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Mirrors > Home > MPE Home > Th. List > divides | Structured version Visualization version Unicode version |
Description: Define the divides relation. means divides into with no remainder. For example, (ex-dvds 27313). As proven in dvdsval3 14987, . See divides 14985 and dvdsval2 14986 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
divides |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4654 | . . 3 | |
2 | df-dvds 14984 | . . . 4 | |
3 | 2 | eleq2i 2693 | . . 3 |
4 | 1, 3 | bitri 264 | . 2 |
5 | oveq2 6658 | . . . . 5 | |
6 | 5 | eqeq1d 2624 | . . . 4 |
7 | 6 | rexbidv 3052 | . . 3 |
8 | eqeq2 2633 | . . . 4 | |
9 | 8 | rexbidv 3052 | . . 3 |
10 | 7, 9 | opelopab2 4996 | . 2 |
11 | 4, 10 | syl5bb 272 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 cop 4183 class class class wbr 4653 copab 4712 (class class class)co 6650 cmul 9941 cz 11377 cdvds 14983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-iota 5851 df-fv 5896 df-ov 6653 df-dvds 14984 |
This theorem is referenced by: dvdsval2 14986 dvds0lem 14992 dvds1lem 14993 dvds2lem 14994 0dvds 15002 dvdsle 15032 divconjdvds 15037 odd2np1 15065 even2n 15066 oddm1even 15067 opeo 15089 omeo 15090 m1exp1 15093 divalglem4 15119 divalglem9 15124 divalgb 15127 modremain 15132 zeqzmulgcd 15232 bezoutlem4 15259 gcddiv 15268 dvdssqim 15273 coprmdvds2 15368 congr 15378 divgcdcoprm0 15379 cncongr2 15382 dvdsnprmd 15403 prmpwdvds 15608 odmulg 17973 gexdvdsi 17998 lgsquadlem2 25106 dvdspw 31636 dvdsrabdioph 37374 jm2.26a 37567 coskpi2 40077 cosknegpi 40080 fourierswlem 40447 dfeven2 41562 |
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