
| iscygd 20101 | Definition of a cyclic gro... |
| iscygodd 20102 | Show that a group with an ... |
| cycsubmcmn 20103 | The set of nonnegative int... |
| cyggrp 20104 | A cyclic group is a group.... |
| cygabl 20105 | A cyclic group is abelian.... |
| cygctb 20106 | A cyclic group is countabl... |
| 0cyg 20107 | The trivial group is cycli... |
| prmcyg 20108 | A group with prime order i... |
| lt6abl 20109 | A group with fewer than ` ... |
| ghmcyg 20110 | The image of a cyclic grou... |
| cyggex2 20111 | The exponent of a cyclic g... |
| cyggex 20112 | The exponent of a finite c... |
| cyggexb 20113 | A finite abelian group is ... |
| giccyg 20114 | Cyclicity is a group prope... |
| cycsubgcyg 20115 | The cyclic subgroup genera... |
| cycsubgcyg2 20116 | The cyclic subgroup genera... |
| gsumval3a 20117 | Value of the group sum ope... |
| gsumval3eu 20118 | The group sum as defined i... |
| gsumval3lem1 20119 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 20120 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 20121 | Value of the group sum ope... |
| gsumcllem 20122 | Lemma for ~ gsumcl and rel... |
| gsumzres 20123 | Extend a finite group sum ... |
| gsumzcl2 20124 | Closure of a finite group ... |
| gsumzcl 20125 | Closure of a finite group ... |
| gsumzf1o 20126 | Re-index a finite group su... |
| gsumres 20127 | Extend a finite group sum ... |
| gsumcl2 20128 | Closure of a finite group ... |
| gsumcl 20129 | Closure of a finite group ... |
| gsumf1o 20130 | Re-index a finite group su... |
| gsumreidx 20131 | Re-index a finite group su... |
| gsumzsubmcl 20132 | Closure of a group sum in ... |
| gsumsubmcl 20133 | Closure of a group sum in ... |
| gsumsubgcl 20134 | Closure of a group sum in ... |
| gsumzaddlem 20135 | The sum of two group sums.... |
| gsumzadd 20136 | The sum of two group sums.... |
| gsumadd 20137 | The sum of two group sums.... |
| gsummptfsadd 20138 | The sum of two group sums ... |
| gsummptfidmadd 20139 | The sum of two group sums ... |
| gsummptfidmadd2 20140 | The sum of two group sums ... |
| gsumzsplit 20141 | Split a group sum into two... |
| gsumsplit 20142 | Split a group sum into two... |
| gsumsplit2 20143 | Split a group sum into two... |
| gsummptfidmsplit 20144 | Split a group sum expresse... |
| gsummptfidmsplitres 20145 | Split a group sum expresse... |
| gsummptfzsplit 20146 | Split a group sum expresse... |
| gsummptfzsplitl 20147 | Split a group sum expresse... |
| gsumconst 20148 | Sum of a constant series. ... |
| gsumconstf 20149 | Sum of a constant series. ... |
| gsummptshft 20150 | Index shift of a finite gr... |
| gsumzmhm 20151 | Apply a group homomorphism... |
| gsummhm 20152 | Apply a group homomorphism... |
| gsummhm2 20153 | Apply a group homomorphism... |
| gsummptmhm 20154 | Apply a group homomorphism... |
| gsummulglem 20155 | Lemma for ~ gsummulg and ~... |
| gsummulg 20156 | Nonnegative multiple of a ... |
| gsummulgz 20157 | Integer multiple of a grou... |
| gsumzoppg 20158 | The opposite of a group su... |
| gsumzinv 20159 | Inverse of a group sum. (... |
| gsuminv 20160 | Inverse of a group sum. (... |
| gsummptfidminv 20161 | Inverse of a group sum exp... |
| gsumsub 20162 | The difference of two grou... |
| gsummptfssub 20163 | The difference of two grou... |
| gsummptfidmsub 20164 | The difference of two grou... |
| gsumsnfd 20165 | Group sum of a singleton, ... |
| gsumsnd 20166 | Group sum of a singleton, ... |
| gsumsnf 20167 | Group sum of a singleton, ... |
| gsumsn 20168 | Group sum of a singleton. ... |
| gsumpr 20169 | Group sum of a pair. (Con... |
| gsumzunsnd 20170 | Append an element to a fin... |
| gsumunsnfd 20171 | Append an element to a fin... |
| gsumunsnd 20172 | Append an element to a fin... |
| gsumunsnf 20173 | Append an element to a fin... |
| gsumunsn 20174 | Append an element to a fin... |
| gsumdifsnd 20175 | Extract a summand from a f... |
| gsumpt 20176 | Sum of a family that is no... |
| gsummptf1o 20177 | Re-index a finite group su... |
| gsummptun 20178 | Group sum of a disjoint un... |
| gsummpt1n0 20179 | If only one summand in a f... |
| gsummptif1n0 20180 | If only one summand in a f... |
| gsummptcl 20181 | Closure of a finite group ... |
| gsummptfif1o 20182 | Re-index a finite group su... |
| gsummptfzcl 20183 | Closure of a finite group ... |
| gsum2dlem1 20184 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 20185 | Lemma for ~ gsum2d . (Con... |
| gsum2d 20186 | Write a sum over a two-dim... |
| gsum2d2lem 20187 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 20188 | Write a group sum over a t... |
| gsumcom2 20189 | Two-dimensional commutatio... |
| gsumxp 20190 | Write a group sum over a c... |
| gsumcom 20191 | Commute the arguments of a... |
| gsumcom3 20192 | A commutative law for fini... |
| gsumcom3fi 20193 | A commutative law for fini... |
| gsumxp2 20194 | Write a group sum over a c... |
| prdsgsum 20195 | Finite commutative sums in... |
| pwsgsum 20196 | Finite commutative sums in... |
| fsfnn0gsumfsffz 20197 | Replacing a finitely suppo... |
| nn0gsumfz 20198 | Replacing a finitely suppo... |
| nn0gsumfz0 20199 | Replacing a finitely suppo... |
| gsummptnn0fz 20200 | A final group sum over a f... |
| gsummptnn0fzfv 20201 | A final group sum over a f... |
| telgsumfzslem 20202 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 20203 | Telescoping group sum rang... |
| telgsumfz 20204 | Telescoping group sum rang... |
| telgsumfz0s 20205 | Telescoping finite group s... |
| telgsumfz0 20206 | Telescoping finite group s... |
| telgsums 20207 | Telescoping finitely suppo... |
| telgsum 20208 | Telescoping finitely suppo... |
| reldmdprd 20213 | The domain of the internal... |
| dmdprd 20214 | The domain of definition o... |
| dmdprdd 20215 | Show that a given family i... |
| dprddomprc 20216 | A family of subgroups inde... |
| dprddomcld 20217 | If a family of subgroups i... |
| dprdval0prc 20218 | The internal direct produc... |
| dprdval 20219 | The value of the internal ... |
| eldprd 20220 | A class ` A ` is an intern... |
| dprdgrp 20221 | Reverse closure for the in... |
| dprdf 20222 | The function ` S ` is a fa... |
| dprdf2 20223 | The function ` S ` is a fa... |
| dprdcntz 20224 | The function ` S ` is a fa... |
| dprddisj 20225 | The function ` S ` is a fa... |
| dprdw 20226 | The property of being a fi... |
| dprdwd 20227 | A mapping being a finitely... |
| dprdff 20228 | A finitely supported funct... |
| dprdfcl 20229 | A finitely supported funct... |
| dprdffsupp 20230 | A finitely supported funct... |
| dprdfcntz 20231 | A function on the elements... |
| dprdssv 20232 | The internal direct produc... |
| dprdfid 20233 | A function mapping all but... |
| eldprdi 20234 | The domain of definition o... |
| dprdfinv 20235 | Take the inverse of a grou... |
| dprdfadd 20236 | Take the sum of group sums... |
| dprdfsub 20237 | Take the difference of gro... |
| dprdfeq0 20238 | The zero function is the o... |
| dprdf11 20239 | Two group sums over a dire... |
| dprdsubg 20240 | The internal direct produc... |
| dprdub 20241 | Each factor is a subset of... |
| dprdlub 20242 | The direct product is smal... |
| dprdspan 20243 | The direct product is the ... |
| dprdres 20244 | Restriction of a direct pr... |
| dprdss 20245 | Create a direct product by... |
| dprdz 20246 | A family consisting entire... |
| dprd0 20247 | The empty family is an int... |
| dprdf1o 20248 | Rearrange the index set of... |
| dprdf1 20249 | Rearrange the index set of... |
| subgdmdprd 20250 | A direct product in a subg... |
| subgdprd 20251 | A direct product in a subg... |
| dprdsn 20252 | A singleton family is an i... |
| dmdprdsplitlem 20253 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20254 | The function ` S ` is a fa... |
| dprddisj2 20255 | The function ` S ` is a fa... |
| dprd2dlem2 20256 | The direct product of a co... |
| dprd2dlem1 20257 | The direct product of a co... |
| dprd2da 20258 | The direct product of a co... |
| dprd2db 20259 | The direct product of a co... |
| dprd2d2 20260 | The direct product of a co... |
| dmdprdsplit2lem 20261 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20262 | The direct product splits ... |
| dmdprdsplit 20263 | The direct product splits ... |
| dprdsplit 20264 | The direct product is the ... |
| dmdprdpr 20265 | A singleton family is an i... |
| dprdpr 20266 | A singleton family is an i... |
| dpjlem 20267 | Lemma for theorems about d... |
| dpjcntz 20268 | The two subgroups that app... |
| dpjdisj 20269 | The two subgroups that app... |
| dpjlsm 20270 | The two subgroups that app... |
| dpjfval 20271 | Value of the direct produc... |
| dpjval 20272 | Value of the direct produc... |
| dpjf 20273 | The ` X ` -th index projec... |
| dpjidcl 20274 | The key property of projec... |
| dpjeq 20275 | Decompose a group sum into... |
| dpjid 20276 | The key property of projec... |
| dpjlid 20277 | The ` X ` -th index projec... |
| dpjrid 20278 | The ` Y ` -th index projec... |
| dpjghm 20279 | The direct product is the ... |
| dpjghm2 20280 | The direct product is the ... |
| ablfacrplem 20281 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20282 | A finite abelian group who... |
| ablfacrp2 20283 | The factors ` K , L ` of ~... |
| ablfac1lem 20284 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20285 | The factors of ~ ablfac1b ... |
| ablfac1b 20286 | Any abelian group is the d... |
| ablfac1c 20287 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20288 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20289 | The factorization of ~ abl... |
| pgpfac1lem1 20290 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20291 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20292 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20293 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20294 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20295 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20296 | Factorization of a finite ... |
| pgpfaclem1 20297 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20298 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20299 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20300 | Full factorization of a fi... |
| ablfaclem1 20301 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20302 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20303 | Lemma for ~ ablfac . (Con... |
| ablfac 20304 | The Fundamental Theorem of... |
| ablfac2 20305 | Choose generators for each... |
| issimpg 20308 | The predicate "is a simple... |
| issimpgd 20309 | Deduce a simple group from... |
| simpggrp 20310 | A simple group is a group.... |
| simpggrpd 20311 | A simple group is a group.... |
| simpg2nsg 20312 | A simple group has two nor... |
| trivnsimpgd 20313 | Trivial groups are not sim... |
| simpgntrivd 20314 | Simple groups are nontrivi... |
| simpgnideld 20315 | A simple group contains a ... |
| simpgnsgd 20316 | The only normal subgroups ... |
| simpgnsgeqd 20317 | A normal subgroup of a sim... |
| 2nsgsimpgd 20318 | If any normal subgroup of ... |
| simpgnsgbid 20319 | A nontrivial group is simp... |
| ablsimpnosubgd 20320 | A subgroup of an abelian s... |
| ablsimpg1gend 20321 | An abelian simple group is... |
| ablsimpgcygd 20322 | An abelian simple group is... |
| ablsimpgfindlem1 20323 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20324 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20325 | Relationship between the o... |
| ablsimpgfind 20326 | An abelian simple group is... |
| fincygsubgd 20327 | The subgroup referenced in... |
| fincygsubgodd 20328 | Calculate the order of a s... |
| fincygsubgodexd 20329 | A finite cyclic group has ... |
| prmgrpsimpgd 20330 | A group of prime order is ... |
| ablsimpgprmd 20331 | An abelian simple group ha... |
| ablsimpgd 20332 | An abelian group is simple... |
| isomnd 20337 | A (left) ordered monoid is... |
| isogrp 20338 | A (left-)ordered group is ... |
| ogrpgrp 20339 | A left-ordered group is a ... |
| omndmnd 20340 | A left-ordered monoid is a... |
| omndtos 20341 | A left-ordered monoid is a... |
| omndadd 20342 | In an ordered monoid, the ... |
| omndaddr 20343 | In a right ordered monoid,... |
| omndadd2d 20344 | In a commutative left orde... |
| omndadd2rd 20345 | In a left- and right- orde... |
| submomnd 20346 | A submonoid of an ordered ... |
| omndmul2 20347 | In an ordered monoid, the ... |
| omndmul3 20348 | In an ordered monoid, the ... |
| omndmul 20349 | In a commutative ordered m... |
| ogrpinv0le 20350 | In an ordered group, the o... |
| ogrpsub 20351 | In an ordered group, the o... |
| ogrpaddlt 20352 | In an ordered group, stric... |
| ogrpaddltbi 20353 | In a right ordered group, ... |
| ogrpaddltrd 20354 | In a right ordered group, ... |
| ogrpaddltrbid 20355 | In a right ordered group, ... |
| ogrpsublt 20356 | In an ordered group, stric... |
| ogrpinv0lt 20357 | In an ordered group, the o... |
| ogrpinvlt 20358 | In an ordered group, the o... |
| gsumle 20359 | A finite sum in an ordered... |
| fnmgp 20362 | The multiplicative group o... |
| mgpval 20363 | Value of the multiplicatio... |
| mgpplusg 20364 | Value of the group operati... |
| mgpbas 20365 | Base set of the multiplica... |
| mgpsca 20366 | The multiplication monoid ... |
| mgptset 20367 | Topology component of the ... |
| mgptopn 20368 | Topology of the multiplica... |
| mgpds 20369 | Distance function of the m... |
| mgpress 20370 | Subgroup commutes with the... |
| prdsmgp 20371 | The multiplicative monoid ... |
| elmgplsm 20372 | Membership in a product of... |
| elmgplsmd 20373 | Membership in a product of... |
| isrng 20376 | The predicate "is a non-un... |
| rngabl 20377 | A non-unital ring is an (a... |
| rngmgp 20378 | A non-unital ring is a sem... |
| rngmgpf 20379 | Restricted functionality o... |
| rnggrp 20380 | A non-unital ring is a (ad... |
| rngass 20381 | Associative law for the mu... |
| rngdi 20382 | Distributive law for the m... |
| rngdir 20383 | Distributive law for the m... |
| rngacl 20384 | Closure of the addition op... |
| rng0cl 20385 | The zero element of a non-... |
| rngcl 20386 | Closure of the multiplicat... |
| rnglz 20387 | The zero of a non-unital r... |
| rngrz 20388 | The zero of a non-unital r... |
| rngmneg1 20389 | Negation of a product in a... |
| rngmneg2 20390 | Negation of a product in a... |
| rngm2neg 20391 | Double negation of a produ... |
| rngansg 20392 | Every additive subgroup of... |
| rngsubdi 20393 | Ring multiplication distri... |
| rngsubdir 20394 | Ring multiplication distri... |
| isrngd 20395 | Properties that determine ... |
| rngpropd 20396 | If two structures have the... |
| prdsmulrngcl 20397 | Closure of the multiplicat... |
| prdsrngd 20398 | A product of non-unital ri... |
| imasrng 20399 | The image structure of a n... |
| imasrngf1 20400 | The image of a non-unital ... |
| xpsrngd 20401 | A product of two non-unita... |
| qusrng 20402 | The quotient structure of ... |
| rng1zrlem 20403 | Lemma for ~ rng1zr and ~ s... |
| rng1zr 20404 | The only ring with a base ... |
| rngen1zr 20405 | The only ring with one ele... |
| rngen1zr0 20406 | The only ring with one ele... |
| ringidval 20409 | The value of the unity ele... |
| dfur2 20410 | The multiplicative identit... |
| ringurd 20411 | Deduce the unity element o... |
| issrg 20414 | The predicate "is a semiri... |
| srgcmn 20415 | A semiring is a commutativ... |
| srgmnd 20416 | A semiring is a monoid. (... |
| srgmgp 20417 | A semiring is a monoid und... |
| srgdilem 20418 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20419 | Closure of the multiplicat... |
| srgass 20420 | Associative law for the mu... |
| srgideu 20421 | The unity element of a sem... |
| srgfcl 20422 | Functionality of the multi... |
| srgdi 20423 | Distributive law for the m... |
| srgdir 20424 | Distributive law for the m... |
| srgidcl 20425 | The unity element of a sem... |
| srg0cl 20426 | The zero element of a semi... |
| srgidmlem 20427 | Lemma for ~ srglidm and ~ ... |
| srglidm 20428 | The unity element of a sem... |
| srgridm 20429 | The unity element of a sem... |
| issrgid 20430 | Properties showing that an... |
| srgacl 20431 | Closure of the addition op... |
| srgcom 20432 | Commutativity of the addit... |
| srgrz 20433 | The zero of a semiring is ... |
| srglz 20434 | The zero of a semiring is ... |
| srgisid 20435 | In a semiring, the only le... |
| o2timesd 20436 | An element of a ring-like ... |
| rglcom4d 20437 | Restricted commutativity o... |
| srgo2times 20438 | A semiring element plus it... |
| srgcom4lem 20439 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20440 | Restricted commutativity o... |
| srg1zr 20441 | The only semiring with a b... |
| srgen1zr0 20442 | The only semiring with one... |
| srgmulgass 20443 | An associative property be... |
| srgpcomp 20444 | If two elements of a semir... |
| srgpcompp 20445 | If two elements of a semir... |
| srgpcomppsc 20446 | If two elements of a semir... |
| srglmhm 20447 | Left-multiplication in a s... |
| srgrmhm 20448 | Right-multiplication in a ... |
| srgsummulcr 20449 | A finite semiring sum mult... |
| sgsummulcl 20450 | A finite semiring sum mult... |
| srg1expzeq1 20451 | The exponentiation (by a n... |
| srgbinomlem1 20452 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20453 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20454 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20455 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20456 | Lemma for ~ srgbinom . In... |
| srgbinom 20457 | The binomial theorem for c... |
| csrgbinom 20458 | The binomial theorem for c... |
| isring 20463 | The predicate "is a (unita... |
| ringgrp 20464 | A ring is a group. (Contr... |
| ringmgp 20465 | A ring is a monoid under m... |
| iscrng 20466 | A commutative ring is a ri... |
| crngmgp 20467 | A commutative ring's multi... |
| ringbn0 20468 | The base set of a ring is ... |
| ringgrpd 20469 | A ring is a group. (Contr... |
| ringmnd 20470 | A ring is a monoid under a... |
| ringmgm 20471 | A ring is a magma. (Contr... |
| crngring 20472 | A commutative ring is a ri... |
| crngringd 20473 | A commutative ring is a ri... |
| crnggrpd 20474 | A commutative ring is a gr... |
| mgpf 20475 | Restricted functionality o... |
| ringdilem 20476 | Properties of a unital rin... |
| ringcl 20477 | Closure of the multiplicat... |
| crngcom 20478 | A commutative ring's multi... |
| iscrng2 20479 | A commutative ring is a ri... |
| ringass 20480 | Associative law for multip... |
| ringideu 20481 | The unity element of a rin... |
| crngcomd 20482 | Multiplication is commutat... |
| crngbascntr 20483 | The base set of a commutat... |
| ringcld 20484 | Closure of the multiplicat... |
| ringassd 20485 | Associative law for multip... |
| crng12d 20486 | Commutative/associative la... |
| crng32d 20487 | Commutative/associative la... |
| crng4 20488 | Commutative/associative la... |
| ringdi 20489 | Distributive law for the m... |
| ringdir 20490 | Distributive law for the m... |
| ringdid 20491 | Distributive law for the m... |
| ringdird 20492 | Distributive law for the m... |
| ringdi22 20493 | Expand the product of two ... |
| ringidcl 20494 | The unity element of a rin... |
| ringidcld 20495 | The unity element of a rin... |
| ring0cl 20496 | The zero element of a ring... |
| ringidmlem 20497 | Lemma for ~ ringlidm and ~... |
| ringlidm 20498 | The unity element of a rin... |
| ringridm 20499 | The unity element of a rin... |
| isringid 20500 | Properties showing that an... |
| ringlidmd 20501 | The unity element of a rin... |
| ringridmd 20502 | The unity element of a rin... |
| ringid 20503 | The multiplication operati... |
| ringo2times 20504 | A ring element plus itself... |
| ringadd2 20505 | A ring element plus itself... |
| ringidss 20506 | A subset of the multiplica... |
| ringacl 20507 | Closure of the addition op... |
| ringcomlem 20508 | Lemma for ~ ringcom . Thi... |
| ringcom 20509 | Commutativity of the addit... |
| ringabl 20510 | A ring is an Abelian group... |
| ringcmn 20511 | A ring is a commutative mo... |
| ringabld 20512 | A ring is an Abelian group... |
| ringcmnd 20513 | A ring is a commutative mo... |
| ringrng 20514 | A unital ring is a non-uni... |
| ringssrng 20515 | The unital rings are non-u... |
| isringrng 20516 | The predicate "is a unital... |
| dfring2 20517 | The predicate "is a unital... |
| dfring3 20518 | The predicate "is a (unita... |
| ringpropd 20519 | If two structures have the... |
| crngpropd 20520 | If two structures have the... |
| ringprop 20521 | If two structures have the... |
| isringd 20522 | Properties that determine ... |
| iscrngd 20523 | Properties that determine ... |
| ringlz 20524 | The zero of a unital ring ... |
| ringrz 20525 | The zero of a unital ring ... |
| ringlzd 20526 | The zero of a unital ring ... |
| ringrzd 20527 | The zero of a unital ring ... |
| ringsrg 20528 | Any ring is also a semirin... |
| ring1eq0 20529 | If one and zero are equal,... |
| ring1ne0 20530 | If a ring has at least two... |
| ringinvnz1ne0 20531 | In a unital ring, a left i... |
| ringinvnzdiv 20532 | In a unital ring, a left i... |
| ringnegl 20533 | Negation in a ring is the ... |
| ringnegr 20534 | Negation in a ring is the ... |
| ringmneg1 20535 | Negation of a product in a... |
| ringmneg2 20536 | Negation of a product in a... |
| ringm2neg 20537 | Double negation of a produ... |
| ringsubdi 20538 | Ring multiplication distri... |
| ringsubdir 20539 | Ring multiplication distri... |
| mulgass2 20540 | An associative property be... |
| ring1 20541 | The (smallest) structure r... |
| ringn0 20542 | Rings exist. (Contributed... |
| ringlghm 20543 | Left-multiplication in a r... |
| ringrghm 20544 | Right-multiplication in a ... |
| gsummulc1 20545 | A finite ring sum multipli... |
| gsummulc2 20546 | A finite ring sum multipli... |
| gsummgp0 20547 | If one factor in a finite ... |
| gsumdixp 20548 | Distribute a binary produc... |
| prdsmulrcl 20549 | A structure product of rin... |
| prdsringd 20550 | A product of rings is a ri... |
| prdscrngd 20551 | A product of commutative r... |
| prds1 20552 | Value of the ring unity in... |
| pwsring 20553 | A structure power of a rin... |
| pws1 20554 | Value of the ring unity in... |
| pwscrng 20555 | A structure power of a com... |
| pwsmgp 20556 | The multiplicative group o... |
| pwspjmhmmgpd 20557 | The projection given by ~ ... |
| pwsexpg 20558 | Value of a group exponenti... |
| pwsgprod 20559 | Finite products in a power... |
| imasring 20560 | The image structure of a r... |
| imasringf1 20561 | The image of a ring under ... |
| xpsringd 20562 | A product of two rings is ... |
| xpsring1d 20563 | The multiplicative identit... |
| qusring2 20564 | The quotient structure of ... |
| crngbinom 20565 | The binomial theorem for c... |
| opprval 20568 | Value of the opposite ring... |
| opprmulfval 20569 | Value of the multiplicatio... |
| opprmul 20570 | Value of the multiplicatio... |
| crngoppr 20571 | In a commutative ring, the... |
| opprlem 20572 | Lemma for ~ opprbas and ~ ... |
| opprbas 20573 | Base set of an opposite ri... |
| oppradd 20574 | Addition operation of an o... |
| opprrng 20575 | An opposite non-unital rin... |
| opprrngb 20576 | A class is a non-unital ri... |
| opprring 20577 | An opposite ring is a ring... |
| opprringb 20578 | Bidirectional form of ~ op... |
| oppr0 20579 | Additive identity of an op... |
| oppr1 20580 | Multiplicative identity of... |
| opprneg 20581 | The negative function in a... |
| opprsubg 20582 | Being a subgroup is a symm... |
| mulgass3 20583 | An associative property be... |
| reldvdsr 20590 | The divides relation is a ... |
| dvdsrval 20591 | Value of the divides relat... |
| dvdsr 20592 | Value of the divides relat... |
| dvdsr2 20593 | Value of the divides relat... |
| dvdsrmul 20594 | A left-multiple of ` X ` i... |
| dvdsrcl 20595 | Closure of a dividing elem... |
| dvdsrcl2 20596 | Closure of a dividing elem... |
| dvdsrid 20597 | An element in a (unital) r... |
| dvdsrtr 20598 | Divisibility is transitive... |
| dvdsrmul1 20599 | The divisibility relation ... |
| dvdsrneg 20600 | An element divides its neg... |
| dvdsr01 20601 | In a ring, zero is divisib... |
| dvdsr02 20602 | Only zero is divisible by ... |
| isunit 20603 | Property of being a unit o... |
| 1unit 20604 | The multiplicative identit... |
| unitcl 20605 | A unit is an element of th... |
| unitss 20606 | The set of units is contai... |
| opprunit 20607 | Being a unit is a symmetri... |
| crngunit 20608 | Property of being a unit i... |
| dvdsunit 20609 | A divisor of a unit is a u... |
| unitmulcl 20610 | The product of units is a ... |
| unitmulclb 20611 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20612 | The base set of the group ... |
| unitgrp 20613 | The group of units is a gr... |
| unitabl 20614 | The group of units of a co... |
| unitgrpid 20615 | The identity of the group ... |
| unitsubm 20616 | The group of units is a su... |
| invrfval 20619 | Multiplicative inverse fun... |
| unitinvcl 20620 | The inverse of a unit exis... |
| unitinvinv 20621 | The inverse of the inverse... |
| ringinvcl 20622 | The inverse of a unit is a... |
| unitlinv 20623 | A unit times its inverse i... |
| unitrinv 20624 | A unit times its inverse i... |
| 1rinv 20625 | The inverse of the ring un... |
| 0unit 20626 | The additive identity is a... |
| unitnegcl 20627 | The negative of a unit is ... |
| ringunitnzdiv 20628 | In a unitary ring, a unit ... |
| ring1nzdiv 20629 | In a unitary ring, the rin... |
| dvrfval 20632 | Division operation in a ri... |
| dvrval 20633 | Division operation in a ri... |
| dvrcl 20634 | Closure of division operat... |
| unitdvcl 20635 | The units are closed under... |
| dvrid 20636 | A ring element divided by ... |
| dvr1 20637 | A ring element divided by ... |
| dvrass 20638 | An associative law for div... |
| dvrcan1 20639 | A cancellation law for div... |
| dvrcan3 20640 | A cancellation law for div... |
| dvreq1 20641 | Equality in terms of ratio... |
| dvrdir 20642 | Distributive law for the d... |
| rdivmuldivd 20643 | Multiplication of two rati... |
| ringinvdv 20644 | Write the inverse function... |
| rngidpropd 20645 | The ring unity depends onl... |
| dvdsrpropd 20646 | The divisibility relation ... |
| unitpropd 20647 | The set of units depends o... |
| invrpropd 20648 | The ring inverse function ... |
| isirred 20649 | An irreducible element of ... |
| isnirred 20650 | The property of being a no... |
| isirred2 20651 | Expand out the class diffe... |
| opprirred 20652 | Irreducibility is symmetri... |
| irredn0 20653 | The additive identity is n... |
| irredcl 20654 | An irreducible element is ... |
| irrednu 20655 | An irreducible element is ... |
| irredn1 20656 | The multiplicative identit... |
| irredrmul 20657 | The product of an irreduci... |
| irredlmul 20658 | The product of a unit and ... |
| irredmul 20659 | If product of two elements... |
| irredneg 20660 | The negative of an irreduc... |
| irrednegb 20661 | An element is irreducible ... |
| rnghmrcl 20668 | Reverse closure of a non-u... |
| rnghmfn 20669 | The mapping of two non-uni... |
| rnghmval 20670 | The set of the non-unital ... |
| isrnghm 20671 | A function is a non-unital... |
| isrnghmmul 20672 | A function is a non-unital... |
| rnghmmgmhm 20673 | A non-unital ring homomorp... |
| rnghmval2 20674 | The non-unital ring homomo... |
| isrngim 20675 | An isomorphism of non-unit... |
| rngimrcl 20676 | Reverse closure for an iso... |
| rnghmghm 20677 | A non-unital ring homomorp... |
| rnghmf 20678 | A ring homomorphism is a f... |
| rnghmmul 20679 | A homomorphism of non-unit... |
| isrnghm2d 20680 | Demonstration of non-unita... |
| isrnghmd 20681 | Demonstration of non-unita... |
| rnghmf1o 20682 | A non-unital ring homomorp... |
| isrngim2 20683 | An isomorphism of non-unit... |
| rngimf1o 20684 | An isomorphism of non-unit... |
| rngimrnghm 20685 | An isomorphism of non-unit... |
| rngimcnv 20686 | The converse of an isomorp... |
| rnghmco 20687 | The composition of non-uni... |
| idrnghm 20688 | The identity homomorphism ... |
| c0mgm 20689 | The constant mapping to ze... |
| c0mhm 20690 | The constant mapping to ze... |
| c0ghm 20691 | The constant mapping to ze... |
| c0snmgmhm 20692 | The constant mapping to ze... |
| c0snmhm 20693 | The constant mapping to ze... |
| c0snghm 20694 | The constant mapping to ze... |
| rngisomfv1 20695 | If there is a non-unital r... |
| rngisom1 20696 | If there is a non-unital r... |
| rngisomring 20697 | If there is a non-unital r... |
| rngisomring1 20698 | If there is a non-unital r... |
| dfrhm2 20704 | The property of a ring hom... |
| rhmval0 20705 | The set of ring homomorphi... |
| isrhm0 20706 | The predicate "is a ring h... |
| rhmrcl1 20707 | Reverse closure of a ring ... |
| rhmrcl2 20708 | Reverse closure of a ring ... |
| isrhm 20709 | A function is a ring homom... |
| rhmmhm 20710 | A ring homomorphism is a h... |
| rhmisrnghm 20711 | Each unital ring homomorph... |
| rimrcl 20712 | Reverse closure for an iso... |
| isrim0 20713 | A ring isomorphism is a ho... |
| rhmghm 20714 | A ring homomorphism is an ... |
| rhmf 20715 | A ring homomorphism is a f... |
| rhmcl 20716 | Closure law for a ring hom... |
| rimcnv 20717 | The converse of a ring iso... |
| rhmadd 20718 | Ring homomorphisms preserv... |
| rhmsub 20719 | Ring homomorphisms preserv... |
| rhmmul 20720 | A homomorphism of rings pr... |
| isrhm2d 20721 | Demonstration of ring homo... |
| isrhmd 20722 | Demonstration of ring homo... |
| rhm0 20723 | A ring homomorphism preser... |
| rhm1 20724 | Ring homomorphisms are req... |
| idrhm 20725 | The identity homomorphism ... |
| crngrhmfo 20726 | The image of a surjective ... |
| rhmf1o 20727 | A ring homomorphism is bij... |
| isrim 20728 | An isomorphism of rings is... |
| rimf1o 20729 | An isomorphism of rings is... |
| rimval 20730 | The set of ring isomorphis... |
| rimrhm 20731 | A ring isomorphism is a ho... |
| rimrcl1 20732 | Reverse closure of a ring ... |
| rimrcl2 20733 | Reverse closure of a ring ... |
| rimgim 20734 | An isomorphism of rings is... |
| rimisrngim 20735 | Each unital ring isomorphi... |
| rhmfn 20736 | The mapping of two rings t... |
| rimfn 20737 | The mapping of two rings t... |
| rhmval 20738 | The ring homomorphisms bet... |
| rhmco 20739 | The composition of ring ho... |
| rhmkerinj 20740 | A ring homomorphism is inj... |
| pwsco1rhm 20741 | Right composition with a f... |
| pwsco2rhm 20742 | Left composition with a ri... |
| ricrel 20744 | The domain of the ring iso... |
| brric 20745 | The relation "is isomorphi... |
| brrici 20746 | Prove isomorphic by an exp... |
| rimco 20747 | The composition of ring is... |
| ricref 20748 | Ring isomorphism is reflex... |
| riclcl 20749 | Ring isomorphism implies t... |
| ricrcl 20750 | Ring isomorphism implies t... |
| ricsym 20751 | Ring isomorphism is symmet... |
| rictr 20752 | Ring isomorphism is transi... |
| isbrric2 20753 | The relation "is isomorphi... |
| brric2 20754 | The ring isomorphism relat... |
| ricgic 20755 | If two rings are (ring) is... |
| ricer 20756 | Ring isomorphism is an equ... |
| dfric2 20757 | Alternate definition of th... |
| rhmdvdsr 20758 | A ring homomorphism preser... |
| rhmopp 20759 | A ring homomorphism is als... |
| elrhmunit 20760 | Ring homomorphisms preserv... |
| rhmunitinv 20761 | Ring homomorphisms preserv... |
| isnzr 20764 | Property of a nonzero ring... |
| nzrnz 20765 | One and zero are different... |
| nzrring 20766 | A nonzero ring is a ring. ... |
| nzrringOLD 20767 | Obsolete version of ~ nzrr... |
| isnzr2 20768 | Equivalent characterizatio... |
| drnglidl1ne0 20769 | In a nonzero ring, the zer... |
| isnzr2hash 20770 | Equivalent characterizatio... |
| nzrpropd 20771 | If two structures have the... |
| opprnzrb 20772 | The opposite of a nonzero ... |
| opprnzr 20773 | The opposite of a nonzero ... |
| ringelnzr 20774 | A ring is nonzero if it ha... |
| nzrunit 20775 | A unit is nonzero in any n... |
| 0ringnnzr 20776 | A ring is a zero ring iff ... |
| 0ring 20777 | If a ring has only one ele... |
| 0ringdif 20778 | A zero ring is a ring whic... |
| 0ringbas 20779 | The base set of a zero rin... |
| 0ring01eq 20780 | In a ring with only one el... |
| 01eq0ring 20781 | If the zero and the identi... |
| 01eq0ringOLD 20782 | Obsolete version of ~ 01eq... |
| 0ring01eqbi2 20783 | In a ring, ` 0 = 1 ` iff t... |
| 0ring01eqbi 20784 | In a unital ring the zero ... |
| 0ring1eq0 20785 | In a zero ring, a ring whi... |
| c0rhm 20786 | The constant mapping to ze... |
| c0rnghm 20787 | The constant mapping to ze... |
| zrrnghm 20788 | The constant mapping to ze... |
| nrhmzr 20789 | There is no ring homomorph... |
| islring 20792 | The predicate "is a local ... |
| lringnzr 20793 | A local ring is a nonzero ... |
| lringring 20794 | A local ring is a ring. (... |
| lringnz 20795 | A local ring is a nonzero ... |
| lringuplu 20796 | If the sum of two elements... |
| issubrng 20799 | The subring of non-unital ... |
| subrngss 20800 | A subring is a subset. (C... |
| subrngid 20801 | Every non-unital ring is a... |
| subrngrng 20802 | A subring is a non-unital ... |
| subrngrcl 20803 | Reverse closure for a subr... |
| subrngsubg 20804 | A subring is a subgroup. ... |
| subrngringnsg 20805 | A subring is a normal subg... |
| subrngbas 20806 | Base set of a subring stru... |
| subrng0 20807 | A subring always has the s... |
| subrngacl 20808 | A subring is closed under ... |
| subrngmcl 20809 | A subring is closed under ... |
| issubrng2 20810 | Characterize the subrings ... |
| opprsubrng 20811 | Being a subring is a symme... |
| subrngint 20812 | The intersection of a none... |
| subrngin 20813 | The intersection of two su... |
| subrngmre 20814 | The subrings of a non-unit... |
| subsubrng 20815 | A subring of a subring is ... |
| subsubrng2 20816 | The set of subrings of a s... |
| rhmimasubrnglem 20817 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20818 | The homomorphic image of a... |
| cntzsubrng 20819 | Centralizers in a non-unit... |
| subrngpropd 20820 | If two structures have the... |
| issubrg 20823 | The subring predicate. (C... |
| subrgss 20824 | A subring is a subset. (C... |
| subrgid 20825 | Every ring is a subring of... |
| subrgring 20826 | A subring is a ring. (Con... |
| subrgcrng 20827 | A subring of a commutative... |
| subrgrcl 20828 | Reverse closure for a subr... |
| subrgsubg 20829 | A subring is a subgroup. ... |
| subrgsubrng 20830 | A subring of a unital ring... |
| subrg0 20831 | A subring always has the s... |
| subrg1cl 20832 | A subring contains the mul... |
| subrgbas 20833 | Base set of a subring stru... |
| subrg1 20834 | A subring always has the s... |
| subrgacl 20835 | A subring is closed under ... |
| subrgmcl 20836 | A subring is closed under ... |
| subrgsubm 20837 | A subring is a submonoid o... |
| subrgdvds 20838 | If an element divides anot... |
| subrguss 20839 | A unit of a subring is a u... |
| subrginv 20840 | A subring always has the s... |
| subrgdv 20841 | A subring always has the s... |
| subrgunit 20842 | An element of a ring is a ... |
| subrgugrp 20843 | The units of a subring for... |
| issubrg2 20844 | Characterize the subrings ... |
| opprsubrg 20845 | Being a subring is a symme... |
| subrgnzr 20846 | A subring of a nonzero rin... |
| subrgint 20847 | The intersection of a none... |
| subrgin 20848 | The intersection of two su... |
| subrgmre 20849 | The subrings of a ring are... |
| subsubrg 20850 | A subring of a subring is ... |
| subsubrg2 20851 | The set of subrings of a s... |
| issubrg3 20852 | A subring is an additive s... |
| resrhm 20853 | Restriction of a ring homo... |
| resrhm2b 20854 | Restriction of the codomai... |
| rhmeql 20855 | The equalizer of two ring ... |
| rhmima 20856 | The homomorphic image of a... |
| rnrhmsubrg 20857 | The range of a ring homomo... |
| cntzsubr 20858 | Centralizers in a ring are... |
| pwsdiagrhm 20859 | Diagonal homomorphism into... |
| subrgpropd 20860 | If two structures have the... |
| rhmpropd 20861 | Ring homomorphism depends ... |
| rgspnval 20864 | Value of the ring-span of ... |
| rgspncl 20865 | The ring-span of a set is ... |
| rgspnssid 20866 | The ring-span of a set con... |
| rgspnmin 20867 | The ring-span is contained... |
| rngcval 20870 | Value of the category of n... |
| rnghmresfn 20871 | The class of non-unital ri... |
| rnghmresel 20872 | An element of the non-unit... |
| rngcbas 20873 | Set of objects of the cate... |
| rngchomfval 20874 | Set of arrows of the categ... |
| rngchom 20875 | Set of arrows of the categ... |
| elrngchom 20876 | A morphism of non-unital r... |
| rngchomfeqhom 20877 | The functionalized Hom-set... |
| rngccofval 20878 | Composition in the categor... |
| rngcco 20879 | Composition in the categor... |
| dfrngc2 20880 | Alternate definition of th... |
| rnghmsscmap2 20881 | The non-unital ring homomo... |
| rnghmsscmap 20882 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20883 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20884 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20885 | The non-unital ring homomo... |
| rngccat 20886 | The category of non-unital... |
| rngcid 20887 | The identity arrow in the ... |
| rngcsect 20888 | A section in the category ... |
| rngcinv 20889 | An inverse in the category... |
| rngciso 20890 | An isomorphism in the cate... |
| rngcifuestrc 20891 | The "inclusion functor" fr... |
| funcrngcsetc 20892 | The "natural forgetful fun... |
| funcrngcsetcALT 20893 | Alternate proof of ~ funcr... |
| zrinitorngc 20894 | The zero ring is an initia... |
| zrtermorngc 20895 | The zero ring is a termina... |
| zrzeroorngc 20896 | The zero ring is a zero ob... |
| ringcval 20899 | Value of the category of u... |
| rhmresfn 20900 | The class of unital ring h... |
| rhmresel 20901 | An element of the unital r... |
| ringcbas 20902 | Set of objects of the cate... |
| ringchomfval 20903 | Set of arrows of the categ... |
| ringchom 20904 | Set of arrows of the categ... |
| elringchom 20905 | A morphism of unital rings... |
| ringchomfeqhom 20906 | The functionalized Hom-set... |
| ringccofval 20907 | Composition in the categor... |
| ringcco 20908 | Composition in the categor... |
| dfringc2 20909 | Alternate definition of th... |
| rhmsscmap2 20910 | The unital ring homomorphi... |
| rhmsscmap 20911 | The unital ring homomorphi... |
| rhmsubcsetclem1 20912 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20913 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20914 | The unital ring homomorphi... |
| ringccat 20915 | The category of unital rin... |
| ringcid 20916 | The identity arrow in the ... |
| rhmsscrnghm 20917 | The unital ring homomorphi... |
| rhmsubcrngclem1 20918 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20919 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20920 | The unital ring homomorphi... |
| rngcresringcat 20921 | The restriction of the cat... |
| ringcsect 20922 | A section in the category ... |
| ringcinv 20923 | An inverse in the category... |
| ringciso 20924 | An isomorphism in the cate... |
| ringcbasbas 20925 | An element of the base set... |
| funcringcsetc 20926 | The "natural forgetful fun... |
| zrtermoringc 20927 | The zero ring is a termina... |
| zrninitoringc 20928 | The zero ring is not an in... |
| srhmsubclem1 20929 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20930 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20931 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20932 | According to ~ df-subc , t... |
| sringcat 20933 | The restriction of the cat... |
| crhmsubc 20934 | According to ~ df-subc , t... |
| cringcat 20935 | The restriction of the cat... |
| rngcrescrhm 20936 | The category of non-unital... |
| rhmsubclem1 20937 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20938 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20939 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20940 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20941 | According to ~ df-subc , t... |
| rhmsubccat 20942 | The restriction of the cat... |
| rrgval 20949 | Value of the set or left-r... |
| isrrg 20950 | Membership in the set of l... |
| rrgeq0i 20951 | Property of a left-regular... |
| rrgeq0 20952 | Left-multiplication by a l... |
| rrgsupp 20953 | Left multiplication by a l... |
| rrgss 20954 | Left-regular elements are ... |
| unitrrg 20955 | Units are regular elements... |
| rrgnz 20956 | In a nonzero ring, the zer... |
| isdomn 20957 | Expand definition of a dom... |
| domnnzr 20958 | A domain is a nonzero ring... |
| domnring 20959 | A domain is a ring. (Cont... |
| domneq0 20960 | In a domain, a product is ... |
| domnmuln0 20961 | In a domain, a product of ... |
| isdomn5 20962 | The equivalence between th... |
| isdomn2 20963 | A ring is a domain iff all... |
| domnrrg 20964 | In a domain, a nonzero ele... |
| isdomn6 20965 | A ring is a domain iff the... |
| isdomn3 20966 | Nonzero elements form a mu... |
| isdomn4 20967 | A ring is a domain iff it ... |
| opprdomnb 20968 | A class is a domain if and... |
| opprdomn 20969 | The opposite of a domain i... |
| isdomn4r 20970 | A ring is a domain iff it ... |
| domnlcanb 20971 | Left-cancellation law for ... |
| domnlcan 20972 | Left-cancellation law for ... |
| domnrcanb 20973 | Right-cancellation law for... |
| domnrcan 20974 | Right-cancellation law for... |
| domneq0r 20975 | Right multiplication by a ... |
| isidom 20976 | An integral domain is a co... |
| idomdomd 20977 | An integral domain is a do... |
| idomcringd 20978 | An integral domain is a co... |
| idomringd 20979 | An integral domain is a ri... |
| isdrng 20984 | The predicate "is a divisi... |
| drngunit 20985 | Elementhood in the set of ... |
| drngui 20986 | The set of units of a divi... |
| drngring 20987 | A division ring is a ring.... |
| drngringd 20988 | A division ring is a ring.... |
| drnggrpd 20989 | A division ring is a group... |
| drnggrp 20990 | A division ring is a group... |
| ringinveu 20991 | If a ring unit element ` X... |
| isdrng4 20992 | A division ring is a ring ... |
| isfld 20993 | A field is a commutative d... |
| flddrngd 20994 | A field is a division ring... |
| fldcrngd 20995 | A field is a commutative r... |
| drngprops 20996 | Properties of a division r... |
| isdrng2 20997 | A division ring can equiva... |
| drngprop 20998 | If two structures have the... |
| drngmgp 20999 | A division ring contains a... |
| drngid 21000 | A division ring's unity is... |
| drngunz 21001 | A division ring's unity is... |
| drngnzr 21002 | A division ring is a nonze... |
| drngdomn 21003 | A division ring is a domai... |
| isdrng3lem0 21004 | Lemma for ~ isdrng3 : The... |
| isdrng3lem1 21005 | Lemma for ~ isdrng3 (for t... |
| isdrng3lem2 21006 | Lemma for ~ isdrng3 (for t... |
| isdrng3 21007 | A division ring is a ring ... |
| isdrng5 21008 | A division ring is a ring ... |
| drngmcl 21009 | The product of two nonzero... |
| drngid2 21010 | Properties showing that an... |
| drnginvrcl 21011 | Closure of the multiplicat... |
| drnginvrn0 21012 | The multiplicative inverse... |
| drnginvrcld 21013 | Closure of the multiplicat... |
| drnginvrl 21014 | Property of the multiplica... |
| drnginvrr 21015 | Property of the multiplica... |
| drnginvrld 21016 | Property of the multiplica... |
| drnginvrrd 21017 | Property of the multiplica... |
| drngmul0or 21018 | A product is zero iff one ... |
| drngmulne0 21019 | A product is nonzero iff b... |
| drngmuleq0 21020 | An element is zero iff its... |
| opprdrng 21021 | The opposite of a division... |
| isdrngd 21022 | Properties that characteri... |
| isdrngrd 21023 | Properties that characteri... |
| isdrngdOLD 21024 | Obsolete version of ~ isdr... |
| isdrngrdOLD 21025 | Obsolete version of ~ isdr... |
| zrdrng 21026 | A zero ring is not a divis... |
| drngpropd 21027 | If two structures have the... |
| fldpropd 21028 | If two structures have the... |
| fldidom 21029 | A field is an integral dom... |
| fidomndrnglem 21030 | Lemma for ~ fidomndrng . ... |
| fidomndrng 21031 | A finite domain is a divis... |
| fiidomfld 21032 | A finite integral domain i... |
| rng1nnzr 21033 | The (smallest) structure r... |
| ring1zr 21034 | The only unital ring with ... |
| ringen1zr0 21035 | The only unital ring with ... |
| rng1nfld 21036 | The zero ring is not a fie... |
| issubdrg 21037 | Characterize the subfields... |
| drhmsubc 21038 | According to ~ df-subc , t... |
| drngcat 21039 | The restriction of the cat... |
| fldcat 21040 | The restriction of the cat... |
| fldc 21041 | The restriction of the cat... |
| fldhmsubc 21042 | According to ~ df-subc , t... |
| issdrg 21045 | Property of a division sub... |
| sdrgrcl 21046 | Reverse closure for a sub-... |
| sdrgdrng 21047 | A sub-division-ring is a d... |
| sdrgsubrg 21048 | A sub-division-ring is a s... |
| sdrgid 21049 | Every division ring is a d... |
| sdrgss 21050 | A division subring is a su... |
| sdrgbas 21051 | Base set of a sub-division... |
| issdrg2 21052 | Property of a division sub... |
| sdrgunit 21053 | A unit of a sub-division-r... |
| imadrhmcl 21054 | The image of a (nontrivial... |
| fldsdrgfld 21055 | A sub-division-ring of a f... |
| acsfn1p 21056 | Construction of a closure ... |
| subrgacs 21057 | Closure property of subrin... |
| sdrgacs 21058 | Closure property of divisi... |
| cntzsdrg 21059 | Centralizers in division r... |
| subdrgint 21060 | The intersection of a none... |
| sdrgint 21061 | The intersection of a none... |
| primefld 21062 | The smallest sub division ... |
| primefld0cl 21063 | The prime field contains t... |
| primefld1cl 21064 | The prime field contains t... |
| abvfval 21067 | Value of the set of absolu... |
| isabv 21068 | Elementhood in the set of ... |
| isabvd 21069 | Properties that determine ... |
| abvrcl 21070 | Reverse closure for the ab... |
| abvfge0 21071 | An absolute value is a fun... |
| abvf 21072 | An absolute value is a fun... |
| abvcl 21073 | An absolute value is a fun... |
| abvge0 21074 | The absolute value of a nu... |
| abveq0 21075 | The value of an absolute v... |
| abvne0 21076 | The absolute value of a no... |
| abvgt0 21077 | The absolute value of a no... |
| abvmul 21078 | An absolute value distribu... |
| abvtri 21079 | An absolute value satisfie... |
| abv0 21080 | The absolute value of zero... |
| abv1z 21081 | The absolute value of one ... |
| abv1 21082 | The absolute value of one ... |
| abvneg 21083 | The absolute value of a ne... |
| abvsubtri 21084 | An absolute value satisfie... |
| abvrec 21085 | The absolute value distrib... |
| abvdiv 21086 | The absolute value distrib... |
| abvdom 21087 | Any ring with an absolute ... |
| abvres 21088 | The restriction of an abso... |
| abvtrivd 21089 | The trivial absolute value... |
| abvtrivg 21090 | The trivial absolute value... |
| abvtriv 21091 | The trivial absolute value... |
| abvpropd 21092 | If two structures have the... |
| abvn0b 21093 | Another characterization o... |
| staffval 21098 | The functionalization of t... |
| stafval 21099 | The functionalization of t... |
| staffn 21100 | The functionalization is e... |
| issrng 21101 | The predicate "is a star r... |
| srngrhm 21102 | The involution function in... |
| srngring 21103 | A star ring is a ring. (C... |
| srngcnv 21104 | The involution function in... |
| srngf1o 21105 | The involution function in... |
| srngcl 21106 | The involution function in... |
| srngnvl 21107 | The involution function in... |
| srngadd 21108 | The involution function in... |
| srngmul 21109 | The involution function in... |
| srng1 21110 | The conjugate of the ring ... |
| srng0 21111 | The conjugate of the ring ... |
| issrngd 21112 | Properties that determine ... |
| idsrngd 21113 | A commutative ring is a st... |
| isorng 21118 | An ordered ring is a ring ... |
| orngring 21119 | An ordered ring is a ring.... |
| orngogrp 21120 | An ordered ring is an orde... |
| isofld 21121 | An ordered field is a fiel... |
| orngmul 21122 | In an ordered ring, the or... |
| orngsqr 21123 | In an ordered ring, all sq... |
| ornglmulle 21124 | In an ordered ring, multip... |
| orngrmulle 21125 | In an ordered ring, multip... |
| ornglmullt 21126 | In an ordered ring, multip... |
| orngrmullt 21127 | In an ordered ring, multip... |
| orngmullt 21128 | In an ordered ring, the st... |
| ofldfld 21129 | An ordered field is a fiel... |
| ofldtos 21130 | An ordered field is a tota... |
| orng0le1 21131 | In an ordered ring, the ri... |
| ofldlt1 21132 | In an ordered field, the r... |
| suborng 21133 | Every subring of an ordere... |
| subofld 21134 | Every subfield of an order... |
| islmod 21139 | The predicate "is a left m... |
| lmodlema 21140 | Lemma for properties of a ... |
| islmodd 21141 | Properties that determine ... |
| lmodgrp 21142 | A left module is a group. ... |
| lmodring 21143 | The scalar component of a ... |
| lmodfgrp 21144 | The scalar component of a ... |
| lmodgrpd 21145 | A left module is a group. ... |
| lmodbn0 21146 | The base set of a left mod... |
| lmodacl 21147 | Closure of ring addition f... |
| lmodmcl 21148 | Closure of ring multiplica... |
| lmodsn0 21149 | The set of scalars in a le... |
| lmodvacl 21150 | Closure of vector addition... |
| lmodass 21151 | Left module vector sum is ... |
| lmodlcan 21152 | Left cancellation law for ... |
| lmodvscl 21153 | Closure of scalar product ... |
| lmodvscld 21154 | Closure of scalar product ... |
| scaffval 21155 | The scalar multiplication ... |
| scafval 21156 | The scalar multiplication ... |
| scafeq 21157 | If the scalar multiplicati... |
| scaffn 21158 | The scalar multiplication ... |
| lmodscaf 21159 | The scalar multiplication ... |
| lmodvsdi 21160 | Distributive law for scala... |
| lmodvsdir 21161 | Distributive law for scala... |
| lmodvsass 21162 | In a module, scalar multip... |
| lmod0cl 21163 | The ring zero in a left mo... |
| lmod1cl 21164 | The ring unity in a left m... |
| lmodvs1 21165 | Scalar product with the ri... |
| lmod0vcl 21166 | The zero vector is a vecto... |
| lmod0vlid 21167 | Left identity law for the ... |
| lmod0vrid 21168 | Right identity law for the... |
| lmod0vid 21169 | Identity equivalent to the... |
| lmod0vs 21170 | Zero times a vector is the... |
| lmodvs0 21171 | Anything times the zero ve... |
| lmodvsmmulgdi 21172 | Distributive law for a gro... |
| lmodfopnelem1 21173 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 21174 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 21175 | The (functionalized) opera... |
| lcomf 21176 | A linear-combination sum i... |
| lcomfsupp 21177 | A linear-combination sum i... |
| lmodvnegcl 21178 | Closure of vector negative... |
| lmodvnegid 21179 | Addition of a vector with ... |
| lmodvneg1 21180 | Minus 1 times a vector is ... |
| lmodvsneg 21181 | Multiplication of a vector... |
| lmodvsubcl 21182 | Closure of vector subtract... |
| lmodcom 21183 | Left module vector sum is ... |
| lmodabl 21184 | A left module is an abelia... |
| lmodcmn 21185 | A left module is a commuta... |
| lmodnegadd 21186 | Distribute negation throug... |
| lmod4 21187 | Commutative/associative la... |
| lmodvsubadd 21188 | Relationship between vecto... |
| lmodvaddsub4 21189 | Vector addition/subtractio... |
| lmodvpncan 21190 | Addition/subtraction cance... |
| lmodvnpcan 21191 | Cancellation law for vecto... |
| lmodvsubval2 21192 | Value of vector subtractio... |
| lmodsubvs 21193 | Subtraction of a scalar pr... |
| lmodsubdi 21194 | Scalar multiplication dist... |
| lmodsubdir 21195 | Scalar multiplication dist... |
| lmodsubeq0 21196 | If the difference between ... |
| lmodsubid 21197 | Subtraction of a vector fr... |
| lmodvsghm 21198 | Scalar multiplication of t... |
| lmodprop2d 21199 | If two structures have the... |
| lmodpropd 21200 | If two structures have the... |
| gsumvsmul 21201 | Pull a scalar multiplicati... |
| mptscmfsupp0 21202 | A mapping to a scalar prod... |
| mptscmfsuppd 21203 | A function mapping to a sc... |
| rmodislmodlem 21204 | Lemma for ~ rmodislmod . ... |
| rmodislmod 21205 | The right module ` R ` ind... |
| lssset 21208 | The set of all (not necess... |
| islss 21209 | The predicate "is a subspa... |
| islssd 21210 | Properties that determine ... |
| lssss 21211 | A subspace is a set of vec... |
| lssel 21212 | A subspace member is a vec... |
| lss1 21213 | The set of vectors in a le... |
| lssuni 21214 | The union of all subspaces... |
| lssn0 21215 | A subspace is not empty. ... |
| 00lss 21216 | The empty structure has no... |
| lsscl 21217 | Closure property of a subs... |
| lssvacl 21218 | Closure of vector addition... |
| lssvsubcl 21219 | Closure of vector subtract... |
| lssvancl1 21220 | Non-closure: if one vector... |
| lssvancl2 21221 | Non-closure: if one vector... |
| lss0cl 21222 | The zero vector belongs to... |
| lsssn0 21223 | The singleton of the zero ... |
| lss0ss 21224 | The zero subspace is inclu... |
| lssle0 21225 | No subspace is smaller tha... |
| lssne0 21226 | A nonzero subspace has a n... |
| lssvneln0 21227 | A vector ` X ` which doesn... |
| lssneln0 21228 | A vector ` X ` which doesn... |
| lssssr 21229 | Conclude subspace ordering... |
| lssvscl 21230 | Closure of scalar product ... |
| lssvnegcl 21231 | Closure of negative vector... |
| lsssubg 21232 | All subspaces are subgroup... |
| lsssssubg 21233 | All subspaces are subgroup... |
| islss3 21234 | A linear subspace of a mod... |
| lsslmod 21235 | A submodule is a module. ... |
| lsslss 21236 | The subspaces of a subspac... |
| islss4 21237 | A linear subspace is a sub... |
| lss1d 21238 | One-dimensional subspace (... |
| lssintcl 21239 | The intersection of a none... |
| lssincl 21240 | The intersection of two su... |
| lssmre 21241 | The subspaces of a module ... |
| lssacs 21242 | Submodules are an algebrai... |
| prdsvscacl 21243 | Pointwise scalar multiplic... |
| prdslmodd 21244 | The product of a family of... |
| pwslmod 21245 | A structure power of a lef... |
| lspfval 21248 | The span function for a le... |
| lspf 21249 | The span function on a lef... |
| lspval 21250 | The span of a set of vecto... |
| lspcl 21251 | The span of a set of vecto... |
| lspsncl 21252 | The span of a singleton is... |
| lspprcl 21253 | The span of a pair is a su... |
| lsptpcl 21254 | The span of an unordered t... |
| lspsnsubg 21255 | The span of a singleton is... |
| 00lsp 21256 | ~ fvco4i lemma for linear ... |
| lspid 21257 | The span of a subspace is ... |
| lspssv 21258 | A span is a set of vectors... |
| lspss 21259 | Span preserves subset orde... |
| lspssid 21260 | A set of vectors is a subs... |
| lspidm 21261 | The span of a set of vecto... |
| lspun 21262 | The span of union is the s... |
| lspssp 21263 | If a set of vectors is a s... |
| mrclsp 21264 | Moore closure generalizes ... |
| lspsnss 21265 | The span of the singleton ... |
| ellspsn3 21266 | A member of the span of th... |
| lspprss 21267 | The span of a pair of vect... |
| lspsnid 21268 | A vector belongs to the sp... |
| ellspsn6 21269 | Relationship between a vec... |
| ellspsn5b 21270 | Relationship between a vec... |
| ellspsn5 21271 | Relationship between a vec... |
| lspprid1 21272 | A member of a pair of vect... |
| lspprid2 21273 | A member of a pair of vect... |
| lspprvacl 21274 | The sum of two vectors bel... |
| lssats2 21275 | A way to express atomistic... |
| ellspsni 21276 | A scalar product with a ve... |
| lspsn 21277 | Span of the singleton of a... |
| ellspsn 21278 | Member of span of the sing... |
| lspsnvsi 21279 | Span of a scalar product o... |
| lspsnss2 21280 | Comparable spans of single... |
| lspsnneg 21281 | Negation does not change t... |
| lspsnsub 21282 | Swapping subtraction order... |
| lspsn0 21283 | Span of the singleton of t... |
| lsp0 21284 | Span of the empty set. (C... |
| lspuni0 21285 | Union of the span of the e... |
| lspun0 21286 | The span of a union with t... |
| lspsneq0 21287 | Span of the singleton is t... |
| lspsneq0b 21288 | Equal singleton spans impl... |
| lmodindp1 21289 | Two independent (non-colin... |
| lsslsp 21290 | Spans in submodules corres... |
| lss0v 21291 | The zero vector in a submo... |
| lsspropd 21292 | If two structures have the... |
| lsppropd 21293 | If two structures have the... |
| reldmlmhm 21300 | Lemma for module homomorph... |
| lmimfn 21301 | Lemma for module isomorphi... |
| islmhm 21302 | Property of being a homomo... |
| islmhm3 21303 | Property of a module homom... |
| lmhmlem 21304 | Non-quantified consequence... |
| lmhmsca 21305 | A homomorphism of left mod... |
| lmghm 21306 | A homomorphism of left mod... |
| lmhmlmod2 21307 | A homomorphism of left mod... |
| lmhmlmod1 21308 | A homomorphism of left mod... |
| lmhmf 21309 | A homomorphism of left mod... |
| lmhmlin 21310 | A homomorphism of left mod... |
| lmodvsinv 21311 | Multiplication of a vector... |
| lmodvsinv2 21312 | Multiplying a negated vect... |
| islmhm2 21313 | A one-equation proof of li... |
| islmhmd 21314 | Deduction for a module hom... |
| 0lmhm 21315 | The constant zero linear f... |
| idlmhm 21316 | The identity function on a... |
| invlmhm 21317 | The negative function on a... |
| lmhmco 21318 | The composition of two mod... |
| lmhmplusg 21319 | The pointwise sum of two l... |
| lmhmvsca 21320 | The pointwise scalar produ... |
| lmhmf1o 21321 | A bijective module homomor... |
| lmhmima 21322 | The image of a subspace un... |
| lmhmpreima 21323 | The inverse image of a sub... |
| lmhmlsp 21324 | Homomorphisms preserve spa... |
| lmhmrnlss 21325 | The range of a homomorphis... |
| lmhmkerlss 21326 | The kernel of a homomorphi... |
| reslmhm 21327 | Restriction of a homomorph... |
| reslmhm2 21328 | Expansion of the codomain ... |
| reslmhm2b 21329 | Expansion of the codomain ... |
| lmhmeql 21330 | The equalizer of two modul... |
| lspextmo 21331 | A linear function is compl... |
| pwsdiaglmhm 21332 | Diagonal homomorphism into... |
| pwssplit0 21333 | Splitting for structure po... |
| pwssplit1 21334 | Splitting for structure po... |
| pwssplit2 21335 | Splitting for structure po... |
| pwssplit3 21336 | Splitting for structure po... |
| islmim 21337 | An isomorphism of left mod... |
| lmimf1o 21338 | An isomorphism of left mod... |
| lmimlmhm 21339 | An isomorphism of modules ... |
| lmimgim 21340 | An isomorphism of modules ... |
| islmim2 21341 | An isomorphism of left mod... |
| lmimcnv 21342 | The converse of a bijectiv... |
| brlmic 21343 | The relation "is isomorphi... |
| brlmici 21344 | Prove isomorphic by an exp... |
| lmiclcl 21345 | Isomorphism implies the le... |
| lmicrcl 21346 | Isomorphism implies the ri... |
| lmicsym 21347 | Module isomorphism is symm... |
| lmhmpropd 21348 | Module homomorphism depend... |
| islbs 21351 | The predicate " ` B ` is a... |
| lbsss 21352 | A basis is a set of vector... |
| lbsel 21353 | An element of a basis is a... |
| lbssp 21354 | The span of a basis is the... |
| lbsind 21355 | A basis is linearly indepe... |
| lbsind2 21356 | A basis is linearly indepe... |
| lbspss 21357 | No proper subset of a basi... |
| lsmcl 21358 | The sum of two subspaces i... |
| lsmspsn 21359 | Member of subspace sum of ... |
| lsmelval2 21360 | Subspace sum membership in... |
| lsmsp 21361 | Subspace sum in terms of s... |
| lsmsp2 21362 | Subspace sum of spans of s... |
| lsmssspx 21363 | Subspace sum (in its exten... |
| lsmpr 21364 | The span of a pair of vect... |
| lsppreli 21365 | A vector expressed as a su... |
| lsmelpr 21366 | Two ways to say that a vec... |
| lsppr0 21367 | The span of a vector paire... |
| lsppr 21368 | Span of a pair of vectors.... |
| lspprel 21369 | Member of the span of a pa... |
| lspprabs 21370 | Absorption of vector sum i... |
| lspvadd 21371 | The span of a vector sum i... |
| lspsntri 21372 | Triangle-type inequality f... |
| lspsntrim 21373 | Triangle-type inequality f... |
| lbspropd 21374 | If two structures have the... |
| pj1lmhm 21375 | The left projection functi... |
| pj1lmhm2 21376 | The left projection functi... |
| islvec 21379 | The predicate "is a left v... |
| lvecdrng 21380 | The set of scalars of a le... |
| lveclmod 21381 | A left vector space is a l... |
| lveclmodd 21382 | A vector space is a left m... |
| lvecgrpd 21383 | A vector space is a group.... |
| lsslvec 21384 | A vector subspace is a vec... |
| lmhmlvec 21385 | The property for modules t... |
| lvecvs0or 21386 | If a scalar product is zer... |
| lvecvsn0 21387 | A scalar product is nonzer... |
| lssvs0or 21388 | If a scalar product belong... |
| lvecvscan 21389 | Cancellation law for scala... |
| lvecvscan2 21390 | Cancellation law for scala... |
| lvecinv 21391 | Invert coefficient of scal... |
| lspsnvs 21392 | A nonzero scalar product d... |
| lspsneleq 21393 | Membership relation that i... |
| lspsncmp 21394 | Comparable spans of nonzer... |
| lspsnne1 21395 | Two ways to express that v... |
| lspsnne2 21396 | Two ways to express that v... |
| lspsnnecom 21397 | Swap two vectors with diff... |
| lspabs2 21398 | Absorption law for span of... |
| lspabs3 21399 | Absorption law for span of... |
| lspsneq 21400 | Equal spans of singletons ... |
| lspsneu 21401 | Nonzero vectors with equal... |
| ellspsn4 21402 | A member of the span of th... |
| lspdisj 21403 | The span of a vector not i... |
| lspdisjb 21404 | A nonzero vector is not in... |
| lspdisj2 21405 | Unequal spans are disjoint... |
| lspfixed 21406 | Show membership in the spa... |
| lspexch 21407 | Exchange property for span... |
| lspexchn1 21408 | Exchange property for span... |
| lspexchn2 21409 | Exchange property for span... |
| lspindpi 21410 | Partial independence prope... |
| lspindp1 21411 | Alternate way to say 3 vec... |
| lspindp2l 21412 | Alternate way to say 3 vec... |
| lspindp2 21413 | Alternate way to say 3 vec... |
| lspindp3 21414 | Independence of 2 vectors ... |
| lspindp4 21415 | (Partial) independence of ... |
| lvecindp 21416 | Compute the ` X ` coeffici... |
| lvecindp2 21417 | Sums of independent vector... |
| lspsnsubn0 21418 | Unequal singleton spans im... |
| lsmcv 21419 | Subspace sum has the cover... |
| lspsolvlem 21420 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21421 | If ` X ` is in the span of... |
| lssacsex 21422 | In a vector space, subspac... |
| lspsnat 21423 | There is no subspace stric... |
| lspsncv0 21424 | The span of a singleton co... |
| lsppratlem1 21425 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21426 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21427 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21428 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21429 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21430 | Lemma for ~ lspprat . Neg... |
| lspprat 21431 | A proper subspace of the s... |
| islbs2 21432 | An equivalent formulation ... |
| islbs3 21433 | An equivalent formulation ... |
| lbsacsbs 21434 | Being a basis in a vector ... |
| lvecdim 21435 | The dimension theorem for ... |
| lbsextlem1 21436 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21437 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21438 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21439 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21440 | For any linearly independe... |
| lbsext 21441 | For any linearly independe... |
| lbsexg 21442 | Every vector space has a b... |
| lbsex 21443 | Every vector space has a b... |
| lvecprop2d 21444 | If two structures have the... |
| lvecpropd 21445 | If two structures have the... |
| sraval 21450 | Lemma for ~ srabase throug... |
| sralem 21451 | Lemma for ~ srabase and si... |
| srabase 21452 | Base set of a subring alge... |
| sraaddg 21453 | Additive operation of a su... |
| sramulr 21454 | Multiplicative operation o... |
| srasca 21455 | The set of scalars of a su... |
| sravsca 21456 | The scalar product operati... |
| sraip 21457 | The inner product operatio... |
| sratset 21458 | Topology component of a su... |
| sratopn 21459 | Topology component of a su... |
| srads 21460 | Distance function of a sub... |
| sraring 21461 | Condition for a subring al... |
| sralmod 21462 | The subring algebra is a l... |
| sralmod0 21463 | The subring module inherit... |
| issubrgd 21464 | Prove a subring by closure... |
| rlmfn 21465 | ` ringLMod ` is a function... |
| rlmval 21466 | Value of the ring module. ... |
| rlmval2 21467 | Value of the ring module e... |
| rlmbas 21468 | Base set of the ring modul... |
| rlmplusg 21469 | Vector addition in the rin... |
| rlm0 21470 | Zero vector in the ring mo... |
| rlmsub 21471 | Subtraction in the ring mo... |
| rlmmulr 21472 | Ring multiplication in the... |
| rlmsca 21473 | Scalars in the ring module... |
| rlmsca2 21474 | Scalars in the ring module... |
| rlmvsca 21475 | Scalar multiplication in t... |
| rlmtopn 21476 | Topology component of the ... |
| rlmds 21477 | Metric component of the ri... |
| rlmlmod 21478 | The ring module is a modul... |
| rlmlvec 21479 | The ring module over a div... |
| rlmlsm 21480 | Subgroup sum of the ring m... |
| rlmvneg 21481 | Vector negation in the rin... |
| rlmscaf 21482 | Functionalized scalar mult... |
| ixpsnbasval 21483 | The value of an infinite C... |
| lidlval 21488 | Value of the set of ring i... |
| rspval 21489 | Value of the ring span fun... |
| lidlss 21490 | An ideal is a subset of th... |
| lidlbasel 21491 | An element of an ideal is ... |
| lidlssbas 21492 | The base set of the restri... |
| lidlbas 21493 | A (left) ideal of a ring i... |
| islidl 21494 | Predicate of being a (left... |
| rnglidlmcl 21495 | A (left) ideal containing ... |
| rngridlmcl 21496 | A right ideal (which is a ... |
| dflidl2rng 21497 | Alternate (the usual textb... |
| isridlrng 21498 | A right ideal is a left id... |
| lidl0cl 21499 | An ideal contains 0. (Con... |
| lidlacl 21500 | An ideal is closed under a... |
| lidlnegcl 21501 | An ideal contains negative... |
| lidlsubg 21502 | An ideal is a subgroup of ... |
| lidlsubcl 21503 | An ideal is closed under s... |
| lidlmcl 21504 | An ideal is closed under l... |
| lidl1el 21505 | An ideal contains 1 iff it... |
| lidlmcld 21506 | An ideal is closed under l... |
| dflidl2 21507 | Alternate (the usual textb... |
| lidl0ALT 21508 | Alternate proof for ~ lidl... |
| rnglidl0 21509 | Every non-unital ring cont... |
| lidl0 21510 | Every ring contains a zero... |
| lidl1ALT 21511 | Alternate proof for ~ lidl... |
| rnglidl1 21512 | The base set of every non-... |
| lidl1 21513 | Every ring contains a unit... |
| 0ringidl 21514 | The zero ideal is the only... |
| lidlunin0 21515 | The union of a nonempty su... |
| unichnlidl 21516 | The union of a nonempty ch... |
| lidlacs 21517 | The ideal system is an alg... |
| rspcl 21518 | The span of a set of ring ... |
| rspssid 21519 | The span of a set of ring ... |
| rsp1 21520 | The span of the identity e... |
| rsp0 21521 | The span of the zero eleme... |
| rspssp 21522 | The ideal span of a set of... |
| rspvalint 21523 | The ideal generated by a s... |
| rspprop 21524 | Properties of a class to b... |
| elrspsn 21525 | Membership in a principal ... |
| rspsn0 21526 | A principal ideal (an idea... |
| rspsnid 21527 | A principal ideal contains... |
| pidlnz 21528 | A principal ideal generate... |
| mrcrsp 21529 | Moore closure generalizes ... |
| lidlnz 21530 | A nonzero ideal contains a... |
| drngnidl 21531 | A division ring has only t... |
| lidlrsppropd 21532 | The left ideals and ring s... |
| rnglidlmmgm 21533 | The multiplicative group o... |
| rnglidlmsgrp 21534 | The multiplicative group o... |
| rnglidlrng 21535 | A (left) ideal of a non-un... |
| lidlnsg 21536 | An ideal is a normal subgr... |
| lsmidllsp 21537 | The sum of two ideals is t... |
| lsmidl 21538 | The sum of two ideals is a... |
| drngidl 21539 | A nonzero ring is a divisi... |
| isfieldidl 21540 | Determine if a ring is a f... |
| isfieldidl2 21541 | Determine if a ring is a f... |
| 2idlval 21544 | Definition of a two-sided ... |
| isridl 21545 | A right ideal is a left id... |
| 2idlelb 21546 | Membership in a two-sided ... |
| 2idllidld 21547 | A two-sided ideal is a lef... |
| 2idlridld 21548 | A two-sided ideal is a rig... |
| 2idl1el 21549 | A two-sided ideal contains... |
| df2idl2rng 21550 | Alternate (the usual textb... |
| df2idl2 21551 | Alternate (the usual textb... |
| ridl0 21552 | Every ring contains a zero... |
| ridl1 21553 | Every ring contains a unit... |
| 2idl0 21554 | Every ring contains a zero... |
| 2idl1 21555 | Every ring contains a unit... |
| 2idlss 21556 | A two-sided ideal is a sub... |
| 2idlbas 21557 | The base set of a two-side... |
| 2idlelbas 21558 | The base set of a two-side... |
| rng2idlsubrng 21559 | A two-sided ideal of a non... |
| rng2idlnsg 21560 | A two-sided ideal of a non... |
| rng2idl0 21561 | The zero (additive identit... |
| rng2idlsubgsubrng 21562 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21563 | A two-sided ideal of a non... |
| rng2idlsubg0 21564 | The zero (additive identit... |
| 2idlcpblrng 21565 | The coset equivalence rela... |
| 2idlcpbl 21566 | The coset equivalence rela... |
| qus2idrng 21567 | The quotient of a non-unit... |
| qus1 21568 | The multiplicative identit... |
| qusring 21569 | If ` S ` is a two-sided id... |
| qusrhm 21570 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21571 | The preimage of an ideal b... |
| kerlidl 21572 | The kernel of a ring homom... |
| ker2idl 21573 | The kernel of a ring homom... |
| qusmul2idl 21574 | Value of the ring operatio... |
| crngridl 21575 | In a commutative ring, the... |
| crng2idl 21576 | In a commutative ring, a t... |
| df2idl2crng 21577 | The predicate "is an ideal... |
| qusmulrng 21578 | Value of the multiplicatio... |
| quscrng 21579 | The quotient of a commutat... |
| qusmulcrng 21580 | Value of the ring operatio... |
| rhmqusnsg 21581 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21582 | The ring unity of a two-si... |
| rngqiprngghmlem1 21583 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21584 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21585 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21586 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21587 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21588 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21589 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21590 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21591 | The base set of the produc... |
| rngqiprng 21592 | The product of the quotien... |
| rngqiprngimf 21593 | ` F ` is a function from (... |
| rngqiprngimfv 21594 | The value of the function ... |
| rngqiprngghm 21595 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21596 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21597 | ` F ` is a function from (... |
| rngqiprnglin 21598 | ` F ` is linear with respe... |
| rngqiprngho 21599 | ` F ` is a homomorphism of... |
| rngqiprngim 21600 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21601 | The unity of a two-sided i... |
| rngringbdlem1 21602 | In a unital ring, the quot... |
| rngringbdlem2 21603 | A non-unital ring is unita... |
| rngringbd 21604 | A non-unital ring is unita... |
| ring2idlqus 21605 | For every unital ring ther... |
| ring2idlqusb 21606 | A non-unital ring is unita... |
| rngqiprngfulem1 21607 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21608 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21609 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21610 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21611 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21612 | The ring unity of the prod... |
| rngqiprngfu 21613 | The function value of ` F ... |
| rngqiprngu 21614 | If a non-unital ring has a... |
| ring2idlqus1 21615 | If a non-unital ring has a... |
| prmidlval 21618 | The class of prime ideals ... |
| isprmidl 21619 | The predicate "is a prime ... |
| prmidlnr 21620 | A prime ideal is a proper ... |
| prmidl 21621 | The main property of a pri... |
| prmidl2 21622 | A condition that shows an ... |
| idlmulssprm 21623 | Let ` P ` be a prime ideal... |
| pridln1 21624 | A proper ideal cannot cont... |
| prmidlidl 21625 | A prime ideal is an ideal.... |
| prmidlssidl 21626 | Prime ideals as a subset o... |
| cringm4 21627 | Commutative/associative la... |
| isprmidlc 21628 | The predicate "is prime id... |
| prmidlc 21629 | Property of a prime ideal ... |
| prmidlc2 21630 | Property of a prime ideal ... |
| cmprmidlmcl 21631 | The complement of a prime ... |
| prmidlprop 21632 | Property of prime ideals. ... |
| 0ringprmidl 21633 | The trivial ring does not ... |
| prmidl0 21634 | The zero ideal of a commut... |
| rhmpreimaprmidl 21635 | The preimage of a prime id... |
| qsidomlem1 21636 | If the quotient ring of a ... |
| qsidomlem2 21637 | A quotient by a prime idea... |
| qsidom 21638 | An ideal ` I ` in the comm... |
| qsnzr 21639 | A quotient of a nonzero ri... |
| ssdifidllem 21640 | Lemma for ~ ssdifidl : Th... |
| ssdifidl 21641 | Let ` R ` be a ring, and l... |
| ssdifidlprm 21642 | If the set ` S ` of ~ ssdi... |
| prmidlsubm 21643 | The complement of a prime ... |
| lpival 21648 | Value of the set of princi... |
| islpidl 21649 | Property of being a princi... |
| lpi0 21650 | The zero ideal is always p... |
| lpi1 21651 | The unit ideal is always p... |
| islpir 21652 | Principal ideal rings are ... |
| lpiss 21653 | Principal ideals are a sub... |
| islpir2 21654 | Principal ideal rings are ... |
| lpirring 21655 | Principal ideal rings are ... |
| drnglpir 21656 | Division rings are princip... |
| rspsn 21657 | Membership in principal id... |
| lidldvgen 21658 | An element generates an id... |
| lpigen 21659 | An ideal is principal iff ... |
| cnfldstr 21680 | The field of complex numbe... |
| cnfldex 21681 | The field of complex numbe... |
| cnfldbas 21682 | The base set of the field ... |
| mpocnfldadd 21683 | The addition operation of ... |
| cnfldadd 21684 | The addition operation of ... |
| mpocnfldmul 21685 | The multiplication operati... |
| cnfldmul 21686 | The multiplication operati... |
| cnfldcj 21687 | The conjugation operation ... |
| cnfldtset 21688 | The topology component of ... |
| cnfldle 21689 | The ordering of the field ... |
| cnfldds 21690 | The metric of the field of... |
| cnfldunif 21691 | The uniform structure comp... |
| cnfldfun 21692 | The field of complex numbe... |
| cnfldfunALT 21693 | The field of complex numbe... |
| xrsstr 21694 | The extended real structur... |
| xrsex 21695 | The extended real structur... |
| xrsadd 21696 | The addition operation of ... |
| xrsmul 21697 | The multiplication operati... |
| xrstset 21698 | The topology component of ... |
| cncrng 21699 | The complex numbers form a... |
| cnring 21700 | The complex numbers form a... |
| xrsmcmn 21701 | The "multiplicative group"... |
| cnfld0 21702 | Zero is the zero element o... |
| cnfld1 21703 | One is the unity element o... |
| cnfldneg 21704 | The additive inverse in th... |
| cnfldplusf 21705 | The functionalized additio... |
| cnfldsub 21706 | The subtraction operator i... |
| cndrng 21707 | The complex numbers form a... |
| cnflddiv 21708 | The division operation in ... |
| cnfldinv 21709 | The multiplicative inverse... |
| cnfldmulg 21710 | The group multiple functio... |
| cnfldexp 21711 | The exponentiation operato... |
| cnsrng 21712 | The complex numbers form a... |
| xrsmgm 21713 | The "additive group" of th... |
| xrsnsgrp 21714 | The "additive group" of th... |
| xrsmgmdifsgrp 21715 | The "additive group" of th... |
| xrsds 21716 | The metric of the extended... |
| xrsdsval 21717 | The metric of the extended... |
| xrsdsreval 21718 | The metric of the extended... |
| xrsdsreclblem 21719 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21720 | The metric of the extended... |
| cnsubmlem 21721 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21722 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21723 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 21724 | Lemma for ~ resubdrg and f... |
| qsubdrg 21725 | The rational numbers form ... |
| zsubrg 21726 | The integers form a subrin... |
| gzsubrg 21727 | The gaussian integers form... |
| nn0subm 21728 | The nonnegative integers f... |
| rege0subm 21729 | The nonnegative reals form... |
| absabv 21730 | The regular absolute value... |
| zsssubrg 21731 | The integers are a subset ... |
| qsssubdrg 21732 | The rational numbers are a... |
| cnsubrg 21733 | There are no subrings of t... |
| cnmgpabl 21734 | The unit group of the comp... |
| cnmgpid 21735 | The group identity element... |
| cnmsubglem 21736 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21737 | The positive reals form a ... |
| gzrngunitlem 21738 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21739 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21740 | Relate a group sum on ` CC... |
| regsumfsum 21741 | Relate a group sum on ` ( ... |
| expmhm 21742 | Exponentiation is a monoid... |
| nn0srg 21743 | The nonnegative integers f... |
| rge0srg 21744 | The nonnegative real numbe... |
| xrge0plusg 21745 | The additive law of the ex... |
| xrs1mnd 21746 | The extended real numbers,... |
| xrs10 21747 | The zero of the extended r... |
| xrs1cmn 21748 | The extended real numbers ... |
| xrge0subm 21749 | The nonnegative extended r... |
| xrge0cmn 21750 | The nonnegative extended r... |
| xrge0omnd 21751 | The nonnegative extended r... |
| zringcrng 21754 | The ring of integers is a ... |
| zringring 21755 | The ring of integers is a ... |
| zringrng 21756 | The ring of integers is a ... |
| zringabl 21757 | The ring of integers is an... |
| zringgrp 21758 | The ring of integers is an... |
| zringbas 21759 | The integers are the base ... |
| zringplusg 21760 | The addition operation of ... |
| zringsub 21761 | The subtraction of element... |
| zringmulg 21762 | The multiplication (group ... |
| zringmulr 21763 | The multiplication operati... |
| zring0 21764 | The zero element of the ri... |
| zring1 21765 | The unity element of the r... |
| zringnzr 21766 | The ring of integers is a ... |
| dvdsrzring 21767 | Ring divisibility in the r... |
| zringlpirlem1 21768 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21769 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21770 | Lemma for ~ zringlpir . A... |
| zringinvg 21771 | The additive inverse of an... |
| zringunit 21772 | The units of ` ZZ ` are th... |
| zringlpir 21773 | The integers are a princip... |
| zringndrg 21774 | The integers are not a div... |
| zringcyg 21775 | The integers are a cyclic ... |
| zringsubgval 21776 | Subtraction in the ring of... |
| zringmpg 21777 | The multiplicative group o... |
| prmirredlem 21778 | A positive integer is irre... |
| dfprm2 21779 | The positive irreducible e... |
| prmirred 21780 | The irreducible elements o... |
| expghm 21781 | Exponentiation is a group ... |
| mulgghm2 21782 | The powers of a group elem... |
| mulgrhm 21783 | The powers of the element ... |
| mulgrhm2 21784 | The powers of the element ... |
| irinitoringc 21785 | The ring of integers is an... |
| nzerooringczr 21786 | There is no zero object in... |
| pzriprnglem1 21787 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21788 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21789 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21790 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21791 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21792 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21793 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21794 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21795 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21796 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21797 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21798 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21799 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21800 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21801 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21802 | The ring unity of the ring... |
| pzriprng 21803 | The non-unital ring ` ( ZZ... |
| pzriprng1 21804 | The ring unity of the ring... |
| zrhval 21813 | Define the unique homomorp... |
| zrhval2 21814 | Alternate value of the ` Z... |
| zrhmulg 21815 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21816 | The ` ZRHom ` homomorphism... |
| zrhrhm 21817 | The ` ZRHom ` homomorphism... |
| zrh1 21818 | Interpretation of 1 in a r... |
| zrh0 21819 | Interpretation of 0 in a r... |
| zrhpropd 21820 | The ` ZZ ` ring homomorphi... |
| zlmval 21821 | Augment an abelian group w... |
| zlmlem 21822 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21823 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21824 | Group operation of a ` ZZ ... |
| zlmmulr 21825 | Ring operation of a ` ZZ `... |
| zlmsca 21826 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21827 | Scalar multiplication oper... |
| zlmlmod 21828 | The ` ZZ ` -module operati... |
| chrval 21829 | Definition substitution of... |
| chrcl 21830 | Closure of the characteris... |
| chrid 21831 | The canonical ` ZZ ` ring ... |
| chrdvds 21832 | The ` ZZ ` ring homomorphi... |
| chrcong 21833 | If two integers are congru... |
| dvdschrmulg 21834 | In a ring, any multiple of... |
| fermltlchr 21835 | A generalization of Fermat... |
| chrnzr 21836 | Nonzero rings are precisel... |
| chrrhm 21837 | The characteristic restric... |
| domnchr 21838 | The characteristic of a do... |
| znlidl 21839 | The set ` n ZZ ` is an ide... |
| zncrng2 21840 | Making a commutative ring ... |
| znval 21841 | The value of the ` Z/nZ ` ... |
| znle 21842 | The value of the ` Z/nZ ` ... |
| znval2 21843 | Self-referential expressio... |
| znbaslem 21844 | Lemma for ~ znbas . (Cont... |
| znbas2 21845 | The base set of ` Z/nZ ` i... |
| znadd 21846 | The additive structure of ... |
| znmul 21847 | The multiplicative structu... |
| znzrh 21848 | The ` ZZ ` ring homomorphi... |
| znbas 21849 | The base set of ` Z/nZ ` s... |
| zncrng 21850 | ` Z/nZ ` is a commutative ... |
| znzrh2 21851 | The ` ZZ ` ring homomorphi... |
| znzrhval 21852 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21853 | The ` ZZ ` ring homomorphi... |
| zncyg 21854 | The group ` ZZ / n ZZ ` is... |
| zndvds 21855 | Express equality of equiva... |
| zndvds0 21856 | Special case of ~ zndvds w... |
| znf1o 21857 | The function ` F ` enumera... |
| zzngim 21858 | The ` ZZ ` ring homomorphi... |
| znle2 21859 | The ordering of the ` Z/nZ... |
| znleval 21860 | The ordering of the ` Z/nZ... |
| znleval2 21861 | The ordering of the ` Z/nZ... |
| zntoslem 21862 | Lemma for ~ zntos . (Cont... |
| zntos 21863 | The ` Z/nZ ` structure is ... |
| znhash 21864 | The ` Z/nZ ` structure has... |
| znfi 21865 | The ` Z/nZ ` structure is ... |
| znfld 21866 | The ` Z/nZ ` structure is ... |
| znidomb 21867 | The ` Z/nZ ` structure is ... |
| znchr 21868 | Cyclic rings are defined b... |
| znunit 21869 | The units of ` Z/nZ ` are ... |
| znunithash 21870 | The size of the unit group... |
| znrrg 21871 | The regular elements of ` ... |
| cygznlem1 21872 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21873 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21874 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21875 | A cyclic group with ` n ` ... |
| cygzn 21876 | A cyclic group with ` n ` ... |
| cygth 21877 | The "fundamental theorem o... |
| cyggic 21878 | Cyclic groups are isomorph... |
| frgpcyg 21879 | A free group is cyclic iff... |
| freshmansdream 21880 | For a prime number ` P ` ,... |
| frobrhm 21881 | In a commutative ring with... |
| ofldchr 21882 | The characteristic of an o... |
| cnmsgnsubg 21883 | The signs form a multiplic... |
| cnmsgnbas 21884 | The base set of the sign s... |
| cnmsgngrp 21885 | The group of signs under m... |
| psgnghm 21886 | The sign is a homomorphism... |
| psgnghm2 21887 | The sign is a homomorphism... |
| psgninv 21888 | The sign of a permutation ... |
| psgnco 21889 | Multiplicativity of the pe... |
| zrhpsgnmhm 21890 | Embedding of permutation s... |
| zrhpsgninv 21891 | The embedded sign of a per... |
| evpmss 21892 | Even permutations are perm... |
| psgnevpmb 21893 | A class is an even permuta... |
| psgnodpm 21894 | A permutation which is odd... |
| psgnevpm 21895 | A permutation which is eve... |
| psgnodpmr 21896 | If a permutation has sign ... |
| zrhpsgnevpm 21897 | The sign of an even permut... |
| zrhpsgnodpm 21898 | The sign of an odd permuta... |
| cofipsgn 21899 | Composition of any class `... |
| zrhpsgnelbas 21900 | Embedding of permutation s... |
| zrhcopsgnelbas 21901 | Embedding of permutation s... |
| evpmodpmf1o 21902 | The function for performin... |
| pmtrodpm 21903 | A transposition is an odd ... |
| psgnfix1 21904 | A permutation of a finite ... |
| psgnfix2 21905 | A permutation of a finite ... |
| psgndiflemB 21906 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21907 | Lemma 2 for ~ psgndif . (... |
| psgndif 21908 | Embedding of permutation s... |
| copsgndif 21909 | Embedding of permutation s... |
| rebase 21912 | The base of the field of r... |
| remulg 21913 | The multiplication (group ... |
| resubdrg 21914 | The real numbers form a di... |
| resubgval 21915 | Subtraction in the field o... |
| replusg 21916 | The addition operation of ... |
| remulr 21917 | The multiplication operati... |
| re0g 21918 | The zero element of the fi... |
| re1r 21919 | The unity element of the f... |
| rele2 21920 | The ordering relation of t... |
| relt 21921 | The ordering relation of t... |
| reds 21922 | The distance of the field ... |
| redvr 21923 | The division operation of ... |
| retos 21924 | The real numbers are a tot... |
| refld 21925 | The real numbers form a fi... |
| refldcj 21926 | The conjugation operation ... |
| resrng 21927 | The real numbers form a st... |
| regsumsupp 21928 | The group sum over the rea... |
| rzgrp 21929 | The quotient group ` RR / ... |
| isphl 21934 | The predicate "is a genera... |
| phllvec 21935 | A pre-Hilbert space is a l... |
| phllmod 21936 | A pre-Hilbert space is a l... |
| phlsrng 21937 | The scalar ring of a pre-H... |
| phllmhm 21938 | The inner product of a pre... |
| ipcl 21939 | Closure of the inner produ... |
| ipcj 21940 | Conjugate of an inner prod... |
| iporthcom 21941 | Orthogonality (meaning inn... |
| ip0l 21942 | Inner product with a zero ... |
| ip0r 21943 | Inner product with a zero ... |
| ipeq0 21944 | The inner product of a vec... |
| ipdir 21945 | Distributive law for inner... |
| ipdi 21946 | Distributive law for inner... |
| ip2di 21947 | Distributive law for inner... |
| ipsubdir 21948 | Distributive law for inner... |
| ipsubdi 21949 | Distributive law for inner... |
| ip2subdi 21950 | Distributive law for inner... |
| ipass 21951 | Associative law for inner ... |
| ipassr 21952 | "Associative" law for seco... |
| ipassr2 21953 | "Associative" law for inne... |
| ipffval 21954 | The inner product operatio... |
| ipfval 21955 | The inner product operatio... |
| ipfeq 21956 | If the inner product opera... |
| ipffn 21957 | The inner product operatio... |
| phlipf 21958 | The inner product operatio... |
| ip2eq 21959 | Two vectors are equal iff ... |
| isphld 21960 | Properties that determine ... |
| phlpropd 21961 | If two structures have the... |
| ssipeq 21962 | The inner product on a sub... |
| phssipval 21963 | The inner product on a sub... |
| phssip 21964 | The inner product (as a fu... |
| phlssphl 21965 | A subspace of an inner pro... |
| ocvfval 21972 | The orthocomplement operat... |
| ocvval 21973 | Value of the orthocompleme... |
| elocv 21974 | Elementhood in the orthoco... |
| ocvi 21975 | Property of a member of th... |
| ocvss 21976 | The orthocomplement of a s... |
| ocvocv 21977 | A set is contained in its ... |
| ocvlss 21978 | The orthocomplement of a s... |
| ocv2ss 21979 | Orthocomplements reverse s... |
| ocvin 21980 | An orthocomplement has tri... |
| ocvsscon 21981 | Two ways to say that ` S `... |
| ocvlsp 21982 | The orthocomplement of a l... |
| ocv0 21983 | The orthocomplement of the... |
| ocvz 21984 | The orthocomplement of the... |
| ocv1 21985 | The orthocomplement of the... |
| unocv 21986 | The orthocomplement of a u... |
| iunocv 21987 | The orthocomplement of an ... |
| cssval 21988 | The set of closed subspace... |
| iscss 21989 | The predicate "is a closed... |
| cssi 21990 | Property of a closed subsp... |
| cssss 21991 | A closed subspace is a sub... |
| iscss2 21992 | It is sufficient to prove ... |
| ocvcss 21993 | The orthocomplement of any... |
| cssincl 21994 | The zero subspace is a clo... |
| css0 21995 | The zero subspace is a clo... |
| css1 21996 | The whole space is a close... |
| csslss 21997 | A closed subspace of a pre... |
| lsmcss 21998 | A subset of a pre-Hilbert ... |
| cssmre 21999 | The closed subspaces of a ... |
| mrccss 22000 | The Moore closure correspo... |
| thlval 22001 | Value of the Hilbert latti... |
| thlbas 22002 | Base set of the Hilbert la... |
| thlle 22003 | Ordering on the Hilbert la... |
| thlleval 22004 | Ordering on the Hilbert la... |
| thloc 22005 | Orthocomplement on the Hil... |
| pjfval 22012 | The value of the projectio... |
| pjdm 22013 | A subspace is in the domai... |
| pjpm 22014 | The projection map is a pa... |
| pjfval2 22015 | Value of the projection ma... |
| pjval 22016 | Value of the projection ma... |
| pjdm2 22017 | A subspace is in the domai... |
| pjff 22018 | A projection is a linear o... |
| pjf 22019 | A projection is a function... |
| pjf2 22020 | A projection is a function... |
| pjfo 22021 | A projection is a surjecti... |
| pjcss 22022 | A projection subspace is a... |
| ocvpj 22023 | The orthocomplement of a p... |
| ishil 22024 | The predicate "is a Hilber... |
| ishil2 22025 | The predicate "is a Hilber... |
| isobs 22026 | The predicate "is an ortho... |
| obsip 22027 | The inner product of two e... |
| obsipid 22028 | A basis element has length... |
| obsrcl 22029 | Reverse closure for an ort... |
| obsss 22030 | An orthonormal basis is a ... |
| obsne0 22031 | A basis element is nonzero... |
| obsocv 22032 | An orthonormal basis has t... |
| obs2ocv 22033 | The double orthocomplement... |
| obselocv 22034 | A basis element is in the ... |
| obs2ss 22035 | A basis has no proper subs... |
| obslbs 22036 | An orthogonal basis is a l... |
| reldmdsmm 22039 | The direct sum is a well-b... |
| dsmmval 22040 | Value of the module direct... |
| dsmmbase 22041 | Base set of the module dir... |
| dsmmval2 22042 | Self-referential definitio... |
| dsmmbas2 22043 | Base set of the direct sum... |
| dsmmfi 22044 | For finite products, the d... |
| dsmmelbas 22045 | Membership in the finitely... |
| dsmm0cl 22046 | The all-zero vector is con... |
| dsmmacl 22047 | The finite hull is closed ... |
| prdsinvgd2 22048 | Negation of a single coord... |
| dsmmsubg 22049 | The finite hull of a produ... |
| dsmmlss 22050 | The finite hull of a produ... |
| dsmmlmod 22051 | The direct sum of a family... |
| frlmval 22054 | Value of the "free module"... |
| frlmlmod 22055 | The free module is a modul... |
| frlmpws 22056 | The free module as a restr... |
| frlmlss 22057 | The base set of the free m... |
| frlmpwsfi 22058 | The finite free module is ... |
| frlmsca 22059 | The ring of scalars of a f... |
| frlm0 22060 | Zero in a free module (rin... |
| frlmbas 22061 | Base set of the free modul... |
| frlmelbas 22062 | Membership in the base set... |
| frlmrcl 22063 | If a free module is inhabi... |
| frlmbasfsupp 22064 | Elements of the free modul... |
| frlmbasmap 22065 | Elements of the free modul... |
| frlmbasf 22066 | Elements of the free modul... |
| frlmlvec 22067 | The free module over a div... |
| frlmfibas 22068 | The base set of the finite... |
| elfrlmbasn0 22069 | If the dimension of a free... |
| frlmplusgval 22070 | Addition in a free module.... |
| frlmsubgval 22071 | Subtraction in a free modu... |
| frlmvscafval 22072 | Scalar multiplication in a... |
| frlmvplusgvalc 22073 | Coordinates of a sum with ... |
| frlmvscaval 22074 | Coordinates of a scalar mu... |
| frlmplusgvalb 22075 | Addition in a free module ... |
| frlmvscavalb 22076 | Scalar multiplication in a... |
| frlmvplusgscavalb 22077 | Addition combined with sca... |
| frlmgsum 22078 | Finite commutative sums in... |
| frlmsplit2 22079 | Restriction is homomorphic... |
| frlmsslss 22080 | A subset of a free module ... |
| frlmsslss2 22081 | A subset of a free module ... |
| frlmbas3 22082 | An element of the base set... |
| mpofrlmd 22083 | Elements of the free modul... |
| frlmip 22084 | The inner product of a fre... |
| frlmipval 22085 | The inner product of a fre... |
| frlmphllem 22086 | Lemma for ~ frlmphl . (Co... |
| frlmphl 22087 | Conditions for a free modu... |
| uvcfval 22090 | Value of the unit-vector g... |
| uvcval 22091 | Value of a single unit vec... |
| uvcvval 22092 | Value of a unit vector coo... |
| uvcvvcl 22093 | A coordinate of a unit vec... |
| uvcvvcl2 22094 | A unit vector coordinate i... |
| uvcvv1 22095 | The unit vector is one at ... |
| uvcvv0 22096 | The unit vector is zero at... |
| uvcff 22097 | Domain and codomain of the... |
| uvcf1 22098 | In a nonzero ring, each un... |
| uvcresum 22099 | Any element of a free modu... |
| frlmssuvc1 22100 | A scalar multiple of a uni... |
| frlmssuvc2 22101 | A nonzero scalar multiple ... |
| frlmsslsp 22102 | A subset of a free module ... |
| frlmlbs 22103 | The unit vectors comprise ... |
| frlmup1 22104 | Any assignment of unit vec... |
| frlmup2 22105 | The evaluation map has the... |
| frlmup3 22106 | The range of such an evalu... |
| frlmup4 22107 | Universal property of the ... |
| ellspd 22108 | The elements of the span o... |
| elfilspd 22109 | Simplified version of ~ el... |
| rellindf 22114 | The independent-family pre... |
| islinds 22115 | Property of an independent... |
| linds1 22116 | An independent set of vect... |
| linds2 22117 | An independent set of vect... |
| islindf 22118 | Property of an independent... |
| islinds2 22119 | Expanded property of an in... |
| islindf2 22120 | Property of an independent... |
| lindff 22121 | Functional property of a l... |
| lindfind 22122 | A linearly independent fam... |
| lindsind 22123 | A linearly independent set... |
| lindfind2 22124 | In a linearly independent ... |
| lindsind2 22125 | In a linearly independent ... |
| lindff1 22126 | A linearly independent fam... |
| lindfrn 22127 | The range of an independen... |
| f1lindf 22128 | Rearranging and deleting e... |
| lindfres 22129 | Any restriction of an inde... |
| lindsss 22130 | Any subset of an independe... |
| f1linds 22131 | A family constructed from ... |
| islindf3 22132 | In a nonzero ring, indepen... |
| lindfmm 22133 | Linear independence of a f... |
| lindsmm 22134 | Linear independence of a s... |
| lindsmm2 22135 | The monomorphic image of a... |
| lsslindf 22136 | Linear independence is unc... |
| lsslinds 22137 | Linear independence is unc... |
| islbs4 22138 | A basis is an independent ... |
| lbslinds 22139 | A basis is independent. (... |
| islinds3 22140 | A subset is linearly indep... |
| islinds4 22141 | A set is independent in a ... |
| lmimlbs 22142 | The isomorphic image of a ... |
| lmiclbs 22143 | Having a basis is an isomo... |
| islindf4 22144 | A family is independent if... |
| islindf5 22145 | A family is independent if... |
| indlcim 22146 | An independent, spanning f... |
| lbslcic 22147 | A module with a basis is i... |
| lmisfree 22148 | A module has a basis iff i... |
| lvecisfrlm 22149 | Every vector space is isom... |
| lmimco 22150 | The composition of two iso... |
| lmictra 22151 | Module isomorphism is tran... |
| uvcf1o 22152 | In a nonzero ring, the map... |
| uvcendim 22153 | In a nonzero ring, the num... |
| frlmisfrlm 22154 | A free module is isomorphi... |
| frlmiscvec 22155 | Every free module is isomo... |
| lindsdom 22156 | A linearly independent set... |
| lindsenlbs 22157 | A maximal linearly indepen... |
| isassa 22164 | The properties of an assoc... |
| assalem 22165 | The properties of an assoc... |
| assaass 22166 | Left-associative property ... |
| assaassr 22167 | Right-associative property... |
| assalmod 22168 | An associative algebra is ... |
| assaring 22169 | An associative algebra is ... |
| assasca 22170 | The scalars of an associat... |
| assa2ass 22171 | Left- and right-associativ... |
| assa2ass2 22172 | Left- and right-associativ... |
| isassad 22173 | Sufficient condition for b... |
| issubassa3 22174 | A subring that is also a s... |
| issubassa 22175 | The subalgebras of an asso... |
| sraassab 22176 | A subring algebra is an as... |
| sraassa 22177 | The subring algebra over a... |
| rlmassa 22178 | The ring module over a com... |
| assapropd 22179 | If two structures have the... |
| aspval 22180 | Value of the algebraic clo... |
| asplss 22181 | The algebraic span of a se... |
| aspid 22182 | The algebraic span of a su... |
| aspsubrg 22183 | The algebraic span of a se... |
| aspss 22184 | Span preserves subset orde... |
| aspssid 22185 | A set of vectors is a subs... |
| asclfval 22186 | Function value of the alge... |
| asclval 22187 | Value of a mapped algebra ... |
| asclfn 22188 | Unconditional functionalit... |
| asclf 22189 | The algebra scalar lifting... |
| asclghm 22190 | The algebra scalar lifting... |
| asclelbas 22191 | Lifted scalars are in the ... |
| ascl0 22192 | The scalar 0 embedded into... |
| ascl1 22193 | The scalar 1 embedded into... |
| asclmul1 22194 | Left multiplication by a l... |
| asclmul2 22195 | Right multiplication by a ... |
| ascldimul 22196 | The algebra scalar lifting... |
| asclinvg 22197 | The group inverse (negatio... |
| asclrhm 22198 | The algebra scalar lifting... |
| rnascl 22199 | The set of lifted scalars ... |
| issubassa2 22200 | A subring of a unital alge... |
| rnasclsubrg 22201 | The scalar multiples of th... |
| rnasclmulcl 22202 | (Vector) multiplication is... |
| rnasclassa 22203 | The scalar multiples of th... |
| ressascl 22204 | The lifting of scalars is ... |
| asclpropd 22205 | If two structures have the... |
| aspval2 22206 | The algebraic closure is t... |
| assamulgscmlem1 22207 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 22208 | Lemma for ~ assamulgscm (i... |
| assamulgscm 22209 | Exponentiation of a scalar... |
| asclmulg 22210 | Apply group multiplication... |
| zlmassa 22211 | The ` ZZ ` -module operati... |
| reldmpsr 22222 | The multivariate power ser... |
| psrval 22223 | Value of the multivariate ... |
| psrvalstr 22224 | The multivariate power ser... |
| psrbag 22225 | Elementhood in the set of ... |
| psrbagf 22226 | A finite bag is a function... |
| psrbagfsupp 22227 | Finite bags have finite su... |
| snifpsrbag 22228 | A bag containing one eleme... |
| fczpsrbag 22229 | The constant function equa... |
| psrbaglesupp 22230 | The support of a dominated... |
| psrbaglecl 22231 | The set of finite bags is ... |
| psrbagaddcl 22232 | The sum of two finite bags... |
| psrbagcon 22233 | The analogue of the statem... |
| psrbaglefi 22234 | There are finitely many ba... |
| psrbagconcl 22235 | The complement of a bag is... |
| psrbagleadd1 22236 | The analogue of " ` X <_ F... |
| psrbagconf1o 22237 | Bag complementation is a b... |
| psrbagres 22238 | Restrict a bag of variable... |
| gsumbagdiaglem 22239 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 22240 | Two-dimensional commutatio... |
| psrass1lem 22241 | A group sum commutation us... |
| psrbas 22242 | The base set of the multiv... |
| psrelbas 22243 | An element of the set of p... |
| psrelbasfun 22244 | An element of the set of p... |
| psrplusg 22245 | The addition operation of ... |
| psradd 22246 | The addition operation of ... |
| psraddcl 22247 | Closure of the power serie... |
| rhmpsrlem1 22248 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 22249 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 22250 | The multiplication operati... |
| psrmulfval 22251 | The multiplication operati... |
| psrmulval 22252 | The multiplication operati... |
| psrmulcllem 22253 | Closure of the power serie... |
| psrmulcl 22254 | Closure of the power serie... |
| psrsca 22255 | The scalar field of the mu... |
| psrvscafval 22256 | The scalar multiplication ... |
| psrvsca 22257 | The scalar multiplication ... |
| psrvscaval 22258 | The scalar multiplication ... |
| psrvscacl 22259 | Closure of the power serie... |
| psr0cl 22260 | The zero element of the ri... |
| psr0lid 22261 | The zero element of the ri... |
| psrnegcl 22262 | The negative function in t... |
| psrlinv 22263 | The negative function in t... |
| psrgrp 22264 | The ring of power series i... |
| psr0 22265 | The zero element of the ri... |
| psrneg 22266 | The negative function of t... |
| psrlmod 22267 | The ring of power series i... |
| psr1cl 22268 | The identity element of th... |
| psrlidm 22269 | The identity element of th... |
| psrridm 22270 | The identity element of th... |
| psrass1 22271 | Associative identity for t... |
| psrdi 22272 | Distributive law for the r... |
| psrdir 22273 | Distributive law for the r... |
| psrass23l 22274 | Associative identity for t... |
| psrcom 22275 | Commutative law for the ri... |
| psrass23 22276 | Associative identities for... |
| psrring 22277 | The ring of power series i... |
| psr1 22278 | The identity element of th... |
| psrcrng 22279 | The ring of power series i... |
| psrassa 22280 | The ring of power series i... |
| resspsrbas 22281 | A restricted power series ... |
| resspsradd 22282 | A restricted power series ... |
| resspsrmul 22283 | A restricted power series ... |
| resspsrvsca 22284 | A restricted power series ... |
| subrgpsr 22285 | A subring of the base ring... |
| psrascl 22286 | Value of the scalar inject... |
| psrasclcl 22287 | A scalar is lifted into a ... |
| mvrfval 22288 | Value of the generating el... |
| mvrval 22289 | Value of the generating el... |
| mvrval2 22290 | Value of the generating el... |
| mvrid 22291 | The ` X i ` -th coefficien... |
| mvrf 22292 | The power series variable ... |
| mvrf1 22293 | The power series variable ... |
| mvrcl2 22294 | A power series variable is... |
| reldmmpl 22295 | The multivariate polynomia... |
| mplval 22296 | Value of the set of multiv... |
| mplbas 22297 | Base set of the set of mul... |
| mplelbas 22298 | Property of being a polyno... |
| mvrcl 22299 | A power series variable is... |
| mvrf2 22300 | The power series/polynomia... |
| mplrcl 22301 | Reverse closure for the po... |
| mplelsfi 22302 | A polynomial treated as a ... |
| mplval2 22303 | Self-referential expressio... |
| mplbasss 22304 | The set of polynomials is ... |
| mplelf 22305 | A polynomial is defined as... |
| mplsubglem 22306 | If ` A ` is an ideal of se... |
| mpllsslem 22307 | If ` A ` is an ideal of su... |
| mplsubglem2 22308 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 22309 | The set of polynomials is ... |
| mpllss 22310 | The set of polynomials is ... |
| mplsubrglem 22311 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 22312 | The set of polynomials is ... |
| mpl0 22313 | The zero polynomial. (Con... |
| mplplusg 22314 | Value of addition in a pol... |
| mplmulr 22315 | Value of multiplication in... |
| mpladd 22316 | The addition operation on ... |
| mplneg 22317 | The negative function on m... |
| mplmul 22318 | The multiplication operati... |
| mpl1 22319 | The identity element of th... |
| mplsca 22320 | The scalar field of a mult... |
| mplvsca2 22321 | The scalar multiplication ... |
| mplvsca 22322 | The scalar multiplication ... |
| mplvscaval 22323 | The scalar multiplication ... |
| mplgrp 22324 | The polynomial ring is a g... |
| mpllmod 22325 | The polynomial ring is a l... |
| mplring 22326 | The polynomial ring is a r... |
| mpllvec 22327 | The polynomial ring is a v... |
| mplcrng 22328 | The polynomial ring is a c... |
| mplassa 22329 | The polynomial ring is an ... |
| mplringd 22330 | The polynomial ring is a r... |
| mplcrngd 22331 | The polynomial ring is a c... |
| mpllmodd 22332 | The polynomial ring is a l... |
| mplascl0 22333 | The zero scalar as a polyn... |
| mplascl1 22334 | The one scalar as a polyno... |
| ressmplbas2 22335 | The base set of a restrict... |
| ressmplbas 22336 | A restricted polynomial al... |
| ressmpladd 22337 | A restricted polynomial al... |
| ressmplmul 22338 | A restricted polynomial al... |
| ressmplvsca 22339 | A restricted power series ... |
| subrgmpl 22340 | A subring of the base ring... |
| mplsubrgcl 22341 | An element of a polynomial... |
| subrgmvr 22342 | The variables in a subring... |
| subrgmvrf 22343 | The variables in a polynom... |
| mplmon 22344 | A monomial is a polynomial... |
| mplmonmul 22345 | The product of two monomia... |
| mplcoe1 22346 | Decompose a polynomial int... |
| mplcoe3 22347 | Decompose a monomial in on... |
| mplcoe5lem 22348 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22349 | Decompose a monomial into ... |
| mplcoe2 22350 | Decompose a monomial into ... |
| mplbas2 22351 | An alternative expression ... |
| ltbval 22352 | Value of the well-order on... |
| ltbwe 22353 | The finite bag order is a ... |
| reldmopsr 22354 | Lemma for ordered power se... |
| opsrval 22355 | The value of the "ordered ... |
| opsrle 22356 | An alternative expression ... |
| opsrval2 22357 | Self-referential expressio... |
| opsrbaslem 22358 | Get a component of the ord... |
| opsrbas 22359 | The base set of the ordere... |
| opsrplusg 22360 | The addition operation of ... |
| opsrmulr 22361 | The multiplication operati... |
| opsrvsca 22362 | The scalar product operati... |
| opsrsca 22363 | The scalar ring of the ord... |
| opsrtoslem1 22364 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22365 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22366 | The ordered power series s... |
| opsrso 22367 | The ordered power series s... |
| opsrcrng 22368 | The ring of ordered power ... |
| opsrassa 22369 | The ring of ordered power ... |
| mplmon2 22370 | Express a scaled monomial.... |
| psrbag0 22371 | The empty bag is a bag. (... |
| psrbagsn 22372 | A singleton bag is a bag. ... |
| mplascl 22373 | Value of the scalar inject... |
| mplasclf 22374 | The scalar injection is a ... |
| subrgascl 22375 | The scalar injection funct... |
| subrgasclcl 22376 | The scalars in a polynomia... |
| mplmon2cl 22377 | A scaled monomial is a pol... |
| mplmon2mul 22378 | Product of scaled monomial... |
| mplind 22379 | Prove a property of polyno... |
| mplcoe4 22380 | Decompose a polynomial int... |
| evlslem4 22385 | The support of a tensor pr... |
| psrbagev1 22386 | A bag of multipliers provi... |
| psrbagev2 22387 | Closure of a sum using a b... |
| evlslem2 22388 | A linear function on the p... |
| evlslem3 22389 | Lemma for ~ evlseu . Poly... |
| evlslem6 22390 | Lemma for ~ evlseu . Fini... |
| evlslem1 22391 | Lemma for ~ evlseu , give ... |
| evlseu 22392 | For a given interpretation... |
| reldmevls 22393 | Well-behaved binary operat... |
| mpfrcl 22394 | Reverse closure for the se... |
| evlsval 22395 | Value of the polynomial ev... |
| evlsval2 22396 | Characterizing properties ... |
| evlsrhm 22397 | Polynomial evaluation is a... |
| evlsval3 22398 | Give a formula for the pol... |
| evlsvval 22399 | Give a formula for the eva... |
| evlsvvvallem 22400 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22401 | Lemma for theorems using ~... |
| evlsvvval 22402 | Give a formula for the eva... |
| evlssca 22403 | Polynomial evaluation maps... |
| evlsvar 22404 | Polynomial evaluation maps... |
| evlsgsumadd 22405 | Polynomial evaluation maps... |
| evlsgsummul 22406 | Polynomial evaluation maps... |
| evlspw 22407 | Polynomial evaluation for ... |
| evlsvarpw 22408 | Polynomial evaluation for ... |
| evlval 22409 | Value of the simple/same r... |
| evlrhm 22410 | The simple evaluation map ... |
| evlcl 22411 | A polynomial over the ring... |
| evladdval 22412 | Polynomial evaluation buil... |
| evlmulval 22413 | Polynomial evaluation buil... |
| evlsscasrng 22414 | The evaluation of a scalar... |
| evlsca 22415 | Simple polynomial evaluati... |
| evlsvarsrng 22416 | The evaluation of the vari... |
| evlvar 22417 | Simple polynomial evaluati... |
| mpfconst 22418 | Constants are multivariate... |
| mpfproj 22419 | Projections are multivaria... |
| mpfsubrg 22420 | Polynomial functions are a... |
| mpff 22421 | Polynomial functions are f... |
| mpfaddcl 22422 | The sum of multivariate po... |
| mpfmulcl 22423 | The product of multivariat... |
| mpfind 22424 | Prove a property of polyno... |
| selvffval 22427 | Value of the "variable sel... |
| selvfval 22428 | Value of the "variable sel... |
| selvval 22429 | Value of the "variable sel... |
| mhmcompl 22430 | The composition of a monoi... |
| mplmapghm 22431 | The function ` H ` mapping... |
| mhmcoaddmpl 22432 | Show that the ring homomor... |
| rhmcomulmpl 22433 | Show that the ring homomor... |
| evlscl 22434 | A polynomial over the ring... |
| evlsscaval 22435 | Polynomial evaluation buil... |
| evlsvarval 22436 | Polynomial evaluation buil... |
| evlsexpval 22437 | Polynomial evaluation buil... |
| evlsaddval 22438 | Polynomial evaluation buil... |
| evlsmulval 22439 | Polynomial evaluation buil... |
| evlsmaprhm 22440 | The function ` F ` mapping... |
| evlsevl 22441 | Evaluation in a subring is... |
| evlvvval 22442 | Give a formula for the eva... |
| selvcllem1 22443 | ` T ` is an associative al... |
| selvcllem2 22444 | ` D ` is a ring homomorphi... |
| selvcllem3 22445 | The third argument passed ... |
| selvcllemh 22446 | Apply the third argument (... |
| selvcllem4 22447 | The fourth argument passed... |
| selvcllem5 22448 | The fifth argument passed ... |
| selvcl 22449 | Closure of the "variable s... |
| selvval2 22450 | Value of the "variable sel... |
| selvvvval 22451 | Recover the original polyn... |
| selvadd 22452 | The "variable selection" f... |
| selvmul 22453 | The "variable selection" f... |
| reldmmhp 22458 | The domain of the homogene... |
| mhpfval 22459 | Value of the "homogeneous ... |
| mhpval 22460 | Value of the "homogeneous ... |
| ismhp 22461 | Property of being a homoge... |
| ismhp2 22462 | Deduce a homogeneous polyn... |
| ismhp3 22463 | A polynomial is homogeneou... |
| mhprcl 22464 | Reverse closure for homoge... |
| mhpmpl 22465 | A homogeneous polynomial i... |
| mhpdeg 22466 | All nonzero terms of a hom... |
| mhp0cl 22467 | The zero polynomial is hom... |
| mhpsclcl 22468 | A scalar (or constant) pol... |
| mhpvarcl 22469 | A power series variable is... |
| mhpmulcl 22470 | A product of homogeneous p... |
| mhppwdeg 22471 | Degree of a homogeneous po... |
| mhpaddcl 22472 | Homogeneous polynomials ar... |
| mhpinvcl 22473 | Homogeneous polynomials ar... |
| mhpsubg 22474 | Homogeneous polynomials fo... |
| mhpvscacl 22475 | Homogeneous polynomials ar... |
| mhplss 22476 | Homogeneous polynomials fo... |
| psdffval 22478 | Value of the power series ... |
| psdfval 22479 | Give a map between power s... |
| psdval 22480 | Evaluate the partial deriv... |
| psdcoef 22481 | Coefficient of a term of t... |
| psdcl 22482 | The derivative of a power ... |
| psdmplcl 22483 | The derivative of a polyno... |
| psdadd 22484 | The derivative of a sum is... |
| psdvsca 22485 | The derivative of a scaled... |
| psdmullem 22486 | Lemma for ~ psdmul . Tran... |
| psdmul 22487 | Product rule for power ser... |
| psd1 22488 | The derivative of one is z... |
| psdascl 22489 | The derivative of a consta... |
| psdmvr 22490 | The partial derivative of ... |
| psdpw 22491 | Power rule for partial der... |
| psr1baslem 22503 | The set of finite bags on ... |
| psr1val 22504 | Value of the ring of univa... |
| psr1crng 22505 | The ring of univariate pow... |
| psr1assa 22506 | The ring of univariate pow... |
| psr1tos 22507 | The ordered power series s... |
| psr1bas2 22508 | The base set of the ring o... |
| psr1bas 22509 | The base set of the ring o... |
| vr1val 22510 | The value of the generator... |
| vr1cl2 22511 | The variable ` X ` is a me... |
| ply1val 22512 | The value of the set of un... |
| ply1bas 22513 | The value of the base set ... |
| ply1lss 22514 | Univariate polynomials for... |
| ply1subrg 22515 | Univariate polynomials for... |
| ply1crng 22516 | The ring of univariate pol... |
| ply1assa 22517 | The ring of univariate pol... |
| psr1bascl 22518 | A univariate power series ... |
| psr1basf 22519 | Univariate power series ba... |
| ply1basf 22520 | Univariate polynomial base... |
| ply1bascl 22521 | A univariate polynomial is... |
| ply1bascl2 22522 | A univariate polynomial is... |
| coe1fval 22523 | Value of the univariate po... |
| coe1fv 22524 | Value of an evaluated coef... |
| fvcoe1 22525 | Value of a multivariate co... |
| coe1fval3 22526 | Univariate power series co... |
| coe1f2 22527 | Functionality of univariat... |
| coe1fval2 22528 | Univariate polynomial coef... |
| coe1f 22529 | Functionality of univariat... |
| coe1fvalcl 22530 | A coefficient of a univari... |
| coe1sfi 22531 | Finite support of univaria... |
| coe1fsupp 22532 | The coefficient vector of ... |
| mptcoe1fsupp 22533 | A mapping involving coeffi... |
| coe1ae0 22534 | The coefficient vector of ... |
| vr1cl 22535 | The generator of a univari... |
| opsr0 22536 | Zero in the ordered power ... |
| opsr1 22537 | One in the ordered power s... |
| psr1plusg 22538 | Value of addition in a uni... |
| psr1vsca 22539 | Value of scalar multiplica... |
| psr1mulr 22540 | Value of multiplication in... |
| ply1plusg 22541 | Value of addition in a uni... |
| ply1vsca 22542 | Value of scalar multiplica... |
| ply1mulr 22543 | Value of multiplication in... |
| ply1ass23l 22544 | Associative identity with ... |
| ressply1bas2 22545 | The base set of a restrict... |
| ressply1bas 22546 | A restricted polynomial al... |
| ressply1add 22547 | A restricted polynomial al... |
| ressply1mul 22548 | A restricted polynomial al... |
| ressply1vsca 22549 | A restricted power series ... |
| subrgply1 22550 | A subring of the base ring... |
| gsumply1subr 22551 | Evaluate a group sum in a ... |
| psrbaspropd 22552 | Property deduction for pow... |
| psrplusgpropd 22553 | Property deduction for pow... |
| mplbaspropd 22554 | Property deduction for pol... |
| psropprmul 22555 | Reversing multiplication i... |
| ply1opprmul 22556 | Reversing multiplication i... |
| 00ply1bas 22557 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22558 | Protection compatibility o... |
| ply1plusgfvi 22559 | Protection compatibility o... |
| ply1baspropd 22560 | Property deduction for uni... |
| ply1plusgpropd 22561 | Property deduction for uni... |
| opsrring 22562 | Ordered power series form ... |
| opsrlmod 22563 | Ordered power series form ... |
| psr1ring 22564 | Univariate power series fo... |
| ply1ring 22565 | Univariate polynomials for... |
| psr1lmod 22566 | Univariate power series fo... |
| psr1sca 22567 | Scalars of a univariate po... |
| psr1sca2 22568 | Scalars of a univariate po... |
| ply1lmod 22569 | Univariate polynomials for... |
| ply1sca 22570 | Scalars of a univariate po... |
| ply1sca2 22571 | Scalars of a univariate po... |
| ply1ascl0 22572 | The zero scalar as a polyn... |
| ply1ascl1 22573 | The multiplicative identit... |
| ply1mpl0 22574 | The univariate polynomial ... |
| ply10s0 22575 | Zero times a univariate po... |
| ply1mpl1 22576 | The univariate polynomial ... |
| ply1ascl 22577 | The univariate polynomial ... |
| subrg1ascl 22578 | The scalar injection funct... |
| subrg1asclcl 22579 | The scalars in a polynomia... |
| subrgvr1 22580 | The variables in a subring... |
| subrgvr1cl 22581 | The variables in a polynom... |
| coe1z 22582 | The coefficient vector of ... |
| coe1add 22583 | The coefficient vector of ... |
| coe1addfv 22584 | A particular coefficient o... |
| coe1subfv 22585 | A particular coefficient o... |
| coe1mul2lem1 22586 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22587 | An equivalence for ~ coe1m... |
| coe1mul2 22588 | The coefficient vector of ... |
| coe1mul 22589 | The coefficient vector of ... |
| ply1moncl 22590 | Closure of the expression ... |
| ply1tmcl 22591 | Closure of the expression ... |
| coe1tm 22592 | Coefficient vector of a po... |
| coe1tmfv1 22593 | Nonzero coefficient of a p... |
| coe1tmfv2 22594 | Zero coefficient of a poly... |
| coe1tmmul2 22595 | Coefficient vector of a po... |
| coe1tmmul 22596 | Coefficient vector of a po... |
| coe1tmmul2fv 22597 | Function value of a right-... |
| coe1pwmul 22598 | Coefficient vector of a po... |
| coe1pwmulfv 22599 | Function value of a right-... |
| ply1scltm 22600 | A scalar is a term with ze... |
| coe1sclmul 22601 | Coefficient vector of a po... |
| coe1sclmulfv 22602 | A single coefficient of a ... |
| coe1sclmul2 22603 | Coefficient vector of a po... |
| ply1sclf 22604 | A scalar polynomial is a p... |
| ply1sclcl 22605 | The value of the algebra s... |
| coe1scl 22606 | Coefficient vector of a sc... |
| ply1sclid 22607 | Recover the base scalar fr... |
| ply1sclf1 22608 | The polynomial scalar func... |
| ply1scl0 22609 | The zero scalar is zero. ... |
| ply1scln0 22610 | Nonzero scalars create non... |
| ply1scl1 22611 | The one scalar is the unit... |
| coe1id 22612 | Coefficient vector of the ... |
| ply1idvr1 22613 | The identity of a polynomi... |
| cply1mul 22614 | The product of two constan... |
| ply1coefsupp 22615 | The decomposition of a uni... |
| ply1coe 22616 | Decompose a univariate pol... |
| eqcoe1ply1eq 22617 | Two polynomials over the s... |
| ply1coe1eq 22618 | Two polynomials over the s... |
| cply1coe0 22619 | All but the first coeffici... |
| cply1coe0bi 22620 | A polynomial is constant (... |
| coe1fzgsumdlem 22621 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22622 | Value of an evaluated coef... |
| ply1scleq 22623 | Equality of a constant pol... |
| ply1chr 22624 | The characteristic of a po... |
| gsumsmonply1 22625 | A finite group sum of scal... |
| gsummoncoe1 22626 | A coefficient of the polyn... |
| gsumply1eq 22627 | Two univariate polynomials... |
| lply1binom 22628 | The binomial theorem for l... |
| lply1binomsc 22629 | The binomial theorem for l... |
| ply1fermltlchr 22630 | Fermat's little theorem fo... |
| reldmevls1 22635 | Well-behaved binary operat... |
| ply1frcl 22636 | Reverse closure for the se... |
| evls1fval 22637 | Value of the univariate po... |
| evls1val 22638 | Value of the univariate po... |
| evls1rhmlem 22639 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22640 | Polynomial evaluation is a... |
| evls1sca 22641 | Univariate polynomial eval... |
| evls1gsumadd 22642 | Univariate polynomial eval... |
| evls1gsummul 22643 | Univariate polynomial eval... |
| evls1pw 22644 | Univariate polynomial eval... |
| evls1varpw 22645 | Univariate polynomial eval... |
| evl1fval 22646 | Value of the simple/same r... |
| evl1val 22647 | Value of the simple/same r... |
| evl1fval1lem 22648 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22649 | Value of the simple/same r... |
| evl1rhm 22650 | Polynomial evaluation is a... |
| fveval1fvcl 22651 | The function value of the ... |
| evl1sca 22652 | Polynomial evaluation maps... |
| evl1scad 22653 | Polynomial evaluation buil... |
| evl1var 22654 | Polynomial evaluation maps... |
| evl1vard 22655 | Polynomial evaluation buil... |
| evls1var 22656 | Univariate polynomial eval... |
| evls1scasrng 22657 | The evaluation of a scalar... |
| evls1varsrng 22658 | The evaluation of the vari... |
| evl1addd 22659 | Polynomial evaluation buil... |
| evl1subd 22660 | Polynomial evaluation buil... |
| evl1muld 22661 | Polynomial evaluation buil... |
| evl1vsd 22662 | Polynomial evaluation buil... |
| evl1expd 22663 | Polynomial evaluation buil... |
| pf1const 22664 | Constants are polynomial f... |
| pf1id 22665 | The identity is a polynomi... |
| pf1subrg 22666 | Polynomial functions are a... |
| pf1rcl 22667 | Reverse closure for the se... |
| pf1f 22668 | Polynomial functions are f... |
| mpfpf1 22669 | Convert a multivariate pol... |
| pf1mpf 22670 | Convert a univariate polyn... |
| pf1addcl 22671 | The sum of multivariate po... |
| pf1mulcl 22672 | The product of multivariat... |
| pf1ind 22673 | Prove a property of polyno... |
| evl1gsumdlem 22674 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22675 | Polynomial evaluation buil... |
| evl1gsumadd 22676 | Univariate polynomial eval... |
| evl1gsumaddval 22677 | Value of a univariate poly... |
| evl1gsummul 22678 | Univariate polynomial eval... |
| evl1varpw 22679 | Univariate polynomial eval... |
| evl1varpwval 22680 | Value of a univariate poly... |
| evl1scvarpw 22681 | Univariate polynomial eval... |
| evl1scvarpwval 22682 | Value of a univariate poly... |
| evl1gsummon 22683 | Value of a univariate poly... |
| evls1scafv 22684 | Value of the univariate po... |
| evls1expd 22685 | Univariate polynomial eval... |
| evls1varpwval 22686 | Univariate polynomial eval... |
| evls1fpws 22687 | Evaluation of a univariate... |
| ressply1evl 22688 | Evaluation of a univariate... |
| evls1addd 22689 | Univariate polynomial eval... |
| evls1muld 22690 | Univariate polynomial eval... |
| evls1vsca 22691 | Univariate polynomial eval... |
| asclply1subcl 22692 | Closure of the algebra sca... |
| evls1fvcl 22693 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22694 | The function ` F ` mapping... |
| evls1maplmhm 22695 | The function ` F ` mapping... |
| evls1maprnss 22696 | The function ` F ` mapping... |
| evl1maprhm 22697 | The function ` F ` mapping... |
| rhmmpl 22698 | Provide a ring homomorphis... |
| ply1vscl 22699 | Closure of scalar multipli... |
| mhmcoply1 22700 | The composition of a monoi... |
| rhmply1 22701 | Provide a ring homomorphis... |
| rhmply1vr1 22702 | A ring homomorphism betwee... |
| rhmply1vsca 22703 | Apply a ring homomorphism ... |
| rhmply1mon 22704 | Apply a ring homomorphism ... |
| mamufval 22707 | Functional value of the ma... |
| mamuval 22708 | Multiplication of two matr... |
| mamufv 22709 | A cell in the multiplicati... |
| mamudm 22710 | The domain of the matrix m... |
| mamufacex 22711 | Every solution of the equa... |
| mamures 22712 | Rows in a matrix product a... |
| grpvlinv 22713 | Tuple-wise left inverse in... |
| grpvrinv 22714 | Tuple-wise right inverse i... |
| ringvcl 22715 | Tuple-wise multiplication ... |
| mamucl 22716 | Operation closure of matri... |
| mamuass 22717 | Matrix multiplication is a... |
| mamudi 22718 | Matrix multiplication dist... |
| mamudir 22719 | Matrix multiplication dist... |
| mamuvs1 22720 | Matrix multiplication dist... |
| mamuvs2 22721 | Matrix multiplication dist... |
| matbas0pc 22724 | There is no matrix with a ... |
| matbas0 22725 | There is no matrix for a n... |
| matval 22726 | Value of the matrix algebr... |
| matrcl 22727 | Reverse closure for the ma... |
| matbas 22728 | The matrix ring has the sa... |
| matplusg 22729 | The matrix ring has the sa... |
| matsca 22730 | The matrix ring has the sa... |
| matvsca 22731 | The matrix ring has the sa... |
| mat0 22732 | The matrix ring has the sa... |
| matinvg 22733 | The matrix ring has the sa... |
| mat0op 22734 | Value of a zero matrix as ... |
| matsca2 22735 | The scalars of the matrix ... |
| matbas2 22736 | The base set of the matrix... |
| matbas2i 22737 | A matrix is a function. (... |
| matbas2d 22738 | The base set of the matrix... |
| eqmat 22739 | Two square matrices of the... |
| matecl 22740 | Each entry (according to W... |
| matecld 22741 | Each entry (according to W... |
| matplusg2 22742 | Addition in the matrix rin... |
| matvsca2 22743 | Scalar multiplication in t... |
| matlmod 22744 | The matrix ring is a linea... |
| matgrp 22745 | The matrix ring is a group... |
| matvscl 22746 | Closure of the scalar mult... |
| matsubg 22747 | The matrix ring has the sa... |
| matplusgcell 22748 | Addition in the matrix rin... |
| matsubgcell 22749 | Subtraction in the matrix ... |
| matinvgcell 22750 | Additive inversion in the ... |
| matvscacell 22751 | Scalar multiplication in t... |
| matgsum 22752 | Finite commutative sums in... |
| matmulr 22753 | Multiplication in the matr... |
| mamumat1cl 22754 | The identity matrix (as op... |
| mat1comp 22755 | The components of the iden... |
| mamulid 22756 | The identity matrix (as op... |
| mamurid 22757 | The identity matrix (as op... |
| matring 22758 | Existence of the matrix ri... |
| matassa 22759 | Existence of the matrix al... |
| matmulcell 22760 | Multiplication in the matr... |
| mpomatmul 22761 | Multiplication of two N x ... |
| mat1 22762 | Value of an identity matri... |
| mat1ov 22763 | Entries of an identity mat... |
| mat1bas 22764 | The identity matrix is a m... |
| matsc 22765 | The identity matrix multip... |
| ofco2 22766 | Distribution law for the f... |
| oftpos 22767 | The transposition of the v... |
| mattposcl 22768 | The transpose of a square ... |
| mattpostpos 22769 | The transpose of the trans... |
| mattposvs 22770 | The transposition of a mat... |
| mattpos1 22771 | The transposition of the i... |
| tposmap 22772 | The transposition of an I ... |
| mamutpos 22773 | Behavior of transposes in ... |
| mattposm 22774 | Multiplying two transposed... |
| matgsumcl 22775 | Closure of a group sum ove... |
| madetsumid 22776 | The identity summand in th... |
| matepmcl 22777 | Each entry of a matrix wit... |
| matepm2cl 22778 | Each entry of a matrix wit... |
| madetsmelbas 22779 | A summand of the determina... |
| madetsmelbas2 22780 | A summand of the determina... |
| mat0dimbas0 22781 | The empty set is the one a... |
| mat0dim0 22782 | The zero of the algebra of... |
| mat0dimid 22783 | The identity of the algebr... |
| mat0dimscm 22784 | The scalar multiplication ... |
| mat0dimcrng 22785 | The algebra of matrices wi... |
| mat1dimelbas 22786 | A matrix with dimension 1 ... |
| mat1dimbas 22787 | A matrix with dimension 1 ... |
| mat1dim0 22788 | The zero of the algebra of... |
| mat1dimid 22789 | The identity of the algebr... |
| mat1dimscm 22790 | The scalar multiplication ... |
| mat1dimmul 22791 | The ring multiplication in... |
| mat1dimcrng 22792 | The algebra of matrices wi... |
| mat1f1o 22793 | There is a 1-1 function fr... |
| mat1rhmval 22794 | The value of the ring homo... |
| mat1rhmelval 22795 | The value of the ring homo... |
| mat1rhmcl 22796 | The value of the ring homo... |
| mat1f 22797 | There is a function from a... |
| mat1ghm 22798 | There is a group homomorph... |
| mat1mhm 22799 | There is a monoid homomorp... |
| mat1rhm 22800 | There is a ring homomorphi... |
| mat1rngiso 22801 | There is a ring isomorphis... |
| mat1ric 22802 | A ring is isomorphic to th... |
| dmatval 22807 | The set of ` N ` x ` N ` d... |
| dmatel 22808 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22809 | An ` N ` x ` N ` diagonal ... |
| dmatid 22810 | The identity matrix is a d... |
| dmatelnd 22811 | An extradiagonal entry of ... |
| dmatmul 22812 | The product of two diagona... |
| dmatsubcl 22813 | The difference of two diag... |
| dmatsgrp 22814 | The set of diagonal matric... |
| dmatmulcl 22815 | The product of two diagona... |
| dmatsrng 22816 | The set of diagonal matric... |
| dmatcrng 22817 | The subring of diagonal ma... |
| dmatscmcl 22818 | The multiplication of a di... |
| scmatval 22819 | The set of ` N ` x ` N ` s... |
| scmatel 22820 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22821 | A scalar matrix can be exp... |
| scmatscmide 22822 | An entry of a scalar matri... |
| scmatscmiddistr 22823 | Distributive law for scala... |
| scmatmat 22824 | An ` N ` x ` N ` scalar ma... |
| scmate 22825 | An entry of an ` N ` x ` N... |
| scmatmats 22826 | The set of an ` N ` x ` N ... |
| scmateALT 22827 | Alternate proof of ~ scmat... |
| scmatscm 22828 | The multiplication of a ma... |
| scmatid 22829 | The identity matrix is a s... |
| scmatdmat 22830 | A scalar matrix is a diago... |
| scmataddcl 22831 | The sum of two scalar matr... |
| scmatsubcl 22832 | The difference of two scal... |
| scmatmulcl 22833 | The product of two scalar ... |
| scmatsgrp 22834 | The set of scalar matrices... |
| scmatsrng 22835 | The set of scalar matrices... |
| scmatcrng 22836 | The subring of scalar matr... |
| scmatsgrp1 22837 | The set of scalar matrices... |
| scmatsrng1 22838 | The set of scalar matrices... |
| smatvscl 22839 | Closure of the scalar mult... |
| scmatlss 22840 | The set of scalar matrices... |
| scmatstrbas 22841 | The set of scalar matrices... |
| scmatrhmval 22842 | The value of the ring homo... |
| scmatrhmcl 22843 | The value of the ring homo... |
| scmatf 22844 | There is a function from a... |
| scmatfo 22845 | There is a function from a... |
| scmatf1 22846 | There is a 1-1 function fr... |
| scmatf1o 22847 | There is a bijection betwe... |
| scmatghm 22848 | There is a group homomorph... |
| scmatmhm 22849 | There is a monoid homomorp... |
| scmatrhm 22850 | There is a ring homomorphi... |
| scmatrngiso 22851 | There is a ring isomorphis... |
| scmatric 22852 | A ring is isomorphic to ev... |
| mat0scmat 22853 | The empty matrix over a ri... |
| mat1scmat 22854 | A 1-dimensional matrix ove... |
| mvmulfval 22857 | Functional value of the ma... |
| mvmulval 22858 | Multiplication of a vector... |
| mvmulfv 22859 | A cell/element in the vect... |
| mavmulval 22860 | Multiplication of a vector... |
| mavmulfv 22861 | A cell/element in the vect... |
| mavmulcl 22862 | Multiplication of an NxN m... |
| 1mavmul 22863 | Multiplication of the iden... |
| mavmulass 22864 | Associativity of the multi... |
| mavmuldm 22865 | The domain of the matrix v... |
| mavmulsolcl 22866 | Every solution of the equa... |
| mavmul0 22867 | Multiplication of a 0-dime... |
| mavmul0g 22868 | The result of the 0-dimens... |
| mvmumamul1 22869 | The multiplication of an M... |
| mavmumamul1 22870 | The multiplication of an N... |
| marrepfval 22875 | First substitution for the... |
| marrepval0 22876 | Second substitution for th... |
| marrepval 22877 | Third substitution for the... |
| marrepeval 22878 | An entry of a matrix with ... |
| marrepcl 22879 | Closure of the row replace... |
| marepvfval 22880 | First substitution for the... |
| marepvval0 22881 | Second substitution for th... |
| marepvval 22882 | Third substitution for the... |
| marepveval 22883 | An entry of a matrix with ... |
| marepvcl 22884 | Closure of the column repl... |
| ma1repvcl 22885 | Closure of the column repl... |
| ma1repveval 22886 | An entry of an identity ma... |
| mulmarep1el 22887 | Element by element multipl... |
| mulmarep1gsum1 22888 | The sum of element by elem... |
| mulmarep1gsum2 22889 | The sum of element by elem... |
| 1marepvmarrepid 22890 | Replacing the ith row by 0... |
| submabas 22893 | Any subset of the index se... |
| submafval 22894 | First substitution for a s... |
| submaval0 22895 | Second substitution for a ... |
| submaval 22896 | Third substitution for a s... |
| submaeval 22897 | An entry of a submatrix of... |
| 1marepvsma1 22898 | The submatrix of the ident... |
| mdetfval 22901 | First substitution for the... |
| mdetleib 22902 | Full substitution of our d... |
| mdetleib2 22903 | Leibniz' formula can also ... |
| nfimdetndef 22904 | The determinant is not def... |
| mdetfval1 22905 | First substitution of an a... |
| mdetleib1 22906 | Full substitution of an al... |
| mdet0pr 22907 | The determinant function f... |
| mdet0f1o 22908 | The determinant function f... |
| mdet0fv0 22909 | The determinant of the emp... |
| mdetf 22910 | Functionality of the deter... |
| mdetcl 22911 | The determinant evaluates ... |
| m1detdiag 22912 | The determinant of a 1-dim... |
| mdetdiaglem 22913 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22914 | The determinant of a diago... |
| mdetdiagid 22915 | The determinant of a diago... |
| mdet1 22916 | The determinant of the ide... |
| mdetrlin 22917 | The determinant function i... |
| mdetrsca 22918 | The determinant function i... |
| mdetrsca2 22919 | The determinant function i... |
| mdetr0 22920 | The determinant of a matri... |
| mdet0 22921 | The determinant of the zer... |
| mdetrlin2 22922 | The determinant function i... |
| mdetralt 22923 | The determinant function i... |
| mdetralt2 22924 | The determinant function i... |
| mdetero 22925 | The determinant function i... |
| mdettpos 22926 | Determinant is invariant u... |
| mdetunilem1 22927 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22928 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22929 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22930 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22931 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22932 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22933 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22934 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22935 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22936 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22937 | According to the definitio... |
| mdetmul 22938 | Multiplicativity of the de... |
| m2detleiblem1 22939 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22940 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22941 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22942 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22943 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22944 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22945 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22946 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22951 | Lemma for ~ maducoeval2 . ... |
| madufval 22952 | First substitution for the... |
| maduval 22953 | Second substitution for th... |
| maducoeval 22954 | An entry of the adjunct (c... |
| maducoeval2 22955 | An entry of the adjunct (c... |
| maduf 22956 | Creating the adjunct of ma... |
| madutpos 22957 | The adjuct of a transposed... |
| madugsum 22958 | The determinant of a matri... |
| madurid 22959 | Multiplying a matrix with ... |
| madulid 22960 | Multiplying the adjunct of... |
| minmar1fval 22961 | First substitution for the... |
| minmar1val0 22962 | Second substitution for th... |
| minmar1val 22963 | Third substitution for the... |
| minmar1eval 22964 | An entry of a matrix for a... |
| minmar1marrep 22965 | The minor matrix is a spec... |
| minmar1cl 22966 | Closure of the row replace... |
| maducoevalmin1 22967 | The coefficients of an adj... |
| symgmatr01lem 22968 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22969 | Applying a permutation tha... |
| gsummatr01lem1 22970 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22971 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22972 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22973 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22974 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22975 | Replacing a row of a squar... |
| smadiadetlem0 22976 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22977 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22978 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22979 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22980 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22981 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22982 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22983 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22984 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22985 | The determinant of a subma... |
| smadiadetglem1 22986 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22987 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22988 | The determinant of a squar... |
| smadiadetg0 22989 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22990 | The determinant of a squar... |
| invrvald 22991 | If a matrix multiplied wit... |
| matinv 22992 | The inverse of a matrix is... |
| matunit 22993 | A matrix is a unit in the ... |
| matunitlindflem1 22994 | One direction of ~ matunit... |
| matunitlindflem2 22995 | One direction of ~ matunit... |
| matunitlindf 22996 | A matrix over a field is i... |
| slesolvec 22997 | Every solution of a system... |
| slesolinv 22998 | The solution of a system o... |
| slesolinvbi 22999 | The solution of a system o... |
| slesolex 23000 | Every system of linear equ... |
| cramerimplem1 23001 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 23002 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 23003 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 23004 | One direction of Cramer's ... |
| cramerlem1 23005 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 23006 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 23007 | Lemma 3 for ~ cramer . (C... |
| cramer0 23008 | Special case of Cramer's r... |
| cramer 23009 | Cramer's rule. According ... |
| pmatring 23010 | The set of polynomial matr... |
| pmatlmod 23011 | The set of polynomial matr... |
| pmatassa 23012 | The set of polynomial matr... |
| pmat0op 23013 | The zero polynomial matrix... |
| pmat1op 23014 | The identity polynomial ma... |
| pmat1ovd 23015 | Entries of the identity po... |
| pmat0opsc 23016 | The zero polynomial matrix... |
| pmat1opsc 23017 | The identity polynomial ma... |
| pmat1ovscd 23018 | Entries of the identity po... |
| pmatcoe1fsupp 23019 | For a polynomial matrix th... |
| 1pmatscmul 23020 | The scalar product of the ... |
| cpmat 23027 | Value of the constructor o... |
| cpmatpmat 23028 | A constant polynomial matr... |
| cpmatel 23029 | Property of a constant pol... |
| cpmatelimp 23030 | Implication of a set being... |
| cpmatel2 23031 | Another property of a cons... |
| cpmatelimp2 23032 | Another implication of a s... |
| 1elcpmat 23033 | The identity of the ring o... |
| cpmatacl 23034 | The set of all constant po... |
| cpmatinvcl 23035 | The set of all constant po... |
| cpmatmcllem 23036 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 23037 | The set of all constant po... |
| cpmatsubgpmat 23038 | The set of all constant po... |
| cpmatsrgpmat 23039 | The set of all constant po... |
| 0elcpmat 23040 | The zero of the ring of al... |
| mat2pmatfval 23041 | Value of the matrix transf... |
| mat2pmatval 23042 | The result of a matrix tra... |
| mat2pmatvalel 23043 | A (matrix) element of the ... |
| mat2pmatbas 23044 | The result of a matrix tra... |
| mat2pmatbas0 23045 | The result of a matrix tra... |
| mat2pmatf 23046 | The matrix transformation ... |
| mat2pmatf1 23047 | The matrix transformation ... |
| mat2pmatghm 23048 | The transformation of matr... |
| mat2pmatmul 23049 | The transformation of matr... |
| mat2pmat1 23050 | The transformation of the ... |
| mat2pmatmhm 23051 | The transformation of matr... |
| mat2pmatrhm 23052 | The transformation of matr... |
| mat2pmatlin 23053 | The transformation of matr... |
| 0mat2pmat 23054 | The transformed zero matri... |
| idmatidpmat 23055 | The transformed identity m... |
| d0mat2pmat 23056 | The transformed empty set ... |
| d1mat2pmat 23057 | The transformation of a ma... |
| mat2pmatscmxcl 23058 | A transformed matrix multi... |
| m2cpm 23059 | The result of a matrix tra... |
| m2cpmf 23060 | The matrix transformation ... |
| m2cpmf1 23061 | The matrix transformation ... |
| m2cpmghm 23062 | The transformation of matr... |
| m2cpmmhm 23063 | The transformation of matr... |
| m2cpmrhm 23064 | The transformation of matr... |
| m2pmfzmap 23065 | The transformed values of ... |
| m2pmfzgsumcl 23066 | Closure of the sum of scal... |
| cpm2mfval 23067 | Value of the inverse matri... |
| cpm2mval 23068 | The result of an inverse m... |
| cpm2mvalel 23069 | A (matrix) element of the ... |
| cpm2mf 23070 | The inverse matrix transfo... |
| m2cpminvid 23071 | The inverse transformation... |
| m2cpminvid2lem 23072 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 23073 | The transformation applied... |
| m2cpmfo 23074 | The matrix transformation ... |
| m2cpmf1o 23075 | The matrix transformation ... |
| m2cpmrngiso 23076 | The transformation of matr... |
| matcpmric 23077 | The ring of matrices over ... |
| m2cpminv 23078 | The inverse matrix transfo... |
| m2cpminv0 23079 | The inverse matrix transfo... |
| decpmatval0 23082 | The matrix consisting of t... |
| decpmatval 23083 | The matrix consisting of t... |
| decpmate 23084 | An entry of the matrix con... |
| decpmatcl 23085 | Closure of the decompositi... |
| decpmataa0 23086 | The matrix consisting of t... |
| decpmatfsupp 23087 | The mapping to the matrice... |
| decpmatid 23088 | The matrix consisting of t... |
| decpmatmullem 23089 | Lemma for ~ decpmatmul . ... |
| decpmatmul 23090 | The matrix consisting of t... |
| decpmatmulsumfsupp 23091 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 23092 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 23093 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 23094 | Write a polynomial matrix ... |
| pmatcollpw2lem 23095 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 23096 | Write a polynomial matrix ... |
| monmatcollpw 23097 | The matrix consisting of t... |
| pmatcollpwlem 23098 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 23099 | Write a polynomial matrix ... |
| pmatcollpwfi 23100 | Write a polynomial matrix ... |
| pmatcollpw3lem 23101 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 23102 | Write a polynomial matrix ... |
| pmatcollpw3fi 23103 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 23104 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 23105 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 23106 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 23107 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 23108 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 23109 | Write a scalar matrix over... |
| pm2mpf1lem 23112 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 23113 | Value of the transformatio... |
| pm2mpfval 23114 | A polynomial matrix transf... |
| pm2mpcl 23115 | The transformation of poly... |
| pm2mpf 23116 | The transformation of poly... |
| pm2mpf1 23117 | The transformation of poly... |
| pm2mpcoe1 23118 | A coefficient of the polyn... |
| idpm2idmp 23119 | The transformation of the ... |
| mptcoe1matfsupp 23120 | The mapping extracting the... |
| mply1topmatcllem 23121 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 23122 | A polynomial over matrices... |
| mply1topmatcl 23123 | A polynomial over matrices... |
| mp2pm2mplem1 23124 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 23125 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 23126 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 23127 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 23128 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 23129 | A polynomial over matrices... |
| pm2mpghmlem2 23130 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 23131 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 23132 | The transformation of poly... |
| pm2mpf1o 23133 | The transformation of poly... |
| pm2mpghm 23134 | The transformation of poly... |
| pm2mpgrpiso 23135 | The transformation of poly... |
| pm2mpmhmlem1 23136 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 23137 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 23138 | The transformation of poly... |
| pm2mprhm 23139 | The transformation of poly... |
| pm2mprngiso 23140 | The transformation of poly... |
| pmmpric 23141 | The ring of polynomial mat... |
| monmat2matmon 23142 | The transformation of a po... |
| pm2mp 23143 | The transformation of a su... |
| chmatcl 23146 | Closure of the characteris... |
| chmatval 23147 | The entries of the charact... |
| chpmatfval 23148 | Value of the characteristi... |
| chpmatval 23149 | The characteristic polynom... |
| chpmatply1 23150 | The characteristic polynom... |
| chpmatval2 23151 | The characteristic polynom... |
| chpmat0d 23152 | The characteristic polynom... |
| chpmat1dlem 23153 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 23154 | The characteristic polynom... |
| chpdmatlem0 23155 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 23156 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 23157 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 23158 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 23159 | The characteristic polynom... |
| chpscmat 23160 | The characteristic polynom... |
| chpscmat0 23161 | The characteristic polynom... |
| chpscmatgsumbin 23162 | The characteristic polynom... |
| chpscmatgsummon 23163 | The characteristic polynom... |
| chp0mat 23164 | The characteristic polynom... |
| chpidmat 23165 | The characteristic polynom... |
| chmaidscmat 23166 | The characteristic polynom... |
| fvmptnn04if 23167 | The function values of a m... |
| fvmptnn04ifa 23168 | The function value of a ma... |
| fvmptnn04ifb 23169 | The function value of a ma... |
| fvmptnn04ifc 23170 | The function value of a ma... |
| fvmptnn04ifd 23171 | The function value of a ma... |
| chfacfisf 23172 | The "characteristic factor... |
| chfacfisfcpmat 23173 | The "characteristic factor... |
| chfacffsupp 23174 | The "characteristic factor... |
| chfacfscmulcl 23175 | Closure of a scaled value ... |
| chfacfscmul0 23176 | A scaled value of the "cha... |
| chfacfscmulfsupp 23177 | A mapping of scaled values... |
| chfacfscmulgsum 23178 | Breaking up a sum of value... |
| chfacfpmmulcl 23179 | Closure of the value of th... |
| chfacfpmmul0 23180 | The value of the "characte... |
| chfacfpmmulfsupp 23181 | A mapping of values of the... |
| chfacfpmmulgsum 23182 | Breaking up a sum of value... |
| chfacfpmmulgsum2 23183 | Breaking up a sum of value... |
| cayhamlem1 23184 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 23185 | The right-hand fundamental... |
| cpmidgsum 23186 | Representation of the iden... |
| cpmidgsumm2pm 23187 | Representation of the iden... |
| cpmidpmatlem1 23188 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 23189 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 23190 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 23191 | Representation of the iden... |
| cpmadugsumlemB 23192 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 23193 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 23194 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 23195 | The product of the charact... |
| cpmadugsum 23196 | The product of the charact... |
| cpmidgsum2 23197 | Representation of the iden... |
| cpmidg2sum 23198 | Equality of two sums repre... |
| cpmadumatpolylem1 23199 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 23200 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 23201 | The product of the charact... |
| cayhamlem2 23202 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 23203 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 23204 | The coefficients of the ch... |
| cayhamlem3 23205 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 23206 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 23207 | The Cayley-Hamilton theore... |
| cayleyhamilton 23208 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 23209 | Alternate proof of ~ cayle... |
| cayleyhamilton1 23210 | The Cayley-Hamilton theore... |
| istopg 23213 | Express the predicate " ` ... |
| istop2g 23214 | Express the predicate " ` ... |
| uniopn 23215 | The union of a subset of a... |
| iunopn 23216 | The indexed union of a sub... |
| inopn 23217 | The intersection of two op... |
| fitop 23218 | A topology is closed under... |
| fiinopn 23219 | The intersection of a none... |
| iinopn 23220 | The intersection of a none... |
| unopn 23221 | The union of two open sets... |
| 0opn 23222 | The empty set is an open s... |
| 0ntop 23223 | The empty set is not a top... |
| topopn 23224 | The underlying set of a to... |
| eltopss 23225 | A member of a topology is ... |
| riinopn 23226 | A finite indexed relative ... |
| rintopn 23227 | A finite relative intersec... |
| istopon 23230 | Property of being a topolo... |
| topontop 23231 | A topology on a given base... |
| toponuni 23232 | The base set of a topology... |
| topontopi 23233 | A topology on a given base... |
| toponunii 23234 | The base set of a topology... |
| toptopon 23235 | Alternative definition of ... |
| toptopon2 23236 | A topology is the same thi... |
| topontopon 23237 | A topology on a set is a t... |
| funtopon 23238 | The class ` TopOn ` is a f... |
| toponrestid 23239 | Given a topology on a set,... |
| toponsspwpw 23240 | The set of topologies on a... |
| dmtopon 23241 | The domain of ` TopOn ` is... |
| fntopon 23242 | The class ` TopOn ` is a f... |
| toprntopon 23243 | A topology is the same thi... |
| toponmax 23244 | The base set of a topology... |
| toponss 23245 | A member of a topology is ... |
| toponcom 23246 | If ` K ` is a topology on ... |
| toponcomb 23247 | Biconditional form of ~ to... |
| topgele 23248 | The topologies over the sa... |
| topsn 23249 | The only topology on a sin... |
| istps 23252 | Express the predicate "is ... |
| istps2 23253 | Express the predicate "is ... |
| tpsuni 23254 | The base set of a topologi... |
| tpstop 23255 | The topology extractor on ... |
| tpspropd 23256 | A topological space depend... |
| tpsprop2d 23257 | A topological space depend... |
| topontopn 23258 | Express the predicate "is ... |
| tsettps 23259 | If the topology component ... |
| istpsi 23260 | Properties that determine ... |
| eltpsg 23261 | Properties that determine ... |
| eltpsi 23262 | Properties that determine ... |
| isbasisg 23265 | Express the predicate "the... |
| isbasis2g 23266 | Express the predicate "the... |
| isbasis3g 23267 | Express the predicate "the... |
| basis1 23268 | Property of a basis. (Con... |
| basis2 23269 | Property of a basis. (Con... |
| fiinbas 23270 | If a set is closed under f... |
| basdif0 23271 | A basis is not affected by... |
| baspartn 23272 | A disjoint system of sets ... |
| tgval 23273 | The topology generated by ... |
| tgval2 23274 | Definition of a topology g... |
| eltg 23275 | Membership in a topology g... |
| eltg2 23276 | Membership in a topology g... |
| eltg2b 23277 | Membership in a topology g... |
| eltg4i 23278 | An open set in a topology ... |
| eltg3i 23279 | The union of a set of basi... |
| eltg3 23280 | Membership in a topology g... |
| tgval3 23281 | Alternate expression for t... |
| tg1 23282 | Property of a member of a ... |
| tg2 23283 | Property of a member of a ... |
| bastg 23284 | A member of a basis is a s... |
| unitg 23285 | The topology generated by ... |
| tgss 23286 | Subset relation for genera... |
| tgcl 23287 | Show that a basis generate... |
| tgclb 23288 | The property ~ tgcl can be... |
| tgtopon 23289 | A basis generates a topolo... |
| topbas 23290 | A topology is its own basi... |
| tgtop 23291 | A topology is its own basi... |
| eltop 23292 | Membership in a topology, ... |
| eltop2 23293 | Membership in a topology. ... |
| eltop3 23294 | Membership in a topology. ... |
| fibas 23295 | A collection of finite int... |
| tgdom 23296 | A space has no more open s... |
| tgiun 23297 | The indexed union of a set... |
| tgidm 23298 | The topology generator fun... |
| bastop 23299 | Two ways to express that a... |
| tgtop11 23300 | The topology generation fu... |
| 0top 23301 | The singleton of the empty... |
| en1top 23302 | ` { (/) } ` is the only to... |
| en2top 23303 | If a topology has two elem... |
| tgss3 23304 | A criterion for determinin... |
| tgss2 23305 | A criterion for determinin... |
| basgen 23306 | Given a topology ` J ` , s... |
| basgen2 23307 | Given a topology ` J ` , s... |
| 2basgen 23308 | Conditions that determine ... |
| tgfiss 23309 | If a subbase is included i... |
| tgdif0 23310 | A generated topology is no... |
| bastop1 23311 | A subset of a topology is ... |
| bastop2 23312 | A version of ~ bastop1 tha... |
| distop 23313 | The discrete topology on a... |
| topnex 23314 | The class of all topologie... |
| distopon 23315 | The discrete topology on a... |
| sn0topon 23316 | The singleton of the empty... |
| sn0top 23317 | The singleton of the empty... |
| indislem 23318 | A lemma to eliminate some ... |
| indistopon 23319 | The indiscrete topology on... |
| indistop 23320 | The indiscrete topology on... |
| indisuni 23321 | The base set of the indisc... |
| fctop 23322 | The finite complement topo... |
| fctop2 23323 | The finite complement topo... |
| cctop 23324 | The countable complement t... |
| ppttop 23325 | The particular point topol... |
| pptbas 23326 | The particular point topol... |
| epttop 23327 | The excluded point topolog... |
| indistpsx 23328 | The indiscrete topology on... |
| indistps 23329 | The indiscrete topology on... |
| indistps2 23330 | The indiscrete topology on... |
| indistpsALT 23331 | The indiscrete topology on... |
| indistps2ALT 23332 | The indiscrete topology on... |
| distps 23333 | The discrete topology on a... |
| fncld 23340 | The closed-set generator i... |
| cldval 23341 | The set of closed sets of ... |
| ntrfval 23342 | The interior function on t... |
| clsfval 23343 | The closure function on th... |
| cldrcl 23344 | Reverse closure of the clo... |
| iscld 23345 | The predicate "the class `... |
| iscld2 23346 | A subset of the underlying... |
| cldss 23347 | A closed set is a subset o... |
| cldss2 23348 | The set of closed sets is ... |
| cldopn 23349 | The complement of a closed... |
| isopn2 23350 | A subset of the underlying... |
| opncld 23351 | The complement of an open ... |
| difopn 23352 | The difference of a closed... |
| topcld 23353 | The underlying set of a to... |
| ntrval 23354 | The interior of a subset o... |
| clsval 23355 | The closure of a subset of... |
| 0cld 23356 | The empty set is closed. ... |
| iincld 23357 | The indexed intersection o... |
| intcld 23358 | The intersection of a set ... |
| uncld 23359 | The union of two closed se... |
| cldcls 23360 | A closed subset equals its... |
| incld 23361 | The intersection of two cl... |
| riincld 23362 | An indexed relative inters... |
| iuncld 23363 | A finite indexed union of ... |
| unicld 23364 | A finite union of closed s... |
| clscld 23365 | The closure of a subset of... |
| clsf 23366 | The closure function is a ... |
| ntropn 23367 | The interior of a subset o... |
| clsval2 23368 | Express closure in terms o... |
| ntrval2 23369 | Interior expressed in term... |
| ntrdif 23370 | An interior of a complemen... |
| clsdif 23371 | A closure of a complement ... |
| clsss 23372 | Subset relationship for cl... |
| ntrss 23373 | Subset relationship for in... |
| sscls 23374 | A subset of a topology's u... |
| ntrss2 23375 | A subset includes its inte... |
| ssntr 23376 | An open subset of a set is... |
| clsss3 23377 | The closure of a subset of... |
| ntrss3 23378 | The interior of a subset o... |
| ntrin 23379 | A pairwise intersection of... |
| cmclsopn 23380 | The complement of a closur... |
| cmntrcld 23381 | The complement of an inter... |
| iscld3 23382 | A subset is closed iff it ... |
| iscld4 23383 | A subset is closed iff it ... |
| isopn3 23384 | A subset is open iff it eq... |
| clsidm 23385 | The closure operation is i... |
| ntridm 23386 | The interior operation is ... |
| clstop 23387 | The closure of a topology'... |
| ntrtop 23388 | The interior of a topology... |
| 0ntr 23389 | A subset with an empty int... |
| clsss2 23390 | If a subset is included in... |
| elcls 23391 | Membership in a closure. ... |
| elcls2 23392 | Membership in a closure. ... |
| clsndisj 23393 | Any open set containing a ... |
| ntrcls0 23394 | A subset whose closure has... |
| ntreq0 23395 | Two ways to say that a sub... |
| cldmre 23396 | The closed sets of a topol... |
| mrccls 23397 | Moore closure generalizes ... |
| cls0 23398 | The closure of the empty s... |
| ntr0 23399 | The interior of the empty ... |
| isopn3i 23400 | An open subset equals its ... |
| elcls3 23401 | Membership in a closure in... |
| opncldf1 23402 | A bijection useful for con... |
| opncldf2 23403 | The values of the open-clo... |
| opncldf3 23404 | The values of the converse... |
| isclo 23405 | A set ` A ` is clopen iff ... |
| isclo2 23406 | A set ` A ` is clopen iff ... |
| discld 23407 | The open sets of a discret... |
| sn0cld 23408 | The closed sets of the top... |
| indiscld 23409 | The closed sets of an indi... |
| mretopd 23410 | A Moore collection which i... |
| toponmre 23411 | The topologies over a give... |
| cldmreon 23412 | The closed sets of a topol... |
| iscldtop 23413 | A family is the closed set... |
| mreclatdemoBAD 23414 | The closed subspaces of a ... |
| neifval 23417 | Value of the neighborhood ... |
| neif 23418 | The neighborhood function ... |
| neiss2 23419 | A set with a neighborhood ... |
| neival 23420 | Value of the set of neighb... |
| isnei 23421 | The predicate "the class `... |
| neiint 23422 | An intuitive definition of... |
| isneip 23423 | The predicate "the class `... |
| neii1 23424 | A neighborhood is included... |
| neisspw 23425 | The neighborhoods of any s... |
| neii2 23426 | Property of a neighborhood... |
| neiss 23427 | Any neighborhood of a set ... |
| ssnei 23428 | A set is included in any o... |
| elnei 23429 | A point belongs to any of ... |
| 0nnei 23430 | The empty set is not a nei... |
| neips 23431 | A neighborhood of a set is... |
| opnneissb 23432 | An open set is a neighborh... |
| opnssneib 23433 | Any superset of an open se... |
| ssnei2 23434 | Any subset ` M ` of ` X ` ... |
| neindisj 23435 | Any neighborhood of an ele... |
| opnneiss 23436 | An open set is a neighborh... |
| opnneip 23437 | An open set is a neighborh... |
| opnnei 23438 | A set is open iff it is a ... |
| tpnei 23439 | The underlying set of a to... |
| neiuni 23440 | The union of the neighborh... |
| neindisj2 23441 | A point ` P ` belongs to t... |
| topssnei 23442 | A finer topology has more ... |
| innei 23443 | The intersection of two ne... |
| opnneiid 23444 | Only an open set is a neig... |
| neissex 23445 | For any neighborhood ` N `... |
| 0nei 23446 | The empty set is a neighbo... |
| neipeltop 23447 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23448 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23449 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23450 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23451 | If, to each element ` P ` ... |
| lpfval 23456 | The limit point function o... |
| lpval 23457 | The set of limit points of... |
| islp 23458 | The predicate "the class `... |
| lpsscls 23459 | The limit points of a subs... |
| lpss 23460 | The limit points of a subs... |
| lpdifsn 23461 | ` P ` is a limit point of ... |
| lpss3 23462 | Subset relationship for li... |
| islp2 23463 | The predicate " ` P ` is a... |
| islp3 23464 | The predicate " ` P ` is a... |
| maxlp 23465 | A point is a limit point o... |
| clslp 23466 | The closure of a subset of... |
| islpi 23467 | A point belonging to a set... |
| cldlp 23468 | A subset of a topological ... |
| isperf 23469 | Definition of a perfect sp... |
| isperf2 23470 | Definition of a perfect sp... |
| isperf3 23471 | A perfect space is a topol... |
| perflp 23472 | The limit points of a perf... |
| perfi 23473 | Property of a perfect spac... |
| perftop 23474 | A perfect space is a topol... |
| restrcl 23475 | Reverse closure for the su... |
| restbas 23476 | A subspace topology basis ... |
| tgrest 23477 | A subspace can be generate... |
| resttop 23478 | A subspace topology is a t... |
| resttopon 23479 | A subspace topology is a t... |
| restuni 23480 | The underlying set of a su... |
| stoig 23481 | The topological space buil... |
| restco 23482 | Composition of subspaces. ... |
| restabs 23483 | Equivalence of being a sub... |
| restin 23484 | When the subspace region i... |
| restuni2 23485 | The underlying set of a su... |
| resttopon2 23486 | The underlying set of a su... |
| rest0 23487 | The subspace topology indu... |
| restsn 23488 | The only subspace topology... |
| restsn2 23489 | The subspace topology indu... |
| restcld 23490 | A closed set of a subspace... |
| restcldi 23491 | A closed set is closed in ... |
| restcldr 23492 | A set which is closed in t... |
| restopnb 23493 | If ` B ` is an open subset... |
| ssrest 23494 | If ` K ` is a finer topolo... |
| restopn2 23495 | If ` A ` is open, then ` B... |
| restdis 23496 | A subspace of a discrete t... |
| restfpw 23497 | The restriction of the set... |
| neitr 23498 | The neighborhood of a trac... |
| restcls 23499 | A closure in a subspace to... |
| restntr 23500 | An interior in a subspace ... |
| restlp 23501 | The limit points of a subs... |
| restperf 23502 | Perfection of a subspace. ... |
| perfopn 23503 | An open subset of a perfec... |
| resstopn 23504 | The topology of a restrict... |
| resstps 23505 | A restricted topological s... |
| ordtbaslem 23506 | Lemma for ~ ordtbas . In ... |
| ordtval 23507 | Value of the order topolog... |
| ordtuni 23508 | Value of the order topolog... |
| ordtbas2 23509 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23510 | In a total order, the fini... |
| ordttopon 23511 | Value of the order topolog... |
| ordtopn1 23512 | An upward ray ` ( P , +oo ... |
| ordtopn2 23513 | A downward ray ` ( -oo , P... |
| ordtopn3 23514 | An open interval ` ( A , B... |
| ordtcld1 23515 | A downward ray ` ( -oo , P... |
| ordtcld2 23516 | An upward ray ` [ P , +oo ... |
| ordtcld3 23517 | A closed interval ` [ A , ... |
| ordttop 23518 | The order topology is a to... |
| ordtcnv 23519 | The order dual generates t... |
| ordtrest 23520 | The subspace topology of a... |
| ordtrest2lem 23521 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23522 | An interval-closed set ` A... |
| letopon 23523 | The topology of the extend... |
| letop 23524 | The topology of the extend... |
| letopuni 23525 | The topology of the extend... |
| xrstopn 23526 | The topology component of ... |
| xrstps 23527 | The extended real number s... |
| leordtvallem1 23528 | Lemma for ~ leordtval . (... |
| leordtvallem2 23529 | Lemma for ~ leordtval . (... |
| leordtval2 23530 | The topology of the extend... |
| leordtval 23531 | The topology of the extend... |
| iccordt 23532 | A closed interval is close... |
| iocpnfordt 23533 | An unbounded above open in... |
| icomnfordt 23534 | An unbounded above open in... |
| iooordt 23535 | An open interval is open i... |
| reordt 23536 | The real numbers are an op... |
| lecldbas 23537 | The set of closed interval... |
| pnfnei 23538 | A neighborhood of ` +oo ` ... |
| mnfnei 23539 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23540 | The restriction of the les... |
| ordtresticc 23541 | The restriction of the les... |
| lmrel 23548 | The topological space conv... |
| lmrcl 23549 | Reverse closure for the co... |
| lmfval 23550 | The relation "sequence ` f... |
| cnfval 23551 | The set of all continuous ... |
| cnpfval 23552 | The function mapping the p... |
| iscn 23553 | The predicate "the class `... |
| cnpval 23554 | The set of all functions f... |
| iscnp 23555 | The predicate "the class `... |
| iscn2 23556 | The predicate "the class `... |
| iscnp2 23557 | The predicate "the class `... |
| cntop1 23558 | Reverse closure for a cont... |
| cntop2 23559 | Reverse closure for a cont... |
| cnptop1 23560 | Reverse closure for a func... |
| cnptop2 23561 | Reverse closure for a func... |
| iscnp3 23562 | The predicate "the class `... |
| cnprcl 23563 | Reverse closure for a func... |
| cnf 23564 | A continuous function is a... |
| cnpf 23565 | A continuous function at p... |
| cnpcl 23566 | The value of a continuous ... |
| cnf2 23567 | A continuous function is a... |
| cnpf2 23568 | A continuous function at p... |
| cnprcl2 23569 | Reverse closure for a func... |
| tgcn 23570 | The continuity predicate w... |
| tgcnp 23571 | The "continuous at a point... |
| subbascn 23572 | The continuity predicate w... |
| ssidcn 23573 | The identity function is a... |
| cnpimaex 23574 | Property of a function con... |
| idcn 23575 | A restricted identity func... |
| lmbr 23576 | Express the binary relatio... |
| lmbr2 23577 | Express the binary relatio... |
| lmbrf 23578 | Express the binary relatio... |
| lmconst 23579 | A constant sequence conver... |
| lmcvg 23580 | Convergence property of a ... |
| iscnp4 23581 | The predicate "the class `... |
| cnpnei 23582 | A condition for continuity... |
| cnima 23583 | An open subset of the codo... |
| cnco 23584 | The composition of two con... |
| cnpco 23585 | The composition of a funct... |
| cnclima 23586 | A closed subset of the cod... |
| iscncl 23587 | A characterization of a co... |
| cncls2i 23588 | Property of the preimage o... |
| cnntri 23589 | Property of the preimage o... |
| cnclsi 23590 | Property of the image of a... |
| cncls2 23591 | Continuity in terms of clo... |
| cncls 23592 | Continuity in terms of clo... |
| cnntr 23593 | Continuity in terms of int... |
| cnss1 23594 | If the topology ` K ` is f... |
| cnss2 23595 | If the topology ` K ` is f... |
| cncnpi 23596 | A continuous function is c... |
| cnsscnp 23597 | The set of continuous func... |
| cncnp 23598 | A continuous function is c... |
| cncnp2 23599 | A continuous function is c... |
| cnnei 23600 | Continuity in terms of nei... |
| cnconst2 23601 | A constant function is con... |
| cnconst 23602 | A constant function is con... |
| cnrest 23603 | Continuity of a restrictio... |
| cnrest2 23604 | Equivalence of continuity ... |
| cnrest2r 23605 | Equivalence of continuity ... |
| cnpresti 23606 | One direction of ~ cnprest... |
| cnprest 23607 | Equivalence of continuity ... |
| cnprest2 23608 | Equivalence of point-conti... |
| cndis 23609 | Every function is continuo... |
| cnindis 23610 | Every function is continuo... |
| cnpdis 23611 | If ` A ` is an isolated po... |
| paste 23612 | Pasting lemma. If ` A ` a... |
| lmfpm 23613 | If ` F ` converges, then `... |
| lmfss 23614 | Inclusion of a function ha... |
| lmcl 23615 | Closure of a limit. (Cont... |
| lmss 23616 | Limit on a subspace. (Con... |
| sslm 23617 | A finer topology has fewer... |
| lmres 23618 | A function converges iff i... |
| lmff 23619 | If ` F ` converges, there ... |
| lmcls 23620 | Any convergent sequence of... |
| lmcld 23621 | Any convergent sequence of... |
| lmcnp 23622 | The image of a convergent ... |
| lmcn 23623 | The image of a convergent ... |
| ist0 23638 | The predicate "is a T_0 sp... |
| ist1 23639 | The predicate "is a T_1 sp... |
| ishaus 23640 | The predicate "is a Hausdo... |
| iscnrm 23641 | The property of being comp... |
| t0sep 23642 | Any two topologically indi... |
| t0dist 23643 | Any two distinct points in... |
| t1sncld 23644 | In a T_1 space, singletons... |
| t1ficld 23645 | In a T_1 space, finite set... |
| hausnei 23646 | Neighborhood property of a... |
| t0top 23647 | A T_0 space is a topologic... |
| t1top 23648 | A T_1 space is a topologic... |
| haustop 23649 | A Hausdorff space is a top... |
| isreg 23650 | The predicate "is a regula... |
| regtop 23651 | A regular space is a topol... |
| regsep 23652 | In a regular space, every ... |
| isnrm 23653 | The predicate "is a normal... |
| nrmtop 23654 | A normal space is a topolo... |
| cnrmtop 23655 | A completely normal space ... |
| iscnrm2 23656 | The property of being comp... |
| ispnrm 23657 | The property of being perf... |
| pnrmnrm 23658 | A perfectly normal space i... |
| pnrmtop 23659 | A perfectly normal space i... |
| pnrmcld 23660 | A closed set in a perfectl... |
| pnrmopn 23661 | An open set in a perfectly... |
| ist0-2 23662 | The predicate "is a T_0 sp... |
| ist0-3 23663 | The predicate "is a T_0 sp... |
| cnt0 23664 | The preimage of a T_0 topo... |
| ist1-2 23665 | An alternate characterizat... |
| t1t0 23666 | A T_1 space is a T_0 space... |
| ist1-3 23667 | A space is T_1 iff every p... |
| cnt1 23668 | The preimage of a T_1 topo... |
| ishaus2 23669 | Express the predicate " ` ... |
| haust1 23670 | A Hausdorff space is a T_1... |
| hausnei2 23671 | The Hausdorff condition st... |
| cnhaus 23672 | The preimage of a Hausdorf... |
| nrmsep3 23673 | In a normal space, given a... |
| nrmsep2 23674 | In a normal space, any two... |
| nrmsep 23675 | In a normal space, disjoin... |
| isnrm2 23676 | An alternate characterizat... |
| isnrm3 23677 | A topological space is nor... |
| cnrmi 23678 | A subspace of a completely... |
| cnrmnrm 23679 | A completely normal space ... |
| restcnrm 23680 | A subspace of a completely... |
| resthauslem 23681 | Lemma for ~ resthaus and s... |
| lpcls 23682 | The limit points of the cl... |
| perfcls 23683 | A subset of a perfect spac... |
| restt0 23684 | A subspace of a T_0 topolo... |
| restt1 23685 | A subspace of a T_1 topolo... |
| resthaus 23686 | A subspace of a Hausdorff ... |
| t1sep2 23687 | Any two points in a T_1 sp... |
| t1sep 23688 | Any two distinct points in... |
| sncld 23689 | A singleton is closed in a... |
| sshauslem 23690 | Lemma for ~ sshaus and sim... |
| sst0 23691 | A topology finer than a T_... |
| sst1 23692 | A topology finer than a T_... |
| sshaus 23693 | A topology finer than a Ha... |
| regsep2 23694 | In a regular space, a clos... |
| isreg2 23695 | A topological space is reg... |
| dnsconst 23696 | If a continuous mapping to... |
| ordtt1 23697 | The order topology is T_1 ... |
| lmmo 23698 | A sequence in a Hausdorff ... |
| lmfun 23699 | The convergence relation i... |
| dishaus 23700 | A discrete topology is Hau... |
| ordthauslem 23701 | Lemma for ~ ordthaus . (C... |
| ordthaus 23702 | The order topology of a to... |
| xrhaus 23703 | The topology of the extend... |
| iscmp 23706 | The predicate "is a compac... |
| cmpcov 23707 | An open cover of a compact... |
| cmpcov2 23708 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23709 | Combine ~ cmpcov with ~ ac... |
| cncmp 23710 | Compactness is respected b... |
| fincmp 23711 | A finite topology is compa... |
| 0cmp 23712 | The singleton of the empty... |
| cmptop 23713 | A compact topology is a to... |
| rncmp 23714 | The image of a compact set... |
| imacmp 23715 | The image of a compact set... |
| discmp 23716 | A discrete topology is com... |
| cmpsublem 23717 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23718 | Two equivalent ways of des... |
| tgcmp 23719 | A topology generated by a ... |
| cmpcld 23720 | A closed subset of a compa... |
| uncmp 23721 | The union of two compact s... |
| fiuncmp 23722 | A finite union of compact ... |
| sscmp 23723 | A subset of a compact topo... |
| hauscmplem 23724 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23725 | A compact subspace of a T2... |
| cmpfi 23726 | If a topology is compact a... |
| cmpfii 23727 | In a compact topology, a s... |
| bwth 23728 | The glorious Bolzano-Weier... |
| isconn 23731 | The predicate ` J ` is a c... |
| isconn2 23732 | The predicate ` J ` is a c... |
| connclo 23733 | The only nonempty clopen s... |
| conndisj 23734 | If a topology is connected... |
| conntop 23735 | A connected topology is a ... |
| indisconn 23736 | The indiscrete topology (o... |
| dfconn2 23737 | An alternate definition of... |
| connsuba 23738 | Connectedness for a subspa... |
| connsub 23739 | Two equivalent ways of say... |
| cnconn 23740 | Connectedness is respected... |
| nconnsubb 23741 | Disconnectedness for a sub... |
| connsubclo 23742 | If a clopen set meets a co... |
| connima 23743 | The image of a connected s... |
| conncn 23744 | A continuous function from... |
| iunconnlem 23745 | Lemma for ~ iunconn . (Co... |
| iunconn 23746 | The indexed union of conne... |
| unconn 23747 | The union of two connected... |
| clsconn 23748 | The closure of a connected... |
| conncompid 23749 | The connected component co... |
| conncompconn 23750 | The connected component co... |
| conncompss 23751 | The connected component co... |
| conncompcld 23752 | The connected component co... |
| conncompclo 23753 | The connected component co... |
| t1connperf 23754 | A connected T_1 space is p... |
| is1stc 23759 | The predicate "is a first-... |
| is1stc2 23760 | An equivalent way of sayin... |
| 1stctop 23761 | A first-countable topology... |
| 1stcclb 23762 | A property of points in a ... |
| 1stcfb 23763 | For any point ` A ` in a f... |
| is2ndc 23764 | The property of being seco... |
| 2ndctop 23765 | A second-countable topolog... |
| 2ndci 23766 | A countable basis generate... |
| 2ndcsb 23767 | Having a countable subbase... |
| 2ndcredom 23768 | A second-countable space h... |
| 2ndc1stc 23769 | A second-countable space i... |
| 1stcrestlem 23770 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23771 | A subspace of a first-coun... |
| 2ndcrest 23772 | A subspace of a second-cou... |
| abrexct 23773 | An image set of a countabl... |
| 2ndcctbss 23774 | If a topology is second-co... |
| 2ndcdisj 23775 | Any disjoint family of ope... |
| 2ndcdisj2 23776 | Any disjoint collection of... |
| 2ndcomap 23777 | A surjective continuous op... |
| 2ndcsep 23778 | A second-countable topolog... |
| dis2ndc 23779 | A discrete space is second... |
| 1stcelcls 23780 | A point belongs to the clo... |
| 1stccnp 23781 | A mapping is continuous at... |
| 1stccn 23782 | A mapping ` X --> Y ` , wh... |
| islly 23787 | The property of being a lo... |
| isnlly 23788 | The property of being an n... |
| llyeq 23789 | Equality theorem for the `... |
| nllyeq 23790 | Equality theorem for the `... |
| llytop 23791 | A locally ` A ` space is a... |
| nllytop 23792 | A locally ` A ` space is a... |
| llyi 23793 | The property of a locally ... |
| nllyi 23794 | The property of an n-local... |
| nlly2i 23795 | Eliminate the neighborhood... |
| llynlly 23796 | A locally ` A ` space is n... |
| llyssnlly 23797 | A locally ` A ` space is n... |
| llyss 23798 | The "locally" predicate re... |
| nllyss 23799 | The "n-locally" predicate ... |
| subislly 23800 | The property of a subspace... |
| restnlly 23801 | If the property ` A ` pass... |
| restlly 23802 | If the property ` A ` pass... |
| islly2 23803 | An alternative expression ... |
| llyrest 23804 | An open subspace of a loca... |
| nllyrest 23805 | An open subspace of an n-l... |
| loclly 23806 | If ` A ` is a local proper... |
| llyidm 23807 | Idempotence of the "locall... |
| nllyidm 23808 | Idempotence of the "n-loca... |
| toplly 23809 | A topology is locally a to... |
| topnlly 23810 | A topology is n-locally a ... |
| hauslly 23811 | A Hausdorff space is local... |
| hausnlly 23812 | A Hausdorff space is n-loc... |
| hausllycmp 23813 | A compact Hausdorff space ... |
| cldllycmp 23814 | A closed subspace of a loc... |
| lly1stc 23815 | First-countability is a lo... |
| dislly 23816 | The discrete space ` ~P X ... |
| disllycmp 23817 | A discrete space is locall... |
| dis1stc 23818 | A discrete space is first-... |
| hausmapdom 23819 | If ` X ` is a first-counta... |
| hauspwdom 23820 | Simplify the cardinal ` A ... |
| refrel 23827 | Refinement is a relation. ... |
| isref 23828 | The property of being a re... |
| refbas 23829 | A refinement covers the sa... |
| refssex 23830 | Every set in a refinement ... |
| ssref 23831 | A subcover is a refinement... |
| refref 23832 | Reflexivity of refinement.... |
| reftr 23833 | Refinement is transitive. ... |
| refun0 23834 | Adding the empty set prese... |
| isptfin 23835 | The statement "is a point-... |
| islocfin 23836 | The statement "is a locall... |
| finptfin 23837 | A finite cover is a point-... |
| ptfinfin 23838 | A point covered by a point... |
| finlocfin 23839 | A finite cover of a topolo... |
| locfintop 23840 | A locally finite cover cov... |
| locfinbas 23841 | A locally finite cover mus... |
| locfinnei 23842 | A point covered by a local... |
| lfinpfin 23843 | A locally finite cover is ... |
| lfinun 23844 | Adding a finite set preser... |
| locfincmp 23845 | For a compact space, the l... |
| unisngl 23846 | Taking the union of the se... |
| dissnref 23847 | The set of singletons is a... |
| dissnlocfin 23848 | The set of singletons is l... |
| locfindis 23849 | The locally finite covers ... |
| locfincf 23850 | A locally finite cover in ... |
| comppfsc 23851 | A space where every open c... |
| kgenval 23854 | Value of the compact gener... |
| elkgen 23855 | Value of the compact gener... |
| kgeni 23856 | Property of the open sets ... |
| kgentopon 23857 | The compact generator gene... |
| kgenuni 23858 | The base set of the compac... |
| kgenftop 23859 | The compact generator gene... |
| kgenf 23860 | The compact generator is a... |
| kgentop 23861 | A compactly generated spac... |
| kgenss 23862 | The compact generator gene... |
| kgenhaus 23863 | The compact generator gene... |
| kgencmp 23864 | The compact generator topo... |
| kgencmp2 23865 | The compact generator topo... |
| kgenidm 23866 | The compact generator is i... |
| iskgen2 23867 | A space is compactly gener... |
| iskgen3 23868 | Derive the usual definitio... |
| llycmpkgen2 23869 | A locally compact space is... |
| cmpkgen 23870 | A compact space is compact... |
| llycmpkgen 23871 | A locally compact space is... |
| 1stckgenlem 23872 | The one-point compactifica... |
| 1stckgen 23873 | A first-countable space is... |
| kgen2ss 23874 | The compact generator pres... |
| kgencn 23875 | A function from a compactl... |
| kgencn2 23876 | A function ` F : J --> K `... |
| kgencn3 23877 | The set of continuous func... |
| kgen2cn 23878 | A continuous function is a... |
| txval 23883 | Value of the binary topolo... |
| txuni2 23884 | The underlying set of the ... |
| txbasex 23885 | The basis for the product ... |
| txbas 23886 | The set of Cartesian produ... |
| eltx 23887 | A set in a product is open... |
| txtop 23888 | The product of two topolog... |
| ptval 23889 | The value of the product t... |
| ptpjpre1 23890 | The preimage of a projecti... |
| elpt 23891 | Elementhood in the bases o... |
| elptr 23892 | A basic open set in the pr... |
| elptr2 23893 | A basic open set in the pr... |
| ptbasid 23894 | The base set of the produc... |
| ptuni2 23895 | The base set for the produ... |
| ptbasin 23896 | The basis for a product to... |
| ptbasin2 23897 | The basis for a product to... |
| ptbas 23898 | The basis for a product to... |
| ptpjpre2 23899 | The basis for a product to... |
| ptbasfi 23900 | The basis for the product ... |
| pttop 23901 | The product topology is a ... |
| ptopn 23902 | A basic open set in the pr... |
| ptopn2 23903 | A sub-basic open set in th... |
| xkotf 23904 | Functionality of function ... |
| xkobval 23905 | Alternative expression for... |
| xkoval 23906 | Value of the compact-open ... |
| xkotop 23907 | The compact-open topology ... |
| xkoopn 23908 | A basic open set of the co... |
| txtopi 23909 | The product of two topolog... |
| txtopon 23910 | The underlying set of the ... |
| txuni 23911 | The underlying set of the ... |
| txunii 23912 | The underlying set of the ... |
| ptuni 23913 | The base set for the produ... |
| ptunimpt 23914 | Base set of a product topo... |
| pttopon 23915 | The base set for the produ... |
| pttoponconst 23916 | The base set for a product... |
| ptuniconst 23917 | The base set for a product... |
| xkouni 23918 | The base set of the compac... |
| xkotopon 23919 | The base set of the compac... |
| ptval2 23920 | The value of the product t... |
| txopn 23921 | The product of two open se... |
| txcld 23922 | The product of two closed ... |
| txcls 23923 | Closure of a rectangle in ... |
| txss12 23924 | Subset property of the top... |
| txbasval 23925 | It is sufficient to consid... |
| neitx 23926 | The Cartesian product of t... |
| txcnpi 23927 | Continuity of a two-argume... |
| tx1cn 23928 | Continuity of the first pr... |
| tx2cn 23929 | Continuity of the second p... |
| ptpjcn 23930 | Continuity of a projection... |
| ptpjopn 23931 | The projection map is an o... |
| ptcld 23932 | A closed box in the produc... |
| ptcldmpt 23933 | A closed box in the produc... |
| ptclsg 23934 | The closure of a box in th... |
| ptcls 23935 | The closure of a box in th... |
| dfac14lem 23936 | Lemma for ~ dfac14 . By e... |
| dfac14 23937 | Theorem ~ ptcls is an equi... |
| xkoccn 23938 | The "constant function" fu... |
| txcnp 23939 | If two functions are conti... |
| ptcnplem 23940 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23941 | If every projection of a f... |
| upxp 23942 | Universal property of the ... |
| txcnmpt 23943 | A map into the product of ... |
| uptx 23944 | Universal property of the ... |
| txcn 23945 | A map into the product of ... |
| ptcn 23946 | If every projection of a f... |
| prdstopn 23947 | Topology of a structure pr... |
| prdstps 23948 | A structure product of top... |
| pwstps 23949 | A structure power of a top... |
| txrest 23950 | The subspace of a topologi... |
| txdis 23951 | The topological product of... |
| txindislem 23952 | Lemma for ~ txindis . (Co... |
| txindis 23953 | The topological product of... |
| txdis1cn 23954 | A function is jointly cont... |
| txlly 23955 | If the property ` A ` is p... |
| txnlly 23956 | If the property ` A ` is p... |
| pthaus 23957 | The product of a collectio... |
| ptrescn 23958 | Restriction is a continuou... |
| txtube 23959 | The "tube lemma". If ` X ... |
| txcmplem1 23960 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23961 | Lemma for ~ txcmp . (Cont... |
| txcmp 23962 | The topological product of... |
| txcmpb 23963 | The topological product of... |
| hausdiag 23964 | A topology is Hausdorff if... |
| hauseqlcld 23965 | In a Hausdorff topology, t... |
| txhaus 23966 | The topological product of... |
| txlm 23967 | Two sequences converge iff... |
| lmcn2 23968 | The image of a convergent ... |
| tx1stc 23969 | The topological product of... |
| tx2ndc 23970 | The topological product of... |
| txkgen 23971 | The topological product of... |
| xkohaus 23972 | If the codomain space is H... |
| xkoptsub 23973 | The compact-open topology ... |
| xkopt 23974 | The compact-open topology ... |
| xkopjcn 23975 | Continuity of a projection... |
| xkoco1cn 23976 | If ` F ` is a continuous f... |
| xkoco2cn 23977 | If ` F ` is a continuous f... |
| xkococnlem 23978 | Continuity of the composit... |
| xkococn 23979 | Continuity of the composit... |
| cnmptid 23980 | The identity function is c... |
| cnmptc 23981 | A constant function is con... |
| cnmpt11 23982 | The composition of continu... |
| cnmpt11f 23983 | The composition of continu... |
| cnmpt1t 23984 | The composition of continu... |
| cnmpt12f 23985 | The composition of continu... |
| cnmpt12 23986 | The composition of continu... |
| cnmpt1st 23987 | The projection onto the fi... |
| cnmpt2nd 23988 | The projection onto the se... |
| cnmpt2c 23989 | A constant function is con... |
| cnmpt21 23990 | The composition of continu... |
| cnmpt21f 23991 | The composition of continu... |
| cnmpt2t 23992 | The composition of continu... |
| cnmpt22 23993 | The composition of continu... |
| cnmpt22f 23994 | The composition of continu... |
| cnmpt1res 23995 | The restriction of a conti... |
| cnmpt2res 23996 | The restriction of a conti... |
| cnmptcom 23997 | The argument converse of a... |
| cnmptkc 23998 | The curried first projecti... |
| cnmptkp 23999 | The evaluation of the inne... |
| cnmptk1 24000 | The composition of a curri... |
| cnmpt1k 24001 | The composition of a one-a... |
| cnmptkk 24002 | The composition of two cur... |
| xkofvcn 24003 | Joint continuity of the fu... |
| cnmptk1p 24004 | The evaluation of a currie... |
| cnmptk2 24005 | The uncurrying of a currie... |
| xkoinjcn 24006 | Continuity of "injection",... |
| cnmpt2k 24007 | The currying of a two-argu... |
| txconn 24008 | The topological product of... |
| imasnopn 24009 | If a relation graph is ope... |
| imasncld 24010 | If a relation graph is clo... |
| imasncls 24011 | If a relation graph is clo... |
| qtopval 24014 | Value of the quotient topo... |
| qtopval2 24015 | Value of the quotient topo... |
| elqtop 24016 | Value of the quotient topo... |
| qtopres 24017 | The quotient topology is u... |
| qtoptop2 24018 | The quotient topology is a... |
| qtoptop 24019 | The quotient topology is a... |
| elqtop2 24020 | Value of the quotient topo... |
| qtopuni 24021 | The base set of the quotie... |
| elqtop3 24022 | Value of the quotient topo... |
| qtoptopon 24023 | The base set of the quotie... |
| qtopid 24024 | A quotient map is a contin... |
| idqtop 24025 | The quotient topology indu... |
| qtopcmplem 24026 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 24027 | A quotient of a compact sp... |
| qtopconn 24028 | A quotient of a connected ... |
| qtopkgen 24029 | A quotient of a compactly ... |
| basqtop 24030 | An injection maps bases to... |
| tgqtop 24031 | An injection maps generate... |
| qtopcld 24032 | The property of being a cl... |
| qtopcn 24033 | Universal property of a qu... |
| qtopss 24034 | A surjective continuous fu... |
| qtopeu 24035 | Universal property of the ... |
| qtoprest 24036 | If ` A ` is a saturated op... |
| qtopomap 24037 | If ` F ` is a surjective c... |
| qtopcmap 24038 | If ` F ` is a surjective c... |
| imastopn 24039 | The topology of an image s... |
| imastps 24040 | The image of a topological... |
| qustps 24041 | A quotient structure is a ... |
| kqfval 24042 | Value of the function appe... |
| kqfeq 24043 | Two points in the Kolmogor... |
| kqffn 24044 | The topological indistingu... |
| kqval 24045 | Value of the quotient topo... |
| kqtopon 24046 | The Kolmogorov quotient is... |
| kqid 24047 | The topological indistingu... |
| ist0-4 24048 | The topological indistingu... |
| kqfvima 24049 | When the image set is open... |
| kqsat 24050 | Any open set is saturated ... |
| kqdisj 24051 | A version of ~ imain for t... |
| kqcldsat 24052 | Any closed set is saturate... |
| kqopn 24053 | The topological indistingu... |
| kqcld 24054 | The topological indistingu... |
| kqt0lem 24055 | Lemma for ~ kqt0 . (Contr... |
| isr0 24056 | The property " ` J ` is an... |
| r0cld 24057 | The analogue of the T_1 ax... |
| regr1lem 24058 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 24059 | A Kolmogorov quotient of a... |
| kqreglem1 24060 | A Kolmogorov quotient of a... |
| kqreglem2 24061 | If the Kolmogorov quotient... |
| kqnrmlem1 24062 | A Kolmogorov quotient of a... |
| kqnrmlem2 24063 | If the Kolmogorov quotient... |
| kqtop 24064 | The Kolmogorov quotient is... |
| kqt0 24065 | The Kolmogorov quotient is... |
| kqf 24066 | The Kolmogorov quotient is... |
| r0sep 24067 | The separation property of... |
| nrmr0reg 24068 | A normal R_0 space is also... |
| regr1 24069 | A regular space is R_1, wh... |
| kqreg 24070 | The Kolmogorov quotient of... |
| kqnrm 24071 | The Kolmogorov quotient of... |
| hmeofn 24076 | The set of homeomorphisms ... |
| hmeofval 24077 | The set of all the homeomo... |
| ishmeo 24078 | The predicate F is a homeo... |
| hmeocn 24079 | A homeomorphism is continu... |
| hmeocnvcn 24080 | The converse of a homeomor... |
| hmeocnv 24081 | The converse of a homeomor... |
| hmeof1o2 24082 | A homeomorphism is a 1-1-o... |
| hmeof1o 24083 | A homeomorphism is a 1-1-o... |
| hmeoima 24084 | The image of an open set b... |
| hmeoopn 24085 | Homeomorphisms preserve op... |
| hmeocld 24086 | Homeomorphisms preserve cl... |
| hmeocls 24087 | Homeomorphisms preserve cl... |
| hmeontr 24088 | Homeomorphisms preserve in... |
| hmeoimaf1o 24089 | The function mapping open ... |
| hmeores 24090 | The restriction of a homeo... |
| hmeoco 24091 | The composite of two homeo... |
| idhmeo 24092 | The identity function is a... |
| hmeocnvb 24093 | The converse of a homeomor... |
| hmeoqtop 24094 | A homeomorphism is a quoti... |
| hmph 24095 | Express the predicate ` J ... |
| hmphi 24096 | If there is a homeomorphis... |
| hmphtop 24097 | Reverse closure for the ho... |
| hmphtop1 24098 | The relation "being homeom... |
| hmphtop2 24099 | The relation "being homeom... |
| hmphref 24100 | "Is homeomorphic to" is re... |
| hmphsym 24101 | "Is homeomorphic to" is sy... |
| hmphtr 24102 | "Is homeomorphic to" is tr... |
| hmpher 24103 | "Is homeomorphic to" is an... |
| hmphen 24104 | Homeomorphisms preserve th... |
| hmphsymb 24105 | "Is homeomorphic to" is sy... |
| haushmphlem 24106 | Lemma for ~ haushmph and s... |
| cmphmph 24107 | Compactness is a topologic... |
| connhmph 24108 | Connectedness is a topolog... |
| t0hmph 24109 | T_0 is a topological prope... |
| t1hmph 24110 | T_1 is a topological prope... |
| haushmph 24111 | Hausdorff-ness is a topolo... |
| reghmph 24112 | Regularity is a topologica... |
| nrmhmph 24113 | Normality is a topological... |
| hmph0 24114 | A topology homeomorphic to... |
| hmphdis 24115 | Homeomorphisms preserve to... |
| hmphindis 24116 | Homeomorphisms preserve to... |
| indishmph 24117 | Equinumerous sets equipped... |
| hmphen2 24118 | Homeomorphisms preserve th... |
| cmphaushmeo 24119 | A continuous bijection fro... |
| ordthmeolem 24120 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 24121 | An order isomorphism is a ... |
| txhmeo 24122 | Lift a pair of homeomorphi... |
| txswaphmeolem 24123 | Show inverse for the "swap... |
| txswaphmeo 24124 | There is a homeomorphism f... |
| pt1hmeo 24125 | The canonical homeomorphis... |
| ptuncnv 24126 | Exhibit the converse funct... |
| ptunhmeo 24127 | Define a homeomorphism fro... |
| xpstopnlem1 24128 | The function ` F ` used in... |
| xpstps 24129 | A binary product of topolo... |
| xpstopnlem2 24130 | Lemma for ~ xpstopn . (Co... |
| xpstopn 24131 | The topology on a binary p... |
| ptcmpfi 24132 | A topological product of f... |
| xkocnv 24133 | The inverse of the "curryi... |
| xkohmeo 24134 | The Exponential Law for to... |
| qtopf1 24135 | If a quotient map is injec... |
| qtophmeo 24136 | If two functions on a base... |
| t0kq 24137 | A topological space is T_0... |
| kqhmph 24138 | A topological space is T_0... |
| ist1-5lem 24139 | Lemma for ~ ist1-5 and sim... |
| t1r0 24140 | A T_1 space is R_0. That ... |
| ist1-5 24141 | A topological space is T_1... |
| ishaus3 24142 | A topological space is Hau... |
| nrmreg 24143 | A normal T_1 space is regu... |
| reghaus 24144 | A regular T_0 space is Hau... |
| nrmhaus 24145 | A T_1 normal space is Haus... |
| elmptrab 24146 | Membership in a one-parame... |
| elmptrab2 24147 | Membership in a one-parame... |
| isfbas 24148 | The predicate " ` F ` is a... |
| fbasne0 24149 | There are no empty filter ... |
| 0nelfb 24150 | No filter base contains th... |
| fbsspw 24151 | A filter base on a set is ... |
| fbelss 24152 | An element of the filter b... |
| fbdmn0 24153 | The domain of a filter bas... |
| isfbas2 24154 | The predicate " ` F ` is a... |
| fbasssin 24155 | A filter base contains sub... |
| fbssfi 24156 | A filter base contains sub... |
| fbssint 24157 | A filter base contains sub... |
| fbncp 24158 | A filter base does not con... |
| fbun 24159 | A necessary and sufficient... |
| fbfinnfr 24160 | No filter base containing ... |
| opnfbas 24161 | The collection of open sup... |
| trfbas2 24162 | Conditions for the trace o... |
| trfbas 24163 | Conditions for the trace o... |
| isfil 24166 | The predicate "is a filter... |
| filfbas 24167 | A filter is a filter base.... |
| 0nelfil 24168 | The empty set doesn't belo... |
| fileln0 24169 | An element of a filter is ... |
| filsspw 24170 | A filter is a subset of th... |
| filelss 24171 | An element of a filter is ... |
| filss 24172 | A filter is closed under t... |
| filin 24173 | A filter is closed under t... |
| filtop 24174 | The underlying set belongs... |
| isfil2 24175 | Derive the standard axioms... |
| isfildlem 24176 | Lemma for ~ isfild . (Con... |
| isfild 24177 | Sufficient condition for a... |
| filfi 24178 | A filter is closed under t... |
| filinn0 24179 | The intersection of two el... |
| filintn0 24180 | A filter has the finite in... |
| filn0 24181 | The empty set is not a fil... |
| infil 24182 | The intersection of two fi... |
| snfil 24183 | A singleton is a filter. ... |
| fbasweak 24184 | A filter base on any set i... |
| snfbas 24185 | Condition for a singleton ... |
| fsubbas 24186 | A condition for a set to g... |
| fbasfip 24187 | A filter base has the fini... |
| fbunfip 24188 | A helpful lemma for showin... |
| fgval 24189 | The filter generating clas... |
| elfg 24190 | A condition for elements o... |
| ssfg 24191 | A filter base is a subset ... |
| fgss 24192 | A bigger base generates a ... |
| fgss2 24193 | A condition for a filter t... |
| fgfil 24194 | A filter generates itself.... |
| elfilss 24195 | An element belongs to a fi... |
| filfinnfr 24196 | No filter containing a fin... |
| fgcl 24197 | A generated filter is a fi... |
| fgabs 24198 | Absorption law for filter ... |
| neifil 24199 | The neighborhoods of a non... |
| filunibas 24200 | Recover the base set from ... |
| filunirn 24201 | Two ways to express a filt... |
| filconn 24202 | A filter gives rise to a c... |
| fbasrn 24203 | Given a filter on a domain... |
| filuni 24204 | The union of a nonempty se... |
| trfil1 24205 | Conditions for the trace o... |
| trfil2 24206 | Conditions for the trace o... |
| trfil3 24207 | Conditions for the trace o... |
| trfilss 24208 | If ` A ` is a member of th... |
| fgtr 24209 | If ` A ` is a member of th... |
| trfg 24210 | The trace operation and th... |
| trnei 24211 | The trace, over a set ` A ... |
| cfinfil 24212 | Relative complements of th... |
| csdfil 24213 | The set of all elements wh... |
| supfil 24214 | The supersets of a nonempt... |
| zfbas 24215 | The set of upper sets of i... |
| uzrest 24216 | The restriction of the set... |
| uzfbas 24217 | The set of upper sets of i... |
| isufil 24222 | The property of being an u... |
| ufilfil 24223 | An ultrafilter is a filter... |
| ufilss 24224 | For any subset of the base... |
| ufilb 24225 | The complement is in an ul... |
| ufilmax 24226 | Any filter finer than an u... |
| isufil2 24227 | The maximal property of an... |
| ufprim 24228 | An ultrafilter is a prime ... |
| trufil 24229 | Conditions for the trace o... |
| filssufilg 24230 | A filter is contained in s... |
| filssufil 24231 | A filter is contained in s... |
| isufl 24232 | Define the (strong) ultraf... |
| ufli 24233 | Property of a set that sat... |
| numufl 24234 | Consequence of ~ filssufil... |
| fiufl 24235 | A finite set satisfies the... |
| acufl 24236 | The axiom of choice implie... |
| ssufl 24237 | If ` Y ` is a subset of ` ... |
| ufileu 24238 | If the ultrafilter contain... |
| filufint 24239 | A filter is equal to the i... |
| uffix 24240 | Lemma for ~ fixufil and ~ ... |
| fixufil 24241 | The condition describing a... |
| uffixfr 24242 | An ultrafilter is either f... |
| uffix2 24243 | A classification of fixed ... |
| uffixsn 24244 | The singleton of the gener... |
| ufildom1 24245 | An ultrafilter is generate... |
| uffinfix 24246 | An ultrafilter containing ... |
| cfinufil 24247 | An ultrafilter is free iff... |
| ufinffr 24248 | An infinite subset is cont... |
| ufilen 24249 | Any infinite set has an ul... |
| ufildr 24250 | An ultrafilter gives rise ... |
| fin1aufil 24251 | There are no definable fre... |
| fmval 24262 | Introduce a function that ... |
| fmfil 24263 | A mapping filter is a filt... |
| fmf 24264 | Pushing-forward via a func... |
| fmss 24265 | A finer filter produces a ... |
| elfm 24266 | An element of a mapping fi... |
| elfm2 24267 | An element of a mapping fi... |
| fmfg 24268 | The image filter of a filt... |
| elfm3 24269 | An alternate formulation o... |
| imaelfm 24270 | An image of a filter eleme... |
| rnelfmlem 24271 | Lemma for ~ rnelfm . (Con... |
| rnelfm 24272 | A condition for a filter t... |
| fmfnfmlem1 24273 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 24274 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 24275 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 24276 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 24277 | A filter finer than an ima... |
| fmufil 24278 | An image filter of an ultr... |
| fmid 24279 | The filter map applied to ... |
| fmco 24280 | Composition of image filte... |
| ufldom 24281 | The ultrafilter lemma prop... |
| flimval 24282 | The set of limit points of... |
| elflim2 24283 | The predicate "is a limit ... |
| flimtop 24284 | Reverse closure for the li... |
| flimneiss 24285 | A filter contains the neig... |
| flimnei 24286 | A filter contains all of t... |
| flimelbas 24287 | A limit point of a filter ... |
| flimfil 24288 | Reverse closure for the li... |
| flimtopon 24289 | Reverse closure for the li... |
| elflim 24290 | The predicate "is a limit ... |
| flimss2 24291 | A limit point of a filter ... |
| flimss1 24292 | A limit point of a filter ... |
| neiflim 24293 | A point is a limit point o... |
| flimopn 24294 | The condition for being a ... |
| fbflim 24295 | A condition for a filter t... |
| fbflim2 24296 | A condition for a filter b... |
| flimclsi 24297 | The convergent points of a... |
| hausflimlem 24298 | If ` A ` and ` B ` are bot... |
| hausflimi 24299 | One direction of ~ hausfli... |
| hausflim 24300 | A condition for a topology... |
| flimcf 24301 | Fineness is properly chara... |
| flimrest 24302 | The set of limit points in... |
| flimclslem 24303 | Lemma for ~ flimcls . (Co... |
| flimcls 24304 | Closure in terms of filter... |
| flimsncls 24305 | If ` A ` is a limit point ... |
| hauspwpwf1 24306 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 24307 | If ` X ` is a Hausdorff sp... |
| flffval 24308 | Given a topology and a fil... |
| flfval 24309 | Given a function from a fi... |
| flfnei 24310 | The property of being a li... |
| flfneii 24311 | A neighborhood of a limit ... |
| isflf 24312 | The property of being a li... |
| flfelbas 24313 | A limit point of a functio... |
| flffbas 24314 | Limit points of a function... |
| flftg 24315 | Limit points of a function... |
| hausflf 24316 | If a function has its valu... |
| hausflf2 24317 | If a convergent function h... |
| cnpflfi 24318 | Forward direction of ~ cnp... |
| cnpflf2 24319 | ` F ` is continuous at poi... |
| cnpflf 24320 | Continuity of a function a... |
| cnflf 24321 | A function is continuous i... |
| cnflf2 24322 | A function is continuous i... |
| flfcnp 24323 | A continuous function pres... |
| lmflf 24324 | The topological limit rela... |
| txflf 24325 | Two sequences converge in ... |
| flfcnp2 24326 | The image of a convergent ... |
| fclsval 24327 | The set of all cluster poi... |
| isfcls 24328 | A cluster point of a filte... |
| fclsfil 24329 | Reverse closure for the cl... |
| fclstop 24330 | Reverse closure for the cl... |
| fclstopon 24331 | Reverse closure for the cl... |
| isfcls2 24332 | A cluster point of a filte... |
| fclsopn 24333 | Write the cluster point co... |
| fclsopni 24334 | An open neighborhood of a ... |
| fclselbas 24335 | A cluster point is in the ... |
| fclsneii 24336 | A neighborhood of a cluste... |
| fclssscls 24337 | The set of cluster points ... |
| fclsnei 24338 | Cluster points in terms of... |
| supnfcls 24339 | The filter of supersets of... |
| fclsbas 24340 | Cluster points in terms of... |
| fclsss1 24341 | A finer topology has fewer... |
| fclsss2 24342 | A finer filter has fewer c... |
| fclsrest 24343 | The set of cluster points ... |
| fclscf 24344 | Characterization of finene... |
| flimfcls 24345 | A limit point is a cluster... |
| fclsfnflim 24346 | A filter clusters at a poi... |
| flimfnfcls 24347 | A filter converges to a po... |
| fclscmpi 24348 | Forward direction of ~ fcl... |
| fclscmp 24349 | A space is compact iff eve... |
| uffclsflim 24350 | The cluster points of an u... |
| ufilcmp 24351 | A space is compact iff eve... |
| fcfval 24352 | The set of cluster points ... |
| isfcf 24353 | The property of being a cl... |
| fcfnei 24354 | The property of being a cl... |
| fcfelbas 24355 | A cluster point of a funct... |
| fcfneii 24356 | A neighborhood of a cluste... |
| flfssfcf 24357 | A limit point of a functio... |
| uffcfflf 24358 | If the domain filter is an... |
| cnpfcfi 24359 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24360 | A function ` F ` is contin... |
| cnfcf 24361 | Continuity of a function i... |
| flfcntr 24362 | A continuous function's va... |
| alexsublem 24363 | Lemma for ~ alexsub . (Co... |
| alexsub 24364 | The Alexander Subbase Theo... |
| alexsubb 24365 | Biconditional form of the ... |
| alexsubALTlem1 24366 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24367 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24368 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24369 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24370 | The Alexander Subbase Theo... |
| ptcmplem1 24371 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24372 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24373 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24374 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24375 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24376 | Tychonoff's theorem: The ... |
| ptcmp 24377 | Tychonoff's theorem: The ... |
| cnextval 24380 | The function applying cont... |
| cnextfval 24381 | The continuous extension o... |
| cnextrel 24382 | In the general case, a con... |
| cnextfun 24383 | If the target space is Hau... |
| cnextfvval 24384 | The value of the continuou... |
| cnextf 24385 | Extension by continuity. ... |
| cnextcn 24386 | Extension by continuity. ... |
| cnextfres1 24387 | ` F ` and its extension by... |
| cnextfres 24388 | ` F ` and its extension by... |
| istmd 24393 | The predicate "is a topolo... |
| tmdmnd 24394 | A topological monoid is a ... |
| tmdtps 24395 | A topological monoid is a ... |
| istgp 24396 | The predicate "is a topolo... |
| tgpgrp 24397 | A topological group is a g... |
| tgptmd 24398 | A topological group is a t... |
| tgptps 24399 | A topological group is a t... |
| tmdtopon 24400 | The topology of a topologi... |
| tgptopon 24401 | The topology of a topologi... |
| tmdcn 24402 | In a topological monoid, t... |
| tgpcn 24403 | In a topological group, th... |
| tgpinv 24404 | In a topological group, th... |
| grpinvhmeo 24405 | The inverse function in a ... |
| cnmpt1plusg 24406 | Continuity of the group su... |
| cnmpt2plusg 24407 | Continuity of the group su... |
| tmdcn2 24408 | Write out the definition o... |
| tgpsubcn 24409 | In a topological group, th... |
| istgp2 24410 | A group with a topology is... |
| tmdmulg 24411 | In a topological monoid, t... |
| tgpmulg 24412 | In a topological group, th... |
| tgpmulg2 24413 | In a topological monoid, t... |
| tmdgsum 24414 | In a topological monoid, t... |
| tmdgsum2 24415 | For any neighborhood ` U `... |
| oppgtmd 24416 | The opposite of a topologi... |
| oppgtgp 24417 | The opposite of a topologi... |
| distgp 24418 | Any group equipped with th... |
| indistgp 24419 | Any group equipped with th... |
| efmndtmd 24420 | The monoid of endofunction... |
| tmdlactcn 24421 | The left group action of e... |
| tgplacthmeo 24422 | The left group action of e... |
| submtmd 24423 | A submonoid of a topologic... |
| subgtgp 24424 | A subgroup of a topologica... |
| symgtgp 24425 | The symmetric group is a t... |
| subgntr 24426 | A subgroup of a topologica... |
| opnsubg 24427 | An open subgroup of a topo... |
| clssubg 24428 | The closure of a subgroup ... |
| clsnsg 24429 | The closure of a normal su... |
| cldsubg 24430 | A subgroup of finite index... |
| tgpconncompeqg 24431 | The connected component co... |
| tgpconncomp 24432 | The identity component, th... |
| tgpconncompss 24433 | The identity component is ... |
| ghmcnp 24434 | A group homomorphism on to... |
| snclseqg 24435 | The coset of the closure o... |
| tgphaus 24436 | A topological group is Hau... |
| tgpt1 24437 | Hausdorff and T1 are equiv... |
| tgpt0 24438 | Hausdorff and T0 are equiv... |
| qustgpopn 24439 | A quotient map in a topolo... |
| qustgplem 24440 | Lemma for ~ qustgp . (Con... |
| qustgp 24441 | The quotient of a topologi... |
| qustgphaus 24442 | The quotient of a topologi... |
| prdstmdd 24443 | The product of a family of... |
| prdstgpd 24444 | The product of a family of... |
| tsmsfbas 24447 | The collection of all sets... |
| tsmslem1 24448 | The finite partial sums of... |
| tsmsval2 24449 | Definition of the topologi... |
| tsmsval 24450 | Definition of the topologi... |
| tsmspropd 24451 | The group sum depends only... |
| eltsms 24452 | The property of being a su... |
| tsmsi 24453 | The property of being a su... |
| tsmscl 24454 | A sum in a topological gro... |
| haustsms 24455 | In a Hausdorff topological... |
| haustsms2 24456 | In a Hausdorff topological... |
| tsmscls 24457 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24458 | The convergent points of a... |
| tsmsid 24459 | If a sum is finite, the us... |
| haustsmsid 24460 | In a Hausdorff topological... |
| tsms0 24461 | The sum of zero is zero. ... |
| tsmssubm 24462 | Evaluate an infinite group... |
| tsmsres 24463 | Extend an infinite group s... |
| tsmsf1o 24464 | Re-index an infinite group... |
| tsmsmhm 24465 | Apply a continuous group h... |
| tsmsadd 24466 | The sum of two infinite gr... |
| tsmsinv 24467 | Inverse of an infinite gro... |
| tsmssub 24468 | The difference of two infi... |
| tgptsmscls 24469 | A sum in a topological gro... |
| tgptsmscld 24470 | The set of limit points to... |
| tsmssplit 24471 | Split a topological group ... |
| tsmsxplem1 24472 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24473 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24474 | Write a sum over a two-dim... |
| istrg 24483 | Express the predicate " ` ... |
| trgtmd 24484 | The multiplicative monoid ... |
| istdrg 24485 | Express the predicate " ` ... |
| tdrgunit 24486 | The unit group of a topolo... |
| trgtgp 24487 | A topological ring is a to... |
| trgtmd2 24488 | A topological ring is a to... |
| trgtps 24489 | A topological ring is a to... |
| trgring 24490 | A topological ring is a ri... |
| trggrp 24491 | A topological ring is a gr... |
| tdrgtrg 24492 | A topological division rin... |
| tdrgdrng 24493 | A topological division rin... |
| tdrgring 24494 | A topological division rin... |
| tdrgtmd 24495 | A topological division rin... |
| tdrgtps 24496 | A topological division rin... |
| istdrg2 24497 | A topological-ring divisio... |
| mulrcn 24498 | The functionalization of t... |
| invrcn2 24499 | The multiplicative inverse... |
| invrcn 24500 | The multiplicative inverse... |
| cnmpt1mulr 24501 | Continuity of ring multipl... |
| cnmpt2mulr 24502 | Continuity of ring multipl... |
| dvrcn 24503 | The division function is c... |
| istlm 24504 | The predicate " ` W ` is a... |
| vscacn 24505 | The scalar multiplication ... |
| tlmtmd 24506 | A topological module is a ... |
| tlmtps 24507 | A topological module is a ... |
| tlmlmod 24508 | A topological module is a ... |
| tlmtrg 24509 | The scalar ring of a topol... |
| tlmscatps 24510 | The scalar ring of a topol... |
| istvc 24511 | A topological vector space... |
| tvctdrg 24512 | The scalar field of a topo... |
| cnmpt1vsca 24513 | Continuity of scalar multi... |
| cnmpt2vsca 24514 | Continuity of scalar multi... |
| tlmtgp 24515 | A topological vector space... |
| tvctlm 24516 | A topological vector space... |
| tvclmod 24517 | A topological vector space... |
| tvclvec 24518 | A topological vector space... |
| ustfn 24521 | The defined uniform struct... |
| ustval 24522 | The class of all uniform s... |
| isust 24523 | The predicate " ` U ` is a... |
| ustssxp 24524 | Entourages are subsets of ... |
| ustssel 24525 | A uniform structure is upw... |
| ustbasel 24526 | The full set is always an ... |
| ustincl 24527 | A uniform structure is clo... |
| ustdiag 24528 | The diagonal set is includ... |
| ustinvel 24529 | If ` V ` is an entourage, ... |
| ustexhalf 24530 | For each entourage ` V ` t... |
| ustrel 24531 | The elements of uniform st... |
| ustfilxp 24532 | A uniform structure on a n... |
| ustne0 24533 | A uniform structure cannot... |
| ustssco 24534 | In an uniform structure, a... |
| ustexsym 24535 | In an uniform structure, f... |
| ustex2sym 24536 | In an uniform structure, f... |
| ustex3sym 24537 | In an uniform structure, f... |
| ustref 24538 | Any element of the base se... |
| ust0 24539 | The unique uniform structu... |
| ustn0 24540 | The empty set is not an un... |
| ustund 24541 | If two intersecting sets `... |
| ustelimasn 24542 | Any point ` A ` is near en... |
| ustneism 24543 | For a point ` A ` in ` X `... |
| ustbas2 24544 | Second direction for ~ ust... |
| ustuni 24545 | The set union of a uniform... |
| ustbas 24546 | Recover the base of an uni... |
| ustimasn 24547 | Lemma for ~ ustuqtop . (C... |
| trust 24548 | The trace of a uniform str... |
| utopval 24551 | The topology induced by a ... |
| elutop 24552 | Open sets in the topology ... |
| utoptop 24553 | The topology induced by a ... |
| utopbas 24554 | The base of the topology i... |
| utoptopon 24555 | Topology induced by a unif... |
| restutop 24556 | Restriction of a topology ... |
| restutopopn 24557 | The restriction of the top... |
| ustuqtoplem 24558 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24559 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24560 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24561 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24562 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24563 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24564 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24565 | For a given uniform struct... |
| utopsnneiplem 24566 | The neighborhoods of a poi... |
| utopsnneip 24567 | The neighborhoods of a poi... |
| utopsnnei 24568 | Images of singletons by en... |
| utop2nei 24569 | For any symmetrical entour... |
| utop3cls 24570 | Relation between a topolog... |
| utopreg 24571 | All Hausdorff uniform spac... |
| ussval 24578 | The uniform structure on u... |
| ussid 24579 | In case the base of the ` ... |
| isusp 24580 | The predicate ` W ` is a u... |
| ressuss 24581 | Value of the uniform struc... |
| ressust 24582 | The uniform structure of a... |
| ressusp 24583 | The restriction of a unifo... |
| tusval 24584 | The value of the uniform s... |
| tuslem 24585 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24586 | The base set of a construc... |
| tusunif 24587 | The uniform structure of a... |
| tususs 24588 | The uniform structure of a... |
| tustopn 24589 | The topology induced by a ... |
| tususp 24590 | A constructed uniform spac... |
| tustps 24591 | A constructed uniform spac... |
| uspreg 24592 | If a uniform space is Haus... |
| ucnval 24595 | The set of all uniformly c... |
| isucn 24596 | The predicate " ` F ` is a... |
| isucn2 24597 | The predicate " ` F ` is a... |
| ucnimalem 24598 | Reformulate the ` G ` func... |
| ucnima 24599 | An equivalent statement of... |
| ucnprima 24600 | The preimage by a uniforml... |
| iducn 24601 | The identity is uniformly ... |
| cstucnd 24602 | A constant function is uni... |
| ucncn 24603 | Uniform continuity implies... |
| iscfilu 24606 | The predicate " ` F ` is a... |
| cfilufbas 24607 | A Cauchy filter base is a ... |
| cfiluexsm 24608 | For a Cauchy filter base a... |
| fmucndlem 24609 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24610 | The image of a Cauchy filt... |
| cfilufg 24611 | The filter generated by a ... |
| trcfilu 24612 | Condition for the trace of... |
| cfiluweak 24613 | A Cauchy filter base is al... |
| neipcfilu 24614 | In an uniform space, a nei... |
| iscusp 24617 | The predicate " ` W ` is a... |
| cuspusp 24618 | A complete uniform space i... |
| cuspcvg 24619 | In a complete uniform spac... |
| iscusp2 24620 | The predicate " ` W ` is a... |
| cnextucn 24621 | Extension by continuity. ... |
| ucnextcn 24622 | Extension by continuity. ... |
| ispsmet 24623 | Express the predicate " ` ... |
| psmetdmdm 24624 | Recover the base set from ... |
| psmetf 24625 | The distance function of a... |
| psmetcl 24626 | Closure of the distance fu... |
| psmet0 24627 | The distance function of a... |
| psmettri2 24628 | Triangle inequality for th... |
| psmetsym 24629 | The distance function of a... |
| psmettri 24630 | Triangle inequality for th... |
| psmetge0 24631 | The distance function of a... |
| psmetxrge0 24632 | The distance function of a... |
| psmetres2 24633 | Restriction of a pseudomet... |
| psmetlecl 24634 | Real closure of an extende... |
| distspace 24635 | A set ` X ` together with ... |
| ismet 24642 | Express the predicate " ` ... |
| isxmet 24643 | Express the predicate " ` ... |
| ismeti 24644 | Properties that determine ... |
| isxmetd 24645 | Properties that determine ... |
| isxmet2d 24646 | It is safe to only require... |
| metflem 24647 | Lemma for ~ metf and other... |
| xmetf 24648 | Mapping of the distance fu... |
| metf 24649 | Mapping of the distance fu... |
| xmetcl 24650 | Closure of the distance fu... |
| metcl 24651 | Closure of the distance fu... |
| ismet2 24652 | An extended metric is a me... |
| metxmet 24653 | A metric is an extended me... |
| xmetdmdm 24654 | Recover the base set from ... |
| metdmdm 24655 | Recover the base set from ... |
| xmetunirn 24656 | Two ways to express an ext... |
| xmeteq0 24657 | The value of an extended m... |
| meteq0 24658 | The value of a metric is z... |
| xmettri2 24659 | Triangle inequality for th... |
| mettri2 24660 | Triangle inequality for th... |
| xmet0 24661 | The distance function of a... |
| met0 24662 | The distance function of a... |
| xmetge0 24663 | The distance function of a... |
| metge0 24664 | The distance function of a... |
| xmetlecl 24665 | Real closure of an extende... |
| xmetsym 24666 | The distance function of a... |
| xmetpsmet 24667 | An extended metric is a ps... |
| xmettpos 24668 | The distance function of a... |
| metsym 24669 | The distance function of a... |
| xmettri 24670 | Triangle inequality for th... |
| mettri 24671 | Triangle inequality for th... |
| xmettri3 24672 | Triangle inequality for th... |
| mettri3 24673 | Triangle inequality for th... |
| xmetrtri 24674 | One half of the reverse tr... |
| xmetrtri2 24675 | The reverse triangle inequ... |
| metrtri 24676 | Reverse triangle inequalit... |
| xmetgt0 24677 | The distance function of a... |
| metgt0 24678 | The distance function of a... |
| metn0 24679 | A metric space is nonempty... |
| xmetres2 24680 | Restriction of an extended... |
| metreslem 24681 | Lemma for ~ metres . (Con... |
| metres2 24682 | Lemma for ~ metres . (Con... |
| xmetres 24683 | A restriction of an extend... |
| metres 24684 | A restriction of a metric ... |
| 0met 24685 | The empty metric. (Contri... |
| prdsdsf 24686 | The product metric is a fu... |
| prdsxmetlem 24687 | The product metric is an e... |
| prdsxmet 24688 | The product metric is an e... |
| prdsmet 24689 | The product metric is a me... |
| ressprdsds 24690 | Restriction of a product m... |
| resspwsds 24691 | Restriction of a power met... |
| imasdsf1olem 24692 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24693 | The distance function is t... |
| imasf1oxmet 24694 | The image of an extended m... |
| imasf1omet 24695 | The image of a metric is a... |
| xpsdsfn 24696 | Closure of the metric in a... |
| xpsdsfn2 24697 | Closure of the metric in a... |
| xpsxmetlem 24698 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24699 | A product metric of extend... |
| xpsdsval 24700 | Value of the metric in a b... |
| xpsmet 24701 | The direct product of two ... |
| blfvalps 24702 | The value of the ball func... |
| blfval 24703 | The value of the ball func... |
| blvalps 24704 | The ball around a point ` ... |
| blval 24705 | The ball around a point ` ... |
| elblps 24706 | Membership in a ball. (Co... |
| elbl 24707 | Membership in a ball. (Co... |
| elbl2ps 24708 | Membership in a ball. (Co... |
| elbl2 24709 | Membership in a ball. (Co... |
| elbl3ps 24710 | Membership in a ball, with... |
| elbl3 24711 | Membership in a ball, with... |
| blcomps 24712 | Commute the arguments to t... |
| blcom 24713 | Commute the arguments to t... |
| xblpnfps 24714 | The infinity ball in an ex... |
| xblpnf 24715 | The infinity ball in an ex... |
| blpnf 24716 | The infinity ball in a sta... |
| bldisj 24717 | Two balls are disjoint if ... |
| blgt0 24718 | A nonempty ball implies th... |
| bl2in 24719 | Two balls are disjoint if ... |
| xblss2ps 24720 | One ball is contained in a... |
| xblss2 24721 | One ball is contained in a... |
| blss2ps 24722 | One ball is contained in a... |
| blss2 24723 | One ball is contained in a... |
| blhalf 24724 | A ball of radius ` R / 2 `... |
| blfps 24725 | Mapping of a ball. (Contr... |
| blf 24726 | Mapping of a ball. (Contr... |
| blrnps 24727 | Membership in the range of... |
| blrn 24728 | Membership in the range of... |
| xblcntrps 24729 | A ball contains its center... |
| xblcntr 24730 | A ball contains its center... |
| blcntrps 24731 | A ball contains its center... |
| blcntr 24732 | A ball contains its center... |
| xbln0 24733 | A ball is nonempty iff the... |
| bln0 24734 | A ball is not empty. (Con... |
| blelrnps 24735 | A ball belongs to the set ... |
| blelrn 24736 | A ball belongs to the set ... |
| blssm 24737 | A ball is a subset of the ... |
| unirnblps 24738 | The union of the set of ba... |
| unirnbl 24739 | The union of the set of ba... |
| blin 24740 | The intersection of two ba... |
| ssblps 24741 | The size of a ball increas... |
| ssbl 24742 | The size of a ball increas... |
| blssps 24743 | Any point ` P ` in a ball ... |
| blss 24744 | Any point ` P ` in a ball ... |
| blssexps 24745 | Two ways to express the ex... |
| blssex 24746 | Two ways to express the ex... |
| ssblex 24747 | A nested ball exists whose... |
| blin2 24748 | Given any two balls and a ... |
| blbas 24749 | The balls of a metric spac... |
| blres 24750 | A ball in a restricted met... |
| xmeterval 24751 | Value of the "finitely sep... |
| xmeter 24752 | The "finitely separated" r... |
| xmetec 24753 | The equivalence classes un... |
| blssec 24754 | A ball centered at ` P ` i... |
| blpnfctr 24755 | The infinity ball in an ex... |
| xmetresbl 24756 | An extended metric restric... |
| mopnval 24757 | An open set is a subset of... |
| mopntopon 24758 | The set of open sets of a ... |
| mopntop 24759 | The set of open sets of a ... |
| mopnuni 24760 | The union of all open sets... |
| elmopn 24761 | The defining property of a... |
| mopnfss 24762 | The family of open sets of... |
| mopnm 24763 | The base set of a metric s... |
| elmopn2 24764 | A defining property of an ... |
| mopnss 24765 | An open set of a metric sp... |
| isxms 24766 | Express the predicate " ` ... |
| isxms2 24767 | Express the predicate " ` ... |
| isms 24768 | Express the predicate " ` ... |
| isms2 24769 | Express the predicate " ` ... |
| xmstopn 24770 | The topology component of ... |
| mstopn 24771 | The topology component of ... |
| xmstps 24772 | An extended metric space i... |
| msxms 24773 | A metric space is an exten... |
| mstps 24774 | A metric space is a topolo... |
| xmsxmet 24775 | The distance function, sui... |
| msmet 24776 | The distance function, sui... |
| msf 24777 | The distance function of a... |
| xmsxmet2 24778 | The distance function, sui... |
| msmet2 24779 | The distance function, sui... |
| mscl 24780 | Closure of the distance fu... |
| xmscl 24781 | Closure of the distance fu... |
| xmsge0 24782 | The distance function in a... |
| xmseq0 24783 | The distance between two p... |
| xmssym 24784 | The distance function in a... |
| xmstri2 24785 | Triangle inequality for th... |
| mstri2 24786 | Triangle inequality for th... |
| xmstri 24787 | Triangle inequality for th... |
| mstri 24788 | Triangle inequality for th... |
| xmstri3 24789 | Triangle inequality for th... |
| mstri3 24790 | Triangle inequality for th... |
| msrtri 24791 | Reverse triangle inequalit... |
| xmspropd 24792 | Property deduction for an ... |
| mspropd 24793 | Property deduction for a m... |
| setsmsbas 24794 | The base set of a construc... |
| setsmsds 24795 | The distance function of a... |
| setsmstset 24796 | The topology of a construc... |
| setsmstopn 24797 | The topology of a construc... |
| setsxms 24798 | The constructed metric spa... |
| setsms 24799 | The constructed metric spa... |
| tmsval 24800 | For any metric there is an... |
| tmslem 24801 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24802 | The base set of a construc... |
| tmsds 24803 | The metric of a constructe... |
| tmstopn 24804 | The topology of a construc... |
| tmsxms 24805 | The constructed metric spa... |
| tmsms 24806 | The constructed metric spa... |
| imasf1obl 24807 | The image of a metric spac... |
| imasf1oxms 24808 | The image of a metric spac... |
| imasf1oms 24809 | The image of a metric spac... |
| prdsbl 24810 | A ball in the product metr... |
| mopni 24811 | An open set of a metric sp... |
| mopni2 24812 | An open set of a metric sp... |
| mopni3 24813 | An open set of a metric sp... |
| blssopn 24814 | The balls of a metric spac... |
| unimopn 24815 | The union of a collection ... |
| mopnin 24816 | The intersection of two op... |
| mopn0 24817 | The empty set is an open s... |
| rnblopn 24818 | A ball of a metric space i... |
| blopn 24819 | A ball of a metric space i... |
| neibl 24820 | The neighborhoods around a... |
| blnei 24821 | A ball around a point is a... |
| lpbl 24822 | Every ball around a limit ... |
| blsscls2 24823 | A smaller closed ball is c... |
| blcld 24824 | A "closed ball" in a metri... |
| blcls 24825 | The closure of an open bal... |
| blsscls 24826 | If two concentric balls ha... |
| metss 24827 | Two ways of saying that me... |
| metequiv 24828 | Two ways of saying that tw... |
| metequiv2 24829 | If there is a sequence of ... |
| metss2lem 24830 | Lemma for ~ metss2 . (Con... |
| metss2 24831 | If the metric ` D ` is "st... |
| comet 24832 | The composition of an exte... |
| stdbdmetval 24833 | Value of the standard boun... |
| stdbdxmet 24834 | The standard bounded metri... |
| stdbdmet 24835 | The standard bounded metri... |
| stdbdbl 24836 | The standard bounded metri... |
| stdbdmopn 24837 | The standard bounded metri... |
| mopnex 24838 | The topology generated by ... |
| methaus 24839 | The topology generated by ... |
| met1stc 24840 | The topology generated by ... |
| met2ndci 24841 | A separable metric space (... |
| met2ndc 24842 | A metric space is second-c... |
| metrest 24843 | Two alternate formulations... |
| ressxms 24844 | The restriction of a metri... |
| ressms 24845 | The restriction of a metri... |
| prdsmslem1 24846 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24847 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24848 | Lemma for ~ prdsxms . The... |
| prdsxms 24849 | The indexed product struct... |
| prdsms 24850 | The indexed product struct... |
| pwsxms 24851 | A power of an extended met... |
| pwsms 24852 | A power of a metric space ... |
| xpsxms 24853 | A binary product of metric... |
| xpsms 24854 | A binary product of metric... |
| tmsxps 24855 | Express the product of two... |
| tmsxpsmopn 24856 | Express the product of two... |
| tmsxpsval 24857 | Value of the product of tw... |
| tmsxpsval2 24858 | Value of the product of tw... |
| metcnp3 24859 | Two ways to express that `... |
| metcnp 24860 | Two ways to say a mapping ... |
| metcnp2 24861 | Two ways to say a mapping ... |
| metcn 24862 | Two ways to say a mapping ... |
| metcnpi 24863 | Epsilon-delta property of ... |
| metcnpi2 24864 | Epsilon-delta property of ... |
| metcnpi3 24865 | Epsilon-delta property of ... |
| txmetcnp 24866 | Continuity of a binary ope... |
| txmetcn 24867 | Continuity of a binary ope... |
| metuval 24868 | Value of the uniform struc... |
| metustel 24869 | Define a filter base ` F `... |
| metustss 24870 | Range of the elements of t... |
| metustrel 24871 | Elements of the filter bas... |
| metustto 24872 | Any two elements of the fi... |
| metustid 24873 | The identity diagonal is i... |
| metustsym 24874 | Elements of the filter bas... |
| metustexhalf 24875 | For any element ` A ` of t... |
| metustfbas 24876 | The filter base generated ... |
| metust 24877 | The uniform structure gene... |
| cfilucfil 24878 | Given a metric ` D ` and a... |
| metuust 24879 | The uniform structure gene... |
| cfilucfil2 24880 | Given a metric ` D ` and a... |
| blval2 24881 | The ball around a point ` ... |
| elbl4 24882 | Membership in a ball, alte... |
| metuel 24883 | Elementhood in the uniform... |
| metuel2 24884 | Elementhood in the uniform... |
| metustbl 24885 | The "section" image of an ... |
| psmetutop 24886 | The topology induced by a ... |
| xmetutop 24887 | The topology induced by a ... |
| xmsusp 24888 | If the uniform set of a me... |
| restmetu 24889 | The uniform structure gene... |
| metucn 24890 | Uniform continuity in metr... |
| dscmet 24891 | The discrete metric on any... |
| dscopn 24892 | The discrete metric genera... |
| nrmmetd 24893 | Show that a group norm gen... |
| abvmet 24894 | An absolute value ` F ` ge... |
| nmfval 24907 | The value of the norm func... |
| nmval 24908 | The value of the norm as t... |
| nmfval0 24909 | The value of the norm func... |
| nmfval2 24910 | The value of the norm func... |
| nmval2 24911 | The value of the norm on a... |
| nmf2 24912 | The norm on a metric group... |
| nmpropd 24913 | Weak property deduction fo... |
| nmpropd2 24914 | Strong property deduction ... |
| isngp 24915 | The property of being a no... |
| isngp2 24916 | The property of being a no... |
| isngp3 24917 | The property of being a no... |
| ngpgrp 24918 | A normed group is a group.... |
| ngpms 24919 | A normed group is a metric... |
| ngpxms 24920 | A normed group is an exten... |
| ngptps 24921 | A normed group is a topolo... |
| ngpmet 24922 | The (induced) metric of a ... |
| ngpds 24923 | Value of the distance func... |
| ngpdsr 24924 | Value of the distance func... |
| ngpds2 24925 | Write the distance between... |
| ngpds2r 24926 | Write the distance between... |
| ngpds3 24927 | Write the distance between... |
| ngpds3r 24928 | Write the distance between... |
| ngprcan 24929 | Cancel right addition insi... |
| ngplcan 24930 | Cancel left addition insid... |
| isngp4 24931 | Express the property of be... |
| ngpinvds 24932 | Two elements are the same ... |
| ngpsubcan 24933 | Cancel right subtraction i... |
| nmf 24934 | The norm on a normed group... |
| nmcl 24935 | The norm of a normed group... |
| nmge0 24936 | The norm of a normed group... |
| nmeq0 24937 | The identity is the only e... |
| nmne0 24938 | The norm of a nonzero elem... |
| nmrpcl 24939 | The norm of a nonzero elem... |
| nminv 24940 | The norm of a negated elem... |
| nmmtri 24941 | The triangle inequality fo... |
| nmsub 24942 | The norm of the difference... |
| nmrtri 24943 | Reverse triangle inequalit... |
| nm2dif 24944 | Inequality for the differe... |
| nmtri 24945 | The triangle inequality fo... |
| nmtri2 24946 | Triangle inequality for th... |
| ngpi 24947 | The properties of a normed... |
| nm0 24948 | Norm of the identity eleme... |
| nmgt0 24949 | The norm of a nonzero elem... |
| sgrim 24950 | The induced metric on a su... |
| sgrimval 24951 | The induced metric on a su... |
| subgnm 24952 | The norm in a subgroup. (... |
| subgnm2 24953 | A substructure assigns the... |
| subgngp 24954 | A normed group restricted ... |
| ngptgp 24955 | A normed abelian group is ... |
| ngppropd 24956 | Property deduction for a n... |
| reldmtng 24957 | The function ` toNrmGrp ` ... |
| tngval 24958 | Value of the function whic... |
| tnglem 24959 | Lemma for ~ tngbas and sim... |
| tngbas 24960 | The base set of a structur... |
| tngplusg 24961 | The group addition of a st... |
| tng0 24962 | The group identity of a st... |
| tngmulr 24963 | The ring multiplication of... |
| tngsca 24964 | The scalar ring of a struc... |
| tngvsca 24965 | The scalar multiplication ... |
| tngip 24966 | The inner product operatio... |
| tngds 24967 | The metric function of a s... |
| tngtset 24968 | The topology generated by ... |
| tngtopn 24969 | The topology generated by ... |
| tngnm 24970 | The topology generated by ... |
| tngngp2 24971 | A norm turns a group into ... |
| tngngpd 24972 | Derive the axioms for a no... |
| tngngp 24973 | Derive the axioms for a no... |
| tnggrpr 24974 | If a structure equipped wi... |
| tngngp3 24975 | Alternate definition of a ... |
| nrmtngdist 24976 | The augmentation of a norm... |
| nrmtngnrm 24977 | The augmentation of a norm... |
| tngngpim 24978 | The induced metric of a no... |
| isnrg 24979 | A normed ring is a ring wi... |
| nrgabv 24980 | The norm of a normed ring ... |
| nrgngp 24981 | A normed ring is a normed ... |
| nrgring 24982 | A normed ring is a ring. ... |
| nmmul 24983 | The norm of a product in a... |
| nrgdsdi 24984 | Distribute a distance calc... |
| nrgdsdir 24985 | Distribute a distance calc... |
| nm1 24986 | The norm of one in a nonze... |
| unitnmn0 24987 | The norm of a unit is nonz... |
| nminvr 24988 | The norm of an inverse in ... |
| nmdvr 24989 | The norm of a division in ... |
| nrgdomn 24990 | A nonzero normed ring is a... |
| nrgtgp 24991 | A normed ring is a topolog... |
| subrgnrg 24992 | A normed ring restricted t... |
| tngnrg 24993 | Given any absolute value o... |
| isnlm 24994 | A normed (left) module is ... |
| nmvs 24995 | Defining property of a nor... |
| nlmngp 24996 | A normed module is a norme... |
| nlmlmod 24997 | A normed module is a left ... |
| nlmnrg 24998 | The scalar component of a ... |
| nlmngp2 24999 | The scalar component of a ... |
| nlmdsdi 25000 | Distribute a distance calc... |
| nlmdsdir 25001 | Distribute a distance calc... |
| nlmmul0or 25002 | If a scalar product is zer... |
| sranlm 25003 | The subring algebra over a... |
| nlmvscnlem2 25004 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 25005 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 25006 | The scalar multiplication ... |
| rlmnlm 25007 | The ring module over a nor... |
| rlmnm 25008 | The norm function in the r... |
| nrgtrg 25009 | A normed ring is a topolog... |
| nrginvrcnlem 25010 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 25011 | The ring inverse function ... |
| nrgtdrg 25012 | A normed division ring is ... |
| nlmtlm 25013 | A normed module is a topol... |
| isnvc 25014 | A normed vector space is j... |
| nvcnlm 25015 | A normed vector space is a... |
| nvclvec 25016 | A normed vector space is a... |
| nvclmod 25017 | A normed vector space is a... |
| isnvc2 25018 | A normed vector space is j... |
| nvctvc 25019 | A normed vector space is a... |
| lssnlm 25020 | A subspace of a normed mod... |
| lssnvc 25021 | A subspace of a normed vec... |
| rlmnvc 25022 | The ring module over a nor... |
| ngpocelbl 25023 | Membership of an off-cente... |
| nmoffn 25030 | The function producing ope... |
| reldmnghm 25031 | Lemma for normed group hom... |
| reldmnmhm 25032 | Lemma for module homomorph... |
| nmofval 25033 | Value of the operator norm... |
| nmoval 25034 | Value of the operator norm... |
| nmogelb 25035 | Property of the operator n... |
| nmolb 25036 | Any upper bound on the val... |
| nmolb2d 25037 | Any upper bound on the val... |
| nmof 25038 | The operator norm is a fun... |
| nmocl 25039 | The operator norm of an op... |
| nmoge0 25040 | The operator norm of an op... |
| nghmfval 25041 | A normed group homomorphis... |
| isnghm 25042 | A normed group homomorphis... |
| isnghm2 25043 | A normed group homomorphis... |
| isnghm3 25044 | A normed group homomorphis... |
| bddnghm 25045 | A bounded group homomorphi... |
| nghmcl 25046 | A normed group homomorphis... |
| nmoi 25047 | The operator norm achieves... |
| nmoix 25048 | The operator norm is a bou... |
| nmoi2 25049 | The operator norm is a bou... |
| nmoleub 25050 | The operator norm, defined... |
| nghmrcl1 25051 | Reverse closure for a norm... |
| nghmrcl2 25052 | Reverse closure for a norm... |
| nghmghm 25053 | A normed group homomorphis... |
| nmo0 25054 | The operator norm of the z... |
| nmoeq0 25055 | The operator norm is zero ... |
| nmoco 25056 | An upper bound on the oper... |
| nghmco 25057 | The composition of normed ... |
| nmotri 25058 | Triangle inequality for th... |
| nghmplusg 25059 | The sum of two bounded lin... |
| 0nghm 25060 | The zero operator is a nor... |
| nmoid 25061 | The operator norm of the i... |
| idnghm 25062 | The identity operator is a... |
| nmods 25063 | Upper bound for the distan... |
| nghmcn 25064 | A normed group homomorphis... |
| isnmhm 25065 | A normed module homomorphi... |
| nmhmrcl1 25066 | Reverse closure for a norm... |
| nmhmrcl2 25067 | Reverse closure for a norm... |
| nmhmlmhm 25068 | A normed module homomorphi... |
| nmhmnghm 25069 | A normed module homomorphi... |
| nmhmghm 25070 | A normed module homomorphi... |
| isnmhm2 25071 | A normed module homomorphi... |
| nmhmcl 25072 | A normed module homomorphi... |
| idnmhm 25073 | The identity operator is a... |
| 0nmhm 25074 | The zero operator is a bou... |
| nmhmco 25075 | The composition of bounded... |
| nmhmplusg 25076 | The sum of two bounded lin... |
| qtopbaslem 25077 | The set of open intervals ... |
| qtopbas 25078 | The set of open intervals ... |
| retopbas 25079 | A basis for the standard t... |
| retop 25080 | The standard topology on t... |
| uniretop 25081 | The underlying set of the ... |
| retopon 25082 | The standard topology on t... |
| retps 25083 | The standard topological s... |
| iooretop 25084 | Open intervals are open se... |
| icccld 25085 | Closed intervals are close... |
| icopnfcld 25086 | Right-unbounded closed int... |
| iocmnfcld 25087 | Left-unbounded closed inte... |
| qdensere 25088 | ` QQ ` is dense in the sta... |
| cnmetdval 25089 | Value of the distance func... |
| cnmet 25090 | The absolute value metric ... |
| cnxmet 25091 | The absolute value metric ... |
| cnbl0 25092 | Two ways to write the open... |
| cnblcld 25093 | Two ways to write the clos... |
| cnfldms 25094 | The complex number field i... |
| cnfldxms 25095 | The complex number field i... |
| cnfldtps 25096 | The complex number field i... |
| cnfldnm 25097 | The norm of the field of c... |
| cnngp 25098 | The complex numbers form a... |
| cnnrg 25099 | The complex numbers form a... |
| cnfldtopn 25100 | The topology of the comple... |
| cnfldtopon 25101 | The topology of the comple... |
| cnfldtop 25102 | The topology of the comple... |
| cnfldhaus 25103 | The topology of the comple... |
| unicntop 25104 | The underlying set of the ... |
| cnopn 25105 | The set of complex numbers... |
| cnn0opn 25106 | The set of nonzero complex... |
| zringnrg 25107 | The ring of integers is a ... |
| remetdval 25108 | Value of the distance func... |
| remet 25109 | The absolute value metric ... |
| rexmet 25110 | The absolute value metric ... |
| bl2ioo 25111 | A ball in terms of an open... |
| ioo2bl 25112 | An open interval of reals ... |
| ioo2blex 25113 | An open interval of reals ... |
| blssioo 25114 | The balls of the standard ... |
| tgioo 25115 | The topology generated by ... |
| qdensere2 25116 | ` QQ ` is dense in ` RR ` ... |
| blcvx 25117 | An open ball in the comple... |
| rehaus 25118 | The standard topology on t... |
| tgqioo 25119 | The topology generated by ... |
| re2ndc 25120 | The standard topology on t... |
| resubmet 25121 | The subspace topology indu... |
| tgioo2 25122 | The standard topology on t... |
| rerest 25123 | The subspace topology indu... |
| tgioo4 25124 | The standard topology on t... |
| tgioo3 25125 | The standard topology on t... |
| xrtgioo 25126 | The topology on the extend... |
| xrrest 25127 | The subspace topology indu... |
| xrrest2 25128 | The subspace topology indu... |
| xrsxmet 25129 | The metric on the extended... |
| xrsdsre 25130 | The metric on the extended... |
| xrsblre 25131 | Any ball of the metric of ... |
| xrsmopn 25132 | The metric on the extended... |
| zcld 25133 | The integers are a closed ... |
| recld2 25134 | The real numbers are a clo... |
| zcld2 25135 | The integers are a closed ... |
| zdis 25136 | The integers are a discret... |
| sszcld 25137 | Every subset of the intege... |
| reperflem 25138 | A subset of the real numbe... |
| reperf 25139 | The real numbers are a per... |
| cnperf 25140 | The complex numbers are a ... |
| iccntr 25141 | The interior of a closed i... |
| icccmplem1 25142 | Lemma for ~ icccmp . (Con... |
| icccmplem2 25143 | Lemma for ~ icccmp . (Con... |
| icccmplem3 25144 | Lemma for ~ icccmp . (Con... |
| icccmp 25145 | A closed interval in ` RR ... |
| reconnlem1 25146 | Lemma for ~ reconn . Conn... |
| reconnlem2 25147 | Lemma for ~ reconn . (Con... |
| reconn 25148 | A subset of the reals is c... |
| retopconn 25149 | Corollary of ~ reconn . T... |
| iccconn 25150 | A closed interval is conne... |
| opnreen 25151 | Every nonempty open set is... |
| rectbntr0 25152 | A countable subset of the ... |
| xrge0gsumle 25153 | A finite sum in the nonneg... |
| xrge0tsms 25154 | Any finite or infinite sum... |
| xrge0tsms2 25155 | Any finite or infinite sum... |
| metdcnlem 25156 | The metric function of a m... |
| xmetdcn2 25157 | The metric function of an ... |
| xmetdcn 25158 | The metric function of an ... |
| metdcn2 25159 | The metric function of a m... |
| metdcn 25160 | The metric function of a m... |
| msdcn 25161 | The metric function of a m... |
| cnmpt1ds 25162 | Continuity of the metric f... |
| cnmpt2ds 25163 | Continuity of the metric f... |
| nmcn 25164 | The norm of a normed group... |
| ngnmcncn 25165 | The norm of a normed group... |
| abscn 25166 | The absolute value functio... |
| metdsval 25167 | Value of the "distance to ... |
| metdsf 25168 | The distance from a point ... |
| metdsge 25169 | The distance from the poin... |
| metds0 25170 | If a point is in a set, it... |
| metdstri 25171 | A generalization of the tr... |
| metdsle 25172 | The distance from a point ... |
| metdsre 25173 | The distance from a point ... |
| metdseq0 25174 | The distance from a point ... |
| metdscnlem 25175 | Lemma for ~ metdscn . (Co... |
| metdscn 25176 | The function ` F ` which g... |
| metdscn2 25177 | The function ` F ` which g... |
| metnrmlem1a 25178 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 25179 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 25180 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 25181 | Lemma for ~ metnrm . (Con... |
| metnrm 25182 | A metric space is normal. ... |
| metreg 25183 | A metric space is regular.... |
| addcnlem 25184 | Lemma for ~ addcn , ~ subc... |
| addcn 25185 | Complex number addition is... |
| subcn 25186 | Complex number subtraction... |
| mulcn 25187 | Complex number multiplicat... |
| mpomulcn 25188 | Complex number multiplicat... |
| divcn 25189 | Complex number division is... |
| cnfldtgp 25190 | The complex numbers form a... |
| fsumcn 25191 | A finite sum of functions ... |
| fsum2cn 25192 | Version of ~ fsumcn for tw... |
| expcn 25193 | The power function on comp... |
| divccn 25194 | Division by a nonzero cons... |
| sqcn 25195 | The square function on com... |
| iitopon 25200 | The unit interval is a top... |
| iitop 25201 | The unit interval is a top... |
| iiuni 25202 | The base set of the unit i... |
| dfii2 25203 | Alternate definition of th... |
| dfii3 25204 | Alternate definition of th... |
| dfii4 25205 | Alternate definition of th... |
| dfii5 25206 | The unit interval expresse... |
| iicmp 25207 | The unit interval is compa... |
| iiconn 25208 | The unit interval is conne... |
| cncfval 25209 | The value of the continuou... |
| elcncf 25210 | Membership in the set of c... |
| elcncf2 25211 | Version of ~ elcncf with a... |
| cncfrss 25212 | Reverse closure of the con... |
| cncfrss2 25213 | Reverse closure of the con... |
| cncff 25214 | A continuous complex funct... |
| cncfi 25215 | Defining property of a con... |
| elcncf1di 25216 | Membership in the set of c... |
| elcncf1ii 25217 | Membership in the set of c... |
| rescncf 25218 | A continuous complex funct... |
| cncfcdm 25219 | Change the codomain of a c... |
| cncfss 25220 | The set of continuous func... |
| climcncf 25221 | Image of a limit under a c... |
| abscncf 25222 | Absolute value is continuo... |
| recncf 25223 | Real part is continuous. ... |
| imcncf 25224 | Imaginary part is continuo... |
| cjcncf 25225 | Complex conjugate is conti... |
| mulc1cncf 25226 | Multiplication by a consta... |
| divccncf 25227 | Division by a constant is ... |
| cncfco 25228 | The composition of two con... |
| cncfcompt2 25229 | Composition of continuous ... |
| cncfmet 25230 | Relate complex function co... |
| cncfcn 25231 | Relate complex function co... |
| cncfcn1 25232 | Relate complex function co... |
| cncfmptc 25233 | A constant function is a c... |
| cncfmptid 25234 | The identity function is a... |
| cncfmpt1f 25235 | Composition of continuous ... |
| cncfmpt2f 25236 | Composition of continuous ... |
| cncfmpt2ss 25237 | Composition of continuous ... |
| addccncf 25238 | Adding a constant is a con... |
| idcncf 25239 | The identity function is a... |
| sub1cncf 25240 | Subtracting a constant is ... |
| sub2cncf 25241 | Subtraction from a constan... |
| cdivcncf 25242 | Division with a constant n... |
| negcncf 25243 | The negative function is c... |
| negfcncf 25244 | The negative of a continuo... |
| abscncfALT 25245 | Absolute value is continuo... |
| cncfcnvcn 25246 | Rewrite ~ cmphaushmeo for ... |
| expcncf 25247 | The power function on comp... |
| cnmptre 25248 | Lemma for ~ iirevcn and re... |
| cnmpopc 25249 | Piecewise definition of a ... |
| iirev 25250 | Reverse the unit interval.... |
| iirevcn 25251 | The reversion function is ... |
| iihalf1 25252 | Map the first half of ` II... |
| iihalf1cn 25253 | The first half function is... |
| iihalf2 25254 | Map the second half of ` I... |
| iihalf2cn 25255 | The second half function i... |
| elii1 25256 | Divide the unit interval i... |
| elii2 25257 | Divide the unit interval i... |
| iimulcl 25258 | The unit interval is close... |
| iimulcn 25259 | Multiplication is a contin... |
| icoopnst 25260 | A half-open interval start... |
| iocopnst 25261 | A half-open interval endin... |
| icchmeo 25262 | The natural bijection from... |
| icopnfcnv 25263 | Define a bijection from ` ... |
| icopnfhmeo 25264 | The defined bijection from... |
| iccpnfcnv 25265 | Define a bijection from ` ... |
| iccpnfhmeo 25266 | The defined bijection from... |
| xrhmeo 25267 | The bijection from ` [ -u ... |
| xrhmph 25268 | The extended reals are hom... |
| xrcmp 25269 | The topology of the extend... |
| xrconn 25270 | The topology of the extend... |
| icccvx 25271 | A linear combination of tw... |
| oprpiece1res1 25272 | Restriction to the first p... |
| oprpiece1res2 25273 | Restriction to the second ... |
| cnrehmeo 25274 | The canonical bijection fr... |
| cnheiborlem 25275 | Lemma for ~ cnheibor . (C... |
| cnheibor 25276 | Heine-Borel theorem for co... |
| cnllycmp 25277 | The topology on the comple... |
| rellycmp 25278 | The topology on the reals ... |
| bndth 25279 | The Boundedness Theorem. ... |
| evth 25280 | The Extreme Value Theorem.... |
| evth2 25281 | The Extreme Value Theorem,... |
| lebnumlem1 25282 | Lemma for ~ lebnum . The ... |
| lebnumlem2 25283 | Lemma for ~ lebnum . As a... |
| lebnumlem3 25284 | Lemma for ~ lebnum . By t... |
| lebnum 25285 | The Lebesgue number lemma,... |
| xlebnum 25286 | Generalize ~ lebnum to ext... |
| lebnumii 25287 | Specialize the Lebesgue nu... |
| ishtpy 25293 | Membership in the class of... |
| htpycn 25294 | A homotopy is a continuous... |
| htpyi 25295 | A homotopy evaluated at it... |
| ishtpyd 25296 | Deduction for membership i... |
| htpycom 25297 | Given a homotopy from ` F ... |
| htpyid 25298 | A homotopy from a function... |
| htpyco1 25299 | Compose a homotopy with a ... |
| htpyco2 25300 | Compose a homotopy with a ... |
| htpycc 25301 | Concatenate two homotopies... |
| isphtpy 25302 | Membership in the class of... |
| phtpyhtpy 25303 | A path homotopy is a homot... |
| phtpycn 25304 | A path homotopy is a conti... |
| phtpyi 25305 | Membership in the class of... |
| phtpy01 25306 | Two path-homotopic paths h... |
| isphtpyd 25307 | Deduction for membership i... |
| isphtpy2d 25308 | Deduction for membership i... |
| phtpycom 25309 | Given a homotopy from ` F ... |
| phtpyid 25310 | A homotopy from a path to ... |
| phtpyco2 25311 | Compose a path homotopy wi... |
| phtpycc 25312 | Concatenate two path homot... |
| phtpcrel 25314 | The path homotopy relation... |
| isphtpc 25315 | The relation "is path homo... |
| phtpcer 25316 | Path homotopy is an equiva... |
| phtpc01 25317 | Path homotopic paths have ... |
| reparphti 25318 | Lemma for ~ reparpht . (C... |
| reparpht 25319 | Reparametrization lemma. ... |
| phtpcco2 25320 | Compose a path homotopy wi... |
| pcofval 25331 | The value of the path conc... |
| pcoval 25332 | The concatenation of two p... |
| pcovalg 25333 | Evaluate the concatenation... |
| pcoval1 25334 | Evaluate the concatenation... |
| pco0 25335 | The starting point of a pa... |
| pco1 25336 | The ending point of a path... |
| pcoval2 25337 | Evaluate the concatenation... |
| pcocn 25338 | The concatenation of two p... |
| copco 25339 | The composition of a conca... |
| pcohtpylem 25340 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25341 | Homotopy invariance of pat... |
| pcoptcl 25342 | A constant function is a p... |
| pcopt 25343 | Concatenation with a point... |
| pcopt2 25344 | Concatenation with a point... |
| pcoass 25345 | Order of concatenation doe... |
| pcorevcl 25346 | Closure for a reversed pat... |
| pcorevlem 25347 | Lemma for ~ pcorev . Prov... |
| pcorev 25348 | Concatenation with the rev... |
| pcorev2 25349 | Concatenation with the rev... |
| pcophtb 25350 | The path homotopy equivale... |
| om1val 25351 | The definition of the loop... |
| om1bas 25352 | The base set of the loop s... |
| om1elbas 25353 | Elementhood in the base se... |
| om1addcl 25354 | Closure of the group opera... |
| om1plusg 25355 | The group operation (which... |
| om1tset 25356 | The topology of the loop s... |
| om1opn 25357 | The topology of the loop s... |
| pi1val 25358 | The definition of the fund... |
| pi1bas 25359 | The base set of the fundam... |
| pi1blem 25360 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25361 | Another way to write the l... |
| pi1bas2 25362 | The base set of the fundam... |
| pi1eluni 25363 | Elementhood in the base se... |
| pi1bas3 25364 | The base set of the fundam... |
| pi1cpbl 25365 | The group operation, loop ... |
| elpi1 25366 | The elements of the fundam... |
| elpi1i 25367 | The elements of the fundam... |
| pi1addf 25368 | The group operation of ` p... |
| pi1addval 25369 | The concatenation of two p... |
| pi1grplem 25370 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25371 | The fundamental group is a... |
| pi1id 25372 | The identity element of th... |
| pi1inv 25373 | An inverse in the fundamen... |
| pi1xfrf 25374 | Functionality of the loop ... |
| pi1xfrval 25375 | The value of the loop tran... |
| pi1xfr 25376 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25377 | Given a path ` F ` between... |
| pi1xfrcnv 25378 | Given a path ` F ` between... |
| pi1xfrgim 25379 | The mapping ` G ` between ... |
| pi1cof 25380 | Functionality of the loop ... |
| pi1coval 25381 | The value of the loop tran... |
| pi1coghm 25382 | The mapping ` G ` between ... |
| isclm 25385 | A subcomplex module is a l... |
| clmsca 25386 | The ring of scalars ` F ` ... |
| clmsubrg 25387 | The base set of the ring o... |
| clmlmod 25388 | A subcomplex module is a l... |
| clmgrp 25389 | A subcomplex module is an ... |
| clmabl 25390 | A subcomplex module is an ... |
| clmring 25391 | The scalar ring of a subco... |
| clmfgrp 25392 | The scalar ring of a subco... |
| clm0 25393 | The zero of the scalar rin... |
| clm1 25394 | The identity of the scalar... |
| clmadd 25395 | The addition of the scalar... |
| clmmul 25396 | The multiplication of the ... |
| clmcj 25397 | The conjugation of the sca... |
| isclmi 25398 | Reverse direction of ~ isc... |
| clmzss 25399 | The scalar ring of a subco... |
| clmsscn 25400 | The scalar ring of a subco... |
| clmsub 25401 | Subtraction in the scalar ... |
| clmneg 25402 | Negation in the scalar rin... |
| clmneg1 25403 | Minus one is in the scalar... |
| clmabs 25404 | Norm in the scalar ring of... |
| clmacl 25405 | Closure of ring addition f... |
| clmmcl 25406 | Closure of ring multiplica... |
| clmsubcl 25407 | Closure of ring subtractio... |
| lmhmclm 25408 | The domain of a linear ope... |
| clmvscl 25409 | Closure of scalar product ... |
| clmvsass 25410 | Scalar multiplication is a... |
| clmvscom 25411 | Commutative law for the sc... |
| clmvsdir 25412 | Distributive law for scala... |
| clmvsdi 25413 | Distributive law for scala... |
| clmvs1 25414 | Scalar product with ring u... |
| clmvs2 25415 | A vector plus itself is tw... |
| clm0vs 25416 | Zero times a vector is the... |
| clmopfne 25417 | The (functionalized) opera... |
| isclmp 25418 | The predicate "is a subcom... |
| isclmi0 25419 | Properties that determine ... |
| clmvneg1 25420 | Minus 1 times a vector is ... |
| clmvsneg 25421 | Multiplication of a vector... |
| clmmulg 25422 | The group multiple functio... |
| clmsubdir 25423 | Scalar multiplication dist... |
| clmpm1dir 25424 | Subtractive distributive l... |
| clmnegneg 25425 | Double negative of a vecto... |
| clmnegsubdi2 25426 | Distribution of negative o... |
| clmsub4 25427 | Rearrangement of 4 terms i... |
| clmvsrinv 25428 | A vector minus itself. (C... |
| clmvslinv 25429 | Minus a vector plus itself... |
| clmvsubval 25430 | Value of vector subtractio... |
| clmvsubval2 25431 | Value of vector subtractio... |
| clmvz 25432 | Two ways to express the ne... |
| zlmclm 25433 | The ` ZZ ` -module operati... |
| clmzlmvsca 25434 | The scalar product of a su... |
| nmoleub2lem 25435 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25436 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25437 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25438 | The operator norm is the s... |
| nmoleub2b 25439 | The operator norm is the s... |
| nmoleub3 25440 | The operator norm is the s... |
| nmhmcn 25441 | A linear operator over a n... |
| cmodscexp 25442 | The powers of ` _i ` belon... |
| cmodscmulexp 25443 | The scalar product of a ve... |
| cvslvec 25446 | A subcomplex vector space ... |
| cvsclm 25447 | A subcomplex vector space ... |
| iscvs 25448 | A subcomplex vector space ... |
| iscvsp 25449 | The predicate "is a subcom... |
| iscvsi 25450 | Properties that determine ... |
| cvsi 25451 | The properties of a subcom... |
| cvsunit 25452 | Unit group of the scalar r... |
| cvsdiv 25453 | Division of the scalar rin... |
| cvsdivcl 25454 | The scalar field of a subc... |
| cvsmuleqdivd 25455 | An equality involving rati... |
| cvsdiveqd 25456 | An equality involving rati... |
| cnlmodlem1 25457 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25458 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25459 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25460 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25461 | The set of complex numbers... |
| cnstrcvs 25462 | The set of complex numbers... |
| cnrbas 25463 | The set of complex numbers... |
| cnrlmod 25464 | The complex left module of... |
| cnrlvec 25465 | The complex left module of... |
| cncvs 25466 | The complex left module of... |
| recvs 25467 | The field of the real numb... |
| qcvs 25468 | The field of rational numb... |
| zclmncvs 25469 | The ring of integers as le... |
| isncvsngp 25470 | A normed subcomplex vector... |
| isncvsngpd 25471 | Properties that determine ... |
| ncvsi 25472 | The properties of a normed... |
| ncvsprp 25473 | Proportionality property o... |
| ncvsge0 25474 | The norm of a scalar produ... |
| ncvsm1 25475 | The norm of the opposite o... |
| ncvsdif 25476 | The norm of the difference... |
| ncvspi 25477 | The norm of a vector plus ... |
| ncvs1 25478 | From any nonzero vector of... |
| cnrnvc 25479 | The module of complex numb... |
| cnncvs 25480 | The module of complex numb... |
| cnnm 25481 | The norm of the normed sub... |
| ncvspds 25482 | Value of the distance func... |
| cnindmet 25483 | The metric induced on the ... |
| cnncvsaddassdemo 25484 | Derive the associative law... |
| cnncvsmulassdemo 25485 | Derive the associative law... |
| cnncvsabsnegdemo 25486 | Derive the absolute value ... |
| iscph 25491 | A subcomplex pre-Hilbert s... |
| cphphl 25492 | A subcomplex pre-Hilbert s... |
| cphnlm 25493 | A subcomplex pre-Hilbert s... |
| cphngp 25494 | A subcomplex pre-Hilbert s... |
| cphlmod 25495 | A subcomplex pre-Hilbert s... |
| cphlvec 25496 | A subcomplex pre-Hilbert s... |
| cphnvc 25497 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25498 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25499 | Lemma for ~ cphreccl . (C... |
| cphsca 25500 | A subcomplex pre-Hilbert s... |
| cphsubrg 25501 | The scalar field of a subc... |
| cphreccl 25502 | The scalar field of a subc... |
| cphdivcl 25503 | The scalar field of a subc... |
| cphcjcl 25504 | The scalar field of a subc... |
| cphsqrtcl 25505 | The scalar field of a subc... |
| cphabscl 25506 | The scalar field of a subc... |
| cphsqrtcl2 25507 | The scalar field of a subc... |
| cphsqrtcl3 25508 | If the scalar field of a s... |
| cphqss 25509 | The scalar field of a subc... |
| cphclm 25510 | A subcomplex pre-Hilbert s... |
| cphnmvs 25511 | Norm of a scalar product. ... |
| cphipcl 25512 | An inner product is a memb... |
| cphnmfval 25513 | The value of the norm in a... |
| cphnm 25514 | The square of the norm is ... |
| nmsq 25515 | The square of the norm is ... |
| cphnmf 25516 | The norm of a vector is a ... |
| cphnmcl 25517 | The norm of a vector is a ... |
| reipcl 25518 | An inner product of an ele... |
| ipge0 25519 | The inner product in a sub... |
| cphipcj 25520 | Conjugate of an inner prod... |
| cphipipcj 25521 | An inner product times its... |
| cphorthcom 25522 | Orthogonality (meaning inn... |
| cphip0l 25523 | Inner product with a zero ... |
| cphip0r 25524 | Inner product with a zero ... |
| cphipeq0 25525 | The inner product of a vec... |
| cphdir 25526 | Distributive law for inner... |
| cphdi 25527 | Distributive law for inner... |
| cph2di 25528 | Distributive law for inner... |
| cphsubdir 25529 | Distributive law for inner... |
| cphsubdi 25530 | Distributive law for inner... |
| cph2subdi 25531 | Distributive law for inner... |
| cphass 25532 | Associative law for inner ... |
| cphassr 25533 | "Associative" law for seco... |
| cph2ass 25534 | Move scalar multiplication... |
| cphassi 25535 | Associative law for the fi... |
| cphassir 25536 | "Associative" law for the ... |
| cphpyth 25537 | The pythagorean theorem fo... |
| tcphex 25538 | Lemma for ~ tcphbas and si... |
| tcphval 25539 | Define a function to augme... |
| tcphbas 25540 | The base set of a subcompl... |
| tchplusg 25541 | The addition operation of ... |
| tcphsub 25542 | The subtraction operation ... |
| tcphmulr 25543 | The ring operation of a su... |
| tcphsca 25544 | The scalar field of a subc... |
| tcphvsca 25545 | The scalar multiplication ... |
| tcphip 25546 | The inner product of a sub... |
| tcphtopn 25547 | The topology of a subcompl... |
| tcphphl 25548 | Augmentation of a subcompl... |
| tchnmfval 25549 | The norm of a subcomplex p... |
| tcphnmval 25550 | The norm of a subcomplex p... |
| cphtcphnm 25551 | The norm of a norm-augment... |
| tcphds 25552 | The distance of a pre-Hilb... |
| phclm 25553 | A pre-Hilbert space whose ... |
| tcphcphlem3 25554 | Lemma for ~ tcphcph : real... |
| ipcau2 25555 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25556 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25557 | Lemma for ~ tcphcph : homo... |
| tcphcph 25558 | The standard definition of... |
| ipcau 25559 | The Cauchy-Schwarz inequal... |
| nmparlem 25560 | Lemma for ~ nmpar . (Cont... |
| nmpar 25561 | A subcomplex pre-Hilbert s... |
| cphipval2 25562 | Value of the inner product... |
| 4cphipval2 25563 | Four times the inner produ... |
| cphipval 25564 | Value of the inner product... |
| ipcnlem2 25565 | The inner product operatio... |
| ipcnlem1 25566 | The inner product operatio... |
| ipcn 25567 | The inner product operatio... |
| cnmpt1ip 25568 | Continuity of inner produc... |
| cnmpt2ip 25569 | Continuity of inner produc... |
| csscld 25570 | A "closed subspace" in a s... |
| clsocv 25571 | The orthogonal complement ... |
| cphsscph 25572 | A subspace of a subcomplex... |
| lmmbr 25579 | Express the binary relatio... |
| lmmbr2 25580 | Express the binary relatio... |
| lmmbr3 25581 | Express the binary relatio... |
| lmmcvg 25582 | Convergence property of a ... |
| lmmbrf 25583 | Express the binary relatio... |
| lmnn 25584 | A condition that implies c... |
| cfilfval 25585 | The set of Cauchy filters ... |
| iscfil 25586 | The property of being a Ca... |
| iscfil2 25587 | The property of being a Ca... |
| cfilfil 25588 | A Cauchy filter is a filte... |
| cfili 25589 | Property of a Cauchy filte... |
| cfil3i 25590 | A Cauchy filter contains b... |
| cfilss 25591 | A filter finer than a Cauc... |
| fgcfil 25592 | The Cauchy filter conditio... |
| fmcfil 25593 | The Cauchy filter conditio... |
| iscfil3 25594 | A filter is Cauchy iff it ... |
| cfilfcls 25595 | Similar to ultrafilters ( ... |
| caufval 25596 | The set of Cauchy sequence... |
| iscau 25597 | Express the property " ` F... |
| iscau2 25598 | Express the property " ` F... |
| iscau3 25599 | Express the Cauchy sequenc... |
| iscau4 25600 | Express the property " ` F... |
| iscauf 25601 | Express the property " ` F... |
| caun0 25602 | A metric with a Cauchy seq... |
| caufpm 25603 | Inclusion of a Cauchy sequ... |
| caucfil 25604 | A Cauchy sequence predicat... |
| iscmet 25605 | The property " ` D ` is a ... |
| cmetcvg 25606 | The convergence of a Cauch... |
| cmetmet 25607 | A complete metric space is... |
| cmetmeti 25608 | A complete metric space is... |
| cmetcaulem 25609 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25610 | The convergence of a Cauch... |
| iscmet3lem3 25611 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25612 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25613 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25614 | The property " ` D ` is a ... |
| iscmet2 25615 | A metric ` D ` is complete... |
| cfilresi 25616 | A Cauchy filter on a metri... |
| cfilres 25617 | Cauchy filter on a metric ... |
| caussi 25618 | Cauchy sequence on a metri... |
| causs 25619 | Cauchy sequence on a metri... |
| equivcfil 25620 | If the metric ` D ` is "st... |
| equivcau 25621 | If the metric ` D ` is "st... |
| lmle 25622 | If the distance from each ... |
| nglmle 25623 | If the norm of each member... |
| lmclim 25624 | Relate a limit on the metr... |
| lmclimf 25625 | Relate a limit on the metr... |
| metelcls 25626 | A point belongs to the clo... |
| metcld 25627 | A subset of a metric space... |
| metcld2 25628 | A subset of a metric space... |
| caubl 25629 | Sufficient condition to en... |
| caublcls 25630 | The convergent point of a ... |
| metcnp4 25631 | Two ways to say a mapping ... |
| metcn4 25632 | Two ways to say a mapping ... |
| iscmet3i 25633 | Properties that determine ... |
| lmcau 25634 | Every convergent sequence ... |
| flimcfil 25635 | Every convergent filter in... |
| metsscmetcld 25636 | A complete subspace of a m... |
| cmetss 25637 | A subspace of a complete m... |
| equivcmet 25638 | If two metrics are strongl... |
| relcmpcmet 25639 | If ` D ` is a metric space... |
| cmpcmet 25640 | A compact metric space is ... |
| cfilucfil3 25641 | Given a metric ` D ` and a... |
| cfilucfil4 25642 | Given a metric ` D ` and a... |
| cncmet 25643 | The set of complex numbers... |
| recmet 25644 | The real numbers are a com... |
| bcthlem1 25645 | Lemma for ~ bcth . Substi... |
| bcthlem2 25646 | Lemma for ~ bcth . The ba... |
| bcthlem3 25647 | Lemma for ~ bcth . The li... |
| bcthlem4 25648 | Lemma for ~ bcth . Given ... |
| bcthlem5 25649 | Lemma for ~ bcth . The pr... |
| bcth 25650 | Baire's Category Theorem. ... |
| bcth2 25651 | Baire's Category Theorem, ... |
| bcth3 25652 | Baire's Category Theorem, ... |
| isbn 25659 | A Banach space is a normed... |
| bnsca 25660 | The scalar field of a Bana... |
| bnnvc 25661 | A Banach space is a normed... |
| bnnlm 25662 | A Banach space is a normed... |
| bnngp 25663 | A Banach space is a normed... |
| bnlmod 25664 | A Banach space is a left m... |
| bncms 25665 | A Banach space is a comple... |
| iscms 25666 | A complete metric space is... |
| cmscmet 25667 | The induced metric on a co... |
| bncmet 25668 | The induced metric on Bana... |
| cmsms 25669 | A complete metric space is... |
| cmspropd 25670 | Property deduction for a c... |
| cmssmscld 25671 | The restriction of a metri... |
| cmsss 25672 | The restriction of a compl... |
| lssbn 25673 | A subspace of a Banach spa... |
| cmetcusp1 25674 | If the uniform set of a co... |
| cmetcusp 25675 | The uniform space generate... |
| cncms 25676 | The field of complex numbe... |
| cnflduss 25677 | The uniform structure of t... |
| cnfldcusp 25678 | The field of complex numbe... |
| resscdrg 25679 | The real numbers are a sub... |
| cncdrg 25680 | The only complete subfield... |
| srabn 25681 | The subring algebra over a... |
| rlmbn 25682 | The ring module over a com... |
| ishl 25683 | The predicate "is a subcom... |
| hlbn 25684 | Every subcomplex Hilbert s... |
| hlcph 25685 | Every subcomplex Hilbert s... |
| hlphl 25686 | Every subcomplex Hilbert s... |
| hlcms 25687 | Every subcomplex Hilbert s... |
| hlprlem 25688 | Lemma for ~ hlpr . (Contr... |
| hlress 25689 | The scalar field of a subc... |
| hlpr 25690 | The scalar field of a subc... |
| ishl2 25691 | A Hilbert space is a compl... |
| cphssphl 25692 | A Banach subspace of a sub... |
| cmslssbn 25693 | A complete linear subspace... |
| cmscsscms 25694 | A closed subspace of a com... |
| bncssbn 25695 | A closed subspace of a Ban... |
| cssbn 25696 | A complete subspace of a n... |
| csschl 25697 | A complete subspace of a c... |
| cmslsschl 25698 | A complete linear subspace... |
| chlcsschl 25699 | A closed subspace of a sub... |
| retopn 25700 | The topology of the real n... |
| recms 25701 | The real numbers form a co... |
| reust 25702 | The Uniform structure of t... |
| recusp 25703 | The real numbers form a co... |
| rrxval 25708 | Value of the generalized E... |
| rrxbase 25709 | The base of the generalize... |
| rrxprds 25710 | Expand the definition of t... |
| rrxip 25711 | The inner product of the g... |
| rrxnm 25712 | The norm of the generalize... |
| rrxcph 25713 | Generalized Euclidean real... |
| rrxds 25714 | The distance over generali... |
| rrxvsca 25715 | The scalar product over ge... |
| rrxplusgvscavalb 25716 | The result of the addition... |
| rrxsca 25717 | The field of real numbers ... |
| rrx0 25718 | The zero ("origin") in a g... |
| rrx0el 25719 | The zero ("origin") in a g... |
| csbren 25720 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25721 | Triangle inequality in R^n... |
| rrxf 25722 | Euclidean vectors as funct... |
| rrxfsupp 25723 | Euclidean vectors are of f... |
| rrxsuppss 25724 | Support of Euclidean vecto... |
| rrxmvallem 25725 | Support of the function us... |
| rrxmval 25726 | The value of the Euclidean... |
| rrxmfval 25727 | The value of the Euclidean... |
| rrxmetlem 25728 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25729 | Euclidean space is a metri... |
| rrxdstprj1 25730 | The distance between two p... |
| rrxbasefi 25731 | The base of the generalize... |
| rrxdsfi 25732 | The distance over generali... |
| rrxmetfi 25733 | Euclidean space is a metri... |
| rrxdsfival 25734 | The value of the Euclidean... |
| ehlval 25735 | Value of the Euclidean spa... |
| ehlbase 25736 | The base of the Euclidean ... |
| ehl0base 25737 | The base of the Euclidean ... |
| ehl0 25738 | The Euclidean space of dim... |
| ehleudis 25739 | The Euclidean distance fun... |
| ehleudisval 25740 | The value of the Euclidean... |
| ehl1eudis 25741 | The Euclidean distance fun... |
| ehl1eudisval 25742 | The value of the Euclidean... |
| ehl2eudis 25743 | The Euclidean distance fun... |
| ehl2eudisval 25744 | The value of the Euclidean... |
| minveclem1 25745 | Lemma for ~ minvec . The ... |
| minveclem4c 25746 | Lemma for ~ minvec . The ... |
| minveclem2 25747 | Lemma for ~ minvec . Any ... |
| minveclem3a 25748 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25749 | Lemma for ~ minvec . The ... |
| minveclem3 25750 | Lemma for ~ minvec . The ... |
| minveclem4a 25751 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25752 | Lemma for ~ minvec . The ... |
| minveclem4 25753 | Lemma for ~ minvec . The ... |
| minveclem5 25754 | Lemma for ~ minvec . Disc... |
| minveclem6 25755 | Lemma for ~ minvec . Any ... |
| minveclem7 25756 | Lemma for ~ minvec . Sinc... |
| minvec 25757 | Minimizing vector theorem,... |
| pjthlem1 25758 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25759 | Lemma for ~ pjth . (Contr... |
| pjth 25760 | Projection Theorem: Any H... |
| pjth2 25761 | Projection Theorem with ab... |
| cldcss 25762 | Corollary of the Projectio... |
| cldcss2 25763 | Corollary of the Projectio... |
| hlhil 25764 | Corollary of the Projectio... |
| addcncf 25765 | The addition of two contin... |
| subcncf 25766 | The subtraction of two con... |
| mulcncf 25767 | The multiplication of two ... |
| divcncf 25768 | The quotient of two contin... |
| pmltpclem1 25769 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25770 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25771 | Any function on the reals ... |
| ivthlem1 25772 | Lemma for ~ ivth . The se... |
| ivthlem2 25773 | Lemma for ~ ivth . Show t... |
| ivthlem3 25774 | Lemma for ~ ivth , the int... |
| ivth 25775 | The intermediate value the... |
| ivth2 25776 | The intermediate value the... |
| ivthle 25777 | The intermediate value the... |
| ivthle2 25778 | The intermediate value the... |
| ivthicc 25779 | The interval between any t... |
| evthicc 25780 | Specialization of the Extr... |
| evthicc2 25781 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25782 | A continuous function on a... |
| ovolfcl 25787 | Closure for the interval e... |
| ovolfioo 25788 | Unpack the interval coveri... |
| ovolficc 25789 | Unpack the interval coveri... |
| ovolficcss 25790 | Any (closed) interval cove... |
| ovolfsval 25791 | The value of the interval ... |
| ovolfsf 25792 | Closure for the interval l... |
| ovolsf 25793 | Closure for the partial su... |
| ovolval 25794 | The value of the outer mea... |
| elovolmlem 25795 | Lemma for ~ elovolm and re... |
| elovolm 25796 | Elementhood in the set ` M... |
| elovolmr 25797 | Sufficient condition for e... |
| ovolmge0 25798 | The set ` M ` is composed ... |
| ovolcl 25799 | The volume of a set is an ... |
| ovollb 25800 | The outer volume is a lowe... |
| ovolgelb 25801 | The outer volume is the gr... |
| ovolge0 25802 | The volume of a set is alw... |
| ovolf 25803 | The domain and codomain of... |
| ovollecl 25804 | If an outer volume is boun... |
| ovolsslem 25805 | Lemma for ~ ovolss . (Con... |
| ovolss 25806 | The volume of a set is mon... |
| ovolsscl 25807 | If a set is contained in a... |
| ovolssnul 25808 | A subset of a nullset is n... |
| ovollb2lem 25809 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25810 | It is often more convenien... |
| ovolctb 25811 | The volume of a denumerabl... |
| ovolq 25812 | The rational numbers have ... |
| ovolctb2 25813 | The volume of a countable ... |
| ovol0 25814 | The empty set has 0 outer ... |
| ovolfi 25815 | A finite set has 0 outer L... |
| ovolsn 25816 | A singleton has 0 outer Le... |
| ovolunlem1a 25817 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25818 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25819 | Lemma for ~ ovolun . (Con... |
| ovolun 25820 | The Lebesgue outer measure... |
| ovolunnul 25821 | Adding a nullset does not ... |
| ovolfiniun 25822 | The Lebesgue outer measure... |
| ovoliunlem1 25823 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25824 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25825 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25826 | The Lebesgue outer measure... |
| ovoliun2 25827 | The Lebesgue outer measure... |
| ovoliunnul 25828 | A countable union of nulls... |
| shft2rab 25829 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25830 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25831 | Lemma for ~ ovolshft . (C... |
| ovolshft 25832 | The Lebesgue outer measure... |
| sca2rab 25833 | If ` B ` is a scale of ` A... |
| ovolscalem1 25834 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25835 | Lemma for ~ ovolshft . (C... |
| ovolsca 25836 | The Lebesgue outer measure... |
| ovolicc1 25837 | The measure of a closed in... |
| ovolicc2lem1 25838 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25839 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25840 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25841 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25842 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25843 | The measure of a closed in... |
| ovolicc 25844 | The measure of a closed in... |
| ovolicopnf 25845 | The measure of a right-unb... |
| ovolre 25846 | The measure of the real nu... |
| ismbl 25847 | The predicate " ` A ` is L... |
| ismbl2 25848 | From ~ ovolun , it suffice... |
| volres 25849 | A self-referencing abbrevi... |
| volf 25850 | The domain and codomain of... |
| mblvol 25851 | The volume of a measurable... |
| mblss 25852 | A measurable set is a subs... |
| mblsplit 25853 | The defining property of m... |
| volss 25854 | The Lebesgue measure is mo... |
| cmmbl 25855 | The complement of a measur... |
| nulmbl 25856 | A nullset is measurable. ... |
| nulmbl2 25857 | A set of outer measure zer... |
| unmbl 25858 | A union of measurable sets... |
| shftmbl 25859 | A shift of a measurable se... |
| 0mbl 25860 | The empty set is measurabl... |
| rembl 25861 | The set of all real number... |
| unidmvol 25862 | The union of the Lebesgue ... |
| inmbl 25863 | An intersection of measura... |
| difmbl 25864 | A difference of measurable... |
| finiunmbl 25865 | A finite union of measurab... |
| volun 25866 | The Lebesgue measure funct... |
| volinun 25867 | Addition of non-disjoint s... |
| volfiniun 25868 | The volume of a disjoint f... |
| iundisj 25869 | Rewrite a countable union ... |
| iundisj2 25870 | A disjoint union is disjoi... |
| voliunlem1 25871 | Lemma for ~ voliun . (Con... |
| voliunlem2 25872 | Lemma for ~ voliun . (Con... |
| voliunlem3 25873 | Lemma for ~ voliun . (Con... |
| iunmbl 25874 | The measurable sets are cl... |
| voliun 25875 | The Lebesgue measure funct... |
| volsuplem 25876 | Lemma for ~ volsup . (Con... |
| volsup 25877 | The volume of the limit of... |
| iunmbl2 25878 | The measurable sets are cl... |
| ioombl1lem1 25879 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25880 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25881 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25882 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25883 | An open right-unbounded in... |
| icombl1 25884 | A closed unbounded-above i... |
| icombl 25885 | A closed-below, open-above... |
| ioombl 25886 | An open real interval is m... |
| iccmbl 25887 | A closed real interval is ... |
| iccvolcl 25888 | A closed real interval has... |
| ovolioo 25889 | The measure of an open int... |
| volioo 25890 | The measure of an open int... |
| ioovolcl 25891 | An open real interval has ... |
| ovolfs2 25892 | Alternative expression for... |
| ioorcl2 25893 | An open interval with fini... |
| ioorf 25894 | Define a function from ope... |
| ioorval 25895 | Define a function from ope... |
| ioorinv2 25896 | The function ` F ` is an "... |
| ioorinv 25897 | The function ` F ` is an "... |
| ioorcl 25898 | The function ` F ` does no... |
| uniiccdif 25899 | A union of closed interval... |
| uniioovol 25900 | A disjoint union of open i... |
| uniiccvol 25901 | An almost-disjoint union o... |
| uniioombllem1 25902 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25903 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25904 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25905 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25906 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25907 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25908 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25909 | Lemma for ~ uniioombl . (... |
| uniioombl 25910 | A disjoint union of open i... |
| uniiccmbl 25911 | An almost-disjoint union o... |
| dyadf 25912 | The function ` F ` returns... |
| dyadval 25913 | Value of the dyadic ration... |
| dyadovol 25914 | Volume of a dyadic rationa... |
| dyadss 25915 | Two closed dyadic rational... |
| dyaddisjlem 25916 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25917 | Two closed dyadic rational... |
| dyadmaxlem 25918 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25919 | Any nonempty set of dyadic... |
| dyadmbllem 25920 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25921 | Any union of dyadic ration... |
| opnmbllem 25922 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25923 | All open sets are measurab... |
| opnmblALT 25924 | All open sets are measurab... |
| subopnmbl 25925 | Sets which are open in a m... |
| volsup2 25926 | The volume of ` A ` is the... |
| volcn 25927 | The function formed by res... |
| volivth 25928 | The Intermediate Value The... |
| vitalilem1 25929 | Lemma for ~ vitali . (Con... |
| vitalilem2 25930 | Lemma for ~ vitali . (Con... |
| vitalilem3 25931 | Lemma for ~ vitali . (Con... |
| vitalilem4 25932 | Lemma for ~ vitali . (Con... |
| vitalilem5 25933 | Lemma for ~ vitali . (Con... |
| vitali 25934 | If the reals can be well-o... |
| ismbf1 25945 | The predicate " ` F ` is a... |
| mbff 25946 | A measurable function is a... |
| mbfdm 25947 | The domain of a measurable... |
| mbfconstlem 25948 | Lemma for ~ mbfconst and r... |
| ismbf 25949 | The predicate " ` F ` is a... |
| ismbfcn 25950 | A complex function is meas... |
| mbfima 25951 | Definitional property of a... |
| mbfimaicc 25952 | The preimage of any closed... |
| mbfimasn 25953 | The preimage of a point un... |
| mbfconst 25954 | A constant function is mea... |
| mbf0 25955 | The empty function is meas... |
| mbfid 25956 | The identity function is m... |
| mbfmptcl 25957 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25958 | The domain of a measurable... |
| ismbfcn2 25959 | A complex function is meas... |
| ismbfd 25960 | Deduction to prove measura... |
| ismbf2d 25961 | Deduction to prove measura... |
| mbfeqalem1 25962 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25963 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25964 | If two functions are equal... |
| mbfres 25965 | The restriction of a measu... |
| mbfres2 25966 | Measurability of a piecewi... |
| mbfss 25967 | Change the domain of a mea... |
| mbfmulc2lem 25968 | Multiplication by a consta... |
| mbfmulc2re 25969 | Multiplication by a consta... |
| mbfmax 25970 | The maximum of two functio... |
| mbfneg 25971 | The negative of a measurab... |
| mbfpos 25972 | The positive part of a mea... |
| mbfposr 25973 | Converse to ~ mbfpos . (C... |
| mbfposb 25974 | A function is measurable i... |
| ismbf3d 25975 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25976 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25977 | The preimage of any open s... |
| mbfimaopn2 25978 | The preimage of any set op... |
| cncombf 25979 | The composition of a conti... |
| cnmbf 25980 | A continuous function is m... |
| mbfaddlem 25981 | The sum of two measurable ... |
| mbfadd 25982 | The sum of two measurable ... |
| mbfsub 25983 | The difference of two meas... |
| mbfmulc2 25984 | A complex constant times a... |
| mbfsup 25985 | The supremum of a sequence... |
| mbfinf 25986 | The infimum of a sequence ... |
| mbflimsup 25987 | The limit supremum of a se... |
| mbflimlem 25988 | The pointwise limit of a s... |
| mbflim 25989 | The pointwise limit of a s... |
| 0pval 25992 | The zero function evaluate... |
| 0plef 25993 | Two ways to say that the f... |
| 0pledm 25994 | Adjust the domain of the l... |
| isi1f 25995 | The predicate " ` F ` is a... |
| i1fmbf 25996 | Simple functions are measu... |
| i1ff 25997 | A simple function is a fun... |
| i1frn 25998 | A simple function has fini... |
| i1fima 25999 | Any preimage of a simple f... |
| i1fima2 26000 | Any preimage of a simple f... |
| i1fima2sn 26001 | Preimage of a singleton. ... |
| i1fd 26002 | A simplified set of assump... |
| i1f0rn 26003 | Any simple function takes ... |
| itg1val 26004 | The value of the integral ... |
| itg1val2 26005 | The value of the integral ... |
| itg1cl 26006 | Closure of the integral on... |
| itg1ge0 26007 | Closure of the integral on... |
| i1f0 26008 | The zero function is simpl... |
| itg10 26009 | The zero function has zero... |
| i1f1lem 26010 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 26011 | Base case simple functions... |
| itg11 26012 | The integral of an indicat... |
| itg1addlem1 26013 | Decompose a preimage, whic... |
| i1faddlem 26014 | Decompose the preimage of ... |
| i1fmullem 26015 | Decompose the preimage of ... |
| i1fadd 26016 | The sum of two simple func... |
| i1fmul 26017 | The pointwise product of t... |
| itg1addlem2 26018 | Lemma for ~ itg1add . The... |
| itg1addlem3 26019 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 26020 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 26021 | Lemma for ~ itg1add . (Co... |
| itg1add 26022 | The integral of a sum of s... |
| i1fmulclem 26023 | Decompose the preimage of ... |
| i1fmulc 26024 | A nonnegative constant tim... |
| itg1mulc 26025 | The integral of a constant... |
| i1fres 26026 | The "restriction" of a sim... |
| i1fpos 26027 | The positive part of a sim... |
| i1fposd 26028 | Deduction form of ~ i1fpos... |
| i1fsub 26029 | The difference of two simp... |
| itg1sub 26030 | The integral of a differen... |
| itg10a 26031 | The integral of a simple f... |
| itg1ge0a 26032 | The integral of an almost ... |
| itg1lea 26033 | Approximate version of ~ i... |
| itg1le 26034 | If one simple function dom... |
| itg1climres 26035 | Restricting the simple fun... |
| mbfi1fseqlem1 26036 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 26037 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 26038 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 26039 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 26040 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 26041 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 26042 | A characterization of meas... |
| mbfi1flimlem 26043 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 26044 | Any real measurable functi... |
| mbfmullem2 26045 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 26046 | Lemma for ~ mbfmul . (Con... |
| mbfmul 26047 | The product of two measura... |
| itg2lcl 26048 | The set of lower sums is a... |
| itg2val 26049 | Value of the integral on n... |
| itg2l 26050 | Elementhood in the set ` L... |
| itg2lr 26051 | Sufficient condition for e... |
| xrge0f 26052 | A real function is a nonne... |
| itg2cl 26053 | The integral of a nonnegat... |
| itg2ub 26054 | The integral of a nonnegat... |
| itg2leub 26055 | Any upper bound on the int... |
| itg2ge0 26056 | The integral of a nonnegat... |
| itg2itg1 26057 | The integral of a nonnegat... |
| itg20 26058 | The integral of the zero f... |
| itg2lecl 26059 | If an ` S.2 ` integral is ... |
| itg2le 26060 | If one function dominates ... |
| itg2const 26061 | Integral of a constant fun... |
| itg2const2 26062 | When the base set of a con... |
| itg2seq 26063 | Definitional property of t... |
| itg2uba 26064 | Approximate version of ~ i... |
| itg2lea 26065 | Approximate version of ~ i... |
| itg2eqa 26066 | Approximate equality of in... |
| itg2mulclem 26067 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 26068 | The integral of a nonnegat... |
| itg2splitlem 26069 | Lemma for ~ itg2split . (... |
| itg2split 26070 | The ` S.2 ` integral split... |
| itg2monolem1 26071 | Lemma for ~ itg2mono . We... |
| itg2monolem2 26072 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 26073 | Lemma for ~ itg2mono . (C... |
| itg2mono 26074 | The Monotone Convergence T... |
| itg2i1fseqle 26075 | Subject to the conditions ... |
| itg2i1fseq 26076 | Subject to the conditions ... |
| itg2i1fseq2 26077 | In an extension to the res... |
| itg2i1fseq3 26078 | Special case of ~ itg2i1fs... |
| itg2addlem 26079 | Lemma for ~ itg2add . (Co... |
| itg2add 26080 | The ` S.2 ` integral is li... |
| itg2gt0 26081 | If the function ` F ` is s... |
| itg2cnlem1 26082 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 26083 | Lemma for ~ itgcn . (Cont... |
| itg2cn 26084 | A sort of absolute continu... |
| ibllem 26085 | Conditioned equality theor... |
| isibl 26086 | The predicate " ` F ` is i... |
| isibl2 26087 | The predicate " ` F ` is i... |
| iblmbf 26088 | An integrable function is ... |
| iblitg 26089 | If a function is integrabl... |
| dfitg 26090 | Evaluate the class substit... |
| itgex 26091 | An integral is a set. (Co... |
| itgeq1f 26092 | Equality theorem for an in... |
| itgeq1 26093 | Equality theorem for an in... |
| nfitg1 26094 | Bound-variable hypothesis ... |
| nfitg 26095 | Bound-variable hypothesis ... |
| cbvitg 26096 | Change bound variable in a... |
| cbvitgv 26097 | Change bound variable in a... |
| itgeq2 26098 | Equality theorem for an in... |
| itgresr 26099 | The domain of an integral ... |
| itg0 26100 | The integral of anything o... |
| itgz 26101 | The integral of zero on an... |
| itgeq2dv 26102 | Equality theorem for an in... |
| itgmpt 26103 | Change bound variable in a... |
| itgcl 26104 | The integral of an integra... |
| itgvallem 26105 | Substitution lemma. (Cont... |
| itgvallem3 26106 | Lemma for ~ itgposval and ... |
| ibl0 26107 | The zero function is integ... |
| iblcnlem1 26108 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 26109 | Expand out the universal q... |
| itgcnlem 26110 | Expand out the sum in ~ df... |
| iblrelem 26111 | Integrability of a real fu... |
| iblposlem 26112 | Lemma for ~ iblpos . (Con... |
| iblpos 26113 | Integrability of a nonnega... |
| iblre 26114 | Integrability of a real fu... |
| itgrevallem1 26115 | Lemma for ~ itgposval and ... |
| itgposval 26116 | The integral of a nonnegat... |
| itgreval 26117 | Decompose the integral of ... |
| itgrecl 26118 | Real closure of an integra... |
| iblcn 26119 | Integrability of a complex... |
| itgcnval 26120 | Decompose the integral of ... |
| itgre 26121 | Real part of an integral. ... |
| itgim 26122 | Imaginary part of an integ... |
| iblneg 26123 | The negative of an integra... |
| itgneg 26124 | Negation of an integral. ... |
| iblss 26125 | A subset of an integrable ... |
| iblss2 26126 | Change the domain of an in... |
| itgitg2 26127 | Transfer an integral using... |
| i1fibl 26128 | A simple function is integ... |
| itgitg1 26129 | Transfer an integral using... |
| itgle 26130 | Monotonicity of an integra... |
| itgge0 26131 | The integral of a positive... |
| itgss 26132 | Expand the set of an integ... |
| itgss2 26133 | Expand the set of an integ... |
| itgeqa 26134 | Approximate equality of in... |
| itgss3 26135 | Expand the set of an integ... |
| itgioo 26136 | Equality of integrals on o... |
| itgless 26137 | Expand the integral of a n... |
| iblconst 26138 | A constant function is int... |
| itgconst 26139 | Integral of a constant fun... |
| ibladdlem 26140 | Lemma for ~ ibladd . (Con... |
| ibladd 26141 | Add two integrals over the... |
| iblsub 26142 | Subtract two integrals ove... |
| itgaddlem1 26143 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 26144 | Lemma for ~ itgadd . (Con... |
| itgadd 26145 | Add two integrals over the... |
| itgsub 26146 | Subtract two integrals ove... |
| itgfsum 26147 | Take a finite sum of integ... |
| iblabslem 26148 | Lemma for ~ iblabs . (Con... |
| iblabs 26149 | The absolute value of an i... |
| iblabsr 26150 | A measurable function is i... |
| iblmulc2 26151 | Multiply an integral by a ... |
| itgmulc2lem1 26152 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 26153 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 26154 | Multiply an integral by a ... |
| itgabs 26155 | The triangle inequality fo... |
| itgsplit 26156 | The ` S. ` integral splits... |
| itgspliticc 26157 | The ` S. ` integral splits... |
| itgsplitioo 26158 | The ` S. ` integral splits... |
| bddmulibl 26159 | A bounded function times a... |
| bddibl 26160 | A bounded function is inte... |
| cniccibl 26161 | A continuous function on a... |
| bddiblnc 26162 | Choice-free proof of ~ bdd... |
| cnicciblnc 26163 | Choice-free proof of ~ cni... |
| itggt0 26164 | The integral of a strictly... |
| itgcn 26165 | Transfer ~ itg2cn to the f... |
| ditgeq1 26168 | Equality theorem for the d... |
| ditgeq2 26169 | Equality theorem for the d... |
| ditgeq3 26170 | Equality theorem for the d... |
| ditgeq3dv 26171 | Equality theorem for the d... |
| ditgex 26172 | A directed integral is a s... |
| ditg0 26173 | Value of the directed inte... |
| cbvditg 26174 | Change bound variable in a... |
| cbvditgv 26175 | Change bound variable in a... |
| ditgpos 26176 | Value of the directed inte... |
| ditgneg 26177 | Value of the directed inte... |
| ditgcl 26178 | Closure of a directed inte... |
| ditgswap 26179 | Reverse a directed integra... |
| ditgsplitlem 26180 | Lemma for ~ ditgsplit . (... |
| ditgsplit 26181 | This theorem is the raison... |
| reldv 26190 | The derivative function is... |
| limcvallem 26191 | Lemma for ~ ellimc . (Con... |
| limcfval 26192 | Value and set bounds on th... |
| ellimc 26193 | Value of the limit predica... |
| limcrcl 26194 | Reverse closure for the li... |
| limccl 26195 | Closure of the limit opera... |
| limcdif 26196 | It suffices to consider fu... |
| ellimc2 26197 | Write the definition of a ... |
| limcnlp 26198 | If ` B ` is not a limit po... |
| ellimc3 26199 | Write the epsilon-delta de... |
| limcflflem 26200 | Lemma for ~ limcflf . (Co... |
| limcflf 26201 | The limit operator can be ... |
| limcmo 26202 | If ` B ` is a limit point ... |
| limcmpt 26203 | Express the limit operator... |
| limcmpt2 26204 | Express the limit operator... |
| limcresi 26205 | Any limit of ` F ` is also... |
| limcres 26206 | If ` B ` is an interior po... |
| cnplimc 26207 | A function is continuous a... |
| cnlimc 26208 | ` F ` is a continuous func... |
| cnlimci 26209 | If ` F ` is a continuous f... |
| cnmptlimc 26210 | If ` F ` is a continuous f... |
| limccnp 26211 | If the limit of ` F ` at `... |
| limccnp2 26212 | The image of a convergent ... |
| limcco 26213 | Composition of two limits.... |
| limciun 26214 | A point is a limit of ` F ... |
| limcun 26215 | A point is a limit of ` F ... |
| dvlem 26216 | Closure for a difference q... |
| dvfval 26217 | Value and set bounds on th... |
| eldv 26218 | The differentiable predica... |
| dvcl 26219 | The derivative function ta... |
| dvbssntr 26220 | The set of differentiable ... |
| dvbss 26221 | The set of differentiable ... |
| dvbsss 26222 | The set of differentiable ... |
| perfdvf 26223 | The derivative is a functi... |
| recnprss 26224 | Both ` RR ` and ` CC ` are... |
| recnperf 26225 | Both ` RR ` and ` CC ` are... |
| dvfg 26226 | Explicitly write out the f... |
| dvf 26227 | The derivative is a functi... |
| dvfcn 26228 | The derivative is a functi... |
| dvreslem 26229 | Lemma for ~ dvres . (Cont... |
| dvres2lem 26230 | Lemma for ~ dvres2 . (Con... |
| dvres 26231 | Restriction of a derivativ... |
| dvres2 26232 | Restriction of the base se... |
| dvres3 26233 | Restriction of a complex d... |
| dvres3a 26234 | Restriction of a complex d... |
| dvidlem 26235 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 26236 | Derivative of a function r... |
| dvconst 26237 | Derivative of a constant f... |
| dvid 26238 | Derivative of the identity... |
| dvcnp 26239 | The difference quotient is... |
| dvcnp2 26240 | A function is continuous a... |
| dvcn 26241 | A differentiable function ... |
| dvnfval 26242 | Value of the iterated deri... |
| dvnff 26243 | The iterated derivative is... |
| dvn0 26244 | Zero times iterated deriva... |
| dvnp1 26245 | Successor iterated derivat... |
| dvn1 26246 | One times iterated derivat... |
| dvnf 26247 | The N-times derivative is ... |
| dvnbss 26248 | The set of N-times differe... |
| dvnadd 26249 | The ` N ` -th derivative o... |
| dvn2bss 26250 | An N-times differentiable ... |
| dvnres 26251 | Multiple derivative versio... |
| cpnfval 26252 | Condition for n-times cont... |
| fncpn 26253 | The ` C^n ` object is a fu... |
| elcpn 26254 | Condition for n-times cont... |
| cpnord 26255 | ` C^n ` conditions are ord... |
| cpncn 26256 | A ` C^n ` function is cont... |
| cpnres 26257 | The restriction of a ` C^n... |
| dvaddbr 26258 | The sum rule for derivativ... |
| dvmulbr 26259 | The product rule for deriv... |
| dvadd 26260 | The sum rule for derivativ... |
| dvmul 26261 | The product rule for deriv... |
| dvaddf 26262 | The sum rule for everywher... |
| dvmulf 26263 | The product rule for every... |
| dvcmul 26264 | The product rule when one ... |
| dvcmulf 26265 | The product rule when one ... |
| dvcobr 26266 | The chain rule for derivat... |
| dvco 26267 | The chain rule for derivat... |
| dvcof 26268 | The chain rule for everywh... |
| dvcjbr 26269 | The derivative of the conj... |
| dvcj 26270 | The derivative of the conj... |
| dvfre 26271 | The derivative of a real f... |
| dvnfre 26272 | The ` N ` -th derivative o... |
| dvexp 26273 | Derivative of a power func... |
| dvexp2 26274 | Derivative of an exponenti... |
| dvrec 26275 | Derivative of the reciproc... |
| dvmptres3 26276 | Function-builder for deriv... |
| dvmptid 26277 | Function-builder for deriv... |
| dvmptc 26278 | Function-builder for deriv... |
| dvmptcl 26279 | Closure lemma for ~ dvmptc... |
| dvmptadd 26280 | Function-builder for deriv... |
| dvmptmul 26281 | Function-builder for deriv... |
| dvmptres2 26282 | Function-builder for deriv... |
| dvmptres 26283 | Function-builder for deriv... |
| dvmptcmul 26284 | Function-builder for deriv... |
| dvmptdivc 26285 | Function-builder for deriv... |
| dvmptneg 26286 | Function-builder for deriv... |
| dvmptsub 26287 | Function-builder for deriv... |
| dvmptcj 26288 | Function-builder for deriv... |
| dvmptre 26289 | Function-builder for deriv... |
| dvmptim 26290 | Function-builder for deriv... |
| dvmptntr 26291 | Function-builder for deriv... |
| dvmptco 26292 | Function-builder for deriv... |
| dvrecg 26293 | Derivative of the reciproc... |
| dvmptdiv 26294 | Function-builder for deriv... |
| dvmptfsum 26295 | Function-builder for deriv... |
| dvcnvlem 26296 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 26297 | A weak version of ~ dvcnvr... |
| dvexp3 26298 | Derivative of an exponenti... |
| dveflem 26299 | Derivative of the exponent... |
| dvef 26300 | Derivative of the exponent... |
| dvsincos 26301 | Derivative of the sine and... |
| dvsin 26302 | Derivative of the sine fun... |
| dvcos 26303 | Derivative of the cosine f... |
| dvferm1lem 26304 | Lemma for ~ dvferm . (Con... |
| dvferm1 26305 | One-sided version of ~ dvf... |
| dvferm2lem 26306 | Lemma for ~ dvferm . (Con... |
| dvferm2 26307 | One-sided version of ~ dvf... |
| dvferm 26308 | Fermat's theorem on statio... |
| rollelem 26309 | Lemma for ~ rolle . (Cont... |
| rolle 26310 | Rolle's theorem. If ` F `... |
| cmvth 26311 | Cauchy's Mean Value Theore... |
| mvth 26312 | The Mean Value Theorem. I... |
| dvlip 26313 | A function with derivative... |
| dvlipcn 26314 | A complex function with de... |
| dvlip2 26315 | Combine the results of ~ d... |
| c1liplem1 26316 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 26317 | C^1 functions are Lipschit... |
| c1lip2 26318 | C^1 functions are Lipschit... |
| c1lip3 26319 | C^1 functions are Lipschit... |
| dveq0 26320 | If a continuous function h... |
| dv11cn 26321 | Two functions defined on a... |
| dvgt0lem1 26322 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 26323 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 26324 | A function on a closed int... |
| dvlt0 26325 | A function on a closed int... |
| dvge0 26326 | A function on a closed int... |
| dvle 26327 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 26328 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 26329 | Lemma for ~ dvivth . (Con... |
| dvivth 26330 | Darboux' theorem, or the i... |
| dvne0 26331 | A function on a closed int... |
| dvne0f1 26332 | A function on a closed int... |
| lhop1lem 26333 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26334 | L'Hôpital's Rule for... |
| lhop2 26335 | L'Hôpital's Rule for... |
| lhop 26336 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26337 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26338 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26339 | The derivative rule for in... |
| dvcvx 26340 | A real function with stric... |
| dvfsumle 26341 | Compare a finite sum to an... |
| dvfsumge 26342 | Compare a finite sum to an... |
| dvfsumabs 26343 | Compare a finite sum to an... |
| dvmptrecl 26344 | Real closure of a derivati... |
| dvfsumrlimf 26345 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26346 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26347 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 26348 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26349 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26350 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26351 | Compare a finite sum to an... |
| dvfsumrlim2 26352 | Compare a finite sum to an... |
| dvfsumrlim3 26353 | Conjoin the statements of ... |
| dvfsum2 26354 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26355 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26356 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26357 | The Fundamental Theorem of... |
| ftc1lem3 26358 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26359 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26360 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26361 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26362 | The Fundamental Theorem of... |
| ftc1cn 26363 | Strengthen the assumptions... |
| ftc2 26364 | The Fundamental Theorem of... |
| ftc2ditglem 26365 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26366 | Directed integral analogue... |
| itgparts 26367 | Integration by parts. If ... |
| itgsubstlem 26368 | Lemma for ~ itgsubst . (C... |
| itgsubst 26369 | Integration by ` u ` -subs... |
| itgpowd 26370 | The integral of a monomial... |
| reldmmdeg 26375 | Multivariate degree is a b... |
| tdeglem1 26376 | Functionality of the total... |
| tdeglem3 26377 | Additivity of the total de... |
| tdeglem4 26378 | There is only one multi-in... |
| tdeglem2 26379 | Simplification of total de... |
| mdegfval 26380 | Value of the multivariate ... |
| mdegval 26381 | Value of the multivariate ... |
| mdegleb 26382 | Property of being of limit... |
| mdeglt 26383 | If there is an upper limit... |
| mdegldg 26384 | A nonzero polynomial has s... |
| mdegxrcl 26385 | Closure of polynomial degr... |
| mdegxrf 26386 | Functionality of polynomia... |
| mdegcl 26387 | Sharp closure for multivar... |
| mdeg0 26388 | Degree of the zero polynom... |
| mdegnn0cl 26389 | Degree of a nonzero polyno... |
| degltlem1 26390 | Theorem on arithmetic of e... |
| degltp1le 26391 | Theorem on arithmetic of e... |
| mdegaddle 26392 | The degree of a sum is at ... |
| mdegvscale 26393 | The degree of a scalar mul... |
| mdegvsca 26394 | The degree of a scalar mul... |
| mdegle0 26395 | A polynomial has nonpositi... |
| mdegmullem 26396 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26397 | The multivariate degree of... |
| deg1fval 26398 | Relate univariate polynomi... |
| deg1xrf 26399 | Functionality of univariat... |
| deg1xrcl 26400 | Closure of univariate poly... |
| deg1cl 26401 | Sharp closure of univariat... |
| mdegpropd 26402 | Property deduction for pol... |
| deg1fvi 26403 | Univariate polynomial degr... |
| deg1propd 26404 | Property deduction for pol... |
| deg1z 26405 | Degree of the zero univari... |
| deg1nn0cl 26406 | Degree of a nonzero univar... |
| deg1n0ima 26407 | Degree image of a set of p... |
| deg1nn0clb 26408 | A polynomial is nonzero if... |
| deg1lt0 26409 | A polynomial is zero iff i... |
| deg1ldg 26410 | A nonzero univariate polyn... |
| deg1ldgn 26411 | An index at which a polyno... |
| deg1ldgdomn 26412 | A nonzero univariate polyn... |
| deg1leb 26413 | Property of being of limit... |
| deg1val 26414 | Value of the univariate de... |
| deg1lt 26415 | If the degree of a univari... |
| deg1ge 26416 | Conversely, a nonzero coef... |
| coe1mul3 26417 | The coefficient vector of ... |
| coe1mul4 26418 | Value of the "leading" coe... |
| deg1addle 26419 | The degree of a sum is at ... |
| deg1addle2 26420 | If both factors have degre... |
| deg1add 26421 | Exact degree of a sum of t... |
| deg1vscale 26422 | The degree of a scalar tim... |
| deg1vsca 26423 | The degree of a scalar tim... |
| deg1invg 26424 | The degree of the negated ... |
| deg1suble 26425 | The degree of a difference... |
| deg1sub 26426 | Exact degree of a differen... |
| deg1mulle2 26427 | Produce a bound on the pro... |
| deg1sublt 26428 | Subtraction of two polynom... |
| deg1le0 26429 | A polynomial has nonpositi... |
| deg1sclle 26430 | A scalar polynomial has no... |
| deg1scl 26431 | A nonzero scalar polynomia... |
| deg1mul2 26432 | Degree of multiplication o... |
| deg1mul 26433 | Degree of multiplication o... |
| deg1mul3 26434 | Degree of multiplication o... |
| deg1mul3le 26435 | Degree of multiplication o... |
| deg1tmle 26436 | Limiting degree of a polyn... |
| deg1tm 26437 | Exact degree of a polynomi... |
| deg1pwle 26438 | Limiting degree of a varia... |
| deg1pw 26439 | Exact degree of a variable... |
| ply1nz 26440 | Univariate polynomials ove... |
| ply1nzb 26441 | Univariate polynomials are... |
| ply1domn 26442 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26443 | The ring of univariate pol... |
| ply1divmo 26454 | Uniqueness of a quotient i... |
| ply1divex 26455 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26456 | The division algorithm for... |
| ply1divalg2 26457 | Reverse the order of multi... |
| uc1pval 26458 | Value of the set of unitic... |
| isuc1p 26459 | Being a unitic polynomial.... |
| mon1pval 26460 | Value of the set of monic ... |
| ismon1p 26461 | Being a monic polynomial. ... |
| uc1pcl 26462 | Unitic polynomials are pol... |
| mon1pcl 26463 | Monic polynomials are poly... |
| uc1pn0 26464 | Unitic polynomials are not... |
| mon1pn0 26465 | Monic polynomials are not ... |
| uc1pdeg 26466 | Unitic polynomials have no... |
| uc1pldg 26467 | Unitic polynomials have un... |
| mon1pldg 26468 | Unitic polynomials have on... |
| mon1puc1p 26469 | Monic polynomials are unit... |
| uc1pmon1p 26470 | Make a unitic polynomial m... |
| deg1submon1p 26471 | The difference of two moni... |
| mon1pid 26472 | Monicity and degree of the... |
| q1pval 26473 | Value of the univariate po... |
| q1peqb 26474 | Characterizing property of... |
| q1pcl 26475 | Closure of the quotient by... |
| r1pval 26476 | Value of the polynomial re... |
| r1pcl 26477 | Closure of remainder follo... |
| r1pdeglt 26478 | The remainder has a degree... |
| r1pid 26479 | Express the original polyn... |
| r1pid2 26480 | Identity law for polynomia... |
| dvdsq1p 26481 | Divisibility in a polynomi... |
| dvdsr1p 26482 | Divisibility in a polynomi... |
| ply1remlem 26483 | A term of the form ` x - N... |
| ply1rem 26484 | The polynomial remainder t... |
| facth1 26485 | The factor theorem and its... |
| fta1glem1 26486 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26487 | Lemma for ~ fta1g . (Cont... |
| fta1g 26488 | The one-sided fundamental ... |
| fta1blem 26489 | Lemma for ~ fta1b . (Cont... |
| fta1b 26490 | The assumption that ` R ` ... |
| idomrootle 26491 | No element of an integral ... |
| drnguc1p 26492 | Over a division ring, all ... |
| ig1peu 26493 | There is a unique monic po... |
| ig1pval 26494 | Substitutions for the poly... |
| ig1pval2 26495 | Generator of the zero idea... |
| ig1pval3 26496 | Characterizing properties ... |
| ig1pcl 26497 | The monic generator of an ... |
| ig1pdvds 26498 | The monic generator of an ... |
| ig1prsp 26499 | Any ideal of polynomials o... |
| ply1lpir 26500 | The ring of polynomials ov... |
| ply1pid 26501 | The polynomials over a fie... |
| plyco0 26510 | Two ways to say that a fun... |
| plyval 26511 | Value of the polynomial se... |
| plybss 26512 | Reverse closure of the par... |
| elply 26513 | Definition of a polynomial... |
| elply2 26514 | The coefficient function c... |
| plyun0 26515 | The set of polynomials is ... |
| plyf 26516 | A polynomial is a function... |
| plyss 26517 | The polynomial set functio... |
| plyssc 26518 | Every polynomial ring is c... |
| elplyr 26519 | Sufficient condition for e... |
| elplyd 26520 | Sufficient condition for e... |
| ply1termlem 26521 | Lemma for ~ ply1term . (C... |
| ply1term 26522 | A one-term polynomial. (C... |
| plypow 26523 | A power is a polynomial. ... |
| plyconst 26524 | A constant function is a p... |
| ne0p 26525 | A test to show that a poly... |
| ply0 26526 | The zero function is a pol... |
| plyid 26527 | The identity function is a... |
| idpfv 26528 | Value of the identity poly... |
| plyeq0lem 26529 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26530 | If a polynomial is zero at... |
| plypf1 26531 | Write the set of complex p... |
| plyaddlem1 26532 | Derive the coefficient fun... |
| plymullem1 26533 | Derive the coefficient fun... |
| plyaddlem 26534 | Lemma for ~ plyadd . (Con... |
| plymullem 26535 | Lemma for ~ plymul . (Con... |
| plyadd 26536 | The sum of two polynomials... |
| plymul 26537 | The product of two polynom... |
| plysub 26538 | The difference of two poly... |
| plyaddcl 26539 | The sum of two polynomials... |
| plymulcl 26540 | The product of two polynom... |
| plysubcl 26541 | The difference of two poly... |
| coeval 26542 | Value of the coefficient f... |
| coeeulem 26543 | Lemma for ~ coeeu . (Cont... |
| coeeu 26544 | Uniqueness of the coeffici... |
| coelem 26545 | Lemma for properties of th... |
| coeeq 26546 | If ` A ` satisfies the pro... |
| dgrval 26547 | Value of the degree functi... |
| dgrlem 26548 | Lemma for ~ dgrcl and simi... |
| coef 26549 | The domain and codomain of... |
| coef2 26550 | The domain and codomain of... |
| coef3 26551 | The domain and codomain of... |
| dgrcl 26552 | The degree of any polynomi... |
| dgrub 26553 | If the ` M ` -th coefficie... |
| dgrub2 26554 | All the coefficients above... |
| dgrlb 26555 | If all the coefficients ab... |
| coeidlem 26556 | Lemma for ~ coeid . (Cont... |
| coeid 26557 | Reconstruct a polynomial a... |
| coeid2 26558 | Reconstruct a polynomial a... |
| coeid3 26559 | Reconstruct a polynomial a... |
| plyco 26560 | The composition of two pol... |
| coeeq2 26561 | Compute the coefficient fu... |
| dgrle 26562 | Given an explicit expressi... |
| dgreq 26563 | If the highest term in a p... |
| 0dgr 26564 | A constant function has de... |
| 0dgrb 26565 | A function has degree zero... |
| dgrnznn 26566 | A nonzero polynomial with ... |
| coefv0 26567 | The result of evaluating a... |
| coeaddlem 26568 | Lemma for ~ coeadd and ~ d... |
| coemullem 26569 | Lemma for ~ coemul and ~ d... |
| coeadd 26570 | The coefficient function o... |
| coemul 26571 | A coefficient of a product... |
| coe11 26572 | The coefficient function i... |
| coemulhi 26573 | The leading coefficient of... |
| coemulc 26574 | The coefficient function i... |
| coe0 26575 | The coefficients of the ze... |
| coesub 26576 | The coefficient function o... |
| coe1termlem 26577 | The coefficient function o... |
| coe1term 26578 | The coefficient function o... |
| dgr1term 26579 | The degree of a monomial. ... |
| plycn 26580 | A polynomial is a continuo... |
| dgr0 26581 | The degree of the zero pol... |
| coeidp 26582 | The coefficients of the id... |
| dgrid 26583 | The degree of the identity... |
| dgreq0 26584 | The leading coefficient of... |
| dgrlt 26585 | Two ways to say that the d... |
| dgradd 26586 | The degree of a sum of pol... |
| dgradd2 26587 | The degree of a sum of pol... |
| dgrmul2 26588 | The degree of a product of... |
| dgrmul 26589 | The degree of a product of... |
| dgrmulc 26590 | Scalar multiplication by a... |
| dgrsub 26591 | The degree of a difference... |
| dgrcolem1 26592 | The degree of a compositio... |
| dgrcolem2 26593 | Lemma for ~ dgrco . (Cont... |
| dgrco 26594 | The degree of a compositio... |
| plycjlem 26595 | Lemma for ~ plycj and ~ co... |
| plycj 26596 | The double conjugation of ... |
| coecj 26597 | Double conjugation of a po... |
| plyrecj 26598 | A polynomial with real coe... |
| plymul0or 26599 | Polynomial multiplication ... |
| ofmulrt 26600 | The set of roots of a prod... |
| plymul02 26601 | Product of a polynomial wi... |
| plyn0mulidp 26602 | Coefficients of a non-zero... |
| plymulidp 26603 | Coefficients of a polynomi... |
| plyreres 26604 | Real-coefficient polynomia... |
| dvply1 26605 | Derivative of a polynomial... |
| dvply2g 26606 | The derivative of a polyno... |
| dvply2 26607 | The derivative of a polyno... |
| dvnply2 26608 | Polynomials have polynomia... |
| dvnply 26609 | Polynomials have polynomia... |
| plycpn 26610 | Polynomials are smooth. (... |
| quotval 26613 | Value of the quotient func... |
| plydivlem1 26614 | Lemma for ~ plydivalg . (... |
| plydivlem2 26615 | Lemma for ~ plydivalg . (... |
| plydivlem3 26616 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26617 | Lemma for ~ plydivex . In... |
| plydivex 26618 | Lemma for ~ plydivalg . (... |
| plydiveu 26619 | Lemma for ~ plydivalg . (... |
| plydivalg 26620 | The division algorithm on ... |
| quotlem 26621 | Lemma for properties of th... |
| quotcl 26622 | The quotient of two polyno... |
| quotcl2 26623 | Closure of the quotient fu... |
| quotdgr 26624 | Remainder property of the ... |
| plyremlem 26625 | Closure of a linear factor... |
| plyrem 26626 | The polynomial remainder t... |
| facth 26627 | The factor theorem. If a ... |
| fta1lem 26628 | Lemma for ~ fta1 . (Contr... |
| fta1 26629 | The easy direction of the ... |
| rnplynfin 26630 | The range of a nonconstant... |
| plyconz 26631 | The composition of a nonze... |
| quotcan 26632 | Exact division with a mult... |
| vieta1lem1 26633 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26634 | Lemma for ~ vieta1 : induc... |
| vieta1 26635 | The first-order Vieta's fo... |
| plyexmo 26636 | An infinite set of values ... |
| elaa 26639 | Elementhood in the set of ... |
| aacn 26640 | An algebraic number is a c... |
| aasscn 26641 | The algebraic numbers are ... |
| elqaalem1 26642 | Lemma for ~ elqaa . The f... |
| elqaalem2 26643 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26644 | Lemma for ~ elqaa . (Cont... |
| elqaa 26645 | The set of numbers generat... |
| preimaaa 26646 | An element of the preimage... |
| qaa 26647 | Every rational number is a... |
| qssaa 26648 | The rational numbers are c... |
| 0aa 26649 | Zero is algebraic. (Contr... |
| 1aa 26650 | One is algebraic. (Contri... |
| iaa 26651 | The imaginary unit is alge... |
| iaaOLD 26652 | Obsolete version of ~ iaa ... |
| aareccl 26653 | The reciprocal of an algeb... |
| aacjcl 26654 | The conjugate of an algebr... |
| aannenlem1 26655 | Lemma for ~ aannen . (Con... |
| aannenlem2 26656 | Lemma for ~ aannen . (Con... |
| aannenlem3 26657 | The algebraic numbers are ... |
| aannen 26658 | The algebraic numbers are ... |
| aalioulem1 26659 | Lemma for ~ aaliou . An i... |
| aalioulem2 26660 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26661 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26662 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26663 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26664 | Lemma for ~ aaliou . (Con... |
| aaliou 26665 | Liouville's theorem on dio... |
| geolim3 26666 | Geometric series convergen... |
| aaliou2 26667 | Liouville's approximation ... |
| aaliou2b 26668 | Liouville's approximation ... |
| aaliou3lem1 26669 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26670 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26671 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26672 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26673 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26674 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26675 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26676 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26677 | Example of a "Liouville nu... |
| aaliou3 26678 | Example of a "Liouville nu... |
| aaliou3r 26679 | The sum presented above is... |
| taylfvallem1 26684 | Lemma for ~ taylfval . (C... |
| taylfvallem 26685 | Lemma for ~ taylfval . (C... |
| taylfval 26686 | Define the Taylor polynomi... |
| eltayl 26687 | Value of the Taylor series... |
| taylf 26688 | The Taylor series defines ... |
| tayl0 26689 | The Taylor series is alway... |
| taylplem1 26690 | Lemma for ~ taylpfval and ... |
| taylplem2 26691 | Lemma for ~ taylpfval and ... |
| taylpfval 26692 | Define the Taylor polynomi... |
| taylpf 26693 | The Taylor polynomial is a... |
| taylpval 26694 | Value of the Taylor polyno... |
| taylply2 26695 | The Taylor polynomial is a... |
| taylply 26696 | The Taylor polynomial is a... |
| dvtaylp 26697 | The derivative of the Tayl... |
| dvntaylp 26698 | The ` M ` -th derivative o... |
| dvntaylp0 26699 | The first ` N ` derivative... |
| taylthlem1 26700 | Lemma for ~ taylth . This... |
| taylthlem2 26701 | Lemma for ~ taylth . (Con... |
| taylth 26702 | Taylor's theorem. The Tay... |
| ulmrel 26705 | The uniform limit relation... |
| ulmscl 26706 | Closure of the base set in... |
| ulmval 26707 | Express the predicate: Th... |
| ulmcl 26708 | Closure of a uniform limit... |
| ulmf 26709 | Closure of a uniform limit... |
| ulmpm 26710 | Closure of a uniform limit... |
| ulmf2 26711 | Closure of a uniform limit... |
| ulm2 26712 | Simplify ~ ulmval when ` F... |
| ulmi 26713 | The uniform limit property... |
| ulmclm 26714 | A uniform limit of functio... |
| ulmres 26715 | A sequence of functions co... |
| ulmshftlem 26716 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26717 | A sequence of functions co... |
| ulm0 26718 | Every function converges u... |
| ulmuni 26719 | A sequence of functions un... |
| ulmdm 26720 | Two ways to express that a... |
| ulmcaulem 26721 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26722 | A sequence of functions co... |
| ulmcau2 26723 | A sequence of functions co... |
| ulmss 26724 | A uniform limit of functio... |
| ulmbdd 26725 | A uniform limit of bounded... |
| ulmcn 26726 | A uniform limit of continu... |
| ulmdvlem1 26727 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26728 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26729 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26730 | If ` F ` is a sequence of ... |
| mtest 26731 | The Weierstrass M-test. I... |
| mtestbdd 26732 | Given the hypotheses of th... |
| mbfulm 26733 | A uniform limit of measura... |
| iblulm 26734 | A uniform limit of integra... |
| itgulm 26735 | A uniform limit of integra... |
| itgulm2 26736 | A uniform limit of integra... |
| pserval 26737 | Value of the function ` G ... |
| pserval2 26738 | Value of the function ` G ... |
| psergf 26739 | The sequence of terms in t... |
| radcnvlem1 26740 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26741 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26742 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26743 | Zero is always a convergen... |
| radcnvcl 26744 | The radius of convergence ... |
| radcnvlt1 26745 | If ` X ` is within the ope... |
| radcnvlt2 26746 | If ` X ` is within the ope... |
| radcnvle 26747 | If ` X ` is a convergent p... |
| dvradcnv 26748 | The radius of convergence ... |
| pserulm 26749 | If ` S ` is a region conta... |
| psercn2 26750 | Since by ~ pserulm the ser... |
| psercnlem2 26751 | Lemma for ~ psercn . (Con... |
| psercnlem1 26752 | Lemma for ~ psercn . (Con... |
| psercn 26753 | An infinite series converg... |
| pserdvlem1 26754 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26755 | Lemma for ~ pserdv . (Con... |
| pserdv 26756 | The derivative of a power ... |
| pserdv2 26757 | The derivative of a power ... |
| abelthlem1 26758 | Lemma for ~ abelth . (Con... |
| abelthlem2 26759 | Lemma for ~ abelth . The ... |
| abelthlem3 26760 | Lemma for ~ abelth . (Con... |
| abelthlem4 26761 | Lemma for ~ abelth . (Con... |
| abelthlem5 26762 | Lemma for ~ abelth . (Con... |
| abelthlem6 26763 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26764 | Lemma for ~ abelth . (Con... |
| abelthlem7 26765 | Lemma for ~ abelth . (Con... |
| abelthlem8 26766 | Lemma for ~ abelth . (Con... |
| abelthlem9 26767 | Lemma for ~ abelth . By a... |
| abelth 26768 | Abel's theorem. If the po... |
| abelth2 26769 | Abel's theorem, restricted... |
| efcn 26770 | The exponential function i... |
| sincn 26771 | Sine is continuous. (Cont... |
| coscn 26772 | Cosine is continuous. (Co... |
| reeff1olem 26773 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26774 | The real exponential funct... |
| reefiso 26775 | The exponential function o... |
| efcvx 26776 | The exponential function o... |
| reefgim 26777 | The exponential function i... |
| pilem1 26778 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26779 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26780 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26781 | ` _pi ` is between 2 and 4... |
| sinpi 26782 | The sine of ` _pi ` is 0. ... |
| pire 26783 | ` _pi ` is a real number. ... |
| 2pire 26784 | ` ( 2 x. _pi ) ` is a real... |
| picn 26785 | ` _pi ` is a complex numbe... |
| 2picn 26786 | ` ( 2 x. _pi ) ` is a comp... |
| pipos 26787 | ` _pi ` is positive. (Con... |
| pige0 26788 | ` _pi ` is nonnegative. (... |
| pine0 26789 | ` _pi ` is nonzero. (Cont... |
| pirp 26790 | ` _pi ` is a positive real... |
| negpicn 26791 | ` -u _pi ` is a complex nu... |
| sinhalfpilem 26792 | Lemma for ~ sinhalfpi and ... |
| halfpire 26793 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26794 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26795 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26796 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26797 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26798 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26799 | The cosine of ` -u _pi / 2... |
| efhalfpi 26800 | The exponential of ` _i _p... |
| cospi 26801 | The cosine of ` _pi ` is `... |
| efipi 26802 | The exponential of ` _i x.... |
| eulerid 26803 | Euler's identity. (Contri... |
| sin2pi 26804 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26805 | The cosine of ` 2 _pi ` is... |
| ef2pi 26806 | The exponential of ` 2 _pi... |
| ef2kpi 26807 | If ` K ` is an integer, th... |
| efper 26808 | The exponential function i... |
| sinperlem 26809 | Lemma for ~ sinper and ~ c... |
| sinper 26810 | The sine function is perio... |
| cosper 26811 | The cosine function is per... |
| sin2kpi 26812 | If ` K ` is an integer, th... |
| cos2kpi 26813 | If ` K ` is an integer, th... |
| sin2pim 26814 | Sine of a number subtracte... |
| cos2pim 26815 | Cosine of a number subtrac... |
| sinmpi 26816 | Sine of a number less ` _p... |
| cosmpi 26817 | Cosine of a number less ` ... |
| sinppi 26818 | Sine of a number plus ` _p... |
| cosppi 26819 | Cosine of a number plus ` ... |
| efimpi 26820 | The exponential function a... |
| sinhalfpip 26821 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26822 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26823 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26824 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26825 | Ptolemy's Theorem. This t... |
| sincosq1lem 26826 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26827 | The signs of the sine and ... |
| sincosq2sgn 26828 | The signs of the sine and ... |
| sincosq3sgn 26829 | The signs of the sine and ... |
| sincosq4sgn 26830 | The signs of the sine and ... |
| coseq00topi 26831 | Location of the zeroes of ... |
| coseq0negpitopi 26832 | Location of the zeroes of ... |
| tanrpcl 26833 | Positive real closure of t... |
| tangtx 26834 | The tangent function is gr... |
| tanabsge 26835 | The tangent function is gr... |
| sinq12gt0 26836 | The sine of a number stric... |
| sinq12ge0 26837 | The sine of a number betwe... |
| sinq34lt0t 26838 | The sine of a number stric... |
| cosq14gt0 26839 | The cosine of a number str... |
| cosq14ge0 26840 | The cosine of a number bet... |
| sincosq1eq 26841 | Complementarity of the sin... |
| sincos4thpi 26842 | The sine and cosine of ` _... |
| tan4thpi 26843 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26844 | The sine and cosine of ` _... |
| sincos3rdpi 26845 | The sine and cosine of ` _... |
| pigt3 26846 | ` _pi ` is greater than 3.... |
| pige3 26847 | ` _pi ` is greater than or... |
| pige3ALT 26848 | Alternate proof of ~ pige3... |
| abssinper 26849 | The absolute value of sine... |
| sinkpi 26850 | The sine of an integer mul... |
| coskpi 26851 | The absolute value of the ... |
| sineq0 26852 | A complex number whose sin... |
| coseq1 26853 | A complex number whose cos... |
| cos02pilt1 26854 | Cosine is less than one be... |
| cosq34lt1 26855 | Cosine is less than one in... |
| efeq1 26856 | A complex number whose exp... |
| cosne0 26857 | The cosine function has no... |
| cosordlem 26858 | Lemma for ~ cosord . (Con... |
| cosord 26859 | Cosine is decreasing over ... |
| cos0pilt1 26860 | Cosine is between minus on... |
| cos11 26861 | Cosine is one-to-one over ... |
| sinord 26862 | Sine is increasing over th... |
| recosf1o 26863 | The cosine function is a b... |
| resinf1o 26864 | The sine function is a bij... |
| tanord1 26865 | The tangent function is st... |
| tanord 26866 | The tangent function is st... |
| tanregt0 26867 | The real part of the tange... |
| negpitopissre 26868 | The interval ` ( -u _pi (,... |
| efgh 26869 | The exponential function o... |
| efif1olem1 26870 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26871 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26872 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26873 | The exponential function o... |
| efif1o 26874 | The exponential function o... |
| efifo 26875 | The exponential function o... |
| eff1olem 26876 | The exponential function m... |
| eff1o 26877 | The exponential function m... |
| efabl 26878 | The image of a subgroup of... |
| efsubm 26879 | The image of a subgroup of... |
| circgrp 26880 | The circle group ` T ` is ... |
| circsubm 26881 | The circle group ` T ` is ... |
| logrn 26886 | The range of the natural l... |
| ellogrn 26887 | Write out the property ` A... |
| dflog2 26888 | The natural logarithm func... |
| relogrn 26889 | The range of the natural l... |
| logrncn 26890 | The range of the natural l... |
| eff1o2 26891 | The exponential function r... |
| logf1o 26892 | The natural logarithm func... |
| dfrelog 26893 | The natural logarithm func... |
| relogf1o 26894 | The natural logarithm func... |
| logrncl 26895 | Closure of the natural log... |
| logcl 26896 | Closure of the natural log... |
| logimcl 26897 | Closure of the imaginary p... |
| logcld 26898 | The logarithm of a nonzero... |
| logimcld 26899 | The imaginary part of the ... |
| logimclad 26900 | The imaginary part of the ... |
| abslogimle 26901 | The imaginary part of the ... |
| logrnaddcl 26902 | The range of the natural l... |
| relogcl 26903 | Closure of the natural log... |
| eflog 26904 | Relationship between the n... |
| logeq0im1 26905 | If the logarithm of a numb... |
| logccne0 26906 | The logarithm isn't 0 if i... |
| logne0 26907 | Logarithm of a non-1 posit... |
| reeflog 26908 | Relationship between the n... |
| logef 26909 | Relationship between the n... |
| relogef 26910 | Relationship between the n... |
| logeftb 26911 | Relationship between the n... |
| relogeftb 26912 | Relationship between the n... |
| log1 26913 | The natural logarithm of `... |
| loge 26914 | The natural logarithm of `... |
| logi 26915 | The natural logarithm of `... |
| logneg 26916 | The natural logarithm of a... |
| logm1 26917 | The natural logarithm of n... |
| lognegb 26918 | If a number has imaginary ... |
| relogoprlem 26919 | Lemma for ~ relogmul and ~... |
| relogmul 26920 | The natural logarithm of t... |
| relogdiv 26921 | The natural logarithm of t... |
| explog 26922 | Exponentiation of a nonzer... |
| reexplog 26923 | Exponentiation of a positi... |
| relogexp 26924 | The natural logarithm of p... |
| relog 26925 | Real part of a logarithm. ... |
| relogiso 26926 | The natural logarithm func... |
| reloggim 26927 | The natural logarithm is a... |
| logltb 26928 | The natural logarithm func... |
| logfac 26929 | The logarithm of a factori... |
| eflogeq 26930 | Solve an equation involvin... |
| logleb 26931 | Natural logarithm preserve... |
| rplogcl 26932 | Closure of the logarithm f... |
| logge0 26933 | The logarithm of a number ... |
| logcj 26934 | The natural logarithm dist... |
| efiarg 26935 | The exponential of the "ar... |
| cosargd 26936 | The cosine of the argument... |
| cosarg0d 26937 | The cosine of the argument... |
| argregt0 26938 | Closure of the argument of... |
| argrege0 26939 | Closure of the argument of... |
| argimgt0 26940 | Closure of the argument of... |
| argimlt0 26941 | Closure of the argument of... |
| logimul 26942 | Multiplying a number by ` ... |
| logneg2 26943 | The logarithm of the negat... |
| logmul2 26944 | Generalization of ~ relogm... |
| logdiv2 26945 | Generalization of ~ relogd... |
| abslogle 26946 | Bound on the magnitude of ... |
| tanarg 26947 | The basic relation between... |
| logdivlti 26948 | The ` log x / x ` function... |
| logdivlt 26949 | The ` log x / x ` function... |
| logdivle 26950 | The ` log x / x ` function... |
| relogcld 26951 | Closure of the natural log... |
| reeflogd 26952 | Relationship between the n... |
| relogmuld 26953 | The natural logarithm of t... |
| relogdivd 26954 | The natural logarithm of t... |
| logled 26955 | Natural logarithm preserve... |
| relogefd 26956 | Relationship between the n... |
| rplogcld 26957 | Closure of the logarithm f... |
| logge0d 26958 | The logarithm of a number ... |
| logge0b 26959 | The logarithm of a number ... |
| loggt0b 26960 | The logarithm of a number ... |
| logle1b 26961 | The logarithm of a number ... |
| loglt1b 26962 | The logarithm of a number ... |
| divlogrlim 26963 | The inverse logarithm func... |
| logno1 26964 | The logarithm function is ... |
| dvrelog 26965 | The derivative of the real... |
| relogcn 26966 | The real logarithm functio... |
| ellogdm 26967 | Elementhood in the "contin... |
| logdmn0 26968 | A number in the continuous... |
| logdmnrp 26969 | A number in the continuous... |
| logdmss 26970 | The continuity domain of `... |
| logcnlem2 26971 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26972 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26973 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26974 | Lemma for ~ logcn . (Cont... |
| logcn 26975 | The logarithm function is ... |
| dvloglem 26976 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26977 | The "continuous domain" of... |
| logf1o2 26978 | The logarithm maps its con... |
| dvlog 26979 | The derivative of the comp... |
| dvlog2lem 26980 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26981 | The derivative of the comp... |
| advlog 26982 | The antiderivative of the ... |
| advlogexp 26983 | The antiderivative of a po... |
| efopnlem1 26984 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26985 | Lemma for ~ efopn . (Cont... |
| efopn 26986 | The exponential map is an ... |
| logtayllem 26987 | Lemma for ~ logtayl . (Co... |
| logtayl 26988 | The Taylor series for ` -u... |
| logtaylsum 26989 | The Taylor series for ` -u... |
| logtayl2 26990 | Power series expression fo... |
| logccv 26991 | The natural logarithm func... |
| cxpval 26992 | Value of the complex power... |
| cxpef 26993 | Value of the complex power... |
| 0cxp 26994 | Value of the complex power... |
| cxpexpz 26995 | Relate the complex power f... |
| cxpexp 26996 | Relate the complex power f... |
| logcxp 26997 | Logarithm of a complex pow... |
| cxp0 26998 | Value of the complex power... |
| cxp1 26999 | Value of the complex power... |
| 1cxp 27000 | Value of the complex power... |
| ecxp 27001 | Write the exponential func... |
| cxpcl 27002 | Closure of the complex pow... |
| recxpcl 27003 | Real closure of the comple... |
| rpcxpcl 27004 | Positive real closure of t... |
| cxpne0 27005 | Complex exponentiation is ... |
| cxpeq0 27006 | Complex exponentiation is ... |
| cxpadd 27007 | Sum of exponents law for c... |
| cxpp1 27008 | Value of a nonzero complex... |
| cxpneg 27009 | Value of a complex number ... |
| cxpsub 27010 | Exponent subtraction law f... |
| cxpge0 27011 | Nonnegative exponentiation... |
| mulcxplem 27012 | Lemma for ~ mulcxp . (Con... |
| mulcxp 27013 | Complex exponentiation of ... |
| cxprec 27014 | Complex exponentiation of ... |
| divcxp 27015 | Complex exponentiation of ... |
| cxpmul 27016 | Product of exponents law f... |
| cxpmul2 27017 | Product of exponents law f... |
| cxproot 27018 | The complex power function... |
| cxpmul2z 27019 | Generalize ~ cxpmul2 to ne... |
| abscxp 27020 | Absolute value of a power,... |
| abscxp2 27021 | Absolute value of a power,... |
| cxplt 27022 | Ordering property for comp... |
| cxple 27023 | Ordering property for comp... |
| cxplea 27024 | Ordering property for comp... |
| cxple2 27025 | Ordering property for comp... |
| cxplt2 27026 | Ordering property for comp... |
| cxple2a 27027 | Ordering property for comp... |
| cxplt3 27028 | Ordering property for comp... |
| cxple3 27029 | Ordering property for comp... |
| cxpsqrtlem 27030 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 27031 | The complex exponential fu... |
| logsqrt 27032 | Logarithm of a square root... |
| cxp0d 27033 | Value of the complex power... |
| cxp1d 27034 | Value of the complex power... |
| 1cxpd 27035 | Value of the complex power... |
| cxpcld 27036 | Closure of the complex pow... |
| cxpmul2d 27037 | Product of exponents law f... |
| 0cxpd 27038 | Value of the complex power... |
| cxpexpzd 27039 | Relate the complex power f... |
| cxpefd 27040 | Value of the complex power... |
| cxpne0d 27041 | Complex exponentiation is ... |
| cxpp1d 27042 | Value of a nonzero complex... |
| cxpnegd 27043 | Value of a complex number ... |
| cxpmul2zd 27044 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 27045 | Sum of exponents law for c... |
| cxpsubd 27046 | Exponent subtraction law f... |
| cxpltd 27047 | Ordering property for comp... |
| cxpled 27048 | Ordering property for comp... |
| cxplead 27049 | Ordering property for comp... |
| divcxpd 27050 | Complex exponentiation of ... |
| recxpcld 27051 | Positive real closure of t... |
| cxpge0d 27052 | Nonnegative exponentiation... |
| cxple2ad 27053 | Ordering property for comp... |
| cxplt2d 27054 | Ordering property for comp... |
| cxple2d 27055 | Ordering property for comp... |
| mulcxpd 27056 | Complex exponentiation of ... |
| recxpf1lem 27057 | Complex exponentiation on ... |
| cxpsqrtth 27058 | Square root theorem over t... |
| 2irrexpq 27059 | There exist irrational num... |
| cxprecd 27060 | Complex exponentiation of ... |
| rpcxpcld 27061 | Positive real closure of t... |
| logcxpd 27062 | Logarithm of a complex pow... |
| cxplt3d 27063 | Ordering property for comp... |
| cxple3d 27064 | Ordering property for comp... |
| cxpmuld 27065 | Product of exponents law f... |
| cxpgt0d 27066 | A positive real raised to ... |
| cxpcom 27067 | Commutative law for real e... |
| dvcxp1 27068 | The derivative of a comple... |
| dvcxp2 27069 | The derivative of a comple... |
| dvsqrt 27070 | The derivative of the real... |
| dvcncxp1 27071 | Derivative of complex powe... |
| dvcnsqrt 27072 | Derivative of square root ... |
| cxpcn 27073 | Domain of continuity of th... |
| cxpcn2 27074 | Continuity of the complex ... |
| cxpcn3lem 27075 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 27076 | Extend continuity of the c... |
| resqrtcn 27077 | Continuity of the real squ... |
| sqrtcn 27078 | Continuity of the square r... |
| cxpaddlelem 27079 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 27080 | Ordering property for comp... |
| abscxpbnd 27081 | Bound on the absolute valu... |
| root1id 27082 | Property of an ` N ` -th r... |
| root1eq1 27083 | The only powers of an ` N ... |
| root1cj 27084 | Within the ` N ` -th roots... |
| cxpeq 27085 | Solve an equation involvin... |
| zrtelqelz 27086 | If the ` N ` -th root of a... |
| zrtdvds 27087 | A positive integer root di... |
| rtprmirr 27088 | The root of a prime number... |
| loglesqrt 27089 | An upper bound on the loga... |
| logreclem 27090 | Symmetry of the natural lo... |
| logrec 27091 | Logarithm of a reciprocal ... |
| logbval 27094 | Define the value of the ` ... |
| logbcl 27095 | General logarithm closure.... |
| logbid1 27096 | General logarithm is 1 whe... |
| logb1 27097 | The logarithm of ` 1 ` to ... |
| elogb 27098 | The general logarithm of a... |
| logbchbase 27099 | Change of base for logarit... |
| relogbval 27100 | Value of the general logar... |
| relogbcl 27101 | Closure of the general log... |
| relogbzcl 27102 | Closure of the general log... |
| relogbreexp 27103 | Power law for the general ... |
| relogbzexp 27104 | Power law for the general ... |
| relogbmul 27105 | The logarithm of the produ... |
| relogbmulexp 27106 | The logarithm of the produ... |
| relogbdiv 27107 | The logarithm of the quoti... |
| relogbexp 27108 | Identity law for general l... |
| nnlogbexp 27109 | Identity law for general l... |
| logbrec 27110 | Logarithm of a reciprocal ... |
| logbleb 27111 | The general logarithm func... |
| logblt 27112 | The general logarithm func... |
| relogbcxp 27113 | Identity law for the gener... |
| cxplogb 27114 | Identity law for the gener... |
| relogbcxpb 27115 | The logarithm is the inver... |
| logbmpt 27116 | The general logarithm to a... |
| logbf 27117 | The general logarithm to a... |
| logbfval 27118 | The general logarithm of a... |
| relogbf 27119 | The general logarithm to a... |
| logblog 27120 | The general logarithm to t... |
| logbgt0b 27121 | The logarithm of a positiv... |
| logbgcd1irr 27122 | The logarithm of an intege... |
| 2logb9irr 27123 | Example for ~ logbgcd1irr ... |
| logbprmirr 27124 | The logarithm of a prime t... |
| 2logb3irr 27125 | Example for ~ logbprmirr .... |
| 2logb9irrALT 27126 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 27127 | The square root of two to ... |
| 2irrexpqALT 27128 | Alternate proof of ~ 2irre... |
| angval 27129 | Define the angle function,... |
| angcan 27130 | Cancel a constant multipli... |
| angneg 27131 | Cancel a negative sign in ... |
| angvald 27132 | The (signed) angle between... |
| angcld 27133 | The (signed) angle between... |
| angrteqvd 27134 | Two vectors are at a right... |
| cosangneg2d 27135 | The cosine of the angle be... |
| angrtmuld 27136 | Perpendicularity of two ve... |
| ang180lem1 27137 | Lemma for ~ ang180 . Show... |
| ang180lem2 27138 | Lemma for ~ ang180 . Show... |
| ang180lem3 27139 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 27140 | Lemma for ~ ang180 . Redu... |
| ang180lem5 27141 | Lemma for ~ ang180 : Redu... |
| ang180 27142 | The sum of angles ` m A B ... |
| lawcoslem1 27143 | Lemma for ~ lawcos . Here... |
| lawcos 27144 | Law of cosines (also known... |
| pythag 27145 | Pythagorean theorem. Give... |
| isosctrlem1 27146 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 27147 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 27148 | Lemma for ~ isosctr . Cor... |
| isosctr 27149 | Isosceles triangle theorem... |
| ssscongptld 27150 | If two triangles have equa... |
| affineequiv 27151 | Equivalence between two wa... |
| affineequiv2 27152 | Equivalence between two wa... |
| affineequiv3 27153 | Equivalence between two wa... |
| affineequiv4 27154 | Equivalence between two wa... |
| affineequivne 27155 | Equivalence between two wa... |
| angpieqvdlem 27156 | Equivalence used in the pr... |
| angpieqvdlem2 27157 | Equivalence used in ~ angp... |
| angpined 27158 | If the angle at ABC is ` _... |
| angpieqvd 27159 | The angle ABC is ` _pi ` i... |
| chordthmlem 27160 | If ` M ` is the midpoint o... |
| chordthmlem2 27161 | If M is the midpoint of AB... |
| chordthmlem3 27162 | If M is the midpoint of AB... |
| chordthmlem4 27163 | If P is on the segment AB ... |
| chordthmlem5 27164 | If P is on the segment AB ... |
| chordthm 27165 | The intersecting chords th... |
| heron 27166 | Heron's formula gives the ... |
| quad2 27167 | The quadratic equation, wi... |
| quad 27168 | The quadratic equation. (... |
| 1cubrlem 27169 | The cube roots of unity. ... |
| 1cubr 27170 | The cube roots of unity. ... |
| dcubic1lem 27171 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 27172 | Reverse direction of ~ dcu... |
| dcubic1 27173 | Forward direction of ~ dcu... |
| dcubic 27174 | Solutions to the depressed... |
| mcubic 27175 | Solutions to a monic cubic... |
| cubic2 27176 | The solution to the genera... |
| cubic 27177 | The cubic equation, which ... |
| binom4 27178 | Work out a quartic binomia... |
| dquartlem1 27179 | Lemma for ~ dquart . (Con... |
| dquartlem2 27180 | Lemma for ~ dquart . (Con... |
| dquart 27181 | Solve a depressed quartic ... |
| quart1cl 27182 | Closure lemmas for ~ quart... |
| quart1lem 27183 | Lemma for ~ quart1 . (Con... |
| quart1 27184 | Depress a quartic equation... |
| quartlem1 27185 | Lemma for ~ quart . (Cont... |
| quartlem2 27186 | Closure lemmas for ~ quart... |
| quartlem3 27187 | Closure lemmas for ~ quart... |
| quartlem4 27188 | Closure lemmas for ~ quart... |
| quart 27189 | The quartic equation, writ... |
| asinlem 27196 | The argument to the logari... |
| asinlem2 27197 | The argument to the logari... |
| asinlem3a 27198 | Lemma for ~ asinlem3 . (C... |
| asinlem3 27199 | The argument to the logari... |
| asinf 27200 | Domain and codomain of the... |
| asincl 27201 | Closure for the arcsin fun... |
| acosf 27202 | Domain and codoamin of the... |
| acoscl 27203 | Closure for the arccos fun... |
| atandm 27204 | Since the property is a li... |
| atandm2 27205 | This form of ~ atandm is a... |
| atandm3 27206 | A compact form of ~ atandm... |
| atandm4 27207 | A compact form of ~ atandm... |
| atanf 27208 | Domain and codoamin of the... |
| atancl 27209 | Closure for the arctan fun... |
| asinval 27210 | Value of the arcsin functi... |
| acosval 27211 | Value of the arccos functi... |
| atanval 27212 | Value of the arctan functi... |
| atanre 27213 | A real number is in the do... |
| asinneg 27214 | The arcsine function is od... |
| acosneg 27215 | The negative symmetry rela... |
| efiasin 27216 | The exponential of the arc... |
| sinasin 27217 | The arcsine function is an... |
| cosacos 27218 | The arccosine function is ... |
| asinsinlem 27219 | Lemma for ~ asinsin . (Co... |
| asinsin 27220 | The arcsine function compo... |
| acoscos 27221 | The arccosine function is ... |
| asin1 27222 | The arcsine of ` 1 ` is ` ... |
| acos1 27223 | The arccosine of ` 1 ` is ... |
| reasinsin 27224 | The arcsine function compo... |
| asinsinb 27225 | Relationship between sine ... |
| acoscosb 27226 | Relationship between cosin... |
| asinbnd 27227 | The arcsine function has r... |
| acosbnd 27228 | The arccosine function has... |
| asinrebnd 27229 | Bounds on the arcsine func... |
| asinrecl 27230 | The arcsine function is re... |
| acosrecl 27231 | The arccosine function is ... |
| cosasin 27232 | The cosine of the arcsine ... |
| sinacos 27233 | The sine of the arccosine ... |
| atandmneg 27234 | The domain of the arctange... |
| atanneg 27235 | The arctangent function is... |
| atan0 27236 | The arctangent of zero is ... |
| atandmcj 27237 | The arctangent function di... |
| atancj 27238 | The arctangent function di... |
| atanrecl 27239 | The arctangent function is... |
| efiatan 27240 | Value of the exponential o... |
| atanlogaddlem 27241 | Lemma for ~ atanlogadd . ... |
| atanlogadd 27242 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 27243 | Lemma for ~ atanlogsub . ... |
| atanlogsub 27244 | A variation on ~ atanlogad... |
| efiatan2 27245 | Value of the exponential o... |
| 2efiatan 27246 | Value of the exponential o... |
| tanatan 27247 | The arctangent function is... |
| atandmtan 27248 | The tangent function has r... |
| cosatan 27249 | The cosine of an arctangen... |
| cosatanne0 27250 | The arctangent function ha... |
| atantan 27251 | The arctangent function is... |
| atantanb 27252 | Relationship between tange... |
| atanbndlem 27253 | Lemma for ~ atanbnd . (Co... |
| atanbnd 27254 | The arctangent function is... |
| atanord 27255 | The arctangent function is... |
| atan1 27256 | The arctangent of ` 1 ` is... |
| bndatandm 27257 | A point in the open unit d... |
| atans 27258 | The "domain of continuity"... |
| atans2 27259 | It suffices to show that `... |
| atansopn 27260 | The domain of continuity o... |
| atansssdm 27261 | The domain of continuity o... |
| ressatans 27262 | The real number line is a ... |
| dvatan 27263 | The derivative of the arct... |
| atancn 27264 | The arctangent is a contin... |
| atantayl 27265 | The Taylor series for ` ar... |
| atantayl2 27266 | The Taylor series for ` ar... |
| atantayl3 27267 | The Taylor series for ` ar... |
| leibpilem1 27268 | Lemma for ~ leibpi . (Con... |
| leibpilem2 27269 | The Leibniz formula for ` ... |
| leibpi 27270 | The Leibniz formula for ` ... |
| leibpisum 27271 | The Leibniz formula for ` ... |
| log2cnv 27272 | Using the Taylor series fo... |
| log2tlbnd 27273 | Bound the error term in th... |
| log2ublem1 27274 | Lemma for ~ log2ub . The ... |
| log2ublem2 27275 | Lemma for ~ log2ub . (Con... |
| log2ublem3 27276 | Lemma for ~ log2ub . In d... |
| log2ub 27277 | ` log 2 ` is less than ` 2... |
| log2le1 27278 | ` log 2 ` is less than ` 1... |
| birthdaylem1 27279 | Lemma for ~ birthday . (C... |
| birthdaylem2 27280 | For general ` N ` and ` K ... |
| birthdaylem3 27281 | For general ` N ` and ` K ... |
| birthday 27282 | The Birthday Problem. The... |
| dmarea 27285 | The domain of the area fun... |
| areambl 27286 | The fibers of a measurable... |
| areass 27287 | A measurable region is a s... |
| dfarea 27288 | Rewrite ~ df-area self-ref... |
| areaf 27289 | Area measurement is a func... |
| areacl 27290 | The area of a measurable r... |
| areage0 27291 | The area of a measurable r... |
| areaval 27292 | The area of a measurable r... |
| rlimcnp 27293 | Relate a limit of a real-v... |
| rlimcnp2 27294 | Relate a limit of a real-v... |
| rlimcnp3 27295 | Relate a limit of a real-v... |
| xrlimcnp 27296 | Relate a limit of a real-v... |
| efrlim 27297 | The limit of the sequence ... |
| dfef2 27298 | The limit of the sequence ... |
| cxplim 27299 | A power to a negative expo... |
| sqrtlim 27300 | The inverse square root fu... |
| rlimcxp 27301 | Any power to a positive ex... |
| o1cxp 27302 | An eventually bounded func... |
| cxp2limlem 27303 | A linear factor grows slow... |
| cxp2lim 27304 | Any power grows slower tha... |
| cxploglim 27305 | The logarithm grows slower... |
| cxploglim2 27306 | Every power of the logarit... |
| divsqrtsumlem 27307 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 27308 | The function ` F ` used in... |
| divsqrsum 27309 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 27310 | A bound on the distance of... |
| divsqrtsumo1 27311 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 27312 | Closure of a 0-1 linear co... |
| scvxcvx 27313 | A strictly convex function... |
| jensenlem1 27314 | Lemma for ~ jensen . (Con... |
| jensenlem2 27315 | Lemma for ~ jensen . (Con... |
| jensen 27316 | Jensen's inequality, a fin... |
| amgmlem 27317 | Lemma for ~ amgm . (Contr... |
| amgm 27318 | Inequality of arithmetic a... |
| logdifbnd 27321 | Bound on the difference of... |
| logdiflbnd 27322 | Lower bound on the differe... |
| emcllem1 27323 | Lemma for ~ emcl . The se... |
| emcllem2 27324 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 27325 | Lemma for ~ emcl . The fu... |
| emcllem4 27326 | Lemma for ~ emcl . The di... |
| emcllem5 27327 | Lemma for ~ emcl . The pa... |
| emcllem6 27328 | Lemma for ~ emcl . By the... |
| emcllem7 27329 | Lemma for ~ emcl and ~ har... |
| emcl 27330 | Closure and bounds for the... |
| harmonicbnd 27331 | A bound on the harmonic se... |
| harmonicbnd2 27332 | A bound on the harmonic se... |
| emre 27333 | The Euler-Mascheroni const... |
| emgt0 27334 | The Euler-Mascheroni const... |
| harmonicbnd3 27335 | A bound on the harmonic se... |
| harmoniclbnd 27336 | A bound on the harmonic se... |
| harmonicubnd 27337 | A bound on the harmonic se... |
| harmonicbnd4 27338 | The asymptotic behavior of... |
| fsumharmonic 27339 | Bound a finite sum based o... |
| zetacvg 27342 | The zeta series is converg... |
| eldmgm 27349 | Elementhood in the set of ... |
| dmgmaddn0 27350 | If ` A ` is not a nonposit... |
| dmlogdmgm 27351 | If ` A ` is in the continu... |
| rpdmgm 27352 | A positive real number is ... |
| dmgmn0 27353 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27354 | If ` A ` is not a nonposit... |
| dmgmdivn0 27355 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27356 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27357 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27358 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27359 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27360 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27361 | The series ` G ` is unifor... |
| lgamgulm 27362 | The series ` G ` is unifor... |
| lgamgulm2 27363 | Rewrite the limit of the s... |
| lgambdd 27364 | The log-Gamma function is ... |
| lgamucov 27365 | The ` U ` regions used in ... |
| lgamucov2 27366 | The ` U ` regions used in ... |
| lgamcvglem 27367 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27368 | The log-Gamma function is ... |
| lgamf 27369 | The log-Gamma function is ... |
| gamf 27370 | The Gamma function is a co... |
| gamcl 27371 | The exponential of the log... |
| eflgam 27372 | The exponential of the log... |
| gamne0 27373 | The Gamma function is neve... |
| igamval 27374 | Value of the inverse Gamma... |
| igamz 27375 | Value of the inverse Gamma... |
| igamgam 27376 | Value of the inverse Gamma... |
| igamlgam 27377 | Value of the inverse Gamma... |
| igamf 27378 | Closure of the inverse Gam... |
| igamcl 27379 | Closure of the inverse Gam... |
| gamigam 27380 | The Gamma function is the ... |
| lgamcvg 27381 | The series ` G ` converges... |
| lgamcvg2 27382 | The series ` G ` converges... |
| gamcvg 27383 | The pointwise exponential ... |
| lgamp1 27384 | The functional equation of... |
| gamp1 27385 | The functional equation of... |
| gamcvg2lem 27386 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27387 | An infinite product expres... |
| regamcl 27388 | The Gamma function is real... |
| relgamcl 27389 | The log-Gamma function is ... |
| rpgamcl 27390 | The log-Gamma function is ... |
| lgam1 27391 | The log-Gamma function at ... |
| gam1 27392 | The log-Gamma function at ... |
| facgam 27393 | The Gamma function general... |
| gamfac 27394 | The Gamma function general... |
| wilthlem1 27395 | The only elements that are... |
| wilthlem2 27396 | Lemma for ~ wilth : induct... |
| wilthlem3 27397 | Lemma for ~ wilth . Here ... |
| wilth 27398 | Wilson's theorem. A numbe... |
| wilthimp 27399 | The forward implication of... |
| ftalem1 27400 | Lemma for ~ fta : "growth... |
| ftalem2 27401 | Lemma for ~ fta . There e... |
| ftalem3 27402 | Lemma for ~ fta . There e... |
| ftalem4 27403 | Lemma for ~ fta : Closure... |
| ftalem5 27404 | Lemma for ~ fta : Main pr... |
| ftalem6 27405 | Lemma for ~ fta : Dischar... |
| ftalem7 27406 | Lemma for ~ fta . Shift t... |
| fta 27407 | The Fundamental Theorem of... |
| basellem1 27408 | Lemma for ~ basel . Closu... |
| basellem2 27409 | Lemma for ~ basel . Show ... |
| basellem3 27410 | Lemma for ~ basel . Using... |
| basellem4 27411 | Lemma for ~ basel . By ~ ... |
| basellem5 27412 | Lemma for ~ basel . Using... |
| basellem6 27413 | Lemma for ~ basel . The f... |
| basellem7 27414 | Lemma for ~ basel . The f... |
| basellem8 27415 | Lemma for ~ basel . The f... |
| basellem9 27416 | Lemma for ~ basel . Since... |
| basel 27417 | The sum of the inverse squ... |
| efnnfsumcl 27430 | Finite sum closure in the ... |
| ppisval 27431 | The set of primes less tha... |
| ppisval2 27432 | The set of primes less tha... |
| ppifi 27433 | The set of primes less tha... |
| prmdvdsfi 27434 | The set of prime divisors ... |
| chtf 27435 | Domain and codoamin of the... |
| chtcl 27436 | Real closure of the Chebys... |
| chtval 27437 | Value of the Chebyshev fun... |
| efchtcl 27438 | The Chebyshev function is ... |
| chtge0 27439 | The Chebyshev function is ... |
| vmaval 27440 | Value of the von Mangoldt ... |
| isppw 27441 | Two ways to say that ` A `... |
| isppw2 27442 | Two ways to say that ` A `... |
| vmappw 27443 | Value of the von Mangoldt ... |
| vmaprm 27444 | Value of the von Mangoldt ... |
| vmacl 27445 | Closure for the von Mangol... |
| vmaf 27446 | Functionality of the von M... |
| efvmacl 27447 | The von Mangoldt is closed... |
| vmage0 27448 | The von Mangoldt function ... |
| chpval 27449 | Value of the second Chebys... |
| chpf 27450 | Functionality of the secon... |
| chpcl 27451 | Closure for the second Che... |
| efchpcl 27452 | The second Chebyshev funct... |
| chpge0 27453 | The second Chebyshev funct... |
| ppival 27454 | Value of the prime-countin... |
| ppival2 27455 | Value of the prime-countin... |
| ppival2g 27456 | Value of the prime-countin... |
| ppif 27457 | Domain and codomain of the... |
| ppicl 27458 | Real closure of the prime-... |
| muval 27459 | The value of the Möbi... |
| muval1 27460 | The value of the Möbi... |
| muval2 27461 | The value of the Möbi... |
| isnsqf 27462 | Two ways to say that a num... |
| issqf 27463 | Two ways to say that a num... |
| sqfpc 27464 | The prime count of a squar... |
| dvdssqf 27465 | A divisor of a squarefree ... |
| sqf11 27466 | A squarefree number is com... |
| muf 27467 | The Möbius function i... |
| mucl 27468 | Closure of the Möbius... |
| sgmval 27469 | The value of the divisor f... |
| sgmval2 27470 | The value of the divisor f... |
| 0sgm 27471 | The value of the sum-of-di... |
| sgmf 27472 | The divisor function is a ... |
| sgmcl 27473 | Closure of the divisor fun... |
| sgmnncl 27474 | Closure of the divisor fun... |
| mule1 27475 | The Möbius function t... |
| chtfl 27476 | The Chebyshev function doe... |
| chpfl 27477 | The second Chebyshev funct... |
| ppiprm 27478 | The prime-counting functio... |
| ppinprm 27479 | The prime-counting functio... |
| chtprm 27480 | The Chebyshev function at ... |
| chtnprm 27481 | The Chebyshev function at ... |
| chpp1 27482 | The second Chebyshev funct... |
| chtwordi 27483 | The Chebyshev function is ... |
| chpwordi 27484 | The second Chebyshev funct... |
| chtdif 27485 | The difference of the Cheb... |
| efchtdvds 27486 | The exponentiated Chebyshe... |
| ppifl 27487 | The prime-counting functio... |
| ppip1le 27488 | The prime-counting functio... |
| ppiwordi 27489 | The prime-counting functio... |
| ppidif 27490 | The difference of the prim... |
| ppi1 27491 | The prime-counting functio... |
| cht1 27492 | The Chebyshev function at ... |
| vma1 27493 | The von Mangoldt function ... |
| chp1 27494 | The second Chebyshev funct... |
| ppi1i 27495 | Inference form of ~ ppiprm... |
| ppi2i 27496 | Inference form of ~ ppinpr... |
| ppi2 27497 | The prime-counting functio... |
| ppi3 27498 | The prime-counting functio... |
| cht2 27499 | The Chebyshev function at ... |
| cht3 27500 | The Chebyshev function at ... |
| ppinncl 27501 | Closure of the prime-count... |
| chtrpcl 27502 | Closure of the Chebyshev f... |
| ppieq0 27503 | The prime-counting functio... |
| ppiltx 27504 | The prime-counting functio... |
| prmorcht 27505 | Relate the primorial (prod... |
| mumullem1 27506 | Lemma for ~ mumul . A mul... |
| mumullem2 27507 | Lemma for ~ mumul . The p... |
| mumul 27508 | The Möbius function i... |
| sqff1o 27509 | There is a bijection from ... |
| fsumdvdsdiaglem 27510 | A "diagonal commutation" o... |
| fsumdvdsdiag 27511 | A "diagonal commutation" o... |
| fsumdvdscom 27512 | A double commutation of di... |
| dvdsppwf1o 27513 | A bijection between the di... |
| dvdsflf1o 27514 | A bijection from the numbe... |
| dvdsflsumcom 27515 | A sum commutation from ` s... |
| fsumfldivdiaglem 27516 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27517 | The right-hand side of ~ d... |
| musum 27518 | The sum of the Möbius... |
| musumsum 27519 | Evaluate a collapsing sum ... |
| muinv 27520 | The Möbius inversion ... |
| mpodvdsmulf1o 27521 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27522 | Product of two divisor sum... |
| dvdsmulf1o 27523 | If ` M ` and ` N ` are two... |
| sgmppw 27524 | The value of the divisor f... |
| 0sgmppw 27525 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27526 | The sum of divisors for a ... |
| 1sgm2ppw 27527 | The sum of the divisors of... |
| sgmmul 27528 | The divisor function for f... |
| ppiublem1 27529 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27530 | A prime greater than ` 3 `... |
| ppiub 27531 | An upper bound on the prim... |
| vmalelog 27532 | The von Mangoldt function ... |
| chtlepsi 27533 | The first Chebyshev functi... |
| chprpcl 27534 | Closure of the second Cheb... |
| chpeq0 27535 | The second Chebyshev funct... |
| chteq0 27536 | The first Chebyshev functi... |
| chtleppi 27537 | Upper bound on the ` theta... |
| chtublem 27538 | Lemma for ~ chtub . (Cont... |
| chtub 27539 | An upper bound on the Cheb... |
| fsumvma 27540 | Rewrite a sum over the von... |
| fsumvma2 27541 | Apply ~ fsumvma for the co... |
| pclogsum 27542 | The logarithmic analogue o... |
| vmasum 27543 | The sum of the von Mangold... |
| logfac2 27544 | Another expression for the... |
| chpval2 27545 | Express the second Chebysh... |
| chpchtsum 27546 | The second Chebyshev funct... |
| chpub 27547 | An upper bound on the seco... |
| logfacubnd 27548 | A simple upper bound on th... |
| logfaclbnd 27549 | A lower bound on the logar... |
| logfacbnd3 27550 | Show the stronger statemen... |
| logfacrlim 27551 | Combine the estimates ~ lo... |
| logexprlim 27552 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27553 | Write out ~ logfacrlim as ... |
| mersenne 27554 | A Mersenne prime is a prim... |
| perfect1 27555 | Euclid's contribution to t... |
| perfectlem1 27556 | Lemma for ~ perfect . (Co... |
| perfectlem2 27557 | Lemma for ~ perfect . (Co... |
| perfect 27558 | The Euclid-Euler theorem, ... |
| dchrval 27561 | Value of the group of Diri... |
| dchrbas 27562 | Base set of the group of D... |
| dchrelbas 27563 | A Dirichlet character is a... |
| dchrelbas2 27564 | A Dirichlet character is a... |
| dchrelbas3 27565 | A Dirichlet character is a... |
| dchrelbasd 27566 | A Dirichlet character is a... |
| dchrrcl 27567 | Reverse closure for a Diri... |
| dchrmhm 27568 | A Dirichlet character is a... |
| dchrf 27569 | A Dirichlet character is a... |
| dchrelbas4 27570 | A Dirichlet character is a... |
| dchrzrh1 27571 | Value of a Dirichlet chara... |
| dchrzrhcl 27572 | A Dirichlet character take... |
| dchrzrhmul 27573 | A Dirichlet character is c... |
| dchrplusg 27574 | Group operation on the gro... |
| dchrmul 27575 | Group operation on the gro... |
| dchrmulcl 27576 | Closure of the group opera... |
| dchrn0 27577 | A Dirichlet character is n... |
| dchr1cl 27578 | Closure of the principal D... |
| dchrmullid 27579 | Left identity for the prin... |
| dchrinvcl 27580 | Closure of the group inver... |
| dchrabl 27581 | The set of Dirichlet chara... |
| dchrfi 27582 | The group of Dirichlet cha... |
| dchrghm 27583 | A Dirichlet character rest... |
| dchr1 27584 | Value of the principal Dir... |
| dchreq 27585 | A Dirichlet character is d... |
| dchrresb 27586 | A Dirichlet character is d... |
| dchrabs 27587 | A Dirichlet character take... |
| dchrinv 27588 | The inverse of a Dirichlet... |
| dchrabs2 27589 | A Dirichlet character take... |
| dchr1re 27590 | The principal Dirichlet ch... |
| dchrptlem1 27591 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27592 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27593 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27594 | For any element other than... |
| dchrsum2 27595 | An orthogonality relation ... |
| dchrsum 27596 | An orthogonality relation ... |
| sumdchr2 27597 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27598 | There are exactly ` phi ( ... |
| sumdchr 27599 | An orthogonality relation ... |
| dchr2sum 27600 | An orthogonality relation ... |
| sum2dchr 27601 | An orthogonality relation ... |
| bcctr 27602 | Value of the central binom... |
| pcbcctr 27603 | Prime count of a central b... |
| bcmono 27604 | The binomial coefficient i... |
| bcmax 27605 | The binomial coefficient t... |
| bcp1ctr 27606 | Ratio of two central binom... |
| bclbnd 27607 | A bound on the binomial co... |
| efexple 27608 | Convert a bound on a power... |
| bpos1lem 27609 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27610 | Bertrand's postulate, chec... |
| bposlem1 27611 | An upper bound on the prim... |
| bposlem2 27612 | There are no odd primes in... |
| bposlem3 27613 | Lemma for ~ bpos . Since ... |
| bposlem4 27614 | Lemma for ~ bpos . (Contr... |
| bposlem5 27615 | Lemma for ~ bpos . Bound ... |
| bposlem6 27616 | Lemma for ~ bpos . By usi... |
| bposlem7 27617 | Lemma for ~ bpos . The fu... |
| bposlem8 27618 | Lemma for ~ bpos . Show t... |
| bposlem9 27619 | Lemma for ~ bpos . Derive... |
| bpos 27620 | Bertrand's postulate: ther... |
| zabsle1 27623 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27624 | When ` a ` is coprime to t... |
| lgslem2 27625 | The set ` Z ` of all integ... |
| lgslem3 27626 | The set ` Z ` of all integ... |
| lgslem4 27627 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27628 | Value of the Legendre symb... |
| lgsfval 27629 | Value of the function ` F ... |
| lgsfcl2 27630 | The function ` F ` is clos... |
| lgscllem 27631 | The Legendre symbol is an ... |
| lgsfcl 27632 | Closure of the function ` ... |
| lgsfle1 27633 | The function ` F ` has mag... |
| lgsval2lem 27634 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27635 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27636 | The Legendre symbol is an ... |
| lgs0 27637 | The Legendre symbol when t... |
| lgscl 27638 | The Legendre symbol is an ... |
| lgsle1 27639 | The Legendre symbol has ab... |
| lgsval2 27640 | The Legendre symbol at a p... |
| lgs2 27641 | The Legendre symbol at ` 2... |
| lgsval3 27642 | The Legendre symbol at an ... |
| lgsvalmod 27643 | The Legendre symbol is equ... |
| lgsval4 27644 | Restate ~ lgsval for nonze... |
| lgsfcl3 27645 | Closure of the function ` ... |
| lgsval4a 27646 | Same as ~ lgsval4 for posi... |
| lgscl1 27647 | The value of the Legendre ... |
| lgsneg 27648 | The Legendre symbol is eit... |
| lgsneg1 27649 | The Legendre symbol for no... |
| lgsmod 27650 | The Legendre (Jacobi) symb... |
| lgsdilem 27651 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27652 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27653 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27654 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27655 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27656 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27657 | The Legendre symbol is com... |
| lgsdirprm 27658 | The Legendre symbol is com... |
| lgsdir 27659 | The Legendre symbol is com... |
| lgsdilem2 27660 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27661 | The Legendre symbol is com... |
| lgsne0 27662 | The Legendre symbol is non... |
| lgsabs1 27663 | The Legendre symbol is non... |
| lgssq 27664 | The Legendre symbol at a s... |
| lgssq2 27665 | The Legendre symbol at a s... |
| lgsprme0 27666 | The Legendre symbol at any... |
| 1lgs 27667 | The Legendre symbol at ` 1... |
| lgs1 27668 | The Legendre symbol at ` 1... |
| lgsmodeq 27669 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27670 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27671 | Variation on ~ lgsdir vali... |
| lgsdinn0 27672 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27673 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27674 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27675 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27676 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27677 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27678 | The Legendre symbol for od... |
| lgsqrmod 27679 | If the Legendre symbol of ... |
| lgsqrmodndvds 27680 | If the Legendre symbol of ... |
| lgsdchrval 27681 | The Legendre symbol functi... |
| lgsdchr 27682 | The Legendre symbol functi... |
| gausslemma2dlem0a 27683 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27684 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27685 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27686 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27687 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27688 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27689 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27690 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27691 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27692 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27693 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27694 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27695 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27696 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27697 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27698 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27699 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27700 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27701 | Gauss' Lemma (see also the... |
| lgseisenlem1 27702 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27703 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27704 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27705 | Lemma for ~ lgseisen . (C... |
| lgseisen 27706 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27707 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27708 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27709 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27710 | The Law of Quadratic Recip... |
| lgsquad2lem1 27711 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27712 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27713 | Extend ~ lgsquad to coprim... |
| lgsquad3 27714 | Extend ~ lgsquad2 to integ... |
| m1lgs 27715 | The first supplement to th... |
| 2lgslem1a1 27716 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27717 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27718 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27719 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27720 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27721 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27722 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27723 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27724 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27725 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27726 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27727 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27728 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27729 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27730 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27731 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27732 | The Legendre symbol for ` ... |
| 2lgslem4 27733 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27734 | The second supplement to t... |
| 2lgsoddprmlem1 27735 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27736 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27737 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27738 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27739 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27740 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27741 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27742 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27743 | The second supplement to t... |
| 2sqlem1 27744 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27745 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27746 | Fibonacci's identity (actu... |
| 2sqlem3 27747 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27748 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27749 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27750 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27751 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27752 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27753 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27754 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27755 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27756 | Lemma for ~ 2sq . (Contri... |
| 2sq 27757 | All primes of the form ` 4... |
| 2sqblem 27758 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27759 | The converse to ~ 2sq . (... |
| 2sq2 27760 | ` 2 ` is the sum of square... |
| 2sqn0 27761 | If the sum of two squares ... |
| 2sqcoprm 27762 | If the sum of two squares ... |
| 2sqmod 27763 | Given two decompositions o... |
| 2sqmo 27764 | There exists at most one d... |
| 2sqnn0 27765 | All primes of the form ` 4... |
| 2sqnn 27766 | All primes of the form ` 4... |
| addsq2reu 27767 | For each complex number ` ... |
| addsqn2reu 27768 | For each complex number ` ... |
| addsqrexnreu 27769 | For each complex number, t... |
| addsqnreup 27770 | There is no unique decompo... |
| addsq2nreurex 27771 | For each complex number ` ... |
| addsqn2reurex2 27772 | For each complex number ` ... |
| 2sqreulem1 27773 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27774 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27775 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27776 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27777 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27778 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27779 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27780 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27781 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27782 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27783 | There exists a unique deco... |
| 2sqreunn 27784 | There exists a unique deco... |
| 2sqreult 27785 | There exists a unique deco... |
| 2sqreultb 27786 | There exists a unique deco... |
| 2sqreunnlt 27787 | There exists a unique deco... |
| 2sqreunnltb 27788 | There exists a unique deco... |
| 2sqreuop 27789 | There exists a unique deco... |
| 2sqreuopnn 27790 | There exists a unique deco... |
| 2sqreuoplt 27791 | There exists a unique deco... |
| 2sqreuopltb 27792 | There exists a unique deco... |
| 2sqreuopnnlt 27793 | There exists a unique deco... |
| 2sqreuopnnltb 27794 | There exists a unique deco... |
| 2sqreuopb 27795 | There exists a unique deco... |
| chebbnd1lem1 27796 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27797 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27798 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27799 | The Chebyshev bound: The ... |
| chtppilimlem1 27800 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27801 | Lemma for ~ chtppilim . (... |
| chtppilim 27802 | The ` theta ` function is ... |
| chto1ub 27803 | The ` theta ` function is ... |
| chebbnd2 27804 | The Chebyshev bound, part ... |
| chto1lb 27805 | The ` theta ` function is ... |
| chpchtlim 27806 | The ` psi ` and ` theta ` ... |
| chpo1ub 27807 | The ` psi ` function is up... |
| chpo1ubb 27808 | The ` psi ` function is up... |
| vmadivsum 27809 | The sum of the von Mangold... |
| vmadivsumb 27810 | Give a total bound on the ... |
| rplogsumlem1 27811 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27812 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27813 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27814 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27815 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27816 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27817 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27818 | Lemma for ~ dchrisum . Le... |
| dchrisum 27819 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27820 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27821 | The sum of the Möbius... |
| dchrvmasumlem1 27822 | An alternative expression ... |
| dchrvmasum2lem 27823 | Give an expression for ` l... |
| dchrvmasum2if 27824 | Combine the results of ~ d... |
| dchrvmasumlem2 27825 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27826 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27827 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27828 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27829 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27830 | An asymptotic approximatio... |
| dchrvmaeq0 27831 | The set ` W ` is the colle... |
| dchrisum0fval 27832 | Value of the function ` F ... |
| dchrisum0fmul 27833 | The function ` F ` , the d... |
| dchrisum0ff 27834 | The function ` F ` is a re... |
| dchrisum0flblem1 27835 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27836 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27837 | The divisor sum of a real ... |
| dchrisum0fno1 27838 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27839 | A partial result along the... |
| dchrisum0re 27840 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27841 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27842 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27843 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27844 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27845 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27846 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27847 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27848 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27849 | The sum of the Möbius... |
| dchrvmasumlem 27850 | The sum of the Möbius... |
| dchrmusum 27851 | The sum of the Möbius... |
| dchrvmasum 27852 | The sum of the von Mangold... |
| rpvmasum 27853 | The sum of the von Mangold... |
| rplogsum 27854 | The sum of ` log p / p ` o... |
| dirith2 27855 | Dirichlet's theorem: there... |
| dirith 27856 | Dirichlet's theorem: there... |
| mudivsum 27857 | Asymptotic formula for ` s... |
| mulogsumlem 27858 | Lemma for ~ mulogsum . (C... |
| mulogsum 27859 | Asymptotic formula for ... |
| logdivsum 27860 | Asymptotic analysis of ... |
| mulog2sumlem1 27861 | Asymptotic formula for ... |
| mulog2sumlem2 27862 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27863 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27864 | Asymptotic formula for ... |
| vmalogdivsum2 27865 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27866 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27867 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27868 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27869 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27870 | The Möbius inverse of... |
| log2sumbnd 27871 | Bound on the difference be... |
| selberglem1 27872 | Lemma for ~ selberg . Est... |
| selberglem2 27873 | Lemma for ~ selberg . (Co... |
| selberglem3 27874 | Lemma for ~ selberg . Est... |
| selberg 27875 | Selberg's symmetry formula... |
| selbergb 27876 | Convert eventual boundedne... |
| selberg2lem 27877 | Lemma for ~ selberg2 . Eq... |
| selberg2 27878 | Selberg's symmetry formula... |
| selberg2b 27879 | Convert eventual boundedne... |
| chpdifbndlem1 27880 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27881 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27882 | A bound on the difference ... |
| logdivbnd 27883 | A bound on a sum of logs, ... |
| selberg3lem1 27884 | Introduce a log weighting ... |
| selberg3lem2 27885 | Lemma for ~ selberg3 . Eq... |
| selberg3 27886 | Introduce a log weighting ... |
| selberg4lem1 27887 | Lemma for ~ selberg4 . Eq... |
| selberg4 27888 | The Selberg symmetry formu... |
| pntrval 27889 | Define the residual of the... |
| pntrf 27890 | Functionality of the resid... |
| pntrmax 27891 | There is a bound on the re... |
| pntrsumo1 27892 | A bound on a sum over ` R ... |
| pntrsumbnd 27893 | A bound on a sum over ` R ... |
| pntrsumbnd2 27894 | A bound on a sum over ` R ... |
| selbergr 27895 | Selberg's symmetry formula... |
| selberg3r 27896 | Selberg's symmetry formula... |
| selberg4r 27897 | Selberg's symmetry formula... |
| selberg34r 27898 | The sum of ~ selberg3r and... |
| pntsval 27899 | Define the "Selberg functi... |
| pntsf 27900 | Functionality of the Selbe... |
| selbergs 27901 | Selberg's symmetry formula... |
| selbergsb 27902 | Selberg's symmetry formula... |
| pntsval2 27903 | The Selberg function can b... |
| pntrlog2bndlem1 27904 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27905 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27906 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27907 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27908 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27909 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27910 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27911 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27912 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27913 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27914 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27915 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27916 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27917 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27918 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27919 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27920 | Lemma for ~ pnt . Establi... |
| pntlemd 27921 | Lemma for ~ pnt . Closure... |
| pntlemc 27922 | Lemma for ~ pnt . Closure... |
| pntlema 27923 | Lemma for ~ pnt . Closure... |
| pntlemb 27924 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27925 | Lemma for ~ pnt . Closure... |
| pntlemh 27926 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27927 | Lemma for ~ pnt . The "na... |
| pntlemq 27928 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27929 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27930 | Lemma for ~ pnt . The ind... |
| pntlemi 27931 | Lemma for ~ pnt . Elimina... |
| pntlemf 27932 | Lemma for ~ pnt . Add up ... |
| pntlemk 27933 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27934 | Lemma for ~ pnt . Combine... |
| pntleme 27935 | Lemma for ~ pnt . Package... |
| pntlem3 27936 | Lemma for ~ pnt . Equatio... |
| pntlemp 27937 | Lemma for ~ pnt . Wrappin... |
| pntleml 27938 | Lemma for ~ pnt . Equatio... |
| pnt3 27939 | The Prime Number Theorem, ... |
| pnt2 27940 | The Prime Number Theorem, ... |
| pnt 27941 | The Prime Number Theorem: ... |
| abvcxp 27942 | Raising an absolute value ... |
| padicfval 27943 | Value of the p-adic absolu... |
| padicval 27944 | Value of the p-adic absolu... |
| ostth2lem1 27945 | Lemma for ~ ostth2 , altho... |
| qrngbas 27946 | The base set of the field ... |
| qdrng 27947 | The rationals form a divis... |
| qrng0 27948 | The zero element of the fi... |
| qrng1 27949 | The unity element of the f... |
| qrngneg 27950 | The additive inverse in th... |
| qrngdiv 27951 | The division operation in ... |
| qabvle 27952 | By using induction on ` N ... |
| qabvexp 27953 | Induct the product rule ~ ... |
| ostthlem1 27954 | Lemma for ~ ostth . If tw... |
| ostthlem2 27955 | Lemma for ~ ostth . Refin... |
| qabsabv 27956 | The regular absolute value... |
| padicabv 27957 | The p-adic absolute value ... |
| padicabvf 27958 | The p-adic absolute value ... |
| padicabvcxp 27959 | All positive powers of the... |
| ostth1 27960 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27961 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27962 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27963 | Lemma for ~ ostth2 . (Con... |
| ostth2 27964 | - Lemma for ~ ostth : regu... |
| ostth3 27965 | - Lemma for ~ ostth : p-ad... |
| ostth 27966 | Ostrowski's theorem, which... |
| infdesc 27967 | Infinite descent. The hyp... |
| fltdiv 27968 | A counterexample to FLT st... |
| flt0 27969 | A counterexample for FLT d... |
| fltdvdsabdvdsc 27970 | Any factor of both ` A ` a... |
| fltabcoprmex 27971 | A counterexample to FLT im... |
| fltaccoprm 27972 | A counterexample to FLT wi... |
| fltbccoprm 27973 | A counterexample to FLT wi... |
| fltabcoprm 27974 | A counterexample to FLT wi... |
| fltne 27975 | If a counterexample to FLT... |
| flt4lem1 27976 | Satisfy the antecedent use... |
| flt4lem2 27977 | If ` A ` is even, ` B ` is... |
| flt4lem3 27978 | Equivalent to ~ pythagtrip... |
| flt4lem4 27979 | If the product of two copr... |
| flt4lem5 27980 | In the context of the lemm... |
| flt4lem5elem 27981 | Version of ~ fltaccoprm an... |
| flt4lem5a 27982 | Part 1 of Equation 1 of ... |
| flt4lem5b 27983 | Part 2 of Equation 1 of ... |
| flt4lem5c 27984 | Part 2 of Equation 2 of ... |
| flt4lem5d 27985 | Part 3 of Equation 2 of ... |
| flt4lem5e 27986 | Satisfy the hypotheses of ... |
| flt4lem5f 27987 | Final equation of ~... |
| flt4lem6 27988 | Remove shared factors in a... |
| flt4lem7 27989 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 27990 | Strengthening of Fermat's ... |
| flt4 27991 | Fermat's last theorem for ... |
| flt4ALT 27992 | Fermat's last theorem for ... |
| fltoprmlem1 27993 | Lemma 1 for ~ fltoprm : E... |
| fltoprmlem2 27994 | Lemma 2 for ~ fltoprm : I... |
| fltoprm 27995 | Fermat's last theorem hold... |
| fltoprmgt3 27996 | Fermat's last theorem hold... |
| elno 28003 | Membership in the surreals... |
| ltsval 28004 | The value of the surreal l... |
| bdayval 28005 | The value of the birthday ... |
| nofun 28006 | A surreal is a function. ... |
| nodmon 28007 | The domain of a surreal is... |
| norn 28008 | The range of a surreal is ... |
| nofnbday 28009 | A surreal is a function ov... |
| nodmord 28010 | The domain of a surreal ha... |
| elno2 28011 | An alternative condition f... |
| elno3 28012 | Another condition for memb... |
| ltsval2 28013 | Alternate expression for s... |
| nofv 28014 | The function value of a su... |
| nosgnn0 28015 | ` (/) ` is not a surreal s... |
| nosgnn0i 28016 | If ` X ` is a surreal sign... |
| noreson 28017 | The restriction of a surre... |
| ltsintdifex 28018 |
If ` A |
| ltsres 28019 | If the restrictions of two... |
| noxp1o 28020 | The Cartesian product of a... |
| noseponlem 28021 | Lemma for ~ nosepon . Con... |
| nosepon 28022 | Given two unequal surreals... |
| noextend 28023 | Extending a surreal by one... |
| noextendseq 28024 | Extend a surreal by a sequ... |
| noextenddif 28025 | Calculate the place where ... |
| noextendlt 28026 | Extending a surreal with a... |
| noextendgt 28027 | Extending a surreal with a... |
| nolesgn2o 28028 | Given ` A ` less-than or e... |
| nolesgn2ores 28029 | Given ` A ` less-than or e... |
| nogesgn1o 28030 | Given ` A ` greater than o... |
| nogesgn1ores 28031 | Given ` A ` greater than o... |
| ltssolem1 28032 | Lemma for ~ ltsso . The "... |
| ltsso 28033 | Less-than totally orders t... |
| bdayfo 28034 | The birthday function maps... |
| fvnobday 28035 | The value of a surreal at ... |
| nosepnelem 28036 | Lemma for ~ nosepne . (Co... |
| nosepne 28037 | The value of two non-equal... |
| nosep1o 28038 | If the value of a surreal ... |
| nosep2o 28039 | If the value of a surreal ... |
| nosepdmlem 28040 | Lemma for ~ nosepdm . (Co... |
| nosepdm 28041 | The first place two surrea... |
| nosepeq 28042 | The values of two surreals... |
| nosepssdm 28043 | Given two non-equal surrea... |
| nodenselem4 28044 | Lemma for ~ nodense . Sho... |
| nodenselem5 28045 | Lemma for ~ nodense . If ... |
| nodenselem6 28046 | The restriction of a surre... |
| nodenselem7 28047 | Lemma for ~ nodense . ` A ... |
| nodenselem8 28048 | Lemma for ~ nodense . Giv... |
| nodense 28049 | Given two distinct surreal... |
| bdayimaon 28050 | Lemma for full-eta propert... |
| nolt02olem 28051 | Lemma for ~ nolt02o . If ... |
| nolt02o 28052 | Given ` A ` less-than ` B ... |
| nogt01o 28053 | Given ` A ` greater than `... |
| noresle 28054 | Restriction law for surrea... |
| nomaxmo 28055 | A class of surreals has at... |
| nominmo 28056 | A class of surreals has at... |
| nosupprefixmo 28057 | In any class of surreals, ... |
| noinfprefixmo 28058 | In any class of surreals, ... |
| nosupcbv 28059 | Lemma to change bound vari... |
| nosupno 28060 | The next several theorems ... |
| nosupdm 28061 | The domain of the surreal ... |
| nosupbday 28062 | Birthday bounding law for ... |
| nosupfv 28063 | The value of surreal supre... |
| nosupres 28064 | A restriction law for surr... |
| nosupbnd1lem1 28065 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 28066 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 28067 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 28068 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 28069 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 28070 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 28071 | Bounding law from below fo... |
| nosupbnd2lem1 28072 | Bounding law from above wh... |
| nosupbnd2 28073 | Bounding law from above fo... |
| noinfcbv 28074 | Change bound variables for... |
| noinfno 28075 | The next several theorems ... |
| noinfdm 28076 | Next, we calculate the dom... |
| noinfbday 28077 | Birthday bounding law for ... |
| noinffv 28078 | The value of surreal infim... |
| noinfres 28079 | The restriction of surreal... |
| noinfbnd1lem1 28080 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 28081 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 28082 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 28083 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 28084 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 28085 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 28086 | Bounding law from above fo... |
| noinfbnd2lem1 28087 | Bounding law from below wh... |
| noinfbnd2 28088 | Bounding law from below fo... |
| nosupinfsep 28089 | Given two sets of surreals... |
| noetasuplem1 28090 | Lemma for ~ noeta . Estab... |
| noetasuplem2 28091 | Lemma for ~ noeta . The r... |
| noetasuplem3 28092 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 28093 | Lemma for ~ noeta . When ... |
| noetainflem1 28094 | Lemma for ~ noeta . Estab... |
| noetainflem2 28095 | Lemma for ~ noeta . The r... |
| noetainflem3 28096 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 28097 | Lemma for ~ noeta . If ` ... |
| noetalem1 28098 | Lemma for ~ noeta . Eithe... |
| noetalem2 28099 | Lemma for ~ noeta . The f... |
| noeta 28100 | The full-eta axiom for the... |
| ltsirr 28103 | Surreal less-than is irref... |
| ltstr 28104 | Surreal less-than is trans... |
| ltsasym 28105 | Surreal less-than is asymm... |
| ltslin 28106 | Surreal less-than obeys tr... |
| ltstrieq2 28107 | Trichotomy law for surreal... |
| ltstrine 28108 | Trichotomy law for surreal... |
| lenlts 28109 | Surreal less-than or equal... |
| ltnles 28110 | Surreal less-than in terms... |
| lesloe 28111 | Surreal less-than or equal... |
| lestri3 28112 | Trichotomy law for surreal... |
| lesnltd 28113 | Surreal less-than or equal... |
| ltsnled 28114 | Surreal less-than in terms... |
| lesloed 28115 | Surreal less-than or equal... |
| lestri3d 28116 | Trichotomy law for surreal... |
| ltlestr 28117 | Surreal transitive law. (... |
| leltstr 28118 | Surreal transitive law. (... |
| lestr 28119 | Surreal transitive law. (... |
| ltstrd 28120 | Surreal less-than is trans... |
| ltlestrd 28121 | Surreal less-than is trans... |
| leltstrd 28122 | Surreal less-than is trans... |
| lestrd 28123 | Surreal less-than or equal... |
| lesid 28124 | Surreal less-than or equal... |
| lestric 28125 | Surreal trichotomy law. (... |
| maxs1 28126 | A surreal is less than or ... |
| maxs2 28127 | A surreal is less than or ... |
| mins1 28128 | The minimum of two surreal... |
| mins2 28129 | The minimum of two surreal... |
| ltlesd 28130 | Surreal less-than implies ... |
| ltsne 28131 | Surreal less-than implies ... |
| ltlesnd 28132 | Surreal less-than in terms... |
| bdayfun 28133 | The birthday function is a... |
| bdayfn 28134 | The birthday function is a... |
| bdaydm 28135 | The birthday function's do... |
| bdaydmOLD 28136 | Obsolete version of ~ bday... |
| bdayrn 28137 | The birthday function's ra... |
| bdayon 28138 | The value of the birthday ... |
| nobdaymin 28139 | Any non-empty class of sur... |
| nocvxminlem 28140 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 28141 | Given a nonempty convex cl... |
| noprc 28142 | The surreal numbers are a ... |
| noeta2 28147 | A version of ~ noeta with ... |
| brslts 28148 | Binary relation form of th... |
| sltsex1 28149 | The first argument of surr... |
| sltsex2 28150 | The second argument of sur... |
| sltsss1 28151 | The first argument of surr... |
| sltsss2 28152 | The second argument of sur... |
| sltssep 28153 | The separation property of... |
| sltsd 28154 | Deduce surreal set less-th... |
| sltssnb 28155 | Surreal set less-than of t... |
| sltssn 28156 | Surreal set less-than of t... |
| sltssepc 28157 | Two elements of separated ... |
| sltssepcd 28158 | Two elements of separated ... |
| ssslts1 28159 | Relation between surreal s... |
| ssslts2 28160 | Relation between surreal s... |
| nulslts 28161 | The empty set is less-than... |
| nulsgts 28162 | The empty set is greater t... |
| nulsltsd 28163 | The empty set is less-than... |
| nulsgtsd 28164 | The empty set is greater t... |
| conway 28165 | Conway's Simplicity Theore... |
| cutsval 28166 | The value of the surreal c... |
| cutcuts 28167 | Cut properties of the surr... |
| cutscl 28168 | Closure law for surreal cu... |
| cutscld 28169 | Closure law for surreal cu... |
| cutbday 28170 | The birthday of the surrea... |
| eqcuts 28171 | Condition for equality to ... |
| eqcuts2 28172 | Condition for equality to ... |
| sltstr 28173 | Transitive law for surreal... |
| sltsun1 28174 | Union law for surreal set ... |
| sltsun2 28175 | Union law for surreal set ... |
| cutsun12 28176 | Union law for surreal cuts... |
| dmcuts 28177 | The domain of the surreal ... |
| cutsf 28178 | Functionality statement fo... |
| etaslts 28179 | A restatement of ~ noeta u... |
| etaslts2 28180 | A version of ~ etaslts wit... |
| cutbdaybnd 28181 | An upper bound on the birt... |
| cutbdaybnd2 28182 | An upper bound on the birt... |
| cutbdaybnd2lim 28183 | An upper bound on the birt... |
| cutbdaylt 28184 | If a surreal lies in a gap... |
| lesrec 28185 | A comparison law for surre... |
| lesrecd 28186 | A comparison law for surre... |
| ltsrec 28187 | A comparison law for surre... |
| ltsrecd 28188 | A comparison law for surre... |
| sltsdisj 28189 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 28190 | A variant of the simplicit... |
| 0no 28195 | Surreal zero is a surreal.... |
| 1no 28196 | Surreal one is a surreal. ... |
| bday0 28197 | Calculate the birthday of ... |
| 0lt1s 28198 | Surreal zero is less than ... |
| bday0b 28199 | The only surreal with birt... |
| bday1 28200 | The birthday of surreal on... |
| cuteq0 28201 | Condition for a surreal cu... |
| cutneg 28202 | The simplest number greate... |
| cuteq1 28203 | Condition for a surreal cu... |
| gt0ne0s 28204 | A positive surreal is not ... |
| gt0ne0sd 28205 | A positive surreal is not ... |
| 1ne0s 28206 | Surreal zero does not equa... |
| rightge0 28207 | A surreal is non-negative ... |
| madeval 28218 | The value of the made by f... |
| madeval2 28219 | Alternative characterizati... |
| oldval 28220 | The value of the old optio... |
| newval 28221 | The value of the new optio... |
| madef 28222 | The made function is a fun... |
| oldf 28223 | The older function is a fu... |
| newf 28224 | The new function is a func... |
| old0 28225 | No surreal is older than `... |
| madessno 28226 | Made sets are surreals. (... |
| oldssno 28227 | Old sets are surreals. (C... |
| newssno 28228 | New sets are surreals. (C... |
| madeno 28229 | An element of a made set i... |
| oldno 28230 | An element of an old set i... |
| newno 28231 | An element of a new set is... |
| madenod 28232 | An element of a made set i... |
| oldnod 28233 | An element of an old set i... |
| newnod 28234 | An element of a new set is... |
| leftval 28235 | The value of the left opti... |
| rightval 28236 | The value of the right opt... |
| elleft 28237 | Membership in the left set... |
| elright 28238 | Membership in the right se... |
| leftlt 28239 | A member of a surreal's le... |
| rightgt 28240 | A member of a surreal's ri... |
| leftf 28241 | The functionality of the l... |
| rightf 28242 | The functionality of the r... |
| elmade 28243 | Membership in the made fun... |
| elmade2 28244 | Membership in the made fun... |
| elold 28245 | Membership in an old set. ... |
| sltsleft 28246 | A surreal is greater than ... |
| sltsright 28247 | A surreal is less than its... |
| lltr 28248 | The left options of a surr... |
| made0 28249 | The only surreal made on d... |
| new0 28250 | The only surreal new on da... |
| old1 28251 | The only surreal older tha... |
| madess 28252 | If ` A ` is less than or e... |
| oldssmade 28253 | The older-than set is a su... |
| oldmade 28254 | An element of an old set i... |
| oldmaded 28255 | An element of an old set i... |
| oldss 28256 | If ` A ` is less than or e... |
| leftssold 28257 | The left options are a sub... |
| rightssold 28258 | The right options are a su... |
| leftssno 28259 | The left set of a surreal ... |
| rightssno 28260 | The right set of a surreal... |
| leftold 28261 | An element of a left set i... |
| rightold 28262 | An element of a right set ... |
| leftno 28263 | An element of a left set i... |
| rightno 28264 | An element of a right set ... |
| leftoldd 28265 | An element of a left set i... |
| leftnod 28266 | An element of a left set i... |
| rightoldd 28267 | An element of a right set ... |
| rightnod 28268 | An element of a right set ... |
| madecut 28269 | Given a section that is a ... |
| madeun 28270 | The made set is the union ... |
| madeoldsuc 28271 | The made set is the old se... |
| oldsuc 28272 | The value of the old set a... |
| oldlim 28273 | The value of the old set a... |
| madebdayim 28274 | If a surreal is a member o... |
| oldbdayim 28275 | If ` X ` is in the old set... |
| oldirr 28276 | No surreal is a member of ... |
| leftirr 28277 | No surreal is a member of ... |
| rightirr 28278 | No surreal is a member of ... |
| left0s 28279 | The left set of ` 0s ` is ... |
| right0s 28280 | The right set of ` 0s ` is... |
| left1s 28281 | The left set of ` 1s ` is ... |
| right1s 28282 | The right set of ` 1s ` is... |
| lrold 28283 | The union of the left and ... |
| madebdaylemold 28284 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 28285 | Lemma for ~ madebday . If... |
| madebday 28286 | A surreal is part of the s... |
| oldbday 28287 | A surreal is part of the s... |
| newbday 28288 | A surreal is an element of... |
| newbdayim 28289 | One direction of the bicon... |
| lrcut 28290 | A surreal is equal to the ... |
| cutsfo 28291 | The surreal cut function i... |
| ltsn0 28292 | If ` X ` is less than ` Y ... |
| lruneq 28293 | If two surreals share a bi... |
| ltslpss 28294 | If two surreals share a bi... |
| leslss 28295 | If two surreals ` A ` and ... |
| 0elold 28296 | Zero is in the old set of ... |
| 0elleft 28297 | Zero is in the left set of... |
| 0elright 28298 | Zero is in the right set o... |
| madefi 28299 | The made set of an ordinal... |
| oldfi 28300 | The old set of an ordinal ... |
| bdayiun 28301 | The birthday of a surreal ... |
| bdayle 28302 | A condition for bounding a... |
| sltsbday 28303 | Birthday comparison rule f... |
| cofslts 28304 | If every element of ` A ` ... |
| coinitslts 28305 | If ` B ` is coinitial with... |
| cofcut1 28306 | If ` C ` is cofinal with `... |
| cofcut1d 28307 | If ` C ` is cofinal with `... |
| cofcut2 28308 | If ` A ` and ` C ` are mut... |
| cofcut2d 28309 | If ` A ` and ` C ` are mut... |
| cofcutr 28310 | If ` X ` is the cut of ` A... |
| cofcutr1d 28311 | If ` X ` is the cut of ` A... |
| cofcutr2d 28312 | If ` X ` is the cut of ` A... |
| cofcutrtime 28313 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 28314 | If ` X ` is a timely cut o... |
| cofcutrtime2d 28315 | If ` X ` is a timely cut o... |
| cofss 28316 | Cofinality for a subset. ... |
| coiniss 28317 | Coinitiality for a subset.... |
| cutlt 28318 | Eliminating all elements b... |
| cutpos 28319 | Reduce the elements of a c... |
| cutmax 28320 | If ` A ` has a maximum, th... |
| cutmin 28321 | If ` B ` has a minimum, th... |
| cutminmax 28322 | If the left set of ` X ` h... |
| lrrecval 28325 | The next step in the devel... |
| lrrecval2 28326 | Next, we establish an alte... |
| lrrecpo 28327 | Now, we establish that ` R... |
| lrrecse 28328 | Next, we show that ` R ` i... |
| lrrecfr 28329 | Now we show that ` R ` is ... |
| lrrecpred 28330 | Finally, we calculate the ... |
| noinds 28331 | Induction principle for a ... |
| norecfn 28332 | Surreal recursion over one... |
| norecov 28333 | Calculate the value of the... |
| noxpordpo 28336 | To get through most of the... |
| noxpordfr 28337 | Next we establish the foun... |
| noxpordse 28338 | Next we establish the set-... |
| noxpordpred 28339 | Next we calculate the pred... |
| no2indlesm 28340 | Double induction on surrea... |
| no2inds 28341 | Double induction on surrea... |
| norec2fn 28342 | The double-recursion opera... |
| norec2ov 28343 | The value of the double-re... |
| no3inds 28344 | Triple induction over surr... |
| addsfn 28347 | Surreal addition is a func... |
| addsval 28348 | The value of surreal addit... |
| addsval2 28349 | The value of surreal addit... |
| addsrid 28350 | Surreal addition to zero i... |
| addsridd 28351 | Surreal addition to zero i... |
| addscom 28352 | Surreal addition is commut... |
| addscomd 28353 | Surreal addition is commut... |
| addslid 28354 | Surreal addition to zero i... |
| addsproplem1 28355 | Lemma for surreal addition... |
| addsproplem2 28356 | Lemma for surreal addition... |
| addsproplem3 28357 | Lemma for surreal addition... |
| addsproplem4 28358 | Lemma for surreal addition... |
| addsproplem5 28359 | Lemma for surreal addition... |
| addsproplem6 28360 | Lemma for surreal addition... |
| addsproplem7 28361 | Lemma for surreal addition... |
| addsprop 28362 | Inductively show that surr... |
| addcutslem 28363 | Lemma for ~ addcuts . Sho... |
| addcuts 28364 | Demonstrate the cut proper... |
| addcuts2 28365 | Show that the cut involved... |
| addscld 28366 | Surreal numbers are closed... |
| addscl 28367 | Surreal numbers are closed... |
| addsf 28368 | Function statement for sur... |
| addsfo 28369 | Surreal addition is onto. ... |
| peano2no 28370 | A theorem for surreals tha... |
| ltadds1im 28371 | Surreal less-than is prese... |
| ltadds2im 28372 | Surreal less-than is prese... |
| leadds1im 28373 | Surreal less-than or equal... |
| leadds2im 28374 | Surreal less-than or equal... |
| leadds1 28375 | Addition to both sides of ... |
| leadds2 28376 | Addition to both sides of ... |
| ltadds2 28377 | Addition to both sides of ... |
| ltadds1 28378 | Addition to both sides of ... |
| addscan2 28379 | Cancellation law for surre... |
| addscan1 28380 | Cancellation law for surre... |
| leadds1d 28381 | Addition to both sides of ... |
| leadds2d 28382 | Addition to both sides of ... |
| ltadds2d 28383 | Addition to both sides of ... |
| ltadds1d 28384 | Addition to both sides of ... |
| addscan2d 28385 | Cancellation law for surre... |
| addscan1d 28386 | Cancellation law for surre... |
| addsuniflem 28387 | Lemma for ~ addsunif . St... |
| addsunif 28388 | Uniformity theorem for sur... |
| addsasslem1 28389 | Lemma for addition associa... |
| addsasslem2 28390 | Lemma for addition associa... |
| addsass 28391 | Surreal addition is associ... |
| addsassd 28392 | Surreal addition is associ... |
| adds32d 28393 | Commutative/associative la... |
| adds12d 28394 | Commutative/associative la... |
| adds4d 28395 | Rearrangement of four term... |
| adds42d 28396 | Rearrangement of four term... |
| ltaddspos1d 28397 | Addition of a positive num... |
| ltaddspos2d 28398 | Addition of a positive num... |
| lt2addsd 28399 | Adding both sides of two s... |
| addsgt0d 28400 | The sum of two positive su... |
| ltsp1d 28401 | A surreal is less than its... |
| addsge01d 28402 | A surreal is less-than or ... |
| addbdaylem 28403 | Lemma for ~ addbday . (Co... |
| addbday 28404 | The birthday of the sum of... |
| negsfn 28409 | Surreal negation is a func... |
| subsfn 28410 | Surreal subtraction is a f... |
| negsval 28411 | The value of the surreal n... |
| neg0s 28412 | Negative surreal zero is s... |
| neg1s 28413 | An expression for negative... |
| negsproplem1 28414 | Lemma for surreal negation... |
| negsproplem2 28415 | Lemma for surreal negation... |
| negsproplem3 28416 | Lemma for surreal negation... |
| negsproplem4 28417 | Lemma for surreal negation... |
| negsproplem5 28418 | Lemma for surreal negation... |
| negsproplem6 28419 | Lemma for surreal negation... |
| negsproplem7 28420 | Lemma for surreal negation... |
| negsprop 28421 | Show closure and ordering ... |
| negscl 28422 | The surreals are closed un... |
| negscld 28423 | The surreals are closed un... |
| ltnegsim 28424 | The forward direction of t... |
| negcut 28425 | The cut properties of surr... |
| negcut2 28426 | The cut that defines surre... |
| negsid 28427 | Surreal addition of a numb... |
| negsidd 28428 | Surreal addition of a numb... |
| negsex 28429 | Every surreal has a negati... |
| negnegs 28430 | A surreal is equal to the ... |
| ltnegs 28431 | Negative of both sides of ... |
| lenegs 28432 | Negative of both sides of ... |
| ltnegsd 28433 | Negative of both sides of ... |
| lenegsd 28434 | Negative of both sides of ... |
| negs11 28435 | Surreal negation is one-to... |
| negsdi 28436 | Distribution of surreal ne... |
| lt0negs2d 28437 | Comparison of a surreal an... |
| negsf 28438 | Function statement for sur... |
| negsfo 28439 | Function statement for sur... |
| negsf1o 28440 | Surreal negation is a bije... |
| negsunif 28441 | Uniformity property for su... |
| negbdaylem 28442 | Lemma for ~ negbday . Bou... |
| negbday 28443 | Negation of a surreal numb... |
| negleft 28444 | The left set of the negati... |
| negright 28445 | The right set of the negat... |
| subsval 28446 | The value of surreal subtr... |
| subsvald 28447 | The value of surreal subtr... |
| subscl 28448 | Closure law for surreal su... |
| subscld 28449 | Closure law for surreal su... |
| subsf 28450 | Function statement for sur... |
| subsfo 28451 | Surreal subtraction is an ... |
| negsval2 28452 | Surreal negation in terms ... |
| negsval2d 28453 | Surreal negation in terms ... |
| subsid1 28454 | Identity law for subtracti... |
| subsid 28455 | Subtraction of a surreal f... |
| subadds 28456 | Relationship between addit... |
| subaddsd 28457 | Relationship between addit... |
| pncans 28458 | Cancellation law for surre... |
| pncan3s 28459 | Subtraction and addition o... |
| pncan2s 28460 | Cancellation law for surre... |
| npcans 28461 | Cancellation law for surre... |
| ltsubs1 28462 | Subtraction from both side... |
| ltsubs2 28463 | Subtraction from both side... |
| ltsubs1d 28464 | Subtraction from both side... |
| ltsubs2d 28465 | Subtraction from both side... |
| negsubsdi2d 28466 | Distribution of negative o... |
| addsubsassd 28467 | Associative-type law for s... |
| addsubsd 28468 | Law for surreal addition a... |
| ltsubsubsbd 28469 | Equivalence for the surrea... |
| ltsubsubs2bd 28470 | Equivalence for the surrea... |
| ltsubsubs3bd 28471 | Equivalence for the surrea... |
| lesubsubsbd 28472 | Equivalence for the surrea... |
| lesubsubs2bd 28473 | Equivalence for the surrea... |
| lesubsubs3bd 28474 | Equivalence for the surrea... |
| ltsubaddsd 28475 | Surreal less-than relation... |
| ltsubadds2d 28476 | Surreal less-than relation... |
| ltaddsubsd 28477 | Surreal less-than relation... |
| ltaddsubs2d 28478 | Surreal less-than relation... |
| lesubaddsd 28479 | Surreal less-than or equal... |
| subsubs4d 28480 | Law for double surreal sub... |
| subsubs2d 28481 | Law for double surreal sub... |
| lesubsd 28482 | Swap subtrahends in a surr... |
| nncansd 28483 | Cancellation law for surre... |
| posdifsd 28484 | Comparison of two surreals... |
| ltsubsposd 28485 | Subtraction of a positive ... |
| subsge0d 28486 | Non-negative subtraction. ... |
| addsubs4d 28487 | Rearrangement of four term... |
| ltsm1d 28488 | A surreal is greater than ... |
| subscan1d 28489 | Cancellation law for surre... |
| subscan2d 28490 | Cancellation law for surre... |
| subseq0d 28491 | The difference between two... |
| mulsfn 28494 | Surreal multiplication is ... |
| mulsval 28495 | The value of surreal multi... |
| mulsval2lem 28496 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28497 | The value of surreal multi... |
| muls01 28498 | Surreal multiplication by ... |
| mulsrid 28499 | Surreal one is a right ide... |
| mulsridd 28500 | Surreal one is a right ide... |
| mulsproplemcbv 28501 | Lemma for surreal multipli... |
| mulsproplem1 28502 | Lemma for surreal multipli... |
| mulsproplem2 28503 | Lemma for surreal multipli... |
| mulsproplem3 28504 | Lemma for surreal multipli... |
| mulsproplem4 28505 | Lemma for surreal multipli... |
| mulsproplem5 28506 | Lemma for surreal multipli... |
| mulsproplem6 28507 | Lemma for surreal multipli... |
| mulsproplem7 28508 | Lemma for surreal multipli... |
| mulsproplem8 28509 | Lemma for surreal multipli... |
| mulsproplem9 28510 | Lemma for surreal multipli... |
| mulsproplem10 28511 | Lemma for surreal multipli... |
| mulsproplem11 28512 | Lemma for surreal multipli... |
| mulsproplem12 28513 | Lemma for surreal multipli... |
| mulsproplem13 28514 | Lemma for surreal multipli... |
| mulsproplem14 28515 | Lemma for surreal multipli... |
| mulsprop 28516 | Surreals are closed under ... |
| mulcutlem 28517 | Lemma for ~ mulcut . Stat... |
| mulcut 28518 | Show the cut properties of... |
| mulcut2 28519 | Show that the cut involved... |
| mulscl 28520 | The surreals are closed un... |
| mulscld 28521 | The surreals are closed un... |
| ltmuls 28522 | An ordering relationship f... |
| ltmulsd 28523 | An ordering relationship f... |
| lemulsd 28524 | An ordering relationship f... |
| mulscom 28525 | Surreal multiplication is ... |
| mulscomd 28526 | Surreal multiplication is ... |
| muls02 28527 | Surreal multiplication by ... |
| mulslid 28528 | Surreal one is a left iden... |
| mulslidd 28529 | Surreal one is a left iden... |
| mulsgt0 28530 | The product of two positiv... |
| mulsgt0d 28531 | The product of two positiv... |
| mulsge0d 28532 | The product of two non-neg... |
| sltmuls1 28533 | One surreal set less-than ... |
| sltmuls2 28534 | One surreal set less-than ... |
| mulsuniflem 28535 | Lemma for ~ mulsunif . St... |
| mulsunif 28536 | Surreal multiplication has... |
| addsdilem1 28537 | Lemma for surreal distribu... |
| addsdilem2 28538 | Lemma for surreal distribu... |
| addsdilem3 28539 | Lemma for ~ addsdi . Show... |
| addsdilem4 28540 | Lemma for ~ addsdi . Show... |
| addsdi 28541 | Distributive law for surre... |
| addsdid 28542 | Distributive law for surre... |
| addsdird 28543 | Distributive law for surre... |
| subsdid 28544 | Distribution of surreal mu... |
| subsdird 28545 | Distribution of surreal mu... |
| mulnegs1d 28546 | Product with negative is n... |
| mulnegs2d 28547 | Product with negative is n... |
| mul2negsd 28548 | Surreal product of two neg... |
| mulsasslem1 28549 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28550 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28551 | Lemma for ~ mulsass . Dem... |
| mulsass 28552 | Associative law for surrea... |
| mulsassd 28553 | Associative law for surrea... |
| muls4d 28554 | Rearrangement of four surr... |
| mulsunif2lem 28555 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28556 | Alternate expression for s... |
| ltmuls2 28557 | Multiplication of both sid... |
| ltmuls2d 28558 | Multiplication of both sid... |
| ltmuls1d 28559 | Multiplication of both sid... |
| lemuls2d 28560 | Multiplication of both sid... |
| lemuls1d 28561 | Multiplication of both sid... |
| ltmulnegs1d 28562 | Multiplication of both sid... |
| ltmulnegs2d 28563 | Multiplication of both sid... |
| mulscan2dlem 28564 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28565 | Cancellation of surreal mu... |
| mulscan1d 28566 | Cancellation of surreal mu... |
| muls12d 28567 | Commutative/associative la... |
| lemuls1ad 28568 | Multiplication of both sid... |
| ltmuls12ad 28569 | Comparison of the product ... |
| divsmo 28570 | Uniqueness of surreal inve... |
| muls0ord 28571 | If a surreal product is ze... |
| mulsne0bd 28572 | The product of two nonzero... |
| divsval 28575 | The value of surreal divis... |
| norecdiv 28576 | If a surreal has a recipro... |
| noreceuw 28577 | If a surreal has a recipro... |
| recsne0 28578 | If a surreal has a recipro... |
| divmulsw 28579 | Relationship between surre... |
| divmulswd 28580 | Relationship between surre... |
| divsclw 28581 | Weak division closure law.... |
| divsclwd 28582 | Weak division closure law.... |
| divscan2wd 28583 | A weak cancellation law fo... |
| divscan1wd 28584 | A weak cancellation law fo... |
| ltdivmulswd 28585 | Surreal less-than relation... |
| ltdivmuls2wd 28586 | Surreal less-than relation... |
| ltmuldivswd 28587 | Surreal less-than relation... |
| ltmuldivs2wd 28588 | Surreal less-than relation... |
| divsasswd 28589 | An associative law for sur... |
| divs1 28590 | A surreal divided by one i... |
| divs1d 28591 | A surreal divided by one i... |
| precsexlemcbv 28592 | Lemma for surreal reciproc... |
| precsexlem1 28593 | Lemma for surreal reciproc... |
| precsexlem2 28594 | Lemma for surreal reciproc... |
| precsexlem3 28595 | Lemma for surreal reciproc... |
| precsexlem4 28596 | Lemma for surreal reciproc... |
| precsexlem5 28597 | Lemma for surreal reciproc... |
| precsexlem6 28598 | Lemma for surreal reciproc... |
| precsexlem7 28599 | Lemma for surreal reciproc... |
| precsexlem8 28600 | Lemma for surreal reciproc... |
| precsexlem9 28601 | Lemma for surreal reciproc... |
| precsexlem10 28602 | Lemma for surreal reciproc... |
| precsexlem11 28603 | Lemma for surreal reciproc... |
| precsex 28604 | Every positive surreal has... |
| recsex 28605 | A nonzero surreal has a re... |
| recsexd 28606 | A nonzero surreal has a re... |
| divmuls 28607 | Relationship between surre... |
| divmulsd 28608 | Relationship between surre... |
| divscl 28609 | Surreal division closure l... |
| divscld 28610 | Surreal division closure l... |
| divscan2d 28611 | A cancellation law for sur... |
| divscan1d 28612 | A cancellation law for sur... |
| ltdivmulsd 28613 | Surreal less-than relation... |
| ltdivmuls2d 28614 | Surreal less-than relation... |
| ltmuldivsd 28615 | Surreal less-than relation... |
| ltmuldivs2d 28616 | Surreal less-than relation... |
| divsassd 28617 | An associative law for sur... |
| divmuldivsd 28618 | Multiplication of two surr... |
| divdivs1d 28619 | Surreal division into a fr... |
| divsrecd 28620 | Relationship between surre... |
| divsdird 28621 | Distribution of surreal di... |
| divscan3d 28622 | A cancellation law for sur... |
| abssval 28625 | The value of surreal absol... |
| absscl 28626 | Closure law for surreal ab... |
| abssid 28627 | The absolute value of a no... |
| abs0s 28628 | The absolute value of surr... |
| abssnid 28629 | For a negative surreal, it... |
| absmuls 28630 | Surreal absolute value dis... |
| abssge0 28631 | The absolute value of a su... |
| abssor 28632 | The absolute value of a su... |
| absnegs 28633 | Surreal absolute value of ... |
| leabss 28634 | A surreal is less than or ... |
| abslts 28635 | Surreal absolute value and... |
| abssubs 28636 | Swapping order of surreal ... |
| elons 28639 | Membership in the class of... |
| onssno 28640 | The surreal ordinals are a... |
| onno 28641 | A surreal ordinal is a sur... |
| 0ons 28642 | Surreal zero is a surreal ... |
| 1ons 28643 | Surreal one is a surreal o... |
| elons2 28644 | A surreal is ordinal iff i... |
| elons2d 28645 | The cut of any set of surr... |
| onleft 28646 | The left set of a surreal ... |
| ltonold 28647 | The class of ordinals less... |
| ltonsex 28648 | The class of ordinals less... |
| oncutleft 28649 | A surreal ordinal is equal... |
| oncutlt 28650 | A surreal ordinal is the s... |
| bday11on 28651 | The birthday function is o... |
| onnolt 28652 | If a surreal ordinal is le... |
| onlts 28653 | Less-than is the same as b... |
| onles 28654 | Less-than or equal is the ... |
| onltsd 28655 | Less-than is the same as b... |
| onlesd 28656 | Less-than or equal is the ... |
| oniso 28657 | The birthday function rest... |
| onswe 28658 | Surreal less-than well-ord... |
| onsse 28659 | Surreal less-than is set-l... |
| onsis 28660 | Transfinite induction sche... |
| ons2ind 28661 | Double induction schema fo... |
| bdayons 28662 | The birthday of a surreal ... |
| onaddscl 28663 | The surreal ordinals are c... |
| onmulscl 28664 | The surreal ordinals are c... |
| addonbday 28665 | The birthday of the sum of... |
| peano2ons 28666 | The successor of a surreal... |
| onsbnd 28667 | The surreals of a given bi... |
| onsbnd2 28668 | The surreals of a given bi... |
| seqsex 28671 | Existence of the surreal s... |
| seqseq123d 28672 | Equality deduction for the... |
| nfseqs 28673 | Hypothesis builder for the... |
| seqsval 28674 | The value of the surreal s... |
| noseqex 28675 | The next several theorems ... |
| noseq0 28676 | The surreal ` A ` is a mem... |
| noseqp1 28677 | One plus an element of ` Z... |
| noseqind 28678 | Peano's inductive postulat... |
| noseqinds 28679 | Induction schema for surre... |
| noseqssno 28680 | A surreal sequence is a su... |
| noseqno 28681 | An element of a surreal se... |
| om2noseq0 28682 | The mapping ` G ` is a one... |
| om2noseqsuc 28683 | The value of ` G ` at a su... |
| om2noseqfo 28684 | Function statement for ` G... |
| om2noseqlt 28685 | Surreal less-than relation... |
| om2noseqlt2 28686 | The mapping ` G ` preserve... |
| om2noseqf1o 28687 | ` G ` is a bijection. (Co... |
| om2noseqiso 28688 | ` G ` is an isomorphism fr... |
| om2noseqoi 28689 | An alternative definition ... |
| om2noseqrdg 28690 | A helper lemma for the val... |
| noseqrdglem 28691 | A helper lemma for the val... |
| noseqrdgfn 28692 | The recursive definition g... |
| noseqrdg0 28693 | Initial value of a recursi... |
| noseqrdgsuc 28694 | Successor value of a recur... |
| seqsfn 28695 | The surreal sequence build... |
| seqs1 28696 | The value of the surreal s... |
| seqsp1 28697 | The value of the surreal s... |
| n0sexg 28702 | The set of all non-negativ... |
| n0sex 28703 | The set of all non-negativ... |
| nnsex 28704 | The set of all positive su... |
| peano5n0s 28705 | Peano's inductive postulat... |
| n0ssno 28706 | The non-negative surreal i... |
| nnssn0s 28707 | The positive surreal integ... |
| nnssno 28708 | The positive surreal integ... |
| n0no 28709 | A non-negative surreal int... |
| nnno 28710 | A positive surreal integer... |
| n0nod 28711 | A non-negative surreal int... |
| nnnod 28712 | A positive surreal integer... |
| nnn0s 28713 | A positive surreal integer... |
| nnn0sd 28714 | A positive surreal integer... |
| 0n0s 28715 | Peano postulate: ` 0s ` is... |
| peano2n0s 28716 | Peano postulate: the succe... |
| peano2n0sd 28717 | Peano postulate: the succe... |
| dfn0s2 28718 | Alternate definition of th... |
| n0sind 28719 | Principle of Mathematical ... |
| n0cut 28720 | A cut form for non-negativ... |
| n0cut2 28721 | A cut form for the success... |
| n0on 28722 | A surreal natural is a sur... |
| nnne0s 28723 | A surreal positive integer... |
| n0sge0 28724 | A non-negative integer is ... |
| nnsgt0 28725 | A positive integer is grea... |
| elnns 28726 | Membership in the positive... |
| elnns2 28727 | A positive surreal integer... |
| n0s0suc 28728 | A non-negative surreal int... |
| nnsge1 28729 | A positive surreal integer... |
| n0addscl 28730 | The non-negative surreal i... |
| n0mulscl 28731 | The non-negative surreal i... |
| nnaddscl 28732 | The positive surreal integ... |
| nnmulscl 28733 | The positive surreal integ... |
| 1n0s 28734 | Surreal one is a non-negat... |
| 1nns 28735 | Surreal one is a positive ... |
| peano2nns 28736 | Peano postulate for positi... |
| nnsrecgt0d 28737 | The reciprocal of a positi... |
| n0bday 28738 | A non-negative surreal int... |
| n0ssoldg 28739 | The non-negative surreal i... |
| n0ssold 28740 | The non-negative surreal i... |
| n0fincut 28741 | The simplest number greate... |
| onsfi 28742 | A surreal ordinal with a f... |
| eln0s2 28743 | A non-negative surreal int... |
| onltn0s 28744 | A surreal ordinal that is ... |
| n0cutlt 28745 | A non-negative surreal int... |
| seqn0sfn 28746 | The surreal sequence build... |
| eln0s 28747 | A non-negative surreal int... |
| n0s0m1 28748 | Every non-negative surreal... |
| n0subs 28749 | Subtraction of non-negativ... |
| n0subs2 28750 | Subtraction of non-negativ... |
| n0ltsp1le 28751 | Non-negative surreal order... |
| n0lesltp1 28752 | Non-negative surreal order... |
| n0lesm1lt 28753 | Non-negative surreal order... |
| n0lts1e0 28754 | A non-negative surreal int... |
| bdayn0p1 28755 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28756 | The birthday function rest... |
| n0p1nns 28757 | One plus a non-negative su... |
| dfnns2 28758 | Alternate definition of th... |
| nnsind 28759 | Principle of Mathematical ... |
| nn1m1nns 28760 | Every positive surreal int... |
| nnm1n0s 28761 | A positive surreal integer... |
| eucliddivs 28762 | Euclid's division lemma fo... |
| oldfib 28763 | The old set of an ordinal ... |
| zsex 28766 | The surreal integers form ... |
| zssno 28767 | The surreal integers are a... |
| zno 28768 | A surreal integer is a sur... |
| znod 28769 | A surreal integer is a sur... |
| elzs 28770 | Membership in the set of s... |
| nnzsubs 28771 | The difference of two surr... |
| nnzs 28772 | A positive surreal integer... |
| nnzsd 28773 | A positive surreal integer... |
| 0zs 28774 | Zero is a surreal integer.... |
| n0zs 28775 | A non-negative surreal int... |
| n0zsd 28776 | A non-negative surreal int... |
| 1zs 28777 | One is a surreal integer. ... |
| znegscl 28778 | The surreal integers are c... |
| znegscld 28779 | The surreal integers are c... |
| zaddscl 28780 | The surreal integers are c... |
| zaddscld 28781 | The surreal integers are c... |
| zsubscld 28782 | The surreal integers are c... |
| zmulscld 28783 | The surreal integers are c... |
| elzn0s 28784 | A surreal integer is a sur... |
| elzs2 28785 | A surreal integer is eithe... |
| eln0zs 28786 | Non-negative surreal integ... |
| elnnzs 28787 | Positive surreal integer p... |
| elznns 28788 | Surreal integer property e... |
| zn0subs 28789 | The non-negative differenc... |
| peano5uzs 28790 | Peano's inductive postulat... |
| uzsind 28791 | Induction on the upper sur... |
| zsbday 28792 | A surreal integer has a fi... |
| zcuts 28793 | A cut expression for surre... |
| zcuts0 28794 | Either the left or right s... |
| zsoring 28795 | The surreal integers form ... |
| 1p1e2s 28802 | One plus one is two. Surr... |
| no2times 28803 | Version of ~ 2times for su... |
| 2nns 28804 | Surreal two is a surreal n... |
| 2no 28805 | Surreal two is a surreal n... |
| 2ne0s 28806 | Surreal two is nonzero. (... |
| n0seo 28807 | A non-negative surreal int... |
| zseo 28808 | A surreal integer is eithe... |
| twocut 28809 | Two times the cut of zero ... |
| nohalf 28810 | An explicit expression for... |
| expsval 28811 | The value of surreal expon... |
| expnnsval 28812 | Value of surreal exponenti... |
| exps0 28813 | Surreal exponentiation to ... |
| exps1 28814 | Surreal exponentiation to ... |
| expsp1 28815 | Value of a surreal number ... |
| expscllem 28816 | Lemma for proving non-nega... |
| expscl 28817 | Closure law for surreal ex... |
| n0expscl 28818 | Closure law for non-negati... |
| nnexpscl 28819 | Closure law for positive s... |
| zexpscl 28820 | Closure law for surreal in... |
| expadds 28821 | Sum of exponents law for s... |
| expsne0 28822 | A non-negative surreal int... |
| expsgt0 28823 | A non-negative surreal int... |
| pw2recs 28824 | Any power of two has a mul... |
| pw2divscld 28825 | Division closure for power... |
| pw2divmulsd 28826 | Relationship between surre... |
| pw2divscan3d 28827 | Cancellation law for surre... |
| pw2divscan2d 28828 | A cancellation law for sur... |
| pw2divsassd 28829 | An associative law for div... |
| pw2divscan4d 28830 | Cancellation law for divis... |
| pw2gt0divsd 28831 | Division of a positive sur... |
| pw2ge0divsd 28832 | Divison of a non-negative ... |
| pw2divsrecd 28833 | Relationship between surre... |
| pw2divsdird 28834 | Distribution of surreal di... |
| pw2divsnegd 28835 | Move negative sign inside ... |
| pw2ltdivmulsd 28836 | Surreal less-than relation... |
| pw2ltmuldivs2d 28837 | Surreal less-than relation... |
| pw2ltsdiv1d 28838 | Surreal less-than relation... |
| avglts1d 28839 | Ordering property for aver... |
| avglts2d 28840 | Ordering property for aver... |
| pw2divs0d 28841 | Division into zero is zero... |
| pw2divsidd 28842 | Identity law for division ... |
| pw2ltdivmuls2d 28843 | Surreal less-than relation... |
| halfcut 28844 | Relate the cut of twice of... |
| addhalfcut 28845 | The cut of a surreal non-n... |
| pw2cut 28846 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28847 | Simplify ~ pw2cut in the c... |
| pw2cut2 28848 | Cut expression for powers ... |
| bdaypw2n0bndlem 28849 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28850 | Upper bound for the birthd... |
| bdaypw2bnd 28851 | Birthday bounding rule for... |
| bdayfinbndcbv 28852 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28853 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28854 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28855 | Given a non-negative integ... |
| z12bdaylem1 28856 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28857 | Lemma for ~ z12bday . Sho... |
| elz12s 28858 | Membership in the dyadic f... |
| elz12si 28859 | Inference form of membersh... |
| z12sex 28860 | The class of dyadic fracti... |
| zz12s 28861 | A surreal integer is a dya... |
| z12no 28862 | A dyadic is a surreal. (C... |
| z12addscl 28863 | The dyadics are closed und... |
| z12negscl 28864 | The dyadics are closed und... |
| z12subscl 28865 | The dyadics are closed und... |
| z12shalf 28866 | Half of a dyadic is a dyad... |
| z12negsclb 28867 | A surreal is a dyadic frac... |
| z12zsodd 28868 | A dyadic fraction is eithe... |
| z12sge0 28869 | An expression for non-nega... |
| z12bdaylem 28870 | Lemma for ~ z12bday . Han... |
| z12bday 28871 | A dyadic fraction has a fi... |
| bdayfinlem 28872 | Lemma for ~ bdayfin . Han... |
| bdayfin 28873 | A surreal has a finite bir... |
| dfz12s2 28874 | The set of dyadic fraction... |
| elreno 28877 | Membership in the set of s... |
| reno 28878 | A surreal real is a surrea... |
| renod 28879 | A surreal real is a surrea... |
| recut 28880 | The cut involved in defini... |
| elreno2 28881 | Alternate characterization... |
| 0reno 28882 | Surreal zero is a surreal ... |
| 1reno 28883 | Surreal one is a surreal r... |
| renegscl 28884 | The surreal reals are clos... |
| readdscl 28885 | The surreal reals are clos... |
| remulscllem1 28886 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28887 | Lemma for ~ remulscl . Bo... |
| remulscl 28888 | The surreal reals are clos... |
| itvndx 28899 | Index value of the Interva... |
| lngndx 28900 | Index value of the "line" ... |
| itvid 28901 | Utility theorem: index-ind... |
| lngid 28902 | Utility theorem: index-ind... |
| slotsinbpsd 28903 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28904 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28905 | The slot for the line is n... |
| trkgstr 28906 | Functionality of a Tarski ... |
| trkgbas 28907 | The base set of a Tarski g... |
| trkgdist 28908 | The measure of a distance ... |
| trkgitv 28909 | The congruence relation in... |
| istrkgc 28916 | Property of being a Tarski... |
| istrkgb 28917 | Property of being a Tarski... |
| istrkgcb 28918 | Property of being a Tarski... |
| istrkge 28919 | Property of fulfilling Euc... |
| istrkgl 28920 | Building lines from the se... |
| istrkgld 28921 | Property of fulfilling the... |
| istrkg2ld 28922 | Property of fulfilling the... |
| istrkg3ld 28923 | Property of fulfilling the... |
| axtgcgrrflx 28924 | Axiom of reflexivity of co... |
| axtgcgrid 28925 | Axiom of identity of congr... |
| axtgsegcon 28926 | Axiom of segment construct... |
| axtg5seg 28927 | Five segments axiom, Axiom... |
| axtgbtwnid 28928 | Identity of Betweenness. ... |
| axtgpasch 28929 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28930 | Axiom of Continuity. Axio... |
| axtgcont 28931 | Axiom of Continuity. Axio... |
| axtglowdim2 28932 | Lower dimension axiom for ... |
| axtgupdim2 28933 | Upper dimension axiom for ... |
| axtgeucl 28934 | Euclid's Axiom. Axiom A10... |
| tgjustf 28935 | Given any function ` F ` ,... |
| tgjustr 28936 | Given any equivalence rela... |
| tgjustc1 28937 | A justification for using ... |
| tgjustc2 28938 | A justification for using ... |
| tgcgrcomimp 28939 | Congruence commutes on the... |
| tgcgrcomr 28940 | Congruence commutes on the... |
| tgcgrcoml 28941 | Congruence commutes on the... |
| tgcgrcomlr 28942 | Congruence commutes on bot... |
| tgcgreqb 28943 | Congruence and equality. ... |
| tgcgreq 28944 | Congruence and equality. ... |
| tgcgrneq 28945 | Congruence and equality. ... |
| tgcgrtriv 28946 | Degenerate segments are co... |
| tgcgrextend 28947 | Link congruence over a pai... |
| tgsegconeq 28948 | Two points that satisfy th... |
| tgsegconeu 28949 | The point constructed in ~... |
| tgbtwntriv2 28950 | Betweenness always holds f... |
| tgbtwncom 28951 | Betweenness commutes. The... |
| tgbtwncomb 28952 | Betweenness commutes, bico... |
| tgbtwnne 28953 | Betweenness and inequality... |
| tgbtwntriv1 28954 | Betweenness always holds f... |
| tgbtwnswapid 28955 | If you can swap the first ... |
| tgbtwnintr 28956 | Inner transitivity law for... |
| tgbtwnexch3 28957 | Exchange the first endpoin... |
| tgbtwnouttr2 28958 | Outer transitivity law for... |
| tgbtwnexch2 28959 | Exchange the outer point o... |
| tgbtwnouttr 28960 | Outer transitivity law for... |
| tgbtwnexch 28961 | Outer transitivity law for... |
| tgtrisegint 28962 | A line segment between two... |
| tglowdim1 28963 | Lower dimension axiom for ... |
| tglowdim1i 28964 | Lower dimension axiom for ... |
| tgldimor 28965 | Excluded-middle like state... |
| tgldim0eq 28966 | In dimension zero, any two... |
| tgldim0itv 28967 | In dimension zero, any two... |
| tgldim0cgr 28968 | In dimension zero, any two... |
| tgbtwndiff 28969 | There is always a ` c ` di... |
| tgdim01 28970 | In geometries of dimension... |
| tgifscgr 28971 | Inner five segment congrue... |
| tgcgrsub 28972 | Removing identical parts f... |
| iscgrg 28975 | The congruence property fo... |
| iscgrgd 28976 | The property for two seque... |
| iscgrglt 28977 | The property for two seque... |
| trgcgrg 28978 | The property for two trian... |
| trgcgr 28979 | Triangle congruence. (Con... |
| ercgrg 28980 | The shape congruence relat... |
| tgcgrxfr 28981 | A line segment can be divi... |
| cgr3id 28982 | Reflexivity law for three-... |
| cgr3simp1 28983 | Deduce segment congruence ... |
| cgr3simp2 28984 | Deduce segment congruence ... |
| cgr3simp3 28985 | Deduce segment congruence ... |
| cgr3swap12 28986 | Permutation law for three-... |
| cgr3swap23 28987 | Permutation law for three-... |
| cgr3swap13 28988 | Permutation law for three-... |
| cgr3rotr 28989 | Permutation law for three-... |
| cgr3rotl 28990 | Permutation law for three-... |
| trgcgrcom 28991 | Commutative law for three-... |
| cgr3tr 28992 | Transitivity law for three... |
| tgbtwnxfr 28993 | A condition for extending ... |
| tgcgr4 28994 | Two quadrilaterals to be c... |
| isismt 28997 | Property of being an isome... |
| ismot 28998 | Property of being an isome... |
| motcgr 28999 | Property of a motion: dist... |
| idmot 29000 | The identity is a motion. ... |
| motf1o 29001 | Motions are bijections. (... |
| motcl 29002 | Closure of motions. (Cont... |
| motco 29003 | The composition of two mot... |
| cnvmot 29004 | The converse of a motion i... |
| motplusg 29005 | The operation for motions ... |
| motgrp 29006 | The motions of a geometry ... |
| motcgrg 29007 | Property of a motion: dist... |
| motcgr3 29008 | Property of a motion: dist... |
| tglng 29009 | Lines of a Tarski Geometry... |
| tglnfn 29010 | Lines as functions. (Cont... |
| tglnunirn 29011 | Lines are sets of points. ... |
| tglnpt 29012 | Lines are sets of points. ... |
| tglngne 29013 | It takes two different poi... |
| tglngval 29014 | The line going through poi... |
| tglnssp 29015 | Lines are subset of the ge... |
| tgellng 29016 | Property of lying on the l... |
| tgcolg 29017 | We choose the notation ` (... |
| btwncolg1 29018 | Betweenness implies coline... |
| btwncolg2 29019 | Betweenness implies coline... |
| btwncolg3 29020 | Betweenness implies coline... |
| colcom 29021 | Swapping the points defini... |
| colrot1 29022 | Rotating the points defini... |
| colrot2 29023 | Rotating the points defini... |
| ncolcom 29024 | Swapping non-colinear poin... |
| ncolrot1 29025 | Rotating non-colinear poin... |
| ncolrot2 29026 | Rotating non-colinear poin... |
| tgdim01ln 29027 | In geometries of dimension... |
| ncoltgdim2 29028 | If there are three non-col... |
| lnxfr 29029 | Transfer law for colineari... |
| lnext 29030 | Extend a line with a missi... |
| tgfscgr 29031 | Congruence law for the gen... |
| lncgr 29032 | Congruence rule for lines.... |
| lnid 29033 | Identity law for points on... |
| tgidinside 29034 | Law for finding a point in... |
| tgbtwnconn1lem1 29035 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 29036 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 29037 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 29038 | Connectivity law for betwe... |
| tgbtwnconn2 29039 | Another connectivity law f... |
| tgbtwnconn3 29040 | Inner connectivity law for... |
| tgbtwnconnln3 29041 | Derive colinearity from be... |
| tgbtwnconn22 29042 | Double connectivity law fo... |
| tgbtwnconnln1 29043 | Derive colinearity from be... |
| tgbtwnconnln2 29044 | Derive colinearity from be... |
| legval 29047 | Value of the less-than rel... |
| legov 29048 | Value of the less-than rel... |
| legov2 29049 | An equivalent definition o... |
| legid 29050 | Reflexivity of the less-th... |
| btwnleg 29051 | Betweenness implies less-t... |
| legtrd 29052 | Transitivity of the less-t... |
| legtri3 29053 | Equality from the less-tha... |
| legtrid 29054 | Trichotomy law for the les... |
| leg0 29055 | Degenerated (zero-length) ... |
| legeq 29056 | Deduce equality from "less... |
| legbtwn 29057 | Deduce betweenness from "l... |
| tgcgrsub2 29058 | Removing identical parts f... |
| ltgseg 29059 | The set ` E ` denotes the ... |
| ltgov 29060 | Strict "shorter than" geom... |
| legov3 29061 | An equivalent definition o... |
| legso 29062 | The "shorter than" relatio... |
| ishlg2 29065 | Alternate version of ~ ish... |
| hlgrcl1 29066 | Reverse closure for rays. ... |
| hlgrcl2 29067 | Reverse closure for rays. ... |
| ishlg 29068 | Rays : Definition 6.1 of ... |
| hlcomb 29069 | The half-line relation is ... |
| hlcomd 29070 | The half-line relation is ... |
| hlne1 29071 | The half-line relation imp... |
| hlne2 29072 | The half-line relation imp... |
| hlln 29073 | The half-line relation imp... |
| hleqnid 29074 | The endpoint does not belo... |
| hlid 29075 | The half-line relation is ... |
| hltr 29076 | The half-line relation is ... |
| hlbtwn 29077 | Betweenness is a sufficien... |
| btwnhl1 29078 | Deduce half-line from betw... |
| btwnhl2 29079 | Deduce half-line from betw... |
| btwnhl 29080 | Swap betweenness for a hal... |
| lnhl 29081 | Either a point ` C ` on th... |
| hlcgrex 29082 | Construct a point on a hal... |
| hlcgreulem 29083 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 29084 | The point constructed in ~... |
| hlcgreq 29085 | A constructed point on a h... |
| tghlsub 29086 | Removing identical parts f... |
| btwnlng1 29087 | Betweenness implies coline... |
| btwnlng2 29088 | Betweenness implies coline... |
| btwnlng3 29089 | Betweenness implies coline... |
| lncom 29090 | Swapping the points defini... |
| lnrot1 29091 | Rotating the points defini... |
| lnrot2 29092 | Rotating the points defini... |
| ncolne1 29093 | Non-colinear points are di... |
| ncolne2 29094 | Non-colinear points are di... |
| tgisline 29095 | The property of being a pr... |
| tglnne 29096 | It takes two different poi... |
| tglndim0 29097 | There are no lines in dime... |
| tgelrnln 29098 | The property of being a pr... |
| tglineeltr 29099 | Transitivity law for lines... |
| tglineelsb2 29100 | If ` S ` lies on PQ , then... |
| tglinerflx1 29101 | Reflexivity law for line m... |
| tglinerflx2 29102 | Reflexivity law for line m... |
| tglinecom 29103 | Commutativity law for line... |
| tglinethru 29104 | If ` A ` is a line contain... |
| tghilberti1 29105 | There is a line through an... |
| tghilberti2 29106 | There is at most one line ... |
| tglinethrueu 29107 | There is a unique line goi... |
| tglinesseq 29108 | If a line is a subset of a... |
| tglnne0 29109 | A line ` A ` has at least ... |
| tglineintmo 29110 | Two distinct lines interse... |
| tglineineq 29111 | Two distinct lines interse... |
| tglineinsn 29112 | If two distinct lines inte... |
| tglineneq 29113 | Given three non-colinear p... |
| tglineinteq 29114 | Two distinct lines interse... |
| ncolncol 29115 | Deduce non-colinearity fro... |
| coltr 29116 | A transitivity law for col... |
| coltr3 29117 | A transitivity law for col... |
| colline 29118 | Three points are colinear ... |
| tglowdim2l 29119 | Reformulation of the lower... |
| tglowdim2ln 29120 | There is always one point ... |
| tglnpt2 29121 | Find a second point on a l... |
| tglnpt3 29122 | Find a third point on a li... |
| tglnpt4 29123 | Find a second point on a l... |
| mirreu3 29126 | Existential uniqueness of ... |
| mirval 29127 | Value of the point inversi... |
| mirfv 29128 | Value of the point inversi... |
| mircgr 29129 | Property of the image by t... |
| mirbtwn 29130 | Property of the image by t... |
| ismir 29131 | Property of the image by t... |
| mirf 29132 | Point inversion as functio... |
| mircl 29133 | Closure of the point inver... |
| mirmir 29134 | The point inversion functi... |
| mircom 29135 | Variation on ~ mirmir . (... |
| mirreu 29136 | Any point has a unique ant... |
| mireq 29137 | Equality deduction for poi... |
| mirinv 29138 | The only invariant point o... |
| mirne 29139 | Mirror of non-center point... |
| mircinv 29140 | The center point is invari... |
| mirf1o 29141 | The point inversion functi... |
| miriso 29142 | The point inversion functi... |
| mirbtwni 29143 | Point inversion preserves ... |
| mirbtwnb 29144 | Point inversion preserves ... |
| mircgrs 29145 | Point inversion preserves ... |
| mirmir2 29146 | Point inversion of a point... |
| mirmot 29147 | Point investion is a motio... |
| mirln 29148 | If two points are on the s... |
| mirln2 29149 | If a point and its mirror ... |
| mirconn 29150 | Point inversion of connect... |
| mirhl 29151 | If two points ` X ` and ` ... |
| mirbtwnhl 29152 | If the center of the point... |
| mirhl2 29153 | Deduce half-line relation ... |
| mircgrextend 29154 | Link congruence over a pai... |
| mirtrcgr 29155 | Point inversion of one poi... |
| mirauto 29156 | Point inversion preserves ... |
| miduniq 29157 | Uniqueness of the middle p... |
| miduniq1 29158 | Uniqueness of the middle p... |
| miduniq2 29159 | If two point inversions co... |
| colmid 29160 | Colinearity and equidistan... |
| symquadlem 29161 | Lemma of the symmetrical q... |
| krippenlem 29162 | Lemma for ~ krippen . We ... |
| krippen 29163 | Krippenlemma (German for c... |
| midexlem 29164 | Lemma for the existence of... |
| symquadprlnglem 29165 | Lemma for ~ symquadprlngle... |
| mirleqb 29166 | Equality theorem for point... |
| mirlni 29167 | The mirror of a point ` X ... |
| israg 29172 | Property for 3 points A, B... |
| ragcom 29173 | Commutative rule for right... |
| ragcol 29174 | The right angle property i... |
| ragmir 29175 | Right angle property is pr... |
| mirrag 29176 | Right angle is conserved b... |
| ragtrivb 29177 | Trivial right angle. Theo... |
| ragflat2 29178 | Deduce equality from two r... |
| ragflat 29179 | Deduce equality from two r... |
| ragtriva 29180 | Trivial right angle. Theo... |
| ragflat3 29181 | Right angle and colinearit... |
| ragcgr 29182 | Right angle and colinearit... |
| motrag 29183 | Right angles are preserved... |
| ragncol 29184 | Right angle implies non-co... |
| perpln1 29185 | Derive a line from perpend... |
| perpln2 29186 | Derive a line from perpend... |
| isperp 29187 | Property for 2 lines A, B ... |
| perpcom 29188 | The "perpendicular" relati... |
| perpneq 29189 | Two perpendicular lines ar... |
| isperp2 29190 | Property for 2 lines A, B,... |
| isperp2d 29191 | One direction of ~ isperp2... |
| ragperp 29192 | Deduce that two lines are ... |
| footexALT 29193 | Alternative version of ~ f... |
| footexlem1 29194 | Lemma for ~ footex . (Con... |
| footexlem2 29195 | Lemma for ~ footex . (Con... |
| footex 29196 | From a point ` C ` outside... |
| foot 29197 | From a point ` C ` outside... |
| footne 29198 | Uniqueness of the foot poi... |
| footeq 29199 | Uniqueness of the foot poi... |
| perpin 29200 | If two lines ` A ` and ` B... |
| hlperpnel 29201 | A point on a half-line whi... |
| perprag 29202 | Deduce a right angle from ... |
| perpdragALT 29203 | Deduce a right angle from ... |
| perpdrag 29204 | Deduce a right angle from ... |
| colperp 29205 | Deduce a perpendicularity ... |
| colperpexlem1 29206 | Lemma for ~ colperp . Fir... |
| colperpexlem2 29207 | Lemma for ~ colperpex . S... |
| colperpexlem3 29208 | Lemma for ~ colperpex . C... |
| colperpex 29209 | In dimension 2 and above, ... |
| mideulem2 29210 | Lemma for ~ opphllem , whi... |
| opphllem 29211 | Lemma 8.24 of [Schwabhause... |
| mideulem 29212 | Lemma for ~ mideu . We ca... |
| midex 29213 | Existence of the midpoint,... |
| mideu 29214 | Existence and uniqueness o... |
| islnopp 29215 | The property for two point... |
| islnoppd 29216 | Deduce that ` A ` and ` B ... |
| oppne1 29217 | Points lying on opposite s... |
| oppne2 29218 | Points lying on opposite s... |
| oppne3 29219 | Points lying on opposite s... |
| oppcom 29220 | Commutativity rule for "op... |
| opptgdim2 29221 | If two points opposite to ... |
| oppnid 29222 | The "opposite to a line" r... |
| opphllem1 29223 | Lemma for ~ opphl . (Cont... |
| opphllem2 29224 | Lemma for ~ opphl . Lemma... |
| opphllem3 29225 | Lemma for ~ opphl : We as... |
| opphllem4 29226 | Lemma for ~ opphl . (Cont... |
| opphllem5 29227 | Second part of Lemma 9.4 o... |
| opphllem6 29228 | First part of Lemma 9.4 of... |
| oppperpex 29229 | Restating ~ colperpex usin... |
| opphl 29230 | If two points ` A ` and ` ... |
| lnoppinn0 29231 | The segment between two po... |
| oppmir 29232 | The mirror point with rega... |
| outpasch 29233 | Axiom of Pasch, outer form... |
| hlpasch 29234 | An application of the axio... |
| ishpg 29237 | Value of the half-plane re... |
| hpgbr 29238 | Half-planes : property for... |
| hpgne1 29239 | Points on the open half pl... |
| hpgne2 29240 | Points on the open half pl... |
| lnopp2hpgb 29241 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 29242 | If two points lie on the o... |
| hpgerlem 29243 | Lemma for the proof that t... |
| hpgid 29244 | The half-plane relation is... |
| hpgcom 29245 | The half-plane relation is... |
| hpgtr 29246 | The half-plane relation is... |
| colopp 29247 | Opposite sides of a line f... |
| colhp 29248 | Half-plane relation for co... |
| hphl 29249 | If two points are on the s... |
| hlopp 29250 | If two points ` X ` and ` ... |
| tgplnfn 29253 | The plane generating funct... |
| tgelrnpln 29254 | The property of being a pl... |
| plngval 29255 | The plane defined by a lin... |
| isplng 29256 | The property of being a pl... |
| plngrnssp 29257 | Planes are sets of points.... |
| elplng 29258 | Elementhood in the plane d... |
| plngssp 29259 | Planes are sets of points.... |
| elplngid 29260 | The point ` R ` is itself ... |
| elplnglnid 29261 | The line ` A ` itself is a... |
| lnincplng 29262 | If two lines ` A ` and ` B... |
| plngcplem 29263 | Lemma for ~ plngcp . (Con... |
| plngcp 29264 | The plane defined by a lin... |
| plngrotlem1 29265 | Lemma for ~ plngrot . (Co... |
| plngrotlem2 29266 | Lemma for ~ plngrot . (Co... |
| plngrotlem3 29267 | Lemma for ~ plngrot . (Co... |
| plngrot 29268 | The plane defined by a lin... |
| lnssplnglem 29269 | Lemma for ~ lnssplng . (C... |
| lnssplng 29270 | A line defined by two poin... |
| lnssplng1 29271 | A line defined by two poin... |
| plngmiropp 29272 | Given a line ` A ` and a p... |
| mirplncl 29273 | The mirror of a point with... |
| hpgssplng 29274 | Any point ` X ` on a half ... |
| plng3p 29275 | If ` H ` is a plane contai... |
| nhpmirhp 29276 | If a point ` Z ` is on the... |
| midf 29281 | Midpoint as a function. (... |
| midcl 29282 | Closure of the midpoint. ... |
| ismidb 29283 | Property of the midpoint. ... |
| midbtwn 29284 | Betweenness of midpoint. ... |
| midcgr 29285 | Congruence of midpoint. (... |
| midid 29286 | Midpoint of a null segment... |
| midcom 29287 | Commutativity rule for the... |
| mirmid 29288 | Point inversion preserves ... |
| lmieu 29289 | Uniqueness of the line mir... |
| lmif 29290 | Line mirror as a function.... |
| lmicl 29291 | Closure of the line mirror... |
| islmib 29292 | Property of the line mirro... |
| lmicom 29293 | The line mirroring functio... |
| lmilmi 29294 | Line mirroring is an invol... |
| lmireu 29295 | Any point has a unique ant... |
| lmieq 29296 | Equality deduction for lin... |
| lmiinv 29297 | The invariants of the line... |
| lmicinv 29298 | The mirroring line is an i... |
| lmimid 29299 | If we have a right angle, ... |
| lmif1o 29300 | The line mirroring functio... |
| lmiisolem 29301 | Lemma for ~ lmiiso . (Con... |
| lmiiso 29302 | The line mirroring functio... |
| lmimot 29303 | Line mirroring is a motion... |
| symquadmid 29304 | In a symmetrical quadrilat... |
| hypcgrlem1 29305 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 29306 | Lemma for ~ hypcgr , case ... |
| hypcgr 29307 | If the catheti of two righ... |
| lmiopp 29308 | Line mirroring produces po... |
| lnperpex 29309 | Existence of a perpendicul... |
| lnperpexs 29310 | Existence of a perpendicul... |
| trgcopy 29311 | Triangle construction: a c... |
| trgcopyeulem 29312 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 29313 | Triangle construction: a c... |
| iscgra 29316 | Property for two angles AB... |
| iscgra1 29317 | A special version of ~ isc... |
| iscgrad 29318 | Sufficient conditions for ... |
| cgrane1 29319 | Angles imply inequality. ... |
| cgrane2 29320 | Angles imply inequality. ... |
| cgrane3 29321 | Angles imply inequality. ... |
| cgrane4 29322 | Angles imply inequality. ... |
| cgrahl1 29323 | Angle congruence is indepe... |
| cgrahl2 29324 | Angle congruence is indepe... |
| cgracgr 29325 | First direction of proposi... |
| cgraid 29326 | Angle congruence is reflex... |
| cgraswap 29327 | Swap rays in a congruence ... |
| cgrcgra 29328 | Triangle congruence implie... |
| cgracom 29329 | Angle congruence commutes.... |
| cgratr 29330 | Angle congruence is transi... |
| zerocgra 29331 | Zero angles are congruent.... |
| flatcgra 29332 | Flat angles are congruent.... |
| cgraswaplr 29333 | Swap both side of angle co... |
| cgrabtwn 29334 | Angle congruence preserves... |
| cgrahl 29335 | Angle congruence preserves... |
| cgracol 29336 | Angle congruence preserves... |
| cgrancol 29337 | Angle congruence preserves... |
| dfcgra2 29338 | This is the full statement... |
| sacgr 29339 | Supplementary angles of co... |
| oacgr 29340 | Vertical angle theorem. V... |
| acopy 29341 | Angle construction. Theor... |
| acopyeu 29342 | Angle construction. Theor... |
| ragcgra 29343 | Right angles are congruent... |
| cgrarag 29344 | Any angle ` <" A B C "> ` ... |
| ragsupplcgra 29345 | An angle ` <" X Y Z "> ` i... |
| ragraghl 29346 | Drawing two right angles a... |
| perpeqlem 29347 | Lemma for ~ perpeq . (Con... |
| perpeq 29348 | Uniqueness of the perpendi... |
| tgaaddcpbllem1 29349 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbllem2 29350 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbllem3 29351 | Lemma for ~ tgaaddcpbl . ... |
| tgaaddcpbl 29352 | The angular addition is co... |
| tgaaddcpbl2 29353 | The angular addition is co... |
| isinag 29357 | Property for point ` X ` t... |
| isinagd 29358 | Sufficient conditions for ... |
| inagflat 29359 | Any point lies in a flat a... |
| inagswap 29360 | Swap the order of the half... |
| inagne1 29361 | Deduce inequality from the... |
| inagne2 29362 | Deduce inequality from the... |
| inagne3 29363 | Deduce inequality from the... |
| inaghl 29364 | The "point lie in angle" r... |
| isleag 29366 | Geometrical "less than" pr... |
| isleagd 29367 | Sufficient condition for "... |
| leagne1 29368 | Deduce inequality from the... |
| leagne2 29369 | Deduce inequality from the... |
| leagne3 29370 | Deduce inequality from the... |
| leagne4 29371 | Deduce inequality from the... |
| cgrg3col4 29372 | Lemma 11.28 of [Schwabhaus... |
| elcgrabasi 29375 | Helper theorem for the mem... |
| elcgrabasrd 29376 | Helper theorem for the mem... |
| cgraer 29377 | The angle congruence relat... |
| cgrabasimass 29378 | The angle congruence relat... |
| angmgmaddeu1 29379 | There exists a unique poin... |
| angmgmaddeu2 29380 | Existence of a unique poin... |
| angmgmaddeu3 29381 | Existence of a unique poin... |
| angmgmaddeu4 29382 | There exists a unique poin... |
| angmgmaddeu5 29383 | There exists a unique poin... |
| angmgmaddeu6 29384 | There exists a unique poin... |
| angmgmaddeu7 29385 | There exists a unique poin... |
| angmgmaddov1lem 29386 | Lemma for ~ angmgmaddov1 .... |
| angmgmaddov2lem 29387 | Lemma for ~ angmgmaddov2 .... |
| angmgmaddov1 29388 | Value of the addition oper... |
| angmgmaddov2 29389 | Value of the addition oper... |
| angmgmaddcpbl 29390 | Addition of angles is comp... |
| angmgmaddcl 29391 | Closure of the addition of... |
| angmgmaddlid 29392 | The left identity element ... |
| angmgmaddrid 29393 | The right identity element... |
| angmgmval 29394 | Explicit the value of the ... |
| angmgmlem 29395 | Lemma for ~ angmgm . (Con... |
| angmgm0g 29396 | The identity element of th... |
| angmgm 29397 | The angle addition magma i... |
| angmgmbas 29398 | The base set of the angle ... |
| tgsas1 29399 | First congruence theorem: ... |
| tgsas 29400 | First congruence theorem: ... |
| tgsas2 29401 | First congruence theorem: ... |
| tgsas3 29402 | First congruence theorem: ... |
| tgasa1 29403 | Second congruence theorem:... |
| tgasa 29404 | Second congruence theorem:... |
| tgsss1 29405 | Third congruence theorem: ... |
| tgsss2 29406 | Third congruence theorem: ... |
| tgsss3 29407 | Third congruence theorem: ... |
| dfcgrg2 29408 | Congruence for two triangl... |
| isoas 29409 | Congruence theorem for iso... |
| iseqlg 29412 | Property of a triangle bei... |
| iseqlgd 29413 | Condition for a triangle t... |
| brprlng 29416 | Property of two lines ` A ... |
| prlngd 29417 | Deduce parallelism between... |
| prlngref 29418 | Parallelism is reflexive. ... |
| prlngsym 29419 | Parallelism is symmetric. ... |
| prlngrcl1 29420 | Reverse closure for parall... |
| prlngrcl2 29421 | Reverse closure for parall... |
| prlngin0 29422 | Two parallel lines do not ... |
| prlngpln 29423 | Two parallel lines are on ... |
| prlnghpg 29424 | If two lines ` A ` and ` B... |
| dfprlng2 29425 | Alternate definition of (s... |
| dfprlng3 29426 | Alternate definition of (s... |
| prlngpln3 29427 | Two parallel lines are on ... |
| perpprlng 29428 | If two lines ` A ` and ` B... |
| prlngex 29429 | There exists at least one ... |
| prlngmolem1 29430 | Lemma for ~ prlngmo : Con... |
| prlngmolem2 29431 | Lemma for ~ prlngmo . (Co... |
| prlngmo 29432 | Playfair's axiom. Given a... |
| prlngeu 29433 | Given a line ` A ` and a p... |
| prlngmo2 29434 | Playfair's axiom, without ... |
| prlngeq 29435 | Playfair's axiom, written ... |
| prlngpln4 29436 | Building a parallel line c... |
| prlngplngtr 29437 | Transitivity of parallelis... |
| prlnginn0 29438 | A line ` C ` intersecting ... |
| prlngmid2 29439 | If the midpoints of two se... |
| symquadprlng 29440 | Symmetrical quadrilaterals... |
| prlngsymquadlem 29441 | Lemma for ~ prlngsymquad .... |
| prlngsymquad 29442 | All parallelograms are sym... |
| prlngsymquadopp 29443 | In parallelograms, opposin... |
| quadcgrprlng 29444 | Nontrivial quadrilaterals ... |
| tgaltai 29445 | Parallelism implies altern... |
| f1otrgds 29446 | Convenient lemma for ~ f1o... |
| f1otrgitv 29447 | Convenient lemma for ~ f1o... |
| f1otrg 29448 | A bijection between bases ... |
| f1otrge 29449 | A bijection between bases ... |
| ttgval 29452 | Define a function to augme... |
| ttglem 29453 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 29454 | The base set of a subcompl... |
| ttgplusg 29455 | The addition operation of ... |
| ttgsub 29456 | The subtraction operation ... |
| ttgvsca 29457 | The scalar product of a su... |
| ttgds 29458 | The metric of a subcomplex... |
| ttgitvval 29459 | Betweenness for a subcompl... |
| ttgelitv 29460 | Betweenness for a subcompl... |
| ttgbtwnid 29461 | Any subcomplex module equi... |
| ttgcontlem1 29462 | Lemma for % ttgcont . (Co... |
| xmstrkgc 29463 | Any metric space fulfills ... |
| cchhllem 29464 | Lemma for chlbas and chlvs... |
| elee 29471 | Membership in a Euclidean ... |
| mptelee 29472 | A condition for a mapping ... |
| mpteleeOLD 29473 | Obsolete version of ~ mpte... |
| eleenn 29474 | If ` A ` is in ` ( EE `` N... |
| eleei 29475 | The forward direction of ~... |
| eedimeq 29476 | A point belongs to at most... |
| brbtwn 29477 | The binary relation form o... |
| brcgr 29478 | The binary relation form o... |
| fveere 29479 | The function value of a po... |
| fveecn 29480 | The function value of a po... |
| eqeefv 29481 | Two points are equal iff t... |
| eqeelen 29482 | Two points are equal iff t... |
| brbtwn2 29483 | Alternate characterization... |
| colinearalglem1 29484 | Lemma for ~ colinearalg . ... |
| colinearalglem2 29485 | Lemma for ~ colinearalg . ... |
| colinearalglem3 29486 | Lemma for ~ colinearalg . ... |
| colinearalglem4 29487 | Lemma for ~ colinearalg . ... |
| colinearalg 29488 | An algebraic characterizat... |
| eleesub 29489 | Membership of a subtractio... |
| eleesubd 29490 | Membership of a subtractio... |
| axdimuniq 29491 | The unique dimension axiom... |
| axcgrrflx 29492 | ` A ` is as far from ` B `... |
| axcgrtr 29493 | Congruence is transitive. ... |
| axcgrid 29494 | If there is no distance be... |
| axsegconlem1 29495 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 29496 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 29497 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 29498 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 29499 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 29500 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 29501 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 29502 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 29503 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 29504 | Lemma for ~ axsegcon . Sh... |
| axsegcon 29505 | Any segment ` A B ` can be... |
| ax5seglem1 29506 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 29507 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 29508 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 29509 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 29510 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 29511 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 29512 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 29513 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 29514 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 29515 | Lemma for ~ ax5seg . Take... |
| ax5seg 29516 | The five segment axiom. T... |
| axbtwnid 29517 | Points are indivisible. T... |
| axpaschlem 29518 | Lemma for ~ axpasch . Set... |
| axpasch 29519 | The inner Pasch axiom. Ta... |
| axlowdimlem1 29520 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29521 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29522 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29523 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29524 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29525 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29526 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29527 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29528 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29529 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29530 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29531 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29532 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29533 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29534 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29535 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29536 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29537 | The lower dimension axiom ... |
| axlowdim2 29538 | The lower two-dimensional ... |
| axlowdim 29539 | The general lower dimensio... |
| axeuclidlem 29540 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29541 | Euclid's axiom. Take an a... |
| axcontlem1 29542 | Lemma for ~ axcont . Chan... |
| axcontlem2 29543 | Lemma for ~ axcont . The ... |
| axcontlem3 29544 | Lemma for ~ axcont . Give... |
| axcontlem4 29545 | Lemma for ~ axcont . Give... |
| axcontlem5 29546 | Lemma for ~ axcont . Comp... |
| axcontlem6 29547 | Lemma for ~ axcont . Stat... |
| axcontlem7 29548 | Lemma for ~ axcont . Give... |
| axcontlem8 29549 | Lemma for ~ axcont . A po... |
| axcontlem9 29550 | Lemma for ~ axcont . Give... |
| axcontlem10 29551 | Lemma for ~ axcont . Give... |
| axcontlem11 29552 | Lemma for ~ axcont . Elim... |
| axcontlem12 29553 | Lemma for ~ axcont . Elim... |
| axcont 29554 | The axiom of continuity. ... |
| eengv 29557 | The value of the Euclidean... |
| eengstr 29558 | The Euclidean geometry as ... |
| eengbas 29559 | The Base of the Euclidean ... |
| ebtwntg 29560 | The betweenness relation u... |
| ecgrtg 29561 | The congruence relation us... |
| elntg 29562 | The line definition in the... |
| elntg2 29563 | The line definition in the... |
| eengtrkg 29564 | The geometry structure for... |
| eengtrkge 29565 | The geometry structure for... |
| edgfid 29568 | Utility theorem: index-ind... |
| edgfndx 29569 | Index value of the ~ df-ed... |
| edgfndxnn 29570 | The index value of the edg... |
| edgfndxid 29571 | The value of the edge func... |
| basendxltedgfndx 29572 | The index value of the ` B... |
| basendxnedgfndx 29573 | The slots ` Base ` and ` .... |
| vtxval 29578 | The set of vertices of a g... |
| iedgval 29579 | The set of indexed edges o... |
| 1vgrex 29580 | A graph with at least one ... |
| opvtxval 29581 | The set of vertices of a g... |
| opvtxfv 29582 | The set of vertices of a g... |
| opvtxov 29583 | The set of vertices of a g... |
| opiedgval 29584 | The set of indexed edges o... |
| opiedgfv 29585 | The set of indexed edges o... |
| opiedgov 29586 | The set of indexed edges o... |
| opvtxfvi 29587 | The set of vertices of a g... |
| opiedgfvi 29588 | The set of indexed edges o... |
| funvtxdmge2val 29589 | The set of vertices of an ... |
| funiedgdmge2val 29590 | The set of indexed edges o... |
| funvtxdm2val 29591 | The set of vertices of an ... |
| funiedgdm2val 29592 | The set of indexed edges o... |
| funvtxval0 29593 | The set of vertices of an ... |
| basvtxval 29594 | The set of vertices of a g... |
| edgfiedgval 29595 | The set of indexed edges o... |
| funvtxval 29596 | The set of vertices of a g... |
| funiedgval 29597 | The set of indexed edges o... |
| structvtxvallem 29598 | Lemma for ~ structvtxval a... |
| structvtxval 29599 | The set of vertices of an ... |
| structiedg0val 29600 | The set of indexed edges o... |
| structgrssvtxlem 29601 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29602 | The set of vertices of a g... |
| structgrssiedg 29603 | The set of indexed edges o... |
| struct2grstr 29604 | A graph represented as an ... |
| struct2grvtx 29605 | The set of vertices of a g... |
| struct2griedg 29606 | The set of indexed edges o... |
| graop 29607 | Any representation of a gr... |
| grastruct 29608 | Any representation of a gr... |
| gropd 29609 | If any representation of a... |
| grstructd 29610 | If any representation of a... |
| gropeld 29611 | If any representation of a... |
| grstructeld 29612 | If any representation of a... |
| setsvtx 29613 | The vertices of a structur... |
| setsiedg 29614 | The (indexed) edges of a s... |
| snstrvtxval 29615 | The set of vertices of a g... |
| snstriedgval 29616 | The set of indexed edges o... |
| vtxval0 29617 | Degenerated case 1 for ver... |
| iedgval0 29618 | Degenerated case 1 for edg... |
| vtxvalsnop 29619 | Degenerated case 2 for ver... |
| iedgvalsnop 29620 | Degenerated case 2 for edg... |
| vtxval3sn 29621 | Degenerated case 3 for ver... |
| iedgval3sn 29622 | Degenerated case 3 for edg... |
| vtxvalprc 29623 | Degenerated case 4 for ver... |
| iedgvalprc 29624 | Degenerated case 4 for edg... |
| edgval 29627 | The edges of a graph. (Co... |
| iedgedg 29628 | An indexed edge is an edge... |
| edgopval 29629 | The edges of a graph repre... |
| edgov 29630 | The edges of a graph repre... |
| edgstruct 29631 | The edges of a graph repre... |
| edgiedgb 29632 | A set is an edge iff it is... |
| edg0iedg0 29633 | There is no edge in a grap... |
| isuhgr 29638 | The predicate "is an undir... |
| isushgr 29639 | The predicate "is an undir... |
| uhgrf 29640 | The edge function of an un... |
| ushgrf 29641 | The edge function of an un... |
| uhgrss 29642 | An edge is a subset of ver... |
| uhgreq12g 29643 | If two sets have the same ... |
| uhgrfun 29644 | The edge function of an un... |
| uhgrn0 29645 | An edge is a nonempty subs... |
| lpvtx 29646 | The endpoints of a loop (w... |
| ushgruhgr 29647 | An undirected simple hyper... |
| isuhgrop 29648 | The property of being an u... |
| uhgr0e 29649 | The empty graph, with vert... |
| uhgr0vb 29650 | The null graph, with no ve... |
| uhgr0 29651 | The null graph represented... |
| uhgrun 29652 | The union ` U ` of two (un... |
| uhgrunop 29653 | The union of two (undirect... |
| ushgrun 29654 | The union ` U ` of two (un... |
| ushgrunop 29655 | The union of two (undirect... |
| uhgrstrrepe 29656 | Replacing (or adding) the ... |
| incistruhgr 29657 | An _incidence structure_ `... |
| isupgr 29662 | The property of being an u... |
| wrdupgr 29663 | The property of being an u... |
| upgrf 29664 | The edge function of an un... |
| upgrfn 29665 | The edge function of an un... |
| upgrss 29666 | An edge is a subset of ver... |
| upgrn0 29667 | An edge is a nonempty subs... |
| upgrle 29668 | An edge of an undirected p... |
| upgrfi 29669 | An edge is a finite subset... |
| upgrex 29670 | An edge is an unordered pa... |
| upgrbi 29671 | Show that an unordered pai... |
| upgrop 29672 | A pseudograph represented ... |
| isumgr 29673 | The property of being an u... |
| isumgrs 29674 | The simplified property of... |
| wrdumgr 29675 | The property of being an u... |
| umgrf 29676 | The edge function of an un... |
| umgrfn 29677 | The edge function of an un... |
| umgredg2 29678 | An edge of a multigraph ha... |
| umgrbi 29679 | Show that an unordered pai... |
| upgruhgr 29680 | An undirected pseudograph ... |
| umgrupgr 29681 | An undirected multigraph i... |
| umgruhgr 29682 | An undirected multigraph i... |
| upgrle2 29683 | An edge of an undirected p... |
| umgrnloopv 29684 | In a multigraph, there is ... |
| umgredgprv 29685 | In a multigraph, an edge i... |
| umgrnloop 29686 | In a multigraph, there is ... |
| umgrnloop0 29687 | A multigraph has no loops.... |
| umgr0e 29688 | The empty graph, with vert... |
| upgr0e 29689 | The empty graph, with vert... |
| upgr1elem 29690 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29691 | A pseudograph with one edg... |
| upgr0eop 29692 | The empty graph, with vert... |
| upgr1eop 29693 | A pseudograph with one edg... |
| upgr0eopALT 29694 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29695 | Alternate proof of ~ upgr1... |
| upgrun 29696 | The union ` U ` of two pse... |
| upgrunop 29697 | The union of two pseudogra... |
| umgrun 29698 | The union ` U ` of two mul... |
| umgrunop 29699 | The union of two multigrap... |
| umgrislfupgrlem 29700 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29701 | A multigraph is a loop-fre... |
| lfgredgge2 29702 | An edge of a loop-free gra... |
| lfgrnloop 29703 | A loop-free graph has no l... |
| uhgredgiedgb 29704 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29705 | A hypergraph has no edges ... |
| uhgredgn0 29706 | An edge of a hypergraph is... |
| edguhgr 29707 | An edge of a hypergraph is... |
| uhgredgrnv 29708 | An edge of a hypergraph co... |
| uhgredgss 29709 | The set of edges of a hype... |
| upgredgss 29710 | The set of edges of a pseu... |
| umgredgss 29711 | The set of edges of a mult... |
| edgupgr 29712 | Properties of an edge of a... |
| edgumgr 29713 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29714 | In a hypergraph, a vertex ... |
| upgredg 29715 | For each edge in a pseudog... |
| umgredg 29716 | For each edge in a multigr... |
| upgrpredgv 29717 | An edge of a pseudograph a... |
| umgrpredgv 29718 | An edge of a multigraph al... |
| upgredg2vtx 29719 | For a vertex incident to a... |
| upgredgpr 29720 | If a proper pair (of verti... |
| edglnl 29721 | The edges incident with a ... |
| numedglnl 29722 | The number of edges incide... |
| umgredgne 29723 | An edge of a multigraph al... |
| umgrnloop2 29724 | A multigraph has no loops.... |
| umgredgnlp 29725 | An edge of a multigraph is... |
| lfuhgr 29726 | A hypergraph is loop-free ... |
| lfuhgr2 29727 | A hypergraph is loop-free ... |
| lfuhgr3 29728 | A hypergraph is loop-free ... |
| isuspgr 29733 | The property of being a si... |
| isusgr 29734 | The property of being a si... |
| uspgrf 29735 | The edge function of a sim... |
| usgrf 29736 | The edge function of a sim... |
| isusgrs 29737 | The property of being a si... |
| usgrfs 29738 | The edge function of a sim... |
| usgrfun 29739 | The edge function of a sim... |
| usgredgss 29740 | The set of edges of a simp... |
| edgusgr 29741 | An edge of a simple graph ... |
| isuspgrop 29742 | The property of being an u... |
| isusgrop 29743 | The property of being an u... |
| usgrop 29744 | A simple graph represented... |
| isausgr 29745 | The property of an ordered... |
| ausgrusgrb 29746 | The equivalence of the def... |
| usgrausgri 29747 | A simple graph represented... |
| ausgrumgri 29748 | If an alternatively define... |
| ausgrusgri 29749 | The equivalence of the def... |
| usgrausgrb 29750 | The equivalence of the def... |
| usgredgop 29751 | An edge of a simple graph ... |
| usgrf1o 29752 | The edge function of a sim... |
| usgrf1 29753 | The edge function of a sim... |
| uspgrf1oedg 29754 | The edge function of a sim... |
| usgrss 29755 | An edge is a subset of ver... |
| uspgredgiedg 29756 | In a simple pseudograph, f... |
| uspgriedgedg 29757 | In a simple pseudograph, f... |
| uspgrushgr 29758 | A simple pseudograph is an... |
| uspgrupgr 29759 | A simple pseudograph is an... |
| uspgrupgrushgr 29760 | A graph is a simple pseudo... |
| usgruspgr 29761 | A simple graph is a simple... |
| usgrumgr 29762 | A simple graph is an undir... |
| usgrumgruspgr 29763 | A graph is a simple graph ... |
| usgruspgrb 29764 | A class is a simple graph ... |
| uspgruhgr 29765 | An undirected simple pseud... |
| usgrupgr 29766 | A simple graph is an undir... |
| usgruhgr 29767 | A simple graph is an undir... |
| usgrislfuspgr 29768 | A simple graph is a loop-f... |
| uspgrun 29769 | The union ` U ` of two sim... |
| uspgrunop 29770 | The union of two simple ps... |
| usgrun 29771 | The union ` U ` of two sim... |
| usgrunop 29772 | The union of two simple gr... |
| usgredg2 29773 | The value of the "edge fun... |
| usgredg2ALT 29774 | Alternate proof of ~ usgre... |
| usgredgprv 29775 | In a simple graph, an edge... |
| usgredgprvALT 29776 | Alternate proof of ~ usgre... |
| usgredgppr 29777 | An edge of a simple graph ... |
| usgrpredgv 29778 | An edge of a simple graph ... |
| edgssv2 29779 | An edge of a simple graph ... |
| usgredg 29780 | For each edge in a simple ... |
| usgrnloopv 29781 | In a simple graph, there i... |
| usgrnloopvALT 29782 | Alternate proof of ~ usgrn... |
| usgrnloop 29783 | In a simple graph, there i... |
| usgrnloopALT 29784 | Alternate proof of ~ usgrn... |
| usgrnloop0 29785 | A simple graph has no loop... |
| usgrnloop0ALT 29786 | Alternate proof of ~ usgrn... |
| usgredgne 29787 | An edge of a simple graph ... |
| usgrf1oedg 29788 | The edge function of a sim... |
| uhgr2edg 29789 | If a vertex is adjacent to... |
| umgr2edg 29790 | If a vertex is adjacent to... |
| usgr2edg 29791 | If a vertex is adjacent to... |
| umgr2edg1 29792 | If a vertex is adjacent to... |
| usgr2edg1 29793 | If a vertex is adjacent to... |
| umgrvad2edg 29794 | If a vertex is adjacent to... |
| umgr2edgneu 29795 | If a vertex is adjacent to... |
| usgrsizedg 29796 | In a simple graph, the siz... |
| usgredg3 29797 | The value of the "edge fun... |
| usgredg4 29798 | For a vertex incident to a... |
| usgredgreu 29799 | For a vertex incident to a... |
| usgredg2vtx 29800 | For a vertex incident to a... |
| uspgredg2vtxeu 29801 | For a vertex incident to a... |
| usgredg2vtxeu 29802 | For a vertex incident to a... |
| usgredg2vtxeuALT 29803 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29804 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29805 | In a simple pseudograph, t... |
| usgredg2vlem1 29806 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29807 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29808 | In a simple graph, the map... |
| usgriedgleord 29809 | Alternate version of ~ usg... |
| ushgredgedg 29810 | In a simple hypergraph the... |
| usgredgedg 29811 | In a simple graph there is... |
| ushgredgedgloop 29812 | In a simple hypergraph the... |
| uspgredgleord 29813 | In a simple pseudograph th... |
| usgredgleord 29814 | In a simple graph the numb... |
| usgredgleordALT 29815 | Alternate proof for ~ usgr... |
| usgrstrrepe 29816 | Replacing (or adding) the ... |
| usgr0e 29817 | The empty graph, with vert... |
| usgr0vb 29818 | The null graph, with no ve... |
| uhgr0v0e 29819 | The null graph, with no ve... |
| uhgr0vsize0 29820 | The size of a hypergraph w... |
| uhgr0edgfi 29821 | A graph of order 0 (i.e. w... |
| usgr0v 29822 | The null graph, with no ve... |
| uhgr0vusgr 29823 | The null graph, with no ve... |
| usgr0 29824 | The null graph represented... |
| uspgr1e 29825 | A simple pseudograph with ... |
| usgr1e 29826 | A simple graph with one ed... |
| usgr0eop 29827 | The empty graph, with vert... |
| uspgr1eop 29828 | A simple pseudograph with ... |
| uspgr1ewop 29829 | A simple pseudograph with ... |
| uspgr1v1eop 29830 | A simple pseudograph with ... |
| usgr1eop 29831 | A simple graph with (at le... |
| uspgr2v1e2w 29832 | A simple pseudograph with ... |
| usgr2v1e2w 29833 | A simple graph with two ve... |
| edg0usgr 29834 | A class without edges is a... |
| lfuhgr1v0e 29835 | A loop-free hypergraph wit... |
| usgr1vr 29836 | A simple graph with one ve... |
| usgr1v 29837 | A class with one (or no) v... |
| usgr1v0edg 29838 | A class with one (or no) v... |
| usgrexmpldifpr 29839 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29840 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29841 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29842 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29843 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29844 | ` G ` is a simple graph of... |
| griedg0prc 29845 | The class of empty graphs ... |
| griedg0ssusgr 29846 | The class of all simple gr... |
| usgrprc 29847 | The class of simple graphs... |
| relsubgr 29850 | The class of the subgraph ... |
| subgrv 29851 | If a class is a subgraph o... |
| issubgr 29852 | The property of a set to b... |
| issubgr2 29853 | The property of a set to b... |
| subgrprop 29854 | The properties of a subgra... |
| subgrprop2 29855 | The properties of a subgra... |
| uhgrissubgr 29856 | The property of a hypergra... |
| subgrprop3 29857 | The properties of a subgra... |
| egrsubgr 29858 | An empty graph consisting ... |
| 0grsubgr 29859 | The null graph (represente... |
| 0uhgrsubgr 29860 | The null graph (as hypergr... |
| uhgrsubgrself 29861 | A hypergraph is a subgraph... |
| subgrfun 29862 | The edge function of a sub... |
| subgruhgrfun 29863 | The edge function of a sub... |
| subgreldmiedg 29864 | An element of the domain o... |
| subgruhgredgd 29865 | An edge of a subgraph of a... |
| subumgredg2 29866 | An edge of a subgraph of a... |
| subuhgr 29867 | A subgraph of a hypergraph... |
| subupgr 29868 | A subgraph of a pseudograp... |
| subumgr 29869 | A subgraph of a multigraph... |
| subusgr 29870 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29871 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29872 | A spanning subgraph ` S ` ... |
| uhgrspan 29873 | A spanning subgraph ` S ` ... |
| upgrspan 29874 | A spanning subgraph ` S ` ... |
| umgrspan 29875 | A spanning subgraph ` S ` ... |
| usgrspan 29876 | A spanning subgraph ` S ` ... |
| uhgrspanop 29877 | A spanning subgraph of a h... |
| upgrspanop 29878 | A spanning subgraph of a p... |
| umgrspanop 29879 | A spanning subgraph of a m... |
| usgrspanop 29880 | A spanning subgraph of a s... |
| uhgrspan1lem1 29881 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29882 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29883 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29884 | The induced subgraph ` S `... |
| upgrreslem 29885 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29886 | Lemma for ~ umgrres and ~ ... |
| upgrres 29887 | A subgraph obtained by rem... |
| umgrres 29888 | A subgraph obtained by rem... |
| usgrres 29889 | A subgraph obtained by rem... |
| upgrres1lem1 29890 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29891 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29892 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29893 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29894 | A pseudograph obtained by ... |
| umgrres1 29895 | A multigraph obtained by r... |
| usgrres1 29896 | Restricting a simple graph... |
| isfusgr 29899 | The property of being a fi... |
| fusgrvtxfi 29900 | A finite simple graph has ... |
| isfusgrf1 29901 | The property of being a fi... |
| isfusgrcl 29902 | The property of being a fi... |
| fusgrusgr 29903 | A finite simple graph is a... |
| opfusgr 29904 | A finite simple graph repr... |
| usgredgffibi 29905 | The number of edges in a s... |
| fusgredgfi 29906 | In a finite simple graph t... |
| usgr1v0e 29907 | The size of a (finite) sim... |
| usgrfilem 29908 | In a finite simple graph, ... |
| fusgrfisbase 29909 | Induction base for ~ fusgr... |
| fusgrfisstep 29910 | Induction step in ~ fusgrf... |
| fusgrfis 29911 | A finite simple graph is o... |
| fusgrfupgrfs 29912 | A finite simple graph is a... |
| nbgrprc0 29915 | The set of neighbors is em... |
| nbgrcl 29916 | If a class ` X ` has at le... |
| nbgrval 29917 | The set of neighbors of a ... |
| dfnbgr2 29918 | Alternate definition of th... |
| dfnbgr3 29919 | Alternate definition of th... |
| nbgrnvtx0 29920 | If a class ` X ` is not a ... |
| nbgrel 29921 | Characterization of a neig... |
| nbgrisvtx 29922 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29923 | The neighbors of a vertex ... |
| nbuhgr 29924 | The set of neighbors of a ... |
| nbupgr 29925 | The set of neighbors of a ... |
| nbupgrel 29926 | A neighbor of a vertex in ... |
| nbumgrvtx 29927 | The set of neighbors of a ... |
| nbumgr 29928 | The set of neighbors of an... |
| nbusgrvtx 29929 | The set of neighbors of a ... |
| nbusgr 29930 | The set of neighbors of an... |
| nbgr2vtx1edg 29931 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29932 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29933 | If a hypergraph has two ve... |
| nbusgreledg 29934 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29935 | A vertex which is not endp... |
| nbgr0vtx 29936 | In a null graph (with no v... |
| nbgr0edglem 29937 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29938 | In an empty graph (with no... |
| nbgr1vtx 29939 | In a graph with one vertex... |
| nbgrnself 29940 | A vertex in a graph is not... |
| nbgrnself2 29941 | A class ` X ` is not a nei... |
| nbgrssovtx 29942 | The neighbors of a vertex ... |
| nbgrssvwo2 29943 | The neighbors of a vertex ... |
| nbgrsym 29944 | In a graph, the neighborho... |
| nbupgrres 29945 | The neighborhood of a vert... |
| usgrnbcnvfv 29946 | Applying the edge function... |
| nbusgredgeu 29947 | For each neighbor of a ver... |
| edgnbusgreu 29948 | For each edge incident to ... |
| nbusgredgeu0 29949 | For each neighbor of a ver... |
| nbusgrf1o0 29950 | The mapping of neighbors o... |
| nbusgrf1o1 29951 | The set of neighbors of a ... |
| nbusgrf1o 29952 | The set of neighbors of a ... |
| nbedgusgr 29953 | The number of neighbors of... |
| edgusgrnbfin 29954 | The number of neighbors of... |
| nbusgrfi 29955 | The class of neighbors of ... |
| nbfiusgrfi 29956 | The class of neighbors of ... |
| hashnbusgrnn0 29957 | The number of neighbors of... |
| nbfusgrlevtxm1 29958 | The number of neighbors of... |
| nbfusgrlevtxm2 29959 | If there is a vertex which... |
| nbusgrvtxm1 29960 | If the number of neighbors... |
| nb3grprlem1 29961 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29962 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29963 | The neighbors of a vertex ... |
| nb3grpr2 29964 | The neighbors of a vertex ... |
| nb3gr2nb 29965 | If the neighbors of two ve... |
| uvtxval 29968 | The set of all universal v... |
| uvtxel 29969 | A universal vertex, i.e. a... |
| uvtxisvtx 29970 | A universal vertex is a ve... |
| uvtxssvtx 29971 | The set of the universal v... |
| vtxnbuvtx 29972 | A universal vertex has all... |
| uvtxnbgrss 29973 | A universal vertex has all... |
| uvtxnbgrvtx 29974 | A universal vertex is neig... |
| uvtx0 29975 | There is no universal vert... |
| isuvtx 29976 | The set of all universal v... |
| uvtxel1 29977 | Characterization of a univ... |
| uvtx01vtx 29978 | If a graph/class has no ed... |
| uvtx2vtx1edg 29979 | If a graph has two vertice... |
| uvtx2vtx1edgb 29980 | If a hypergraph has two ve... |
| uvtxnbgr 29981 | A universal vertex has all... |
| uvtxnbgrb 29982 | A vertex is universal iff ... |
| uvtxusgr 29983 | The set of all universal v... |
| uvtxusgrel 29984 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29985 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29986 | If the number of neighbors... |
| uvtxnbvtxm1 29987 | A universal vertex has ` n... |
| nbupgruvtxres 29988 | The neighborhood of a univ... |
| uvtxupgrres 29989 | A universal vertex is univ... |
| cplgruvtxb 29994 | A graph ` G ` is complete ... |
| prcliscplgr 29995 | A proper class (representi... |
| iscplgr 29996 | The property of being a co... |
| iscplgrnb 29997 | A graph is complete iff al... |
| iscplgredg 29998 | A graph ` G ` is complete ... |
| iscusgr 29999 | The property of being a co... |
| cusgrusgr 30000 | A complete simple graph is... |
| cusgrcplgr 30001 | A complete simple graph is... |
| iscusgrvtx 30002 | A simple graph is complete... |
| cusgruvtxb 30003 | A simple graph is complete... |
| iscusgredg 30004 | A simple graph is complete... |
| cusgredg 30005 | In a complete simple graph... |
| cplgr0 30006 | The null graph (with no ve... |
| cusgr0 30007 | The null graph (with no ve... |
| cplgr0v 30008 | A null graph (with no vert... |
| cusgr0v 30009 | A graph with no vertices a... |
| cplgr1vlem 30010 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 30011 | A graph with one vertex is... |
| cusgr1v 30012 | A graph with one vertex an... |
| cplgr2v 30013 | An undirected hypergraph w... |
| cplgr2vpr 30014 | An undirected hypergraph w... |
| nbcplgr 30015 | In a complete graph, each ... |
| cplgr3v 30016 | A pseudograph with three (... |
| cusgr3vnbpr 30017 | The neighbors of a vertex ... |
| cplgrop 30018 | A complete graph represent... |
| cusgrop 30019 | A complete simple graph re... |
| cusgrexilem1 30020 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 30021 | Lemma for ~ usgrexi . (Co... |
| usgrexi 30022 | An arbitrary set regarded ... |
| cusgrexilem2 30023 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 30024 | An arbitrary set ` V ` reg... |
| cusgrexg 30025 | For each set there is a se... |
| structtousgr 30026 | Any (extensible) structure... |
| structtocusgr 30027 | Any (extensible) structure... |
| cffldtocusgr 30028 | The field of complex numbe... |
| cusgrres 30029 | Restricting a complete sim... |
| cusgrsizeindb0 30030 | Base case of the induction... |
| cusgrsizeindb1 30031 | Base case of the induction... |
| cusgrsizeindslem 30032 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 30033 | Part 1 of induction step i... |
| cusgrsize2inds 30034 | Induction step in ~ cusgrs... |
| cusgrsize 30035 | The size of a finite compl... |
| cusgrfilem1 30036 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 30037 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 30038 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 30039 | If the size of a complete ... |
| usgredgsscusgredg 30040 | A simple graph is a subgra... |
| usgrsscusgr 30041 | A simple graph is a subgra... |
| sizusglecusglem1 30042 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 30043 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 30044 | The size of a simple graph... |
| fusgrmaxsize 30045 | The maximum size of a fini... |
| vtxdgfval 30048 | The value of the vertex de... |
| vtxdgval 30049 | The degree of a vertex. (... |
| vtxdgfival 30050 | The degree of a vertex for... |
| vtxdgop 30051 | The vertex degree expresse... |
| vtxdgf 30052 | The vertex degree function... |
| vtxdgelxnn0 30053 | The degree of a vertex is ... |
| vtxdg0v 30054 | The degree of a vertex in ... |
| vtxdg0e 30055 | The degree of a vertex in ... |
| vtxdgfisnn0 30056 | The degree of a vertex in ... |
| vtxdgfisf 30057 | The vertex degree function... |
| vtxdeqd 30058 | Equality theorem for the v... |
| vtxduhgr0e 30059 | The degree of a vertex in ... |
| vtxdlfuhgr1v 30060 | The degree of the vertex i... |
| vdumgr0 30061 | A vertex in a multigraph h... |
| vtxdun 30062 | The degree of a vertex in ... |
| vtxdfiun 30063 | The degree of a vertex in ... |
| vtxduhgrun 30064 | The degree of a vertex in ... |
| vtxduhgrfiun 30065 | The degree of a vertex in ... |
| vtxdlfgrval 30066 | The value of the vertex de... |
| vtxdumgrval 30067 | The value of the vertex de... |
| vtxdusgrval 30068 | The value of the vertex de... |
| vtxd0nedgb 30069 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 30070 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 30071 | The value of the vertex de... |
| vtxdusgrfvedg 30072 | The value of the vertex de... |
| vtxduhgr0nedg 30073 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 30074 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 30075 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 30076 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 30077 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 30078 | The vertex degree function... |
| vtxdgfusgr 30079 | In a finite simple graph, ... |
| fusgrn0degnn0 30080 | In a nonempty, finite grap... |
| 1loopgruspgr 30081 | A graph with one edge whic... |
| 1loopgredg 30082 | The set of edges in a grap... |
| 1loopgrnb0 30083 | In a graph (simple pseudog... |
| 1loopgrvd2 30084 | The vertex degree of a one... |
| 1loopgrvd0 30085 | The vertex degree of a one... |
| 1hevtxdg0 30086 | The vertex degree of verte... |
| 1hevtxdg1 30087 | The vertex degree of verte... |
| 1hegrvtxdg1 30088 | The vertex degree of a gra... |
| 1hegrvtxdg1r 30089 | The vertex degree of a gra... |
| 1egrvtxdg1 30090 | The vertex degree of a one... |
| 1egrvtxdg1r 30091 | The vertex degree of a one... |
| 1egrvtxdg0 30092 | The vertex degree of a one... |
| p1evtxdeqlem 30093 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 30094 | If an edge ` E ` which doe... |
| p1evtxdp1 30095 | If an edge ` E ` (not bein... |
| uspgrloopvtx 30096 | The set of vertices in a g... |
| uspgrloopvtxel 30097 | A vertex in a graph (simpl... |
| uspgrloopiedg 30098 | The set of edges in a grap... |
| uspgrloopedg 30099 | The set of edges in a grap... |
| uspgrloopnb0 30100 | In a graph (simple pseudog... |
| uspgrloopvd2 30101 | The vertex degree of a one... |
| umgr2v2evtx 30102 | The set of vertices in a m... |
| umgr2v2evtxel 30103 | A vertex in a multigraph w... |
| umgr2v2eiedg 30104 | The edge function in a mul... |
| umgr2v2eedg 30105 | The set of edges in a mult... |
| umgr2v2e 30106 | A multigraph with two edge... |
| umgr2v2enb1 30107 | In a multigraph with two e... |
| umgr2v2evd2 30108 | In a multigraph with two e... |
| hashnbusgrvd 30109 | In a simple graph, the num... |
| usgruvtxvdb 30110 | In a finite simple graph w... |
| vdiscusgrb 30111 | A finite simple graph with... |
| vdiscusgr 30112 | In a finite complete simpl... |
| vtxdusgradjvtx 30113 | The degree of a vertex in ... |
| usgrvd0nedg 30114 | If a vertex in a simple gr... |
| uhgrvd00 30115 | If every vertex in a hyper... |
| usgrvd00 30116 | If every vertex in a simpl... |
| vdegp1ai 30117 | The induction step for a v... |
| vdegp1bi 30118 | The induction step for a v... |
| vdegp1ci 30119 | The induction step for a v... |
| vtxdginducedm1lem1 30120 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 30121 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 30122 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 30123 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 30124 | The degree of a vertex ` v... |
| vtxdginducedm1fi 30125 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 30126 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 30127 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 30128 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 30129 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 30130 | Induction step of ~ finsum... |
| finsumvtxdg2size 30131 | The sum of the degrees of ... |
| fusgr1th 30132 | The sum of the degrees of ... |
| finsumvtxdgeven 30133 | The sum of the degrees of ... |
| vtxdgoddnumeven 30134 | The number of vertices of ... |
| fusgrvtxdgonume 30135 | The number of vertices of ... |
| isrgr 30140 | The property of a class be... |
| rgrprop 30141 | The properties of a k-regu... |
| isrusgr 30142 | The property of being a k-... |
| rusgrprop 30143 | The properties of a k-regu... |
| rusgrrgr 30144 | A k-regular simple graph i... |
| rusgrusgr 30145 | A k-regular simple graph i... |
| finrusgrfusgr 30146 | A finite regular simple gr... |
| isrusgr0 30147 | The property of being a k-... |
| rusgrprop0 30148 | The properties of a k-regu... |
| usgreqdrusgr 30149 | If all vertices in a simpl... |
| fusgrregdegfi 30150 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 30151 | If all vertices in a nonem... |
| frusgrnn0 30152 | In a nonempty finite k-reg... |
| 0edg0rgr 30153 | A graph is 0-regular if it... |
| uhgr0edg0rgr 30154 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 30155 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 30156 | A simple graph is 0-regula... |
| 0vtxrgr 30157 | A null graph (with no vert... |
| 0vtxrusgr 30158 | A graph with no vertices a... |
| 0uhgrrusgr 30159 | The null graph as hypergra... |
| 0grrusgr 30160 | The null graph represented... |
| 0grrgr 30161 | The null graph represented... |
| cusgrrusgr 30162 | A complete simple graph wi... |
| cusgrm1rusgr 30163 | A finite simple graph with... |
| rusgrpropnb 30164 | The properties of a k-regu... |
| rusgrpropedg 30165 | The properties of a k-regu... |
| rusgrpropadjvtx 30166 | The properties of a k-regu... |
| rusgrnumwrdl2 30167 | In a k-regular simple grap... |
| rusgr1vtxlem 30168 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 30169 | If a k-regular simple grap... |
| rgrusgrprc 30170 | The class of 0-regular sim... |
| rusgrprc 30171 | The class of 0-regular sim... |
| rgrprc 30172 | The class of 0-regular gra... |
| rgrprcx 30173 | The class of 0-regular gra... |
| rgrx0ndm 30174 | 0 is not in the domain of ... |
| rgrx0nd 30175 | The potentially alternativ... |
| ewlksfval 30182 | The set of s-walks of edge... |
| isewlk 30183 | Conditions for a function ... |
| ewlkprop 30184 | Properties of an s-walk of... |
| ewlkinedg 30185 | The intersection (common v... |
| ewlkle 30186 | An s-walk of edges is also... |
| upgrewlkle2 30187 | In a pseudograph, there is... |
| wkslem1 30188 | Lemma 1 for walks to subst... |
| wkslem2 30189 | Lemma 2 for walks to subst... |
| wksfval 30190 | The set of walks (in an un... |
| iswlk 30191 | Properties of a pair of fu... |
| wlkprop 30192 | Properties of a walk. (Co... |
| wlkv 30193 | The classes involved in a ... |
| iswlkg 30194 | Generalization of ~ iswlk ... |
| wlkf 30195 | The mapping enumerating th... |
| wlkcl 30196 | A walk has length ` # ( F ... |
| wlkp 30197 | The mapping enumerating th... |
| wlkpwrd 30198 | The sequence of vertices o... |
| wlklenvp1 30199 | The number of vertices of ... |
| wksv 30200 | The class of walks is a se... |
| wlkn0 30201 | The sequence of vertices o... |
| wlklenvm1 30202 | The number of edges of a w... |
| ifpsnprss 30203 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 30204 | Each pair of adjacent vert... |
| wlkvtxiedg 30205 | The vertices of a walk are... |
| relwlk 30206 | The set ` ( Walks `` G ) `... |
| wlkvv 30207 | If there is at least one w... |
| wlkop 30208 | A walk is an ordered pair.... |
| wlkcpr 30209 | A walk as class with two c... |
| wlk2f 30210 | If there is a walk ` W ` t... |
| wlkcomp 30211 | A walk expressed by proper... |
| wlkcompim 30212 | Implications for the prope... |
| wlkelwrd 30213 | The components of a walk a... |
| wlkeq 30214 | Conditions for two walks (... |
| edginwlk 30215 | The value of the edge func... |
| upgredginwlk 30216 | The value of the edge func... |
| iedginwlk 30217 | The value of the edge func... |
| wlkl1loop 30218 | A walk of length 1 from a ... |
| wlk1walk 30219 | A walk is a 1-walk "on the... |
| wlk1ewlk 30220 | A walk is an s-walk "on th... |
| upgriswlk 30221 | Properties of a pair of fu... |
| upgrwlkedg 30222 | The edges of a walk in a p... |
| upgrwlkcompim 30223 | Implications for the prope... |
| wlkvtxedg 30224 | The vertices of a walk are... |
| upgrwlkvtxedg 30225 | The pairs of connected ver... |
| uspgr2wlkeq 30226 | Conditions for two walks w... |
| uspgr2wlkeq2 30227 | Conditions for two walks w... |
| uspgr2wlkeqi 30228 | Conditions for two walks w... |
| umgrwlknloop 30229 | In a multigraph, each walk... |
| wlkv0 30230 | If there is a walk in the ... |
| g0wlk0 30231 | There is no walk in a null... |
| 0wlk0 30232 | There is no walk for the e... |
| wlk0prc 30233 | There is no walk in a null... |
| wlklenvclwlk 30234 | The number of vertices in ... |
| wlkson 30235 | The set of walks between t... |
| iswlkon 30236 | Properties of a pair of fu... |
| wlkonprop 30237 | Properties of a walk betwe... |
| wlkpvtx 30238 | A walk connects vertices. ... |
| wlkepvtx 30239 | The endpoints of a walk ar... |
| wlkoniswlk 30240 | A walk between two vertice... |
| wlkonwlk 30241 | A walk is a walk between i... |
| wlkonwlk1l 30242 | A walk is a walk from its ... |
| wlksoneq1eq2 30243 | Two walks with identical s... |
| wlkonl1iedg 30244 | If there is a walk between... |
| wlkon2n0 30245 | The length of a walk betwe... |
| 2wlklem 30246 | Lemma for theorems for wal... |
| upgr2wlk 30247 | Properties of a pair of fu... |
| wlkreslem 30248 | Lemma for ~ wlkres . (Con... |
| wlkres 30249 | The restriction ` <. H , Q... |
| redwlklem 30250 | Lemma for ~ redwlk . (Con... |
| redwlk 30251 | A walk ending at the last ... |
| wlkp1lem1 30252 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 30253 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 30254 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 30255 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 30256 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 30257 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 30258 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 30259 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 30260 | Append one path segment (e... |
| wlkdlem1 30261 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 30262 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 30263 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 30264 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 30265 | Two words representing a w... |
| pfxwlk 30266 | A prefix of a walk is a wa... |
| revwlk 30267 | The reverse of a walk is a... |
| swrdwlk 30268 | Two matching subwords of a... |
| subgrwlk 30269 | If a walk exists in a subg... |
| lfgrwlkprop 30270 | Two adjacent vertices in a... |
| lfgriswlk 30271 | Conditions for a pair of f... |
| lfgrwlknloop 30272 | In a loop-free graph, each... |
| reltrls 30277 | The set ` ( Trails `` G ) ... |
| trlsfval 30278 | The set of trails (in an u... |
| istrl 30279 | Conditions for a pair of c... |
| trliswlk 30280 | A trail is a walk. (Contr... |
| trlf1 30281 | The enumeration ` F ` of a... |
| trlreslem 30282 | Lemma for ~ trlres . Form... |
| trlres 30283 | The restriction ` <. H , Q... |
| upgrtrls 30284 | The set of trails in a pse... |
| upgristrl 30285 | Properties of a pair of fu... |
| upgrf1istrl 30286 | Properties of a pair of a ... |
| wksonproplem 30287 | Lemma for theorems for pro... |
| trlsonfval 30288 | The set of trails between ... |
| istrlson 30289 | Properties of a pair of fu... |
| trlsonprop 30290 | Properties of a trail betw... |
| trlsonistrl 30291 | A trail between two vertic... |
| trlsonwlkon 30292 | A trail between two vertic... |
| trlontrl 30293 | A trail is a trail between... |
| subgrtrl 30294 | If a trail exists in a sub... |
| relpths 30303 | The set ` ( Paths `` G ) `... |
| pthsfval 30304 | The set of paths (in an un... |
| spthsfval 30305 | The set of simple paths (i... |
| ispth 30306 | Conditions for a pair of c... |
| isspth 30307 | Conditions for a pair of c... |
| pthistrl 30308 | A path is a trail (in an u... |
| spthispth 30309 | A simple path is a path (i... |
| pthiswlk 30310 | A path is a walk (in an un... |
| spthiswlk 30311 | A simple path is a walk (i... |
| pthdivtx 30312 | The inner vertices of a pa... |
| pthdadjvtx 30313 | The adjacent vertices of a... |
| dfpth2 30314 | Alternate definition for a... |
| pthhashvtx 30315 | A graph containing a path ... |
| pthdifv 30316 | The vertices of a path are... |
| 2pthnloop 30317 | A path of length at least ... |
| upgr2pthnlp 30318 | A path of length at least ... |
| spthdifv 30319 | The vertices of a simple p... |
| spthdep 30320 | A simple path (at least of... |
| pthdepisspth 30321 | A path with different star... |
| upgrwlkdvdelem 30322 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 30323 | In a pseudograph, all edge... |
| upgrspthswlk 30324 | The set of simple paths in... |
| upgrwlkdvspth 30325 | A walk consisting of diffe... |
| pthsonfval 30326 | The set of paths between t... |
| spthson 30327 | The set of simple paths be... |
| ispthson 30328 | Properties of a pair of fu... |
| isspthson 30329 | Properties of a pair of fu... |
| pthsonprop 30330 | Properties of a path betwe... |
| spthonprop 30331 | Properties of a simple pat... |
| pthonispth 30332 | A path between two vertice... |
| pthontrlon 30333 | A path between two vertice... |
| pthonpth 30334 | A path is a path between i... |
| isspthonpth 30335 | A pair of functions is a s... |
| spthonisspth 30336 | A simple path between to v... |
| spthonpthon 30337 | A simple path between two ... |
| spthonepeq 30338 | The endpoints of a simple ... |
| uhgrwkspthlem1 30339 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 30340 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 30341 | Any walk of length 1 betwe... |
| usgr2wlkneq 30342 | The vertices and edges are... |
| usgr2wlkspthlem1 30343 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 30344 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 30345 | In a simple graph, any wal... |
| usgr2trlncl 30346 | In a simple graph, any tra... |
| usgr2trlspth 30347 | In a simple graph, any tra... |
| usgr2pthspth 30348 | In a simple graph, any pat... |
| usgr2pthlem 30349 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 30350 | In a simple graph, there i... |
| usgr2pth0 30351 | In a simply graph, there i... |
| pthdlem1 30352 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 30353 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 30354 | Lemma 2 for ~ pthd . (Con... |
| pthd 30355 | Two words representing a t... |
| subgrpth 30356 | If a path exists in a subg... |
| clwlks 30359 | The set of closed walks (i... |
| isclwlk 30360 | A pair of functions repres... |
| clwlkiswlk 30361 | A closed walk is a walk (i... |
| clwlkwlk 30362 | Closed walks are walks (in... |
| clwlkswks 30363 | Closed walks are walks (in... |
| isclwlke 30364 | Properties of a pair of fu... |
| isclwlkupgr 30365 | Properties of a pair of fu... |
| clwlkcomp 30366 | A closed walk expressed by... |
| clwlkcompim 30367 | Implications for the prope... |
| upgrclwlkcompim 30368 | Implications for the prope... |
| clwlkcompbp 30369 | Basic properties of the co... |
| clwlkl1loop 30370 | A closed walk of length 1 ... |
| crcts 30375 | The set of circuits (in an... |
| cycls 30376 | The set of cycles (in an u... |
| iscrct 30377 | Sufficient and necessary c... |
| iscycl 30378 | Sufficient and necessary c... |
| crctprop 30379 | The properties of a circui... |
| cyclprop 30380 | The properties of a cycle:... |
| crctisclwlk 30381 | A circuit is a closed walk... |
| crctistrl 30382 | A circuit is a trail. (Co... |
| crctiswlk 30383 | A circuit is a walk. (Con... |
| cyclispth 30384 | A cycle is a path. (Contr... |
| subgrcycl 30385 | If a cycle exists in a sub... |
| cycliswlk 30386 | A cycle is a walk. (Contr... |
| cycliscrct 30387 | A cycle is a circuit. (Co... |
| cyclnumvtx 30388 | The number of vertices of ... |
| cyclnspth 30389 | A (non-trivial) cycle is n... |
| pthisspthorcycl 30390 | A path is either a simple ... |
| pthspthcyc 30391 | A pair ` <. F , P >. ` rep... |
| spthcycl 30392 | A walk is a trivial path i... |
| cyclispthon 30393 | A cycle is a path starting... |
| lfgrn1cycl 30394 | In a loop-free graph there... |
| usgr2trlncrct 30395 | In a simple graph, any tra... |
| umgrn1cycl 30396 | In a multigraph graph (wit... |
| uspgrn2crct 30397 | In a simple pseudograph th... |
| usgrn2cycl 30398 | In a simple graph there ar... |
| crctcshwlkn0lem1 30399 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 30400 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 30401 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 30402 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 30403 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 30404 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 30405 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 30406 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 30407 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 30408 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 30409 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 30410 | Cyclically shifting the in... |
| crctcshwlk 30411 | Cyclically shifting the in... |
| crctcshtrl 30412 | Cyclically shifting the in... |
| crctcsh 30413 | Cyclically shifting the in... |
| wwlks 30424 | The set of walks (in an un... |
| iswwlks 30425 | A word over the set of ver... |
| wwlksn 30426 | The set of walks (in an un... |
| iswwlksn 30427 | A word over the set of ver... |
| wwlksnprcl 30428 | Derivation of the length o... |
| iswwlksnx 30429 | Properties of a word to re... |
| wwlkbp 30430 | Basic properties of a walk... |
| wwlknbp 30431 | Basic properties of a walk... |
| wwlknp 30432 | Properties of a set being ... |
| wwlknbp1 30433 | Other basic properties of ... |
| wwlknvtx 30434 | The symbols of a word ` W ... |
| wwlknllvtx 30435 | If a word ` W ` represents... |
| wwlknlsw 30436 | If a word represents a wal... |
| wspthsn 30437 | The set of simple paths of... |
| iswspthn 30438 | An element of the set of s... |
| wspthnp 30439 | Properties of a set being ... |
| wwlksnon 30440 | The set of walks of a fixe... |
| wspthsnon 30441 | The set of simple paths of... |
| iswwlksnon 30442 | The set of walks of a fixe... |
| wwlksnon0 30443 | Sufficient conditions for ... |
| wwlksonvtx 30444 | If a word ` W ` represents... |
| iswspthsnon 30445 | The set of simple paths of... |
| wwlknon 30446 | An element of the set of w... |
| wspthnon 30447 | An element of the set of s... |
| wspthnonp 30448 | Properties of a set being ... |
| wspthneq1eq2 30449 | Two simple paths with iden... |
| wwlksn0s 30450 | The set of all walks as wo... |
| wwlkssswrd 30451 | Walks (represented by word... |
| wwlksn0 30452 | A walk of length 0 is repr... |
| 0enwwlksnge1 30453 | In graphs without edges, t... |
| wwlkswwlksn 30454 | A walk of a fixed length a... |
| wwlkssswwlksn 30455 | The walks of a fixed lengt... |
| wlkiswwlks1 30456 | The sequence of vertices i... |
| wlklnwwlkln1 30457 | The sequence of vertices i... |
| wlkiswwlks2lem1 30458 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 30459 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 30460 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 30461 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 30462 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 30463 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 30464 | A walk as word corresponds... |
| wlkiswwlks 30465 | A walk as word corresponds... |
| wlkiswwlksupgr2 30466 | A walk as word corresponds... |
| wlkiswwlkupgr 30467 | A walk as word corresponds... |
| wlkswwlksf1o 30468 | The mapping of (ordinary) ... |
| wlkswwlksen 30469 | The set of walks as words ... |
| wwlksm1edg 30470 | Removing the trailing edge... |
| wlklnwwlkln2lem 30471 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 30472 | A walk of length ` N ` as ... |
| wlklnwwlkn 30473 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 30474 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 30475 | A walk of length ` N ` as ... |
| wlknewwlksn 30476 | If a walk in a pseudograph... |
| wlknwwlksnbij 30477 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 30478 | In a simple pseudograph, t... |
| wlknwwlksneqs 30479 | The set of walks of a fixe... |
| wwlkseq 30480 | Equality of two walks (as ... |
| wwlksnred 30481 | Reduction of a walk (as wo... |
| wwlksnext 30482 | Extension of a walk (as wo... |
| wwlksnextbi 30483 | Extension of a walk (as wo... |
| wwlksnredwwlkn 30484 | For each walk (as word) of... |
| wwlksnredwwlkn0 30485 | For each walk (as word) of... |
| wwlksnextwrd 30486 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 30487 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 30488 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 30489 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 30490 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 30491 | There is a bijection betwe... |
| wwlksnexthasheq 30492 | The number of the extensio... |
| disjxwwlksn 30493 | Sets of walks (as words) e... |
| wwlksnndef 30494 | Conditions for ` WWalksN `... |
| wwlksnfi 30495 | The number of walks repres... |
| wlksnfi 30496 | The number of walks of fix... |
| wlksnwwlknvbij 30497 | There is a bijection betwe... |
| wwlksnextproplem1 30498 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 30499 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 30500 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 30501 | Adding additional properti... |
| disjxwwlkn 30502 | Sets of walks (as words) e... |
| hashwwlksnext 30503 | Number of walks (as words)... |
| wwlksnwwlksnon 30504 | A walk of fixed length is ... |
| wspthsnwspthsnon 30505 | A simple path of fixed len... |
| wspthsnonn0vne 30506 | If the set of simple paths... |
| wspthsswwlkn 30507 | The set of simple paths of... |
| wspthnfi 30508 | In a finite graph, the set... |
| wwlksnonfi 30509 | In a finite graph, the set... |
| wspthsswwlknon 30510 | The set of simple paths of... |
| wspthnonfi 30511 | In a finite graph, the set... |
| wspniunwspnon 30512 | The set of nonempty simple... |
| wspn0 30513 | If there are no vertices, ... |
| 2wlkdlem1 30514 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 30515 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 30516 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 30517 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 30518 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 30519 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 30520 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 30521 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 30522 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 30523 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 30524 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 30525 | Construction of a walk fro... |
| 2wlkond 30526 | A walk of length 2 from on... |
| 2trld 30527 | Construction of a trail fr... |
| 2trlond 30528 | A trail of length 2 from o... |
| 2pthd 30529 | A path of length 2 from on... |
| 2spthd 30530 | A simple path of length 2 ... |
| 2pthond 30531 | A simple path of length 2 ... |
| 2pthon3v 30532 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30533 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30534 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30535 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30536 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30537 | In a multigraph, two adjac... |
| umgr2wlk 30538 | In a multigraph, there is ... |
| umgr2wlkon 30539 | For each pair of adjacent ... |
| elwwlks2s3 30540 | A walk of length 2 as word... |
| midwwlks2s3 30541 | There is a vertex between ... |
| wwlks2onv 30542 | If a length 3 string repre... |
| elwwlks2ons3im 30543 | A walk as word of length 2... |
| elwwlks2ons3 30544 | For each walk of length 2 ... |
| s3wwlks2on 30545 | A length 3 string which re... |
| sps3wwlks2on 30546 | A length 3 string which re... |
| usgrwwlks2on 30547 | A walk of length 2 between... |
| umgrwwlks2on 30548 | A walk of length 2 between... |
| wwlks2onsym 30549 | There is a walk of length ... |
| elwwlks2on 30550 | A walk of length 2 between... |
| elwspths2on 30551 | A simple path of length 2 ... |
| elwspths2onw 30552 | A simple path of length 2 ... |
| wpthswwlks2on 30553 | For two different vertices... |
| 2wspdisj 30554 | All simple paths of length... |
| 2wspiundisj 30555 | All simple paths of length... |
| usgr2wspthons3 30556 | A simple path of length 2 ... |
| usgr2wspthon 30557 | A simple path of length 2 ... |
| elwwlks2 30558 | A walk of length 2 between... |
| elwspths2spth 30559 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30560 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30561 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30562 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30563 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30564 | Induction base 1 for ~ rus... |
| rusgr0edg 30565 | Special case for graphs wi... |
| rusgrnumwwlks 30566 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30567 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30568 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30569 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30570 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30571 | In a ` K `-regular graph, ... |
| clwwlk 30574 | The set of closed walks (i... |
| isclwwlk 30575 | Properties of a word to re... |
| clwwlkbp 30576 | Basic properties of a clos... |
| clwwlkgt0 30577 | There is no empty closed w... |
| clwwlksswrd 30578 | Closed walks (represented ... |
| clwwlk1loop 30579 | A closed walk of length 1 ... |
| clwwlkccatlem 30580 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30581 | The concatenation of two w... |
| umgrclwwlkge2 30582 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30583 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30584 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30585 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30586 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30587 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30588 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30589 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30590 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30591 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30592 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30593 | A closed walk as word of l... |
| clwlkclwwlk2 30594 | A closed walk corresponds ... |
| clwlkclwwlkflem 30595 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30596 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30597 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30598 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30599 | ` F ` is a function from t... |
| clwlkclwwlkfo 30600 | ` F ` is a function from t... |
| clwlkclwwlkf1 30601 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30602 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30603 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30604 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30605 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30606 | Cyclically shifting a clos... |
| clwwisshclwwsn 30607 | Cyclically shifting a clos... |
| erclwwlkrel 30608 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30609 | Two classes are equivalent... |
| erclwwlkeqlen 30610 | If two classes are equival... |
| erclwwlkref 30611 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30612 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30613 | ` .~ ` is a transitive rel... |
| erclwwlk 30614 | ` .~ ` is an equivalence r... |
| clwwlkn 30617 | The set of closed walks of... |
| isclwwlkn 30618 | A word over the set of ver... |
| clwwlkn0 30619 | There is no closed walk of... |
| clwwlkneq0 30620 | Sufficient conditions for ... |
| clwwlkclwwlkn 30621 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30622 | The closed walks of a fixe... |
| clwwlknlen 30623 | The length of a word repre... |
| clwwlknnn 30624 | The length of a closed wal... |
| clwwlknwrd 30625 | A closed walk of a fixed l... |
| clwwlknbp 30626 | Basic properties of a clos... |
| isclwwlknx 30627 | Characterization of a word... |
| clwwlknp 30628 | Properties of a set being ... |
| clwwlknwwlksn 30629 | A word representing a clos... |
| clwwlknlbonbgr1 30630 | The last but one vertex in... |
| clwwlkinwwlk 30631 | If the initial vertex of a... |
| clwwlkn1 30632 | A closed walk of length 1 ... |
| loopclwwlkn1b 30633 | The singleton word consist... |
| clwwlkn1loopb 30634 | A word represents a closed... |
| clwwlkn2 30635 | A closed walk of length 2 ... |
| clwwlknfi 30636 | If there is only a finite ... |
| clwwlkel 30637 | Obtaining a closed walk (a... |
| clwwlkf 30638 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30639 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30640 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30641 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30642 | F is a 1-1 onto function, ... |
| clwwlken 30643 | The set of closed walks of... |
| clwwlknwwlkncl 30644 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30645 | A nonempty word over verti... |
| clwwlknwwlksnb 30646 | A word over vertices repre... |
| clwwlkext2edg 30647 | If a word concatenated wit... |
| wwlksext2clwwlk 30648 | If a word represents a wal... |
| wwlksubclwwlk 30649 | Any prefix of a word repre... |
| clwwnisshclwwsn 30650 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30651 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30652 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30653 | The set of cyclical shifts... |
| clwwlknccat 30654 | The concatenation of two w... |
| umgr2cwwk2dif 30655 | If a word represents a clo... |
| umgr2cwwkdifex 30656 | If a word represents a clo... |
| erclwwlknrel 30657 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30658 | Two classes are equivalent... |
| erclwwlkneqlen 30659 | If two classes are equival... |
| erclwwlknref 30660 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30661 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30662 | ` .~ ` is a transitive rel... |
| erclwwlkn 30663 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30664 | The quotient set of the se... |
| hashclwwlkn0 30665 | The number of closed walks... |
| eclclwwlkn1 30666 | An equivalence class accor... |
| eleclclwwlkn 30667 | A member of an equivalence... |
| hashecclwwlkn1 30668 | The size of every equivale... |
| umgrhashecclwwlk 30669 | The size of every equivale... |
| fusgrhashclwwlkn 30670 | The size of the set of clo... |
| clwwlkndivn 30671 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30672 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30673 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30674 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30675 | There is a one-to-one onto... |
| clwlkssizeeq 30676 | The size of the set of clo... |
| clwlksndivn 30677 | The size of the set of clo... |
| clwwlknonmpo 30680 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30681 | The set of closed walks on... |
| isclwwlknon 30682 | A word over the set of ver... |
| clwwlk0on0 30683 | There is no word over the ... |
| clwwlknon0 30684 | Sufficient conditions for ... |
| clwwlknonfin 30685 | In a finite graph ` G ` , ... |
| clwwlknonel 30686 | Characterization of a word... |
| clwwlknonccat 30687 | The concatenation of two w... |
| clwwlknon1 30688 | The set of closed walks on... |
| clwwlknon1loop 30689 | If there is a loop at vert... |
| clwwlknon1nloop 30690 | If there is no loop at ver... |
| clwwlknon1sn 30691 | The set of (closed) walks ... |
| clwwlknon1le1 30692 | There is at most one (clos... |
| clwwlknon2 30693 | The set of closed walks on... |
| clwwlknon2x 30694 | The set of closed walks on... |
| s2elclwwlknon2 30695 | Sufficient conditions of a... |
| clwwlknon2num 30696 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30697 | A word over vertices repre... |
| clwwlknonex2lem1 30698 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30699 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30700 | Extending a closed walk ` ... |
| clwwlknonex2e 30701 | Extending a closed walk ` ... |
| clwwlknondisj 30702 | The sets of closed walks o... |
| clwwlknun 30703 | The set of closed walks of... |
| clwwlkvbij 30704 | There is a bijection betwe... |
| 0ewlk 30705 | The empty set (empty seque... |
| 1ewlk 30706 | A sequence of 1 edge is an... |
| 0wlk 30707 | A pair of an empty set (of... |
| is0wlk 30708 | A pair of an empty set (of... |
| 0wlkonlem1 30709 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30710 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30711 | A walk of length 0 from a ... |
| 0wlkons1 30712 | A walk of length 0 from a ... |
| 0trl 30713 | A pair of an empty set (of... |
| is0trl 30714 | A pair of an empty set (of... |
| 0trlon 30715 | A trail of length 0 from a... |
| 0pth 30716 | A pair of an empty set (of... |
| 0spth 30717 | A pair of an empty set (of... |
| 0pthon 30718 | A path of length 0 from a ... |
| 0pthon1 30719 | A path of length 0 from a ... |
| 0pthonv 30720 | For each vertex there is a... |
| 0clwlk 30721 | A pair of an empty set (of... |
| 0clwlkv 30722 | Any vertex (more precisely... |
| 0clwlk0 30723 | There is no closed walk in... |
| 0crct 30724 | A pair of an empty set (of... |
| 0cycl 30725 | A pair of an empty set (of... |
| 1pthdlem1 30726 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30727 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30728 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30729 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30730 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30731 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30732 | In a graph with two vertic... |
| 1trld 30733 | In a graph with two vertic... |
| 1pthd 30734 | In a graph with two vertic... |
| 1pthond 30735 | In a graph with two vertic... |
| upgr1wlkdlem1 30736 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30737 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30738 | In a pseudograph with two ... |
| upgr1trld 30739 | In a pseudograph with two ... |
| upgr1pthd 30740 | In a pseudograph with two ... |
| upgr1pthond 30741 | In a pseudograph with two ... |
| lppthon 30742 | A loop (which is an edge a... |
| lp1cycl 30743 | A loop (which is an edge a... |
| loop1cycl 30744 | A hypergraph has a cycle o... |
| 2cycld 30745 | Construction of a 2-cycle ... |
| umgr2cycllem 30746 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 30747 | A multigraph with two dist... |
| dfacycgr1 30750 | An alternate definition of... |
| isacycgr 30751 | The property of being an a... |
| isacycgr1 30752 | The property of being an a... |
| acycgrcycl 30753 | Any cycle in an acyclic gr... |
| 1pthon2v 30754 | For each pair of adjacent ... |
| 1pthon2ve 30755 | For each pair of adjacent ... |
| wlk2v2elem1 30756 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30757 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30758 | In a graph with two vertic... |
| ntrl2v2e 30759 | A walk which is not a trai... |
| 3wlkdlem1 30760 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30761 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30762 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30763 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30764 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30765 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30766 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30767 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30768 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30769 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30770 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30771 | Construction of a walk fro... |
| 3wlkond 30772 | A walk of length 3 from on... |
| 3trld 30773 | Construction of a trail fr... |
| 3trlond 30774 | A trail of length 3 from o... |
| 3pthd 30775 | A path of length 3 from on... |
| 3pthond 30776 | A path of length 3 from on... |
| 3spthd 30777 | A simple path of length 3 ... |
| 3spthond 30778 | A simple path of length 3 ... |
| 3cycld 30779 | Construction of a 3-cycle ... |
| 3cyclpd 30780 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30781 | If there is a cycle of len... |
| uhgr3cyclexlem 30782 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30783 | If there are three differe... |
| umgr3cyclex 30784 | If there are three (differ... |
| umgr3v3e3cycl 30785 | If and only if there is a ... |
| upgr4cycl4dv4e 30786 | If there is a cycle of len... |
| dfconngr1 30789 | Alternative definition of ... |
| isconngr 30790 | The property of being a co... |
| isconngr1 30791 | The property of being a co... |
| cusconngr 30792 | A complete hypergraph is c... |
| 0conngr 30793 | A graph without vertices i... |
| 0vconngr 30794 | A graph without vertices i... |
| 1conngr 30795 | A graph with (at most) one... |
| conngrv2edg 30796 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30797 | A vertex in a connected mu... |
| releupth 30800 | The set ` ( EulerPaths `` ... |
| eupths 30801 | The Eulerian paths on the ... |
| iseupth 30802 | The property " ` <. F , P ... |
| iseupthf1o 30803 | The property " ` <. F , P ... |
| eupthi 30804 | Properties of an Eulerian ... |
| eupthf1o 30805 | The ` F ` function in an E... |
| eupthfi 30806 | Any graph with an Eulerian... |
| eupthseg 30807 | The ` N ` -th edge in an e... |
| upgriseupth 30808 | The property " ` <. F , P ... |
| upgreupthi 30809 | Properties of an Eulerian ... |
| upgreupthseg 30810 | The ` N ` -th edge in an e... |
| eupthcl 30811 | An Eulerian path has lengt... |
| eupthistrl 30812 | An Eulerian path is a trai... |
| eupthiswlk 30813 | An Eulerian path is a walk... |
| eupthpf 30814 | The ` P ` function in an E... |
| eupth0 30815 | There is an Eulerian path ... |
| eupthres 30816 | The restriction ` <. H , Q... |
| eupthp1 30817 | Append one path segment to... |
| eupth2eucrct 30818 | Append one path segment to... |
| eupth2lem1 30819 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30820 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30821 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30822 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30823 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30824 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30825 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30826 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30827 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30828 | The effect on vertex degre... |
| eupth2lem3lem1 30829 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30830 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30831 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30832 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30833 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30834 | Formerly part of proof of ... |
| eupth2lem3lem7 30835 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30836 | Formerly part of proof of ... |
| eupth2lem3 30837 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30838 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30839 | Lemma for ~ eupth2 (induct... |
| eupth2 30840 | The only vertices of odd d... |
| eulerpathpr 30841 | A graph with an Eulerian p... |
| eulerpath 30842 | A pseudograph with an Eule... |
| eulercrct 30843 | A pseudograph with an Eule... |
| eucrctshift 30844 | Cyclically shifting the in... |
| eucrct2eupth1 30845 | Removing one edge ` ( I ``... |
| eucrct2eupth 30846 | Removing one edge ` ( I ``... |
| konigsbergvtx 30847 | The set of vertices of the... |
| konigsbergiedg 30848 | The indexed edges of the K... |
| konigsbergiedgw 30849 | The indexed edges of the K... |
| konigsbergssiedgwpr 30850 | Each subset of the indexed... |
| konigsbergssiedgw 30851 | Each subset of the indexed... |
| konigsbergumgr 30852 | The Königsberg graph ... |
| konigsberglem1 30853 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30854 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30855 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30856 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30857 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30858 | The Königsberg Bridge... |
| isfrgr 30861 | The property of being a fr... |
| frgrusgr 30862 | A friendship graph is a si... |
| frgr0v 30863 | Any null graph (set with n... |
| frgr0vb 30864 | Any null graph (without ve... |
| frgruhgr0v 30865 | Any null graph (without ve... |
| frgr0 30866 | The null graph (graph with... |
| frcond1 30867 | The friendship condition: ... |
| frcond2 30868 | The friendship condition: ... |
| frgreu 30869 | Variant of ~ frcond2 : An... |
| frcond3 30870 | The friendship condition, ... |
| frcond4 30871 | The friendship condition, ... |
| frgr1v 30872 | Any graph with (at most) o... |
| nfrgr2v 30873 | Any graph with two (differ... |
| frgr3vlem1 30874 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30875 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30876 | Any graph with three verti... |
| 1vwmgr 30877 | Every graph with one verte... |
| 3vfriswmgrlem 30878 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30879 | Every friendship graph wit... |
| 1to2vfriswmgr 30880 | Every friendship graph wit... |
| 1to3vfriswmgr 30881 | Every friendship graph wit... |
| 1to3vfriendship 30882 | The friendship theorem for... |
| 2pthfrgrrn 30883 | Between any two (different... |
| 2pthfrgrrn2 30884 | Between any two (different... |
| 2pthfrgr 30885 | Between any two (different... |
| 3cyclfrgrrn1 30886 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30887 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30888 | Every vertex in a friendsh... |
| 3cyclfrgr 30889 | Every vertex in a friendsh... |
| 4cycl2v2nb 30890 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30891 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30892 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30893 | A graph with a 4-cycle is ... |
| frgrnbnb 30894 | If two neighbors ` U ` and... |
| frgrconngr 30895 | A friendship graph is conn... |
| vdgn0frgrv2 30896 | A vertex in a friendship g... |
| vdgn1frgrv2 30897 | Any vertex in a friendship... |
| vdgn1frgrv3 30898 | Any vertex in a friendship... |
| vdgfrgrgt2 30899 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30900 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30901 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30902 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30903 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30904 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30905 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30906 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30907 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30908 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30909 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30910 | In a friendship graph, two... |
| frgrwopreglem4a 30911 | In a friendship graph any ... |
| frgrwopreglem5a 30912 | If a friendship graph has ... |
| frgrwopreglem1 30913 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30914 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30915 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30916 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30917 | According to statement 5 i... |
| frgrwopregbsn 30918 | According to statement 5 i... |
| frgrwopreg1 30919 | According to statement 5 i... |
| frgrwopreg2 30920 | According to statement 5 i... |
| frgrwopreglem5lem 30921 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30922 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30923 | Alternate direct proof of ... |
| frgrwopreg 30924 | In a friendship graph ther... |
| frgrregorufr0 30925 | In a friendship graph ther... |
| frgrregorufr 30926 | If there is a vertex havin... |
| frgrregorufrg 30927 | If there is a vertex havin... |
| frgr2wwlkeu 30928 | For two different vertices... |
| frgr2wwlkn0 30929 | In a friendship graph, the... |
| frgr2wwlk1 30930 | In a friendship graph, the... |
| frgr2wsp1 30931 | In a friendship graph, the... |
| frgr2wwlkeqm 30932 | If there is a (simple) pat... |
| frgrhash2wsp 30933 | The number of simple paths... |
| fusgreg2wsplem 30934 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30935 | The set of paths of length... |
| fusgreghash2wspv 30936 | According to statement 7 i... |
| fusgreg2wsp 30937 | In a finite simple graph, ... |
| 2wspmdisj 30938 | The sets of paths of lengt... |
| fusgreghash2wsp 30939 | In a finite k-regular grap... |
| frrusgrord0lem 30940 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30941 | If a nonempty finite frien... |
| frrusgrord 30942 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30943 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30944 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30945 | If the initial vertex of a... |
| clwwnonrepclwwnon 30946 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30947 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30948 | Value of operation ` C ` ,... |
| 2clwwlk2 30949 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30950 | Characterization of an ele... |
| 2clwwlk2clwwlk 30951 | An element of the value of... |
| numclwwlk1lem2foalem 30952 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30953 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30954 | Characterization of an ele... |
| numclwwlk1lem2foa 30955 | Going forth and back from ... |
| numclwwlk1lem2f 30956 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30957 | Value of the function ` T ... |
| numclwwlk1lem2f1 30958 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30959 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30960 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30961 | The set of double loops of... |
| numclwwlk1 30962 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30963 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30964 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30965 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30966 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30967 | The sets of the two repres... |
| wlkl0 30968 | There is exactly one walk ... |
| clwlknon2num 30969 | There are k walks of lengt... |
| numclwlk1lem1 30970 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30971 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30972 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30973 | Value of operation ` H ` ,... |
| numclwwlkovh 30974 | Value of operation ` H ` ,... |
| numclwwlkovq 30975 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30976 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30977 | In a friendship graph, for... |
| numclwlk2lem2f 30978 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30979 | Value of the function ` R ... |
| numclwlk2lem2f1o 30980 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30981 | In a friendship graph, the... |
| numclwwlk2 30982 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30983 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30984 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30985 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30986 | Statement 12 in [Huneke] p... |
| numclwwlk4 30987 | The total number of closed... |
| numclwwlk5lem 30988 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30989 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30990 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30991 | For a prime divisor ` P ` ... |
| numclwwlk7 30992 | Statement 14 in [Huneke] p... |
| numclwwlk8 30993 | The size of the set of clo... |
| frgrreggt1 30994 | If a finite nonempty frien... |
| frgrreg 30995 | If a finite nonempty frien... |
| frgrregord013 30996 | If a finite friendship gra... |
| frgrregord13 30997 | If a nonempty finite frien... |
| frgrogt3nreg 30998 | If a finite friendship gra... |
| friendshipgt3 30999 | The friendship theorem for... |
| friendship 31000 | The friendship theorem: I... |
| conventions 31001 |
H... |
| conventions-labels 31002 |
... |
| conventions-comments 31003 |
... |
| natded 31004 | Here are typical n... |
| ex-natded5.2 31005 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 31006 | A more efficient proof of ... |
| ex-natded5.2i 31007 | The same as ~ ex-natded5.2... |
| ex-natded5.3 31008 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 31009 | A more efficient proof of ... |
| ex-natded5.3i 31010 | The same as ~ ex-natded5.3... |
| ex-natded5.5 31011 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 31012 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 31013 | A more efficient proof of ... |
| ex-natded5.8 31014 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 31015 | A more efficient proof of ... |
| ex-natded5.13 31016 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 31017 | A more efficient proof of ... |
| ex-natded9.20 31018 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 31019 | A more efficient proof of ... |
| ex-natded9.26 31020 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 31021 | A more efficient proof of ... |
| ex-or 31022 | Example for ~ df-or . Exa... |
| ex-an 31023 | Example for ~ df-an . Exa... |
| ex-dif 31024 | Example for ~ df-dif . Ex... |
| ex-un 31025 | Example for ~ df-un . Exa... |
| ex-in 31026 | Example for ~ df-in . Exa... |
| ex-uni 31027 | Example for ~ df-uni . Ex... |
| ex-ss 31028 | Example for ~ df-ss . Exa... |
| ex-pss 31029 | Example for ~ df-pss . Ex... |
| ex-pw 31030 | Example for ~ df-pw . Exa... |
| ex-pr 31031 | Example for ~ df-pr . (Co... |
| ex-br 31032 | Example for ~ df-br . Exa... |
| ex-opab 31033 | Example for ~ df-opab . E... |
| ex-eprel 31034 | Example for ~ df-eprel . ... |
| ex-id 31035 | Example for ~ df-id . Exa... |
| ex-po 31036 | Example for ~ df-po . Exa... |
| ex-xp 31037 | Example for ~ df-xp . Exa... |
| ex-cnv 31038 | Example for ~ df-cnv . Ex... |
| ex-co 31039 | Example for ~ df-co . Exa... |
| ex-dm 31040 | Example for ~ df-dm . Exa... |
| ex-rn 31041 | Example for ~ df-rn . Exa... |
| ex-res 31042 | Example for ~ df-res . Ex... |
| ex-ima 31043 | Example for ~ df-ima . Ex... |
| ex-fv 31044 | Example for ~ df-fv . Exa... |
| ex-1st 31045 | Example for ~ df-1st . Ex... |
| ex-2nd 31046 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 31047 | Example for ~ df-dec , 100... |
| ex-fl 31048 | Example for ~ df-fl . Exa... |
| ex-ceil 31049 | Example for ~ df-ceil . (... |
| ex-mod 31050 | Example for ~ df-mod . (C... |
| ex-exp 31051 | Example for ~ df-exp . (C... |
| ex-fac 31052 | Example for ~ df-fac . (C... |
| ex-bc 31053 | Example for ~ df-bc . (Co... |
| ex-hash 31054 | Example for ~ df-hash . (... |
| ex-sqrt 31055 | Example for ~ df-sqrt . (... |
| ex-abs 31056 | Example for ~ df-abs . (C... |
| ex-dvds 31057 | Example for ~ df-dvds : 3 ... |
| ex-gcd 31058 | Example for ~ df-gcd . (C... |
| ex-lcm 31059 | Example for ~ df-lcm . (C... |
| ex-prmo 31060 | Example for ~ df-prmo : ` ... |
| aevdemo 31061 | Proof illustrating the com... |
| ex-ind-dvds 31062 | Example of a proof by indu... |
| ex-fpar 31063 | Formalized example provide... |
| avril1 31064 | Poisson d'Avril's Theorem.... |
| 2bornot2b 31065 | The law of excluded middle... |
| helloworld 31066 | The classic "Hello world" ... |
| 1p1e2apr1 31067 | One plus one equals two. ... |
| eqid1 31068 | Law of identity (reflexivi... |
| 1div0apr 31069 | Division by zero is forbid... |
| topnfbey 31070 | Nothing seems to be imposs... |
| 9p10ne21 31071 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 31072 | 9 + 10 equals 21. This as... |
| nrt2irr 31074 | The ` N ` -th root of 2 is... |
| nowisdomv 31075 | One's wisdom on matters of... |
| isplig 31078 | The predicate "is a planar... |
| ispligb 31079 | The predicate "is a planar... |
| tncp 31080 | In any planar incidence ge... |
| l2p 31081 | For any line in a planar i... |
| lpni 31082 | For any line in a planar i... |
| nsnlplig 31083 | There is no "one-point lin... |
| nsnlpligALT 31084 | Alternate version of ~ nsn... |
| n0lplig 31085 | There is no "empty line" i... |
| n0lpligALT 31086 | Alternate version of ~ n0l... |
| eulplig 31087 | Through two distinct point... |
| pliguhgr 31088 | Any planar incidence geome... |
| dummylink 31089 | Alias for ~ a1ii that may ... |
| id1 31090 | Alias for ~ idALT that may... |
| isgrpo 31099 | The predicate "is a group ... |
| isgrpoi 31100 | Properties that determine ... |
| grpofo 31101 | A group operation maps ont... |
| grpocl 31102 | Closure law for a group op... |
| grpolidinv 31103 | A group has a left identit... |
| grpon0 31104 | The base set of a group is... |
| grpoass 31105 | A group operation is assoc... |
| grpoidinvlem1 31106 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 31107 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 31108 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 31109 | Lemma for ~ grpoidinv . (... |
| grpoidinv 31110 | A group has a left and rig... |
| grpoideu 31111 | The left identity element ... |
| grporndm 31112 | A group's range in terms o... |
| 0ngrp 31113 | The empty set is not a gro... |
| gidval 31114 | The value of the identity ... |
| grpoidval 31115 | Lemma for ~ grpoidcl and o... |
| grpoidcl 31116 | The identity element of a ... |
| grpoidinv2 31117 | A group's properties using... |
| grpolid 31118 | The identity element of a ... |
| grporid 31119 | The identity element of a ... |
| grporcan 31120 | Right cancellation law for... |
| grpoinveu 31121 | The left inverse element o... |
| grpoid 31122 | Two ways of saying that an... |
| grporn 31123 | The range of a group opera... |
| grpoinvfval 31124 | The inverse function of a ... |
| grpoinvval 31125 | The inverse of a group ele... |
| grpoinvcl 31126 | A group element's inverse ... |
| grpoinv 31127 | The properties of a group ... |
| grpolinv 31128 | The left inverse of a grou... |
| grporinv 31129 | The right inverse of a gro... |
| grpoinvid1 31130 | The inverse of a group ele... |
| grpoinvid2 31131 | The inverse of a group ele... |
| grpolcan 31132 | Left cancellation law for ... |
| grpo2inv 31133 | Double inverse law for gro... |
| grpoinvf 31134 | Mapping of the inverse fun... |
| grpoinvop 31135 | The inverse of the group o... |
| grpodivfval 31136 | Group division (or subtrac... |
| grpodivval 31137 | Group division (or subtrac... |
| grpodivinv 31138 | Group division by an inver... |
| grpoinvdiv 31139 | Inverse of a group divisio... |
| grpodivf 31140 | Mapping for group division... |
| grpodivcl 31141 | Closure of group division ... |
| grpodivdiv 31142 | Double group division. (C... |
| grpomuldivass 31143 | Associative-type law for m... |
| grpodivid 31144 | Division of a group member... |
| grponpcan 31145 | Cancellation law for group... |
| isablo 31148 | The predicate "is an Abeli... |
| ablogrpo 31149 | An Abelian group operation... |
| ablocom 31150 | An Abelian group operation... |
| ablo32 31151 | Commutative/associative la... |
| ablo4 31152 | Commutative/associative la... |
| isabloi 31153 | Properties that determine ... |
| ablomuldiv 31154 | Law for group multiplicati... |
| ablodivdiv 31155 | Law for double group divis... |
| ablodivdiv4 31156 | Law for double group divis... |
| ablodiv32 31157 | Swap the second and third ... |
| ablonncan 31158 | Cancellation law for group... |
| ablonnncan1 31159 | Cancellation law for group... |
| vcrel 31162 | The class of all complex v... |
| vciOLD 31163 | Obsolete version of ~ cvsi... |
| vcsm 31164 | Functionality of the scala... |
| vccl 31165 | Closure of the scalar prod... |
| vcidOLD 31166 | Identity element for the s... |
| vcdi 31167 | Distributive law for the s... |
| vcdir 31168 | Distributive law for the s... |
| vcass 31169 | Associative law for the sc... |
| vc2OLD 31170 | A vector plus itself is tw... |
| vcablo 31171 | Vector addition is an Abel... |
| vcgrp 31172 | Vector addition is a group... |
| vclcan 31173 | Left cancellation law for ... |
| vczcl 31174 | The zero vector is a vecto... |
| vc0rid 31175 | The zero vector is a right... |
| vc0 31176 | Zero times a vector is the... |
| vcz 31177 | Anything times the zero ve... |
| vcm 31178 | Minus 1 times a vector is ... |
| isvclem 31179 | Lemma for ~ isvcOLD . (Co... |
| vcex 31180 | The components of a comple... |
| isvcOLD 31181 | The predicate "is a comple... |
| isvciOLD 31182 | Properties that determine ... |
| cnaddabloOLD 31183 | Obsolete version of ~ cnad... |
| cnidOLD 31184 | Obsolete version of ~ cnad... |
| cncvcOLD 31185 | Obsolete version of ~ cncv... |
| nvss 31195 | Structure of the class of ... |
| nvvcop 31196 | A normed complex vector sp... |
| nvrel 31204 | The class of all normed co... |
| vafval 31205 | Value of the function for ... |
| bafval 31206 | Value of the function for ... |
| smfval 31207 | Value of the function for ... |
| 0vfval 31208 | Value of the function for ... |
| nmcvfval 31209 | Value of the norm function... |
| nvop2 31210 | A normed complex vector sp... |
| nvvop 31211 | The vector space component... |
| isnvlem 31212 | Lemma for ~ isnv . (Contr... |
| nvex 31213 | The components of a normed... |
| isnv 31214 | The predicate "is a normed... |
| isnvi 31215 | Properties that determine ... |
| nvi 31216 | The properties of a normed... |
| nvvc 31217 | The vector space component... |
| nvablo 31218 | The vector addition operat... |
| nvgrp 31219 | The vector addition operat... |
| nvgf 31220 | Mapping for the vector add... |
| nvsf 31221 | Mapping for the scalar mul... |
| nvgcl 31222 | Closure law for the vector... |
| nvcom 31223 | The vector addition (group... |
| nvass 31224 | The vector addition (group... |
| nvadd32 31225 | Commutative/associative la... |
| nvrcan 31226 | Right cancellation law for... |
| nvadd4 31227 | Rearrangement of 4 terms i... |
| nvscl 31228 | Closure law for the scalar... |
| nvsid 31229 | Identity element for the s... |
| nvsass 31230 | Associative law for the sc... |
| nvscom 31231 | Commutative law for the sc... |
| nvdi 31232 | Distributive law for the s... |
| nvdir 31233 | Distributive law for the s... |
| nv2 31234 | A vector plus itself is tw... |
| vsfval 31235 | Value of the function for ... |
| nvzcl 31236 | Closure law for the zero v... |
| nv0rid 31237 | The zero vector is a right... |
| nv0lid 31238 | The zero vector is a left ... |
| nv0 31239 | Zero times a vector is the... |
| nvsz 31240 | Anything times the zero ve... |
| nvinv 31241 | Minus 1 times a vector is ... |
| nvinvfval 31242 | Function for the negative ... |
| nvm 31243 | Vector subtraction in term... |
| nvmval 31244 | Value of vector subtractio... |
| nvmval2 31245 | Value of vector subtractio... |
| nvmfval 31246 | Value of the function for ... |
| nvmf 31247 | Mapping for the vector sub... |
| nvmcl 31248 | Closure law for the vector... |
| nvnnncan1 31249 | Cancellation law for vecto... |
| nvmdi 31250 | Distributive law for scala... |
| nvnegneg 31251 | Double negative of a vecto... |
| nvmul0or 31252 | If a scalar product is zer... |
| nvrinv 31253 | A vector minus itself. (C... |
| nvlinv 31254 | Minus a vector plus itself... |
| nvpncan2 31255 | Cancellation law for vecto... |
| nvpncan 31256 | Cancellation law for vecto... |
| nvaddsub 31257 | Commutative/associative la... |
| nvnpcan 31258 | Cancellation law for a nor... |
| nvaddsub4 31259 | Rearrangement of 4 terms i... |
| nvmeq0 31260 | The difference between two... |
| nvmid 31261 | A vector minus itself is t... |
| nvf 31262 | Mapping for the norm funct... |
| nvcl 31263 | The norm of a normed compl... |
| nvcli 31264 | The norm of a normed compl... |
| nvs 31265 | Proportionality property o... |
| nvsge0 31266 | The norm of a scalar produ... |
| nvm1 31267 | The norm of the negative o... |
| nvdif 31268 | The norm of the difference... |
| nvpi 31269 | The norm of a vector plus ... |
| nvz0 31270 | The norm of a zero vector ... |
| nvz 31271 | The norm of a vector is ze... |
| nvtri 31272 | Triangle inequality for th... |
| nvmtri 31273 | Triangle inequality for th... |
| nvabs 31274 | Norm difference property o... |
| nvge0 31275 | The norm of a normed compl... |
| nvgt0 31276 | A nonzero norm is positive... |
| nv1 31277 | From any nonzero vector, c... |
| nvop 31278 | A complex inner product sp... |
| cnnv 31279 | The set of complex numbers... |
| cnnvg 31280 | The vector addition (group... |
| cnnvba 31281 | The base set of the normed... |
| cnnvs 31282 | The scalar product operati... |
| cnnvnm 31283 | The norm operation of the ... |
| cnnvm 31284 | The vector subtraction ope... |
| elimnv 31285 | Hypothesis elimination lem... |
| elimnvu 31286 | Hypothesis elimination lem... |
| imsval 31287 | Value of the induced metri... |
| imsdval 31288 | Value of the induced metri... |
| imsdval2 31289 | Value of the distance func... |
| nvnd 31290 | The norm of a normed compl... |
| imsdf 31291 | Mapping for the induced me... |
| imsmetlem 31292 | Lemma for ~ imsmet . (Con... |
| imsmet 31293 | The induced metric of a no... |
| imsxmet 31294 | The induced metric of a no... |
| cnims 31295 | The metric induced on the ... |
| vacn 31296 | Vector addition is jointly... |
| nmcvcn 31297 | The norm of a normed compl... |
| nmcnc 31298 | The norm of a normed compl... |
| smcnlem 31299 | Lemma for ~ smcn . (Contr... |
| smcn 31300 | Scalar multiplication is j... |
| vmcn 31301 | Vector subtraction is join... |
| dipfval 31304 | The inner product function... |
| ipval 31305 | Value of the inner product... |
| ipval2lem2 31306 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 31307 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 31308 | Lemma for ~ ipval3 . (Con... |
| ipval2 31309 | Expansion of the inner pro... |
| 4ipval2 31310 | Four times the inner produ... |
| ipval3 31311 | Expansion of the inner pro... |
| ipidsq 31312 | The inner product of a vec... |
| ipnm 31313 | Norm expressed in terms of... |
| dipcl 31314 | An inner product is a comp... |
| ipf 31315 | Mapping for the inner prod... |
| dipcj 31316 | The complex conjugate of a... |
| ipipcj 31317 | An inner product times its... |
| diporthcom 31318 | Orthogonality (meaning inn... |
| dip0r 31319 | Inner product with a zero ... |
| dip0l 31320 | Inner product with a zero ... |
| ipz 31321 | The inner product of a vec... |
| dipcn 31322 | Inner product is jointly c... |
| sspval 31325 | The set of all subspaces o... |
| isssp 31326 | The predicate "is a subspa... |
| sspid 31327 | A normed complex vector sp... |
| sspnv 31328 | A subspace is a normed com... |
| sspba 31329 | The base set of a subspace... |
| sspg 31330 | Vector addition on a subsp... |
| sspgval 31331 | Vector addition on a subsp... |
| ssps 31332 | Scalar multiplication on a... |
| sspsval 31333 | Scalar multiplication on a... |
| sspmlem 31334 | Lemma for ~ sspm and other... |
| sspmval 31335 | Vector addition on a subsp... |
| sspm 31336 | Vector subtraction on a su... |
| sspz 31337 | The zero vector of a subsp... |
| sspn 31338 | The norm on a subspace is ... |
| sspnval 31339 | The norm on a subspace in ... |
| sspimsval 31340 | The induced metric on a su... |
| sspims 31341 | The induced metric on a su... |
| lnoval 31354 | The set of linear operator... |
| islno 31355 | The predicate "is a linear... |
| lnolin 31356 | Basic linearity property o... |
| lnof 31357 | A linear operator is a map... |
| lno0 31358 | The value of a linear oper... |
| lnocoi 31359 | The composition of two lin... |
| lnoadd 31360 | Addition property of a lin... |
| lnosub 31361 | Subtraction property of a ... |
| lnomul 31362 | Scalar multiplication prop... |
| nvo00 31363 | Two ways to express a zero... |
| nmoofval 31364 | The operator norm function... |
| nmooval 31365 | The operator norm function... |
| nmosetre 31366 | The set in the supremum of... |
| nmosetn0 31367 | The set in the supremum of... |
| nmoxr 31368 | The norm of an operator is... |
| nmooge0 31369 | The norm of an operator is... |
| nmorepnf 31370 | The norm of an operator is... |
| nmoreltpnf 31371 | The norm of any operator i... |
| nmogtmnf 31372 | The norm of an operator is... |
| nmoolb 31373 | A lower bound for an opera... |
| nmoubi 31374 | An upper bound for an oper... |
| nmoub3i 31375 | An upper bound for an oper... |
| nmoub2i 31376 | An upper bound for an oper... |
| nmobndi 31377 | Two ways to express that a... |
| nmounbi 31378 | Two ways two express that ... |
| nmounbseqi 31379 | An unbounded operator dete... |
| nmounbseqiALT 31380 | Alternate shorter proof of... |
| nmobndseqi 31381 | A bounded sequence determi... |
| nmobndseqiALT 31382 | Alternate shorter proof of... |
| bloval 31383 | The class of bounded linea... |
| isblo 31384 | The predicate "is a bounde... |
| isblo2 31385 | The predicate "is a bounde... |
| bloln 31386 | A bounded operator is a li... |
| blof 31387 | A bounded operator is an o... |
| nmblore 31388 | The norm of a bounded oper... |
| 0ofval 31389 | The zero operator between ... |
| 0oval 31390 | Value of the zero operator... |
| 0oo 31391 | The zero operator is an op... |
| 0lno 31392 | The zero operator is linea... |
| nmoo0 31393 | The operator norm of the z... |
| 0blo 31394 | The zero operator is a bou... |
| nmlno0lem 31395 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 31396 | The norm of a linear opera... |
| nmlno0 31397 | The norm of a linear opera... |
| nmlnoubi 31398 | An upper bound for the ope... |
| nmlnogt0 31399 | The norm of a nonzero line... |
| lnon0 31400 | The domain of a nonzero li... |
| nmblolbii 31401 | A lower bound for the norm... |
| nmblolbi 31402 | A lower bound for the norm... |
| isblo3i 31403 | The predicate "is a bounde... |
| blo3i 31404 | Properties that determine ... |
| blometi 31405 | Upper bound for the distan... |
| blocnilem 31406 | Lemma for ~ blocni and ~ l... |
| blocni 31407 | A linear operator is conti... |
| lnocni 31408 | If a linear operator is co... |
| blocn 31409 | A linear operator is conti... |
| blocn2 31410 | A bounded linear operator ... |
| ajfval 31411 | The adjoint function. (Co... |
| hmoval 31412 | The set of Hermitian (self... |
| ishmo 31413 | The predicate "is a hermit... |
| phnv 31416 | Every complex inner produc... |
| phrel 31417 | The class of all complex i... |
| phnvi 31418 | Every complex inner produc... |
| isphg 31419 | The predicate "is a comple... |
| phop 31420 | A complex inner product sp... |
| cncph 31421 | The set of complex numbers... |
| elimph 31422 | Hypothesis elimination lem... |
| elimphu 31423 | Hypothesis elimination lem... |
| isph 31424 | The predicate "is an inner... |
| phpar2 31425 | The parallelogram law for ... |
| phpar 31426 | The parallelogram law for ... |
| ip0i 31427 | A slight variant of Equati... |
| ip1ilem 31428 | Lemma for ~ ip1i . (Contr... |
| ip1i 31429 | Equation 6.47 of [Ponnusam... |
| ip2i 31430 | Equation 6.48 of [Ponnusam... |
| ipdirilem 31431 | Lemma for ~ ipdiri . (Con... |
| ipdiri 31432 | Distributive law for inner... |
| ipasslem1 31433 | Lemma for ~ ipassi . Show... |
| ipasslem2 31434 | Lemma for ~ ipassi . Show... |
| ipasslem3 31435 | Lemma for ~ ipassi . Show... |
| ipasslem4 31436 | Lemma for ~ ipassi . Show... |
| ipasslem5 31437 | Lemma for ~ ipassi . Show... |
| ipasslem7 31438 | Lemma for ~ ipassi . Show... |
| ipasslem8 31439 | Lemma for ~ ipassi . By ~... |
| ipasslem9 31440 | Lemma for ~ ipassi . Conc... |
| ipasslem10 31441 | Lemma for ~ ipassi . Show... |
| ipasslem11 31442 | Lemma for ~ ipassi . Show... |
| ipassi 31443 | Associative law for inner ... |
| dipdir 31444 | Distributive law for inner... |
| dipdi 31445 | Distributive law for inner... |
| ip2dii 31446 | Inner product of two sums.... |
| dipass 31447 | Associative law for inner ... |
| dipassr 31448 | "Associative" law for seco... |
| dipassr2 31449 | "Associative" law for inne... |
| dipsubdir 31450 | Distributive law for inner... |
| dipsubdi 31451 | Distributive law for inner... |
| pythi 31452 | The Pythagorean theorem fo... |
| siilem1 31453 | Lemma for ~ sii . (Contri... |
| siilem2 31454 | Lemma for ~ sii . (Contri... |
| siii 31455 | Inference from ~ sii . (C... |
| sii 31456 | Obsolete version of ~ ipca... |
| ipblnfi 31457 | A function ` F ` generated... |
| ip2eqi 31458 | Two vectors are equal iff ... |
| phoeqi 31459 | A condition implying that ... |
| ajmoi 31460 | Every operator has at most... |
| ajfuni 31461 | The adjoint function is a ... |
| ajfun 31462 | The adjoint function is a ... |
| ajval 31463 | Value of the adjoint funct... |
| iscbn 31466 | A complex Banach space is ... |
| cbncms 31467 | The induced metric on comp... |
| bnnv 31468 | Every complex Banach space... |
| bnrel 31469 | The class of all complex B... |
| bnsscmcl 31470 | A subspace of a Banach spa... |
| cnbn 31471 | The set of complex numbers... |
| ubthlem1 31472 | Lemma for ~ ubth . The fu... |
| ubthlem2 31473 | Lemma for ~ ubth . Given ... |
| ubthlem3 31474 | Lemma for ~ ubth . Prove ... |
| ubth 31475 | Uniform Boundedness Theore... |
| minvecolem1 31476 | Lemma for ~ minveco . The... |
| minvecolem2 31477 | Lemma for ~ minveco . Any... |
| minvecolem3 31478 | Lemma for ~ minveco . The... |
| minvecolem4a 31479 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 31480 | Lemma for ~ minveco . The... |
| minvecolem4c 31481 | Lemma for ~ minveco . The... |
| minvecolem4 31482 | Lemma for ~ minveco . The... |
| minvecolem5 31483 | Lemma for ~ minveco . Dis... |
| minvecolem6 31484 | Lemma for ~ minveco . Any... |
| minvecolem7 31485 | Lemma for ~ minveco . Sin... |
| minveco 31486 | Minimizing vector theorem,... |
| ishlo 31489 | The predicate "is a comple... |
| hlobn 31490 | Every complex Hilbert spac... |
| hlph 31491 | Every complex Hilbert spac... |
| hlrel 31492 | The class of all complex H... |
| hlnv 31493 | Every complex Hilbert spac... |
| hlnvi 31494 | Every complex Hilbert spac... |
| hlvc 31495 | Every complex Hilbert spac... |
| hlcmet 31496 | The induced metric on a co... |
| hlmet 31497 | The induced metric on a co... |
| hlpar2 31498 | The parallelogram law sati... |
| hlpar 31499 | The parallelogram law sati... |
| hlex 31500 | The base set of a Hilbert ... |
| hladdf 31501 | Mapping for Hilbert space ... |
| hlcom 31502 | Hilbert space vector addit... |
| hlass 31503 | Hilbert space vector addit... |
| hl0cl 31504 | The Hilbert space zero vec... |
| hladdid 31505 | Hilbert space addition wit... |
| hlmulf 31506 | Mapping for Hilbert space ... |
| hlmulid 31507 | Hilbert space scalar multi... |
| hlmulass 31508 | Hilbert space scalar multi... |
| hldi 31509 | Hilbert space scalar multi... |
| hldir 31510 | Hilbert space scalar multi... |
| hlmul0 31511 | Hilbert space scalar multi... |
| hlipf 31512 | Mapping for Hilbert space ... |
| hlipcj 31513 | Conjugate law for Hilbert ... |
| hlipdir 31514 | Distributive law for Hilbe... |
| hlipass 31515 | Associative law for Hilber... |
| hlipgt0 31516 | The inner product of a Hil... |
| hlcompl 31517 | Completeness of a Hilbert ... |
| cnchl 31518 | The set of complex numbers... |
| htthlem 31519 | Lemma for ~ htth . The co... |
| htth 31520 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31576 | The group (addition) opera... |
| h2hsm 31577 | The scalar product operati... |
| h2hnm 31578 | The norm function of Hilbe... |
| h2hvs 31579 | The vector subtraction ope... |
| h2hmetdval 31580 | Value of the distance func... |
| h2hcau 31581 | The Cauchy sequences of Hi... |
| h2hlm 31582 | The limit sequences of Hil... |
| axhilex-zf 31583 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31584 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31585 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31586 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31587 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31588 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31589 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31590 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31591 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31592 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31593 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31594 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31595 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31596 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31597 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31598 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31599 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31600 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31613 | The Hilbert space scalar p... |
| hvaddcl 31614 | Closure of vector addition... |
| hvmulcl 31615 | Closure of scalar multipli... |
| hvmulcli 31616 | Closure inference for scal... |
| hvsubf 31617 | Mapping domain and codomai... |
| hvsubval 31618 | Value of vector subtractio... |
| hvsubcl 31619 | Closure of vector subtract... |
| hvaddcli 31620 | Closure of vector addition... |
| hvcomi 31621 | Commutation of vector addi... |
| hvsubvali 31622 | Value of vector subtractio... |
| hvsubcli 31623 | Closure of vector subtract... |
| ifhvhv0 31624 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31625 | Addition with the zero vec... |
| hvmul0 31626 | Scalar multiplication with... |
| hvmul0or 31627 | If a scalar product is zer... |
| hvsubid 31628 | Subtraction of a vector fr... |
| hvnegid 31629 | Addition of negative of a ... |
| hv2neg 31630 | Two ways to express the ne... |
| hvaddlidi 31631 | Addition with the zero vec... |
| hvnegidi 31632 | Addition of negative of a ... |
| hv2negi 31633 | Two ways to express the ne... |
| hvm1neg 31634 | Convert minus one times a ... |
| hvaddsubval 31635 | Value of vector addition i... |
| hvadd32 31636 | Commutative/associative la... |
| hvadd12 31637 | Commutative/associative la... |
| hvadd4 31638 | Hilbert vector space addit... |
| hvsub4 31639 | Hilbert vector space addit... |
| hvaddsub12 31640 | Commutative/associative la... |
| hvpncan 31641 | Addition/subtraction cance... |
| hvpncan2 31642 | Addition/subtraction cance... |
| hvaddsubass 31643 | Associativity of sum and d... |
| hvpncan3 31644 | Subtraction and addition o... |
| hvmulcom 31645 | Scalar multiplication comm... |
| hvsubass 31646 | Hilbert vector space assoc... |
| hvsub32 31647 | Hilbert vector space commu... |
| hvmulassi 31648 | Scalar multiplication asso... |
| hvmulcomi 31649 | Scalar multiplication comm... |
| hvmul2negi 31650 | Double negative in scalar ... |
| hvsubdistr1 31651 | Scalar multiplication dist... |
| hvsubdistr2 31652 | Scalar multiplication dist... |
| hvdistr1i 31653 | Scalar multiplication dist... |
| hvsubdistr1i 31654 | Scalar multiplication dist... |
| hvassi 31655 | Hilbert vector space assoc... |
| hvadd32i 31656 | Hilbert vector space commu... |
| hvsubassi 31657 | Hilbert vector space assoc... |
| hvsub32i 31658 | Hilbert vector space commu... |
| hvadd12i 31659 | Hilbert vector space commu... |
| hvadd4i 31660 | Hilbert vector space addit... |
| hvsubsub4i 31661 | Hilbert vector space addit... |
| hvsubsub4 31662 | Hilbert vector space addit... |
| hv2times 31663 | Two times a vector. (Cont... |
| hvnegdii 31664 | Distribution of negative o... |
| hvsubeq0i 31665 | If the difference between ... |
| hvsubcan2i 31666 | Vector cancellation law. ... |
| hvaddcani 31667 | Cancellation law for vecto... |
| hvsubaddi 31668 | Relationship between vecto... |
| hvnegdi 31669 | Distribution of negative o... |
| hvsubeq0 31670 | If the difference between ... |
| hvaddeq0 31671 | If the sum of two vectors ... |
| hvaddcan 31672 | Cancellation law for vecto... |
| hvaddcan2 31673 | Cancellation law for vecto... |
| hvmulcan 31674 | Cancellation law for scala... |
| hvmulcan2 31675 | Cancellation law for scala... |
| hvsubcan 31676 | Cancellation law for vecto... |
| hvsubcan2 31677 | Cancellation law for vecto... |
| hvsub0 31678 | Subtraction of a zero vect... |
| hvsubadd 31679 | Relationship between vecto... |
| hvaddsub4 31680 | Hilbert vector space addit... |
| hicl 31682 | Closure of inner product. ... |
| hicli 31683 | Closure inference for inne... |
| his5 31688 | Associative law for inner ... |
| his52 31689 | Associative law for inner ... |
| his35 31690 | Move scalar multiplication... |
| his35i 31691 | Move scalar multiplication... |
| his7 31692 | Distributive law for inner... |
| hiassdi 31693 | Distributive/associative l... |
| his2sub 31694 | Distributive law for inner... |
| his2sub2 31695 | Distributive law for inner... |
| hire 31696 | A necessary and sufficient... |
| hiidrcl 31697 | Real closure of inner prod... |
| hi01 31698 | Inner product with the 0 v... |
| hi02 31699 | Inner product with the 0 v... |
| hiidge0 31700 | Inner product with self is... |
| his6 31701 | Zero inner product with se... |
| his1i 31702 | Conjugate law for inner pr... |
| abshicom 31703 | Commuted inner products ha... |
| hial0 31704 | A vector whose inner produ... |
| hial02 31705 | A vector whose inner produ... |
| hisubcomi 31706 | Two vector subtractions si... |
| hi2eq 31707 | Lemma used to prove equali... |
| hial2eq 31708 | Two vectors whose inner pr... |
| hial2eq2 31709 | Two vectors whose inner pr... |
| orthcom 31710 | Orthogonality commutes. (... |
| normlem0 31711 | Lemma used to derive prope... |
| normlem1 31712 | Lemma used to derive prope... |
| normlem2 31713 | Lemma used to derive prope... |
| normlem3 31714 | Lemma used to derive prope... |
| normlem4 31715 | Lemma used to derive prope... |
| normlem5 31716 | Lemma used to derive prope... |
| normlem6 31717 | Lemma used to derive prope... |
| normlem7 31718 | Lemma used to derive prope... |
| normlem8 31719 | Lemma used to derive prope... |
| normlem9 31720 | Lemma used to derive prope... |
| normlem7tALT 31721 | Lemma used to derive prope... |
| bcseqi 31722 | Equality case of Bunjakova... |
| normlem9at 31723 | Lemma used to derive prope... |
| dfhnorm2 31724 | Alternate definition of th... |
| normf 31725 | The norm function maps fro... |
| normval 31726 | The value of the norm of a... |
| normcl 31727 | Real closure of the norm o... |
| normge0 31728 | The norm of a vector is no... |
| normgt0 31729 | The norm of nonzero vector... |
| norm0 31730 | The norm of a zero vector.... |
| norm-i 31731 | Theorem 3.3(i) of [Beran] ... |
| normne0 31732 | A norm is nonzero iff its ... |
| normcli 31733 | Real closure of the norm o... |
| normsqi 31734 | The square of a norm. (Co... |
| norm-i-i 31735 | Theorem 3.3(i) of [Beran] ... |
| normsq 31736 | The square of a norm. (Co... |
| normsub0i 31737 | Two vectors are equal iff ... |
| normsub0 31738 | Two vectors are equal iff ... |
| norm-ii-i 31739 | Triangle inequality for no... |
| norm-ii 31740 | Triangle inequality for no... |
| norm-iii-i 31741 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31742 | Theorem 3.3(iii) of [Beran... |
| normsubi 31743 | Negative doesn't change th... |
| normpythi 31744 | Analogy to Pythagorean the... |
| normsub 31745 | Swapping order of subtract... |
| normneg 31746 | The norm of a vector equal... |
| normpyth 31747 | Analogy to Pythagorean the... |
| normpyc 31748 | Corollary to Pythagorean t... |
| norm3difi 31749 | Norm of differences around... |
| norm3adifii 31750 | Norm of differences around... |
| norm3lem 31751 | Lemma involving norm of di... |
| norm3dif 31752 | Norm of differences around... |
| norm3dif2 31753 | Norm of differences around... |
| norm3lemt 31754 | Lemma involving norm of di... |
| norm3adifi 31755 | Norm of differences around... |
| normpari 31756 | Parallelogram law for norm... |
| normpar 31757 | Parallelogram law for norm... |
| normpar2i 31758 | Corollary of parallelogram... |
| polid2i 31759 | Generalized polarization i... |
| polidi 31760 | Polarization identity. Re... |
| polid 31761 | Polarization identity. Re... |
| hilablo 31762 | Hilbert space vector addit... |
| hilid 31763 | The group identity element... |
| hilvc 31764 | Hilbert space is a complex... |
| hilnormi 31765 | Hilbert space norm in term... |
| hilhhi 31766 | Deduce the structure of Hi... |
| hhnv 31767 | Hilbert space is a normed ... |
| hhva 31768 | The group (addition) opera... |
| hhba 31769 | The base set of Hilbert sp... |
| hh0v 31770 | The zero vector of Hilbert... |
| hhsm 31771 | The scalar product operati... |
| hhvs 31772 | The vector subtraction ope... |
| hhnm 31773 | The norm function of Hilbe... |
| hhims 31774 | The induced metric of Hilb... |
| hhims2 31775 | Hilbert space distance met... |
| hhmet 31776 | The induced metric of Hilb... |
| hhxmet 31777 | The induced metric of Hilb... |
| hhmetdval 31778 | Value of the distance func... |
| hhip 31779 | The inner product operatio... |
| hhph 31780 | The Hilbert space of the H... |
| bcsiALT 31781 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31782 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31783 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31784 | Corollary of the Bunjakova... |
| bcs3 31785 | Corollary of the Bunjakova... |
| hcau 31786 | Member of the set of Cauch... |
| hcauseq 31787 | A Cauchy sequences on a Hi... |
| hcaucvg 31788 | A Cauchy sequence on a Hil... |
| seq1hcau 31789 | A sequence on a Hilbert sp... |
| hlimi 31790 | Express the predicate: Th... |
| hlimseqi 31791 | A sequence with a limit on... |
| hlimveci 31792 | Closure of the limit of a ... |
| hlimconvi 31793 | Convergence of a sequence ... |
| hlim2 31794 | The limit of a sequence on... |
| hlimadd 31795 | Limit of the sum of two se... |
| hilmet 31796 | The Hilbert space norm det... |
| hilxmet 31797 | The Hilbert space norm det... |
| hilmetdval 31798 | Value of the distance func... |
| hilims 31799 | Hilbert space distance met... |
| hhcau 31800 | The Cauchy sequences of Hi... |
| hhlm 31801 | The limit sequences of Hil... |
| hhcmpl 31802 | Lemma used for derivation ... |
| hilcompl 31803 | Lemma used for derivation ... |
| hhcms 31805 | The Hilbert space induced ... |
| hhhl 31806 | The Hilbert space structur... |
| hilcms 31807 | The Hilbert space norm det... |
| hilhl 31808 | The Hilbert space of the H... |
| issh 31810 | Subspace ` H ` of a Hilber... |
| issh2 31811 | Subspace ` H ` of a Hilber... |
| shss 31812 | A subspace is a subset of ... |
| shel 31813 | A member of a subspace of ... |
| shex 31814 | The set of subspaces of a ... |
| shssii 31815 | A closed subspace of a Hil... |
| sheli 31816 | A member of a subspace of ... |
| shelii 31817 | A member of a subspace of ... |
| sh0 31818 | The zero vector belongs to... |
| shaddcl 31819 | Closure of vector addition... |
| shmulcl 31820 | Closure of vector scalar m... |
| issh3 31821 | Subspace ` H ` of a Hilber... |
| shsubcl 31822 | Closure of vector subtract... |
| isch 31824 | Closed subspace ` H ` of a... |
| isch2 31825 | Closed subspace ` H ` of a... |
| chsh 31826 | A closed subspace is a sub... |
| chsssh 31827 | Closed subspaces are subsp... |
| chex 31828 | The set of closed subspace... |
| chshii 31829 | A closed subspace is a sub... |
| ch0 31830 | The zero vector belongs to... |
| chss 31831 | A closed subspace of a Hil... |
| chel 31832 | A member of a closed subsp... |
| chssii 31833 | A closed subspace of a Hil... |
| cheli 31834 | A member of a closed subsp... |
| chelii 31835 | A member of a closed subsp... |
| chlimi 31836 | The limit property of a cl... |
| hlim0 31837 | The zero sequence in Hilbe... |
| hlimcaui 31838 | If a sequence in Hilbert s... |
| hlimf 31839 | Function-like behavior of ... |
| hlimuni 31840 | A Hilbert space sequence c... |
| hlimreui 31841 | The limit of a Hilbert spa... |
| hlimeui 31842 | The limit of a Hilbert spa... |
| isch3 31843 | A Hilbert subspace is clos... |
| chcompl 31844 | Completeness of a closed s... |
| helch 31845 | The Hilbert lattice one (w... |
| ifchhv 31846 | Prove ` if ( A e. CH , A ,... |
| helsh 31847 | Hilbert space is a subspac... |
| shsspwh 31848 | Subspaces are subsets of H... |
| chsspwh 31849 | Closed subspaces are subse... |
| hsn0elch 31850 | The zero subspace belongs ... |
| norm1 31851 | From any nonzero Hilbert s... |
| norm1exi 31852 | A normalized vector exists... |
| norm1hex 31853 | A normalized vector can ex... |
| elch0 31856 | Membership in zero for clo... |
| h0elch 31857 | The zero subspace is a clo... |
| h0elsh 31858 | The zero subspace is a sub... |
| hhssva 31859 | The vector addition operat... |
| hhsssm 31860 | The scalar multiplication ... |
| hhssnm 31861 | The norm operation on a su... |
| issubgoilem 31862 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31863 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31864 | Abelian group property of ... |
| hhssablo 31865 | Abelian group property of ... |
| hhssnv 31866 | Normed complex vector spac... |
| hhssnvt 31867 | Normed complex vector spac... |
| hhsst 31868 | A member of ` SH ` is a su... |
| hhshsslem1 31869 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31870 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31871 | The predicate " ` H ` is a... |
| hhsssh2 31872 | The predicate " ` H ` is a... |
| hhssba 31873 | The base set of a subspace... |
| hhssvs 31874 | The vector subtraction ope... |
| hhssvsf 31875 | Mapping of the vector subt... |
| hhssims 31876 | Induced metric of a subspa... |
| hhssims2 31877 | Induced metric of a subspa... |
| hhssmet 31878 | Induced metric of a subspa... |
| hhssmetdval 31879 | Value of the distance func... |
| hhsscms 31880 | The induced metric of a cl... |
| hhssbnOLD 31881 | Obsolete version of ~ cssb... |
| ocval 31882 | Value of orthogonal comple... |
| ocel 31883 | Membership in orthogonal c... |
| shocel 31884 | Membership in orthogonal c... |
| ocsh 31885 | The orthogonal complement ... |
| shocsh 31886 | The orthogonal complement ... |
| ocss 31887 | An orthogonal complement i... |
| shocss 31888 | An orthogonal complement i... |
| occon 31889 | Contraposition law for ort... |
| occon2 31890 | Double contraposition for ... |
| occon2i 31891 | Double contraposition for ... |
| oc0 31892 | The zero vector belongs to... |
| ocorth 31893 | Members of a subset and it... |
| shocorth 31894 | Members of a subspace and ... |
| ococss 31895 | Inclusion in complement of... |
| shococss 31896 | Inclusion in complement of... |
| shorth 31897 | Members of orthogonal subs... |
| ocin 31898 | Intersection of a Hilbert ... |
| occon3 31899 | Hilbert lattice contraposi... |
| ocnel 31900 | A nonzero vector in the co... |
| chocvali 31901 | Value of the orthogonal co... |
| shuni 31902 | Two subspaces with trivial... |
| chocunii 31903 | Lemma for uniqueness part ... |
| pjhthmo 31904 | Projection Theorem, unique... |
| occllem 31905 | Lemma for ~ occl . (Contr... |
| occl 31906 | Closure of complement of H... |
| shoccl 31907 | Closure of complement of H... |
| choccl 31908 | Closure of complement of H... |
| choccli 31909 | Closure of ` CH ` orthocom... |
| shsval 31914 | Value of subspace sum of t... |
| shsss 31915 | The subspace sum is a subs... |
| shsel 31916 | Membership in the subspace... |
| shsel3 31917 | Membership in the subspace... |
| shseli 31918 | Membership in subspace sum... |
| shscli 31919 | Closure of subspace sum. ... |
| shscl 31920 | Closure of subspace sum. ... |
| shscom 31921 | Commutative law for subspa... |
| shsva 31922 | Vector sum belongs to subs... |
| shsel1 31923 | A subspace sum contains a ... |
| shsel2 31924 | A subspace sum contains a ... |
| shsvs 31925 | Vector subtraction belongs... |
| shsub1 31926 | Subspace sum is an upper b... |
| shsub2 31927 | Subspace sum is an upper b... |
| choc0 31928 | The orthocomplement of the... |
| choc1 31929 | The orthocomplement of the... |
| chocnul 31930 | Orthogonal complement of t... |
| shintcli 31931 | Closure of intersection of... |
| shintcl 31932 | The intersection of a none... |
| chintcli 31933 | The intersection of a none... |
| chintcl 31934 | The intersection (infimum)... |
| spanval 31935 | Value of the linear span o... |
| hsupval 31936 | Value of supremum of set o... |
| chsupval 31937 | The value of the supremum ... |
| spancl 31938 | The span of a subset of Hi... |
| elspancl 31939 | A member of a span is a ve... |
| shsupcl 31940 | Closure of the subspace su... |
| hsupcl 31941 | Closure of supremum of set... |
| chsupcl 31942 | Closure of supremum of sub... |
| hsupss 31943 | Subset relation for suprem... |
| chsupss 31944 | Subset relation for suprem... |
| hsupunss 31945 | The union of a set of Hilb... |
| chsupunss 31946 | The union of a set of clos... |
| spanss2 31947 | A subset of Hilbert space ... |
| shsupunss 31948 | The union of a set of subs... |
| spanid 31949 | A subspace of Hilbert spac... |
| spanss 31950 | Ordering relationship for ... |
| spanssoc 31951 | The span of a subset of Hi... |
| sshjval 31952 | Value of join for subsets ... |
| shjval 31953 | Value of join in ` SH ` . ... |
| chjval 31954 | Value of join in ` CH ` . ... |
| chjvali 31955 | Value of join in ` CH ` . ... |
| sshjval3 31956 | Value of join for subsets ... |
| sshjcl 31957 | Closure of join for subset... |
| shjcl 31958 | Closure of join in ` SH ` ... |
| chjcl 31959 | Closure of join in ` CH ` ... |
| shjcom 31960 | Commutative law for Hilber... |
| shless 31961 | Subset implies subset of s... |
| shlej1 31962 | Add disjunct to both sides... |
| shlej2 31963 | Add disjunct to both sides... |
| shincli 31964 | Closure of intersection of... |
| shscomi 31965 | Commutative law for subspa... |
| shsvai 31966 | Vector sum belongs to subs... |
| shsel1i 31967 | A subspace sum contains a ... |
| shsel2i 31968 | A subspace sum contains a ... |
| shsvsi 31969 | Vector subtraction belongs... |
| shunssi 31970 | Union is smaller than subs... |
| shunssji 31971 | Union is smaller than Hilb... |
| shsleji 31972 | Subspace sum is smaller th... |
| shjcomi 31973 | Commutative law for join i... |
| shsub1i 31974 | Subspace sum is an upper b... |
| shsub2i 31975 | Subspace sum is an upper b... |
| shub1i 31976 | Hilbert lattice join is an... |
| shjcli 31977 | Closure of ` CH ` join. (... |
| shjshcli 31978 | ` SH ` closure of join. (... |
| shlessi 31979 | Subset implies subset of s... |
| shlej1i 31980 | Add disjunct to both sides... |
| shlej2i 31981 | Add disjunct to both sides... |
| shslej 31982 | Subspace sum is smaller th... |
| shincl 31983 | Closure of intersection of... |
| shub1 31984 | Hilbert lattice join is an... |
| shub2 31985 | A subspace is a subset of ... |
| shsidmi 31986 | Idempotent law for Hilbert... |
| shslubi 31987 | The least upper bound law ... |
| shlesb1i 31988 | Hilbert lattice ordering i... |
| shsval2i 31989 | An alternate way to expres... |
| shsval3i 31990 | An alternate way to expres... |
| shmodsi 31991 | The modular law holds for ... |
| shmodi 31992 | The modular law is implied... |
| pjhthlem1 31993 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31994 | Lemma for ~ pjhth . (Cont... |
| pjhth 31995 | Projection Theorem: Any H... |
| pjhtheu 31996 | Projection Theorem: Any H... |
| pjhfval 31998 | The value of the projectio... |
| pjhval 31999 | Value of a projection. (C... |
| pjpreeq 32000 | Equality with a projection... |
| pjeq 32001 | Equality with a projection... |
| axpjcl 32002 | Closure of a projection in... |
| pjhcl 32003 | Closure of a projection in... |
| omlsilem 32004 | Lemma for orthomodular law... |
| omlsii 32005 | Subspace inference form of... |
| omlsi 32006 | Subspace form of orthomodu... |
| ococi 32007 | Complement of complement o... |
| ococ 32008 | Complement of complement o... |
| dfch2 32009 | Alternate definition of th... |
| ococin 32010 | The double complement is t... |
| hsupval2 32011 | Alternate definition of su... |
| chsupval2 32012 | The value of the supremum ... |
| sshjval2 32013 | Value of join in the set o... |
| chsupid 32014 | A subspace is the supremum... |
| chsupsn 32015 | Value of supremum of subse... |
| shlub 32016 | Hilbert lattice join is th... |
| shlubi 32017 | Hilbert lattice join is th... |
| pjhtheu2 32018 | Uniqueness of ` y ` for th... |
| pjcli 32019 | Closure of a projection in... |
| pjhcli 32020 | Closure of a projection in... |
| pjpjpre 32021 | Decomposition of a vector ... |
| axpjpj 32022 | Decomposition of a vector ... |
| pjclii 32023 | Closure of a projection in... |
| pjhclii 32024 | Closure of a projection in... |
| pjpj0i 32025 | Decomposition of a vector ... |
| pjpji 32026 | Decomposition of a vector ... |
| pjpjhth 32027 | Projection Theorem: Any H... |
| pjpjhthi 32028 | Projection Theorem: Any H... |
| pjop 32029 | Orthocomplement projection... |
| pjpo 32030 | Projection in terms of ort... |
| pjopi 32031 | Orthocomplement projection... |
| pjpoi 32032 | Projection in terms of ort... |
| pjoc1i 32033 | Projection of a vector in ... |
| pjchi 32034 | Projection of a vector in ... |
| pjoccl 32035 | The part of a vector that ... |
| pjoc1 32036 | Projection of a vector in ... |
| pjomli 32037 | Subspace form of orthomodu... |
| pjoml 32038 | Subspace form of orthomodu... |
| pjococi 32039 | Proof of orthocomplement t... |
| pjoc2i 32040 | Projection of a vector in ... |
| pjoc2 32041 | Projection of a vector in ... |
| sh0le 32042 | The zero subspace is the s... |
| ch0le 32043 | The zero subspace is the s... |
| shle0 32044 | No subspace is smaller tha... |
| chle0 32045 | No Hilbert lattice element... |
| chnlen0 32046 | A Hilbert lattice element ... |
| ch0pss 32047 | The zero subspace is a pro... |
| orthin 32048 | The intersection of orthog... |
| ssjo 32049 | The lattice join of a subs... |
| shne0i 32050 | A nonzero subspace has a n... |
| shs0i 32051 | Hilbert subspace sum with ... |
| shs00i 32052 | Two subspaces are zero iff... |
| ch0lei 32053 | The closed subspace zero i... |
| chle0i 32054 | No Hilbert closed subspace... |
| chne0i 32055 | A nonzero closed subspace ... |
| chocini 32056 | Intersection of a closed s... |
| chj0i 32057 | Join with lattice zero in ... |
| chm1i 32058 | Meet with lattice one in `... |
| chjcli 32059 | Closure of ` CH ` join. (... |
| chsleji 32060 | Subspace sum is smaller th... |
| chseli 32061 | Membership in subspace sum... |
| chincli 32062 | Closure of Hilbert lattice... |
| chsscon3i 32063 | Hilbert lattice contraposi... |
| chsscon1i 32064 | Hilbert lattice contraposi... |
| chsscon2i 32065 | Hilbert lattice contraposi... |
| chcon2i 32066 | Hilbert lattice contraposi... |
| chcon1i 32067 | Hilbert lattice contraposi... |
| chcon3i 32068 | Hilbert lattice contraposi... |
| chunssji 32069 | Union is smaller than ` CH... |
| chjcomi 32070 | Commutative law for join i... |
| chub1i 32071 | ` CH ` join is an upper bo... |
| chub2i 32072 | ` CH ` join is an upper bo... |
| chlubi 32073 | Hilbert lattice join is th... |
| chlubii 32074 | Hilbert lattice join is th... |
| chlej1i 32075 | Add join to both sides of ... |
| chlej2i 32076 | Add join to both sides of ... |
| chlej12i 32077 | Add join to both sides of ... |
| chlejb1i 32078 | Hilbert lattice ordering i... |
| chdmm1i 32079 | De Morgan's law for meet i... |
| chdmm2i 32080 | De Morgan's law for meet i... |
| chdmm3i 32081 | De Morgan's law for meet i... |
| chdmm4i 32082 | De Morgan's law for meet i... |
| chdmj1i 32083 | De Morgan's law for join i... |
| chdmj2i 32084 | De Morgan's law for join i... |
| chdmj3i 32085 | De Morgan's law for join i... |
| chdmj4i 32086 | De Morgan's law for join i... |
| chnlei 32087 | Equivalent expressions for... |
| chjassi 32088 | Associative law for Hilber... |
| chj00i 32089 | Two Hilbert lattice elemen... |
| chjoi 32090 | The join of a closed subsp... |
| chj1i 32091 | Join with Hilbert lattice ... |
| chm0i 32092 | Meet with Hilbert lattice ... |
| chm0 32093 | Meet with Hilbert lattice ... |
| shjshsi 32094 | Hilbert lattice join equal... |
| shjshseli 32095 | A closed subspace sum equa... |
| chne0 32096 | A nonzero closed subspace ... |
| chocin 32097 | Intersection of a closed s... |
| chssoc 32098 | A closed subspace less tha... |
| chj0 32099 | Join with Hilbert lattice ... |
| chslej 32100 | Subspace sum is smaller th... |
| chincl 32101 | Closure of Hilbert lattice... |
| chsscon3 32102 | Hilbert lattice contraposi... |
| chsscon1 32103 | Hilbert lattice contraposi... |
| chsscon2 32104 | Hilbert lattice contraposi... |
| chpsscon3 32105 | Hilbert lattice contraposi... |
| chpsscon1 32106 | Hilbert lattice contraposi... |
| chpsscon2 32107 | Hilbert lattice contraposi... |
| chjcom 32108 | Commutative law for Hilber... |
| chub1 32109 | Hilbert lattice join is gr... |
| chub2 32110 | Hilbert lattice join is gr... |
| chlub 32111 | Hilbert lattice join is th... |
| chlej1 32112 | Add join to both sides of ... |
| chlej2 32113 | Add join to both sides of ... |
| chlejb1 32114 | Hilbert lattice ordering i... |
| chlejb2 32115 | Hilbert lattice ordering i... |
| chnle 32116 | Equivalent expressions for... |
| chjo 32117 | The join of a closed subsp... |
| chabs1 32118 | Hilbert lattice absorption... |
| chabs2 32119 | Hilbert lattice absorption... |
| chabs1i 32120 | Hilbert lattice absorption... |
| chabs2i 32121 | Hilbert lattice absorption... |
| chjidm 32122 | Idempotent law for Hilbert... |
| chjidmi 32123 | Idempotent law for Hilbert... |
| chj12i 32124 | A rearrangement of Hilbert... |
| chj4i 32125 | Rearrangement of the join ... |
| chjjdiri 32126 | Hilbert lattice join distr... |
| chdmm1 32127 | De Morgan's law for meet i... |
| chdmm2 32128 | De Morgan's law for meet i... |
| chdmm3 32129 | De Morgan's law for meet i... |
| chdmm4 32130 | De Morgan's law for meet i... |
| chdmj1 32131 | De Morgan's law for join i... |
| chdmj2 32132 | De Morgan's law for join i... |
| chdmj3 32133 | De Morgan's law for join i... |
| chdmj4 32134 | De Morgan's law for join i... |
| chjass 32135 | Associative law for Hilber... |
| chj12 32136 | A rearrangement of Hilbert... |
| chj4 32137 | Rearrangement of the join ... |
| ledii 32138 | An ortholattice is distrib... |
| lediri 32139 | An ortholattice is distrib... |
| lejdii 32140 | An ortholattice is distrib... |
| lejdiri 32141 | An ortholattice is distrib... |
| ledi 32142 | An ortholattice is distrib... |
| spansn0 32143 | The span of the singleton ... |
| span0 32144 | The span of the empty set ... |
| elspani 32145 | Membership in the span of ... |
| spanuni 32146 | The span of a union is the... |
| spanun 32147 | The span of a union is the... |
| sshhococi 32148 | The join of two Hilbert sp... |
| hne0 32149 | Hilbert space has a nonzer... |
| chsup0 32150 | The supremum of the empty ... |
| h1deoi 32151 | Membership in orthocomplem... |
| h1dei 32152 | Membership in 1-dimensiona... |
| h1did 32153 | A generating vector belong... |
| h1dn0 32154 | A nonzero vector generates... |
| h1de2i 32155 | Membership in 1-dimensiona... |
| h1de2bi 32156 | Membership in 1-dimensiona... |
| h1de2ctlem 32157 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 32158 | Membership in 1-dimensiona... |
| spansni 32159 | The span of a singleton in... |
| elspansni 32160 | Membership in the span of ... |
| spansn 32161 | The span of a singleton in... |
| spansnch 32162 | The span of a Hilbert spac... |
| spansnsh 32163 | The span of a Hilbert spac... |
| spansnchi 32164 | The span of a singleton in... |
| spansnid 32165 | A vector belongs to the sp... |
| spansnmul 32166 | A scalar product with a ve... |
| elspansncl 32167 | A member of a span of a si... |
| elspansn 32168 | Membership in the span of ... |
| elspansn2 32169 | Membership in the span of ... |
| spansncol 32170 | The singletons of collinea... |
| spansneleqi 32171 | Membership relation implie... |
| spansneleq 32172 | Membership relation that i... |
| spansnss 32173 | The span of the singleton ... |
| elspansn3 32174 | A member of the span of th... |
| elspansn4 32175 | A span membership conditio... |
| elspansn5 32176 | A vector belonging to both... |
| spansnss2 32177 | The span of the singleton ... |
| normcan 32178 | Cancellation-type law that... |
| pjspansn 32179 | A projection on the span o... |
| spansnpji 32180 | A subset of Hilbert space ... |
| spanunsni 32181 | The span of the union of a... |
| spanpr 32182 | The span of a pair of vect... |
| h1datomi 32183 | A 1-dimensional subspace i... |
| h1datom 32184 | A 1-dimensional subspace i... |
| cmbr 32186 | Binary relation expressing... |
| pjoml2i 32187 | Variation of orthomodular ... |
| pjoml3i 32188 | Variation of orthomodular ... |
| pjoml4i 32189 | Variation of orthomodular ... |
| pjoml5i 32190 | The orthomodular law. Rem... |
| pjoml6i 32191 | An equivalent of the ortho... |
| cmbri 32192 | Binary relation expressing... |
| cmcmlem 32193 | Commutation is symmetric. ... |
| cmcmi 32194 | Commutation is symmetric. ... |
| cmcm2i 32195 | Commutation with orthocomp... |
| cmcm3i 32196 | Commutation with orthocomp... |
| cmcm4i 32197 | Commutation with orthocomp... |
| cmbr2i 32198 | Alternate definition of th... |
| cmcmii 32199 | Commutation is symmetric. ... |
| cmcm2ii 32200 | Commutation with orthocomp... |
| cmcm3ii 32201 | Commutation with orthocomp... |
| cmbr3i 32202 | Alternate definition for t... |
| cmbr4i 32203 | Alternate definition for t... |
| lecmi 32204 | Comparable Hilbert lattice... |
| lecmii 32205 | Comparable Hilbert lattice... |
| cmj1i 32206 | A Hilbert lattice element ... |
| cmj2i 32207 | A Hilbert lattice element ... |
| cmm1i 32208 | A Hilbert lattice element ... |
| cmm2i 32209 | A Hilbert lattice element ... |
| cmbr3 32210 | Alternate definition for t... |
| cm0 32211 | The zero Hilbert lattice e... |
| cmidi 32212 | The commutes relation is r... |
| pjoml2 32213 | Variation of orthomodular ... |
| pjoml3 32214 | Variation of orthomodular ... |
| pjoml5 32215 | The orthomodular law. Rem... |
| cmcm 32216 | Commutation is symmetric. ... |
| cmcm3 32217 | Commutation with orthocomp... |
| cmcm2 32218 | Commutation with orthocomp... |
| lecm 32219 | Comparable Hilbert lattice... |
| fh1 32220 | Foulis-Holland Theorem. I... |
| fh2 32221 | Foulis-Holland Theorem. I... |
| cm2j 32222 | A lattice element that com... |
| fh1i 32223 | Foulis-Holland Theorem. I... |
| fh2i 32224 | Foulis-Holland Theorem. I... |
| fh3i 32225 | Variation of the Foulis-Ho... |
| fh4i 32226 | Variation of the Foulis-Ho... |
| cm2ji 32227 | A lattice element that com... |
| cm2mi 32228 | A lattice element that com... |
| qlax1i 32229 | One of the equations showi... |
| qlax2i 32230 | One of the equations showi... |
| qlax3i 32231 | One of the equations showi... |
| qlax4i 32232 | One of the equations showi... |
| qlax5i 32233 | One of the equations showi... |
| qlaxr1i 32234 | One of the conditions show... |
| qlaxr2i 32235 | One of the conditions show... |
| qlaxr4i 32236 | One of the conditions show... |
| qlaxr5i 32237 | One of the conditions show... |
| qlaxr3i 32238 | A variation of the orthomo... |
| chscllem1 32239 | Lemma for ~ chscl . (Cont... |
| chscllem2 32240 | Lemma for ~ chscl . (Cont... |
| chscllem3 32241 | Lemma for ~ chscl . (Cont... |
| chscllem4 32242 | Lemma for ~ chscl . (Cont... |
| chscl 32243 | The subspace sum of two cl... |
| osumi 32244 | If two closed subspaces of... |
| osumcori 32245 | Corollary of ~ osumi . (C... |
| osumcor2i 32246 | Corollary of ~ osumi , sho... |
| osum 32247 | If two closed subspaces of... |
| spansnji 32248 | The subspace sum of a clos... |
| spansnj 32249 | The subspace sum of a clos... |
| spansnscl 32250 | The subspace sum of a clos... |
| sumspansn 32251 | The sum of two vectors bel... |
| spansnm0i 32252 | The meet of different one-... |
| nonbooli 32253 | A Hilbert lattice with two... |
| spansncvi 32254 | Hilbert space has the cove... |
| spansncv 32255 | Hilbert space has the cove... |
| 5oalem1 32256 | Lemma for orthoarguesian l... |
| 5oalem2 32257 | Lemma for orthoarguesian l... |
| 5oalem3 32258 | Lemma for orthoarguesian l... |
| 5oalem4 32259 | Lemma for orthoarguesian l... |
| 5oalem5 32260 | Lemma for orthoarguesian l... |
| 5oalem6 32261 | Lemma for orthoarguesian l... |
| 5oalem7 32262 | Lemma for orthoarguesian l... |
| 5oai 32263 | Orthoarguesian law 5OA. Th... |
| 3oalem1 32264 | Lemma for 3OA (weak) ortho... |
| 3oalem2 32265 | Lemma for 3OA (weak) ortho... |
| 3oalem3 32266 | Lemma for 3OA (weak) ortho... |
| 3oalem4 32267 | Lemma for 3OA (weak) ortho... |
| 3oalem5 32268 | Lemma for 3OA (weak) ortho... |
| 3oalem6 32269 | Lemma for 3OA (weak) ortho... |
| 3oai 32270 | 3OA (weak) orthoarguesian ... |
| pjorthi 32271 | Projection components on o... |
| pjch1 32272 | Property of identity proje... |
| pjo 32273 | The orthogonal projection.... |
| pjcompi 32274 | Component of a projection.... |
| pjidmi 32275 | A projection is idempotent... |
| pjadjii 32276 | A projection is self-adjoi... |
| pjaddii 32277 | Projection of vector sum i... |
| pjinormii 32278 | The inner product of a pro... |
| pjmulii 32279 | Projection of (scalar) pro... |
| pjsubii 32280 | Projection of vector diffe... |
| pjsslem 32281 | Lemma for subset relations... |
| pjss2i 32282 | Subset relationship for pr... |
| pjssmii 32283 | Projection meet property. ... |
| pjssge0ii 32284 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 32285 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 32286 | The projection on a subspa... |
| pjadji 32287 | A projection is self-adjoi... |
| pjaddi 32288 | Projection of vector sum i... |
| pjinormi 32289 | The inner product of a pro... |
| pjsubi 32290 | Projection of vector diffe... |
| pjmuli 32291 | Projection of scalar produ... |
| pjige0i 32292 | The inner product of a pro... |
| pjige0 32293 | The inner product of a pro... |
| pjcjt2 32294 | The projection on a subspa... |
| pj0i 32295 | The projection of the zero... |
| pjch 32296 | Projection of a vector in ... |
| pjid 32297 | The projection of a vector... |
| pjvec 32298 | The set of vectors belongi... |
| pjocvec 32299 | The set of vectors belongi... |
| pjocini 32300 | Membership of projection i... |
| pjini 32301 | Membership of projection i... |
| pjjsi 32302 | A sufficient condition for... |
| pjfni 32303 | Functionality of a project... |
| pjrni 32304 | The range of a projection.... |
| pjfoi 32305 | A projection maps onto its... |
| pjfi 32306 | The mapping of a projectio... |
| pjvi 32307 | The value of a projection ... |
| pjhfo 32308 | A projection maps onto its... |
| pjrn 32309 | The range of a projection.... |
| pjhf 32310 | The mapping of a projectio... |
| pjfn 32311 | Functionality of a project... |
| pjsumi 32312 | The projection on a subspa... |
| pj11i 32313 | One-to-one correspondence ... |
| pjdsi 32314 | Vector decomposition into ... |
| pjds3i 32315 | Vector decomposition into ... |
| pj11 32316 | One-to-one correspondence ... |
| pjmfn 32317 | Functionality of the proje... |
| pjmf1 32318 | The projector function map... |
| pjoi0 32319 | The inner product of proje... |
| pjoi0i 32320 | The inner product of proje... |
| pjopythi 32321 | Pythagorean theorem for pr... |
| pjopyth 32322 | Pythagorean theorem for pr... |
| pjnormi 32323 | The norm of the projection... |
| pjpythi 32324 | Pythagorean theorem for pr... |
| pjneli 32325 | If a vector does not belon... |
| pjnorm 32326 | The norm of the projection... |
| pjpyth 32327 | Pythagorean theorem for pr... |
| pjnel 32328 | If a vector does not belon... |
| pjnorm2 32329 | A vector belongs to the su... |
| mayete3i 32330 | Mayet's equation E_3. Par... |
| mayetes3i 32331 | Mayet's equation E^*_3, de... |
| hosmval 32337 | Value of the sum of two Hi... |
| hommval 32338 | Value of the scalar produc... |
| hodmval 32339 | Value of the difference of... |
| hfsmval 32340 | Value of the sum of two Hi... |
| hfmmval 32341 | Value of the scalar produc... |
| hosval 32342 | Value of the sum of two Hi... |
| homval 32343 | Value of the scalar produc... |
| hodval 32344 | Value of the difference of... |
| hfsval 32345 | Value of the sum of two Hi... |
| hfmval 32346 | Value of the scalar produc... |
| hoscl 32347 | Closure of the sum of two ... |
| homcl 32348 | Closure of the scalar prod... |
| hodcl 32349 | Closure of the difference ... |
| ho0val 32352 | Value of the zero Hilbert ... |
| ho0f 32353 | Functionality of the zero ... |
| df0op2 32354 | Alternate definition of Hi... |
| dfiop2 32355 | Alternate definition of Hi... |
| hoif 32356 | Functionality of the Hilbe... |
| hoival 32357 | The value of the Hilbert s... |
| hoico1 32358 | Composition with the Hilbe... |
| hoico2 32359 | Composition with the Hilbe... |
| hoaddcl 32360 | The sum of Hilbert space o... |
| homulcl 32361 | The scalar product of a Hi... |
| hoeq 32362 | Equality of Hilbert space ... |
| hoeqi 32363 | Equality of Hilbert space ... |
| hoscli 32364 | Closure of Hilbert space o... |
| hodcli 32365 | Closure of Hilbert space o... |
| hocoi 32366 | Composition of Hilbert spa... |
| hococli 32367 | Closure of composition of ... |
| hocofi 32368 | Mapping of composition of ... |
| hocofni 32369 | Functionality of compositi... |
| hoaddcli 32370 | Mapping of sum of Hilbert ... |
| hosubcli 32371 | Mapping of difference of H... |
| hoaddfni 32372 | Functionality of sum of Hi... |
| hosubfni 32373 | Functionality of differenc... |
| hoaddcomi 32374 | Commutativity of sum of Hi... |
| hosubcl 32375 | Mapping of difference of H... |
| hoaddcom 32376 | Commutativity of sum of Hi... |
| hodsi 32377 | Relationship between Hilbe... |
| hoaddassi 32378 | Associativity of sum of Hi... |
| hoadd12i 32379 | Commutative/associative la... |
| hoadd32i 32380 | Commutative/associative la... |
| hocadddiri 32381 | Distributive law for Hilbe... |
| hocsubdiri 32382 | Distributive law for Hilbe... |
| ho2coi 32383 | Double composition of Hilb... |
| hoaddass 32384 | Associativity of sum of Hi... |
| hoadd32 32385 | Commutative/associative la... |
| hoadd4 32386 | Rearrangement of 4 terms i... |
| hocsubdir 32387 | Distributive law for Hilbe... |
| hoaddridi 32388 | Sum of a Hilbert space ope... |
| hodidi 32389 | Difference of a Hilbert sp... |
| ho0coi 32390 | Composition of the zero op... |
| hoid1i 32391 | Composition of Hilbert spa... |
| hoid1ri 32392 | Composition of Hilbert spa... |
| hoaddrid 32393 | Sum of a Hilbert space ope... |
| hodid 32394 | Difference of a Hilbert sp... |
| hon0 32395 | A Hilbert space operator i... |
| hodseqi 32396 | Subtraction and addition o... |
| ho0subi 32397 | Subtraction of Hilbert spa... |
| honegsubi 32398 | Relationship between Hilbe... |
| ho0sub 32399 | Subtraction of Hilbert spa... |
| hosubid1 32400 | The zero operator subtract... |
| honegsub 32401 | Relationship between Hilbe... |
| homullid 32402 | An operator equals its sca... |
| homco1 32403 | Associative law for scalar... |
| homulass 32404 | Scalar product associative... |
| hoadddi 32405 | Scalar product distributiv... |
| hoadddir 32406 | Scalar product reverse dis... |
| homul12 32407 | Swap first and second fact... |
| honegneg 32408 | Double negative of a Hilbe... |
| hosubneg 32409 | Relationship between opera... |
| hosubdi 32410 | Scalar product distributiv... |
| honegdi 32411 | Distribution of negative o... |
| honegsubdi 32412 | Distribution of negative o... |
| honegsubdi2 32413 | Distribution of negative o... |
| hosubsub2 32414 | Law for double subtraction... |
| hosub4 32415 | Rearrangement of 4 terms i... |
| hosubadd4 32416 | Rearrangement of 4 terms i... |
| hoaddsubass 32417 | Associative-type law for a... |
| hoaddsub 32418 | Law for operator addition ... |
| hosubsub 32419 | Law for double subtraction... |
| hosubsub4 32420 | Law for double subtraction... |
| ho2times 32421 | Two times a Hilbert space ... |
| hoaddsubassi 32422 | Associativity of sum and d... |
| hoaddsubi 32423 | Law for sum and difference... |
| hosd1i 32424 | Hilbert space operator sum... |
| hosd2i 32425 | Hilbert space operator sum... |
| hopncani 32426 | Hilbert space operator can... |
| honpcani 32427 | Hilbert space operator can... |
| hosubeq0i 32428 | If the difference between ... |
| honpncani 32429 | Hilbert space operator can... |
| ho01i 32430 | A condition implying that ... |
| ho02i 32431 | A condition implying that ... |
| hoeq1 32432 | A condition implying that ... |
| hoeq2 32433 | A condition implying that ... |
| adjmo 32434 | Every Hilbert space operat... |
| adjsym 32435 | Symmetry property of an ad... |
| eigrei 32436 | A necessary and sufficient... |
| eigre 32437 | A necessary and sufficient... |
| eigposi 32438 | A sufficient condition (fi... |
| eigorthi 32439 | A necessary and sufficient... |
| eigorth 32440 | A necessary and sufficient... |
| nmopval 32458 | Value of the norm of a Hil... |
| elcnop 32459 | Property defining a contin... |
| ellnop 32460 | Property defining a linear... |
| lnopf 32461 | A linear Hilbert space ope... |
| elbdop 32462 | Property defining a bounde... |
| bdopln 32463 | A bounded linear Hilbert s... |
| bdopf 32464 | A bounded linear Hilbert s... |
| nmopsetretALT 32465 | The set in the supremum of... |
| nmopsetretHIL 32466 | The set in the supremum of... |
| nmopsetn0 32467 | The set in the supremum of... |
| nmopxr 32468 | The norm of a Hilbert spac... |
| nmoprepnf 32469 | The norm of a Hilbert spac... |
| nmopgtmnf 32470 | The norm of a Hilbert spac... |
| nmopreltpnf 32471 | The norm of a Hilbert spac... |
| nmopre 32472 | The norm of a bounded oper... |
| elbdop2 32473 | Property defining a bounde... |
| elunop 32474 | Property defining a unitar... |
| elhmop 32475 | Property defining a Hermit... |
| hmopf 32476 | A Hermitian operator is a ... |
| hmopex 32477 | The class of Hermitian ope... |
| nmfnval 32478 | Value of the norm of a Hil... |
| nmfnsetre 32479 | The set in the supremum of... |
| nmfnsetn0 32480 | The set in the supremum of... |
| nmfnxr 32481 | The norm of any Hilbert sp... |
| nmfnrepnf 32482 | The norm of a Hilbert spac... |
| nlfnval 32483 | Value of the null space of... |
| elcnfn 32484 | Property defining a contin... |
| ellnfn 32485 | Property defining a linear... |
| lnfnf 32486 | A linear Hilbert space fun... |
| dfadj2 32487 | Alternate definition of th... |
| funadj 32488 | Functionality of the adjoi... |
| dmadjss 32489 | The domain of the adjoint ... |
| dmadjop 32490 | A member of the domain of ... |
| adjeu 32491 | Elementhood in the domain ... |
| adjval 32492 | Value of the adjoint funct... |
| adjval2 32493 | Value of the adjoint funct... |
| cnvadj 32494 | The adjoint function equal... |
| funcnvadj 32495 | The converse of the adjoin... |
| adj1o 32496 | The adjoint function maps ... |
| dmadjrn 32497 | The adjoint of an operator... |
| eigvecval 32498 | The set of eigenvectors of... |
| eigvalfval 32499 | The eigenvalues of eigenve... |
| specval 32500 | The value of the spectrum ... |
| speccl 32501 | The spectrum of an operato... |
| hhlnoi 32502 | The linear operators of Hi... |
| hhnmoi 32503 | The norm of an operator in... |
| hhbloi 32504 | A bounded linear operator ... |
| hh0oi 32505 | The zero operator in Hilbe... |
| hhcno 32506 | The continuous operators o... |
| hhcnf 32507 | The continuous functionals... |
| dmadjrnb 32508 | The adjoint of an operator... |
| nmoplb 32509 | A lower bound for an opera... |
| nmopub 32510 | An upper bound for an oper... |
| nmopub2tALT 32511 | An upper bound for an oper... |
| nmopub2tHIL 32512 | An upper bound for an oper... |
| nmopge0 32513 | The norm of any Hilbert sp... |
| nmopgt0 32514 | A linear Hilbert space ope... |
| cnopc 32515 | Basic continuity property ... |
| lnopl 32516 | Basic linearity property o... |
| unop 32517 | Basic inner product proper... |
| unopf1o 32518 | A unitary operator in Hilb... |
| unopnorm 32519 | A unitary operator is idem... |
| cnvunop 32520 | The inverse (converse) of ... |
| unopadj 32521 | The inverse (converse) of ... |
| unoplin 32522 | A unitary operator is line... |
| counop 32523 | The composition of two uni... |
| hmop 32524 | Basic inner product proper... |
| hmopre 32525 | The inner product of the v... |
| nmfnlb 32526 | A lower bound for a functi... |
| nmfnleub 32527 | An upper bound for the nor... |
| nmfnleub2 32528 | An upper bound for the nor... |
| nmfnge0 32529 | The norm of any Hilbert sp... |
| elnlfn 32530 | Membership in the null spa... |
| elnlfn2 32531 | Membership in the null spa... |
| cnfnc 32532 | Basic continuity property ... |
| lnfnl 32533 | Basic linearity property o... |
| adjcl 32534 | Closure of the adjoint of ... |
| adj1 32535 | Property of an adjoint Hil... |
| adj2 32536 | Property of an adjoint Hil... |
| adjeq 32537 | A property that determines... |
| adjadj 32538 | Double adjoint. Theorem 3... |
| adjvalval 32539 | Value of the value of the ... |
| unopadj2 32540 | The adjoint of a unitary o... |
| hmopadj 32541 | A Hermitian operator is se... |
| hmdmadj 32542 | Every Hermitian operator h... |
| hmopadj2 32543 | An operator is Hermitian i... |
| hmoplin 32544 | A Hermitian operator is li... |
| brafval 32545 | The bra of a vector, expre... |
| braval 32546 | A bra-ket juxtaposition, e... |
| braadd 32547 | Linearity property of bra ... |
| bramul 32548 | Linearity property of bra ... |
| brafn 32549 | The bra function is a func... |
| bralnfn 32550 | The Dirac bra function is ... |
| bracl 32551 | Closure of the bra functio... |
| bra0 32552 | The Dirac bra of the zero ... |
| brafnmul 32553 | Anti-linearity property of... |
| kbfval 32554 | The outer product of two v... |
| kbop 32555 | The outer product of two v... |
| kbval 32556 | The value of the operator ... |
| kbmul 32557 | Multiplication property of... |
| kbpj 32558 | If a vector ` A ` has norm... |
| eleigvec 32559 | Membership in the set of e... |
| eleigvec2 32560 | Membership in the set of e... |
| eleigveccl 32561 | Closure of an eigenvector ... |
| eigvalval 32562 | The eigenvalue of an eigen... |
| eigvalcl 32563 | An eigenvalue is a complex... |
| eigvec1 32564 | Property of an eigenvector... |
| eighmre 32565 | The eigenvalues of a Hermi... |
| eighmorth 32566 | Eigenvectors of a Hermitia... |
| nmopnegi 32567 | Value of the norm of the n... |
| lnop0 32568 | The value of a linear Hilb... |
| lnopmul 32569 | Multiplicative property of... |
| lnopli 32570 | Basic scalar product prope... |
| lnopfi 32571 | A linear Hilbert space ope... |
| lnop0i 32572 | The value of a linear Hilb... |
| lnopaddi 32573 | Additive property of a lin... |
| lnopmuli 32574 | Multiplicative property of... |
| lnopaddmuli 32575 | Sum/product property of a ... |
| lnopsubi 32576 | Subtraction property for a... |
| lnopsubmuli 32577 | Subtraction/product proper... |
| lnopmulsubi 32578 | Product/subtraction proper... |
| homco2 32579 | Move a scalar product out ... |
| idunop 32580 | The identity function (res... |
| 0cnop 32581 | The identically zero funct... |
| 0cnfn 32582 | The identically zero funct... |
| idcnop 32583 | The identity function (res... |
| idhmop 32584 | The Hilbert space identity... |
| 0hmop 32585 | The identically zero funct... |
| 0lnop 32586 | The identically zero funct... |
| 0lnfn 32587 | The identically zero funct... |
| nmop0 32588 | The norm of the zero opera... |
| nmfn0 32589 | The norm of the identicall... |
| hmopbdoptHIL 32590 | A Hermitian operator is a ... |
| hoddii 32591 | Distributive law for Hilbe... |
| hoddi 32592 | Distributive law for Hilbe... |
| nmop0h 32593 | The norm of any operator o... |
| idlnop 32594 | The identity function (res... |
| 0bdop 32595 | The identically zero opera... |
| adj0 32596 | Adjoint of the zero operat... |
| nmlnop0iALT 32597 | A linear operator with a z... |
| nmlnop0iHIL 32598 | A linear operator with a z... |
| nmlnopgt0i 32599 | A linear Hilbert space ope... |
| nmlnop0 32600 | A linear operator with a z... |
| nmlnopne0 32601 | A linear operator with a n... |
| lnopmi 32602 | The scalar product of a li... |
| lnophsi 32603 | The sum of two linear oper... |
| lnophdi 32604 | The difference of two line... |
| lnopcoi 32605 | The composition of two lin... |
| lnopco0i 32606 | The composition of a linea... |
| lnopeq0lem1 32607 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 32608 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 32609 | A condition implying that ... |
| lnopeqi 32610 | Two linear Hilbert space o... |
| lnopeq 32611 | Two linear Hilbert space o... |
| lnopunilem1 32612 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 32613 | Lemma for ~ lnopunii . (C... |
| lnopunii 32614 | If a linear operator (whos... |
| elunop2 32615 | An operator is unitary iff... |
| nmopun 32616 | Norm of a unitary Hilbert ... |
| unopbd 32617 | A unitary operator is a bo... |
| lnophmlem1 32618 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 32619 | Lemma for ~ lnophmi . (Co... |
| lnophmi 32620 | A linear operator is Hermi... |
| lnophm 32621 | A linear operator is Hermi... |
| hmops 32622 | The sum of two Hermitian o... |
| hmopm 32623 | The scalar product of a He... |
| hmopd 32624 | The difference of two Herm... |
| hmopco 32625 | The composition of two com... |
| nmbdoplbi 32626 | A lower bound for the norm... |
| nmbdoplb 32627 | A lower bound for the norm... |
| nmcexi 32628 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 32629 | The norm of a continuous l... |
| nmcoplbi 32630 | A lower bound for the norm... |
| nmcopex 32631 | The norm of a continuous l... |
| nmcoplb 32632 | A lower bound for the norm... |
| nmophmi 32633 | The norm of the scalar pro... |
| bdophmi 32634 | The scalar product of a bo... |
| lnconi 32635 | Lemma for ~ lnopconi and ~... |
| lnopconi 32636 | A condition equivalent to ... |
| lnopcon 32637 | A condition equivalent to ... |
| lnopcnbd 32638 | A linear operator is conti... |
| lncnopbd 32639 | A continuous linear operat... |
| lncnbd 32640 | A continuous linear operat... |
| lnopcnre 32641 | A linear operator is conti... |
| lnfnli 32642 | Basic property of a linear... |
| lnfnfi 32643 | A linear Hilbert space fun... |
| lnfn0i 32644 | The value of a linear Hilb... |
| lnfnaddi 32645 | Additive property of a lin... |
| lnfnmuli 32646 | Multiplicative property of... |
| lnfnaddmuli 32647 | Sum/product property of a ... |
| lnfnsubi 32648 | Subtraction property for a... |
| lnfn0 32649 | The value of a linear Hilb... |
| lnfnmul 32650 | Multiplicative property of... |
| nmbdfnlbi 32651 | A lower bound for the norm... |
| nmbdfnlb 32652 | A lower bound for the norm... |
| nmcfnexi 32653 | The norm of a continuous l... |
| nmcfnlbi 32654 | A lower bound for the norm... |
| nmcfnex 32655 | The norm of a continuous l... |
| nmcfnlb 32656 | A lower bound of the norm ... |
| lnfnconi 32657 | A condition equivalent to ... |
| lnfncon 32658 | A condition equivalent to ... |
| lnfncnbd 32659 | A linear functional is con... |
| imaelshi 32660 | The image of a subspace un... |
| rnelshi 32661 | The range of a linear oper... |
| nlelshi 32662 | The null space of a linear... |
| nlelchi 32663 | The null space of a contin... |
| riesz3i 32664 | A continuous linear functi... |
| riesz4i 32665 | A continuous linear functi... |
| riesz4 32666 | A continuous linear functi... |
| riesz1 32667 | Part 1 of the Riesz repres... |
| riesz2 32668 | Part 2 of the Riesz repres... |
| cnlnadjlem1 32669 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 32670 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 32671 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 32672 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 32673 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 32674 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 32675 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 32676 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 32677 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 32678 | Every continuous linear op... |
| cnlnadjeui 32679 | Every continuous linear op... |
| cnlnadjeu 32680 | Every continuous linear op... |
| cnlnadj 32681 | Every continuous linear op... |
| cnlnssadj 32682 | Every continuous linear Hi... |
| bdopssadj 32683 | Every bounded linear Hilbe... |
| bdopadj 32684 | Every bounded linear Hilbe... |
| adjbdln 32685 | The adjoint of a bounded l... |
| adjbdlnb 32686 | An operator is bounded and... |
| adjbd1o 32687 | The mapping of adjoints of... |
| adjlnop 32688 | The adjoint of an operator... |
| adjsslnop 32689 | Every operator with an adj... |
| nmopadjlei 32690 | Property of the norm of an... |
| nmopadjlem 32691 | Lemma for ~ nmopadji . (C... |
| nmopadji 32692 | Property of the norm of an... |
| adjeq0 32693 | An operator is zero iff it... |
| adjmul 32694 | The adjoint of the scalar ... |
| adjadd 32695 | The adjoint of the sum of ... |
| nmoptrii 32696 | Triangle inequality for th... |
| nmopcoi 32697 | Upper bound for the norm o... |
| bdophsi 32698 | The sum of two bounded lin... |
| bdophdi 32699 | The difference between two... |
| bdopcoi 32700 | The composition of two bou... |
| nmoptri2i 32701 | Triangle-type inequality f... |
| adjcoi 32702 | The adjoint of a compositi... |
| nmopcoadji 32703 | The norm of an operator co... |
| nmopcoadj2i 32704 | The norm of an operator co... |
| nmopcoadj0i 32705 | An operator composed with ... |
| unierri 32706 | If we approximate a chain ... |
| branmfn 32707 | The norm of the bra functi... |
| brabn 32708 | The bra of a vector is a b... |
| rnbra 32709 | The set of bras equals the... |
| bra11 32710 | The bra function maps vect... |
| bracnln 32711 | A bra is a continuous line... |
| cnvbraval 32712 | Value of the converse of t... |
| cnvbracl 32713 | Closure of the converse of... |
| cnvbrabra 32714 | The converse bra of the br... |
| bracnvbra 32715 | The bra of the converse br... |
| bracnlnval 32716 | The vector that a continuo... |
| cnvbramul 32717 | Multiplication property of... |
| kbass1 32718 | Dirac bra-ket associative ... |
| kbass2 32719 | Dirac bra-ket associative ... |
| kbass3 32720 | Dirac bra-ket associative ... |
| kbass4 32721 | Dirac bra-ket associative ... |
| kbass5 32722 | Dirac bra-ket associative ... |
| kbass6 32723 | Dirac bra-ket associative ... |
| leopg 32724 | Ordering relation for posi... |
| leop 32725 | Ordering relation for oper... |
| leop2 32726 | Ordering relation for oper... |
| leop3 32727 | Operator ordering in terms... |
| leoppos 32728 | Binary relation defining a... |
| leoprf2 32729 | The ordering relation for ... |
| leoprf 32730 | The ordering relation for ... |
| leopsq 32731 | The square of a Hermitian ... |
| 0leop 32732 | The zero operator is a pos... |
| idleop 32733 | The identity operator is a... |
| leopadd 32734 | The sum of two positive op... |
| leopmuli 32735 | The scalar product of a no... |
| leopmul 32736 | The scalar product of a po... |
| leopmul2i 32737 | Scalar product applied to ... |
| leoptri 32738 | The positive operator orde... |
| leoptr 32739 | The positive operator orde... |
| leopnmid 32740 | A bounded Hermitian operat... |
| nmopleid 32741 | A nonzero, bounded Hermiti... |
| opsqrlem1 32742 | Lemma for opsqri . (Contr... |
| opsqrlem2 32743 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32744 | Lemma for opsqri . (Contr... |
| opsqrlem4 32745 | Lemma for opsqri . (Contr... |
| opsqrlem5 32746 | Lemma for opsqri . (Contr... |
| opsqrlem6 32747 | Lemma for opsqri . (Contr... |
| pjhmopi 32748 | A projector is a Hermitian... |
| pjlnopi 32749 | A projector is a linear op... |
| pjnmopi 32750 | The operator norm of a pro... |
| pjbdlni 32751 | A projector is a bounded l... |
| pjhmop 32752 | A projection is a Hermitia... |
| hmopidmchi 32753 | An idempotent Hermitian op... |
| hmopidmpji 32754 | An idempotent Hermitian op... |
| hmopidmch 32755 | An idempotent Hermitian op... |
| hmopidmpj 32756 | An idempotent Hermitian op... |
| pjsdii 32757 | Distributive law for Hilbe... |
| pjddii 32758 | Distributive law for Hilbe... |
| pjsdi2i 32759 | Chained distributive law f... |
| pjcoi 32760 | Composition of projections... |
| pjcocli 32761 | Closure of composition of ... |
| pjcohcli 32762 | Closure of composition of ... |
| pjadjcoi 32763 | Adjoint of composition of ... |
| pjcofni 32764 | Functionality of compositi... |
| pjss1coi 32765 | Subset relationship for pr... |
| pjss2coi 32766 | Subset relationship for pr... |
| pjssmi 32767 | Projection meet property. ... |
| pjssge0i 32768 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32769 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32770 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32771 | Composition of projections... |
| pjscji 32772 | The projection of orthogon... |
| pjssumi 32773 | The projection on a subspa... |
| pjssposi 32774 | Projector ordering can be ... |
| pjordi 32775 | The definition of projecto... |
| pjssdif2i 32776 | The projection subspace of... |
| pjssdif1i 32777 | A necessary and sufficient... |
| pjimai 32778 | The image of a projection.... |
| pjidmcoi 32779 | A projection is idempotent... |
| pjoccoi 32780 | Composition of projections... |
| pjtoi 32781 | Subspace sum of projection... |
| pjoci 32782 | Projection of orthocomplem... |
| pjidmco 32783 | A projection operator is i... |
| dfpjop 32784 | Definition of projection o... |
| pjhmopidm 32785 | Two ways to express the se... |
| elpjidm 32786 | A projection operator is i... |
| elpjhmop 32787 | A projection operator is H... |
| 0leopj 32788 | A projector is a positive ... |
| pjadj2 32789 | A projector is self-adjoin... |
| pjadj3 32790 | A projector is self-adjoin... |
| elpjch 32791 | Reconstruction of the subs... |
| elpjrn 32792 | Reconstruction of the subs... |
| pjinvari 32793 | A closed subspace ` H ` wi... |
| pjin1i 32794 | Lemma for Theorem 1.22 of ... |
| pjin2i 32795 | Lemma for Theorem 1.22 of ... |
| pjin3i 32796 | Lemma for Theorem 1.22 of ... |
| pjclem1 32797 | Lemma for projection commu... |
| pjclem2 32798 | Lemma for projection commu... |
| pjclem3 32799 | Lemma for projection commu... |
| pjclem4a 32800 | Lemma for projection commu... |
| pjclem4 32801 | Lemma for projection commu... |
| pjci 32802 | Two subspaces commute iff ... |
| pjcmul1i 32803 | A necessary and sufficient... |
| pjcmul2i 32804 | The projection subspace of... |
| pjcohocli 32805 | Closure of composition of ... |
| pjadj2coi 32806 | Adjoint of double composit... |
| pj2cocli 32807 | Closure of double composit... |
| pj3lem1 32808 | Lemma for projection tripl... |
| pj3si 32809 | Stronger projection triple... |
| pj3i 32810 | Projection triplet theorem... |
| pj3cor1i 32811 | Projection triplet corolla... |
| pjs14i 32812 | Theorem S-14 of Watanabe, ... |
| isst 32815 | Property of a state. (Con... |
| ishst 32816 | Property of a complex Hilb... |
| sticl 32817 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32818 | Real closure of the value ... |
| hstcl 32819 | Closure of the value of a ... |
| hst1a 32820 | Unit value of a Hilbert-sp... |
| hstel2 32821 | Properties of a Hilbert-sp... |
| hstorth 32822 | Orthogonality property of ... |
| hstosum 32823 | Orthogonal sum property of... |
| hstoc 32824 | Sum of a Hilbert-space-val... |
| hstnmoc 32825 | Sum of norms of a Hilbert-... |
| stge0 32826 | The value of a state is no... |
| stle1 32827 | The value of a state is le... |
| hstle1 32828 | The norm of the value of a... |
| hst1h 32829 | The norm of a Hilbert-spac... |
| hst0h 32830 | The norm of a Hilbert-spac... |
| hstpyth 32831 | Pythagorean property of a ... |
| hstle 32832 | Ordering property of a Hil... |
| hstles 32833 | Ordering property of a Hil... |
| hstoh 32834 | A Hilbert-space-valued sta... |
| hst0 32835 | A Hilbert-space-valued sta... |
| sthil 32836 | The value of a state at th... |
| stj 32837 | The value of a state on a ... |
| sto1i 32838 | The state of a subspace pl... |
| sto2i 32839 | The state of the orthocomp... |
| stge1i 32840 | If a state is greater than... |
| stle0i 32841 | If a state is less than or... |
| stlei 32842 | Ordering law for states. ... |
| stlesi 32843 | Ordering law for states. ... |
| stji1i 32844 | Join of components of Sasa... |
| stm1i 32845 | State of component of unit... |
| stm1ri 32846 | State of component of unit... |
| stm1addi 32847 | Sum of states whose meet i... |
| staddi 32848 | If the sum of 2 states is ... |
| stm1add3i 32849 | Sum of states whose meet i... |
| stadd3i 32850 | If the sum of 3 states is ... |
| st0 32851 | The state of the zero subs... |
| strlem1 32852 | Lemma for strong state the... |
| strlem2 32853 | Lemma for strong state the... |
| strlem3a 32854 | Lemma for strong state the... |
| strlem3 32855 | Lemma for strong state the... |
| strlem4 32856 | Lemma for strong state the... |
| strlem5 32857 | Lemma for strong state the... |
| strlem6 32858 | Lemma for strong state the... |
| stri 32859 | Strong state theorem. The... |
| strb 32860 | Strong state theorem (bidi... |
| hstrlem2 32861 | Lemma for strong set of CH... |
| hstrlem3a 32862 | Lemma for strong set of CH... |
| hstrlem3 32863 | Lemma for strong set of CH... |
| hstrlem4 32864 | Lemma for strong set of CH... |
| hstrlem5 32865 | Lemma for strong set of CH... |
| hstrlem6 32866 | Lemma for strong set of CH... |
| hstri 32867 | Hilbert space admits a str... |
| hstrbi 32868 | Strong CH-state theorem (b... |
| largei 32869 | A Hilbert lattice admits a... |
| jplem1 32870 | Lemma for Jauch-Piron theo... |
| jplem2 32871 | Lemma for Jauch-Piron theo... |
| jpi 32872 | The function ` S ` , that ... |
| golem1 32873 | Lemma for Godowski's equat... |
| golem2 32874 | Lemma for Godowski's equat... |
| goeqi 32875 | Godowski's equation, shown... |
| stcltr1i 32876 | Property of a strong class... |
| stcltr2i 32877 | Property of a strong class... |
| stcltrlem1 32878 | Lemma for strong classical... |
| stcltrlem2 32879 | Lemma for strong classical... |
| stcltrthi 32880 | Theorem for classically st... |
| cvbr 32884 | Binary relation expressing... |
| cvbr2 32885 | Binary relation expressing... |
| cvcon3 32886 | Contraposition law for the... |
| cvpss 32887 | The covers relation implie... |
| cvnbtwn 32888 | The covers relation implie... |
| cvnbtwn2 32889 | The covers relation implie... |
| cvnbtwn3 32890 | The covers relation implie... |
| cvnbtwn4 32891 | The covers relation implie... |
| cvnsym 32892 | The covers relation is not... |
| cvnref 32893 | The covers relation is not... |
| cvntr 32894 | The covers relation is not... |
| spansncv2 32895 | Hilbert space has the cove... |
| mdbr 32896 | Binary relation expressing... |
| mdi 32897 | Consequence of the modular... |
| mdbr2 32898 | Binary relation expressing... |
| mdbr3 32899 | Binary relation expressing... |
| mdbr4 32900 | Binary relation expressing... |
| dmdbr 32901 | Binary relation expressing... |
| dmdmd 32902 | The dual modular pair prop... |
| mddmd 32903 | The modular pair property ... |
| dmdi 32904 | Consequence of the dual mo... |
| dmdbr2 32905 | Binary relation expressing... |
| dmdi2 32906 | Consequence of the dual mo... |
| dmdbr3 32907 | Binary relation expressing... |
| dmdbr4 32908 | Binary relation expressing... |
| dmdi4 32909 | Consequence of the dual mo... |
| dmdbr5 32910 | Binary relation expressing... |
| mddmd2 32911 | Relationship between modul... |
| mdsl0 32912 | A sublattice condition tha... |
| ssmd1 32913 | Ordering implies the modul... |
| ssmd2 32914 | Ordering implies the modul... |
| ssdmd1 32915 | Ordering implies the dual ... |
| ssdmd2 32916 | Ordering implies the dual ... |
| dmdsl3 32917 | Sublattice mapping for a d... |
| mdsl3 32918 | Sublattice mapping for a m... |
| mdslle1i 32919 | Order preservation of the ... |
| mdslle2i 32920 | Order preservation of the ... |
| mdslj1i 32921 | Join preservation of the o... |
| mdslj2i 32922 | Meet preservation of the r... |
| mdsl1i 32923 | If the modular pair proper... |
| mdsl2i 32924 | If the modular pair proper... |
| mdsl2bi 32925 | If the modular pair proper... |
| cvmdi 32926 | The covering property impl... |
| mdslmd1lem1 32927 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32928 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32929 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32930 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32931 | Preservation of the modula... |
| mdslmd2i 32932 | Preservation of the modula... |
| mdsldmd1i 32933 | Preservation of the dual m... |
| mdslmd3i 32934 | Modular pair conditions th... |
| mdslmd4i 32935 | Modular pair condition tha... |
| csmdsymi 32936 | Cross-symmetry implies M-s... |
| mdexchi 32937 | An exchange lemma for modu... |
| cvmd 32938 | The covering property impl... |
| cvdmd 32939 | The covering property impl... |
| ela 32941 | Atoms in a Hilbert lattice... |
| elat2 32942 | Expanded membership relati... |
| elatcv0 32943 | A Hilbert lattice element ... |
| atcv0 32944 | An atom covers the zero su... |
| atssch 32945 | Atoms are a subset of the ... |
| atelch 32946 | An atom is a Hilbert latti... |
| atne0 32947 | An atom is not the Hilbert... |
| atss 32948 | A lattice element smaller ... |
| atsseq 32949 | Two atoms in a subset rela... |
| atcveq0 32950 | A Hilbert lattice element ... |
| h1da 32951 | A 1-dimensional subspace i... |
| spansna 32952 | The span of the singleton ... |
| sh1dle 32953 | A 1-dimensional subspace i... |
| ch1dle 32954 | A 1-dimensional subspace i... |
| atom1d 32955 | The 1-dimensional subspace... |
| superpos 32956 | Superposition Principle. ... |
| chcv1 32957 | The Hilbert lattice has th... |
| chcv2 32958 | The Hilbert lattice has th... |
| chjatom 32959 | The join of a closed subsp... |
| shatomici 32960 | The lattice of Hilbert sub... |
| hatomici 32961 | The Hilbert lattice is ato... |
| hatomic 32962 | A Hilbert lattice is atomi... |
| shatomistici 32963 | The lattice of Hilbert sub... |
| hatomistici 32964 | ` CH ` is atomistic, i.e. ... |
| chpssati 32965 | Two Hilbert lattice elemen... |
| chrelati 32966 | The Hilbert lattice is rel... |
| chrelat2i 32967 | A consequence of relative ... |
| cvati 32968 | If a Hilbert lattice eleme... |
| cvbr4i 32969 | An alternate way to expres... |
| cvexchlem 32970 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32971 | The Hilbert lattice satisf... |
| chrelat2 32972 | A consequence of relative ... |
| chrelat3 32973 | A consequence of relative ... |
| chrelat3i 32974 | A consequence of the relat... |
| chrelat4i 32975 | A consequence of relative ... |
| cvexch 32976 | The Hilbert lattice satisf... |
| cvp 32977 | The Hilbert lattice satisf... |
| atnssm0 32978 | The meet of a Hilbert latt... |
| atnemeq0 32979 | The meet of distinct atoms... |
| atssma 32980 | The meet with an atom's su... |
| atcv0eq 32981 | Two atoms covering the zer... |
| atcv1 32982 | Two atoms covering the zer... |
| atexch 32983 | The Hilbert lattice satisf... |
| atomli 32984 | An assertion holding in at... |
| atoml2i 32985 | An assertion holding in at... |
| atordi 32986 | An ordering law for a Hilb... |
| atcvatlem 32987 | Lemma for ~ atcvati . (Co... |
| atcvati 32988 | A nonzero Hilbert lattice ... |
| atcvat2i 32989 | A Hilbert lattice element ... |
| atord 32990 | An ordering law for a Hilb... |
| atcvat2 32991 | A Hilbert lattice element ... |
| chirredlem1 32992 | Lemma for ~ chirredi . (C... |
| chirredlem2 32993 | Lemma for ~ chirredi . (C... |
| chirredlem3 32994 | Lemma for ~ chirredi . (C... |
| chirredlem4 32995 | Lemma for ~ chirredi . (C... |
| chirredi 32996 | The Hilbert lattice is irr... |
| chirred 32997 | The Hilbert lattice is irr... |
| atcvat3i 32998 | A condition implying that ... |
| atcvat4i 32999 | A condition implying exist... |
| atdmd 33000 | Two Hilbert lattice elemen... |
| atmd 33001 | Two Hilbert lattice elemen... |
| atmd2 33002 | Two Hilbert lattice elemen... |
| atabsi 33003 | Absorption of an incompara... |
| atabs2i 33004 | Absorption of an incompara... |
| mdsymlem1 33005 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 33006 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 33007 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 33008 | Lemma for ~ mdsymi . This... |
| mdsymlem5 33009 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 33010 | Lemma for ~ mdsymi . This... |
| mdsymlem7 33011 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 33012 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 33013 | M-symmetry of the Hilbert ... |
| mdsym 33014 | M-symmetry of the Hilbert ... |
| dmdsym 33015 | Dual M-symmetry of the Hil... |
| atdmd2 33016 | Two Hilbert lattice elemen... |
| sumdmdii 33017 | If the subspace sum of two... |
| cmmdi 33018 | Commuting subspaces form a... |
| cmdmdi 33019 | Commuting subspaces form a... |
| sumdmdlem 33020 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 33021 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 33022 | The subspace sum of two Hi... |
| dmdbr4ati 33023 | Dual modular pair property... |
| dmdbr5ati 33024 | Dual modular pair property... |
| dmdbr6ati 33025 | Dual modular pair property... |
| dmdbr7ati 33026 | Dual modular pair property... |
| mdoc1i 33027 | Orthocomplements form a mo... |
| mdoc2i 33028 | Orthocomplements form a mo... |
| dmdoc1i 33029 | Orthocomplements form a du... |
| dmdoc2i 33030 | Orthocomplements form a du... |
| mdcompli 33031 | A condition equivalent to ... |
| dmdcompli 33032 | A condition equivalent to ... |
| mddmdin0i 33033 | If dual modular implies mo... |
| cdjreui 33034 | A member of the sum of dis... |
| cdj1i 33035 | Two ways to express " ` A ... |
| cdj3lem1 33036 | A property of " ` A ` and ... |
| cdj3lem2 33037 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 33038 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 33039 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 33040 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 33041 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 33042 | Lemma for ~ cdj3i . The s... |
| cdj3i 33043 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 33044 | (_This theorem is a dummy ... |
| tgacne 33045 | Non-collinearity of triang... |
| tgbcne 33046 | Non-collinearity of triang... |
| cgrahlb 33047 | Two null angles are congru... |
| inagbtwn 33048 | A flat sub-angle forces th... |
| inagcgrasubhl 33049 | In-angle transport, degene... |
| inagcgrasubbtwn 33050 | In-angle transport, degene... |
| inagcgrabtwn 33051 | In-angle transport, degene... |
| iscgrag 33052 | Ray normalization: an angl... |
| cgrgptxfr 33053 | Transfer a segment point a... |
| inagcgratrgmain 33054 | In-angle transport, the no... |
| inagcgratrgw 33055 | ~ inagcgratrgmain with the... |
| inagcgratrgxfr 33056 | In-angle transport with th... |
| inagcgratrgncol 33057 | In-angle transport once th... |
| inagcgratrgn 33058 | In-angle transport, non-co... |
| inaglea 33059 | If ` C ` lies inside angle... |
| cgrainag 33060 | Target-side in-angle trans... |
| leawittrg 33061 | Target-side ` leA ` witnes... |
| leacgra2 33062 | Target-side ` leA ` transp... |
| leacgra1 33063 | Source-side ` leA ` transp... |
| leacgra 33064 | GeoCoq l11__30 analogue: `... |
| tgmircgrg 33065 | Reflecting ` B ` through t... |
| tgmircgra 33066 | The angle congruence the e... |
| tgextamirn 33067 | The midpoint-reflected poi... |
| tgextancd 33068 | The target-side non-collin... |
| tgextahpl 33069 | The generic open half-plan... |
| tgextalem1inag 33070 | The point obtained by refl... |
| tgextale 33071 | The non-strict exterior-an... |
| tgextamirnhl 33072 | The midpoint-reflected poi... |
| tgextahp 33073 | The reflected point and ` ... |
| tgextancgra 33074 | The exterior angle is NOT ... |
| tgextaoa 33075 | The two exterior angles at... |
| tgextalt1 33076 | ~ tgexta 's FIRST conjunct... |
| tgextalt2 33077 | ~ tgexta 's SECOND conjunc... |
| tgexta 33078 | The exterior-angle theorem... |
| tgextan 33079 | ~ tgexta 's strictness can... |
| tgaasncol 33080 | Slide a non-collinearity a... |
| tgaasexc 33081 | The contradiction branch o... |
| tgaasne 33082 | Two DISTINCT points ` F ` ... |
| tgaascgra 33083 | The comparison triangle of... |
| tgaas1h 33084 | ~ tgaas1 with the construc... |
| tgaas1 33085 | Under the AAS hypotheses, ... |
| tgaas 33086 | Angle-angle-side: two tria... |
| sa-abvi 33087 | A theorem about the univer... |
| xfree 33088 | A partial converse to ~ 19... |
| xfree2 33089 | A partial converse to ~ 19... |
| addltmulALT 33090 | A proof readability experi... |
| ad11antr 33091 | Deduction adding 11 conjun... |
| simp-12l 33092 | Simplification of a conjun... |
| simp-12r 33093 | Simplification of a conjun... |
| an52ds 33094 | Inference exchanging the l... |
| an62ds 33095 | Inference exchanging the l... |
| an72ds 33096 | Inference exchanging the l... |
| an82ds 33097 | Inference exchanging the l... |
| syl22anbrc 33098 | Syllogism inference. (Con... |
| or3di 33099 | Distributive law for disju... |
| or3dir 33100 | Distributive law for disju... |
| 3o1cs 33101 | Deduction eliminating disj... |
| 3o2cs 33102 | Deduction eliminating disj... |
| 3o3cs 33103 | Deduction eliminating disj... |
| sbc2iedf 33104 | Conversion of implicit sub... |
| rspc2daf 33105 | Double restricted speciali... |
| ralcom4f 33106 | Commutation of restricted ... |
| rexcom4f 33107 | Commutation of restricted ... |
| 19.9d2rf 33108 | A deduction version of one... |
| 19.9d2r 33109 | A deduction version of one... |
| r19.29ffa 33110 | A commonly used pattern ba... |
| reu6dv 33111 | A condition which implies ... |
| eqtrb 33112 | A transposition of equalit... |
| eqelbid 33113 | A variable elimination law... |
| opsbc2ie 33114 | Conversion of implicit sub... |
| opreu2reuALT 33115 | Correspondence between uni... |
| 2reucom 33118 | Double restricted existent... |
| 2reu2rex1 33119 | Double restricted existent... |
| 2reureurex 33120 | Double restricted existent... |
| 2reu2reu2 33121 | Double restricted existent... |
| opreu2reu1 33122 | Equivalent definition of t... |
| sq2reunnltb 33123 | There exists a unique deco... |
| addsqnot2reu 33124 | For each complex number ` ... |
| sbceqbidf 33125 | Equality theorem for class... |
| sbcies 33126 | A special version of class... |
| mo5f 33127 | Alternate definition of "a... |
| nmo 33128 | Negation of "at most one".... |
| reuxfrdf 33129 | Transfer existential uniqu... |
| rexunirn 33130 | Restricted existential qua... |
| rmoxfrd 33131 | Transfer "at most one" res... |
| rmoun 33132 | "At most one" restricted e... |
| rmounid 33133 | A case where an "at most o... |
| riotaeqbidva 33134 | Equivalent wff's yield equ... |
| dmrab 33135 | Domain of a restricted cla... |
| difrab2 33136 | Difference of two restrict... |
| rabexgfGS 33137 | Separation Scheme in terms... |
| rabsnel 33138 | Truth implied by equality ... |
| rabsspr 33139 | Conditions for a restricte... |
| rabsstp 33140 | Conditions for a restricte... |
| 3unrab 33141 | Union of three restricted ... |
| foresf1o 33142 | From a surjective function... |
| rabfodom 33143 | Domination relation for re... |
| rabrexfi 33144 | Conditions for a class abs... |
| abrexdomjm 33145 | An indexed set is dominate... |
| abrexdom2jm 33146 | An indexed set is dominate... |
| abrexexd 33147 | Existence of a class abstr... |
| elabreximd 33148 | Class substitution in an i... |
| elabreximdv 33149 | Class substitution in an i... |
| abrexss 33150 | A necessary condition for ... |
| nelun 33151 | Negated membership for a u... |
| snsssng 33152 | If a singleton is a subset... |
| n0nsnel 33153 | If a class with one elemen... |
| inin 33154 | Intersection with an inter... |
| difininv 33155 | Condition for the intersec... |
| difeq 33156 | Rewriting an equation with... |
| eqdif 33157 | If both set differences of... |
| indifbi 33158 | Two ways to express equali... |
| diffib 33159 | Case where ~ diffi is a bi... |
| difxp1ss 33160 | Difference law for Cartesi... |
| difxp2ss 33161 | Difference law for Cartesi... |
| indifundif 33162 | A remarkable equation with... |
| elpwincl1 33163 | Closure of intersection wi... |
| elpwdifcl 33164 | Closure of class differenc... |
| elpwiuncl 33165 | Closure of indexed union w... |
| elpreq 33166 | Equality wihin a pair. (C... |
| prssad 33167 | If a pair is a subset of a... |
| prssbd 33168 | If a pair is a subset of a... |
| nelpr 33169 | A set ` A ` not in a pair ... |
| inpr0 33170 | Rewrite an empty intersect... |
| neldifpr1 33171 | The first element of a pai... |
| neldifpr2 33172 | The second element of a pa... |
| unidifsnel 33173 | The other element of a pai... |
| unidifsnne 33174 | The other element of a pai... |
| tpssg 33175 | An unordered triple of ele... |
| tpssd 33176 | Deduction version of tpssi... |
| tpssad 33177 | If an ordered triple is a ... |
| tpssbd 33178 | If an ordered triple is a ... |
| tpsscd 33179 | If an ordered triple is a ... |
| ifeqeqx 33180 | An equality theorem tailor... |
| elimifd 33181 | Elimination of a condition... |
| elim2if 33182 | Elimination of two conditi... |
| elim2ifim 33183 | Elimination of two conditi... |
| ifeq3da 33184 | Given an expression ` C ` ... |
| ifnetrue 33185 | Deduce truth from a condit... |
| ifnefals 33186 | Deduce falsehood from a co... |
| ifnebib 33187 | The converse of ~ ifbi hol... |
| ififcom 33188 | Commute two nested conditi... |
| uniinn0 33189 | Sufficient and necessary c... |
| difuncomp 33190 | Express a class difference... |
| elpwunicl 33191 | Closure of a set union wit... |
| cbviunf 33192 | Rule used to change the bo... |
| iuneq12daf 33193 | Equality deduction for ind... |
| iunin1f 33194 | Indexed union of intersect... |
| ssiun3 33195 | Subset equivalence for an ... |
| ssiun2sf 33196 | Subset relationship for an... |
| iuninc 33197 | The union of an increasing... |
| iundifdifd 33198 | The intersection of a set ... |
| iundifdif 33199 | The intersection of a set ... |
| iunrdx 33200 | Re-index an indexed union.... |
| iunrnmptss 33201 | A subset relation for an i... |
| iunxunsn 33202 | Appending a set to an inde... |
| iunxunpr 33203 | Appending two sets to an i... |
| iunxpssiun1 33204 | Provide an upper bound for... |
| iinabrex 33205 | Rewriting an indexed inter... |
| disjnf 33206 | In case ` x ` is not free ... |
| cbvdisjf 33207 | Change bound variables in ... |
| disjss1f 33208 | A subset of a disjoint col... |
| disjeq1f 33209 | Equality theorem for disjo... |
| disjxun0 33210 | Simplify a disjoint union.... |
| disjdifprg 33211 | A trivial partition into a... |
| disjdifprg2 33212 | A trivial partition of a s... |
| disji2f 33213 | Property of a disjoint col... |
| disjif 33214 | Property of a disjoint col... |
| disjorf 33215 | Two ways to say that a col... |
| disjorsf 33216 | Two ways to say that a col... |
| disjif2 33217 | Property of a disjoint col... |
| disjabrex 33218 | Rewriting a disjoint colle... |
| disjabrexf 33219 | Rewriting a disjoint colle... |
| disjpreima 33220 | A preimage of a disjoint s... |
| disjrnmpt 33221 | Rewriting a disjoint colle... |
| disjin 33222 | If a collection is disjoin... |
| disjin2 33223 | If a collection is disjoin... |
| disjxpin 33224 | Derive a disjunction over ... |
| iundisjf 33225 | Rewrite a countable union ... |
| iundisj2f 33226 | A disjoint union is disjoi... |
| disjrdx 33227 | Re-index a disjunct collec... |
| disjex 33228 | Two ways to say that two c... |
| disjexc 33229 | A variant of ~ disjex , ap... |
| disjunsn 33230 | Append an element to a dis... |
| disjun0 33231 | Adding the empty element p... |
| disjiunel 33232 | A set of elements B of a d... |
| disjuniel 33233 | A set of elements B of a d... |
| xpdisjres 33234 | Restriction of a constant ... |
| opeldifid 33235 | Ordered pair elementhood o... |
| difres 33236 | Case when class difference... |
| imadifxp 33237 | Image of the difference wi... |
| relfi 33238 | A relation (set) is finite... |
| 0res 33239 | Restriction of the empty f... |
| fcoinver 33240 | Build an equivalence relat... |
| fcoinvbr 33241 | Binary relation for the eq... |
| brabgaf 33242 | The law of concretion for ... |
| brelg 33243 | Two things in a binary rel... |
| br8d 33244 | Substitution for an eight-... |
| fnfvor 33245 | Relation between two funct... |
| ofrco 33246 | Function relation between ... |
| opabdm 33247 | Domain of an ordered-pair ... |
| opabrn 33248 | Range of an ordered-pair c... |
| opabssi 33249 | Sufficient condition for a... |
| opabid2ss 33250 | One direction of ~ opabid2... |
| ssrelf 33251 | A subclass relationship de... |
| eqrelrd2 33252 | A version of ~ eqrelrdv2 w... |
| erbr3b 33253 | Biconditional for equivale... |
| iunsnima 33254 | Image of a singleton by an... |
| iunsnima2 33255 | Version of ~ iunsnima with... |
| fconst7v 33256 | An alternative way to expr... |
| constcof 33257 | Composition with a constan... |
| ac6sf2 33258 | Alternate version of ~ ac6... |
| ac6mapd 33259 | Axiom of choice equivalent... |
| fnresin 33260 | Restriction of a function ... |
| fresunsn 33261 | Recover the original funct... |
| f1o3d 33262 | Describe an implicit one-t... |
| eldmne0 33263 | A function of nonempty dom... |
| f1rnen 33264 | Equinumerosity of the rang... |
| f1oeq3dd 33265 | Equality deduction for one... |
| rinvf1o 33266 | Sufficient conditions for ... |
| fresf1o 33267 | Conditions for a restricti... |
| nfpconfp 33268 | The set of fixed points of... |
| fmptco1f1o 33269 | The action of composing (t... |
| cofmpt2 33270 | Express composition of a m... |
| f1mptrn 33271 | Express injection for a ma... |
| dfimafnf 33272 | Alternate definition of th... |
| funimass4f 33273 | Membership relation for th... |
| suppss2f 33274 | Show that the support of a... |
| ofrn 33275 | The range of the function ... |
| ofrn2 33276 | The range of the function ... |
| off2 33277 | The function operation pro... |
| ofresid 33278 | Applying an operation rest... |
| unipreima 33279 | Preimage of a class union.... |
| opfv 33280 | Value of a function produc... |
| xppreima 33281 | The preimage of a Cartesia... |
| 2ndimaxp 33282 | Image of a cartesian produ... |
| dmdju 33283 | Domain of a disjoint union... |
| djussxp2 33284 | Stronger version of ~ djus... |
| 2ndresdju 33285 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 33286 | The ` 2nd ` function restr... |
| xppreima2 33287 | The preimage of a Cartesia... |
| abfmpunirn 33288 | Membership in a union of a... |
| rabfmpunirn 33289 | Membership in a union of a... |
| abfmpeld 33290 | Membership in an element o... |
| abfmpel 33291 | Membership in an element o... |
| fmptdf2 33292 | Domain and codomain of the... |
| fmptcof2 33293 | Composition of two functio... |
| fcomptf 33294 | Express composition of two... |
| acunirnmpt 33295 | Axiom of choice for the un... |
| acunirnmpt2 33296 | Axiom of choice for the un... |
| acunirnmpt2f 33297 | Axiom of choice for the un... |
| aciunf1lem 33298 | Choice in an index union. ... |
| aciunf1 33299 | Choice in an index union. ... |
| ofoprabco 33300 | Function operation as a co... |
| ofpreima 33301 | Express the preimage of a ... |
| ofpreima2 33302 | Express the preimage of a ... |
| funcnv5mpt 33303 | Two ways to say that a fun... |
| funcnv4mpt 33304 | Two ways to say that a fun... |
| preimane 33305 | Different elements have di... |
| fnpreimac 33306 | Choose a set ` x ` contain... |
| fgreu 33307 | Exactly one point of a fun... |
| fcnvgreu 33308 | If the converse of a relat... |
| rnmposs 33309 | The range of an operation ... |
| mptssALT 33310 | Deduce subset relation of ... |
| dfcnv2 33311 | Alternative definition of ... |
| partfun2 33312 | Rewrite a function defined... |
| rnressnsn 33313 | The range of a restriction... |
| mpomptxf 33314 | Express a two-argument fun... |
| of0r 33315 | Function operation with th... |
| suppovss 33316 | A bound for the support of... |
| elsuppfnd 33317 | Deduce membership in the s... |
| fisuppov1 33318 | Formula building theorem f... |
| suppun2 33319 | The support of a union is ... |
| fdifsupp 33320 | Express the support of a f... |
| suppiniseg 33321 | Relation between the suppo... |
| fsuppinisegfi 33322 | The initial segment ` ( ``... |
| fressupp 33323 | The restriction of a funct... |
| fdifsuppconst 33324 | A function is a zero const... |
| ressupprn 33325 | The range of a function re... |
| supppreima 33326 | Express the support of a f... |
| fsupprnfi 33327 | Finite support implies fin... |
| mptiffisupp 33328 | Conditions for a mapping f... |
| cosnopne 33329 | Composition of two ordered... |
| cosnop 33330 | Composition of two ordered... |
| cnvprop 33331 | Converse of a pair of orde... |
| brprop 33332 | Binary relation for a pair... |
| mptprop 33333 | Rewrite pairs of ordered p... |
| coprprop 33334 | Composition of two pairs o... |
| fmptunsnop 33335 | Two ways to express a func... |
| gtiso 33336 | Two ways to write a strict... |
| isoun 33337 | Infer an isomorphism from ... |
| disjdsct 33338 | A disjoint collection is d... |
| df1stres 33339 | Definition for a restricti... |
| df2ndres 33340 | Definition for a restricti... |
| 1stpreimas 33341 | The preimage of a singleto... |
| 1stpreima 33342 | The preimage by ` 1st ` is... |
| 2ndpreima 33343 | The preimage by ` 2nd ` is... |
| curry2ima 33344 | The image of a curried fun... |
| preiman0 33345 | The preimage of a nonempty... |
| intimafv 33346 | The intersection of an ima... |
| snct 33347 | A singleton is countable. ... |
| prct 33348 | An unordered pair is count... |
| mpocti 33349 | An operation is countable ... |
| mptctf 33350 | A countable mapping set is... |
| abrexctf 33351 | An image set of a countabl... |
| padct 33352 | Index a countable set with... |
| f1od2 33353 | Sufficient condition for a... |
| fcobij 33354 | Composing functions with a... |
| fcobijfs 33355 | Composing finitely support... |
| fcobijfs2 33356 | Composing finitely support... |
| suppss3 33357 | Deduce a function's suppor... |
| fsuppcurry1 33358 | Finite support of a currie... |
| fsuppcurry2 33359 | Finite support of a currie... |
| offinsupp1 33360 | Finite support for a funct... |
| ffs2 33361 | Rewrite a function's suppo... |
| ffsrn 33362 | The range of a finitely su... |
| cocnvf1o 33363 | Composing with the inverse... |
| resf1o 33364 | Restriction of functions t... |
| maprnin 33365 | Restricting the range of t... |
| fpwrelmapffslem 33366 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 33367 | Define a canonical mapping... |
| fpwrelmapffs 33368 | Define a canonical mapping... |
| sgnval2 33369 | Value of the signum of a r... |
| creq0 33370 | The real representation of... |
| 1nei 33371 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 33372 | An integer unit times itse... |
| nnmulge 33373 | Multiplying by a positive ... |
| submuladdd 33374 | The product of a differenc... |
| binom2subadd 33375 | The difference of the squa... |
| cjsubd 33376 | Complex conjugate distribu... |
| re0cj 33377 | The conjugate of a pure im... |
| receqid 33378 | Real numbers equal to thei... |
| pythagreim 33379 | A simplified version of th... |
| efiargd 33380 | The exponential of the "ar... |
| arginv 33381 | The argument of the invers... |
| argcj 33382 | The argument of the conjug... |
| quad3d 33383 | Variant of quadratic equat... |
| lt2addrd 33384 | If the right-hand side of ... |
| nn0mnfxrd 33385 | Nonnegative integers or mi... |
| xrlelttric 33386 | Trichotomy law for extende... |
| xaddeq0 33387 | Two extended reals which a... |
| rexmul2 33388 | If the result ` A ` of an ... |
| xrinfm 33389 | The extended real numbers ... |
| le2halvesd 33390 | A sum is less than the who... |
| xraddge02 33391 | A number is less than or e... |
| xrge0addge 33392 | A number is less than or e... |
| xlt2addrd 33393 | If the right-hand side of ... |
| xrge0infss 33394 | Any subset of nonnegative ... |
| xrge0infssd 33395 | Inequality deduction for i... |
| xrge0addcld 33396 | Nonnegative extended reals... |
| xrge0subcld 33397 | Condition for closure of n... |
| infxrge0lb 33398 | A member of a set of nonne... |
| infxrge0glb 33399 | The infimum of a set of no... |
| infxrge0gelb 33400 | The infimum of a set of no... |
| xrofsup 33401 | The supremum is preserved ... |
| supxrnemnf 33402 | The supremum of a nonempty... |
| xnn0gt0 33403 | Nonzero extended nonnegati... |
| xnn01gt 33404 | An extended nonnegative in... |
| nn0xmulclb 33405 | Finite multiplication in t... |
| xnn0nn0d 33406 | Conditions for an extended... |
| xnn0nnd 33407 | Conditions for an extended... |
| joiniooico 33408 | Disjoint joining an open i... |
| ubico 33409 | A right-open interval does... |
| xeqlelt 33410 | Equality in terms of 'less... |
| eliccelico 33411 | Relate elementhood to a cl... |
| elicoelioo 33412 | Relate elementhood to a cl... |
| iocinioc2 33413 | Intersection between two o... |
| xrdifh 33414 | Class difference of a half... |
| iocinif 33415 | Relate intersection of two... |
| difioo 33416 | The difference between two... |
| difico 33417 | The difference between two... |
| uzssico 33418 | Upper integer sets are a s... |
| fz2ssnn0 33419 | A finite set of sequential... |
| nndiffz1 33420 | Upper set of the positive ... |
| ssnnssfz 33421 | For any finite subset of `... |
| fzm1ne1 33422 | Elementhood of an integer ... |
| fzspl 33423 | Split the last element of ... |
| fzdif2 33424 | Split the last element of ... |
| fzodif2 33425 | Split the last element of ... |
| fzodif1 33426 | Set difference of two half... |
| fzsplit3 33427 | Split a finite interval of... |
| nn0diffz0 33428 | Upper set of the nonnegati... |
| bcm1n 33429 | The proportion of one bino... |
| iundisjfi 33430 | Rewrite a countable union ... |
| iundisj2fi 33431 | A disjoint union is disjoi... |
| iundisjcnt 33432 | Rewrite a countable union ... |
| iundisj2cnt 33433 | A countable disjoint union... |
| f1ocnt 33434 | Given a countable set ` A ... |
| fz1nnct 33435 | NN and integer ranges star... |
| fz1nntr 33436 | NN and integer ranges star... |
| fzo0opth 33437 | Equality for a half open i... |
| nn0difffzod 33438 | A nonnegative integer that... |
| suppssnn0 33439 | Show that the support of a... |
| hashunif 33440 | The cardinality of a disjo... |
| hashxpe 33441 | The size of the Cartesian ... |
| hashgt1 33442 | Restate "set contains at l... |
| hashne0 33443 | Deduce that the size of a ... |
| hashimaf1 33444 | Taking the image of a set ... |
| elq2 33445 | Elementhood in the rationa... |
| znumd 33446 | Numerator of an integer. ... |
| zdend 33447 | Denominator of an integer.... |
| numdenneg 33448 | Numerator and denominator ... |
| divnumden2 33449 | Calculate the reduced form... |
| expgt0b 33450 | A real number ` A ` raised... |
| nn0split01 33451 | Split 0 and 1 from the non... |
| nn0disj01 33452 | The pair ` { 0 , 1 } ` doe... |
| nnindf 33453 | Principle of Mathematical ... |
| nn0min 33454 | Extracting the minimum pos... |
| subne0nn 33455 | A nonnegative difference i... |
| ltesubnnd 33456 | Subtracting an integer num... |
| fprodeq02 33457 | If one of the factors is z... |
| fprodex01 33458 | A product of factors equal... |
| prodpr 33459 | A product over a pair is t... |
| prodtp 33460 | A product over a triple is... |
| fsumub 33461 | An upper bound for a term ... |
| fsumiunle 33462 | Upper bound for a sum of n... |
| dfdec100 33463 | Split the hundreds from a ... |
| sgnsgn 33464 | Signum is idempotent. (Co... |
| sgnmulsgp 33465 | If two real numbers are of... |
| nexple 33466 | A lower bound for an expon... |
| 2exple2exp 33467 | If a nonnegative integer `... |
| expevenpos 33468 | Even powers are positive. ... |
| oexpled 33469 | Odd power monomials are mo... |
| indsumin 33470 | Finite sum of a product wi... |
| prodindf 33471 | The product of indicators ... |
| indsn 33472 | The indicator function of ... |
| indf1o 33473 | The bijection between a po... |
| indpreima 33474 | A function with range ` { ... |
| indf1ofs 33475 | The bijection between fini... |
| indsupp 33476 | The support of the indicat... |
| indfsd 33477 | The indicator function of ... |
| indfsid 33478 | Conditions for a function ... |
| dp2eq1 33481 | Equality theorem for the d... |
| dp2eq2 33482 | Equality theorem for the d... |
| dp2eq1i 33483 | Equality theorem for the d... |
| dp2eq2i 33484 | Equality theorem for the d... |
| dp2eq12i 33485 | Equality theorem for the d... |
| dp20u 33486 | Add a zero in the tenths (... |
| dp20h 33487 | Add a zero in the unit pla... |
| dp2cl 33488 | Closure for the decimal fr... |
| dp2clq 33489 | Closure for a decimal frac... |
| rpdp2cl 33490 | Closure for a decimal frac... |
| rpdp2cl2 33491 | Closure for a decimal frac... |
| dp2lt10 33492 | Decimal fraction builds re... |
| dp2lt 33493 | Comparing two decimal frac... |
| dp2ltsuc 33494 | Comparing a decimal fracti... |
| dp2ltc 33495 | Comparing two decimal expa... |
| dpval 33498 | Define the value of the de... |
| dpcl 33499 | Prove that the closure of ... |
| dpfrac1 33500 | Prove a simple equivalence... |
| dpval2 33501 | Value of the decimal point... |
| dpval3 33502 | Value of the decimal point... |
| dpmul10 33503 | Multiply by 10 a decimal e... |
| decdiv10 33504 | Divide a decimal number by... |
| dpmul100 33505 | Multiply by 100 a decimal ... |
| dp3mul10 33506 | Multiply by 10 a decimal e... |
| dpmul1000 33507 | Multiply by 1000 a decimal... |
| dpval3rp 33508 | Value of the decimal point... |
| dp0u 33509 | Add a zero in the tenths p... |
| dp0h 33510 | Remove a zero in the units... |
| rpdpcl 33511 | Closure of the decimal poi... |
| dplt 33512 | Comparing two decimal expa... |
| dplti 33513 | Comparing a decimal expans... |
| dpgti 33514 | Comparing a decimal expans... |
| dpltc 33515 | Comparing two decimal inte... |
| dpexpp1 33516 | Add one zero to the mantis... |
| 0dp2dp 33517 | Multiply by 10 a decimal e... |
| dpadd2 33518 | Addition with one decimal,... |
| dpadd 33519 | Addition with one decimal.... |
| dpadd3 33520 | Addition with two decimals... |
| dpmul 33521 | Multiplication with one de... |
| dpmul4 33522 | An upper bound to multipli... |
| threehalves 33523 | Example theorem demonstrat... |
| 1mhdrd 33524 | Example theorem demonstrat... |
| xdivval 33527 | Value of division: the (un... |
| xrecex 33528 | Existence of reciprocal of... |
| xmulcand 33529 | Cancellation law for exten... |
| xreceu 33530 | Existential uniqueness of ... |
| xdivcld 33531 | Closure law for the extend... |
| xdivcl 33532 | Closure law for the extend... |
| xdivmul 33533 | Relationship between divis... |
| rexdiv 33534 | The extended real division... |
| xdivrec 33535 | Relationship between divis... |
| xdivid 33536 | A number divided by itself... |
| xdiv0 33537 | Division into zero is zero... |
| xdiv0rp 33538 | Division into zero is zero... |
| eliccioo 33539 | Membership in a closed int... |
| elxrge02 33540 | Elementhood in the set of ... |
| xdivpnfrp 33541 | Plus infinity divided by a... |
| rpxdivcld 33542 | Closure law for extended d... |
| xrpxdivcld 33543 | Closure law for extended d... |
| wrdres 33544 | Condition for the restrict... |
| wrdsplex 33545 | Existence of a split of a ... |
| wrdfsupp 33546 | A word has finite support.... |
| wrdpmcl 33547 | Closure of a word with per... |
| pfx1s2 33548 | The prefix of length 1 of ... |
| pfxrn2 33549 | The range of a prefix of a... |
| pfxrn3 33550 | Express the range of a pre... |
| pfxf1 33551 | Condition for a prefix to ... |
| s2f1 33552 | Conditions for a length 2 ... |
| s3f1 33553 | Conditions for a length 3 ... |
| s3clhash 33554 | Closure of the words of le... |
| pfxlsw2ccat 33555 | Reconstruct a word from it... |
| ccatws1f1o 33556 | Conditions for the concate... |
| ccatws1f1olast 33557 | Two ways to reorder symbol... |
| wrdt2ind 33558 | Perform an induction over ... |
| swrdrn2 33559 | The range of a subword is ... |
| swrdrndisj 33560 | Condition for the range of... |
| splfv3 33561 | Symbols to the right of a ... |
| 1cshid 33562 | Cyclically shifting a sing... |
| cshw1s2 33563 | Cyclically shifting a leng... |
| cshwrnid 33564 | Cyclically shifting a word... |
| cshf1o 33565 | Condition for the cyclic s... |
| ressplusf 33566 | The group operation functi... |
| ressnm 33567 | The norm in a restricted s... |
| abvpropd2 33568 | Weaker version of ~ abvpro... |
| ressprs 33569 | The restriction of a prose... |
| posrasymb 33570 | A poset ordering is asymme... |
| odutos 33571 | Being a toset is a self-du... |
| tlt2 33572 | In a Toset, two elements m... |
| tlt3 33573 | In a Toset, two elements m... |
| trleile 33574 | In a Toset, two elements m... |
| toslublem 33575 | Lemma for ~ toslub and ~ x... |
| toslub 33576 | In a toset, the lowest upp... |
| tosglblem 33577 | Lemma for ~ tosglb and ~ x... |
| tosglb 33578 | Same theorem as ~ toslub ,... |
| clatp0cl 33579 | The poset zero of a comple... |
| clatp1cl 33580 | The poset one of a complet... |
| mntoval 33585 | Operation value of the mon... |
| ismnt 33586 | Express the statement " ` ... |
| ismntd 33587 | Property of being a monoto... |
| mntf 33588 | A monotone function is a f... |
| mgcoval 33589 | Operation value of the mon... |
| mgcval 33590 | Monotone Galois connection... |
| mgcf1 33591 | The lower adjoint ` F ` of... |
| mgcf2 33592 | The upper adjoint ` G ` of... |
| mgccole1 33593 | An inequality for the kern... |
| mgccole2 33594 | Inequality for the closure... |
| mgcmnt1 33595 | The lower adjoint ` F ` of... |
| mgcmnt2 33596 | The upper adjoint ` G ` of... |
| mgcmntco 33597 | A Galois connection like s... |
| dfmgc2lem 33598 | Lemma for dfmgc2, backward... |
| dfmgc2 33599 | Alternate definition of th... |
| mgcmnt1d 33600 | Galois connection implies ... |
| mgcmnt2d 33601 | Galois connection implies ... |
| mgccnv 33602 | The inverse Galois connect... |
| pwrssmgc 33603 | Given a function ` F ` , e... |
| mgcf1olem1 33604 | Property of a Galois conne... |
| mgcf1olem2 33605 | Property of a Galois conne... |
| mgcf1o 33606 | Given a Galois connection,... |
| xrs0 33609 | The zero of the extended r... |
| xrslt 33610 | The "strictly less than" r... |
| xrsinvgval 33611 | The inversion operation in... |
| xrsmulgzz 33612 | The "multiple" function in... |
| xrstos 33613 | The extended real numbers ... |
| xrsclat 33614 | The extended real numbers ... |
| xrsp0 33615 | The poset 0 of the extende... |
| xrsp1 33616 | The poset 1 of the extende... |
| xrge00 33617 | The zero of the extended n... |
| xrge0mulgnn0 33618 | The group multiple functio... |
| xrge0addass 33619 | Associativity of extended ... |
| xrge0addgt0 33620 | The sum of nonnegative and... |
| xrge0adddir 33621 | Right-distributivity of ex... |
| xrge0adddi 33622 | Left-distributivity of ext... |
| xrge0npcan 33623 | Extended nonnegative real ... |
| fsumrp0cl 33624 | Closure of a finite sum of... |
| mndcld 33625 | Closure of the operation o... |
| mndassd 33626 | A monoid operation is asso... |
| mndlrinv 33627 | In a monoid, if an element... |
| mndlrinvb 33628 | In a monoid, if an element... |
| mndlactf1 33629 | If an element ` X ` of a m... |
| mndlactfo 33630 | An element ` X ` of a mono... |
| mndractf1 33631 | If an element ` X ` of a m... |
| mndractfo 33632 | An element ` X ` of a mono... |
| mndlactf1o 33633 | An element ` X ` of a mono... |
| mndractf1o 33634 | An element ` X ` of a mono... |
| cmn4d 33635 | Commutative/associative la... |
| cmn246135 33636 | Rearrange terms in a commu... |
| cmn145236 33637 | Rearrange terms in a commu... |
| abliso 33638 | The image of an Abelian gr... |
| lmhmghmd 33639 | A module homomorphism is a... |
| mhmimasplusg 33640 | Value of the operation of ... |
| lmhmimasvsca 33641 | Value of the scalar produc... |
| grpidcld 33642 | The identity element of a ... |
| grpinvinvd 33643 | Double inverse law for gro... |
| grpsubcld 33644 | Closure of group subtracti... |
| subgsubcld 33645 | A subgroup is closed under... |
| subgmulgcld 33646 | Closure of the group multi... |
| ressmulgnn0d 33647 | Values for the group multi... |
| ablcomd 33648 | An abelian group operation... |
| gsumsubg 33649 | The group sum in a subgrou... |
| gsumsra 33650 | The group sum in a subring... |
| gsummpt2co 33651 | Split a finite sum into a ... |
| gsummpt2d 33652 | Express a finite sum over ... |
| lmodvslmhm 33653 | Scalar multiplication in a... |
| gsumvsmul1 33654 | Pull a scalar multiplicati... |
| gsummptres 33655 | Extend a finite group sum ... |
| gsummptres2 33656 | Extend a finite group sum ... |
| gsummptfsres 33657 | Extend a finitely supporte... |
| gsummptf1od 33658 | Re-index a finite group su... |
| gsummptrev 33659 | Revert ordering in a group... |
| gsummptp1 33660 | Reindex a zero-based sum a... |
| gsummptfzsplitra 33661 | Split a group sum expresse... |
| gsummptfzsplitla 33662 | Split a group sum expresse... |
| gsummptfsf1o 33663 | Re-index a finite group su... |
| gsumfs2d 33664 | Express a finite sum over ... |
| gsumzresunsn 33665 | Append an element to a fin... |
| gsumpart 33666 | Express a group sum as a d... |
| gsumtp 33667 | Group sum of an unordered ... |
| gsumzrsum 33668 | Relate a group sum on ` ZZ... |
| gsummulgc2 33669 | A finite group sum multipl... |
| gsumhashmul 33670 | Express a group sum by gro... |
| gsummulsubdishift1 33671 | Distribute a subtraction o... |
| gsummulsubdishift2 33672 | Distribute a subtraction o... |
| gsummulsubdishift1s 33673 | Distribute a subtraction o... |
| gsummulsubdishift2s 33674 | Distribute a subtraction o... |
| suppgsumssiun 33675 | The support of a function ... |
| xrge0tsmsd 33676 | Any finite or infinite sum... |
| xrge0tsmsbi 33677 | Any limit of a finite or i... |
| xrge0tsmseq 33678 | Any limit of a finite or i... |
| gsumwun 33679 | In a commutative ring, a g... |
| gsumwrd2dccatlem 33680 | Lemma for ~ gsumwrd2dccat ... |
| gsumwrd2dccat 33681 | Rewrite a sum ranging over... |
| cntzun 33682 | The centralizer of a union... |
| cntzsnid 33683 | The centralizer of the ide... |
| cntrcrng 33684 | The center of a ring is a ... |
| symgfcoeu 33685 | Uniqueness property of per... |
| symgcom 33686 | Two permutations ` X ` and... |
| symgcom2 33687 | Two permutations ` X ` and... |
| symgcntz 33688 | All elements of a (finite)... |
| odpmco 33689 | The composition of two odd... |
| symgsubg 33690 | The value of the group sub... |
| pmtrprfv2 33691 | In a transposition of two ... |
| pmtrcnel 33692 | Composing a permutation ` ... |
| pmtrcnel2 33693 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 33694 | Composing a permutation ` ... |
| fzo0pmtrlast 33695 | Reorder a half-open intege... |
| wrdpmtrlast 33696 | Reorder a word, so that th... |
| pmtridf1o 33697 | Transpositions of ` X ` an... |
| pmtridfv1 33698 | Value at X of the transpos... |
| pmtridfv2 33699 | Value at Y of the transpos... |
| psgnid 33700 | Permutation sign of the id... |
| psgndmfi 33701 | For a finite base set, the... |
| pmtrto1cl 33702 | Useful lemma for the follo... |
| psgnfzto1stlem 33703 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 33704 | Value of our permutation `... |
| fzto1st1 33705 | Special case where the per... |
| fzto1st 33706 | The function moving one el... |
| fzto1stinvn 33707 | Value of the inverse of ou... |
| psgnfzto1st 33708 | The permutation sign for m... |
| tocycval 33711 | Value of the cycle builder... |
| tocycfv 33712 | Function value of a permut... |
| tocycfvres1 33713 | A cyclic permutation is a ... |
| tocycfvres2 33714 | A cyclic permutation is th... |
| cycpmfvlem 33715 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 33716 | Value of a cycle function ... |
| cycpmfv2 33717 | Value of a cycle function ... |
| cycpmfv3 33718 | Values outside of the orbi... |
| cycpmcl 33719 | Cyclic permutations are pe... |
| tocycf 33720 | The permutation cycle buil... |
| tocyc01 33721 | Permutation cycles built f... |
| cycpm2tr 33722 | A cyclic permutation of 2 ... |
| cycpm2cl 33723 | Closure for the 2-cycles. ... |
| cyc2fv1 33724 | Function value of a 2-cycl... |
| cyc2fv2 33725 | Function value of a 2-cycl... |
| trsp2cyc 33726 | Exhibit the word a transpo... |
| cycpmco2f1 33727 | The word U used in ~ cycpm... |
| cycpmco2rn 33728 | The orbit of the compositi... |
| cycpmco2lem1 33729 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 33730 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 33731 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 33732 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 33733 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 33734 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 33735 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 33736 | The composition of a cycli... |
| cyc2fvx 33737 | Function value of a 2-cycl... |
| cycpm3cl 33738 | Closure of the 3-cycles in... |
| cycpm3cl2 33739 | Closure of the 3-cycles in... |
| cyc3fv1 33740 | Function value of a 3-cycl... |
| cyc3fv2 33741 | Function value of a 3-cycl... |
| cyc3fv3 33742 | Function value of a 3-cycl... |
| cyc3co2 33743 | Represent a 3-cycle as a c... |
| cycpmconjvlem 33744 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 33745 | A formula for computing co... |
| cycpmrn 33746 | The range of the word used... |
| tocyccntz 33747 | All elements of a (finite)... |
| evpmval 33748 | Value of the set of even p... |
| cnmsgn0g 33749 | The neutral element of the... |
| evpmsubg 33750 | The alternating group is a... |
| evpmid 33751 | The identity is an even pe... |
| altgnsg 33752 | The alternating group ` ( ... |
| cyc3evpm 33753 | 3-Cycles are even permutat... |
| cyc3genpmlem 33754 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 33755 | The alternating group ` A ... |
| cycpmgcl 33756 | Cyclic permutations are pe... |
| cycpmconjslem1 33757 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 33758 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 33759 | All cycles of the same len... |
| cyc3conja 33760 | All 3-cycles are conjugate... |
| sgnsv 33763 | The sign mapping. (Contri... |
| sgnsval 33764 | The sign value. (Contribu... |
| sgnsf 33765 | The sign function. (Contr... |
| fxpval 33768 | Value of the set of fixed ... |
| fxpss 33769 | The set of fixed points is... |
| fxpgaval 33770 | Value of the set of fixed ... |
| isfxp 33771 | Property of being a fixed ... |
| fxpgaeq 33772 | A fixed point ` X ` is inv... |
| conjga 33773 | Group conjugation induces ... |
| cntrval2 33774 | Express the center ` Z ` o... |
| fxpsubm 33775 | Provided the group action ... |
| fxpsubg 33776 | The fixed points of a grou... |
| fxpsubrg 33777 | The fixed points of a grou... |
| fxpsdrg 33778 | The fixed points of a grou... |
| inftmrel 33783 | The infinitesimal relation... |
| isinftm 33784 | Express ` x ` is infinites... |
| isarchi 33785 | Express the predicate " ` ... |
| pnfinf 33786 | Plus infinity is an infini... |
| xrnarchi 33787 | The completed real line is... |
| isarchi2 33788 | Alternative way to express... |
| submarchi 33789 | A submonoid is archimedean... |
| isarchi3 33790 | This is the usual definiti... |
| archirng 33791 | Property of Archimedean or... |
| archirngz 33792 | Property of Archimedean le... |
| archiexdiv 33793 | In an Archimedean group, g... |
| archiabllem1a 33794 | Lemma for ~ archiabl : In... |
| archiabllem1b 33795 | Lemma for ~ archiabl . (C... |
| archiabllem1 33796 | Archimedean ordered groups... |
| archiabllem2a 33797 | Lemma for ~ archiabl , whi... |
| archiabllem2c 33798 | Lemma for ~ archiabl . (C... |
| archiabllem2b 33799 | Lemma for ~ archiabl . (C... |
| archiabllem2 33800 | Archimedean ordered groups... |
| archiabl 33801 | Archimedean left- and righ... |
| isarchiofld 33802 | Axiom of Archimedes : a ch... |
| isslmd 33805 | The predicate "is a semimo... |
| slmdlema 33806 | Lemma for properties of a ... |
| lmodslmd 33807 | Left semimodules generaliz... |
| slmdcmn 33808 | A semimodule is a commutat... |
| slmdmnd 33809 | A semimodule is a monoid. ... |
| slmdsrg 33810 | The scalar component of a ... |
| slmdbn0 33811 | The base set of a semimodu... |
| slmdacl 33812 | Closure of ring addition f... |
| slmdmcl 33813 | Closure of ring multiplica... |
| slmdsn0 33814 | The set of scalars in a se... |
| slmdvacl 33815 | Closure of vector addition... |
| slmdass 33816 | Semiring left module vecto... |
| slmdvscl 33817 | Closure of scalar product ... |
| slmdvsdi 33818 | Distributive law for scala... |
| slmdvsdir 33819 | Distributive law for scala... |
| slmdvsass 33820 | Associative law for scalar... |
| slmd0cl 33821 | The ring zero in a semimod... |
| slmd1cl 33822 | The ring unity in a semiri... |
| slmdvs1 33823 | Scalar product with ring u... |
| slmd0vcl 33824 | The zero vector is a vecto... |
| slmd0vlid 33825 | Left identity law for the ... |
| slmd0vrid 33826 | Right identity law for the... |
| slmd0vs 33827 | Zero times a vector is the... |
| slmdvs0 33828 | Anything times the zero ve... |
| gsumvsca1 33829 | Scalar product of a finite... |
| gsumvsca2 33830 | Scalar product of a finite... |
| prmsimpcyc 33831 | A group of prime order is ... |
| ringrngd 33832 | A unital ring is a non-uni... |
| urpropd 33833 | Sufficient condition for r... |
| subrgmcld 33834 | A subring is closed under ... |
| ress1r 33835 | ` 1r ` is unaffected by re... |
| ringm1expp1 33836 | Ring exponentiation of min... |
| ringinvval 33837 | The ring inverse expressed... |
| dvrcan5 33838 | Cancellation law for commo... |
| subrgchr 33839 | If ` A ` is a subring of `... |
| rmfsupp2 33840 | A mapping of a multiplicat... |
| unitnz 33841 | In a nonzero ring, a unit ... |
| isunit2 33842 | Alternate definition of be... |
| isunit3 33843 | Alternate definition of be... |
| isunitc 33844 | Characterize units in a co... |
| elrgspnlem1 33845 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem2 33846 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem3 33847 | Lemma for ~ elrgspn . (Co... |
| elrgspnlem4 33848 | Lemma for ~ elrgspn . (Co... |
| elrgspn 33849 | Membership in the subring ... |
| elrgspnsubrunlem1 33850 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrunlem2 33851 | Lemma for ~ elrgspnsubrun ... |
| elrgspnsubrun 33852 | Membership in the ring spa... |
| irrednzr 33853 | A ring with an irreducible... |
| 0ringsubrg 33854 | A subring of a zero ring i... |
| 0ringcring 33855 | The zero ring is commutati... |
| reldmrloc 33860 | Ring localization is a pro... |
| erlval 33861 | Value of the ring localiza... |
| rlocval 33862 | Expand the value of the ri... |
| erlcl1 33863 | Closure for the ring local... |
| erlcl2 33864 | Closure for the ring local... |
| erldi 33865 | Main property of the ring ... |
| erlbrd 33866 | Deduce the ring localizati... |
| erlbr2d 33867 | Deduce the ring localizati... |
| erler 33868 | The relation used to build... |
| erld2 33869 | Main property of the ring ... |
| elrlocbasi 33870 | Membership in the basis of... |
| rlocbas 33871 | The base set of a ring loc... |
| rlocaddval 33872 | Value of the addition in t... |
| rlocmulval 33873 | Value of the addition in t... |
| rloccring 33874 | The ring localization ` L ... |
| rloc0g 33875 | The zero of a ring localiz... |
| rloc1r 33876 | The multiplicative identit... |
| rlocf1 33877 | The embedding ` F ` of a r... |
| rlocinvunit 33878 | In the localization of a r... |
| rlocisunit 33879 | Characterize the units of ... |
| domnmuln0rd 33880 | In a domain, factors of a ... |
| domnprodn0 33881 | In a domain, a finite prod... |
| domnprodeq0 33882 | A product over a domain is... |
| domnpropd 33883 | If two structures have the... |
| idompropd 33884 | If two structures have the... |
| idomrcan 33885 | Right-cancellation law for... |
| 1rrg 33886 | The multiplicative identit... |
| rrgsubm 33887 | The left regular elements ... |
| subrdom 33888 | A subring of a domain is a... |
| subridom 33889 | A subring of an integral d... |
| subrfld 33890 | A subring of a field is an... |
| ricnzr1 33891 | A ring isomorphism maps a ... |
| ricdomn1 33892 | A ring isomorphism maps a ... |
| ricdomn 33893 | A ring is a domain if and ... |
| eufndx 33896 | Index value of the Euclide... |
| eufid 33897 | Utility theorem: index-ind... |
| rndrhmcl 33900 | The image of a division ri... |
| qfld 33901 | The field of rational numb... |
| subsdrg 33902 | A subring of a sub-divisio... |
| sdrgdvcl 33903 | A sub-division-ring is clo... |
| sdrginvcl 33904 | A sub-division-ring is clo... |
| primefldchr 33905 | The characteristic of a pr... |
| fracval 33908 | Value of the field of frac... |
| fracbas 33909 | The base of the field of f... |
| fracerl 33910 | Rewrite the ring localizat... |
| fracf1 33911 | The embedding of a commuta... |
| fracfld 33912 | The field of fractions of ... |
| idomsubr 33913 | Every integral domain is i... |
| fldgenval 33916 | Value of the field generat... |
| fldgenssid 33917 | The field generated by a s... |
| fldgensdrg 33918 | A generated subfield is a ... |
| fldgenssv 33919 | A generated subfield is a ... |
| fldgenss 33920 | Generated subfields preser... |
| fldgenidfld 33921 | The subfield generated by ... |
| fldgenssp 33922 | The field generated by a s... |
| fldgenid 33923 | The subfield of a field ` ... |
| fldgenfld 33924 | A generated subfield is a ... |
| primefldgen1 33925 | The prime field of a divis... |
| 1fldgenq 33926 | The field of rational numb... |
| rhmdvd 33927 | A ring homomorphism preser... |
| kerunit 33928 | If a unit element lies in ... |
| reldmresv 33931 | The scalar restriction is ... |
| resvval 33932 | Value of structure restric... |
| resvid2 33933 | General behavior of trivia... |
| resvval2 33934 | Value of nontrivial struct... |
| resvsca 33935 | Base set of a structure re... |
| resvlem 33936 | Other elements of a scalar... |
| resvbas 33937 | ` Base ` is unaffected by ... |
| resvplusg 33938 | ` +g ` is unaffected by sc... |
| resvvsca 33939 | ` .s ` is unaffected by sc... |
| resvmulr 33940 | ` .r ` is unaffected by sc... |
| resv0g 33941 | ` 0g ` is unaffected by sc... |
| resv1r 33942 | ` 1r ` is unaffected by sc... |
| resvcmn 33943 | Scalar restriction preserv... |
| gzcrng 33944 | The gaussian integers form... |
| cnfldfld 33945 | The complex numbers form a... |
| reofld 33946 | The real numbers form an o... |
| nn0omnd 33947 | The nonnegative integers f... |
| gsumind 33948 | The group sum of an indica... |
| rearchi 33949 | The field of the real numb... |
| nn0archi 33950 | The monoid of the nonnegat... |
| xrge0slmod 33951 | The extended nonnegative r... |
| qusker 33952 | The kernel of a quotient m... |
| eqgvscpbl 33953 | The left coset equivalence... |
| qusvscpbl 33954 | The quotient map distribut... |
| qusvsval 33955 | Value of the scalar multip... |
| imaslmod 33956 | The image structure of a l... |
| imasmhm 33957 | Given a function ` F ` wit... |
| imasghm 33958 | Given a function ` F ` wit... |
| imasrhm 33959 | Given a function ` F ` wit... |
| imaslmhm 33960 | Given a function ` F ` wit... |
| quslmod 33961 | If ` G ` is a submodule in... |
| quslmhm 33962 | If ` G ` is a submodule of... |
| quslvec 33963 | If ` S ` is a vector subsp... |
| znfermltl 33964 | Fermat's little theorem in... |
| islinds5 33965 | A set is linearly independ... |
| ellspds 33966 | Variation on ~ ellspd . (... |
| 0ellsp 33967 | Zero is in all spans. (Co... |
| 0nellinds 33968 | The group identity cannot ... |
| elrsp 33969 | Write the elements of a ri... |
| ellpi 33970 | Elementhood in a left prin... |
| lpirlidllpi 33971 | In a principal ideal ring,... |
| rspidlid 33972 | The ideal span of an ideal... |
| rsp2idlid 33973 | The ideal span of a two-si... |
| lbslsp 33974 | Any element of a left modu... |
| lindssn 33975 | Any singleton of a nonzero... |
| lindflbs 33976 | Conditions for an independ... |
| islbs5 33977 | An equivalent formulation ... |
| linds2eq 33978 | Deduce equality of element... |
| lindfpropd 33979 | Property deduction for lin... |
| lindspropd 33980 | Property deduction for lin... |
| dvdsruassoi 33981 | If two elements ` X ` and ... |
| dvdsruasso 33982 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33983 | A reformulation of ~ dvdsr... |
| dvdsrspss 33984 | In a ring, an element ` X ... |
| rspsnasso 33985 | Two elements ` X ` and ` Y... |
| unitprodclb 33986 | A finite product is a unit... |
| elgrplsmsn 33987 | Membership in a sumset wit... |
| lsmsnorb 33988 | The sumset of a group with... |
| lsmsnorb2 33989 | The sumset of a single ele... |
| ringlsmss 33990 | Closure of the product of ... |
| ringlsmss1 33991 | The product of an ideal ` ... |
| ringlsmss2 33992 | The product with an ideal ... |
| lsmsnpridl 33993 | The product of the ring wi... |
| lsmsnidl 33994 | The product of the ring wi... |
| lsmssass 33995 | Group sum is associative, ... |
| grplsm0l 33996 | Sumset with the identity s... |
| grplsmid 33997 | The direct sum of an eleme... |
| quslsm 33998 | Express the image by the q... |
| qusbas2 33999 | Alternate definition of th... |
| qus0g 34000 | The identity element of a ... |
| qusima 34001 | The image of a subgroup by... |
| qusrn 34002 | The natural map from eleme... |
| nsgqus0 34003 | A normal subgroup ` N ` is... |
| nsgmgclem 34004 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 34005 | There is a monotone Galois... |
| nsgqusf1olem1 34006 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 34007 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 34008 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 34009 | The canonical projection h... |
| lmhmqusker 34010 | A surjective module homomo... |
| lmicqusker 34011 | The image ` H ` of a modul... |
| intlidl 34012 | The intersection of a none... |
| inlidl 34013 | The intersection of two id... |
| pidlnzb 34014 | A principal ideal is nonze... |
| lidlunitel 34015 | If an ideal ` I ` contains... |
| unitpidl1 34016 | The ideal ` I ` generated ... |
| rhmquskerlem 34017 | The mapping ` J ` induced ... |
| rhmqusker 34018 | A surjective ring homomorp... |
| ricqusker 34019 | The image ` H ` of a ring ... |
| elrspunidl 34020 | Elementhood in the span of... |
| elrspunsn 34021 | Membership to the span of ... |
| lidlincl 34022 | Ideals are closed under in... |
| idlinsubrg 34023 | The intersection between a... |
| rhmimaidl 34024 | The image of an ideal ` I ... |
| drngidlhash 34025 | A ring is a division ring ... |
| mxidlval 34028 | The set of maximal ideals ... |
| ismxidl 34029 | The predicate "is a maxima... |
| mxidlidl 34030 | A maximal ideal is an idea... |
| mxidlnr 34031 | A maximal ideal is proper.... |
| mxidlmax 34032 | A maximal ideal is a maxim... |
| mxidln1 34033 | One is not contained in an... |
| mxidlnzr 34034 | A ring with a maximal idea... |
| mxidlmaxv 34035 | An ideal ` I ` strictly co... |
| crngmxidl 34036 | In a commutative ring, max... |
| mxidlprm 34037 | Every maximal ideal is pri... |
| mxidlirredi 34038 | In an integral domain, the... |
| mxidlirred 34039 | In a principal ideal domai... |
| ssmxidllem 34040 | The set ` P ` used in the ... |
| ssmxidl 34041 | Let ` R ` be a ring, and l... |
| drng0mxidl 34042 | In a division ring, the ze... |
| drngmxidl 34043 | The zero ideal is the only... |
| drngmxidlr 34044 | If a ring's only maximal i... |
| krull 34045 | Krull's theorem: Any nonz... |
| mxidlnzrb 34046 | A ring is nonzero if and o... |
| krullndrng 34047 | Krull's theorem for non-di... |
| opprabs 34048 | The opposite ring of the o... |
| oppreqg 34049 | Group coset equivalence re... |
| opprnsg 34050 | Normal subgroups of the op... |
| opprlidlabs 34051 | The ideals of the opposite... |
| oppr2idl 34052 | Two sided ideal of the opp... |
| opprmxidlabs 34053 | The maximal ideal of the o... |
| opprqusbas 34054 | The base of the quotient o... |
| opprqusplusg 34055 | The group operation of the... |
| opprqus0g 34056 | The group identity element... |
| opprqusmulr 34057 | The multiplication operati... |
| opprqus1r 34058 | The ring unity of the quot... |
| opprqusdrng 34059 | The quotient of the opposi... |
| qsdrngilem 34060 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 34061 | A quotient by a maximal le... |
| qsdrnglem2 34062 | Lemma for ~ qsdrng . (Con... |
| qsdrng 34063 | An ideal ` M ` is both lef... |
| qsfld 34064 | An ideal ` M ` in the comm... |
| mxidlprmALT 34065 | Every maximal ideal is pri... |
| drnglring 34066 | A division ring is a local... |
| dflring2 34067 | Alternate definition of a ... |
| dflringlem 34068 | Lemma for ~ dflring3 . If... |
| dflringlem2 34069 | Lemma for ~ dflring3 . In... |
| dflringlem3 34070 | Lemma for ~ dflring3 . In... |
| dflring3 34071 | Alternate definition of a ... |
| dflring4 34072 | Alternate definition of a ... |
| fldlring 34073 | A field is a local ring. ... |
| idlsrgstr 34076 | A constructed semiring of ... |
| idlsrgval 34077 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 34078 | Base of the ideals of a ri... |
| idlsrgplusg 34079 | Additive operation of the ... |
| idlsrg0g 34080 | The zero ideal is the addi... |
| idlsrgmulr 34081 | Multiplicative operation o... |
| idlsrgtset 34082 | Topology component of the ... |
| idlsrgmulrval 34083 | Value of the ring multipli... |
| idlsrgmulrcl 34084 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 34085 | In a commutative ring, the... |
| idlsrgmulrss2 34086 | The product of two ideals ... |
| idlsrgmulrssin 34087 | In a commutative ring, the... |
| idlsrgmnd 34088 | The ideals of a ring form ... |
| idlsrgcmnd 34089 | The ideals of a ring form ... |
| rprmval 34090 | The prime elements of a ri... |
| isrprm 34091 | Property for ` P ` to be a... |
| rprmcl 34092 | A ring prime is an element... |
| rprmdvds 34093 | If a ring prime ` Q ` divi... |
| rprmnz 34094 | A ring prime is nonzero. ... |
| rprmnunit 34095 | A ring prime is not a unit... |
| rsprprmprmidl 34096 | In a commutative ring, ide... |
| rsprprmprmidlb 34097 | An ideal generated by a si... |
| rprmndvdsr1 34098 | A ring prime element does ... |
| rprmasso 34099 | In an integral domain, the... |
| rprmasso2 34100 | In an integral domain, if ... |
| rprmasso3 34101 | In an integral domain, if ... |
| unitmulrprm 34102 | A ring unit multiplied by ... |
| rprmndvdsru 34103 | A ring prime element does ... |
| rprmirredlem 34104 | Lemma for ~ rprmirred . (... |
| rprmirred 34105 | In an integral domain, rin... |
| rprmirredb 34106 | In a principal ideal domai... |
| rprmdvdspow 34107 | If a prime element divides... |
| rprmdvdsprod 34108 | If a prime element ` Q ` d... |
| 1arithidomlem1 34109 | Lemma for ~ 1arithidom . ... |
| 1arithidomlem2 34110 | Lemma for ~ 1arithidom : i... |
| 1arithidom 34111 | Uniqueness of prime factor... |
| isufd 34114 | The property of being a Un... |
| ufdprmidl 34115 | In a unique factorization ... |
| ufdidom 34116 | A nonzero unique factoriza... |
| pidufd 34117 | Every principal ideal doma... |
| 1arithufdlem1 34118 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem2 34119 | Lemma for ~ 1arithufd . T... |
| 1arithufdlem3 34120 | Lemma for ~ 1arithufd . I... |
| 1arithufdlem4 34121 | Lemma for ~ 1arithufd . N... |
| 1arithufd 34122 | Existence of a factorizati... |
| dfufd2lem 34123 | Lemma for ~ dfufd2 . (Con... |
| dfufd2 34124 | Alternative definition of ... |
| zringidom 34125 | The ring of integers is an... |
| zringpid 34126 | The ring of integers is a ... |
| dfprm3 34127 | The (positive) prime eleme... |
| zringfrac 34128 | The field of fractions of ... |
| assaassd 34129 | Left-associative property ... |
| assaassrd 34130 | Right-associative property... |
| 0ringmon1p 34131 | There are no monic polynom... |
| fply1 34132 | Conditions for a function ... |
| ply1lvec 34133 | In a division ring, the un... |
| evls1fn 34134 | Functionality of the subri... |
| evls1dm 34135 | The domain of the subring ... |
| evls1fvf 34136 | The subring evaluation fun... |
| evl1fvf 34137 | The univariate polynomial ... |
| evl1fpws 34138 | Evaluation of a univariate... |
| ressply1evls1 34139 | Subring evaluation of a un... |
| ressdeg1 34140 | The degree of a univariate... |
| ressply10g 34141 | A restricted polynomial al... |
| ressply1mon1p 34142 | The monic polynomials of a... |
| ressply1invg 34143 | An element of a restricted... |
| ressply1sub 34144 | A restricted polynomial al... |
| ressasclcl 34145 | Closure of the univariate ... |
| evls1subd 34146 | Univariate polynomial eval... |
| deg1le0eq0 34147 | A polynomial with nonposit... |
| ply1asclunit 34148 | A nonzero scalar polynomia... |
| ply1unit 34149 | In a field ` F ` , a polyn... |
| evl1deg1 34150 | Evaluation of a univariate... |
| evl1deg2 34151 | Evaluation of a univariate... |
| evl1deg3 34152 | Evaluation of a univariate... |
| evls1monply1 34153 | Subring evaluation of a sc... |
| ply1dg1rt 34154 | Express the root ` - B / A... |
| ply1dg1rtn0 34155 | Polynomials of degree 1 ov... |
| ply1mulrtss 34156 | The roots of a factor ` F ... |
| deg1prod 34157 | Degree of a product of pol... |
| ply1dg3rt0irred 34158 | If a cubic polynomial over... |
| m1pmeq 34159 | If two monic polynomials `... |
| ply1fermltl 34160 | Fermat's little theorem fo... |
| coe1mon 34161 | Coefficient vector of a mo... |
| ply1moneq 34162 | Two monomials are equal if... |
| ply1coedeg 34163 | Decompose a univariate pol... |
| coe1zfv 34164 | The coefficients of the ze... |
| coe1vr1 34165 | Polynomial coefficient of ... |
| deg1vr 34166 | The degree of the variable... |
| vr1nz 34167 | A univariate polynomial va... |
| ply1degltel 34168 | Characterize elementhood i... |
| ply1degleel 34169 | Characterize elementhood i... |
| ply1degltlss 34170 | The space ` S ` of the uni... |
| gsummoncoe1fzo 34171 | A coefficient of the polyn... |
| gsummoncoe1fz 34172 | A coefficient of the polyn... |
| ply1gsumz 34173 | If a polynomial given as a... |
| deg1addlt 34174 | If both factors have degre... |
| ig1pnunit 34175 | The polynomial ideal gener... |
| ig1pmindeg 34176 | The polynomial ideal gener... |
| q1pdir 34177 | Distribution of univariate... |
| q1pvsca 34178 | Scalar multiplication prop... |
| r1pvsca 34179 | Scalar multiplication prop... |
| r1p0 34180 | Polynomial remainder opera... |
| r1pcyc 34181 | The polynomial remainder o... |
| r1padd1 34182 | Addition property of the p... |
| r1plmhm 34183 | The univariate polynomial ... |
| r1pquslmic 34184 | The univariate polynomial ... |
| psrbasfsupp 34185 | Rewrite a finite support f... |
| psrnzr 34186 | The ring of power series o... |
| mplnzr 34187 | The multivariate polynomia... |
| 0mplrim 34188 | Build a ring isomorphism b... |
| 0mplric 34189 | Multivariate polynomials w... |
| mplasclco 34190 | Case where composing an al... |
| selvascl 34191 | The "variable selection" f... |
| selvply1rhmlema 34192 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlemb 34193 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem1 34194 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem2 34195 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem3 34196 | Lemma for ~ selvply1rhm . ... |
| selvply1rhmlem4 34197 | Lemma for ~ selvply1rhm : ... |
| selvply1rhmlem5 34198 | Lemma for ~ selvply1rhm . ... |
| selvply1rhm 34199 | Build a ring homomorphism ... |
| selvply1rhm0 34200 | The ring homomorphism ` H ... |
| mplidomlem 34201 | Lemma for ~ mplidom . (Co... |
| mplidom 34202 | The multivariate polynomia... |
| extvval 34205 | Value of the "variable ext... |
| extvfval 34206 | The "variable extension" f... |
| extvfv 34207 | The "variable extension" f... |
| extvfvv 34208 | The "variable extension" f... |
| extvfvvcl 34209 | Closure for the "variable ... |
| extvfvcl 34210 | Closure for the "variable ... |
| extvfvalf 34211 | The "variable extension" f... |
| mvrvalind 34212 | Value of the generating el... |
| mplmulmvr 34213 | Multiply a polynomial ` F ... |
| evlscaval 34214 | Polynomial evaluation for ... |
| evlvarval 34215 | Polynomial evaluation buil... |
| evlextv 34216 | Evaluating a variable-exte... |
| mplvrpmlem 34217 | Lemma for ~ mplvrpmga and ... |
| mplvrpmfgalem 34218 | Permuting variables in a m... |
| mplvrpmga 34219 | The action of permuting va... |
| mplvrpmmhm 34220 | The action of permuting va... |
| mplvrpmrhm 34221 | The action of permuting va... |
| psrgsum 34222 | Finite commutative sums of... |
| psrmon 34223 | A monomial is a power seri... |
| psrmonmul 34224 | The product of two power s... |
| psrmonmul2 34225 | The product of two power s... |
| psrmonprod 34226 | Finite product of bags of ... |
| mplgsum 34227 | Finite commutative sums of... |
| mplmonprod 34228 | Finite product of monomial... |
| splyval 34233 | The symmetric polynomials ... |
| splysubrg 34234 | The symmetric polynomials ... |
| issply 34235 | Conditions for being a sym... |
| esplyval 34236 | The elementary polynomials... |
| esplyfval 34237 | The ` K ` -th elementary p... |
| esplyfval0 34238 | The ` 0 ` -th elementary s... |
| esplyfval2 34239 | When ` K ` is out-of-bound... |
| esplylem 34240 | Lemma for ~ esplyfv and ot... |
| esplympl 34241 | Elementary symmetric polyn... |
| esplymhp 34242 | The ` K ` -th elementary s... |
| esplyfv1 34243 | Coefficient for the ` K ` ... |
| esplyfv 34244 | Coefficient for the ` K ` ... |
| esplysply 34245 | The ` K ` -th elementary s... |
| esplyfval3 34246 | Alternate expression for t... |
| esplyfval1 34247 | The first elementary symme... |
| esplyfvaln 34248 | The last elementary symmet... |
| esplyind 34249 | A recursive formula for th... |
| esplyindfv 34250 | A recursive formula for th... |
| esplyfvn 34251 | Express the last elementar... |
| vietadeg1 34252 | The degree of a product of... |
| vietalem 34253 | Lemma for ~ vieta : induct... |
| vieta 34254 | Vieta's Formulas: Coeffic... |
| sra1r 34255 | The unity element of a sub... |
| sradrng 34256 | Condition for a subring al... |
| sraidom 34257 | Condition for a subring al... |
| srasubrg 34258 | A subring of the original ... |
| sralvec 34259 | Given a sub division ring ... |
| srafldlvec 34260 | Given a subfield ` F ` of ... |
| resssra 34261 | The subring algebra of a r... |
| lsssra 34262 | A subring is a subspace of... |
| srapwov 34263 | The "power" operation on a... |
| drgext0g 34264 | The additive neutral eleme... |
| drgextvsca 34265 | The scalar multiplication ... |
| drgext0gsca 34266 | The additive neutral eleme... |
| drgextsubrg 34267 | The scalar field is a subr... |
| drgextlsp 34268 | The scalar field is a subs... |
| drgextgsum 34269 | Group sum in a division ri... |
| lvecdimfi 34270 | Finite version of ~ lvecdi... |
| exsslsb 34271 | Any finite generating set ... |
| lbslelsp 34272 | The size of a basis ` X ` ... |
| dimval 34275 | The dimension of a vector ... |
| dimvalfi 34276 | The dimension of a vector ... |
| dimcl 34277 | Closure of the vector spac... |
| lmimdim 34278 | Module isomorphisms preser... |
| lmicdim 34279 | Module isomorphisms preser... |
| lvecdim0i 34280 | A vector space of dimensio... |
| lvecdim0 34281 | A vector space of dimensio... |
| lssdimle 34282 | The dimension of a linear ... |
| dimpropd 34283 | If two structures have the... |
| rlmdim 34284 | The left vector space indu... |
| frlmdim 34285 | Dimension of a free left m... |
| tnglvec 34286 | Augmenting a structure wit... |
| tngdim 34287 | Dimension of a left vector... |
| rrxdim 34288 | Dimension of the generaliz... |
| matdim 34289 | Dimension of the space of ... |
| lbslsat 34290 | A nonzero vector ` X ` is ... |
| lsatdim 34291 | A line, spanned by a nonze... |
| drngdimgt0 34292 | The dimension of a vector ... |
| lmhmlvec2 34293 | A homomorphism of left vec... |
| kerlmhm 34294 | The kernel of a vector spa... |
| imlmhm 34295 | The image of a vector spac... |
| ply1degltdimlem 34296 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 34297 | The space ` S ` of the uni... |
| lindsunlem 34298 | Lemma for ~ lindsun . (Co... |
| lindsun 34299 | Condition for the union of... |
| lbsdiflsp0 34300 | The linear spans of two di... |
| dimkerim 34301 | Given a linear map ` F ` b... |
| qusdimsum 34302 | Let ` W ` be a vector spac... |
| fedgmullem1 34303 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 34304 | Lemma for ~ fedgmul . (Co... |
| fedgmul 34305 | The multiplicativity formu... |
| dimlssid 34306 | If the dimension of a line... |
| lvecendof1f1o 34307 | If an endomorphism ` U ` o... |
| lactlmhm 34308 | In an associative algebra ... |
| assalactf1o 34309 | In an associative algebra ... |
| assarrginv 34310 | If an element ` X ` of an ... |
| assafld 34311 | If an algebra ` A ` of fin... |
| relfldext 34318 | The field extension is a r... |
| brfldext 34319 | The field extension relati... |
| ccfldextrr 34320 | The field of the complex n... |
| fldextfld1 34321 | A field extension is only ... |
| fldextfld2 34322 | A field extension is only ... |
| fldextsubrg 34323 | Field extension implies a ... |
| sdrgfldext 34324 | A field ` E ` and any sub-... |
| fldextress 34325 | Field extension implies a ... |
| brfinext 34326 | The finite field extension... |
| extdgval 34327 | Value of the field extensi... |
| fldextsdrg 34328 | Deduce sub-division-ring f... |
| fldextsralvec 34329 | The subring algebra associ... |
| extdgcl 34330 | Closure of the field exten... |
| extdggt0 34331 | Degrees of field extension... |
| fldexttr 34332 | Field extension is a trans... |
| fldextid 34333 | The field extension relati... |
| extdgid 34334 | A trivial field extension ... |
| fldsdrgfldext 34335 | A sub-division-ring of a f... |
| fldsdrgfldext2 34336 | A sub-sub-division-ring of... |
| extdgmul 34337 | The multiplicativity formu... |
| finextfldext 34338 | A finite field extension i... |
| finexttrb 34339 | The extension ` E ` of ` K... |
| extdg1id 34340 | If the degree of the exten... |
| extdg1b 34341 | The degree of the extensio... |
| fldgenfldext 34342 | A subfield ` F ` extended ... |
| fldextchr 34343 | The characteristic of a su... |
| evls1fldgencl 34344 | Closure of the subring pol... |
| ccfldsrarelvec 34345 | The subring algebra of the... |
| ccfldextdgrr 34346 | The degree of the field ex... |
| fldextrspunlsplem 34347 | Lemma for ~ fldextrspunlsp... |
| fldextrspunlsp 34348 | Lemma for ~ fldextrspunfld... |
| fldextrspunlem1 34349 | Lemma for ~ fldextrspunfld... |
| fldextrspunfld 34350 | The ring generated by the ... |
| fldextrspunlem2 34351 | Part of the proof of Propo... |
| fldextrspundgle 34352 | Inequality involving the d... |
| fldextrspundglemul 34353 | Given two field extensions... |
| fldextrspundgdvdslem 34354 | Lemma for ~ fldextrspundgd... |
| fldextrspundgdvds 34355 | Given two finite extension... |
| fldext2rspun 34356 | Given two field extensions... |
| irngval 34359 | The elements of a field ` ... |
| elirng 34360 | Property for an element ` ... |
| irngss 34361 | All elements of a subring ... |
| irngssv 34362 | An integral element is an ... |
| 0ringirng 34363 | A zero ring ` R ` has no i... |
| irngnzply1lem 34364 | In the case of a field ` E... |
| irngnzply1 34365 | In the case of a field ` E... |
| extdgfialglem1 34366 | Lemma for ~ extdgfialg . ... |
| extdgfialglem2 34367 | Lemma for ~ extdgfialg . ... |
| extdgfialg 34368 | A finite field extension `... |
| bralgext 34371 | Express the fact that a fi... |
| finextalg 34372 | A finite field extension i... |
| ply1annidllem 34375 | Write the set ` Q ` of pol... |
| ply1annidl 34376 | The set ` Q ` of polynomia... |
| ply1annnr 34377 | The set ` Q ` of polynomia... |
| ply1annig1p 34378 | The ideal ` Q ` of polynom... |
| minplyval 34379 | Expand the value of the mi... |
| minplycl 34380 | The minimal polynomial is ... |
| ply1annprmidl 34381 | The set ` Q ` of polynomia... |
| minplymindeg 34382 | The minimal polynomial of ... |
| minplyann 34383 | The minimal polynomial for... |
| minplyirredlem 34384 | Lemma for ~ minplyirred . ... |
| minplyirred 34385 | A nonzero minimal polynomi... |
| irngnminplynz 34386 | Integral elements have non... |
| minplym1p 34387 | A minimal polynomial is mo... |
| minplynzm1p 34388 | If a minimal polynomial is... |
| minplyelirng 34389 | If the minimal polynomial ... |
| irredminply 34390 | An irreducible, monic, ann... |
| algextdeglem1 34391 | Lemma for ~ algextdeg . (... |
| algextdeglem2 34392 | Lemma for ~ algextdeg . B... |
| algextdeglem3 34393 | Lemma for ~ algextdeg . T... |
| algextdeglem4 34394 | Lemma for ~ algextdeg . B... |
| algextdeglem5 34395 | Lemma for ~ algextdeg . T... |
| algextdeglem6 34396 | Lemma for ~ algextdeg . B... |
| algextdeglem7 34397 | Lemma for ~ algextdeg . T... |
| algextdeglem8 34398 | Lemma for ~ algextdeg . T... |
| algextdeg 34399 | The degree of an algebraic... |
| rtelextdg2lem 34400 | Lemma for ~ rtelextdg2 : ... |
| rtelextdg2 34401 | If an element ` X ` is a s... |
| fldext2chn 34402 | In a non-empty chain ` T `... |
| constrrtll 34405 | In the construction of con... |
| constrrtlc1 34406 | In the construction of con... |
| constrrtlc2 34407 | In the construction of con... |
| constrrtcclem 34408 | In the construction of con... |
| constrrtcc 34409 | In the construction of con... |
| isconstr 34410 | Property of being a constr... |
| constr0 34411 | The first step of the cons... |
| constrsuc 34412 | Membership in the successo... |
| constrlim 34413 | Limit step of the construc... |
| constrsscn 34414 | Closure of the constructib... |
| constrsslem 34415 | Lemma for ~ constrss . Th... |
| constr01 34416 | ` 0 ` and ` 1 ` are in all... |
| constrss 34417 | Constructed points are in ... |
| constrmon 34418 | The construction of constr... |
| constrconj 34419 | If a point ` X ` of the co... |
| constrfin 34420 | Each step of the construct... |
| constrelextdg2 34421 | If the ` N ` -th step ` ( ... |
| constrextdg2lem 34422 | Lemma for ~ constrextdg2 .... |
| constrextdg2 34423 | Any step ` ( C `` N ) ` of... |
| constrext2chnlem 34424 | Lemma for ~ constrext2chn ... |
| constrfiss 34425 | For any finite set ` A ` o... |
| constrllcllem 34426 | Constructible numbers are ... |
| constrlccllem 34427 | Constructible numbers are ... |
| constrcccllem 34428 | Constructible numbers are ... |
| constrcbvlem 34429 | Technical lemma for elimin... |
| constrllcl 34430 | Constructible numbers are ... |
| constrlccl 34431 | Constructible numbers are ... |
| constrcccl 34432 | Constructible numbers are ... |
| constrext2chn 34433 | If a constructible number ... |
| constrcn 34434 | Constructible numbers are ... |
| nn0constr 34435 | Nonnegative integers are c... |
| constraddcl 34436 | Constructive numbers are c... |
| constrnegcl 34437 | Constructible numbers are ... |
| zconstr 34438 | Integers are constructible... |
| constrdircl 34439 | Constructible numbers are ... |
| iconstr 34440 | The imaginary unit ` _i ` ... |
| constrremulcl 34441 | If two real numbers ` X ` ... |
| constrcjcl 34442 | Constructible numbers are ... |
| constrrecl 34443 | Constructible numbers are ... |
| constrimcl 34444 | Constructible numbers are ... |
| constrmulcl 34445 | Constructible numbers are ... |
| constrreinvcl 34446 | If a real number ` X ` is ... |
| constrinvcl 34447 | Constructible numbers are ... |
| constrcon 34448 | Contradiction of construct... |
| constrsdrg 34449 | Constructible numbers form... |
| constrfld 34450 | The constructible numbers ... |
| constrresqrtcl 34451 | If a positive real number ... |
| constrabscl 34452 | Constructible numbers are ... |
| constrsqrtcl 34453 | Constructible numbers are ... |
| 2sqr3minply 34454 | The polynomial ` ( ( X ^ 3... |
| 2sqr3nconstr 34455 | Doubling the cube is an im... |
| cos9thpiminplylem1 34456 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem2 34457 | The polynomial ` ( ( X ^ 3... |
| cos9thpiminplylem3 34458 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem4 34459 | Lemma for ~ cos9thpiminply... |
| cos9thpiminplylem5 34460 | The constructed complex nu... |
| cos9thpiminplylem6 34461 | Evaluation of the polynomi... |
| cos9thpiminply 34462 | The polynomial ` ( ( X ^ 3... |
| cos9thpinconstrlem1 34463 | The complex number ` O ` ,... |
| cos9thpinconstrlem2 34464 | The complex number ` A ` i... |
| cos9thpinconstr 34465 | Trisecting an angle is an ... |
| trisecnconstr 34466 | Not all angles can be tris... |
| smatfval 34469 | Value of the submatrix. (... |
| smatrcl 34470 | Closure of the rectangular... |
| smatlem 34471 | Lemma for the next theorem... |
| smattl 34472 | Entries of a submatrix, to... |
| smattr 34473 | Entries of a submatrix, to... |
| smatbl 34474 | Entries of a submatrix, bo... |
| smatbr 34475 | Entries of a submatrix, bo... |
| smatcl 34476 | Closure of the square subm... |
| matmpo 34477 | Write a square matrix as a... |
| 1smat1 34478 | The submatrix of the ident... |
| submat1n 34479 | One case where the submatr... |
| submatres 34480 | Special case where the sub... |
| submateqlem1 34481 | Lemma for ~ submateq . (C... |
| submateqlem2 34482 | Lemma for ~ submateq . (C... |
| submateq 34483 | Sufficient condition for t... |
| submatminr1 34484 | If we take a submatrix by ... |
| lmatval 34487 | Value of the literal matri... |
| lmatfval 34488 | Entries of a literal matri... |
| lmatfvlem 34489 | Useful lemma to extract li... |
| lmatcl 34490 | Closure of the literal mat... |
| lmat22lem 34491 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 34492 | Entry of a 2x2 literal mat... |
| lmat22e12 34493 | Entry of a 2x2 literal mat... |
| lmat22e21 34494 | Entry of a 2x2 literal mat... |
| lmat22e22 34495 | Entry of a 2x2 literal mat... |
| lmat22det 34496 | The determinant of a liter... |
| mdetpmtr1 34497 | The determinant of a matri... |
| mdetpmtr2 34498 | The determinant of a matri... |
| mdetpmtr12 34499 | The determinant of a matri... |
| mdetlap1 34500 | A Laplace expansion of the... |
| madjusmdetlem1 34501 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 34502 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 34503 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 34504 | Lemma for ~ madjusmdet . ... |
| madjusmdet 34505 | Express the cofactor of th... |
| mdetlap 34506 | Laplace expansion of the d... |
| ist0cld 34507 | The predicate "is a T_0 sp... |
| txomap 34508 | Given two open maps ` F ` ... |
| qtopt1 34509 | If every equivalence class... |
| qtophaus 34510 | If an open map's graph in ... |
| circtopn 34511 | The topology of the unit c... |
| circcn 34512 | The function gluing the re... |
| reff 34513 | For any cover refinement, ... |
| locfinreflem 34514 | A locally finite refinemen... |
| locfinref 34515 | A locally finite refinemen... |
| iscref 34518 | The property that every op... |
| crefeq 34519 | Equality theorem for the "... |
| creftop 34520 | A space where every open c... |
| crefi 34521 | The property that every op... |
| crefdf 34522 | A formulation of ~ crefi e... |
| crefss 34523 | The "every open cover has ... |
| cmpcref 34524 | Equivalent definition of c... |
| cmpfiref 34525 | Every open cover of a Comp... |
| ldlfcntref 34528 | Every open cover of a Lind... |
| ispcmp 34531 | The predicate "is a paraco... |
| cmppcmp 34532 | Every compact space is par... |
| dispcmp 34533 | Every discrete space is pa... |
| pcmplfin 34534 | Given a paracompact topolo... |
| pcmplfinf 34535 | Given a paracompact topolo... |
| rspecval 34538 | Value of the spectrum of t... |
| rspecbas 34539 | The prime ideals form the ... |
| rspectset 34540 | Topology component of the ... |
| rspectopn 34541 | The topology component of ... |
| zarcls0 34542 | The closure of the identit... |
| zarcls1 34543 | The unit ideal ` B ` is th... |
| zarclsun 34544 | The union of two closed se... |
| zarclsiin 34545 | In a Zariski topology, the... |
| zarclsint 34546 | The intersection of a fami... |
| zarclssn 34547 | The closed points of Zaris... |
| zarcls 34548 | The open sets of the Zaris... |
| zartopn 34549 | The Zariski topology is a ... |
| zartop 34550 | The Zariski topology is a ... |
| zartopon 34551 | The points of the Zariski ... |
| zar0ring 34552 | The Zariski Topology of th... |
| zart0 34553 | The Zariski topology is T_... |
| zarmxt1 34554 | The Zariski topology restr... |
| zarcmplem 34555 | Lemma for ~ zarcmp . (Con... |
| zarcmp 34556 | The Zariski topology is co... |
| rspectps 34557 | The spectrum of a ring ` R... |
| rhmpreimacnlem 34558 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 34559 | The function mapping a pri... |
| metidval 34564 | Value of the metric identi... |
| metidss 34565 | As a relation, the metric ... |
| metidv 34566 | ` A ` and ` B ` identify b... |
| metideq 34567 | Basic property of the metr... |
| metider 34568 | The metric identification ... |
| pstmval 34569 | Value of the metric induce... |
| pstmfval 34570 | Function value of the metr... |
| pstmxmet 34571 | The metric induced by a ps... |
| hauseqcn 34572 | In a Hausdorff topology, t... |
| elunitge0 34573 | An element of the closed u... |
| unitssxrge0 34574 | The closed unit interval i... |
| unitdivcld 34575 | Necessary conditions for a... |
| iistmd 34576 | The closed unit interval f... |
| unicls 34577 | The union of the closed se... |
| tpr2tp 34578 | The usual topology on ` ( ... |
| tpr2uni 34579 | The usual topology on ` ( ... |
| xpinpreima 34580 | Rewrite the cartesian prod... |
| xpinpreima2 34581 | Rewrite the cartesian prod... |
| sqsscirc1 34582 | The complex square of side... |
| sqsscirc2 34583 | The complex square of side... |
| cnre2csqlem 34584 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 34585 | Image of a centered square... |
| tpr2rico 34586 | For any point of an open s... |
| cnvordtrestixx 34587 | The restriction of the 'gr... |
| prsdm 34588 | Domain of the relation of ... |
| prsrn 34589 | Range of the relation of a... |
| prsss 34590 | Relation of a subproset. ... |
| prsssdm 34591 | Domain of a subproset rela... |
| ordtprsval 34592 | Value of the order topolog... |
| ordtprsuni 34593 | Value of the order topolog... |
| ordtcnvNEW 34594 | The order dual generates t... |
| ordtrestNEW 34595 | The subspace topology of a... |
| ordtrest2NEWlem 34596 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 34597 | An interval-closed set ` A... |
| ordtconnlem1 34598 | Connectedness in the order... |
| ordtconn 34599 | Connectedness in the order... |
| mndpluscn 34600 | A mapping that is both a h... |
| mhmhmeotmd 34601 | Deduce a Topological Monoi... |
| rmulccn 34602 | Multiplication by a real c... |
| raddcn 34603 | Addition in the real numbe... |
| xrmulc1cn 34604 | The operation multiplying ... |
| fmcncfil 34605 | The image of a Cauchy filt... |
| xrge0hmph 34606 | The extended nonnegative r... |
| xrge0iifcnv 34607 | Define a bijection from ` ... |
| xrge0iifcv 34608 | The defined function's val... |
| xrge0iifiso 34609 | The defined bijection from... |
| xrge0iifhmeo 34610 | Expose a homeomorphism fro... |
| xrge0iifhom 34611 | The defined function from ... |
| xrge0iif1 34612 | Condition for the defined ... |
| xrge0iifmhm 34613 | The defined function from ... |
| xrge0pluscn 34614 | The addition operation of ... |
| xrge0mulc1cn 34615 | The operation multiplying ... |
| xrge0tps 34616 | The extended nonnegative r... |
| xrge0topn 34617 | The topology of the extend... |
| xrge0haus 34618 | The topology of the extend... |
| xrge0tmd 34619 | The extended nonnegative r... |
| xrge0tmdALT 34620 | Alternate proof of ~ xrge0... |
| lmlim 34621 | Relate a limit in a given ... |
| lmlimxrge0 34622 | Relate a limit in the nonn... |
| rge0scvg 34623 | Implication of convergence... |
| fsumcvg4 34624 | A serie with finite suppor... |
| pnfneige0 34625 | A neighborhood of ` +oo ` ... |
| lmxrge0 34626 | Express "sequence ` F ` co... |
| lmdvg 34627 | If a monotonic sequence of... |
| lmdvglim 34628 | If a monotonic real number... |
| pl1cn 34629 | A univariate polynomial is... |
| zringnm 34632 | The norm (function) for a ... |
| zzsnm 34633 | The norm of the ring of th... |
| zlm0 34634 | Zero of a ` ZZ ` -module. ... |
| zlm1 34635 | Unity element of a ` ZZ ` ... |
| zlmds 34636 | Distance in a ` ZZ ` -modu... |
| zlmtset 34637 | Topology in a ` ZZ ` -modu... |
| zlmnm 34638 | Norm of a ` ZZ ` -module (... |
| zhmnrg 34639 | The ` ZZ ` -module built f... |
| nmmulg 34640 | The norm of a group produc... |
| zrhnm 34641 | The norm of the image by `... |
| cnzh 34642 | The ` ZZ ` -module of ` CC... |
| rezh 34643 | The ` ZZ ` -module of ` RR... |
| qqhval 34646 | Value of the canonical hom... |
| zrhf1ker 34647 | The kernel of the homomorp... |
| zrhchr 34648 | The kernel of the homomorp... |
| zrhker 34649 | The kernel of the homomorp... |
| zrhunitpreima 34650 | The preimage by ` ZRHom ` ... |
| elzrhunit 34651 | Condition for the image by... |
| zrhneg 34652 | The canonical homomorphism... |
| zrhcntr 34653 | The canonical representati... |
| elzdif0 34654 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 34655 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 34656 | Value of the canonical hom... |
| qqhvval 34657 | Value of the canonical hom... |
| qqh0 34658 | The image of ` 0 ` by the ... |
| qqh1 34659 | The image of ` 1 ` by the ... |
| qqhf 34660 | ` QQHom ` as a function. ... |
| qqhvq 34661 | The image of a quotient by... |
| qqhghm 34662 | The ` QQHom ` homomorphism... |
| qqhrhm 34663 | The ` QQHom ` homomorphism... |
| qqhnm 34664 | The norm of the image by `... |
| qqhcn 34665 | The ` QQHom ` homomorphism... |
| qqhucn 34666 | The ` QQHom ` homomorphism... |
| rrhval 34670 | Value of the canonical hom... |
| rrhcn 34671 | If the topology of ` R ` i... |
| rrhf 34672 | If the topology of ` R ` i... |
| isrrext 34674 | Express the property " ` R... |
| rrextnrg 34675 | An extension of ` RR ` is ... |
| rrextdrg 34676 | An extension of ` RR ` is ... |
| rrextnlm 34677 | The norm of an extension o... |
| rrextchr 34678 | The ring characteristic of... |
| rrextcusp 34679 | An extension of ` RR ` is ... |
| rrexttps 34680 | An extension of ` RR ` is ... |
| rrexthaus 34681 | The topology of an extensi... |
| rrextust 34682 | The uniformity of an exten... |
| rerrext 34683 | The field of the real numb... |
| cnrrext 34684 | The field of the complex n... |
| qqtopn 34685 | The topology of the field ... |
| rrhfe 34686 | If ` R ` is an extension o... |
| rrhcne 34687 | If ` R ` is an extension o... |
| rrhqima 34688 | The ` RRHom ` homomorphism... |
| rrh0 34689 | The image of ` 0 ` by the ... |
| xrhval 34692 | The value of the embedding... |
| zrhre 34693 | The ` ZRHom ` homomorphism... |
| qqhre 34694 | The ` QQHom ` homomorphism... |
| rrhre 34695 | The ` RRHom ` homomorphism... |
| relmntop 34698 | Manifold is a relation. (... |
| ismntoplly 34699 | Property of being a manifo... |
| ismntop 34700 | Property of being a manifo... |
| esumex 34703 | An extended sum is a set b... |
| esumcl 34704 | Closure for extended sum i... |
| esumeq12dvaf 34705 | Equality deduction for ext... |
| esumeq12dva 34706 | Equality deduction for ext... |
| esumeq12d 34707 | Equality deduction for ext... |
| esumeq1 34708 | Equality theorem for an ex... |
| esumeq1d 34709 | Equality theorem for an ex... |
| esumeq2 34710 | Equality theorem for exten... |
| esumeq2d 34711 | Equality deduction for ext... |
| esumeq2dv 34712 | Equality deduction for ext... |
| esumeq2sdv 34713 | Equality deduction for ext... |
| nfesum1 34714 | Bound-variable hypothesis ... |
| nfesum2 34715 | Bound-variable hypothesis ... |
| cbvesum 34716 | Change bound variable in a... |
| cbvesumv 34717 | Change bound variable in a... |
| esumid 34718 | Identify the extended sum ... |
| esumgsum 34719 | A finite extended sum is t... |
| esumval 34720 | Develop the value of the e... |
| esumel 34721 | The extended sum is a limi... |
| esumnul 34722 | Extended sum over the empt... |
| esum0 34723 | Extended sum of zero. (Co... |
| esumf1o 34724 | Re-index an extended sum u... |
| esumc 34725 | Convert from the collectio... |
| esumrnmpt 34726 | Rewrite an extended sum in... |
| esumsplit 34727 | Split an extended sum into... |
| esummono 34728 | Extended sum is monotonic.... |
| esumpad 34729 | Extend an extended sum by ... |
| esumpad2 34730 | Remove zeroes from an exte... |
| esumadd 34731 | Addition of infinite sums.... |
| esumle 34732 | If all of the terms of an ... |
| gsumesum 34733 | Relate a group sum on ` ( ... |
| esumlub 34734 | The extended sum is the lo... |
| esumaddf 34735 | Addition of infinite sums.... |
| esumlef 34736 | If all of the terms of an ... |
| esumcst 34737 | The extended sum of a cons... |
| esumsnf 34738 | The extended sum of a sing... |
| esumsn 34739 | The extended sum of a sing... |
| esumpr 34740 | Extended sum over a pair. ... |
| esumpr2 34741 | Extended sum over a pair, ... |
| esumrnmpt2 34742 | Rewrite an extended sum in... |
| esumfzf 34743 | Formulating a partial exte... |
| esumfsup 34744 | Formulating an extended su... |
| esumfsupre 34745 | Formulating an extended su... |
| esumss 34746 | Change the index set to a ... |
| esumpinfval 34747 | The value of the extended ... |
| esumpfinvallem 34748 | Lemma for ~ esumpfinval . ... |
| esumpfinval 34749 | The value of the extended ... |
| esumpfinvalf 34750 | Same as ~ esumpfinval , mi... |
| esumpinfsum 34751 | The value of the extended ... |
| esumpcvgval 34752 | The value of the extended ... |
| esumpmono 34753 | The partial sums in an ext... |
| esumcocn 34754 | Lemma for ~ esummulc2 and ... |
| esummulc1 34755 | An extended sum multiplied... |
| esummulc2 34756 | An extended sum multiplied... |
| esumdivc 34757 | An extended sum divided by... |
| hashf2 34758 | Lemma for ~ hasheuni . (C... |
| hasheuni 34759 | The cardinality of a disjo... |
| esumcvg 34760 | The sequence of partial su... |
| esumcvg2 34761 | Simpler version of ~ esumc... |
| esumcvgsum 34762 | The value of the extended ... |
| esumsup 34763 | Express an extended sum as... |
| esumgect 34764 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 34765 | All terms of a converging ... |
| esum2dlem 34766 | Lemma for ~ esum2d (finite... |
| esum2d 34767 | Write a double extended su... |
| esumiun 34768 | Sum over a nonnecessarily ... |
| ofceq 34771 | Equality theorem for funct... |
| ofcfval 34772 | Value of an operation appl... |
| ofcval 34773 | Evaluate a function/consta... |
| ofcfn 34774 | The function operation pro... |
| ofcfeqd2 34775 | Equality theorem for funct... |
| ofcfval3 34776 | General value of ` ( F oFC... |
| ofcf 34777 | The function/constant oper... |
| ofcfval2 34778 | The function operation exp... |
| ofcfval4 34779 | The function/constant oper... |
| ofcc 34780 | Left operation by a consta... |
| ofcof 34781 | Relate function operation ... |
| sigaex 34784 | Lemma for ~ issiga and ~ i... |
| sigaval 34785 | The set of sigma-algebra w... |
| issiga 34786 | An alternative definition ... |
| isrnsiga 34787 | The property of being a si... |
| 0elsiga 34788 | A sigma-algebra contains t... |
| baselsiga 34789 | A sigma-algebra contains i... |
| sigasspw 34790 | A sigma-algebra is a set o... |
| sigaclcu 34791 | A sigma-algebra is closed ... |
| sigaclcuni 34792 | A sigma-algebra is closed ... |
| sigaclfu 34793 | A sigma-algebra is closed ... |
| sigaclcu2 34794 | A sigma-algebra is closed ... |
| sigaclfu2 34795 | A sigma-algebra is closed ... |
| sigaclcu3 34796 | A sigma-algebra is closed ... |
| issgon 34797 | Property of being a sigma-... |
| sgon 34798 | A sigma-algebra is a sigma... |
| elsigass 34799 | An element of a sigma-alge... |
| elrnsiga 34800 | Dropping the base informat... |
| isrnsigau 34801 | The property of being a si... |
| unielsiga 34802 | A sigma-algebra contains i... |
| dmvlsiga 34803 | Lebesgue-measurable subset... |
| pwsiga 34804 | Any power set forms a sigm... |
| prsiga 34805 | The smallest possible sigm... |
| sigaclci 34806 | A sigma-algebra is closed ... |
| difunielsiga 34807 | A sigma-algebra is closed ... |
| unelsiga 34808 | A sigma-algebra is closed ... |
| difelsiga 34809 | A sigma-algebra is closed ... |
| inelsiga 34810 | A sigma-algebra is closed ... |
| sigainb 34811 | Building a sigma-algebra f... |
| insiga 34812 | The intersection of a coll... |
| sigagenval 34815 | Value of the generated sig... |
| sigagensiga 34816 | A generated sigma-algebra ... |
| sgsiga 34817 | A generated sigma-algebra ... |
| unisg 34818 | The sigma-algebra generate... |
| dmsigagen 34819 | A sigma-algebra can be gen... |
| sssigagen 34820 | A set is a subset of the s... |
| sssigagen2 34821 | A subset of the generating... |
| elsigagen 34822 | Any element of a set is al... |
| elsigagen2 34823 | Any countable union of ele... |
| sigagenss 34824 | The generated sigma-algebr... |
| sigagenss2 34825 | Sufficient condition for i... |
| sigagenid 34826 | The sigma-algebra generate... |
| ispisys 34827 | The property of being a pi... |
| ispisys2 34828 | The property of being a pi... |
| inelpisys 34829 | Pi-systems are closed unde... |
| sigapisys 34830 | All sigma-algebras are pi-... |
| isldsys 34831 | The property of being a la... |
| pwldsys 34832 | The power set of the unive... |
| unelldsys 34833 | Lambda-systems are closed ... |
| sigaldsys 34834 | All sigma-algebras are lam... |
| ldsysgenld 34835 | The intersection of all la... |
| sigapildsyslem 34836 | Lemma for ~ sigapildsys . ... |
| sigapildsys 34837 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 34838 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 34839 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 34840 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 34841 | The lambda system ` E ` ge... |
| dynkin 34842 | Dynkin's lambda-pi theorem... |
| isros 34843 | The property of being a ri... |
| rossspw 34844 | A ring of sets is a collec... |
| 0elros 34845 | A ring of sets contains th... |
| unelros 34846 | A ring of sets is closed u... |
| difelros 34847 | A ring of sets is closed u... |
| inelros 34848 | A ring of sets is closed u... |
| fiunelros 34849 | A ring of sets is closed u... |
| issros 34850 | The property of being a se... |
| srossspw 34851 | A semiring of sets is a co... |
| 0elsros 34852 | A semiring of sets contain... |
| inelsros 34853 | A semiring of sets is clos... |
| diffiunisros 34854 | In semiring of sets, compl... |
| rossros 34855 | Rings of sets are semiring... |
| brsiga 34858 | The Borel Algebra on real ... |
| brsigarn 34859 | The Borel Algebra is a sig... |
| brsigasspwrn 34860 | The Borel Algebra is a set... |
| unibrsiga 34861 | The union of the Borel Alg... |
| cldssbrsiga 34862 | A Borel Algebra contains a... |
| sxval 34865 | Value of the product sigma... |
| sxsiga 34866 | A product sigma-algebra is... |
| sxsigon 34867 | A product sigma-algebra is... |
| sxuni 34868 | The base set of a product ... |
| elsx 34869 | The cartesian product of t... |
| measbase 34872 | The base set of a measure ... |
| measval 34873 | The value of the ` measure... |
| ismeas 34874 | The property of being a me... |
| isrnmeas 34875 | The property of being a me... |
| dmmeas 34876 | The domain of a measure is... |
| measbasedom 34877 | The base set of a measure ... |
| measfrge0 34878 | A measure is a function ov... |
| measfn 34879 | A measure is a function on... |
| measvxrge0 34880 | The values of a measure ar... |
| measvnul 34881 | The measure of the empty s... |
| measge0 34882 | A measure is nonnegative. ... |
| measle0 34883 | If the measure of a given ... |
| measvun 34884 | The measure of a countable... |
| measxun2 34885 | The measure the union of t... |
| measun 34886 | The measure the union of t... |
| measvunilem 34887 | Lemma for ~ measvuni . (C... |
| measvunilem0 34888 | Lemma for ~ measvuni . (C... |
| measvuni 34889 | The measure of a countable... |
| measssd 34890 | A measure is monotone with... |
| measunl 34891 | A measure is sub-additive ... |
| measiuns 34892 | The measure of the union o... |
| measiun 34893 | A measure is sub-additive.... |
| meascnbl 34894 | A measure is continuous fr... |
| measinblem 34895 | Lemma for ~ measinb . (Co... |
| measinb 34896 | Building a measure restric... |
| measres 34897 | Building a measure restric... |
| measinb2 34898 | Building a measure restric... |
| measdivcst 34899 | Division of a measure by a... |
| measdivcstALTV 34900 | Alternate version of ~ mea... |
| cntmeas 34901 | The Counting measure is a ... |
| pwcntmeas 34902 | The counting measure is a ... |
| cntnevol 34903 | Counting and Lebesgue meas... |
| voliune 34904 | The Lebesgue measure funct... |
| volfiniune 34905 | The Lebesgue measure funct... |
| volmeas 34906 | The Lebesgue measure is a ... |
| ddeval1 34909 | Value of the delta measure... |
| ddeval0 34910 | Value of the delta measure... |
| ddemeas 34911 | The Dirac delta measure is... |
| relae 34915 | 'almost everywhere' is a r... |
| brae 34916 | 'almost everywhere' relati... |
| braew 34917 | 'almost everywhere' relati... |
| truae 34918 | A truth holds almost every... |
| aean 34919 | A conjunction holds almost... |
| faeval 34921 | Value of the 'almost every... |
| relfae 34922 | The 'almost everywhere' bu... |
| brfae 34923 | 'almost everywhere' relati... |
| ismbfm 34926 | The predicate " ` F ` is a... |
| elunirnmbfm 34927 | The property of being a me... |
| mbfmfun 34928 | A measurable function is a... |
| mbfmf 34929 | A measurable function as a... |
| mbfmcnvima 34930 | The preimage by a measurab... |
| isanmbfm 34931 | The predicate to be a meas... |
| mbfmbfm 34932 | A measurable function to a... |
| mbfmcst 34933 | A constant function is mea... |
| 1stmbfm 34934 | The first projection map i... |
| 2ndmbfm 34935 | The second projection map ... |
| imambfm 34936 | If the sigma-algebra in th... |
| cnmbfm 34937 | A continuous function is m... |
| mbfmco 34938 | The composition of two mea... |
| mbfmco2 34939 | The pair building of two m... |
| mbfmvolf 34940 | Measurable functions with ... |
| elmbfmvol2 34941 | Measurable functions with ... |
| mbfmcnt 34942 | All functions are measurab... |
| br2base 34943 | The base set for the gener... |
| dya2ub 34944 | An upper bound for a dyadi... |
| sxbrsigalem0 34945 | The closed half-spaces of ... |
| sxbrsigalem3 34946 | The sigma-algebra generate... |
| dya2iocival 34947 | The function ` I ` returns... |
| dya2iocress 34948 | Dyadic intervals are subse... |
| dya2iocbrsiga 34949 | Dyadic intervals are Borel... |
| dya2icobrsiga 34950 | Dyadic intervals are Borel... |
| dya2icoseg 34951 | For any point and any clos... |
| dya2icoseg2 34952 | For any point and any open... |
| dya2iocrfn 34953 | The function returning dya... |
| dya2iocct 34954 | The dyadic rectangle set i... |
| dya2iocnrect 34955 | For any point of an open r... |
| dya2iocnei 34956 | For any point of an open s... |
| dya2iocuni 34957 | Every open set of ` ( RR X... |
| dya2iocucvr 34958 | The dyadic rectangular set... |
| sxbrsigalem1 34959 | The Borel algebra on ` ( R... |
| sxbrsigalem2 34960 | The sigma-algebra generate... |
| sxbrsigalem4 34961 | The Borel algebra on ` ( R... |
| sxbrsigalem5 34962 | First direction for ~ sxbr... |
| sxbrsigalem6 34963 | First direction for ~ sxbr... |
| sxbrsiga 34964 | The product sigma-algebra ... |
| omsval 34967 | Value of the function mapp... |
| omsfval 34968 | Value of the outer measure... |
| omscl 34969 | A closure lemma for the co... |
| omsf 34970 | A constructed outer measur... |
| oms0 34971 | A constructed outer measur... |
| omsmon 34972 | A constructed outer measur... |
| omssubaddlem 34973 | For any small margin ` E `... |
| omssubadd 34974 | A constructed outer measur... |
| carsgval 34977 | Value of the Caratheodory ... |
| carsgcl 34978 | Closure of the Caratheodor... |
| elcarsg 34979 | Property of being a Carath... |
| baselcarsg 34980 | The universe set, ` O ` , ... |
| 0elcarsg 34981 | The empty set is Caratheod... |
| carsguni 34982 | The union of all Caratheod... |
| elcarsgss 34983 | Caratheodory measurable se... |
| difelcarsg 34984 | The Caratheodory measurabl... |
| inelcarsg 34985 | The Caratheodory measurabl... |
| unelcarsg 34986 | The Caratheodory-measurabl... |
| difelcarsg2 34987 | The Caratheodory-measurabl... |
| carsgmon 34988 | Utility lemma: Apply mono... |
| carsgsigalem 34989 | Lemma for the following th... |
| fiunelcarsg 34990 | The Caratheodory measurabl... |
| carsgclctunlem1 34991 | Lemma for ~ carsgclctun . ... |
| carsggect 34992 | The outer measure is count... |
| carsgclctunlem2 34993 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34994 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34995 | The Caratheodory measurabl... |
| carsgsiga 34996 | The Caratheodory measurabl... |
| omsmeas 34997 | The restriction of a const... |
| pmeasmono 34998 | This theorem's hypotheses ... |
| pmeasadd 34999 | A premeasure on a ring of ... |
| itgeq12dv 35000 | Equality theorem for an in... |
| sitgval 35006 | Value of the simple functi... |
| issibf 35007 | The predicate " ` F ` is a... |
| sibf0 35008 | The constant zero function... |
| sibfmbl 35009 | A simple function is measu... |
| sibff 35010 | A simple function is a fun... |
| sibfrn 35011 | A simple function has fini... |
| sibfima 35012 | Any preimage of a singleto... |
| sibfinima 35013 | The measure of the interse... |
| sibfof 35014 | Applying function operatio... |
| sitgfval 35015 | Value of the Bochner integ... |
| sitgclg 35016 | Closure of the Bochner int... |
| sitgclbn 35017 | Closure of the Bochner int... |
| sitgclcn 35018 | Closure of the Bochner int... |
| sitgclre 35019 | Closure of the Bochner int... |
| sitg0 35020 | The integral of the consta... |
| sitgf 35021 | The integral for simple fu... |
| sitgaddlemb 35022 | Lemma for * sitgadd . (Co... |
| sitmval 35023 | Value of the simple functi... |
| sitmfval 35024 | Value of the integral dist... |
| sitmcl 35025 | Closure of the integral di... |
| sitmf 35026 | The integral metric as a f... |
| oddpwdc 35028 | Lemma for ~ eulerpart . T... |
| oddpwdcv 35029 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 35030 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 35031 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 35032 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 35033 | Lemma for ~ eulerpart . (... |
| eulerpartlems 35034 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 35035 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 35036 | Lemma for ~ eulerpart . (... |
| eulerpartleme 35037 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 35038 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 35039 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 35040 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 35041 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 35042 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 35043 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 35044 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 35045 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 35046 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 35047 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 35048 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 35049 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 35050 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 35051 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 35052 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 35053 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 35054 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 35055 | Lemma for ~ eulerpart . (... |
| eulerpart 35056 | Euler's theorem on partiti... |
| subiwrd 35059 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 35060 | Length of a subword of an ... |
| iwrdsplit 35061 | Lemma for ~ sseqp1 . (Con... |
| sseqval 35062 | Value of the strong sequen... |
| sseqfv1 35063 | Value of the strong sequen... |
| sseqfn 35064 | A strong recursive sequenc... |
| sseqmw 35065 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 35066 | A strong recursive sequenc... |
| sseqfres 35067 | The first elements in the ... |
| sseqfv2 35068 | Value of the strong sequen... |
| sseqp1 35069 | Value of the strong sequen... |
| fiblem 35072 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 35073 | Value of the Fibonacci seq... |
| fib1 35074 | Value of the Fibonacci seq... |
| fibp1 35075 | Value of the Fibonacci seq... |
| fib2 35076 | Value of the Fibonacci seq... |
| fib3 35077 | Value of the Fibonacci seq... |
| fib4 35078 | Value of the Fibonacci seq... |
| fib5 35079 | Value of the Fibonacci seq... |
| fib6 35080 | Value of the Fibonacci seq... |
| elprob 35083 | The property of being a pr... |
| domprobmeas 35084 | A probability measure is a... |
| domprobsiga 35085 | The domain of a probabilit... |
| probtot 35086 | The probability of the uni... |
| prob01 35087 | A probability is an elemen... |
| probnul 35088 | The probability of the emp... |
| unveldomd 35089 | The universe is an element... |
| unveldom 35090 | The universe is an element... |
| nuleldmp 35091 | The empty set is an elemen... |
| probcun 35092 | The probability of the uni... |
| probun 35093 | The probability of the uni... |
| probdif 35094 | The probability of the dif... |
| probinc 35095 | A probability law is incre... |
| probdsb 35096 | The probability of the com... |
| probmeasd 35097 | A probability measure is a... |
| probvalrnd 35098 | The value of a probability... |
| probtotrnd 35099 | The probability of the uni... |
| totprobd 35100 | Law of total probability, ... |
| totprob 35101 | Law of total probability. ... |
| probfinmeasb 35102 | Build a probability measur... |
| probfinmeasbALTV 35103 | Alternate version of ~ pro... |
| probmeasb 35104 | Build a probability from a... |
| cndprobval 35107 | The value of the condition... |
| cndprobin 35108 | An identity linking condit... |
| cndprob01 35109 | The conditional probabilit... |
| cndprobtot 35110 | The conditional probabilit... |
| cndprobnul 35111 | The conditional probabilit... |
| cndprobprob 35112 | The conditional probabilit... |
| bayesth 35113 | Bayes Theorem. (Contribut... |
| rrvmbfm 35116 | A real-valued random varia... |
| isrrvv 35117 | Elementhood to the set of ... |
| rrvvf 35118 | A real-valued random varia... |
| rrvfn 35119 | A real-valued random varia... |
| rrvdm 35120 | The domain of a random var... |
| rrvrnss 35121 | The range of a random vari... |
| rrvf2 35122 | A real-valued random varia... |
| rrvdmss 35123 | The domain of a random var... |
| rrvfinvima 35124 | For a real-value random va... |
| 0rrv 35125 | The constant function equa... |
| rrvadd 35126 | The sum of two random vari... |
| rrvmulc 35127 | A random variable multipli... |
| rrvsum 35128 | An indexed sum of random v... |
| boolesineq 35129 | Boole's inequality (union ... |
| orvcval 35132 | Value of the preimage mapp... |
| orvcval2 35133 | Another way to express the... |
| elorvc 35134 | Elementhood of a preimage.... |
| orvcval4 35135 | The value of the preimage ... |
| orvcoel 35136 | If the relation produces o... |
| orvccel 35137 | If the relation produces c... |
| elorrvc 35138 | Elementhood of a preimage ... |
| orrvcval4 35139 | The value of the preimage ... |
| orrvcoel 35140 | If the relation produces o... |
| orrvccel 35141 | If the relation produces c... |
| orvcgteel 35142 | Preimage maps produced by ... |
| orvcelval 35143 | Preimage maps produced by ... |
| orvcelel 35144 | Preimage maps produced by ... |
| dstrvval 35145 | The value of the distribut... |
| dstrvprob 35146 | The distribution of a rand... |
| orvclteel 35147 | Preimage maps produced by ... |
| dstfrvel 35148 | Elementhood of preimage ma... |
| dstfrvunirn 35149 | The limit of all preimage ... |
| orvclteinc 35150 | Preimage maps produced by ... |
| dstfrvinc 35151 | A cumulative distribution ... |
| dstfrvclim1 35152 | The limit of the cumulativ... |
| coinfliplem 35153 | Division in the extended r... |
| coinflipprob 35154 | The ` P ` we defined for c... |
| coinflipspace 35155 | The space of our coin-flip... |
| coinflipuniv 35156 | The universe of our coin-f... |
| coinfliprv 35157 | The ` X ` we defined for c... |
| coinflippv 35158 | The probability of heads i... |
| coinflippvt 35159 | The probability of tails i... |
| ballotlemoex 35160 | ` O ` is a set. (Contribu... |
| ballotlem1 35161 | The size of the universe i... |
| ballotlemelo 35162 | Elementhood in ` O ` . (C... |
| ballotlem2 35163 | The probability that the f... |
| ballotlemfval 35164 | The value of ` F ` . (Con... |
| ballotlemfelz 35165 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 35166 | If the ` J ` th ballot is ... |
| ballotlemfc0 35167 | ` F ` takes value 0 betwee... |
| ballotlemfcc 35168 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 35169 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 35170 | ` ( F `` C ) ` always star... |
| ballotleme 35171 | Elements of ` E ` . (Cont... |
| ballotlemodife 35172 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 35173 | If the first pick is a vot... |
| ballotlem5 35174 | If A is not ahead througho... |
| ballotlemi 35175 | Value of ` I ` for a given... |
| ballotlemiex 35176 | Properties of ` ( I `` C )... |
| ballotlemi1 35177 | The first tie cannot be re... |
| ballotlemii 35178 | The first tie cannot be re... |
| ballotlemsup 35179 | The set of zeroes of ` F `... |
| ballotlemimin 35180 | ` ( I `` C ) ` is the firs... |
| ballotlemic 35181 | If the first vote is for B... |
| ballotlem1c 35182 | If the first vote is for A... |
| ballotlemsval 35183 | Value of ` S ` . (Contrib... |
| ballotlemsv 35184 | Value of ` S ` evaluated a... |
| ballotlemsgt1 35185 | ` S ` maps values less tha... |
| ballotlemsdom 35186 | Domain of ` S ` for a give... |
| ballotlemsel1i 35187 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 35188 | The defined ` S ` is a bij... |
| ballotlemsi 35189 | The image by ` S ` of the ... |
| ballotlemsima 35190 | The image by ` S ` of an i... |
| ballotlemieq 35191 | If two countings share the... |
| ballotlemrval 35192 | Value of ` R ` . (Contrib... |
| ballotlemscr 35193 | The image of ` ( R `` C ) ... |
| ballotlemrv 35194 | Value of ` R ` evaluated a... |
| ballotlemrv1 35195 | Value of ` R ` before the ... |
| ballotlemrv2 35196 | Value of ` R ` after the t... |
| ballotlemro 35197 | Range of ` R ` is included... |
| ballotlemgval 35198 | Expand the value of ` .^ `... |
| ballotlemgun 35199 | A property of the defined ... |
| ballotlemfg 35200 | Express the value of ` ( F... |
| ballotlemfrc 35201 | Express the value of ` ( F... |
| ballotlemfrci 35202 | Reverse counting preserves... |
| ballotlemfrceq 35203 | Value of ` F ` for a rever... |
| ballotlemfrcn0 35204 | Value of ` F ` for a rever... |
| ballotlemrc 35205 | Range of ` R ` . (Contrib... |
| ballotlemirc 35206 | Applying ` R ` does not ch... |
| ballotlemrinv0 35207 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 35208 | ` R ` is its own inverse :... |
| ballotlem1ri 35209 | When the vote on the first... |
| ballotlem7 35210 | ` R ` is a bijection betwe... |
| ballotlem8 35211 | There are as many counting... |
| ballotth 35212 | Bertrand's ballot problem ... |
| fzssfzo 35213 | Condition for an integer i... |
| gsumncl 35214 | Closure of a group sum in ... |
| gsumnunsn 35215 | Closure of a group sum in ... |
| ccatmulgnn0dir 35216 | Concatenation of words fol... |
| ofcccat 35217 | Letterwise operations on w... |
| ofcs1 35218 | Letterwise operations on a... |
| ofcs2 35219 | Letterwise operations on a... |
| plyrecld 35220 | Closure of a polynomial wi... |
| signsplypnf 35221 | The quotient of a polynomi... |
| signsply0 35222 | Lemma for the rule of sign... |
| signspval 35223 | The value of the skipping ... |
| signsw0glem 35224 | Neutral element property o... |
| signswbase 35225 | The base of ` W ` is the u... |
| signswplusg 35226 | The operation of ` W ` . ... |
| signsw0g 35227 | The neutral element of ` W... |
| signswmnd 35228 | ` W ` is a monoid structur... |
| signswrid 35229 | The zero-skipping operatio... |
| signswlid 35230 | The zero-skipping operatio... |
| signswn0 35231 | The zero-skipping operatio... |
| signswch 35232 | The zero-skipping operatio... |
| signslema 35233 | Computational part of ~~? ... |
| signstfv 35234 | Value of the zero-skipping... |
| signstfval 35235 | Value of the zero-skipping... |
| signstcl 35236 | Closure of the zero skippi... |
| signstf 35237 | The zero skipping sign wor... |
| signstlen 35238 | Length of the zero skippin... |
| signstf0 35239 | Sign of a single letter wo... |
| signstfvn 35240 | Zero-skipping sign in a wo... |
| signsvtn0 35241 | If the last letter is nonz... |
| signstfvp 35242 | Zero-skipping sign in a wo... |
| signstfvneq0 35243 | In case the first letter i... |
| signstfvcl 35244 | Closure of the zero skippi... |
| signstfvc 35245 | Zero-skipping sign in a wo... |
| signstres 35246 | Restriction of a zero skip... |
| signstfveq0a 35247 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 35248 | In case the last letter is... |
| signsvvfval 35249 | The value of ` V ` , which... |
| signsvvf 35250 | ` V ` is a function. (Con... |
| signsvf0 35251 | There is no change of sign... |
| signsvf1 35252 | In a single-letter word, w... |
| signsvfn 35253 | Number of changes in a wor... |
| signsvtp 35254 | Adding a letter of the sam... |
| signsvtn 35255 | Adding a letter of a diffe... |
| signsvfpn 35256 | Adding a letter of the sam... |
| signsvfnn 35257 | Adding a letter of a diffe... |
| signlem0 35258 | Adding a zero as the highe... |
| signshf 35259 | ` H ` , corresponding to t... |
| signshwrd 35260 | ` H ` , corresponding to t... |
| signshlen 35261 | Length of ` H ` , correspo... |
| signshnz 35262 | ` H ` is not the empty wor... |
| iblidicc 35263 | The identity function is i... |
| rpsqrtcn 35264 | Continuity of the real pos... |
| divsqrtid 35265 | A real number divided by i... |
| cxpcncf1 35266 | The power function on comp... |
| efmul2picn 35267 | Multiplying by ` ( _i x. (... |
| fct2relem 35268 | Lemma for ~ ftc2re . (Con... |
| ftc2re 35269 | The Fundamental Theorem of... |
| fdvposlt 35270 | Functions with a positive ... |
| fdvneggt 35271 | Functions with a negative ... |
| fdvposle 35272 | Functions with a nonnegati... |
| fdvnegge 35273 | Functions with a nonpositi... |
| prodfzo03 35274 | A product of three factors... |
| actfunsnf1o 35275 | The action ` F ` of extend... |
| actfunsnrndisj 35276 | The action ` F ` of extend... |
| itgexpif 35277 | The basis for the circle m... |
| fsum2dsub 35278 | Lemma for ~ breprexp - Re-... |
| reprval 35281 | Value of the representatio... |
| repr0 35282 | There is exactly one repre... |
| reprf 35283 | Members of the representat... |
| reprsum 35284 | Sums of values of the memb... |
| reprle 35285 | Upper bound to the terms i... |
| reprsuc 35286 | Express the representation... |
| reprfi 35287 | Bounded representations ar... |
| reprss 35288 | Representations with terms... |
| reprinrn 35289 | Representations with term ... |
| reprlt 35290 | There are no representatio... |
| hashreprin 35291 | Express a sum of represent... |
| reprgt 35292 | There are no representatio... |
| reprinfz1 35293 | For the representation of ... |
| reprfi2 35294 | Corollary of ~ reprinfz1 .... |
| reprfz1 35295 | Corollary of ~ reprinfz1 .... |
| hashrepr 35296 | Develop the number of repr... |
| reprpmtf1o 35297 | Transposing ` 0 ` and ` X ... |
| reprdifc 35298 | Express the representation... |
| chpvalz 35299 | Value of the second Chebys... |
| chtvalz 35300 | Value of the Chebyshev fun... |
| breprexplema 35301 | Lemma for ~ breprexp (indu... |
| breprexplemb 35302 | Lemma for ~ breprexp (clos... |
| breprexplemc 35303 | Lemma for ~ breprexp (indu... |
| breprexp 35304 | Express the ` S ` th power... |
| breprexpnat 35305 | Express the ` S ` th power... |
| vtsval 35308 | Value of the Vinogradov tr... |
| vtscl 35309 | Closure of the Vinogradov ... |
| vtsprod 35310 | Express the Vinogradov tri... |
| circlemeth 35311 | The Hardy, Littlewood and ... |
| circlemethnat 35312 | The Hardy, Littlewood and ... |
| circlevma 35313 | The Circle Method, where t... |
| circlemethhgt 35314 | The circle method, where t... |
| hgt750lemc 35318 | An upper bound to the summ... |
| hgt750lemd 35319 | An upper bound to the summ... |
| hgt749d 35320 | A deduction version of ~ a... |
| logdivsqrle 35321 | Conditions for ` ( ( log `... |
| hgt750lem 35322 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 35323 | Decimal multiplication gal... |
| hgt750lemf 35324 | Lemma for the statement 7.... |
| hgt750lemg 35325 | Lemma for the statement 7.... |
| oddprm2 35326 | Two ways to write the set ... |
| hgt750lemb 35327 | An upper bound on the cont... |
| hgt750lema 35328 | An upper bound on the cont... |
| hgt750leme 35329 | An upper bound on the cont... |
| tgoldbachgnn 35330 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 35331 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 35332 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 35333 | Odd integers greater than ... |
| tgoldbachgt 35334 | Odd integers greater than ... |
| istrkg2d 35337 | Property of fulfilling dim... |
| axtglowdim2ALTV 35338 | Alternate version of ~ axt... |
| axtgupdim2ALTV 35339 | Alternate version of ~ axt... |
| cgranbtwn 35340 | Null angle implies between... |
| btwnlng13 35341 | If ` Z ` is between ` X ` ... |
| morleylemrneab 35342 | Lemma for morley . (Contr... |
| afsval 35345 | Value of the AFS relation ... |
| brafs 35346 | Binary relation form of th... |
| tg5segofs 35347 | Rephrase ~ axtg5seg using ... |
| lpadval 35350 | Value of the ` leftpad ` f... |
| lpadlem1 35351 | Lemma for the ` leftpad ` ... |
| lpadlem3 35352 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 35353 | Length of a left-padded wo... |
| lpadlem2 35354 | Lemma for the ` leftpad ` ... |
| lpadlen2 35355 | Length of a left-padded wo... |
| lpadmax 35356 | Length of a left-padded wo... |
| lpadleft 35357 | The contents of prefix of ... |
| lpadright 35358 | The suffix of a left-padde... |
| bnj170 35371 | ` /\ ` -manipulation. (Co... |
| bnj240 35372 | ` /\ ` -manipulation. (Co... |
| bnj248 35373 | ` /\ ` -manipulation. (Co... |
| bnj250 35374 | ` /\ ` -manipulation. (Co... |
| bnj251 35375 | ` /\ ` -manipulation. (Co... |
| bnj252 35376 | ` /\ ` -manipulation. (Co... |
| bnj253 35377 | ` /\ ` -manipulation. (Co... |
| bnj255 35378 | ` /\ ` -manipulation. (Co... |
| bnj256 35379 | ` /\ ` -manipulation. (Co... |
| bnj257 35380 | ` /\ ` -manipulation. (Co... |
| bnj258 35381 | ` /\ ` -manipulation. (Co... |
| bnj268 35382 | ` /\ ` -manipulation. (Co... |
| bnj290 35383 | ` /\ ` -manipulation. (Co... |
| bnj291 35384 | ` /\ ` -manipulation. (Co... |
| bnj312 35385 | ` /\ ` -manipulation. (Co... |
| bnj334 35386 | ` /\ ` -manipulation. (Co... |
| bnj345 35387 | ` /\ ` -manipulation. (Co... |
| bnj422 35388 | ` /\ ` -manipulation. (Co... |
| bnj432 35389 | ` /\ ` -manipulation. (Co... |
| bnj446 35390 | ` /\ ` -manipulation. (Co... |
| bnj23 35391 | First-order logic and set ... |
| bnj31 35392 | First-order logic and set ... |
| bnj62 35393 | First-order logic and set ... |
| bnj89 35394 | First-order logic and set ... |
| bnj90 35395 | First-order logic and set ... |
| bnj101 35396 | First-order logic and set ... |
| bnj105 35397 | First-order logic and set ... |
| bnj115 35398 | First-order logic and set ... |
| bnj132 35399 | First-order logic and set ... |
| bnj133 35400 | First-order logic and set ... |
| bnj156 35401 | First-order logic and set ... |
| bnj158 35402 | First-order logic and set ... |
| bnj168 35403 | First-order logic and set ... |
| bnj206 35404 | First-order logic and set ... |
| bnj216 35405 | First-order logic and set ... |
| bnj219 35406 | First-order logic and set ... |
| bnj226 35407 | First-order logic and set ... |
| bnj228 35408 | First-order logic and set ... |
| bnj519 35409 | First-order logic and set ... |
| bnj524 35410 | First-order logic and set ... |
| bnj525 35411 | First-order logic and set ... |
| bnj534 35412 | First-order logic and set ... |
| bnj538 35413 | First-order logic and set ... |
| bnj529 35414 | First-order logic and set ... |
| bnj551 35415 | First-order logic and set ... |
| bnj563 35416 | First-order logic and set ... |
| bnj564 35417 | First-order logic and set ... |
| bnj593 35418 | First-order logic and set ... |
| bnj596 35419 | First-order logic and set ... |
| bnj610 35420 | Pass from equality ( ` x =... |
| bnj642 35421 | ` /\ ` -manipulation. (Co... |
| bnj643 35422 | ` /\ ` -manipulation. (Co... |
| bnj645 35423 | ` /\ ` -manipulation. (Co... |
| bnj658 35424 | ` /\ ` -manipulation. (Co... |
| bnj667 35425 | ` /\ ` -manipulation. (Co... |
| bnj705 35426 | ` /\ ` -manipulation. (Co... |
| bnj706 35427 | ` /\ ` -manipulation. (Co... |
| bnj707 35428 | ` /\ ` -manipulation. (Co... |
| bnj708 35429 | ` /\ ` -manipulation. (Co... |
| bnj721 35430 | ` /\ ` -manipulation. (Co... |
| bnj832 35431 | ` /\ ` -manipulation. (Co... |
| bnj835 35432 | ` /\ ` -manipulation. (Co... |
| bnj836 35433 | ` /\ ` -manipulation. (Co... |
| bnj837 35434 | ` /\ ` -manipulation. (Co... |
| bnj769 35435 | ` /\ ` -manipulation. (Co... |
| bnj770 35436 | ` /\ ` -manipulation. (Co... |
| bnj771 35437 | ` /\ ` -manipulation. (Co... |
| bnj887 35438 | ` /\ ` -manipulation. (Co... |
| bnj918 35439 | First-order logic and set ... |
| bnj919 35440 | First-order logic and set ... |
| bnj923 35441 | First-order logic and set ... |
| bnj927 35442 | First-order logic and set ... |
| bnj931 35443 | First-order logic and set ... |
| bnj937 35444 | First-order logic and set ... |
| bnj941 35445 | First-order logic and set ... |
| bnj945 35446 | Technical lemma for ~ bnj6... |
| bnj946 35447 | First-order logic and set ... |
| bnj951 35448 | ` /\ ` -manipulation. (Co... |
| bnj956 35449 | First-order logic and set ... |
| bnj976 35450 | First-order logic and set ... |
| bnj982 35451 | First-order logic and set ... |
| bnj1019 35452 | First-order logic and set ... |
| bnj1023 35453 | First-order logic and set ... |
| bnj1095 35454 | First-order logic and set ... |
| bnj1096 35455 | First-order logic and set ... |
| bnj1098 35456 | First-order logic and set ... |
| bnj1101 35457 | First-order logic and set ... |
| bnj1113 35458 | First-order logic and set ... |
| bnj1109 35459 | First-order logic and set ... |
| bnj1131 35460 | First-order logic and set ... |
| bnj1138 35461 | First-order logic and set ... |
| bnj1143 35462 | First-order logic and set ... |
| bnj1146 35463 | First-order logic and set ... |
| bnj1149 35464 | First-order logic and set ... |
| bnj1185 35465 | First-order logic and set ... |
| bnj1196 35466 | First-order logic and set ... |
| bnj1198 35467 | First-order logic and set ... |
| bnj1209 35468 | First-order logic and set ... |
| bnj1211 35469 | First-order logic and set ... |
| bnj1213 35470 | First-order logic and set ... |
| bnj1212 35471 | First-order logic and set ... |
| bnj1219 35472 | First-order logic and set ... |
| bnj1224 35473 | First-order logic and set ... |
| bnj1230 35474 | First-order logic and set ... |
| bnj1232 35475 | First-order logic and set ... |
| bnj1235 35476 | First-order logic and set ... |
| bnj1239 35477 | First-order logic and set ... |
| bnj1238 35478 | First-order logic and set ... |
| bnj1241 35479 | First-order logic and set ... |
| bnj1247 35480 | First-order logic and set ... |
| bnj1254 35481 | First-order logic and set ... |
| bnj1262 35482 | First-order logic and set ... |
| bnj1266 35483 | First-order logic and set ... |
| bnj1265 35484 | First-order logic and set ... |
| bnj1275 35485 | First-order logic and set ... |
| bnj1276 35486 | First-order logic and set ... |
| bnj1292 35487 | First-order logic and set ... |
| bnj1293 35488 | First-order logic and set ... |
| bnj1294 35489 | First-order logic and set ... |
| bnj1299 35490 | First-order logic and set ... |
| bnj1304 35491 | First-order logic and set ... |
| bnj1316 35492 | First-order logic and set ... |
| bnj1317 35493 | First-order logic and set ... |
| bnj1322 35494 | First-order logic and set ... |
| bnj1340 35495 | First-order logic and set ... |
| bnj1345 35496 | First-order logic and set ... |
| bnj1350 35497 | First-order logic and set ... |
| bnj1351 35498 | First-order logic and set ... |
| bnj1352 35499 | First-order logic and set ... |
| bnj1361 35500 | First-order logic and set ... |
| bnj1366 35501 | First-order logic and set ... |
| bnj1379 35502 | First-order logic and set ... |
| bnj1383 35503 | First-order logic and set ... |
| bnj1385 35504 | First-order logic and set ... |
| bnj1386 35505 | First-order logic and set ... |
| bnj1397 35506 | First-order logic and set ... |
| bnj1400 35507 | First-order logic and set ... |
| bnj1405 35508 | First-order logic and set ... |
| bnj1422 35509 | First-order logic and set ... |
| bnj1424 35510 | First-order logic and set ... |
| bnj1436 35511 | First-order logic and set ... |
| bnj1441 35512 | First-order logic and set ... |
| bnj1441g 35513 | First-order logic and set ... |
| bnj1454 35514 | First-order logic and set ... |
| bnj1459 35515 | First-order logic and set ... |
| bnj1464 35516 | Conversion of implicit sub... |
| bnj1465 35517 | First-order logic and set ... |
| bnj1468 35518 | Conversion of implicit sub... |
| bnj1476 35519 | First-order logic and set ... |
| bnj1502 35520 | First-order logic and set ... |
| bnj1503 35521 | First-order logic and set ... |
| bnj1517 35522 | First-order logic and set ... |
| bnj1521 35523 | First-order logic and set ... |
| bnj1533 35524 | First-order logic and set ... |
| bnj1534 35525 | First-order logic and set ... |
| bnj1536 35526 | First-order logic and set ... |
| bnj1538 35527 | First-order logic and set ... |
| bnj1541 35528 | First-order logic and set ... |
| bnj1542 35529 | First-order logic and set ... |
| bnj110 35530 | Well-founded induction res... |
| bnj157 35531 | Well-founded induction res... |
| bnj66 35532 | Technical lemma for ~ bnj6... |
| bnj91 35533 | First-order logic and set ... |
| bnj92 35534 | First-order logic and set ... |
| bnj93 35535 | Technical lemma for ~ bnj9... |
| bnj95 35536 | Technical lemma for ~ bnj1... |
| bnj96 35537 | Technical lemma for ~ bnj1... |
| bnj97 35538 | Technical lemma for ~ bnj1... |
| bnj98 35539 | Technical lemma for ~ bnj1... |
| bnj106 35540 | First-order logic and set ... |
| bnj118 35541 | First-order logic and set ... |
| bnj121 35542 | First-order logic and set ... |
| bnj124 35543 | Technical lemma for ~ bnj1... |
| bnj125 35544 | Technical lemma for ~ bnj1... |
| bnj126 35545 | Technical lemma for ~ bnj1... |
| bnj130 35546 | Technical lemma for ~ bnj1... |
| bnj149 35547 | Technical lemma for ~ bnj1... |
| bnj150 35548 | Technical lemma for ~ bnj1... |
| bnj151 35549 | Technical lemma for ~ bnj1... |
| bnj154 35550 | Technical lemma for ~ bnj1... |
| bnj155 35551 | Technical lemma for ~ bnj1... |
| bnj153 35552 | Technical lemma for ~ bnj8... |
| bnj207 35553 | Technical lemma for ~ bnj8... |
| bnj213 35554 | First-order logic and set ... |
| bnj222 35555 | Technical lemma for ~ bnj2... |
| bnj229 35556 | Technical lemma for ~ bnj5... |
| bnj517 35557 | Technical lemma for ~ bnj5... |
| bnj518 35558 | Technical lemma for ~ bnj8... |
| bnj523 35559 | Technical lemma for ~ bnj8... |
| bnj526 35560 | Technical lemma for ~ bnj8... |
| bnj528 35561 | Technical lemma for ~ bnj8... |
| bnj535 35562 | Technical lemma for ~ bnj8... |
| bnj539 35563 | Technical lemma for ~ bnj8... |
| bnj540 35564 | Technical lemma for ~ bnj8... |
| bnj543 35565 | Technical lemma for ~ bnj8... |
| bnj544 35566 | Technical lemma for ~ bnj8... |
| bnj545 35567 | Technical lemma for ~ bnj8... |
| bnj546 35568 | Technical lemma for ~ bnj8... |
| bnj548 35569 | Technical lemma for ~ bnj8... |
| bnj553 35570 | Technical lemma for ~ bnj8... |
| bnj554 35571 | Technical lemma for ~ bnj8... |
| bnj556 35572 | Technical lemma for ~ bnj8... |
| bnj557 35573 | Technical lemma for ~ bnj8... |
| bnj558 35574 | Technical lemma for ~ bnj8... |
| bnj561 35575 | Technical lemma for ~ bnj8... |
| bnj562 35576 | Technical lemma for ~ bnj8... |
| bnj570 35577 | Technical lemma for ~ bnj8... |
| bnj571 35578 | Technical lemma for ~ bnj8... |
| bnj605 35579 | Technical lemma. This lem... |
| bnj581 35580 | Technical lemma for ~ bnj5... |
| bnj589 35581 | Technical lemma for ~ bnj8... |
| bnj590 35582 | Technical lemma for ~ bnj8... |
| bnj591 35583 | Technical lemma for ~ bnj8... |
| bnj594 35584 | Technical lemma for ~ bnj8... |
| bnj580 35585 | Technical lemma for ~ bnj5... |
| bnj579 35586 | Technical lemma for ~ bnj8... |
| bnj602 35587 | Equality theorem for the `... |
| bnj607 35588 | Technical lemma for ~ bnj8... |
| bnj609 35589 | Technical lemma for ~ bnj8... |
| bnj611 35590 | Technical lemma for ~ bnj8... |
| bnj600 35591 | Technical lemma for ~ bnj8... |
| bnj601 35592 | Technical lemma for ~ bnj8... |
| bnj852 35593 | Technical lemma for ~ bnj6... |
| bnj864 35594 | Technical lemma for ~ bnj6... |
| bnj865 35595 | Technical lemma for ~ bnj6... |
| bnj873 35596 | Technical lemma for ~ bnj6... |
| bnj849 35597 | Technical lemma for ~ bnj6... |
| bnj882 35598 | Definition (using hypothes... |
| bnj18eq1 35599 | Equality theorem for trans... |
| bnj893 35600 | Property of ` _trCl ` . U... |
| bnj900 35601 | Technical lemma for ~ bnj6... |
| bnj906 35602 | Property of ` _trCl ` . (... |
| bnj908 35603 | Technical lemma for ~ bnj6... |
| bnj911 35604 | Technical lemma for ~ bnj6... |
| bnj916 35605 | Technical lemma for ~ bnj6... |
| bnj917 35606 | Technical lemma for ~ bnj6... |
| bnj934 35607 | Technical lemma for ~ bnj6... |
| bnj929 35608 | Technical lemma for ~ bnj6... |
| bnj938 35609 | Technical lemma for ~ bnj6... |
| bnj944 35610 | Technical lemma for ~ bnj6... |
| bnj953 35611 | Technical lemma for ~ bnj6... |
| bnj958 35612 | Technical lemma for ~ bnj6... |
| bnj1000 35613 | Technical lemma for ~ bnj8... |
| bnj965 35614 | Technical lemma for ~ bnj8... |
| bnj964 35615 | Technical lemma for ~ bnj6... |
| bnj966 35616 | Technical lemma for ~ bnj6... |
| bnj967 35617 | Technical lemma for ~ bnj6... |
| bnj969 35618 | Technical lemma for ~ bnj6... |
| bnj970 35619 | Technical lemma for ~ bnj6... |
| bnj910 35620 | Technical lemma for ~ bnj6... |
| bnj978 35621 | Technical lemma for ~ bnj6... |
| bnj981 35622 | Technical lemma for ~ bnj6... |
| bnj983 35623 | Technical lemma for ~ bnj6... |
| bnj984 35624 | Technical lemma for ~ bnj6... |
| bnj985v 35625 | Version of ~ bnj985 with a... |
| bnj985 35626 | Technical lemma for ~ bnj6... |
| bnj986 35627 | Technical lemma for ~ bnj6... |
| bnj996 35628 | Technical lemma for ~ bnj6... |
| bnj998 35629 | Technical lemma for ~ bnj6... |
| bnj999 35630 | Technical lemma for ~ bnj6... |
| bnj1001 35631 | Technical lemma for ~ bnj6... |
| bnj1006 35632 | Technical lemma for ~ bnj6... |
| bnj1014 35633 | Technical lemma for ~ bnj6... |
| bnj1015 35634 | Technical lemma for ~ bnj6... |
| bnj1018g 35635 | Version of ~ bnj1018 with ... |
| bnj1018 35636 | Technical lemma for ~ bnj6... |
| bnj1020 35637 | Technical lemma for ~ bnj6... |
| bnj1021 35638 | Technical lemma for ~ bnj6... |
| bnj907 35639 | Technical lemma for ~ bnj6... |
| bnj1029 35640 | Property of ` _trCl ` . (... |
| bnj1033 35641 | Technical lemma for ~ bnj6... |
| bnj1034 35642 | Technical lemma for ~ bnj6... |
| bnj1039 35643 | Technical lemma for ~ bnj6... |
| bnj1040 35644 | Technical lemma for ~ bnj6... |
| bnj1047 35645 | Technical lemma for ~ bnj6... |
| bnj1049 35646 | Technical lemma for ~ bnj6... |
| bnj1052 35647 | Technical lemma for ~ bnj6... |
| bnj1053 35648 | Technical lemma for ~ bnj6... |
| bnj1071 35649 | Technical lemma for ~ bnj6... |
| bnj1083 35650 | Technical lemma for ~ bnj6... |
| bnj1090 35651 | Technical lemma for ~ bnj6... |
| bnj1093 35652 | Technical lemma for ~ bnj6... |
| bnj1097 35653 | Technical lemma for ~ bnj6... |
| bnj1110 35654 | Technical lemma for ~ bnj6... |
| bnj1112 35655 | Technical lemma for ~ bnj6... |
| bnj1118 35656 | Technical lemma for ~ bnj6... |
| bnj1121 35657 | Technical lemma for ~ bnj6... |
| bnj1123 35658 | Technical lemma for ~ bnj6... |
| bnj1030 35659 | Technical lemma for ~ bnj6... |
| bnj1124 35660 | Property of ` _trCl ` . (... |
| bnj1133 35661 | Technical lemma for ~ bnj6... |
| bnj1128 35662 | Technical lemma for ~ bnj6... |
| bnj1127 35663 | Property of ` _trCl ` . (... |
| bnj1125 35664 | Property of ` _trCl ` . (... |
| bnj1145 35665 | Technical lemma for ~ bnj6... |
| bnj1147 35666 | Property of ` _trCl ` . (... |
| bnj1137 35667 | Property of ` _trCl ` . (... |
| bnj1148 35668 | Property of ` _pred ` . (... |
| bnj1136 35669 | Technical lemma for ~ bnj6... |
| bnj1152 35670 | Technical lemma for ~ bnj6... |
| bnj1154 35671 | Property of ` Fr ` . (Con... |
| bnj1171 35672 | Technical lemma for ~ bnj6... |
| bnj1172 35673 | Technical lemma for ~ bnj6... |
| bnj1173 35674 | Technical lemma for ~ bnj6... |
| bnj1174 35675 | Technical lemma for ~ bnj6... |
| bnj1175 35676 | Technical lemma for ~ bnj6... |
| bnj1176 35677 | Technical lemma for ~ bnj6... |
| bnj1177 35678 | Technical lemma for ~ bnj6... |
| bnj1186 35679 | Technical lemma for ~ bnj6... |
| bnj1190 35680 | Technical lemma for ~ bnj6... |
| bnj1189 35681 | Technical lemma for ~ bnj6... |
| bnj69 35682 | Existence of a minimal ele... |
| bnj1228 35683 | Existence of a minimal ele... |
| bnj1204 35684 | Well-founded induction. T... |
| bnj1234 35685 | Technical lemma for ~ bnj6... |
| bnj1245 35686 | Technical lemma for ~ bnj6... |
| bnj1256 35687 | Technical lemma for ~ bnj6... |
| bnj1259 35688 | Technical lemma for ~ bnj6... |
| bnj1253 35689 | Technical lemma for ~ bnj6... |
| bnj1279 35690 | Technical lemma for ~ bnj6... |
| bnj1286 35691 | Technical lemma for ~ bnj6... |
| bnj1280 35692 | Technical lemma for ~ bnj6... |
| bnj1296 35693 | Technical lemma for ~ bnj6... |
| bnj1309 35694 | Technical lemma for ~ bnj6... |
| bnj1307 35695 | Technical lemma for ~ bnj6... |
| bnj1311 35696 | Technical lemma for ~ bnj6... |
| bnj1318 35697 | Technical lemma for ~ bnj6... |
| bnj1326 35698 | Technical lemma for ~ bnj6... |
| bnj1321 35699 | Technical lemma for ~ bnj6... |
| bnj1364 35700 | Property of ` _FrSe ` . (... |
| bnj1371 35701 | Technical lemma for ~ bnj6... |
| bnj1373 35702 | Technical lemma for ~ bnj6... |
| bnj1374 35703 | Technical lemma for ~ bnj6... |
| bnj1384 35704 | Technical lemma for ~ bnj6... |
| bnj1388 35705 | Technical lemma for ~ bnj6... |
| bnj1398 35706 | Technical lemma for ~ bnj6... |
| bnj1413 35707 | Property of ` _trCl ` . (... |
| bnj1408 35708 | Technical lemma for ~ bnj1... |
| bnj1414 35709 | Property of ` _trCl ` . (... |
| bnj1415 35710 | Technical lemma for ~ bnj6... |
| bnj1416 35711 | Technical lemma for ~ bnj6... |
| bnj1418 35712 | Property of ` _pred ` . (... |
| bnj1417 35713 | Technical lemma for ~ bnj6... |
| bnj1421 35714 | Technical lemma for ~ bnj6... |
| bnj1444 35715 | Technical lemma for ~ bnj6... |
| bnj1445 35716 | Technical lemma for ~ bnj6... |
| bnj1446 35717 | Technical lemma for ~ bnj6... |
| bnj1447 35718 | Technical lemma for ~ bnj6... |
| bnj1448 35719 | Technical lemma for ~ bnj6... |
| bnj1449 35720 | Technical lemma for ~ bnj6... |
| bnj1442 35721 | Technical lemma for ~ bnj6... |
| bnj1450 35722 | Technical lemma for ~ bnj6... |
| bnj1423 35723 | Technical lemma for ~ bnj6... |
| bnj1452 35724 | Technical lemma for ~ bnj6... |
| bnj1466 35725 | Technical lemma for ~ bnj6... |
| bnj1467 35726 | Technical lemma for ~ bnj6... |
| bnj1463 35727 | Technical lemma for ~ bnj6... |
| bnj1489 35728 | Technical lemma for ~ bnj6... |
| bnj1491 35729 | Technical lemma for ~ bnj6... |
| bnj1312 35730 | Technical lemma for ~ bnj6... |
| bnj1493 35731 | Technical lemma for ~ bnj6... |
| bnj1497 35732 | Technical lemma for ~ bnj6... |
| bnj1498 35733 | Technical lemma for ~ bnj6... |
| bnj60 35734 | Well-founded recursion, pa... |
| bnj1514 35735 | Technical lemma for ~ bnj1... |
| bnj1518 35736 | Technical lemma for ~ bnj1... |
| bnj1519 35737 | Technical lemma for ~ bnj1... |
| bnj1520 35738 | Technical lemma for ~ bnj1... |
| bnj1501 35739 | Technical lemma for ~ bnj1... |
| bnj1500 35740 | Well-founded recursion, pa... |
| bnj1525 35741 | Technical lemma for ~ bnj1... |
| bnj1529 35742 | Technical lemma for ~ bnj1... |
| bnj1523 35743 | Technical lemma for ~ bnj1... |
| bnj1522 35744 | Well-founded recursion, pa... |
| nfan1c 35745 | Variant of ~ nfan and comm... |
| cbvex1v 35746 | Rule used to change bound ... |
| dvelimalcased 35747 | Eliminate a disjoint varia... |
| dvelimalcasei 35748 | Eliminate a disjoint varia... |
| dvelimexcased 35749 | Eliminate a disjoint varia... |
| dvelimexcasei 35750 | Eliminate a disjoint varia... |
| inv2 35751 | The intersection of the un... |
| jaeqifi 35752 | Inference combining four e... |
| exdifsn 35753 | There exists an element in... |
| axnulALT2 35754 | Alternate proof of ~ axnul... |
| f1preimaex 35755 | If the image under a one-t... |
| fnfvintima 35756 | Condition for a function v... |
| ordprcon 35757 | If an ordinal class is not... |
| xoromon 35758 | ` _om ` is either an ordin... |
| soinfdom 35759 | A strict order relation on... |
| abweex 35760 | The class of well-orders o... |
| ordtypeon 35761 | A proper class with a set-... |
| fnrelpredd 35762 | A function that preserves ... |
| cardpred 35763 | The cardinality function p... |
| nummin 35764 | Every nonempty class of nu... |
| 1enumen 35765 | The Fundamental Theorem of... |
| 1enumcard 35766 | The Fundamental Theorem of... |
| r11 35767 | Value of the cumulative hi... |
| r12 35768 | Value of the cumulative hi... |
| onrankid 35769 | The rank of an ordinal num... |
| rankfilimb 35770 | The rank of a finite well-... |
| r1filim 35771 | A finite set appears in th... |
| r1omfi 35772 | Obsolete theorem, use ~ hf... |
| r1omhf 35773 | A set is hereditarily fini... |
| r1ssel 35774 | A set is a subset of the v... |
| axnulALT3 35775 | Alternate proof of ~ axnul... |
| axprALT2 35776 | Alternate proof of ~ axpr ... |
| rankfo 35777 | The rank function maps the... |
| rankfn 35778 | The rank function is a fun... |
| trssfir1om 35779 | If every element in a tran... |
| r1omhfb 35780 | The class of all hereditar... |
| werankwe 35781 | Construct a well-order giv... |
| scotteqi 35782 | Equality theorem for the S... |
| elscott 35783 | Membership in a Scott's tr... |
| dfscott2 35784 | Alternate definition of a ... |
| dfscott3 35785 | Alternate definition of a ... |
| elscott2 35786 | Membership in a Scott's tr... |
| elscottrank 35787 | The rank of an element in ... |
| elscottrankeq 35788 | Elements in a Scott's tric... |
| elscottrankss 35789 | Relationship between the r... |
| scottrankeqel 35790 | If a member of the input s... |
| nelscottrankgt 35791 | If a member of the input s... |
| scottsn 35792 | Applying Scott's trick to ... |
| scott0bOLD 35793 | Obsolete version of ~ scot... |
| rankscott 35794 | The rank of a nonempty Sco... |
| rankscottu 35795 | An upper bound on the rank... |
| scottssr1 35796 | Relationship between a Sco... |
| acnum 35797 | The Axiom of Choice implie... |
| weexenwe 35798 | If a well-ordering of an i... |
| infinfnum 35799 | Equivalence between two in... |
| rncardr1prc 35800 | The Axiom of Choice implie... |
| acwer1prclem 35801 | Lemma for ~ acwer1prc . (... |
| acwer1prc 35802 | The class of all well-orde... |
| 5on 35813 | Ordinal 5 is an ordinal nu... |
| 6on 35814 | Ordinal 6 is an ordinal nu... |
| 7on 35815 | Ordinal 7 is an ordinal nu... |
| 8on 35816 | Ordinal 8 is an ordinal nu... |
| 9on 35817 | Ordinal 9 is an ordinal nu... |
| 5onn 35818 | The ordinal 5 is a natural... |
| 6onn 35819 | The ordinal 6 is a natural... |
| 7onn 35820 | The ordinal 7 is a natural... |
| 8onn 35821 | The ordinal 8 is a natural... |
| 9onn 35822 | The ordinal 9 is a natural... |
| prcinf 35823 | Any proper class is litera... |
| fineqvrep 35824 | If all sets are finite, th... |
| fineqvpow 35825 | If all sets are finite, th... |
| fineqvac 35826 | If all sets are finite, th... |
| fineqvacALT 35827 | Shorter proof of ~ fineqva... |
| fineqvomon 35828 | If all sets are finite, th... |
| fineqvomonb 35829 | All sets are finite iff al... |
| omprcomonb 35830 | The class of all finite or... |
| fineqvnttrclselem1 35831 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem2 35832 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclselem3 35833 | Lemma for ~ fineqvnttrclse... |
| fineqvnttrclse 35834 | A counterexample demonstra... |
| fineqvinfep 35835 | A counterexample demonstra... |
| axreg 35837 | Derivation of ~ ax-reg fro... |
| axregscl 35838 | A version of ~ ax-regs wit... |
| axregszf 35839 | Derivation of ~ zfregs usi... |
| setindregs 35840 | Set (epsilon) induction. ... |
| setinds2regs 35841 | Principle of set induction... |
| noinfepfnregs 35842 | There are no infinite desc... |
| noinfepregs 35843 | There are no infinite desc... |
| tz9.1regs 35844 | Every set has a transitive... |
| unir1regs 35845 | The cumulative hierarchy o... |
| trssfir1omregs 35846 | If every element in a tran... |
| r1omhfbregs 35847 | The class of all hereditar... |
| fineqvr1ombregs 35848 | All sets are finite iff al... |
| axregs 35849 | Derivation of ~ ax-regs fr... |
| axsepg2 35850 | A generalization of ~ ax-s... |
| axsepg3 35851 | A generalization of ~ ax-s... |
| axsepg3ALT 35852 | Alternate proof of ~ axsep... |
| axsepg4 35853 | A generalization of ~ ax-s... |
| axsepg5 35854 | A generalization of ~ ax-s... |
| axnulg 35855 | A generalization of ~ ax-n... |
| axpowg 35856 | A generalization of ~ ax-p... |
| axpowg2 35857 | A generalization of ~ ax-p... |
| axpowg3 35858 | A generalization of ~ ax-p... |
| kardfn 35861 | The ` kard ` class is a fu... |
| kardval 35862 | The value of the ` kard ` ... |
| kardval2 35863 | The value of the ` kard ` ... |
| kard0 35864 | The ` kard ` cardinality o... |
| elkarden 35865 | Any member of the ` kard `... |
| kardeq0 35866 | Applying ` kard ` to a cla... |
| kardeng 35867 | Two sets are equinumerous ... |
| kardenir 35868 | If two sets are equinumero... |
| kard0b 35869 | The empty set is the only ... |
| kardsn 35870 | A singleton has cardinalit... |
| karddom 35871 | One set dominates another ... |
| kardsdom 35872 | One set strictly dominates... |
| kardexen 35873 | One set is equinumerous to... |
| kardcard2a 35874 | If two sets have equal non... |
| kardcard2b 35875 | If two sets have equal ` k... |
| kardcard2 35876 | Two numerable sets have eq... |
| ackardcard 35877 | The Axiom of Choice implie... |
| kardcard 35878 | Two sets have equal ` kard... |
| kardnnfi 35879 | The ` kard ` cardinal numb... |
| kardfi 35880 | The ` kard ` cardinal numb... |
| rankkardu 35881 | An upper bound on the rank... |
| 1enumkard 35882 | The Fundamental Theorem of... |
| gblacfnacd 35923 | If ` G ` is a global choic... |
| onvf1odlem1 35924 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem2 35925 | Lemma for ~ onvf1od . (Co... |
| onvf1odlem3 35926 | Lemma for ~ onvf1od . The... |
| onvf1odlem4 35927 | Lemma for ~ onvf1od . If ... |
| onvf1od 35928 | If ` G ` is a global choic... |
| vonf1wev 35929 | If ` F ` maps the universe... |
| vonf1owev 35930 | If ` F ` is a bijection fr... |
| vonf1owevOLD 35931 | Obsolete version of ~ vonf... |
| wevgblacfn 35932 | If ` R ` is a well-orderin... |
| vonf1osev 35933 | If ` F ` is a bijection fr... |
| wevonprcf1o 35934 | If ` R ` is a set-like wel... |
| vonf1oonf1 35935 | If ` F ` is a bijection fr... |
| vonf1onprcf1ac 35936 | If ` F ` maps the universe... |
| onprcf1acwevdlem1 35937 | Lemma for ~ onprcf1acwevd ... |
| onprcf1acwevdlem2 35938 | Lemma for ~ onprcf1acwevd ... |
| onprcf1acwevd 35939 | If ` F ` maps the ordinals... |
| vonf1oonfo 35940 | If ` F ` is a bijection fr... |
| onvfowev 35941 | If ` F ` maps the ordinals... |
| zltp1ne 35942 | Integer ordering relation.... |
| nnltp1ne 35943 | Positive integer ordering ... |
| nn0ltp1ne 35944 | Nonnegative integer orderi... |
| fisshasheq 35945 | A finite set is equal to i... |
| 1enum 35946 | The Fundamental Theorem of... |
| cplgredgex 35947 | Any two (distinct) vertice... |
| cusgredgex 35948 | Any two (distinct) vertice... |
| cusgredgex2 35949 | Any two distinct vertices ... |
| revwlkb 35950 | Two words represent a walk... |
| usgrgt2cycl 35951 | A non-trivial cycle in a s... |
| usgrcyclgt2v 35952 | A simple graph with a non-... |
| cusgr3cyclex 35953 | Every complete simple grap... |
| 2cycl2d 35954 | Construction of a 2-cycle ... |
| acycgr0v 35955 | A null graph (with no vert... |
| acycgr1v 35956 | A multigraph with one vert... |
| acycgr2v 35957 | A simple graph with two ve... |
| prclisacycgr 35958 | A proper class (representi... |
| acycgrislfgr 35959 | An acyclic hypergraph is a... |
| upgracycumgr 35960 | An acyclic pseudograph is ... |
| umgracycusgr 35961 | An acyclic multigraph is a... |
| upgracycusgr 35962 | An acyclic pseudograph is ... |
| cusgracyclt3v 35963 | A complete simple graph is... |
| pthacycspth 35964 | A path in an acyclic graph... |
| acycgrsubgr 35965 | The subgraph of an acyclic... |
| quartfull 35972 | The quartic equation, writ... |
| deranglem 35973 | Lemma for derangements. (... |
| derangval 35974 | Define the derangement fun... |
| derangf 35975 | The derangement number is ... |
| derang0 35976 | The derangement number of ... |
| derangsn 35977 | The derangement number of ... |
| derangenlem 35978 | One half of ~ derangen . ... |
| derangen 35979 | The derangement number is ... |
| subfacval 35980 | The subfactorial is define... |
| derangen2 35981 | Write the derangement numb... |
| subfacf 35982 | The subfactorial is a func... |
| subfaclefac 35983 | The subfactorial is less t... |
| subfac0 35984 | The subfactorial at zero. ... |
| subfac1 35985 | The subfactorial at one. ... |
| subfacp1lem1 35986 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 35987 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 35988 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 35989 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 35990 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 35991 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 35992 | Lemma for ~ subfacp1 . By... |
| subfacp1 35993 | A two-term recurrence for ... |
| subfacval2 35994 | A closed-form expression f... |
| subfaclim 35995 | The subfactorial converges... |
| subfacval3 35996 | Another closed form expres... |
| derangfmla 35997 | The derangements formula, ... |
| erdszelem1 35998 | Lemma for ~ erdsze . (Con... |
| erdszelem2 35999 | Lemma for ~ erdsze . (Con... |
| erdszelem3 36000 | Lemma for ~ erdsze . (Con... |
| erdszelem4 36001 | Lemma for ~ erdsze . (Con... |
| erdszelem5 36002 | Lemma for ~ erdsze . (Con... |
| erdszelem6 36003 | Lemma for ~ erdsze . (Con... |
| erdszelem7 36004 | Lemma for ~ erdsze . (Con... |
| erdszelem8 36005 | Lemma for ~ erdsze . (Con... |
| erdszelem9 36006 | Lemma for ~ erdsze . (Con... |
| erdszelem10 36007 | Lemma for ~ erdsze . (Con... |
| erdszelem11 36008 | Lemma for ~ erdsze . (Con... |
| erdsze 36009 | The Erdős-Szekeres th... |
| erdsze2lem1 36010 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 36011 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 36012 | Generalize the statement o... |
| kur14lem1 36013 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 36014 | Lemma for ~ kur14 . Write... |
| kur14lem3 36015 | Lemma for ~ kur14 . A clo... |
| kur14lem4 36016 | Lemma for ~ kur14 . Compl... |
| kur14lem5 36017 | Lemma for ~ kur14 . Closu... |
| kur14lem6 36018 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 36019 | Lemma for ~ kur14 : main p... |
| kur14lem8 36020 | Lemma for ~ kur14 . Show ... |
| kur14lem9 36021 | Lemma for ~ kur14 . Since... |
| kur14lem10 36022 | Lemma for ~ kur14 . Disch... |
| kur14 36023 | Kuratowski's closure-compl... |
| ispconn 36030 | The property of being a pa... |
| pconncn 36031 | The property of being a pa... |
| pconntop 36032 | A simply connected space i... |
| issconn 36033 | The property of being a si... |
| sconnpconn 36034 | A simply connected space i... |
| sconntop 36035 | A simply connected space i... |
| sconnpht 36036 | A closed path in a simply ... |
| cnpconn 36037 | An image of a path-connect... |
| pconnconn 36038 | A path-connected space is ... |
| txpconn 36039 | The topological product of... |
| ptpconn 36040 | The topological product of... |
| indispconn 36041 | The indiscrete topology (o... |
| connpconn 36042 | A connected and locally pa... |
| qtoppconn 36043 | A quotient of a path-conne... |
| pconnpi1 36044 | All fundamental groups in ... |
| sconnpht2 36045 | Any two paths in a simply ... |
| sconnpi1 36046 | A path-connected topologic... |
| txsconnlem 36047 | Lemma for ~ txsconn . (Co... |
| txsconn 36048 | The topological product of... |
| cvxpconn 36049 | A convex subset of the com... |
| cvxsconn 36050 | A convex subset of the com... |
| blsconn 36051 | An open ball in the comple... |
| cnllysconn 36052 | The topology of the comple... |
| resconn 36053 | A subset of ` RR ` is simp... |
| ioosconn 36054 | An open interval is simply... |
| iccsconn 36055 | A closed interval is simpl... |
| retopsconn 36056 | The real numbers are simpl... |
| iccllysconn 36057 | A closed interval is local... |
| rellysconn 36058 | The real numbers are local... |
| iisconn 36059 | The unit interval is simpl... |
| iillysconn 36060 | The unit interval is local... |
| iinllyconn 36061 | The unit interval is local... |
| fncvm 36064 | Lemma for covering maps. ... |
| cvmscbv 36065 | Change bound variables in ... |
| iscvm 36066 | The property of being a co... |
| cvmtop1 36067 | Reverse closure for a cove... |
| cvmtop2 36068 | Reverse closure for a cove... |
| cvmcn 36069 | A covering map is a contin... |
| cvmcov 36070 | Property of a covering map... |
| cvmsrcl 36071 | Reverse closure for an eve... |
| cvmsi 36072 | One direction of ~ cvmsval... |
| cvmsval 36073 | Elementhood in the set ` S... |
| cvmsss 36074 | An even covering is a subs... |
| cvmsn0 36075 | An even covering is nonemp... |
| cvmsuni 36076 | An even covering of ` U ` ... |
| cvmsdisj 36077 | An even covering of ` U ` ... |
| cvmshmeo 36078 | Every element of an even c... |
| cvmsf1o 36079 | ` F ` , localized to an el... |
| cvmscld 36080 | The sets of an even coveri... |
| cvmsss2 36081 | An open subset of an evenl... |
| cvmcov2 36082 | The covering map property ... |
| cvmseu 36083 | Every element in ` U. T ` ... |
| cvmsiota 36084 | Identify the unique elemen... |
| cvmopnlem 36085 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 36086 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 36087 | A covering map is an open ... |
| cvmliftmolem1 36088 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 36089 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 36090 | A lift of a continuous fun... |
| cvmliftmo 36091 | A lift of a continuous fun... |
| cvmliftlem1 36092 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 36093 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 36094 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 36095 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 36096 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 36097 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 36098 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 36099 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 36100 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 36101 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 36102 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 36103 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 36104 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 36105 | Lemma for ~ cvmlift . Dis... |
| cvmlift 36106 | One of the important prope... |
| cvmfo 36107 | A covering map is an onto ... |
| cvmliftiota 36108 | Write out a function ` H `... |
| cvmlift2lem1 36109 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 36110 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 36111 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 36112 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 36113 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 36114 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 36115 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 36116 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 36117 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 36118 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 36119 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 36120 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 36121 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 36122 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 36123 | A two-dimensional version ... |
| cvmliftphtlem 36124 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 36125 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 36126 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 36127 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 36128 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 36129 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 36130 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 36131 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 36132 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 36133 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 36134 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 36135 | A general version of ~ cvm... |
| snmlff 36136 | The function ` F ` from ~ ... |
| snmlfval 36137 | The function ` F ` from ~ ... |
| snmlval 36138 | The property " ` A ` is si... |
| snmlflim 36139 | If ` A ` is simply normal,... |
| goel 36154 | A "Godel-set of membership... |
| goelel3xp 36155 | A "Godel-set of membership... |
| goeleq12bg 36156 | Two "Godel-set of membersh... |
| gonafv 36157 | The "Godel-set for the She... |
| goaleq12d 36158 | Equality of the "Godel-set... |
| gonanegoal 36159 | The Godel-set for the Shef... |
| satf 36160 | The satisfaction predicate... |
| satfsucom 36161 | The satisfaction predicate... |
| satfn 36162 | The satisfaction predicate... |
| satom 36163 | The satisfaction predicate... |
| satfvsucom 36164 | The satisfaction predicate... |
| satfv0 36165 | The value of the satisfact... |
| satfvsuclem1 36166 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 36167 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 36168 | The value of the satisfact... |
| satfv1lem 36169 | Lemma for ~ satfv1 . (Con... |
| satfv1 36170 | The value of the satisfact... |
| satfsschain 36171 | The binary relation of a s... |
| satfvsucsuc 36172 | The satisfaction predicate... |
| satfbrsuc 36173 | The binary relation of a s... |
| satfrel 36174 | The value of the satisfact... |
| satfdmlem 36175 | Lemma for ~ satfdm . (Con... |
| satfdm 36176 | The domain of the satisfac... |
| satfrnmapom 36177 | The range of the satisfact... |
| satfv0fun 36178 | The value of the satisfact... |
| satf0 36179 | The satisfaction predicate... |
| satf0sucom 36180 | The satisfaction predicate... |
| satf00 36181 | The value of the satisfact... |
| satf0suclem 36182 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 36183 | The value of the satisfact... |
| satf0op 36184 | An element of a value of t... |
| satf0n0 36185 | The value of the satisfact... |
| sat1el2xp 36186 | The first component of an ... |
| fmlafv 36187 | The valid Godel formulas o... |
| fmla 36188 | The set of all valid Godel... |
| fmla0 36189 | The valid Godel formulas o... |
| fmla0xp 36190 | The valid Godel formulas o... |
| fmlasuc0 36191 | The valid Godel formulas o... |
| fmlafvel 36192 | A class is a valid Godel f... |
| fmlasuc 36193 | The valid Godel formulas o... |
| fmla1 36194 | The valid Godel formulas o... |
| isfmlasuc 36195 | The characterization of a ... |
| fmlasssuc 36196 | The Godel formulas of heig... |
| fmlaomn0 36197 | The empty set is not a God... |
| fmlan0 36198 | The empty set is not a God... |
| gonan0 36199 | The "Godel-set of NAND" is... |
| goaln0 36200 | The "Godel-set of universa... |
| gonarlem 36201 | Lemma for ~ gonar (inducti... |
| gonar 36202 | If the "Godel-set of NAND"... |
| goalrlem 36203 | Lemma for ~ goalr (inducti... |
| goalr 36204 | If the "Godel-set of unive... |
| fmla0disjsuc 36205 | The set of valid Godel for... |
| fmlasucdisj 36206 | The valid Godel formulas o... |
| satfdmfmla 36207 | The domain of the satisfac... |
| satffunlem 36208 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 36209 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 36210 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 36211 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 36212 | The domain of an ordered p... |
| satffunlem2lem2 36213 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 36214 | Lemma 1 for ~ satffun : in... |
| satffunlem2 36215 | Lemma 2 for ~ satffun : in... |
| satffun 36216 | The value of the satisfact... |
| satff 36217 | The satisfaction predicate... |
| satfun 36218 | The satisfaction predicate... |
| satfvel 36219 | An element of the value of... |
| satfv0fvfmla0 36220 | The value of the satisfact... |
| satefv 36221 | The simplified satisfactio... |
| sate0 36222 | The simplified satisfactio... |
| satef 36223 | The simplified satisfactio... |
| sate0fv0 36224 | A simplified satisfaction ... |
| satefvfmla0 36225 | The simplified satisfactio... |
| sategoelfvb 36226 | Characterization of a valu... |
| sategoelfv 36227 | Condition of a valuation `... |
| ex-sategoelel 36228 | Example of a valuation of ... |
| ex-sategoel 36229 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 36230 | The value of the satisfact... |
| 2goelgoanfmla1 36231 | Two Godel-sets of membersh... |
| satefvfmla1 36232 | The simplified satisfactio... |
| ex-sategoelelomsuc 36233 | Example of a valuation of ... |
| ex-sategoelel12 36234 | Example of a valuation of ... |
| prv 36235 | The "proves" relation on a... |
| elnanelprv 36236 | The wff ` ( A e. B -/\ B e... |
| prv0 36237 | Every wff encoded as ` U `... |
| prv1n 36238 | No wff encoded as a Godel-... |
| mvtval 36307 | The set of variable typeco... |
| mrexval 36308 | The set of "raw expression... |
| mexval 36309 | The set of expressions, wh... |
| mexval2 36310 | The set of expressions, wh... |
| mdvval 36311 | The set of disjoint variab... |
| mvrsval 36312 | The set of variables in an... |
| mvrsfpw 36313 | The set of variables in an... |
| mrsubffval 36314 | The substitution of some v... |
| mrsubfval 36315 | The substitution of some v... |
| mrsubval 36316 | The substitution of some v... |
| mrsubcv 36317 | The value of a substituted... |
| mrsubvr 36318 | The value of a substituted... |
| mrsubff 36319 | A substitution is a functi... |
| mrsubrn 36320 | Although it is defined for... |
| mrsubff1 36321 | When restricted to complet... |
| mrsubff1o 36322 | When restricted to complet... |
| mrsub0 36323 | The value of the substitut... |
| mrsubf 36324 | A substitution is a functi... |
| mrsubccat 36325 | Substitution distributes o... |
| mrsubcn 36326 | A substitution does not ch... |
| elmrsubrn 36327 | Characterization of the su... |
| mrsubco 36328 | The composition of two sub... |
| mrsubvrs 36329 | The set of variables in a ... |
| msubffval 36330 | A substitution applied to ... |
| msubfval 36331 | A substitution applied to ... |
| msubval 36332 | A substitution applied to ... |
| msubrsub 36333 | A substitution applied to ... |
| msubty 36334 | The type of a substituted ... |
| elmsubrn 36335 | Characterization of substi... |
| msubrn 36336 | Although it is defined for... |
| msubff 36337 | A substitution is a functi... |
| msubco 36338 | The composition of two sub... |
| msubf 36339 | A substitution is a functi... |
| mvhfval 36340 | Value of the function mapp... |
| mvhval 36341 | Value of the function mapp... |
| mpstval 36342 | A pre-statement is an orde... |
| elmpst 36343 | Property of being a pre-st... |
| msrfval 36344 | Value of the reduct of a p... |
| msrval 36345 | Value of the reduct of a p... |
| mpstssv 36346 | A pre-statement is an orde... |
| mpst123 36347 | Decompose a pre-statement ... |
| mpstrcl 36348 | The elements of a pre-stat... |
| msrf 36349 | The reduct of a pre-statem... |
| msrrcl 36350 | If ` X ` and ` Y ` have th... |
| mstaval 36351 | Value of the set of statem... |
| msrid 36352 | The reduct of a statement ... |
| msrfo 36353 | The reduct of a pre-statem... |
| mstapst 36354 | A statement is a pre-state... |
| elmsta 36355 | Property of being a statem... |
| ismfs 36356 | A formal system is a tuple... |
| mfsdisj 36357 | The constants and variable... |
| mtyf2 36358 | The type function maps var... |
| mtyf 36359 | The type function maps var... |
| mvtss 36360 | The set of variable typeco... |
| maxsta 36361 | An axiom is a statement. ... |
| mvtinf 36362 | Each variable typecode has... |
| msubff1 36363 | When restricted to complet... |
| msubff1o 36364 | When restricted to complet... |
| mvhf 36365 | The function mapping varia... |
| mvhf1 36366 | The function mapping varia... |
| msubvrs 36367 | The set of variables in a ... |
| mclsrcl 36368 | Reverse closure for the cl... |
| mclsssvlem 36369 | Lemma for ~ mclsssv . (Co... |
| mclsval 36370 | The function mapping varia... |
| mclsssv 36371 | The closure of a set of ex... |
| ssmclslem 36372 | Lemma for ~ ssmcls . (Con... |
| vhmcls 36373 | All variable hypotheses ar... |
| ssmcls 36374 | The original expressions a... |
| ss2mcls 36375 | The closure is monotonic u... |
| mclsax 36376 | The closure is closed unde... |
| mclsind 36377 | Induction theorem for clos... |
| mppspstlem 36378 | Lemma for ~ mppspst . (Co... |
| mppsval 36379 | Definition of a provable p... |
| elmpps 36380 | Definition of a provable p... |
| mppspst 36381 | A provable pre-statement i... |
| mthmval 36382 | A theorem is a pre-stateme... |
| elmthm 36383 | A theorem is a pre-stateme... |
| mthmi 36384 | A statement whose reduct i... |
| mthmsta 36385 | A theorem is a pre-stateme... |
| mppsthm 36386 | A provable pre-statement i... |
| mthmblem 36387 | Lemma for ~ mthmb . (Cont... |
| mthmb 36388 | If two statements have the... |
| mthmpps 36389 | Given a theorem, there is ... |
| mclsppslem 36390 | The closure is closed unde... |
| mclspps 36391 | The closure is closed unde... |
| rexxfr3d 36445 | Transfer existential quant... |
| rexxfr3dALT 36446 | Longer proof of ~ rexxfr3d... |
| rspssbasd 36447 | The span of a set of ring ... |
| ellcsrspsn 36448 | Membership in a left coset... |
| ply1divalg3 36449 | Uniqueness of polynomial r... |
| r1peuqusdeg1 36450 | Uniqueness of polynomial r... |
| problem1 36472 | Practice problem 1. Clues... |
| problem2 36473 | Practice problem 2. Clues... |
| problem3 36474 | Practice problem 3. Clues... |
| problem4 36475 | Practice problem 4. Clues... |
| problem5 36476 | Practice problem 5. Clues... |
| quad3 36477 | Variant of quadratic equat... |
| climuzcnv 36478 | Utility lemma to convert b... |
| sinccvglem 36479 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 36480 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 36481 | The circumference of a cir... |
| elfzm12 36482 | Membership in a curtailed ... |
| nn0seqcvg 36483 | A strictly-decreasing nonn... |
| lediv2aALT 36484 | Division of both sides of ... |
| abs2sqlei 36485 | The absolute values of two... |
| abs2sqlti 36486 | The absolute values of two... |
| abs2sqle 36487 | The absolute values of two... |
| abs2sqlt 36488 | The absolute values of two... |
| abs2difi 36489 | Difference of absolute val... |
| abs2difabsi 36490 | Absolute value of differen... |
| 2thALT 36491 | Alternate proof of ~ 2th .... |
| orbi2iALT 36492 | Alternate proof of ~ orbi2... |
| pm3.48ALT 36493 | Alternate proof of ~ pm3.4... |
| 3jcadALT 36494 | Alternate proof of ~ 3jcad... |
| currybi 36495 | Biconditional version of C... |
| antnest 36496 | Suppose ` ph ` , ` ps ` ar... |
| antnestlaw3lem 36497 | Lemma for ~ antnestlaw3 . ... |
| antnestlaw1 36498 | A law of nested antecedent... |
| antnestlaw2 36499 | A law of nested antecedent... |
| antnestlaw3 36500 | A law of nested antecedent... |
| antnestALT 36501 | Alternative proof of ~ ant... |
| axextprim 36508 | ~ ax-ext without distinct ... |
| axrepprim 36509 | ~ ax-rep without distinct ... |
| axunprim 36510 | ~ ax-un without distinct v... |
| axpowprim 36511 | ~ ax-pow without distinct ... |
| axregprim 36512 | ~ ax-reg without distinct ... |
| axinfprim 36513 | ~ ax-inf without distinct ... |
| axacprim 36514 | ~ ax-ac without distinct v... |
| untelirr 36515 | We call a class "untanged"... |
| untuni 36516 | The union of a class is un... |
| untsucf 36517 | If a class is untangled, t... |
| unt0 36518 | The empty set is untangled... |
| untint 36519 | If there is an untangled e... |
| efrunt 36520 | If ` A ` is well-founded b... |
| untangtr 36521 | A transitive class is unta... |
| 3jaodd 36522 | Double deduction form of ~... |
| 3orit 36523 | Closed form of ~ 3ori . (... |
| biimpexp 36524 | A biconditional in the ant... |
| nepss 36525 | Two classes are unequal if... |
| 3ccased 36526 | Triple disjunction form of... |
| dfso3 36527 | Expansion of the definitio... |
| brtpid1 36528 | A binary relation involvin... |
| brtpid2 36529 | A binary relation involvin... |
| brtpid3 36530 | A binary relation involvin... |
| iota5f 36531 | A method for computing iot... |
| jath 36532 | Closed form of ~ ja . Pro... |
| xpab 36533 | Cartesian product of two c... |
| nnuni 36534 | The union of a finite ordi... |
| sqdivzi 36535 | Distribution of square ove... |
| supfz 36536 | The supremum of a finite s... |
| inffz 36537 | The infimum of a finite se... |
| fz0n 36538 | The sequence ` ( 0 ... ( N... |
| shftvalg 36539 | Value of a sequence shifte... |
| divcnvlin 36540 | Limit of the ratio of two ... |
| climlec3 36541 | Comparison of a constant t... |
| iexpire 36542 | ` _i ` raised to itself is... |
| bcneg1 36543 | The binomial coefficient o... |
| bcm1nt 36544 | The proportion of one bino... |
| bcprod 36545 | A product identity for bin... |
| bccolsum 36546 | A column-sum rule for bino... |
| iprodefisumlem 36547 | Lemma for ~ iprodefisum . ... |
| iprodefisum 36548 | Applying the exponential f... |
| iprodgam 36549 | An infinite product versio... |
| faclimlem1 36550 | Lemma for ~ faclim . Clos... |
| faclimlem2 36551 | Lemma for ~ faclim . Show... |
| faclimlem3 36552 | Lemma for ~ faclim . Alge... |
| faclim 36553 | An infinite product expres... |
| iprodfac 36554 | An infinite product expres... |
| faclim2 36555 | Another factorial limit du... |
| gcd32 36556 | Swap the second and third ... |
| gcdabsorb 36557 | Absorption law for gcd. (... |
| dftr6 36558 | A potential definition of ... |
| coep 36559 | Composition with the membe... |
| coepr 36560 | Composition with the conve... |
| dffr5 36561 | A quantifier-free definiti... |
| dfso2 36562 | Quantifier-free definition... |
| br8 36563 | Substitution for an eight-... |
| br6 36564 | Substitution for a six-pla... |
| br4 36565 | Substitution for a four-pl... |
| cnvco1 36566 | Another distributive law o... |
| cnvco2 36567 | Another distributive law o... |
| eldm3 36568 | Quantifier-free definition... |
| elrn3 36569 | Quantifier-free definition... |
| pocnv 36570 | The converse of a partial ... |
| socnv 36571 | The converse of a strict o... |
| elintfv 36572 | Membership in an intersect... |
| funpsstri 36573 | A condition for subset tri... |
| fundmpss 36574 | If a class ` F ` is a prop... |
| funsseq 36575 | Given two functions with e... |
| funbreq 36576 | An equality condition for ... |
| br1steq 36577 | Uniqueness condition for t... |
| br2ndeq 36578 | Uniqueness condition for t... |
| dfdm5 36579 | Definition of domain in te... |
| dfrn5 36580 | Definition of range in ter... |
| opelco3 36581 | Alternate way of saying th... |
| elima4 36582 | Quantifier-free expression... |
| fv1stcnv 36583 | The value of the converse ... |
| fv2ndcnv 36584 | The value of the converse ... |
| elpotr 36585 | A class of transitive sets... |
| dford5reg 36586 | Given ~ ax-reg , an ordina... |
| dfon2lem1 36587 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 36588 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 36589 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 36590 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 36591 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 36592 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 36593 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 36594 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 36595 | Lemma for ~ dfon2 . A cla... |
| dfon2 36596 | ` On ` consists of all set... |
| rdgprc0 36597 | The value of the recursive... |
| rdgprc 36598 | The value of the recursive... |
| dfrdg2 36599 | Alternate definition of th... |
| dfrdg3 36600 | Generalization of ~ dfrdg2... |
| axextdfeq 36601 | A version of ~ ax-ext for ... |
| ax8dfeq 36602 | A version of ~ ax-8 for us... |
| axextdist 36603 | ~ ax-ext with distinctors ... |
| axextbdist 36604 | ~ axextb with distinctors ... |
| 19.12b 36605 | Version of ~ 19.12vv with ... |
| exnel 36606 | There is always a set not ... |
| distel 36607 | Distinctors in terms of me... |
| axextndbi 36608 | ~ axextnd as a bicondition... |
| hbntg 36609 | A more general form of ~ h... |
| hbimtg 36610 | A more general and closed ... |
| hbaltg 36611 | A more general and closed ... |
| hbng 36612 | A more general form of ~ h... |
| hbimg 36613 | A more general form of ~ h... |
| wsuceq123 36618 | Equality theorem for well-... |
| wsuceq1 36619 | Equality theorem for well-... |
| wsuceq2 36620 | Equality theorem for well-... |
| wsuceq3 36621 | Equality theorem for well-... |
| nfwsuc 36622 | Bound-variable hypothesis ... |
| wlimeq12 36623 | Equality theorem for the l... |
| wlimeq1 36624 | Equality theorem for the l... |
| wlimeq2 36625 | Equality theorem for the l... |
| nfwlim 36626 | Bound-variable hypothesis ... |
| elwlim 36627 | Membership in the limit cl... |
| wzel 36628 | The zero of a well-founded... |
| wsuclem 36629 | Lemma for the supremum pro... |
| wsucex 36630 | Existence theorem for well... |
| wsuccl 36631 | If ` X ` is a set with an ... |
| wsuclb 36632 | A well-founded successor i... |
| wlimss 36633 | The class of limit points ... |
| txpss3v 36682 | A tail Cartesian product i... |
| txprel 36683 | A tail Cartesian product i... |
| brtxp 36684 | Characterize a ternary rel... |
| brtxp2 36685 | The binary relation over a... |
| dfpprod2 36686 | Expanded definition of par... |
| pprodcnveq 36687 | A converse law for paralle... |
| pprodss4v 36688 | The parallel product is a ... |
| brpprod 36689 | Characterize a quaternary ... |
| brpprod3a 36690 | Condition for parallel pro... |
| brpprod3b 36691 | Condition for parallel pro... |
| relsset 36692 | The subset class is a bina... |
| brsset 36693 | For sets, the ` SSet ` bin... |
| idsset 36694 | ` _I ` is equal to the int... |
| eltrans 36695 | Membership in the class of... |
| dfon3 36696 | A quantifier-free definiti... |
| dfon4 36697 | Another quantifier-free de... |
| brtxpsd 36698 | Expansion of a common form... |
| brtxpsd2 36699 | Another common abbreviatio... |
| brtxpsd3 36700 | A third common abbreviatio... |
| relbigcup 36701 | The ` Bigcup ` relationshi... |
| brbigcup 36702 | Binary relation over ` Big... |
| dfbigcup2 36703 | ` Bigcup ` using maps-to n... |
| fobigcup 36704 | ` Bigcup ` maps the univer... |
| fnbigcup 36705 | ` Bigcup ` is a function o... |
| fvbigcup 36706 | For sets, ` Bigcup ` yield... |
| elfix 36707 | Membership in the fixpoint... |
| elfix2 36708 | Alternative membership in ... |
| dffix2 36709 | The fixpoints of a class i... |
| fixssdm 36710 | The fixpoints of a class a... |
| fixssrn 36711 | The fixpoints of a class a... |
| fixcnv 36712 | The fixpoints of a class a... |
| fixun 36713 | The fixpoint operator dist... |
| ellimits 36714 | Membership in the class of... |
| limitssson 36715 | The class of all limit ord... |
| dfom5b 36716 | A quantifier-free definiti... |
| sscoid 36717 | A condition for subset and... |
| dffun10 36718 | Another potential definiti... |
| elfuns 36719 | Membership in the class of... |
| elfunsg 36720 | Closed form of ~ elfuns . ... |
| brsingle 36721 | The binary relation form o... |
| elsingles 36722 | Membership in the class of... |
| fnsingle 36723 | The singleton relationship... |
| fvsingle 36724 | The value of the singleton... |
| dfsingles2 36725 | Alternate definition of th... |
| snelsingles 36726 | A singleton is a member of... |
| dfiota3 36727 | A definition of iota using... |
| dffv5 36728 | Another quantifier-free de... |
| unisnif 36729 | Express union of singleton... |
| brimage 36730 | Binary relation form of th... |
| brimageg 36731 | Closed form of ~ brimage .... |
| funimage 36732 | ` Image A ` is a function.... |
| fnimage 36733 | ` Image R ` is a function ... |
| imageval 36734 | The image functor in maps-... |
| fvimage 36735 | Value of the image functor... |
| brcart 36736 | Binary relation form of th... |
| brdomain 36737 | Binary relation form of th... |
| brrange 36738 | Binary relation form of th... |
| brdomaing 36739 | Closed form of ~ brdomain ... |
| brrangeg 36740 | Closed form of ~ brrange .... |
| brimg 36741 | Binary relation form of th... |
| brapply 36742 | Binary relation form of th... |
| brcup 36743 | Binary relation form of th... |
| brcap 36744 | Binary relation form of th... |
| lemsuccf 36745 | Lemma for unfolding differ... |
| brsuccf 36746 | Binary relation form of th... |
| dfsuccf2 36747 | Alternate definition of Sc... |
| funpartlem 36748 | Lemma for ~ funpartfun . ... |
| funpartfun 36749 | The functional part of ` F... |
| funpartss 36750 | The functional part of ` F... |
| funpartfv 36751 | The function value of the ... |
| fullfunfnv 36752 | The full functional part o... |
| fullfunfv 36753 | The function value of the ... |
| brfullfun 36754 | A binary relation form con... |
| brrestrict 36755 | Binary relation form of th... |
| dfrecs2 36756 | A quantifier-free definiti... |
| dfrdg4 36757 | A quantifier-free definiti... |
| dfint3 36758 | Quantifier-free definition... |
| imagesset 36759 | The Image functor applied ... |
| brub 36760 | Binary relation form of th... |
| brlb 36761 | Binary relation form of th... |
| dffr7 36762 | Alternate quantifier-free ... |
| altopex 36767 | Alternative ordered pairs ... |
| altopthsn 36768 | Two alternate ordered pair... |
| altopeq12 36769 | Equality for alternate ord... |
| altopeq1 36770 | Equality for alternate ord... |
| altopeq2 36771 | Equality for alternate ord... |
| altopth1 36772 | Equality of the first memb... |
| altopth2 36773 | Equality of the second mem... |
| altopthg 36774 | Alternate ordered pair the... |
| altopthbg 36775 | Alternate ordered pair the... |
| altopth 36776 | The alternate ordered pair... |
| altopthb 36777 | Alternate ordered pair the... |
| altopthc 36778 | Alternate ordered pair the... |
| altopthd 36779 | Alternate ordered pair the... |
| altxpeq1 36780 | Equality for alternate Car... |
| altxpeq2 36781 | Equality for alternate Car... |
| elaltxp 36782 | Membership in alternate Ca... |
| altopelaltxp 36783 | Alternate ordered pair mem... |
| altxpsspw 36784 | An inclusion rule for alte... |
| altxpexg 36785 | The alternate Cartesian pr... |
| rankaltopb 36786 | Compute the rank of an alt... |
| nfaltop 36787 | Bound-variable hypothesis ... |
| sbcaltop 36788 | Distribution of class subs... |
| cgrrflx2d 36791 | Deduction form of ~ axcgrr... |
| cgrtr4d 36792 | Deduction form of ~ axcgrt... |
| cgrtr4and 36793 | Deduction form of ~ axcgrt... |
| cgrrflx 36794 | Reflexivity law for congru... |
| cgrrflxd 36795 | Deduction form of ~ cgrrfl... |
| cgrcomim 36796 | Congruence commutes on the... |
| cgrcom 36797 | Congruence commutes betwee... |
| cgrcomand 36798 | Deduction form of ~ cgrcom... |
| cgrtr 36799 | Transitivity law for congr... |
| cgrtrand 36800 | Deduction form of ~ cgrtr ... |
| cgrtr3 36801 | Transitivity law for congr... |
| cgrtr3and 36802 | Deduction form of ~ cgrtr3... |
| cgrcoml 36803 | Congruence commutes on the... |
| cgrcomr 36804 | Congruence commutes on the... |
| cgrcomlr 36805 | Congruence commutes on bot... |
| cgrcomland 36806 | Deduction form of ~ cgrcom... |
| cgrcomrand 36807 | Deduction form of ~ cgrcom... |
| cgrcomlrand 36808 | Deduction form of ~ cgrcom... |
| cgrtriv 36809 | Degenerate segments are co... |
| cgrid2 36810 | Identity law for congruenc... |
| cgrdegen 36811 | Two congruent segments are... |
| brofs 36812 | Binary relation form of th... |
| 5segofs 36813 | Rephrase ~ ax5seg using th... |
| ofscom 36814 | The outer five segment pre... |
| cgrextend 36815 | Link congruence over a pai... |
| cgrextendand 36816 | Deduction form of ~ cgrext... |
| segconeq 36817 | Two points that satisfy th... |
| segconeu 36818 | Existential uniqueness ver... |
| btwntriv2 36819 | Betweenness always holds f... |
| btwncomim 36820 | Betweenness commutes. Imp... |
| btwncom 36821 | Betweenness commutes. (Co... |
| btwncomand 36822 | Deduction form of ~ btwnco... |
| btwntriv1 36823 | Betweenness always holds f... |
| btwnswapid 36824 | If you can swap the first ... |
| btwnswapid2 36825 | If you can swap arguments ... |
| btwnintr 36826 | Inner transitivity law for... |
| btwnexch3 36827 | Exchange the first endpoin... |
| btwnexch3and 36828 | Deduction form of ~ btwnex... |
| btwnouttr2 36829 | Outer transitivity law for... |
| btwnexch2 36830 | Exchange the outer point o... |
| btwnouttr 36831 | Outer transitivity law for... |
| btwnexch 36832 | Outer transitivity law for... |
| btwnexchand 36833 | Deduction form of ~ btwnex... |
| btwndiff 36834 | There is always a ` c ` di... |
| trisegint 36835 | A line segment between two... |
| funtransport 36838 | The ` TransportTo ` relati... |
| fvtransport 36839 | Calculate the value of the... |
| transportcl 36840 | Closure law for segment tr... |
| transportprops 36841 | Calculate the defining pro... |
| brifs 36850 | Binary relation form of th... |
| ifscgr 36851 | Inner five segment congrue... |
| cgrsub 36852 | Removing identical parts f... |
| brcgr3 36853 | Binary relation form of th... |
| cgr3permute3 36854 | Permutation law for three-... |
| cgr3permute1 36855 | Permutation law for three-... |
| cgr3permute2 36856 | Permutation law for three-... |
| cgr3permute4 36857 | Permutation law for three-... |
| cgr3permute5 36858 | Permutation law for three-... |
| cgr3tr4 36859 | Transitivity law for three... |
| cgr3com 36860 | Commutativity law for thre... |
| cgr3rflx 36861 | Identity law for three-pla... |
| cgrxfr 36862 | A line segment can be divi... |
| btwnxfr 36863 | A condition for extending ... |
| colinrel 36864 | Colinearity is a relations... |
| brcolinear2 36865 | Alternate colinearity bina... |
| brcolinear 36866 | The binary relation form o... |
| colinearex 36867 | The colinear predicate exi... |
| colineardim1 36868 | If ` A ` is colinear with ... |
| colinearperm1 36869 | Permutation law for coline... |
| colinearperm3 36870 | Permutation law for coline... |
| colinearperm2 36871 | Permutation law for coline... |
| colinearperm4 36872 | Permutation law for coline... |
| colinearperm5 36873 | Permutation law for coline... |
| colineartriv1 36874 | Trivial case of colinearit... |
| colineartriv2 36875 | Trivial case of colinearit... |
| btwncolinear1 36876 | Betweenness implies coline... |
| btwncolinear2 36877 | Betweenness implies coline... |
| btwncolinear3 36878 | Betweenness implies coline... |
| btwncolinear4 36879 | Betweenness implies coline... |
| btwncolinear5 36880 | Betweenness implies coline... |
| btwncolinear6 36881 | Betweenness implies coline... |
| colinearxfr 36882 | Transfer law for colineari... |
| lineext 36883 | Extend a line with a missi... |
| brofs2 36884 | Change some conditions for... |
| brifs2 36885 | Change some conditions for... |
| brfs 36886 | Binary relation form of th... |
| fscgr 36887 | Congruence law for the gen... |
| linecgr 36888 | Congruence rule for lines.... |
| linecgrand 36889 | Deduction form of ~ linecg... |
| lineid 36890 | Identity law for points on... |
| idinside 36891 | Law for finding a point in... |
| endofsegid 36892 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 36893 | Deduction form of ~ endofs... |
| btwnconn1lem1 36894 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 36895 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 36896 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 36897 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 36898 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 36899 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 36900 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 36901 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 36902 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 36903 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 36904 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 36905 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 36906 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 36907 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 36908 | Connectitivy law for betwe... |
| btwnconn2 36909 | Another connectivity law f... |
| btwnconn3 36910 | Inner connectivity law for... |
| midofsegid 36911 | If two points fall in the ... |
| segcon2 36912 | Generalization of ~ axsegc... |
| brsegle 36915 | Binary relation form of th... |
| brsegle2 36916 | Alternate characterization... |
| seglecgr12im 36917 | Substitution law for segme... |
| seglecgr12 36918 | Substitution law for segme... |
| seglerflx 36919 | Segment comparison is refl... |
| seglemin 36920 | Any segment is at least as... |
| segletr 36921 | Segment less than is trans... |
| segleantisym 36922 | Antisymmetry law for segme... |
| seglelin 36923 | Linearity law for segment ... |
| btwnsegle 36924 | If ` B ` falls between ` A... |
| colinbtwnle 36925 | Given three colinear point... |
| broutsideof 36928 | Binary relation form of ` ... |
| broutsideof2 36929 | Alternate form of ` Outsid... |
| outsidene1 36930 | Outsideness implies inequa... |
| outsidene2 36931 | Outsideness implies inequa... |
| btwnoutside 36932 | A principle linking outsid... |
| broutsideof3 36933 | Characterization of outsid... |
| outsideofrflx 36934 | Reflexivity of outsideness... |
| outsideofcom 36935 | Commutativity law for outs... |
| outsideoftr 36936 | Transitivity law for outsi... |
| outsideofeq 36937 | Uniqueness law for ` Outsi... |
| outsideofeu 36938 | Given a nondegenerate ray,... |
| outsidele 36939 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 36940 | Outside of implies colinea... |
| funray 36947 | Show that the ` Ray ` rela... |
| fvray 36948 | Calculate the value of the... |
| funline 36949 | Show that the ` Line ` rel... |
| linedegen 36950 | When ` Line ` is applied w... |
| fvline 36951 | Calculate the value of the... |
| liness 36952 | A line is a subset of the ... |
| fvline2 36953 | Alternate definition of a ... |
| lineunray 36954 | A line is composed of a po... |
| lineelsb2 36955 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 36956 | Reflexivity law for line m... |
| linecom 36957 | Commutativity law for line... |
| linerflx2 36958 | Reflexivity law for line m... |
| ellines 36959 | Membership in the set of a... |
| linethru 36960 | If ` A ` is a line contain... |
| hilbert1.1 36961 | There is a line through an... |
| hilbert1.2 36962 | There is at most one line ... |
| linethrueu 36963 | There is a unique line goi... |
| lineintmo 36964 | Two distinct lines interse... |
| fwddifval 36969 | Calculate the value of the... |
| fwddifnval 36970 | The value of the forward d... |
| fwddifn0 36971 | The value of the n-iterate... |
| fwddifnp1 36972 | The value of the n-iterate... |
| rank0 36973 | The rank of the empty set ... |
| rankeq1o 36974 | The only set with rank ` 1... |
| hftr 36975 | The class of all hereditar... |
| hfext 36976 | Extensionality for HF sets... |
| hfninf 36977 | ` _om ` is not hereditaril... |
| nmulfn 36980 | Natural multiplication is ... |
| nmulprop 36981 | Show closure and value of ... |
| nmulcl 36982 | Closure law for natural mu... |
| nmulval 36983 | Show the value of natural ... |
| nmulcld 36984 | Closure law for natural mu... |
| nmulcom 36985 | Natural multiplication is ... |
| nmulr0 36986 | Natural multiplication by ... |
| nmull0 36987 | Natural multiplication by ... |
| nmulrid 36988 | Identity law for natural m... |
| nmullid 36989 | Identity law for natural m... |
| onelond 36990 | An element of an ordinal n... |
| ontr2d 36991 | Transitive law for ordinal... |
| onelssd 36992 | An element of an ordinal n... |
| nmulr0d 36993 | Natural multiplication by ... |
| nmull0d 36994 | Natural multiplication by ... |
| nmulridd 36995 | Identity law for natural m... |
| nmullidd 36996 | Identity law for natural m... |
| nmulcomd 36997 | Natural multiplication com... |
| naddridd 36998 | Identity law for natural a... |
| naddlidd 36999 | Identity law for natural a... |
| naddcomd 37000 | Natural addition commutes.... |
| naddassd 37001 | Natural addition associate... |
| nadd32d 37002 | Commutative/associative la... |
| nmuladdel 37003 | Ordering relationship for ... |
| nmuladdss 37004 | Ordering relationship for ... |
| nmulss1 37005 | Natural multiplication pre... |
| nmulel1 37006 | Natural multiplication by ... |
| ltnmul 37007 | Characterize less-than a n... |
| nmulle 37008 | A condition for bounding a... |
| ltnadd 37009 | Condition for bounding a n... |
| naddle 37010 | Condition for bounding nat... |
| nadddilem1 37011 | Lemma for ~ nadddi . Prov... |
| nadddilem2 37012 | Lemma for ~ nadddi . Prov... |
| nadddilem3 37013 | Lemma for ~ nadddi . Prov... |
| nadddilem4 37014 | Lemma for ~ nadddi . Prov... |
| nadddi 37015 | Natural multiplication dis... |
| nadddid 37016 | Natural multiplication dis... |
| nadddird 37017 | Natural multiplication dis... |
| rmoeqi 37018 | Equality inference for res... |
| rmoeqbii 37019 | Equality inference for res... |
| reueqi 37020 | Equality inference for res... |
| reueqbii 37021 | Equality inference for res... |
| sbceqbii 37022 | Formula-building inference... |
| disjeq1i 37023 | Equality theorem for disjo... |
| disjeq12i 37024 | Equality theorem for disjo... |
| rabeqbii 37025 | Equality theorem for restr... |
| iuneq12i 37026 | Equality theorem for index... |
| iineq1i 37027 | Equality theorem for index... |
| iineq12i 37028 | Equality theorem for index... |
| riotaeqbii 37029 | Equivalent wff's and equal... |
| riotaeqi 37030 | Equal domains yield equal ... |
| ixpeq1i 37031 | Equality inference for inf... |
| ixpeq12i 37032 | Equality inference for inf... |
| sumeq2si 37033 | Equality inference for sum... |
| sumeq12si 37034 | Equality inference for sum... |
| prodeq2si 37035 | Equality inference for pro... |
| prodeq12si 37036 | Equality inference for pro... |
| itgeq12i 37037 | Equality inference for an ... |
| itgeq1i 37038 | Equality inference for an ... |
| itgeq2i 37039 | Equality inference for an ... |
| ditgeq123i 37040 | Equality inference for the... |
| ditgeq12i 37041 | Equality inference for the... |
| ditgeq3i 37042 | Equality inference for the... |
| rmoeqdv 37043 | Formula-building rule for ... |
| rmoeqbidv 37044 | Formula-building rule for ... |
| sbequbidv 37045 | Deduction substituting bot... |
| disjeq12dv 37046 | Equality theorem for disjo... |
| ixpeq12dv 37047 | Equality theorem for infin... |
| sumeq12sdv 37048 | Equality deduction for sum... |
| prodeq12sdv 37049 | Equality deduction for pro... |
| itgeq12sdv 37050 | Equality theorem for an in... |
| itgeq2sdv 37051 | Equality theorem for an in... |
| ditgeq123dv 37052 | Equality theorem for the d... |
| ditgeq12d 37053 | Equality theorem for the d... |
| ditgeq3sdv 37054 | Equality theorem for the d... |
| in-ax8 37055 | A proof of ~ ax-8 that doe... |
| ss-ax8 37056 | A proof of ~ ax-8 that doe... |
| cbvralvw2 37057 | Change bound variable and ... |
| cbvrexvw2 37058 | Change bound variable and ... |
| cbvrmovw2 37059 | Change bound variable and ... |
| cbvreuvw2 37060 | Change bound variable and ... |
| cbvsbcvw2 37061 | Change bound variable of a... |
| cbvcsbvw2 37062 | Change bound variable of a... |
| cbviunvw2 37063 | Change bound variable and ... |
| cbviinvw2 37064 | Change bound variable and ... |
| cbvmptvw2 37065 | Change bound variable and ... |
| cbvdisjvw2 37066 | Change bound variable and ... |
| cbvriotavw2 37067 | Change bound variable and ... |
| cbvoprab1vw 37068 | Change the first bound var... |
| cbvoprab2vw 37069 | Change the second bound va... |
| cbvoprab123vw 37070 | Change all bound variables... |
| cbvoprab23vw 37071 | Change the second and thir... |
| cbvoprab13vw 37072 | Change the first and third... |
| cbvmpovw2 37073 | Change bound variables and... |
| cbvmpo1vw2 37074 | Change domains and the fir... |
| cbvmpo2vw2 37075 | Change domains and the sec... |
| cbvixpvw2 37076 | Change bound variable and ... |
| cbvsumvw2 37077 | Change bound variable and ... |
| cbvprodvw2 37078 | Change bound variable and ... |
| cbvitgvw2 37079 | Change bound variable and ... |
| cbvditgvw2 37080 | Change bound variable and ... |
| cbvmodavw 37081 | Change bound variable in t... |
| cbveudavw 37082 | Change bound variable in t... |
| cbvrmodavw 37083 | Change bound variable in t... |
| cbvreudavw 37084 | Change bound variable in t... |
| cbvsbdavw 37085 | Change bound variable in p... |
| cbvsbdavw2 37086 | Change bound variable in p... |
| cbvabdavw 37087 | Change bound variable in c... |
| cbvsbcdavw 37088 | Change bound variable of a... |
| cbvsbcdavw2 37089 | Change bound variable of a... |
| cbvcsbdavw 37090 | Change bound variable of a... |
| cbvcsbdavw2 37091 | Change bound variable of a... |
| cbvrabdavw 37092 | Change bound variable in r... |
| cbviundavw 37093 | Change bound variable in i... |
| cbviindavw 37094 | Change bound variable in i... |
| cbvopab1davw 37095 | Change the first bound var... |
| cbvopab2davw 37096 | Change the second bound va... |
| cbvopabdavw 37097 | Change bound variables in ... |
| cbvmptdavw 37098 | Change bound variable in a... |
| cbvdisjdavw 37099 | Change bound variable in a... |
| cbviotadavw 37100 | Change bound variable in a... |
| cbvriotadavw 37101 | Change bound variable in a... |
| cbvoprab1davw 37102 | Change the first bound var... |
| cbvoprab2davw 37103 | Change the second bound va... |
| cbvoprab3davw 37104 | Change the third bound var... |
| cbvoprab123davw 37105 | Change all bound variables... |
| cbvoprab12davw 37106 | Change the first and secon... |
| cbvoprab23davw 37107 | Change the second and thir... |
| cbvoprab13davw 37108 | Change the first and third... |
| cbvixpdavw 37109 | Change bound variable in a... |
| cbvsumdavw 37110 | Change bound variable in a... |
| cbvproddavw 37111 | Change bound variable in a... |
| cbvitgdavw 37112 | Change bound variable in a... |
| cbvditgdavw 37113 | Change bound variable in a... |
| cbvrmodavw2 37114 | Change bound variable and ... |
| cbvreudavw2 37115 | Change bound variable and ... |
| cbvrabdavw2 37116 | Change bound variable and ... |
| cbviundavw2 37117 | Change bound variable and ... |
| cbviindavw2 37118 | Change bound variable and ... |
| cbvmptdavw2 37119 | Change bound variable and ... |
| cbvdisjdavw2 37120 | Change bound variable and ... |
| cbvriotadavw2 37121 | Change bound variable and ... |
| cbvmpodavw2 37122 | Change bound variable and ... |
| cbvmpo1davw2 37123 | Change first bound variabl... |
| cbvmpo2davw2 37124 | Change second bound variab... |
| cbvixpdavw2 37125 | Change bound variable and ... |
| cbvsumdavw2 37126 | Change bound variable and ... |
| cbvproddavw2 37127 | Change bound variable and ... |
| cbvitgdavw2 37128 | Change bound variable and ... |
| cbvditgdavw2 37129 | Change bound variable and ... |
| mpomulnzcnf 37130 | Multiplication maps nonzer... |
| a1i14 37131 | Add two antecedents to a w... |
| a1i24 37132 | Add two antecedents to a w... |
| exp5d 37133 | An exportation inference. ... |
| exp5g 37134 | An exportation inference. ... |
| exp5k 37135 | An exportation inference. ... |
| exp56 37136 | An exportation inference. ... |
| exp58 37137 | An exportation inference. ... |
| exp510 37138 | An exportation inference. ... |
| exp511 37139 | An exportation inference. ... |
| exp512 37140 | An exportation inference. ... |
| 3com12d 37141 | Commutation in consequent.... |
| imp5p 37142 | A triple importation infer... |
| imp5q 37143 | A triple importation infer... |
| subtr 37144 | Transitivity of implicit s... |
| subtr2 37145 | Transitivity of implicit s... |
| trer 37146 | A relation intersected wit... |
| elicc3 37147 | An equivalent membership c... |
| finminlem 37148 | A useful lemma about finit... |
| gtinf 37149 | Any number greater than an... |
| opnrebl 37150 | A set is open in the stand... |
| opnrebl2 37151 | A set is open in the stand... |
| nn0prpwlem 37152 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 37153 | Two nonnegative integers a... |
| topbnd 37154 | Two equivalent expressions... |
| opnbnd 37155 | A set is open iff it is di... |
| cldbnd 37156 | A set is closed iff it con... |
| ntruni 37157 | A union of interiors is a ... |
| clsun 37158 | A pairwise union of closur... |
| clsint2 37159 | The closure of an intersec... |
| opnregcld 37160 | A set is regularly closed ... |
| cldregopn 37161 | A set if regularly open if... |
| neiin 37162 | Two neighborhoods intersec... |
| hmeoclda 37163 | Homeomorphisms preserve cl... |
| hmeocldb 37164 | Homeomorphisms preserve cl... |
| ivthALT 37165 | An alternate proof of the ... |
| fnerel 37168 | Fineness is a relation. (... |
| isfne 37169 | The predicate " ` B ` is f... |
| isfne4 37170 | The predicate " ` B ` is f... |
| isfne4b 37171 | A condition for a topology... |
| isfne2 37172 | The predicate " ` B ` is f... |
| isfne3 37173 | The predicate " ` B ` is f... |
| fnebas 37174 | A finer cover covers the s... |
| fnetg 37175 | A finer cover generates a ... |
| fnessex 37176 | If ` B ` is finer than ` A... |
| fneuni 37177 | If ` B ` is finer than ` A... |
| fneint 37178 | If a cover is finer than a... |
| fness 37179 | A cover is finer than its ... |
| fneref 37180 | Reflexivity of the finenes... |
| fnetr 37181 | Transitivity of the finene... |
| fneval 37182 | Two covers are finer than ... |
| fneer 37183 | Fineness intersected with ... |
| topfne 37184 | Fineness for covers corres... |
| topfneec 37185 | A cover is equivalent to a... |
| topfneec2 37186 | A topology is precisely id... |
| fnessref 37187 | A cover is finer iff it ha... |
| refssfne 37188 | A cover is a refinement if... |
| neibastop1 37189 | A collection of neighborho... |
| neibastop2lem 37190 | Lemma for ~ neibastop2 . ... |
| neibastop2 37191 | In the topology generated ... |
| neibastop3 37192 | The topology generated by ... |
| topmtcl 37193 | The meet of a collection o... |
| topmeet 37194 | Two equivalent formulation... |
| topjoin 37195 | Two equivalent formulation... |
| fnemeet1 37196 | The meet of a collection o... |
| fnemeet2 37197 | The meet of equivalence cl... |
| fnejoin1 37198 | Join of equivalence classe... |
| fnejoin2 37199 | Join of equivalence classe... |
| fgmin 37200 | Minimality property of a g... |
| neifg 37201 | The neighborhood filter of... |
| tailfval 37202 | The tail function for a di... |
| tailval 37203 | The tail of an element in ... |
| eltail 37204 | An element of a tail. (Co... |
| tailf 37205 | The tail function of a dir... |
| tailini 37206 | A tail contains its initia... |
| tailfb 37207 | The collection of tails of... |
| filnetlem1 37208 | Lemma for ~ filnet . Chan... |
| filnetlem2 37209 | Lemma for ~ filnet . The ... |
| filnetlem3 37210 | Lemma for ~ filnet . (Con... |
| filnetlem4 37211 | Lemma for ~ filnet . (Con... |
| filnet 37212 | A filter has the same conv... |
| tb-ax1 37213 | The first of three axioms ... |
| tb-ax2 37214 | The second of three axioms... |
| tb-ax3 37215 | The third of three axioms ... |
| tbsyl 37216 | The weak syllogism from Ta... |
| re1ax2lem 37217 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 37218 | ~ ax-2 rederived from the ... |
| naim1 37219 | Constructor theorem for ` ... |
| naim2 37220 | Constructor theorem for ` ... |
| naim1i 37221 | Constructor rule for ` -/\... |
| naim2i 37222 | Constructor rule for ` -/\... |
| naim12i 37223 | Constructor rule for ` -/\... |
| nabi1i 37224 | Constructor rule for ` -/\... |
| nabi2i 37225 | Constructor rule for ` -/\... |
| nabi12i 37226 | Constructor rule for ` -/\... |
| df3nandALT1 37229 | The double nand expressed ... |
| df3nandALT2 37230 | The double nand expressed ... |
| andnand1 37231 | Double and in terms of dou... |
| imnand2 37232 | An ` -> ` nand relation. ... |
| nalfal 37233 | Not all sets hold ` F. ` a... |
| nexntru 37234 | There does not exist a set... |
| nexfal 37235 | There does not exist a set... |
| neufal 37236 | There does not exist exact... |
| neutru 37237 | There does not exist exact... |
| nmotru 37238 | There does not exist at mo... |
| mofal 37239 | There exist at most one se... |
| nrmo 37240 | "At most one" restricted e... |
| meran1 37241 | A single axiom for proposi... |
| meran2 37242 | A single axiom for proposi... |
| meran3 37243 | A single axiom for proposi... |
| waj-ax 37244 | A single axiom for proposi... |
| lukshef-ax2 37245 | A single axiom for proposi... |
| arg-ax 37246 | A single axiom for proposi... |
| negsym1 37247 | In the paper "On Variable ... |
| imsym1 37248 | A symmetry with ` -> ` . ... |
| bisym1 37249 | A symmetry with ` <-> ` . ... |
| consym1 37250 | A symmetry with ` /\ ` . ... |
| dissym1 37251 | A symmetry with ` \/ ` . ... |
| nandsym1 37252 | A symmetry with ` -/\ ` . ... |
| unisym1 37253 | A symmetry with ` A. ` . ... |
| exisym1 37254 | A symmetry with ` E. ` . ... |
| unqsym1 37255 | A symmetry with ` E! ` . ... |
| amosym1 37256 | A symmetry with ` E* ` . ... |
| subsym1 37257 | A symmetry with ` [ x / y ... |
| ontopbas 37258 | An ordinal number is a top... |
| onsstopbas 37259 | The class of ordinal numbe... |
| onpsstopbas 37260 | The class of ordinal numbe... |
| ontgval 37261 | The topology generated fro... |
| ontgsucval 37262 | The topology generated fro... |
| onsuctop 37263 | A successor ordinal number... |
| onsuctopon 37264 | One of the topologies on a... |
| ordtoplem 37265 | Membership of the class of... |
| ordtop 37266 | An ordinal is a topology i... |
| onsucconni 37267 | A successor ordinal number... |
| onsucconn 37268 | A successor ordinal number... |
| ordtopconn 37269 | An ordinal topology is con... |
| onintopssconn 37270 | An ordinal topology is con... |
| onsuct0 37271 | A successor ordinal number... |
| ordtopt0 37272 | An ordinal topology is T_0... |
| onsucsuccmpi 37273 | The successor of a success... |
| onsucsuccmp 37274 | The successor of a success... |
| limsucncmpi 37275 | The successor of a limit o... |
| limsucncmp 37276 | The successor of a limit o... |
| ordcmp 37277 | An ordinal topology is com... |
| ssoninhaus 37278 | The ordinal topologies ` 1... |
| onint1 37279 | The ordinal T_1 spaces are... |
| oninhaus 37280 | The ordinal Hausdorff spac... |
| fveleq 37281 | Please add description her... |
| findfvcl 37282 | Please add description her... |
| findreccl 37283 | Please add description her... |
| findabrcl 37284 | Please add description her... |
| nnssi2 37285 | Convert a theorem for real... |
| nnssi3 37286 | Convert a theorem for real... |
| nndivsub 37287 | Please add description her... |
| nndivlub 37288 | A factor of a positive int... |
| ee7.2aOLD 37291 | Lemma for Euclid's Element... |
| weiunval 37292 | Value of the relation cons... |
| weiunlem 37293 | Lemma for ~ weiunpo , ~ we... |
| weiunfrlem 37294 | Lemma for ~ weiunfr . (Co... |
| weiunpo 37295 | A partial ordering on an i... |
| weiunso 37296 | A strict ordering on an in... |
| weiunfr 37297 | A well-founded relation on... |
| weiunse 37298 | The relation constructed i... |
| weiunwe 37299 | A well-ordering on an inde... |
| numiunnum 37300 | An indexed union of sets i... |
| axtco 37301 | Axiom of Transitive Contai... |
| axtco1 37303 | Strong form of the Axiom o... |
| axtco2 37304 | Weak form of the Axiom of ... |
| axtco1from2 37305 | Strong form ~ axtco1 of th... |
| axtco1g 37306 | Strong form of the Axiom o... |
| axtco2g 37307 | Weak form of the Axiom of ... |
| axtcond 37308 | A version of the Axiom of ... |
| axuntco 37309 | Derivation of ~ ax-un from... |
| axnulregtco 37310 | Derivation of ~ ax-nul fro... |
| elALTtco 37311 | Derivation of ~ el from ~ ... |
| tz9.1ctco 37312 | Version of ~ tz9.1c derive... |
| tz9.1tco 37313 | Version of ~ tz9.1 derived... |
| tr0elw 37314 | Every nonempty transitive ... |
| tr0el 37315 | Every nonempty transitive ... |
| ttceq 37318 | Equality theorem for trans... |
| ttceqi 37319 | Equality inference for tra... |
| ttceqd 37320 | Equality deduction for tra... |
| nfttc 37321 | Bound-variable hypothesis ... |
| ttcid 37322 | The transitive closure con... |
| ttctr 37323 | The transitive closure of ... |
| ttctr2 37324 | The transitive closure of ... |
| ttctr3 37325 | The transitive closure of ... |
| ttcmin 37326 | The transitive closure of ... |
| ttcexrg 37327 | If the transitive closure ... |
| ttcss 37328 | A transitive closure conta... |
| ttcss2 37329 | The subclass relationship ... |
| ttcel 37330 | A transitive closure conta... |
| ttcel2 37331 | Elements turn into subclas... |
| ttctrid 37332 | The transitive closure of ... |
| ttcidm 37333 | The transitive closure ope... |
| ssttctr 37334 | Transitivity of ` A C_ TC+... |
| elttctr 37335 | Transitivity of ` A e. TC+... |
| dfttc2g 37336 | A shorter expression for t... |
| ttc0 37337 | The transitive closure of ... |
| ttc00 37338 | A class has an empty trans... |
| csbttc 37339 | Distribute proper substitu... |
| ttcuniun 37340 | Relationship between ` TC+... |
| ttciunun 37341 | Relationship between ` TC+... |
| ttcun 37342 | Distribute union of two cl... |
| ttcuni 37343 | Distribute union of a clas... |
| ttciun 37344 | Distribute indexed union t... |
| ttcpwss 37345 | The transitive closure of ... |
| ttcsnssg 37346 | The transitive closure is ... |
| ttcsnidg 37347 | The singleton transitive c... |
| ttcsnmin 37348 | The singleton transitive c... |
| ttcsng 37349 | Relationship between ` TC+... |
| ttcsnexg 37350 | If the transitive closure ... |
| ttcsnexbig 37351 | The transitive closure of ... |
| ttcsntrsucg 37352 | The singleton transitive c... |
| dfttc3gw 37353 | If the transitive closure ... |
| ttcwf 37354 | A set is well-founded iff ... |
| ttcwf2 37355 | If a transitive closure cl... |
| ttcwf3 37356 | The sets whose transitive ... |
| ttc0elw 37357 | If a transitive closure is... |
| dfttc4lem1 37358 | Lemma for ~ dfttc4 . (Con... |
| dfttc4lem2 37359 | Lemma for ~ dfttc4 . (Con... |
| dfttc4 37360 | An alternative expression ... |
| elttcirr 37361 | Irreflexivity of ` A e. TC... |
| ttcexg 37362 | The transitive closure of ... |
| ttcexbi 37363 | A class is a set iff its t... |
| dfttc3g 37364 | The transitive closure of ... |
| ttc0el 37365 | A transitive closure conta... |
| mh-setind 37366 | Principle of set induction... |
| mh-setindnd 37367 | A version of ~ mh-setind w... |
| regsfromregtco 37368 | Derivation of ~ ax-regs fr... |
| regsfromsetind 37369 | Derivation of ~ ax-regs fr... |
| regsfromunir1 37370 | Derivation of ~ ax-regs fr... |
| mh-inf3f1 37371 | A variant of ~ inf3 . If ... |
| mh-inf3sn 37372 | Version of ~ inf3 for the ... |
| mh-prprimbi 37373 | Shortest possible version ... |
| mh-unprimbi 37374 | Shortest possible version ... |
| mh-regprimbi 37375 | Shortest possible version ... |
| mh-infprim1bi 37376 | Shortest possible axiom of... |
| mh-infprim2bi 37377 | Shortest possible axiom of... |
| mh-infprim3bi 37378 | An axiom of infinity in pr... |
| dnival 37379 | Value of the "distance to ... |
| dnicld1 37380 | Closure theorem for the "d... |
| dnicld2 37381 | Closure theorem for the "d... |
| dnif 37382 | The "distance to nearest i... |
| dnizeq0 37383 | The distance to nearest in... |
| dnizphlfeqhlf 37384 | The distance to nearest in... |
| rddif2 37385 | Variant of ~ rddif . (Con... |
| dnibndlem1 37386 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 37387 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 37388 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 37389 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 37390 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 37391 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 37392 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 37393 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 37394 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 37395 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 37396 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 37397 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 37398 | Lemma for ~ dnibnd . (Con... |
| dnibnd 37399 | The "distance to nearest i... |
| dnicn 37400 | The "distance to nearest i... |
| knoppcnlem1 37401 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 37402 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 37403 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 37404 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 37405 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 37406 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 37407 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 37408 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 37409 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 37410 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 37411 | Lemma for ~ knoppcn . (Co... |
| knoppcn 37412 | The continuous nowhere dif... |
| knoppcld 37413 | Closure theorem for Knopp'... |
| unblimceq0lem 37414 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 37415 | If ` F ` is unbounded near... |
| unbdqndv1 37416 | If the difference quotient... |
| unbdqndv2lem1 37417 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 37418 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 37419 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 37420 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 37421 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 37422 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 37423 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 37424 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 37425 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 37426 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 37427 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 37428 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 37429 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 37430 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 37431 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 37432 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 37433 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 37434 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 37435 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 37436 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 37437 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 37438 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 37439 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 37440 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 37441 | Lemma for ~ knoppndv . (C... |
| knoppndv 37442 | The continuous nowhere dif... |
| knoppf 37443 | Knopp's function is a func... |
| knoppcn2 37444 | Variant of ~ knoppcn with ... |
| cnndvlem1 37445 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 37446 | Lemma for ~ cnndv . (Cont... |
| cnndv 37447 | There exists a continuous ... |
| bj-mp2c 37448 | A double _modus ponens_ in... |
| bj-mp2d 37449 | A double _modus ponens_ in... |
| bj-0 37450 | A syntactic theorem. See ... |
| bj-1 37451 | In this proof, the use of ... |
| bj-poni 37452 | Inference associated with ... |
| bj-nnclav 37453 | When ` F. ` is substituted... |
| bj-nnclavi 37454 | Inference associated with ... |
| bj-nnclavc 37455 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 37456 | Inference associated with ... |
| bj-jarrii 37457 | Inference associated with ... |
| bj-imim21 37458 | The propositional function... |
| bj-imim21i 37459 | The propositional function... |
| bj-imim11 37460 | The propositional function... |
| bj-imim11i 37461 | The propositional function... |
| bj-peircestab 37462 | Over minimal implicational... |
| bj-stabpeirce 37463 | This minimal implicational... |
| bj-bisimpl 37464 | Implication from equivalen... |
| bj-bisimpr 37465 | Implication from equivalen... |
| bj-syl66ib 37466 | A mixed syllogism inferenc... |
| bj-orim2 37467 | Proof of ~ orim2 from the ... |
| bj-currypeirce 37468 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 37469 | Peirce's axiom ~ peirce im... |
| bj-animbi 37470 | Conjunction in terms of im... |
| bj-currypara 37471 | Curry's paradox. The proo... |
| bj-con2com 37472 | A commuted form of the con... |
| bj-con2comi 37473 | Inference associated with ... |
| bj-nimn 37474 | If a formula is true, then... |
| bj-nimni 37475 | Inference associated with ... |
| bj-peircei 37476 | Inference associated with ... |
| bj-looinvi 37477 | Inference associated with ... |
| bj-looinvii 37478 | Inference associated with ... |
| bj-mt2bi 37479 | Version of ~ mt2 where the... |
| bj-fal 37480 | Shortening of ~ fal using ... |
| bj-ntrufal 37481 | The negation of a theorem ... |
| bj-dfnul2 37482 | Alternate definition of th... |
| bj-jaoi1 37483 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 37484 | Shortens ~ consensus (110>... |
| bj-dfbi4 37485 | Alternate definition of th... |
| bj-dfbi5 37486 | Alternate definition of th... |
| bj-dfbi6 37487 | Alternate definition of th... |
| bj-bijust0ALT 37488 | Alternate proof of ~ bijus... |
| bj-bijust00 37489 | A self-implication does no... |
| bj-consensus 37490 | Version of ~ consensus exp... |
| bj-consensusALT 37491 | Alternate proof of ~ bj-co... |
| bj-df-ifc 37492 | Candidate definition for t... |
| bj-dfif 37493 | Alternate definition of th... |
| bj-ififc 37494 | A biconditional connecting... |
| bj-imbi12 37495 | Uncurried (imported) form ... |
| bj-falor 37496 | Dual of ~ truan (which has... |
| bj-falor2 37497 | Dual of ~ truan . (Contri... |
| bj-bibibi 37498 | A property of the bicondit... |
| bj-imn3ani 37499 | Duplication of ~ bnj1224 .... |
| bj-andnotim 37500 | Two ways of expressing a c... |
| bj-bi3ant 37501 | This used to be in the mai... |
| bj-bisym 37502 | This used to be in the mai... |
| bj-bixor 37503 | Equivalence of two ternary... |
| bj-axdd2 37504 | This implication, proved u... |
| bj-axd2d 37505 | This implication, proved u... |
| bj-axtd 37506 | This implication, proved f... |
| bj-gl4 37507 | In a normal modal logic, t... |
| bj-axc4 37508 | Over minimal calculus, the... |
| prvlem1 37513 | An elementary property of ... |
| prvlem2 37514 | An elementary property of ... |
| bj-babygodel 37515 | See the section header com... |
| bj-babylob 37516 | See the section header com... |
| bj-godellob 37517 | Proof of Gödel's theo... |
| bj-exexalal 37518 | A lemma for changing bound... |
| bj-genr 37519 | Generalization rule on the... |
| bj-genl 37520 | Generalization rule on the... |
| bj-genan 37521 | Generalization rule on a c... |
| bj-mpgs 37522 | From a closed form theorem... |
| bj-almp 37523 | A quantified form of ~ ax-... |
| bj-sylggt 37524 | Stronger form of ~ sylgt ,... |
| bj-alrimg 37525 | The general form of the *a... |
| bj-sylgt2 37526 | Uncurried (imported) form ... |
| bj-nexdh 37527 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 37528 | Uncurried (imported) form ... |
| bj-alimii 37529 | Inference associated with ... |
| bj-ala1i 37530 | Add an antecedent in a uni... |
| bj-almpi 37531 | A quantified form of ~ mpi... |
| bj-almpig 37532 | A partially quantified for... |
| bj-alsyl 37533 | Syllogism under the univer... |
| bj-2alim 37534 | Closed form of ~ 2alimi . ... |
| bj-alimdh 37535 | General instance of ~ alim... |
| bj-alrimdh 37536 | Deduction form of Theorem ... |
| bj-alrimd 37537 | A slightly more general ~ ... |
| bj-exa1i 37538 | Add an antecedent in an ex... |
| bj-alanim 37539 | Closed form of ~ alanimi .... |
| bj-2albi 37540 | Closed form of ~ 2albii . ... |
| bj-notalbii 37541 | Equivalence of universal q... |
| bj-2exim 37542 | Closed form of ~ 2eximi . ... |
| bj-2exbi 37543 | Closed form of ~ 2exbii . ... |
| bj-3exbi 37544 | Closed form of ~ 3exbii . ... |
| bj-sylget 37545 | Dual statement of ~ sylgt ... |
| bj-sylget2 37546 | Uncurried (imported) form ... |
| bj-exlimg 37547 | The general form of the *e... |
| bj-sylge 37548 | Dual statement of ~ sylg (... |
| bj-exlimd 37549 | A slightly more general ~ ... |
| bj-nfimexal 37550 | A weak from of nonfreeness... |
| bj-exim 37551 | Theorem 19.22 of [Margaris... |
| bj-alexim 37552 | Closed form of ~ aleximi .... |
| bj-aleximiALT 37553 | Alternate proof of ~ alexi... |
| bj-hbxfrbi 37554 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 37555 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 37556 | Distribute quantifiers ove... |
| bj-exalimi 37557 | An inference for distribut... |
| bj-eximcom 37558 | A commuted form of ~ exim ... |
| bj-exalims 37559 | Distributing quantifiers o... |
| bj-exalimsi 37560 | An inference for distribut... |
| bj-axdd2ALT 37561 | Alternate proof of ~ bj-ax... |
| bj-ax12ig 37562 | A lemma used to prove a we... |
| bj-ax12i 37563 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 37564 | Closed form of ~ nfim and ... |
| bj-spimnfe 37565 | A universal specification ... |
| bj-spimenfa 37566 | An existential generalizat... |
| bj-spim 37567 | A lemma for universal spec... |
| bj-spime 37568 | A lemma for existential ge... |
| bj-cbvalimd0 37569 | A lemma for alpha-renaming... |
| bj-cbvalimdlem 37570 | A lemma for alpha-renaming... |
| bj-cbveximdlem 37571 | A lemma for alpha-renaming... |
| bj-cbvalimd 37572 | A lemma for alpha-renaming... |
| bj-cbveximd 37573 | A lemma for alpha-renaming... |
| bj-cbvalimdv 37574 | A lemma for alpha-renaming... |
| bj-cbveximdv 37575 | A lemma for alpha-renaming... |
| bj-spvw 37576 | Version of ~ spvw and ~ 19... |
| bj-spvew 37577 | Version of ~ 19.8v and ~ 1... |
| bj-alextruim 37578 | An equivalent expression f... |
| bj-exextruan 37579 | An equivalent expression f... |
| bj-cbvalvv 37580 | Universally quantifying ov... |
| bj-cbvexvv 37581 | Existentially quantifying ... |
| bj-cbvaw 37582 | Universally quantifying ov... |
| bj-cbvew 37583 | Existentially quantifying ... |
| bj-cbveaw 37584 | Universally quantifying ov... |
| bj-cbvaew 37585 | Exixtentially quantifying ... |
| bj-ax12wlem 37586 | A lemma used to prove a we... |
| bj-cbval 37587 | Changing a bound variable ... |
| bj-cbvex 37588 | Changing a bound variable ... |
| bj-df-sb 37591 | Proposed definition to rep... |
| bj-sbcex 37592 | Proof of ~ sbcex when taki... |
| bj-dfsbc 37593 | Proof of ~ df-sbc when tak... |
| bj-ssbeq 37594 | Substitution in an equalit... |
| bj-ssblem1 37595 | A lemma for the definiens ... |
| bj-ssblem2 37596 | An instance of ~ ax-11 pro... |
| bj-ax12v 37597 | A weaker form of ~ ax-12 a... |
| bj-ax12 37598 | Remove a DV condition from... |
| bj-ax12ssb 37599 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 37600 | Special case of ~ 19.41 pr... |
| bj-equsexval 37601 | Special case of ~ equsexv ... |
| bj-subst 37602 | Proof of ~ sbalex from cor... |
| bj-ssbid2 37603 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 37604 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 37605 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 37606 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 37607 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 37608 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 37609 | Proof of ~ ax6e (hence ~ a... |
| bj-spim0 37610 | A universal specialization... |
| bj-spimvwt 37611 | Closed form of ~ spimvw . ... |
| bj-spnfw 37612 | Theorem close to a closed ... |
| bj-cbvexiw 37613 | Change bound variable. Th... |
| bj-cbvexivw 37614 | Change bound variable. Th... |
| bj-modald 37615 | A short form of the axiom ... |
| bj-denot 37616 | A weakening of ~ ax-6 and ... |
| bj-eqs 37617 | A lemma for substitutions,... |
| bj-cbvexw 37618 | Change bound variable. Th... |
| bj-ax12w 37619 | The general statement that... |
| bj-ax89 37620 | A theorem which could be u... |
| bj-cleljusti 37621 | One direction of ~ cleljus... |
| bj-alcomexcom 37622 | Commutation of two existen... |
| bj-hbald 37623 | General statement that ~ h... |
| bj-hbalt 37624 | Closed form of (general in... |
| bj-hbal 37625 | More general instance of ~... |
| axc11n11 37626 | Proof of ~ axc11n from { ~... |
| axc11n11r 37627 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 37628 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 37629 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 37630 | Alternate proof of ~ bj-ax... |
| bj-sb 37631 | A weak variant of ~ sbid2 ... |
| bj-modalbe 37632 | The predicate-calculus ver... |
| bj-spst 37633 | Closed form of ~ sps . On... |
| bj-19.21bit 37634 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 37635 | Closed form of ~ 19.23bi .... |
| bj-nexrt 37636 | Closed form of ~ nexr . C... |
| bj-alrim 37637 | Closed form of ~ alrimi . ... |
| bj-alrim2 37638 | Uncurried (imported) form ... |
| bj-nfdt0 37639 | A theorem close to a close... |
| bj-nfdt 37640 | Closed form of ~ nf5d and ... |
| bj-nexdt 37641 | Closed form of ~ nexd . (... |
| bj-nexdvt 37642 | Closed form of ~ nexdv . ... |
| bj-alexbiex 37643 | Adding a second quantifier... |
| bj-exexbiex 37644 | Adding a second quantifier... |
| bj-alalbial 37645 | Adding a second quantifier... |
| bj-exalbial 37646 | Adding a second quantifier... |
| bj-19.9htbi 37647 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 37648 | Strengthening ~ hbnt by re... |
| bj-biexal1 37649 | A general FOL biconditiona... |
| bj-biexal2 37650 | When ` ph ` is substituted... |
| bj-biexal3 37651 | When ` ph ` is substituted... |
| bj-bialal 37652 | When ` ph ` is substituted... |
| bj-biexex 37653 | When ` ph ` is substituted... |
| bj-hbexd 37654 | A more general instance of... |
| bj-hbext 37655 | Closed form of ~ bj-hbex a... |
| bj-hbex 37656 | A more general instance of... |
| bj-nfalt 37657 | Closed form of ~ nfal . (... |
| bj-nfext 37658 | Closed form of ~ nfex . (... |
| bj-eeanvw 37659 | Version of ~ exdistrv with... |
| bj-modal4 37660 | First-order logic form of ... |
| bj-modal4e 37661 | First-order logic form of ... |
| bj-modalb 37662 | A short form of the axiom ... |
| bj-wnf1 37663 | When ` ph ` is substituted... |
| bj-wnf2 37664 | When ` ph ` is substituted... |
| bj-wnfanf 37665 | When ` ph ` is substituted... |
| bj-wnfenf 37666 | When ` ph ` is substituted... |
| bj-19.12 37667 | See ~ 19.12 . Could be la... |
| bj-substax12 37668 | Equivalent form of the axi... |
| bj-substw 37669 | Weak form of the LHS of ~ ... |
| bj-nnfa 37672 | Nonfreeness implies the eq... |
| bj-nnfad 37673 | Nonfreeness implies the eq... |
| bj-nnfai 37674 | Nonfreeness implies the eq... |
| bj-nnfe 37675 | Nonfreeness implies the eq... |
| bj-nnfed 37676 | Nonfreeness implies the eq... |
| bj-nnfei 37677 | Nonfreeness implies the eq... |
| bj-nnfea 37678 | Nonfreeness implies the eq... |
| bj-nnfead 37679 | Nonfreeness implies the eq... |
| bj-nnfeai 37680 | Nonfreeness implies the eq... |
| bj-alnnf 37681 | In deduction-style proofs,... |
| bj-alnnf2 37682 | If a proposition holds, th... |
| bj-dfnnf2 37683 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 37684 | New nonfreeness implies ol... |
| bj-nnfim1 37685 | A consequence of nonfreene... |
| bj-nnfim2 37686 | A consequence of nonfreene... |
| bj-nnftht 37687 | A variable is nonfree in a... |
| bj-nnfth 37688 | A variable is nonfree in a... |
| bj-nnf-alrim 37689 | Proof of the closed form o... |
| bj-stdpc5t 37690 | Alias of ~ bj-nnf-alrim fo... |
| bj-nnfbi 37691 | If two formulas are equiva... |
| bj-nnfbd0 37692 | If two formulas are equiva... |
| bj-nnfbii 37693 | If two formulas are equiva... |
| bj-nnfnt 37694 | A variable is nonfree in a... |
| bj-nnfnth 37695 | A variable is nonfree in t... |
| bj-nnfim 37696 | Nonfreeness in the anteced... |
| bj-nnfimd 37697 | Nonfreeness in the anteced... |
| bj-nnfan 37698 | Nonfreeness in both conjun... |
| bj-nnfand 37699 | Nonfreeness in both conjun... |
| bj-nnfor 37700 | Nonfreeness in both disjun... |
| bj-nnford 37701 | Nonfreeness in both disjun... |
| bj-nnfbit 37702 | Nonfreeness in both sides ... |
| bj-nnfbid 37703 | Nonfreeness in both sides ... |
| bj-nnf-exlim 37704 | Proof of the closed form o... |
| bj-19.21t 37705 | Statement ~ 19.21t proved ... |
| bj-19.23t 37706 | Statement ~ 19.23t proved ... |
| bj-19.36im 37707 | One direction of ~ 19.36 f... |
| bj-19.37im 37708 | One direction of ~ 19.37 f... |
| bj-19.42t 37709 | Closed form of ~ 19.42 fro... |
| bj-19.41t 37710 | Closed form of ~ 19.41 fro... |
| bj-pm11.53vw 37711 | Version of ~ pm11.53v with... |
| bj-nnfv 37712 | A non-occurring variable i... |
| bj-nnfbd 37713 | If two formulas are equiva... |
| bj-pm11.53a 37714 | A variant of ~ pm11.53v . ... |
| bj-equsvt 37715 | A variant of ~ equsv . (C... |
| bj-equsalvwd 37716 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 37717 | Variant of ~ equsexvw . (... |
| bj-nnf-spim 37718 | A universal specialization... |
| bj-nnf-spime 37719 | An existential generalizat... |
| bj-nnf-cbvaliv 37720 | The only DV conditions are... |
| bj-sbievwd 37721 | Variant of ~ sbievw . (Co... |
| bj-sbft 37722 | Version of ~ sbft using ` ... |
| bj-nnf-cbvali 37723 | Compared with ~ bj-nnf-cbv... |
| bj-nnf-cbval 37724 | Compared with ~ cbvalv1 , ... |
| bj-dfnnf3 37725 | Alternate definition of no... |
| bj-nfnnfTEMP 37726 | New nonfreeness is equival... |
| bj-wnfnf 37727 | When ` ph ` is substituted... |
| bj-nnfa1 37728 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 37729 | See ~ nfe1 . (Contributed... |
| bj-nnflemaa 37730 | One of four lemmas for non... |
| bj-nnflemee 37731 | One of four lemmas for non... |
| bj-nnflemae 37732 | One of four lemmas for non... |
| bj-nnflemea 37733 | One of four lemmas for non... |
| bj-nnfalt 37734 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 37735 | See ~ nfex and ~ bj-nfext ... |
| bj-pm11.53v 37736 | Version of ~ pm11.53v with... |
| bj-axc10 37737 | Alternate proof of ~ axc10... |
| bj-alequex 37738 | A fol lemma. See ~ aleque... |
| bj-spimt2 37739 | A step in the proof of ~ s... |
| bj-cbv3ta 37740 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 37741 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 37742 | A theorem close to a close... |
| bj-hbsb3 37743 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 37744 | A theorem close to a close... |
| bj-nfs1t2 37745 | A theorem close to a close... |
| bj-nfs1 37746 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 37747 | Version of ~ axc10 with a ... |
| bj-spimtv 37748 | Version of ~ spimt with a ... |
| bj-cbv3hv2 37749 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 37750 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 37751 | Version of ~ cbv2h with a ... |
| bj-cbv2v 37752 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 37753 | Version of ~ cbvald with a... |
| bj-cbvexdv 37754 | Version of ~ cbvexd with a... |
| bj-cbval2vv 37755 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 37756 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 37757 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 37758 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 37759 | Version of ~ cbvex4v with ... |
| bj-equsalhv 37760 | Version of ~ equsalh with ... |
| bj-axc11nv 37761 | Version of ~ axc11n with a... |
| bj-aecomsv 37762 | Version of ~ aecoms with a... |
| bj-axc11v 37763 | Version of ~ axc11 with a ... |
| bj-drnf2v 37764 | Version of ~ drnf2 with a ... |
| bj-equs45fv 37765 | Version of ~ equs45f with ... |
| bj-hbs1 37766 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 37767 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 37768 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 37769 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 37770 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 37771 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 37772 | Biconditional version of a... |
| bj-hbaeb 37773 | Biconditional version of ~... |
| bj-hbnaeb 37774 | Biconditional version of ~... |
| bj-dvv 37775 | A special instance of ~ bj... |
| bj-equsal1t 37776 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 37777 | Inference associated with ... |
| bj-equsal1 37778 | One direction of ~ equsal ... |
| bj-equsal2 37779 | One direction of ~ equsal ... |
| bj-equsal 37780 | Shorter proof of ~ equsal ... |
| stdpc5t 37781 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 37782 | More direct proof of ~ std... |
| 2stdpc5 37783 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 37784 | Proof of ~ 19.21t from ~ s... |
| exlimii 37785 | Inference associated with ... |
| ax11-pm 37786 | Proof of ~ ax-11 similar t... |
| ax6er 37787 | Commuted form of ~ ax6e . ... |
| exlimiieq1 37788 | Inferring a theorem when i... |
| exlimiieq2 37789 | Inferring a theorem when i... |
| ax11-pm2 37790 | Proof of ~ ax-11 from the ... |
| bj-sbsb 37791 | Biconditional showing two ... |
| bj-dfsb2 37792 | Alternate (dual) definitio... |
| bj-sbf3 37793 | Substitution has no effect... |
| bj-sbf4 37794 | Substitution has no effect... |
| bj-eu3f 37795 | Version of ~ eu3v where th... |
| bj-sblem1 37796 | Lemma for substitution. (... |
| bj-sblem2 37797 | Lemma for substitution. (... |
| bj-sblem 37798 | Lemma for substitution. (... |
| bj-sbievw1 37799 | Lemma for substitution. (... |
| bj-sbievw2 37800 | Lemma for substitution. (... |
| bj-sbievw 37801 | Lemma for substitution. C... |
| bj-sbievv 37802 | Version of ~ sbie with a s... |
| bj-moeub 37803 | Uniqueness is equivalent t... |
| bj-sbidmOLD 37804 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 37805 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 37806 | Curried (exported) form of... |
| bj-dvelimv 37807 | A version of ~ dvelim usin... |
| bj-nfeel2 37808 | Nonfreeness in a membershi... |
| bj-axc14nf 37809 | Proof of a version of ~ ax... |
| bj-axc14 37810 | Alternate proof of ~ axc14... |
| mobidvALT 37811 | Alternate proof of ~ mobid... |
| sbn1ALT 37812 | Alternate proof of ~ sbn1 ... |
| eliminable1 37813 | A theorem used to prove th... |
| eliminable2a 37814 | A theorem used to prove th... |
| eliminable2b 37815 | A theorem used to prove th... |
| eliminable2c 37816 | A theorem used to prove th... |
| eliminable3a 37817 | A theorem used to prove th... |
| eliminable3b 37818 | A theorem used to prove th... |
| eliminable-velab 37819 | A theorem used to prove th... |
| eliminable-veqab 37820 | A theorem used to prove th... |
| eliminable-abeqv 37821 | A theorem used to prove th... |
| eliminable-abeqab 37822 | A theorem used to prove th... |
| eliminable-abelv 37823 | A theorem used to prove th... |
| eliminable-abelab 37824 | A theorem used to prove th... |
| bj-denoteslem 37825 | Duplicate of ~ issettru an... |
| bj-denotesALTV 37826 | Moved to main as ~ iseqset... |
| bj-issettruALTV 37827 | Moved to main as ~ issettr... |
| bj-elabtru 37828 | This is as close as we can... |
| bj-issetwt 37829 | Closed form of ~ bj-issetw... |
| bj-issetw 37830 | The closest one can get to... |
| bj-issetiv 37831 | Version of ~ bj-isseti wit... |
| bj-isseti 37832 | Version of ~ isseti with a... |
| bj-ralvw 37833 | A weak version of ~ ralv n... |
| bj-rexvw 37834 | A weak version of ~ rexv n... |
| bj-rababw 37835 | A weak version of ~ rabab ... |
| bj-rexcom4bv 37836 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 37837 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 37838 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 37839 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 37840 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 37841 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 37842 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 37843 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 37844 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 37845 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 37846 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 37847 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 37848 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 37849 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 37850 | Remove from ~ spcimdv depe... |
| elelb 37851 | Equivalence between two co... |
| bj-pwvrelb 37852 | Characterization of the el... |
| bj-nfcsym 37853 | The nonfreeness quantifier... |
| bj-sbeqALT 37854 | Substitution in an equalit... |
| bj-sbeq 37855 | Distribute proper substitu... |
| bj-sbceqgALT 37856 | Distribute proper substitu... |
| bj-csbsnlem 37857 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 37858 | Substitution in a singleto... |
| bj-sbel1 37859 | Version of ~ sbcel1g when ... |
| bj-abv 37860 | The class of sets verifyin... |
| bj-abvALT 37861 | Alternate version of ~ bj-... |
| bj-ab0 37862 | The class of sets verifyin... |
| bj-abf 37863 | Shorter proof of ~ abf (wh... |
| bj-csbprc 37864 | More direct proof of ~ csb... |
| bj-exlimvmpi 37865 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 37866 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 37867 | Lemma for theorems of the ... |
| bj-exlimmpbir 37868 | Lemma for theorems of the ... |
| bj-vtoclf 37869 | Remove dependency on ~ ax-... |
| bj-vtocl 37870 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 37871 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 37872 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 37873 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 37874 | A version of ~ vtoclg with... |
| bj-rabeqbid 37875 | Version of ~ rabeqbidv wit... |
| bj-seex 37876 | Version of ~ seex with a d... |
| bj-nfcf 37877 | Version of ~ df-nfc with a... |
| bj-sepg 37878 | Version of ~ sepg which do... |
| bj-inex1gALT 37879 | Proof of ~ inex1g from ~ s... |
| bj-elabd2ALT 37880 | Alternate proof of ~ elabd... |
| bj-unrab 37881 | Generalization of ~ unrab ... |
| bj-inrab 37882 | Generalization of ~ inrab ... |
| bj-inrab2 37883 | Shorter proof of ~ inrab .... |
| bj-inrab3 37884 | Generalization of ~ dfrab3... |
| bj-rabtr 37885 | Restricted class abstracti... |
| bj-rabtrALT 37886 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 37887 | Proof of ~ bj-rabtr found ... |
| bj-gabss 37890 | Inclusion of generalized c... |
| bj-gabssd 37891 | Inclusion of generalized c... |
| bj-gabeqd 37892 | Equality of generalized cl... |
| bj-gabeqis 37893 | Equality of generalized cl... |
| bj-elgab 37894 | Elements of a generalized ... |
| bj-gabima 37895 | Generalized class abstract... |
| bj-ru1 37898 | A version of Russell's par... |
| bj-ru 37899 | Remove dependency on ~ ax-... |
| currysetlem 37900 | Lemma for ~ currysetlem , ... |
| curryset 37901 | Curry's paradox in set the... |
| currysetlem1 37902 | Lemma for ~ currysetALT . ... |
| currysetlem2 37903 | Lemma for ~ currysetALT . ... |
| currysetlem3 37904 | Lemma for ~ currysetALT . ... |
| currysetALT 37905 | Alternate proof of ~ curry... |
| bj-n0i 37906 | Inference associated with ... |
| bj-disjsn01 37907 | Disjointness of the single... |
| bj-0nel1 37908 | The empty set does not bel... |
| bj-1nel0 37909 | ` 1o ` does not belong to ... |
| bj-xpimasn 37910 | The image of a singleton, ... |
| bj-xpima1sn 37911 | The image of a singleton b... |
| bj-xpima1snALT 37912 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 37913 | The image of a singleton b... |
| bj-xpnzex 37914 | If the first factor of a p... |
| bj-xpexg2 37915 | Curried (exported) form of... |
| bj-xpnzexb 37916 | If the first factor of a p... |
| bj-cleq 37917 | Substitution property for ... |
| bj-snsetex 37918 | The class of sets "whose s... |
| bj-clexab 37919 | Sethood of certain classes... |
| bj-sngleq 37922 | Substitution property for ... |
| bj-elsngl 37923 | Characterization of the el... |
| bj-snglc 37924 | Characterization of the el... |
| bj-snglss 37925 | The singletonization of a ... |
| bj-0nelsngl 37926 | The empty set is not a mem... |
| bj-snglinv 37927 | Inverse of singletonizatio... |
| bj-snglex 37928 | A class is a set if and on... |
| bj-tageq 37931 | Substitution property for ... |
| bj-eltag 37932 | Characterization of the el... |
| bj-0eltag 37933 | The empty set belongs to t... |
| bj-tagn0 37934 | The tagging of a class is ... |
| bj-tagss 37935 | The tagging of a class is ... |
| bj-snglsstag 37936 | The singletonization is in... |
| bj-sngltagi 37937 | The singletonization is in... |
| bj-sngltag 37938 | The singletonization and t... |
| bj-tagci 37939 | Characterization of the el... |
| bj-tagcg 37940 | Characterization of the el... |
| bj-taginv 37941 | Inverse of tagging. (Cont... |
| bj-tagex 37942 | A class is a set if and on... |
| bj-xtageq 37943 | The products of a given cl... |
| bj-xtagex 37944 | The product of a set and t... |
| bj-projeq 37947 | Substitution property for ... |
| bj-projeq2 37948 | Substitution property for ... |
| bj-projun 37949 | The class projection on a ... |
| bj-projex 37950 | Sethood of the class proje... |
| bj-projval 37951 | Value of the class project... |
| bj-1upleq 37954 | Substitution property for ... |
| bj-pr1eq 37957 | Substitution property for ... |
| bj-pr1un 37958 | The first projection prese... |
| bj-pr1val 37959 | Value of the first project... |
| bj-pr11val 37960 | Value of the first project... |
| bj-pr1ex 37961 | Sethood of the first proje... |
| bj-1uplth 37962 | The characteristic propert... |
| bj-1uplex 37963 | A monuple is a set if and ... |
| bj-1upln0 37964 | A monuple is nonempty. (C... |
| bj-2upleq 37967 | Substitution property for ... |
| bj-pr21val 37968 | Value of the first project... |
| bj-pr2eq 37971 | Substitution property for ... |
| bj-pr2un 37972 | The second projection pres... |
| bj-pr2val 37973 | Value of the second projec... |
| bj-pr22val 37974 | Value of the second projec... |
| bj-pr2ex 37975 | Sethood of the second proj... |
| bj-2uplth 37976 | The characteristic propert... |
| bj-2uplex 37977 | A couple is a set if and o... |
| bj-2upln0 37978 | A couple is nonempty. (Co... |
| bj-2upln1upl 37979 | A couple is never equal to... |
| bj-rcleqf 37980 | Relative version of ~ cleq... |
| bj-rcleq 37981 | Relative version of ~ dfcl... |
| bj-reabeq 37982 | Relative form of ~ eqabb .... |
| bj-disj2r 37983 | Relative version of ~ ssdi... |
| bj-sscon 37984 | Contraposition law for rel... |
| bj-abex 37985 | Two ways of stating that t... |
| bj-clex 37986 | Two ways of stating that a... |
| bj-axsn 37987 | Two ways of stating the ax... |
| bj-snexg 37989 | A singleton built on a set... |
| bj-snex 37990 | A singleton is a set. See... |
| bj-axbun 37991 | Two ways of stating the ax... |
| bj-unexg 37993 | Existence of binary unions... |
| bj-prexg 37994 | Existence of unordered pai... |
| bj-prex 37995 | Existence of unordered pai... |
| bj-axadj 37996 | Two ways of stating the ax... |
| bj-adjg1 37998 | Existence of the result of... |
| bj-snfromadj 37999 | Singleton from adjunction ... |
| bj-prfromadj 38000 | Unordered pair from adjunc... |
| bj-adjfrombun 38001 | Adjunction from singleton ... |
| elco 38002 | Membership in a compositio... |
| coi1in 38003 | Precomposition with the id... |
| eleq2w2ALT 38004 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 38005 | Alternate proof of ~ clel3... |
| bj-pw0ALT 38006 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 38007 | Quantitative version of ~ ... |
| bj-unirel 38008 | Quantitative version of ~ ... |
| bj-elpwg 38009 | If the intersection of two... |
| bj-velpwALT 38010 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 38011 | Alternate proof of ~ elpwg... |
| bj-vjust 38012 | Justification theorem for ... |
| bj-nul 38013 | Two formulations of the ax... |
| bj-nuliota 38014 | Definition of the empty se... |
| bj-nuliotaALT 38015 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 38016 | Alternate proof of ~ vtocl... |
| bj-elsn12g 38017 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 38018 | Biconditional version of ~... |
| bj-pwcfsdom 38019 | Remove hypothesis from ~ p... |
| bj-grur1 38020 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 38021 | The extension of a predica... |
| bj-dfid2ALT 38022 | Alternate version of ~ dfi... |
| bj-0nelopab 38023 | The empty set is not an el... |
| bj-brrelex12ALT 38024 | Two classes related by a b... |
| bj-epelg 38025 | The membership relation an... |
| bj-epelb 38026 | Two classes are related by... |
| bj-nsnid 38027 | A set does not contain the... |
| bj-rdg0gALT 38028 | Alternate proof of ~ rdg0g... |
| bj-vn0ALT 38029 | Alternate proof of ~ vn0 w... |
| bj-axnul 38030 | Over the base theory ~ ax-... |
| bj-rep 38031 | Version of the axiom of re... |
| bj-axseprep 38032 | Axiom of separation (unive... |
| bj-axreprepsep 38033 | Strong axiom of replacemen... |
| bj-evaleq 38034 | Equality theorem for the `... |
| bj-evalfun 38035 | The evaluation at a class ... |
| bj-evalfn 38036 | The evaluation at a class ... |
| bj-evalf 38037 | The evaluation at a class ... |
| bj-evalval 38038 | Value of the evaluation at... |
| bj-evalid 38039 | The evaluation at a set of... |
| bj-ndxarg 38040 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 38041 | Closed general form of ~ s... |
| bj-rest00 38044 | An elementwise intersectio... |
| bj-restsn 38045 | An elementwise intersectio... |
| bj-restsnss 38046 | Special case of ~ bj-rests... |
| bj-restsnss2 38047 | Special case of ~ bj-rests... |
| bj-restsn0 38048 | An elementwise intersectio... |
| bj-restsn10 38049 | Special case of ~ bj-rests... |
| bj-restsnid 38050 | The elementwise intersecti... |
| bj-rest10 38051 | An elementwise intersectio... |
| bj-rest10b 38052 | Alternate version of ~ bj-... |
| bj-restn0 38053 | An elementwise intersectio... |
| bj-restn0b 38054 | Alternate version of ~ bj-... |
| bj-restpw 38055 | The elementwise intersecti... |
| bj-rest0 38056 | An elementwise intersectio... |
| bj-restb 38057 | An elementwise intersectio... |
| bj-restv 38058 | An elementwise intersectio... |
| bj-resta 38059 | An elementwise intersectio... |
| bj-restuni 38060 | The union of an elementwis... |
| bj-restuni2 38061 | The union of an elementwis... |
| bj-restreg 38062 | A reformulation of the axi... |
| bj-raldifsn 38063 | All elements of a set sati... |
| bj-0int 38064 | If ` A ` is a collection o... |
| bj-mooreset 38065 | A Moore collection is a se... |
| bj-ismoore 38068 | Characterization of Moore ... |
| bj-ismoored0 38069 | Necessary condition to be ... |
| bj-ismoored 38070 | Necessary condition to be ... |
| bj-ismoored2 38071 | Necessary condition to be ... |
| bj-ismooredr 38072 | Sufficient condition to be... |
| bj-ismooredr2 38073 | Sufficient condition to be... |
| bj-discrmoore 38074 | The powerclass ` ~P A ` is... |
| bj-0nmoore 38075 | The empty set is not a Moo... |
| bj-snmoore 38076 | A singleton is a Moore col... |
| bj-snmooreb 38077 | A singleton is a Moore col... |
| bj-prmoore 38078 | A pair formed of two neste... |
| bj-pr0moore 38079 | A pair containing the empt... |
| bj-0nelmpt 38080 | The empty set is not an el... |
| bj-mptval 38081 | Value of a function given ... |
| bj-dfmpoa 38082 | An equivalent definition o... |
| bj-mpomptALT 38083 | Alternate proof of ~ mpomp... |
| setsstrset 38098 | Relation between ~ df-sets... |
| bj-nfald 38099 | Variant of ~ nfald . (Con... |
| bj-nfexd 38100 | Variant of ~ nfexd . (Con... |
| cgsex2gd 38101 | Implicit substitution infe... |
| copsex2gd 38102 | Implicit substitution infe... |
| copsex2d 38103 | Implicit substitution dedu... |
| copsex2b 38104 | Biconditional form of ~ co... |
| opelopabd 38105 | Membership of an ordered p... |
| opelopabb 38106 | Membership of an ordered p... |
| opelopabbv 38107 | Membership of an ordered p... |
| bj-opelrelex 38108 | The coordinates of an orde... |
| bj-opelresdm 38109 | If an ordered pair is in a... |
| bj-brresdm 38110 | If two classes are related... |
| brabd0 38111 | Expressing that two sets a... |
| brabd 38112 | Expressing that two sets a... |
| bj-brab2a1 38113 | "Unbounded" version of ~ b... |
| bj-opabssvv 38114 | A variant of ~ relopabiv (... |
| bj-funidres 38115 | The restricted identity re... |
| bj-opelidb 38116 | Characterization of the or... |
| bj-opelidb1 38117 | Characterization of the or... |
| bj-inexeqex 38118 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 38119 | If the intersection of two... |
| bj-opelid 38120 | Characterization of the or... |
| bj-ideqg 38121 | Characterization of the cl... |
| bj-ideqgALT 38122 | Alternate proof of ~ bj-id... |
| bj-ideqb 38123 | Characterization of classe... |
| bj-idres 38124 | Alternate expression for t... |
| bj-opelidres 38125 | Characterization of the or... |
| bj-idreseq 38126 | Sufficient condition for t... |
| bj-idreseqb 38127 | Characterization for two c... |
| bj-ideqg1 38128 | For sets, the identity rel... |
| bj-ideqg1ALT 38129 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 38130 | Characterization of the co... |
| bj-elid3 38131 | Characterization of the co... |
| bj-elid4 38132 | Characterization of the el... |
| bj-elid5 38133 | Characterization of the el... |
| bj-elid6 38134 | Characterization of the el... |
| bj-elid7 38135 | Characterization of the el... |
| bj-diagval 38138 | Value of the functionalize... |
| bj-diagval2 38139 | Value of the functionalize... |
| bj-eldiag 38140 | Characterization of the el... |
| bj-eldiag2 38141 | Characterization of the el... |
| bj-imdirvallem 38144 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 38145 | Value of the functionalize... |
| bj-imdirval2lem 38146 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 38147 | Value of the functionalize... |
| bj-imdirval3 38148 | Value of the functionalize... |
| bj-imdiridlem 38149 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 38150 | Functorial property of the... |
| bj-opelopabid 38151 | Membership in an ordered-p... |
| bj-opabco 38152 | Composition of ordered-pai... |
| bj-xpcossxp 38153 | The composition of two Car... |
| bj-imdirco 38154 | Functorial property of the... |
| bj-iminvval 38157 | Value of the functionalize... |
| bj-iminvval2 38158 | Value of the functionalize... |
| bj-iminvid 38159 | Functorial property of the... |
| bj-inftyexpitaufo 38166 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 38169 | An element of the circle a... |
| bj-inftyexpiinv 38172 | Utility theorem for the in... |
| bj-inftyexpiinj 38173 | Injectivity of the paramet... |
| bj-inftyexpidisj 38174 | An element of the circle a... |
| bj-ccinftydisj 38177 | The circle at infinity is ... |
| bj-elccinfty 38178 | A lemma for infinite exten... |
| bj-ccssccbar 38181 | Complex numbers are extend... |
| bj-ccinftyssccbar 38182 | Infinite extended complex ... |
| bj-pinftyccb 38185 | The class ` pinfty ` is an... |
| bj-pinftynrr 38186 | The extended complex numbe... |
| bj-minftyccb 38189 | The class ` minfty ` is an... |
| bj-minftynrr 38190 | The extended complex numbe... |
| bj-pinftynminfty 38191 | The extended complex numbe... |
| bj-rrhatsscchat 38200 | The real projective line i... |
| bj-imafv 38215 | If the direct image of a s... |
| bj-funun 38216 | Value of a function expres... |
| bj-fununsn1 38217 | Value of a function expres... |
| bj-fununsn2 38218 | Value of a function expres... |
| bj-fvsnun1 38219 | The value of a function wi... |
| bj-fvsnun2 38220 | The value of a function wi... |
| bj-fvmptunsn1 38221 | Value of a function expres... |
| bj-fvmptunsn2 38222 | Value of a function expres... |
| bj-iomnnom 38223 | The canonical bijection fr... |
| bj-smgrpssmgm 38232 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 38233 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 38234 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 38235 | Monoids are semigroups (el... |
| bj-cmnssmnd 38236 | Commutative monoids are mo... |
| bj-cmnssmndel 38237 | Commutative monoids are mo... |
| bj-grpssmnd 38238 | Groups are monoids. (Cont... |
| bj-grpssmndel 38239 | Groups are monoids (elemen... |
| bj-ablssgrp 38240 | Abelian groups are groups.... |
| bj-ablssgrpel 38241 | Abelian groups are groups ... |
| bj-ablsscmn 38242 | Abelian groups are commuta... |
| bj-ablsscmnel 38243 | Abelian groups are commuta... |
| bj-modssabl 38244 | (The additive groups of) m... |
| bj-vecssmod 38245 | Vector spaces are modules.... |
| bj-vecssmodel 38246 | Vector spaces are modules ... |
| bj-finsumval0 38249 | Value of a finite sum. (C... |
| bj-fvimacnv0 38250 | Variant of ~ fvimacnv wher... |
| bj-isvec 38251 | The predicate "is a vector... |
| bj-fldssdrng 38252 | Fields are division rings.... |
| bj-flddrng 38253 | Fields are division rings ... |
| bj-rrdrg 38254 | The field of real numbers ... |
| bj-isclm 38255 | The predicate "is a subcom... |
| bj-isrvec 38258 | The predicate "is a real v... |
| bj-rvecmod 38259 | Real vector spaces are mod... |
| bj-rvecssmod 38260 | Real vector spaces are mod... |
| bj-rvecrr 38261 | The field of scalars of a ... |
| bj-isrvecd 38262 | The predicate "is a real v... |
| bj-rvecvec 38263 | Real vector spaces are vec... |
| bj-isrvec2 38264 | The predicate "is a real v... |
| bj-rvecssvec 38265 | Real vector spaces are vec... |
| bj-rveccmod 38266 | Real vector spaces are sub... |
| bj-rvecsscmod 38267 | Real vector spaces are sub... |
| bj-rvecsscvec 38268 | Real vector spaces are sub... |
| bj-rveccvec 38269 | Real vector spaces are sub... |
| bj-rvecssabl 38270 | (The additive groups of) r... |
| bj-rvecabl 38271 | (The additive groups of) r... |
| bj-subcom 38272 | A consequence of commutati... |
| bj-lineqi 38273 | Solution of a (scalar) lin... |
| bj-bary1lem 38274 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 38275 | Lemma for ~ bj-bary1 : com... |
| bj-bary1 38276 | Barycentric coordinates in... |
| bj-endval 38279 | Value of the monoid of end... |
| bj-endbase 38280 | Base set of the monoid of ... |
| bj-endcomp 38281 | Composition law of the mon... |
| bj-endmnd 38282 | The monoid of endomorphism... |
| taupilem3 38283 | Lemma for tau-related theo... |
| taupilemrplb 38284 | A set of positive reals ha... |
| taupilem1 38285 | Lemma for ~ taupi . A pos... |
| taupilem2 38286 | Lemma for ~ taupi . The s... |
| taupi 38287 | Relationship between ` _ta... |
| dfgcd3 38288 | Alternate definition of th... |
| irrdifflemf 38289 | Lemma for ~ irrdiff . The... |
| irrdiff 38290 | The irrationals are exactl... |
| qdiff 38291 | The rationals are exactly ... |
| qdiffALT 38292 | Alternate proof of ~ qdiff... |
| iccioo01 38293 | The closed unit interval i... |
| csbrecsg 38294 | Move class substitution in... |
| csbrdgg 38295 | Move class substitution in... |
| csboprabg 38296 | Move class substitution in... |
| csbmpo123 38297 | Move class substitution in... |
| con1bii2 38298 | A contraposition inference... |
| con2bii2 38299 | A contraposition inference... |
| vtoclefex 38300 | Implicit substitution of a... |
| rnmptsn 38301 | The range of a function ma... |
| f1omptsnlem 38302 | This is the core of the pr... |
| f1omptsn 38303 | A function mapping to sing... |
| mptsnunlem 38304 | This is the core of the pr... |
| mptsnun 38305 | A class ` B ` is equal to ... |
| dissneqlem 38306 | This is the core of the pr... |
| dissneq 38307 | Any topology that contains... |
| exlimim 38308 | Closed form of ~ exlimimd ... |
| exlimimd 38309 | Existential elimination ru... |
| exellim 38310 | Closed form of ~ exellimdd... |
| exellimddv 38311 | Eliminate an antecedent wh... |
| topdifinfindis 38312 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 38313 | This is the core of the pr... |
| topdifinffin 38314 | Part of Exercise 3 of [Mun... |
| topdifinf 38315 | Part of Exercise 3 of [Mun... |
| topdifinfeq 38316 | Two different ways of defi... |
| icorempo 38317 | Closed-below, open-above i... |
| icoreresf 38318 | Closed-below, open-above i... |
| icoreval 38319 | Value of the closed-below,... |
| icoreelrnab 38320 | Elementhood in the set of ... |
| isbasisrelowllem1 38321 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 38322 | Lemma for ~ isbasisrelowl ... |
| icoreclin 38323 | The set of closed-below, o... |
| isbasisrelowl 38324 | The set of all closed-belo... |
| icoreunrn 38325 | The union of all closed-be... |
| istoprelowl 38326 | The set of all closed-belo... |
| icoreelrn 38327 | A class abstraction which ... |
| iooelexlt 38328 | An element of an open inte... |
| relowlssretop 38329 | The lower limit topology o... |
| relowlpssretop 38330 | The lower limit topology o... |
| sucneqond 38331 | Inequality of an ordinal s... |
| sucneqoni 38332 | Inequality of an ordinal s... |
| onsucuni3 38333 | If an ordinal number has a... |
| 1oequni2o 38334 | The ordinal number ` 1o ` ... |
| rdgsucuni 38335 | If an ordinal number has a... |
| rdgeqoa 38336 | If a recursive function wi... |
| elxp8 38337 | Membership in a Cartesian ... |
| cbveud 38338 | Deduction used to change b... |
| cbvreud 38339 | Deduction used to change b... |
| difunieq 38340 | The difference of unions i... |
| inunissunidif 38341 | Theorem about subsets of t... |
| rdgellim 38342 | Elementhood in a recursive... |
| rdglimss 38343 | A recursive definition at ... |
| rdgssun 38344 | In a recursive definition ... |
| exrecfnlem 38345 | Lemma for ~ exrecfn . (Co... |
| exrecfn 38346 | Theorem about the existenc... |
| exrecfnpw 38347 | For any base set, a set wh... |
| finorwe 38348 | If the Axiom of Infinity i... |
| dffinxpf 38351 | This theorem is the same a... |
| finxpeq1 38352 | Equality theorem for Carte... |
| finxpeq2 38353 | Equality theorem for Carte... |
| csbfinxpg 38354 | Distribute proper substitu... |
| finxpreclem1 38355 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 38356 | Lemma for ` ^^ ` recursion... |
| finxp0 38357 | The value of Cartesian exp... |
| finxp1o 38358 | The value of Cartesian exp... |
| finxpreclem3 38359 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 38360 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 38361 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 38362 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 38363 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 38364 | The value of Cartesian exp... |
| finxp2o 38365 | The value of Cartesian exp... |
| finxp3o 38366 | The value of Cartesian exp... |
| finxpnom 38367 | Cartesian exponentiation w... |
| finxp00 38368 | Cartesian exponentiation o... |
| iunctb2 38369 | Using the axiom of countab... |
| domalom 38370 | A class which dominates ev... |
| isinf2 38371 | The converse of ~ isinf . ... |
| ctbssinf 38372 | Using the axiom of choice,... |
| ralssiun 38373 | The index set of an indexe... |
| nlpineqsn 38374 | For every point ` p ` of a... |
| nlpfvineqsn 38375 | Given a subset ` A ` of ` ... |
| fvineqsnf1 38376 | A theorem about functions ... |
| fvineqsneu 38377 | A theorem about functions ... |
| fvineqsneq 38378 | A theorem about functions ... |
| pibp16 38379 | Property P000016 of pi-bas... |
| pibp19 38380 | Property P000019 of pi-bas... |
| pibp21 38381 | Property P000021 of pi-bas... |
| pibt1 38382 | Theorem T000001 of pi-base... |
| pibt2 38383 | Theorem T000002 of pi-base... |
| wl-section-prop 38384 | Intuitionistic logic is no... |
| wl-section-boot 38388 | In this section, I provide... |
| wl-luk-imim1i 38389 | Inference adding common co... |
| wl-luk-syl 38390 | An inference version of th... |
| wl-luk-imtrid 38391 | A syllogism rule of infere... |
| wl-luk-pm2.18d 38392 | Deduction based on reducti... |
| wl-luk-con4i 38393 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 38394 | Inference rule. Copy of ~... |
| wl-luk-a1i 38395 | Inference rule. Copy of ~... |
| wl-luk-mpi 38396 | A nested _modus ponens_ in... |
| wl-luk-imim2i 38397 | Inference adding common an... |
| wl-luk-imtrdi 38398 | A syllogism rule of infere... |
| wl-luk-ax3 38399 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 38400 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 38401 | This theorem, called "Asse... |
| wl-luk-com12 38402 | Inference that swaps (comm... |
| wl-luk-pm2.21 38403 | From a wff and its negatio... |
| wl-luk-con1i 38404 | A contraposition inference... |
| wl-luk-ja 38405 | Inference joining the ante... |
| wl-luk-imim2 38406 | A closed form of syllogism... |
| wl-luk-a1d 38407 | Deduction introducing an e... |
| wl-luk-ax2 38408 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 38409 | Principle of identity. Th... |
| wl-luk-notnotr 38410 | Converse of double negatio... |
| wl-luk-pm2.04 38411 | Swap antecedents. Theorem... |
| wl-section-impchain 38412 | An implication like ` ( ps... |
| wl-impchain-mp-x 38413 | This series of theorems pr... |
| wl-impchain-mp-0 38414 | This theorem is the start ... |
| wl-impchain-mp-1 38415 | This theorem is in fact a ... |
| wl-impchain-mp-2 38416 | This theorem is in fact a ... |
| wl-impchain-com-1.x 38417 | It is often convenient to ... |
| wl-impchain-com-1.1 38418 | A degenerate form of antec... |
| wl-impchain-com-1.2 38419 | This theorem is in fact a ... |
| wl-impchain-com-1.3 38420 | This theorem is in fact a ... |
| wl-impchain-com-1.4 38421 | This theorem is in fact a ... |
| wl-impchain-com-n.m 38422 | This series of theorems al... |
| wl-impchain-com-2.3 38423 | This theorem is in fact a ... |
| wl-impchain-com-2.4 38424 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 38425 | This theorem is in fact a ... |
| wl-impchain-a1-x 38426 | If an implication chain is... |
| wl-impchain-a1-1 38427 | Inference rule, a copy of ... |
| wl-impchain-a1-2 38428 | Inference rule, a copy of ... |
| wl-impchain-a1-3 38429 | Inference rule, a copy of ... |
| wl-ifp-ncond1 38430 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 38431 | If one case of an ` if- ` ... |
| wl-ifpimpr 38432 | If one case of an ` if- ` ... |
| wl-ifp4impr 38433 | If one case of an ` if- ` ... |
| wl-df-3xor 38434 | Alternative definition of ... |
| wl-df3xor2 38435 | Alternative definition of ... |
| wl-df3xor3 38436 | Alternative form of ~ wl-d... |
| wl-3xortru 38437 | If the first input is true... |
| wl-3xorfal 38438 | If the first input is fals... |
| wl-3xorbi 38439 | Triple xor can be replaced... |
| wl-3xorbi2 38440 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 38441 | Equivalence theorem for tr... |
| wl-3xorbi123i 38442 | Equivalence theorem for tr... |
| wl-3xorrot 38443 | Rotation law for triple xo... |
| wl-3xorcoma 38444 | Commutative law for triple... |
| wl-3xorcomb 38445 | Commutative law for triple... |
| wl-3xornot1 38446 | Flipping the first input f... |
| wl-3xornot 38447 | Triple xor distributes ove... |
| wl-1xor 38448 | In the recursive scheme ... |
| wl-2xor 38449 | In the recursive scheme ... |
| wl-df-3mintru2 38450 | Alternative definition of ... |
| wl-df2-3mintru2 38451 | The adder carry in disjunc... |
| wl-df3-3mintru2 38452 | The adder carry in conjunc... |
| wl-df4-3mintru2 38453 | An alternative definition ... |
| wl-1mintru1 38454 | Using the recursion formul... |
| wl-1mintru2 38455 | Using the recursion formul... |
| wl-2mintru1 38456 | Using the recursion formul... |
| wl-2mintru2 38457 | Using the recursion formul... |
| wl-df3maxtru1 38458 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 38460 | A version of ~ ax-wl-13v w... |
| wl-cleq-0 38461 |
Disclaimer: |
| wl-cleq-1 38462 |
Disclaimer: |
| wl-cleq-2 38463 |
Disclaimer: |
| wl-cleq-3 38464 |
Disclaimer: |
| wl-cleq-4 38465 |
Disclaimer: |
| wl-cleq-5 38466 |
Disclaimer: |
| wl-cleq-6 38467 |
Disclaimer: |
| wl-df-clab 38470 | Disclaimer: The material ... |
| wl-isseteq 38471 | A class equal to a set var... |
| wl-ax12v2cl 38472 | The class version of ~ ax1... |
| wl-df.clab 38473 | Define class abstractions,... |
| wl-df.cleq 38474 | Define the equality connec... |
| wl-dfcleq.basic 38475 | This theorem is a conserva... |
| wl-dfcleq.just 38476 | The hypotheses added to th... |
| wl-df.clel 38477 | Define the membership conn... |
| wl-dfclel.basic 38478 | This theorem gives a conse... |
| wl-dfclel.just 38479 | Add a hypothesis to ~ wl-d... |
| wl-dfcleq 38480 | The defining characterizat... |
| wl-dfclel 38481 | The defining characterizat... |
| wl-mps 38482 | Replacing a nested consequ... |
| wl-syls1 38483 | Replacing a nested consequ... |
| wl-syls2 38484 | Replacing a nested anteced... |
| wl-embant 38485 | A true wff can always be a... |
| wl-orel12 38486 | In a conjunctive normal fo... |
| wl-cases2-dnf 38487 | A particular instance of ~... |
| wl-cbvmotv 38488 | Change bound variable. Us... |
| wl-moteq 38489 | Change bound variable. Us... |
| wl-motae 38490 | Change bound variable. Us... |
| wl-moae 38491 | Two ways to express "at mo... |
| wl-euae 38492 | Two ways to express "exact... |
| wl-nax6im 38493 | The following series of th... |
| wl-hbae1 38494 | This specialization of ~ h... |
| wl-naevhba1v 38495 | An instance of ~ hbn1w app... |
| wl-spae 38496 | Prove an instance of ~ sp ... |
| wl-speqv 38497 | Under the assumption ` -. ... |
| wl-19.8eqv 38498 | Under the assumption ` -. ... |
| wl-19.2reqv 38499 | Under the assumption ` -. ... |
| wl-nfalv 38500 | If ` x ` is not present in... |
| wl-nfimf1 38501 | An antecedent is irrelevan... |
| wl-nfae1 38502 | Unlike ~ nfae , this speci... |
| wl-nfnae1 38503 | Unlike ~ nfnae , this spec... |
| wl-aetr 38504 | A transitive law for varia... |
| wl-axc11r 38505 | Same as ~ axc11r , but usi... |
| wl-dral1d 38506 | A version of ~ dral1 with ... |
| wl-cbvalnaed 38507 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 38508 | A more general version of ... |
| wl-exeq 38509 | The semantics of ` E. x y ... |
| wl-aleq 38510 | The semantics of ` A. x y ... |
| wl-nfeqfb 38511 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 38512 | If ` y ` is not free in ` ... |
| wl-equsalvw 38513 | Version of ~ equsalv with ... |
| wl-equsald 38514 | Deduction version of ~ equ... |
| wl-equsaldv 38515 | Deduction version of ~ equ... |
| wl-equsal 38516 | A useful equivalence relat... |
| wl-equsal1t 38517 | The expression ` x = y ` i... |
| wl-equsalcom 38518 | This simple equivalence ea... |
| wl-equsal1i 38519 | The antecedent ` x = y ` i... |
| wl-sbid2ft 38520 | A more general version of ... |
| wl-cbvalsbi 38521 | Change bounded variables i... |
| wl-sbrimt 38522 | Substitution with a variab... |
| wl-sblimt 38523 | Substitution with a variab... |
| wl-sb9v 38524 | Commutation of quantificat... |
| wl-sb8ft 38525 | Substitution of variable i... |
| wl-sb8eft 38526 | Substitution of variable i... |
| wl-sb8t 38527 | Substitution of variable i... |
| wl-sb8et 38528 | Substitution of variable i... |
| wl-sbhbt 38529 | Closed form of ~ sbhb . C... |
| wl-sbnf1 38530 | Two ways expressing that `... |
| wl-equsb3 38531 | ~ equsb3 with a distinctor... |
| wl-equsb4 38532 | Substitution applied to an... |
| wl-2sb6d 38533 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 38534 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 38535 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 38536 | Version of ~ sbcom2 with a... |
| wl-sbalnae 38537 | A theorem used in eliminat... |
| wl-sbal1 38538 | A theorem used in eliminat... |
| wl-sbal2 38539 | Move quantifier in and out... |
| wl-2spsbbi 38540 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 38541 | This theorem provides a ba... |
| wl-lem-nexmo 38542 | This theorem provides a ba... |
| wl-lem-moexsb 38543 | The antecedent ` A. x ( ph... |
| wl-alanbii 38544 | This theorem extends ~ ala... |
| wl-mo2df 38545 | Version of ~ mof with a co... |
| wl-mo2tf 38546 | Closed form of ~ mof with ... |
| wl-eudf 38547 | Version of ~ eu6 with a co... |
| wl-eutf 38548 | Closed form of ~ eu6 with ... |
| wl-euequf 38549 | ~ euequ proved with a dist... |
| wl-mo2t 38550 | Closed form of ~ mof . (C... |
| wl-mo3t 38551 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 38552 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 38553 | Substitution of variable i... |
| wl-sb8eutv 38554 | Substitution of variable i... |
| wl-sb8mot 38555 | Substitution of variable i... |
| wl-sb8motv 38556 | Substitution of variable i... |
| wl-issetft 38557 | A closed form of ~ issetf ... |
| wl-axc11rc11 38558 | Proving ~ axc11r from ~ ax... |
| wl-clabv 38559 | Variant of ~ df-clab , whe... |
| wl-dfclab 38560 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 38561 | Using class abstraction in... |
| wl-clabt 38562 | Using class abstraction in... |
| wl-eujustlem1 38563 | Version of ~ cbvexvw with ... |
| rabiun 38564 | Abstraction restricted to ... |
| iundif1 38565 | Indexed union of class dif... |
| imadifss 38566 | The difference of images i... |
| unceq 38567 | Equality theorem for uncur... |
| curunc 38568 | Currying of uncurrying. (... |
| unccur 38569 | Uncurrying of currying. (... |
| phpreu 38570 | Theorem related to pigeonh... |
| finixpnum 38571 | A finite Cartesian product... |
| fin2solem 38572 | Lemma for ~ fin2so . (Con... |
| fin2so 38573 | Any totally ordered Tarski... |
| ltflcei 38574 | Theorem to move the floor ... |
| leceifl 38575 | Theorem to move the floor ... |
| sin2h 38576 | Half-angle rule for sine. ... |
| cos2h 38577 | Half-angle rule for cosine... |
| tan2h 38578 | Half-angle rule for tangen... |
| lindsadd 38579 | In a vector space, the uni... |
| ptrest 38580 | Expressing a restriction o... |
| ptrecube 38581 | Any point in an open set o... |
| poimirlem1 38582 | Lemma for ~ poimir - the v... |
| poimirlem2 38583 | Lemma for ~ poimir - conse... |
| poimirlem3 38584 | Lemma for ~ poimir to add ... |
| poimirlem4 38585 | Lemma for ~ poimir connect... |
| poimirlem5 38586 | Lemma for ~ poimir to esta... |
| poimirlem6 38587 | Lemma for ~ poimir establi... |
| poimirlem7 38588 | Lemma for ~ poimir , simil... |
| poimirlem8 38589 | Lemma for ~ poimir , estab... |
| poimirlem9 38590 | Lemma for ~ poimir , estab... |
| poimirlem10 38591 | Lemma for ~ poimir establi... |
| poimirlem11 38592 | Lemma for ~ poimir connect... |
| poimirlem12 38593 | Lemma for ~ poimir connect... |
| poimirlem13 38594 | Lemma for ~ poimir - for a... |
| poimirlem14 38595 | Lemma for ~ poimir - for a... |
| poimirlem15 38596 | Lemma for ~ poimir , that ... |
| poimirlem16 38597 | Lemma for ~ poimir establi... |
| poimirlem17 38598 | Lemma for ~ poimir establi... |
| poimirlem18 38599 | Lemma for ~ poimir stating... |
| poimirlem19 38600 | Lemma for ~ poimir establi... |
| poimirlem20 38601 | Lemma for ~ poimir establi... |
| poimirlem21 38602 | Lemma for ~ poimir stating... |
| poimirlem22 38603 | Lemma for ~ poimir , that ... |
| poimirlem23 38604 | Lemma for ~ poimir , two w... |
| poimirlem24 38605 | Lemma for ~ poimir , two w... |
| poimirlem25 38606 | Lemma for ~ poimir stating... |
| poimirlem26 38607 | Lemma for ~ poimir showing... |
| poimirlem27 38608 | Lemma for ~ poimir showing... |
| poimirlem28 38609 | Lemma for ~ poimir , a var... |
| poimirlem29 38610 | Lemma for ~ poimir connect... |
| poimirlem30 38611 | Lemma for ~ poimir combini... |
| poimirlem31 38612 | Lemma for ~ poimir , assig... |
| poimirlem32 38613 | Lemma for ~ poimir , combi... |
| poimir 38614 | Poincare-Miranda theorem. ... |
| broucube 38615 | Brouwer - or as Kulpa call... |
| heicant 38616 | Heine-Cantor theorem: a co... |
| opnmbllem0 38617 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 38618 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 38619 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 38620 | The difference between two... |
| mblfinlem4 38621 | Backward direction of ~ is... |
| ismblfin 38622 | Measurability in terms of ... |
| ovoliunnfl 38623 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 38624 | Demonstration of ~ ovoliun... |
| voliunnfl 38625 | ~ voliun is incompatible w... |
| volsupnfl 38626 | ~ volsup is incompatible w... |
| mbfresfi 38627 | Measurability of a piecewi... |
| mbfposadd 38628 | If the sum of two measurab... |
| cnambfre 38629 | A real-valued, a.e. contin... |
| dvtanlem 38630 | Lemma for ~ dvtan - the do... |
| dvtan 38631 | Derivative of tangent. (C... |
| itg2addnclem 38632 | An alternate expression fo... |
| itg2addnclem2 38633 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 38634 | Lemma incomprehensible in ... |
| itg2addnc 38635 | Alternate proof of ~ itg2a... |
| itg2gt0cn 38636 | ~ itg2gt0 holds on functio... |
| ibladdnclem 38637 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 38638 | Choice-free analogue of ~ ... |
| itgaddnclem1 38639 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 38640 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 38641 | Choice-free analogue of ~ ... |
| iblsubnc 38642 | Choice-free analogue of ~ ... |
| itgsubnc 38643 | Choice-free analogue of ~ ... |
| iblabsnclem 38644 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 38645 | Choice-free analogue of ~ ... |
| iblmulc2nc 38646 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 38647 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 38648 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 38649 | Choice-free analogue of ~ ... |
| itgabsnc 38650 | Choice-free analogue of ~ ... |
| itggt0cn 38651 | ~ itggt0 holds for continu... |
| ftc1cnnclem 38652 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 38653 | Choice-free proof of ~ ftc... |
| ftc1anclem1 38654 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 38655 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 38656 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 38657 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 38658 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 38659 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 38660 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 38661 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 38662 | ~ ftc1a holds for function... |
| ftc2nc 38663 | Choice-free proof of ~ ftc... |
| asindmre 38664 | Real part of domain of dif... |
| dvasin 38665 | Derivative of arcsine. (C... |
| dvacos 38666 | Derivative of arccosine. ... |
| dvreasin 38667 | Real derivative of arcsine... |
| dvreacos 38668 | Real derivative of arccosi... |
| areacirclem1 38669 | Antiderivative of cross-se... |
| areacirclem2 38670 | Endpoint-inclusive continu... |
| areacirclem3 38671 | Integrability of cross-sec... |
| areacirclem4 38672 | Endpoint-inclusive continu... |
| areacirclem5 38673 | Finding the cross-section ... |
| areacirc 38674 | The area of a circle of ra... |
| findcard4 38675 | Schema for strong inductio... |
| dfproplem 38684 | Given a set A, the set of ... |
| varprop 38685 | Variables encoded as natur... |
| negprop 38686 | The negation of a sentence... |
| impprop 38687 | The implication between tw... |
| dfprop1 38688 | The set of variables encod... |
| dfprop2 38689 | Every sentence of proposit... |
| dfprop 38690 | The set of sentences of pr... |
| unirep 38691 | Define a quantity whose de... |
| cover2 38692 | Two ways of expressing the... |
| cover2g 38693 | Two ways of expressing the... |
| brabg2 38694 | Relation by a binary relat... |
| opelopab3 38695 | Ordered pair membership in... |
| cocanfo 38696 | Cancellation of a surjecti... |
| brresi2 38697 | Restriction of a binary re... |
| fnopabeqd 38698 | Equality deduction for fun... |
| fvopabf4g 38699 | Function value of an opera... |
| fnopabco 38700 | Composition of a function ... |
| opropabco 38701 | Composition of an operator... |
| cocnv 38702 | Composition with a functio... |
| f1ocan1fv 38703 | Cancel a composition by a ... |
| f1ocan2fv 38704 | Cancel a composition by th... |
| inixp 38705 | Intersection of Cartesian ... |
| upixp 38706 | Universal property of the ... |
| abrexdom 38707 | An indexed set is dominate... |
| abrexdom2 38708 | An indexed set is dominate... |
| ac6gf 38709 | Axiom of Choice. (Contrib... |
| indexa 38710 | If for every element of an... |
| indexdom 38711 | If for every element of an... |
| frinfm 38712 | A subset of a well-founded... |
| welb 38713 | A nonempty subset of a wel... |
| supex2g 38714 | Existence of supremum. (C... |
| supclt 38715 | Closure of supremum. (Con... |
| supubt 38716 | Upper bound property of su... |
| filbcmb 38717 | Combine a finite set of lo... |
| fzmul 38718 | Membership of a product in... |
| sdclem2 38719 | Lemma for ~ sdc . (Contri... |
| sdclem1 38720 | Lemma for ~ sdc . (Contri... |
| sdc 38721 | Strong dependent choice. ... |
| fdc 38722 | Finite version of dependen... |
| fdc1 38723 | Variant of ~ fdc with no s... |
| seqpo 38724 | Two ways to say that a seq... |
| incsequz 38725 | An increasing sequence of ... |
| incsequz2 38726 | An increasing sequence of ... |
| nnubfi 38727 | A bounded above set of pos... |
| nninfnub 38728 | An infinite set of positiv... |
| subspopn 38729 | An open set is open in the... |
| neificl 38730 | Neighborhoods are closed u... |
| lpss2 38731 | Limit points of a subset a... |
| metf1o 38732 | Use a bijection with a met... |
| blssp 38733 | A ball in the subspace met... |
| mettrifi 38734 | Generalized triangle inequ... |
| lmclim2 38735 | A sequence in a metric spa... |
| geomcau 38736 | If the distance between co... |
| caures 38737 | The restriction of a Cauch... |
| caushft 38738 | A shifted Cauchy sequence ... |
| constcncf 38739 | A constant function is a c... |
| cnres2 38740 | The restriction of a conti... |
| cnresima 38741 | A continuous function is c... |
| cncfres 38742 | A continuous function on c... |
| istotbnd 38746 | The predicate "is a totall... |
| istotbnd2 38747 | The predicate "is a totall... |
| istotbnd3 38748 | A metric space is totally ... |
| totbndmet 38749 | The predicate "totally bou... |
| 0totbnd 38750 | The metric (there is only ... |
| sstotbnd2 38751 | Condition for a subset of ... |
| sstotbnd 38752 | Condition for a subset of ... |
| sstotbnd3 38753 | Use a net that is not nece... |
| totbndss 38754 | A subset of a totally boun... |
| equivtotbnd 38755 | If the metric ` M ` is "st... |
| isbnd 38757 | The predicate "is a bounde... |
| bndmet 38758 | A bounded metric space is ... |
| isbndx 38759 | A "bounded extended metric... |
| isbnd2 38760 | The predicate "is a bounde... |
| isbnd3 38761 | A metric space is bounded ... |
| isbnd3b 38762 | A metric space is bounded ... |
| bndss 38763 | A subset of a bounded metr... |
| blbnd 38764 | A ball is bounded. (Contr... |
| ssbnd 38765 | A subset of a metric space... |
| totbndbnd 38766 | A totally bounded metric s... |
| equivbnd 38767 | If the metric ` M ` is "st... |
| bnd2lem 38768 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 38769 | If balls are totally bound... |
| prdsbnd 38770 | The product metric over fi... |
| prdstotbnd 38771 | The product metric over fi... |
| prdsbnd2 38772 | If balls are totally bound... |
| cntotbnd 38773 | A subset of the complex nu... |
| cnpwstotbnd 38774 | A subset of ` A ^ I ` , wh... |
| ismtyval 38777 | The set of isometries betw... |
| isismty 38778 | The condition "is an isome... |
| ismtycnv 38779 | The inverse of an isometry... |
| ismtyima 38780 | The image of a ball under ... |
| ismtyhmeolem 38781 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 38782 | An isometry is a homeomorp... |
| ismtybndlem 38783 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 38784 | Isometries preserve bounde... |
| ismtyres 38785 | A restriction of an isomet... |
| heibor1lem 38786 | Lemma for ~ heibor1 . A c... |
| heibor1 38787 | One half of ~ heibor , tha... |
| heiborlem1 38788 | Lemma for ~ heibor . We w... |
| heiborlem2 38789 | Lemma for ~ heibor . Subs... |
| heiborlem3 38790 | Lemma for ~ heibor . Usin... |
| heiborlem4 38791 | Lemma for ~ heibor . Usin... |
| heiborlem5 38792 | Lemma for ~ heibor . The ... |
| heiborlem6 38793 | Lemma for ~ heibor . Sinc... |
| heiborlem7 38794 | Lemma for ~ heibor . Sinc... |
| heiborlem8 38795 | Lemma for ~ heibor . The ... |
| heiborlem9 38796 | Lemma for ~ heibor . Disc... |
| heiborlem10 38797 | Lemma for ~ heibor . The ... |
| heibor 38798 | Generalized Heine-Borel Th... |
| bfplem1 38799 | Lemma for ~ bfp . The seq... |
| bfplem2 38800 | Lemma for ~ bfp . Using t... |
| bfp 38801 | Banach fixed point theorem... |
| rrnval 38804 | The n-dimensional Euclidea... |
| rrnmval 38805 | The value of the Euclidean... |
| rrnmet 38806 | Euclidean space is a metri... |
| rrndstprj1 38807 | The distance between two p... |
| rrndstprj2 38808 | Bound on the distance betw... |
| rrncmslem 38809 | Lemma for ~ rrncms . (Con... |
| rrncms 38810 | Euclidean space is complet... |
| repwsmet 38811 | The supremum metric on ` R... |
| rrnequiv 38812 | The supremum metric on ` R... |
| rrntotbnd 38813 | A set in Euclidean space i... |
| rrnheibor 38814 | Heine-Borel theorem for Eu... |
| ismrer1 38815 | An isometry between ` RR `... |
| reheibor 38816 | Heine-Borel theorem for re... |
| iccbnd 38817 | A closed interval in ` RR ... |
| icccmpALT 38818 | A closed interval in ` RR ... |
| isass 38823 | The predicate "is an assoc... |
| isexid 38824 | The predicate ` G ` has a ... |
| ismgmOLD 38827 | Obsolete version of ~ ismg... |
| clmgmOLD 38828 | Obsolete version of ~ mgmc... |
| opidonOLD 38829 | Obsolete version of ~ mgmi... |
| rngopidOLD 38830 | Obsolete version of ~ mgmi... |
| opidon2OLD 38831 | Obsolete version of ~ mgmf... |
| isexid2 38832 | Obsolete theorem. If ` G ... |
| exidu1 38833 | Obsolete theorem, use ~ mg... |
| idrval 38834 | Obsolete theorem, use ~ id... |
| iorlid 38835 | Obsolete theorem, use ~ mg... |
| cmpidelt 38836 | Obsolete theorem, use ~ mg... |
| smgrpismgmOLD 38839 | Obsolete version of ~ sgrp... |
| issmgrpOLD 38840 | Obsolete version of ~ issg... |
| smgrpmgm 38841 | Obsolete theorem, use ~ sg... |
| smgrpassOLD 38842 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 38845 | Obsolete version of ~ mnds... |
| mndoisexid 38846 | Obsolete theorem, use ~ mn... |
| mndoismgmOLD 38847 | Obsolete version of ~ mndm... |
| mndomgmid 38848 | Obsolete theorem, use ~ mn... |
| ismndo 38849 | Obsolete theorem, use ~ is... |
| ismndo1 38850 | Obsolete theorem, use ~ is... |
| ismndo2 38851 | Obsolete theorem, use ~ is... |
| grpomndo 38852 | Obsolete theorem, use ~ gr... |
| exidcl 38853 | Obsolete theorem, use ~ mg... |
| exidreslem 38854 | Obsolete theorem, use ~ 0g... |
| exidres 38855 | Obsolete theorem, use ~ id... |
| exidresid 38856 | Obsolete theorem, use ~ id... |
| ablo4pnp 38857 | Obsolete theorem, use ~ ab... |
| grpoeqdivid 38858 | Obsolete theorem, use ~ gr... |
| grposnOLD 38859 | The group operation for th... |
| elghomlem1OLD 38862 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 38863 | Obsolete as of 15-Mar-2020... |
| elghomOLD 38864 | Obsolete version of ~ isgh... |
| ghomlinOLD 38865 | Obsolete version of ~ ghml... |
| ghomidOLD 38866 | Obsolete version of ~ ghmi... |
| ghomf 38867 | Obsolete theorem, use ~ gh... |
| ghomco 38868 | Obsolete theorem, use ~ gh... |
| ghomdiv 38869 | Obsolete theorem, use ~ gh... |
| grpokerinj 38870 | Obsolete theorem, use ~ ke... |
| relrngo 38873 | Obsolete theorem. The cla... |
| isrngo 38874 | Obsolete theorem, use ~ df... |
| isrngod 38875 | Obsolete theorem, use ~ is... |
| rngoi 38876 | Obsolete theorem, use ~ df... |
| rngosm 38877 | Obsolete theorem. Functio... |
| rngocl 38878 | Obsolete theorem, use ~ ri... |
| rngoid 38879 | Obsolete theorem, use ~ ri... |
| rngoideu 38880 | Obsolete theorem, use ~ ri... |
| rngodi 38881 | Obsolete theorem, use ~ ri... |
| rngodir 38882 | Obsolete theorem, use ~ ri... |
| rngoass 38883 | Obsolete theorem, use ~ ri... |
| rngo2 38884 | Obsolete theorem, use ~ ri... |
| rngoablo 38885 | Obsolete theorem, use ~ ri... |
| rngoablo2 38886 | Obsolete theorem, use ~ ri... |
| rngogrpo 38887 | Obsolete theorem, use ~ ri... |
| rngone0 38888 | Obsolete theorem, use ~ ri... |
| rngogcl 38889 | Obsolete theorem, use ~ ri... |
| rngocom 38890 | Obsolete theorem, use ~ ri... |
| rngoaass 38891 | Obsolete theorem, use ~ ri... |
| rngoa32 38892 | Obsolete theorem, use ~ ri... |
| rngoa4 38893 | Obsolete theorem, use ~ ri... |
| rngorcan 38894 | Obsolete theorem, use ~ ri... |
| rngolcan 38895 | Obsolete theorem, use ~ ri... |
| rngo0cl 38896 | Obsolete theorem, use ~ ri... |
| rngo0rid 38897 | Obsolete theorem, use ~ ri... |
| rngo0lid 38898 | Obsolete theorem, use ~ ri... |
| rngolz 38899 | Obsolete theorem, use ~ ri... |
| rngorz 38900 | Obsolete theorem, use ~ ri... |
| rngosn3 38901 | Obsolete as of 25-Jan-2020... |
| rngosn4 38902 | Obsolete as of 25-Jan-2020... |
| rngosn6 38903 | Obsolete as of 25-Jan-2020... |
| rngonegcl 38904 | Obsolete theorem, use ~ ri... |
| rngoaddneg1 38905 | Obsolete theorem, use ~ ri... |
| rngoaddneg2 38906 | Obsolete theorem, use ~ ri... |
| rngosub 38907 | Obsolete theorem, use ~ ri... |
| rngmgmbs4 38908 | Obsolete theorem. The ran... |
| rngodm1dm2 38909 | Obsolete theorem. In a un... |
| rngorn1 38910 | Obsolete theorem. In a un... |
| rngorn1eq 38911 | Obsolete theorem. In a un... |
| rngomndo 38912 | Obsolete theorem, use ~ ri... |
| rngoidmlem 38913 | Obsolete theorem, use ~ ri... |
| rngolidm 38914 | Obsolete theorem, use ~ ri... |
| rngoridm 38915 | Obsolete theorem, use ~ ri... |
| rngo1cl 38916 | Obsolete theorem, use ~ ri... |
| rngoueqz 38917 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 38918 | Obsolete theorem, use ~ ri... |
| rngonegmn1r 38919 | Obsolete theorem, use ~ ri... |
| rngoneglmul 38920 | Obsolete theorem, use ~ ri... |
| rngonegrmul 38921 | Obsolete theorem, use ~ ri... |
| rngosubdi 38922 | Obsolete theorem, use ~ ri... |
| rngosubdir 38923 | Obsolete theorem, use ~ ri... |
| zerdivemp1x 38924 | Obsolete theorem, use ~ ri... |
| isdivrngo 38927 | Obsolete theorem, use ~ is... |
| drngoi 38928 | Obsolete theorem, use ~ dr... |
| gidsn 38929 | Obsolete as of 23-Jan-2020... |
| zrdivrng 38930 | Obsolete theorem, use ~ zr... |
| dvrunz 38931 | Obsolete theorem, use ~ dr... |
| isgrpda 38932 | Obsolete theorem, use ~ is... |
| isdrngo1 38933 | Obsolete theorem, use ~ is... |
| divrngcl 38934 | Obsolete theorem, use ~ dr... |
| isdrngo2 38935 | Obsolete theorem, use ~ is... |
| isdrngo3 38936 | Obsolete theorem, use ~ is... |
| rngohomval 38941 | Obsolete theorem, use ~ rh... |
| isrngohom 38942 | Obsolete theorem, use ~ is... |
| rngohomf 38943 | Obsolete theorem, use ~ rh... |
| rngohomcl 38944 | Obsolete theorem, use ~ rh... |
| rngohom1 38945 | Obsolete theorem, use ~ rh... |
| rngohomadd 38946 | Obsolete theorem, use ~ rh... |
| rngohommul 38947 | Obsolete theorem, use ~ rh... |
| rngogrphom 38948 | Obsolete theorem, use ~ rh... |
| rngohom0 38949 | Obsolete theorem, use ~ rh... |
| rngohomsub 38950 | Obsolete theorem, use ~ rh... |
| rngohomco 38951 | Obsolete theorem, use ~ rh... |
| rngokerinj 38952 | Obsolete theorem, use ~ rh... |
| rngoisoval 38954 | Obsolete theorem, use ~ ri... |
| isrngoiso 38955 | Obsolete theorem, use ~ is... |
| rngoiso1o 38956 | Obsolete theorem, use ~ ri... |
| rngoisohom 38957 | Obsolete theorem, use ~ ri... |
| rngoisocnv 38958 | Obsolete theorem, use ~ ri... |
| rngoisoco 38959 | Obsolete theorem, use ~ ri... |
| isriscg 38961 | Obsolete theorem, use ~ is... |
| isrisc 38962 | Obsolete theorem, use ~ is... |
| risc 38963 | Obsolete theorem, use ~ br... |
| risci 38964 | Obsolete theorem, use ~ br... |
| riscer 38965 | Obsolete theorem, use ~ ri... |
| iscom2 38972 | Obsolete theorem, used (as... |
| iscrngo 38973 | Obsolete theorem, use ~ is... |
| iscrngo2 38974 | Obsolete theorem, use ~ is... |
| iscringd 38975 | Obsolete theorem, use ~ is... |
| flddivrng 38976 | Obsolete theorem, use ~ fl... |
| crngorngo 38977 | Obsolete theorem, use ~ cr... |
| crngocom 38978 | Obsolete theorem, use ~ cr... |
| crngm23 38979 | Obsolete theorem, use ~ cr... |
| crngm4 38980 | Obsolete theorem, use ~ cr... |
| fldcrngo 38981 | Obsolete theorem, use ~ fl... |
| isfld2 38982 | Obsolete theorem, use ~ is... |
| crngohomfo 38983 | Obsolete theorem, use ~ cr... |
| idlval 38990 | Obsolete theorem, use ~ 2i... |
| isidl 38991 | Obsolete theorem, use ~ df... |
| isidlc 38992 | Obsolete theorem, use ~ df... |
| idlss 38993 | Obsolete theorem, use ~ 2i... |
| idlcl 38994 | Obsolete theorem, use ~ 2i... |
| idl0cl 38995 | Obsolete theorem, use ~ ri... |
| idladdcl 38996 | Obsolete theorem, use ~ 2i... |
| idllmulcl 38997 | Obsolete theorem, use ~ 2i... |
| idlrmulcl 38998 | Obsolete theorem, use ~ 2i... |
| idlnegcl 38999 | Obsolete theorem, use ~ 2i... |
| idlsubcl 39000 | Obsolete theorem, use ~ 2i... |
| rngoidl 39001 | Obsolete theorem, use ~ 2i... |
| 0idl 39002 | Obsolete theorem, use ~ 2i... |
| 1idl 39003 | Obsolete theorem, use ~ 2i... |
| 0rngo 39004 | Obsolete theorem, use ~ 0r... |
| divrngidl 39005 | Obsolete theorem, use ~ dr... |
| intidl 39006 | Obsolete theorem, use ~ in... |
| inidl 39007 | Obsolete theorem, use ~ in... |
| unichnidl 39008 | Obsolete theorem, use ~ un... |
| keridl 39009 | Obsolete theorem, use ~ ke... |
| pridlval 39010 | Obsolete theorem, use ~ pr... |
| ispridl 39011 | Obsolete theorem, use ~ is... |
| pridlidl 39012 | Obsolete theorem, use ~ pr... |
| pridlnr 39013 | Obsolete theorem, use ~ pr... |
| pridl 39014 | Obsolete theorem, use ~ is... |
| ispridl2 39015 | Obsolete theorem, use ~ pr... |
| maxidlval 39016 | Obsolete theorem, use ~ mx... |
| ismaxidl 39017 | Obsolete theorem, use ~ is... |
| maxidlidl 39018 | Obsolete theorem, use ~ mx... |
| maxidlnr 39019 | Obsolete theorem, use ~ mx... |
| maxidlmax 39020 | Obsolete theorem, use ~ mx... |
| maxidln1 39021 | Obsolete theorem, use ~ mx... |
| maxidln0 39022 | Obsolete theorem, use ~ mx... |
| isprrngo 39027 | Obsolete theorem, use ~ is... |
| prrngorngo 39028 | Obsolete theorem, use ~ pr... |
| smprngopr 39029 | Obsolete theorem, use ~ sm... |
| divrngpr 39030 | Obsolete theorem, use ~ dr... |
| isdmn 39031 | Obsolete theorem, use ~ is... |
| isdmn2 39032 | Obsolete theorem, use ~ is... |
| dmncrng 39033 | Obsolete theorem, use ~ id... |
| dmnrngo 39034 | Obsolete theorem, use ~ id... |
| flddmn 39035 | Obsolete theorem, use ~ fl... |
| igenval 39038 | Obsolete theorem, use ~ rs... |
| igenss 39039 | Obsolete theorem, use ~ rs... |
| igenidl 39040 | Obsolete theorem, use ~ rs... |
| igenmin 39041 | Obsolete theorem, use ~ rs... |
| igenidl2 39042 | Obsolete theorem, use ~ rs... |
| igenval2 39043 | Obsolete theorem, use ~ rs... |
| prnc 39044 | Obsolete theorem, use ~ rs... |
| isfldidl 39045 | Obsolete theorem, use ~ is... |
| isfldidl2 39046 | Obsolete theorem, use ~ is... |
| ispridlc 39047 | Obsolete theorem, use ~ is... |
| pridlc 39048 | Obsolete theorem, use ~ pr... |
| pridlc2 39049 | Obsolete theorem, use ~ pr... |
| pridlc3 39050 | Obsolete theorem, use ~ cm... |
| isdmn3 39051 | Obsolete theorem, use ~ is... |
| dmnnzd 39052 | Obsolete theorem, use ~ id... |
| dmncan1 39053 | Obsolete theorem, use ~ id... |
| dmncan2 39054 | Obsolete theorem, use ~ id... |
| efald2 39055 | A proof by contradiction. ... |
| notbinot1 39056 | Simplification rule of neg... |
| bicontr 39057 | Biconditional of its own n... |
| impor 39058 | An equivalent formula for ... |
| orfa 39059 | The falsum ` F. ` can be r... |
| notbinot2 39060 | Commutation rule between n... |
| biimpor 39061 | A rewriting rule for bicon... |
| orfa1 39062 | Add a contradicting disjun... |
| orfa2 39063 | Remove a contradicting dis... |
| bifald 39064 | Infer the equivalence to a... |
| cnf1dd 39065 | A lemma for Conjunctive No... |
| cnf2dd 39066 | A lemma for Conjunctive No... |
| cnfn1dd 39067 | A lemma for Conjunctive No... |
| cnfn2dd 39068 | A lemma for Conjunctive No... |
| or32dd 39069 | A rearrangement of disjunc... |
| notornotel1 39070 | A lemma for not-or-not eli... |
| notornotel2 39071 | A lemma for not-or-not eli... |
| contrd 39072 | A proof by contradiction, ... |
| an12i 39073 | An inference from commutin... |
| exmid2 39074 | An excluded middle law. (... |
| selconj 39075 | An inference for selecting... |
| truconj 39076 | Add true as a conjunct. (... |
| orel 39077 | An inference for disjuncti... |
| negel 39078 | An inference for negation ... |
| botel 39079 | An inference for bottom el... |
| tradd 39080 | Add top ad a conjunct. (C... |
| gm-sbtru 39081 | Substitution does not chan... |
| sbfal 39082 | Substitution does not chan... |
| sbcani 39083 | Distribution of class subs... |
| sbcori 39084 | Distribution of class subs... |
| sbcimi 39085 | Distribution of class subs... |
| sbcni 39086 | Move class substitution in... |
| sbali 39087 | Discard class substitution... |
| sbexi 39088 | Discard class substitution... |
| sbcalf 39089 | Move universal quantifier ... |
| sbcexf 39090 | Move existential quantifie... |
| sbcalfi 39091 | Move universal quantifier ... |
| sbcexfi 39092 | Move existential quantifie... |
| spsbcdi 39093 | A lemma for eliminating a ... |
| alrimii 39094 | A lemma for introducing a ... |
| spesbcdi 39095 | A lemma for introducing an... |
| exlimddvf 39096 | A lemma for eliminating an... |
| exlimddvfi 39097 | A lemma for eliminating an... |
| sbceq1ddi 39098 | A lemma for eliminating in... |
| sbccom2lem 39099 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 39100 | Commutative law for double... |
| sbccom2f 39101 | Commutative law for double... |
| sbccom2fi 39102 | Commutative law for double... |
| csbcom2fi 39103 | Commutative law for double... |
| fald 39104 | Refutation of falsity, in ... |
| tsim1 39105 | A Tseitin axiom for logica... |
| tsim2 39106 | A Tseitin axiom for logica... |
| tsim3 39107 | A Tseitin axiom for logica... |
| tsbi1 39108 | A Tseitin axiom for logica... |
| tsbi2 39109 | A Tseitin axiom for logica... |
| tsbi3 39110 | A Tseitin axiom for logica... |
| tsbi4 39111 | A Tseitin axiom for logica... |
| tsxo1 39112 | A Tseitin axiom for logica... |
| tsxo2 39113 | A Tseitin axiom for logica... |
| tsxo3 39114 | A Tseitin axiom for logica... |
| tsxo4 39115 | A Tseitin axiom for logica... |
| tsan1 39116 | A Tseitin axiom for logica... |
| tsan2 39117 | A Tseitin axiom for logica... |
| tsan3 39118 | A Tseitin axiom for logica... |
| tsna1 39119 | A Tseitin axiom for logica... |
| tsna2 39120 | A Tseitin axiom for logica... |
| tsna3 39121 | A Tseitin axiom for logica... |
| tsor1 39122 | A Tseitin axiom for logica... |
| tsor2 39123 | A Tseitin axiom for logica... |
| tsor3 39124 | A Tseitin axiom for logica... |
| ts3an1 39125 | A Tseitin axiom for triple... |
| ts3an2 39126 | A Tseitin axiom for triple... |
| ts3an3 39127 | A Tseitin axiom for triple... |
| ts3or1 39128 | A Tseitin axiom for triple... |
| ts3or2 39129 | A Tseitin axiom for triple... |
| ts3or3 39130 | A Tseitin axiom for triple... |
| iuneq2f 39131 | Equality deduction for ind... |
| rabeq12f 39132 | Equality deduction for res... |
| csbeq12 39133 | Equality deduction for sub... |
| sbeqi 39134 | Equality deduction for sub... |
| ralbi12f 39135 | Equality deduction for res... |
| oprabbi 39136 | Equality deduction for cla... |
| mpobi123f 39137 | Equality deduction for map... |
| iuneq12f 39138 | Equality deduction for ind... |
| iineq12f 39139 | Equality deduction for ind... |
| opabbi 39140 | Equality deduction for cla... |
| mptbi12f 39141 | Equality deduction for map... |
| orcomdd 39142 | Commutativity of logic dis... |
| scottexf 39143 | A version of ~ scottex wit... |
| scott0f 39144 | A version of ~ scott0b wit... |
| scottn0f 39145 | A version of ~ scott0f wit... |
| ac6s3f 39146 | Generalization of the Axio... |
| ac6s6 39147 | Generalization of the Axio... |
| ac6s6f 39148 | Generalization of the Axio... |
| el2v1 39204 | New way ( ~ elv , and the ... |
| el3v1 39205 | New way ( ~ elv , and the ... |
| el3v2 39206 | New way ( ~ elv , and the ... |
| el3v12 39207 | New way ( ~ elv , and the ... |
| el3v13 39208 | New way ( ~ elv , and the ... |
| el3v23 39209 | New way ( ~ elv , and the ... |
| anan 39210 | Multiple commutations in c... |
| triantru3 39211 | A wff is equivalent to its... |
| biorfd 39212 | A wff is equivalent to its... |
| eqbrtr 39213 | Substitution of equal clas... |
| eqbrb 39214 | Substitution of equal clas... |
| eqeltr 39215 | Substitution of equal clas... |
| eqelb 39216 | Substitution of equal clas... |
| eqeqan2d 39217 | Implication of introducing... |
| disjresin 39218 | The restriction to a disjo... |
| disjresdisj 39219 | The intersection of restri... |
| disjresdif 39220 | The difference between res... |
| disjresundif 39221 | Lemma for ~ ressucdifsn2 .... |
| inres2 39222 | Two ways of expressing the... |
| coideq 39223 | Equality theorem for compo... |
| nexmo1 39224 | If there is no case where ... |
| eqab2 39225 | Implication of a class abs... |
| r2alan 39226 | Double restricted universa... |
| ssrabi 39227 | Inference of restricted ab... |
| rabimbieq 39228 | Restricted equivalent wff'... |
| abeqin 39229 | Intersection with class ab... |
| abeqinbi 39230 | Intersection with class ab... |
| eqrabi 39231 | Class element of a restric... |
| rabeqel 39232 | Class element of a restric... |
| eqrelf 39233 | The equality connective be... |
| br1cnvinxp 39234 | Binary relation on the con... |
| releleccnv 39235 | Elementhood in a converse ... |
| releccnveq 39236 | Equality of converse ` R `... |
| xpv 39237 | Cartesian product of a cla... |
| vxp 39238 | Cartesian product of the u... |
| opelvvdif 39239 | Negated elementhood of ord... |
| vvdifopab 39240 | Ordered-pair class abstrac... |
| brvdif 39241 | Binary relation with unive... |
| brvdif2 39242 | Binary relation with unive... |
| brvvdif 39243 | Binary relation with the c... |
| brvbrvvdif 39244 | Binary relation with the c... |
| brcnvep 39245 | The converse of the binary... |
| elecALTV 39246 | Elementhood in the ` R ` -... |
| brcnvepres 39247 | Restricted converse epsilo... |
| brres2 39248 | Binary relation on a restr... |
| br1cnvres 39249 | Binary relation on the con... |
| elec1cnvres 39250 | Elementhood in the convers... |
| ec1cnvres 39251 | Converse restricted coset ... |
| eldmres 39252 | Elementhood in the domain ... |
| elrnres 39253 | Element of the range of a ... |
| eldmressnALTV 39254 | Element of the domain of a... |
| elrnressn 39255 | Element of the range of a ... |
| eldm4 39256 | Elementhood in a domain. ... |
| eldmres2 39257 | Elementhood in the domain ... |
| eldmres3 39258 | Elementhood in the domain ... |
| eceq1i 39259 | Equality theorem for ` C `... |
| ecres 39260 | Restricted coset of ` B ` ... |
| eccnvepres 39261 | Restricted converse epsilo... |
| eleccnvep 39262 | Elementhood in the convers... |
| eccnvep 39263 | The converse epsilon coset... |
| extep 39264 | Property of epsilon relati... |
| disjeccnvep 39265 | Property of the epsilon re... |
| eccnvepres2 39266 | The restricted converse ep... |
| eccnvepres3 39267 | Condition for a restricted... |
| eldmqsres 39268 | Elementhood in a restricte... |
| eldmqsres2 39269 | Elementhood in a restricte... |
| qsss1 39270 | Subclass theorem for quoti... |
| qseq1i 39271 | Equality theorem for quoti... |
| brinxprnres 39272 | Binary relation on a restr... |
| inxprnres 39273 | Restriction of a class as ... |
| dfres4 39274 | Alternate definition of th... |
| exan3 39275 | Equivalent expressions wit... |
| exanres 39276 | Equivalent expressions wit... |
| exanres3 39277 | Equivalent expressions wit... |
| exanres2 39278 | Equivalent expressions wit... |
| cnvepres 39279 | Restricted converse epsilo... |
| eqrel2 39280 | Equality of relations. (C... |
| rncnv 39281 | Range of converse is the d... |
| dfdm6 39282 | Alternate definition of do... |
| dfrn6 39283 | Alternate definition of ra... |
| rncnvepres 39284 | The range of the restricte... |
| dmecd 39285 | Equality of the coset of `... |
| dmec2d 39286 | Equality of the coset of `... |
| brid 39287 | Property of the identity b... |
| ideq2 39288 | For sets, the identity bin... |
| idresssidinxp 39289 | Condition for the identity... |
| idreseqidinxp 39290 | Condition for the identity... |
| extid 39291 | Property of identity relat... |
| inxpss 39292 | Two ways to say that an in... |
| idinxpss 39293 | Two ways to say that an in... |
| ref5 39294 | Two ways to say that an in... |
| inxpss3 39295 | Two ways to say that an in... |
| inxpss2 39296 | Two ways to say that inter... |
| inxpssidinxp 39297 | Two ways to say that inter... |
| idinxpssinxp 39298 | Two ways to say that inter... |
| idinxpssinxp2 39299 | Identity intersection with... |
| idinxpssinxp3 39300 | Identity intersection with... |
| idinxpssinxp4 39301 | Identity intersection with... |
| relcnveq3 39302 | Two ways of saying a relat... |
| relcnveq 39303 | Two ways of saying a relat... |
| relcnveq2 39304 | Two ways of saying a relat... |
| relcnveq4 39305 | Two ways of saying a relat... |
| qsresid 39306 | Simplification of a specia... |
| n0elqs 39307 | Two ways of expressing tha... |
| n0elqs2 39308 | Two ways of expressing tha... |
| rnresequniqs 39309 | The range of a restriction... |
| n0el2 39310 | Two ways of expressing tha... |
| cnvepresex 39311 | Sethood condition for the ... |
| cnvepima 39312 | The image of converse epsi... |
| inex3 39313 | Sufficient condition for t... |
| inxpex 39314 | Sufficient condition for a... |
| eqres 39315 | Converting a class constan... |
| brrabga 39316 | The law of concretion for ... |
| brcnvrabga 39317 | The law of concretion for ... |
| opideq 39318 | Equality conditions for or... |
| iss2 39319 | A subclass of the identity... |
| eldmcnv 39320 | Elementhood in a domain of... |
| dfrel5 39321 | Alternate definition of th... |
| dfrel6 39322 | Alternate definition of th... |
| cnvresrn 39323 | Converse restricted to ran... |
| relssinxpdmrn 39324 | Subset of restriction, spe... |
| cnvref4 39325 | Two ways to say that a rel... |
| cnvref5 39326 | Two ways to say that a rel... |
| ecin0 39327 | Two ways of saying that th... |
| ecinn0 39328 | Two ways of saying that th... |
| ineleq 39329 | Equivalence of restricted ... |
| inecmo 39330 | Equivalence of a double re... |
| inecmo2 39331 | Equivalence of a double re... |
| ineccnvmo 39332 | Equivalence of a double re... |
| alrmomorn 39333 | Equivalence of an "at most... |
| alrmomodm 39334 | Equivalence of an "at most... |
| ralmo 39335 | "At most one" can be restr... |
| ralrnmo 39336 | On the range, "at most one... |
| dmqsex 39337 | Sethood of the domain quot... |
| raldmqsmo 39338 | On the quotient carrier, "... |
| ralrmo3 39339 | Pull a restricted universa... |
| raldmqseu 39340 | Equivalence between "exact... |
| rsp3 39341 | From a restricted universa... |
| rsp3eq 39342 | From a restricted universa... |
| ineccnvmo2 39343 | Equivalence of a double un... |
| inecmo3 39344 | Equivalence of a double un... |
| moeu2 39345 | Uniqueness is equivalent t... |
| mopickr 39346 | "At most one" picks a vari... |
| moantr 39347 | Sufficient condition for t... |
| brabidgaw 39348 | The law of concretion for ... |
| brabidga 39349 | The law of concretion for ... |
| inxp2 39350 | Intersection with a Cartes... |
| opabf 39351 | A class abstraction of a c... |
| ec0 39352 | The empty-coset of a class... |
| brcnvin 39353 | Intersection with a conver... |
| ssdmral 39354 | Subclass of a domain. (Co... |
| xrnss3v 39356 | A range Cartesian product ... |
| xrnrel 39357 | A range Cartesian product ... |
| brxrn 39358 | Characterize a ternary rel... |
| brxrn2 39359 | A characterization of the ... |
| dfxrn2 39360 | Alternate definition of th... |
| brxrncnvep 39361 | The range product with con... |
| dmxrn 39362 | Domain of the range produc... |
| dmcnvep 39363 | Domain of converse epsilon... |
| dmxrncnvep 39364 | Domain of the range produc... |
| dmcnvepres 39365 | Domain of the restricted c... |
| dmuncnvepres 39366 | Domain of the union with t... |
| dmxrnuncnvepres 39367 | Domain of the combined rel... |
| ecun 39368 | The union coset of ` A ` .... |
| ecunres 39369 | The restricted union coset... |
| ecuncnvepres 39370 | The restricted union with ... |
| xrneq1 39371 | Equality theorem for the r... |
| xrneq1i 39372 | Equality theorem for the r... |
| xrneq1d 39373 | Equality theorem for the r... |
| xrneq2 39374 | Equality theorem for the r... |
| xrneq2i 39375 | Equality theorem for the r... |
| xrneq2d 39376 | Equality theorem for the r... |
| xrneq12 39377 | Equality theorem for the r... |
| xrneq12i 39378 | Equality theorem for the r... |
| xrneq12d 39379 | Equality theorem for the r... |
| elecxrn 39380 | Elementhood in the ` ( R |... |
| ecxrn 39381 | The ` ( R |X. S ) ` -coset... |
| relecxrn 39382 | The ` ( R |X. S ) ` -coset... |
| ecxrn2 39383 | The ` ( R |X. S ) ` -coset... |
| ecxrncnvep 39384 | The ` ( R |X. ``' _E ) ` -... |
| ecxrncnvep2 39385 | The ` ( R |X. ``' _E ) ` -... |
| disjressuc2 39386 | Double restricted quantifi... |
| disjecxrn 39387 | Two ways of saying that ` ... |
| disjecxrncnvep 39388 | Two ways of saying that co... |
| disjsuc2 39389 | Double restricted quantifi... |
| xrninxp 39390 | Intersection of a range Ca... |
| xrninxp2 39391 | Intersection of a range Ca... |
| xrninxpex 39392 | Sufficient condition for t... |
| inxpxrn 39393 | Two ways to express the in... |
| br1cnvxrn2 39394 | The converse of a binary r... |
| elec1cnvxrn2 39395 | Elementhood in the convers... |
| rnxrn 39396 | Range of the range Cartesi... |
| rnxrnres 39397 | Range of a range Cartesian... |
| rnxrncnvepres 39398 | Range of a range Cartesian... |
| rnxrnidres 39399 | Range of a range Cartesian... |
| xrnres 39400 | Two ways to express restri... |
| xrnres2 39401 | Two ways to express restri... |
| xrnres3 39402 | Two ways to express restri... |
| xrnres4 39403 | Two ways to express restri... |
| xrnresex 39404 | Sufficient condition for a... |
| xrnidresex 39405 | Sufficient condition for a... |
| xrncnvepresex 39406 | Sufficient condition for a... |
| dmxrncnvepres 39407 | Domain of the range produc... |
| dmxrncnvepres2 39408 | Domain of the range produc... |
| eldmxrncnvepres 39409 | Element of the domain of t... |
| eldmxrncnvepres2 39410 | Element of the domain of t... |
| eceldmqsxrncnvepres 39411 | An ` ( R |X. ( ``' _E |`` ... |
| eceldmqsxrncnvepres2 39412 | An ` ( R |X. ( ``' _E |`` ... |
| brin2 39413 | Binary relation on an inte... |
| brin3 39414 | Binary relation on an inte... |
| elrels2 39416 | The element of the relatio... |
| elrelsrel 39417 | The element of the relatio... |
| elrelsrelim 39418 | The element of the relatio... |
| elrels5 39419 | Equivalent expressions for... |
| elrels6 39420 | Equivalent expressions for... |
| dfqmap2 39422 | Alternate definition of th... |
| dfqmap3 39423 | Alternate definition of th... |
| ecqmap 39424 | ` QMap ` fibers are single... |
| ecqmap2 39425 | Fiber of ` QMap ` equals s... |
| qmapex 39426 | Quotient map exists if ` R... |
| relqmap 39427 | Quotient map is a relation... |
| dmqmap 39428 | ` QMap ` preserves the dom... |
| rnqmap 39429 | The range of the quotient ... |
| dfadjliftmap 39431 | Alternate (expanded) defin... |
| dfadjliftmap2 39432 | Alternate definition of th... |
| blockadjliftmap 39433 | A "two-stage" construction... |
| dfblockliftmap 39435 | Alternate definition of th... |
| dfblockliftmap2 39436 | Alternate definition of th... |
| dfsucmap3 39438 | Alternate definition of th... |
| dfsucmap2 39439 | Alternate definition of th... |
| dfsucmap4 39440 | Alternate definition of th... |
| brsucmap 39441 | Binary relation form of th... |
| relsucmap 39442 | The successor map is a rel... |
| dmsucmap 39443 | The domain of the successo... |
| dfsuccl2 39445 | Alternate definition of th... |
| mopre 39446 | There is at most one prede... |
| exeupre2 39447 | Whenever a predecessor exi... |
| dfsuccl3 39448 | Alternate definition of th... |
| dfsuccl4 39449 | Alternate definition that ... |
| dfpre 39451 | Alternate definition of th... |
| dfpre2 39452 | Alternate definition of th... |
| dfpre3 39453 | Alternate definition of th... |
| dfpred4 39454 | Alternate definition of th... |
| dfpre4 39455 | Alternate definition of th... |
| shiftstableeq2 39458 | Equality theorem for shift... |
| suceqsneq 39459 | One-to-one relationship be... |
| sucdifsn2 39460 | Absorption of union with a... |
| sucdifsn 39461 | The difference between the... |
| ressucdifsn2 39462 | The difference between res... |
| ressucdifsn 39463 | The difference between res... |
| sucmapsuc 39464 | A set is succeeded by its ... |
| sucmapleftuniq 39465 | Left uniqueness of the suc... |
| exeupre 39466 | Whenever a predecessor exi... |
| preex 39467 | The successor-predecessor ... |
| eupre2 39468 | Unique predecessor exists ... |
| eupre 39469 | Unique predecessor exists ... |
| presucmap 39470 | ` pre ` is really a predec... |
| preuniqval 39471 | Uniqueness/canonicity of `... |
| sucpre 39472 | ` suc ` is a right-inverse... |
| presuc 39473 | ` pre ` is a left-inverse ... |
| press 39474 | Predecessor is a subset of... |
| preel 39475 | Predecessor is a subset of... |
| dfcoss2 39478 | Alternate definition of th... |
| dfcoss3 39479 | Alternate definition of th... |
| dfcoss4 39480 | Alternate definition of th... |
| cosscnv 39481 | Class of cosets by the con... |
| coss1cnvres 39482 | Class of cosets by the con... |
| coss2cnvepres 39483 | Special case of ~ coss1cnv... |
| cossex 39484 | If ` A ` is a set then the... |
| cosscnvex 39485 | If ` A ` is a set then the... |
| 1cosscnvepresex 39486 | Sufficient condition for a... |
| 1cossxrncnvepresex 39487 | Sufficient condition for a... |
| relcoss 39488 | Cosets by ` R ` is a relat... |
| relcoels 39489 | Coelements on ` A ` is a r... |
| cossss 39490 | Subclass theorem for the c... |
| cosseq 39491 | Equality theorem for the c... |
| cosseqi 39492 | Equality theorem for the c... |
| cosseqd 39493 | Equality theorem for the c... |
| 1cossres 39494 | The class of cosets by a r... |
| dfcoels 39495 | Alternate definition of th... |
| brcoss 39496 | ` A ` and ` B ` are cosets... |
| brcoss2 39497 | Alternate form of the ` A ... |
| brcoss3 39498 | Alternate form of the ` A ... |
| brcosscnvcoss 39499 | For sets, the ` A ` and ` ... |
| brcoels 39500 | ` B ` and ` C ` are coelem... |
| cocossss 39501 | Two ways of saying that co... |
| cnvcosseq 39502 | The converse of cosets by ... |
| br2coss 39503 | Cosets by ` ,~ R ` binary ... |
| br1cossres 39504 | ` B ` and ` C ` are cosets... |
| br1cossres2 39505 | ` B ` and ` C ` are cosets... |
| brressn 39506 | Binary relation on a restr... |
| ressn2 39507 | A class ' R ' restricted t... |
| refressn 39508 | Any class ' R ' restricted... |
| antisymressn 39509 | Every class ' R ' restrict... |
| trressn 39510 | Any class ' R ' restricted... |
| relbrcoss 39511 | ` A ` and ` B ` are cosets... |
| br1cossinres 39512 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 39513 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 39514 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 39515 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 39516 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 39517 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 39518 | The domain of cosets is th... |
| dmcoss2 39519 | The domain of cosets is th... |
| rncossdmcoss 39520 | The range of cosets is the... |
| dm1cosscnvepres 39521 | The domain of cosets of th... |
| dmcoels 39522 | The domain of coelements i... |
| eldmcoss 39523 | Elementhood in the domain ... |
| eldmcoss2 39524 | Elementhood in the domain ... |
| eldm1cossres 39525 | Elementhood in the domain ... |
| eldm1cossres2 39526 | Elementhood in the domain ... |
| refrelcosslem 39527 | Lemma for the left side of... |
| refrelcoss3 39528 | The class of cosets by ` R... |
| refrelcoss2 39529 | The class of cosets by ` R... |
| symrelcoss3 39530 | The class of cosets by ` R... |
| symrelcoss2 39531 | The class of cosets by ` R... |
| cossssid 39532 | Equivalent expressions for... |
| cossssid2 39533 | Equivalent expressions for... |
| cossssid3 39534 | Equivalent expressions for... |
| cossssid4 39535 | Equivalent expressions for... |
| cossssid5 39536 | Equivalent expressions for... |
| brcosscnv 39537 | ` A ` and ` B ` are cosets... |
| brcosscnv2 39538 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 39539 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 39540 | Cosets by the converse ran... |
| cosscnvssid3 39541 | Equivalent expressions for... |
| cosscnvssid4 39542 | Equivalent expressions for... |
| cosscnvssid5 39543 | Equivalent expressions for... |
| coss0 39544 | Cosets by the empty set ar... |
| cossid 39545 | Cosets by the identity rel... |
| cosscnvid 39546 | Cosets by the converse ide... |
| trcoss 39547 | Sufficient condition for t... |
| eleccossin 39548 | Two ways of saying that th... |
| trcoss2 39549 | Equivalent expressions for... |
| cosselrels 39550 | Cosets of sets are element... |
| cnvelrels 39551 | The converse of a set is a... |
| cosscnvelrels 39552 | Cosets of converse sets ar... |
| dfssr2 39554 | Alternate definition of th... |
| relssr 39555 | The subset relation is a r... |
| brssr 39556 | The subset relation and su... |
| brssrid 39557 | Any set is a subset of its... |
| issetssr 39558 | Two ways of expressing set... |
| brssrres 39559 | Restricted subset binary r... |
| br1cnvssrres 39560 | Restricted converse subset... |
| brcnvssr 39561 | The converse of a subset r... |
| brcnvssrid 39562 | Any set is a converse subs... |
| br1cossxrncnvssrres 39563 | ` <. B , C >. ` and ` <. D... |
| extssr 39564 | Property of subset relatio... |
| dfrefrels2 39568 | Alternate definition of th... |
| dfrefrels3 39569 | Alternate definition of th... |
| dfrefrel2 39570 | Alternate definition of th... |
| dfrefrel3 39571 | Alternate definition of th... |
| dfrefrel5 39572 | Alternate definition of th... |
| elrefrels2 39573 | Element of the class of re... |
| elrefrels3 39574 | Element of the class of re... |
| elrefrelsrel 39575 | For sets, being an element... |
| refreleq 39576 | Equality theorem for refle... |
| refrelid 39577 | Identity relation is refle... |
| refrelcoss 39578 | The class of cosets by ` R... |
| refrelressn 39579 | Any class ' R ' restricted... |
| dfcnvrefrels2 39583 | Alternate definition of th... |
| dfcnvrefrels3 39584 | Alternate definition of th... |
| dfcnvrefrel2 39585 | Alternate definition of th... |
| dfcnvrefrel3 39586 | Alternate definition of th... |
| dfcnvrefrel4 39587 | Alternate definition of th... |
| dfcnvrefrel5 39588 | Alternate definition of th... |
| elcnvrefrels2 39589 | Element of the class of co... |
| elcnvrefrels3 39590 | Element of the class of co... |
| elcnvrefrelsrel 39591 | For sets, being an element... |
| cnvrefrelcoss2 39592 | Necessary and sufficient c... |
| cosselcnvrefrels2 39593 | Necessary and sufficient c... |
| cosselcnvrefrels3 39594 | Necessary and sufficient c... |
| cosselcnvrefrels4 39595 | Necessary and sufficient c... |
| cosselcnvrefrels5 39596 | Necessary and sufficient c... |
| dfsymrels2 39600 | Alternate definition of th... |
| dfsymrels3 39601 | Alternate definition of th... |
| elrelscnveq3 39602 | Two ways of saying a relat... |
| elrelscnveq 39603 | Two ways of saying a relat... |
| elrelscnveq2 39604 | Two ways of saying a relat... |
| elrelscnveq4 39605 | Two ways of saying a relat... |
| dfsymrels4 39606 | Alternate definition of th... |
| dfsymrels5 39607 | Alternate definition of th... |
| dfsymrel2 39608 | Alternate definition of th... |
| dfsymrel3 39609 | Alternate definition of th... |
| dfsymrel4 39610 | Alternate definition of th... |
| dfsymrel5 39611 | Alternate definition of th... |
| elsymrels2 39612 | Element of the class of sy... |
| elsymrels3 39613 | Element of the class of sy... |
| elsymrels4 39614 | Element of the class of sy... |
| elsymrels5 39615 | Element of the class of sy... |
| elsymrelsrel 39616 | For sets, being an element... |
| symreleq 39617 | Equality theorem for symme... |
| symrelim 39618 | Symmetric relation implies... |
| symrelcoss 39619 | The class of cosets by ` R... |
| idsymrel 39620 | The identity relation is s... |
| epnsymrel 39621 | The membership (epsilon) r... |
| symrefref2 39622 | Symmetry is a sufficient c... |
| symrefref3 39623 | Symmetry is a sufficient c... |
| refsymrels2 39624 | Elements of the class of r... |
| refsymrels3 39625 | Elements of the class of r... |
| refsymrel2 39626 | A relation which is reflex... |
| refsymrel3 39627 | A relation which is reflex... |
| elrefsymrels2 39628 | Elements of the class of r... |
| elrefsymrels3 39629 | Elements of the class of r... |
| elrefsymrelsrel 39630 | For sets, being an element... |
| dftrrels2 39634 | Alternate definition of th... |
| dftrrels3 39635 | Alternate definition of th... |
| dftrrel2 39636 | Alternate definition of th... |
| dftrrel3 39637 | Alternate definition of th... |
| eltrrels2 39638 | Element of the class of tr... |
| eltrrels3 39639 | Element of the class of tr... |
| eltrrelsrel 39640 | For sets, being an element... |
| trreleq 39641 | Equality theorem for the t... |
| trrelressn 39642 | Any class ' R ' restricted... |
| dfeqvrels2 39647 | Alternate definition of th... |
| dfeqvrels3 39648 | Alternate definition of th... |
| dfeqvrel2 39649 | Alternate definition of th... |
| dfeqvrel3 39650 | Alternate definition of th... |
| eleqvrels2 39651 | Element of the class of eq... |
| eleqvrels3 39652 | Element of the class of eq... |
| eleqvrelsrel 39653 | For sets, being an element... |
| elcoeleqvrels 39654 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 39655 | For sets, being an element... |
| eqvrelrel 39656 | An equivalence relation is... |
| eqvrelrefrel 39657 | An equivalence relation is... |
| eqvrelsymrel 39658 | An equivalence relation is... |
| eqvreltrrel 39659 | An equivalence relation is... |
| eqvrelim 39660 | Equivalence relation impli... |
| eqvreleq 39661 | Equality theorem for equiv... |
| eqvreleqi 39662 | Equality theorem for equiv... |
| eqvreleqd 39663 | Equality theorem for equiv... |
| eqvrelsym 39664 | An equivalence relation is... |
| eqvrelsymb 39665 | An equivalence relation is... |
| eqvreltr 39666 | An equivalence relation is... |
| eqvreltrd 39667 | A transitivity relation fo... |
| eqvreltr4d 39668 | A transitivity relation fo... |
| eqvrelref 39669 | An equivalence relation is... |
| eqvrelth 39670 | Basic property of equivale... |
| eqvrelcl 39671 | Elementhood in the field o... |
| eqvrelthi 39672 | Basic property of equivale... |
| eqvreldisj 39673 | Equivalence classes do not... |
| qsdisjALTV 39674 | Elements of a quotient set... |
| eqvrelqsel 39675 | If an element of a quotien... |
| eqvrelcoss 39676 | Two ways to express equiva... |
| eqvrelcoss3 39677 | Two ways to express equiva... |
| eqvrelcoss2 39678 | Two ways to express equiva... |
| eqvrelcoss4 39679 | Two ways to express equiva... |
| dfcoeleqvrels 39680 | Alternate definition of th... |
| dfcoeleqvrel 39681 | Alternate definition of th... |
| brredunds 39685 | Binary relation on the cla... |
| brredundsredund 39686 | For sets, binary relation ... |
| redundss3 39687 | Implication of redundancy ... |
| redundeq1 39688 | Equivalence of redundancy ... |
| redundpim3 39689 | Implication of redundancy ... |
| redundpbi1 39690 | Equivalence of redundancy ... |
| refrelsredund4 39691 | The naive version of the c... |
| refrelsredund2 39692 | The naive version of the c... |
| refrelsredund3 39693 | The naive version of the c... |
| refrelredund4 39694 | The naive version of the d... |
| refrelredund2 39695 | The naive version of the d... |
| refrelredund3 39696 | The naive version of the d... |
| dmqseq 39699 | Equality theorem for domai... |
| dmqseqi 39700 | Equality theorem for domai... |
| dmqseqd 39701 | Equality theorem for domai... |
| dmqseqeq1 39702 | Equality theorem for domai... |
| dmqseqeq1i 39703 | Equality theorem for domai... |
| dmqseqeq1d 39704 | Equality theorem for domai... |
| brdmqss 39705 | The domain quotient binary... |
| brdmqssqs 39706 | If ` A ` and ` R ` are set... |
| n0eldmqs 39707 | The empty set is not an el... |
| qseq 39708 | The quotient set equal to ... |
| n0eldmqseq 39709 | The empty set is not an el... |
| n0elim 39710 | Implication of that the em... |
| n0el3 39711 | Two ways of expressing tha... |
| cnvepresdmqss 39712 | The domain quotient binary... |
| cnvepresdmqs 39713 | The domain quotient predic... |
| unidmqs 39714 | The range of a relation is... |
| unidmqseq 39715 | The union of the domain qu... |
| dmqseqim 39716 | If the domain quotient of ... |
| dmqseqim2 39717 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 39718 | Elementhood in the domain ... |
| eldmqs1cossres 39719 | Elementhood in the domain ... |
| releldmqscoss 39720 | Elementhood in the domain ... |
| dmqscoelseq 39721 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 39722 | Two ways to express the eq... |
| brers 39727 | Binary equivalence relatio... |
| dferALTV2 39728 | Equivalence relation with ... |
| erALTVeq1 39729 | Equality theorem for equiv... |
| erALTVeq1i 39730 | Equality theorem for equiv... |
| erALTVeq1d 39731 | Equality theorem for equiv... |
| dfcomember 39732 | Alternate definition of th... |
| dfcomember2 39733 | Alternate definition of th... |
| dfcomember3 39734 | Alternate definition of th... |
| eqvreldmqs 39735 | Two ways to express comemb... |
| eqvreldmqs2 39736 | Two ways to express comemb... |
| brerser 39737 | Binary equivalence relatio... |
| erimeq2 39738 | Equivalence relation on it... |
| erimeq 39739 | Equivalence relation on it... |
| dffunsALTV 39743 | Alternate definition of th... |
| dffunsALTV2 39744 | Alternate definition of th... |
| dffunsALTV3 39745 | Alternate definition of th... |
| dffunsALTV4 39746 | Alternate definition of th... |
| dffunsALTV5 39747 | Alternate definition of th... |
| dffunALTV2 39748 | Alternate definition of th... |
| dffunALTV3 39749 | Alternate definition of th... |
| dffunALTV4 39750 | Alternate definition of th... |
| dffunALTV5 39751 | Alternate definition of th... |
| elfunsALTV 39752 | Elementhood in the class o... |
| elfunsALTV2 39753 | Elementhood in the class o... |
| elfunsALTV3 39754 | Elementhood in the class o... |
| elfunsALTV4 39755 | Elementhood in the class o... |
| elfunsALTV5 39756 | Elementhood in the class o... |
| elfunsALTVfunALTV 39757 | The element of the class o... |
| funALTVfun 39758 | Our definition of the func... |
| funALTVss 39759 | Subclass theorem for funct... |
| funALTVeq 39760 | Equality theorem for funct... |
| funALTVeqi 39761 | Equality inference for the... |
| funALTVeqd 39762 | Equality deduction for the... |
| dfdisjs 39768 | Alternate definition of th... |
| dfdisjs2 39769 | Alternate definition of th... |
| dfdisjs3 39770 | Alternate definition of th... |
| dfdisjs4 39771 | Alternate definition of th... |
| dfdisjs5 39772 | Alternate definition of th... |
| dfdisjALTV 39773 | Alternate definition of th... |
| dfdisjALTV2 39774 | Alternate definition of th... |
| dfdisjALTV3 39775 | Alternate definition of th... |
| dfdisjALTV4 39776 | Alternate definition of th... |
| dfdisjALTV5 39777 | Alternate definition of th... |
| dfdisjALTV5a 39778 | Alternate definition of th... |
| disjimeceqim 39779 | ` Disj ` implies coset-equ... |
| disjimeceqim2 39780 | ` Disj ` implies injectivi... |
| disjimeceqbi 39781 | ` Disj ` gives bicondition... |
| disjimeceqbi2 39782 | Injectivity of the block c... |
| disjimrmoeqec 39783 | Under ` Disj ` , every blo... |
| disjimdmqseq 39784 | Disjointness implies uniqu... |
| dfeldisj2 39785 | Alternate definition of th... |
| dfeldisj3 39786 | Alternate definition of th... |
| dfeldisj4 39787 | Alternate definition of th... |
| dfeldisj5 39788 | Alternate definition of th... |
| dfeldisj5a 39789 | Alternate definition of th... |
| eldisjim3 39790 | ` ElDisj ` elimination (tw... |
| eldisjdmqsim2 39791 | ElDisj of quotient implies... |
| eldisjdmqsim 39792 | Shared output implies equa... |
| suceldisj 39793 | Disjointness of successor ... |
| eldisjs 39794 | Elementhood in the class o... |
| eldisjs2 39795 | Elementhood in the class o... |
| eldisjs3 39796 | Elementhood in the class o... |
| eldisjs4 39797 | Elementhood in the class o... |
| eldisjs5 39798 | Elementhood in the class o... |
| eldisjsdisj 39799 | The element of the class o... |
| qmapeldisjs 39800 | When ` R ` is a set (e.g.,... |
| disjqmap2 39801 | Disjointness of ` QMap ` e... |
| disjqmap 39802 | Disjointness of ` QMap ` e... |
| eleldisjs 39803 | Elementhood in the disjoin... |
| eleldisjseldisj 39804 | The element of the disjoin... |
| disjrel 39805 | Disjoint relation is a rel... |
| disjss 39806 | Subclass theorem for disjo... |
| disjssi 39807 | Subclass theorem for disjo... |
| disjssd 39808 | Subclass theorem for disjo... |
| disjeq 39809 | Equality theorem for disjo... |
| disjeqi 39810 | Equality theorem for disjo... |
| disjeqd 39811 | Equality theorem for disjo... |
| disjdmqseqeq1 39812 | Lemma for the equality the... |
| eldisjss 39813 | Subclass theorem for disjo... |
| eldisjssi 39814 | Subclass theorem for disjo... |
| eldisjssd 39815 | Subclass theorem for disjo... |
| eldisjeq 39816 | Equality theorem for disjo... |
| eldisjeqi 39817 | Equality theorem for disjo... |
| eldisjeqd 39818 | Equality theorem for disjo... |
| disjres 39819 | Disjoint restriction. (Co... |
| eldisjn0elb 39820 | Two forms of disjoint elem... |
| disjxrn 39821 | Two ways of saying that a ... |
| disjxrnres5 39822 | Disjoint range Cartesian p... |
| disjorimxrn 39823 | Disjointness condition for... |
| disjimxrn 39824 | Disjointness condition for... |
| disjimres 39825 | Disjointness condition for... |
| disjimin 39826 | Disjointness condition for... |
| disjiminres 39827 | Disjointness condition for... |
| disjimxrnres 39828 | Disjointness condition for... |
| disjALTV0 39829 | The null class is disjoint... |
| disjALTVid 39830 | The class of identity rela... |
| disjALTVidres 39831 | The class of identity rela... |
| disjALTVinidres 39832 | The intersection with rest... |
| disjALTVxrnidres 39833 | The class of range Cartesi... |
| disjsuc 39834 | Disjoint range Cartesian p... |
| qmapeldisjsim 39835 | Injectivity of coset map f... |
| qmapeldisjsbi 39836 | Injectivity of coset map f... |
| rnqmapeleldisjsim 39837 | Element-disjointness of th... |
| dfantisymrel4 39839 | Alternate definition of th... |
| dfantisymrel5 39840 | Alternate definition of th... |
| antisymrelres 39841 | (Contributed by Peter Mazs... |
| antisymrelressn 39842 | (Contributed by Peter Mazs... |
| dfpart2 39847 | Alternate definition of th... |
| dfmembpart2 39848 | Alternate definition of th... |
| brparts 39849 | Binary partitions relation... |
| brparts2 39850 | Binary partitions relation... |
| brpartspart 39851 | Binary partition and the p... |
| parteq1 39852 | Equality theorem for parti... |
| parteq2 39853 | Equality theorem for parti... |
| parteq12 39854 | Equality theorem for parti... |
| parteq1i 39855 | Equality theorem for parti... |
| parteq1d 39856 | Equality theorem for parti... |
| partsuc2 39857 | Property of the partition.... |
| partsuc 39858 | Property of the partition.... |
| disjim 39859 | The "Divide et Aequivalere... |
| disjimi 39860 | Every disjoint relation ge... |
| detlem 39861 | If a relation is disjoint,... |
| eldisjim 39862 | If the elements of ` A ` a... |
| eldisjim2 39863 | Alternate form of ~ eldisj... |
| eqvrel0 39864 | The null class is an equiv... |
| det0 39865 | The cosets by the null cla... |
| eqvrelcoss0 39866 | The cosets by the null cla... |
| eqvrelid 39867 | The identity relation is a... |
| eqvrel1cossidres 39868 | The cosets by a restricted... |
| eqvrel1cossinidres 39869 | The cosets by an intersect... |
| eqvrel1cossxrnidres 39870 | The cosets by a range Cart... |
| detid 39871 | The cosets by the identity... |
| eqvrelcossid 39872 | The cosets by the identity... |
| detidres 39873 | The cosets by the restrict... |
| detinidres 39874 | The cosets by the intersec... |
| detxrnidres 39875 | The cosets by the range Ca... |
| disjlem14 39876 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 39877 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 39878 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 39879 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 39880 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 39881 | Lemma for ~ disjdmqseq via... |
| disjdmqs 39882 | If a relation is disjoint,... |
| disjdmqseq 39883 | If a relation is disjoint,... |
| eldisjn0el 39884 | Special case of ~ disjdmqs... |
| partim2 39885 | Disjoint relation on its n... |
| partim 39886 | Partition implies equivale... |
| partimeq 39887 | Partition implies that the... |
| eldisjlem19 39888 | Special case of ~ disjlem1... |
| membpartlem19 39889 | Together with ~ disjlem19 ... |
| petlem 39890 | If you can prove that the ... |
| petlemi 39891 | If you can prove disjointn... |
| pet02 39892 | Class ` A ` is a partition... |
| pet0 39893 | Class ` A ` is a partition... |
| petid2 39894 | Class ` A ` is a partition... |
| petid 39895 | A class is a partition by ... |
| petidres2 39896 | Class ` A ` is a partition... |
| petidres 39897 | A class is a partition by ... |
| petinidres2 39898 | Class ` A ` is a partition... |
| petinidres 39899 | A class is a partition by ... |
| petxrnidres2 39900 | Class ` A ` is a partition... |
| petxrnidres 39901 | A class is a partition by ... |
| eqvreldisj1 39902 | The elements of the quotie... |
| eqvreldisj2 39903 | The elements of the quotie... |
| eqvreldisj3 39904 | The elements of the quotie... |
| eqvreldisj4 39905 | Intersection with the conv... |
| eqvreldisj5 39906 | Range Cartesian product wi... |
| eqvrelqseqdisj2 39907 | Implication of ~ eqvreldis... |
| disjimeldisjdmqs 39908 | ` Disj ` implies element-d... |
| eldisjsim1 39909 | An element of the class of... |
| eldisjsim2 39910 | An element of the class of... |
| disjsssrels 39911 | The class of disjoint rela... |
| eldisjsim3 39912 | ` Disjs ` implies element-... |
| eldisjsim4 39913 | ` Disjs ` implies element-... |
| eldisjsim5 39914 | ` Disjs ` is closed under ... |
| eldisjs6 39915 | Elementhood in the class o... |
| eldisjs7 39916 | Elementhood in the class o... |
| dfdisjs6 39917 | Alternate definition of th... |
| dfdisjs7 39918 | Alternate definition of th... |
| fences3 39919 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 39920 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 39921 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 39922 | Lemma for the Partition-Eq... |
| mainer 39923 | The Main Theorem of Equiva... |
| partimcomember 39924 | Partition with general ` R... |
| mpet3 39925 | Member Partition-Equivalen... |
| cpet2 39926 | The conventional form of t... |
| cpet 39927 | The conventional form of M... |
| mpet 39928 | Member Partition-Equivalen... |
| mpet2 39929 | Member Partition-Equivalen... |
| mpets2 39930 | Member Partition-Equivalen... |
| mpets 39931 | Member Partition-Equivalen... |
| mainpart 39932 | Partition with general ` R... |
| fences 39933 | The Theorem of Fences by E... |
| fences2 39934 | The Theorem of Fences by E... |
| mainer2 39935 | The Main Theorem of Equiva... |
| mainerim 39936 | Every equivalence relation... |
| petincnvepres2 39937 | A partition-equivalence th... |
| petincnvepres 39938 | The shortest form of a par... |
| pet2 39939 | Partition-Equivalence Theo... |
| pet 39940 | Partition-Equivalence Theo... |
| pets 39941 | Partition-Equivalence Theo... |
| dmqsblocks 39942 | If the ~ pet span ` ( R |X... |
| dfpetparts2 39947 | Alternate definition of ` ... |
| dfpet2parts2 39948 | Grade stability applied to... |
| dfpeters2 39949 | Alternate definition of ` ... |
| typesafepets 39950 | Type-safe ~ pets scheme. ... |
| petseq 39951 | Generalized partition-equi... |
| pets2eq 39952 | Grade-stable generalized p... |
| prtlem60 39953 | Lemma for ~ prter3 . (Con... |
| bicomdd 39954 | Commute two sides of a bic... |
| jca2r 39955 | Inference conjoining the c... |
| jca3 39956 | Inference conjoining the c... |
| prtlem70 39957 | Lemma for ~ prter3 : a rea... |
| ibdr 39958 | Reverse of ~ ibd . (Contr... |
| prtlem100 39959 | Lemma for ~ prter3 . (Con... |
| prtlem5 39960 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 39961 | Lemma for ~ prter2 . (Con... |
| brabsb2 39962 | A closed form of ~ brabsb ... |
| eqbrrdv2 39963 | Other version of ~ eqbrrdi... |
| prtlem9 39964 | Lemma for ~ prter3 . (Con... |
| prtlem10 39965 | Lemma for ~ prter3 . (Con... |
| prtlem11 39966 | Lemma for ~ prter2 . (Con... |
| prtlem12 39967 | Lemma for ~ prtex and ~ pr... |
| prtlem13 39968 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 39969 | Lemma for ~ prtex , ~ prte... |
| prtlem400 39970 | Lemma for ~ prter2 and als... |
| erprt 39973 | The quotient set of an equ... |
| prtlem14 39974 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 39975 | Lemma for ~ prter1 and ~ p... |
| prtlem17 39976 | Lemma for ~ prter2 . (Con... |
| prtlem18 39977 | Lemma for ~ prter2 . (Con... |
| prtlem19 39978 | Lemma for ~ prter2 . (Con... |
| prter1 39979 | Every partition generates ... |
| prtex 39980 | The equivalence relation g... |
| prter2 39981 | The quotient set of the eq... |
| prter3 39982 | For every partition there ... |
| axc5 39993 | This theorem repeats ~ sp ... |
| ax4fromc4 39994 | Rederivation of Axiom ~ ax... |
| ax10fromc7 39995 | Rederivation of Axiom ~ ax... |
| ax6fromc10 39996 | Rederivation of Axiom ~ ax... |
| hba1-o 39997 | The setvar ` x ` is not fr... |
| axc4i-o 39998 | Inference version of ~ ax-... |
| equid1 39999 | Proof of ~ equid from our ... |
| equcomi1 40000 | Proof of ~ equcomi from ~ ... |
| aecom-o 40001 | Commutation law for identi... |
| aecoms-o 40002 | A commutation rule for ide... |
| hbae-o 40003 | All variables are effectiv... |
| dral1-o 40004 | Formula-building lemma for... |
| ax12fromc15 40005 | Rederivation of Axiom ~ ax... |
| ax13fromc9 40006 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 40007 | Axiom to quantify a variab... |
| sps-o 40008 | Generalization of antecede... |
| hbequid 40009 | Bound-variable hypothesis ... |
| nfequid-o 40010 | Bound-variable hypothesis ... |
| axc5c7 40011 | Proof of a single axiom th... |
| axc5c7toc5 40012 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 40013 | Rederivation of ~ ax-c7 fr... |
| axc711 40014 | Proof of a single axiom th... |
| nfa1-o 40015 | ` x ` is not free in ` A. ... |
| axc711toc7 40016 | Rederivation of ~ ax-c7 fr... |
| axc711to11 40017 | Rederivation of ~ ax-11 fr... |
| axc5c711 40018 | Proof of a single axiom th... |
| axc5c711toc5 40019 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 40020 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 40021 | Rederivation of ~ ax-11 fr... |
| equidqe 40022 | ~ equid with existential q... |
| axc5sp1 40023 | A special case of ~ ax-c5 ... |
| equidq 40024 | ~ equid with universal qua... |
| equid1ALT 40025 | Alternate proof of ~ equid... |
| axc11nfromc11 40026 | Rederivation of ~ ax-c11n ... |
| naecoms-o 40027 | A commutation rule for dis... |
| hbnae-o 40028 | All variables are effectiv... |
| dvelimf-o 40029 | Proof of ~ dvelimh that us... |
| dral2-o 40030 | Formula-building lemma for... |
| aev-o 40031 | A "distinctor elimination"... |
| ax5eq 40032 | Theorem to add distinct qu... |
| dveeq2-o 40033 | Quantifier introduction wh... |
| axc16g-o 40034 | A generalization of Axiom ... |
| dveeq1-o 40035 | Quantifier introduction wh... |
| dveeq1-o16 40036 | Version of ~ dveeq1 using ... |
| ax5el 40037 | Theorem to add distinct qu... |
| axc11n-16 40038 | This theorem shows that, g... |
| dveel2ALT 40039 | Alternate proof of ~ dveel... |
| ax12f 40040 | Basis step for constructin... |
| ax12eq 40041 | Basis step for constructin... |
| ax12el 40042 | Basis step for constructin... |
| ax12indn 40043 | Induction step for constru... |
| ax12indi 40044 | Induction step for constru... |
| ax12indalem 40045 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 40046 | Alternate proof of ~ ax12i... |
| ax12inda2 40047 | Induction step for constru... |
| ax12inda 40048 | Induction step for constru... |
| ax12v2-o 40049 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 40050 | Derive ~ ax-c15 from a hyp... |
| axc11-o 40051 | Show that ~ ax-c11 can be ... |
| fsumshftd 40052 | Index shift of a finite su... |
| riotaclbgBAD 40054 | Closure of restricted iota... |
| riotaclbBAD 40055 | Closure of restricted iota... |
| riotasvd 40056 | Deduction version of ~ rio... |
| riotasv2d 40057 | Value of description binde... |
| riotasv2s 40058 | The value of description b... |
| riotasv 40059 | Value of description binde... |
| riotasv3d 40060 | A property ` ch ` holding ... |
| elimhyps 40061 | A version of ~ elimhyp usi... |
| dedths 40062 | A version of weak deductio... |
| renegclALT 40063 | Closure law for negative o... |
| elimhyps2 40064 | Generalization of ~ elimhy... |
| dedths2 40065 | Generalization of ~ dedths... |
| nfcxfrdf 40066 | A utility lemma to transfe... |
| nfded 40067 | A deduction theorem that c... |
| nfded2 40068 | A deduction theorem that c... |
| nfunidALT2 40069 | Deduction version of ~ nfu... |
| nfunidALT 40070 | Deduction version of ~ nfu... |
| nfopdALT 40071 | Deduction version of bound... |
| cnaddcom 40072 | Recover the commutative la... |
| toycom 40073 | Show the commutative law f... |
| lshpset 40078 | The set of all hyperplanes... |
| islshp 40079 | The predicate "is a hyperp... |
| islshpsm 40080 | Hyperplane properties expr... |
| lshplss 40081 | A hyperplane is a subspace... |
| lshpne 40082 | A hyperplane is not equal ... |
| lshpnel 40083 | A hyperplane's generating ... |
| lshpnelb 40084 | The subspace sum of a hype... |
| lshpnel2N 40085 | Condition that determines ... |
| lshpne0 40086 | The member of the span in ... |
| lshpdisj 40087 | A hyperplane and the span ... |
| lshpcmp 40088 | If two hyperplanes are com... |
| lshpinN 40089 | The intersection of two di... |
| lsatset 40090 | The set of all 1-dim subsp... |
| islsat 40091 | The predicate "is a 1-dim ... |
| lsatlspsn2 40092 | The span of a nonzero sing... |
| lsatlspsn 40093 | The span of a nonzero sing... |
| islsati 40094 | A 1-dim subspace (atom) (o... |
| lsateln0 40095 | A 1-dim subspace (atom) (o... |
| lsatlss 40096 | The set of 1-dim subspaces... |
| lsatlssel 40097 | An atom is a subspace. (C... |
| lsatssv 40098 | An atom is a set of vector... |
| lsatn0 40099 | A 1-dim subspace (atom) of... |
| lsatspn0 40100 | The span of a vector is an... |
| lsator0sp 40101 | The span of a vector is ei... |
| lsatssn0 40102 | A subspace (or any class) ... |
| lsatcmp 40103 | If two atoms are comparabl... |
| lsatcmp2 40104 | If an atom is included in ... |
| lsatel 40105 | A nonzero vector in an ato... |
| lsatelbN 40106 | A nonzero vector in an ato... |
| lsat2el 40107 | Two atoms sharing a nonzer... |
| lsmsat 40108 | Convert comparison of atom... |
| lsatfixedN 40109 | Show equality with the spa... |
| lsmsatcv 40110 | Subspace sum has the cover... |
| lssatomic 40111 | The lattice of subspaces i... |
| lssats 40112 | The lattice of subspaces i... |
| lpssat 40113 | Two subspaces in a proper ... |
| lrelat 40114 | Subspaces are relatively a... |
| lssatle 40115 | The ordering of two subspa... |
| lssat 40116 | Two subspaces in a proper ... |
| islshpat 40117 | Hyperplane properties expr... |
| lcvfbr 40120 | The covers relation for a ... |
| lcvbr 40121 | The covers relation for a ... |
| lcvbr2 40122 | The covers relation for a ... |
| lcvbr3 40123 | The covers relation for a ... |
| lcvpss 40124 | The covers relation implie... |
| lcvnbtwn 40125 | The covers relation implie... |
| lcvntr 40126 | The covers relation is not... |
| lcvnbtwn2 40127 | The covers relation implie... |
| lcvnbtwn3 40128 | The covers relation implie... |
| lsmcv2 40129 | Subspace sum has the cover... |
| lcvat 40130 | If a subspace covers anoth... |
| lsatcv0 40131 | An atom covers the zero su... |
| lsatcveq0 40132 | A subspace covered by an a... |
| lsat0cv 40133 | A subspace is an atom iff ... |
| lcvexchlem1 40134 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 40135 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 40136 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 40137 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 40138 | Lemma for ~ lcvexch . (Co... |
| lcvexch 40139 | Subspaces satisfy the exch... |
| lcvp 40140 | Covering property of Defin... |
| lcv1 40141 | Covering property of a sub... |
| lcv2 40142 | Covering property of a sub... |
| lsatexch 40143 | The atom exchange property... |
| lsatnle 40144 | The meet of a subspace and... |
| lsatnem0 40145 | The meet of distinct atoms... |
| lsatexch1 40146 | The atom exch1ange propert... |
| lsatcv0eq 40147 | If the sum of two atoms co... |
| lsatcv1 40148 | Two atoms covering the zer... |
| lsatcvatlem 40149 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 40150 | A nonzero subspace less th... |
| lsatcvat2 40151 | A subspace covered by the ... |
| lsatcvat3 40152 | A condition implying that ... |
| islshpcv 40153 | Hyperplane properties expr... |
| l1cvpat 40154 | A subspace covered by the ... |
| l1cvat 40155 | Create an atom under an el... |
| lshpat 40156 | Create an atom under a hyp... |
| lflset 40159 | The set of linear function... |
| islfl 40160 | The predicate "is a linear... |
| lfli 40161 | Property of a linear funct... |
| islfld 40162 | Properties that determine ... |
| lflf 40163 | A linear functional is a f... |
| lflcl 40164 | A linear functional value ... |
| lfl0 40165 | A linear functional is zer... |
| lfladd 40166 | Property of a linear funct... |
| lflsub 40167 | Property of a linear funct... |
| lflmul 40168 | Property of a linear funct... |
| lfl0f 40169 | The zero function is a fun... |
| lfl1 40170 | A nonzero functional has a... |
| lfladdcl 40171 | Closure of addition of two... |
| lfladdcom 40172 | Commutativity of functiona... |
| lfladdass 40173 | Associativity of functiona... |
| lfladd0l 40174 | Functional addition with t... |
| lflnegcl 40175 | Closure of the negative of... |
| lflnegl 40176 | A functional plus its nega... |
| lflvscl 40177 | Closure of a scalar produc... |
| lflvsdi1 40178 | Distributive law for (righ... |
| lflvsdi2 40179 | Reverse distributive law f... |
| lflvsdi2a 40180 | Reverse distributive law f... |
| lflvsass 40181 | Associative law for (right... |
| lfl0sc 40182 | The (right vector space) s... |
| lflsc0N 40183 | The scalar product with th... |
| lfl1sc 40184 | The (right vector space) s... |
| lkrfval 40187 | The kernel of a functional... |
| lkrval 40188 | Value of the kernel of a f... |
| ellkr 40189 | Membership in the kernel o... |
| lkrval2 40190 | Value of the kernel of a f... |
| ellkr2 40191 | Membership in the kernel o... |
| lkrcl 40192 | A member of the kernel of ... |
| lkrf0 40193 | The value of a functional ... |
| lkr0f 40194 | The kernel of the zero fun... |
| lkrlss 40195 | The kernel of a linear fun... |
| lkrssv 40196 | The kernel of a linear fun... |
| lkrsc 40197 | The kernel of a nonzero sc... |
| lkrscss 40198 | The kernel of a scalar pro... |
| eqlkr 40199 | Two functionals with the s... |
| eqlkr2 40200 | Two functionals with the s... |
| eqlkr3 40201 | Two functionals with the s... |
| lkrlsp 40202 | The subspace sum of a kern... |
| lkrlsp2 40203 | The subspace sum of a kern... |
| lkrlsp3 40204 | The subspace sum of a kern... |
| lkrshp 40205 | The kernel of a nonzero fu... |
| lkrshp3 40206 | The kernels of nonzero fun... |
| lkrshpor 40207 | The kernel of a functional... |
| lkrshp4 40208 | A kernel is a hyperplane i... |
| lshpsmreu 40209 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 40210 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 40211 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 40212 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 40213 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 40214 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 40215 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 40216 | The set ` G ` defined by h... |
| lshpkr 40217 | The kernel of functional `... |
| lshpkrex 40218 | There exists a functional ... |
| lshpset2N 40219 | The set of all hyperplanes... |
| islshpkrN 40220 | The predicate "is a hyperp... |
| lfl1dim 40221 | Equivalent expressions for... |
| lfl1dim2N 40222 | Equivalent expressions for... |
| ldualset 40225 | Define the (left) dual of ... |
| ldualvbase 40226 | The vectors of a dual spac... |
| ldualelvbase 40227 | Utility theorem for conver... |
| ldualfvadd 40228 | Vector addition in the dua... |
| ldualvadd 40229 | Vector addition in the dua... |
| ldualvaddcl 40230 | The value of vector additi... |
| ldualvaddval 40231 | The value of the value of ... |
| ldualsca 40232 | The ring of scalars of the... |
| ldualsbase 40233 | Base set of scalar ring fo... |
| ldualsaddN 40234 | Scalar addition for the du... |
| ldualsmul 40235 | Scalar multiplication for ... |
| ldualfvs 40236 | Scalar product operation f... |
| ldualvs 40237 | Scalar product operation v... |
| ldualvsval 40238 | Value of scalar product op... |
| ldualvscl 40239 | The scalar product operati... |
| ldualvaddcom 40240 | Commutative law for vector... |
| ldualvsass 40241 | Associative law for scalar... |
| ldualvsass2 40242 | Associative law for scalar... |
| ldualvsdi1 40243 | Distributive law for scala... |
| ldualvsdi2 40244 | Reverse distributive law f... |
| ldualgrplem 40245 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 40246 | The dual of a vector space... |
| ldual0 40247 | The zero scalar of the dua... |
| ldual1 40248 | The unit scalar of the dua... |
| ldualneg 40249 | The negative of a scalar o... |
| ldual0v 40250 | The zero vector of the dua... |
| ldual0vcl 40251 | The dual zero vector is a ... |
| lduallmodlem 40252 | Lemma for ~ lduallmod . (... |
| lduallmod 40253 | The dual of a left module ... |
| lduallvec 40254 | The dual of a left vector ... |
| ldualvsub 40255 | The value of vector subtra... |
| ldualvsubcl 40256 | Closure of vector subtract... |
| ldualvsubval 40257 | The value of the value of ... |
| ldualssvscl 40258 | Closure of scalar product ... |
| ldualssvsubcl 40259 | Closure of vector subtract... |
| ldual0vs 40260 | Scalar zero times a functi... |
| lkr0f2 40261 | The kernel of the zero fun... |
| lduallkr3 40262 | The kernels of nonzero fun... |
| lkrpssN 40263 | Proper subset relation bet... |
| lkrin 40264 | Intersection of the kernel... |
| eqlkr4 40265 | Two functionals with the s... |
| ldual1dim 40266 | Equivalent expressions for... |
| ldualkrsc 40267 | The kernel of a nonzero sc... |
| lkrss 40268 | The kernel of a scalar pro... |
| lkrss2N 40269 | Two functionals with kerne... |
| lkreqN 40270 | Proportional functionals h... |
| lkrlspeqN 40271 | Condition for colinear fun... |
| isopos 40280 | The predicate "is an ortho... |
| opposet 40281 | Every orthoposet is a pose... |
| oposlem 40282 | Lemma for orthoposet prope... |
| op01dm 40283 | Conditions necessary for z... |
| op0cl 40284 | An orthoposet has a zero e... |
| op1cl 40285 | An orthoposet has a unity ... |
| op0le 40286 | Orthoposet zero is less th... |
| ople0 40287 | An element less than or eq... |
| opnlen0 40288 | An element not less than a... |
| lub0N 40289 | The least upper bound of t... |
| opltn0 40290 | A lattice element greater ... |
| ople1 40291 | Any element is less than t... |
| op1le 40292 | If the orthoposet unity is... |
| glb0N 40293 | The greatest lower bound o... |
| opoccl 40294 | Closure of orthocomplement... |
| opococ 40295 | Double negative law for or... |
| opcon3b 40296 | Contraposition law for ort... |
| opcon2b 40297 | Orthocomplement contraposi... |
| opcon1b 40298 | Orthocomplement contraposi... |
| oplecon3 40299 | Contraposition law for ort... |
| oplecon3b 40300 | Contraposition law for ort... |
| oplecon1b 40301 | Contraposition law for str... |
| opoc1 40302 | Orthocomplement of orthopo... |
| opoc0 40303 | Orthocomplement of orthopo... |
| opltcon3b 40304 | Contraposition law for str... |
| opltcon1b 40305 | Contraposition law for str... |
| opltcon2b 40306 | Contraposition law for str... |
| opexmid 40307 | Law of excluded middle for... |
| opnoncon 40308 | Law of contradiction for o... |
| riotaocN 40309 | The orthocomplement of the... |
| cmtfvalN 40310 | Value of commutes relation... |
| cmtvalN 40311 | Equivalence for commutes r... |
| isolat 40312 | The predicate "is an ortho... |
| ollat 40313 | An ortholattice is a latti... |
| olop 40314 | An ortholattice is an orth... |
| olposN 40315 | An ortholattice is a poset... |
| isolatiN 40316 | Properties that determine ... |
| oldmm1 40317 | De Morgan's law for meet i... |
| oldmm2 40318 | De Morgan's law for meet i... |
| oldmm3N 40319 | De Morgan's law for meet i... |
| oldmm4 40320 | De Morgan's law for meet i... |
| oldmj1 40321 | De Morgan's law for join i... |
| oldmj2 40322 | De Morgan's law for join i... |
| oldmj3 40323 | De Morgan's law for join i... |
| oldmj4 40324 | De Morgan's law for join i... |
| olj01 40325 | An ortholattice element jo... |
| olj02 40326 | An ortholattice element jo... |
| olm11 40327 | The meet of an ortholattic... |
| olm12 40328 | The meet of an ortholattic... |
| latmassOLD 40329 | Ortholattice meet is assoc... |
| latm12 40330 | A rearrangement of lattice... |
| latm32 40331 | A rearrangement of lattice... |
| latmrot 40332 | Rotate lattice meet of 3 c... |
| latm4 40333 | Rearrangement of lattice m... |
| latmmdiN 40334 | Lattice meet distributes o... |
| latmmdir 40335 | Lattice meet distributes o... |
| olm01 40336 | Meet with lattice zero is ... |
| olm02 40337 | Meet with lattice zero is ... |
| isoml 40338 | The predicate "is an ortho... |
| isomliN 40339 | Properties that determine ... |
| omlol 40340 | An orthomodular lattice is... |
| omlop 40341 | An orthomodular lattice is... |
| omllat 40342 | An orthomodular lattice is... |
| omllaw 40343 | The orthomodular law. (Co... |
| omllaw2N 40344 | Variation of orthomodular ... |
| omllaw3 40345 | Orthomodular law equivalen... |
| omllaw4 40346 | Orthomodular law equivalen... |
| omllaw5N 40347 | The orthomodular law. Rem... |
| cmtcomlemN 40348 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 40349 | Commutation is symmetric. ... |
| cmt2N 40350 | Commutation with orthocomp... |
| cmt3N 40351 | Commutation with orthocomp... |
| cmt4N 40352 | Commutation with orthocomp... |
| cmtbr2N 40353 | Alternate definition of th... |
| cmtbr3N 40354 | Alternate definition for t... |
| cmtbr4N 40355 | Alternate definition for t... |
| lecmtN 40356 | Ordered elements commute. ... |
| cmtidN 40357 | Any element commutes with ... |
| omlfh1N 40358 | Foulis-Holland Theorem, pa... |
| omlfh3N 40359 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 40360 | Analogue of modular law ~ ... |
| omlspjN 40361 | Contraction of a Sasaki pr... |
| cvrfval 40368 | Value of covers relation "... |
| cvrval 40369 | Binary relation expressing... |
| cvrlt 40370 | The covers relation implie... |
| cvrnbtwn 40371 | There is no element betwee... |
| ncvr1 40372 | No element covers the latt... |
| cvrletrN 40373 | Property of an element abo... |
| cvrval2 40374 | Binary relation expressing... |
| cvrnbtwn2 40375 | The covers relation implie... |
| cvrnbtwn3 40376 | The covers relation implie... |
| cvrcon3b 40377 | Contraposition law for the... |
| cvrle 40378 | The covers relation implie... |
| cvrnbtwn4 40379 | The covers relation implie... |
| cvrnle 40380 | The covers relation implie... |
| cvrne 40381 | The covers relation implie... |
| cvrnrefN 40382 | The covers relation is not... |
| cvrcmp 40383 | If two lattice elements th... |
| cvrcmp2 40384 | If two lattice elements co... |
| pats 40385 | The set of atoms in a pose... |
| isat 40386 | The predicate "is an atom"... |
| isat2 40387 | The predicate "is an atom"... |
| atcvr0 40388 | An atom covers zero. ( ~ ... |
| atbase 40389 | An atom is a member of the... |
| atssbase 40390 | The set of atoms is a subs... |
| 0ltat 40391 | An atom is greater than ze... |
| leatb 40392 | A poset element less than ... |
| leat 40393 | A poset element less than ... |
| leat2 40394 | A nonzero poset element le... |
| leat3 40395 | A poset element less than ... |
| meetat 40396 | The meet of any element wi... |
| meetat2 40397 | The meet of any element wi... |
| isatl 40399 | The predicate "is an atomi... |
| atllat 40400 | An atomic lattice is a lat... |
| atlpos 40401 | An atomic lattice is a pos... |
| atl0dm 40402 | Condition necessary for ze... |
| atl0cl 40403 | An atomic lattice has a ze... |
| atl0le 40404 | Orthoposet zero is less th... |
| atlle0 40405 | An element less than or eq... |
| atlltn0 40406 | A lattice element greater ... |
| isat3 40407 | The predicate "is an atom"... |
| atn0 40408 | An atom is not zero. ( ~ ... |
| atnle0 40409 | An atom is not less than o... |
| atlen0 40410 | A lattice element is nonze... |
| atcmp 40411 | If two atoms are comparabl... |
| atncmp 40412 | Frequently-used variation ... |
| atnlt 40413 | Two atoms cannot satisfy t... |
| atcvreq0 40414 | An element covered by an a... |
| atncvrN 40415 | Two atoms cannot satisfy t... |
| atlex 40416 | Every nonzero element of a... |
| atnle 40417 | Two ways of expressing "an... |
| atnem0 40418 | The meet of distinct atoms... |
| atlatmstc 40419 | An atomic, complete, ortho... |
| atlatle 40420 | The ordering of two Hilber... |
| atlrelat1 40421 | An atomistic lattice with ... |
| iscvlat 40423 | The predicate "is an atomi... |
| iscvlat2N 40424 | The predicate "is an atomi... |
| cvlatl 40425 | An atomic lattice with the... |
| cvllat 40426 | An atomic lattice with the... |
| cvlposN 40427 | An atomic lattice with the... |
| cvlexch1 40428 | An atomic covering lattice... |
| cvlexch2 40429 | An atomic covering lattice... |
| cvlexchb1 40430 | An atomic covering lattice... |
| cvlexchb2 40431 | An atomic covering lattice... |
| cvlexch3 40432 | An atomic covering lattice... |
| cvlexch4N 40433 | An atomic covering lattice... |
| cvlatexchb1 40434 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 40435 | A version of ~ cvlexchb2 f... |
| cvlatexch1 40436 | Atom exchange property. (... |
| cvlatexch2 40437 | Atom exchange property. (... |
| cvlatexch3 40438 | Atom exchange property. (... |
| cvlcvr1 40439 | The covering property. Pr... |
| cvlcvrp 40440 | A Hilbert lattice satisfie... |
| cvlatcvr1 40441 | An atom is covered by its ... |
| cvlatcvr2 40442 | An atom is covered by its ... |
| cvlsupr2 40443 | Two equivalent ways of exp... |
| cvlsupr3 40444 | Two equivalent ways of exp... |
| cvlsupr4 40445 | Consequence of superpositi... |
| cvlsupr5 40446 | Consequence of superpositi... |
| cvlsupr6 40447 | Consequence of superpositi... |
| cvlsupr7 40448 | Consequence of superpositi... |
| cvlsupr8 40449 | Consequence of superpositi... |
| ishlat1 40452 | The predicate "is a Hilber... |
| ishlat2 40453 | The predicate "is a Hilber... |
| ishlat3N 40454 | The predicate "is a Hilber... |
| ishlatiN 40455 | Properties that determine ... |
| hlomcmcv 40456 | A Hilbert lattice is ortho... |
| hloml 40457 | A Hilbert lattice is ortho... |
| hlclat 40458 | A Hilbert lattice is compl... |
| hlcvl 40459 | A Hilbert lattice is an at... |
| hlatl 40460 | A Hilbert lattice is atomi... |
| hlol 40461 | A Hilbert lattice is an or... |
| hlop 40462 | A Hilbert lattice is an or... |
| hllat 40463 | A Hilbert lattice is a lat... |
| hllatd 40464 | Deduction form of ~ hllat ... |
| hlomcmat 40465 | A Hilbert lattice is ortho... |
| hlpos 40466 | A Hilbert lattice is a pos... |
| hlatjcl 40467 | Closure of join operation.... |
| hlatjcom 40468 | Commutatitivity of join op... |
| hlatjidm 40469 | Idempotence of join operat... |
| hlatjass 40470 | Lattice join is associativ... |
| hlatj12 40471 | Swap 1st and 2nd members o... |
| hlatj32 40472 | Swap 2nd and 3rd members o... |
| hlatjrot 40473 | Rotate lattice join of 3 c... |
| hlatj4 40474 | Rearrangement of lattice j... |
| hlatlej1 40475 | A join's first argument is... |
| hlatlej2 40476 | A join's second argument i... |
| glbconN 40477 | De Morgan's law for GLB an... |
| glbconxN 40478 | De Morgan's law for GLB an... |
| atnlej1 40479 | If an atom is not less tha... |
| atnlej2 40480 | If an atom is not less tha... |
| hlsuprexch 40481 | A Hilbert lattice has the ... |
| hlexch1 40482 | A Hilbert lattice has the ... |
| hlexch2 40483 | A Hilbert lattice has the ... |
| hlexchb1 40484 | A Hilbert lattice has the ... |
| hlexchb2 40485 | A Hilbert lattice has the ... |
| hlsupr 40486 | A Hilbert lattice has the ... |
| hlsupr2 40487 | A Hilbert lattice has the ... |
| hlhgt4 40488 | A Hilbert lattice has a he... |
| hlhgt2 40489 | A Hilbert lattice has a he... |
| hl0lt1N 40490 | Lattice 0 is less than lat... |
| hlexch3 40491 | A Hilbert lattice has the ... |
| hlexch4N 40492 | A Hilbert lattice has the ... |
| hlatexchb1 40493 | A version of ~ hlexchb1 fo... |
| hlatexchb2 40494 | A version of ~ hlexchb2 fo... |
| hlatexch1 40495 | Atom exchange property. (... |
| hlatexch2 40496 | Atom exchange property. (... |
| hlatmstcOLDN 40497 | An atomic, complete, ortho... |
| hlatle 40498 | The ordering of two Hilber... |
| hlateq 40499 | The equality of two Hilber... |
| hlrelat1 40500 | An atomistic lattice with ... |
| hlrelat5N 40501 | An atomistic lattice with ... |
| hlrelat 40502 | A Hilbert lattice is relat... |
| hlrelat2 40503 | A consequence of relative ... |
| exatleN 40504 | A condition for an atom to... |
| hl2at 40505 | A Hilbert lattice has at l... |
| atex 40506 | At least one atom exists. ... |
| intnatN 40507 | If the intersection with a... |
| 2llnne2N 40508 | Condition implying that tw... |
| 2llnneN 40509 | Condition implying that tw... |
| cvr1 40510 | A Hilbert lattice has the ... |
| cvr2N 40511 | Less-than and covers equiv... |
| hlrelat3 40512 | The Hilbert lattice is rel... |
| cvrval3 40513 | Binary relation expressing... |
| cvrval4N 40514 | Binary relation expressing... |
| cvrval5 40515 | Binary relation expressing... |
| cvrp 40516 | A Hilbert lattice satisfie... |
| atcvr1 40517 | An atom is covered by its ... |
| atcvr2 40518 | An atom is covered by its ... |
| cvrexchlem 40519 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 40520 | A Hilbert lattice satisfie... |
| cvratlem 40521 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 40522 | A nonzero Hilbert lattice ... |
| ltltncvr 40523 | A chained strong ordering ... |
| ltcvrntr 40524 | Non-transitive condition f... |
| cvrntr 40525 | The covers relation is not... |
| atcvr0eq 40526 | The covers relation is not... |
| lnnat 40527 | A line (the join of two di... |
| atcvrj0 40528 | Two atoms covering the zer... |
| cvrat2 40529 | A Hilbert lattice element ... |
| atcvrneN 40530 | Inequality derived from at... |
| atcvrj1 40531 | Condition for an atom to b... |
| atcvrj2b 40532 | Condition for an atom to b... |
| atcvrj2 40533 | Condition for an atom to b... |
| atleneN 40534 | Inequality derived from at... |
| atltcvr 40535 | An equivalence of less-tha... |
| atle 40536 | Any nonzero element has an... |
| atlt 40537 | Two atoms are unequal iff ... |
| atlelt 40538 | Transfer less-than relatio... |
| 2atlt 40539 | Given an atom less than an... |
| atexchcvrN 40540 | Atom exchange property. V... |
| atexchltN 40541 | Atom exchange property. V... |
| cvrat3 40542 | A condition implying that ... |
| cvrat4 40543 | A condition implying exist... |
| cvrat42 40544 | Commuted version of ~ cvra... |
| 2atjm 40545 | The meet of a line (expres... |
| atbtwn 40546 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 40547 | There exists a 3rd atom ` ... |
| atbtwnex 40548 | Given atoms ` P ` in ` X `... |
| 3noncolr2 40549 | Two ways to express 3 non-... |
| 3noncolr1N 40550 | Two ways to express 3 non-... |
| hlatcon3 40551 | Atom exchange combined wit... |
| hlatcon2 40552 | Atom exchange combined wit... |
| 4noncolr3 40553 | A way to express 4 non-col... |
| 4noncolr2 40554 | A way to express 4 non-col... |
| 4noncolr1 40555 | A way to express 4 non-col... |
| athgt 40556 | A Hilbert lattice, whose h... |
| 3dim0 40557 | There exists a 3-dimension... |
| 3dimlem1 40558 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 40559 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 40560 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 40561 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 40562 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 40563 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 40564 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 40565 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 40566 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 40567 | Construct a 3-dimensional ... |
| 3dim2 40568 | Construct 2 new layers on ... |
| 3dim3 40569 | Construct a new layer on t... |
| 2dim 40570 | Generate a height-3 elemen... |
| 1dimN 40571 | An atom is covered by a he... |
| 1cvrco 40572 | The orthocomplement of an ... |
| 1cvratex 40573 | There exists an atom less ... |
| 1cvratlt 40574 | An atom less than or equal... |
| 1cvrjat 40575 | An element covered by the ... |
| 1cvrat 40576 | Create an atom under an el... |
| ps-1 40577 | The join of two atoms ` R ... |
| ps-2 40578 | Lattice analogue for the p... |
| 2atjlej 40579 | Two atoms are different if... |
| hlatexch3N 40580 | Rearrange join of atoms in... |
| hlatexch4 40581 | Exchange 2 atoms. (Contri... |
| ps-2b 40582 | Variation of projective ge... |
| 3atlem1 40583 | Lemma for ~ 3at . (Contri... |
| 3atlem2 40584 | Lemma for ~ 3at . (Contri... |
| 3atlem3 40585 | Lemma for ~ 3at . (Contri... |
| 3atlem4 40586 | Lemma for ~ 3at . (Contri... |
| 3atlem5 40587 | Lemma for ~ 3at . (Contri... |
| 3atlem6 40588 | Lemma for ~ 3at . (Contri... |
| 3atlem7 40589 | Lemma for ~ 3at . (Contri... |
| 3at 40590 | Any three non-colinear ato... |
| llnset 40605 | The set of lattice lines i... |
| islln 40606 | The predicate "is a lattic... |
| islln4 40607 | The predicate "is a lattic... |
| llni 40608 | Condition implying a latti... |
| llnbase 40609 | A lattice line is a lattic... |
| islln3 40610 | The predicate "is a lattic... |
| islln2 40611 | The predicate "is a lattic... |
| llni2 40612 | The join of two different ... |
| llnnleat 40613 | An atom cannot majorize a ... |
| llnneat 40614 | A lattice line is not an a... |
| 2atneat 40615 | The join of two distinct a... |
| llnn0 40616 | A lattice line is nonzero.... |
| islln2a 40617 | The predicate "is a lattic... |
| llnle 40618 | Any element greater than 0... |
| atcvrlln2 40619 | An atom under a line is co... |
| atcvrlln 40620 | An element covering an ato... |
| llnexatN 40621 | Given an atom on a line, t... |
| llncmp 40622 | If two lattice lines are c... |
| llnnlt 40623 | Two lattice lines cannot s... |
| 2llnmat 40624 | Two intersecting lines int... |
| 2at0mat0 40625 | Special case of ~ 2atmat0 ... |
| 2atmat0 40626 | The meet of two unequal li... |
| 2atm 40627 | An atom majorized by two d... |
| ps-2c 40628 | Variation of projective ge... |
| lplnset 40629 | The set of lattice planes ... |
| islpln 40630 | The predicate "is a lattic... |
| islpln4 40631 | The predicate "is a lattic... |
| lplni 40632 | Condition implying a latti... |
| islpln3 40633 | The predicate "is a lattic... |
| lplnbase 40634 | A lattice plane is a latti... |
| islpln5 40635 | The predicate "is a lattic... |
| islpln2 40636 | The predicate "is a lattic... |
| lplni2 40637 | The join of 3 different at... |
| lvolex3N 40638 | There is an atom outside o... |
| llnmlplnN 40639 | The intersection of a line... |
| lplnle 40640 | Any element greater than 0... |
| lplnnle2at 40641 | A lattice line (or atom) c... |
| lplnnleat 40642 | A lattice plane cannot maj... |
| lplnnlelln 40643 | A lattice plane is not les... |
| 2atnelpln 40644 | The join of two atoms is n... |
| lplnneat 40645 | No lattice plane is an ato... |
| lplnnelln 40646 | No lattice plane is a latt... |
| lplnn0N 40647 | A lattice plane is nonzero... |
| islpln2a 40648 | The predicate "is a lattic... |
| islpln2ah 40649 | The predicate "is a lattic... |
| lplnriaN 40650 | Property of a lattice plan... |
| lplnribN 40651 | Property of a lattice plan... |
| lplnric 40652 | Property of a lattice plan... |
| lplnri1 40653 | Property of a lattice plan... |
| lplnri2N 40654 | Property of a lattice plan... |
| lplnri3N 40655 | Property of a lattice plan... |
| lplnllnneN 40656 | Two lattice lines defined ... |
| llncvrlpln2 40657 | A lattice line under a lat... |
| llncvrlpln 40658 | An element covering a latt... |
| 2lplnmN 40659 | If the join of two lattice... |
| 2llnmj 40660 | The meet of two lattice li... |
| 2atmat 40661 | The meet of two intersecti... |
| lplncmp 40662 | If two lattice planes are ... |
| lplnexatN 40663 | Given a lattice line on a ... |
| lplnexllnN 40664 | Given an atom on a lattice... |
| lplnnlt 40665 | Two lattice planes cannot ... |
| 2llnjaN 40666 | The join of two different ... |
| 2llnjN 40667 | The join of two different ... |
| 2llnm2N 40668 | The meet of two different ... |
| 2llnm3N 40669 | Two lattice lines in a lat... |
| 2llnm4 40670 | Two lattice lines that maj... |
| 2llnmeqat 40671 | An atom equals the interse... |
| lvolset 40672 | The set of 3-dim lattice v... |
| islvol 40673 | The predicate "is a 3-dim ... |
| islvol4 40674 | The predicate "is a 3-dim ... |
| lvoli 40675 | Condition implying a 3-dim... |
| islvol3 40676 | The predicate "is a 3-dim ... |
| lvoli3 40677 | Condition implying a 3-dim... |
| lvolbase 40678 | A 3-dim lattice volume is ... |
| islvol5 40679 | The predicate "is a 3-dim ... |
| islvol2 40680 | The predicate "is a 3-dim ... |
| lvoli2 40681 | The join of 4 different at... |
| lvolnle3at 40682 | A lattice plane (or lattic... |
| lvolnleat 40683 | An atom cannot majorize a ... |
| lvolnlelln 40684 | A lattice line cannot majo... |
| lvolnlelpln 40685 | A lattice plane cannot maj... |
| 3atnelvolN 40686 | The join of 3 atoms is not... |
| 2atnelvolN 40687 | The join of two atoms is n... |
| lvolneatN 40688 | No lattice volume is an at... |
| lvolnelln 40689 | No lattice volume is a lat... |
| lvolnelpln 40690 | No lattice volume is a lat... |
| lvoln0N 40691 | A lattice volume is nonzer... |
| islvol2aN 40692 | The predicate "is a lattic... |
| 4atlem0a 40693 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 40694 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 40695 | Lemma for ~ 4at . (Contri... |
| 4atlem3 40696 | Lemma for ~ 4at . Break i... |
| 4atlem3a 40697 | Lemma for ~ 4at . Break i... |
| 4atlem3b 40698 | Lemma for ~ 4at . Break i... |
| 4atlem4a 40699 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 40700 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 40701 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 40702 | Lemma for ~ 4at . Frequen... |
| 4atlem9 40703 | Lemma for ~ 4at . Substit... |
| 4atlem10a 40704 | Lemma for ~ 4at . Substit... |
| 4atlem10b 40705 | Lemma for ~ 4at . Substit... |
| 4atlem10 40706 | Lemma for ~ 4at . Combine... |
| 4atlem11a 40707 | Lemma for ~ 4at . Substit... |
| 4atlem11b 40708 | Lemma for ~ 4at . Substit... |
| 4atlem11 40709 | Lemma for ~ 4at . Combine... |
| 4atlem12a 40710 | Lemma for ~ 4at . Substit... |
| 4atlem12b 40711 | Lemma for ~ 4at . Substit... |
| 4atlem12 40712 | Lemma for ~ 4at . Combine... |
| 4at 40713 | Four atoms determine a lat... |
| 4at2 40714 | Four atoms determine a lat... |
| lplncvrlvol2 40715 | A lattice line under a lat... |
| lplncvrlvol 40716 | An element covering a latt... |
| lvolcmp 40717 | If two lattice planes are ... |
| lvolnltN 40718 | Two lattice volumes cannot... |
| 2lplnja 40719 | The join of two different ... |
| 2lplnj 40720 | The join of two different ... |
| 2lplnm2N 40721 | The meet of two different ... |
| 2lplnmj 40722 | The meet of two lattice pl... |
| dalemkehl 40723 | Lemma for ~ dath . Freque... |
| dalemkelat 40724 | Lemma for ~ dath . Freque... |
| dalemkeop 40725 | Lemma for ~ dath . Freque... |
| dalempea 40726 | Lemma for ~ dath . Freque... |
| dalemqea 40727 | Lemma for ~ dath . Freque... |
| dalemrea 40728 | Lemma for ~ dath . Freque... |
| dalemsea 40729 | Lemma for ~ dath . Freque... |
| dalemtea 40730 | Lemma for ~ dath . Freque... |
| dalemuea 40731 | Lemma for ~ dath . Freque... |
| dalemyeo 40732 | Lemma for ~ dath . Freque... |
| dalemzeo 40733 | Lemma for ~ dath . Freque... |
| dalemclpjs 40734 | Lemma for ~ dath . Freque... |
| dalemclqjt 40735 | Lemma for ~ dath . Freque... |
| dalemclrju 40736 | Lemma for ~ dath . Freque... |
| dalem-clpjq 40737 | Lemma for ~ dath . Freque... |
| dalemceb 40738 | Lemma for ~ dath . Freque... |
| dalempeb 40739 | Lemma for ~ dath . Freque... |
| dalemqeb 40740 | Lemma for ~ dath . Freque... |
| dalemreb 40741 | Lemma for ~ dath . Freque... |
| dalemseb 40742 | Lemma for ~ dath . Freque... |
| dalemteb 40743 | Lemma for ~ dath . Freque... |
| dalemueb 40744 | Lemma for ~ dath . Freque... |
| dalempjqeb 40745 | Lemma for ~ dath . Freque... |
| dalemsjteb 40746 | Lemma for ~ dath . Freque... |
| dalemtjueb 40747 | Lemma for ~ dath . Freque... |
| dalemqrprot 40748 | Lemma for ~ dath . Freque... |
| dalemyeb 40749 | Lemma for ~ dath . Freque... |
| dalemcnes 40750 | Lemma for ~ dath . Freque... |
| dalempnes 40751 | Lemma for ~ dath . Freque... |
| dalemqnet 40752 | Lemma for ~ dath . Freque... |
| dalempjsen 40753 | Lemma for ~ dath . Freque... |
| dalemply 40754 | Lemma for ~ dath . Freque... |
| dalemsly 40755 | Lemma for ~ dath . Freque... |
| dalemswapyz 40756 | Lemma for ~ dath . Swap t... |
| dalemrot 40757 | Lemma for ~ dath . Rotate... |
| dalemrotyz 40758 | Lemma for ~ dath . Rotate... |
| dalem1 40759 | Lemma for ~ dath . Show t... |
| dalemcea 40760 | Lemma for ~ dath . Freque... |
| dalem2 40761 | Lemma for ~ dath . Show t... |
| dalemdea 40762 | Lemma for ~ dath . Freque... |
| dalemeea 40763 | Lemma for ~ dath . Freque... |
| dalem3 40764 | Lemma for ~ dalemdnee . (... |
| dalem4 40765 | Lemma for ~ dalemdnee . (... |
| dalemdnee 40766 | Lemma for ~ dath . Axis o... |
| dalem5 40767 | Lemma for ~ dath . Atom `... |
| dalem6 40768 | Lemma for ~ dath . Analog... |
| dalem7 40769 | Lemma for ~ dath . Analog... |
| dalem8 40770 | Lemma for ~ dath . Plane ... |
| dalem-cly 40771 | Lemma for ~ dalem9 . Cent... |
| dalem9 40772 | Lemma for ~ dath . Since ... |
| dalem10 40773 | Lemma for ~ dath . Atom `... |
| dalem11 40774 | Lemma for ~ dath . Analog... |
| dalem12 40775 | Lemma for ~ dath . Analog... |
| dalem13 40776 | Lemma for ~ dalem14 . (Co... |
| dalem14 40777 | Lemma for ~ dath . Planes... |
| dalem15 40778 | Lemma for ~ dath . The ax... |
| dalem16 40779 | Lemma for ~ dath . The at... |
| dalem17 40780 | Lemma for ~ dath . When p... |
| dalem18 40781 | Lemma for ~ dath . Show t... |
| dalem19 40782 | Lemma for ~ dath . Show t... |
| dalemccea 40783 | Lemma for ~ dath . Freque... |
| dalemddea 40784 | Lemma for ~ dath . Freque... |
| dalem-ccly 40785 | Lemma for ~ dath . Freque... |
| dalem-ddly 40786 | Lemma for ~ dath . Freque... |
| dalemccnedd 40787 | Lemma for ~ dath . Freque... |
| dalemclccjdd 40788 | Lemma for ~ dath . Freque... |
| dalemcceb 40789 | Lemma for ~ dath . Freque... |
| dalemswapyzps 40790 | Lemma for ~ dath . Swap t... |
| dalemrotps 40791 | Lemma for ~ dath . Rotate... |
| dalemcjden 40792 | Lemma for ~ dath . Show t... |
| dalem20 40793 | Lemma for ~ dath . Show t... |
| dalem21 40794 | Lemma for ~ dath . Show t... |
| dalem22 40795 | Lemma for ~ dath . Show t... |
| dalem23 40796 | Lemma for ~ dath . Show t... |
| dalem24 40797 | Lemma for ~ dath . Show t... |
| dalem25 40798 | Lemma for ~ dath . Show t... |
| dalem27 40799 | Lemma for ~ dath . Show t... |
| dalem28 40800 | Lemma for ~ dath . Lemma ... |
| dalem29 40801 | Lemma for ~ dath . Analog... |
| dalem30 40802 | Lemma for ~ dath . Analog... |
| dalem31N 40803 | Lemma for ~ dath . Analog... |
| dalem32 40804 | Lemma for ~ dath . Analog... |
| dalem33 40805 | Lemma for ~ dath . Analog... |
| dalem34 40806 | Lemma for ~ dath . Analog... |
| dalem35 40807 | Lemma for ~ dath . Analog... |
| dalem36 40808 | Lemma for ~ dath . Analog... |
| dalem37 40809 | Lemma for ~ dath . Analog... |
| dalem38 40810 | Lemma for ~ dath . Plane ... |
| dalem39 40811 | Lemma for ~ dath . Auxili... |
| dalem40 40812 | Lemma for ~ dath . Analog... |
| dalem41 40813 | Lemma for ~ dath . (Contr... |
| dalem42 40814 | Lemma for ~ dath . Auxili... |
| dalem43 40815 | Lemma for ~ dath . Planes... |
| dalem44 40816 | Lemma for ~ dath . Dummy ... |
| dalem45 40817 | Lemma for ~ dath . Dummy ... |
| dalem46 40818 | Lemma for ~ dath . Analog... |
| dalem47 40819 | Lemma for ~ dath . Analog... |
| dalem48 40820 | Lemma for ~ dath . Analog... |
| dalem49 40821 | Lemma for ~ dath . Analog... |
| dalem50 40822 | Lemma for ~ dath . Analog... |
| dalem51 40823 | Lemma for ~ dath . Constr... |
| dalem52 40824 | Lemma for ~ dath . Lines ... |
| dalem53 40825 | Lemma for ~ dath . The au... |
| dalem54 40826 | Lemma for ~ dath . Line `... |
| dalem55 40827 | Lemma for ~ dath . Lines ... |
| dalem56 40828 | Lemma for ~ dath . Analog... |
| dalem57 40829 | Lemma for ~ dath . Axis o... |
| dalem58 40830 | Lemma for ~ dath . Analog... |
| dalem59 40831 | Lemma for ~ dath . Analog... |
| dalem60 40832 | Lemma for ~ dath . ` B ` i... |
| dalem61 40833 | Lemma for ~ dath . Show t... |
| dalem62 40834 | Lemma for ~ dath . Elimin... |
| dalem63 40835 | Lemma for ~ dath . Combin... |
| dath 40836 | Desargues's theorem of pro... |
| dath2 40837 | Version of Desargues's the... |
| lineset 40838 | The set of lines in a Hilb... |
| isline 40839 | The predicate "is a line".... |
| islinei 40840 | Condition implying "is a l... |
| pointsetN 40841 | The set of points in a Hil... |
| ispointN 40842 | The predicate "is a point"... |
| atpointN 40843 | The singleton of an atom i... |
| psubspset 40844 | The set of projective subs... |
| ispsubsp 40845 | The predicate "is a projec... |
| ispsubsp2 40846 | The predicate "is a projec... |
| psubspi 40847 | Property of a projective s... |
| psubspi2N 40848 | Property of a projective s... |
| 0psubN 40849 | The empty set is a project... |
| snatpsubN 40850 | The singleton of an atom i... |
| pointpsubN 40851 | A point (singleton of an a... |
| linepsubN 40852 | A line is a projective sub... |
| atpsubN 40853 | The set of all atoms is a ... |
| psubssat 40854 | A projective subspace cons... |
| psubatN 40855 | A member of a projective s... |
| pmapfval 40856 | The projective map of a Hi... |
| pmapval 40857 | Value of the projective ma... |
| elpmap 40858 | Member of a projective map... |
| pmapssat 40859 | The projective map of a Hi... |
| pmapssbaN 40860 | A weakening of ~ pmapssat ... |
| pmaple 40861 | The projective map of a Hi... |
| pmap11 40862 | The projective map of a Hi... |
| pmapat 40863 | The projective map of an a... |
| elpmapat 40864 | Member of the projective m... |
| pmap0 40865 | Value of the projective ma... |
| pmapeq0 40866 | A projective map value is ... |
| pmap1N 40867 | Value of the projective ma... |
| pmapsub 40868 | The projective map of a Hi... |
| pmapglbx 40869 | The projective map of the ... |
| pmapglb 40870 | The projective map of the ... |
| pmapglb2N 40871 | The projective map of the ... |
| pmapglb2xN 40872 | The projective map of the ... |
| pmapmeet 40873 | The projective map of a me... |
| isline2 40874 | Definition of line in term... |
| linepmap 40875 | A line described with a pr... |
| isline3 40876 | Definition of line in term... |
| isline4N 40877 | Definition of line in term... |
| lneq2at 40878 | A line equals the join of ... |
| lnatexN 40879 | There is an atom in a line... |
| lnjatN 40880 | Given an atom in a line, t... |
| lncvrelatN 40881 | A lattice element covered ... |
| lncvrat 40882 | A line covers the atoms it... |
| lncmp 40883 | If two lines are comparabl... |
| 2lnat 40884 | Two intersecting lines int... |
| 2atm2atN 40885 | Two joins with a common at... |
| 2llnma1b 40886 | Generalization of ~ 2llnma... |
| 2llnma1 40887 | Two different intersecting... |
| 2llnma3r 40888 | Two different intersecting... |
| 2llnma2 40889 | Two different intersecting... |
| 2llnma2rN 40890 | Two different intersecting... |
| cdlema1N 40891 | A condition for required f... |
| cdlema2N 40892 | A condition for required f... |
| cdlemblem 40893 | Lemma for ~ cdlemb . (Con... |
| cdlemb 40894 | Given two atoms not less t... |
| paddfval 40897 | Projective subspace sum op... |
| paddval 40898 | Projective subspace sum op... |
| elpadd 40899 | Member of a projective sub... |
| elpaddn0 40900 | Member of projective subsp... |
| paddvaln0N 40901 | Projective subspace sum op... |
| elpaddri 40902 | Condition implying members... |
| elpaddatriN 40903 | Condition implying members... |
| elpaddat 40904 | Membership in a projective... |
| elpaddatiN 40905 | Consequence of membership ... |
| elpadd2at 40906 | Membership in a projective... |
| elpadd2at2 40907 | Membership in a projective... |
| paddunssN 40908 | Projective subspace sum in... |
| elpadd0 40909 | Member of projective subsp... |
| paddval0 40910 | Projective subspace sum wi... |
| padd01 40911 | Projective subspace sum wi... |
| padd02 40912 | Projective subspace sum wi... |
| paddcom 40913 | Projective subspace sum co... |
| paddssat 40914 | A projective subspace sum ... |
| sspadd1 40915 | A projective subspace sum ... |
| sspadd2 40916 | A projective subspace sum ... |
| paddss1 40917 | Subset law for projective ... |
| paddss2 40918 | Subset law for projective ... |
| paddss12 40919 | Subset law for projective ... |
| paddasslem1 40920 | Lemma for ~ paddass . (Co... |
| paddasslem2 40921 | Lemma for ~ paddass . (Co... |
| paddasslem3 40922 | Lemma for ~ paddass . Res... |
| paddasslem4 40923 | Lemma for ~ paddass . Com... |
| paddasslem5 40924 | Lemma for ~ paddass . Sho... |
| paddasslem6 40925 | Lemma for ~ paddass . (Co... |
| paddasslem7 40926 | Lemma for ~ paddass . Com... |
| paddasslem8 40927 | Lemma for ~ paddass . (Co... |
| paddasslem9 40928 | Lemma for ~ paddass . Com... |
| paddasslem10 40929 | Lemma for ~ paddass . Use... |
| paddasslem11 40930 | Lemma for ~ paddass . The... |
| paddasslem12 40931 | Lemma for ~ paddass . The... |
| paddasslem13 40932 | Lemma for ~ paddass . The... |
| paddasslem14 40933 | Lemma for ~ paddass . Rem... |
| paddasslem15 40934 | Lemma for ~ paddass . Use... |
| paddasslem16 40935 | Lemma for ~ paddass . Use... |
| paddasslem17 40936 | Lemma for ~ paddass . The... |
| paddasslem18 40937 | Lemma for ~ paddass . Com... |
| paddass 40938 | Projective subspace sum is... |
| padd12N 40939 | Commutative/associative la... |
| padd4N 40940 | Rearrangement of 4 terms i... |
| paddidm 40941 | Projective subspace sum is... |
| paddclN 40942 | The projective sum of two ... |
| paddssw1 40943 | Subset law for projective ... |
| paddssw2 40944 | Subset law for projective ... |
| paddss 40945 | Subset law for projective ... |
| pmodlem1 40946 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 40947 | Lemma for ~ pmod1i . (Con... |
| pmod1i 40948 | The modular law holds in a... |
| pmod2iN 40949 | Dual of the modular law. ... |
| pmodN 40950 | The modular law for projec... |
| pmodl42N 40951 | Lemma derived from modular... |
| pmapjoin 40952 | The projective map of the ... |
| pmapjat1 40953 | The projective map of the ... |
| pmapjat2 40954 | The projective map of the ... |
| pmapjlln1 40955 | The projective map of the ... |
| hlmod1i 40956 | A version of the modular l... |
| atmod1i1 40957 | Version of modular law ~ p... |
| atmod1i1m 40958 | Version of modular law ~ p... |
| atmod1i2 40959 | Version of modular law ~ p... |
| llnmod1i2 40960 | Version of modular law ~ p... |
| atmod2i1 40961 | Version of modular law ~ p... |
| atmod2i2 40962 | Version of modular law ~ p... |
| llnmod2i2 40963 | Version of modular law ~ p... |
| atmod3i1 40964 | Version of modular law tha... |
| atmod3i2 40965 | Version of modular law tha... |
| atmod4i1 40966 | Version of modular law tha... |
| atmod4i2 40967 | Version of modular law tha... |
| llnexchb2lem 40968 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 40969 | Line exchange property (co... |
| llnexch2N 40970 | Line exchange property (co... |
| dalawlem1 40971 | Lemma for ~ dalaw . Speci... |
| dalawlem2 40972 | Lemma for ~ dalaw . Utili... |
| dalawlem3 40973 | Lemma for ~ dalaw . First... |
| dalawlem4 40974 | Lemma for ~ dalaw . Secon... |
| dalawlem5 40975 | Lemma for ~ dalaw . Speci... |
| dalawlem6 40976 | Lemma for ~ dalaw . First... |
| dalawlem7 40977 | Lemma for ~ dalaw . Secon... |
| dalawlem8 40978 | Lemma for ~ dalaw . Speci... |
| dalawlem9 40979 | Lemma for ~ dalaw . Speci... |
| dalawlem10 40980 | Lemma for ~ dalaw . Combi... |
| dalawlem11 40981 | Lemma for ~ dalaw . First... |
| dalawlem12 40982 | Lemma for ~ dalaw . Secon... |
| dalawlem13 40983 | Lemma for ~ dalaw . Speci... |
| dalawlem14 40984 | Lemma for ~ dalaw . Combi... |
| dalawlem15 40985 | Lemma for ~ dalaw . Swap ... |
| dalaw 40986 | Desargues's law, derived f... |
| pclfvalN 40989 | The projective subspace cl... |
| pclvalN 40990 | Value of the projective su... |
| pclclN 40991 | Closure of the projective ... |
| elpclN 40992 | Membership in the projecti... |
| elpcliN 40993 | Implication of membership ... |
| pclssN 40994 | Ordering is preserved by s... |
| pclssidN 40995 | A set of atoms is included... |
| pclidN 40996 | The projective subspace cl... |
| pclbtwnN 40997 | A projective subspace sand... |
| pclunN 40998 | The projective subspace cl... |
| pclun2N 40999 | The projective subspace cl... |
| pclfinN 41000 | The projective subspace cl... |
| pclcmpatN 41001 | The set of projective subs... |
| polfvalN 41004 | The projective subspace po... |
| polvalN 41005 | Value of the projective su... |
| polval2N 41006 | Alternate expression for v... |
| polsubN 41007 | The polarity of a set of a... |
| polssatN 41008 | The polarity of a set of a... |
| pol0N 41009 | The polarity of the empty ... |
| pol1N 41010 | The polarity of the whole ... |
| 2pol0N 41011 | The closed subspace closur... |
| polpmapN 41012 | The polarity of a projecti... |
| 2polpmapN 41013 | Double polarity of a proje... |
| 2polvalN 41014 | Value of double polarity. ... |
| 2polssN 41015 | A set of atoms is a subset... |
| 3polN 41016 | Triple polarity cancels to... |
| polcon3N 41017 | Contraposition law for pol... |
| 2polcon4bN 41018 | Contraposition law for pol... |
| polcon2N 41019 | Contraposition law for pol... |
| polcon2bN 41020 | Contraposition law for pol... |
| pclss2polN 41021 | The projective subspace cl... |
| pcl0N 41022 | The projective subspace cl... |
| pcl0bN 41023 | The projective subspace cl... |
| pmaplubN 41024 | The LUB of a projective ma... |
| sspmaplubN 41025 | A set of atoms is a subset... |
| 2pmaplubN 41026 | Double projective map of a... |
| paddunN 41027 | The closure of the project... |
| poldmj1N 41028 | De Morgan's law for polari... |
| pmapj2N 41029 | The projective map of the ... |
| pmapocjN 41030 | The projective map of the ... |
| polatN 41031 | The polarity of the single... |
| 2polatN 41032 | Double polarity of the sin... |
| pnonsingN 41033 | The intersection of a set ... |
| psubclsetN 41036 | The set of closed projecti... |
| ispsubclN 41037 | The predicate "is a closed... |
| psubcliN 41038 | Property of a closed proje... |
| psubcli2N 41039 | Property of a closed proje... |
| psubclsubN 41040 | A closed projective subspa... |
| psubclssatN 41041 | A closed projective subspa... |
| pmapidclN 41042 | Projective map of the LUB ... |
| 0psubclN 41043 | The empty set is a closed ... |
| 1psubclN 41044 | The set of all atoms is a ... |
| atpsubclN 41045 | A point (singleton of an a... |
| pmapsubclN 41046 | A projective map value is ... |
| ispsubcl2N 41047 | Alternate predicate for "i... |
| psubclinN 41048 | The intersection of two cl... |
| paddatclN 41049 | The projective sum of a cl... |
| pclfinclN 41050 | The projective subspace cl... |
| linepsubclN 41051 | A line is a closed project... |
| polsubclN 41052 | A polarity is a closed pro... |
| poml4N 41053 | Orthomodular law for proje... |
| poml5N 41054 | Orthomodular law for proje... |
| poml6N 41055 | Orthomodular law for proje... |
| osumcllem1N 41056 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 41057 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 41058 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 41059 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 41060 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 41061 | Lemma for ~ osumclN . Use... |
| osumcllem7N 41062 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 41063 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 41064 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 41065 | Lemma for ~ osumclN . Con... |
| osumcllem11N 41066 | Lemma for ~ osumclN . (Co... |
| osumclN 41067 | Closure of orthogonal sum.... |
| pmapojoinN 41068 | For orthogonal elements, p... |
| pexmidN 41069 | Excluded middle law for cl... |
| pexmidlem1N 41070 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 41071 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 41072 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 41073 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 41074 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 41075 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 41076 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 41077 | Lemma for ~ pexmidN . The... |
| pexmidALTN 41078 | Excluded middle law for cl... |
| pl42lem1N 41079 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 41080 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 41081 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 41082 | Lemma for ~ pl42N . (Cont... |
| pl42N 41083 | Law holding in a Hilbert l... |
| watfvalN 41092 | The W atoms function. (Co... |
| watvalN 41093 | Value of the W atoms funct... |
| iswatN 41094 | The predicate "is a W atom... |
| lhpset 41095 | The set of co-atoms (latti... |
| islhp 41096 | The predicate "is a co-ato... |
| islhp2 41097 | The predicate "is a co-ato... |
| lhpbase 41098 | A co-atom is a member of t... |
| lhp1cvr 41099 | The lattice unity covers a... |
| lhplt 41100 | An atom under a co-atom is... |
| lhp2lt 41101 | The join of two atoms unde... |
| lhpexlt 41102 | There exists an atom less ... |
| lhp0lt 41103 | A co-atom is greater than ... |
| lhpn0 41104 | A co-atom is nonzero. TOD... |
| lhpexle 41105 | There exists an atom under... |
| lhpexnle 41106 | There exists an atom not u... |
| lhpexle1lem 41107 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 41108 | There exists an atom under... |
| lhpexle2lem 41109 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 41110 | There exists atom under a ... |
| lhpexle3lem 41111 | There exists atom under a ... |
| lhpexle3 41112 | There exists atom under a ... |
| lhpex2leN 41113 | There exist at least two d... |
| lhpoc 41114 | The orthocomplement of a c... |
| lhpoc2N 41115 | The orthocomplement of an ... |
| lhpocnle 41116 | The orthocomplement of a c... |
| lhpocat 41117 | The orthocomplement of a c... |
| lhpocnel 41118 | The orthocomplement of a c... |
| lhpocnel2 41119 | The orthocomplement of a c... |
| lhpjat1 41120 | The join of a co-atom (hyp... |
| lhpjat2 41121 | The join of a co-atom (hyp... |
| lhpj1 41122 | The join of a co-atom (hyp... |
| lhpmcvr 41123 | The meet of a lattice hype... |
| lhpmcvr2 41124 | Alternate way to express t... |
| lhpmcvr3 41125 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 41126 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 41127 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 41128 | Specialization of ~ lhpmcv... |
| lhpm0atN 41129 | If the meet of a lattice h... |
| lhpmat 41130 | An element covered by the ... |
| lhpmatb 41131 | An element covered by the ... |
| lhp2at0 41132 | Join and meet with differe... |
| lhp2atnle 41133 | Inequality for 2 different... |
| lhp2atne 41134 | Inequality for joins with ... |
| lhp2at0nle 41135 | Inequality for 2 different... |
| lhp2at0ne 41136 | Inequality for joins with ... |
| lhpelim 41137 | Eliminate an atom not unde... |
| lhpmod2i2 41138 | Modular law for hyperplane... |
| lhpmod6i1 41139 | Modular law for hyperplane... |
| lhprelat3N 41140 | The Hilbert lattice is rel... |
| cdlemb2 41141 | Given two atoms not under ... |
| lhple 41142 | Property of a lattice elem... |
| lhpat 41143 | Create an atom under a co-... |
| lhpat4N 41144 | Property of an atom under ... |
| lhpat2 41145 | Create an atom under a co-... |
| lhpat3 41146 | There is only one atom und... |
| 4atexlemk 41147 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 41148 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 41149 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 41150 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 41151 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 41152 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 41153 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 41154 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 41155 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 41156 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 41157 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 41158 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 41159 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 41160 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 41161 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 41162 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 41163 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 41164 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 41165 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 41166 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 41167 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 41168 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 41169 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 41170 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 41171 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 41172 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 41173 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 41174 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 41175 | Whenever there are at leas... |
| 4atex 41176 | Whenever there are at leas... |
| 4atex2 41177 | More general version of ~ ... |
| 4atex2-0aOLDN 41178 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 41179 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 41180 | Same as ~ 4atex2 except th... |
| 4atex3 41181 | More general version of ~ ... |
| lautset 41182 | The set of lattice automor... |
| islaut 41183 | The predicate "is a lattic... |
| lautle 41184 | Less-than or equal propert... |
| laut1o 41185 | A lattice automorphism is ... |
| laut11 41186 | One-to-one property of a l... |
| lautcl 41187 | A lattice automorphism val... |
| lautcnvclN 41188 | Reverse closure of a latti... |
| lautcnvle 41189 | Less-than or equal propert... |
| lautcnv 41190 | The converse of a lattice ... |
| lautlt 41191 | Less-than property of a la... |
| lautcvr 41192 | Covering property of a lat... |
| lautj 41193 | Meet property of a lattice... |
| lautm 41194 | Meet property of a lattice... |
| lauteq 41195 | A lattice automorphism arg... |
| idlaut 41196 | The identity function is a... |
| lautco 41197 | The composition of two lat... |
| pautsetN 41198 | The set of projective auto... |
| ispautN 41199 | The predicate "is a projec... |
| ldilfset 41208 | The mapping from fiducial ... |
| ldilset 41209 | The set of lattice dilatio... |
| isldil 41210 | The predicate "is a lattic... |
| ldillaut 41211 | A lattice dilation is an a... |
| ldil1o 41212 | A lattice dilation is a on... |
| ldilval 41213 | Value of a lattice dilatio... |
| idldil 41214 | The identity function is a... |
| ldilcnv 41215 | The converse of a lattice ... |
| ldilco 41216 | The composition of two lat... |
| ltrnfset 41217 | The set of all lattice tra... |
| ltrnset 41218 | The set of lattice transla... |
| isltrn 41219 | The predicate "is a lattic... |
| isltrn2N 41220 | The predicate "is a lattic... |
| ltrnu 41221 | Uniqueness property of a l... |
| ltrnldil 41222 | A lattice translation is a... |
| ltrnlaut 41223 | A lattice translation is a... |
| ltrn1o 41224 | A lattice translation is a... |
| ltrncl 41225 | Closure of a lattice trans... |
| ltrn11 41226 | One-to-one property of a l... |
| ltrncnvnid 41227 | If a translation is differ... |
| ltrncoidN 41228 | Two translations are equal... |
| ltrnle 41229 | Less-than or equal propert... |
| ltrncnvleN 41230 | Less-than or equal propert... |
| ltrnm 41231 | Lattice translation of a m... |
| ltrnj 41232 | Lattice translation of a m... |
| ltrncvr 41233 | Covering property of a lat... |
| ltrnval1 41234 | Value of a lattice transla... |
| ltrnid 41235 | A lattice translation is t... |
| ltrnnid 41236 | If a lattice translation i... |
| ltrnatb 41237 | The lattice translation of... |
| ltrncnvatb 41238 | The converse of the lattic... |
| ltrnel 41239 | The lattice translation of... |
| ltrnat 41240 | The lattice translation of... |
| ltrncnvat 41241 | The converse of the lattic... |
| ltrncnvel 41242 | The converse of the lattic... |
| ltrncoelN 41243 | Composition of lattice tra... |
| ltrncoat 41244 | Composition of lattice tra... |
| ltrncoval 41245 | Two ways to express value ... |
| ltrncnv 41246 | The converse of a lattice ... |
| ltrn11at 41247 | Frequently used one-to-one... |
| ltrneq2 41248 | The equality of two transl... |
| ltrneq 41249 | The equality of two transl... |
| idltrn 41250 | The identity function is a... |
| ltrnmw 41251 | Property of lattice transl... |
| dilfsetN 41252 | The mapping from fiducial ... |
| dilsetN 41253 | The set of dilations for a... |
| isdilN 41254 | The predicate "is a dilati... |
| trnfsetN 41255 | The mapping from fiducial ... |
| trnsetN 41256 | The set of translations fo... |
| istrnN 41257 | The predicate "is a transl... |
| trlfset 41260 | The set of all traces of l... |
| trlset 41261 | The set of traces of latti... |
| trlval 41262 | The value of the trace of ... |
| trlval2 41263 | The value of the trace of ... |
| trlcl 41264 | Closure of the trace of a ... |
| trlcnv 41265 | The trace of the converse ... |
| trljat1 41266 | The value of a translation... |
| trljat2 41267 | The value of a translation... |
| trljat3 41268 | The value of a translation... |
| trlat 41269 | If an atom differs from it... |
| trl0 41270 | If an atom not under the f... |
| trlator0 41271 | The trace of a lattice tra... |
| trlatn0 41272 | The trace of a lattice tra... |
| trlnidat 41273 | The trace of a lattice tra... |
| ltrnnidn 41274 | If a lattice translation i... |
| ltrnideq 41275 | Property of the identity l... |
| trlid0 41276 | The trace of the identity ... |
| trlnidatb 41277 | A lattice translation is n... |
| trlid0b 41278 | A lattice translation is t... |
| trlnid 41279 | Different translations wit... |
| ltrn2ateq 41280 | Property of the equality o... |
| ltrnateq 41281 | If any atom (under ` W ` )... |
| ltrnatneq 41282 | If any atom (under ` W ` )... |
| ltrnatlw 41283 | If the value of an atom eq... |
| trlle 41284 | The trace of a lattice tra... |
| trlne 41285 | The trace of a lattice tra... |
| trlnle 41286 | The atom not under the fid... |
| trlval3 41287 | The value of the trace of ... |
| trlval4 41288 | The value of the trace of ... |
| trlval5 41289 | The value of the trace of ... |
| arglem1N 41290 | Lemma for Desargues's law.... |
| cdlemc1 41291 | Part of proof of Lemma C i... |
| cdlemc2 41292 | Part of proof of Lemma C i... |
| cdlemc3 41293 | Part of proof of Lemma C i... |
| cdlemc4 41294 | Part of proof of Lemma C i... |
| cdlemc5 41295 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 41296 | Lemma for ~ cdlemc . (Con... |
| cdlemc 41297 | Lemma C in [Crawley] p. 11... |
| cdlemd1 41298 | Part of proof of Lemma D i... |
| cdlemd2 41299 | Part of proof of Lemma D i... |
| cdlemd3 41300 | Part of proof of Lemma D i... |
| cdlemd4 41301 | Part of proof of Lemma D i... |
| cdlemd5 41302 | Part of proof of Lemma D i... |
| cdlemd6 41303 | Part of proof of Lemma D i... |
| cdlemd7 41304 | Part of proof of Lemma D i... |
| cdlemd8 41305 | Part of proof of Lemma D i... |
| cdlemd9 41306 | Part of proof of Lemma D i... |
| cdlemd 41307 | If two translations agree ... |
| ltrneq3 41308 | Two translations agree at ... |
| cdleme00a 41309 | Part of proof of Lemma E i... |
| cdleme0aa 41310 | Part of proof of Lemma E i... |
| cdleme0a 41311 | Part of proof of Lemma E i... |
| cdleme0b 41312 | Part of proof of Lemma E i... |
| cdleme0c 41313 | Part of proof of Lemma E i... |
| cdleme0cp 41314 | Part of proof of Lemma E i... |
| cdleme0cq 41315 | Part of proof of Lemma E i... |
| cdleme0dN 41316 | Part of proof of Lemma E i... |
| cdleme0e 41317 | Part of proof of Lemma E i... |
| cdleme0fN 41318 | Part of proof of Lemma E i... |
| cdleme0gN 41319 | Part of proof of Lemma E i... |
| cdlemeulpq 41320 | Part of proof of Lemma E i... |
| cdleme01N 41321 | Part of proof of Lemma E i... |
| cdleme02N 41322 | Part of proof of Lemma E i... |
| cdleme0ex1N 41323 | Part of proof of Lemma E i... |
| cdleme0ex2N 41324 | Part of proof of Lemma E i... |
| cdleme0moN 41325 | Part of proof of Lemma E i... |
| cdleme1b 41326 | Part of proof of Lemma E i... |
| cdleme1 41327 | Part of proof of Lemma E i... |
| cdleme2 41328 | Part of proof of Lemma E i... |
| cdleme3b 41329 | Part of proof of Lemma E i... |
| cdleme3c 41330 | Part of proof of Lemma E i... |
| cdleme3d 41331 | Part of proof of Lemma E i... |
| cdleme3e 41332 | Part of proof of Lemma E i... |
| cdleme3fN 41333 | Part of proof of Lemma E i... |
| cdleme3g 41334 | Part of proof of Lemma E i... |
| cdleme3h 41335 | Part of proof of Lemma E i... |
| cdleme3fa 41336 | Part of proof of Lemma E i... |
| cdleme3 41337 | Part of proof of Lemma E i... |
| cdleme4 41338 | Part of proof of Lemma E i... |
| cdleme4a 41339 | Part of proof of Lemma E i... |
| cdleme5 41340 | Part of proof of Lemma E i... |
| cdleme6 41341 | Part of proof of Lemma E i... |
| cdleme7aa 41342 | Part of proof of Lemma E i... |
| cdleme7a 41343 | Part of proof of Lemma E i... |
| cdleme7b 41344 | Part of proof of Lemma E i... |
| cdleme7c 41345 | Part of proof of Lemma E i... |
| cdleme7d 41346 | Part of proof of Lemma E i... |
| cdleme7e 41347 | Part of proof of Lemma E i... |
| cdleme7ga 41348 | Part of proof of Lemma E i... |
| cdleme7 41349 | Part of proof of Lemma E i... |
| cdleme8 41350 | Part of proof of Lemma E i... |
| cdleme9a 41351 | Part of proof of Lemma E i... |
| cdleme9b 41352 | Utility lemma for Lemma E ... |
| cdleme9 41353 | Part of proof of Lemma E i... |
| cdleme10 41354 | Part of proof of Lemma E i... |
| cdleme8tN 41355 | Part of proof of Lemma E i... |
| cdleme9taN 41356 | Part of proof of Lemma E i... |
| cdleme9tN 41357 | Part of proof of Lemma E i... |
| cdleme10tN 41358 | Part of proof of Lemma E i... |
| cdleme16aN 41359 | Part of proof of Lemma E i... |
| cdleme11a 41360 | Part of proof of Lemma E i... |
| cdleme11c 41361 | Part of proof of Lemma E i... |
| cdleme11dN 41362 | Part of proof of Lemma E i... |
| cdleme11e 41363 | Part of proof of Lemma E i... |
| cdleme11fN 41364 | Part of proof of Lemma E i... |
| cdleme11g 41365 | Part of proof of Lemma E i... |
| cdleme11h 41366 | Part of proof of Lemma E i... |
| cdleme11j 41367 | Part of proof of Lemma E i... |
| cdleme11k 41368 | Part of proof of Lemma E i... |
| cdleme11l 41369 | Part of proof of Lemma E i... |
| cdleme11 41370 | Part of proof of Lemma E i... |
| cdleme12 41371 | Part of proof of Lemma E i... |
| cdleme13 41372 | Part of proof of Lemma E i... |
| cdleme14 41373 | Part of proof of Lemma E i... |
| cdleme15a 41374 | Part of proof of Lemma E i... |
| cdleme15b 41375 | Part of proof of Lemma E i... |
| cdleme15c 41376 | Part of proof of Lemma E i... |
| cdleme15d 41377 | Part of proof of Lemma E i... |
| cdleme15 41378 | Part of proof of Lemma E i... |
| cdleme16b 41379 | Part of proof of Lemma E i... |
| cdleme16c 41380 | Part of proof of Lemma E i... |
| cdleme16d 41381 | Part of proof of Lemma E i... |
| cdleme16e 41382 | Part of proof of Lemma E i... |
| cdleme16f 41383 | Part of proof of Lemma E i... |
| cdleme16g 41384 | Part of proof of Lemma E i... |
| cdleme16 41385 | Part of proof of Lemma E i... |
| cdleme17a 41386 | Part of proof of Lemma E i... |
| cdleme17b 41387 | Lemma leading to ~ cdleme1... |
| cdleme17c 41388 | Part of proof of Lemma E i... |
| cdleme17d1 41389 | Part of proof of Lemma E i... |
| cdleme0nex 41390 | Part of proof of Lemma E i... |
| cdleme18a 41391 | Part of proof of Lemma E i... |
| cdleme18b 41392 | Part of proof of Lemma E i... |
| cdleme18c 41393 | Part of proof of Lemma E i... |
| cdleme22gb 41394 | Utility lemma for Lemma E ... |
| cdleme18d 41395 | Part of proof of Lemma E i... |
| cdlemesner 41396 | Part of proof of Lemma E i... |
| cdlemedb 41397 | Part of proof of Lemma E i... |
| cdlemeda 41398 | Part of proof of Lemma E i... |
| cdlemednpq 41399 | Part of proof of Lemma E i... |
| cdlemednuN 41400 | Part of proof of Lemma E i... |
| cdleme20zN 41401 | Part of proof of Lemma E i... |
| cdleme20y 41402 | Part of proof of Lemma E i... |
| cdleme19a 41403 | Part of proof of Lemma E i... |
| cdleme19b 41404 | Part of proof of Lemma E i... |
| cdleme19c 41405 | Part of proof of Lemma E i... |
| cdleme19d 41406 | Part of proof of Lemma E i... |
| cdleme19e 41407 | Part of proof of Lemma E i... |
| cdleme19f 41408 | Part of proof of Lemma E i... |
| cdleme20aN 41409 | Part of proof of Lemma E i... |
| cdleme20bN 41410 | Part of proof of Lemma E i... |
| cdleme20c 41411 | Part of proof of Lemma E i... |
| cdleme20d 41412 | Part of proof of Lemma E i... |
| cdleme20e 41413 | Part of proof of Lemma E i... |
| cdleme20f 41414 | Part of proof of Lemma E i... |
| cdleme20g 41415 | Part of proof of Lemma E i... |
| cdleme20h 41416 | Part of proof of Lemma E i... |
| cdleme20i 41417 | Part of proof of Lemma E i... |
| cdleme20j 41418 | Part of proof of Lemma E i... |
| cdleme20k 41419 | Part of proof of Lemma E i... |
| cdleme20l1 41420 | Part of proof of Lemma E i... |
| cdleme20l2 41421 | Part of proof of Lemma E i... |
| cdleme20l 41422 | Part of proof of Lemma E i... |
| cdleme20m 41423 | Part of proof of Lemma E i... |
| cdleme20 41424 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 41425 | Part of proof of Lemma E i... |
| cdleme21b 41426 | Part of proof of Lemma E i... |
| cdleme21c 41427 | Part of proof of Lemma E i... |
| cdleme21at 41428 | Part of proof of Lemma E i... |
| cdleme21ct 41429 | Part of proof of Lemma E i... |
| cdleme21d 41430 | Part of proof of Lemma E i... |
| cdleme21e 41431 | Part of proof of Lemma E i... |
| cdleme21f 41432 | Part of proof of Lemma E i... |
| cdleme21g 41433 | Part of proof of Lemma E i... |
| cdleme21h 41434 | Part of proof of Lemma E i... |
| cdleme21i 41435 | Part of proof of Lemma E i... |
| cdleme21j 41436 | Combine ~ cdleme20 and ~ c... |
| cdleme21 41437 | Part of proof of Lemma E i... |
| cdleme21k 41438 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 41439 | Part of proof of Lemma E i... |
| cdleme22a 41440 | Part of proof of Lemma E i... |
| cdleme22b 41441 | Part of proof of Lemma E i... |
| cdleme22cN 41442 | Part of proof of Lemma E i... |
| cdleme22d 41443 | Part of proof of Lemma E i... |
| cdleme22e 41444 | Part of proof of Lemma E i... |
| cdleme22eALTN 41445 | Part of proof of Lemma E i... |
| cdleme22f 41446 | Part of proof of Lemma E i... |
| cdleme22f2 41447 | Part of proof of Lemma E i... |
| cdleme22g 41448 | Part of proof of Lemma E i... |
| cdleme23a 41449 | Part of proof of Lemma E i... |
| cdleme23b 41450 | Part of proof of Lemma E i... |
| cdleme23c 41451 | Part of proof of Lemma E i... |
| cdleme24 41452 | Quantified version of ~ cd... |
| cdleme25a 41453 | Lemma for ~ cdleme25b . (... |
| cdleme25b 41454 | Transform ~ cdleme24 . TO... |
| cdleme25c 41455 | Transform ~ cdleme25b . (... |
| cdleme25dN 41456 | Transform ~ cdleme25c . (... |
| cdleme25cl 41457 | Show closure of the unique... |
| cdleme25cv 41458 | Change bound variables in ... |
| cdleme26e 41459 | Part of proof of Lemma E i... |
| cdleme26ee 41460 | Part of proof of Lemma E i... |
| cdleme26eALTN 41461 | Part of proof of Lemma E i... |
| cdleme26fALTN 41462 | Part of proof of Lemma E i... |
| cdleme26f 41463 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 41464 | Part of proof of Lemma E i... |
| cdleme26f2 41465 | Part of proof of Lemma E i... |
| cdleme27cl 41466 | Part of proof of Lemma E i... |
| cdleme27a 41467 | Part of proof of Lemma E i... |
| cdleme27b 41468 | Lemma for ~ cdleme27N . (... |
| cdleme27N 41469 | Part of proof of Lemma E i... |
| cdleme28a 41470 | Lemma for ~ cdleme25b . T... |
| cdleme28b 41471 | Lemma for ~ cdleme25b . T... |
| cdleme28c 41472 | Part of proof of Lemma E i... |
| cdleme28 41473 | Quantified version of ~ cd... |
| cdleme29ex 41474 | Lemma for ~ cdleme29b . (... |
| cdleme29b 41475 | Transform ~ cdleme28 . (C... |
| cdleme29c 41476 | Transform ~ cdleme28b . (... |
| cdleme29cl 41477 | Show closure of the unique... |
| cdleme30a 41478 | Part of proof of Lemma E i... |
| cdleme31so 41479 | Part of proof of Lemma E i... |
| cdleme31sn 41480 | Part of proof of Lemma E i... |
| cdleme31sn1 41481 | Part of proof of Lemma E i... |
| cdleme31se 41482 | Part of proof of Lemma D i... |
| cdleme31se2 41483 | Part of proof of Lemma D i... |
| cdleme31sc 41484 | Part of proof of Lemma E i... |
| cdleme31sde 41485 | Part of proof of Lemma D i... |
| cdleme31snd 41486 | Part of proof of Lemma D i... |
| cdleme31sdnN 41487 | Part of proof of Lemma E i... |
| cdleme31sn1c 41488 | Part of proof of Lemma E i... |
| cdleme31sn2 41489 | Part of proof of Lemma E i... |
| cdleme31fv 41490 | Part of proof of Lemma E i... |
| cdleme31fv1 41491 | Part of proof of Lemma E i... |
| cdleme31fv1s 41492 | Part of proof of Lemma E i... |
| cdleme31fv2 41493 | Part of proof of Lemma E i... |
| cdleme31id 41494 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 41495 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 41496 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 41497 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 41498 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 41499 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 41500 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 41501 | Part of proof of Lemma E i... |
| cdlemefr29exN 41502 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 41503 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 41504 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 41505 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 41506 | TODO fix comment. (Contri... |
| cdlemefr29clN 41507 | Show closure of the unique... |
| cdleme43frv1snN 41508 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 41509 | Part of proof of Lemma E i... |
| cdlemefr32fva1 41510 | Part of proof of Lemma E i... |
| cdlemefr31fv1 41511 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 41512 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 41513 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 41514 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 41515 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 41516 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 41517 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 41518 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 41519 | Show closure of the unique... |
| cdleme43fsv1snlem 41520 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 41521 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 41522 | Part of proof of Lemma E i... |
| cdlemefs32fva1 41523 | Part of proof of Lemma E i... |
| cdlemefs31fv1 41524 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 41525 | Value of f(r) when r is an... |
| cdlemefs44 41526 | Value of f_s(r) when r is ... |
| cdlemefr45 41527 | Value of f(r) when r is an... |
| cdlemefr45e 41528 | Explicit expansion of ~ cd... |
| cdlemefs45 41529 | Value of f_s(r) when r is ... |
| cdlemefs45ee 41530 | Explicit expansion of ~ cd... |
| cdlemefs45eN 41531 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 41532 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 41533 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 41534 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 41535 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 41536 | Show closure of ` [_ R / s... |
| cdleme32fva 41537 | Part of proof of Lemma D i... |
| cdleme32fva1 41538 | Part of proof of Lemma D i... |
| cdleme32fvaw 41539 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 41540 | Part of proof of Lemma D i... |
| cdleme32a 41541 | Part of proof of Lemma D i... |
| cdleme32b 41542 | Part of proof of Lemma D i... |
| cdleme32c 41543 | Part of proof of Lemma D i... |
| cdleme32d 41544 | Part of proof of Lemma D i... |
| cdleme32e 41545 | Part of proof of Lemma D i... |
| cdleme32f 41546 | Part of proof of Lemma D i... |
| cdleme32le 41547 | Part of proof of Lemma D i... |
| cdleme35a 41548 | Part of proof of Lemma E i... |
| cdleme35fnpq 41549 | Part of proof of Lemma E i... |
| cdleme35b 41550 | Part of proof of Lemma E i... |
| cdleme35c 41551 | Part of proof of Lemma E i... |
| cdleme35d 41552 | Part of proof of Lemma E i... |
| cdleme35e 41553 | Part of proof of Lemma E i... |
| cdleme35f 41554 | Part of proof of Lemma E i... |
| cdleme35g 41555 | Part of proof of Lemma E i... |
| cdleme35h 41556 | Part of proof of Lemma E i... |
| cdleme35h2 41557 | Part of proof of Lemma E i... |
| cdleme35sn2aw 41558 | Part of proof of Lemma E i... |
| cdleme35sn3a 41559 | Part of proof of Lemma E i... |
| cdleme36a 41560 | Part of proof of Lemma E i... |
| cdleme36m 41561 | Part of proof of Lemma E i... |
| cdleme37m 41562 | Part of proof of Lemma E i... |
| cdleme38m 41563 | Part of proof of Lemma E i... |
| cdleme38n 41564 | Part of proof of Lemma E i... |
| cdleme39a 41565 | Part of proof of Lemma E i... |
| cdleme39n 41566 | Part of proof of Lemma E i... |
| cdleme40m 41567 | Part of proof of Lemma E i... |
| cdleme40n 41568 | Part of proof of Lemma E i... |
| cdleme40v 41569 | Part of proof of Lemma E i... |
| cdleme40w 41570 | Part of proof of Lemma E i... |
| cdleme42a 41571 | Part of proof of Lemma E i... |
| cdleme42c 41572 | Part of proof of Lemma E i... |
| cdleme42d 41573 | Part of proof of Lemma E i... |
| cdleme41sn3aw 41574 | Part of proof of Lemma E i... |
| cdleme41sn4aw 41575 | Part of proof of Lemma E i... |
| cdleme41snaw 41576 | Part of proof of Lemma E i... |
| cdleme41fva11 41577 | Part of proof of Lemma E i... |
| cdleme42b 41578 | Part of proof of Lemma E i... |
| cdleme42e 41579 | Part of proof of Lemma E i... |
| cdleme42f 41580 | Part of proof of Lemma E i... |
| cdleme42g 41581 | Part of proof of Lemma E i... |
| cdleme42h 41582 | Part of proof of Lemma E i... |
| cdleme42i 41583 | Part of proof of Lemma E i... |
| cdleme42k 41584 | Part of proof of Lemma E i... |
| cdleme42ke 41585 | Part of proof of Lemma E i... |
| cdleme42keg 41586 | Part of proof of Lemma E i... |
| cdleme42mN 41587 | Part of proof of Lemma E i... |
| cdleme42mgN 41588 | Part of proof of Lemma E i... |
| cdleme43aN 41589 | Part of proof of Lemma E i... |
| cdleme43bN 41590 | Lemma for Lemma E in [Craw... |
| cdleme43cN 41591 | Part of proof of Lemma E i... |
| cdleme43dN 41592 | Part of proof of Lemma E i... |
| cdleme46f2g2 41593 | Conversion for ` G ` to re... |
| cdleme46f2g1 41594 | Conversion for ` G ` to re... |
| cdleme17d2 41595 | Part of proof of Lemma E i... |
| cdleme17d3 41596 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 41597 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 41598 | Part of proof of Lemma E i... |
| cdleme48fv 41599 | Part of proof of Lemma D i... |
| cdleme48fvg 41600 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 41601 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 41602 | TODO: fix comment. TODO: ... |
| cdleme48b 41603 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 41604 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 41605 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 41606 | TODO: fix comment. (Contr... |
| cdleme4gfv 41607 | Part of proof of Lemma D i... |
| cdlemeg47b 41608 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 41609 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 41610 | Value of g_s(r) when r is ... |
| cdlemeg49le 41611 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 41612 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 41613 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 41614 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 41615 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 41616 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 41617 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 41618 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 41619 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 41620 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 41621 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 41622 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 41623 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 41624 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 41625 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 41626 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 41627 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 41628 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 41629 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 41630 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 41631 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 41632 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 41633 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 41634 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 41635 | TODO: fix comment. (Contr... |
| cdleme48gfv1 41636 | TODO: fix comment. (Contr... |
| cdleme48gfv 41637 | TODO: fix comment. (Contr... |
| cdleme48fgv 41638 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 41639 | Part of proof of Lemma D i... |
| cdleme50lebi 41640 | Part of proof of Lemma D i... |
| cdleme50eq 41641 | Part of proof of Lemma D i... |
| cdleme50f 41642 | Part of proof of Lemma D i... |
| cdleme50f1 41643 | Part of proof of Lemma D i... |
| cdleme50rnlem 41644 | Part of proof of Lemma D i... |
| cdleme50rn 41645 | Part of proof of Lemma D i... |
| cdleme50f1o 41646 | Part of proof of Lemma D i... |
| cdleme50laut 41647 | Part of proof of Lemma D i... |
| cdleme50ldil 41648 | Part of proof of Lemma D i... |
| cdleme50trn1 41649 | Part of proof that ` F ` i... |
| cdleme50trn2a 41650 | Part of proof that ` F ` i... |
| cdleme50trn2 41651 | Part of proof that ` F ` i... |
| cdleme50trn12 41652 | Part of proof that ` F ` i... |
| cdleme50trn3 41653 | Part of proof that ` F ` i... |
| cdleme50trn123 41654 | Part of proof that ` F ` i... |
| cdleme51finvfvN 41655 | Part of proof of Lemma E i... |
| cdleme51finvN 41656 | Part of proof of Lemma E i... |
| cdleme50ltrn 41657 | Part of proof of Lemma E i... |
| cdleme51finvtrN 41658 | Part of proof of Lemma E i... |
| cdleme50ex 41659 | Part of Lemma E in [Crawle... |
| cdleme 41660 | Lemma E in [Crawley] p. 11... |
| cdlemf1 41661 | Part of Lemma F in [Crawle... |
| cdlemf2 41662 | Part of Lemma F in [Crawle... |
| cdlemf 41663 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 41664 | ~ cdlemf with additional c... |
| cdlemftr3 41665 | Special case of ~ cdlemf s... |
| cdlemftr2 41666 | Special case of ~ cdlemf s... |
| cdlemftr1 41667 | Part of proof of Lemma G o... |
| cdlemftr0 41668 | Special case of ~ cdlemf s... |
| trlord 41669 | The ordering of two Hilber... |
| cdlemg1a 41670 | Shorter expression for ` G... |
| cdlemg1b2 41671 | This theorem can be used t... |
| cdlemg1idlemN 41672 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 41673 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 41674 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 41675 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 41676 | This theorem can be used t... |
| cdlemg1idN 41677 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 41678 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 41679 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 41680 | Version of ~ cdleme51finvt... |
| ltrniotaval 41681 | Value of the unique transl... |
| ltrniotacnvval 41682 | Converse value of the uniq... |
| ltrniotaidvalN 41683 | Value of the unique transl... |
| ltrniotavalbN 41684 | Value of the unique transl... |
| cdlemeiota 41685 | A translation is uniquely ... |
| cdlemg1ci2 41686 | Any function of the form o... |
| cdlemg1cN 41687 | Any translation belongs to... |
| cdlemg1cex 41688 | Any translation is one of ... |
| cdlemg2cN 41689 | Any translation belongs to... |
| cdlemg2dN 41690 | This theorem can be used t... |
| cdlemg2cex 41691 | Any translation is one of ... |
| cdlemg2ce 41692 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 41693 | Part of proof of Lemma E i... |
| cdlemg2fvlem 41694 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 41695 | ~ cdleme42keg with simpler... |
| cdlemg2idN 41696 | Version of ~ cdleme31id wi... |
| cdlemg3a 41697 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 41698 | TODO: Replace this with ~... |
| cdlemg2fv 41699 | Value of a translation in ... |
| cdlemg2fv2 41700 | Value of a translation in ... |
| cdlemg2k 41701 | ~ cdleme42keg with simpler... |
| cdlemg2kq 41702 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 41703 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 41704 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 41705 | TODO: Is there a simpler ... |
| cdlemb3 41706 | Given two atoms not under ... |
| cdlemg7fvbwN 41707 | Properties of a translatio... |
| cdlemg4a 41708 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 41709 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 41710 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 41711 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 41712 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 41713 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 41714 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 41715 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 41716 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 41717 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 41718 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 41719 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 41720 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 41721 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 41722 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 41723 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 41724 | Value of a translation com... |
| cdlemg7aN 41725 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 41726 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 41727 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 41728 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 41729 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 41730 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 41731 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 41732 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 41733 | The triples ` <. P , ( F `... |
| cdlemg9 41734 | The triples ` <. P , ( F `... |
| cdlemg10b 41735 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 41736 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 41737 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 41738 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 41739 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 41740 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 41741 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 41742 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 41743 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 41744 | The triples ` <. P , ( F `... |
| cdlemg12c 41745 | The triples ` <. P , ( F `... |
| cdlemg12d 41746 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 41747 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 41748 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 41749 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 41750 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 41751 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 41752 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 41753 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 41754 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 41755 | Eliminate the ` ( F `` P )... |
| cdlemg15 41756 | Eliminate the ` ( (... |
| cdlemg16 41757 | Part of proof of Lemma G o... |
| cdlemg16ALTN 41758 | This version of ~ cdlemg16... |
| cdlemg16z 41759 | Eliminate ` ( ( F `... |
| cdlemg16zz 41760 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 41761 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 41762 | Part of proof of Lemma G i... |
| cdlemg17dN 41763 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 41764 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 41765 | TODO: fix comment. (Contr... |
| cdlemg17f 41766 | TODO: fix comment. (Contr... |
| cdlemg17g 41767 | TODO: fix comment. (Contr... |
| cdlemg17h 41768 | TODO: fix comment. (Contr... |
| cdlemg17i 41769 | TODO: fix comment. (Contr... |
| cdlemg17ir 41770 | TODO: fix comment. (Contr... |
| cdlemg17j 41771 | TODO: fix comment. (Contr... |
| cdlemg17pq 41772 | Utility theorem for swappi... |
| cdlemg17bq 41773 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 41774 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 41775 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 41776 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 41777 | Part of Lemma G of [Crawle... |
| cdlemg18a 41778 | Show two lines are differe... |
| cdlemg18b 41779 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 41780 | Show two lines intersect a... |
| cdlemg18d 41781 | Show two lines intersect a... |
| cdlemg18 41782 | Show two lines intersect a... |
| cdlemg19a 41783 | Show two lines intersect a... |
| cdlemg19 41784 | Show two lines intersect a... |
| cdlemg20 41785 | Show two lines intersect a... |
| cdlemg21 41786 | Version of cdlemg19 with `... |
| cdlemg22 41787 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 41788 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 41789 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 41790 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 41791 | ~ cdlemg16zz restated for ... |
| cdlemg27a 41792 | For use with case when ` (... |
| cdlemg28a 41793 | Part of proof of Lemma G o... |
| cdlemg31b0N 41794 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 41795 | TODO: Fix comment. (Cont... |
| cdlemg27b 41796 | TODO: Fix comment. (Cont... |
| cdlemg31a 41797 | TODO: fix comment. (Contr... |
| cdlemg31b 41798 | TODO: fix comment. (Contr... |
| cdlemg31c 41799 | Show that when ` N ` is an... |
| cdlemg31d 41800 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 41801 | TODO: Fix comment. (Cont... |
| cdlemg33c0 41802 | TODO: Fix comment. (Cont... |
| cdlemg28b 41803 | Part of proof of Lemma G o... |
| cdlemg28 41804 | Part of proof of Lemma G o... |
| cdlemg29 41805 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 41806 | TODO: Fix comment. (Cont... |
| cdlemg33b 41807 | TODO: Fix comment. (Cont... |
| cdlemg33c 41808 | TODO: Fix comment. (Cont... |
| cdlemg33d 41809 | TODO: Fix comment. (Cont... |
| cdlemg33e 41810 | TODO: Fix comment. (Cont... |
| cdlemg33 41811 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 41812 | Use cdlemg33 to eliminate ... |
| cdlemg35 41813 | TODO: Fix comment. TODO:... |
| cdlemg36 41814 | Use cdlemg35 to eliminate ... |
| cdlemg38 41815 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 41816 | Eliminate ` =/= ` conditio... |
| cdlemg40 41817 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 41818 | Convert ~ cdlemg40 to func... |
| ltrnco 41819 | The composition of two tra... |
| trlcocnv 41820 | Swap the arguments of the ... |
| trlcoabs 41821 | Absorption into a composit... |
| trlcoabs2N 41822 | Absorption of the trace of... |
| trlcoat 41823 | The trace of a composition... |
| trlcocnvat 41824 | Commonly used special case... |
| trlconid 41825 | The composition of two dif... |
| trlcolem 41826 | Lemma for ~ trlco . (Cont... |
| trlco 41827 | The trace of a composition... |
| trlcone 41828 | If two translations have d... |
| cdlemg42 41829 | Part of proof of Lemma G o... |
| cdlemg43 41830 | Part of proof of Lemma G o... |
| cdlemg44a 41831 | Part of proof of Lemma G o... |
| cdlemg44b 41832 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 41833 | Part of proof of Lemma G o... |
| cdlemg47a 41834 | TODO: fix comment. TODO: ... |
| cdlemg46 41835 | Part of proof of Lemma G o... |
| cdlemg47 41836 | Part of proof of Lemma G o... |
| cdlemg48 41837 | Eliminate ` h ` from ~ cdl... |
| ltrncom 41838 | Composition is commutative... |
| ltrnco4 41839 | Rearrange a composition of... |
| trljco 41840 | Trace joined with trace of... |
| trljco2 41841 | Trace joined with trace of... |
| tgrpfset 41844 | The translation group maps... |
| tgrpset 41845 | The translation group for ... |
| tgrpbase 41846 | The base set of the transl... |
| tgrpopr 41847 | The group operation of the... |
| tgrpov 41848 | The group operation value ... |
| tgrpgrplem 41849 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 41850 | The translation group is a... |
| tgrpabl 41851 | The translation group is a... |
| tendofset 41858 | The set of all trace-prese... |
| tendoset 41859 | The set of trace-preservin... |
| istendo 41860 | The predicate "is a trace-... |
| tendotp 41861 | Trace-preserving property ... |
| istendod 41862 | Deduce the predicate "is a... |
| tendof 41863 | Functionality of a trace-p... |
| tendoeq1 41864 | Condition determining equa... |
| tendovalco 41865 | Value of composition of tr... |
| tendocoval 41866 | Value of composition of en... |
| tendocl 41867 | Closure of a trace-preserv... |
| tendoco2 41868 | Distribution of compositio... |
| tendoidcl 41869 | The identity is a trace-pr... |
| tendo1mul 41870 | Multiplicative identity mu... |
| tendo1mulr 41871 | Multiplicative identity mu... |
| tendococl 41872 | The composition of two tra... |
| tendoid 41873 | The identity value of a tr... |
| tendoeq2 41874 | Condition determining equa... |
| tendoplcbv 41875 | Define sum operation for t... |
| tendopl 41876 | Value of endomorphism sum ... |
| tendopl2 41877 | Value of result of endomor... |
| tendoplcl2 41878 | Value of result of endomor... |
| tendoplco2 41879 | Value of result of endomor... |
| tendopltp 41880 | Trace-preserving property ... |
| tendoplcl 41881 | Endomorphism sum is a trac... |
| tendoplcom 41882 | The endomorphism sum opera... |
| tendoplass 41883 | The endomorphism sum opera... |
| tendodi1 41884 | Endomorphism composition d... |
| tendodi2 41885 | Endomorphism composition d... |
| tendo0cbv 41886 | Define additive identity f... |
| tendo02 41887 | Value of additive identity... |
| tendo0co2 41888 | The additive identity trac... |
| tendo0tp 41889 | Trace-preserving property ... |
| tendo0cl 41890 | The additive identity is a... |
| tendo0pl 41891 | Property of the additive i... |
| tendo0plr 41892 | Property of the additive i... |
| tendoicbv 41893 | Define inverse function fo... |
| tendoi 41894 | Value of inverse endomorph... |
| tendoi2 41895 | Value of additive inverse ... |
| tendoicl 41896 | Closure of the additive in... |
| tendoipl 41897 | Property of the additive i... |
| tendoipl2 41898 | Property of the additive i... |
| erngfset 41899 | The division rings on trac... |
| erngset 41900 | The division ring on trace... |
| erngbase 41901 | The base set of the divisi... |
| erngfplus 41902 | Ring addition operation. ... |
| erngplus 41903 | Ring addition operation. ... |
| erngplus2 41904 | Ring addition operation. ... |
| erngfmul 41905 | Ring multiplication operat... |
| erngmul 41906 | Ring addition operation. ... |
| erngfset-rN 41907 | The division rings on trac... |
| erngset-rN 41908 | The division ring on trace... |
| erngbase-rN 41909 | The base set of the divisi... |
| erngfplus-rN 41910 | Ring addition operation. ... |
| erngplus-rN 41911 | Ring addition operation. ... |
| erngplus2-rN 41912 | Ring addition operation. ... |
| erngfmul-rN 41913 | Ring multiplication operat... |
| erngmul-rN 41914 | Ring addition operation. ... |
| cdlemh1 41915 | Part of proof of Lemma H o... |
| cdlemh2 41916 | Part of proof of Lemma H o... |
| cdlemh 41917 | Lemma H of [Crawley] p. 11... |
| cdlemi1 41918 | Part of proof of Lemma I o... |
| cdlemi2 41919 | Part of proof of Lemma I o... |
| cdlemi 41920 | Lemma I of [Crawley] p. 11... |
| cdlemj1 41921 | Part of proof of Lemma J o... |
| cdlemj2 41922 | Part of proof of Lemma J o... |
| cdlemj3 41923 | Part of proof of Lemma J o... |
| tendocan 41924 | Cancellation law: if the v... |
| tendoid0 41925 | A trace-preserving endomor... |
| tendo0mul 41926 | Additive identity multipli... |
| tendo0mulr 41927 | Additive identity multipli... |
| tendo1ne0 41928 | The identity (unity) is no... |
| tendoconid 41929 | The composition (product) ... |
| tendotr 41930 | The trace of the value of ... |
| cdlemk1 41931 | Part of proof of Lemma K o... |
| cdlemk2 41932 | Part of proof of Lemma K o... |
| cdlemk3 41933 | Part of proof of Lemma K o... |
| cdlemk4 41934 | Part of proof of Lemma K o... |
| cdlemk5a 41935 | Part of proof of Lemma K o... |
| cdlemk5 41936 | Part of proof of Lemma K o... |
| cdlemk6 41937 | Part of proof of Lemma K o... |
| cdlemk8 41938 | Part of proof of Lemma K o... |
| cdlemk9 41939 | Part of proof of Lemma K o... |
| cdlemk9bN 41940 | Part of proof of Lemma K o... |
| cdlemki 41941 | Part of proof of Lemma K o... |
| cdlemkvcl 41942 | Part of proof of Lemma K o... |
| cdlemk10 41943 | Part of proof of Lemma K o... |
| cdlemksv 41944 | Part of proof of Lemma K o... |
| cdlemksel 41945 | Part of proof of Lemma K o... |
| cdlemksat 41946 | Part of proof of Lemma K o... |
| cdlemksv2 41947 | Part of proof of Lemma K o... |
| cdlemk7 41948 | Part of proof of Lemma K o... |
| cdlemk11 41949 | Part of proof of Lemma K o... |
| cdlemk12 41950 | Part of proof of Lemma K o... |
| cdlemkoatnle 41951 | Utility lemma. (Contribut... |
| cdlemk13 41952 | Part of proof of Lemma K o... |
| cdlemkole 41953 | Utility lemma. (Contribut... |
| cdlemk14 41954 | Part of proof of Lemma K o... |
| cdlemk15 41955 | Part of proof of Lemma K o... |
| cdlemk16a 41956 | Part of proof of Lemma K o... |
| cdlemk16 41957 | Part of proof of Lemma K o... |
| cdlemk17 41958 | Part of proof of Lemma K o... |
| cdlemk1u 41959 | Part of proof of Lemma K o... |
| cdlemk5auN 41960 | Part of proof of Lemma K o... |
| cdlemk5u 41961 | Part of proof of Lemma K o... |
| cdlemk6u 41962 | Part of proof of Lemma K o... |
| cdlemkj 41963 | Part of proof of Lemma K o... |
| cdlemkuvN 41964 | Part of proof of Lemma K o... |
| cdlemkuel 41965 | Part of proof of Lemma K o... |
| cdlemkuat 41966 | Part of proof of Lemma K o... |
| cdlemkuv2 41967 | Part of proof of Lemma K o... |
| cdlemk18 41968 | Part of proof of Lemma K o... |
| cdlemk19 41969 | Part of proof of Lemma K o... |
| cdlemk7u 41970 | Part of proof of Lemma K o... |
| cdlemk11u 41971 | Part of proof of Lemma K o... |
| cdlemk12u 41972 | Part of proof of Lemma K o... |
| cdlemk21N 41973 | Part of proof of Lemma K o... |
| cdlemk20 41974 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 41975 | Utility lemma. (Contribut... |
| cdlemk13-2N 41976 | Part of proof of Lemma K o... |
| cdlemkole-2N 41977 | Utility lemma. (Contribut... |
| cdlemk14-2N 41978 | Part of proof of Lemma K o... |
| cdlemk15-2N 41979 | Part of proof of Lemma K o... |
| cdlemk16-2N 41980 | Part of proof of Lemma K o... |
| cdlemk17-2N 41981 | Part of proof of Lemma K o... |
| cdlemkj-2N 41982 | Part of proof of Lemma K o... |
| cdlemkuv-2N 41983 | Part of proof of Lemma K o... |
| cdlemkuel-2N 41984 | Part of proof of Lemma K o... |
| cdlemkuv2-2 41985 | Part of proof of Lemma K o... |
| cdlemk18-2N 41986 | Part of proof of Lemma K o... |
| cdlemk19-2N 41987 | Part of proof of Lemma K o... |
| cdlemk7u-2N 41988 | Part of proof of Lemma K o... |
| cdlemk11u-2N 41989 | Part of proof of Lemma K o... |
| cdlemk12u-2N 41990 | Part of proof of Lemma K o... |
| cdlemk21-2N 41991 | Part of proof of Lemma K o... |
| cdlemk20-2N 41992 | Part of proof of Lemma K o... |
| cdlemk22 41993 | Part of proof of Lemma K o... |
| cdlemk30 41994 | Part of proof of Lemma K o... |
| cdlemkuu 41995 | Convert between function a... |
| cdlemk31 41996 | Part of proof of Lemma K o... |
| cdlemk32 41997 | Part of proof of Lemma K o... |
| cdlemkuel-3 41998 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 41999 | Part of proof of Lemma K o... |
| cdlemk18-3N 42000 | Part of proof of Lemma K o... |
| cdlemk22-3 42001 | Part of proof of Lemma K o... |
| cdlemk23-3 42002 | Part of proof of Lemma K o... |
| cdlemk24-3 42003 | Part of proof of Lemma K o... |
| cdlemk25-3 42004 | Part of proof of Lemma K o... |
| cdlemk26b-3 42005 | Part of proof of Lemma K o... |
| cdlemk26-3 42006 | Part of proof of Lemma K o... |
| cdlemk27-3 42007 | Part of proof of Lemma K o... |
| cdlemk28-3 42008 | Part of proof of Lemma K o... |
| cdlemk33N 42009 | Part of proof of Lemma K o... |
| cdlemk34 42010 | Part of proof of Lemma K o... |
| cdlemk29-3 42011 | Part of proof of Lemma K o... |
| cdlemk35 42012 | Part of proof of Lemma K o... |
| cdlemk36 42013 | Part of proof of Lemma K o... |
| cdlemk37 42014 | Part of proof of Lemma K o... |
| cdlemk38 42015 | Part of proof of Lemma K o... |
| cdlemk39 42016 | Part of proof of Lemma K o... |
| cdlemk40 42017 | TODO: fix comment. (Contr... |
| cdlemk40t 42018 | TODO: fix comment. (Contr... |
| cdlemk40f 42019 | TODO: fix comment. (Contr... |
| cdlemk41 42020 | Part of proof of Lemma K o... |
| cdlemkfid1N 42021 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 42022 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 42023 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 42024 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 42025 | TODO: is this useful or sh... |
| cdlemky 42026 | Part of proof of Lemma K o... |
| cdlemkyu 42027 | Convert between function a... |
| cdlemkyuu 42028 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 42029 | Part of proof of Lemma K o... |
| cdlemk19ylem 42030 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 42031 | Part of proof of Lemma K o... |
| cdlemk19y 42032 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 42033 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 42034 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 42035 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 42036 | The value of the tau funct... |
| cdlemk35s 42037 | Substitution version of ~ ... |
| cdlemk35s-id 42038 | Substitution version of ~ ... |
| cdlemk39s 42039 | Substitution version of ~ ... |
| cdlemk39s-id 42040 | Substitution version of ~ ... |
| cdlemk42 42041 | Part of proof of Lemma K o... |
| cdlemk19xlem 42042 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 42043 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 42044 | Part of proof of Lemma K o... |
| cdlemk11tc 42045 | Part of proof of Lemma K o... |
| cdlemk11t 42046 | Part of proof of Lemma K o... |
| cdlemk45 42047 | Part of proof of Lemma K o... |
| cdlemk46 42048 | Part of proof of Lemma K o... |
| cdlemk47 42049 | Part of proof of Lemma K o... |
| cdlemk48 42050 | Part of proof of Lemma K o... |
| cdlemk49 42051 | Part of proof of Lemma K o... |
| cdlemk50 42052 | Part of proof of Lemma K o... |
| cdlemk51 42053 | Part of proof of Lemma K o... |
| cdlemk52 42054 | Part of proof of Lemma K o... |
| cdlemk53a 42055 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 42056 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 42057 | Part of proof of Lemma K o... |
| cdlemk54 42058 | Part of proof of Lemma K o... |
| cdlemk55a 42059 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 42060 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 42061 | Part of proof of Lemma K o... |
| cdlemkyyN 42062 | Part of proof of Lemma K o... |
| cdlemk43N 42063 | Part of proof of Lemma K o... |
| cdlemk35u 42064 | Substitution version of ~ ... |
| cdlemk55u1 42065 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 42066 | Part of proof of Lemma K o... |
| cdlemk39u1 42067 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 42068 | Part of proof of Lemma K o... |
| cdlemk19u1 42069 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 42070 | Part of Lemma K of [Crawle... |
| cdlemk56 42071 | Part of Lemma K of [Crawle... |
| cdlemk19w 42072 | Use a fixed element to eli... |
| cdlemk56w 42073 | Use a fixed element to eli... |
| cdlemk 42074 | Lemma K of [Crawley] p. 11... |
| tendoex 42075 | Generalization of Lemma K ... |
| cdleml1N 42076 | Part of proof of Lemma L o... |
| cdleml2N 42077 | Part of proof of Lemma L o... |
| cdleml3N 42078 | Part of proof of Lemma L o... |
| cdleml4N 42079 | Part of proof of Lemma L o... |
| cdleml5N 42080 | Part of proof of Lemma L o... |
| cdleml6 42081 | Part of proof of Lemma L o... |
| cdleml7 42082 | Part of proof of Lemma L o... |
| cdleml8 42083 | Part of proof of Lemma L o... |
| cdleml9 42084 | Part of proof of Lemma L o... |
| dva1dim 42085 | Two expressions for the 1-... |
| dvhb1dimN 42086 | Two expressions for the 1-... |
| erng1lem 42087 | Value of the endomorphism ... |
| erngdvlem1 42088 | Lemma for ~ eringring . (... |
| erngdvlem2N 42089 | Lemma for ~ eringring . (... |
| erngdvlem3 42090 | Lemma for ~ eringring . (... |
| erngdvlem4 42091 | Lemma for ~ erngdv . (Con... |
| eringring 42092 | An endomorphism ring is a ... |
| erngdv 42093 | An endomorphism ring is a ... |
| erng0g 42094 | The division ring zero of ... |
| erng1r 42095 | The division ring unity of... |
| erngdvlem1-rN 42096 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 42097 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 42098 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 42099 | Lemma for ~ erngdv . (Con... |
| erngring-rN 42100 | An endomorphism ring is a ... |
| erngdv-rN 42101 | An endomorphism ring is a ... |
| dvafset 42104 | The constructed partial ve... |
| dvaset 42105 | The constructed partial ve... |
| dvasca 42106 | The ring base set of the c... |
| dvabase 42107 | The ring base set of the c... |
| dvafplusg 42108 | Ring addition operation fo... |
| dvaplusg 42109 | Ring addition operation fo... |
| dvaplusgv 42110 | Ring addition operation fo... |
| dvafmulr 42111 | Ring multiplication operat... |
| dvamulr 42112 | Ring multiplication operat... |
| dvavbase 42113 | The vectors (vector base s... |
| dvafvadd 42114 | The vector sum operation f... |
| dvavadd 42115 | Ring addition operation fo... |
| dvafvsca 42116 | Ring addition operation fo... |
| dvavsca 42117 | Ring addition operation fo... |
| tendospcl 42118 | Closure of endomorphism sc... |
| tendospass 42119 | Associative law for endomo... |
| tendospdi1 42120 | Forward distributive law f... |
| tendocnv 42121 | Converse of a trace-preser... |
| tendospdi2 42122 | Reverse distributive law f... |
| tendospcanN 42123 | Cancellation law for trace... |
| dvaabl 42124 | The constructed partial ve... |
| dvalveclem 42125 | Lemma for ~ dvalvec . (Co... |
| dvalvec 42126 | The constructed partial ve... |
| dva0g 42127 | The zero vector of partial... |
| diaffval 42130 | The partial isomorphism A ... |
| diafval 42131 | The partial isomorphism A ... |
| diaval 42132 | The partial isomorphism A ... |
| diaelval 42133 | Member of the partial isom... |
| diafn 42134 | Functionality and domain o... |
| diadm 42135 | Domain of the partial isom... |
| diaeldm 42136 | Member of domain of the pa... |
| diadmclN 42137 | A member of domain of the ... |
| diadmleN 42138 | A member of domain of the ... |
| dian0 42139 | The value of the partial i... |
| dia0eldmN 42140 | The lattice zero belongs t... |
| dia1eldmN 42141 | The fiducial hyperplane (t... |
| diass 42142 | The value of the partial i... |
| diael 42143 | A member of the value of t... |
| diatrl 42144 | Trace of a member of the p... |
| diaelrnN 42145 | Any value of the partial i... |
| dialss 42146 | The value of partial isomo... |
| diaord 42147 | The partial isomorphism A ... |
| dia11N 42148 | The partial isomorphism A ... |
| diaf11N 42149 | The partial isomorphism A ... |
| diaclN 42150 | Closure of partial isomorp... |
| diacnvclN 42151 | Closure of partial isomorp... |
| dia0 42152 | The value of the partial i... |
| dia1N 42153 | The value of the partial i... |
| dia1elN 42154 | The largest subspace in th... |
| diaglbN 42155 | Partial isomorphism A of a... |
| diameetN 42156 | Partial isomorphism A of a... |
| diainN 42157 | Inverse partial isomorphis... |
| diaintclN 42158 | The intersection of partia... |
| diasslssN 42159 | The partial isomorphism A ... |
| diassdvaN 42160 | The partial isomorphism A ... |
| dia1dim 42161 | Two expressions for the 1-... |
| dia1dim2 42162 | Two expressions for a 1-di... |
| dia1dimid 42163 | A vector (translation) bel... |
| dia2dimlem1 42164 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 42165 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 42166 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 42167 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 42168 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 42169 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 42170 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 42171 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 42172 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 42173 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 42174 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 42175 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 42176 | Lemma for ~ dia2dim . Eli... |
| dia2dim 42177 | A two-dimensional subspace... |
| dvhfset 42180 | The constructed full vecto... |
| dvhset 42181 | The constructed full vecto... |
| dvhsca 42182 | The ring of scalars of the... |
| dvhbase 42183 | The ring base set of the c... |
| dvhfplusr 42184 | Ring addition operation fo... |
| dvhfmulr 42185 | Ring multiplication operat... |
| dvhmulr 42186 | Ring multiplication operat... |
| dvhvbase 42187 | The vectors (vector base s... |
| dvhelvbasei 42188 | Vector membership in the c... |
| dvhvaddcbv 42189 | Change bound variables to ... |
| dvhvaddval 42190 | The vector sum operation f... |
| dvhfvadd 42191 | The vector sum operation f... |
| dvhvadd 42192 | The vector sum operation f... |
| dvhopvadd 42193 | The vector sum operation f... |
| dvhopvadd2 42194 | The vector sum operation f... |
| dvhvaddcl 42195 | Closure of the vector sum ... |
| dvhvaddcomN 42196 | Commutativity of vector su... |
| dvhvaddass 42197 | Associativity of vector su... |
| dvhvscacbv 42198 | Change bound variables to ... |
| dvhvscaval 42199 | The scalar product operati... |
| dvhfvsca 42200 | Scalar product operation f... |
| dvhvsca 42201 | Scalar product operation f... |
| dvhopvsca 42202 | Scalar product operation f... |
| dvhvscacl 42203 | Closure of the scalar prod... |
| tendoinvcl 42204 | Closure of multiplicative ... |
| tendolinv 42205 | Left multiplicative invers... |
| tendorinv 42206 | Right multiplicative inver... |
| dvhgrp 42207 | The full vector space ` U ... |
| dvhlveclem 42208 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 42209 | The full vector space ` U ... |
| dvhlmod 42210 | The full vector space ` U ... |
| dvh0g 42211 | The zero vector of vector ... |
| dvheveccl 42212 | Properties of a unit vecto... |
| dvhopclN 42213 | Closure of a ` DVecH ` vec... |
| dvhopaddN 42214 | Sum of ` DVecH ` vectors e... |
| dvhopspN 42215 | Scalar product of ` DVecH ... |
| dvhopN 42216 | Decompose a ` DVecH ` vect... |
| dvhopellsm 42217 | Ordered pair membership in... |
| cdlemm10N 42218 | The image of the map ` G `... |
| docaffvalN 42221 | Subspace orthocomplement f... |
| docafvalN 42222 | Subspace orthocomplement f... |
| docavalN 42223 | Subspace orthocomplement f... |
| docaclN 42224 | Closure of subspace orthoc... |
| diaocN 42225 | Value of partial isomorphi... |
| doca2N 42226 | Double orthocomplement of ... |
| doca3N 42227 | Double orthocomplement of ... |
| dvadiaN 42228 | Any closed subspace is a m... |
| diarnN 42229 | Partial isomorphism A maps... |
| diaf1oN 42230 | The partial isomorphism A ... |
| djaffvalN 42233 | Subspace join for ` DVecA ... |
| djafvalN 42234 | Subspace join for ` DVecA ... |
| djavalN 42235 | Subspace join for ` DVecA ... |
| djaclN 42236 | Closure of subspace join f... |
| djajN 42237 | Transfer lattice join to `... |
| dibffval 42240 | The partial isomorphism B ... |
| dibfval 42241 | The partial isomorphism B ... |
| dibval 42242 | The partial isomorphism B ... |
| dibopelvalN 42243 | Member of the partial isom... |
| dibval2 42244 | Value of the partial isomo... |
| dibopelval2 42245 | Member of the partial isom... |
| dibval3N 42246 | Value of the partial isomo... |
| dibelval3 42247 | Member of the partial isom... |
| dibopelval3 42248 | Member of the partial isom... |
| dibelval1st 42249 | Membership in value of the... |
| dibelval1st1 42250 | Membership in value of the... |
| dibelval1st2N 42251 | Membership in value of the... |
| dibelval2nd 42252 | Membership in value of the... |
| dibn0 42253 | The value of the partial i... |
| dibfna 42254 | Functionality and domain o... |
| dibdiadm 42255 | Domain of the partial isom... |
| dibfnN 42256 | Functionality and domain o... |
| dibdmN 42257 | Domain of the partial isom... |
| dibeldmN 42258 | Member of domain of the pa... |
| dibord 42259 | The isomorphism B for a la... |
| dib11N 42260 | The isomorphism B for a la... |
| dibf11N 42261 | The partial isomorphism A ... |
| dibclN 42262 | Closure of partial isomorp... |
| dibvalrel 42263 | The value of partial isomo... |
| dib0 42264 | The value of partial isomo... |
| dib1dim 42265 | Two expressions for the 1-... |
| dibglbN 42266 | Partial isomorphism B of a... |
| dibintclN 42267 | The intersection of partia... |
| dib1dim2 42268 | Two expressions for a 1-di... |
| dibss 42269 | The partial isomorphism B ... |
| diblss 42270 | The value of partial isomo... |
| diblsmopel 42271 | Membership in subspace sum... |
| dicffval 42274 | The partial isomorphism C ... |
| dicfval 42275 | The partial isomorphism C ... |
| dicval 42276 | The partial isomorphism C ... |
| dicopelval 42277 | Membership in value of the... |
| dicelvalN 42278 | Membership in value of the... |
| dicval2 42279 | The partial isomorphism C ... |
| dicelval3 42280 | Member of the partial isom... |
| dicopelval2 42281 | Membership in value of the... |
| dicelval2N 42282 | Membership in value of the... |
| dicfnN 42283 | Functionality and domain o... |
| dicdmN 42284 | Domain of the partial isom... |
| dicvalrelN 42285 | The value of partial isomo... |
| dicssdvh 42286 | The partial isomorphism C ... |
| dicelval1sta 42287 | Membership in value of the... |
| dicelval1stN 42288 | Membership in value of the... |
| dicelval2nd 42289 | Membership in value of the... |
| dicvaddcl 42290 | Membership in value of the... |
| dicvscacl 42291 | Membership in value of the... |
| dicn0 42292 | The value of the partial i... |
| diclss 42293 | The value of partial isomo... |
| diclspsn 42294 | The value of isomorphism C... |
| cdlemn2 42295 | Part of proof of Lemma N o... |
| cdlemn2a 42296 | Part of proof of Lemma N o... |
| cdlemn3 42297 | Part of proof of Lemma N o... |
| cdlemn4 42298 | Part of proof of Lemma N o... |
| cdlemn4a 42299 | Part of proof of Lemma N o... |
| cdlemn5pre 42300 | Part of proof of Lemma N o... |
| cdlemn5 42301 | Part of proof of Lemma N o... |
| cdlemn6 42302 | Part of proof of Lemma N o... |
| cdlemn7 42303 | Part of proof of Lemma N o... |
| cdlemn8 42304 | Part of proof of Lemma N o... |
| cdlemn9 42305 | Part of proof of Lemma N o... |
| cdlemn10 42306 | Part of proof of Lemma N o... |
| cdlemn11a 42307 | Part of proof of Lemma N o... |
| cdlemn11b 42308 | Part of proof of Lemma N o... |
| cdlemn11c 42309 | Part of proof of Lemma N o... |
| cdlemn11pre 42310 | Part of proof of Lemma N o... |
| cdlemn11 42311 | Part of proof of Lemma N o... |
| cdlemn 42312 | Lemma N of [Crawley] p. 12... |
| dihordlem6 42313 | Part of proof of Lemma N o... |
| dihordlem7 42314 | Part of proof of Lemma N o... |
| dihordlem7b 42315 | Part of proof of Lemma N o... |
| dihjustlem 42316 | Part of proof after Lemma ... |
| dihjust 42317 | Part of proof after Lemma ... |
| dihord1 42318 | Part of proof after Lemma ... |
| dihord2a 42319 | Part of proof after Lemma ... |
| dihord2b 42320 | Part of proof after Lemma ... |
| dihord2cN 42321 | Part of proof after Lemma ... |
| dihord11b 42322 | Part of proof after Lemma ... |
| dihord10 42323 | Part of proof after Lemma ... |
| dihord11c 42324 | Part of proof after Lemma ... |
| dihord2pre 42325 | Part of proof after Lemma ... |
| dihord2pre2 42326 | Part of proof after Lemma ... |
| dihord2 42327 | Part of proof after Lemma ... |
| dihffval 42330 | The isomorphism H for a la... |
| dihfval 42331 | Isomorphism H for a lattic... |
| dihval 42332 | Value of isomorphism H for... |
| dihvalc 42333 | Value of isomorphism H for... |
| dihlsscpre 42334 | Closure of isomorphism H f... |
| dihvalcqpre 42335 | Value of isomorphism H for... |
| dihvalcq 42336 | Value of isomorphism H for... |
| dihvalb 42337 | Value of isomorphism H for... |
| dihopelvalbN 42338 | Ordered pair member of the... |
| dihvalcqat 42339 | Value of isomorphism H for... |
| dih1dimb 42340 | Two expressions for a 1-di... |
| dih1dimb2 42341 | Isomorphism H at an atom u... |
| dih1dimc 42342 | Isomorphism H at an atom n... |
| dib2dim 42343 | Extend ~ dia2dim to partia... |
| dih2dimb 42344 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 42345 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 42346 | Ordered pair member of the... |
| dihvalcq2 42347 | Value of isomorphism H for... |
| dihopelvalcpre 42348 | Member of value of isomorp... |
| dihopelvalc 42349 | Member of value of isomorp... |
| dihlss 42350 | The value of isomorphism H... |
| dihss 42351 | The value of isomorphism H... |
| dihssxp 42352 | An isomorphism H value is ... |
| dihopcl 42353 | Closure of an ordered pair... |
| xihopellsmN 42354 | Ordered pair membership in... |
| dihopellsm 42355 | Ordered pair membership in... |
| dihord6apre 42356 | Part of proof that isomorp... |
| dihord3 42357 | The isomorphism H for a la... |
| dihord4 42358 | The isomorphism H for a la... |
| dihord5b 42359 | Part of proof that isomorp... |
| dihord6b 42360 | Part of proof that isomorp... |
| dihord6a 42361 | Part of proof that isomorp... |
| dihord5apre 42362 | Part of proof that isomorp... |
| dihord5a 42363 | Part of proof that isomorp... |
| dihord 42364 | The isomorphism H is order... |
| dih11 42365 | The isomorphism H is one-t... |
| dihf11lem 42366 | Functionality of the isomo... |
| dihf11 42367 | The isomorphism H for a la... |
| dihfn 42368 | Functionality and domain o... |
| dihdm 42369 | Domain of isomorphism H. (... |
| dihcl 42370 | Closure of isomorphism H. ... |
| dihcnvcl 42371 | Closure of isomorphism H c... |
| dihcnvid1 42372 | The converse isomorphism o... |
| dihcnvid2 42373 | The isomorphism of a conve... |
| dihcnvord 42374 | Ordering property for conv... |
| dihcnv11 42375 | The converse of isomorphis... |
| dihsslss 42376 | The isomorphism H maps to ... |
| dihrnlss 42377 | The isomorphism H maps to ... |
| dihrnss 42378 | The isomorphism H maps to ... |
| dihvalrel 42379 | The value of isomorphism H... |
| dih0 42380 | The value of isomorphism H... |
| dih0bN 42381 | A lattice element is zero ... |
| dih0vbN 42382 | A vector is zero iff its s... |
| dih0cnv 42383 | The isomorphism H converse... |
| dih0rn 42384 | The zero subspace belongs ... |
| dih0sb 42385 | A subspace is zero iff the... |
| dih1 42386 | The value of isomorphism H... |
| dih1rn 42387 | The full vector space belo... |
| dih1cnv 42388 | The isomorphism H converse... |
| dihwN 42389 | Value of isomorphism H at ... |
| dihmeetlem1N 42390 | Isomorphism H of a conjunc... |
| dihglblem5apreN 42391 | A conjunction property of ... |
| dihglblem5aN 42392 | A conjunction property of ... |
| dihglblem2aN 42393 | Lemma for isomorphism H of... |
| dihglblem2N 42394 | The GLB of a set of lattic... |
| dihglblem3N 42395 | Isomorphism H of a lattice... |
| dihglblem3aN 42396 | Isomorphism H of a lattice... |
| dihglblem4 42397 | Isomorphism H of a lattice... |
| dihglblem5 42398 | Isomorphism H of a lattice... |
| dihmeetlem2N 42399 | Isomorphism H of a conjunc... |
| dihglbcpreN 42400 | Isomorphism H of a lattice... |
| dihglbcN 42401 | Isomorphism H of a lattice... |
| dihmeetcN 42402 | Isomorphism H of a lattice... |
| dihmeetbN 42403 | Isomorphism H of a lattice... |
| dihmeetbclemN 42404 | Lemma for isomorphism H of... |
| dihmeetlem3N 42405 | Lemma for isomorphism H of... |
| dihmeetlem4preN 42406 | Lemma for isomorphism H of... |
| dihmeetlem4N 42407 | Lemma for isomorphism H of... |
| dihmeetlem5 42408 | Part of proof that isomorp... |
| dihmeetlem6 42409 | Lemma for isomorphism H of... |
| dihmeetlem7N 42410 | Lemma for isomorphism H of... |
| dihjatc1 42411 | Lemma for isomorphism H of... |
| dihjatc2N 42412 | Isomorphism H of join with... |
| dihjatc3 42413 | Isomorphism H of join with... |
| dihmeetlem8N 42414 | Lemma for isomorphism H of... |
| dihmeetlem9N 42415 | Lemma for isomorphism H of... |
| dihmeetlem10N 42416 | Lemma for isomorphism H of... |
| dihmeetlem11N 42417 | Lemma for isomorphism H of... |
| dihmeetlem12N 42418 | Lemma for isomorphism H of... |
| dihmeetlem13N 42419 | Lemma for isomorphism H of... |
| dihmeetlem14N 42420 | Lemma for isomorphism H of... |
| dihmeetlem15N 42421 | Lemma for isomorphism H of... |
| dihmeetlem16N 42422 | Lemma for isomorphism H of... |
| dihmeetlem17N 42423 | Lemma for isomorphism H of... |
| dihmeetlem18N 42424 | Lemma for isomorphism H of... |
| dihmeetlem19N 42425 | Lemma for isomorphism H of... |
| dihmeetlem20N 42426 | Lemma for isomorphism H of... |
| dihmeetALTN 42427 | Isomorphism H of a lattice... |
| dih1dimatlem0 42428 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 42429 | Lemma for ~ dih1dimat . (... |
| dih1dimat 42430 | Any 1-dimensional subspace... |
| dihlsprn 42431 | The span of a vector belon... |
| dihlspsnssN 42432 | A subspace included in a 1... |
| dihlspsnat 42433 | The inverse isomorphism H ... |
| dihatlat 42434 | The isomorphism H of an at... |
| dihat 42435 | There exists at least one ... |
| dihpN 42436 | The value of isomorphism H... |
| dihlatat 42437 | The reverse isomorphism H ... |
| dihatexv 42438 | There is a nonzero vector ... |
| dihatexv2 42439 | There is a nonzero vector ... |
| dihglblem6 42440 | Isomorphism H of a lattice... |
| dihglb 42441 | Isomorphism H of a lattice... |
| dihglb2 42442 | Isomorphism H of a lattice... |
| dihmeet 42443 | Isomorphism H of a lattice... |
| dihintcl 42444 | The intersection of closed... |
| dihmeetcl 42445 | Closure of closed subspace... |
| dihmeet2 42446 | Reverse isomorphism H of a... |
| dochffval 42449 | Subspace orthocomplement f... |
| dochfval 42450 | Subspace orthocomplement f... |
| dochval 42451 | Subspace orthocomplement f... |
| dochval2 42452 | Subspace orthocomplement f... |
| dochcl 42453 | Closure of subspace orthoc... |
| dochlss 42454 | A subspace orthocomplement... |
| dochssv 42455 | A subspace orthocomplement... |
| dochfN 42456 | Domain and codomain of the... |
| dochvalr 42457 | Orthocomplement of a close... |
| doch0 42458 | Orthocomplement of the zer... |
| doch1 42459 | Orthocomplement of the uni... |
| dochoc0 42460 | The zero subspace is close... |
| dochoc1 42461 | The unit subspace (all vec... |
| dochvalr2 42462 | Orthocomplement of a close... |
| dochvalr3 42463 | Orthocomplement of a close... |
| doch2val2 42464 | Double orthocomplement for... |
| dochss 42465 | Subset law for orthocomple... |
| dochocss 42466 | Double negative law for or... |
| dochoc 42467 | Double negative law for or... |
| dochsscl 42468 | If a set of vectors is inc... |
| dochoccl 42469 | A set of vectors is closed... |
| dochord 42470 | Ordering law for orthocomp... |
| dochord2N 42471 | Ordering law for orthocomp... |
| dochord3 42472 | Ordering law for orthocomp... |
| doch11 42473 | Orthocomplement is one-to-... |
| dochsordN 42474 | Strict ordering law for or... |
| dochn0nv 42475 | An orthocomplement is nonz... |
| dihoml4c 42476 | Version of ~ dihoml4 with ... |
| dihoml4 42477 | Orthomodular law for const... |
| dochspss 42478 | The span of a set of vecto... |
| dochocsp 42479 | The span of an orthocomple... |
| dochspocN 42480 | The span of an orthocomple... |
| dochocsn 42481 | The double orthocomplement... |
| dochsncom 42482 | Swap vectors in an orthoco... |
| dochsat 42483 | The double orthocomplement... |
| dochshpncl 42484 | If a hyperplane is not clo... |
| dochlkr 42485 | Equivalent conditions for ... |
| dochkrshp 42486 | The closure of a kernel is... |
| dochkrshp2 42487 | Properties of the closure ... |
| dochkrshp3 42488 | Properties of the closure ... |
| dochkrshp4 42489 | Properties of the closure ... |
| dochdmj1 42490 | De Morgan-like law for sub... |
| dochnoncon 42491 | Law of noncontradiction. ... |
| dochnel2 42492 | A nonzero member of a subs... |
| dochnel 42493 | A nonzero vector doesn't b... |
| djhffval 42496 | Subspace join for ` DVecH ... |
| djhfval 42497 | Subspace join for ` DVecH ... |
| djhval 42498 | Subspace join for ` DVecH ... |
| djhval2 42499 | Value of subspace join for... |
| djhcl 42500 | Closure of subspace join f... |
| djhlj 42501 | Transfer lattice join to `... |
| djhljjN 42502 | Lattice join in terms of `... |
| djhjlj 42503 | ` DVecH ` vector space clo... |
| djhj 42504 | ` DVecH ` vector space clo... |
| djhcom 42505 | Subspace join is commutati... |
| djhspss 42506 | Subspace span of union is ... |
| djhsumss 42507 | Subspace sum is a subset o... |
| dihsumssj 42508 | The subspace sum of two is... |
| djhunssN 42509 | Subspace union is a subset... |
| dochdmm1 42510 | De Morgan-like law for clo... |
| djhexmid 42511 | Excluded middle property o... |
| djh01 42512 | Closed subspace join with ... |
| djh02 42513 | Closed subspace join with ... |
| djhlsmcl 42514 | A closed subspace sum equa... |
| djhcvat42 42515 | A covering property. ( ~ ... |
| dihjatb 42516 | Isomorphism H of lattice j... |
| dihjatc 42517 | Isomorphism H of lattice j... |
| dihjatcclem1 42518 | Lemma for isomorphism H of... |
| dihjatcclem2 42519 | Lemma for isomorphism H of... |
| dihjatcclem3 42520 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 42521 | Lemma for isomorphism H of... |
| dihjatcc 42522 | Isomorphism H of lattice j... |
| dihjat 42523 | Isomorphism H of lattice j... |
| dihprrnlem1N 42524 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 42525 | Lemma for ~ dihprrn . (Co... |
| dihprrn 42526 | The span of a vector pair ... |
| djhlsmat 42527 | The sum of two subspace at... |
| dihjat1lem 42528 | Subspace sum of a closed s... |
| dihjat1 42529 | Subspace sum of a closed s... |
| dihsmsprn 42530 | Subspace sum of a closed s... |
| dihjat2 42531 | The subspace sum of a clos... |
| dihjat3 42532 | Isomorphism H of lattice j... |
| dihjat4 42533 | Transfer the subspace sum ... |
| dihjat6 42534 | Transfer the subspace sum ... |
| dihsmsnrn 42535 | The subspace sum of two si... |
| dihsmatrn 42536 | The subspace sum of a clos... |
| dihjat5N 42537 | Transfer lattice join with... |
| dvh4dimat 42538 | There is an atom that is o... |
| dvh3dimatN 42539 | There is an atom that is o... |
| dvh2dimatN 42540 | Given an atom, there exist... |
| dvh1dimat 42541 | There exists an atom. (Co... |
| dvh1dim 42542 | There exists a nonzero vec... |
| dvh4dimlem 42543 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 42544 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 42545 | There is a vector that is ... |
| dvh3dim 42546 | There is a vector that is ... |
| dvh4dimN 42547 | There is a vector that is ... |
| dvh3dim2 42548 | There is a vector that is ... |
| dvh3dim3N 42549 | There is a vector that is ... |
| dochsnnz 42550 | The orthocomplement of a s... |
| dochsatshp 42551 | The orthocomplement of a s... |
| dochsatshpb 42552 | The orthocomplement of a s... |
| dochsnshp 42553 | The orthocomplement of a n... |
| dochshpsat 42554 | A hyperplane is closed iff... |
| dochkrsat 42555 | The orthocomplement of a k... |
| dochkrsat2 42556 | The orthocomplement of a k... |
| dochsat0 42557 | The orthocomplement of a k... |
| dochkrsm 42558 | The subspace sum of a clos... |
| dochexmidat 42559 | Special case of excluded m... |
| dochexmidlem1 42560 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 42561 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 42562 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 42563 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 42564 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 42565 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 42566 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 42567 | Lemma for ~ dochexmid . T... |
| dochexmid 42568 | Excluded middle law for cl... |
| dochsnkrlem1 42569 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 42570 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 42571 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 42572 | A (closed) kernel expresse... |
| dochsnkr2 42573 | Kernel of the explicit fun... |
| dochsnkr2cl 42574 | The ` X ` determining func... |
| dochflcl 42575 | Closure of the explicit fu... |
| dochfl1 42576 | The value of the explicit ... |
| dochfln0 42577 | The value of a functional ... |
| dochkr1 42578 | A nonzero functional has a... |
| dochkr1OLDN 42579 | A nonzero functional has a... |
| lpolsetN 42582 | The set of polarities of a... |
| islpolN 42583 | The predicate "is a polari... |
| islpoldN 42584 | Properties that determine ... |
| lpolfN 42585 | Functionality of a polarit... |
| lpolvN 42586 | The polarity of the whole ... |
| lpolconN 42587 | Contraposition property of... |
| lpolsatN 42588 | The polarity of an atomic ... |
| lpolpolsatN 42589 | Property of a polarity. (... |
| dochpolN 42590 | The subspace orthocompleme... |
| lcfl1lem 42591 | Property of a functional w... |
| lcfl1 42592 | Property of a functional w... |
| lcfl2 42593 | Property of a functional w... |
| lcfl3 42594 | Property of a functional w... |
| lcfl4N 42595 | Property of a functional w... |
| lcfl5 42596 | Property of a functional w... |
| lcfl5a 42597 | Property of a functional w... |
| lcfl6lem 42598 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 42599 | Lemma for ~ lcfl7N . If t... |
| lcfl6 42600 | Property of a functional w... |
| lcfl7N 42601 | Property of a functional w... |
| lcfl8 42602 | Property of a functional w... |
| lcfl8a 42603 | Property of a functional w... |
| lcfl8b 42604 | Property of a nonzero func... |
| lcfl9a 42605 | Property implying that a f... |
| lclkrlem1 42606 | The set of functionals hav... |
| lclkrlem2a 42607 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 42608 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 42609 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 42610 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 42611 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 42612 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 42613 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 42614 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 42615 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 42616 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 42617 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 42618 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 42619 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 42620 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 42621 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 42622 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 42623 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 42624 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 42625 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 42626 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 42627 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 42628 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 42629 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 42630 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 42631 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 42632 | The set of functionals hav... |
| lclkr 42633 | The set of functionals wit... |
| lcfls1lem 42634 | Property of a functional w... |
| lcfls1N 42635 | Property of a functional w... |
| lcfls1c 42636 | Property of a functional w... |
| lclkrslem1 42637 | The set of functionals hav... |
| lclkrslem2 42638 | The set of functionals hav... |
| lclkrs 42639 | The set of functionals hav... |
| lclkrs2 42640 | The set of functionals wit... |
| lcfrvalsnN 42641 | Reconstruction from the du... |
| lcfrlem1 42642 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 42643 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 42644 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 42645 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 42646 | Lemma for ~ lcfr . The se... |
| lcfrlem6 42647 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 42648 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 42649 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 42650 | Lemma for ~ lcf1o . (This... |
| lcf1o 42651 | Define a function ` J ` th... |
| lcfrlem10 42652 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 42653 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 42654 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 42655 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 42656 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 42657 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 42658 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 42659 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 42660 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 42661 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 42662 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 42663 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 42664 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 42665 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 42666 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 42667 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 42668 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 42669 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 42670 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 42671 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 42672 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 42673 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 42674 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 42675 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 42676 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 42677 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 42678 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 42679 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 42680 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 42681 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 42682 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 42683 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 42684 | Lemma for ~ lcfr . Elimin... |
| lcfr 42685 | Reconstruction of a subspa... |
| lcdfval 42688 | Dual vector space of funct... |
| lcdval 42689 | Dual vector space of funct... |
| lcdval2 42690 | Dual vector space of funct... |
| lcdlvec 42691 | The dual vector space of f... |
| lcdlmod 42692 | The dual vector space of f... |
| lcdvbase 42693 | Vector base set of a dual ... |
| lcdvbasess 42694 | The vector base set of the... |
| lcdvbaselfl 42695 | A vector in the base set o... |
| lcdvbasecl 42696 | Closure of the value of a ... |
| lcdvadd 42697 | Vector addition for the cl... |
| lcdvaddval 42698 | The value of the value of ... |
| lcdsca 42699 | The ring of scalars of the... |
| lcdsbase 42700 | Base set of scalar ring fo... |
| lcdsadd 42701 | Scalar addition for the cl... |
| lcdsmul 42702 | Scalar multiplication for ... |
| lcdvs 42703 | Scalar product for the clo... |
| lcdvsval 42704 | Value of scalar product op... |
| lcdvscl 42705 | The scalar product operati... |
| lcdlssvscl 42706 | Closure of scalar product ... |
| lcdvsass 42707 | Associative law for scalar... |
| lcd0 42708 | The zero scalar of the clo... |
| lcd1 42709 | The unit scalar of the clo... |
| lcdneg 42710 | The unit scalar of the clo... |
| lcd0v 42711 | The zero functional in the... |
| lcd0v2 42712 | The zero functional in the... |
| lcd0vvalN 42713 | Value of the zero function... |
| lcd0vcl 42714 | Closure of the zero functi... |
| lcd0vs 42715 | A scalar zero times a func... |
| lcdvs0N 42716 | A scalar times the zero fu... |
| lcdvsub 42717 | The value of vector subtra... |
| lcdvsubval 42718 | The value of the value of ... |
| lcdlss 42719 | Subspaces of a dual vector... |
| lcdlss2N 42720 | Subspaces of a dual vector... |
| lcdlsp 42721 | Span in the set of functio... |
| lcdlkreqN 42722 | Colinear functionals have ... |
| lcdlkreq2N 42723 | Colinear functionals have ... |
| mapdffval 42726 | Projectivity from vector s... |
| mapdfval 42727 | Projectivity from vector s... |
| mapdval 42728 | Value of projectivity from... |
| mapdvalc 42729 | Value of projectivity from... |
| mapdval2N 42730 | Value of projectivity from... |
| mapdval3N 42731 | Value of projectivity from... |
| mapdval4N 42732 | Value of projectivity from... |
| mapdval5N 42733 | Value of projectivity from... |
| mapdordlem1a 42734 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 42735 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 42736 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 42737 | Lemma for ~ mapdord . Ord... |
| mapdord 42738 | Ordering property of the m... |
| mapd11 42739 | The map defined by ~ df-ma... |
| mapddlssN 42740 | The mapping of a subspace ... |
| mapdsn 42741 | Value of the map defined b... |
| mapdsn2 42742 | Value of the map defined b... |
| mapdsn3 42743 | Value of the map defined b... |
| mapd1dim2lem1N 42744 | Value of the map defined b... |
| mapdrvallem2 42745 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 42746 | Lemma for ~ mapdrval . (C... |
| mapdrval 42747 | Given a dual subspace ` R ... |
| mapd1o 42748 | The map defined by ~ df-ma... |
| mapdrn 42749 | Range of the map defined b... |
| mapdunirnN 42750 | Union of the range of the ... |
| mapdrn2 42751 | Range of the map defined b... |
| mapdcnvcl 42752 | Closure of the converse of... |
| mapdcl 42753 | Closure the value of the m... |
| mapdcnvid1N 42754 | Converse of the value of t... |
| mapdsord 42755 | Strong ordering property o... |
| mapdcl2 42756 | The mapping of a subspace ... |
| mapdcnvid2 42757 | Value of the converse of t... |
| mapdcnvordN 42758 | Ordering property of the c... |
| mapdcnv11N 42759 | The converse of the map de... |
| mapdcv 42760 | Covering property of the c... |
| mapdincl 42761 | Closure of dual subspace i... |
| mapdin 42762 | Subspace intersection is p... |
| mapdlsmcl 42763 | Closure of dual subspace s... |
| mapdlsm 42764 | Subspace sum is preserved ... |
| mapd0 42765 | Projectivity map of the ze... |
| mapdcnvatN 42766 | Atoms are preserved by the... |
| mapdat 42767 | Atoms are preserved by the... |
| mapdspex 42768 | The map of a span equals t... |
| mapdn0 42769 | Transfer nonzero property ... |
| mapdncol 42770 | Transfer non-colinearity f... |
| mapdindp 42771 | Transfer (part of) vector ... |
| mapdpglem1 42772 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 42773 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 42774 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 42775 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 42776 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 42777 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 42778 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 42779 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 42780 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 42781 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 42782 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 42783 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 42784 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 42785 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 42786 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 42787 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 42788 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 42789 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 42790 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 42791 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 42792 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 42793 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 42794 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 42795 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 42796 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 42797 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 42798 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 42799 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 42800 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 42801 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 42802 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 42803 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 42804 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 42805 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 42806 | Part 1 of proof of the fir... |
| baerlem3lem1 42807 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 42808 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 42809 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 42810 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 42811 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 42812 | Lemma for ~ baerlem5b . (... |
| baerlem3 42813 | An equality that holds whe... |
| baerlem5a 42814 | An equality that holds whe... |
| baerlem5b 42815 | An equality that holds whe... |
| baerlem5amN 42816 | An equality that holds whe... |
| baerlem5bmN 42817 | An equality that holds whe... |
| baerlem5abmN 42818 | An equality that holds whe... |
| mapdindp0 42819 | Vector independence lemma.... |
| mapdindp1 42820 | Vector independence lemma.... |
| mapdindp2 42821 | Vector independence lemma.... |
| mapdindp3 42822 | Vector independence lemma.... |
| mapdindp4 42823 | Vector independence lemma.... |
| mapdhval 42824 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 42825 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 42826 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 42827 | Lemmma for ~~? mapdh . (C... |
| mapdheq 42828 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 42829 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 42830 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 42831 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 42832 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 42833 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 42834 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 42835 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 42836 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 42837 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 42838 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 42839 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 42840 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 42841 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 42842 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 42843 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 42844 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 42845 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 42846 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 42847 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 42848 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 42849 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 42850 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 42851 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 42852 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 42853 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 42854 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 42855 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 42858 | Map from nonzero vectors t... |
| hvmapfval 42859 | Map from nonzero vectors t... |
| hvmapval 42860 | Value of map from nonzero ... |
| hvmapvalvalN 42861 | Value of value of map (i.e... |
| hvmapidN 42862 | The value of the vector to... |
| hvmap1o 42863 | The vector to functional m... |
| hvmapclN 42864 | Closure of the vector to f... |
| hvmap1o2 42865 | The vector to functional m... |
| hvmapcl2 42866 | Closure of the vector to f... |
| hvmaplfl 42867 | The vector to functional m... |
| hvmaplkr 42868 | Kernel of the vector to fu... |
| mapdhvmap 42869 | Relationship between ` map... |
| lspindp5 42870 | Obtain an independent vect... |
| hdmaplem1 42871 | Lemma to convert a frequen... |
| hdmaplem2N 42872 | Lemma to convert a frequen... |
| hdmaplem3 42873 | Lemma to convert a frequen... |
| hdmaplem4 42874 | Lemma to convert a frequen... |
| mapdh8a 42875 | Part of Part (8) in [Baer]... |
| mapdh8aa 42876 | Part of Part (8) in [Baer]... |
| mapdh8ab 42877 | Part of Part (8) in [Baer]... |
| mapdh8ac 42878 | Part of Part (8) in [Baer]... |
| mapdh8ad 42879 | Part of Part (8) in [Baer]... |
| mapdh8b 42880 | Part of Part (8) in [Baer]... |
| mapdh8c 42881 | Part of Part (8) in [Baer]... |
| mapdh8d0N 42882 | Part of Part (8) in [Baer]... |
| mapdh8d 42883 | Part of Part (8) in [Baer]... |
| mapdh8e 42884 | Part of Part (8) in [Baer]... |
| mapdh8g 42885 | Part of Part (8) in [Baer]... |
| mapdh8i 42886 | Part of Part (8) in [Baer]... |
| mapdh8j 42887 | Part of Part (8) in [Baer]... |
| mapdh8 42888 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 42889 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 42890 | Lemma for part (9) in [Bae... |
| hdmap1ffval 42895 | Preliminary map from vecto... |
| hdmap1fval 42896 | Preliminary map from vecto... |
| hdmap1vallem 42897 | Value of preliminary map f... |
| hdmap1val 42898 | Value of preliminary map f... |
| hdmap1val0 42899 | Value of preliminary map f... |
| hdmap1val2 42900 | Value of preliminary map f... |
| hdmap1eq 42901 | The defining equation for ... |
| hdmap1cbv 42902 | Frequently used lemma to c... |
| hdmap1valc 42903 | Connect the value of the p... |
| hdmap1cl 42904 | Convert closure theorem ~ ... |
| hdmap1eq2 42905 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 42906 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 42907 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 42908 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 42909 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 42910 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 42911 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 42912 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 42913 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 42914 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 42915 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 42916 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 42917 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 42918 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 42919 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 42920 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 42921 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 42922 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 42923 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 42924 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 42925 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 42926 | Map from vectors to functi... |
| hdmapfval 42927 | Map from vectors to functi... |
| hdmapval 42928 | Value of map from vectors ... |
| hdmapfnN 42929 | Functionality of map from ... |
| hdmapcl 42930 | Closure of map from vector... |
| hdmapval2lem 42931 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 42932 | Value of map from vectors ... |
| hdmapval0 42933 | Value of map from vectors ... |
| hdmapeveclem 42934 | Lemma for ~ hdmapevec . T... |
| hdmapevec 42935 | Value of map from vectors ... |
| hdmapevec2 42936 | The inner product of the r... |
| hdmapval3lemN 42937 | Value of map from vectors ... |
| hdmapval3N 42938 | Value of map from vectors ... |
| hdmap10lem 42939 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 42940 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 42941 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 42942 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 42943 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 42944 | Part of proof of part 12 i... |
| hdmapnzcl 42945 | Nonzero vector closure of ... |
| hdmapneg 42946 | Part of proof of part 12 i... |
| hdmapsub 42947 | Part of proof of part 12 i... |
| hdmap11 42948 | Part of proof of part 12 i... |
| hdmaprnlem1N 42949 | Part of proof of part 12 i... |
| hdmaprnlem3N 42950 | Part of proof of part 12 i... |
| hdmaprnlem3uN 42951 | Part of proof of part 12 i... |
| hdmaprnlem4tN 42952 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 42953 | Part of proof of part 12 i... |
| hdmaprnlem6N 42954 | Part of proof of part 12 i... |
| hdmaprnlem7N 42955 | Part of proof of part 12 i... |
| hdmaprnlem8N 42956 | Part of proof of part 12 i... |
| hdmaprnlem9N 42957 | Part of proof of part 12 i... |
| hdmaprnlem3eN 42958 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 42959 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 42960 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 42961 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 42962 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 42963 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 42964 | Part of proof of part 12 i... |
| hdmapf1oN 42965 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 42966 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 42967 | Prior to part 14 in [Baer]... |
| hdmap14lem1 42968 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 42969 | Prior to part 14 in [Baer]... |
| hdmap14lem3 42970 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 42971 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 42972 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 42973 | Case where ` F ` is zero. ... |
| hdmap14lem7 42974 | Combine cases of ` F ` . ... |
| hdmap14lem8 42975 | Part of proof of part 14 i... |
| hdmap14lem9 42976 | Part of proof of part 14 i... |
| hdmap14lem10 42977 | Part of proof of part 14 i... |
| hdmap14lem11 42978 | Part of proof of part 14 i... |
| hdmap14lem12 42979 | Lemma for proof of part 14... |
| hdmap14lem13 42980 | Lemma for proof of part 14... |
| hdmap14lem14 42981 | Part of proof of part 14 i... |
| hdmap14lem15 42982 | Part of proof of part 14 i... |
| hgmapffval 42985 | Map from the scalar divisi... |
| hgmapfval 42986 | Map from the scalar divisi... |
| hgmapval 42987 | Value of map from the scal... |
| hgmapfnN 42988 | Functionality of scalar si... |
| hgmapcl 42989 | Closure of scalar sigma ma... |
| hgmapdcl 42990 | Closure of the vector spac... |
| hgmapvs 42991 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 42992 | Value of the scalar sigma ... |
| hgmapval1 42993 | Value of the scalar sigma ... |
| hgmapadd 42994 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 42995 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 42996 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 42997 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 42998 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 42999 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 43000 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 43001 | Part of proof of part 16 i... |
| hgmap11 43002 | The scalar sigma map is on... |
| hgmapf1oN 43003 | The scalar sigma map is a ... |
| hgmapeq0 43004 | The scalar sigma map is ze... |
| hdmapipcl 43005 | The inner product (Hermiti... |
| hdmapln1 43006 | Linearity property that wi... |
| hdmaplna1 43007 | Additive property of first... |
| hdmaplns1 43008 | Subtraction property of fi... |
| hdmaplnm1 43009 | Multiplicative property of... |
| hdmaplna2 43010 | Additive property of secon... |
| hdmapglnm2 43011 | g-linear property of secon... |
| hdmapgln2 43012 | g-linear property that wil... |
| hdmaplkr 43013 | Kernel of the vector to du... |
| hdmapellkr 43014 | Membership in the kernel (... |
| hdmapip0 43015 | Zero property that will be... |
| hdmapip1 43016 | Construct a proportional v... |
| hdmapip0com 43017 | Commutation property of Ba... |
| hdmapinvlem1 43018 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 43019 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 43020 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 43021 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 43022 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 43023 | Involution property of sca... |
| hgmapvvlem2 43024 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 43025 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 43026 | Value of a double involuti... |
| hdmapglem7a 43027 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 43028 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 43029 | Lemma for ~ hdmapg . Line... |
| hdmapg 43030 | Apply the scalar sigma fun... |
| hdmapoc 43031 | Express our constructed or... |
| hlhilset 43034 | The final Hilbert space co... |
| hlhilsca 43035 | The scalar of the final co... |
| hlhilbase 43036 | The base set of the final ... |
| hlhilplus 43037 | The vector addition for th... |
| hlhilslem 43038 | Lemma for ~ hlhilsbase etc... |
| hlhilsbase 43039 | The scalar base set of the... |
| hlhilsplus 43040 | Scalar addition for the fi... |
| hlhilsmul 43041 | Scalar multiplication for ... |
| hlhilsbase2 43042 | The scalar base set of the... |
| hlhilsplus2 43043 | Scalar addition for the fi... |
| hlhilsmul2 43044 | Scalar multiplication for ... |
| hlhils0 43045 | The scalar ring zero for t... |
| hlhils1N 43046 | The scalar ring unity for ... |
| hlhilvsca 43047 | The scalar product for the... |
| hlhilip 43048 | Inner product operation fo... |
| hlhilipval 43049 | Value of inner product ope... |
| hlhilnvl 43050 | The involution operation o... |
| hlhillvec 43051 | The final constructed Hilb... |
| hlhildrng 43052 | The star division ring for... |
| hlhilsrnglem 43053 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 43054 | The star division ring for... |
| hlhil0 43055 | The zero vector for the fi... |
| hlhillsm 43056 | The vector sum operation f... |
| hlhilocv 43057 | The orthocomplement for th... |
| hlhillcs 43058 | The closed subspaces of th... |
| hlhilphllem 43059 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 43060 | Lemma for ~ hlhil . (Cont... |
| hlathil 43061 | Construction of a Hilbert ... |
| iscsrg 43064 | A commutative semiring is ... |
| rhmzrhval 43065 | Evaluation of integers acr... |
| zndvdchrrhm 43066 | Construction of a ring hom... |
| relogbcld 43067 | Closure of the general log... |
| relogbexpd 43068 | Identity law for general l... |
| relogbzexpd 43069 | Power law for the general ... |
| logblebd 43070 | The general logarithm is m... |
| uzindd 43071 | Induction on the upper int... |
| fzadd2d 43072 | Membership of a sum in a f... |
| fzne2d 43073 | Elementhood in a finite se... |
| eqfnfv2d2 43074 | Equality of functions is d... |
| fzsplitnd 43075 | Split a finite interval of... |
| fzsplitnr 43076 | Split a finite interval of... |
| addassnni 43077 | Associative law for additi... |
| addcomnni 43078 | Commutative law for additi... |
| mulassnni 43079 | Associative law for multip... |
| mulcomnni 43080 | Commutative law for multip... |
| gcdcomnni 43081 | Commutative law for gcd. ... |
| gcdnegnni 43082 | Negation invariance for gc... |
| neggcdnni 43083 | Negation invariance for gc... |
| bccl2d 43084 | Closure of the binomial co... |
| recbothd 43085 | Take reciprocal on both si... |
| gcdmultiplei 43086 | The GCD of a multiple of a... |
| gcdaddmzz2nni 43087 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 43088 | Adding a multiple of one o... |
| gcdnncli 43089 | Closure of the gcd operato... |
| muldvds1d 43090 | If a product divides an in... |
| muldvds2d 43091 | If a product divides an in... |
| nndivdvdsd 43092 | A positive integer divides... |
| nnproddivdvdsd 43093 | A product of natural numbe... |
| coprmdvds2d 43094 | If an integer is divisible... |
| imadomfi 43095 | An image of a function und... |
| 12gcd5e1 43096 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 43097 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 43098 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 43099 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 43100 | Calculate the least common... |
| 12lcm5e60 43101 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 43102 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 43103 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 43104 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 43105 | Useful equation to calcula... |
| lcm1un 43106 | Least common multiple of n... |
| lcm2un 43107 | Least common multiple of n... |
| lcm3un 43108 | Least common multiple of n... |
| lcm4un 43109 | Least common multiple of n... |
| lcm5un 43110 | Least common multiple of n... |
| lcm6un 43111 | Least common multiple of n... |
| lcm7un 43112 | Least common multiple of n... |
| lcm8un 43113 | Least common multiple of n... |
| 3factsumint1 43114 | Move constants out of inte... |
| 3factsumint2 43115 | Move constants out of inte... |
| 3factsumint3 43116 | Move constants out of inte... |
| 3factsumint4 43117 | Move constants out of inte... |
| 3factsumint 43118 | Helpful equation for lcm i... |
| resopunitintvd 43119 | Restrict continuous functi... |
| resclunitintvd 43120 | Restrict continuous functi... |
| resdvopclptsd 43121 | Restrict derivative on uni... |
| lcmineqlem1 43122 | Part of lcm inequality lem... |
| lcmineqlem2 43123 | Part of lcm inequality lem... |
| lcmineqlem3 43124 | Part of lcm inequality lem... |
| lcmineqlem4 43125 | Part of lcm inequality lem... |
| lcmineqlem5 43126 | Technical lemma for recipr... |
| lcmineqlem6 43127 | Part of lcm inequality lem... |
| lcmineqlem7 43128 | Derivative of 1-x for chai... |
| lcmineqlem8 43129 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 43130 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 43131 | Induction step of ~ lcmine... |
| lcmineqlem11 43132 | Induction step, continuati... |
| lcmineqlem12 43133 | Base case for induction. ... |
| lcmineqlem13 43134 | Induction proof for lcm in... |
| lcmineqlem14 43135 | Technical lemma for inequa... |
| lcmineqlem15 43136 | F times the least common m... |
| lcmineqlem16 43137 | Technical divisibility lem... |
| lcmineqlem17 43138 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 43139 | Technical lemma to shift f... |
| lcmineqlem19 43140 | Dividing implies inequalit... |
| lcmineqlem20 43141 | Inequality for lcm lemma. ... |
| lcmineqlem21 43142 | The lcm inequality lemma w... |
| lcmineqlem22 43143 | The lcm inequality lemma w... |
| lcmineqlem23 43144 | Penultimate step to the lc... |
| lcmineqlem 43145 | The least common multiple ... |
| 3exp7 43146 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 43147 | First inequality in inequa... |
| 3lexlogpow5ineq2 43148 | Second inequality in inequ... |
| 3lexlogpow5ineq4 43149 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 43150 | Combined inequality chain ... |
| 3lexlogpow2ineq1 43151 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 43152 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 43153 | Result for bound in AKS in... |
| intlewftc 43154 | Inequality inference by in... |
| aks4d1lem1 43155 | Technical lemma to reduce ... |
| aks4d1p1p1 43156 | Exponential law for finite... |
| dvrelog2 43157 | The derivative of the loga... |
| dvrelog3 43158 | The derivative of the loga... |
| dvrelog2b 43159 | Derivative of the binary l... |
| 0nonelalab 43160 | Technical lemma for open i... |
| dvrelogpow2b 43161 | Derivative of the power of... |
| aks4d1p1p3 43162 | Bound of a ceiling of the ... |
| aks4d1p1p2 43163 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 43164 | Technical step for inequal... |
| dvle2 43165 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 43166 | Inequality lift to differe... |
| aks4d1p1p7 43167 | Bound of intermediary of i... |
| aks4d1p1p5 43168 | Show inequality for existe... |
| aks4d1p1 43169 | Show inequality for existe... |
| aks4d1p2 43170 | Technical lemma for existe... |
| aks4d1p3 43171 | There exists a small enoug... |
| aks4d1p4 43172 | There exists a small enoug... |
| aks4d1p5 43173 | Show that ` N ` and ` R ` ... |
| aks4d1p6 43174 | The maximal prime power ex... |
| aks4d1p7d1 43175 | Technical step in AKS lemm... |
| aks4d1p7 43176 | Technical step in AKS lemm... |
| aks4d1p8d1 43177 | If a prime divides one num... |
| aks4d1p8d2 43178 | Any prime power dividing a... |
| aks4d1p8d3 43179 | The remainder of a divisio... |
| aks4d1p8 43180 | Show that ` N ` and ` R ` ... |
| aks4d1p9 43181 | Show that the order is bou... |
| aks4d1 43182 | Lemma 4.1 from ~ https://w... |
| fldhmf1 43183 | A field homomorphism is in... |
| isprimroot 43186 | The value of a primitive r... |
| isprimroot2 43187 | Alternative way of creatin... |
| mndmolinv 43188 | An element of a monoid tha... |
| linvh 43189 | If an element has a unique... |
| primrootsunit1 43190 | Primitive roots have left ... |
| primrootsunit 43191 | Primitive roots have left ... |
| primrootscoprmpow 43192 | Coprime powers of primitiv... |
| posbezout 43193 | Bezout's identity restrict... |
| primrootscoprf 43194 | Coprime powers of primitiv... |
| primrootscoprbij 43195 | A bijection between coprim... |
| primrootscoprbij2 43196 | A bijection between coprim... |
| remexz 43197 | Division with rest. (Cont... |
| primrootlekpowne0 43198 | There is no smaller power ... |
| primrootspoweq0 43199 | The power of a ` R ` -th p... |
| aks6d1c1p1 43200 | Definition of the introspe... |
| aks6d1c1p1rcl 43201 | Reverse closure of the int... |
| aks6d1c1p2 43202 | ` P ` and linear factors a... |
| aks6d1c1p3 43203 | In a field with a Frobeniu... |
| aks6d1c1p4 43204 | The product of polynomials... |
| aks6d1c1p5 43205 | The product of exponents i... |
| aks6d1c1p7 43206 | ` X ` is introspective to ... |
| aks6d1c1p6 43207 | If a polynomials ` F ` is ... |
| aks6d1c1p8 43208 | If a number ` E ` is intro... |
| aks6d1c1 43209 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 43210 | Polynomial evaluation buil... |
| aks6d1c2p1 43211 | In the AKS-theorem the sub... |
| aks6d1c2p2 43212 | Injective condition for co... |
| hashscontpowcl 43213 | Closure of E for ~ https:/... |
| hashscontpow1 43214 | Helper lemma for to prove ... |
| hashscontpow 43215 | If a set contains all ` N ... |
| aks6d1c3 43216 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 43217 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 43218 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 43219 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 43220 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 43221 | If the number of elements ... |
| hashnexinjle 43222 | If the number of elements ... |
| aks6d1c2 43223 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 43224 | Special case related to ~ ... |
| idomnnzpownz 43225 | A nonzero power in an inte... |
| idomnnzgmulnz 43226 | A finite product of nonzer... |
| ringexp0nn 43227 | Zero to the power of a pos... |
| aks6d1c5lem0 43228 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 43229 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 43230 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 43231 | Lemma for Claim 5, contrad... |
| aks6d1c5 43232 | Claim 5 of Theorem 6.1 ~ h... |
| deg1gprod 43233 | Degree multiplication is a... |
| deg1pow 43234 | Exact degree of a power of... |
| 5bc2eq10 43235 | The value of 5 choose 2. ... |
| facp2 43236 | The factorial of a success... |
| 2np3bcnp1 43237 | Part of induction step for... |
| 2ap1caineq 43238 | Inequality for Theorem 6.6... |
| sticksstones1 43239 | Different strictly monoton... |
| sticksstones2 43240 | The range function on stri... |
| sticksstones3 43241 | The range function on stri... |
| sticksstones4 43242 | Equinumerosity lemma for s... |
| sticksstones5 43243 | Count the number of strict... |
| sticksstones6 43244 | Function induces an order ... |
| sticksstones7 43245 | Closure property of sticks... |
| sticksstones8 43246 | Establish mapping between ... |
| sticksstones9 43247 | Establish mapping between ... |
| sticksstones10 43248 | Establish mapping between ... |
| sticksstones11 43249 | Establish bijective mappin... |
| sticksstones12a 43250 | Establish bijective mappin... |
| sticksstones12 43251 | Establish bijective mappin... |
| sticksstones13 43252 | Establish bijective mappin... |
| sticksstones14 43253 | Sticks and stones with def... |
| sticksstones15 43254 | Sticks and stones with alm... |
| sticksstones16 43255 | Sticks and stones with col... |
| sticksstones17 43256 | Extend sticks and stones t... |
| sticksstones18 43257 | Extend sticks and stones t... |
| sticksstones19 43258 | Extend sticks and stones t... |
| sticksstones20 43259 | Lift sticks and stones to ... |
| sticksstones21 43260 | Lift sticks and stones to ... |
| sticksstones22 43261 | Non-exhaustive sticks and ... |
| sticksstones23 43262 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 43263 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 43264 | Every primitive root is ro... |
| aks6d1c6lem3 43265 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 43266 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 43267 | Lemma to construct the map... |
| aks6d1c6isolem2 43268 | Lemma to construct the gro... |
| aks6d1c6isolem3 43269 | The preimage of a map send... |
| aks6d1c6lem5 43270 | Eliminate the size hypothe... |
| bcled 43271 | Inequality for binomial co... |
| bcle2d 43272 | Inequality for binomial co... |
| aks6d1c7lem1 43273 | The last set of inequaliti... |
| aks6d1c7lem2 43274 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 43275 | Remove lots of hypotheses ... |
| aks6d1c7lem4 43276 | In the AKS algorithm there... |
| aks6d1c7 43277 | ` N ` is a prime power if ... |
| rhmqusspan 43278 | Ring homomorphism out of a... |
| aks5lem1 43279 | Section 5 of ~ https://www... |
| aks5lem2 43280 | Lemma for section 5 ~ http... |
| ply1asclzrhval 43281 | Transfer results from alge... |
| aks5lem3a 43282 | Lemma for AKS section 5. ... |
| aks5lem4a 43283 | Lemma for AKS section 5, r... |
| aks5lem5a 43284 | Lemma for AKS, section 5, ... |
| aks5lem6 43285 | Connect results of section... |
| indstrd 43286 | Strong induction, deductio... |
| grpods 43287 | Relate sums of elements of... |
| unitscyglem1 43288 | Lemma for unitscyg . (Con... |
| unitscyglem2 43289 | Lemma for unitscyg . (Con... |
| unitscyglem3 43290 | Lemma for unitscyg . (Con... |
| unitscyglem4 43291 | Lemma for unitscyg . (Con... |
| unitscyglem5 43292 | Lemma for unitscyg . (Con... |
| aks5lem7 43293 | Lemma for aks5. We clean ... |
| aks5lem8 43294 | Lemma for aks5. Clean up ... |
| exfinfldd 43296 | For any prime ` P ` and an... |
| aks5 43297 | The AKS Primality test, gi... |
| quadfac 43298 | The solution of a quadrati... |
| 25or6to4 43299 | Question 67 of 68 from a l... |
| jarrii 43300 | Inference associated with ... |
| intnanrt 43301 | Introduction of conjunct i... |
| ioin9i8 43302 | Miscellaneous inference cr... |
| jaodd 43303 | Double deduction form of ~... |
| syl3an12 43304 | A double syllogism inferen... |
| exbiii 43305 | Inference associated with ... |
| sbtd 43306 | A true statement is true u... |
| sbor2 43307 | One direction of ~ sbor , ... |
| sbalexi 43308 | Inference form of ~ sbalex... |
| nfalh 43309 | Version of ~ nfal with an ... |
| nfe2 43310 | An inner existential quant... |
| nfale2 43311 | An inner existential quant... |
| 19.9dev 43312 | ~ 19.9d in the case of an ... |
| 3rspcedvd 43313 | Triple application of ~ rs... |
| sn-axrep5v 43314 | A condensed form of ~ axre... |
| sn-axprlem3 43315 | ~ axprlem3 using only Tars... |
| sn-exelALT 43316 | Alternate proof of ~ exel ... |
| ssabdv 43317 | Deduction of abstraction s... |
| sn-iotalem 43318 | An unused lemma showing th... |
| sn-iotalemcor 43319 | Corollary of ~ sn-iotalem ... |
| abbi1sn 43320 | Originally part of ~ uniab... |
| brif2 43321 | Move a relation inside and... |
| brif12 43322 | Move a relation inside and... |
| pssexg 43323 | The proper subset of a set... |
| pssn0 43324 | A proper superset is nonem... |
| psspwb 43325 | Classes are proper subclas... |
| xppss12 43326 | Proper subset theorem for ... |
| elpwbi 43327 | Membership in a power set,... |
| imaopab 43328 | The image of a class of or... |
| eqresfnbd 43329 | Property of being the rest... |
| fmpocos 43330 | Composition of two functio... |
| ovmpogad 43331 | Value of an operation give... |
| ofun 43332 | A function operation of un... |
| dfqs3 43333 | Alternate definition of qu... |
| qseq12d 43334 | Equality theorem for quoti... |
| qsalrel 43335 | The quotient set is equal ... |
| supinf 43336 | The supremum is the infimu... |
| mapcod 43337 | Compose two mappings. (Co... |
| fisdomnn 43338 | A finite set is dominated ... |
| ltex 43339 | The less-than relation is ... |
| leex 43340 | The less-than-or-equal-to ... |
| subex 43341 | The subtraction operation ... |
| absex 43342 | The absolute value functio... |
| cjex 43343 | The conjugate function is ... |
| fzosumm1 43344 | Separate out the last term... |
| ccatcan2d 43345 | Cancellation law for conca... |
| c0exALT 43346 | Alternate proof of ~ c0ex ... |
| 0cnALT3 43347 | Alternate proof of ~ 0cn u... |
| elre0re 43348 | Specialized version of ~ 0... |
| lttrii 43349 | 'Less than' is transitive.... |
| remulcan2d 43350 | ~ mulcan2d for real number... |
| readdridaddlidd 43351 | Given some real number ` B... |
| 4p4e8ALT 43352 | A shorter proof of ~ 4p4e8... |
| 1p3e4 43353 | 1 + 3 = 4. (Contributed b... |
| 1p4e5 43354 | 1 + 4 = 5. (Contributed b... |
| 1p5e6 43355 | 1 + 5 = 6. (Contributed b... |
| 1p6e7 43356 | 1 + 6 = 7. (Contributed b... |
| 1p7e8 43357 | 1 + 7 = 8. (Contributed b... |
| 1p8e9 43358 | 1 + 8 = 9. (Contributed b... |
| 2p3e5 43359 | 2 + 3 = 5. (Contributed b... |
| 2p4e6 43360 | 2 + 4 = 6. (Contributed b... |
| 2p5e7 43361 | 2 + 5 = 7. (Contributed b... |
| 2p6e8 43362 | 2 + 6 = 8. (Contributed b... |
| 2p7e9 43363 | 2 + 7 = 9. (Contributed b... |
| 3p4e7 43364 | 3 + 4 = 7. (Contributed b... |
| 3p5e8 43365 | 3 + 5 = 8. (Contributed b... |
| 3p6e9 43366 | 3 + 6 = 9. (Contributed b... |
| 4p5e9 43367 | 4 + 5 = 9. (Contributed b... |
| 5ne0 43368 | The number 5 is nonzero. ... |
| 6ne0 43369 | The number 6 is nonzero. ... |
| 7ne0 43370 | The number 7 is nonzero. ... |
| 8ne0 43371 | The number 8 is nonzero. ... |
| 9ne0 43372 | The number 9 is nonzero. ... |
| sn-1ne2 43373 | A proof of ~ 1ne2 without ... |
| nnn1suc 43374 | A positive integer that is... |
| readdrcl2d 43375 | Reverse closure for additi... |
| raddswap12d 43376 | Swap the first two variabl... |
| lsubrotld 43377 | Rotate the variables left ... |
| rsubrotld 43378 | Rotate the variables left ... |
| lsubswap23d 43379 | Swap the second and third ... |
| addsubeq4com 43380 | Relation between sums and ... |
| sqsumi 43381 | A sum squared. (Contribut... |
| negn0nposznnd 43382 | Lemma for ~ dffltz . (Con... |
| sqmid3api 43383 | Value of the square of the... |
| decaddcom 43384 | Commute ones place in addi... |
| sqn5i 43385 | The square of a number end... |
| sqn5ii 43386 | The square of a number end... |
| decpmulnc 43387 | Partial products algorithm... |
| decpmul 43388 | Partial products algorithm... |
| sqdeccom12 43389 | The square of a number in ... |
| sq3deccom12 43390 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 43391 | 4 times 5 equals 20. (Con... |
| 3rdpwhole 43392 | A third of a number plus t... |
| sq4 43393 | The square of 4 is 16. (C... |
| sq5 43394 | The square of 5 is 25. (C... |
| sq6 43395 | The square of 6 is 36. (C... |
| sq7 43396 | The square of 7 is 49. (C... |
| sq8 43397 | The square of 8 is 64. (C... |
| sq9 43398 | The square of 9 is 81. (C... |
| rpsscn 43399 | The positive reals are a s... |
| 4rp 43400 | 4 is a positive real. (Co... |
| 6rp 43401 | 6 is a positive real. (Co... |
| 7rp 43402 | 7 is a positive real. (Co... |
| 8rp 43403 | 8 is a positive real. (Co... |
| 9rp 43404 | 9 is a positive real. (Co... |
| 235t711 43405 | Calculate a product by lon... |
| ex-decpmul 43406 | Example usage of ~ decpmul... |
| eluzp1 43407 | Membership in a successor ... |
| sn-eluzp1l 43408 | Shorter proof of ~ eluzp1l... |
| fz1sumconst 43409 | The sum of ` N ` constant ... |
| fz1sump1 43410 | Add one more term to a sum... |
| oddnumth 43411 | The Odd Number Theorem. T... |
| nicomachus 43412 | Nicomachus's Theorem. The... |
| sumcubes 43413 | The sum of the first ` N `... |
| ine1 43414 | ` _i ` is not 1. (Contrib... |
| 0tie0 43415 | 0 times ` _i ` equals 0. ... |
| it1ei 43416 | ` _i ` times 1 equals ` _i... |
| 1tiei 43417 | 1 times ` _i ` equals ` _i... |
| itrere 43418 | ` _i ` times a real is rea... |
| retire 43419 | A real times ` _i ` is rea... |
| iocioodisjd 43420 | Adjacent intervals where t... |
| rpabsid 43421 | A positive real is its own... |
| oexpreposd 43422 | Lemma for ~ dffltz . For ... |
| explt1d 43423 | A nonnegative real number ... |
| expeq1d 43424 | A nonnegative real number ... |
| expeqidd 43425 | A nonnegative real number ... |
| exp11d 43426 | ~ exp11nnd for nonzero int... |
| 0dvds0 43427 | 0 divides 0. (Contributed... |
| absdvdsabsb 43428 | Divisibility is invariant ... |
| dvdsexpnn0 43429 | ~ dvdsexpnn generalized to... |
| dvdsexpb 43430 | ~ dvdssq generalized to po... |
| zdivgd 43431 | Two ways to express " ` N ... |
| efsubd 43432 | Difference of exponents la... |
| ef11d 43433 | General condition for the ... |
| logccne0d 43434 | The logarithm isn't 0 if i... |
| cxp112d 43435 | General condition for comp... |
| cxp111d 43436 | General condition for comp... |
| cxpi11d 43437 | ` _i ` to the powers of ` ... |
| logne0d 43438 | Deduction form of ~ logne0... |
| rxp112d 43439 | Real exponentiation is one... |
| log11d 43440 | The natural logarithm is o... |
| rplog11d 43441 | The natural logarithm is o... |
| rxp11d 43442 | Real exponentiation is one... |
| tanhalfpim 43443 | The tangent of ` _pi / 2 `... |
| sinpim 43444 | Sine of a number subtracte... |
| cospim 43445 | Cosine of a number subtrac... |
| tan3rdpi 43446 | The tangent of ` _pi / 3 `... |
| sin2t3rdpi 43447 | The sine of ` 2 x. ( _pi /... |
| cos2t3rdpi 43448 | The cosine of ` 2 x. ( _pi... |
| sin4t3rdpi 43449 | The sine of ` 4 x. ( _pi /... |
| cos4t3rdpi 43450 | The cosine of ` 4 x. ( _pi... |
| asin1half 43451 | The arcsine of ` 1 / 2 ` i... |
| acos1half 43452 | The arccosine of ` 1 / 2 `... |
| dvun 43453 | Condition for the union of... |
| redvmptabs 43454 | The derivative of the abso... |
| readvrec2 43455 | The antiderivative of 1/x ... |
| readvrec 43456 | For real numbers, the anti... |
| resuppsinopn 43457 | The support of sin ( ~ df-... |
| readvcot 43458 | Real antiderivative of cot... |
| resubval 43461 | Value of real subtraction,... |
| renegeulemv 43462 | Lemma for ~ renegeu and si... |
| renegeulem 43463 | Lemma for ~ renegeu and si... |
| renegeu 43464 | Existential uniqueness of ... |
| rernegcl 43465 | Closure law for negative r... |
| renegadd 43466 | Relationship between real ... |
| renegid 43467 | Addition of a real number ... |
| reneg0addlid 43468 | Negative zero is a left ad... |
| resubeulem1 43469 | Lemma for ~ resubeu . A v... |
| resubeulem2 43470 | Lemma for ~ resubeu . A v... |
| resubeu 43471 | Existential uniqueness of ... |
| rersubcl 43472 | Closure for real subtracti... |
| resubadd 43473 | Relation between real subt... |
| resubaddd 43474 | Relationship between subtr... |
| resubf 43475 | Real subtraction is an ope... |
| repncan2 43476 | Addition and subtraction o... |
| repncan3 43477 | Addition and subtraction o... |
| readdsub 43478 | Law for addition and subtr... |
| reladdrsub 43479 | Move LHS of a sum into RHS... |
| reltsub1 43480 | Subtraction from both side... |
| reltsubadd2 43481 | 'Less than' relationship b... |
| resubcan2 43482 | Cancellation law for real ... |
| resubsub4 43483 | Law for double subtraction... |
| rennncan2 43484 | Cancellation law for real ... |
| renpncan3 43485 | Cancellation law for real ... |
| repnpcan 43486 | Cancellation law for addit... |
| reppncan 43487 | Cancellation law for mixed... |
| resubidaddlidlem 43488 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 43489 | Any real number subtracted... |
| resubdi 43490 | Distribution of multiplica... |
| re1m1e0m0 43491 | Equality of two left-addit... |
| sn-00idlem1 43492 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 43493 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 43494 | Lemma for ~ sn-00id . (Co... |
| sn-00id 43495 | ~ 00id proven without ~ ax... |
| re0m0e0 43496 | Real number version of ~ 0... |
| readdlid 43497 | Real number version of ~ a... |
| sn-addlid 43498 | ~ addlid without ~ ax-mulc... |
| remul02 43499 | Real number version of ~ m... |
| sn-0ne2 43500 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 43501 | Real number version of ~ m... |
| sn-remul0ord 43502 | A product is zero iff one ... |
| resubid 43503 | Subtraction of a real numb... |
| readdrid 43504 | Real number version of ~ a... |
| resubid1 43505 | Real number version of ~ s... |
| renegneg 43506 | A real number is equal to ... |
| readdcan2 43507 | Commuted version of ~ read... |
| renegid2 43508 | Commuted version of ~ rene... |
| remulneg2d 43509 | Product with negative is n... |
| sn-it0e0 43510 | Proof of ~ it0e0 without ~... |
| sn-negex12 43511 | A combination of ~ cnegex ... |
| sn-negex 43512 | Proof of ~ cnegex without ... |
| sn-negex2 43513 | Proof of ~ cnegex2 without... |
| sn-addcand 43514 | ~ addcand without ~ ax-mul... |
| sn-addrid 43515 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 43516 | ~ addcan2d without ~ ax-mu... |
| reixi 43517 | ~ ixi without ~ ax-mulcom ... |
| rei4 43518 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 43519 | A number that sums to itse... |
| sn-mul01 43520 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 43521 | ~ negeu without ~ ax-mulco... |
| sn-subcl 43522 | ~ subcl without ~ ax-mulco... |
| sn-subf 43523 | ~ subf without ~ ax-mulcom... |
| resubeqsub 43524 | Equivalence between real s... |
| subresre 43525 | Subtraction restricted to ... |
| addinvcom 43526 | A number commutes with its... |
| remulinvcom 43527 | A left multiplicative inve... |
| remullid 43528 | Commuted version of ~ ax-1... |
| sn-1ticom 43529 | Lemma for ~ sn-mullid and ... |
| sn-mullid 43530 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 43531 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 43532 | The multiplicative inverse... |
| remulcand 43533 | Commuted version of ~ remu... |
| redivvald 43536 | Value of real division, wh... |
| rediveud 43537 | Existential uniqueness of ... |
| sn-redivcld 43538 | Closure law for real divis... |
| redivmuld 43539 | Relationship between divis... |
| redivmul2d 43540 | Relationship between divis... |
| redivcan2d 43541 | A cancellation law for div... |
| redivcan3d 43542 | A cancellation law for div... |
| rediveq0d 43543 | A ratio is zero iff the nu... |
| redivne0bd 43544 | The ratio of nonzero numbe... |
| rediveq1d 43545 | Equality in terms of unit ... |
| sn-rediv1d 43546 | A number divided by 1 is i... |
| sn-rediv0d 43547 | Division into zero is zero... |
| sn-redividd 43548 | A number divided by itself... |
| sn-rereccld 43549 | Closure law for reciprocal... |
| rerecne0d 43550 | The reciprocal of a nonzer... |
| rerecidd 43551 | Multiplication of a number... |
| rerecid2d 43552 | Multiplication of a number... |
| rerecrecd 43553 | A number is equal to the r... |
| redivrec2d 43554 | Relationship between divis... |
| rediv23d 43555 | A "commutative"/associativ... |
| redivdird 43556 | Distribution of division o... |
| rediv11d 43557 | One-to-one relationship fo... |
| sn-0tie0 43558 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 43559 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 43560 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 43561 | ~ ltaddneg without ~ ax-mu... |
| reposdif 43562 | Comparison of two numbers ... |
| relt0neg1 43563 | Comparison of a real and i... |
| relt0neg2 43564 | Comparison of a real and i... |
| sn-addlt0d 43565 | The sum of negative number... |
| sn-addgt0d 43566 | The sum of positive number... |
| sn-nnne0 43567 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 43568 | ~ elznn0nn restated using ... |
| nn0addcom 43569 | Addition is commutative fo... |
| zaddcomlem 43570 | Lemma for ~ zaddcom . (Co... |
| zaddcom 43571 | Addition is commutative fo... |
| renegmulnnass 43572 | Move multiplication by a n... |
| nn0mulcom 43573 | Multiplication is commutat... |
| zmulcomlem 43574 | Lemma for ~ zmulcom . (Co... |
| zmulcom 43575 | Multiplication is commutat... |
| mulgt0con1dlem 43576 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 43577 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 43578 | Lemma for ~ mulgt0b1d and ... |
| mulgt0b1d 43579 | Biconditional, deductive f... |
| sn-ltmul2d 43580 | ~ ltmul2d without ~ ax-mul... |
| sn-ltmulgt11d 43581 | ~ ltmulgt11d without ~ ax-... |
| sn-0lt1 43582 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 43583 | ~ ltp1 without ~ ax-mulcom... |
| sn-recgt0d 43584 | The reciprocal of a positi... |
| mulgt0b2d 43585 | Biconditional, deductive f... |
| sn-mulgt1d 43586 | ~ mulgt1d without ~ ax-mul... |
| reneg1lt0 43587 | Negative one is a negative... |
| sn-reclt0d 43588 | The reciprocal of a negati... |
| mulltgt0d 43589 | Negative times positive is... |
| mullt0b1d 43590 | When the first term is neg... |
| mullt0b2d 43591 | When the second term is ne... |
| sn-mullt0d 43592 | The product of two negativ... |
| sn-msqgt0d 43593 | A nonzero square is positi... |
| sn-inelr 43594 | ~ inelr without ~ ax-mulco... |
| sn-itrere 43595 | ` _i ` times a real is rea... |
| sn-retire 43596 | Commuted version of ~ sn-i... |
| cnreeu 43597 | The reals in the expressio... |
| sn-sup2 43598 | ~ sup2 with exactly the sa... |
| sn-sup3d 43599 | ~ sup3 without ~ ax-mulcom... |
| sn-suprcld 43600 | ~ suprcld without ~ ax-mul... |
| sn-suprubd 43601 | ~ suprubd without ~ ax-mul... |
| sn-base0 43602 | Avoid axioms in ~ base0 by... |
| nelsubginvcld 43603 | The inverse of a non-subgr... |
| nelsubgcld 43604 | A non-subgroup-member plus... |
| nelsubgsubcld 43605 | A non-subgroup-member minu... |
| rnasclg 43606 | The set of injected scalar... |
| reldmfrlm 43607 | The domain of the free mod... |
| frlmbasfn 43608 | Elements of the free modul... |
| frlmvscl 43609 | Closure of scalar multipli... |
| frlmfielbas 43610 | The vectors of a finite fr... |
| frlmfzwrd 43611 | A vector of a module with ... |
| frlmfzowrd 43612 | A vector of a module with ... |
| frlmfzolen 43613 | The dimension of a vector ... |
| frlmfzowrdb 43614 | The vectors of a module wi... |
| frlmfzoccat 43615 | The concatenation of two v... |
| frlmvscadiccat 43616 | Scalar multiplication dist... |
| grpasscan2d 43617 | An associative cancellatio... |
| grpcominv1 43618 | If two elements commute, t... |
| grpcominv2 43619 | If two elements commute, t... |
| finsubmsubg 43620 | A submonoid of a finite gr... |
| opprmndb 43621 | A class is a monoid if and... |
| opprgrpb 43622 | A class is a group if and ... |
| opprablb 43623 | A class is an Abelian grou... |
| imacrhmcl 43624 | The image of a commutative... |
| riccrng1 43625 | Ring isomorphism preserves... |
| riccrng 43626 | A ring is commutative if a... |
| domnexpgn0cl 43627 | In a domain, a (nonnegativ... |
| drnginvrn0d 43628 | A multiplicative inverse i... |
| drngmullcan 43629 | Cancellation of a nonzero ... |
| drngmulrcan 43630 | Cancellation of a nonzero ... |
| drnginvmuld 43631 | Inverse of a nonzero produ... |
| lmod0vsd 43632 | Zero times a vector is the... |
| lmodvs0d 43633 | A scalar times the zero ve... |
| lmodvsassd 43634 | In a module, scalar multip... |
| ricdrng1 43635 | A ring isomorphic to a div... |
| ricdrng 43636 | A ring is a division ring ... |
| ricfld 43637 | A ring is a field if and o... |
| asclf1 43638 | Two ways of saying the sca... |
| abvexp 43639 | Move exponentiation in and... |
| fimgmcyclem 43640 | Lemma for ~ fimgmcyc . (C... |
| fimgmcyc 43641 | Version of ~ odcl2 for fin... |
| fidomncyc 43642 | Version of ~ odcl2 for mul... |
| fiabv 43643 | In a finite domain (a fini... |
| lvecgrp 43644 | A vector space is a group.... |
| lvecring 43645 | The scalar component of a ... |
| frlm0vald 43646 | All coordinates of the zer... |
| frlmsnic 43647 | Given a free module with a... |
| uvccl 43648 | A unit vector is a vector.... |
| uvcn0 43649 | A unit vector is nonzero. ... |
| psrmnd 43650 | The ring of power series i... |
| mhmcopsr 43651 | The composition of a monoi... |
| mhmcoaddpsr 43652 | Show that the ring homomor... |
| rhmcomulpsr 43653 | Show that the ring homomor... |
| rhmpsr 43654 | Provide a ring homomorphis... |
| rhmpsr1 43655 | Provide a ring homomorphis... |
| evl0 43656 | The zero polynomial evalua... |
| evlsbagval 43657 | Polynomial evaluation buil... |
| evlvvvallem 43658 | Lemma for theorems using ~... |
| evlselvlem 43659 | Lemma for ~ evlselv . Use... |
| evlselv 43660 | Evaluating a selection of ... |
| fsuppind 43661 | Induction on functions ` F... |
| fsuppssindlem1 43662 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 43663 | Lemma for ~ fsuppssind . ... |
| fsuppssind 43664 | Induction on functions ` F... |
| mhpind 43665 | The homogeneous polynomial... |
| evlsmhpvvval 43666 | Give a formula for the eva... |
| mhphflem 43667 | Lemma for ~ mhphf . Add s... |
| mhphf 43668 | A homogeneous polynomial d... |
| mhphf2 43669 | A homogeneous polynomial d... |
| mhphf3 43670 | A homogeneous polynomial d... |
| mhphf4 43671 | A homogeneous polynomial d... |
| prjspval 43674 | Value of the projective sp... |
| prjsprel 43675 | Utility theorem regarding ... |
| prjspertr 43676 | The relation in ` PrjSp ` ... |
| prjsperref 43677 | The relation in ` PrjSp ` ... |
| prjspersym 43678 | The relation in ` PrjSp ` ... |
| prjsper 43679 | The relation used to defin... |
| prjspreln0 43680 | Two nonzero vectors are eq... |
| prjspvs 43681 | A nonzero multiple of a ve... |
| prjsprellsp 43682 | Two vectors are equivalent... |
| prjspeclsp 43683 | The vectors equivalent to ... |
| prjspval2 43684 | Alternate definition of pr... |
| prjspnval 43687 | Value of the n-dimensional... |
| prjspnerlem 43688 | A lemma showing that the e... |
| prjspnval2 43689 | Value of the n-dimensional... |
| prjspner 43690 | The relation used to defin... |
| prjspnvs 43691 | A nonzero multiple of a ve... |
| prjspnssbas 43692 | A projective space is a se... |
| elprjspnss 43693 | A point of a projective sp... |
| prjspnn0 43694 | A projective point is none... |
| frlmnzcoordex 43695 | Every nonzero vector of a ... |
| frlmnzcoordval 43696 | Value of ` J ` , a functio... |
| frlmnzcoordinf 43697 | The result ` ( J `` V ) ` ... |
| frlmnzcoordcl 43698 | Closure of the first nonze... |
| frlmnzcoordcl2 43699 | The first nonzero coordina... |
| frlmnzcoordn0 43700 | The function ` J ` gives a... |
| frlmnzcoordsca 43701 | In a free module, scaling ... |
| prjspnnormval 43702 | Value of ` F ` , a functio... |
| prjspnequivnorm 43703 | In a free module, a nonzer... |
| prjspnnorm 43704 | In a free module, two nonz... |
| 0prjspnlem 43705 | Lemma for ~ 0prjspn . The... |
| 0prjspnrel 43706 | In a zero-dimensional proj... |
| 0prjspn 43707 | A zero-dimensional project... |
| prjcrvfval 43710 | Value of the projective cu... |
| prjcrvval 43711 | Value of the projective cu... |
| prjcrv0 43712 | The "curve" (zero set) cor... |
| dffltz 43713 | Fermat's Last Theorem (FLT... |
| fltmul 43714 | A counterexample to FLT st... |
| fltltc 43715 | ` ( C ^ N ) ` is the large... |
| fltnltalem 43716 | Lemma for ~ fltnlta . A l... |
| fltnlta 43717 | In a Fermat counterexample... |
| iddii 43718 | Version of ~ a1ii with the... |
| bicomdALT 43719 | Alternate proof of ~ bicom... |
| alan 43720 | Alias for ~ 19.26 for easi... |
| exor 43721 | Alias for ~ 19.43 for easi... |
| rexor 43722 | Alias for ~ r19.43 for eas... |
| ruvALT 43723 | Alternate proof of ~ ruv w... |
| sn-wcdeq 43724 | Alternative to ~ wcdeq and... |
| sq45 43725 | 45 squared is 2025. (Cont... |
| sum9cubes 43726 | The sum of the first nine ... |
| sn-isghm 43727 | Longer proof of ~ isghm , ... |
| aprilfools2025 43728 | An abuse of notation. (Co... |
| nfa1w 43729 | Replace ~ ax-10 in ~ nfa1 ... |
| eu6w 43730 | Replace ~ ax-10 , ~ ax-12 ... |
| abbibw 43731 | Replace ~ ax-10 , ~ ax-11 ... |
| absnw 43732 | Replace ~ ax-10 , ~ ax-11 ... |
| euabsn2w 43733 | Replace ~ ax-10 , ~ ax-11 ... |
| cu3addd 43734 | Cube of sum of three numbe... |
| negexpidd 43735 | The sum of a real number t... |
| rexlimdv3d 43736 | An extended version of ~ r... |
| 3cubeslem1 43737 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 43738 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 43739 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 43740 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 43741 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 43742 | Lemma for ~ 3cubes . This... |
| 3cubes 43743 | Every rational number is a... |
| omexcn 43744 | The axioms of the complex ... |
| rntrclfvOAI 43745 | The range of the transitiv... |
| moxfr 43746 | Transfer at-most-one betwe... |
| imaiinfv 43747 | Indexed intersection of an... |
| elrfi 43748 | Elementhood in a set of re... |
| elrfirn 43749 | Elementhood in a set of re... |
| elrfirn2 43750 | Elementhood in a set of re... |
| cmpfiiin 43751 | In a compact topology, a s... |
| ismrcd1 43752 | Any function from the subs... |
| ismrcd2 43753 | Second half of ~ ismrcd1 .... |
| istopclsd 43754 | A closure function which s... |
| ismrc 43755 | A function is a Moore clos... |
| isnacs 43758 | Expand definition of Noeth... |
| nacsfg 43759 | In a Noetherian-type closu... |
| isnacs2 43760 | Express Noetherian-type cl... |
| mrefg2 43761 | Slight variation on finite... |
| mrefg3 43762 | Slight variation on finite... |
| nacsacs 43763 | A closure system of Noethe... |
| isnacs3 43764 | A choice-free order equiva... |
| incssnn0 43765 | Transitivity induction of ... |
| nacsfix 43766 | An increasing sequence of ... |
| constmap 43767 | A constant (represented wi... |
| mapco2g 43768 | Renaming indices in a tupl... |
| mapco2 43769 | Post-composition (renaming... |
| mapfzcons 43770 | Extending a one-based mapp... |
| mapfzcons1 43771 | Recover prefix mapping fro... |
| mapfzcons1cl 43772 | A nonempty mapping has a p... |
| mapfzcons2 43773 | Recover added element from... |
| mptfcl 43774 | Interpret range of a maps-... |
| mzpclval 43779 | Substitution lemma for ` m... |
| elmzpcl 43780 | Double substitution lemma ... |
| mzpclall 43781 | The set of all functions w... |
| mzpcln0 43782 | Corollary of ~ mzpclall : ... |
| mzpcl1 43783 | Defining property 1 of a p... |
| mzpcl2 43784 | Defining property 2 of a p... |
| mzpcl34 43785 | Defining properties 3 and ... |
| mzpval 43786 | Value of the ` mzPoly ` fu... |
| dmmzp 43787 | ` mzPoly ` is defined for ... |
| mzpincl 43788 | Polynomial closedness is a... |
| mzpconst 43789 | Constant functions are pol... |
| mzpf 43790 | A polynomial function is a... |
| mzpproj 43791 | A projection function is p... |
| mzpadd 43792 | The pointwise sum of two p... |
| mzpmul 43793 | The pointwise product of t... |
| mzpconstmpt 43794 | A constant function expres... |
| mzpaddmpt 43795 | Sum of polynomial function... |
| mzpmulmpt 43796 | Product of polynomial func... |
| mzpsubmpt 43797 | The difference of two poly... |
| mzpnegmpt 43798 | Negation of a polynomial f... |
| mzpexpmpt 43799 | Raise a polynomial functio... |
| mzpindd 43800 | "Structural" induction to ... |
| mzpmfp 43801 | Relationship between multi... |
| mzpsubst 43802 | Substituting polynomials f... |
| mzprename 43803 | Simplified version of ~ mz... |
| mzpresrename 43804 | A polynomial is a polynomi... |
| mzpcompact2lem 43805 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 43806 | Polynomials are finitary o... |
| coeq0i 43807 | ~ coeq0 but without explic... |
| fzsplit1nn0 43808 | Split a finite 1-based set... |
| eldiophb 43811 | Initial expression of Diop... |
| eldioph 43812 | Condition for a set to be ... |
| diophrw 43813 | Renaming and adding unused... |
| eldioph2lem1 43814 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 43815 | Lemma for ~ eldioph2 . Co... |
| eldioph2 43816 | Construct a Diophantine se... |
| eldioph2b 43817 | While Diophantine sets wer... |
| eldiophelnn0 43818 | Remove antecedent on ` B `... |
| eldioph3b 43819 | Define Diophantine sets in... |
| eldioph3 43820 | Inference version of ~ eld... |
| ellz1 43821 | Membership in a lower set ... |
| lzunuz 43822 | The union of a lower set o... |
| fz1eqin 43823 | Express a one-based finite... |
| lzenom 43824 | Lower integers are countab... |
| elmapresaunres2 43825 | ~ fresaunres2 transposed t... |
| diophin 43826 | If two sets are Diophantin... |
| diophun 43827 | If two sets are Diophantin... |
| eldiophss 43828 | Diophantine sets are sets ... |
| diophrex 43829 | Projecting a Diophantine s... |
| eq0rabdioph 43830 | This is the first of a num... |
| eqrabdioph 43831 | Diophantine set builder fo... |
| 0dioph 43832 | The empty set is Diophanti... |
| vdioph 43833 | The "universal" set (as la... |
| anrabdioph 43834 | Diophantine set builder fo... |
| orrabdioph 43835 | Diophantine set builder fo... |
| 3anrabdioph 43836 | Diophantine set builder fo... |
| 3orrabdioph 43837 | Diophantine set builder fo... |
| 2sbcrex 43838 | Exchange an existential qu... |
| sbc2rex 43839 | Exchange a substitution wi... |
| sbc4rex 43840 | Exchange a substitution wi... |
| sbcrot3 43841 | Rotate a sequence of three... |
| sbcrot5 43842 | Rotate a sequence of five ... |
| sbccomieg 43843 | Commute two explicit subst... |
| rexrabdioph 43844 | Diophantine set builder fo... |
| rexfrabdioph 43845 | Diophantine set builder fo... |
| 2rexfrabdioph 43846 | Diophantine set builder fo... |
| 3rexfrabdioph 43847 | Diophantine set builder fo... |
| 4rexfrabdioph 43848 | Diophantine set builder fo... |
| 6rexfrabdioph 43849 | Diophantine set builder fo... |
| 7rexfrabdioph 43850 | Diophantine set builder fo... |
| rabdiophlem1 43851 | Lemma for arithmetic dioph... |
| rabdiophlem2 43852 | Lemma for arithmetic dioph... |
| elnn0rabdioph 43853 | Diophantine set builder fo... |
| rexzrexnn0 43854 | Rewrite an existential qua... |
| lerabdioph 43855 | Diophantine set builder fo... |
| eluzrabdioph 43856 | Diophantine set builder fo... |
| elnnrabdioph 43857 | Diophantine set builder fo... |
| ltrabdioph 43858 | Diophantine set builder fo... |
| nerabdioph 43859 | Diophantine set builder fo... |
| dvdsrabdioph 43860 | Divisibility is a Diophant... |
| eldioph4b 43861 | Membership in ` Dioph ` ex... |
| eldioph4i 43862 | Forward-only version of ~ ... |
| diophren 43863 | Change variables in a Diop... |
| rabrenfdioph 43864 | Change variable numbers in... |
| rabren3dioph 43865 | Change variable numbers in... |
| fphpd 43866 | Pigeonhole principle expre... |
| fphpdo 43867 | Pigeonhole principle for s... |
| ctbnfien 43868 | An infinite subset of a co... |
| fiphp3d 43869 | Infinite pigeonhole princi... |
| rencldnfilem 43870 | Lemma for ~ rencldnfi . (... |
| rencldnfi 43871 | A set of real numbers whic... |
| irrapxlem1 43872 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 43873 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 43874 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 43875 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 43876 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 43877 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 43878 | Dirichlet's approximation ... |
| pellexlem1 43879 | Lemma for ~ pellex . Arit... |
| pellexlem2 43880 | Lemma for ~ pellex . Arit... |
| pellexlem3 43881 | Lemma for ~ pellex . To e... |
| pellexlem4 43882 | Lemma for ~ pellex . Invo... |
| pellexlem5 43883 | Lemma for ~ pellex . Invo... |
| pellexlem6 43884 | Lemma for ~ pellex . Doin... |
| pellex 43885 | Every Pell equation has a ... |
| pell1qrval 43896 | Value of the set of first-... |
| elpell1qr 43897 | Membership in a first-quad... |
| pell14qrval 43898 | Value of the set of positi... |
| elpell14qr 43899 | Membership in the set of p... |
| pell1234qrval 43900 | Value of the set of genera... |
| elpell1234qr 43901 | Membership in the set of g... |
| pell1234qrre 43902 | General Pell solutions are... |
| pell1234qrne0 43903 | No solution to a Pell equa... |
| pell1234qrreccl 43904 | General solutions of the P... |
| pell1234qrmulcl 43905 | General solutions of the P... |
| pell14qrss1234 43906 | A positive Pell solution i... |
| pell14qrre 43907 | A positive Pell solution i... |
| pell14qrne0 43908 | A positive Pell solution i... |
| pell14qrgt0 43909 | A positive Pell solution i... |
| pell14qrrp 43910 | A positive Pell solution i... |
| pell1234qrdich 43911 | A general Pell solution is... |
| elpell14qr2 43912 | A number is a positive Pel... |
| pell14qrmulcl 43913 | Positive Pell solutions ar... |
| pell14qrreccl 43914 | Positive Pell solutions ar... |
| pell14qrdivcl 43915 | Positive Pell solutions ar... |
| pell14qrexpclnn0 43916 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 43917 | Positive Pell solutions ar... |
| pell1qrss14 43918 | First-quadrant Pell soluti... |
| pell14qrdich 43919 | A positive Pell solution i... |
| pell1qrge1 43920 | A Pell solution in the fir... |
| pell1qr1 43921 | 1 is a Pell solution and i... |
| elpell1qr2 43922 | The first quadrant solutio... |
| pell1qrgaplem 43923 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 43924 | First-quadrant Pell soluti... |
| pell14qrgap 43925 | Positive Pell solutions ar... |
| pell14qrgapw 43926 | Positive Pell solutions ar... |
| pellqrexplicit 43927 | Condition for a calculated... |
| infmrgelbi 43928 | Any lower bound of a nonem... |
| pellqrex 43929 | There is a nontrivial solu... |
| pellfundval 43930 | Value of the fundamental s... |
| pellfundre 43931 | The fundamental solution o... |
| pellfundge 43932 | Lower bound on the fundame... |
| pellfundgt1 43933 | Weak lower bound on the Pe... |
| pellfundlb 43934 | A nontrivial first quadran... |
| pellfundglb 43935 | If a real is larger than t... |
| pellfundex 43936 | The fundamental solution a... |
| pellfund14gap 43937 | There are no solutions bet... |
| pellfundrp 43938 | The fundamental Pell solut... |
| pellfundne1 43939 | The fundamental Pell solut... |
| reglogcl 43940 | General logarithm is a rea... |
| reglogltb 43941 | General logarithm preserve... |
| reglogleb 43942 | General logarithm preserve... |
| reglogmul 43943 | Multiplication law for gen... |
| reglogexp 43944 | Power law for general log.... |
| reglogbas 43945 | General log of the base is... |
| reglog1 43946 | General log of 1 is 0. (C... |
| reglogexpbas 43947 | General log of a power of ... |
| pellfund14 43948 | Every positive Pell soluti... |
| pellfund14b 43949 | The positive Pell solution... |
| rmxfval 43954 | Value of the X sequence. ... |
| rmyfval 43955 | Value of the Y sequence. ... |
| rmspecsqrtnq 43956 | The discriminant used to d... |
| rmspecnonsq 43957 | The discriminant used to d... |
| qirropth 43958 | This lemma implements the ... |
| rmspecfund 43959 | The base of exponent used ... |
| rmxyelqirr 43960 | The solutions used to cons... |
| rmxypairf1o 43961 | The function used to extra... |
| rmxyelxp 43962 | Lemma for ~ frmx and ~ frm... |
| frmx 43963 | The X sequence is a nonneg... |
| frmy 43964 | The Y sequence is an integ... |
| rmxyval 43965 | Main definition of the X a... |
| rmspecpos 43966 | The discriminant used to d... |
| rmxycomplete 43967 | The X and Y sequences take... |
| rmxynorm 43968 | The X and Y sequences defi... |
| rmbaserp 43969 | The base of exponentiation... |
| rmxyneg 43970 | Negation law for X and Y s... |
| rmxyadd 43971 | Addition formula for X and... |
| rmxy1 43972 | Value of the X and Y seque... |
| rmxy0 43973 | Value of the X and Y seque... |
| rmxneg 43974 | Negation law (even functio... |
| rmx0 43975 | Value of X sequence at 0. ... |
| rmx1 43976 | Value of X sequence at 1. ... |
| rmxadd 43977 | Addition formula for X seq... |
| rmyneg 43978 | Negation formula for Y seq... |
| rmy0 43979 | Value of Y sequence at 0. ... |
| rmy1 43980 | Value of Y sequence at 1. ... |
| rmyadd 43981 | Addition formula for Y seq... |
| rmxp1 43982 | Special addition-of-1 form... |
| rmyp1 43983 | Special addition of 1 form... |
| rmxm1 43984 | Subtraction of 1 formula f... |
| rmym1 43985 | Subtraction of 1 formula f... |
| rmxluc 43986 | The X sequence is a Lucas ... |
| rmyluc 43987 | The Y sequence is a Lucas ... |
| rmyluc2 43988 | Lucas sequence property of... |
| rmxdbl 43989 | "Double-angle formula" for... |
| rmydbl 43990 | "Double-angle formula" for... |
| monotuz 43991 | A function defined on an u... |
| monotoddzzfi 43992 | A function which is odd an... |
| monotoddzz 43993 | A function (given implicit... |
| oddcomabszz 43994 | An odd function which take... |
| 2nn0ind 43995 | Induction on nonnegative i... |
| zindbi 43996 | Inductively transfer a pro... |
| rmxypos 43997 | For all nonnegative indice... |
| ltrmynn0 43998 | The Y-sequence is strictly... |
| ltrmxnn0 43999 | The X-sequence is strictly... |
| lermxnn0 44000 | The X-sequence is monotoni... |
| rmxnn 44001 | The X-sequence is defined ... |
| ltrmy 44002 | The Y-sequence is strictly... |
| rmyeq0 44003 | Y is zero only at zero. (... |
| rmyeq 44004 | Y is one-to-one. (Contrib... |
| lermy 44005 | Y is monotonic (non-strict... |
| rmynn 44006 | ` rmY ` is positive for po... |
| rmynn0 44007 | ` rmY ` is nonnegative for... |
| rmyabs 44008 | ` rmY ` commutes with ` ab... |
| jm2.24nn 44009 | X(n) is strictly greater t... |
| jm2.17a 44010 | First half of lemma 2.17 o... |
| jm2.17b 44011 | Weak form of the second ha... |
| jm2.17c 44012 | Second half of lemma 2.17 ... |
| jm2.24 44013 | Lemma 2.24 of [JonesMatija... |
| rmygeid 44014 | Y(n) increases faster than... |
| congtr 44015 | A wff of the form ` A || (... |
| congadd 44016 | If two pairs of numbers ar... |
| congmul 44017 | If two pairs of numbers ar... |
| congsym 44018 | Congruence mod ` A ` is a ... |
| congneg 44019 | If two integers are congru... |
| congsub 44020 | If two pairs of numbers ar... |
| congid 44021 | Every integer is congruent... |
| mzpcong 44022 | Polynomials commute with c... |
| congrep 44023 | Every integer is congruent... |
| congabseq 44024 | If two integers are congru... |
| acongid 44025 | A wff like that in this th... |
| acongsym 44026 | Symmetry of alternating co... |
| acongneg2 44027 | Negate right side of alter... |
| acongtr 44028 | Transitivity of alternatin... |
| acongeq12d 44029 | Substitution deduction for... |
| acongrep 44030 | Every integer is alternati... |
| fzmaxdif 44031 | Bound on the difference be... |
| fzneg 44032 | Reflection of a finite ran... |
| acongeq 44033 | Two numbers in the fundame... |
| dvdsacongtr 44034 | Alternating congruence pas... |
| coprmdvdsb 44035 | Multiplication by a coprim... |
| modabsdifz 44036 | Divisibility in terms of m... |
| dvdsabsmod0 44037 | Divisibility in terms of m... |
| jm2.18 44038 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 44039 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 44040 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 44041 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 44042 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 44043 | Lemma 2.19 of [JonesMatija... |
| jm2.21 44044 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 44045 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 44046 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 44047 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 44048 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 44049 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 44050 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 44051 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 44052 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 44053 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 44054 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 44055 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 44056 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 44057 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 44058 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 44059 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 44060 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 44061 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 44062 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 44063 | Lemma for ~ rmydioph . Us... |
| rmydioph 44064 | ~ jm2.27 restated in terms... |
| rmxdiophlem 44065 | X can be expressed in term... |
| rmxdioph 44066 | X is a Diophantine functio... |
| jm3.1lem1 44067 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 44068 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 44069 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 44070 | Diophantine expression for... |
| expdiophlem1 44071 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 44072 | Lemma for ~ expdioph . Ex... |
| expdioph 44073 | The exponential function i... |
| setindtr 44074 | Set induction for sets con... |
| setindtrs 44075 | Set induction scheme witho... |
| dford3lem1 44076 | Lemma for ~ dford3 . (Con... |
| dford3lem2 44077 | Lemma for ~ dford3 . (Con... |
| dford3 44078 | Ordinals are precisely the... |
| dford4 44079 | ~ dford3 expressed in prim... |
| wopprc 44080 | Unrelated: Wiener pairs t... |
| rpnnen3lem 44081 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 44082 | Dedekind cut injection of ... |
| axac10 44083 | Characterization of choice... |
| harinf 44084 | The Hartogs number of an i... |
| wdom2d2 44085 | Deduction for weak dominan... |
| ttac 44086 | Tarski's theorem about cho... |
| pw2f1ocnv 44087 | Define a bijection between... |
| pw2f1o2 44088 | Define a bijection between... |
| pw2f1o2val 44089 | Function value of the ~ pw... |
| pw2f1o2val2 44090 | Membership in a mapped set... |
| limsuc2 44091 | Limit ordinals in the sens... |
| wepwsolem 44092 | Transfer an ordering on ch... |
| wepwso 44093 | A well-ordering induces a ... |
| dnnumch1 44094 | Define an enumeration of a... |
| dnnumch2 44095 | Define an enumeration (wea... |
| dnnumch3lem 44096 | Value of the ordinal injec... |
| dnnumch3 44097 | Define an injection from a... |
| dnwech 44098 | Define a well-ordering fro... |
| aomclem1 44099 | Lemma for ~ dfac11 . This... |
| aomclem2 44100 | Lemma for ~ dfac11 . Succ... |
| aomclem3 44101 | Lemma for ~ dfac11 . Succ... |
| aomclem4 44102 | Lemma for ~ dfac11 . Limi... |
| aomclem5 44103 | Lemma for ~ dfac11 . Comb... |
| aomclem6 44104 | Lemma for ~ dfac11 . Tran... |
| aomclem7 44105 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 44106 | Lemma for ~ dfac11 . Perf... |
| dfac11 44107 | The right-hand side of thi... |
| kelac1 44108 | Kelley's choice, basic for... |
| kelac2lem 44109 | Lemma for ~ kelac2 and ~ d... |
| kelac2 44110 | Kelley's choice, most comm... |
| dfac21 44111 | Tychonoff's theorem is a c... |
| islmodfg 44114 | Property of a finitely gen... |
| islssfg 44115 | Property of a finitely gen... |
| islssfg2 44116 | Property of a finitely gen... |
| islssfgi 44117 | Finitely spanned subspaces... |
| fglmod 44118 | Finitely generated left mo... |
| lsmfgcl 44119 | The sum of two finitely ge... |
| islnm 44122 | Property of being a Noethe... |
| islnm2 44123 | Property of being a Noethe... |
| lnmlmod 44124 | A Noetherian left module i... |
| lnmlssfg 44125 | A submodule of Noetherian ... |
| lnmlsslnm 44126 | All submodules of a Noethe... |
| lnmfg 44127 | A Noetherian left module i... |
| kercvrlsm 44128 | The domain of a linear fun... |
| lmhmfgima 44129 | A homomorphism maps finite... |
| lnmepi 44130 | Epimorphic images of Noeth... |
| lmhmfgsplit 44131 | If the kernel and range of... |
| lmhmlnmsplit 44132 | If the kernel and range of... |
| lnmlmic 44133 | Noetherian is an invariant... |
| pwssplit4 44134 | Splitting for structure po... |
| filnm 44135 | Finite left modules are No... |
| pwslnmlem0 44136 | Zeroeth powers are Noether... |
| pwslnmlem1 44137 | First powers are Noetheria... |
| pwslnmlem2 44138 | A sum of powers is Noether... |
| pwslnm 44139 | Finite powers of Noetheria... |
| unxpwdom3 44140 | Weaker version of ~ unxpwd... |
| pwfi2f1o 44141 | The ~ pw2f1o bijection rel... |
| pwfi2en 44142 | Finitely supported indicat... |
| frlmpwfi 44143 | Formal linear combinations... |
| gicabl 44144 | Being Abelian is a group i... |
| imasgim 44145 | A relabeling of the elemen... |
| isnumbasgrplem1 44146 | A set which is equipollent... |
| harn0 44147 | The Hartogs number of a se... |
| numinfctb 44148 | A numerable infinite set c... |
| isnumbasgrplem2 44149 | If the (to be thought of a... |
| isnumbasgrplem3 44150 | Every nonempty numerable s... |
| isnumbasabl 44151 | A set is numerable iff it ... |
| isnumbasgrp 44152 | A set is numerable iff it ... |
| dfacbasgrp 44153 | A choice equivalent in abs... |
| islnr 44156 | Property of a left-Noether... |
| lnrring 44157 | Left-Noetherian rings are ... |
| lnrlnm 44158 | Left-Noetherian rings have... |
| islnr2 44159 | Property of being a left-N... |
| islnr3 44160 | Relate left-Noetherian rin... |
| lnr2i 44161 | Given an ideal in a left-N... |
| lpirlnr 44162 | Left principal ideal rings... |
| lnrfrlm 44163 | Finite-dimensional free mo... |
| lnrfg 44164 | Finitely-generated modules... |
| lnrfgtr 44165 | A submodule of a finitely ... |
| hbtlem1 44168 | Value of the leading coeff... |
| hbtlem2 44169 | Leading coefficient ideals... |
| hbtlem7 44170 | Functionality of leading c... |
| hbtlem4 44171 | The leading ideal function... |
| hbtlem3 44172 | The leading ideal function... |
| hbtlem5 44173 | The leading ideal function... |
| hbtlem6 44174 | There is a finite set of p... |
| hbt 44175 | The Hilbert Basis Theorem ... |
| dgrsub2 44180 | Subtracting two polynomial... |
| elmnc 44181 | Property of a monic polyno... |
| mncply 44182 | A monic polynomial is a po... |
| mnccoe 44183 | A monic polynomial has lea... |
| mncn0 44184 | A monic polynomial is not ... |
| dgraaval 44189 | Value of the degree functi... |
| dgraalem 44190 | Properties of the degree o... |
| dgraacl 44191 | Closure of the degree func... |
| dgraaf 44192 | Degree function on algebra... |
| dgraaub 44193 | Upper bound on degree of a... |
| dgraa0p 44194 | A rational polynomial of d... |
| mpaaeu 44195 | An algebraic number has ex... |
| mpaaval 44196 | Value of the minimal polyn... |
| mpaalem 44197 | Properties of the minimal ... |
| mpaacl 44198 | Minimal polynomial is a po... |
| mpaadgr 44199 | Minimal polynomial has deg... |
| mpaaroot 44200 | The minimal polynomial of ... |
| mpaamn 44201 | Minimal polynomial is moni... |
| itgoval 44206 | Value of the integral-over... |
| aaitgo 44207 | The standard algebraic num... |
| itgoss 44208 | An integral element is int... |
| itgocn 44209 | All integral elements are ... |
| cnsrexpcl 44210 | Exponentiation is closed i... |
| fsumcnsrcl 44211 | Finite sums are closed in ... |
| cnsrplycl 44212 | Polynomials are closed in ... |
| rgspnid 44213 | The span of a subring is i... |
| rngunsnply 44214 | Adjoining one element to a... |
| flcidc 44215 | Finite linear combinations... |
| algstr 44218 | Lemma to shorten proofs of... |
| algbase 44219 | The base set of a construc... |
| algaddg 44220 | The additive operation of ... |
| algmulr 44221 | The multiplicative operati... |
| algsca 44222 | The set of scalars of a co... |
| algvsca 44223 | The scalar product operati... |
| mendval 44224 | Value of the module endomo... |
| mendbas 44225 | Base set of the module end... |
| mendplusgfval 44226 | Addition in the module end... |
| mendplusg 44227 | A specific addition in the... |
| mendmulrfval 44228 | Multiplication in the modu... |
| mendmulr 44229 | A specific multiplication ... |
| mendsca 44230 | The module endomorphism al... |
| mendvscafval 44231 | Scalar multiplication in t... |
| mendvsca 44232 | A specific scalar multipli... |
| mendring 44233 | The module endomorphism al... |
| mendlmod 44234 | The module endomorphism al... |
| mendassa 44235 | The module endomorphism al... |
| idomodle 44236 | Limit on the number of ` N... |
| fiuneneq 44237 | Two finite sets of equal s... |
| idomsubgmo 44238 | The units of an integral d... |
| proot1mul 44239 | Any primitive ` N ` -th ro... |
| proot1hash 44240 | If an integral domain has ... |
| proot1ex 44241 | The complex field has prim... |
| mon1psubm 44244 | Monic polynomials are a mu... |
| deg1mhm 44245 | Homomorphic property of th... |
| cytpfn 44246 | Functionality of the cyclo... |
| cytpval 44247 | Substitutions for the Nth ... |
| fgraphopab 44248 | Express a function as a su... |
| fgraphxp 44249 | Express a function as a su... |
| hausgraph 44250 | The graph of a continuous ... |
| r1sssucd 44255 | Deductive form of ~ r1sssu... |
| iocunico 44256 | Split an open interval int... |
| iocinico 44257 | The intersection of two se... |
| iocmbl 44258 | An open-below, closed-abov... |
| cnioobibld 44259 | A bounded, continuous func... |
| arearect 44260 | The area of a rectangle wh... |
| areaquad 44261 | The area of a quadrilatera... |
| uniel 44262 | Two ways to say a union is... |
| unielss 44263 | Two ways to say the union ... |
| unielid 44264 | Two ways to say the union ... |
| ssunib 44265 | Two ways to say a class is... |
| rp-intrabeq 44266 | Equality theorem for supre... |
| rp-unirabeq 44267 | Equality theorem for infim... |
| onmaxnelsup 44268 | Two ways to say the maximu... |
| onsupneqmaxlim0 44269 | If the supremum of a class... |
| onsupcl2 44270 | The supremum of a set of o... |
| onuniintrab 44271 | The union of a set of ordi... |
| onintunirab 44272 | The intersection of a non-... |
| onsupnmax 44273 | If the union of a class of... |
| onsupuni 44274 | The supremum of a set of o... |
| onsupuni2 44275 | The supremum of a set of o... |
| onsupintrab 44276 | The supremum of a set of o... |
| onsupintrab2 44277 | The supremum of a set of o... |
| onsupcl3 44278 | The supremum of a set of o... |
| onsupex3 44279 | The supremum of a set of o... |
| onuniintrab2 44280 | The union of a set of ordi... |
| oninfint 44281 | The infimum of a non-empty... |
| oninfunirab 44282 | The infimum of a non-empty... |
| oninfcl2 44283 | The infimum of a non-empty... |
| onsupmaxb 44284 | The union of a class of or... |
| onexgt 44285 | For any ordinal, there is ... |
| onexomgt 44286 | For any ordinal, there is ... |
| omlimcl2 44287 | The product of a limit ord... |
| onexlimgt 44288 | For any ordinal, there is ... |
| onexoegt 44289 | For any ordinal, there is ... |
| oninfex2 44290 | The infimum of a non-empty... |
| onsupeqmax 44291 | Condition when the supremu... |
| onsupeqnmax 44292 | Condition when the supremu... |
| onsuplub 44293 | The supremum of a set of o... |
| onsupnub 44294 | An upper bound of a set of... |
| onfisupcl 44295 | Sufficient condition when ... |
| onelord 44296 | Every element of a ordinal... |
| onepsuc 44297 | Every ordinal is less than... |
| epsoon 44298 | The ordinals are strictly ... |
| epirron 44299 | The strict order on the or... |
| oneptr 44300 | The strict order on the or... |
| oneltr 44301 | The elementhood relation o... |
| oneptri 44302 | The strict, complete (line... |
| ordeldif 44303 | Membership in the differen... |
| ordeldifsucon 44304 | Membership in the differen... |
| ordeldif1o 44305 | Membership in the differen... |
| ordne0gt0 44306 | Ordinal zero is less than ... |
| ondif1i 44307 | Ordinal zero is less than ... |
| onsucelab 44308 | The successor of every ord... |
| dflim6 44309 | A limit ordinal is a nonze... |
| limnsuc 44310 | A limit ordinal is not an ... |
| onsucss 44311 | If one ordinal is less tha... |
| ordnexbtwnsuc 44312 | For any distinct pair of o... |
| orddif0suc 44313 | For any distinct pair of o... |
| onsucf1lem 44314 | For ordinals, the successo... |
| onsucf1olem 44315 | The successor operation is... |
| onsucrn 44316 | The successor operation is... |
| onsucf1o 44317 | The successor operation is... |
| dflim7 44318 | A limit ordinal is a nonze... |
| onov0suclim 44319 | Compactly express rules fo... |
| oa0suclim 44320 | Closed form expression of ... |
| om0suclim 44321 | Closed form expression of ... |
| oe0suclim 44322 | Closed form expression of ... |
| oaomoecl 44323 | The operations of addition... |
| onsupsucismax 44324 | If the union of a set of o... |
| onsssupeqcond 44325 | If for every element of a ... |
| limexissup 44326 | An ordinal which is a limi... |
| limiun 44327 | A limit ordinal is the uni... |
| limexissupab 44328 | An ordinal which is a limi... |
| om1om1r 44329 | Ordinal one is both a left... |
| oe0rif 44330 | Ordinal zero raised to any... |
| oasubex 44331 | While subtraction can't be... |
| nnamecl 44332 | Natural numbers are closed... |
| onsucwordi 44333 | The successor operation pr... |
| oalim2cl 44334 | The ordinal sum of any ord... |
| oaltublim 44335 | Given ` C ` is a limit ord... |
| oaordi3 44336 | Ordinal addition of the sa... |
| oaord3 44337 | When the same ordinal is a... |
| 1oaomeqom 44338 | Ordinal one plus omega is ... |
| oaabsb 44339 | The right addend absorbs t... |
| oaordnrex 44340 | When omega is added on the... |
| oaordnr 44341 | When the same ordinal is a... |
| omge1 44342 | Any nonzero ordinal produc... |
| omge2 44343 | Any nonzero ordinal produc... |
| omlim2 44344 | The nonzero product with a... |
| omord2lim 44345 | Given a limit ordinal, the... |
| omord2i 44346 | Ordinal multiplication of ... |
| omord2com 44347 | When the same nonzero ordi... |
| 2omomeqom 44348 | Ordinal two times omega is... |
| omnord1ex 44349 | When omega is multiplied o... |
| omnord1 44350 | When the same nonzero ordi... |
| oege1 44351 | Any nonzero ordinal power ... |
| oege2 44352 | Any power of an ordinal at... |
| rp-oelim2 44353 | The power of an ordinal at... |
| oeord2lim 44354 | Given a limit ordinal, the... |
| oeord2i 44355 | Ordinal exponentiation of ... |
| oeord2com 44356 | When the same base at leas... |
| nnoeomeqom 44357 | Any natural number at leas... |
| df3o2 44358 | Ordinal 3 is the unordered... |
| df3o3 44359 | Ordinal 3, fully expanded.... |
| oenord1ex 44360 | When ordinals two and thre... |
| oenord1 44361 | When two ordinals (both at... |
| oaomoencom 44362 | Ordinal addition, multipli... |
| oenassex 44363 | Ordinal two raised to two ... |
| oenass 44364 | Ordinal exponentiation is ... |
| cantnftermord 44365 | For terms of the form of a... |
| cantnfub 44366 | Given a finite number of t... |
| cantnfub2 44367 | Given a finite number of t... |
| bropabg 44368 | Equivalence for two classe... |
| cantnfresb 44369 | A Cantor normal form which... |
| cantnf2 44370 | For every ordinal, ` A ` ,... |
| oawordex2 44371 | If ` C ` is between ` A ` ... |
| nnawordexg 44372 | If an ordinal, ` B ` , is ... |
| succlg 44373 | Closure law for ordinal su... |
| dflim5 44374 | A limit ordinal is either ... |
| oacl2g 44375 | Closure law for ordinal ad... |
| onmcl 44376 | If an ordinal is less than... |
| omabs2 44377 | Ordinal multiplication by ... |
| omcl2 44378 | Closure law for ordinal mu... |
| omcl3g 44379 | Closure law for ordinal mu... |
| ordsssucb 44380 | An ordinal number is less ... |
| tfsconcatlem 44381 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 44382 | The concatenation of two t... |
| tfsconcatfn 44383 | The concatenation of two t... |
| tfsconcatfv1 44384 | An early value of the conc... |
| tfsconcatfv2 44385 | A latter value of the conc... |
| tfsconcatfv 44386 | The value of the concatena... |
| tfsconcatrn 44387 | The range of the concatena... |
| tfsconcatfo 44388 | The concatenation of two t... |
| tfsconcatb0 44389 | The concatentation with th... |
| tfsconcat0i 44390 | The concatentation with th... |
| tfsconcat0b 44391 | The concatentation with th... |
| tfsconcat00 44392 | The concatentation of two ... |
| tfsconcatrev 44393 | If the domain of a transfi... |
| tfsconcatrnss12 44394 | The range of the concatena... |
| tfsconcatrnss 44395 | The concatenation of trans... |
| tfsconcatrnsson 44396 | The concatenation of trans... |
| tfsnfin 44397 | A transfinite sequence is ... |
| rp-tfslim 44398 | The limit of a sequence of... |
| ofoafg 44399 | Addition operator for func... |
| ofoaf 44400 | Addition operator for func... |
| ofoafo 44401 | Addition operator for func... |
| ofoacl 44402 | Closure law for component ... |
| ofoaid1 44403 | Identity law for component... |
| ofoaid2 44404 | Identity law for component... |
| ofoaass 44405 | Component-wise addition of... |
| ofoacom 44406 | Component-wise addition of... |
| naddcnff 44407 | Addition operator for Cant... |
| naddcnffn 44408 | Addition operator for Cant... |
| naddcnffo 44409 | Addition of Cantor normal ... |
| naddcnfcl 44410 | Closure law for component-... |
| naddcnfcom 44411 | Component-wise ordinal add... |
| naddcnfid1 44412 | Identity law for component... |
| naddcnfid2 44413 | Identity law for component... |
| naddcnfass 44414 | Component-wise addition of... |
| onsucunifi 44415 | The successor to the union... |
| sucunisn 44416 | The successor to the union... |
| onsucunipr 44417 | The successor to the union... |
| onsucunitp 44418 | The successor to the union... |
| oaun3lem1 44419 | The class of all ordinal s... |
| oaun3lem2 44420 | The class of all ordinal s... |
| oaun3lem3 44421 | The class of all ordinal s... |
| oaun3lem4 44422 | The class of all ordinal s... |
| rp-abid 44423 | Two ways to express a clas... |
| oadif1lem 44424 | Express the set difference... |
| oadif1 44425 | Express the set difference... |
| oaun2 44426 | Ordinal addition as a unio... |
| oaun3 44427 | Ordinal addition as a unio... |
| naddov4 44428 | Alternate expression for n... |
| nadd2rabtr 44429 | The set of ordinals which ... |
| nadd2rabord 44430 | The set of ordinals which ... |
| nadd2rabex 44431 | The class of ordinals whic... |
| nadd2rabon 44432 | The set of ordinals which ... |
| nadd1rabtr 44433 | The set of ordinals which ... |
| nadd1rabord 44434 | The set of ordinals which ... |
| nadd1rabex 44435 | The class of ordinals whic... |
| nadd1rabon 44436 | The set of ordinals which ... |
| nadd1suc 44437 | Natural addition with 1 is... |
| naddass1 44438 | Natural addition of ordina... |
| naddgeoa 44439 | Natural addition results i... |
| naddonnn 44440 | Natural addition with a na... |
| naddwordnexlem0 44441 | When ` A ` is the sum of a... |
| naddwordnexlem1 44442 | When ` A ` is the sum of a... |
| naddwordnexlem2 44443 | When ` A ` is the sum of a... |
| naddwordnexlem3 44444 | When ` A ` is the sum of a... |
| oawordex3 44445 | When ` A ` is the sum of a... |
| naddwordnexlem4 44446 | When ` A ` is the sum of a... |
| ordsssucim 44447 | If an ordinal is less than... |
| insucid 44448 | The intersection of a clas... |
| oaltom 44449 | Multiplication eventually ... |
| oe2 44450 | Two ways to square an ordi... |
| omltoe 44451 | Exponentiation eventually ... |
| abeqabi 44452 | Generalized condition for ... |
| abpr 44453 | Condition for a class abst... |
| abtp 44454 | Condition for a class abst... |
| ralopabb 44455 | Restricted universal quant... |
| fpwfvss 44456 | Functions into a powerset ... |
| sdomne0 44457 | A class that strictly domi... |
| sdomne0d 44458 | A class that strictly domi... |
| safesnsupfiss 44459 | If ` B ` is a finite subse... |
| safesnsupfiub 44460 | If ` B ` is a finite subse... |
| safesnsupfidom1o 44461 | If ` B ` is a finite subse... |
| safesnsupfilb 44462 | If ` B ` is a finite subse... |
| isoeq145d 44463 | Equality deduction for iso... |
| resisoeq45d 44464 | Equality deduction for equ... |
| negslem1 44465 | An equivalence between ide... |
| nvocnvb 44466 | Equivalence to saying the ... |
| rp-brsslt 44467 | Binary relation form of a ... |
| nla0002 44468 | Extending a linear order t... |
| nla0003 44469 | Extending a linear order t... |
| nla0001 44470 | Extending a linear order t... |
| faosnf0.11b 44471 | ` B ` is called a non-limi... |
| dfno2 44472 | A surreal number, in the f... |
| onnoxpg 44473 | Every ordinal maps to a su... |
| onnobdayg 44474 | Every ordinal maps to a su... |
| bdaybndex 44475 | Bounds formed from the bir... |
| bdaybndbday 44476 | Bounds formed from the bir... |
| onnoxp 44477 | Every ordinal maps to a su... |
| onnoxpi 44478 | Every ordinal maps to a su... |
| 0fno 44479 | Ordinal zero maps to a sur... |
| 1fno 44480 | Ordinal one maps to a surr... |
| 2fno 44481 | Ordinal two maps to a surr... |
| 3fno 44482 | Ordinal three maps to a su... |
| 4fno 44483 | Ordinal four maps to a sur... |
| fnimafnex 44484 | The functional image of a ... |
| nlimsuc 44485 | A successor is not a limit... |
| nlim1NEW 44486 | 1 is not a limit ordinal. ... |
| nlim2NEW 44487 | 2 is not a limit ordinal. ... |
| nlim3 44488 | 3 is not a limit ordinal. ... |
| nlim4 44489 | 4 is not a limit ordinal. ... |
| oa1un 44490 | Given ` A e. On ` , let ` ... |
| oa1cl 44491 | ` A +o 1o ` is in ` On ` .... |
| 0finon 44492 | 0 is a finite ordinal. Se... |
| 1finon 44493 | 1 is a finite ordinal. Se... |
| 2finon 44494 | 2 is a finite ordinal. Se... |
| 3finon 44495 | 3 is a finite ordinal. Se... |
| 4finon 44496 | 4 is a finite ordinal. Se... |
| finona1cl 44497 | The finite ordinals are cl... |
| finonex 44498 | The finite ordinals are a ... |
| fzunt 44499 | Union of two adjacent fini... |
| fzuntd 44500 | Union of two adjacent fini... |
| fzunt1d 44501 | Union of two overlapping f... |
| fzuntgd 44502 | Union of two adjacent or o... |
| ifpan123g 44503 | Conjunction of conditional... |
| ifpan23 44504 | Conjunction of conditional... |
| ifpdfor2 44505 | Define or in terms of cond... |
| ifporcor 44506 | Corollary of commutation o... |
| ifpdfan2 44507 | Define and with conditiona... |
| ifpancor 44508 | Corollary of commutation o... |
| ifpdfor 44509 | Define or in terms of cond... |
| ifpdfan 44510 | Define and with conditiona... |
| ifpbi2 44511 | Equivalence theorem for co... |
| ifpbi3 44512 | Equivalence theorem for co... |
| ifpim1 44513 | Restate implication as con... |
| ifpnot 44514 | Restate negated wff as con... |
| ifpid2 44515 | Restate wff as conditional... |
| ifpim2 44516 | Restate implication as con... |
| ifpbi23 44517 | Equivalence theorem for co... |
| ifpbiidcor 44518 | Restatement of ~ biid . (... |
| ifpbicor 44519 | Corollary of commutation o... |
| ifpxorcor 44520 | Corollary of commutation o... |
| ifpbi1 44521 | Equivalence theorem for co... |
| ifpnot23 44522 | Negation of conditional lo... |
| ifpnotnotb 44523 | Factor conditional logic o... |
| ifpnorcor 44524 | Corollary of commutation o... |
| ifpnancor 44525 | Corollary of commutation o... |
| ifpnot23b 44526 | Negation of conditional lo... |
| ifpbiidcor2 44527 | Restatement of ~ biid . (... |
| ifpnot23c 44528 | Negation of conditional lo... |
| ifpnot23d 44529 | Negation of conditional lo... |
| ifpdfnan 44530 | Define nand as conditional... |
| ifpdfxor 44531 | Define xor as conditional ... |
| ifpbi12 44532 | Equivalence theorem for co... |
| ifpbi13 44533 | Equivalence theorem for co... |
| ifpbi123 44534 | Equivalence theorem for co... |
| ifpidg 44535 | Restate wff as conditional... |
| ifpid3g 44536 | Restate wff as conditional... |
| ifpid2g 44537 | Restate wff as conditional... |
| ifpid1g 44538 | Restate wff as conditional... |
| ifpim23g 44539 | Restate implication as con... |
| ifpim3 44540 | Restate implication as con... |
| ifpnim1 44541 | Restate negated implicatio... |
| ifpim4 44542 | Restate implication as con... |
| ifpnim2 44543 | Restate negated implicatio... |
| ifpim123g 44544 | Implication of conditional... |
| ifpim1g 44545 | Implication of conditional... |
| ifp1bi 44546 | Substitute the first eleme... |
| ifpbi1b 44547 | When the first variable is... |
| ifpimimb 44548 | Factor conditional logic o... |
| ifpororb 44549 | Factor conditional logic o... |
| ifpananb 44550 | Factor conditional logic o... |
| ifpnannanb 44551 | Factor conditional logic o... |
| ifpor123g 44552 | Disjunction of conditional... |
| ifpimim 44553 | Consequnce of implication.... |
| ifpbibib 44554 | Factor conditional logic o... |
| ifpxorxorb 44555 | Factor conditional logic o... |
| rp-fakeimass 44556 | A special case where impli... |
| rp-fakeanorass 44557 | A special case where a mix... |
| rp-fakeoranass 44558 | A special case where a mix... |
| rp-fakeinunass 44559 | A special case where a mix... |
| rp-fakeuninass 44560 | A special case where a mix... |
| rp-isfinite5 44561 | A set is said to be finite... |
| rp-isfinite6 44562 | A set is said to be finite... |
| intabssd 44563 | When for each element ` y ... |
| eu0 44564 | There is only one empty se... |
| epelon2 44565 | Over the ordinal numbers, ... |
| ontric3g 44566 | For all ` x , y e. On ` , ... |
| dfsucon 44567 | ` A ` is called a successo... |
| snen1g 44568 | A singleton is equinumerou... |
| snen1el 44569 | A singleton is equinumerou... |
| sn1dom 44570 | A singleton is dominated b... |
| pr2dom 44571 | An unordered pair is domin... |
| tr3dom 44572 | An unordered triple is dom... |
| ensucne0 44573 | A class equinumerous to a ... |
| ensucne0OLD 44574 | A class equinumerous to a ... |
| dfom6 44575 | Let ` _om ` be defined to ... |
| infordmin 44576 | ` _om ` is the smallest in... |
| iscard4 44577 | Two ways to express the pr... |
| minregex 44578 | Given any cardinal number ... |
| minregex2 44579 | Given any cardinal number ... |
| iscard5 44580 | Two ways to express the pr... |
| elrncard 44581 | Let us define a cardinal n... |
| harval3 44582 | ` ( har `` A ) ` is the le... |
| harval3on 44583 | For any ordinal number ` A... |
| omssrncard 44584 | All natural numbers are ca... |
| 0iscard 44585 | 0 is a cardinal number. (... |
| 1iscard 44586 | 1 is a cardinal number. (... |
| omiscard 44587 | ` _om ` is a cardinal numb... |
| sucomisnotcard 44588 | ` _om +o 1o ` is not a car... |
| nna1iscard 44589 | For any natural number, th... |
| har2o 44590 | The least cardinal greater... |
| en2pr 44591 | A class is equinumerous to... |
| pr2cv 44592 | If an unordered pair is eq... |
| pr2el1 44593 | If an unordered pair is eq... |
| pr2cv1 44594 | If an unordered pair is eq... |
| pr2el2 44595 | If an unordered pair is eq... |
| pr2cv2 44596 | If an unordered pair is eq... |
| pren2 44597 | An unordered pair is equin... |
| pr2eldif1 44598 | If an unordered pair is eq... |
| pr2eldif2 44599 | If an unordered pair is eq... |
| pren2d 44600 | A pair of two distinct set... |
| aleph1min 44601 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 44602 | ` aleph ` is a strictly or... |
| alephiso3 44603 | ` aleph ` is a strictly or... |
| pwelg 44604 | The powerclass is an eleme... |
| pwinfig 44605 | The powerclass of an infin... |
| pwinfi2 44606 | The powerclass of an infin... |
| pwinfi3 44607 | The powerclass of an infin... |
| pwinfi 44608 | The powerclass of an infin... |
| fipjust 44609 | A definition of the finite... |
| cllem0 44610 | The class of all sets with... |
| superficl 44611 | The class of all supersets... |
| superuncl 44612 | The class of all supersets... |
| ssficl 44613 | The class of all subsets o... |
| ssuncl 44614 | The class of all subsets o... |
| ssdifcl 44615 | The class of all subsets o... |
| sssymdifcl 44616 | The class of all subsets o... |
| fiinfi 44617 | If two classes have the fi... |
| rababg 44618 | Condition when restricted ... |
| elinintab 44619 | Two ways of saying a set i... |
| elmapintrab 44620 | Two ways to say a set is a... |
| elinintrab 44621 | Two ways of saying a set i... |
| inintabss 44622 | Upper bound on intersectio... |
| inintabd 44623 | Value of the intersection ... |
| xpinintabd 44624 | Value of the intersection ... |
| relintabex 44625 | If the intersection of a c... |
| elcnvcnvintab 44626 | Two ways of saying a set i... |
| relintab 44627 | Value of the intersection ... |
| nonrel 44628 | A non-relation is equal to... |
| elnonrel 44629 | Only an ordered pair where... |
| relnonrel 44630 | The non-relation part of a... |
| cnvnonrel 44631 | The converse of the non-re... |
| brnonrel 44632 | A non-relation cannot rela... |
| dmnonrel 44633 | The domain of the non-rela... |
| rnnonrel 44634 | The range of the non-relat... |
| resnonrel 44635 | A restriction of the non-r... |
| imanonrel 44636 | An image under the non-rel... |
| cononrel1 44637 | Composition with the non-r... |
| cononrel2 44638 | Composition with the non-r... |
| elmapintab 44639 | Two ways to say a set is a... |
| fvnonrel 44640 | The function value of any ... |
| elinlem 44641 | Two ways to say a set is a... |
| elcnvcnvlem 44642 | Two ways to say a set is a... |
| cnvcnvintabd 44643 | Value of the relationship ... |
| elcnvlem 44644 | Two ways to say a set is a... |
| elcnvintab 44645 | Two ways of saying a set i... |
| cnvintabd 44646 | Value of the converse of t... |
| undmrnresiss 44647 | Two ways of saying the ide... |
| reflexg 44648 | Two ways of saying a relat... |
| cnvssco 44649 | A condition weaker than re... |
| refimssco 44650 | Reflexive relations are su... |
| cleq2lem 44651 | Equality implies bijection... |
| cbvcllem 44652 | Change of bound variable i... |
| clublem 44653 | If a superset ` Y ` of ` X... |
| clss2lem 44654 | The closure of a property ... |
| dfid7 44655 | Definition of identity rel... |
| mptrcllem 44656 | Show two versions of a clo... |
| cotrintab 44657 | The intersection of a clas... |
| rclexi 44658 | The reflexive closure of a... |
| rtrclexlem 44659 | Existence of relation impl... |
| rtrclex 44660 | The reflexive-transitive c... |
| trclubgNEW 44661 | If a relation exists then ... |
| trclubNEW 44662 | If a relation exists then ... |
| trclexi 44663 | The transitive closure of ... |
| rtrclexi 44664 | The reflexive-transitive c... |
| clrellem 44665 | When the property ` ps ` h... |
| clcnvlem 44666 | When ` A ` , an upper boun... |
| cnvtrucl0 44667 | The converse of the trivia... |
| cnvrcl0 44668 | The converse of the reflex... |
| cnvtrcl0 44669 | The converse of the transi... |
| dmtrcl 44670 | The domain of the transiti... |
| rntrcl 44671 | The range of the transitiv... |
| dfrtrcl5 44672 | Definition of reflexive-tr... |
| trcleq2lemRP 44673 | Equality implies bijection... |
| sqrtcvallem1 44674 | Two ways of saying a compl... |
| reabsifneg 44675 | Alternate expression for t... |
| reabsifnpos 44676 | Alternate expression for t... |
| reabsifpos 44677 | Alternate expression for t... |
| reabsifnneg 44678 | Alternate expression for t... |
| reabssgn 44679 | Alternate expression for t... |
| sqrtcvallem2 44680 | Equivalent to saying that ... |
| sqrtcvallem3 44681 | Equivalent to saying that ... |
| sqrtcvallem4 44682 | Equivalent to saying that ... |
| sqrtcvallem5 44683 | Equivalent to saying that ... |
| sqrtcval 44684 | Explicit formula for the c... |
| sqrtcval2 44685 | Explicit formula for the c... |
| resqrtval 44686 | Real part of the complex s... |
| imsqrtval 44687 | Imaginary part of the comp... |
| resqrtvalex 44688 | Example for ~ resqrtval . ... |
| imsqrtvalex 44689 | Example for ~ imsqrtval . ... |
| al3im 44690 | Version of ~ ax-4 for a ne... |
| intima0 44691 | Two ways of expressing the... |
| elimaint 44692 | Element of image of inters... |
| cnviun 44693 | Converse of indexed union.... |
| imaiun1 44694 | The image of an indexed un... |
| coiun1 44695 | Composition with an indexe... |
| elintima 44696 | Element of intersection of... |
| intimass 44697 | The image under the inters... |
| intimass2 44698 | The image under the inters... |
| intimag 44699 | Requirement for the image ... |
| intimasn 44700 | Two ways to express the im... |
| intimasn2 44701 | Two ways to express the im... |
| ss2iundf 44702 | Subclass theorem for index... |
| ss2iundv 44703 | Subclass theorem for index... |
| cbviuneq12df 44704 | Rule used to change the bo... |
| cbviuneq12dv 44705 | Rule used to change the bo... |
| conrel1d 44706 | Deduction about compositio... |
| conrel2d 44707 | Deduction about compositio... |
| trrelind 44708 | The intersection of transi... |
| xpintrreld 44709 | The intersection of a tran... |
| restrreld 44710 | The restriction of a trans... |
| trrelsuperreldg 44711 | Concrete construction of a... |
| trficl 44712 | The class of all transitiv... |
| cnvtrrel 44713 | The converse of a transiti... |
| trrelsuperrel2dg 44714 | Concrete construction of a... |
| dfrcl2 44717 | Reflexive closure of a rel... |
| dfrcl3 44718 | Reflexive closure of a rel... |
| dfrcl4 44719 | Reflexive closure of a rel... |
| relexp2 44720 | A set operated on by the r... |
| relexpnul 44721 | If the domain and range of... |
| eliunov2 44722 | Membership in the indexed ... |
| eltrclrec 44723 | Membership in the indexed ... |
| elrtrclrec 44724 | Membership in the indexed ... |
| briunov2 44725 | Two classes related by the... |
| brmptiunrelexpd 44726 | If two elements are connec... |
| fvmptiunrelexplb0d 44727 | If the indexed union range... |
| fvmptiunrelexplb0da 44728 | If the indexed union range... |
| fvmptiunrelexplb1d 44729 | If the indexed union range... |
| brfvid 44730 | If two elements are connec... |
| brfvidRP 44731 | If two elements are connec... |
| fvilbd 44732 | A set is a subset of its i... |
| fvilbdRP 44733 | A set is a subset of its i... |
| brfvrcld 44734 | If two elements are connec... |
| brfvrcld2 44735 | If two elements are connec... |
| fvrcllb0d 44736 | A restriction of the ident... |
| fvrcllb0da 44737 | A restriction of the ident... |
| fvrcllb1d 44738 | A set is a subset of its i... |
| brtrclrec 44739 | Two classes related by the... |
| brrtrclrec 44740 | Two classes related by the... |
| briunov2uz 44741 | Two classes related by the... |
| eliunov2uz 44742 | Membership in the indexed ... |
| ov2ssiunov2 44743 | Any particular operator va... |
| relexp0eq 44744 | The zeroth power of relati... |
| iunrelexp0 44745 | Simplification of zeroth p... |
| relexpxpnnidm 44746 | Any positive power of a Ca... |
| relexpiidm 44747 | Any power of any restricti... |
| relexpss1d 44748 | The relational power of a ... |
| comptiunov2i 44749 | The composition two indexe... |
| corclrcl 44750 | The reflexive closure is i... |
| iunrelexpmin1 44751 | The indexed union of relat... |
| relexpmulnn 44752 | With exponents limited to ... |
| relexpmulg 44753 | With ordered exponents, th... |
| trclrelexplem 44754 | The union of relational po... |
| iunrelexpmin2 44755 | The indexed union of relat... |
| relexp01min 44756 | With exponents limited to ... |
| relexp1idm 44757 | Repeated raising a relatio... |
| relexp0idm 44758 | Repeated raising a relatio... |
| relexp0a 44759 | Absorption law for zeroth ... |
| relexpxpmin 44760 | The composition of powers ... |
| relexpaddss 44761 | The composition of two pow... |
| iunrelexpuztr 44762 | The indexed union of relat... |
| dftrcl3 44763 | Transitive closure of a re... |
| brfvtrcld 44764 | If two elements are connec... |
| fvtrcllb1d 44765 | A set is a subset of its i... |
| trclfvcom 44766 | The transitive closure of ... |
| cnvtrclfv 44767 | The converse of the transi... |
| cotrcltrcl 44768 | The transitive closure is ... |
| trclimalb2 44769 | Lower bound for image unde... |
| brtrclfv2 44770 | Two ways to indicate two e... |
| trclfvdecomr 44771 | The transitive closure of ... |
| trclfvdecoml 44772 | The transitive closure of ... |
| dmtrclfvRP 44773 | The domain of the transiti... |
| rntrclfvRP 44774 | The range of the transitiv... |
| rntrclfv 44775 | The range of the transitiv... |
| dfrtrcl3 44776 | Reflexive-transitive closu... |
| brfvrtrcld 44777 | If two elements are connec... |
| fvrtrcllb0d 44778 | A restriction of the ident... |
| fvrtrcllb0da 44779 | A restriction of the ident... |
| fvrtrcllb1d 44780 | A set is a subset of its i... |
| dfrtrcl4 44781 | Reflexive-transitive closu... |
| corcltrcl 44782 | The composition of the ref... |
| cortrcltrcl 44783 | Composition with the refle... |
| corclrtrcl 44784 | Composition with the refle... |
| cotrclrcl 44785 | The composition of the ref... |
| cortrclrcl 44786 | Composition with the refle... |
| cotrclrtrcl 44787 | Composition with the refle... |
| cortrclrtrcl 44788 | The reflexive-transitive c... |
| frege77d 44789 | If the images of both ` { ... |
| frege81d 44790 | If the image of ` U ` is a... |
| frege83d 44791 | If the image of the union ... |
| frege96d 44792 | If ` C ` follows ` A ` in ... |
| frege87d 44793 | If the images of both ` { ... |
| frege91d 44794 | If ` B ` follows ` A ` in ... |
| frege97d 44795 | If ` A ` contains all elem... |
| frege98d 44796 | If ` C ` follows ` A ` and... |
| frege102d 44797 | If either ` A ` and ` C ` ... |
| frege106d 44798 | If ` B ` follows ` A ` in ... |
| frege108d 44799 | If either ` A ` and ` C ` ... |
| frege109d 44800 | If ` A ` contains all elem... |
| frege114d 44801 | If either ` R ` relates ` ... |
| frege111d 44802 | If either ` A ` and ` C ` ... |
| frege122d 44803 | If ` F ` is a function, ` ... |
| frege124d 44804 | If ` F ` is a function, ` ... |
| frege126d 44805 | If ` F ` is a function, ` ... |
| frege129d 44806 | If ` F ` is a function and... |
| frege131d 44807 | If ` F ` is a function and... |
| frege133d 44808 | If ` F ` is a function and... |
| dfxor4 44809 | Express exclusive-or in te... |
| dfxor5 44810 | Express exclusive-or in te... |
| df3or2 44811 | Express triple-or in terms... |
| df3an2 44812 | Express triple-and in term... |
| nev 44813 | Express that not every set... |
| 0pssin 44814 | Express that an intersecti... |
| dfhe2 44817 | The property of relation `... |
| dfhe3 44818 | The property of relation `... |
| heeq12 44819 | Equality law for relations... |
| heeq1 44820 | Equality law for relations... |
| heeq2 44821 | Equality law for relations... |
| sbcheg 44822 | Distribute proper substitu... |
| hess 44823 | Subclass law for relations... |
| xphe 44824 | Any Cartesian product is h... |
| 0he 44825 | The empty relation is here... |
| 0heALT 44826 | The empty relation is here... |
| he0 44827 | Any relation is hereditary... |
| unhe1 44828 | The union of two relations... |
| snhesn 44829 | Any singleton is hereditar... |
| idhe 44830 | The identity relation is h... |
| psshepw 44831 | The relation between sets ... |
| sshepw 44832 | The relation between sets ... |
| rp-simp2-frege 44835 | Simplification of triple c... |
| rp-simp2 44836 | Simplification of triple c... |
| rp-frege3g 44837 | Add antecedent to ~ ax-fre... |
| frege3 44838 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 44839 | Double-use of ~ ax-frege2 ... |
| rp-frege24 44840 | Introducing an embedded an... |
| rp-frege4g 44841 | Deduction related to distr... |
| frege4 44842 | Special case of closed for... |
| frege5 44843 | A closed form of ~ syl . ... |
| rp-7frege 44844 | Distribute antecedent and ... |
| rp-4frege 44845 | Elimination of a nested an... |
| rp-6frege 44846 | Elimination of a nested an... |
| rp-8frege 44847 | Eliminate antecedent when ... |
| rp-frege25 44848 | Closed form for ~ a1dd . ... |
| frege6 44849 | A closed form of ~ imim2d ... |
| axfrege8 44850 | Swap antecedents. Identic... |
| frege7 44851 | A closed form of ~ syl6 . ... |
| frege26 44853 | Identical to ~ idd . Prop... |
| frege27 44854 | We cannot (at the same tim... |
| frege9 44855 | Closed form of ~ syl with ... |
| frege12 44856 | A closed form of ~ com23 .... |
| frege11 44857 | Elimination of a nested an... |
| frege24 44858 | Closed form for ~ a1d . D... |
| frege16 44859 | A closed form of ~ com34 .... |
| frege25 44860 | Closed form for ~ a1dd . ... |
| frege18 44861 | Closed form of a syllogism... |
| frege22 44862 | A closed form of ~ com45 .... |
| frege10 44863 | Result commuting anteceden... |
| frege17 44864 | A closed form of ~ com3l .... |
| frege13 44865 | A closed form of ~ com3r .... |
| frege14 44866 | Closed form of a deduction... |
| frege19 44867 | A closed form of ~ syl6 . ... |
| frege23 44868 | Syllogism followed by rota... |
| frege15 44869 | A closed form of ~ com4r .... |
| frege21 44870 | Replace antecedent in ante... |
| frege20 44871 | A closed form of ~ syl8 . ... |
| axfrege28 44872 | Contraposition. Identical... |
| frege29 44874 | Closed form of ~ con3d . ... |
| frege30 44875 | Commuted, closed form of ~... |
| axfrege31 44876 | Identical to ~ notnotr . ... |
| frege32 44878 | Deduce ~ con1 from ~ con3 ... |
| frege33 44879 | If ` ph ` or ` ps ` takes ... |
| frege34 44880 | If as a consequence of the... |
| frege35 44881 | Commuted, closed form of ~... |
| frege36 44882 | The case in which ` ps ` i... |
| frege37 44883 | If ` ch ` is a necessary c... |
| frege38 44884 | Identical to ~ pm2.21 . P... |
| frege39 44885 | Syllogism between ~ pm2.18... |
| frege40 44886 | Anything implies ~ pm2.18 ... |
| axfrege41 44887 | Identical to ~ notnot . A... |
| frege42 44889 | Not not ~ id . Propositio... |
| frege43 44890 | If there is a choice only ... |
| frege44 44891 | Similar to a commuted ~ pm... |
| frege45 44892 | Deduce ~ pm2.6 from ~ con1... |
| frege46 44893 | If ` ps ` holds when ` ph ... |
| frege47 44894 | Deduce consequence follows... |
| frege48 44895 | Closed form of syllogism w... |
| frege49 44896 | Closed form of deduction w... |
| frege50 44897 | Closed form of ~ jaoi . P... |
| frege51 44898 | Compare with ~ jaod . Pro... |
| axfrege52a 44899 | Justification for ~ ax-fre... |
| frege52aid 44901 | The case when the content ... |
| frege53aid 44902 | Specialization of ~ frege5... |
| frege53a 44903 | Lemma for ~ frege55a . Pr... |
| axfrege54a 44904 | Justification for ~ ax-fre... |
| frege54cor0a 44906 | Synonym for logical equiva... |
| frege54cor1a 44907 | Reflexive equality. (Cont... |
| frege55aid 44908 | Lemma for ~ frege57aid . ... |
| frege55lem1a 44909 | Necessary deduction regard... |
| frege55lem2a 44910 | Core proof of Proposition ... |
| frege55a 44911 | Proposition 55 of [Frege18... |
| frege55cor1a 44912 | Proposition 55 of [Frege18... |
| frege56aid 44913 | Lemma for ~ frege57aid . ... |
| frege56a 44914 | Proposition 56 of [Frege18... |
| frege57aid 44915 | This is the all important ... |
| frege57a 44916 | Analogue of ~ frege57aid .... |
| axfrege58a 44917 | Identical to ~ anifp . Ju... |
| frege58acor 44919 | Lemma for ~ frege59a . (C... |
| frege59a 44920 | A kind of Aristotelian inf... |
| frege60a 44921 | Swap antecedents of ~ ax-f... |
| frege61a 44922 | Lemma for ~ frege65a . Pr... |
| frege62a 44923 | A kind of Aristotelian inf... |
| frege63a 44924 | Proposition 63 of [Frege18... |
| frege64a 44925 | Lemma for ~ frege65a . Pr... |
| frege65a 44926 | A kind of Aristotelian inf... |
| frege66a 44927 | Swap antecedents of ~ freg... |
| frege67a 44928 | Lemma for ~ frege68a . Pr... |
| frege68a 44929 | Combination of applying a ... |
| axfrege52c 44930 | Justification for ~ ax-fre... |
| frege52b 44932 | The case when the content ... |
| frege53b 44933 | Lemma for frege102 (via ~ ... |
| axfrege54c 44934 | Reflexive equality of clas... |
| frege54b 44936 | Reflexive equality of sets... |
| frege54cor1b 44937 | Reflexive equality. (Cont... |
| frege55lem1b 44938 | Necessary deduction regard... |
| frege55lem2b 44939 | Lemma for ~ frege55b . Co... |
| frege55b 44940 | Lemma for ~ frege57b . Pr... |
| frege56b 44941 | Lemma for ~ frege57b . Pr... |
| frege57b 44942 | Analogue of ~ frege57aid .... |
| axfrege58b 44943 | If ` A. x ph ` is affirmed... |
| frege58bid 44945 | If ` A. x ph ` is affirmed... |
| frege58bcor 44946 | Lemma for ~ frege59b . (C... |
| frege59b 44947 | A kind of Aristotelian inf... |
| frege60b 44948 | Swap antecedents of ~ ax-f... |
| frege61b 44949 | Lemma for ~ frege65b . Pr... |
| frege62b 44950 | A kind of Aristotelian inf... |
| frege63b 44951 | Lemma for ~ frege91 . Pro... |
| frege64b 44952 | Lemma for ~ frege65b . Pr... |
| frege65b 44953 | A kind of Aristotelian inf... |
| frege66b 44954 | Swap antecedents of ~ freg... |
| frege67b 44955 | Lemma for ~ frege68b . Pr... |
| frege68b 44956 | Combination of applying a ... |
| frege53c 44957 | Proposition 53 of [Frege18... |
| frege54cor1c 44958 | Reflexive equality. (Cont... |
| frege55lem1c 44959 | Necessary deduction regard... |
| frege55lem2c 44960 | Core proof of Proposition ... |
| frege55c 44961 | Proposition 55 of [Frege18... |
| frege56c 44962 | Lemma for ~ frege57c . Pr... |
| frege57c 44963 | Swap order of implication ... |
| frege58c 44964 | Principle related to ~ sp ... |
| frege59c 44965 | A kind of Aristotelian inf... |
| frege60c 44966 | Swap antecedents of ~ freg... |
| frege61c 44967 | Lemma for ~ frege65c . Pr... |
| frege62c 44968 | A kind of Aristotelian inf... |
| frege63c 44969 | Analogue of ~ frege63b . ... |
| frege64c 44970 | Lemma for ~ frege65c . Pr... |
| frege65c 44971 | A kind of Aristotelian inf... |
| frege66c 44972 | Swap antecedents of ~ freg... |
| frege67c 44973 | Lemma for ~ frege68c . Pr... |
| frege68c 44974 | Combination of applying a ... |
| dffrege69 44975 | If from the proposition th... |
| frege70 44976 | Lemma for ~ frege72 . Pro... |
| frege71 44977 | Lemma for ~ frege72 . Pro... |
| frege72 44978 | If property ` A ` is hered... |
| frege73 44979 | Lemma for ~ frege87 . Pro... |
| frege74 44980 | If ` X ` has a property ` ... |
| frege75 44981 | If from the proposition th... |
| dffrege76 44982 | If from the two propositio... |
| frege77 44983 | If ` Y ` follows ` X ` in ... |
| frege78 44984 | Commuted form of ~ frege77... |
| frege79 44985 | Distributed form of ~ freg... |
| frege80 44986 | Add additional condition t... |
| frege81 44987 | If ` X ` has a property ` ... |
| frege82 44988 | Closed-form deduction base... |
| frege83 44989 | Apply commuted form of ~ f... |
| frege84 44990 | Commuted form of ~ frege81... |
| frege85 44991 | Commuted form of ~ frege77... |
| frege86 44992 | Conclusion about element o... |
| frege87 44993 | If ` Z ` is a result of an... |
| frege88 44994 | Commuted form of ~ frege87... |
| frege89 44995 | One direction of ~ dffrege... |
| frege90 44996 | Add antecedent to ~ frege8... |
| frege91 44997 | Every result of an applica... |
| frege92 44998 | Inference from ~ frege91 .... |
| frege93 44999 | Necessary condition for tw... |
| frege94 45000 | Looking one past a pair re... |
| frege95 45001 | Looking one past a pair re... |
| frege96 45002 | Every result of an applica... |
| frege97 45003 | The property of following ... |
| frege98 45004 | If ` Y ` follows ` X ` and... |
| dffrege99 45005 | If ` Z ` is identical with... |
| frege100 45006 | One direction of ~ dffrege... |
| frege101 45007 | Lemma for ~ frege102 . Pr... |
| frege102 45008 | If ` Z ` belongs to the ` ... |
| frege103 45009 | Proposition 103 of [Frege1... |
| frege104 45010 | Proposition 104 of [Frege1... |
| frege105 45011 | Proposition 105 of [Frege1... |
| frege106 45012 | Whatever follows ` X ` in ... |
| frege107 45013 | Proposition 107 of [Frege1... |
| frege108 45014 | If ` Y ` belongs to the ` ... |
| frege109 45015 | The property of belonging ... |
| frege110 45016 | Proposition 110 of [Frege1... |
| frege111 45017 | If ` Y ` belongs to the ` ... |
| frege112 45018 | Identity implies belonging... |
| frege113 45019 | Proposition 113 of [Frege1... |
| frege114 45020 | If ` X ` belongs to the ` ... |
| dffrege115 45021 | If from the circumstance t... |
| frege116 45022 | One direction of ~ dffrege... |
| frege117 45023 | Lemma for ~ frege118 . Pr... |
| frege118 45024 | Simplified application of ... |
| frege119 45025 | Lemma for ~ frege120 . Pr... |
| frege120 45026 | Simplified application of ... |
| frege121 45027 | Lemma for ~ frege122 . Pr... |
| frege122 45028 | If ` X ` is a result of an... |
| frege123 45029 | Lemma for ~ frege124 . Pr... |
| frege124 45030 | If ` X ` is a result of an... |
| frege125 45031 | Lemma for ~ frege126 . Pr... |
| frege126 45032 | If ` M ` follows ` Y ` in ... |
| frege127 45033 | Communte antecedents of ~ ... |
| frege128 45034 | Lemma for ~ frege129 . Pr... |
| frege129 45035 | If the procedure ` R ` is ... |
| frege130 45036 | Lemma for ~ frege131 . Pr... |
| frege131 45037 | If the procedure ` R ` is ... |
| frege132 45038 | Lemma for ~ frege133 . Pr... |
| frege133 45039 | If the procedure ` R ` is ... |
| enrelmap 45040 | The set of all possible re... |
| enrelmapr 45041 | The set of all possible re... |
| enmappw 45042 | The set of all mappings fr... |
| enmappwid 45043 | The set of all mappings fr... |
| rfovd 45044 | Value of the operator, ` (... |
| rfovfvd 45045 | Value of the operator, ` (... |
| rfovfvfvd 45046 | Value of the operator, ` (... |
| rfovcnvf1od 45047 | Properties of the operator... |
| rfovcnvd 45048 | Value of the converse of t... |
| rfovf1od 45049 | The value of the operator,... |
| rfovcnvfvd 45050 | Value of the converse of t... |
| fsovd 45051 | Value of the operator, ` (... |
| fsovrfovd 45052 | The operator which gives a... |
| fsovfvd 45053 | Value of the operator, ` (... |
| fsovfvfvd 45054 | Value of the operator, ` (... |
| fsovfd 45055 | The operator, ` ( A O B ) ... |
| fsovcnvlem 45056 | The ` O ` operator, which ... |
| fsovcnvd 45057 | The value of the converse ... |
| fsovcnvfvd 45058 | The value of the converse ... |
| fsovf1od 45059 | The value of ` ( A O B ) `... |
| dssmapfvd 45060 | Value of the duality opera... |
| dssmapfv2d 45061 | Value of the duality opera... |
| dssmapfv3d 45062 | Value of the duality opera... |
| dssmapnvod 45063 | For any base set ` B ` the... |
| dssmapf1od 45064 | For any base set ` B ` the... |
| dssmap2d 45065 | For any base set ` B ` the... |
| or3or 45066 | Decompose disjunction into... |
| andi3or 45067 | Distribute over triple dis... |
| uneqsn 45068 | If a union of classes is e... |
| brfvimex 45069 | If a binary relation holds... |
| brovmptimex 45070 | If a binary relation holds... |
| brovmptimex1 45071 | If a binary relation holds... |
| brovmptimex2 45072 | If a binary relation holds... |
| brcoffn 45073 | Conditions allowing the de... |
| brcofffn 45074 | Conditions allowing the de... |
| brco2f1o 45075 | Conditions allowing the de... |
| brco3f1o 45076 | Conditions allowing the de... |
| ntrclsbex 45077 | If (pseudo-)interior and (... |
| ntrclsrcomplex 45078 | The relative complement of... |
| neik0imk0p 45079 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 45080 | If an interior function is... |
| ntrkbimka 45081 | If the interiors of disjoi... |
| ntrk0kbimka 45082 | If the interiors of disjoi... |
| clsk3nimkb 45083 | If the base set is not emp... |
| clsk1indlem0 45084 | The ansatz closure functio... |
| clsk1indlem2 45085 | The ansatz closure functio... |
| clsk1indlem3 45086 | The ansatz closure functio... |
| clsk1indlem4 45087 | The ansatz closure functio... |
| clsk1indlem1 45088 | The ansatz closure functio... |
| clsk1independent 45089 | For generalized closure fu... |
| neik0pk1imk0 45090 | Kuratowski's K0' and K1 ax... |
| isotone1 45091 | Two different ways to say ... |
| isotone2 45092 | Two different ways to say ... |
| ntrk1k3eqk13 45093 | An interior function is bo... |
| ntrclsf1o 45094 | If (pseudo-)interior and (... |
| ntrclsnvobr 45095 | If (pseudo-)interior and (... |
| ntrclsiex 45096 | If (pseudo-)interior and (... |
| ntrclskex 45097 | If (pseudo-)interior and (... |
| ntrclsfv1 45098 | If (pseudo-)interior and (... |
| ntrclsfv2 45099 | If (pseudo-)interior and (... |
| ntrclselnel1 45100 | If (pseudo-)interior and (... |
| ntrclselnel2 45101 | If (pseudo-)interior and (... |
| ntrclsfv 45102 | The value of the interior ... |
| ntrclsfveq1 45103 | If interior and closure fu... |
| ntrclsfveq2 45104 | If interior and closure fu... |
| ntrclsfveq 45105 | If interior and closure fu... |
| ntrclsss 45106 | If interior and closure fu... |
| ntrclsneine0lem 45107 | If (pseudo-)interior and (... |
| ntrclsneine0 45108 | If (pseudo-)interior and (... |
| ntrclscls00 45109 | If (pseudo-)interior and (... |
| ntrclsiso 45110 | If (pseudo-)interior and (... |
| ntrclsk2 45111 | An interior function is co... |
| ntrclskb 45112 | The interiors of disjoint ... |
| ntrclsk3 45113 | The intersection of interi... |
| ntrclsk13 45114 | The interior of the inters... |
| ntrclsk4 45115 | Idempotence of the interio... |
| ntrneibex 45116 | If (pseudo-)interior and (... |
| ntrneircomplex 45117 | The relative complement of... |
| ntrneif1o 45118 | If (pseudo-)interior and (... |
| ntrneiiex 45119 | If (pseudo-)interior and (... |
| ntrneinex 45120 | If (pseudo-)interior and (... |
| ntrneicnv 45121 | If (pseudo-)interior and (... |
| ntrneifv1 45122 | If (pseudo-)interior and (... |
| ntrneifv2 45123 | If (pseudo-)interior and (... |
| ntrneiel 45124 | If (pseudo-)interior and (... |
| ntrneifv3 45125 | The value of the neighbors... |
| ntrneineine0lem 45126 | If (pseudo-)interior and (... |
| ntrneineine1lem 45127 | If (pseudo-)interior and (... |
| ntrneifv4 45128 | The value of the interior ... |
| ntrneiel2 45129 | Membership in iterated int... |
| ntrneineine0 45130 | If (pseudo-)interior and (... |
| ntrneineine1 45131 | If (pseudo-)interior and (... |
| ntrneicls00 45132 | If (pseudo-)interior and (... |
| ntrneicls11 45133 | If (pseudo-)interior and (... |
| ntrneiiso 45134 | If (pseudo-)interior and (... |
| ntrneik2 45135 | An interior function is co... |
| ntrneix2 45136 | An interior (closure) func... |
| ntrneikb 45137 | The interiors of disjoint ... |
| ntrneixb 45138 | The interiors (closures) o... |
| ntrneik3 45139 | The intersection of interi... |
| ntrneix3 45140 | The closure of the union o... |
| ntrneik13 45141 | The interior of the inters... |
| ntrneix13 45142 | The closure of the union o... |
| ntrneik4w 45143 | Idempotence of the interio... |
| ntrneik4 45144 | Idempotence of the interio... |
| clsneibex 45145 | If (pseudo-)closure and (p... |
| clsneircomplex 45146 | The relative complement of... |
| clsneif1o 45147 | If a (pseudo-)closure func... |
| clsneicnv 45148 | If a (pseudo-)closure func... |
| clsneikex 45149 | If closure and neighborhoo... |
| clsneinex 45150 | If closure and neighborhoo... |
| clsneiel1 45151 | If a (pseudo-)closure func... |
| clsneiel2 45152 | If a (pseudo-)closure func... |
| clsneifv3 45153 | Value of the neighborhoods... |
| clsneifv4 45154 | Value of the closure (inte... |
| neicvgbex 45155 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 45156 | The relative complement of... |
| neicvgf1o 45157 | If neighborhood and conver... |
| neicvgnvo 45158 | If neighborhood and conver... |
| neicvgnvor 45159 | If neighborhood and conver... |
| neicvgmex 45160 | If the neighborhoods and c... |
| neicvgnex 45161 | If the neighborhoods and c... |
| neicvgel1 45162 | A subset being an element ... |
| neicvgel2 45163 | The complement of a subset... |
| neicvgfv 45164 | The value of the neighborh... |
| ntrrn 45165 | The range of the interior ... |
| ntrf 45166 | The interior function of a... |
| ntrf2 45167 | The interior function is a... |
| ntrelmap 45168 | The interior function is a... |
| clsf2 45169 | The closure function is a ... |
| clselmap 45170 | The closure function is a ... |
| dssmapntrcls 45171 | The interior and closure o... |
| dssmapclsntr 45172 | The closure and interior o... |
| gneispa 45173 | Each point ` p ` of the ne... |
| gneispb 45174 | Given a neighborhood ` N `... |
| gneispace2 45175 | The predicate that ` F ` i... |
| gneispace3 45176 | The predicate that ` F ` i... |
| gneispace 45177 | The predicate that ` F ` i... |
| gneispacef 45178 | A generic neighborhood spa... |
| gneispacef2 45179 | A generic neighborhood spa... |
| gneispacefun 45180 | A generic neighborhood spa... |
| gneispacern 45181 | A generic neighborhood spa... |
| gneispacern2 45182 | A generic neighborhood spa... |
| gneispace0nelrn 45183 | A generic neighborhood spa... |
| gneispace0nelrn2 45184 | A generic neighborhood spa... |
| gneispace0nelrn3 45185 | A generic neighborhood spa... |
| gneispaceel 45186 | Every neighborhood of a po... |
| gneispaceel2 45187 | Every neighborhood of a po... |
| gneispacess 45188 | All supersets of a neighbo... |
| gneispacess2 45189 | All supersets of a neighbo... |
| k0004lem1 45190 | Application of ~ ssin to r... |
| k0004lem2 45191 | A mapping with a particula... |
| k0004lem3 45192 | When the value of a mappin... |
| k0004val 45193 | The topological simplex of... |
| k0004ss1 45194 | The topological simplex of... |
| k0004ss2 45195 | The topological simplex of... |
| k0004ss3 45196 | The topological simplex of... |
| k0004val0 45197 | The topological simplex of... |
| inductionexd 45198 | Simple induction example. ... |
| wwlemuld 45199 | Natural deduction form of ... |
| leeq1d 45200 | Specialization of ~ breq1d... |
| leeq2d 45201 | Specialization of ~ breq2d... |
| absmulrposd 45202 | Specialization of absmuld ... |
| imadisjld 45203 | Natural dduction form of o... |
| wnefimgd 45204 | The image of a mapping fro... |
| fco2d 45205 | Natural deduction form of ... |
| wfximgfd 45206 | The value of a function on... |
| extoimad 45207 | If |f(x)| <= C for all x t... |
| imo72b2lem0 45208 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 45209 | Natural deduction form of ... |
| imo72b2lem2 45210 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 45211 | Natural deduction form of ... |
| imo72b2lem1 45212 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 45213 | 'Less than or equal to' re... |
| lemuldiv4d 45214 | 'Less than or equal to' re... |
| imo72b2 45215 | IMO 1972 B2. (14th Intern... |
| int-addcomd 45216 | AdditionCommutativity gene... |
| int-addassocd 45217 | AdditionAssociativity gene... |
| int-addsimpd 45218 | AdditionSimplification gen... |
| int-mulcomd 45219 | MultiplicationCommutativit... |
| int-mulassocd 45220 | MultiplicationAssociativit... |
| int-mulsimpd 45221 | MultiplicationSimplificati... |
| int-leftdistd 45222 | AdditionMultiplicationLeft... |
| int-rightdistd 45223 | AdditionMultiplicationRigh... |
| int-sqdefd 45224 | SquareDefinition generator... |
| int-mul11d 45225 | First MultiplicationOne ge... |
| int-mul12d 45226 | Second MultiplicationOne g... |
| int-add01d 45227 | First AdditionZero generat... |
| int-add02d 45228 | Second AdditionZero genera... |
| int-sqgeq0d 45229 | SquareGEQZero generator ru... |
| int-eqprincd 45230 | PrincipleOfEquality genera... |
| int-eqtransd 45231 | EqualityTransitivity gener... |
| int-eqmvtd 45232 | EquMoveTerm generator rule... |
| int-eqineqd 45233 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 45234 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 45235 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 45236 | SecondPrincipleOfInequalit... |
| int-ineqtransd 45237 | InequalityTransitivity gen... |
| unitadd 45238 | Theorem used in conjunctio... |
| gsumws3 45239 | Valuation of a length 3 wo... |
| gsumws4 45240 | Valuation of a length 4 wo... |
| amgm2d 45241 | Arithmetic-geometric mean ... |
| amgm3d 45242 | Arithmetic-geometric mean ... |
| amgm4d 45243 | Arithmetic-geometric mean ... |
| spALT 45244 | ~ sp can be proven from th... |
| rr-spce 45245 | Prove an existential. (Co... |
| rexlimdvaacbv 45246 | Unpack a restricted existe... |
| rexlimddvcbvw 45247 | Unpack a restricted existe... |
| rexlimddvcbv 45248 | Unpack a restricted existe... |
| rr-elrnmpt3d 45249 | Elementhood in an image se... |
| rr-phpd 45250 | Equivalent of ~ php withou... |
| tfindsd 45251 | Deduction associated with ... |
| mnringvald 45254 | Value of the monoid ring f... |
| mnringnmulrd 45255 | Components of a monoid rin... |
| mnringbased 45256 | The base set of a monoid r... |
| mnringbaserd 45257 | The base set of a monoid r... |
| mnringelbased 45258 | Membership in the base set... |
| mnringbasefd 45259 | Elements of a monoid ring ... |
| mnringbasefsuppd 45260 | Elements of a monoid ring ... |
| mnringaddgd 45261 | The additive operation of ... |
| mnring0gd 45262 | The additive identity of a... |
| mnring0g2d 45263 | The additive identity of a... |
| mnringmulrd 45264 | The ring product of a mono... |
| mnringscad 45265 | The scalar ring of a monoi... |
| mnringvscad 45266 | The scalar product of a mo... |
| mnringlmodd 45267 | Monoid rings are left modu... |
| mnringmulrvald 45268 | Value of multiplication in... |
| mnringmulrcld 45269 | Monoid rings are closed un... |
| gru0eld 45270 | A nonempty Grothendieck un... |
| grusucd 45271 | Grothendieck universes are... |
| r1rankcld 45272 | Any rank of the cumulative... |
| grur1cld 45273 | Grothendieck universes are... |
| grurankcld 45274 | Grothendieck universes are... |
| grurankrcld 45275 | If a Grothendieck universe... |
| gruscottcld 45276 | If a Grothendieck universe... |
| dfcoll2 45279 | Alternate definition of th... |
| colleq12d 45280 | Equality theorem for the c... |
| colleq1 45281 | Equality theorem for the c... |
| colleq2 45282 | Equality theorem for the c... |
| nfcoll 45283 | Bound-variable hypothesis ... |
| collexd 45284 | The output of the collecti... |
| cpcolld 45285 | Property of the collection... |
| cpcoll2d 45286 | ~ cpcolld with an extra ex... |
| grucollcld 45287 | A Grothendieck universe co... |
| ismnu 45288 | The hypothesis of this the... |
| mnuop123d 45289 | Operations of a minimal un... |
| mnussd 45290 | Minimal universes are clos... |
| mnuss2d 45291 | ~ mnussd with arguments pr... |
| mnu0eld 45292 | A nonempty minimal univers... |
| mnuop23d 45293 | Second and third operation... |
| mnupwd 45294 | Minimal universes are clos... |
| mnusnd 45295 | Minimal universes are clos... |
| mnuprssd 45296 | A minimal universe contain... |
| mnuprss2d 45297 | Special case of ~ mnuprssd... |
| mnuop3d 45298 | Third operation of a minim... |
| mnuprdlem1 45299 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 45300 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 45301 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 45302 | Lemma for ~ mnuprd . Gene... |
| mnuprd 45303 | Minimal universes are clos... |
| mnuunid 45304 | Minimal universes are clos... |
| mnuund 45305 | Minimal universes are clos... |
| mnutrcld 45306 | Minimal universes contain ... |
| mnutrd 45307 | Minimal universes are tran... |
| mnurndlem1 45308 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 45309 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 45310 | Minimal universes contain ... |
| mnugrud 45311 | Minimal universes are Grot... |
| grumnudlem 45312 | Lemma for ~ grumnud . (Co... |
| grumnud 45313 | Grothendieck universes are... |
| grumnueq 45314 | The class of Grothendieck ... |
| expandan 45315 | Expand conjunction to prim... |
| expandexn 45316 | Expand an existential quan... |
| expandral 45317 | Expand a restricted univer... |
| expandrexn 45318 | Expand a restricted existe... |
| expandrex 45319 | Expand a restricted existe... |
| expanduniss 45320 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 45321 | Express the predicate on `... |
| rr-grothprimbi 45322 | Express "every set is cont... |
| inagrud 45323 | Inaccessible levels of the... |
| inaex 45324 | Assuming the Tarski-Grothe... |
| gruex 45325 | Assuming the Tarski-Grothe... |
| rr-groth 45326 | An equivalent of ~ ax-grot... |
| rr-grothprim 45327 | An equivalent of ~ ax-grot... |
| ismnushort 45328 | Express the predicate on `... |
| dfuniv2 45329 | Alternative definition of ... |
| rr-grothshortbi 45330 | Express "every set is cont... |
| rr-grothshort 45331 | A shorter equivalent of ~ ... |
| nanorxor 45332 | 'nand' is equivalent to th... |
| undisjrab 45333 | Union of two disjoint rest... |
| iso0 45334 | The empty set is an ` R , ... |
| ssrecnpr 45335 | ` RR ` is a subset of both... |
| seff 45336 | Let set ` S ` be the real ... |
| sblpnf 45337 | The infinity ball in the a... |
| prmunb2 45338 | The primes are unbounded. ... |
| dvgrat 45339 | Ratio test for divergence ... |
| cvgdvgrat 45340 | Ratio test for convergence... |
| radcnvrat 45341 | Let ` L ` be the limit, if... |
| reldvds 45342 | The divides relation is in... |
| nznngen 45343 | All positive integers in t... |
| nzss 45344 | The set of multiples of _m... |
| nzin 45345 | The intersection of the se... |
| nzprmdif 45346 | Subtract one prime's multi... |
| hashnzfz 45347 | Special case of ~ hashdvds... |
| hashnzfz2 45348 | Special case of ~ hashnzfz... |
| hashnzfzclim 45349 | As the upper bound ` K ` o... |
| caofcan 45350 | Transfer a cancellation la... |
| ofsubid 45351 | Function analogue of ~ sub... |
| ofmul12 45352 | Function analogue of ~ mul... |
| ofdivrec 45353 | Function analogue of ~ div... |
| ofdivcan4 45354 | Function analogue of ~ div... |
| ofdivdiv2 45355 | Function analogue of ~ div... |
| lhe4.4ex1a 45356 | Example of the Fundamental... |
| dvsconst 45357 | Derivative of a constant f... |
| dvsid 45358 | Derivative of the identity... |
| dvsef 45359 | Derivative of the exponent... |
| expgrowthi 45360 | Exponential growth and dec... |
| dvconstbi 45361 | The derivative of a functi... |
| expgrowth 45362 | Exponential growth and dec... |
| bccval 45365 | Value of the generalized b... |
| bcccl 45366 | Closure of the generalized... |
| bcc0 45367 | The generalized binomial c... |
| bccp1k 45368 | Generalized binomial coeff... |
| bccm1k 45369 | Generalized binomial coeff... |
| bccn0 45370 | Generalized binomial coeff... |
| bccn1 45371 | Generalized binomial coeff... |
| bccbc 45372 | The binomial coefficient a... |
| uzmptshftfval 45373 | When ` F ` is a maps-to fu... |
| dvradcnv2 45374 | The radius of convergence ... |
| binomcxplemwb 45375 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 45376 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 45377 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 45378 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 45379 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 45380 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 45381 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 45382 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 45383 | Lemma for ~ binomcxp . Wh... |
| binomcxp 45384 | Generalize the binomial th... |
| pm10.12 45385 | Theorem *10.12 in [Whitehe... |
| pm10.14 45386 | Theorem *10.14 in [Whitehe... |
| pm10.251 45387 | Theorem *10.251 in [Whiteh... |
| pm10.252 45388 | Theorem *10.252 in [Whiteh... |
| pm10.253 45389 | Theorem *10.253 in [Whiteh... |
| albitr 45390 | Theorem *10.301 in [Whiteh... |
| pm10.42 45391 | Theorem *10.42 in [Whitehe... |
| pm10.52 45392 | Theorem *10.52 in [Whitehe... |
| pm10.53 45393 | Theorem *10.53 in [Whitehe... |
| pm10.541 45394 | Theorem *10.541 in [Whiteh... |
| pm10.542 45395 | Theorem *10.542 in [Whiteh... |
| pm10.55 45396 | Theorem *10.55 in [Whitehe... |
| pm10.56 45397 | Theorem *10.56 in [Whitehe... |
| pm10.57 45398 | Theorem *10.57 in [Whitehe... |
| 2alanimi 45399 | Removes two universal quan... |
| 2al2imi 45400 | Removes two universal quan... |
| pm11.11 45401 | Theorem *11.11 in [Whitehe... |
| pm11.12 45402 | Theorem *11.12 in [Whitehe... |
| 19.21vv 45403 | Compare Theorem *11.3 in [... |
| 2alim 45404 | Theorem *11.32 in [Whitehe... |
| 2albi 45405 | Theorem *11.33 in [Whitehe... |
| 2exim 45406 | Theorem *11.34 in [Whitehe... |
| 2exbi 45407 | Theorem *11.341 in [Whiteh... |
| spsbce-2 45408 | Theorem *11.36 in [Whitehe... |
| 19.33-2 45409 | Theorem *11.421 in [Whiteh... |
| 19.36vv 45410 | Theorem *11.43 in [Whitehe... |
| 19.31vv 45411 | Theorem *11.44 in [Whitehe... |
| 19.37vv 45412 | Theorem *11.46 in [Whitehe... |
| 19.28vv 45413 | Theorem *11.47 in [Whitehe... |
| pm11.52 45414 | Theorem *11.52 in [Whitehe... |
| aaanv 45415 | Theorem *11.56 in [Whitehe... |
| pm11.57 45416 | Theorem *11.57 in [Whitehe... |
| pm11.58 45417 | Theorem *11.58 in [Whitehe... |
| pm11.59 45418 | Theorem *11.59 in [Whitehe... |
| pm11.6 45419 | Theorem *11.6 in [Whitehea... |
| pm11.61 45420 | Theorem *11.61 in [Whitehe... |
| pm11.62 45421 | Theorem *11.62 in [Whitehe... |
| pm11.63 45422 | Theorem *11.63 in [Whitehe... |
| pm11.7 45423 | Theorem *11.7 in [Whitehea... |
| pm11.71 45424 | Theorem *11.71 in [Whitehe... |
| sbeqal1 45425 | If ` x = y ` always implie... |
| sbeqal1i 45426 | Suppose you know ` x = y `... |
| sbeqal2i 45427 | If ` x = y ` implies ` x =... |
| axc5c4c711 45428 | Proof of a theorem that ca... |
| axc5c4c711toc5 45429 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 45430 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 45431 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 45432 | Rederivation of ~ ax-11 fr... |
| axc11next 45433 | This theorem shows that, g... |
| pm13.13a 45434 | One result of theorem *13.... |
| pm13.13b 45435 | Theorem *13.13 in [Whitehe... |
| pm13.14 45436 | Theorem *13.14 in [Whitehe... |
| pm13.192 45437 | Theorem *13.192 in [Whiteh... |
| pm13.193 45438 | Theorem *13.193 in [Whiteh... |
| pm13.194 45439 | Theorem *13.194 in [Whiteh... |
| pm13.195 45440 | Theorem *13.195 in [Whiteh... |
| pm13.196a 45441 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 45442 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 45443 | Theorem *13.22 in [Whitehe... |
| iotain 45444 | Equivalence between two di... |
| iotaexeu 45445 | The iota class exists. Th... |
| iotasbc 45446 | Definition *14.01 in [Whit... |
| iotasbc2 45447 | Theorem *14.111 in [Whiteh... |
| pm14.12 45448 | Theorem *14.12 in [Whitehe... |
| pm14.122a 45449 | Theorem *14.122 in [Whiteh... |
| pm14.122b 45450 | Theorem *14.122 in [Whiteh... |
| pm14.122c 45451 | Theorem *14.122 in [Whiteh... |
| pm14.123a 45452 | Theorem *14.123 in [Whiteh... |
| pm14.123b 45453 | Theorem *14.123 in [Whiteh... |
| pm14.123c 45454 | Theorem *14.123 in [Whiteh... |
| pm14.18 45455 | Theorem *14.18 in [Whitehe... |
| iotaequ 45456 | Theorem *14.2 in [Whitehea... |
| iotavalb 45457 | Theorem *14.202 in [Whiteh... |
| iotasbc5 45458 | Theorem *14.205 in [Whiteh... |
| pm14.24 45459 | Theorem *14.24 in [Whitehe... |
| iotavalsb 45460 | Theorem *14.242 in [Whiteh... |
| sbiota1 45461 | Theorem *14.25 in [Whitehe... |
| sbaniota 45462 | Theorem *14.26 in [Whitehe... |
| iotasbcq 45463 | Theorem *14.272 in [Whiteh... |
| elnev 45464 | Any set that contains one ... |
| rusbcALT 45465 | A version of Russell's par... |
| compeq 45466 | Equality between two ways ... |
| compne 45467 | The complement of ` A ` is... |
| compab 45468 | Two ways of saying "the co... |
| conss2 45469 | Contrapositive law for sub... |
| conss1 45470 | Contrapositive law for sub... |
| ralbidar 45471 | More general form of ~ ral... |
| rexbidar 45472 | More general form of ~ rex... |
| dropab1 45473 | Theorem to aid use of the ... |
| dropab2 45474 | Theorem to aid use of the ... |
| ipo0 45475 | If the identity relation p... |
| ifr0 45476 | A class that is founded by... |
| fvsb 45477 | Explicit substitution of a... |
| fveqsb 45478 | Implicit substitution of a... |
| xpexb 45479 | A Cartesian product exists... |
| trelpss 45480 | An element of a transitive... |
| addcomgi 45481 | Generalization of commutat... |
| addrval 45491 | Value of the operation of ... |
| subrval 45492 | Value of the operation of ... |
| mulvval 45493 | Value of the operation of ... |
| addrfv 45494 | Vector addition at a value... |
| subrfv 45495 | Vector subtraction at a va... |
| mulvfv 45496 | Scalar multiplication at a... |
| addrfn 45497 | Vector addition produces a... |
| subrfn 45498 | Vector subtraction produce... |
| mulvfn 45499 | Scalar multiplication prod... |
| addrcom 45500 | Vector addition is commuta... |
| idiALT 45504 | Placeholder for ~ idi . T... |
| exbir 45505 | Exportation implication al... |
| 3impexpbicom 45506 | Version of ~ 3impexp where... |
| 3impexpbicomi 45507 | Inference associated with ... |
| bi1imp 45508 | Importation inference simi... |
| bi2imp 45509 | Importation inference simi... |
| bi3impb 45510 | Similar to ~ 3impb with im... |
| bi3impa 45511 | Similar to ~ 3impa with im... |
| bi23impib 45512 | ~ 3impib with the inner im... |
| bi13impib 45513 | ~ 3impib with the outer im... |
| bi123impib 45514 | ~ 3impib with the implicat... |
| bi13impia 45515 | ~ 3impia with the outer im... |
| bi123impia 45516 | ~ 3impia with the implicat... |
| bi33imp12 45517 | ~ 3imp with innermost impl... |
| bi13imp23 45518 | ~ 3imp with outermost impl... |
| bi13imp2 45519 | Similar to ~ 3imp except t... |
| bi12imp3 45520 | Similar to ~ 3imp except a... |
| bi23imp1 45521 | Similar to ~ 3imp except a... |
| bi123imp0 45522 | Similar to ~ 3imp except a... |
| 4animp1 45523 | A single hypothesis unific... |
| 4an31 45524 | A rearrangement of conjunc... |
| 4an4132 45525 | A rearrangement of conjunc... |
| expcomdg 45526 | Biconditional form of ~ ex... |
| iidn3 45527 | ~ idn3 without virtual ded... |
| ee222 45528 | ~ e222 without virtual ded... |
| ee3bir 45529 | Right-biconditional form o... |
| ee13 45530 | ~ e13 without virtual dedu... |
| ee121 45531 | ~ e121 without virtual ded... |
| ee122 45532 | ~ e122 without virtual ded... |
| ee333 45533 | ~ e333 without virtual ded... |
| ee323 45534 | ~ e323 without virtual ded... |
| 3ornot23 45535 | If the second and third di... |
| orbi1r 45536 | ~ orbi1 with order of disj... |
| 3orbi123 45537 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 45538 | Closed form of ~ syl5 . D... |
| impexpd 45539 | The following User's Proof... |
| com3rgbi 45540 | The following User's Proof... |
| impexpdcom 45541 | The following User's Proof... |
| ee1111 45542 | Non-virtual deduction form... |
| pm2.43bgbi 45543 | Logical equivalence of a 2... |
| pm2.43cbi 45544 | Logical equivalence of a 3... |
| ee233 45545 | Non-virtual deduction form... |
| imbi13 45546 | Join three logical equival... |
| ee33 45547 | Non-virtual deduction form... |
| con5 45548 | Biconditional contrapositi... |
| con5i 45549 | Inference form of ~ con5 .... |
| exlimexi 45550 | Inference similar to Theor... |
| sb5ALT 45551 | Equivalence for substituti... |
| eexinst01 45552 | ~ exinst01 without virtual... |
| eexinst11 45553 | ~ exinst11 without virtual... |
| vk15.4j 45554 | Excercise 4j of Unit 15 of... |
| notnotrALT 45555 | Converse of double negatio... |
| con3ALT2 45556 | Contraposition. Alternate... |
| ssralv2 45557 | Quantification restricted ... |
| sbc3or 45558 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 45559 | Closed form of ~ alrimi wi... |
| rspsbc2 45560 | ~ rspsbc with two quantify... |
| sbcoreleleq 45561 | Substitution of a setvar v... |
| tratrb 45562 | If a class is transitive a... |
| ordelordALT 45563 | An element of an ordinal c... |
| sbcim2g 45564 | Distribution of class subs... |
| sbcbi 45565 | Implication form of ~ sbcb... |
| trsbc 45566 | Formula-building inference... |
| truniALT 45567 | The union of a class of tr... |
| onfrALTlem5 45568 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 45569 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 45570 | Lemma for ~ onfrALT . (Co... |
| ggen31 45571 | ~ gen31 without virtual de... |
| onfrALTlem2 45572 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 45573 | A theorem pertaining to th... |
| onfrALTlem1 45574 | Lemma for ~ onfrALT . (Co... |
| onfrALT 45575 | The membership relation is... |
| 19.41rg 45576 | Closed form of right-to-le... |
| opelopab4 45577 | Ordered pair membership in... |
| 2pm13.193 45578 | ~ pm13.193 for two variabl... |
| hbntal 45579 | A closed form of ~ hbn . ~... |
| hbimpg 45580 | A closed form of ~ hbim . ... |
| hbalg 45581 | Closed form of ~ hbal . D... |
| hbexg 45582 | Closed form of ~ nfex . D... |
| ax6e2eq 45583 | Alternate form of ~ ax6e f... |
| ax6e2nd 45584 | If at least two sets exist... |
| ax6e2ndeq 45585 | "At least two sets exist" ... |
| 2sb5nd 45586 | Equivalence for double sub... |
| 2uasbanh 45587 | Distribute the unabbreviat... |
| 2uasban 45588 | Distribute the unabbreviat... |
| e2ebind 45589 | Absorption of an existenti... |
| elpwgded 45590 | ~ elpwgdedVD in convention... |
| trelded 45591 | Deduction form of ~ trel .... |
| jaoded 45592 | Deduction form of ~ jao . ... |
| sbtT 45593 | A substitution into a theo... |
| not12an2impnot1 45594 | If a double conjunction is... |
| in1 45597 | Inference form of ~ df-vd1... |
| iin1 45598 | ~ in1 without virtual dedu... |
| dfvd1ir 45599 | Inference form of ~ df-vd1... |
| idn1 45600 | Virtual deduction identity... |
| dfvd1imp 45601 | Left-to-right part of defi... |
| dfvd1impr 45602 | Right-to-left part of defi... |
| dfvd2 45605 | Definition of a 2-hypothes... |
| dfvd2an 45608 | Definition of a 2-hypothes... |
| dfvd2ani 45609 | Inference form of ~ dfvd2a... |
| dfvd2anir 45610 | Right-to-left inference fo... |
| dfvd2i 45611 | Inference form of ~ dfvd2 ... |
| dfvd2ir 45612 | Right-to-left inference fo... |
| dfvd3 45617 | Definition of a 3-hypothes... |
| dfvd3i 45618 | Inference form of ~ dfvd3 ... |
| dfvd3ir 45619 | Right-to-left inference fo... |
| dfvd3an 45620 | Definition of a 3-hypothes... |
| dfvd3ani 45621 | Inference form of ~ dfvd3a... |
| dfvd3anir 45622 | Right-to-left inference fo... |
| vd01 45623 | A virtual hypothesis virtu... |
| vd02 45624 | Two virtual hypotheses vir... |
| vd03 45625 | A theorem is virtually inf... |
| vd12 45626 | A virtual deduction with 1... |
| vd13 45627 | A virtual deduction with 1... |
| vd23 45628 | A virtual deduction with 2... |
| dfvd2imp 45629 | The virtual deduction form... |
| dfvd2impr 45630 | A 2-antecedent nested impl... |
| in2 45631 | The virtual deduction intr... |
| int2 45632 | The virtual deduction intr... |
| iin2 45633 | ~ in2 without virtual dedu... |
| in2an 45634 | The virtual deduction intr... |
| in3 45635 | The virtual deduction intr... |
| iin3 45636 | ~ in3 without virtual dedu... |
| in3an 45637 | The virtual deduction intr... |
| int3 45638 | The virtual deduction intr... |
| idn2 45639 | Virtual deduction identity... |
| iden2 45640 | Virtual deduction identity... |
| idn3 45641 | Virtual deduction identity... |
| gen11 45642 | Virtual deduction generali... |
| gen11nv 45643 | Virtual deduction generali... |
| gen12 45644 | Virtual deduction generali... |
| gen21 45645 | Virtual deduction generali... |
| gen21nv 45646 | Virtual deduction form of ... |
| gen31 45647 | Virtual deduction generali... |
| gen22 45648 | Virtual deduction generali... |
| ggen22 45649 | ~ gen22 without virtual de... |
| exinst 45650 | Existential Instantiation.... |
| exinst01 45651 | Existential Instantiation.... |
| exinst11 45652 | Existential Instantiation.... |
| e1a 45653 | A Virtual deduction elimin... |
| el1 45654 | A Virtual deduction elimin... |
| e1bi 45655 | Biconditional form of ~ e1... |
| e1bir 45656 | Right biconditional form o... |
| e2 45657 | A virtual deduction elimin... |
| e2bi 45658 | Biconditional form of ~ e2... |
| e2bir 45659 | Right biconditional form o... |
| ee223 45660 | ~ e223 without virtual ded... |
| e223 45661 | A virtual deduction elimin... |
| e222 45662 | A virtual deduction elimin... |
| e220 45663 | A virtual deduction elimin... |
| ee220 45664 | ~ e220 without virtual ded... |
| e202 45665 | A virtual deduction elimin... |
| ee202 45666 | ~ e202 without virtual ded... |
| e022 45667 | A virtual deduction elimin... |
| ee022 45668 | ~ e022 without virtual ded... |
| e002 45669 | A virtual deduction elimin... |
| ee002 45670 | ~ e002 without virtual ded... |
| e020 45671 | A virtual deduction elimin... |
| ee020 45672 | ~ e020 without virtual ded... |
| e200 45673 | A virtual deduction elimin... |
| ee200 45674 | ~ e200 without virtual ded... |
| e221 45675 | A virtual deduction elimin... |
| ee221 45676 | ~ e221 without virtual ded... |
| e212 45677 | A virtual deduction elimin... |
| ee212 45678 | ~ e212 without virtual ded... |
| e122 45679 | A virtual deduction elimin... |
| e112 45680 | A virtual deduction elimin... |
| ee112 45681 | ~ e112 without virtual ded... |
| e121 45682 | A virtual deduction elimin... |
| e211 45683 | A virtual deduction elimin... |
| ee211 45684 | ~ e211 without virtual ded... |
| e210 45685 | A virtual deduction elimin... |
| ee210 45686 | ~ e210 without virtual ded... |
| e201 45687 | A virtual deduction elimin... |
| ee201 45688 | ~ e201 without virtual ded... |
| e120 45689 | A virtual deduction elimin... |
| ee120 45690 | Virtual deduction rule ~ e... |
| e021 45691 | A virtual deduction elimin... |
| ee021 45692 | ~ e021 without virtual ded... |
| e012 45693 | A virtual deduction elimin... |
| ee012 45694 | ~ e012 without virtual ded... |
| e102 45695 | A virtual deduction elimin... |
| ee102 45696 | ~ e102 without virtual ded... |
| e22 45697 | A virtual deduction elimin... |
| e22an 45698 | Conjunction form of ~ e22 ... |
| ee22an 45699 | ~ e22an without virtual de... |
| e111 45700 | A virtual deduction elimin... |
| e1111 45701 | A virtual deduction elimin... |
| e110 45702 | A virtual deduction elimin... |
| ee110 45703 | ~ e110 without virtual ded... |
| e101 45704 | A virtual deduction elimin... |
| ee101 45705 | ~ e101 without virtual ded... |
| e011 45706 | A virtual deduction elimin... |
| ee011 45707 | ~ e011 without virtual ded... |
| e100 45708 | A virtual deduction elimin... |
| ee100 45709 | ~ e100 without virtual ded... |
| e010 45710 | A virtual deduction elimin... |
| ee010 45711 | ~ e010 without virtual ded... |
| e001 45712 | A virtual deduction elimin... |
| ee001 45713 | ~ e001 without virtual ded... |
| e11 45714 | A virtual deduction elimin... |
| e11an 45715 | Conjunction form of ~ e11 ... |
| ee11an 45716 | ~ e11an without virtual de... |
| e01 45717 | A virtual deduction elimin... |
| e01an 45718 | Conjunction form of ~ e01 ... |
| ee01an 45719 | ~ e01an without virtual de... |
| e10 45720 | A virtual deduction elimin... |
| e10an 45721 | Conjunction form of ~ e10 ... |
| ee10an 45722 | ~ e10an without virtual de... |
| e02 45723 | A virtual deduction elimin... |
| e02an 45724 | Conjunction form of ~ e02 ... |
| ee02an 45725 | ~ e02an without virtual de... |
| eel021old 45726 | ~ el021old without virtual... |
| el021old 45727 | A virtual deduction elimin... |
| eel000cT 45728 | An elimination deduction. ... |
| eel0TT 45729 | An elimination deduction. ... |
| eelT00 45730 | An elimination deduction. ... |
| eelTTT 45731 | An elimination deduction. ... |
| eelT11 45732 | An elimination deduction. ... |
| eelT1 45733 | Syllogism inference combin... |
| eelT12 45734 | An elimination deduction. ... |
| eelTT1 45735 | An elimination deduction. ... |
| eelT01 45736 | An elimination deduction. ... |
| eel0T1 45737 | An elimination deduction. ... |
| eel12131 45738 | An elimination deduction. ... |
| eel2131 45739 | ~ syl2an with antecedents ... |
| eel3132 45740 | ~ syl2an with antecedents ... |
| eel0321old 45741 | ~ el0321old without virtua... |
| el0321old 45742 | A virtual deduction elimin... |
| eel2122old 45743 | ~ el2122old without virtua... |
| el2122old 45744 | A virtual deduction elimin... |
| eel0000 45745 | Elimination rule similar t... |
| eel00001 45746 | An elimination deduction. ... |
| eel00000 45747 | Elimination rule similar ~... |
| eel11111 45748 | Five-hypothesis eliminatio... |
| e12 45749 | A virtual deduction elimin... |
| e12an 45750 | Conjunction form of ~ e12 ... |
| el12 45751 | Virtual deduction form of ... |
| e20 45752 | A virtual deduction elimin... |
| e20an 45753 | Conjunction form of ~ e20 ... |
| ee20an 45754 | ~ e20an without virtual de... |
| e21 45755 | A virtual deduction elimin... |
| e21an 45756 | Conjunction form of ~ e21 ... |
| ee21an 45757 | ~ e21an without virtual de... |
| e333 45758 | A virtual deduction elimin... |
| e33 45759 | A virtual deduction elimin... |
| e33an 45760 | Conjunction form of ~ e33 ... |
| ee33an 45761 | ~ e33an without virtual de... |
| e3 45762 | Meta-connective form of ~ ... |
| e3bi 45763 | Biconditional form of ~ e3... |
| e3bir 45764 | Right biconditional form o... |
| e03 45765 | A virtual deduction elimin... |
| ee03 45766 | ~ e03 without virtual dedu... |
| e03an 45767 | Conjunction form of ~ e03 ... |
| ee03an 45768 | Conjunction form of ~ ee03... |
| e30 45769 | A virtual deduction elimin... |
| ee30 45770 | ~ e30 without virtual dedu... |
| e30an 45771 | A virtual deduction elimin... |
| ee30an 45772 | Conjunction form of ~ ee30... |
| e13 45773 | A virtual deduction elimin... |
| e13an 45774 | A virtual deduction elimin... |
| ee13an 45775 | ~ e13an without virtual de... |
| e31 45776 | A virtual deduction elimin... |
| ee31 45777 | ~ e31 without virtual dedu... |
| e31an 45778 | A virtual deduction elimin... |
| ee31an 45779 | ~ e31an without virtual de... |
| e23 45780 | A virtual deduction elimin... |
| e23an 45781 | A virtual deduction elimin... |
| ee23an 45782 | ~ e23an without virtual de... |
| e32 45783 | A virtual deduction elimin... |
| ee32 45784 | ~ e32 without virtual dedu... |
| e32an 45785 | A virtual deduction elimin... |
| ee32an 45786 | ~ e33an without virtual de... |
| e123 45787 | A virtual deduction elimin... |
| ee123 45788 | ~ e123 without virtual ded... |
| el123 45789 | A virtual deduction elimin... |
| e233 45790 | A virtual deduction elimin... |
| e323 45791 | A virtual deduction elimin... |
| e000 45792 | A virtual deduction elimin... |
| e00 45793 | Elimination rule identical... |
| e00an 45794 | Elimination rule identical... |
| eel00cT 45795 | An elimination deduction. ... |
| eelTT 45796 | An elimination deduction. ... |
| e0a 45797 | Elimination rule identical... |
| eelT 45798 | An elimination deduction. ... |
| eel0cT 45799 | An elimination deduction. ... |
| eelT0 45800 | An elimination deduction. ... |
| e0bi 45801 | Elimination rule identical... |
| e0bir 45802 | Elimination rule identical... |
| uun0.1 45803 | Convention notation form o... |
| un0.1 45804 | ` T. ` is the constant tru... |
| uunT1 45805 | A deduction unionizing a n... |
| uunT1p1 45806 | A deduction unionizing a n... |
| uunT21 45807 | A deduction unionizing a n... |
| uun121 45808 | A deduction unionizing a n... |
| uun121p1 45809 | A deduction unionizing a n... |
| uun132 45810 | A deduction unionizing a n... |
| uun132p1 45811 | A deduction unionizing a n... |
| anabss7p1 45812 | A deduction unionizing a n... |
| un10 45813 | A unionizing deduction. (... |
| un01 45814 | A unionizing deduction. (... |
| un2122 45815 | A deduction unionizing a n... |
| uun2131 45816 | A deduction unionizing a n... |
| uun2131p1 45817 | A deduction unionizing a n... |
| uunTT1 45818 | A deduction unionizing a n... |
| uunTT1p1 45819 | A deduction unionizing a n... |
| uunTT1p2 45820 | A deduction unionizing a n... |
| uunT11 45821 | A deduction unionizing a n... |
| uunT11p1 45822 | A deduction unionizing a n... |
| uunT11p2 45823 | A deduction unionizing a n... |
| uunT12 45824 | A deduction unionizing a n... |
| uunT12p1 45825 | A deduction unionizing a n... |
| uunT12p2 45826 | A deduction unionizing a n... |
| uunT12p3 45827 | A deduction unionizing a n... |
| uunT12p4 45828 | A deduction unionizing a n... |
| uunT12p5 45829 | A deduction unionizing a n... |
| uun111 45830 | A deduction unionizing a n... |
| 3anidm12p1 45831 | A deduction unionizing a n... |
| 3anidm12p2 45832 | A deduction unionizing a n... |
| uun123 45833 | A deduction unionizing a n... |
| uun123p1 45834 | A deduction unionizing a n... |
| uun123p2 45835 | A deduction unionizing a n... |
| uun123p3 45836 | A deduction unionizing a n... |
| uun123p4 45837 | A deduction unionizing a n... |
| uun2221 45838 | A deduction unionizing a n... |
| uun2221p1 45839 | A deduction unionizing a n... |
| uun2221p2 45840 | A deduction unionizing a n... |
| 3impdirp1 45841 | A deduction unionizing a n... |
| 3impcombi 45842 | A 1-hypothesis proposition... |
| trsspwALT 45843 | Virtual deduction proof of... |
| trsspwALT2 45844 | Virtual deduction proof of... |
| trsspwALT3 45845 | Short predicate calculus p... |
| sspwtr 45846 | Virtual deduction proof of... |
| sspwtrALT 45847 | Virtual deduction proof of... |
| sspwtrALT2 45848 | Short predicate calculus p... |
| pwtrVD 45849 | Virtual deduction proof of... |
| pwtrrVD 45850 | Virtual deduction proof of... |
| suctrALT 45851 | The successor of a transit... |
| snssiALTVD 45852 | Virtual deduction proof of... |
| snssiALT 45853 | If a class is an element o... |
| snsslVD 45854 | Virtual deduction proof of... |
| snssl 45855 | If a singleton is a subcla... |
| snelpwrVD 45856 | Virtual deduction proof of... |
| unipwrVD 45857 | Virtual deduction proof of... |
| unipwr 45858 | A class is a subclass of t... |
| sstrALT2VD 45859 | Virtual deduction proof of... |
| sstrALT2 45860 | Virtual deduction proof of... |
| suctrALT2VD 45861 | Virtual deduction proof of... |
| suctrALT2 45862 | Virtual deduction proof of... |
| elex2VD 45863 | Virtual deduction proof of... |
| elex22VD 45864 | Virtual deduction proof of... |
| eqsbc2VD 45865 | Virtual deduction proof of... |
| zfregs2VD 45866 | Virtual deduction proof of... |
| tpid3gVD 45867 | Virtual deduction proof of... |
| en3lplem1VD 45868 | Virtual deduction proof of... |
| en3lplem2VD 45869 | Virtual deduction proof of... |
| en3lpVD 45870 | Virtual deduction proof of... |
| simplbi2VD 45871 | Virtual deduction proof of... |
| 3ornot23VD 45872 | Virtual deduction proof of... |
| orbi1rVD 45873 | Virtual deduction proof of... |
| bitr3VD 45874 | Virtual deduction proof of... |
| 3orbi123VD 45875 | Virtual deduction proof of... |
| sbc3orgVD 45876 | Virtual deduction proof of... |
| 19.21a3con13vVD 45877 | Virtual deduction proof of... |
| exbirVD 45878 | Virtual deduction proof of... |
| exbiriVD 45879 | Virtual deduction proof of... |
| rspsbc2VD 45880 | Virtual deduction proof of... |
| 3impexpVD 45881 | Virtual deduction proof of... |
| 3impexpbicomVD 45882 | Virtual deduction proof of... |
| 3impexpbicomiVD 45883 | Virtual deduction proof of... |
| sbcoreleleqVD 45884 | Virtual deduction proof of... |
| hbra2VD 45885 | Virtual deduction proof of... |
| tratrbVD 45886 | Virtual deduction proof of... |
| al2imVD 45887 | Virtual deduction proof of... |
| syl5impVD 45888 | Virtual deduction proof of... |
| idiVD 45889 | Virtual deduction proof of... |
| ancomstVD 45890 | Closed form of ~ ancoms . ... |
| ssralv2VD 45891 | Quantification restricted ... |
| ordelordALTVD 45892 | An element of an ordinal c... |
| equncomVD 45893 | If a class equals the unio... |
| equncomiVD 45894 | Inference form of ~ equnco... |
| sucidALTVD 45895 | A set belongs to its succe... |
| sucidALT 45896 | A set belongs to its succe... |
| sucidVD 45897 | A set belongs to its succe... |
| imbi12VD 45898 | Implication form of ~ imbi... |
| imbi13VD 45899 | Join three logical equival... |
| sbcim2gVD 45900 | Distribution of class subs... |
| sbcbiVD 45901 | Implication form of ~ sbcb... |
| trsbcVD 45902 | Formula-building inference... |
| truniALTVD 45903 | The union of a class of tr... |
| ee33VD 45904 | Non-virtual deduction form... |
| trintALTVD 45905 | The intersection of a clas... |
| trintALT 45906 | The intersection of a clas... |
| undif3VD 45907 | The first equality of Exer... |
| sbcssgVD 45908 | Virtual deduction proof of... |
| csbingVD 45909 | Virtual deduction proof of... |
| onfrALTlem5VD 45910 | Virtual deduction proof of... |
| onfrALTlem4VD 45911 | Virtual deduction proof of... |
| onfrALTlem3VD 45912 | Virtual deduction proof of... |
| simplbi2comtVD 45913 | Virtual deduction proof of... |
| onfrALTlem2VD 45914 | Virtual deduction proof of... |
| onfrALTlem1VD 45915 | Virtual deduction proof of... |
| onfrALTVD 45916 | Virtual deduction proof of... |
| csbeq2gVD 45917 | Virtual deduction proof of... |
| csbsngVD 45918 | Virtual deduction proof of... |
| csbxpgVD 45919 | Virtual deduction proof of... |
| csbresgVD 45920 | Virtual deduction proof of... |
| csbrngVD 45921 | Virtual deduction proof of... |
| csbima12gALTVD 45922 | Virtual deduction proof of... |
| csbunigVD 45923 | Virtual deduction proof of... |
| csbfv12gALTVD 45924 | Virtual deduction proof of... |
| con5VD 45925 | Virtual deduction proof of... |
| relopabVD 45926 | Virtual deduction proof of... |
| 19.41rgVD 45927 | Virtual deduction proof of... |
| 2pm13.193VD 45928 | Virtual deduction proof of... |
| hbimpgVD 45929 | Virtual deduction proof of... |
| hbalgVD 45930 | Virtual deduction proof of... |
| hbexgVD 45931 | Virtual deduction proof of... |
| ax6e2eqVD 45932 | The following User's Proof... |
| ax6e2ndVD 45933 | The following User's Proof... |
| ax6e2ndeqVD 45934 | The following User's Proof... |
| 2sb5ndVD 45935 | The following User's Proof... |
| 2uasbanhVD 45936 | The following User's Proof... |
| e2ebindVD 45937 | The following User's Proof... |
| sb5ALTVD 45938 | The following User's Proof... |
| vk15.4jVD 45939 | The following User's Proof... |
| notnotrALTVD 45940 | The following User's Proof... |
| con3ALTVD 45941 | The following User's Proof... |
| elpwgdedVD 45942 | Membership in a power clas... |
| sspwimp 45943 | If a class is a subclass o... |
| sspwimpVD 45944 | The following User's Proof... |
| sspwimpcf 45945 | If a class is a subclass o... |
| sspwimpcfVD 45946 | The following User's Proof... |
| suctrALTcf 45947 | The successor of a transit... |
| suctrALTcfVD 45948 | The following User's Proof... |
| suctrALT3 45949 | The successor of a transit... |
| sspwimpALT 45950 | If a class is a subclass o... |
| unisnALT 45951 | A set equals the union of ... |
| notnotrALT2 45952 | Converse of double negatio... |
| sspwimpALT2 45953 | If a class is a subclass o... |
| e2ebindALT 45954 | Absorption of an existenti... |
| ax6e2ndALT 45955 | If at least two sets exist... |
| ax6e2ndeqALT 45956 | "At least two sets exist" ... |
| 2sb5ndALT 45957 | Equivalence for double sub... |
| chordthmALT 45958 | The intersecting chords th... |
| isosctrlem1ALT 45959 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 45960 | The indexed union of conne... |
| iunconnALT 45961 | The indexed union of conne... |
| sineq0ALT 45962 | A complex number whose sin... |
| rspesbcd 45963 | Restricted quantifier vers... |
| rext0 45964 | Nonempty existential quant... |
| cocanss2 45965 | Cancellation law for compo... |
| cocan2g 45966 | Cancellation law for compo... |
| cocanss1 45967 | Cancellation law for compo... |
| cocan1g 45968 | Cancellation law for compo... |
| dmstructnn 45969 | The domain of an extensibl... |
| dmstructfi 45970 | The domain of an extensibl... |
| rnstructfi 45971 | The range of an extensible... |
| dfbi1ALTa 45972 | Version of ~ dfbi1ALT usin... |
| simprimi 45973 | Inference associated with ... |
| dfbi1ALTb 45974 | Further shorten ~ dfbi1ALT... |
| relpeq1 45977 | Equality theorem for relat... |
| relpeq2 45978 | Equality theorem for relat... |
| relpeq3 45979 | Equality theorem for relat... |
| relpeq4 45980 | Equality theorem for relat... |
| relpeq5 45981 | Equality theorem for relat... |
| nfrelp 45982 | Bound-variable hypothesis ... |
| relpf 45983 | A relation-preserving func... |
| relprel 45984 | A relation-preserving func... |
| relpmin 45985 | A preimage of a minimal el... |
| relpfrlem 45986 | Lemma for ~ relpfr . Prov... |
| relpfr 45987 | If the image of a set unde... |
| orbitex 45988 | Orbits exist. Given a set... |
| orbitinit 45989 | A set is contained in its ... |
| orbitcl 45990 | The orbit under a function... |
| orbitclmpt 45991 | Version of ~ orbitcl using... |
| trwf 45992 | The class of well-founded ... |
| rankrelp 45993 | The rank function preserve... |
| wffr 45994 | The class of well-founded ... |
| trfr 45995 | A transitive class well-fo... |
| tcfr 45996 | A set is well-founded if a... |
| xpwf 45997 | The Cartesian product of t... |
| dmwf 45998 | The domain of a well-found... |
| rnwf 45999 | The range of a well-founde... |
| relwf 46000 | A relation is a well-found... |
| ralabso 46001 | Simplification of restrict... |
| rexabso 46002 | Simplification of restrict... |
| ralabsod 46003 | Deduction form of ~ ralabs... |
| rexabsod 46004 | Deduction form of ~ rexabs... |
| ralabsobidv 46005 | Formula-building lemma for... |
| rexabsobidv 46006 | Formula-building lemma for... |
| ssabso 46007 | The notion " ` x ` is a su... |
| disjabso 46008 | Disjointness is absolute f... |
| n0abso 46009 | Nonemptiness is absolute f... |
| traxext 46010 | A transitive class models ... |
| modelaxreplem1 46011 | Lemma for ~ modelaxrep . ... |
| modelaxreplem2 46012 | Lemma for ~ modelaxrep . ... |
| modelaxreplem3 46013 | Lemma for ~ modelaxrep . ... |
| modelaxrep 46014 | Conditions which guarantee... |
| ssclaxsep 46015 | A class that is closed und... |
| 0elaxnul 46016 | A class that contains the ... |
| pwclaxpow 46017 | Suppose ` M ` is a transit... |
| prclaxpr 46018 | A class that is closed und... |
| uniclaxun 46019 | A class that is closed und... |
| sswfaxreg 46020 | A subclass of the class of... |
| omssaxinf2 46021 | A class that contains all ... |
| omelaxinf2 46022 | A transitive class that co... |
| dfac5prim 46023 | ~ dfac5 expanded into prim... |
| ac8prim 46024 | ~ ac8 expanded into primit... |
| modelac8prim 46025 | If ` M ` is a transitive c... |
| wfaxext 46026 | The class of well-founded ... |
| wfaxrep 46027 | The class of well-founded ... |
| wfaxsep 46028 | The class of well-founded ... |
| wfaxnul 46029 | The class of well-founded ... |
| wfaxpow 46030 | The class of well-founded ... |
| wfaxpr 46031 | The class of well-founded ... |
| wfaxun 46032 | The class of well-founded ... |
| wfaxreg 46033 | The class of well-founded ... |
| wfaxinf2 46034 | The class of well-founded ... |
| wfac8prim 46035 | The class of well-founded ... |
| brpermmodel 46036 | The membership relation in... |
| brpermmodelcnv 46037 | Ordinary membership expres... |
| permaxext 46038 | The Axiom of Extensionalit... |
| permaxrep 46039 | The Axiom of Replacement ~... |
| permaxsep 46040 | The Axiom of Separation ~ ... |
| permaxnul 46041 | The Null Set Axiom ~ ax-nu... |
| permaxpow 46042 | The Axiom of Power Sets ~ ... |
| permaxpr 46043 | The Axiom of Pairing ~ ax-... |
| permaxun 46044 | The Axiom of Union ~ ax-un... |
| permaxinf2lem 46045 | Lemma for ~ permaxinf2 . ... |
| permaxinf2 46046 | The Axiom of Infinity ~ ax... |
| permac8prim 46047 | The Axiom of Choice ~ ac8p... |
| nregmodelf1o 46048 | Define a permutation ` F `... |
| nregmodellem 46049 | Lemma for ~ nregmodel . (... |
| nregmodel 46050 | The Axiom of Regularity ~ ... |
| nregmodelaxext 46051 | The Axiom of Extensionalit... |
| hashnna 46052 | The ` # ` function on ` _o... |
| hashnnsuc 46053 | The ` # ` function on ` _o... |
| hashnnm 46054 | The ` # ` function on ` _o... |
| hashnnlt 46055 | The ` # ` function on ` _o... |
| hashnnltb 46056 | The ` # ` function on ` _o... |
| hashomf1o 46057 | The ` # ` function yields ... |
| hashomiso 46058 | The ` # ` function yields ... |
| hfpr 46059 | A pair whose members are h... |
| hfxp 46060 | The Cartesian product of t... |
| hfdm 46061 | The domain of a hereditari... |
| hfrn 46062 | The range of a hereditaril... |
| hfrel 46063 | A relation is a hereditari... |
| omhf 46064 | Finite ordinals are heredi... |
| omsshf 46065 | The class of finite ordina... |
| hfstructstruct 46068 | An HFStruct is an extensib... |
| hfstructfun 46069 | An HFStruct is a function.... |
| rnhfstructsshf 46070 | The range of an HFStruct c... |
| ishfstruct 46071 | Conditions that imply that... |
| rnhfstructhf 46072 | The range of an HFStruct i... |
| hfstructhf 46073 | A converted HFStruct is a ... |
| hfstructcan 46074 | Two HFStructs are equal if... |
| evth2f 46075 | A version of ~ evth2 using... |
| elunif 46076 | A version of ~ eluni using... |
| rzalf 46077 | A version of ~ rzal using ... |
| fvelrnbf 46078 | A version of ~ fvelrnb usi... |
| rfcnpre1 46079 | If F is a continuous funct... |
| ubelsupr 46080 | If U belongs to A and U is... |
| fsumcnf 46081 | A finite sum of functions ... |
| mulltgt0 46082 | The product of a negative ... |
| rspcegf 46083 | A version of ~ rspcev usin... |
| rabexgf 46084 | A version of ~ rabexg usin... |
| fcnre 46085 | A function continuous with... |
| sumsnd 46086 | A sum of a singleton is th... |
| evthf 46087 | A version of ~ evth using ... |
| cnfex 46088 | The class of continuous fu... |
| fnchoice 46089 | For a finite set, a choice... |
| refsumcn 46090 | A finite sum of continuous... |
| rfcnpre2 46091 | If ` F ` is a continuous f... |
| cncmpmax 46092 | When the hypothesis for th... |
| rfcnpre3 46093 | If F is a continuous funct... |
| rfcnpre4 46094 | If F is a continuous funct... |
| sumpair 46095 | Sum of two distinct comple... |
| rfcnnnub 46096 | Given a real continuous fu... |
| refsum2cnlem1 46097 | This is the core Lemma for... |
| refsum2cn 46098 | The sum of two continuus r... |
| adantlllr 46099 | Deduction adding a conjunc... |
| 3adantlr3 46100 | Deduction adding a conjunc... |
| 3adantll2 46101 | Deduction adding a conjunc... |
| 3adantll3 46102 | Deduction adding a conjunc... |
| ssnel 46103 | If not element of a set, t... |
| sncldre 46104 | A singleton is closed w.r.... |
| n0p 46105 | A polynomial with a nonzer... |
| pm2.65ni 46106 | Inference rule for proof b... |
| iuneq2df 46107 | Equality deduction for ind... |
| nnfoctb 46108 | There exists a mapping fro... |
| elpwinss 46109 | An element of the powerset... |
| unidmex 46110 | If ` F ` is a set, then ` ... |
| ndisj2 46111 | A non-disjointness conditi... |
| zenom 46112 | The set of integer numbers... |
| uzwo4 46113 | Well-ordering principle: a... |
| unisn0 46114 | The union of the singleton... |
| ssin0 46115 | If two classes are disjoin... |
| inabs3 46116 | Absorption law for interse... |
| pwpwuni 46117 | Relationship between power... |
| disjiun2 46118 | In a disjoint collection, ... |
| 0pwfi 46119 | The empty set is in any po... |
| ssinss2d 46120 | Intersection preserves sub... |
| zct 46121 | The set of integer numbers... |
| pwfin0 46122 | A finite set always belong... |
| uzct 46123 | An upper integer set is co... |
| iunxsnf 46124 | A singleton index picks ou... |
| fiiuncl 46125 | If a set is closed under t... |
| iunp1 46126 | The addition of the next s... |
| fiunicl 46127 | If a set is closed under t... |
| ixpeq2d 46128 | Equality theorem for infin... |
| disjxp1 46129 | The sets of a cartesian pr... |
| disjsnxp 46130 | The sets in the cartesian ... |
| eliind 46131 | Membership in indexed inte... |
| rspcef 46132 | Restricted existential spe... |
| ixpssmapc 46133 | An infinite Cartesian prod... |
| elintd 46134 | Membership in class inters... |
| ssdf 46135 | A sufficient condition for... |
| brneqtrd 46136 | Substitution of equal clas... |
| ssnct 46137 | A set containing an uncoun... |
| ssuniint 46138 | Sufficient condition for b... |
| elintdv 46139 | Membership in class inters... |
| ssd 46140 | A sufficient condition for... |
| ralimralim 46141 | Introducing any antecedent... |
| snelmap 46142 | Membership of the element ... |
| xrnmnfpnf 46143 | An extended real that is n... |
| iuneq1i 46144 | Equality theorem for index... |
| ssinc 46145 | Inclusion relation for a m... |
| ssdec 46146 | Inclusion relation for a m... |
| elixpconstg 46147 | Membership in an infinite ... |
| iineq1d 46148 | Equality theorem for index... |
| metpsmet 46149 | A metric is a pseudometric... |
| ixpssixp 46150 | Subclass theorem for infin... |
| ballss3 46151 | A sufficient condition for... |
| iunincfi 46152 | Given a sequence of increa... |
| nsstr 46153 | If it's not a subclass, it... |
| rexanuz3 46154 | Combine two different uppe... |
| cbvmpo2 46155 | Rule to change the second ... |
| cbvmpo1 46156 | Rule to change the first b... |
| eliuniin 46157 | Indexed union of indexed i... |
| ssabf 46158 | Subclass of a class abstra... |
| pssnssi 46159 | A proper subclass does not... |
| rabidim2 46160 | Membership in a restricted... |
| eluni2f 46161 | Membership in class union.... |
| eliin2f 46162 | Membership in indexed inte... |
| nssd 46163 | Negation of subclass relat... |
| iineq12dv 46164 | Equality deduction for ind... |
| supxrcld 46165 | The supremum of an arbitra... |
| elrestd 46166 | A sufficient condition for... |
| eliuniincex 46167 | Counterexample to show tha... |
| eliincex 46168 | Counterexample to show tha... |
| eliinid 46169 | Membership in an indexed i... |
| abssf 46170 | Class abstraction in a sub... |
| supxrubd 46171 | A member of a set of exten... |
| ssrabf 46172 | Subclass of a restricted c... |
| ssrabdf 46173 | Subclass of a restricted c... |
| eliin2 46174 | Membership in indexed inte... |
| ssrab2f 46175 | Subclass relation for a re... |
| restuni3 46176 | The underlying set of a su... |
| rabssf 46177 | Restricted class abstracti... |
| eliuniin2 46178 | Indexed union of indexed i... |
| restuni4 46179 | The underlying set of a su... |
| restuni6 46180 | The underlying set of a su... |
| restuni5 46181 | The underlying set of a su... |
| unirestss 46182 | The union of an elementwis... |
| iniin1 46183 | Indexed intersection of in... |
| iniin2 46184 | Indexed intersection of in... |
| cbvrabv2 46185 | A more general version of ... |
| cbvrabv2w 46186 | A more general version of ... |
| iinssiin 46187 | Subset implication for an ... |
| eliind2 46188 | Membership in indexed inte... |
| iinssd 46189 | Subset implication for an ... |
| rabbida2 46190 | Equivalent wff's yield equ... |
| iinexd 46191 | The existence of an indexe... |
| rabexf 46192 | Separation Scheme in terms... |
| rabbida3 46193 | Equivalent wff's yield equ... |
| r19.36vf 46194 | Restricted quantifier vers... |
| raleqd 46195 | Equality deduction for res... |
| iinssf 46196 | Subset implication for an ... |
| iinssdf 46197 | Subset implication for an ... |
| resabs2i 46198 | Absorption law for restric... |
| ssdf2 46199 | A sufficient condition for... |
| rabssd 46200 | Restricted class abstracti... |
| rexnegd 46201 | Minus a real number. (Con... |
| rexlimd3 46202 | * Inference from Theorem 1... |
| nel1nelini 46203 | Membership in an intersect... |
| nel2nelini 46204 | Membership in an intersect... |
| eliunid 46205 | Membership in indexed unio... |
| reximdd 46206 | Deduction from Theorem 19.... |
| inopnd 46207 | The intersection of two op... |
| ss2rabdf 46208 | Deduction of restricted ab... |
| restopn3 46209 | If ` A ` is open, then ` A... |
| restopnssd 46210 | A topology restricted to a... |
| restsubel 46211 | A subset belongs in the sp... |
| toprestsubel 46212 | A subset is open in the to... |
| rabidd 46213 | An "identity" law of concr... |
| iunssdf 46214 | Subset theorem for an inde... |
| iinss2d 46215 | Subset implication for an ... |
| r19.3rzf 46216 | Restricted quantification ... |
| r19.28zf 46217 | Restricted quantifier vers... |
| iindif2f 46218 | Indexed intersection of cl... |
| ralfal 46219 | Two ways of expressing emp... |
| archd 46220 | Archimedean property of re... |
| nimnbi 46221 | If an implication is false... |
| nimnbi2 46222 | If an implication is false... |
| notbicom 46223 | Commutative law for the ne... |
| rexeqif 46224 | Equality inference for res... |
| rspced 46225 | Restricted existential spe... |
| fnresdmss 46226 | A function does not change... |
| fmptsnxp 46227 | Maps-to notation and Carte... |
| fvmpt2bd 46228 | Value of a function given ... |
| rnmptfi 46229 | The range of a function wi... |
| fresin2 46230 | Restriction of a function ... |
| ffi 46231 | A function with finite dom... |
| suprnmpt 46232 | An explicit bound for the ... |
| rnffi 46233 | The range of a function wi... |
| mptelpm 46234 | A function in maps-to nota... |
| rnmptpr 46235 | Range of a function define... |
| resmpti 46236 | Restriction of the mapping... |
| founiiun 46237 | Union expressed as an inde... |
| rnresun 46238 | Distribution law for range... |
| elrnmptf 46239 | The range of a function in... |
| rnmptssrn 46240 | Inclusion relation for two... |
| disjf1 46241 | A 1 to 1 mapping built fro... |
| rnsnf 46242 | The range of a function wh... |
| wessf1ornlem 46243 | Given a function ` F ` on ... |
| wessf1orn 46244 | Given a function ` F ` on ... |
| nelrnres 46245 | If ` A ` is not in the ran... |
| disjrnmpt2 46246 | Disjointness of the range ... |
| elrnmpt1sf 46247 | Elementhood in an image se... |
| founiiun0 46248 | Union expressed as an inde... |
| disjf1o 46249 | A bijection built from dis... |
| disjinfi 46250 | Only a finite number of di... |
| fvovco 46251 | Value of the composition o... |
| ssnnf1octb 46252 | There exists a bijection b... |
| nnf1oxpnn 46253 | There is a bijection betwe... |
| projf1o 46254 | A biijection from a set to... |
| fvmap 46255 | Function value for a membe... |
| fvixp2 46256 | Projection of a factor of ... |
| choicefi 46257 | For a finite set, a choice... |
| mpct 46258 | The exponentiation of a co... |
| cnmetcoval 46259 | Value of the distance func... |
| fcomptss 46260 | Express composition of two... |
| elmapsnd 46261 | Membership in a set expone... |
| mapss2 46262 | Subset inheritance for set... |
| difmap 46263 | Difference of two sets exp... |
| unirnmap 46264 | Given a subset of a set ex... |
| inmap 46265 | Intersection of two sets e... |
| fcoss 46266 | Composition of two mapping... |
| fsneqrn 46267 | Equality condition for two... |
| difmapsn 46268 | Difference of two sets exp... |
| mapssbi 46269 | Subset inheritance for set... |
| unirnmapsn 46270 | Equality theorem for a sub... |
| iunmapss 46271 | The indexed union of set e... |
| ssmapsn 46272 | A subset ` C ` of a set ex... |
| iunmapsn 46273 | The indexed union of set e... |
| absfico 46274 | Mapping domain and codomai... |
| icof 46275 | The set of left-closed rig... |
| elpmrn 46276 | The range of a partial fun... |
| imaexi 46277 | The image of a set is a se... |
| axccdom 46278 | Relax the constraint on ax... |
| dmmptdff 46279 | The domain of the mapping ... |
| dmmptdf 46280 | The domain of the mapping ... |
| elpmi2 46281 | The domain of a partial fu... |
| dmrelrnrel 46282 | A relation preserving func... |
| elrnmpoid 46283 | Membership in the range of... |
| axccd 46284 | An alternative version of ... |
| axccd2 46285 | An alternative version of ... |
| feqresmptf 46286 | Express a restricted funct... |
| dmmptssf 46287 | The domain of a mapping is... |
| dmmptdf2 46288 | The domain of the mapping ... |
| dmuz 46289 | Domain of the upper intege... |
| fmptd2f 46290 | Domain and codomain of the... |
| mpteq1df 46291 | An equality theorem for th... |
| mptexf 46292 | If the domain of a functio... |
| fvmpt4 46293 | Value of a function given ... |
| fmptf 46294 | Functionality of the mappi... |
| resimass 46295 | The image of a restriction... |
| mptssid 46296 | The mapping operation expr... |
| mptfnd 46297 | The maps-to notation defin... |
| rnmptlb 46298 | Boundness below of the ran... |
| rnmptbddlem 46299 | Boundness of the range of ... |
| rnmptbdd 46300 | Boundness of the range of ... |
| funimaeq 46301 | Membership relation for th... |
| rnmptssf 46302 | The range of a function gi... |
| rnmptbd2lem 46303 | Boundness below of the ran... |
| rnmptbd2 46304 | Boundness below of the ran... |
| infnsuprnmpt 46305 | The indexed infimum of rea... |
| suprclrnmpt 46306 | Closure of the indexed sup... |
| suprubrnmpt2 46307 | A member of a nonempty ind... |
| suprubrnmpt 46308 | A member of a nonempty ind... |
| rnmptssdf 46309 | The range of a function gi... |
| rnmptbdlem 46310 | Boundness above of the ran... |
| rnmptbd 46311 | Boundness above of the ran... |
| rnmptss2 46312 | The range of a function gi... |
| elmptima 46313 | The image of a function in... |
| ralrnmpt3 46314 | A restricted quantifier ov... |
| rnmptssbi 46315 | The range of a function gi... |
| imass2d 46316 | Subset theorem for image. ... |
| imassmpt 46317 | Membership relation for th... |
| fpmd 46318 | A total function is a part... |
| fconst7 46319 | An alternative way to expr... |
| fnmptif 46320 | Functionality and domain o... |
| dmmptif 46321 | Domain of the mapping oper... |
| mpteq2dfa 46322 | Slightly more general equa... |
| dmmpt1 46323 | The domain of the mapping ... |
| fmptff 46324 | Functionality of the mappi... |
| fvmptelcdmf 46325 | The value of a function at... |
| fmptdff 46326 | A version of ~ fmptd using... |
| fvmpt2df 46327 | Deduction version of ~ fvm... |
| rn1st 46328 | The range of a function wi... |
| rnmptssff 46329 | The range of a function gi... |
| rnmptssdff 46330 | The range of a function gi... |
| fvmpt4d 46331 | Value of a function given ... |
| sub2times 46332 | Subtracting from a number,... |
| nnxrd 46333 | A natural number is an ext... |
| nnxr 46334 | A natural number is an ext... |
| abssubrp 46335 | The distance of two distin... |
| elfzfzo 46336 | Relationship between membe... |
| oddfl 46337 | Odd number representation ... |
| abscosbd 46338 | Bound for the absolute val... |
| mul13d 46339 | Commutative/associative la... |
| negpilt0 46340 | Negative ` _pi ` is negati... |
| dstregt0 46341 | A complex number ` A ` tha... |
| subadd4b 46342 | Rearrangement of 4 terms i... |
| xrlttri5d 46343 | Not equal and not larger i... |
| zltlesub 46344 | If an integer ` N ` is les... |
| divlt0gt0d 46345 | The ratio of a negative nu... |
| subsub23d 46346 | Swap subtrahend and result... |
| 2timesgt 46347 | Double of a positive real ... |
| reopn 46348 | The reals are open with re... |
| sub31 46349 | Swap the first and third t... |
| nnne1ge2 46350 | A positive integer which i... |
| lefldiveq 46351 | A closed enough, smaller r... |
| negsubdi3d 46352 | Distribution of negative o... |
| ltdiv2dd 46353 | Division of a positive num... |
| abssinbd 46354 | Bound for the absolute val... |
| halffl 46355 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 46356 | Ordering relation for a st... |
| hashssle 46357 | The size of a subset of a ... |
| lttri5d 46358 | Not equal and not larger i... |
| fzisoeu 46359 | A finite ordered set has a... |
| lt3addmuld 46360 | If three real numbers are ... |
| absnpncan2d 46361 | Triangular inequality, com... |
| fperiodmullem 46362 | A function with period ` T... |
| fperiodmul 46363 | A function with period T i... |
| upbdrech 46364 | Choice of an upper bound f... |
| lt4addmuld 46365 | If four real numbers are l... |
| absnpncan3d 46366 | Triangular inequality, com... |
| upbdrech2 46367 | Choice of an upper bound f... |
| ssfiunibd 46368 | A finite union of bounded ... |
| fzdifsuc2 46369 | Remove a successor from th... |
| fzsscn 46370 | A finite sequence of integ... |
| divcan8d 46371 | A cancellation law for div... |
| dmmcand 46372 | Cancellation law for divis... |
| bccld 46373 | A binomial coefficient, in... |
| fzssnn0 46374 | A finite set of sequential... |
| xreqle 46375 | Equality implies 'less tha... |
| xaddlidd 46376 | ` 0 ` is a left identity f... |
| xadd0ge 46377 | A number is less than or e... |
| xrleneltd 46378 | 'Less than or equal to' an... |
| xaddcomd 46379 | The extended real addition... |
| supxrre3 46380 | The supremum of a nonempty... |
| uzfissfz 46381 | For any finite subset of t... |
| xleadd2d 46382 | Addition of extended reals... |
| suprltrp 46383 | The supremum of a nonempty... |
| xleadd1d 46384 | Addition of extended reals... |
| xreqled 46385 | Equality implies 'less tha... |
| xrgepnfd 46386 | An extended real greater t... |
| xrge0nemnfd 46387 | A nonnegative extended rea... |
| supxrgere 46388 | If a real number can be ap... |
| iuneqfzuzlem 46389 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 46390 | If two unions indexed by u... |
| xle2addd 46391 | Adding both side of two in... |
| supxrgelem 46392 | If an extended real number... |
| supxrge 46393 | If an extended real number... |
| suplesup 46394 | If any element of ` A ` ca... |
| infxrglb 46395 | The infimum of a set of ex... |
| xadd0ge2 46396 | A number is less than or e... |
| nepnfltpnf 46397 | An extended real that is n... |
| ltadd12dd 46398 | Addition to both sides of ... |
| nemnftgtmnft 46399 | An extended real that is n... |
| xrgtso 46400 | 'Greater than' is a strict... |
| rpex 46401 | The positive reals form a ... |
| xrge0ge0 46402 | A nonnegative extended rea... |
| xrssre 46403 | A subset of extended reals... |
| ssuzfz 46404 | A finite subset of the upp... |
| absfun 46405 | The absolute value is a fu... |
| infrpge 46406 | The infimum of a nonempty,... |
| xrlexaddrp 46407 | If an extended real number... |
| supsubc 46408 | The supremum function dist... |
| xralrple2 46409 | Show that ` A ` is less th... |
| nnuzdisj 46410 | The first ` N ` elements o... |
| ltdivgt1 46411 | Divsion by a number greate... |
| xrltned 46412 | 'Less than' implies not eq... |
| nnsplit 46413 | Express the set of positiv... |
| divdiv3d 46414 | Division into a fraction. ... |
| abslt2sqd 46415 | Comparison of the square o... |
| qenom 46416 | The set of rational number... |
| qct 46417 | The set of rational number... |
| lenlteq 46418 | 'less than or equal to' bu... |
| xrred 46419 | An extended real that is n... |
| rr2sscn2 46420 | The cartesian square of ` ... |
| infxr 46421 | The infimum of a set of ex... |
| infxrunb2 46422 | The infimum of an unbounde... |
| infxrbnd2 46423 | The infimum of a bounded-b... |
| infleinflem1 46424 | Lemma for ~ infleinf , cas... |
| infleinflem2 46425 | Lemma for ~ infleinf , whe... |
| infleinf 46426 | If any element of ` B ` ca... |
| xralrple4 46427 | Show that ` A ` is less th... |
| xralrple3 46428 | Show that ` A ` is less th... |
| eluzelzd 46429 | A member of an upper set o... |
| suplesup2 46430 | If any element of ` A ` is... |
| recnnltrp 46431 | ` N ` is a natural number ... |
| nnn0 46432 | The set of positive intege... |
| fzct 46433 | A finite set of sequential... |
| rpgtrecnn 46434 | Any positive real number i... |
| fzossuz 46435 | A half-open integer interv... |
| infxrrefi 46436 | The real and extended real... |
| xrralrecnnle 46437 | Show that ` A ` is less th... |
| fzoct 46438 | A finite set of sequential... |
| frexr 46439 | A function taking real val... |
| nnrecrp 46440 | The reciprocal of a positi... |
| reclt0d 46441 | The reciprocal of a negati... |
| lt0neg1dd 46442 | If a number is negative, i... |
| infxrcld 46443 | The infimum of an arbitrar... |
| xrralrecnnge 46444 | Show that ` A ` is less th... |
| reclt0 46445 | The reciprocal of a negati... |
| ltmulneg 46446 | Multiplying by a negative ... |
| allbutfi 46447 | For all but finitely many.... |
| ltdiv23neg 46448 | Swap denominator with othe... |
| xreqnltd 46449 | A consequence of trichotom... |
| mnfnre2 46450 | Minus infinity is not a re... |
| zssxr 46451 | The integers are a subset ... |
| fisupclrnmpt 46452 | A nonempty finite indexed ... |
| supxrunb3 46453 | The supremum of an unbound... |
| fimaxre4 46454 | A nonempty finite set of r... |
| ren0 46455 | The set of reals is nonemp... |
| eluzelz2 46456 | A member of an upper set o... |
| resabs2d 46457 | Absorption law for restric... |
| uzid2 46458 | Membership of the least me... |
| supxrleubrnmpt 46459 | The supremum of a nonempty... |
| uzssre2 46460 | An upper set of integers i... |
| uzssd 46461 | Subset relationship for tw... |
| eluzd 46462 | Membership in an upper set... |
| infxrlbrnmpt2 46463 | A member of a nonempty ind... |
| xrre4 46464 | An extended real is real i... |
| uz0 46465 | The upper integers functio... |
| eluzelz2d 46466 | A member of an upper set o... |
| infleinf2 46467 | If any element in ` B ` is... |
| unb2ltle 46468 | "Unbounded below" expresse... |
| uzidd2 46469 | Membership of the least me... |
| uzssd2 46470 | Subset relationship for tw... |
| rexabslelem 46471 | An indexed set of absolute... |
| rexabsle 46472 | An indexed set of absolute... |
| allbutfiinf 46473 | Given a "for all but finit... |
| supxrrernmpt 46474 | The real and extended real... |
| suprleubrnmpt 46475 | The supremum of a nonempty... |
| infrnmptle 46476 | An indexed infimum of exte... |
| infxrunb3 46477 | The infimum of an unbounde... |
| uzn0d 46478 | The upper integers are all... |
| uzssd3 46479 | Subset relationship for tw... |
| rexabsle2 46480 | An indexed set of absolute... |
| infxrunb3rnmpt 46481 | The infimum of an unbounde... |
| supxrre3rnmpt 46482 | The indexed supremum of a ... |
| uzublem 46483 | A set of reals, indexed by... |
| uzub 46484 | A set of reals, indexed by... |
| ssrexr 46485 | A subset of the reals is a... |
| supxrmnf2 46486 | Removing minus infinity fr... |
| supxrcli 46487 | The supremum of an arbitra... |
| uzid3 46488 | Membership of the least me... |
| infxrlesupxr 46489 | The supremum of a nonempty... |
| xnegeqd 46490 | Equality of two extended n... |
| xnegrecl 46491 | The extended real negative... |
| xnegnegi 46492 | Extended real version of ~... |
| xnegeqi 46493 | Equality of two extended n... |
| nfxnegd 46494 | Deduction version of ~ nfx... |
| xnegnegd 46495 | Extended real version of ~... |
| uzred 46496 | An upper integer is a real... |
| xnegcli 46497 | Closure of extended real n... |
| supminfrnmpt 46498 | The indexed supremum of a ... |
| infxrpnf 46499 | Adding plus infinity to a ... |
| infxrrnmptcl 46500 | The infimum of an arbitrar... |
| leneg2d 46501 | Negative of one side of 'l... |
| supxrltinfxr 46502 | The supremum of the empty ... |
| max1d 46503 | A number is less than or e... |
| supxrleubrnmptf 46504 | The supremum of a nonempty... |
| nleltd 46505 | 'Not less than or equal to... |
| zxrd 46506 | An integer is an extended ... |
| infxrgelbrnmpt 46507 | The infimum of an indexed ... |
| rphalfltd 46508 | Half of a positive real is... |
| uzssz2 46509 | An upper set of integers i... |
| leneg3d 46510 | Negative of one side of 'l... |
| max2d 46511 | A number is less than or e... |
| uzn0bi 46512 | The upper integers functio... |
| xnegrecl2 46513 | If the extended real negat... |
| nfxneg 46514 | Bound-variable hypothesis ... |
| uzxrd 46515 | An upper integer is an ext... |
| infxrpnf2 46516 | Removing plus infinity fro... |
| supminfxr 46517 | The extended real suprema ... |
| infrpgernmpt 46518 | The infimum of a nonempty,... |
| xnegre 46519 | An extended real is real i... |
| xnegrecl2d 46520 | If the extended real negat... |
| uzxr 46521 | An upper integer is an ext... |
| supminfxr2 46522 | The extended real suprema ... |
| xnegred 46523 | An extended real is real i... |
| supminfxrrnmpt 46524 | The indexed supremum of a ... |
| min1d 46525 | The minimum of two numbers... |
| min2d 46526 | The minimum of two numbers... |
| xrnpnfmnf 46527 | An extended real that is n... |
| uzsscn 46528 | An upper set of integers i... |
| absimnre 46529 | The absolute value of the ... |
| uzsscn2 46530 | An upper set of integers i... |
| xrtgcntopre 46531 | The standard topologies on... |
| absimlere 46532 | The absolute value of the ... |
| rpssxr 46533 | The positive reals are a s... |
| monoordxrv 46534 | Ordering relation for a mo... |
| monoordxr 46535 | Ordering relation for a mo... |
| monoord2xrv 46536 | Ordering relation for a mo... |
| monoord2xr 46537 | Ordering relation for a mo... |
| xrpnf 46538 | An extended real is plus i... |
| xlenegcon1 46539 | Extended real version of ~... |
| xlenegcon2 46540 | Extended real version of ~... |
| pimxrneun 46541 | The preimage of a set of e... |
| caucvgbf 46542 | A function is convergent i... |
| cvgcau 46543 | A convergent function is C... |
| cvgcaule 46544 | A convergent function is C... |
| rexanuz2nf 46545 | A simple counterexample re... |
| gtnelioc 46546 | A real number larger than ... |
| ioossioc 46547 | An open interval is a subs... |
| ioondisj2 46548 | A condition for two open i... |
| ioondisj1 46549 | A condition for two open i... |
| ioogtlb 46550 | An element of a closed int... |
| evthiccabs 46551 | Extreme Value Theorem on y... |
| ltnelicc 46552 | A real number smaller than... |
| eliood 46553 | Membership in an open real... |
| iooabslt 46554 | An upper bound for the dis... |
| gtnelicc 46555 | A real number greater than... |
| iooinlbub 46556 | An open interval has empty... |
| iocgtlb 46557 | An element of a left-open ... |
| iocleub 46558 | An element of a left-open ... |
| eliccd 46559 | Membership in a closed rea... |
| eliccre 46560 | A member of a closed inter... |
| eliooshift 46561 | Element of an open interva... |
| eliocd 46562 | Membership in a left-open ... |
| icoltub 46563 | An element of a left-close... |
| eliocre 46564 | A member of a left-open ri... |
| iooltub 46565 | An element of an open inte... |
| ioontr 46566 | The interior of an interva... |
| snunioo1 46567 | The closure of one end of ... |
| lbioc 46568 | A left-open right-closed i... |
| ioomidp 46569 | The midpoint is an element... |
| iccdifioo 46570 | If the open inverval is re... |
| iccdifprioo 46571 | An open interval is the cl... |
| ioossioobi 46572 | Biconditional form of ~ io... |
| iccshift 46573 | A closed interval shifted ... |
| iccsuble 46574 | An upper bound to the dist... |
| iocopn 46575 | A left-open right-closed i... |
| eliccelioc 46576 | Membership in a closed int... |
| iooshift 46577 | An open interval shifted b... |
| iccintsng 46578 | Intersection of two adiace... |
| icoiccdif 46579 | Left-closed right-open int... |
| icoopn 46580 | A left-closed right-open i... |
| icoub 46581 | A left-closed, right-open ... |
| eliccxrd 46582 | Membership in a closed rea... |
| pnfel0pnf 46583 | ` +oo ` is a nonnegative e... |
| eliccnelico 46584 | An element of a closed int... |
| eliccelicod 46585 | A member of a closed inter... |
| ge0xrre 46586 | A nonnegative extended rea... |
| ge0lere 46587 | A nonnegative extended Rea... |
| elicores 46588 | Membership in a left-close... |
| inficc 46589 | The infimum of a nonempty ... |
| qinioo 46590 | The rational numbers are d... |
| lenelioc 46591 | A real number smaller than... |
| ioonct 46592 | A nonempty open interval i... |
| xrgtnelicc 46593 | A real number greater than... |
| iccdificc 46594 | The difference of two clos... |
| iocnct 46595 | A nonempty left-open, righ... |
| iccnct 46596 | A closed interval, with mo... |
| iooiinicc 46597 | A closed interval expresse... |
| iccgelbd 46598 | An element of a closed int... |
| iooltubd 46599 | An element of an open inte... |
| icoltubd 46600 | An element of a left-close... |
| qelioo 46601 | The rational numbers are d... |
| tgqioo2 46602 | Every open set of reals is... |
| iccleubd 46603 | An element of a closed int... |
| elioored 46604 | A member of an open interv... |
| ioogtlbd 46605 | An element of a closed int... |
| ioofun 46606 | ` (,) ` is a function. (C... |
| icomnfinre 46607 | A left-closed, right-open,... |
| sqrlearg 46608 | The square compared with i... |
| ressiocsup 46609 | If the supremum belongs to... |
| ressioosup 46610 | If the supremum does not b... |
| iooiinioc 46611 | A left-open, right-closed ... |
| ressiooinf 46612 | If the infimum does not be... |
| iocleubd 46613 | An element of a left-open ... |
| uzinico 46614 | An upper interval of integ... |
| preimaiocmnf 46615 | Preimage of a right-closed... |
| uzinico2 46616 | An upper interval of integ... |
| uzinico3 46617 | An upper interval of integ... |
| dmico 46618 | The domain of the closed-b... |
| ndmico 46619 | The closed-below, open-abo... |
| uzubioo 46620 | The upper integers are unb... |
| uzubico 46621 | The upper integers are unb... |
| uzubioo2 46622 | The upper integers are unb... |
| uzubico2 46623 | The upper integers are unb... |
| iocgtlbd 46624 | An element of a left-open ... |
| xrtgioo2 46625 | The topology on the extend... |
| fsummulc1f 46626 | Closure of a finite sum of... |
| fsumnncl 46627 | Closure of a nonempty, fin... |
| fsumge0cl 46628 | The finite sum of nonnegat... |
| fsumf1of 46629 | Re-index a finite sum usin... |
| fsumiunss 46630 | Sum over a disjoint indexe... |
| fsumreclf 46631 | Closure of a finite sum of... |
| fsumlessf 46632 | A shorter sum of nonnegati... |
| fsumsupp0 46633 | Finite sum of function val... |
| fsumsermpt 46634 | A finite sum expressed in ... |
| fmul01 46635 | Multiplying a finite numbe... |
| fmulcl 46636 | If ' Y ' is closed under t... |
| fmuldfeqlem1 46637 | induction step for the pro... |
| fmuldfeq 46638 | X and Z are two equivalent... |
| fmul01lt1lem1 46639 | Given a finite multiplicat... |
| fmul01lt1lem2 46640 | Given a finite multiplicat... |
| fmul01lt1 46641 | Given a finite multiplicat... |
| cncfmptss 46642 | A continuous complex funct... |
| rrpsscn 46643 | The positive reals are a s... |
| mulc1cncfg 46644 | A version of ~ mulc1cncf u... |
| infrglb 46645 | The infimum of a nonempty ... |
| expcnfg 46646 | If ` F ` is a complex cont... |
| prodeq2ad 46647 | Equality deduction for pro... |
| fprodsplit1 46648 | Separate out a term in a f... |
| fprodexp 46649 | Positive integer exponenti... |
| fprodabs2 46650 | The absolute value of a fi... |
| fprod0 46651 | A finite product with a ze... |
| mccllem 46652 | * Induction step for ~ mcc... |
| mccl 46653 | A multinomial coefficient,... |
| fprodcnlem 46654 | A finite product of functi... |
| fprodcn 46655 | A finite product of functi... |
| clim1fr1 46656 | A class of sequences of fr... |
| isumneg 46657 | Negation of a converging s... |
| climrec 46658 | Limit of the reciprocal of... |
| climmulf 46659 | A version of ~ climmul usi... |
| climexp 46660 | The limit of natural power... |
| climinf 46661 | A bounded monotonic noninc... |
| climsuselem1 46662 | The subsequence index ` I ... |
| climsuse 46663 | A subsequence ` G ` of a c... |
| climrecf 46664 | A version of ~ climrec usi... |
| climneg 46665 | Complex limit of the negat... |
| climinff 46666 | A version of ~ climinf usi... |
| climdivf 46667 | Limit of the ratio of two ... |
| climreeq 46668 | If ` F ` is a real functio... |
| ellimciota 46669 | An explicit value for the ... |
| climaddf 46670 | A version of ~ climadd usi... |
| mullimc 46671 | Limit of the product of tw... |
| ellimcabssub0 46672 | An equivalent condition fo... |
| limcdm0 46673 | If a function has empty do... |
| islptre 46674 | An equivalence condition f... |
| limccog 46675 | Limit of the composition o... |
| limciccioolb 46676 | The limit of a function at... |
| climf 46677 | Express the predicate: Th... |
| mullimcf 46678 | Limit of the multiplicatio... |
| constlimc 46679 | Limit of constant function... |
| rexlim2d 46680 | Inference removing two res... |
| idlimc 46681 | Limit of the identity func... |
| divcnvg 46682 | The sequence of reciprocal... |
| limcperiod 46683 | If ` F ` is a periodic fun... |
| limcrecl 46684 | If ` F ` is a real-valued ... |
| sumnnodd 46685 | A series indexed by ` NN `... |
| lptioo2 46686 | The upper bound of an open... |
| lptioo1 46687 | The lower bound of an open... |
| limcmptdm 46688 | The domain of a maps-to fu... |
| clim2f 46689 | Express the predicate: Th... |
| limcicciooub 46690 | The limit of a function at... |
| ltmod 46691 | A sufficient condition for... |
| islpcn 46692 | A characterization for a l... |
| lptre2pt 46693 | If a set in the real line ... |
| limsupre 46694 | If a sequence is bounded, ... |
| limcresiooub 46695 | The left limit doesn't cha... |
| limcresioolb 46696 | The right limit doesn't ch... |
| limcleqr 46697 | If the left and the right ... |
| lptioo2cn 46698 | The upper bound of an open... |
| lptioo1cn 46699 | The lower bound of an open... |
| neglimc 46700 | Limit of the negative func... |
| addlimc 46701 | Sum of two limits. (Contr... |
| 0ellimcdiv 46702 | If the numerator converges... |
| clim2cf 46703 | Express the predicate ` F ... |
| limclner 46704 | For a limit point, both fr... |
| sublimc 46705 | Subtraction of two limits.... |
| reclimc 46706 | Limit of the reciprocal of... |
| clim0cf 46707 | Express the predicate ` F ... |
| limclr 46708 | For a limit point, both fr... |
| divlimc 46709 | Limit of the quotient of t... |
| expfac 46710 | Factorial grows faster tha... |
| climconstmpt 46711 | A constant sequence conver... |
| climresmpt 46712 | A function restricted to u... |
| climsubmpt 46713 | Limit of the difference of... |
| climsubc2mpt 46714 | Limit of the difference of... |
| climsubc1mpt 46715 | Limit of the difference of... |
| fnlimfv 46716 | The value of the limit fun... |
| climreclf 46717 | The limit of a convergent ... |
| climeldmeq 46718 | Two functions that are eve... |
| climf2 46719 | Express the predicate: Th... |
| fnlimcnv 46720 | The sequence of function v... |
| climeldmeqmpt 46721 | Two functions that are eve... |
| climfveq 46722 | Two functions that are eve... |
| clim2f2 46723 | Express the predicate: Th... |
| climfveqmpt 46724 | Two functions that are eve... |
| climd 46725 | Express the predicate: Th... |
| clim2d 46726 | The limit of complex numbe... |
| fnlimfvre 46727 | The limit function of real... |
| allbutfifvre 46728 | Given a sequence of real-v... |
| climleltrp 46729 | The limit of complex numbe... |
| fnlimfvre2 46730 | The limit function of real... |
| fnlimf 46731 | The limit function of real... |
| fnlimabslt 46732 | A sequence of function val... |
| climfveqf 46733 | Two functions that are eve... |
| climmptf 46734 | Exhibit a function ` G ` w... |
| climfveqmpt3 46735 | Two functions that are eve... |
| climeldmeqf 46736 | Two functions that are eve... |
| climreclmpt 46737 | The limit of B convergent ... |
| limsupref 46738 | If a sequence is bounded, ... |
| limsupbnd1f 46739 | If a sequence is eventuall... |
| climbddf 46740 | A converging sequence of c... |
| climeqf 46741 | Two functions that are eve... |
| climeldmeqmpt3 46742 | Two functions that are eve... |
| limsupcld 46743 | Closure of the superior li... |
| climfv 46744 | The limit of a convergent ... |
| limsupval3 46745 | The superior limit of an i... |
| climfveqmpt2 46746 | Two functions that are eve... |
| limsup0 46747 | The superior limit of the ... |
| climeldmeqmpt2 46748 | Two functions that are eve... |
| limsupresre 46749 | The supremum limit of a fu... |
| climeqmpt 46750 | Two functions that are eve... |
| climfvd 46751 | The limit of a convergent ... |
| limsuplesup 46752 | An upper bound for the sup... |
| limsupresico 46753 | The superior limit doesn't... |
| limsuppnfdlem 46754 | If the restriction of a fu... |
| limsuppnfd 46755 | If the restriction of a fu... |
| limsupresuz 46756 | If the real part of the do... |
| limsupub 46757 | If the limsup is not ` +oo... |
| limsupres 46758 | The superior limit of a re... |
| climinf2lem 46759 | A convergent, nonincreasin... |
| climinf2 46760 | A convergent, nonincreasin... |
| limsupvaluz 46761 | The superior limit, when t... |
| limsupresuz2 46762 | If the domain of a functio... |
| limsuppnflem 46763 | If the restriction of a fu... |
| limsuppnf 46764 | If the restriction of a fu... |
| limsupubuzlem 46765 | If the limsup is not ` +oo... |
| limsupubuz 46766 | For a real-valued function... |
| climinf2mpt 46767 | A bounded below, monotonic... |
| climinfmpt 46768 | A bounded below, monotonic... |
| climinf3 46769 | A convergent, nonincreasin... |
| limsupvaluzmpt 46770 | The superior limit, when t... |
| limsupequzmpt2 46771 | Two functions that are eve... |
| limsupubuzmpt 46772 | If the limsup is not ` +oo... |
| limsupmnflem 46773 | The superior limit of a fu... |
| limsupmnf 46774 | The superior limit of a fu... |
| limsupequzlem 46775 | Two functions that are eve... |
| limsupequz 46776 | Two functions that are eve... |
| limsupre2lem 46777 | Given a function on the ex... |
| limsupre2 46778 | Given a function on the ex... |
| limsupmnfuzlem 46779 | The superior limit of a fu... |
| limsupmnfuz 46780 | The superior limit of a fu... |
| limsupequzmptlem 46781 | Two functions that are eve... |
| limsupequzmpt 46782 | Two functions that are eve... |
| limsupre2mpt 46783 | Given a function on the ex... |
| limsupequzmptf 46784 | Two functions that are eve... |
| limsupre3lem 46785 | Given a function on the ex... |
| limsupre3 46786 | Given a function on the ex... |
| limsupre3mpt 46787 | Given a function on the ex... |
| limsupre3uzlem 46788 | Given a function on the ex... |
| limsupre3uz 46789 | Given a function on the ex... |
| limsupreuz 46790 | Given a function on the re... |
| limsupvaluz2 46791 | The superior limit, when t... |
| limsupreuzmpt 46792 | Given a function on the re... |
| supcnvlimsup 46793 | If a function on a set of ... |
| supcnvlimsupmpt 46794 | If a function on a set of ... |
| 0cnv 46795 | If ` (/) ` is a complex nu... |
| climuzlem 46796 | Express the predicate: Th... |
| climuz 46797 | Express the predicate: Th... |
| lmbr3v 46798 | Express the binary relatio... |
| climisp 46799 | If a sequence converges to... |
| lmbr3 46800 | Express the binary relatio... |
| climrescn 46801 | A sequence converging w.r.... |
| climxrrelem 46802 | If a sequence ranging over... |
| climxrre 46803 | If a sequence ranging over... |
| limsuplt2 46806 | The defining property of t... |
| liminfgord 46807 | Ordering property of the i... |
| limsupvald 46808 | The superior limit of a se... |
| limsupresicompt 46809 | The superior limit doesn't... |
| limsupcli 46810 | Closure of the superior li... |
| liminfgf 46811 | Closure of the inferior li... |
| liminfval 46812 | The inferior limit of a se... |
| climlimsup 46813 | A sequence of real numbers... |
| limsupge 46814 | The defining property of t... |
| liminfgval 46815 | Value of the inferior limi... |
| liminfcl 46816 | Closure of the inferior li... |
| liminfvald 46817 | The inferior limit of a se... |
| liminfval5 46818 | The inferior limit of an i... |
| limsupresxr 46819 | The superior limit of a fu... |
| liminfresxr 46820 | The inferior limit of a fu... |
| liminfval2 46821 | The superior limit, relati... |
| climlimsupcex 46822 | Counterexample for ~ climl... |
| liminfcld 46823 | Closure of the inferior li... |
| liminfresico 46824 | The inferior limit doesn't... |
| limsup10exlem 46825 | The range of the given fun... |
| limsup10ex 46826 | The superior limit of a fu... |
| liminf10ex 46827 | The inferior limit of a fu... |
| liminflelimsuplem 46828 | The superior limit is grea... |
| liminflelimsup 46829 | The superior limit is grea... |
| limsupgtlem 46830 | For any positive real, the... |
| limsupgt 46831 | Given a sequence of real n... |
| liminfresre 46832 | The inferior limit of a fu... |
| liminfresicompt 46833 | The inferior limit doesn't... |
| liminfltlimsupex 46834 | An example where the ` lim... |
| liminfgelimsup 46835 | The inferior limit is grea... |
| liminfvalxr 46836 | Alternate definition of ` ... |
| liminfresuz 46837 | If the real part of the do... |
| liminflelimsupuz 46838 | The superior limit is grea... |
| liminfvalxrmpt 46839 | Alternate definition of ` ... |
| liminfresuz2 46840 | If the domain of a functio... |
| liminfgelimsupuz 46841 | The inferior limit is grea... |
| liminfval4 46842 | Alternate definition of ` ... |
| liminfval3 46843 | Alternate definition of ` ... |
| liminfequzmpt2 46844 | Two functions that are eve... |
| liminfvaluz 46845 | Alternate definition of ` ... |
| liminf0 46846 | The inferior limit of the ... |
| limsupval4 46847 | Alternate definition of ` ... |
| liminfvaluz2 46848 | Alternate definition of ` ... |
| liminfvaluz3 46849 | Alternate definition of ` ... |
| liminflelimsupcex 46850 | A counterexample for ~ lim... |
| limsupvaluz3 46851 | Alternate definition of ` ... |
| liminfvaluz4 46852 | Alternate definition of ` ... |
| limsupvaluz4 46853 | Alternate definition of ` ... |
| climliminflimsupd 46854 | If a sequence of real numb... |
| liminfreuzlem 46855 | Given a function on the re... |
| liminfreuz 46856 | Given a function on the re... |
| liminfltlem 46857 | Given a sequence of real n... |
| liminflt 46858 | Given a sequence of real n... |
| climliminf 46859 | A sequence of real numbers... |
| liminflimsupclim 46860 | A sequence of real numbers... |
| climliminflimsup 46861 | A sequence of real numbers... |
| climliminflimsup2 46862 | A sequence of real numbers... |
| climliminflimsup3 46863 | A sequence of real numbers... |
| climliminflimsup4 46864 | A sequence of real numbers... |
| limsupub2 46865 | A extended real valued fun... |
| limsupubuz2 46866 | A sequence with values in ... |
| xlimpnfxnegmnf 46867 | A sequence converges to ` ... |
| liminflbuz2 46868 | A sequence with values in ... |
| liminfpnfuz 46869 | The inferior limit of a fu... |
| liminflimsupxrre 46870 | A sequence with values in ... |
| xlimrel 46873 | The limit on extended real... |
| xlimres 46874 | A function converges iff i... |
| xlimcl 46875 | The limit of a sequence of... |
| rexlimddv2 46876 | Restricted existential eli... |
| xlimclim 46877 | Given a sequence of reals,... |
| xlimconst 46878 | A constant sequence conver... |
| climxlim 46879 | A converging sequence in t... |
| xlimbr 46880 | Express the binary relatio... |
| fuzxrpmcn 46881 | A function mapping from an... |
| cnrefiisplem 46882 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 46883 | A non-real, complex number... |
| xlimxrre 46884 | If a sequence ranging over... |
| xlimmnfvlem1 46885 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 46886 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 46887 | A function converges to mi... |
| xlimconst2 46888 | A sequence that eventually... |
| xlimpnfvlem1 46889 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 46890 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 46891 | A function converges to pl... |
| xlimclim2lem 46892 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 46893 | Given a sequence of extend... |
| xlimmnf 46894 | A function converges to mi... |
| xlimpnf 46895 | A function converges to pl... |
| xlimmnfmpt 46896 | A function converges to pl... |
| xlimpnfmpt 46897 | A function converges to pl... |
| climxlim2lem 46898 | In this lemma for ~ climxl... |
| climxlim2 46899 | A sequence of extended rea... |
| dfxlim2v 46900 | An alternative definition ... |
| dfxlim2 46901 | An alternative definition ... |
| climresd 46902 | A function restricted to u... |
| climresdm 46903 | A real function converges ... |
| dmclimxlim 46904 | A real valued sequence tha... |
| xlimmnflimsup2 46905 | A sequence of extended rea... |
| xlimuni 46906 | An infinite sequence conve... |
| xlimclimdm 46907 | A sequence of extended rea... |
| xlimfun 46908 | The convergence relation o... |
| xlimmnflimsup 46909 | If a sequence of extended ... |
| xlimdm 46910 | Two ways to express that a... |
| xlimpnfxnegmnf2 46911 | A sequence converges to ` ... |
| xlimresdm 46912 | A function converges in th... |
| xlimpnfliminf 46913 | If a sequence of extended ... |
| xlimpnfliminf2 46914 | A sequence of extended rea... |
| xlimliminflimsup 46915 | A sequence of extended rea... |
| xlimlimsupleliminf 46916 | A sequence of extended rea... |
| coseq0 46917 | A complex number whose cos... |
| sinmulcos 46918 | Multiplication formula for... |
| coskpi2 46919 | The cosine of an integer m... |
| cosnegpi 46920 | The cosine of negative ` _... |
| sinaover2ne0 46921 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 46922 | The cosine of an integer m... |
| mulcncff 46923 | The multiplication of two ... |
| cncfmptssg 46924 | A continuous complex funct... |
| constcncfg 46925 | A constant function is a c... |
| idcncfg 46926 | The identity function is a... |
| cncfshift 46927 | A periodic continuous func... |
| resincncf 46928 | ` sin ` restricted to real... |
| addccncf2 46929 | Adding a constant is a con... |
| 0cnf 46930 | The empty set is a continu... |
| fsumcncf 46931 | The finite sum of continuo... |
| cncfperiod 46932 | A periodic continuous func... |
| subcncff 46933 | The subtraction of two con... |
| negcncfg 46934 | The opposite of a continuo... |
| cnfdmsn 46935 | A function with a singleto... |
| cncfcompt 46936 | Composition of continuous ... |
| addcncff 46937 | The sum of two continuous ... |
| ioccncflimc 46938 | Limit at the upper bound o... |
| cncfuni 46939 | A complex function on a su... |
| icccncfext 46940 | A continuous function on a... |
| cncficcgt0 46941 | A the absolute value of a ... |
| icocncflimc 46942 | Limit at the lower bound, ... |
| cncfdmsn 46943 | A complex function with a ... |
| divcncff 46944 | The quotient of two contin... |
| cncfshiftioo 46945 | A periodic continuous func... |
| cncfiooicclem1 46946 | A continuous function ` F ... |
| cncfiooicc 46947 | A continuous function ` F ... |
| cncfiooiccre 46948 | A continuous function ` F ... |
| cncfioobdlem 46949 | ` G ` actually extends ` F... |
| cncfioobd 46950 | A continuous function ` F ... |
| jumpncnp 46951 | Jump discontinuity or disc... |
| cxpcncf2 46952 | The complex power function... |
| fprodcncf 46953 | The finite product of cont... |
| add1cncf 46954 | Addition to a constant is ... |
| add2cncf 46955 | Addition to a constant is ... |
| sub1cncfd 46956 | Subtracting a constant is ... |
| sub2cncfd 46957 | Subtraction from a constan... |
| fprodsub2cncf 46958 | ` F ` is continuous. (Con... |
| fprodadd2cncf 46959 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 46960 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 46961 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 46962 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 46963 | The sequence ` S ` of fini... |
| dvsinexp 46964 | The derivative of sin^N . ... |
| dvcosre 46965 | The real derivative of the... |
| dvsinax 46966 | Derivative exercise: the d... |
| dvsubf 46967 | The subtraction rule for e... |
| dvmptconst 46968 | Function-builder for deriv... |
| dvcnre 46969 | From complex differentiati... |
| dvmptidg 46970 | Function-builder for deriv... |
| dvresntr 46971 | Function-builder for deriv... |
| fperdvper 46972 | The derivative of a period... |
| dvasinbx 46973 | Derivative exercise: the d... |
| dvresioo 46974 | Restriction of a derivativ... |
| dvdivf 46975 | The quotient rule for ever... |
| dvdivbd 46976 | A sufficient condition for... |
| dvsubcncf 46977 | A sufficient condition for... |
| dvmulcncf 46978 | A sufficient condition for... |
| dvcosax 46979 | Derivative exercise: the d... |
| dvdivcncf 46980 | A sufficient condition for... |
| dvbdfbdioolem1 46981 | Given a function with boun... |
| dvbdfbdioolem2 46982 | A function on an open inte... |
| dvbdfbdioo 46983 | A function on an open inte... |
| ioodvbdlimc1lem1 46984 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 46985 | Limit at the lower bound o... |
| ioodvbdlimc1 46986 | A real function with bound... |
| ioodvbdlimc2lem 46987 | Limit at the upper bound o... |
| ioodvbdlimc2 46988 | A real function with bound... |
| dvdmsscn 46989 | ` X ` is a subset of ` CC ... |
| dvmptmulf 46990 | Function-builder for deriv... |
| dvnmptdivc 46991 | Function-builder for itera... |
| dvdsn1add 46992 | If ` K ` divides ` N ` but... |
| dvxpaek 46993 | Derivative of the polynomi... |
| dvnmptconst 46994 | The ` N ` -th derivative o... |
| dvnxpaek 46995 | The ` n ` -th derivative o... |
| dvnmul 46996 | Function-builder for the `... |
| dvmptfprodlem 46997 | Induction step for ~ dvmpt... |
| dvmptfprod 46998 | Function-builder for deriv... |
| dvnprodlem1 46999 | ` D ` is bijective. (Cont... |
| dvnprodlem2 47000 | Induction step for ~ dvnpr... |
| dvnprodlem3 47001 | The multinomial formula fo... |
| dvnprod 47002 | The multinomial formula fo... |
| itgsin0pilem1 47003 | Calculation of the integra... |
| ibliccsinexp 47004 | sin^n on a closed interval... |
| itgsin0pi 47005 | Calculation of the integra... |
| iblioosinexp 47006 | sin^n on an open integral ... |
| itgsinexplem1 47007 | Integration by parts is ap... |
| itgsinexp 47008 | A recursive formula for th... |
| iblconstmpt 47009 | A constant function is int... |
| itgeq1d 47010 | Equality theorem for an in... |
| mbfres2cn 47011 | Measurability of a piecewi... |
| vol0 47012 | The measure of the empty s... |
| ditgeqiooicc 47013 | A function ` F ` on an ope... |
| volge0 47014 | The volume of a set is alw... |
| cnbdibl 47015 | A continuous bounded funct... |
| snmbl 47016 | A singleton is measurable.... |
| ditgeq3d 47017 | Equality theorem for the d... |
| iblempty 47018 | The empty function is inte... |
| iblsplit 47019 | The union of two integrabl... |
| volsn 47020 | A singleton has 0 Lebesgue... |
| itgvol0 47021 | If the domani is negligibl... |
| itgcoscmulx 47022 | Exercise: the integral of ... |
| iblsplitf 47023 | A version of ~ iblsplit us... |
| ibliooicc 47024 | If a function is integrabl... |
| volioc 47025 | The measure of a left-open... |
| iblspltprt 47026 | If a function is integrabl... |
| itgsincmulx 47027 | Exercise: the integral of ... |
| itgsubsticclem 47028 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 47029 | Integration by u-substitut... |
| itgioocnicc 47030 | The integral of a piecewis... |
| iblcncfioo 47031 | A continuous function ` F ... |
| itgspltprt 47032 | The ` S. ` integral splits... |
| itgiccshift 47033 | The integral of a function... |
| itgperiod 47034 | The integral of a periodic... |
| itgsbtaddcnst 47035 | Integral substitution, add... |
| volico 47036 | The measure of left-closed... |
| sublevolico 47037 | The Lebesgue measure of a ... |
| dmvolss 47038 | Lebesgue measurable sets a... |
| ismbl3 47039 | The predicate " ` A ` is L... |
| volioof 47040 | The function that assigns ... |
| ovolsplit 47041 | The Lebesgue outer measure... |
| fvvolioof 47042 | The function value of the ... |
| volioore 47043 | The measure of an open int... |
| fvvolicof 47044 | The function value of the ... |
| voliooico 47045 | An open interval and a lef... |
| ismbl4 47046 | The predicate " ` A ` is L... |
| volioofmpt 47047 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 47048 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 47049 | The Lebesgue measure of op... |
| volicofmpt 47050 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 47051 | The Lebesgue measure of a ... |
| voliccico 47052 | A closed interval and a le... |
| mbfdmssre 47053 | The domain of a measurable... |
| stoweidlem1 47054 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 47055 | lemma for ~ stoweid : here... |
| stoweidlem3 47056 | Lemma for ~ stoweid : if `... |
| stoweidlem4 47057 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 47058 | There exists a δ as ... |
| stoweidlem6 47059 | Lemma for ~ stoweid : two ... |
| stoweidlem7 47060 | This lemma is used to prov... |
| stoweidlem8 47061 | Lemma for ~ stoweid : two ... |
| stoweidlem9 47062 | Lemma for ~ stoweid : here... |
| stoweidlem10 47063 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 47064 | This lemma is used to prov... |
| stoweidlem12 47065 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 47066 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 47067 | There exists a ` k ` as in... |
| stoweidlem15 47068 | This lemma is used to prov... |
| stoweidlem16 47069 | Lemma for ~ stoweid . The... |
| stoweidlem17 47070 | This lemma proves that the... |
| stoweidlem18 47071 | This theorem proves Lemma ... |
| stoweidlem19 47072 | If a set of real functions... |
| stoweidlem20 47073 | If a set A of real functio... |
| stoweidlem21 47074 | Once the Stone Weierstrass... |
| stoweidlem22 47075 | If a set of real functions... |
| stoweidlem23 47076 | This lemma is used to prov... |
| stoweidlem24 47077 | This lemma proves that for... |
| stoweidlem25 47078 | This lemma proves that for... |
| stoweidlem26 47079 | This lemma is used to prov... |
| stoweidlem27 47080 | This lemma is used to prov... |
| stoweidlem28 47081 | There exists a δ as ... |
| stoweidlem29 47082 | When the hypothesis for th... |
| stoweidlem30 47083 | This lemma is used to prov... |
| stoweidlem31 47084 | This lemma is used to prov... |
| stoweidlem32 47085 | If a set A of real functio... |
| stoweidlem33 47086 | If a set of real functions... |
| stoweidlem34 47087 | This lemma proves that for... |
| stoweidlem35 47088 | This lemma is used to prov... |
| stoweidlem36 47089 | This lemma is used to prov... |
| stoweidlem37 47090 | This lemma is used to prov... |
| stoweidlem38 47091 | This lemma is used to prov... |
| stoweidlem39 47092 | This lemma is used to prov... |
| stoweidlem40 47093 | This lemma proves that q_n... |
| stoweidlem41 47094 | This lemma is used to prov... |
| stoweidlem42 47095 | This lemma is used to prov... |
| stoweidlem43 47096 | This lemma is used to prov... |
| stoweidlem44 47097 | This lemma is used to prov... |
| stoweidlem45 47098 | This lemma proves that, gi... |
| stoweidlem46 47099 | This lemma proves that set... |
| stoweidlem47 47100 | Subtracting a constant fro... |
| stoweidlem48 47101 | This lemma is used to prov... |
| stoweidlem49 47102 | There exists a function q_... |
| stoweidlem50 47103 | This lemma proves that set... |
| stoweidlem51 47104 | There exists a function x ... |
| stoweidlem52 47105 | There exists a neighborhoo... |
| stoweidlem53 47106 | This lemma is used to prov... |
| stoweidlem54 47107 | There exists a function ` ... |
| stoweidlem55 47108 | This lemma proves the exis... |
| stoweidlem56 47109 | This theorem proves Lemma ... |
| stoweidlem57 47110 | There exists a function x ... |
| stoweidlem58 47111 | This theorem proves Lemma ... |
| stoweidlem59 47112 | This lemma proves that the... |
| stoweidlem60 47113 | This lemma proves that the... |
| stoweidlem61 47114 | This lemma proves that the... |
| stoweidlem62 47115 | This theorem proves the St... |
| stoweid 47116 | This theorem proves the St... |
| stowei 47117 | This theorem proves the St... |
| wallispilem1 47118 | ` I ` is monotone: increas... |
| wallispilem2 47119 | A first set of properties ... |
| wallispilem3 47120 | I maps to real values. (C... |
| wallispilem4 47121 | ` F ` maps to explicit exp... |
| wallispilem5 47122 | The sequence ` H ` converg... |
| wallispi 47123 | Wallis' formula for π :... |
| wallispi2lem1 47124 | An intermediate step betwe... |
| wallispi2lem2 47125 | Two expressions are proven... |
| wallispi2 47126 | An alternative version of ... |
| stirlinglem1 47127 | A simple limit of fraction... |
| stirlinglem2 47128 | ` A ` maps to positive rea... |
| stirlinglem3 47129 | Long but simple algebraic ... |
| stirlinglem4 47130 | Algebraic manipulation of ... |
| stirlinglem5 47131 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 47132 | A series that converges to... |
| stirlinglem7 47133 | Algebraic manipulation of ... |
| stirlinglem8 47134 | If ` A ` converges to ` C ... |
| stirlinglem9 47135 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 47136 | A bound for any B(N)-B(N +... |
| stirlinglem11 47137 | ` B ` is decreasing. (Con... |
| stirlinglem12 47138 | The sequence ` B ` is boun... |
| stirlinglem13 47139 | ` B ` is decreasing and ha... |
| stirlinglem14 47140 | The sequence ` A ` converg... |
| stirlinglem15 47141 | The Stirling's formula is ... |
| stirling 47142 | Stirling's approximation f... |
| stirlingr 47143 | Stirling's approximation f... |
| dirkerval 47144 | The N_th Dirichlet kernel.... |
| dirker2re 47145 | The Dirichlet kernel value... |
| dirkerdenne0 47146 | The Dirichlet kernel denom... |
| dirkerval2 47147 | The N_th Dirichlet kernel ... |
| dirkerre 47148 | The Dirichlet kernel at an... |
| dirkerper 47149 | the Dirichlet kernel has p... |
| dirkerf 47150 | For any natural number ` N... |
| dirkertrigeqlem1 47151 | Sum of an even number of a... |
| dirkertrigeqlem2 47152 | Trigonometric equality lem... |
| dirkertrigeqlem3 47153 | Trigonometric equality lem... |
| dirkertrigeq 47154 | Trigonometric equality for... |
| dirkeritg 47155 | The definite integral of t... |
| dirkercncflem1 47156 | If ` Y ` is a multiple of ... |
| dirkercncflem2 47157 | Lemma used to prove that t... |
| dirkercncflem3 47158 | The Dirichlet kernel is co... |
| dirkercncflem4 47159 | The Dirichlet kernel is co... |
| dirkercncf 47160 | For any natural number ` N... |
| fourierdlem1 47161 | A partition interval is a ... |
| fourierdlem2 47162 | Membership in a partition.... |
| fourierdlem3 47163 | Membership in a partition.... |
| fourierdlem4 47164 | ` E ` is a function that m... |
| fourierdlem5 47165 | ` S ` is a function. (Con... |
| fourierdlem6 47166 | ` X ` is in the periodic p... |
| fourierdlem7 47167 | The difference between the... |
| fourierdlem8 47168 | A partition interval is a ... |
| fourierdlem9 47169 | ` H ` is a complex functio... |
| fourierdlem10 47170 | Condition on the bounds of... |
| fourierdlem11 47171 | If there is a partition, t... |
| fourierdlem12 47172 | A point of a partition is ... |
| fourierdlem13 47173 | Value of ` V ` in terms of... |
| fourierdlem14 47174 | Given the partition ` V ` ... |
| fourierdlem15 47175 | The range of the partition... |
| fourierdlem16 47176 | The coefficients of the fo... |
| fourierdlem17 47177 | The defined ` L ` is actua... |
| fourierdlem18 47178 | The function ` S ` is cont... |
| fourierdlem19 47179 | If two elements of ` D ` h... |
| fourierdlem20 47180 | Every interval in the part... |
| fourierdlem21 47181 | The coefficients of the fo... |
| fourierdlem22 47182 | The coefficients of the fo... |
| fourierdlem23 47183 | If ` F ` is continuous and... |
| fourierdlem24 47184 | A sufficient condition for... |
| fourierdlem25 47185 | If ` C ` is not in the ran... |
| fourierdlem26 47186 | Periodic image of a point ... |
| fourierdlem27 47187 | A partition open interval ... |
| fourierdlem28 47188 | Derivative of ` ( F `` ( X... |
| fourierdlem29 47189 | Explicit function value fo... |
| fourierdlem30 47190 | Sum of three small pieces ... |
| fourierdlem31 47191 | If ` A ` is finite and for... |
| fourierdlem32 47192 | Limit of a continuous func... |
| fourierdlem33 47193 | Limit of a continuous func... |
| fourierdlem34 47194 | A partition is one to one.... |
| fourierdlem35 47195 | There is a single point in... |
| fourierdlem36 47196 | ` F ` is an isomorphism. ... |
| fourierdlem37 47197 | ` I ` is a function that m... |
| fourierdlem38 47198 | The function ` F ` is cont... |
| fourierdlem39 47199 | Integration by parts of ... |
| fourierdlem40 47200 | ` H ` is a continuous func... |
| fourierdlem41 47201 | Lemma used to prove that e... |
| fourierdlem42 47202 | The set of points in a mov... |
| fourierdlem43 47203 | ` K ` is a real function. ... |
| fourierdlem44 47204 | A condition for having ` (... |
| fourierdlem46 47205 | The function ` F ` has a l... |
| fourierdlem47 47206 | For ` r ` large enough, th... |
| fourierdlem48 47207 | The given periodic functio... |
| fourierdlem49 47208 | The given periodic functio... |
| fourierdlem50 47209 | Continuity of ` O ` and it... |
| fourierdlem51 47210 | ` X ` is in the periodic p... |
| fourierdlem52 47211 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 47212 | The limit of ` F ( s ) ` a... |
| fourierdlem54 47213 | Given a partition ` Q ` an... |
| fourierdlem55 47214 | ` U ` is a real function. ... |
| fourierdlem56 47215 | Derivative of the ` K ` fu... |
| fourierdlem57 47216 | The derivative of ` O ` . ... |
| fourierdlem58 47217 | The derivative of ` K ` is... |
| fourierdlem59 47218 | The derivative of ` H ` is... |
| fourierdlem60 47219 | Given a differentiable fun... |
| fourierdlem61 47220 | Given a differentiable fun... |
| fourierdlem62 47221 | The function ` K ` is cont... |
| fourierdlem63 47222 | The upper bound of interva... |
| fourierdlem64 47223 | The partition ` V ` is fin... |
| fourierdlem65 47224 | The distance of two adjace... |
| fourierdlem66 47225 | Value of the ` G ` functio... |
| fourierdlem67 47226 | ` G ` is a function. (Con... |
| fourierdlem68 47227 | The derivative of ` O ` is... |
| fourierdlem69 47228 | A piecewise continuous fun... |
| fourierdlem70 47229 | A piecewise continuous fun... |
| fourierdlem71 47230 | A periodic piecewise conti... |
| fourierdlem72 47231 | The derivative of ` O ` is... |
| fourierdlem73 47232 | A version of the Riemann L... |
| fourierdlem74 47233 | Given a piecewise smooth f... |
| fourierdlem75 47234 | Given a piecewise smooth f... |
| fourierdlem76 47235 | Continuity of ` O ` and it... |
| fourierdlem77 47236 | If ` H ` is bounded, then ... |
| fourierdlem78 47237 | ` G ` is continuous when r... |
| fourierdlem79 47238 | ` E ` projects every inter... |
| fourierdlem80 47239 | The derivative of ` O ` is... |
| fourierdlem81 47240 | The integral of a piecewis... |
| fourierdlem82 47241 | Integral by substitution, ... |
| fourierdlem83 47242 | The fourier partial sum fo... |
| fourierdlem84 47243 | If ` F ` is piecewise cont... |
| fourierdlem85 47244 | Limit of the function ` G ... |
| fourierdlem86 47245 | Continuity of ` O ` and it... |
| fourierdlem87 47246 | The integral of ` G ` goes... |
| fourierdlem88 47247 | Given a piecewise continuo... |
| fourierdlem89 47248 | Given a piecewise continuo... |
| fourierdlem90 47249 | Given a piecewise continuo... |
| fourierdlem91 47250 | Given a piecewise continuo... |
| fourierdlem92 47251 | The integral of a piecewis... |
| fourierdlem93 47252 | Integral by substitution (... |
| fourierdlem94 47253 | For a piecewise smooth fun... |
| fourierdlem95 47254 | Algebraic manipulation of ... |
| fourierdlem96 47255 | limit for ` F ` at the low... |
| fourierdlem97 47256 | ` F ` is continuous on the... |
| fourierdlem98 47257 | ` F ` is continuous on the... |
| fourierdlem99 47258 | limit for ` F ` at the upp... |
| fourierdlem100 47259 | A piecewise continuous fun... |
| fourierdlem101 47260 | Integral by substitution f... |
| fourierdlem102 47261 | For a piecewise smooth fun... |
| fourierdlem103 47262 | The half lower part of the... |
| fourierdlem104 47263 | The half upper part of the... |
| fourierdlem105 47264 | A piecewise continuous fun... |
| fourierdlem106 47265 | For a piecewise smooth fun... |
| fourierdlem107 47266 | The integral of a piecewis... |
| fourierdlem108 47267 | The integral of a piecewis... |
| fourierdlem109 47268 | The integral of a piecewis... |
| fourierdlem110 47269 | The integral of a piecewis... |
| fourierdlem111 47270 | The fourier partial sum fo... |
| fourierdlem112 47271 | Here abbreviations (local ... |
| fourierdlem113 47272 | Fourier series convergence... |
| fourierdlem114 47273 | Fourier series convergence... |
| fourierdlem115 47274 | Fourier serier convergence... |
| fourierd 47275 | Fourier series convergence... |
| fourierclimd 47276 | Fourier series convergence... |
| fourierclim 47277 | Fourier series convergence... |
| fourier 47278 | Fourier series convergence... |
| fouriercnp 47279 | If ` F ` is continuous at ... |
| fourier2 47280 | Fourier series convergence... |
| sqwvfoura 47281 | Fourier coefficients for t... |
| sqwvfourb 47282 | Fourier series ` B ` coeff... |
| fourierswlem 47283 | The Fourier series for the... |
| fouriersw 47284 | Fourier series convergence... |
| fouriercn 47285 | If the derivative of ` F `... |
| elaa2lem 47286 | Elementhood in the set of ... |
| elaa2 47287 | Elementhood in the set of ... |
| etransclem1 47288 | ` H ` is a function. (Con... |
| etransclem2 47289 | Derivative of ` G ` . (Co... |
| etransclem3 47290 | The given ` if ` term is a... |
| etransclem4 47291 | ` F ` expressed as a finit... |
| etransclem5 47292 | A change of bound variable... |
| etransclem6 47293 | A change of bound variable... |
| etransclem7 47294 | The given product is an in... |
| etransclem8 47295 | ` F ` is a function. (Con... |
| etransclem9 47296 | If ` K ` divides ` N ` but... |
| etransclem10 47297 | The given ` if ` term is a... |
| etransclem11 47298 | A change of bound variable... |
| etransclem12 47299 | ` C ` applied to ` N ` . ... |
| etransclem13 47300 | ` F ` applied to ` Y ` . ... |
| etransclem14 47301 | Value of the term ` T ` , ... |
| etransclem15 47302 | Value of the term ` T ` , ... |
| etransclem16 47303 | Every element in the range... |
| etransclem17 47304 | The ` N ` -th derivative o... |
| etransclem18 47305 | The given function is inte... |
| etransclem19 47306 | The ` N ` -th derivative o... |
| etransclem20 47307 | ` H ` is smooth. (Contrib... |
| etransclem21 47308 | The ` N ` -th derivative o... |
| etransclem22 47309 | The ` N ` -th derivative o... |
| etransclem23 47310 | This is the claim proof in... |
| etransclem24 47311 | ` P ` divides the I -th de... |
| etransclem25 47312 | ` P ` factorial divides th... |
| etransclem26 47313 | Every term in the sum of t... |
| etransclem27 47314 | The ` N ` -th derivative o... |
| etransclem28 47315 | ` ( P - 1 ) ` factorial di... |
| etransclem29 47316 | The ` N ` -th derivative o... |
| etransclem30 47317 | The ` N ` -th derivative o... |
| etransclem31 47318 | The ` N ` -th derivative o... |
| etransclem32 47319 | This is the proof for the ... |
| etransclem33 47320 | ` F ` is smooth. (Contrib... |
| etransclem34 47321 | The ` N ` -th derivative o... |
| etransclem35 47322 | ` P ` does not divide the ... |
| etransclem36 47323 | The ` N ` -th derivative o... |
| etransclem37 47324 | ` ( P - 1 ) ` factorial di... |
| etransclem38 47325 | ` P ` divides the I -th de... |
| etransclem39 47326 | ` G ` is a function. (Con... |
| etransclem40 47327 | The ` N ` -th derivative o... |
| etransclem41 47328 | ` P ` does not divide the ... |
| etransclem42 47329 | The ` N ` -th derivative o... |
| etransclem43 47330 | ` G ` is a continuous func... |
| etransclem44 47331 | The given finite sum is no... |
| etransclem45 47332 | ` K ` is an integer. (Con... |
| etransclem46 47333 | This is the proof for equa... |
| etransclem47 47334 | ` _e ` is transcendental. ... |
| etransclem48 47335 | ` _e ` is transcendental. ... |
| etransc 47336 | ` _e ` is transcendental. ... |
| rrxtopn 47337 | The topology of the genera... |
| rrxngp 47338 | Generalized Euclidean real... |
| rrxtps 47339 | Generalized Euclidean real... |
| rrxtopnfi 47340 | The topology of the n-dime... |
| rrxtopon 47341 | The topology on generalize... |
| rrxtop 47342 | The topology on generalize... |
| rrndistlt 47343 | Given two points in the sp... |
| rrxtoponfi 47344 | The topology on n-dimensio... |
| rrxunitopnfi 47345 | The base set of the standa... |
| rrxtopn0 47346 | The topology of the zero-d... |
| qndenserrnbllem 47347 | n-dimensional rational num... |
| qndenserrnbl 47348 | n-dimensional rational num... |
| rrxtopn0b 47349 | The topology of the zero-d... |
| qndenserrnopnlem 47350 | n-dimensional rational num... |
| qndenserrnopn 47351 | n-dimensional rational num... |
| qndenserrn 47352 | n-dimensional rational num... |
| rrxsnicc 47353 | A multidimensional singlet... |
| rrnprjdstle 47354 | The distance between two p... |
| rrndsmet 47355 | ` D ` is a metric for the ... |
| rrndsxmet 47356 | ` D ` is an extended metri... |
| ioorrnopnlem 47357 | The a point in an indexed ... |
| ioorrnopn 47358 | The indexed product of ope... |
| ioorrnopnxrlem 47359 | Given a point ` F ` that b... |
| ioorrnopnxr 47360 | The indexed product of ope... |
| issal 47367 | Express the predicate " ` ... |
| pwsal 47368 | The power set of a given s... |
| salunicl 47369 | SAlg sigma-algebra is clos... |
| saluncl 47370 | The union of two sets in a... |
| prsal 47371 | The pair of the empty set ... |
| saldifcl 47372 | The complement of an eleme... |
| 0sal 47373 | The empty set belongs to e... |
| salgenval 47374 | The sigma-algebra generate... |
| saliunclf 47375 | SAlg sigma-algebra is clos... |
| saliuncl 47376 | SAlg sigma-algebra is clos... |
| salincl 47377 | The intersection of two se... |
| saluni 47378 | A set is an element of any... |
| saliinclf 47379 | SAlg sigma-algebra is clos... |
| saliincl 47380 | SAlg sigma-algebra is clos... |
| saldifcl2 47381 | The difference of two elem... |
| intsaluni 47382 | The union of an arbitrary ... |
| intsal 47383 | The arbitrary intersection... |
| salgenn0 47384 | The set used in the defini... |
| salgencl 47385 | ` SalGen ` actually genera... |
| issald 47386 | Sufficient condition to pr... |
| salexct 47387 | An example of nontrivial s... |
| sssalgen 47388 | A set is a subset of the s... |
| salgenss 47389 | The sigma-algebra generate... |
| salgenuni 47390 | The base set of the sigma-... |
| issalgend 47391 | One side of ~ dfsalgen2 . ... |
| salexct2 47392 | An example of a subset tha... |
| unisalgen 47393 | The union of a set belongs... |
| dfsalgen2 47394 | Alternate characterization... |
| salexct3 47395 | An example of a sigma-alge... |
| salgencntex 47396 | This counterexample shows ... |
| salgensscntex 47397 | This counterexample shows ... |
| issalnnd 47398 | Sufficient condition to pr... |
| dmvolsal 47399 | Lebesgue measurable sets f... |
| saldifcld 47400 | The complement of an eleme... |
| saluncld 47401 | The union of two sets in a... |
| salgencld 47402 | ` SalGen ` actually genera... |
| 0sald 47403 | The empty set belongs to e... |
| iooborel 47404 | An open interval is a Bore... |
| salincld 47405 | The intersection of two se... |
| salunid 47406 | A set is an element of any... |
| unisalgen2 47407 | The union of a set belongs... |
| bor1sal 47408 | The Borel sigma-algebra on... |
| iocborel 47409 | A left-open, right-closed ... |
| subsaliuncllem 47410 | A subspace sigma-algebra i... |
| subsaliuncl 47411 | A subspace sigma-algebra i... |
| subsalsal 47412 | A subspace sigma-algebra i... |
| subsaluni 47413 | A set belongs to the subsp... |
| salrestss 47414 | A sigma-algebra restricted... |
| sge0rnre 47417 | When ` sum^ ` is applied t... |
| fge0icoicc 47418 | If ` F ` maps to nonnegati... |
| sge0val 47419 | The value of the sum of no... |
| fge0npnf 47420 | If ` F ` maps to nonnegati... |
| sge0rnn0 47421 | The range used in the defi... |
| sge0vald 47422 | The value of the sum of no... |
| fge0iccico 47423 | A range of nonnegative ext... |
| gsumge0cl 47424 | Closure of group sum, for ... |
| sge0reval 47425 | Value of the sum of nonneg... |
| sge0pnfval 47426 | If a term in the sum of no... |
| fge0iccre 47427 | A range of nonnegative ext... |
| sge0z 47428 | Any nonnegative extended s... |
| sge00 47429 | The sum of nonnegative ext... |
| fsumlesge0 47430 | Every finite subsum of non... |
| sge0revalmpt 47431 | Value of the sum of nonneg... |
| sge0sn 47432 | A sum of a nonnegative ext... |
| sge0tsms 47433 | ` sum^ ` applied to a nonn... |
| sge0cl 47434 | The arbitrary sum of nonne... |
| sge0f1o 47435 | Re-index a nonnegative ext... |
| sge0snmpt 47436 | A sum of a nonnegative ext... |
| sge0ge0 47437 | The sum of nonnegative ext... |
| sge0xrcl 47438 | The arbitrary sum of nonne... |
| sge0repnf 47439 | The of nonnegative extende... |
| sge0fsum 47440 | The arbitrary sum of a fin... |
| sge0rern 47441 | If the sum of nonnegative ... |
| sge0supre 47442 | If the arbitrary sum of no... |
| sge0fsummpt 47443 | The arbitrary sum of a fin... |
| sge0sup 47444 | The arbitrary sum of nonne... |
| sge0less 47445 | A shorter sum of nonnegati... |
| sge0rnbnd 47446 | The range used in the defi... |
| sge0pr 47447 | Sum of a pair of nonnegati... |
| sge0gerp 47448 | The arbitrary sum of nonne... |
| sge0pnffigt 47449 | If the sum of nonnegative ... |
| sge0ssre 47450 | If a sum of nonnegative ex... |
| sge0lefi 47451 | A sum of nonnegative exten... |
| sge0lessmpt 47452 | A shorter sum of nonnegati... |
| sge0ltfirp 47453 | If the sum of nonnegative ... |
| sge0prle 47454 | The sum of a pair of nonne... |
| sge0gerpmpt 47455 | The arbitrary sum of nonne... |
| sge0resrnlem 47456 | The sum of nonnegative ext... |
| sge0resrn 47457 | The sum of nonnegative ext... |
| sge0ssrempt 47458 | If a sum of nonnegative ex... |
| sge0resplit 47459 | ` sum^ ` splits into two p... |
| sge0le 47460 | If all of the terms of sum... |
| sge0ltfirpmpt 47461 | If the extended sum of non... |
| sge0split 47462 | Split a sum of nonnegative... |
| sge0lempt 47463 | If all of the terms of sum... |
| sge0splitmpt 47464 | Split a sum of nonnegative... |
| sge0ss 47465 | Change the index set to a ... |
| sge0iunmptlemfi 47466 | Sum of nonnegative extende... |
| sge0p1 47467 | The addition of the next t... |
| sge0iunmptlemre 47468 | Sum of nonnegative extende... |
| sge0fodjrnlem 47469 | Re-index a nonnegative ext... |
| sge0fodjrn 47470 | Re-index a nonnegative ext... |
| sge0iunmpt 47471 | Sum of nonnegative extende... |
| sge0iun 47472 | Sum of nonnegative extende... |
| sge0nemnf 47473 | The generalized sum of non... |
| sge0rpcpnf 47474 | The sum of an infinite num... |
| sge0rernmpt 47475 | If the sum of nonnegative ... |
| sge0lefimpt 47476 | A sum of nonnegative exten... |
| nn0ssge0 47477 | Nonnegative integers are n... |
| sge0clmpt 47478 | The generalized sum of non... |
| sge0ltfirpmpt2 47479 | If the extended sum of non... |
| sge0isum 47480 | If a series of nonnegative... |
| sge0xrclmpt 47481 | The generalized sum of non... |
| sge0xp 47482 | Combine two generalized su... |
| sge0isummpt 47483 | If a series of nonnegative... |
| sge0ad2en 47484 | The value of the infinite ... |
| sge0isummpt2 47485 | If a series of nonnegative... |
| sge0xaddlem1 47486 | The extended addition of t... |
| sge0xaddlem2 47487 | The extended addition of t... |
| sge0xadd 47488 | The extended addition of t... |
| sge0fsummptf 47489 | The generalized sum of a f... |
| sge0snmptf 47490 | A sum of a nonnegative ext... |
| sge0ge0mpt 47491 | The sum of nonnegative ext... |
| sge0repnfmpt 47492 | The of nonnegative extende... |
| sge0pnffigtmpt 47493 | If the generalized sum of ... |
| sge0splitsn 47494 | Separate out a term in a g... |
| sge0pnffsumgt 47495 | If the sum of nonnegative ... |
| sge0gtfsumgt 47496 | If the generalized sum of ... |
| sge0uzfsumgt 47497 | If a real number is smalle... |
| sge0pnfmpt 47498 | If a term in the sum of no... |
| sge0seq 47499 | A series of nonnegative re... |
| sge0reuz 47500 | Value of the generalized s... |
| sge0reuzb 47501 | Value of the generalized s... |
| ismea 47504 | Express the predicate " ` ... |
| dmmeasal 47505 | The domain of a measure is... |
| meaf 47506 | A measure is a function th... |
| mea0 47507 | The measure of the empty s... |
| nnfoctbdjlem 47508 | There exists a mapping fro... |
| nnfoctbdj 47509 | There exists a mapping fro... |
| meadjuni 47510 | The measure of the disjoin... |
| meacl 47511 | The measure of a set is a ... |
| iundjiunlem 47512 | The sets in the sequence `... |
| iundjiun 47513 | Given a sequence ` E ` of ... |
| meaxrcl 47514 | The measure of a set is an... |
| meadjun 47515 | The measure of the union o... |
| meassle 47516 | The measure of a set is gr... |
| meaunle 47517 | The measure of the union o... |
| meadjiunlem 47518 | The sum of nonnegative ext... |
| meadjiun 47519 | The measure of the disjoin... |
| ismeannd 47520 | Sufficient condition to pr... |
| meaiunlelem 47521 | The measure of the union o... |
| meaiunle 47522 | The measure of the union o... |
| psmeasurelem 47523 | ` M ` applied to a disjoin... |
| psmeasure 47524 | Point supported measure, R... |
| voliunsge0lem 47525 | The Lebesgue measure funct... |
| voliunsge0 47526 | The Lebesgue measure funct... |
| volmea 47527 | The Lebesgue measure on th... |
| meage0 47528 | If the measure of a measur... |
| meadjunre 47529 | The measure of the union o... |
| meassre 47530 | If the measure of a measur... |
| meale0eq0 47531 | A measure that is less tha... |
| meadif 47532 | The measure of the differe... |
| meaiuninclem 47533 | Measures are continuous fr... |
| meaiuninc 47534 | Measures are continuous fr... |
| meaiuninc2 47535 | Measures are continuous fr... |
| meaiunincf 47536 | Measures are continuous fr... |
| meaiuninc3v 47537 | Measures are continuous fr... |
| meaiuninc3 47538 | Measures are continuous fr... |
| meaiininclem 47539 | Measures are continuous fr... |
| meaiininc 47540 | Measures are continuous fr... |
| meaiininc2 47541 | Measures are continuous fr... |
| caragenval 47546 | The sigma-algebra generate... |
| isome 47547 | Express the predicate " ` ... |
| caragenel 47548 | Membership in the Caratheo... |
| omef 47549 | An outer measure is a func... |
| ome0 47550 | The outer measure of the e... |
| omessle 47551 | The outer measure of a set... |
| omedm 47552 | The domain of an outer mea... |
| caragensplit 47553 | If ` E ` is in the set gen... |
| caragenelss 47554 | An element of the Caratheo... |
| carageneld 47555 | Membership in the Caratheo... |
| omecl 47556 | The outer measure of a set... |
| caragenss 47557 | The sigma-algebra generate... |
| omeunile 47558 | The outer measure of the u... |
| caragen0 47559 | The empty set belongs to a... |
| omexrcl 47560 | The outer measure of a set... |
| caragenunidm 47561 | The base set of an outer m... |
| caragensspw 47562 | The sigma-algebra generate... |
| omessre 47563 | If the outer measure of a ... |
| caragenuni 47564 | The base set of the sigma-... |
| caragenuncllem 47565 | The Caratheodory's constru... |
| caragenuncl 47566 | The Caratheodory's constru... |
| caragendifcl 47567 | The Caratheodory's constru... |
| caragenfiiuncl 47568 | The Caratheodory's constru... |
| omeunle 47569 | The outer measure of the u... |
| omeiunle 47570 | The outer measure of the i... |
| omelesplit 47571 | The outer measure of a set... |
| omeiunltfirp 47572 | If the outer measure of a ... |
| omeiunlempt 47573 | The outer measure of the i... |
| carageniuncllem1 47574 | The outer measure of ` A i... |
| carageniuncllem2 47575 | The Caratheodory's constru... |
| carageniuncl 47576 | The Caratheodory's constru... |
| caragenunicl 47577 | The Caratheodory's constru... |
| caragensal 47578 | Caratheodory's method gene... |
| caratheodorylem1 47579 | Lemma used to prove that C... |
| caratheodorylem2 47580 | Caratheodory's constructio... |
| caratheodory 47581 | Caratheodory's constructio... |
| 0ome 47582 | The map that assigns 0 to ... |
| isomenndlem 47583 | ` O ` is sub-additive w.r.... |
| isomennd 47584 | Sufficient condition to pr... |
| caragenel2d 47585 | Membership in the Caratheo... |
| omege0 47586 | If the outer measure of a ... |
| omess0 47587 | If the outer measure of a ... |
| caragencmpl 47588 | A measure built with the C... |
| vonval 47593 | Value of the Lebesgue meas... |
| ovnval 47594 | Value of the Lebesgue oute... |
| elhoi 47595 | Membership in a multidimen... |
| icoresmbl 47596 | A closed-below, open-above... |
| hoissre 47597 | The projection of a half-o... |
| ovnval2 47598 | Value of the Lebesgue oute... |
| volicorecl 47599 | The Lebesgue measure of a ... |
| hoiprodcl 47600 | The pre-measure of half-op... |
| hoicvr 47601 | ` I ` is a countable set o... |
| hoissrrn 47602 | A half-open interval is a ... |
| ovn0val 47603 | The Lebesgue outer measure... |
| ovnn0val 47604 | The value of a (multidimen... |
| ovnval2b 47605 | Value of the Lebesgue oute... |
| volicorescl 47606 | The Lebesgue measure of a ... |
| ovnprodcl 47607 | The product used in the de... |
| hoiprodcl2 47608 | The pre-measure of half-op... |
| hoicvrrex 47609 | Any subset of the multidim... |
| ovnsupge0 47610 | The set used in the defini... |
| ovnlecvr 47611 | Given a subset of multidim... |
| ovnpnfelsup 47612 | ` +oo ` is an element of t... |
| ovnsslelem 47613 | The (multidimensional, non... |
| ovnssle 47614 | The (multidimensional) Leb... |
| ovnlerp 47615 | The Lebesgue outer measure... |
| ovnf 47616 | The Lebesgue outer measure... |
| ovncvrrp 47617 | The Lebesgue outer measure... |
| ovn0lem 47618 | For any finite dimension, ... |
| ovn0 47619 | For any finite dimension, ... |
| ovncl 47620 | The Lebesgue outer measure... |
| ovn02 47621 | For the zero-dimensional s... |
| ovnxrcl 47622 | The Lebesgue outer measure... |
| ovnsubaddlem1 47623 | The Lebesgue outer measure... |
| ovnsubaddlem2 47624 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 47625 | ` ( voln* `` X ) ` is suba... |
| ovnome 47626 | ` ( voln* `` X ) ` is an o... |
| vonmea 47627 | ` ( voln `` X ) ` is a mea... |
| volicon0 47628 | The measure of a nonempty ... |
| hsphoif 47629 | ` H ` is a function (that ... |
| hoidmvval 47630 | The dimensional volume of ... |
| hoissrrn2 47631 | A half-open interval is a ... |
| hsphoival 47632 | ` H ` is a function (that ... |
| hoiprodcl3 47633 | The pre-measure of half-op... |
| volicore 47634 | The Lebesgue measure of a ... |
| hoidmvcl 47635 | The dimensional volume of ... |
| hoidmv0val 47636 | The dimensional volume of ... |
| hoidmvn0val 47637 | The dimensional volume of ... |
| hsphoidmvle2 47638 | The dimensional volume of ... |
| hsphoidmvle 47639 | The dimensional volume of ... |
| hoidmvval0 47640 | The dimensional volume of ... |
| hoiprodp1 47641 | The dimensional volume of ... |
| sge0hsphoire 47642 | If the generalized sum of ... |
| hoidmvval0b 47643 | The dimensional volume of ... |
| hoidmv1lelem1 47644 | The supremum of ` U ` belo... |
| hoidmv1lelem2 47645 | This is the contradiction ... |
| hoidmv1lelem3 47646 | The dimensional volume of ... |
| hoidmv1le 47647 | The dimensional volume of ... |
| hoidmvlelem1 47648 | The supremum of ` U ` belo... |
| hoidmvlelem2 47649 | This is the contradiction ... |
| hoidmvlelem3 47650 | This is the contradiction ... |
| hoidmvlelem4 47651 | The dimensional volume of ... |
| hoidmvlelem5 47652 | The dimensional volume of ... |
| hoidmvle 47653 | The dimensional volume of ... |
| ovnhoilem1 47654 | The Lebesgue outer measure... |
| ovnhoilem2 47655 | The Lebesgue outer measure... |
| ovnhoi 47656 | The Lebesgue outer measure... |
| dmovn 47657 | The domain of the Lebesgue... |
| hoicoto2 47658 | The half-open interval exp... |
| dmvon 47659 | Lebesgue measurable n-dime... |
| hoi2toco 47660 | The half-open interval exp... |
| hoidifhspval 47661 | ` D ` is a function that r... |
| hspval 47662 | The value of the half-spac... |
| ovnlecvr2 47663 | Given a subset of multidim... |
| ovncvr2 47664 | ` B ` and ` T ` are the le... |
| dmovnsal 47665 | The domain of the Lebesgue... |
| unidmovn 47666 | Base set of the n-dimensio... |
| rrnmbl 47667 | The set of n-dimensional R... |
| hoidifhspval2 47668 | ` D ` is a function that r... |
| hspdifhsp 47669 | A n-dimensional half-open ... |
| unidmvon 47670 | Base set of the n-dimensio... |
| hoidifhspf 47671 | ` D ` is a function that r... |
| hoidifhspval3 47672 | ` D ` is a function that r... |
| hoidifhspdmvle 47673 | The dimensional volume of ... |
| voncmpl 47674 | The Lebesgue measure is co... |
| hoiqssbllem1 47675 | The center of the n-dimens... |
| hoiqssbllem2 47676 | The center of the n-dimens... |
| hoiqssbllem3 47677 | A n-dimensional ball conta... |
| hoiqssbl 47678 | A n-dimensional ball conta... |
| hspmbllem1 47679 | Any half-space of the n-di... |
| hspmbllem2 47680 | Any half-space of the n-di... |
| hspmbllem3 47681 | Any half-space of the n-di... |
| hspmbl 47682 | Any half-space of the n-di... |
| hoimbllem 47683 | Any n-dimensional half-ope... |
| hoimbl 47684 | Any n-dimensional half-ope... |
| opnvonmbllem1 47685 | The half-open interval exp... |
| opnvonmbllem2 47686 | An open subset of the n-di... |
| opnvonmbl 47687 | An open subset of the n-di... |
| opnssborel 47688 | Open sets of a generalized... |
| borelmbl 47689 | All Borel subsets of the n... |
| volicorege0 47690 | The Lebesgue measure of a ... |
| isvonmbl 47691 | The predicate " ` A ` is m... |
| mblvon 47692 | The n-dimensional Lebesgue... |
| vonmblss 47693 | n-dimensional Lebesgue mea... |
| volico2 47694 | The measure of left-closed... |
| vonmblss2 47695 | n-dimensional Lebesgue mea... |
| ovolval2lem 47696 | The value of the Lebesgue ... |
| ovolval2 47697 | The value of the Lebesgue ... |
| ovnsubadd2lem 47698 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 47699 | ` ( voln* `` X ) ` is suba... |
| ovolval3 47700 | The value of the Lebesgue ... |
| ovnsplit 47701 | The n-dimensional Lebesgue... |
| ovolval4lem1 47702 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 47703 | The value of the Lebesgue ... |
| ovolval4 47704 | The value of the Lebesgue ... |
| ovolval5lem1 47705 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 47706 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 47707 | The value of the Lebesgue ... |
| ovolval5 47708 | The value of the Lebesgue ... |
| ovnovollem1 47709 | if ` F ` is a cover of ` B... |
| ovnovollem2 47710 | if ` I ` is a cover of ` (... |
| ovnovollem3 47711 | The 1-dimensional Lebesgue... |
| ovnovol 47712 | The 1-dimensional Lebesgue... |
| vonvolmbllem 47713 | If a subset ` B ` of real ... |
| vonvolmbl 47714 | A subset of Real numbers i... |
| vonvol 47715 | The 1-dimensional Lebesgue... |
| vonvolmbl2 47716 | A subset ` X ` of the spac... |
| vonvol2 47717 | The 1-dimensional Lebesgue... |
| hoimbl2 47718 | Any n-dimensional half-ope... |
| voncl 47719 | The Lebesgue measure of a ... |
| vonhoi 47720 | The Lebesgue outer measure... |
| vonxrcl 47721 | The Lebesgue measure of a ... |
| ioosshoi 47722 | A n-dimensional open inter... |
| vonn0hoi 47723 | The Lebesgue outer measure... |
| von0val 47724 | The Lebesgue measure (for ... |
| vonhoire 47725 | The Lebesgue measure of a ... |
| iinhoiicclem 47726 | A n-dimensional closed int... |
| iinhoiicc 47727 | A n-dimensional closed int... |
| iunhoiioolem 47728 | A n-dimensional open inter... |
| iunhoiioo 47729 | A n-dimensional open inter... |
| ioovonmbl 47730 | Any n-dimensional open int... |
| iccvonmbllem 47731 | Any n-dimensional closed i... |
| iccvonmbl 47732 | Any n-dimensional closed i... |
| vonioolem1 47733 | The sequence of the measur... |
| vonioolem2 47734 | The n-dimensional Lebesgue... |
| vonioo 47735 | The n-dimensional Lebesgue... |
| vonicclem1 47736 | The sequence of the measur... |
| vonicclem2 47737 | The n-dimensional Lebesgue... |
| vonicc 47738 | The n-dimensional Lebesgue... |
| snvonmbl 47739 | A n-dimensional singleton ... |
| vonn0ioo 47740 | The n-dimensional Lebesgue... |
| vonn0icc 47741 | The n-dimensional Lebesgue... |
| ctvonmbl 47742 | Any n-dimensional countabl... |
| vonn0ioo2 47743 | The n-dimensional Lebesgue... |
| vonsn 47744 | The n-dimensional Lebesgue... |
| vonn0icc2 47745 | The n-dimensional Lebesgue... |
| vonct 47746 | The n-dimensional Lebesgue... |
| vitali2 47747 | There are non-measurable s... |
| pimltmnf2f 47750 | Given a real-valued functi... |
| pimltmnf2 47751 | Given a real-valued functi... |
| preimagelt 47752 | The preimage of a right-op... |
| preimalegt 47753 | The preimage of a left-ope... |
| pimconstlt0 47754 | Given a constant function,... |
| pimconstlt1 47755 | Given a constant function,... |
| pimltpnff 47756 | Given a real-valued functi... |
| pimltpnf 47757 | Given a real-valued functi... |
| pimgtpnf2f 47758 | Given a real-valued functi... |
| pimgtpnf2 47759 | Given a real-valued functi... |
| salpreimagelt 47760 | If all the preimages of le... |
| pimrecltpos 47761 | The preimage of an unbound... |
| salpreimalegt 47762 | If all the preimages of ri... |
| pimiooltgt 47763 | The preimage of an open in... |
| preimaicomnf 47764 | Preimage of an open interv... |
| pimltpnf2f 47765 | Given a real-valued functi... |
| pimltpnf2 47766 | Given a real-valued functi... |
| pimgtmnf2 47767 | Given a real-valued functi... |
| pimdecfgtioc 47768 | Given a nonincreasing func... |
| pimincfltioc 47769 | Given a nondecreasing func... |
| pimdecfgtioo 47770 | Given a nondecreasing func... |
| pimincfltioo 47771 | Given a nondecreasing func... |
| preimaioomnf 47772 | Preimage of an open interv... |
| preimageiingt 47773 | A preimage of a left-close... |
| preimaleiinlt 47774 | A preimage of a left-open,... |
| pimgtmnff 47775 | Given a real-valued functi... |
| pimgtmnf 47776 | Given a real-valued functi... |
| pimrecltneg 47777 | The preimage of an unbound... |
| salpreimagtge 47778 | If all the preimages of le... |
| salpreimaltle 47779 | If all the preimages of ri... |
| issmflem 47780 | The predicate " ` F ` is a... |
| issmf 47781 | The predicate " ` F ` is a... |
| salpreimalelt 47782 | If all the preimages of ri... |
| salpreimagtlt 47783 | If all the preimages of le... |
| smfpreimalt 47784 | Given a function measurabl... |
| smff 47785 | A function measurable w.r.... |
| smfdmss 47786 | The domain of a function m... |
| issmff 47787 | The predicate " ` F ` is a... |
| issmfd 47788 | A sufficient condition for... |
| smfpreimaltf 47789 | Given a function measurabl... |
| issmfdf 47790 | A sufficient condition for... |
| sssmf 47791 | The restriction of a sigma... |
| mbfresmf 47792 | A real-valued measurable f... |
| cnfsmf 47793 | A continuous function is m... |
| incsmflem 47794 | A nondecreasing function i... |
| incsmf 47795 | A real-valued, nondecreasi... |
| smfsssmf 47796 | If a function is measurabl... |
| issmflelem 47797 | The predicate " ` F ` is a... |
| issmfle 47798 | The predicate " ` F ` is a... |
| smfpimltmpt 47799 | Given a function measurabl... |
| smfpimltxr 47800 | Given a function measurabl... |
| issmfdmpt 47801 | A sufficient condition for... |
| smfconst 47802 | Given a sigma-algebra over... |
| sssmfmpt 47803 | The restriction of a sigma... |
| cnfrrnsmf 47804 | A function, continuous fro... |
| smfid 47805 | The identity function is B... |
| bormflebmf 47806 | A Borel measurable functio... |
| smfpreimale 47807 | Given a function measurabl... |
| issmfgtlem 47808 | The predicate " ` F ` is a... |
| issmfgt 47809 | The predicate " ` F ` is a... |
| issmfled 47810 | A sufficient condition for... |
| smfpimltxrmptf 47811 | Given a function measurabl... |
| smfpimltxrmpt 47812 | Given a function measurabl... |
| smfmbfcex 47813 | A constant function, with ... |
| issmfgtd 47814 | A sufficient condition for... |
| smfpreimagt 47815 | Given a function measurabl... |
| smfaddlem1 47816 | Given the sum of two funct... |
| smfaddlem2 47817 | The sum of two sigma-measu... |
| smfadd 47818 | The sum of two sigma-measu... |
| decsmflem 47819 | A nonincreasing function i... |
| decsmf 47820 | A real-valued, nonincreasi... |
| smfpreimagtf 47821 | Given a function measurabl... |
| issmfgelem 47822 | The predicate " ` F ` is a... |
| issmfge 47823 | The predicate " ` F ` is a... |
| smflimlem1 47824 | Lemma for the proof that t... |
| smflimlem2 47825 | Lemma for the proof that t... |
| smflimlem3 47826 | The limit of sigma-measura... |
| smflimlem4 47827 | Lemma for the proof that t... |
| smflimlem5 47828 | Lemma for the proof that t... |
| smflimlem6 47829 | Lemma for the proof that t... |
| smflim 47830 | The limit of sigma-measura... |
| nsssmfmbflem 47831 | The sigma-measurable funct... |
| nsssmfmbf 47832 | The sigma-measurable funct... |
| smfpimgtxr 47833 | Given a function measurabl... |
| smfpimgtmpt 47834 | Given a function measurabl... |
| smfpreimage 47835 | Given a function measurabl... |
| mbfpsssmf 47836 | Real-valued measurable fun... |
| smfpimgtxrmptf 47837 | Given a function measurabl... |
| smfpimgtxrmpt 47838 | Given a function measurabl... |
| smfpimioompt 47839 | Given a function measurabl... |
| smfpimioo 47840 | Given a function measurabl... |
| smfresal 47841 | Given a sigma-measurable f... |
| smfrec 47842 | The reciprocal of a sigma-... |
| smfres 47843 | The restriction of sigma-m... |
| smfmullem1 47844 | The multiplication of two ... |
| smfmullem2 47845 | The multiplication of two ... |
| smfmullem3 47846 | The multiplication of two ... |
| smfmullem4 47847 | The multiplication of two ... |
| smfmul 47848 | The multiplication of two ... |
| smfmulc1 47849 | A sigma-measurable functio... |
| smfdiv 47850 | The fraction of two sigma-... |
| smfpimbor1lem1 47851 | Every open set belongs to ... |
| smfpimbor1lem2 47852 | Given a sigma-measurable f... |
| smfpimbor1 47853 | Given a sigma-measurable f... |
| smf2id 47854 | Twice the identity functio... |
| smfco 47855 | The composition of a Borel... |
| smfneg 47856 | The negative of a sigma-me... |
| smffmptf 47857 | A function measurable w.r.... |
| smffmpt 47858 | A function measurable w.r.... |
| smflim2 47859 | The limit of a sequence of... |
| smfpimcclem 47860 | Lemma for ~ smfpimcc given... |
| smfpimcc 47861 | Given a countable set of s... |
| issmfle2d 47862 | A sufficient condition for... |
| smflimmpt 47863 | The limit of a sequence of... |
| smfsuplem1 47864 | The supremum of a countabl... |
| smfsuplem2 47865 | The supremum of a countabl... |
| smfsuplem3 47866 | The supremum of a countabl... |
| smfsup 47867 | The supremum of a countabl... |
| smfsupmpt 47868 | The supremum of a countabl... |
| smfsupxr 47869 | The supremum of a countabl... |
| smfinflem 47870 | The infimum of a countable... |
| smfinf 47871 | The infimum of a countable... |
| smfinfmpt 47872 | The infimum of a countable... |
| smflimsuplem1 47873 | If ` H ` converges, the ` ... |
| smflimsuplem2 47874 | The superior limit of a se... |
| smflimsuplem3 47875 | The limit of the ` ( H `` ... |
| smflimsuplem4 47876 | If ` H ` converges, the ` ... |
| smflimsuplem5 47877 | ` H ` converges to the sup... |
| smflimsuplem6 47878 | The superior limit of a se... |
| smflimsuplem7 47879 | The superior limit of a se... |
| smflimsuplem8 47880 | The superior limit of a se... |
| smflimsup 47881 | The superior limit of a se... |
| smflimsupmpt 47882 | The superior limit of a se... |
| smfliminflem 47883 | The inferior limit of a co... |
| smfliminf 47884 | The inferior limit of a co... |
| smfliminfmpt 47885 | The inferior limit of a co... |
| adddmmbl 47886 | If two functions have doma... |
| adddmmbl2 47887 | If two functions have doma... |
| muldmmbl 47888 | If two functions have doma... |
| muldmmbl2 47889 | If two functions have doma... |
| smfdmmblpimne 47890 | If a measurable function w... |
| smfdivdmmbl 47891 | If a functions and a sigma... |
| smfpimne 47892 | Given a function measurabl... |
| smfpimne2 47893 | Given a function measurabl... |
| smfdivdmmbl2 47894 | If a functions and a sigma... |
| fsupdm 47895 | The domain of the sup func... |
| fsupdm2 47896 | The domain of the sup func... |
| smfsupdmmbllem 47897 | If a countable set of sigm... |
| smfsupdmmbl 47898 | If a countable set of sigm... |
| finfdm 47899 | The domain of the inf func... |
| finfdm2 47900 | The domain of the inf func... |
| smfinfdmmbllem 47901 | If a countable set of sigm... |
| smfinfdmmbl 47902 | If a countable set of sigm... |
| sigarval 47903 | Define the signed area by ... |
| sigarim 47904 | Signed area takes value in... |
| sigarac 47905 | Signed area is anticommuta... |
| sigaraf 47906 | Signed area is additive by... |
| sigarmf 47907 | Signed area is additive (w... |
| sigaras 47908 | Signed area is additive by... |
| sigarms 47909 | Signed area is additive (w... |
| sigarls 47910 | Signed area is linear by t... |
| sigarid 47911 | Signed area of a flat para... |
| sigarexp 47912 | Expand the signed area for... |
| sigarperm 47913 | Signed area ` ( A - C ) G ... |
| sigardiv 47914 | If signed area between vec... |
| sigarimcd 47915 | Signed area takes value in... |
| sigariz 47916 | If signed area is zero, th... |
| sigarcol 47917 | Given three points ` A ` ,... |
| sharhght 47918 | Let ` A B C ` be a triangl... |
| sigaradd 47919 | Subtracting (double) area ... |
| cevathlem1 47920 | Ceva's theorem first lemma... |
| cevathlem2 47921 | Ceva's theorem second lemm... |
| cevath 47922 | Ceva's theorem. Let ` A B... |
| simpcntrab 47923 | The center of a simple gro... |
| et-ltneverrefl 47924 | Less-than class is never r... |
| et-equeucl 47925 | Alternative proof that equ... |
| et-sqrtnegnre 47926 | The square root of a negat... |
| quantgodel 47927 | There can be no formula as... |
| quantgodelALT 47928 | There can be no formula as... |
| ormklocald 47929 | If elements of a certain s... |
| ormkglobd 47930 | If all adjacent elements o... |
| chnsubseqword 47931 | A subsequence of a chain i... |
| chnsubseqwl 47932 | A subsequence of a chain h... |
| chnsubseq 47933 | An order-preserving subseq... |
| chnsuslle 47934 | Length of a subsequence is... |
| chnerlem1 47935 | In a chain constructed on ... |
| chnerlem2 47936 | Lemma for ~ chner where th... |
| chnerlem3 47937 | Lemma for ~ chner - tricho... |
| chner 47938 | Any two elements are equiv... |
| wrddin 47939 | A word in two alphabets is... |
| wrddrin 47940 | A word whose alphabet is i... |
| wrddin2 47941 | Distribution of word class... |
| wrddun 47942 | Words in either of two alp... |
| wrddun2 47943 | Superadditivity of word co... |
| chndin 47944 | A chain in two alphabets a... |
| chndrin 47945 | A chain whose alphabet is ... |
| chndin2 47946 | Distribution of chain clas... |
| chndun 47947 | Chains in either of two al... |
| chndun2 47948 | Superaddivity of chain con... |
| chnrin 47949 | Satisfying two chain relat... |
| chnrrin 47950 | A chain of elements satisf... |
| chnrin2 47951 | Distribution of chain clas... |
| chnrun 47952 | Satisfying either of two c... |
| chnrun2 47953 | Superadditivity of chain c... |
| evenwodadd 47954 | If an integer is multiplie... |
| squeezedltsq 47955 | If a real value is squeeze... |
| sqrtnnaa 47956 | Square root of a natural n... |
| sqrtnzqaa 47957 | Square root of a nonzero r... |
| sqrtqaa 47958 | Square root of a rational ... |
| numtowerdt 47959 | Certain number sets and fi... |
| sin3t 47960 | Triple-angle formula for s... |
| cos3t 47961 | Triple-angle formula for c... |
| sin5tlem1 47962 | Lemma 1 for quintupled ang... |
| sin5tlem2 47963 | Lemma 2 for quintupled ang... |
| sin5tlem3 47964 | Lemma 3 for quintupled ang... |
| sin5tlem4 47965 | Lemma 4 for quintupled ang... |
| sin5tlem5 47966 | Lemma 5 for quintupled ang... |
| sin5t 47967 | Five-times-angle formula f... |
| cos5t 47968 | Five-times-angle formula f... |
| cos5teq 47969 | Five-times-angle formula f... |
| goldpolyfactor 47970 | Factorization of a polynom... |
| goldrarr 47971 | The golden ratio is a real... |
| goldrasin 47972 | Alternative trigonometric ... |
| goldrapos 47973 | Golden ratio is positive. ... |
| goldrarp 47974 | The golden ratio is a posi... |
| goldracos5teq 47975 | Lemma 1 for determining th... |
| goldratmolem2 47976 | Lemma 2 for determining th... |
| goldratmolem3 47977 | Lemma 3 for determining th... |
| goldratmolem4 47978 | Lemma 4 for determining th... |
| goldratval 47979 | Value of the golden ratio.... |
| lambert0 47980 | A value of Lambert W (prod... |
| lamberte 47981 | A value of Lambert W (prod... |
| cjnpoly 47982 | Complex conjugation operat... |
| tannpoly 47983 | The tangent function is no... |
| sinnpoly 47984 | Sine function is not a pol... |
| sqrtrrnpoly 47985 | Real square root is not a ... |
| sqrtnpoly 47986 | Square root function is no... |
| tmachlem-extapes 47987 | The class of all tapes is ... |
| tmachlem-finscan 47988 | Execution on any tape only... |
| tmachlem-agreeself 47989 | Any tape belongs to its ow... |
| tmachlem-agreeprod 47990 | Agreement set can be writt... |
| tmachlem-tpcomp 47991 | Product (discrete) topolog... |
| tmachlem-tpbase 47992 | The base set of product to... |
| tmachlem-tpitem 47993 | Topology lemma. (Contribu... |
| tmachlem-tpopen 47994 | Agreement sets are open in... |
| tmachlem-tpopen2 47995 | Variable-renaming lemma co... |
| tmachlem-uassst 47996 | Union of all agreement set... |
| tmachlem-exlargecover 47997 | Product topology of tapes ... |
| tmachlem-extpcover 47998 | Product topology of tapes ... |
| tmachlem-exagreecover 47999 | Particular properties of t... |
| tmachlem-agreesn 48000 | Scans for all tapes of a s... |
| tmachlem-agreefin 48001 | Any agreement set has a fi... |
| tmachlem-franscan 48002 | There is a finite number o... |
| tmachlem-fssscan 48003 | Any scan set is finite. (... |
| tmachfullfin 48004 | Folk theorem. For any alg... |
| hirstL-ax3 48005 | The third axiom of a syste... |
| ax3h 48006 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 48007 | A closed form showing (a i... |
| aibandbiaiaiffb 48008 | A closed form showing (a i... |
| notatnand 48009 | Do not use. Use intnanr i... |
| aistia 48010 | Given a is equivalent to `... |
| aisfina 48011 | Given a is equivalent to `... |
| bothtbothsame 48012 | Given both a, b are equiva... |
| bothfbothsame 48013 | Given both a, b are equiva... |
| aiffbbtat 48014 | Given a is equivalent to b... |
| aisbbisfaisf 48015 | Given a is equivalent to b... |
| axorbtnotaiffb 48016 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 48017 | Given a is equivalent to (... |
| axorbciffatcxorb 48018 | Given a is equivalent to (... |
| aibnbna 48019 | Given a implies b, (not b)... |
| aibnbaif 48020 | Given a implies b, not b, ... |
| aiffbtbat 48021 | Given a is equivalent to b... |
| astbstanbst 48022 | Given a is equivalent to T... |
| aistbistaandb 48023 | Given a is equivalent to T... |
| aisbnaxb 48024 | Given a is equivalent to b... |
| atbiffatnnb 48025 | If a implies b, then a imp... |
| bisaiaisb 48026 | Application of bicom1 with... |
| atbiffatnnbalt 48027 | If a implies b, then a imp... |
| abnotbtaxb 48028 | Assuming a, not b, there e... |
| abnotataxb 48029 | Assuming not a, b, there e... |
| conimpf 48030 | Assuming a, not b, and a i... |
| conimpfalt 48031 | Assuming a, not b, and a i... |
| aistbisfiaxb 48032 | Given a is equivalent to T... |
| aisfbistiaxb 48033 | Given a is equivalent to F... |
| aifftbifffaibif 48034 | Given a is equivalent to T... |
| aifftbifffaibifff 48035 | Given a is equivalent to T... |
| atnaiana 48036 | Given a, it is not the cas... |
| ainaiaandna 48037 | Given a, a implies it is n... |
| abcdta 48038 | Given (((a and b) and c) a... |
| abcdtb 48039 | Given (((a and b) and c) a... |
| abcdtc 48040 | Given (((a and b) and c) a... |
| abcdtd 48041 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 48042 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 48043 | Operands in a biconditiona... |
| nabctnabc 48044 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 48045 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 48046 | From one negated implicati... |
| twonotinotbothi 48047 | From these two negated imp... |
| clifte 48048 | show d is the same as an i... |
| cliftet 48049 | show d is the same as an i... |
| clifteta 48050 | show d is the same as an i... |
| cliftetb 48051 | show d is the same as an i... |
| confun 48052 | Given the hypotheses there... |
| confun2 48053 | Confun simplified to two p... |
| confun3 48054 | Confun's more complex form... |
| confun4 48055 | An attempt at derivative. ... |
| confun5 48056 | An attempt at derivative. ... |
| plcofph 48057 | Given, a,b and a "definiti... |
| pldofph 48058 | Given, a,b c, d, "definiti... |
| plvcofph 48059 | Given, a,b,d, and "definit... |
| plvcofphax 48060 | Given, a,b,d, and "definit... |
| plvofpos 48061 | rh is derivable because ON... |
| mdandyv0 48062 | Given the equivalences set... |
| mdandyv1 48063 | Given the equivalences set... |
| mdandyv2 48064 | Given the equivalences set... |
| mdandyv3 48065 | Given the equivalences set... |
| mdandyv4 48066 | Given the equivalences set... |
| mdandyv5 48067 | Given the equivalences set... |
| mdandyv6 48068 | Given the equivalences set... |
| mdandyv7 48069 | Given the equivalences set... |
| mdandyv8 48070 | Given the equivalences set... |
| mdandyv9 48071 | Given the equivalences set... |
| mdandyv10 48072 | Given the equivalences set... |
| mdandyv11 48073 | Given the equivalences set... |
| mdandyv12 48074 | Given the equivalences set... |
| mdandyv13 48075 | Given the equivalences set... |
| mdandyv14 48076 | Given the equivalences set... |
| mdandyv15 48077 | Given the equivalences set... |
| mdandyvr0 48078 | Given the equivalences set... |
| mdandyvr1 48079 | Given the equivalences set... |
| mdandyvr2 48080 | Given the equivalences set... |
| mdandyvr3 48081 | Given the equivalences set... |
| mdandyvr4 48082 | Given the equivalences set... |
| mdandyvr5 48083 | Given the equivalences set... |
| mdandyvr6 48084 | Given the equivalences set... |
| mdandyvr7 48085 | Given the equivalences set... |
| mdandyvr8 48086 | Given the equivalences set... |
| mdandyvr9 48087 | Given the equivalences set... |
| mdandyvr10 48088 | Given the equivalences set... |
| mdandyvr11 48089 | Given the equivalences set... |
| mdandyvr12 48090 | Given the equivalences set... |
| mdandyvr13 48091 | Given the equivalences set... |
| mdandyvr14 48092 | Given the equivalences set... |
| mdandyvr15 48093 | Given the equivalences set... |
| mdandyvrx0 48094 | Given the exclusivities se... |
| mdandyvrx1 48095 | Given the exclusivities se... |
| mdandyvrx2 48096 | Given the exclusivities se... |
| mdandyvrx3 48097 | Given the exclusivities se... |
| mdandyvrx4 48098 | Given the exclusivities se... |
| mdandyvrx5 48099 | Given the exclusivities se... |
| mdandyvrx6 48100 | Given the exclusivities se... |
| mdandyvrx7 48101 | Given the exclusivities se... |
| mdandyvrx8 48102 | Given the exclusivities se... |
| mdandyvrx9 48103 | Given the exclusivities se... |
| mdandyvrx10 48104 | Given the exclusivities se... |
| mdandyvrx11 48105 | Given the exclusivities se... |
| mdandyvrx12 48106 | Given the exclusivities se... |
| mdandyvrx13 48107 | Given the exclusivities se... |
| mdandyvrx14 48108 | Given the exclusivities se... |
| mdandyvrx15 48109 | Given the exclusivities se... |
| H15NH16TH15IH16 48110 | Given 15 hypotheses and a ... |
| dandysum2p2e4 48111 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 48112 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 48113 | Replacement of a nested an... |
| adh-minim 48114 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 48115 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 48116 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 48117 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 48118 | Fourth lemma for the deriv... |
| adh-minim-ax1 48119 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 48120 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 48121 | Sixth lemma for the deriva... |
| adh-minim-ax2c 48122 | Derivation of a commuted f... |
| adh-minim-ax2 48123 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 48124 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 48125 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 48126 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 48127 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 48128 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 48129 | Third lemma for the deriva... |
| adh-minimp-sylsimp 48130 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 48131 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 48132 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 48133 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 48134 | Fourth lemma for the deriv... |
| adh-minimp-ax2 48135 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 48136 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 48137 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 48138 | If a class with one elemen... |
| eusnsn 48139 | There is a unique element ... |
| absnsb 48140 | If the class abstraction `... |
| euabsneu 48141 | Another way to express exi... |
| elprneb 48142 | An element of a proper uno... |
| oppr 48143 | Equality for ordered pairs... |
| opprb 48144 | Equality for unordered pai... |
| or2expropbilem1 48145 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 48146 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 48147 | If two classes are strictl... |
| eubrv 48148 | If there is a unique set w... |
| eubrdm 48149 | If there is a unique set w... |
| eldmressn 48150 | Element of the domain of a... |
| iota0def 48151 | Example for a defined iota... |
| iota0ndef 48152 | Example for an undefined i... |
| fveqvfvv 48153 | If a function's value at a... |
| fnresfnco 48154 | Composition of two functio... |
| funcoressn 48155 | A composition restricted t... |
| funressnfv 48156 | A restriction to a singlet... |
| funressndmfvrn 48157 | The value of a function ` ... |
| funressnvmo 48158 | A function restricted to a... |
| funressnmo 48159 | A function restricted to a... |
| funressneu 48160 | There is exactly one value... |
| fresfo 48161 | Conditions for a restricti... |
| fsetsniunop 48162 | The class of all functions... |
| fsetabsnop 48163 | The class of all functions... |
| fsetsnf 48164 | The mapping of an element ... |
| fsetsnf1 48165 | The mapping of an element ... |
| fsetsnfo 48166 | The mapping of an element ... |
| fsetsnf1o 48167 | The mapping of an element ... |
| fsetsnprcnex 48168 | The class of all functions... |
| cfsetssfset 48169 | The class of constant func... |
| cfsetsnfsetfv 48170 | The function value of the ... |
| cfsetsnfsetf 48171 | The mapping of the class o... |
| cfsetsnfsetf1 48172 | The mapping of the class o... |
| cfsetsnfsetfo 48173 | The mapping of the class o... |
| cfsetsnfsetf1o 48174 | The mapping of the class o... |
| fsetprcnexALT 48175 | First version of proof for... |
| fcoreslem1 48176 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 48177 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 48178 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 48179 | Lemma 4 for ~ fcores . (C... |
| fcores 48180 | Every composite function `... |
| fcoresf1lem 48181 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 48182 | If a composition is inject... |
| fcoresf1b 48183 | A composition is injective... |
| fcoresfo 48184 | If a composition is surjec... |
| fcoresfob 48185 | A composition is surjectiv... |
| fcoresf1ob 48186 | A composition is bijective... |
| f1cof1blem 48187 | Lemma for ~ f1cof1b and ~ ... |
| 3f1oss1 48188 | The composition of three b... |
| 3f1oss2 48189 | The composition of three b... |
| f1cof1b 48190 | If the range of ` F ` equa... |
| funfocofob 48191 | If the domain of a functio... |
| fnfocofob 48192 | If the domain of a functio... |
| focofob 48193 | If the domain of a functio... |
| f1ocof1ob 48194 | If the range of ` F ` equa... |
| f1ocof1ob2 48195 | If the range of ` F ` equa... |
| aiotajust 48197 | Soundness justification th... |
| dfaiota2 48199 | Alternate definition of th... |
| reuabaiotaiota 48200 | The iota and the alternate... |
| reuaiotaiota 48201 | The iota and the alternate... |
| aiotaexb 48202 | The alternate iota over a ... |
| aiotavb 48203 | The alternate iota over a ... |
| aiotaint 48204 | This is to ~ df-aiota what... |
| dfaiota3 48205 | Alternate definition of ` ... |
| iotan0aiotaex 48206 | If the iota over a wff ` p... |
| aiotaexaiotaiota 48207 | The alternate iota over a ... |
| aiotaval 48208 | Theorem 8.19 in [Quine] p.... |
| aiota0def 48209 | Example for a defined alte... |
| aiota0ndef 48210 | Example for an undefined a... |
| r19.32 48211 | Theorem 19.32 of [Margaris... |
| rexsb 48212 | An equivalent expression f... |
| rexrsb 48213 | An equivalent expression f... |
| 2rexsb 48214 | An equivalent expression f... |
| 2rexrsb 48215 | An equivalent expression f... |
| cbvral2 48216 | Change bound variables of ... |
| cbvrex2 48217 | Change bound variables of ... |
| ralndv1 48218 | Example for a theorem abou... |
| ralndv2 48219 | Second example for a theor... |
| reuf1odnf 48220 | There is exactly one eleme... |
| reuf1od 48221 | There is exactly one eleme... |
| euoreqb 48222 | There is a set which is eq... |
| 2reu3 48223 | Double restricted existent... |
| 2reu7 48224 | Two equivalent expressions... |
| 2reu8 48225 | Two equivalent expressions... |
| 2reu8i 48226 | Implication of a double re... |
| 2reuimp0 48227 | Implication of a double re... |
| 2reuimp 48228 | Implication of a double re... |
| ralbinrald 48235 | Elemination of a restricte... |
| nvelim 48236 | If a class is the universa... |
| alneu 48237 | If a statement holds for a... |
| eu2ndop1stv 48238 | If there is a unique secon... |
| dfateq12d 48239 | Equality deduction for "de... |
| nfdfat 48240 | Bound-variable hypothesis ... |
| dfdfat2 48241 | Alternate definition of th... |
| fundmdfat 48242 | A function is defined at a... |
| dfatprc 48243 | A function is not defined ... |
| dfatelrn 48244 | The value of a function ` ... |
| dfafv2 48245 | Alternative definition of ... |
| afveq12d 48246 | Equality deduction for fun... |
| afveq1 48247 | Equality theorem for funct... |
| afveq2 48248 | Equality theorem for funct... |
| nfafv 48249 | Bound-variable hypothesis ... |
| csbafv12g 48250 | Move class substitution in... |
| afvfundmfveq 48251 | If a class is a function r... |
| afvnfundmuv 48252 | If a set is not in the dom... |
| ndmafv 48253 | The value of a class outsi... |
| afvvdm 48254 | If the function value of a... |
| nfunsnafv 48255 | If the restriction of a cl... |
| afvvfunressn 48256 | If the function value of a... |
| afvprc 48257 | A function's value at a pr... |
| afvvv 48258 | If a function's value at a... |
| afvpcfv0 48259 | If the value of the altern... |
| afvnufveq 48260 | The value of the alternati... |
| afvvfveq 48261 | The value of the alternati... |
| afv0fv0 48262 | If the value of the altern... |
| afvfvn0fveq 48263 | If the function's value at... |
| afv0nbfvbi 48264 | The function's value at an... |
| afvfv0bi 48265 | The function's value at an... |
| afveu 48266 | The value of a function at... |
| fnbrafvb 48267 | Equivalence of function va... |
| fnopafvb 48268 | Equivalence of function va... |
| funbrafvb 48269 | Equivalence of function va... |
| funopafvb 48270 | Equivalence of function va... |
| funbrafv 48271 | The second argument of a b... |
| funbrafv2b 48272 | Function value in terms of... |
| dfafn5a 48273 | Representation of a functi... |
| dfafn5b 48274 | Representation of a functi... |
| fnrnafv 48275 | The range of a function ex... |
| afvelrnb 48276 | A member of a function's r... |
| afvelrnb0 48277 | A member of a function's r... |
| dfaimafn 48278 | Alternate definition of th... |
| dfaimafn2 48279 | Alternate definition of th... |
| afvelima 48280 | Function value in an image... |
| afvelrn 48281 | A function's value belongs... |
| fnafvelrn 48282 | A function's value belongs... |
| fafvelcdm 48283 | A function's value belongs... |
| ffnafv 48284 | A function maps to a class... |
| afvres 48285 | The value of a restricted ... |
| tz6.12-afv 48286 | Function value. Theorem 6... |
| tz6.12-1-afv 48287 | Function value (Theorem 6.... |
| dmfcoafv 48288 | Domains of a function comp... |
| afvco2 48289 | Value of a function compos... |
| rlimdmafv 48290 | Two ways to express that a... |
| aoveq123d 48291 | Equality deduction for ope... |
| nfaov 48292 | Bound-variable hypothesis ... |
| csbaovg 48293 | Move class substitution in... |
| aovfundmoveq 48294 | If a class is a function r... |
| aovnfundmuv 48295 | If an ordered pair is not ... |
| ndmaov 48296 | The value of an operation ... |
| ndmaovg 48297 | The value of an operation ... |
| aovvdm 48298 | If the operation value of ... |
| nfunsnaov 48299 | If the restriction of a cl... |
| aovvfunressn 48300 | If the operation value of ... |
| aovprc 48301 | The value of an operation ... |
| aovrcl 48302 | Reverse closure for an ope... |
| aovpcov0 48303 | If the alternative value o... |
| aovnuoveq 48304 | The alternative value of t... |
| aovvoveq 48305 | The alternative value of t... |
| aov0ov0 48306 | If the alternative value o... |
| aovovn0oveq 48307 | If the operation's value a... |
| aov0nbovbi 48308 | The operation's value on a... |
| aovov0bi 48309 | The operation's value on a... |
| rspceaov 48310 | A frequently used special ... |
| fnotaovb 48311 | Equivalence of operation v... |
| ffnaov 48312 | An operation maps to a cla... |
| faovcl 48313 | Closure law for an operati... |
| aovmpt4g 48314 | Value of a function given ... |
| aoprssdm 48315 | Domain of closure of an op... |
| ndmaovcl 48316 | The "closure" of an operat... |
| ndmaovrcl 48317 | Reverse closure law, in co... |
| ndmaovcom 48318 | Any operation is commutati... |
| ndmaovass 48319 | Any operation is associati... |
| ndmaovdistr 48320 | Any operation is distribut... |
| dfatafv2iota 48323 | If a function is defined a... |
| ndfatafv2 48324 | The alternate function val... |
| ndfatafv2undef 48325 | The alternate function val... |
| dfatafv2ex 48326 | The alternate function val... |
| afv2ex 48327 | The alternate function val... |
| afv2eq12d 48328 | Equality deduction for fun... |
| afv2eq1 48329 | Equality theorem for funct... |
| afv2eq2 48330 | Equality theorem for funct... |
| nfafv2 48331 | Bound-variable hypothesis ... |
| csbafv212g 48332 | Move class substitution in... |
| fexafv2ex 48333 | The alternate function val... |
| ndfatafv2nrn 48334 | The alternate function val... |
| ndmafv2nrn 48335 | The value of a class outsi... |
| funressndmafv2rn 48336 | The alternate function val... |
| afv2ndefb 48337 | Two ways to say that an al... |
| nfunsnafv2 48338 | If the restriction of a cl... |
| afv2prc 48339 | A function's value at a pr... |
| dfatafv2rnb 48340 | The alternate function val... |
| afv2orxorb 48341 | If a set is in the range o... |
| dmafv2rnb 48342 | The alternate function val... |
| fundmafv2rnb 48343 | The alternate function val... |
| afv2elrn 48344 | An alternate function valu... |
| afv20defat 48345 | If the alternate function ... |
| fnafv2elrn 48346 | An alternate function valu... |
| fafv2elcdm 48347 | An alternate function valu... |
| fafv2elrnb 48348 | An alternate function valu... |
| fcdmvafv2v 48349 | If the codomain of a funct... |
| tz6.12-2-afv2 48350 | Function value when ` F ` ... |
| afv2eu 48351 | The value of a function at... |
| afv2res 48352 | The value of a restricted ... |
| tz6.12-afv2 48353 | Function value (Theorem 6.... |
| tz6.12-1-afv2 48354 | Function value (Theorem 6.... |
| tz6.12c-afv2 48355 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 48356 | Corollary of Theorem 6.12(... |
| funressnbrafv2 48357 | The second argument of a b... |
| dfatbrafv2b 48358 | Equivalence of function va... |
| dfatopafv2b 48359 | Equivalence of function va... |
| funbrafv2 48360 | The second argument of a b... |
| fnbrafv2b 48361 | Equivalence of function va... |
| fnopafv2b 48362 | Equivalence of function va... |
| funbrafv22b 48363 | Equivalence of function va... |
| funopafv2b 48364 | Equivalence of function va... |
| dfatsnafv2 48365 | Singleton of function valu... |
| dfafv23 48366 | A definition of function v... |
| dfatdmfcoafv2 48367 | Domain of a function compo... |
| dfatcolem 48368 | Lemma for ~ dfatco . (Con... |
| dfatco 48369 | The predicate "defined at"... |
| afv2co2 48370 | Value of a function compos... |
| rlimdmafv2 48371 | Two ways to express that a... |
| dfafv22 48372 | Alternate definition of ` ... |
| afv2ndeffv0 48373 | If the alternate function ... |
| dfatafv2eqfv 48374 | If a function is defined a... |
| afv2rnfveq 48375 | If the alternate function ... |
| afv20fv0 48376 | If the alternate function ... |
| afv2fvn0fveq 48377 | If the function's value at... |
| afv2fv0 48378 | If the function's value at... |
| afv2fv0b 48379 | The function's value at an... |
| afv2fv0xorb 48380 | If a set is in the range o... |
| an4com24 48381 | Rearrangement of 4 conjunc... |
| 3an4ancom24 48382 | Commutative law for a conj... |
| 4an21 48383 | Rearrangement of 4 conjunc... |
| dfnelbr2 48386 | Alternate definition of th... |
| nelbr 48387 | The binary relation of a s... |
| nelbrim 48388 | If a set is related to ano... |
| nelbrnel 48389 | A set is related to anothe... |
| nelbrnelim 48390 | If a set is related to ano... |
| ralralimp 48391 | Selecting one of two alter... |
| otiunsndisjX 48392 | The union of singletons co... |
| fvifeq 48393 | Equality of function value... |
| rnfdmpr 48394 | The range of a one-to-one ... |
| imarnf1pr 48395 | The image of the range of ... |
| funop1 48396 | A function is an ordered p... |
| fun2dmnopgexmpl 48397 | A function with a domain c... |
| opabresex0d 48398 | A collection of ordered pa... |
| opabbrfex0d 48399 | A collection of ordered pa... |
| opabresexd 48400 | A collection of ordered pa... |
| opabbrfexd 48401 | A collection of ordered pa... |
| f1oresf1orab 48402 | Build a bijection by restr... |
| f1oresf1o 48403 | Build a bijection by restr... |
| f1oresf1o2 48404 | Build a bijection by restr... |
| fvmptrab 48405 | Value of a function mappin... |
| fvmptrabdm 48406 | Value of a function mappin... |
| cnambpcma 48407 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 48408 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 48409 | The sum of two complex num... |
| leaddsuble 48410 | Addition and subtraction o... |
| 2leaddle2 48411 | If two real numbers are le... |
| ltnltne 48412 | Variant of trichotomy law ... |
| p1lep2 48413 | A real number increasd by ... |
| ltsubsubaddltsub 48414 | If the result of subtracti... |
| zm1nn 48415 | An integer minus 1 is posi... |
| readdcnnred 48416 | The sum of a real number a... |
| resubcnnred 48417 | The difference of a real n... |
| recnmulnred 48418 | The product of a real numb... |
| cndivrenred 48419 | The quotient of an imagina... |
| sqrtnegnre 48420 | The square root of a negat... |
| nn0resubcl 48421 | Closure law for subtractio... |
| zgeltp1eq 48422 | If an integer is between a... |
| 1t10e1p1e11 48423 | 11 is 1 times 10 to the po... |
| deccarry 48424 | Add 1 to a 2 digit number ... |
| eluzge0nn0 48425 | If an integer is greater t... |
| nltle2tri 48426 | Negated extended trichotom... |
| ssfz12 48427 | Subset relationship for fi... |
| elfz2z 48428 | Membership of an integer i... |
| 2elfz3nn0 48429 | If there are two elements ... |
| fz0addcom 48430 | The addition of two member... |
| 2elfz2melfz 48431 | If the sum of two integers... |
| fz0addge0 48432 | The sum of two integers in... |
| elfzlble 48433 | Membership of an integer i... |
| elfzelfzlble 48434 | Membership of an element o... |
| elfz2nn 48435 | A member of a finite set o... |
| fzopred 48436 | Join a predecessor to the ... |
| fzopredsuc 48437 | Join a predecessor and a s... |
| 1fzopredsuc 48438 | Join 0 and a successor to ... |
| el1fzopredsuc 48439 | An element of an open inte... |
| subsubelfzo0 48440 | Subtracting a difference f... |
| 2ffzoeq 48441 | Two functions over a half-... |
| elfzo2nn 48442 | A member of a half-open ra... |
| nnmul2 48443 | If one factor of a product... |
| nnmul2b 48444 | A factor of a product of i... |
| 2ltceilhalf 48445 | The ceiling of half of an ... |
| ceilhalfgt1 48446 | The ceiling of half of an ... |
| ceilhalfelfzo1 48447 | A positive integer less th... |
| gpgedgvtx1lem 48448 | Lemma for ~ gpgedgvtx1 . ... |
| 2tceilhalfelfzo1 48449 | Two times a positive integ... |
| ceilbi 48450 | A condition equivalent to ... |
| ceilhalf1 48451 | The ceiling of one half is... |
| rehalfge1 48452 | Half of a real number grea... |
| ceilhalfnn 48453 | The ceiling of half of a p... |
| 1elfzo1ceilhalf1 48454 | 1 is in the half-open inte... |
| nnge2recfl0 48455 | The floor of the reciproca... |
| flmrecm1 48456 | The floor of an integer mi... |
| fldivmod 48457 | Expressing the floor of a ... |
| ceildivmod 48458 | Expressing the ceiling of ... |
| ceil5half3 48459 | The ceiling of half of 5 i... |
| submodaddmod 48460 | Subtraction and addition m... |
| difltmodne 48461 | Two nonnegative integers a... |
| zplusmodne 48462 | A nonnegative integer is n... |
| addmodne 48463 | The sum of a nonnegative i... |
| plusmod5ne 48464 | A nonnegative integer is n... |
| zp1modne 48465 | An integer is not itself p... |
| p1modne 48466 | A nonnegative integer is n... |
| m1modne 48467 | A nonnegative integer is n... |
| minusmod5ne 48468 | A nonnegative integer is n... |
| submodlt 48469 | The difference of an eleme... |
| submodneaddmod 48470 | An integer minus ` B ` is ... |
| m1modnep2mod 48471 | A nonnegative integer minu... |
| minusmodnep2tmod 48472 | A nonnegative integer minu... |
| m1mod0mod1 48473 | An integer decreased by 1 ... |
| elmod2 48474 | An integer modulo 2 is eit... |
| mod0mul 48475 | If an integer is 0 modulo ... |
| modn0mul 48476 | If an integer is not 0 mod... |
| m1modmmod 48477 | An integer decreased by 1 ... |
| difmodm1lt 48478 | The difference between an ... |
| 8mod5e3 48479 | 8 modulo 5 is 3. (Contrib... |
| modmkpkne 48480 | If an integer minus a cons... |
| modmknepk 48481 | A nonnegative integer less... |
| modlt0b 48482 | An integer with an absolut... |
| mod2addne 48483 | The sums of a nonnegative ... |
| modm1nep1 48484 | A nonnegative integer less... |
| modm2nep1 48485 | A nonnegative integer less... |
| modp2nep1 48486 | A nonnegative integer less... |
| modm1nep2 48487 | A nonnegative integer less... |
| modm1nem2 48488 | A nonnegative integer less... |
| modm1p1ne 48489 | If an integer minus one eq... |
| smonoord 48490 | Ordering relation for a st... |
| 2timesltsq 48491 | Two times an integer great... |
| 2timesltsqm1 48492 | Two times an integer great... |
| fsummsndifre 48493 | A finite sum with one of i... |
| fsumsplitsndif 48494 | Separate out a term in a f... |
| fsummmodsndifre 48495 | A finite sum of summands m... |
| fsummmodsnunz 48496 | A finite sum of summands m... |
| nndivides2 48497 | Definition of the divides ... |
| facnn0dvdsfac 48498 | The factorial of a nonnega... |
| muldvdsfacgt 48499 | The product of two differe... |
| muldvdsfacm1 48500 | The product of two differe... |
| setsidel 48501 | The injected slot is an el... |
| setsnidel 48502 | The injected slot is an el... |
| setsv 48503 | The value of the structure... |
| preimafvsnel 48504 | The preimage of a function... |
| preimafvn0 48505 | The preimage of a function... |
| uniimafveqt 48506 | The union of the image of ... |
| uniimaprimaeqfv 48507 | The union of the image of ... |
| setpreimafvex 48508 | The class ` P ` of all pre... |
| elsetpreimafvb 48509 | The characterization of an... |
| elsetpreimafv 48510 | An element of the class ` ... |
| elsetpreimafvssdm 48511 | An element of the class ` ... |
| fvelsetpreimafv 48512 | There is an element in a p... |
| preimafvelsetpreimafv 48513 | The preimage of a function... |
| preimafvsspwdm 48514 | The class ` P ` of all pre... |
| 0nelsetpreimafv 48515 | The empty set is not an el... |
| elsetpreimafvbi 48516 | An element of the preimage... |
| elsetpreimafveqfv 48517 | The elements of the preima... |
| eqfvelsetpreimafv 48518 | If an element of the domai... |
| elsetpreimafvrab 48519 | An element of the preimage... |
| imaelsetpreimafv 48520 | The image of an element of... |
| uniimaelsetpreimafv 48521 | The union of the image of ... |
| elsetpreimafveq 48522 | If two preimages of functi... |
| fundcmpsurinjlem1 48523 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 48524 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 48525 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 48526 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 48527 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 48528 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 48529 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 48530 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 48531 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 48532 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 48533 | Every function ` F : A -->... |
| fundcmpsurinj 48534 | Every function ` F : A -->... |
| fundcmpsurbijinj 48535 | Every function ` F : A -->... |
| fundcmpsurinjimaid 48536 | Every function ` F : A -->... |
| fundcmpsurinjALT 48537 | Alternate proof of ~ fundc... |
| iccpval 48540 | Partition consisting of a ... |
| iccpart 48541 | A special partition. Corr... |
| iccpartimp 48542 | Implications for a class b... |
| iccpartres 48543 | The restriction of a parti... |
| iccpartxr 48544 | If there is a partition, t... |
| iccpartgtprec 48545 | If there is a partition, t... |
| iccpartipre 48546 | If there is a partition, t... |
| iccpartiltu 48547 | If there is a partition, t... |
| iccpartigtl 48548 | If there is a partition, t... |
| iccpartlt 48549 | If there is a partition, t... |
| iccpartltu 48550 | If there is a partition, t... |
| iccpartgtl 48551 | If there is a partition, t... |
| iccpartgt 48552 | If there is a partition, t... |
| iccpartleu 48553 | If there is a partition, t... |
| iccpartgel 48554 | If there is a partition, t... |
| iccpartrn 48555 | If there is a partition, t... |
| iccpartf 48556 | The range of the partition... |
| iccpartel 48557 | If there is a partition, t... |
| iccelpart 48558 | An element of any partitio... |
| iccpartiun 48559 | A half-open interval of ex... |
| icceuelpartlem 48560 | Lemma for ~ icceuelpart . ... |
| icceuelpart 48561 | An element of a partitione... |
| iccpartdisj 48562 | The segments of a partitio... |
| iccpartnel 48563 | A point of a partition is ... |
| fargshiftfv 48564 | If a class is a function, ... |
| fargshiftf 48565 | If a class is a function, ... |
| fargshiftf1 48566 | If a function is 1-1, then... |
| fargshiftfo 48567 | If a function is onto, the... |
| fargshiftfva 48568 | The values of a shifted fu... |
| lswn0 48569 | The last symbol of a nonem... |
| nfich1 48572 | The first interchangeable ... |
| nfich2 48573 | The second interchangeable... |
| ichv 48574 | Setvar variables are inter... |
| ichf 48575 | Setvar variables are inter... |
| ichid 48576 | A setvar variable is alway... |
| icht 48577 | A theorem is interchangeab... |
| ichbidv 48578 | Formula building rule for ... |
| ichcircshi 48579 | The setvar variables are i... |
| ichan 48580 | If two setvar variables ar... |
| ichn 48581 | Negation does not affect i... |
| ichim 48582 | Formula building rule for ... |
| dfich2 48583 | Alternate definition of th... |
| ichcom 48584 | The interchangeability of ... |
| ichbi12i 48585 | Equivalence for interchang... |
| icheqid 48586 | In an equality for the sam... |
| icheq 48587 | In an equality of setvar v... |
| ichnfimlem 48588 | Lemma for ~ ichnfim : A s... |
| ichnfim 48589 | If in an interchangeabilit... |
| ichnfb 48590 | If ` x ` and ` y ` are int... |
| ichal 48591 | Move a universal quantifie... |
| ich2al 48592 | Two setvar variables are a... |
| ich2ex 48593 | Two setvar variables are a... |
| ichexmpl1 48594 | Example for interchangeabl... |
| ichexmpl2 48595 | Example for interchangeabl... |
| ich2exprop 48596 | If the setvar variables ar... |
| ichnreuop 48597 | If the setvar variables ar... |
| ichreuopeq 48598 | If the setvar variables ar... |
| sprid 48599 | Two identical representati... |
| elsprel 48600 | An unordered pair is an el... |
| spr0nelg 48601 | The empty set is not an el... |
| sprval 48604 | The set of all unordered p... |
| sprvalpw 48605 | The set of all unordered p... |
| sprssspr 48606 | The set of all unordered p... |
| spr0el 48607 | The empty set is not an un... |
| sprvalpwn0 48608 | The set of all unordered p... |
| sprel 48609 | An element of the set of a... |
| prssspr 48610 | An element of a subset of ... |
| prelspr 48611 | An unordered pair of eleme... |
| prsprel 48612 | The elements of a pair fro... |
| prsssprel 48613 | The elements of a pair fro... |
| sprvalpwle2 48614 | The set of all unordered p... |
| sprsymrelfvlem 48615 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 48616 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 48617 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 48618 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 48619 | The value of the function ... |
| sprsymrelf 48620 | The mapping ` F ` is a fun... |
| sprsymrelf1 48621 | The mapping ` F ` is a one... |
| sprsymrelfo 48622 | The mapping ` F ` is a fun... |
| sprsymrelf1o 48623 | The mapping ` F ` is a bij... |
| sprbisymrel 48624 | There is a bijection betwe... |
| sprsymrelen 48625 | The class ` P ` of subsets... |
| prpair 48626 | Characterization of a prop... |
| prproropf1olem0 48627 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 48628 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 48629 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 48630 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 48631 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 48632 | There is a bijection betwe... |
| prproropen 48633 | The set of proper pairs an... |
| prproropreud 48634 | There is exactly one order... |
| pairreueq 48635 | Two equivalent representat... |
| paireqne 48636 | Two sets are not equal iff... |
| prprval 48639 | The set of all proper unor... |
| prprvalpw 48640 | The set of all proper unor... |
| prprelb 48641 | An element of the set of a... |
| prprelprb 48642 | A set is an element of the... |
| prprspr2 48643 | The set of all proper unor... |
| prprsprreu 48644 | There is a unique proper u... |
| prprreueq 48645 | There is a unique proper u... |
| sbcpr 48646 | The proper substitution of... |
| reupr 48647 | There is a unique unordere... |
| reuprpr 48648 | There is a unique proper u... |
| poprelb 48649 | Equality for unordered pai... |
| 2exopprim 48650 | The existence of an ordere... |
| reuopreuprim 48651 | There is a unique unordere... |
| nprmmul1 48652 | Special factorization of a... |
| nprmmul2 48653 | Special factorization of a... |
| nprmmul3 48654 | Special factorization of a... |
| fmtno 48657 | The ` N ` th Fermat number... |
| fmtnoge3 48658 | Each Fermat number is grea... |
| fmtnonn 48659 | Each Fermat number is a po... |
| fmtnom1nn 48660 | A Fermat number minus one ... |
| fmtnoodd 48661 | Each Fermat number is odd.... |
| fmtnorn 48662 | A Fermat number is a funct... |
| fmtnof1 48663 | The enumeration of the Fer... |
| fmtnoinf 48664 | The set of Fermat numbers ... |
| fmtnorec1 48665 | The first recurrence relat... |
| sqrtpwpw2p 48666 | The floor of the square ro... |
| fmtnosqrt 48667 | The floor of the square ro... |
| fmtno0 48668 | The ` 0 ` th Fermat number... |
| fmtno1 48669 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 48670 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 48671 | The second recurrence rela... |
| fmtnodvds 48672 | Any Fermat number divides ... |
| goldbachthlem1 48673 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 48674 | Lemma 2 for ~ goldbachth .... |
| goldbachth 48675 | Goldbach's theorem: Two d... |
| fmtnorec3 48676 | The third recurrence relat... |
| fmtnorec4 48677 | The fourth recurrence rela... |
| fmtno2 48678 | The ` 2 ` nd Fermat number... |
| fmtno3 48679 | The ` 3 ` rd Fermat number... |
| fmtno4 48680 | The ` 4 ` th Fermat number... |
| fmtno5lem1 48681 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 48682 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 48683 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 48684 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 48685 | The ` 5 ` th Fermat number... |
| fmtno0prm 48686 | The ` 0 ` th Fermat number... |
| fmtno1prm 48687 | The ` 1 ` st Fermat number... |
| fmtno2prm 48688 | The ` 2 ` nd Fermat number... |
| 257prm 48689 | 257 is a prime number (the... |
| fmtno3prm 48690 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 48691 | Any power of two is coprim... |
| fmtnoprmfac1lem 48692 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 48693 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 48694 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 48695 | Divisor of Fermat number (... |
| fmtnofac2lem 48696 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 48697 | Divisor of Fermat number (... |
| fmtnofac1 48698 | Divisor of Fermat number (... |
| fmtno4sqrt 48699 | The floor of the square ro... |
| fmtno4prmfac 48700 | If P was a (prime) factor ... |
| fmtno4prmfac193 48701 | If P was a (prime) factor ... |
| fmtno4nprmfac193 48702 | 193 is not a (prime) facto... |
| fmtno4prm 48703 | The ` 4 `-th Fermat number... |
| 65537prm 48704 | 65537 is a prime number (t... |
| fmtnofz04prm 48705 | The first five Fermat numb... |
| fmtnole4prm 48706 | The first five Fermat numb... |
| fmtno5faclem1 48707 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 48708 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 48709 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 48710 | The factorization of the `... |
| fmtno5nprm 48711 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 48712 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 48713 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 48714 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 48715 | The mapping of a Fermat nu... |
| prminf2 48716 | The set of prime numbers i... |
| 2pwp1prm 48717 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 48718 | Every prime number of the ... |
| m2prm 48719 | The second Mersenne number... |
| m3prm 48720 | The third Mersenne number ... |
| flsqrt 48721 | A condition equivalent to ... |
| flsqrt5 48722 | The floor of the square ro... |
| 3ndvds4 48723 | 3 does not divide 4. (Con... |
| 139prmALT 48724 | 139 is a prime number. In... |
| 31prm 48725 | 31 is a prime number. In ... |
| m5prm 48726 | The fifth Mersenne number ... |
| 127prm 48727 | 127 is a prime number. (C... |
| m7prm 48728 | The seventh Mersenne numbe... |
| m11nprm 48729 | The eleventh Mersenne numb... |
| mod42tp1mod8 48730 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 48731 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 48732 | If ` P ` is a Sophie Germa... |
| lighneallem1 48733 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 48734 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 48735 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 48736 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 48737 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 48738 | Lemma 3 for ~ lighneal . ... |
| lighneal 48739 | If a power of a prime ` P ... |
| modexp2m1d 48740 | The square of an integer w... |
| proththdlem 48741 | Lemma for ~ proththd . (C... |
| proththd 48742 | Proth's theorem (1878). I... |
| 5tcu2e40 48743 | 5 times the cube of 2 is 4... |
| 3exp4mod41 48744 | 3 to the fourth power is -... |
| 41prothprmlem1 48745 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 48746 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 48747 | 41 is a _Proth prime_. (C... |
| nprmdvdsfacm1lem1 48748 | Lemma 1 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem2 48749 | Lemma 2 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem3 48750 | Lemma 3 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1lem4 48751 | Lemma 4 for ~ nprmdvdsfacm... |
| nprmdvdsfacm1 48752 | A non-prime integer greate... |
| ppivalnnprm 48753 | Value of a term of the pri... |
| ppivalnnnprmge6 48754 | Value of a term of the pri... |
| ppivalnn4 48755 | Value of the term of the p... |
| ppivalnnnprm 48756 | Value of a term of the pri... |
| indprm 48757 | An indicator function for ... |
| indprmfz 48758 | An indicator function for ... |
| ppi1sum 48759 | Value of the prime-countin... |
| ppivalnn 48760 | Value of the prime-countin... |
| quad1 48761 | A condition for a quadrati... |
| requad01 48762 | A condition for a quadrati... |
| requad1 48763 | A condition for a quadrati... |
| requad2 48764 | A condition for a quadrati... |
| iseven 48769 | The predicate "is an even ... |
| isodd 48770 | The predicate "is an odd n... |
| evenz 48771 | An even number is an integ... |
| oddz 48772 | An odd number is an intege... |
| evendiv2z 48773 | The result of dividing an ... |
| oddp1div2z 48774 | The result of dividing an ... |
| oddm1div2z 48775 | The result of dividing an ... |
| isodd2 48776 | The predicate "is an odd n... |
| dfodd2 48777 | Alternate definition for o... |
| dfodd6 48778 | Alternate definition for o... |
| dfeven4 48779 | Alternate definition for e... |
| evenm1odd 48780 | The predecessor of an even... |
| evenp1odd 48781 | The successor of an even n... |
| oddp1eveni 48782 | The successor of an odd nu... |
| oddm1eveni 48783 | The predecessor of an odd ... |
| evennodd 48784 | An even number is not an o... |
| oddneven 48785 | An odd number is not an ev... |
| enege 48786 | The negative of an even nu... |
| onego 48787 | The negative of an odd num... |
| m1expevenALTV 48788 | Exponentiation of -1 by an... |
| m1expoddALTV 48789 | Exponentiation of -1 by an... |
| dfeven2 48790 | Alternate definition for e... |
| dfodd3 48791 | Alternate definition for o... |
| iseven2 48792 | The predicate "is an even ... |
| isodd3 48793 | The predicate "is an odd n... |
| 2dvdseven 48794 | 2 divides an even number. ... |
| m2even 48795 | A multiple of 2 is an even... |
| 2ndvdsodd 48796 | 2 does not divide an odd n... |
| 2dvdsoddp1 48797 | 2 divides an odd number in... |
| 2dvdsoddm1 48798 | 2 divides an odd number de... |
| dfeven3 48799 | Alternate definition for e... |
| dfodd4 48800 | Alternate definition for o... |
| dfodd5 48801 | Alternate definition for o... |
| zefldiv2ALTV 48802 | The floor of an even numbe... |
| zofldiv2ALTV 48803 | The floor of an odd number... |
| oddflALTV 48804 | Odd number representation ... |
| iseven5 48805 | The predicate "is an even ... |
| isodd7 48806 | The predicate "is an odd n... |
| dfeven5 48807 | Alternate definition for e... |
| dfodd7 48808 | Alternate definition for o... |
| gcd2odd1 48809 | The greatest common diviso... |
| zneoALTV 48810 | No even integer equals an ... |
| zeoALTV 48811 | An integer is even or odd.... |
| zeo2ALTV 48812 | An integer is even or odd ... |
| nneoALTV 48813 | A positive integer is even... |
| nneoiALTV 48814 | A positive integer is even... |
| odd2np1ALTV 48815 | An integer is odd iff it i... |
| oddm1evenALTV 48816 | An integer is odd iff its ... |
| oddp1evenALTV 48817 | An integer is odd iff its ... |
| oexpnegALTV 48818 | The exponential of the neg... |
| oexpnegnz 48819 | The exponential of the neg... |
| bits0ALTV 48820 | Value of the zeroth bit. ... |
| bits0eALTV 48821 | The zeroth bit of an even ... |
| bits0oALTV 48822 | The zeroth bit of an odd n... |
| divgcdoddALTV 48823 | Either ` A / ( A gcd B ) `... |
| opoeALTV 48824 | The sum of two odds is eve... |
| opeoALTV 48825 | The sum of an odd and an e... |
| omoeALTV 48826 | The difference of two odds... |
| omeoALTV 48827 | The difference of an odd a... |
| oddprmALTV 48828 | A prime not equal to ` 2 `... |
| 0evenALTV 48829 | 0 is an even number. (Con... |
| 0noddALTV 48830 | 0 is not an odd number. (... |
| 1oddALTV 48831 | 1 is an odd number. (Cont... |
| 1nevenALTV 48832 | 1 is not an even number. ... |
| 2evenALTV 48833 | 2 is an even number. (Con... |
| 2noddALTV 48834 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 48835 | An odd nonnegative integer... |
| nnoALTV 48836 | An alternate characterizat... |
| nn0oALTV 48837 | An alternate characterizat... |
| nn0e 48838 | An alternate characterizat... |
| nneven 48839 | An alternate characterizat... |
| nn0onn0exALTV 48840 | For each odd nonnegative i... |
| nn0enn0exALTV 48841 | For each even nonnegative ... |
| nnennexALTV 48842 | For each even positive int... |
| nnpw2evenALTV 48843 | 2 to the power of a positi... |
| epoo 48844 | The sum of an even and an ... |
| emoo 48845 | The difference of an even ... |
| epee 48846 | The sum of two even number... |
| emee 48847 | The difference of two even... |
| evensumeven 48848 | If a summand is even, the ... |
| 3odd 48849 | 3 is an odd number. (Cont... |
| 4even 48850 | 4 is an even number. (Con... |
| 5odd 48851 | 5 is an odd number. (Cont... |
| 6even 48852 | 6 is an even number. (Con... |
| 7odd 48853 | 7 is an odd number. (Cont... |
| 8even 48854 | 8 is an even number. (Con... |
| evenprm2 48855 | A prime number is even iff... |
| oddprmne2 48856 | Every prime number not bei... |
| oddprmuzge3 48857 | A prime number which is od... |
| evenltle 48858 | If an even number is great... |
| odd2prm2 48859 | If an odd number is the su... |
| even3prm2 48860 | If an even number is the s... |
| mogoldbblem 48861 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 48862 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 48863 | Lemma for ~ perfectALTV . ... |
| perfectALTV 48864 | The Euclid-Euler theorem, ... |
| fppr 48867 | The set of Fermat pseudopr... |
| fpprmod 48868 | The set of Fermat pseudopr... |
| fpprel 48869 | A Fermat pseudoprime to th... |
| fpprbasnn 48870 | The base of a Fermat pseud... |
| fpprnn 48871 | A Fermat pseudoprime to th... |
| fppr2odd 48872 | A Fermat pseudoprime to th... |
| 11t31e341 48873 | 341 is the product of 11 a... |
| 2exp340mod341 48874 | Eight to the eighth power ... |
| 341fppr2 48875 | 341 is the (smallest) _Pou... |
| 4fppr1 48876 | 4 is the (smallest) Fermat... |
| 8exp8mod9 48877 | Eight to the eighth power ... |
| 9fppr8 48878 | 9 is the (smallest) Fermat... |
| dfwppr 48879 | Alternate definition of a ... |
| fpprwppr 48880 | A Fermat pseudoprime to th... |
| fpprwpprb 48881 | An integer ` X ` which is ... |
| fpprel2 48882 | An alternate definition fo... |
| nfermltl8rev 48883 | Fermat's little theorem wi... |
| nfermltl2rev 48884 | Fermat's little theorem wi... |
| nfermltlrev 48885 | Fermat's little theorem re... |
| isgbe 48892 | The predicate "is an even ... |
| isgbow 48893 | The predicate "is a weak o... |
| isgbo 48894 | The predicate "is an odd G... |
| gbeeven 48895 | An even Goldbach number is... |
| gbowodd 48896 | A weak odd Goldbach number... |
| gbogbow 48897 | A (strong) odd Goldbach nu... |
| gboodd 48898 | An odd Goldbach number is ... |
| gbepos 48899 | Any even Goldbach number i... |
| gbowpos 48900 | Any weak odd Goldbach numb... |
| gbopos 48901 | Any odd Goldbach number is... |
| gbegt5 48902 | Any even Goldbach number i... |
| gbowgt5 48903 | Any weak odd Goldbach numb... |
| gbowge7 48904 | Any weak odd Goldbach numb... |
| gboge9 48905 | Any odd Goldbach number is... |
| gbege6 48906 | Any even Goldbach number i... |
| gbpart6 48907 | The Goldbach partition of ... |
| gbpart7 48908 | The (weak) Goldbach partit... |
| gbpart8 48909 | The Goldbach partition of ... |
| gbpart9 48910 | The (strong) Goldbach part... |
| gbpart11 48911 | The (strong) Goldbach part... |
| 6gbe 48912 | 6 is an even Goldbach numb... |
| 7gbow 48913 | 7 is a weak odd Goldbach n... |
| 8gbe 48914 | 8 is an even Goldbach numb... |
| 9gbo 48915 | 9 is an odd Goldbach numbe... |
| 11gbo 48916 | 11 is an odd Goldbach numb... |
| stgoldbwt 48917 | If the strong ternary Gold... |
| sbgoldbwt 48918 | If the strong binary Goldb... |
| sbgoldbst 48919 | If the strong binary Goldb... |
| sbgoldbaltlem1 48920 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 48921 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 48922 | An alternate (related to t... |
| sbgoldbb 48923 | If the strong binary Goldb... |
| sgoldbeven3prm 48924 | If the binary Goldbach con... |
| sbgoldbm 48925 | If the strong binary Goldb... |
| mogoldbb 48926 | If the modern version of t... |
| sbgoldbmb 48927 | The strong binary Goldbach... |
| sbgoldbo 48928 | If the strong binary Goldb... |
| nnsum3primes4 48929 | 4 is the sum of at most 3 ... |
| nnsum4primes4 48930 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 48931 | Every prime is "the sum of... |
| nnsum4primesprm 48932 | Every prime is "the sum of... |
| nnsum3primesgbe 48933 | Any even Goldbach number i... |
| nnsum4primesgbe 48934 | Any even Goldbach number i... |
| nnsum3primesle9 48935 | Every integer greater than... |
| nnsum4primesle9 48936 | Every integer greater than... |
| nnsum4primesodd 48937 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 48938 | If the (strong) ternary Go... |
| evengpop3 48939 | If the (weak) ternary Gold... |
| evengpoap3 48940 | If the (strong) ternary Go... |
| nnsum4primeseven 48941 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 48942 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 48943 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 48944 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 48945 | If the binary Goldbach con... |
| bgoldbtbndlem1 48946 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 48947 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 48948 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 48949 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 48950 | If the binary Goldbach con... |
| tgoldbachgtALTV 48953 | Variant of Thierry Arnoux'... |
| bgoldbachlt 48954 | The binary Goldbach conjec... |
| tgblthelfgott 48956 | The ternary Goldbach conje... |
| tgoldbachlt 48957 | The ternary Goldbach conje... |
| tgoldbach 48958 | The ternary Goldbach conje... |
| clnbgrprc0 48961 | The closed neighborhood is... |
| clnbgrcl 48962 | If a class ` X ` has at le... |
| clnbgrval 48963 | The closed neighborhood of... |
| dfclnbgr2 48964 | Alternate definition of th... |
| dfclnbgr4 48965 | Alternate definition of th... |
| elclnbgrelnbgr 48966 | An element of the closed n... |
| dfclnbgr3 48967 | Alternate definition of th... |
| clnbgrnvtx0 48968 | If a class ` X ` is not a ... |
| clnbgrel 48969 | Characterization of a memb... |
| clnbgrvtxel 48970 | Every vertex ` K ` is a me... |
| clnbgrisvtx 48971 | Every member ` N ` of the ... |
| clnbgrssvtx 48972 | The closed neighborhood of... |
| clnbgrn0 48973 | The closed neighborhood of... |
| clnbupgr 48974 | The closed neighborhood of... |
| clnbupgrel 48975 | A member of the closed nei... |
| clnbupgreli 48976 | A member of the closed nei... |
| clnbgr0vtx 48977 | In a null graph (with no v... |
| clnbgr0edg 48978 | In an empty graph (with no... |
| clnbgrsym 48979 | In a graph, the closed nei... |
| predgclnbgrel 48980 | If a (not necessarily prop... |
| clnbgredg 48981 | A vertex connected by an e... |
| clnbgrssedg 48982 | The vertices connected by ... |
| edgusgrclnbfin 48983 | The size of the closed nei... |
| clnbusgrfi 48984 | The closed neighborhood of... |
| clnbfiusgrfi 48985 | The closed neighborhood of... |
| clnbgrlevtx 48986 | The size of the closed nei... |
| dfsclnbgr2 48987 | Alternate definition of th... |
| sclnbgrel 48988 | Characterization of a memb... |
| sclnbgrelself 48989 | A vertex ` N ` is a member... |
| sclnbgrisvtx 48990 | Every member ` X ` of the ... |
| dfclnbgr5 48991 | Alternate definition of th... |
| dfnbgr5 48992 | Alternate definition of th... |
| dfnbgrss 48993 | Subset chain for different... |
| dfvopnbgr2 48994 | Alternate definition of th... |
| vopnbgrel 48995 | Characterization of a memb... |
| vopnbgrelself 48996 | A vertex ` N ` is a member... |
| dfclnbgr6 48997 | Alternate definition of th... |
| dfnbgr6 48998 | Alternate definition of th... |
| dfsclnbgr6 48999 | Alternate definition of a ... |
| dfnbgrss2 49000 | Subset chain for different... |
| isisubgr 49003 | The subgraph induced by a ... |
| isubgriedg 49004 | The edges of an induced su... |
| isubgrvtxuhgr 49005 | The subgraph induced by th... |
| isubgredgss 49006 | The edges of an induced su... |
| isubgredg 49007 | An edge of an induced subg... |
| isubgrvtx 49008 | The vertices of an induced... |
| isubgruhgr 49009 | An induced subgraph of a h... |
| isubgrsubgr 49010 | An induced subgraph of a h... |
| isubgrupgr 49011 | An induced subgraph of a p... |
| isubgrumgr 49012 | An induced subgraph of a m... |
| isubgrusgr 49013 | An induced subgraph of a s... |
| isubgr0uhgr 49014 | The subgraph induced by an... |
| grimfn 49020 | The graph isomorphism func... |
| grimdmrel 49021 | The domain of the graph is... |
| isgrim 49023 | An isomorphism of graphs i... |
| grimprop 49024 | Properties of an isomorphi... |
| grimf1o 49025 | An isomorphism of graphs i... |
| grimidvtxedg 49026 | The identity relation rest... |
| grimid 49027 | The identity relation rest... |
| grimuhgr 49028 | If there is a graph isomor... |
| grimcnv 49029 | The converse of a graph is... |
| grimco 49030 | The composition of graph i... |
| uhgrimedgi 49031 | An isomorphism between gra... |
| uhgrimedg 49032 | An isomorphism between gra... |
| uhgrimprop 49033 | An isomorphism between hyp... |
| isuspgrim0lem 49034 | An isomorphism of simple p... |
| isuspgrim0 49035 | An isomorphism of simple p... |
| isuspgrimlem 49036 | Lemma for ~ isuspgrim . (... |
| isuspgrim 49037 | A class is an isomorphism ... |
| upgrimwlklem1 49038 | Lemma 1 for ~ upgrimwlk an... |
| upgrimwlklem2 49039 | Lemma 2 for ~ upgrimwlk . ... |
| upgrimwlklem3 49040 | Lemma 3 for ~ upgrimwlk . ... |
| upgrimwlklem4 49041 | Lemma 4 for ~ upgrimwlk . ... |
| upgrimwlklem5 49042 | Lemma 5 for ~ upgrimwlk . ... |
| upgrimwlk 49043 | Graph isomorphisms between... |
| upgrimwlklen 49044 | Graph isomorphisms between... |
| upgrimtrlslem1 49045 | Lemma 1 for ~ upgrimtrls .... |
| upgrimtrlslem2 49046 | Lemma 2 for ~ upgrimtrls .... |
| upgrimtrls 49047 | Graph isomorphisms between... |
| upgrimpthslem1 49048 | Lemma 1 for ~ upgrimpths .... |
| upgrimpthslem2 49049 | Lemma 2 for ~ upgrimpths .... |
| upgrimpths 49050 | Graph isomorphisms between... |
| upgrimspths 49051 | Graph isomorphisms between... |
| upgrimcycls 49052 | Graph isomorphisms between... |
| brgric 49053 | The relation "is isomorphi... |
| brgrici 49054 | Prove that two graphs are ... |
| gricrcl 49055 | Reverse closure of the "is... |
| dfgric2 49056 | Alternate, explicit defini... |
| gricbri 49057 | Implications of two graphs... |
| gricushgr 49058 | The "is isomorphic to" rel... |
| gricuspgr 49059 | The "is isomorphic to" rel... |
| gricrel 49060 | The "is isomorphic to" rel... |
| gricref 49061 | Graph isomorphism is refle... |
| gricsym 49062 | Graph isomorphism is symme... |
| gricsymb 49063 | Graph isomorphism is symme... |
| grictr 49064 | Graph isomorphism is trans... |
| gricer 49065 | Isomorphism is an equivale... |
| gricen 49066 | Isomorphic graphs have equ... |
| opstrgric 49067 | A graph represented as an ... |
| ushggricedg 49068 | A simple hypergraph (with ... |
| cycldlenngric 49069 | Two simple pseudographs ar... |
| isubgrgrim 49070 | Isomorphic subgraphs induc... |
| uhgrimisgrgriclem 49071 | Lemma for ~ uhgrimisgrgric... |
| uhgrimisgrgric 49072 | For isomorphic hypergraphs... |
| clnbgrisubgrgrim 49073 | Isomorphic subgraphs induc... |
| clnbgrgrimlem 49074 | Lemma for ~ clnbgrgrim : ... |
| clnbgrgrim 49075 | Graph isomorphisms between... |
| grimedg 49076 | For two isomorphic graphs,... |
| grimedgi 49077 | Graph isomorphisms map edg... |
| grtriproplem 49080 | Lemma for ~ grtriprop . (... |
| grtri 49081 | The triangles in a graph. ... |
| grtriprop 49082 | The properties of a triang... |
| grtrif1o 49083 | Any bijection onto a trian... |
| isgrtri 49084 | A triangle in a graph. (C... |
| grtrissvtx 49085 | A triangle is a subset of ... |
| grtriclwlk3 49086 | A triangle induces a close... |
| cycl3grtrilem 49087 | Lemma for ~ cycl3grtri . ... |
| cycl3grtri 49088 | The vertices of a cycle of... |
| grtrimap 49089 | Conditions for mapping tri... |
| grimgrtri 49090 | Graph isomorphisms map tri... |
| usgrgrtrirex 49091 | Conditions for a simple gr... |
| stgrfv 49094 | The star graph S_N. (Contr... |
| stgrvtx 49095 | The vertices of the star g... |
| stgriedg 49096 | The indexed edges of the s... |
| stgredg 49097 | The edges of the star grap... |
| stgredgel 49098 | An edge of the star graph ... |
| stgredgiun 49099 | The edges of the star grap... |
| stgrusgra 49100 | The star graph S_N is a si... |
| stgr0 49101 | The star graph S_0 consist... |
| stgr1 49102 | The star graph S_1 consist... |
| stgrvtx0 49103 | The center ("internal node... |
| stgrorder 49104 | The order of a star graph ... |
| stgrnbgr0 49105 | All vertices of a star gra... |
| stgrclnbgr0 49106 | All vertices of a star gra... |
| isubgr3stgrlem1 49107 | Lemma 1 for ~ isubgr3stgr ... |
| isubgr3stgrlem2 49108 | Lemma 2 for ~ isubgr3stgr ... |
| isubgr3stgrlem3 49109 | Lemma 3 for ~ isubgr3stgr ... |
| isubgr3stgrlem4 49110 | Lemma 4 for ~ isubgr3stgr ... |
| isubgr3stgrlem5 49111 | Lemma 5 for ~ isubgr3stgr ... |
| isubgr3stgrlem6 49112 | Lemma 6 for ~ isubgr3stgr ... |
| isubgr3stgrlem7 49113 | Lemma 7 for ~ isubgr3stgr ... |
| isubgr3stgrlem8 49114 | Lemma 8 for ~ isubgr3stgr ... |
| isubgr3stgrlem9 49115 | Lemma 9 for ~ isubgr3stgr ... |
| isubgr3stgr 49116 | If a vertex of a simple gr... |
| grlimfn 49120 | The graph local isomorphis... |
| grlimdmrel 49121 | The domain of the graph lo... |
| isgrlim 49123 | A local isomorphism of gra... |
| isgrlim2 49124 | A local isomorphism of gra... |
| grlimprop 49125 | Properties of a local isom... |
| grlimf1o 49126 | A local isomorphism of gra... |
| grlimprop2 49127 | Properties of a local isom... |
| uhgrimgrlim 49128 | An isomorphism of hypergra... |
| uspgrlimlem1 49129 | Lemma 1 for ~ uspgrlim . ... |
| uspgrlimlem2 49130 | Lemma 2 for ~ uspgrlim . ... |
| uspgrlimlem3 49131 | Lemma 3 for ~ uspgrlim . ... |
| uspgrlimlem4 49132 | Lemma 4 for ~ uspgrlim . ... |
| uspgrlim 49133 | A local isomorphism of sim... |
| usgrlimprop 49134 | Properties of a local isom... |
| clnbgrvtxedg 49135 | An edge ` E ` containing a... |
| grlimedgclnbgr 49136 | For two locally isomorphic... |
| grlimprclnbgr 49137 | For two locally isomorphic... |
| grlimprclnbgredg 49138 | For two locally isomorphic... |
| grlimpredg 49139 | For two locally isomorphic... |
| grlimprclnbgrvtx 49140 | For two locally isomorphic... |
| grlimgredgex 49141 | Local isomorphisms between... |
| grlimgrtrilem1 49142 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtrilem2 49143 | Lemma 3 for ~ grlimgrtri .... |
| grlimgrtri 49144 | If one of two locally isom... |
| brgrlic 49145 | The relation "is locally i... |
| brgrilci 49146 | Prove that two graphs are ... |
| grlicrel 49147 | The "is locally isomorphic... |
| grlicrcl 49148 | Reverse closure of the "is... |
| dfgrlic2 49149 | Alternate, explicit defini... |
| grilcbri 49150 | Implications of two graphs... |
| dfgrlic3 49151 | Alternate, explicit defini... |
| grilcbri2 49152 | Implications of two graphs... |
| grlicref 49153 | Graph local isomorphism is... |
| grlicsym 49154 | Graph local isomorphism is... |
| grlicsymb 49155 | Graph local isomorphism is... |
| grlictr 49156 | Graph local isomorphism is... |
| grlicer 49157 | Local isomorphism is an eq... |
| grlicen 49158 | Locally isomorphic graphs ... |
| gricgrlic 49159 | Isomorphic hypergraphs are... |
| clnbgr3stgrgrlim 49160 | If all (closed) neighborho... |
| clnbgr3stgrgrlic 49161 | If all (closed) neighborho... |
| usgrexmpl1lem 49162 | Lemma for ~ usgrexmpl1 . ... |
| usgrexmpl1 49163 | ` G ` is a simple graph of... |
| usgrexmpl1vtx 49164 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl1edg 49165 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl1tri 49166 | ` G ` contains a triangle ... |
| usgrexmpl2lem 49167 | Lemma for ~ usgrexmpl2 . ... |
| usgrexmpl2 49168 | ` G ` is a simple graph of... |
| usgrexmpl2vtx 49169 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpl2edg 49170 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl2nblem 49171 | Lemma for ~ usgrexmpl2nb0 ... |
| usgrexmpl2nb0 49172 | The neighborhood of the fi... |
| usgrexmpl2nb1 49173 | The neighborhood of the se... |
| usgrexmpl2nb2 49174 | The neighborhood of the th... |
| usgrexmpl2nb3 49175 | The neighborhood of the fo... |
| usgrexmpl2nb4 49176 | The neighborhood of the fi... |
| usgrexmpl2nb5 49177 | The neighborhood of the si... |
| usgrexmpl2trifr 49178 | ` G ` is triangle-free. (... |
| usgrexmpl12ngric 49179 | The graphs ` H ` and ` G `... |
| usgrexmpl12ngrlic 49180 | The graphs ` H ` and ` G `... |
| gpgov 49183 | The generalized Petersen g... |
| gpgvtx 49184 | The vertices of the genera... |
| gpgiedg 49185 | The indexed edges of the g... |
| gpgedg 49186 | The edges of the generaliz... |
| gpgiedgdmellem 49187 | Lemma for ~ gpgiedgdmel an... |
| gpgvtxel 49188 | A vertex in a generalized ... |
| gpgvtxel2 49189 | The second component of a ... |
| gpgiedgdmel 49190 | An index of edges of the g... |
| gpgedgel 49191 | An edge in a generalized P... |
| gpgprismgriedgdmel 49192 | An index of edges of the g... |
| gpgprismgriedgdmss 49193 | A subset of the index of e... |
| gpgvtx0 49194 | The outside vertices in a ... |
| gpgvtx1 49195 | The inside vertices in a g... |
| opgpgvtx 49196 | A vertex in a generalized ... |
| gpgusgralem 49197 | Lemma for ~ gpgusgra . (C... |
| gpgusgra 49198 | The generalized Petersen g... |
| gpgprismgrusgra 49199 | The generalized Petersen g... |
| gpgorder 49200 | The order of the generaliz... |
| gpg5order 49201 | The order of a generalized... |
| gpgedgvtx0 49202 | The edges starting at an o... |
| gpgedgvtx1 49203 | The edges starting at an i... |
| gpgvtxedg0 49204 | The edges starting at an o... |
| gpgvtxedg1 49205 | The edges starting at an i... |
| gpgedgiov 49206 | The edges of the generaliz... |
| gpgedg2ov 49207 | The edges of the generaliz... |
| gpgedg2iv 49208 | The edges of the generaliz... |
| gpg5nbgrvtx03starlem1 49209 | Lemma 1 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem2 49210 | Lemma 2 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx03starlem3 49211 | Lemma 3 for ~ gpg5nbgrvtx0... |
| gpg5nbgrvtx13starlem1 49212 | Lemma 1 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem2 49213 | Lemma 2 for ~ gpg5nbgr3sta... |
| gpg5nbgrvtx13starlem3 49214 | Lemma 3 for ~ gpg5nbgr3sta... |
| gpgnbgrvtx0 49215 | The (open) neighborhood of... |
| gpgnbgrvtx1 49216 | The (open) neighborhood of... |
| gpg3nbgrvtx0 49217 | In a generalized Petersen ... |
| gpg3nbgrvtx0ALT 49218 | In a generalized Petersen ... |
| gpg3nbgrvtx1 49219 | In a generalized Petersen ... |
| gpgcubic 49220 | Every generalized Petersen... |
| gpg5nbgrvtx03star 49221 | In a generalized Petersen ... |
| gpg5nbgr3star 49222 | In a generalized Petersen ... |
| gpgvtxdg3 49223 | Every vertex in a generali... |
| gpg3kgrtriexlem1 49224 | Lemma 1 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem2 49225 | Lemma 2 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem3 49226 | Lemma 3 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem4 49227 | Lemma 4 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem5 49228 | Lemma 5 for ~ gpg3kgrtriex... |
| gpg3kgrtriexlem6 49229 | Lemma 6 for ~ gpg3kgrtriex... |
| gpg3kgrtriex 49230 | All generalized Petersen g... |
| gpg5gricstgr3 49231 | Each closed neighborhood i... |
| pglem 49232 | Lemma for theorems about P... |
| pgjsgr 49233 | A Petersen graph is a simp... |
| gpg5grlim 49234 | A local isomorphism betwee... |
| gpg5grlic 49235 | The two generalized Peters... |
| gpgprismgr4cycllem1 49236 | Lemma 1 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem2 49237 | Lemma 2 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem3 49238 | Lemma 3 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem4 49239 | Lemma 4 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem5 49240 | Lemma 5 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem6 49241 | Lemma 6 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem7 49242 | Lemma 7 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem8 49243 | Lemma 8 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem9 49244 | Lemma 9 for ~ gpgprismgr4c... |
| gpgprismgr4cycllem10 49245 | Lemma 10 for ~ gpgprismgr4... |
| gpgprismgr4cycllem11 49246 | Lemma 11 for ~ gpgprismgr4... |
| gpgprismgr4cycl0 49247 | The generalized Petersen g... |
| gpgprismgr4cyclex 49248 | The generalized Petersen g... |
| pgnioedg1 49249 | An inside and an outside v... |
| pgnioedg2 49250 | An inside and an outside v... |
| pgnioedg3 49251 | An inside and an outside v... |
| pgnioedg4 49252 | An inside and an outside v... |
| pgnioedg5 49253 | An inside and an outside v... |
| pgnbgreunbgrlem1 49254 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem1 49255 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem2 49256 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2lem3 49257 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem2 49258 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem3 49259 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem4 49260 | Lemma 4 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem1 49261 | Lemma 1 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem2 49262 | Lemma 2 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5lem3 49263 | Lemma 3 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem5 49264 | Lemma 5 for ~ pgnbgreunbgr... |
| pgnbgreunbgrlem6 49265 | Lemma 6 for ~ pgnbgreunbgr... |
| pgnbgreunbgr 49266 | In a Petersen graph, two d... |
| pgn4cyclex 49267 | A cycle in a Petersen grap... |
| pg4cyclnex 49268 | In the Petersen graph G(5,... |
| gpg5ngric 49269 | The two generalized Peters... |
| lgricngricex 49270 | There are two different lo... |
| gpg5edgnedg 49271 | Two consecutive (according... |
| grlimedgnedg 49272 | In general, the image of a... |
| 1hegrlfgr 49273 | A graph ` G ` with one hyp... |
| upwlksfval 49276 | The set of simple walks (i... |
| isupwlk 49277 | Properties of a pair of fu... |
| isupwlkg 49278 | Generalization of ~ isupwl... |
| upwlkbprop 49279 | Basic properties of a simp... |
| upwlkwlk 49280 | A simple walk is a walk. ... |
| upgrwlkupwlk 49281 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 49282 | In a pseudograph, the defi... |
| upgrisupwlkALT 49283 | Alternate proof of ~ upgri... |
| upgredgssspr 49284 | The set of edges of a pseu... |
| uspgropssxp 49285 | The set ` G ` of "simple p... |
| uspgrsprfv 49286 | The value of the function ... |
| uspgrsprf 49287 | The mapping ` F ` is a fun... |
| uspgrsprf1 49288 | The mapping ` F ` is a one... |
| uspgrsprfo 49289 | The mapping ` F ` is a fun... |
| uspgrsprf1o 49290 | The mapping ` F ` is a bij... |
| uspgrex 49291 | The class ` G ` of all "si... |
| uspgrbispr 49292 | There is a bijection betwe... |
| uspgrspren 49293 | The set ` G ` of the "simp... |
| uspgrymrelen 49294 | The set ` G ` of the "simp... |
| uspgrbisymrel 49295 | There is a bijection betwe... |
| uspgrbisymrelALT 49296 | Alternate proof of ~ uspgr... |
| ovn0dmfun 49297 | If a class operation value... |
| xpsnopab 49298 | A Cartesian product with a... |
| xpiun 49299 | A Cartesian product expres... |
| fnxpdmdm 49300 | The domain of the domain o... |
| cnfldsrngbas 49301 | The base set of a subring ... |
| cnfldsrngadd 49302 | The group addition operati... |
| cnfldsrngmul 49303 | The ring multiplication op... |
| plusfreseq 49304 | If the empty set is not co... |
| mgmplusfreseq 49305 | If the empty set is not co... |
| 0mgm 49306 | A set with an empty base s... |
| opmpoismgm 49307 | A structure with a group a... |
| copissgrp 49308 | A structure with a constan... |
| copisnmnd 49309 | A structure with a constan... |
| 0nodd 49310 | 0 is not an odd integer. ... |
| 1odd 49311 | 1 is an odd integer. (Con... |
| 2nodd 49312 | 2 is not an odd integer. ... |
| oddibas 49313 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 49314 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 49315 | The structure of all odd i... |
| nnsgrpmgm 49316 | The structure of positive ... |
| nnsgrp 49317 | The structure of positive ... |
| nnsgrpnmnd 49318 | The structure of positive ... |
| nn0mnd 49319 | The set of nonnegative int... |
| gsumsplit2f 49320 | Split a group sum into two... |
| gsumdifsndf 49321 | Extract a summand from a f... |
| gsumfsupp 49322 | A group sum of a family ca... |
| iscllaw 49329 | The predicate "is a closed... |
| iscomlaw 49330 | The predicate "is a commut... |
| clcllaw 49331 | Closure of a closed operat... |
| isasslaw 49332 | The predicate "is an assoc... |
| asslawass 49333 | Associativity of an associ... |
| mgmplusgiopALT 49334 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 49335 | Slot 2 (group operation) o... |
| intopval 49342 | The internal (binary) oper... |
| intop 49343 | An internal (binary) opera... |
| clintopval 49344 | The closed (internal binar... |
| assintopval 49345 | The associative (closed in... |
| assintopmap 49346 | The associative (closed in... |
| isclintop 49347 | The predicate "is a closed... |
| clintop 49348 | A closed (internal binary)... |
| assintop 49349 | An associative (closed int... |
| isassintop 49350 | The predicate "is an assoc... |
| clintopcllaw 49351 | The closure law holds for ... |
| assintopcllaw 49352 | The closure low holds for ... |
| assintopasslaw 49353 | The associative low holds ... |
| assintopass 49354 | An associative (closed int... |
| ismgmALT 49363 | The predicate "is a magma"... |
| iscmgmALT 49364 | The predicate "is a commut... |
| issgrpALT 49365 | The predicate "is a semigr... |
| iscsgrpALT 49366 | The predicate "is a commut... |
| mgm2mgm 49367 | Equivalence of the two def... |
| sgrp2sgrp 49368 | Equivalence of the two def... |
| lmod0rng 49369 | If the scalar ring of a mo... |
| nzrneg1ne0 49370 | The additive inverse of th... |
| lidldomn1 49371 | If a (left) ideal (which i... |
| lidlabl 49372 | A (left) ideal of a ring i... |
| lidlrng 49373 | A (left) ideal of a ring i... |
| zlidlring 49374 | The zero (left) ideal of a... |
| uzlidlring 49375 | Only the zero (left) ideal... |
| lidldomnnring 49376 | A (left) ideal of a domain... |
| 0even 49377 | 0 is an even integer. (Co... |
| 1neven 49378 | 1 is not an even integer. ... |
| 2even 49379 | 2 is an even integer. (Co... |
| 2zlidl 49380 | The even integers are a (l... |
| 2zrng 49381 | The ring of integers restr... |
| 2zrngbas 49382 | The base set of R is the s... |
| 2zrngadd 49383 | The group addition operati... |
| 2zrng0 49384 | The additive identity of R... |
| 2zrngamgm 49385 | R is an (additive) magma. ... |
| 2zrngasgrp 49386 | R is an (additive) semigro... |
| 2zrngamnd 49387 | R is an (additive) monoid.... |
| 2zrngacmnd 49388 | R is a commutative (additi... |
| 2zrngagrp 49389 | R is an (additive) group. ... |
| 2zrngaabl 49390 | R is an (additive) abelian... |
| 2zrngmul 49391 | The ring multiplication op... |
| 2zrngmmgm 49392 | R is a (multiplicative) ma... |
| 2zrngmsgrp 49393 | R is a (multiplicative) se... |
| 2zrngALT 49394 | The ring of integers restr... |
| 2zrngnmlid 49395 | R has no multiplicative (l... |
| 2zrngnmrid 49396 | R has no multiplicative (r... |
| 2zrngnmlid2 49397 | R has no multiplicative (l... |
| 2zrngnring 49398 | R is not a unital ring. (... |
| cznrnglem 49399 | Lemma for ~ cznrng : The ... |
| cznabel 49400 | The ring constructed from ... |
| cznrng 49401 | The ring constructed from ... |
| cznnring 49402 | The ring constructed from ... |
| rngcvalALTV 49405 | Value of the category of n... |
| rngcbasALTV 49406 | Set of objects of the cate... |
| rngchomfvalALTV 49407 | Set of arrows of the categ... |
| rngchomALTV 49408 | Set of arrows of the categ... |
| elrngchomALTV 49409 | A morphism of non-unital r... |
| rngccofvalALTV 49410 | Composition in the categor... |
| rngccoALTV 49411 | Composition in the categor... |
| rngccatidALTV 49412 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 49413 | The category of non-unital... |
| rngcidALTV 49414 | The identity arrow in the ... |
| rngcsectALTV 49415 | A section in the category ... |
| rngcinvALTV 49416 | An inverse in the category... |
| rngcisoALTV 49417 | An isomorphism in the cate... |
| rngchomffvalALTV 49418 | The value of the functiona... |
| rngchomrnghmresALTV 49419 | The value of the functiona... |
| rngcrescrhmALTV 49420 | The category of non-unital... |
| rhmsubcALTVlem1 49421 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 49422 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 49423 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 49424 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 49425 | According to ~ df-subc , t... |
| rhmsubcALTVcat 49426 | The restriction of the cat... |
| ringcvalALTV 49429 | Value of the category of r... |
| funcringcsetcALTV2lem1 49430 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 49431 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 49432 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 49433 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 49434 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 49435 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 49436 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 49437 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 49438 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 49439 | The "natural forgetful fun... |
| ringcbasALTV 49440 | Set of objects of the cate... |
| ringchomfvalALTV 49441 | Set of arrows of the categ... |
| ringchomALTV 49442 | Set of arrows of the categ... |
| elringchomALTV 49443 | A morphism of rings is a f... |
| ringccofvalALTV 49444 | Composition in the categor... |
| ringccoALTV 49445 | Composition in the categor... |
| ringccatidALTV 49446 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 49447 | The category of rings is a... |
| ringcidALTV 49448 | The identity arrow in the ... |
| ringcsectALTV 49449 | A section in the category ... |
| ringcinvALTV 49450 | An inverse in the category... |
| ringcisoALTV 49451 | An isomorphism in the cate... |
| ringcbasbasALTV 49452 | An element of the base set... |
| funcringcsetclem1ALTV 49453 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 49454 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 49455 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 49456 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 49457 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 49458 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 49459 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 49460 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 49461 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 49462 | The "natural forgetful fun... |
| srhmsubcALTVlem1 49463 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 49464 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 49465 | According to ~ df-subc , t... |
| sringcatALTV 49466 | The restriction of the cat... |
| crhmsubcALTV 49467 | According to ~ df-subc , t... |
| cringcatALTV 49468 | The restriction of the cat... |
| drhmsubcALTV 49469 | According to ~ df-subc , t... |
| drngcatALTV 49470 | The restriction of the cat... |
| fldcatALTV 49471 | The restriction of the cat... |
| fldcALTV 49472 | The restriction of the cat... |
| fldhmsubcALTV 49473 | According to ~ df-subc , t... |
| isprmrng 49476 | The predicate "is a prime ... |
| prmringnzring 49477 | A prime ring is a nonzero ... |
| prmrngring 49478 | A prime ring is a ring. (... |
| smprngprmrng 49479 | A simple ring (a nonzero r... |
| drngprmrng 49480 | A division ring is a prime... |
| crngprmringidom 49481 | A commutative ring is a pr... |
| crngprmringdom 49482 | A commutative ring is a pr... |
| dfidom2 49483 | Alternate definition of th... |
| isidom2 49484 | The predicate "is an integ... |
| isidom3 49485 | The predicate "is a domain... |
| idomnzd 49486 | A domain has no zero-divis... |
| idomcanl 49487 | Cancellation law for domai... |
| idomcanr 49488 | Cancellation law for domai... |
| eliunxp2 49489 | Membership in a union of C... |
| mpomptx2 49490 | Express a two-argument fun... |
| cbvmpox2 49491 | Rule to change the bound v... |
| dmmpossx2 49492 | The domain of a mapping is... |
| mpoexxg2 49493 | Existence of an operation ... |
| ovmpordxf 49494 | Value of an operation give... |
| ovmpordx 49495 | Value of an operation give... |
| ovmpox2 49496 | The value of an operation ... |
| fdmdifeqresdif 49497 | The restriction of a condi... |
| ofaddmndmap 49498 | The function operation app... |
| mapsnop 49499 | A singleton of an ordered ... |
| fprmappr 49500 | A function with a domain o... |
| mapprop 49501 | An unordered pair containi... |
| ztprmneprm 49502 | A prime is not an integer ... |
| 2t6m3t4e0 49503 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 49504 | For any finite subset of `... |
| nn0sumltlt 49505 | If the sum of two nonnegat... |
| bcpascm1 49506 | Pascal's rule for the bino... |
| altgsumbc 49507 | The sum of binomial coeffi... |
| altgsumbcALT 49508 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 49509 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 49510 | An element of the (base se... |
| zlmodzxz0 49511 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 49512 | The scalar multiplication ... |
| zlmodzxzadd 49513 | The addition of the ` ZZ `... |
| zlmodzxzsubm 49514 | The subtraction of the ` Z... |
| zlmodzxzsub 49515 | The subtraction of the ` Z... |
| mgpsumunsn 49516 | Extract a summand/factor f... |
| mgpsumz 49517 | If the group sum for the m... |
| mgpsumn 49518 | If the group sum for the m... |
| exple2lt6 49519 | A nonnegative integer to t... |
| pgrple2abl 49520 | Every symmetric group on a... |
| pgrpgt2nabl 49521 | Every symmetric group on a... |
| invginvrid 49522 | Identity for a multiplicat... |
| rmsupp0 49523 | The support of a mapping o... |
| domnmsuppn0 49524 | The support of a mapping o... |
| rmsuppss 49525 | The support of a mapping o... |
| scmsuppss 49526 | The support of a mapping o... |
| rmsuppfi 49527 | The support of a mapping o... |
| rmfsupp 49528 | A mapping of a multiplicat... |
| scmsuppfi 49529 | The support of a mapping o... |
| scmfsupp 49530 | A mapping of a scalar mult... |
| suppmptcfin 49531 | The support of a mapping w... |
| mptcfsupp 49532 | A mapping with value 0 exc... |
| fsuppmptdmf 49533 | A mapping with a finite do... |
| lmodvsmdi 49534 | Multiple distributive law ... |
| gsumlsscl 49535 | Closure of a group sum in ... |
| assaascl0 49536 | The scalar 0 embedded into... |
| assaascl1 49537 | The scalar 1 embedded into... |
| ply1vr1smo 49538 | The variable in a polynomi... |
| ply1sclrmsm 49539 | The ring multiplication of... |
| coe1sclmulval 49540 | The value of the coefficie... |
| ply1mulgsumlem1 49541 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 49542 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 49543 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 49544 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 49545 | The product of two polynom... |
| evl1at0 49546 | Polynomial evaluation for ... |
| evl1at1 49547 | Polynomial evaluation for ... |
| linply1 49548 | A term of the form ` x - C... |
| lineval 49549 | A term of the form ` x - C... |
| linevalexample 49550 | The polynomial ` x - 3 ` o... |
| dmatALTval 49555 | The algebra of ` N ` x ` N... |
| dmatALTbas 49556 | The base set of the algebr... |
| dmatALTbasel 49557 | An element of the base set... |
| dmatbas 49558 | The set of all ` N ` x ` N... |
| lincop 49563 | A linear combination as op... |
| lincval 49564 | The value of a linear comb... |
| dflinc2 49565 | Alternative definition of ... |
| lcoop 49566 | A linear combination as op... |
| lcoval 49567 | The value of a linear comb... |
| lincfsuppcl 49568 | A linear combination of ve... |
| linccl 49569 | A linear combination of ve... |
| lincval0 49570 | The value of an empty line... |
| lincvalsng 49571 | The linear combination ove... |
| lincvalsn 49572 | The linear combination ove... |
| lincvalpr 49573 | The linear combination ove... |
| lincval1 49574 | The linear combination ove... |
| lcosn0 49575 | Properties of a linear com... |
| lincvalsc0 49576 | The linear combination whe... |
| lcoc0 49577 | Properties of a linear com... |
| linc0scn0 49578 | If a set contains the zero... |
| lincdifsn 49579 | A vector is a linear combi... |
| linc1 49580 | A vector is a linear combi... |
| lincellss 49581 | A linear combination of a ... |
| lco0 49582 | The set of empty linear co... |
| lcoel0 49583 | The zero vector is always ... |
| lincsum 49584 | The sum of two linear comb... |
| lincscm 49585 | A linear combinations mult... |
| lincsumcl 49586 | The sum of two linear comb... |
| lincscmcl 49587 | The multiplication of a li... |
| lincsumscmcl 49588 | The sum of a linear combin... |
| lincolss 49589 | According to the statement... |
| ellcoellss 49590 | Every linear combination o... |
| lcoss 49591 | A set of vectors of a modu... |
| lspsslco 49592 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 49593 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 49594 | Equivalence of a _span_ of... |
| rellininds 49598 | The class defining the rel... |
| linindsv 49600 | The classes of the module ... |
| islininds 49601 | The property of being a li... |
| linindsi 49602 | The implications of being ... |
| linindslinci 49603 | The implications of being ... |
| islinindfis 49604 | The property of being a li... |
| islinindfiss 49605 | The property of being a li... |
| linindscl 49606 | A linearly independent set... |
| lindepsnlininds 49607 | A linearly dependent subse... |
| islindeps 49608 | The property of being a li... |
| lincext1 49609 | Property 1 of an extension... |
| lincext2 49610 | Property 2 of an extension... |
| lincext3 49611 | Property 3 of an extension... |
| lindslinindsimp1 49612 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 49613 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 49614 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 49615 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 49616 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 49617 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 49618 | Implication 2 for ~ lindsl... |
| lindslininds 49619 | Equivalence of definitions... |
| linds0 49620 | The empty set is always a ... |
| el0ldep 49621 | A set containing the zero ... |
| el0ldepsnzr 49622 | A set containing the zero ... |
| lindsrng01 49623 | Any subset of a module is ... |
| lindszr 49624 | Any subset of a module ove... |
| snlindsntorlem 49625 | Lemma for ~ snlindsntor . ... |
| snlindsntor 49626 | A singleton is linearly in... |
| ldepsprlem 49627 | Lemma for ~ ldepspr . (Co... |
| ldepspr 49628 | If a vector is a scalar mu... |
| lincresunit3lem3 49629 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 49630 | Lemma 1 for properties of ... |
| lincresunitlem2 49631 | Lemma for properties of a ... |
| lincresunit1 49632 | Property 1 of a specially ... |
| lincresunit2 49633 | Property 2 of a specially ... |
| lincresunit3lem1 49634 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 49635 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 49636 | Property 3 of a specially ... |
| lincreslvec3 49637 | Property 3 of a specially ... |
| islindeps2 49638 | Conditions for being a lin... |
| islininds2 49639 | Implication of being a lin... |
| isldepslvec2 49640 | Alternative definition of ... |
| lindssnlvec 49641 | A singleton not containing... |
| lmod1lem1 49642 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 49643 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 49644 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 49645 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 49646 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 49647 | The (smallest) structure r... |
| lmod1zr 49648 | The (smallest) structure r... |
| lmod1zrnlvec 49649 | There is a (left) module (... |
| lmodn0 49650 | Left modules exist. (Cont... |
| zlmodzxzequa 49651 | Example of an equation wit... |
| zlmodzxznm 49652 | Example of a linearly depe... |
| zlmodzxzldeplem 49653 | A and B are not equal. (C... |
| zlmodzxzequap 49654 | Example of an equation wit... |
| zlmodzxzldeplem1 49655 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 49656 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 49657 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 49658 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 49659 | { A , B } is a linearly de... |
| ldepsnlinclem1 49660 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 49661 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 49662 | The class of all (left) ve... |
| ldepsnlinc 49663 | The reverse implication of... |
| ldepslinc 49664 | For (left) vector spaces, ... |
| suppdm 49665 | If the range of a function... |
| eluz2cnn0n1 49666 | An integer greater than 1 ... |
| divge1b 49667 | The ratio of a real number... |
| divgt1b 49668 | The ratio of a real number... |
| ltsubaddb 49669 | Equivalence for the "less ... |
| ltsubsubb 49670 | Equivalence for the "less ... |
| ltsubadd2b 49671 | Equivalence for the "less ... |
| divsub1dir 49672 | Distribution of division o... |
| expnegico01 49673 | An integer greater than 1 ... |
| elfzolborelfzop1 49674 | An element of a half-open ... |
| pw2m1lepw2m1 49675 | 2 to the power of a positi... |
| zgtp1leeq 49676 | If an integer is between a... |
| flsubz 49677 | An integer can be moved in... |
| nn0onn0ex 49678 | For each odd nonnegative i... |
| nn0enn0ex 49679 | For each even nonnegative ... |
| nnennex 49680 | For each even positive int... |
| nneop 49681 | A positive integer is even... |
| nneom 49682 | A positive integer is even... |
| nn0eo 49683 | A nonnegative integer is e... |
| nnpw2even 49684 | 2 to the power of a positi... |
| zefldiv2 49685 | The floor of an even integ... |
| zofldiv2 49686 | The floor of an odd intege... |
| nn0ofldiv2 49687 | The floor of an odd nonneg... |
| flnn0div2ge 49688 | The floor of a positive in... |
| flnn0ohalf 49689 | The floor of the half of a... |
| logcxp0 49690 | Logarithm of a complex pow... |
| regt1loggt0 49691 | The natural logarithm for ... |
| fdivval 49694 | The quotient of two functi... |
| fdivmpt 49695 | The quotient of two functi... |
| fdivmptf 49696 | The quotient of two functi... |
| refdivmptf 49697 | The quotient of two functi... |
| fdivpm 49698 | The quotient of two functi... |
| refdivpm 49699 | The quotient of two functi... |
| fdivmptfv 49700 | The function value of a qu... |
| refdivmptfv 49701 | The function value of a qu... |
| bigoval 49704 | Set of functions of order ... |
| elbigofrcl 49705 | Reverse closure of the "bi... |
| elbigo 49706 | Properties of a function o... |
| elbigo2 49707 | Properties of a function o... |
| elbigo2r 49708 | Sufficient condition for a... |
| elbigof 49709 | A function of order G(x) i... |
| elbigodm 49710 | The domain of a function o... |
| elbigoimp 49711 | The defining property of a... |
| elbigolo1 49712 | A function (into the posit... |
| rege1logbrege0 49713 | The general logarithm, wit... |
| rege1logbzge0 49714 | The general logarithm, wit... |
| fllogbd 49715 | A real number is between t... |
| relogbmulbexp 49716 | The logarithm of the produ... |
| relogbdivb 49717 | The logarithm of the quoti... |
| logbge0b 49718 | The logarithm of a number ... |
| logblt1b 49719 | The logarithm of a number ... |
| fldivexpfllog2 49720 | The floor of a positive re... |
| nnlog2ge0lt1 49721 | A positive integer is 1 if... |
| logbpw2m1 49722 | The floor of the binary lo... |
| fllog2 49723 | The floor of the binary lo... |
| blenval 49726 | The binary length of an in... |
| blen0 49727 | The binary length of 0. (... |
| blenn0 49728 | The binary length of a "nu... |
| blenre 49729 | The binary length of a pos... |
| blennn 49730 | The binary length of a pos... |
| blennnelnn 49731 | The binary length of a pos... |
| blennn0elnn 49732 | The binary length of a non... |
| blenpw2 49733 | The binary length of a pow... |
| blenpw2m1 49734 | The binary length of a pow... |
| nnpw2blen 49735 | A positive integer is betw... |
| nnpw2blenfzo 49736 | A positive integer is betw... |
| nnpw2blenfzo2 49737 | A positive integer is eith... |
| nnpw2pmod 49738 | Every positive integer can... |
| blen1 49739 | The binary length of 1. (... |
| blen2 49740 | The binary length of 2. (... |
| nnpw2p 49741 | Every positive integer can... |
| nnpw2pb 49742 | A number is a positive int... |
| blen1b 49743 | The binary length of a non... |
| blennnt2 49744 | The binary length of a pos... |
| nnolog2flm1 49745 | The floor of the binary lo... |
| blennn0em1 49746 | The binary length of the h... |
| blennngt2o2 49747 | The binary length of an od... |
| blengt1fldiv2p1 49748 | The binary length of an in... |
| blennn0e2 49749 | The binary length of an ev... |
| digfval 49752 | Operation to obtain the ` ... |
| digval 49753 | The ` K ` th digit of a no... |
| digvalnn0 49754 | The ` K ` th digit of a no... |
| nn0digval 49755 | The ` K ` th digit of a no... |
| dignn0fr 49756 | The digits of the fraction... |
| dignn0ldlem 49757 | Lemma for ~ dignnld . (Co... |
| dignnld 49758 | The leading digits of a po... |
| dig2nn0ld 49759 | The leading digits of a po... |
| dig2nn1st 49760 | The first (relevant) digit... |
| dig0 49761 | All digits of 0 are 0. (C... |
| digexp 49762 | The ` K ` th digit of a po... |
| dig1 49763 | All but one digits of 1 ar... |
| 0dig1 49764 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 49765 | The integers 0 and 1 corre... |
| dig2nn0 49766 | A digit of a nonnegative i... |
| 0dig2nn0e 49767 | The last bit of an even in... |
| 0dig2nn0o 49768 | The last bit of an odd int... |
| dig2bits 49769 | The ` K ` th digit of a no... |
| dignn0flhalflem1 49770 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 49771 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 49772 | The digits of the half of ... |
| dignn0flhalf 49773 | The digits of the rounded ... |
| nn0sumshdiglemA 49774 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 49775 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 49776 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 49777 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 49778 | A nonnegative integer can ... |
| nn0mulfsum 49779 | Trivial algorithm to calcu... |
| nn0mullong 49780 | Standard algorithm (also k... |
| naryfval 49783 | The set of the n-ary (endo... |
| naryfvalixp 49784 | The set of the n-ary (endo... |
| naryfvalel 49785 | An n-ary (endo)function on... |
| naryrcl 49786 | Reverse closure for n-ary ... |
| naryfvalelfv 49787 | The value of an n-ary (end... |
| naryfvalelwrdf 49788 | An n-ary (endo)function on... |
| 0aryfvalel 49789 | A nullary (endo)function o... |
| 0aryfvalelfv 49790 | The value of a nullary (en... |
| 1aryfvalel 49791 | A unary (endo)function on ... |
| fv1arycl 49792 | Closure of a unary (endo)f... |
| 1arympt1 49793 | A unary (endo)function in ... |
| 1arympt1fv 49794 | The value of a unary (endo... |
| 1arymaptfv 49795 | The value of the mapping o... |
| 1arymaptf 49796 | The mapping of unary (endo... |
| 1arymaptf1 49797 | The mapping of unary (endo... |
| 1arymaptfo 49798 | The mapping of unary (endo... |
| 1arymaptf1o 49799 | The mapping of unary (endo... |
| 1aryenef 49800 | The set of unary (endo)fun... |
| 1aryenefmnd 49801 | The set of unary (endo)fun... |
| 2aryfvalel 49802 | A binary (endo)function on... |
| fv2arycl 49803 | Closure of a binary (endo)... |
| 2arympt 49804 | A binary (endo)function in... |
| 2arymptfv 49805 | The value of a binary (end... |
| 2arymaptfv 49806 | The value of the mapping o... |
| 2arymaptf 49807 | The mapping of binary (end... |
| 2arymaptf1 49808 | The mapping of binary (end... |
| 2arymaptfo 49809 | The mapping of binary (end... |
| 2arymaptf1o 49810 | The mapping of binary (end... |
| 2aryenef 49811 | The set of binary (endo)fu... |
| itcoval 49816 | The value of the function ... |
| itcoval0 49817 | A function iterated zero t... |
| itcoval1 49818 | A function iterated once. ... |
| itcoval2 49819 | A function iterated twice.... |
| itcoval3 49820 | A function iterated three ... |
| itcoval0mpt 49821 | A mapping iterated zero ti... |
| itcovalsuc 49822 | The value of the function ... |
| itcovalsucov 49823 | The value of the function ... |
| itcovalendof 49824 | The n-th iterate of an end... |
| itcovalpclem1 49825 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 49826 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 49827 | The value of the function ... |
| itcovalt2lem2lem1 49828 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 49829 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 49830 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 49831 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 49832 | The value of the function ... |
| ackvalsuc1mpt 49833 | The Ackermann function at ... |
| ackvalsuc1 49834 | The Ackermann function at ... |
| ackval0 49835 | The Ackermann function at ... |
| ackval1 49836 | The Ackermann function at ... |
| ackval2 49837 | The Ackermann function at ... |
| ackval3 49838 | The Ackermann function at ... |
| ackendofnn0 49839 | The Ackermann function at ... |
| ackfnnn0 49840 | The Ackermann function at ... |
| ackval0val 49841 | The Ackermann function at ... |
| ackvalsuc0val 49842 | The Ackermann function at ... |
| ackvalsucsucval 49843 | The Ackermann function at ... |
| ackval0012 49844 | The Ackermann function at ... |
| ackval1012 49845 | The Ackermann function at ... |
| ackval2012 49846 | The Ackermann function at ... |
| ackval3012 49847 | The Ackermann function at ... |
| ackval40 49848 | The Ackermann function at ... |
| ackval41a 49849 | The Ackermann function at ... |
| ackval41 49850 | The Ackermann function at ... |
| ackval42 49851 | The Ackermann function at ... |
| ackval42a 49852 | The Ackermann function at ... |
| ackval50 49853 | The Ackermann function at ... |
| fv1prop 49854 | The function value of unor... |
| fv2prop 49855 | The function value of unor... |
| submuladdmuld 49856 | Transformation of a sum of... |
| affinecomb1 49857 | Combination of two real af... |
| affinecomb2 49858 | Combination of two real af... |
| affineid 49859 | Identity of an affine comb... |
| 1subrec1sub 49860 | Subtract the reciprocal of... |
| resum2sqcl 49861 | The sum of two squares of ... |
| resum2sqgt0 49862 | The sum of the square of a... |
| resum2sqrp 49863 | The sum of the square of a... |
| resum2sqorgt0 49864 | The sum of the square of t... |
| reorelicc 49865 | Membership in and outside ... |
| rrx2pxel 49866 | The x-coordinate of a poin... |
| rrx2pyel 49867 | The y-coordinate of a poin... |
| prelrrx2 49868 | An unordered pair of order... |
| prelrrx2b 49869 | An unordered pair of order... |
| rrx2pnecoorneor 49870 | If two different points ` ... |
| rrx2pnedifcoorneor 49871 | If two different points ` ... |
| rrx2pnedifcoorneorr 49872 | If two different points ` ... |
| rrx2xpref1o 49873 | There is a bijection betwe... |
| rrx2xpreen 49874 | The set of points in the t... |
| rrx2plord 49875 | The lexicographical orderi... |
| rrx2plord1 49876 | The lexicographical orderi... |
| rrx2plord2 49877 | The lexicographical orderi... |
| rrx2plordisom 49878 | The set of points in the t... |
| rrx2plordso 49879 | The lexicographical orderi... |
| ehl2eudisval0 49880 | The Euclidean distance of ... |
| ehl2eudis0lt 49881 | An upper bound of the Eucl... |
| lines 49886 | The lines passing through ... |
| line 49887 | The line passing through t... |
| rrxlines 49888 | Definition of lines passin... |
| rrxline 49889 | The line passing through t... |
| rrxlinesc 49890 | Definition of lines passin... |
| rrxlinec 49891 | The line passing through t... |
| eenglngeehlnmlem1 49892 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 49893 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 49894 | The line definition in the... |
| rrx2line 49895 | The line passing through t... |
| rrx2vlinest 49896 | The vertical line passing ... |
| rrx2linest 49897 | The line passing through t... |
| rrx2linesl 49898 | The line passing through t... |
| rrx2linest2 49899 | The line passing through t... |
| elrrx2linest2 49900 | The line passing through t... |
| spheres 49901 | The spheres for given cent... |
| sphere 49902 | A sphere with center ` X `... |
| rrxsphere 49903 | The sphere with center ` M... |
| 2sphere 49904 | The sphere with center ` M... |
| 2sphere0 49905 | The sphere around the orig... |
| line2ylem 49906 | Lemma for ~ line2y . This... |
| line2 49907 | Example for a line ` G ` p... |
| line2xlem 49908 | Lemma for ~ line2x . This... |
| line2x 49909 | Example for a horizontal l... |
| line2y 49910 | Example for a vertical lin... |
| itsclc0lem1 49911 | Lemma for theorems about i... |
| itsclc0lem2 49912 | Lemma for theorems about i... |
| itsclc0lem3 49913 | Lemma for theorems about i... |
| itscnhlc0yqe 49914 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 49915 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 49916 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 49917 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 49918 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 49919 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 49920 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 49921 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 49922 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 49923 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 49924 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 49925 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 49926 | The intersection points of... |
| itsclc0b 49927 | The intersection points of... |
| itsclinecirc0 49928 | The intersection points of... |
| itsclinecirc0b 49929 | The intersection points of... |
| itsclinecirc0in 49930 | The intersection points of... |
| itsclquadb 49931 | Quadratic equation for the... |
| itsclquadeu 49932 | Quadratic equation for the... |
| 2itscplem1 49933 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 49934 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 49935 | Lemma D for ~ 2itscp . (C... |
| 2itscp 49936 | A condition for a quadrati... |
| itscnhlinecirc02plem1 49937 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 49938 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 49939 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 49940 | Intersection of a nonhoriz... |
| inlinecirc02plem 49941 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 49942 | Intersection of a line wit... |
| inlinecirc02preu 49943 | Intersection of a line wit... |
| imbi12d2 49944 | Distribution of implicatio... |
| imbi12d2a 49945 | Variant of ~ imbi12d2 . (... |
| imbi12d3 49946 | Variant of ~ imbi12d2 . (... |
| pm5.32rda 49947 | Distribution of implicatio... |
| pm5.32dar 49948 | Reverse distribution of im... |
| exp12bd 49949 | The import-export theorem ... |
| mpbiran3d 49950 | Equivalence with a conjunc... |
| mpbiran4d 49951 | Equivalence with a conjunc... |
| dtrucor3 49952 | An example of how ~ ax-5 w... |
| ralbidb 49953 | Formula-building rule for ... |
| ralbidc 49954 | Formula-building rule for ... |
| r19.41dv 49955 | A complex deduction form o... |
| rmotru 49956 | Two ways of expressing "at... |
| reutru 49957 | Two ways of expressing "ex... |
| reutruALT 49958 | Alternate proof of ~ reutr... |
| reueqbidva 49959 | Formula-building rule for ... |
| reuxfr1dd 49960 | Transfer existential uniqu... |
| ssdisjd 49961 | Subset preserves disjointn... |
| ssdisjdr 49962 | Subset preserves disjointn... |
| disjdifb 49963 | Relative complement is ant... |
| predisj 49964 | Preimages of disjoint sets... |
| vsn 49965 | The singleton of the unive... |
| mosn 49966 | "At most one" element in a... |
| mo0 49967 | "At most one" element in a... |
| mosssn 49968 | "At most one" element in a... |
| mo0sn 49969 | Two ways of expressing "at... |
| mosssn2 49970 | Two ways of expressing "at... |
| unilbss 49971 | Superclass of the greatest... |
| iuneq0 49972 | An indexed union is empty ... |
| iineq0 49973 | An indexed intersection is... |
| iunlub 49974 | The indexed union is the t... |
| iinglb 49975 | The indexed intersection i... |
| iuneqconst2 49976 | Indexed union of identical... |
| iineqconst2 49977 | Indexed intersection of id... |
| inpw 49978 | Two ways of expressing a c... |
| opth1neg 49979 | Two ordered pairs are not ... |
| opth2neg 49980 | Two ordered pairs are not ... |
| brab2dd 49981 | Expressing that two sets a... |
| brab2ddw 49982 | Expressing that two sets a... |
| brab2ddw2 49983 | Expressing that two sets a... |
| iinxp 49984 | Indexed intersection of Ca... |
| intxpd 49985 | Intersection of Cartesian ... |
| coxp 49986 | Composition with a Cartesi... |
| cosn 49987 | Composition with an ordere... |
| cosni 49988 | Composition with an ordere... |
| inisegn0a 49989 | The inverse image of a sin... |
| dmrnxp 49990 | A Cartesian product is the... |
| mof0 49991 | There is at most one funct... |
| mof02 49992 | A variant of ~ mof0 . (Co... |
| mof0ALT 49993 | Alternate proof of ~ mof0 ... |
| eufsnlem 49994 | There is exactly one funct... |
| eufsn 49995 | There is exactly one funct... |
| eufsn2 49996 | There is exactly one funct... |
| mofsn 49997 | There is at most one funct... |
| mofsn2 49998 | There is at most one funct... |
| mofsssn 49999 | There is at most one funct... |
| mofmo 50000 | There is at most one funct... |
| mofeu 50001 | The uniqueness of a functi... |
| elfvne0 50002 | If a function value has a ... |
| fdomne0 50003 | A function with non-empty ... |
| f1sn2g 50004 | A function that maps a sin... |
| f102g 50005 | A function that maps the e... |
| f1mo 50006 | A function that maps a set... |
| f002 50007 | A function with an empty c... |
| map0cor 50008 | A function exists iff an e... |
| ffvbr 50009 | Relation with function val... |
| xpco2 50010 | Composition of a Cartesian... |
| ovsng 50011 | The operation value of a s... |
| ovsng2 50012 | The operation value of a s... |
| ovsn 50013 | The operation value of a s... |
| ovsn2 50014 | The operation value of a s... |
| ovconstbrd 50015 | Two ways of expressing ` A... |
| ovconstbrn0d 50016 | Two ways of expressing ` A... |
| elovconstbrd 50017 | Two ways of expressing ` A... |
| ovmpt4d 50018 | Deduction version of ~ ovm... |
| eqfnovd 50019 | Deduction for equality of ... |
| fonex 50020 | The domain of a surjection... |
| eloprab1st2nd 50021 | Reconstruction of a nested... |
| resinsnlem 50022 | Lemma for ~ resinsnALT . ... |
| resinsn 50023 | Restriction to the interse... |
| resinsnALT 50024 | Restriction to the interse... |
| dftpos5 50025 | Alternate definition of ` ... |
| dftpos6 50026 | Alternate definition of ` ... |
| dmtposss 50027 | The domain of ` tpos F ` i... |
| tposres0 50028 | The transposition of a set... |
| tposresg 50029 | The transposition restrict... |
| tposrescnv 50030 | The transposition restrict... |
| tposres2 50031 | The transposition restrict... |
| tposres3 50032 | The transposition restrict... |
| tposres 50033 | The transposition restrict... |
| tposresxp 50034 | The transposition restrict... |
| tposf1o 50035 | Condition of a bijective t... |
| tposid 50036 | Swap an ordered pair. (Co... |
| tposidres 50037 | Swap an ordered pair. (Co... |
| tposidf1o 50038 | The swap function, or the ... |
| tposideq 50039 | Two ways of expressing the... |
| tposideq2 50040 | Two ways of expressing the... |
| ixpv 50041 | Infinite Cartesian product... |
| fvconst0ci 50042 | A constant function's valu... |
| fvconstdomi 50043 | A constant function's valu... |
| f1omo 50044 | There is at most one eleme... |
| f1omoOLD 50045 | Obsolete version of ~ f1om... |
| f1omoALT 50046 | There is at most one eleme... |
| iccin 50047 | Intersection of two closed... |
| iccdisj2 50048 | If the upper bound of one ... |
| iccdisj 50049 | If the upper bound of one ... |
| slotresfo 50050 | The condition of a structu... |
| mreuniss 50051 | The union of a collection ... |
| clduni 50052 | The union of closed sets i... |
| opncldbid 50053 | Conditions on open sets ar... |
| opndisj 50054 | Two ways of saying that tw... |
| clddisj 50055 | Two ways of saying that tw... |
| neircl 50056 | Reverse closure of the nei... |
| opnneilem 50057 | Lemma factoring out common... |
| opnneir 50058 | If something is true for a... |
| opnneirv 50059 | A variant of ~ opnneir wit... |
| opnneilv 50060 | The converse of ~ opnneir ... |
| opnneil 50061 | A variant of ~ opnneilv . ... |
| opnneibid 50062 | The equivalence between ne... |
| opnneibid2 50063 | The equivalence between ne... |
| restcls2lem 50064 | A closed set in a subspace... |
| restcls2 50065 | A closed set in a subspace... |
| restclsseplem 50066 | Lemma for ~ restclssep . ... |
| restclssep 50067 | Two disjoint closed sets i... |
| cnneiima 50068 | Given a continuous functio... |
| iooii 50069 | Open intervals are open se... |
| icccldii 50070 | Closed intervals are close... |
| i0oii 50071 | ` ( 0 [,) A ) ` is open in... |
| io1ii 50072 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 50073 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 50074 | Open neighborhood and neig... |
| sepnsepo 50075 | Open neighborhood and neig... |
| sepdisj 50076 | Separated sets are disjoin... |
| seposep 50077 | If two sets are separated ... |
| sepcsepo 50078 | If two sets are separated ... |
| sepfsepc 50079 | If two sets are separated ... |
| seppsepf 50080 | If two sets are precisely ... |
| seppcld 50081 | If two sets are precisely ... |
| isnrm4 50082 | A topological space is nor... |
| dfnrm2 50083 | A topological space is nor... |
| dfnrm3 50084 | A topological space is nor... |
| iscnrm3lem1 50085 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 50086 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem4 50087 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 50088 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 50089 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 50090 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 50091 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 50092 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 50093 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 50094 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 50095 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 50096 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 50097 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 50098 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 50099 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 50100 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 50101 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 50102 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 50103 | A completely normal topolo... |
| iscnrm3v 50104 | A topology is completely n... |
| iscnrm4 50105 | A completely normal topolo... |
| isprsd 50106 | Property of being a preord... |
| lubeldm2 50107 | Member of the domain of th... |
| glbeldm2 50108 | Member of the domain of th... |
| lubeldm2d 50109 | Member of the domain of th... |
| glbeldm2d 50110 | Member of the domain of th... |
| lubsscl 50111 | If a subset of ` S ` conta... |
| glbsscl 50112 | If a subset of ` S ` conta... |
| lubprlem 50113 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 50114 | The set of two comparable ... |
| lubpr 50115 | The LUB of the set of two ... |
| glbprlem 50116 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 50117 | The set of two comparable ... |
| glbpr 50118 | The GLB of the set of two ... |
| joindm2 50119 | The join of any two elemen... |
| joindm3 50120 | The join of any two elemen... |
| meetdm2 50121 | The meet of any two elemen... |
| meetdm3 50122 | The meet of any two elemen... |
| posjidm 50123 | Poset join is idempotent. ... |
| posmidm 50124 | Poset meet is idempotent. ... |
| resiposbas 50125 | Construct a poset ( ~ resi... |
| resipos 50126 | A set equipped with an ord... |
| exbaspos 50127 | There exists a poset for a... |
| exbasprs 50128 | There exists a preordered ... |
| basresposfo 50129 | The base function restrict... |
| basresprsfo 50130 | The base function restrict... |
| posnex 50131 | The class of posets is a p... |
| prsnex 50132 | The class of preordered se... |
| toslat 50133 | A toset is a lattice. (Co... |
| isclatd 50134 | The predicate "is a comple... |
| intubeu 50135 | Existential uniqueness of ... |
| unilbeu 50136 | Existential uniqueness of ... |
| ipolublem 50137 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 50138 | The domain of the LUB of t... |
| ipolub 50139 | The LUB of the inclusion p... |
| ipoglblem 50140 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 50141 | The domain of the GLB of t... |
| ipoglb 50142 | The GLB of the inclusion p... |
| ipolub0 50143 | The LUB of the empty set i... |
| ipolub00 50144 | The LUB of the empty set i... |
| ipoglb0 50145 | The GLB of the empty set i... |
| mrelatlubALT 50146 | Least upper bounds in a Mo... |
| mrelatglbALT 50147 | Greatest lower bounds in a... |
| mreclat 50148 | A Moore space is a complet... |
| topclat 50149 | A topology is a complete l... |
| toplatglb0 50150 | The empty intersection in ... |
| toplatlub 50151 | Least upper bounds in a to... |
| toplatglb 50152 | Greatest lower bounds in a... |
| toplatjoin 50153 | Joins in a topology are re... |
| toplatmeet 50154 | Meets in a topology are re... |
| topdlat 50155 | A topology is a distributi... |
| elmgpcntrd 50156 | The center of a ring. (Co... |
| asclelbasALT 50157 | Alternate proof of ~ ascle... |
| asclcntr 50158 | The algebra scalar lifting... |
| asclcom 50159 | Scalars are commutative af... |
| homf0 50160 | The base is empty iff the ... |
| catprslem 50161 | Lemma for ~ catprs . (Con... |
| catprs 50162 | A preorder can be extracte... |
| catprs2 50163 | A category equipped with t... |
| catprsc 50164 | A construction of the preo... |
| catprsc2 50165 | An alternate construction ... |
| endmndlem 50166 | A diagonal hom-set in a ca... |
| oppccatb 50167 | An opposite category is a ... |
| oppcmndclem 50168 | Lemma for ~ oppcmndc . Ev... |
| oppcendc 50169 | The opposite category of a... |
| oppcmndc 50170 | The opposite category of a... |
| idmon 50171 | An identity arrow, or an i... |
| idepi 50172 | An identity arrow, or an i... |
| sectrcl 50173 | Reverse closure for sectio... |
| sectrcl2 50174 | Reverse closure for sectio... |
| invrcl 50175 | Reverse closure for invers... |
| invrcl2 50176 | Reverse closure for invers... |
| isinv2 50177 | The property " ` F ` is an... |
| isisod 50178 | The predicate "is an isomo... |
| upeu2lem 50179 | Lemma for ~ upeu2 . There... |
| sectfn 50180 | The function value of the ... |
| invfn 50181 | The function value of the ... |
| isofnALT 50182 | The function value of the ... |
| isofval2 50183 | Function value of the func... |
| isorcl 50184 | Reverse closure for isomor... |
| isorcl2 50185 | Reverse closure for isomor... |
| isoval2 50186 | The isomorphisms are the d... |
| sectpropdlem 50187 | Lemma for ~ sectpropd . (... |
| sectpropd 50188 | Two structures with the sa... |
| invpropdlem 50189 | Lemma for ~ invpropd . (C... |
| invpropd 50190 | Two structures with the sa... |
| isopropdlem 50191 | Lemma for ~ isopropd . (C... |
| isopropd 50192 | Two structures with the sa... |
| cicfn 50193 | ` ~=c ` is a function on `... |
| cicrcl2 50194 | Isomorphism implies the st... |
| oppccic 50195 | Isomorphic objects are iso... |
| relcic 50196 | The set of isomorphic obje... |
| cicerALT 50197 | Isomorphism is an equivale... |
| cic1st2nd 50198 | Reconstruction of a pair o... |
| cic1st2ndbr 50199 | Rewrite the predicate of i... |
| cicpropdlem 50200 | Lemma for ~ cicpropd . (C... |
| cicpropd 50201 | Two structures with the sa... |
| oppccicb 50202 | Isomorphic objects are iso... |
| oppcciceq 50203 | The opposite category has ... |
| dmdm 50204 | The double domain of a fun... |
| iinfssclem1 50205 | Lemma for ~ iinfssc . (Co... |
| iinfssclem2 50206 | Lemma for ~ iinfssc . (Co... |
| iinfssclem3 50207 | Lemma for ~ iinfssc . (Co... |
| iinfssc 50208 | Indexed intersection of su... |
| iinfsubc 50209 | Indexed intersection of su... |
| iinfprg 50210 | Indexed intersection of fu... |
| infsubc 50211 | The intersection of two su... |
| infsubc2 50212 | The intersection of two su... |
| infsubc2d 50213 | The intersection of two su... |
| discsubclem 50214 | Lemma for ~ discsubc . (C... |
| discsubc 50215 | A discrete category, whose... |
| iinfconstbaslem 50216 | Lemma for ~ iinfconstbas .... |
| iinfconstbas 50217 | The discrete category is t... |
| nelsubclem 50218 | Lemma for ~ nelsubc . (Co... |
| nelsubc 50219 | An empty "hom-set" for non... |
| nelsubc2 50220 | An empty "hom-set" for non... |
| nelsubc3lem 50221 | Lemma for ~ nelsubc3 . (C... |
| nelsubc3 50222 | Remark 4.2(2) of [Adamek] ... |
| ssccatid 50223 | A category ` C ` restricte... |
| resccatlem 50224 | Lemma for ~ resccat . (Co... |
| resccat 50225 | A class ` C ` restricted b... |
| reldmfunc 50226 | The domain of ` Func ` is ... |
| func1st2nd 50227 | Rewrite the functor predic... |
| func1st 50228 | Extract the first member o... |
| func2nd 50229 | Extract the second member ... |
| funcrcl2 50230 | Reverse closure for a func... |
| funcrcl3 50231 | Reverse closure for a func... |
| funcf2lem 50232 | A utility theorem for prov... |
| funcf2lem2 50233 | A utility theorem for prov... |
| 0funcglem 50234 | Lemma for ~ 0funcg . (Con... |
| 0funcg2 50235 | The functor from the empty... |
| 0funcg 50236 | The functor from the empty... |
| 0funclem 50237 | Lemma for ~ 0funcALT . (C... |
| 0func 50238 | The functor from the empty... |
| 0funcALT 50239 | Alternate proof of ~ 0func... |
| func0g 50240 | The source category of a f... |
| func0g2 50241 | The source category of a f... |
| initc 50242 | Sets with empty base are t... |
| cofu1st2nd 50243 | Rewrite the functor compos... |
| rescofuf 50244 | The restriction of functor... |
| cofu1a 50245 | Value of the object part o... |
| cofu2a 50246 | Value of the morphism part... |
| cofucla 50247 | The composition of two fun... |
| funchomf 50248 | Source categories of a fun... |
| idfurcl 50249 | Reverse closure for an ide... |
| idfu1stf1o 50250 | The identity functor/inclu... |
| idfu1stalem 50251 | Lemma for ~ idfu1sta . (C... |
| idfu1sta 50252 | Value of the object part o... |
| idfu1a 50253 | Value of the object part o... |
| idfu2nda 50254 | Value of the morphism part... |
| imasubclem1 50255 | Lemma for ~ imasubc . (Co... |
| imasubclem2 50256 | Lemma for ~ imasubc . (Co... |
| imasubclem3 50257 | Lemma for ~ imasubc . (Co... |
| imaf1homlem 50258 | Lemma for ~ imaf1hom and o... |
| imaf1hom 50259 | The hom-set of an image of... |
| imaidfu2lem 50260 | Lemma for ~ imaidfu2 . (C... |
| imaidfu 50261 | The image of the identity ... |
| imaidfu2 50262 | The image of the identity ... |
| cofid1a 50263 | Express the object part of... |
| cofid2a 50264 | Express the morphism part ... |
| cofid1 50265 | Express the object part of... |
| cofid2 50266 | Express the morphism part ... |
| cofidvala 50267 | The property " ` F ` is a ... |
| cofidf2a 50268 | If " ` F ` is a section of... |
| cofidf1a 50269 | If " ` F ` is a section of... |
| cofidval 50270 | The property " ` <. F , G ... |
| cofidf2 50271 | If " ` F ` is a section of... |
| cofidf1 50272 | If " ` <. F , G >. ` is a ... |
| oppffn 50275 | ` oppFunc ` is a function ... |
| reldmoppf 50276 | The domain of ` oppFunc ` ... |
| oppfvalg 50277 | Value of the opposite func... |
| oppfrcllem 50278 | Lemma for ~ oppfrcl . (Co... |
| oppfrcl 50279 | If an opposite functor of ... |
| oppfrcl2 50280 | If an opposite functor of ... |
| oppfrcl3 50281 | If an opposite functor of ... |
| oppf1st2nd 50282 | Rewrite the opposite funct... |
| 2oppf 50283 | The double opposite functo... |
| eloppf 50284 | The pre-image of a non-emp... |
| eloppf2 50285 | Both components of a pre-i... |
| oppfvallem 50286 | Lemma for ~ oppfval . (Co... |
| oppfval 50287 | Value of the opposite func... |
| oppfval2 50288 | Value of the opposite func... |
| oppfval3 50289 | Value of the opposite func... |
| oppf1 50290 | Value of the object part o... |
| oppf2 50291 | Value of the morphism part... |
| oppfoppc 50292 | The opposite functor is a ... |
| oppfoppc2 50293 | The opposite functor is a ... |
| funcoppc2 50294 | A functor on opposite cate... |
| funcoppc4 50295 | A functor on opposite cate... |
| funcoppc5 50296 | A functor on opposite cate... |
| 2oppffunc 50297 | The opposite functor of an... |
| funcoppc3 50298 | A functor on opposite cate... |
| oppff1 50299 | The operation generating o... |
| oppff1o 50300 | The operation generating o... |
| cofuoppf 50301 | Composition of opposite fu... |
| imasubc 50302 | An image of a full functor... |
| imasubc2 50303 | An image of a full functor... |
| imassc 50304 | An image of a functor sati... |
| imaid 50305 | An image of a functor pres... |
| imaf1co 50306 | An image of a functor whos... |
| imasubc3 50307 | An image of a functor inje... |
| fthcomf 50308 | Source categories of a fai... |
| idfth 50309 | The inclusion functor is a... |
| idemb 50310 | The inclusion functor is a... |
| idsubc 50311 | The source category of an ... |
| idfullsubc 50312 | The source category of an ... |
| cofidfth 50313 | If " ` F ` is a section of... |
| fulloppf 50314 | The opposite functor of a ... |
| fthoppf 50315 | The opposite functor of a ... |
| ffthoppf 50316 | The opposite functor of a ... |
| upciclem1 50317 | Lemma for ~ upcic , ~ upeu... |
| upciclem2 50318 | Lemma for ~ upciclem3 and ... |
| upciclem3 50319 | Lemma for ~ upciclem4 . (... |
| upciclem4 50320 | Lemma for ~ upcic and ~ up... |
| upcic 50321 | A universal property defin... |
| upeu 50322 | A universal property defin... |
| upeu2 50323 | Generate new universal mor... |
| reldmup 50326 | The domain of ` UP ` is a ... |
| upfval 50327 | Function value of the clas... |
| upfval2 50328 | Function value of the clas... |
| upfval3 50329 | Function value of the clas... |
| isuplem 50330 | Lemma for ~ isup and other... |
| isup 50331 | The predicate "is a univer... |
| uppropd 50332 | If two categories have the... |
| reldmup2 50333 | The domain of ` ( D UP E )... |
| relup 50334 | The set of universal pairs... |
| uprcl 50335 | Reverse closure for the cl... |
| up1st2nd 50336 | Rewrite the universal prop... |
| up1st2ndr 50337 | Combine separated parts in... |
| up1st2ndb 50338 | Combine/separate parts in ... |
| up1st2nd2 50339 | Rewrite the universal prop... |
| uprcl2 50340 | Reverse closure for the cl... |
| uprcl3 50341 | Reverse closure for the cl... |
| uprcl4 50342 | Reverse closure for the cl... |
| uprcl5 50343 | Reverse closure for the cl... |
| uobrcl 50344 | Reverse closure for univer... |
| isup2 50345 | The universal property of ... |
| upeu3 50346 | The universal pair ` <. X ... |
| upeu4 50347 | Generate a new universal m... |
| uptposlem 50348 | Lemma for ~ uptpos . (Con... |
| uptpos 50349 | Rewrite the predicate of u... |
| oppcuprcl4 50350 | Reverse closure for the cl... |
| oppcuprcl3 50351 | Reverse closure for the cl... |
| oppcuprcl5 50352 | Reverse closure for the cl... |
| oppcuprcl2 50353 | Reverse closure for the cl... |
| uprcl2a 50354 | Reverse closure for the cl... |
| oppfuprcl 50355 | Reverse closure for the cl... |
| oppfuprcl2 50356 | Reverse closure for the cl... |
| oppcup3lem 50357 | Lemma for ~ oppcup3 . (Co... |
| oppcup 50358 | The universal pair ` <. X ... |
| oppcup2 50359 | The universal property for... |
| oppcup3 50360 | The universal property for... |
| uptrlem1 50361 | Lemma for ~ uptr . (Contr... |
| uptrlem2 50362 | Lemma for ~ uptr . (Contr... |
| uptrlem3 50363 | Lemma for ~ uptr . (Contr... |
| uptr 50364 | Universal property and ful... |
| uptri 50365 | Universal property and ful... |
| uptra 50366 | Universal property and ful... |
| uptrar 50367 | Universal property and ful... |
| uptrai 50368 | Universal property and ful... |
| uobffth 50369 | A fully faithful functor g... |
| uobeqw 50370 | If a full functor (in fact... |
| uobeq 50371 | If a full functor (in fact... |
| uptr2 50372 | Universal property and ful... |
| uptr2a 50373 | Universal property and ful... |
| isnatd 50374 | Property of being a natura... |
| natrcl2 50375 | Reverse closure for a natu... |
| natrcl3 50376 | Reverse closure for a natu... |
| catbas 50377 | The base of the category s... |
| cathomfval 50378 | The hom-sets of the catego... |
| catcofval 50379 | Composition of the categor... |
| natoppf 50380 | A natural transformation i... |
| natoppf2 50381 | A natural transformation i... |
| natoppfb 50382 | A natural transformation i... |
| initoo2 50383 | An initial object is an ob... |
| termoo2 50384 | A terminal object is an ob... |
| zeroo2 50385 | A zero object is an object... |
| oppcinito 50386 | Initial objects are termin... |
| oppctermo 50387 | Terminal objects are initi... |
| oppczeroo 50388 | Zero objects are zero in t... |
| termoeu2 50389 | Terminal objects are essen... |
| initopropdlemlem 50390 | Lemma for ~ initopropdlem ... |
| initopropdlem 50391 | Lemma for ~ initopropd . ... |
| termopropdlem 50392 | Lemma for ~ termopropd . ... |
| zeroopropdlem 50393 | Lemma for ~ zeroopropd . ... |
| initopropd 50394 | Two structures with the sa... |
| termopropd 50395 | Two structures with the sa... |
| zeroopropd 50396 | Two structures with the sa... |
| reldmxpc 50397 | The binary product of cate... |
| reldmxpcALT 50398 | Alternate proof of ~ reldm... |
| elxpcbasex1 50399 | A non-empty base set of th... |
| elxpcbasex1ALT 50400 | Alternate proof of ~ elxpc... |
| elxpcbasex2 50401 | A non-empty base set of th... |
| elxpcbasex2ALT 50402 | Alternate proof of ~ elxpc... |
| xpcfucbas 50403 | The base set of the produc... |
| xpcfuchomfval 50404 | Set of morphisms of the bi... |
| xpcfuchom 50405 | Set of morphisms of the bi... |
| xpcfuchom2 50406 | Value of the set of morphi... |
| xpcfucco2 50407 | Value of composition in th... |
| xpcfuccocl 50408 | The composition of two nat... |
| xpcfucco3 50409 | Value of composition in th... |
| dfswapf2 50412 | Alternate definition of ` ... |
| swapfval 50413 | Value of the swap functor.... |
| swapfelvv 50414 | A swap functor is an order... |
| swapf2fvala 50415 | The morphism part of the s... |
| swapf2fval 50416 | The morphism part of the s... |
| swapf1vala 50417 | The object part of the swa... |
| swapf1val 50418 | The object part of the swa... |
| swapf2fn 50419 | The morphism part of the s... |
| swapf1a 50420 | The object part of the swa... |
| swapf2vala 50421 | The morphism part of the s... |
| swapf2a 50422 | The morphism part of the s... |
| swapf1 50423 | The object part of the swa... |
| swapf2val 50424 | The morphism part of the s... |
| swapf2 50425 | The morphism part of the s... |
| swapf1f1o 50426 | The object part of the swa... |
| swapf2f1o 50427 | The morphism part of the s... |
| swapf2f1oa 50428 | The morphism part of the s... |
| swapf2f1oaALT 50429 | Alternate proof of ~ swapf... |
| swapfid 50430 | Each identity morphism in ... |
| swapfida 50431 | Each identity morphism in ... |
| swapfcoa 50432 | Composition in the source ... |
| swapffunc 50433 | The swap functor is a func... |
| swapfffth 50434 | The swap functor is a full... |
| swapffunca 50435 | The swap functor is a func... |
| swapfiso 50436 | The swap functor is an iso... |
| swapciso 50437 | The product category is ca... |
| oppc1stflem 50438 | A utility theorem for prov... |
| oppc1stf 50439 | The opposite functor of th... |
| oppc2ndf 50440 | The opposite functor of th... |
| 1stfpropd 50441 | If two categories have the... |
| 2ndfpropd 50442 | If two categories have the... |
| diagpropd 50443 | If two categories have the... |
| cofuswapfcl 50444 | The bifunctor pre-composed... |
| cofuswapf1 50445 | The object part of a bifun... |
| cofuswapf2 50446 | The morphism part of a bif... |
| tposcurf1cl 50447 | The partially evaluated tr... |
| tposcurf11 50448 | Value of the double evalua... |
| tposcurf12 50449 | The partially evaluated tr... |
| tposcurf1 50450 | Value of the object part o... |
| tposcurf2 50451 | Value of the transposed cu... |
| tposcurf2val 50452 | Value of a component of th... |
| tposcurf2cl 50453 | The transposed curry funct... |
| tposcurfcl 50454 | The transposed curry funct... |
| diag1 50455 | The constant functor of ` ... |
| diag1a 50456 | The constant functor of ` ... |
| diag1f1lem 50457 | The object part of the dia... |
| diag1f1 50458 | The object part of the dia... |
| diag2f1lem 50459 | Lemma for ~ diag2f1 . The... |
| diag2f1 50460 | If ` B ` is non-empty, the... |
| fucofulem1 50461 | Lemma for proving functor ... |
| fucofulem2 50462 | Lemma for proving functor ... |
| fuco2el 50463 | Equivalence of product fun... |
| fuco2eld 50464 | Equivalence of product fun... |
| fuco2eld2 50465 | Equivalence of product fun... |
| fuco2eld3 50466 | Equivalence of product fun... |
| fucofvalg 50469 | Value of the function givi... |
| fucofval 50470 | Value of the function givi... |
| fucoelvv 50471 | A functor composition bifu... |
| fuco1 50472 | The object part of the fun... |
| fucof1 50473 | The object part of the fun... |
| fuco2 50474 | The morphism part of the f... |
| fucofn2 50475 | The morphism part of the f... |
| fucofvalne 50476 | Value of the function givi... |
| fuco11 50477 | The object part of the fun... |
| fuco11cl 50478 | The object part of the fun... |
| fuco11a 50479 | The object part of the fun... |
| fuco112 50480 | The object part of the fun... |
| fuco111 50481 | The object part of the fun... |
| fuco111x 50482 | The object part of the fun... |
| fuco112x 50483 | The object part of the fun... |
| fuco112xa 50484 | The object part of the fun... |
| fuco11id 50485 | The identity morphism of t... |
| fuco11idx 50486 | The identity morphism of t... |
| fuco21 50487 | The morphism part of the f... |
| fuco11b 50488 | The object part of the fun... |
| fuco11bALT 50489 | Alternate proof of ~ fuco1... |
| fuco22 50490 | The morphism part of the f... |
| fucofn22 50491 | The morphism part of the f... |
| fuco23 50492 | The morphism part of the f... |
| fuco22natlem1 50493 | Lemma for ~ fuco22nat . T... |
| fuco22natlem2 50494 | Lemma for ~ fuco22nat . T... |
| fuco22natlem3 50495 | Combine ~ fuco22natlem2 wi... |
| fuco22natlem 50496 | The composed natural trans... |
| fuco22nat 50497 | The composed natural trans... |
| fucof21 50498 | The morphism part of the f... |
| fucoid 50499 | Each identity morphism in ... |
| fucoid2 50500 | Each identity morphism in ... |
| fuco22a 50501 | The morphism part of the f... |
| fuco23alem 50502 | The naturality property ( ... |
| fuco23a 50503 | The morphism part of the f... |
| fucocolem1 50504 | Lemma for ~ fucoco . Asso... |
| fucocolem2 50505 | Lemma for ~ fucoco . The ... |
| fucocolem3 50506 | Lemma for ~ fucoco . The ... |
| fucocolem4 50507 | Lemma for ~ fucoco . The ... |
| fucoco 50508 | Composition in the source ... |
| fucoco2 50509 | Composition in the source ... |
| fucofunc 50510 | The functor composition bi... |
| fucofunca 50511 | The functor composition bi... |
| fucolid 50512 | Post-compose a natural tra... |
| fucorid 50513 | Pre-composing a natural tr... |
| fucorid2 50514 | Pre-composing a natural tr... |
| postcofval 50515 | Value of the post-composit... |
| postcofcl 50516 | The post-composition funct... |
| precofvallem 50517 | Lemma for ~ precofval to e... |
| precofval 50518 | Value of the pre-compositi... |
| precofvalALT 50519 | Alternate proof of ~ preco... |
| precofval2 50520 | Value of the pre-compositi... |
| precofcl 50521 | The pre-composition functo... |
| precofval3 50522 | Value of the pre-compositi... |
| precoffunc 50523 | The pre-composition functo... |
| reldmprcof 50526 | The domain of ` -o.F ` is ... |
| prcofvalg 50527 | Value of the pre-compositi... |
| prcofvala 50528 | Value of the pre-compositi... |
| prcofval 50529 | Value of the pre-compositi... |
| prcofpropd 50530 | If the categories have the... |
| prcofelvv 50531 | The pre-composition functo... |
| reldmprcof1 50532 | The domain of the object p... |
| reldmprcof2 50533 | The domain of the morphism... |
| prcoftposcurfuco 50534 | The pre-composition functo... |
| prcoftposcurfucoa 50535 | The pre-composition functo... |
| prcoffunc 50536 | The pre-composition functo... |
| prcoffunca 50537 | The pre-composition functo... |
| prcoffunca2 50538 | The pre-composition functo... |
| prcof1 50539 | The object part of the pre... |
| prcof2a 50540 | The morphism part of the p... |
| prcof2 50541 | The morphism part of the p... |
| prcof21a 50542 | The morphism part of the p... |
| prcof22a 50543 | The morphism part of the p... |
| prcofdiag1 50544 | A constant functor pre-com... |
| prcofdiag 50545 | A diagonal functor post-co... |
| catcrcl 50546 | Reverse closure for the ca... |
| catcrcl2 50547 | Reverse closure for the ca... |
| elcatchom 50548 | A morphism of the category... |
| catcsect 50549 | The property " ` F ` is a ... |
| catcinv 50550 | The property " ` F ` is an... |
| catcisoi 50551 | A functor is an isomorphis... |
| uobeq2 50552 | If a full functor (in fact... |
| uobeq3 50553 | An isomorphism between cat... |
| opf11 50554 | The object part of the op ... |
| opf12 50555 | The object part of the op ... |
| opf2fval 50556 | The morphism part of the o... |
| opf2 50557 | The morphism part of the o... |
| fucoppclem 50558 | Lemma for ~ fucoppc . (Co... |
| fucoppcid 50559 | The opposite category of f... |
| fucoppcco 50560 | The opposite category of f... |
| fucoppc 50561 | The isomorphism from the o... |
| fucoppcffth 50562 | A fully faithful functor f... |
| fucoppcfunc 50563 | A functor from the opposit... |
| fucoppccic 50564 | The opposite category of f... |
| oppfdiag1 50565 | A constant functor for opp... |
| oppfdiag1a 50566 | A constant functor for opp... |
| oppfdiag 50567 | A diagonal functor for opp... |
| isthinc 50570 | The predicate "is a thin c... |
| isthinc2 50571 | A thin category is a categ... |
| isthinc3 50572 | A thin category is a categ... |
| thinccat 50573 | A thin category is a categ... |
| thinccatd 50574 | A thin category is a categ... |
| thincssc 50575 | A thin category is a categ... |
| isthincd2lem1 50576 | Lemma for ~ isthincd2 and ... |
| thincmo2 50577 | Morphisms in the same hom-... |
| thinchom 50578 | A non-empty hom-set of a t... |
| thincmo 50579 | There is at most one morph... |
| thincmoALT 50580 | Alternate proof of ~ thinc... |
| thincmod 50581 | At most one morphism in ea... |
| thincn0eu 50582 | In a thin category, a hom-... |
| thincid 50583 | In a thin category, a morp... |
| thincmon 50584 | In a thin category, all mo... |
| thincepi 50585 | In a thin category, all mo... |
| isthincd2lem2 50586 | Lemma for ~ isthincd2 . (... |
| isthincd 50587 | The predicate "is a thin c... |
| isthincd2 50588 | The predicate " ` C ` is a... |
| oppcthin 50589 | The opposite category of a... |
| oppcthinco 50590 | If the opposite category o... |
| oppcthinendc 50591 | The opposite category of a... |
| oppcthinendcALT 50592 | Alternate proof of ~ oppct... |
| thincpropd 50593 | Two structures with the sa... |
| subthinc 50594 | A subcategory of a thin ca... |
| functhinclem1 50595 | Lemma for ~ functhinc . G... |
| functhinclem2 50596 | Lemma for ~ functhinc . (... |
| functhinclem3 50597 | Lemma for ~ functhinc . T... |
| functhinclem4 50598 | Lemma for ~ functhinc . O... |
| functhinc 50599 | A functor to a thin catego... |
| functhincfun 50600 | A functor to a thin catego... |
| fullthinc 50601 | A functor to a thin catego... |
| fullthinc2 50602 | A full functor to a thin c... |
| thincfth 50603 | A functor from a thin cate... |
| thincciso 50604 | Two thin categories are is... |
| thinccisod 50605 | Two thin categories are is... |
| thincciso2 50606 | Categories isomorphic to a... |
| thincciso3 50607 | Categories isomorphic to a... |
| thincciso4 50608 | Two isomorphic categories ... |
| 0thincg 50609 | Any structure with an empt... |
| 0thinc 50610 | The empty category (see ~ ... |
| indcthing 50611 | An indiscrete category, i.... |
| discthing 50612 | A discrete category, i.e.,... |
| indthinc 50613 | An indiscrete category in ... |
| indthincALT 50614 | An alternate proof of ~ in... |
| prsthinc 50615 | Preordered sets as categor... |
| setcthin 50616 | A category of sets all of ... |
| setc2othin 50617 | The category ` ( SetCat ``... |
| thincsect 50618 | In a thin category, one mo... |
| thincsect2 50619 | In a thin category, ` F ` ... |
| thincinv 50620 | In a thin category, ` F ` ... |
| thinciso 50621 | In a thin category, ` F : ... |
| thinccic 50622 | In a thin category, two ob... |
| istermc 50625 | The predicate "is a termin... |
| istermc2 50626 | The predicate "is a termin... |
| istermc3 50627 | The predicate "is a termin... |
| termcthin 50628 | A terminal category is a t... |
| termcthind 50629 | A terminal category is a t... |
| termccatd 50630 | A terminal category is a c... |
| termcbas 50631 | The base of a terminal cat... |
| termco 50632 | The object of a terminal c... |
| termcbas2 50633 | The base of a terminal cat... |
| termcbasmo 50634 | Two objects in a terminal ... |
| termchomn0 50635 | All hom-sets of a terminal... |
| termchommo 50636 | All morphisms of a termina... |
| termcid 50637 | The morphism of a terminal... |
| termcid2 50638 | The morphism of a terminal... |
| termchom 50639 | The hom-set of a terminal ... |
| termchom2 50640 | The hom-set of a terminal ... |
| setcsnterm 50641 | The category of one set, e... |
| setc1oterm 50642 | The category ` ( SetCat ``... |
| setc1obas 50643 | The base of the trivial ca... |
| setc1ohomfval 50644 | Set of morphisms of the tr... |
| setc1ocofval 50645 | Composition in the trivial... |
| setc1oid 50646 | The identity morphism of t... |
| funcsetc1ocl 50647 | The functor to the trivial... |
| funcsetc1o 50648 | Value of the functor to th... |
| isinito2lem 50649 | The predicate "is an initi... |
| isinito2 50650 | The predicate "is an initi... |
| isinito3 50651 | The predicate "is an initi... |
| dfinito4 50652 | An alternate definition of... |
| dftermo4 50653 | An alternate definition of... |
| termcpropd 50654 | Two structures with the sa... |
| oppctermhom 50655 | The opposite category of a... |
| oppctermco 50656 | The opposite category of a... |
| oppcterm 50657 | The opposite category of a... |
| functermclem 50658 | Lemma for ~ functermc . (... |
| functermc 50659 | Functor to a terminal cate... |
| functermc2 50660 | Functor to a terminal cate... |
| functermceu 50661 | There exists a unique func... |
| fulltermc 50662 | A functor to a terminal ca... |
| fulltermc2 50663 | Given a full functor to a ... |
| termcterm 50664 | A terminal category is a t... |
| termcterm2 50665 | A terminal object of the c... |
| termcterm3 50666 | In the category of small c... |
| termcciso 50667 | A category is isomorphic t... |
| termccisoeu 50668 | The isomorphism between te... |
| termc2 50669 | If there exists a unique f... |
| termc 50670 | Alternate definition of ` ... |
| dftermc2 50671 | Alternate definition of ` ... |
| eufunclem 50672 | If there exists a unique f... |
| eufunc 50673 | If there exists a unique f... |
| idfudiag1lem 50674 | Lemma for ~ idfudiag1bas a... |
| idfudiag1bas 50675 | If the identity functor of... |
| idfudiag1 50676 | If the identity functor of... |
| euendfunc 50677 | If there exists a unique e... |
| euendfunc2 50678 | If there exists a unique e... |
| termcarweu 50679 | There exists a unique disj... |
| arweuthinc 50680 | If a structure has a uniqu... |
| arweutermc 50681 | If a structure has a uniqu... |
| dftermc3 50682 | Alternate definition of ` ... |
| termcfuncval 50683 | The value of a functor fro... |
| diag1f1olem 50684 | To any functor from a term... |
| diag1f1o 50685 | The object part of the dia... |
| termcnatval 50686 | Value of natural transform... |
| diag2f1olem 50687 | Lemma for ~ diag2f1o . (C... |
| diag2f1o 50688 | If ` D ` is terminal, the ... |
| diagffth 50689 | The diagonal functor is a ... |
| diagciso 50690 | The diagonal functor is an... |
| diagcic 50691 | Any category ` C ` is isom... |
| funcsn 50692 | The category of one functo... |
| fucterm 50693 | The category of functors t... |
| 0fucterm 50694 | The category of functors f... |
| termfucterm 50695 | All functors between two t... |
| cofuterm 50696 | Post-compose with a functo... |
| uobeqterm 50697 | Universal objects and term... |
| isinito4 50698 | The predicate "is an initi... |
| isinito4a 50699 | The predicate "is an initi... |
| prstcval 50702 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 50703 | Lemma for ~ prstcnid and ~... |
| prstcnid 50704 | Components other than ` Ho... |
| prstcbas 50705 | The base set is unchanged.... |
| prstcleval 50706 | Value of the less-than-or-... |
| prstcle 50707 | Value of the less-than-or-... |
| prstcocval 50708 | Orthocomplementation is un... |
| prstcoc 50709 | Orthocomplementation is un... |
| prstchomval 50710 | Hom-sets of the constructe... |
| prstcprs 50711 | The category is a preorder... |
| prstcthin 50712 | The preordered set is equi... |
| prstchom 50713 | Hom-sets of the constructe... |
| prstchom2 50714 | Hom-sets of the constructe... |
| prstchom2ALT 50715 | Hom-sets of the constructe... |
| oduoppcbas 50716 | The dual of a preordered s... |
| oduoppcciso 50717 | The dual of a preordered s... |
| postcpos 50718 | The converted category is ... |
| postcposALT 50719 | Alternate proof of ~ postc... |
| postc 50720 | The converted category is ... |
| discsntermlem 50721 | A singlegon is an element ... |
| basrestermcfolem 50722 | An element of the class of... |
| discbas 50723 | A discrete category (a cat... |
| discthin 50724 | A discrete category (a cat... |
| discsnterm 50725 | A discrete category (a cat... |
| basrestermcfo 50726 | The base function restrict... |
| termcnex 50727 | The class of all terminal ... |
| mndtcval 50730 | Value of the category buil... |
| mndtcbasval 50731 | The base set of the catego... |
| mndtcbaseu 50732 | The category built from a ... |
| mndtcob 50733 | Lemma for ~ mndtchom and ~... |
| mndtcobeq 50734 | Two objects in a category ... |
| mndtchom 50735 | The only hom-set of the ca... |
| mndtcco 50736 | The composition of the cat... |
| mndtcco2 50737 | The composition of the cat... |
| mndtccatid 50738 | Lemma for ~ mndtccat and ~... |
| mndtccat 50739 | The function value is a ca... |
| mndtcid 50740 | The identity morphism, or ... |
| oppgoppchom 50741 | The converted opposite mon... |
| oppgoppcco 50742 | The converted opposite mon... |
| oppgoppcid 50743 | The converted opposite mon... |
| grptcmon 50744 | All morphisms in a categor... |
| grptcepi 50745 | All morphisms in a categor... |
| 2arwcatlem1 50746 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem2 50747 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem3 50748 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem4 50749 | Lemma for ~ 2arwcat . (Co... |
| 2arwcatlem5 50750 | Lemma for ~ 2arwcat . (Co... |
| 2arwcat 50751 | The condition for a struct... |
| incat 50752 | Constructing a category wi... |
| setc1onsubc 50753 | Construct a category with ... |
| cnelsubclem 50754 | Lemma for ~ cnelsubc . (C... |
| cnelsubc 50755 | Remark 4.2(2) of [Adamek] ... |
| lanfn 50760 | ` Lan ` is a function on `... |
| ranfn 50761 | ` Ran ` is a function on `... |
| reldmlan 50762 | The domain of ` Lan ` is a... |
| reldmran 50763 | The domain of ` Ran ` is a... |
| lanfval 50764 | Value of the function gene... |
| ranfval 50765 | Value of the function gene... |
| lanpropd 50766 | If the categories have the... |
| ranpropd 50767 | If the categories have the... |
| reldmlan2 50768 | The domain of ` ( P Lan E ... |
| reldmran2 50769 | The domain of ` ( P Ran E ... |
| lanval 50770 | Value of the set of left K... |
| ranval 50771 | Value of the set of right ... |
| lanrcl 50772 | Reverse closure for left K... |
| ranrcl 50773 | Reverse closure for right ... |
| rellan 50774 | The set of left Kan extens... |
| relran 50775 | The set of right Kan exten... |
| islan 50776 | A left Kan extension is a ... |
| islan2 50777 | A left Kan extension is a ... |
| lanval2 50778 | The set of left Kan extens... |
| isran 50779 | A right Kan extension is a... |
| isran2 50780 | A right Kan extension is a... |
| ranval2 50781 | The set of right Kan exten... |
| ranval3 50782 | The set of right Kan exten... |
| lanrcl2 50783 | Reverse closure for left K... |
| lanrcl3 50784 | Reverse closure for left K... |
| lanrcl4 50785 | The first component of a l... |
| lanrcl5 50786 | The second component of a ... |
| ranrcl2 50787 | Reverse closure for right ... |
| ranrcl3 50788 | Reverse closure for right ... |
| ranrcl4lem 50789 | Lemma for ~ ranrcl4 and ~ ... |
| ranrcl4 50790 | The first component of a r... |
| ranrcl5 50791 | The second component of a ... |
| lanup 50792 | The universal property of ... |
| ranup 50793 | The universal property of ... |
| reldmlmd 50798 | The domain of ` Limit ` is... |
| reldmcmd 50799 | The domain of ` Colimit ` ... |
| lmdfval 50800 | Function value of ` Limit ... |
| cmdfval 50801 | Function value of ` Colimi... |
| lmdrcl 50802 | Reverse closure for a limi... |
| cmdrcl 50803 | Reverse closure for a coli... |
| reldmlmd2 50804 | The domain of ` ( C Limit ... |
| reldmcmd2 50805 | The domain of ` ( C Colimi... |
| lmdfval2 50806 | The set of limits of a dia... |
| cmdfval2 50807 | The set of colimits of a d... |
| lmdpropd 50808 | If the categories have the... |
| cmdpropd 50809 | If the categories have the... |
| rellmd 50810 | The set of limits of a dia... |
| relcmd 50811 | The set of colimits of a d... |
| concl 50812 | A natural transformation f... |
| coccl 50813 | A natural transformation t... |
| concom 50814 | A cone to a diagram commut... |
| coccom 50815 | A co-cone to a diagram com... |
| islmd 50816 | The universal property of ... |
| iscmd 50817 | The universal property of ... |
| lmddu 50818 | The duality of limits and ... |
| cmddu 50819 | The duality of limits and ... |
| initocmd 50820 | Initial objects are the ob... |
| termolmd 50821 | Terminal objects are the o... |
| lmdran 50822 | To each limit of a diagram... |
| cmdlan 50823 | To each colimit of a diagr... |
| nfintd 50824 | Bound-variable hypothesis ... |
| nfiund 50825 | Bound-variable hypothesis ... |
| nfiundg 50826 | Bound-variable hypothesis ... |
| iunord 50827 | The indexed union of a col... |
| iunordi 50828 | The indexed union of a col... |
| spd 50829 | Specialization deduction, ... |
| tfis2d 50830 | Transfinite Induction Sche... |
| setrecseq 50831 | Equality theorem for set r... |
| nfsetrecs 50832 | Bound-variable hypothesis ... |
| setrec2mpt 50833 | Version of ~ setrec2 where... |
| setis 50834 | Version of ~ setrec2 expre... |
| elsetrecslem 50835 | Lemma for ~ elsetrecs . A... |
| elsetrecs 50836 | A set ` A ` is an element ... |
| setrecsss 50837 | The ` setrecs ` operator r... |
| setrecsres 50838 | A recursively generated cl... |
| vsetrec 50839 | Construct ` _V ` using set... |
| 0setrec 50840 | If a function sends the em... |
| onsetreclem1 50841 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 50842 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 50843 | Lemma for ~ onsetrec . (C... |
| onsetrec 50844 | Construct ` On ` using set... |
| elpglem1 50847 | Lemma for ~ elpg . (Contr... |
| elpglem2 50848 | Lemma for ~ elpg . (Contr... |
| elpglem3 50849 | Lemma for ~ elpg . (Contr... |
| elpg 50850 | Membership in the class of... |
| pgindlem 50851 | Lemma for ~ pgind . (Cont... |
| pgindnf 50852 | Version of ~ pgind with ex... |
| pgind 50853 | Induction on partizan game... |
| sbidd 50854 | An identity theorem for su... |
| sbidd-misc 50855 | An identity theorem for su... |
| gte-lte 50860 | Simple relationship betwee... |
| gt-lt 50861 | Simple relationship betwee... |
| gte-lteh 50862 | Relationship between ` <_ ... |
| gt-lth 50863 | Relationship between ` < `... |
| ex-gt 50864 | Simple example of ` > ` , ... |
| ex-gte 50865 | Simple example of ` >_ ` ,... |
| sinhval-named 50872 | Value of the named sinh fu... |
| coshval-named 50873 | Value of the named cosh fu... |
| tanhval-named 50874 | Value of the named tanh fu... |
| sinh-conventional 50875 | Conventional definition of... |
| sinhpcosh 50876 | Prove that ` ( sinh `` A )... |
| secval 50883 | Value of the secant functi... |
| cscval 50884 | Value of the cosecant func... |
| cotval 50885 | Value of the cotangent fun... |
| seccl 50886 | The closure of the secant ... |
| csccl 50887 | The closure of the cosecan... |
| cotcl 50888 | The closure of the cotange... |
| reseccl 50889 | The closure of the secant ... |
| recsccl 50890 | The closure of the cosecan... |
| recotcl 50891 | The closure of the cotange... |
| recsec 50892 | The reciprocal of secant i... |
| reccsc 50893 | The reciprocal of cosecant... |
| reccot 50894 | The reciprocal of cotangen... |
| rectan 50895 | The reciprocal of tangent ... |
| sec0 50896 | The value of the secant fu... |
| onetansqsecsq 50897 | Prove the tangent squared ... |
| cotsqcscsq 50898 | Prove the tangent squared ... |
| dvsec 50899 | Derivative of the secant f... |
| dvcsc 50900 | Derivative of the cosecant... |
| dvcot 50901 | Derivative of the cotangen... |
| ifnmfalse 50902 | If A is not a member of B,... |
| logb2aval 50903 | Define the value of the ` ... |
| mvlraddi 50910 | Move the right term in a s... |
| assraddsubi 50911 | Associate RHS addition-sub... |
| joinlmuladdmuli 50912 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 50913 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 50914 | Join AB-CB into (A-C) on L... |
| mvlrmuld 50915 | Move the right term in a p... |
| mvlrmuli 50916 | Move the right term in a p... |
| i2linesi 50917 | Solve for the intersection... |
| i2linesd 50918 | Solve for the intersection... |
| alimp-surprise 50919 | Demonstrate that when usin... |
| alimp-no-surprise 50920 | There is no "surprise" in ... |
| empty-surprise 50921 | Demonstrate that when usin... |
| empty-surprise2 50922 | "Prove" that false is true... |
| eximp-surprise 50923 | Show what implication insi... |
| eximp-surprise2 50924 | Show that "there exists" w... |
| dfrals2 50929 | The bounded "all some" for... |
| alsd 50930 | Introduction rule: "all s... |
| ralsd 50931 | Introduction rule for "all... |
| als1d 50932 | Deduction rule: Given "al... |
| als2d 50933 | Deduction rule: Given "al... |
| rals1d 50934 | Deduction rule: Given "al... |
| rals2d 50935 | Deduction rule: Given "al... |
| ralsn0d 50936 | Deduction rule: Given "al... |
| alsex 50937 | The consequent of an "all ... |
| ralsex 50938 | The consequent of an "all ... |
| alsbii 50939 | Congruence: equivalents ma... |
| ralsbii 50940 | Congruence for "all some" ... |
| alsbid 50941 | Deduction form of ~ alsbii... |
| nfals 50942 | Bound-variable hypothesis ... |
| nfrals 50943 | Bound-variable hypothesis ... |
| cbvals 50944 | Rule used to change bound ... |
| als-no-surprise 50945 | Demonstrate that there is ... |
| rals-no-surprise 50946 | Demonstrate that there is ... |
| ralrals 50947 | If the universal part of a... |
| rexrals 50948 | If a member of ` A ` satis... |
| alsanmo 50949 | An "all some" statement co... |
| ralsanmo 50950 | An "all some" statement re... |
| alsralrex 50951 | The general "all some" qua... |
| alsraln0 50952 | The general "all some" qua... |
| ralals 50953 | If ` ph ` holds for every ... |
| rexals 50954 | If some ` x ` in ` A ` sat... |
| n0als 50955 | If ` A ` is not empty, the... |
| 2alsraln0 50956 | Nested general "all some" ... |
| 2alsraln0id 50957 | Nested general "all some" ... |
| dfralseu2 50962 | The bounded "all some one"... |
| alseuals 50963 | "All some one" implies "al... |
| ralseurals 50964 | "All some one" restricted ... |
| alseud 50965 | Introduction rule: "all s... |
| ralseud 50966 | Introduction rule for "all... |
| alseu1d 50967 | Deduction rule: Given "al... |
| alseu2d 50968 | Deduction rule: Given "al... |
| ralseu1d 50969 | Deduction rule: Given "al... |
| ralseu2d 50970 | Deduction rule: Given "al... |
| alseubii 50971 | Congruence: equivalents ma... |
| ralseubii 50972 | Congruence for "all some o... |
| nfalseu 50973 | Bound-variable hypothesis ... |
| nfralseu 50974 | Bound-variable hypothesis ... |
| dfalseu2 50975 | An "all some one" statemen... |
| alseueu 50976 | "The ` ph ` is ` ps ` " im... |
| alseu-no-surprise 50977 | Demonstrate that there is ... |
| 5m4e1 50978 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 50979 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 50980 | Resolution rule. This is ... |
| testable 50981 | In classical logic all wff... |
| aacllem 50982 | Lemma for other theorems a... |
| wrdf1d 50983 | A one-to-one word maps its... |
| 1ne3 50984 | ` 1 ` is not equal to ` 3 ... |
| 2ne3 50985 | ` 2 ` is not equal to ` 3 ... |
| 1elfz13 50986 | Membership of 1 in the int... |
| 2elfz13 50987 | Membership of 2 in the int... |
| 3elfz13 50988 | Membership of 3 in the int... |
| rr3fvcl 50989 | The components of a 3-dime... |
| rr3fv1cld 50990 | First component of a 3-dim... |
| rr3fv2cld 50991 | Second component of a 3-di... |
| rr3fv3cld 50992 | Third component of a 3-dim... |
| crosspval 50997 | Value of the cross product... |
| crosspcle1d 50998 | Closure of the first compo... |
| crosspcle2d 50999 | Closure of the second comp... |
| crosspcle3d 51000 | Closure of the third compo... |
| crosspclem 51001 | Lemma for ~ crosspcld . C... |
| crosspcld 51002 | Closure of the cross produ... |
| crosspv1d 51003 | Value of the first compone... |
| crosspv2d 51004 | Value of the second compon... |
| crosspv3d 51005 | Value of the third compone... |
| crosspdot0lem 51006 | Lemma for ~ crosspdotd . ... |
| crosspdotsumlem 51007 | Lemma for ~ crosspdotd . ... |
| crosspdotd 51008 | Value of the scalar triple... |
| crosspaltd 51009 | Antisymmetry of the cross ... |
| crossp3d 51010 | The vector triple product ... |
| nellindf 51013 | A nonzero coefficient vect... |
| veronesevald 51014 | Value of the Veronese map ... |
| veronesefvcl 51015 | Every coordinate of the Ve... |
| veronesev1lem 51016 | Lemma for ~ veronesevrowd ... |
| veronesev2lem 51017 | Lemma for ~ veronesevrowd ... |
| veronesev3lem 51018 | Lemma for ~ veronesevrowd ... |
| veronesev4lem 51019 | Lemma for ~ veronesevrowd ... |
| veronesev5lem 51020 | Lemma for ~ veronesevrowd ... |
| veronesev6lem 51021 | Lemma for ~ veronesevrowd ... |
| veronesevrowd 51022 | The Veronese map at a poin... |
| veronesematbasd 51023 | The matrix whose ` i ` -th... |
| veronesematrowd 51024 | Currying the Veronese matr... |
| veronesematrowexpd 51025 | Currying the Veronese matr... |
| veroquadgsumlem 51026 | Lemma for ~ veroquadmodzer... |
| veroquadmodzerod 51027 | The columns of the Verones... |
| veroquadnolindfd 51028 | A nonzero homogeneous quad... |
| veroquaddetzerod 51029 | The Veronese matrix of six... |
| amgmwlem 51030 | Weighted version of ~ amgm... |
| amgmlemALT 51031 | Alternate proof of ~ amgml... |
| amgmw2d 51032 | Weighted arithmetic-geomet... |
| young2d 51033 | Young's inequality for ` n... |
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