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Mirrors > Home > ILE Home > Th. List > divides | Unicode version |
Description: Define the divides relation. means divides into with no remainder. For example, (ex-dvds 10567). As proven in dvdsval3 10199, . See divides 10197 and dvdsval2 10198 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divides |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3786 | . . 3 | |
2 | df-dvds 10196 | . . . 4 | |
3 | 2 | eleq2i 2145 | . . 3 |
4 | 1, 3 | bitri 182 | . 2 |
5 | oveq2 5540 | . . . . 5 | |
6 | 5 | eqeq1d 2089 | . . . 4 |
7 | 6 | rexbidv 2369 | . . 3 |
8 | eqeq2 2090 | . . . 4 | |
9 | 8 | rexbidv 2369 | . . 3 |
10 | 7, 9 | opelopab2 4025 | . 2 |
11 | 4, 10 | syl5bb 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wrex 2349 cop 3401 class class class wbr 3785 copab 3838 (class class class)co 5532 cmul 6986 cz 8351 cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-iota 4887 df-fv 4930 df-ov 5535 df-dvds 10196 |
This theorem is referenced by: dvdsval2 10198 dvds0lem 10205 dvds1lem 10206 dvds2lem 10207 0dvds 10215 dvdsle 10244 divconjdvds 10249 odd2np1 10272 even2n 10273 oddm1even 10274 opeo 10297 omeo 10298 m1exp1 10301 divalgb 10325 modremain 10329 zeqzmulgcd 10362 gcddiv 10408 dvdssqim 10413 coprmdvds2 10475 congr 10482 divgcdcoprm0 10483 cncongr2 10486 dvdsnprmd 10507 |
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