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Theorem divides 13988
Description: Define the divides relation. means divides into with no remainder. For example, (ex-dvds 25169). As proven in dvdsval3 13990, M N<->(N M)=0. See divides 13988 and dvdsval2 13989 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
divides
Distinct variable groups:   ,M   ,N

Proof of Theorem divides
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4453 . . 3
2 df-dvds 13987 . . . 4
32eleq2i 2535 . . 3
41, 3bitri 249 . 2
5 oveq2 6304 . . . . 5
65eqeq1d 2459 . . . 4
76rexbidv 2968 . . 3
8 eqeq2 2472 . . . 4
98rexbidv 2968 . . 3
107, 9opelopab2 4773 . 2
114, 10syl5bb 257 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  E.wrex 2808  <.cop 4035   class class class wbr 4452  {copab 4509  (class class class)co 6296   cmul 9518   cz 10889   cdvds 13986
This theorem is referenced by:  dvdsval2  13989  dvds0lem  13994  dvds1lem  13995  dvds2lem  13996  0dvds  14004  dvdsle  14031  odd2np1  14046  oddm1even  14047  divalglem4  14054  divalglem9  14059  divalgb  14062  bezoutlem4  14179  gcddiv  14187  dvdssqim  14191  coprmdvds2  14244  opeo  14337  omeo  14338  prmpwdvds  14422  odmulg  16578  gexdvdsi  16603  lgsquadlem2  23630  dvdspw  29175  dvdsrabdioph  30743  jm2.26a  30942  coskpi2  31666  cosknegpi  31669  fourierswlem  32013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-iota 5556  df-fv 5601  df-ov 6299  df-dvds 13987
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