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Theorem divides 12782
Description: Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 21605). As proven in dvdsval3 12784, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 12782 and dvdsval2 12783 for other equivalent expressions. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
divides  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. n  e.  ZZ  (
n  x.  M )  =  N ) )
Distinct variable groups:    n, M    n, N

Proof of Theorem divides
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4155 . . 3  |-  ( M 
||  N  <->  <. M ,  N >.  e.  ||  )
2 df-dvds 12781 . . . 4  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x
)  =  y ) }
32eleq2i 2452 . . 3  |-  ( <. M ,  N >.  e. 
|| 
<-> 
<. M ,  N >.  e. 
{ <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x
)  =  y ) } )
41, 3bitri 241 . 2  |-  ( M 
||  N  <->  <. M ,  N >.  e.  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) } )
5 oveq2 6029 . . . . 5  |-  ( x  =  M  ->  (
n  x.  x )  =  ( n  x.  M ) )
65eqeq1d 2396 . . . 4  |-  ( x  =  M  ->  (
( n  x.  x
)  =  y  <->  ( n  x.  M )  =  y ) )
76rexbidv 2671 . . 3  |-  ( x  =  M  ->  ( E. n  e.  ZZ  ( n  x.  x
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  y ) )
8 eqeq2 2397 . . . 4  |-  ( y  =  N  ->  (
( n  x.  M
)  =  y  <->  ( n  x.  M )  =  N ) )
98rexbidv 2671 . . 3  |-  ( y  =  N  ->  ( E. n  e.  ZZ  ( n  x.  M
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  N ) )
107, 9opelopab2 4417 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( <. M ,  N >.  e.  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) }  <->  E. n  e.  ZZ  ( n  x.  M )  =  N ) )
114, 10syl5bb 249 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  N  <->  E. n  e.  ZZ  (
n  x.  M )  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651   <.cop 3761   class class class wbr 4154   {copab 4207  (class class class)co 6021    x. cmul 8929   ZZcz 10215    || cdivides 12780
This theorem is referenced by:  dvdsval2  12783  dvds0lem  12788  dvds1lem  12789  dvds2lem  12790  0dvds  12798  dvdsle  12823  odd2np1  12836  oddm1even  12837  divalglem4  12844  divalglem9  12849  divalgb  12852  bezoutlem4  12969  gcddiv  12977  dvdssqim  12981  coprmdvds2  13031  opeo  13115  omeo  13116  prmpwdvds  13200  odmulg  15120  gexdvdsi  15145  lgsquadlem2  21007  dvdspw  25128  dvdsrabdioph  26562  jm2.26a  26763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-iota 5359  df-fv 5403  df-ov 6024  df-dvds 12781
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