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Theorem divides 11502
Description: Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 12972). As proven in dvdsval3 11504, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 11502 and dvdsval2 11503 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 3930 . . 3  |-  ( M 
||  N  <->  <. M ,  N >.  e.  ||  )
2 df-dvds 11501 . . . 4  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x
)  =  y ) }
32eleq2i 2206 . . 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 183 . 2  |-  ( M 
||  N  <->  <. M ,  N >.  e.  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) } )
5 oveq2 5782 . . . . 5  |-  ( x  =  M  ->  (
n  x.  x )  =  ( n  x.  M ) )
65eqeq1d 2148 . . . 4  |-  ( x  =  M  ->  (
( n  x.  x
)  =  y  <->  ( n  x.  M )  =  y ) )
76rexbidv 2438 . . 3  |-  ( x  =  M  ->  ( E. n  e.  ZZ  ( n  x.  x
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  y ) )
8 eqeq2 2149 . . . 4  |-  ( y  =  N  ->  (
( n  x.  M
)  =  y  <->  ( n  x.  M )  =  N ) )
98rexbidv 2438 . . 3  |-  ( y  =  N  ->  ( E. n  e.  ZZ  ( n  x.  M
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  N ) )
107, 9opelopab2 4192 . 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 191 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    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2417   <.cop 3530   class class class wbr 3929   {copab 3988  (class class class)co 5774    x. cmul 7632   ZZcz 9061    || cdvds 11500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-iota 5088  df-fv 5131  df-ov 5777  df-dvds 11501
This theorem is referenced by:  dvdsval2  11503  dvds0lem  11510  dvds1lem  11511  dvds2lem  11512  0dvds  11520  dvdsle  11549  divconjdvds  11554  odd2np1  11577  even2n  11578  oddm1even  11579  opeo  11601  omeo  11602  m1exp1  11605  divalgb  11629  modremain  11633  zeqzmulgcd  11666  gcddiv  11714  dvdssqim  11719  coprmdvds2  11781  congr  11788  divgcdcoprm0  11789  cncongr2  11792  dvdsnprmd  11813
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