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Theorem divides 12549
Description: Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 20851). As proven in dvdsval3 12551, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 12549 and dvdsval2 12550 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 4040 . . 3  |-  ( M 
||  N  <->  <. M ,  N >.  e.  ||  )
2 df-dvds 12548 . . . 4  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x
)  =  y ) }
32eleq2i 2360 . . 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 240 . 2  |-  ( M 
||  N  <->  <. M ,  N >.  e.  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) } )
5 oveq2 5882 . . . . 5  |-  ( x  =  M  ->  (
n  x.  x )  =  ( n  x.  M ) )
65eqeq1d 2304 . . . 4  |-  ( x  =  M  ->  (
( n  x.  x
)  =  y  <->  ( n  x.  M )  =  y ) )
76rexbidv 2577 . . 3  |-  ( x  =  M  ->  ( E. n  e.  ZZ  ( n  x.  x
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  y ) )
8 eqeq2 2305 . . . 4  |-  ( y  =  N  ->  (
( n  x.  M
)  =  y  <->  ( n  x.  M )  =  N ) )
98rexbidv 2577 . . 3  |-  ( y  =  N  ->  ( E. n  e.  ZZ  ( n  x.  M
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  N ) )
107, 9opelopab2 4301 . 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 248 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039   {copab 4092  (class class class)co 5874    x. cmul 8758   ZZcz 10040    || cdivides 12547
This theorem is referenced by:  dvdsval2  12550  dvds0lem  12555  dvds1lem  12556  dvds2lem  12557  0dvds  12565  dvdsle  12590  odd2np1  12603  oddm1even  12604  divalglem4  12611  divalglem9  12616  divalgb  12619  bezoutlem4  12736  gcddiv  12744  dvdssqim  12748  coprmdvds2  12798  opeo  12882  omeo  12883  prmpwdvds  12967  odmulg  14885  gexdvdsi  14910  lgsquadlem2  20610  dvdspw  24174  dvdsrabdioph  26994  jm2.26a  27196
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-iota 5235  df-fv 5279  df-ov 5877  df-dvds 12548
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