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Theorem divides 10891
Description: Define the divides relation.  M  ||  N means  M divides into  N with no remainder. For example,  3  ||  6 (ex-dvds 11314). As proven in dvdsval3 10893, 
M  ||  N  <->  ( N  mod  M )  =  0. See divides 10891 and dvdsval2 10892 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 3838 . . 3  |-  ( M 
||  N  <->  <. M ,  N >.  e.  ||  )
2 df-dvds 10890 . . . 4  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x
)  =  y ) }
32eleq2i 2154 . . 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 182 . 2  |-  ( M 
||  N  <->  <. M ,  N >.  e.  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. n  e.  ZZ  ( n  x.  x )  =  y ) } )
5 oveq2 5642 . . . . 5  |-  ( x  =  M  ->  (
n  x.  x )  =  ( n  x.  M ) )
65eqeq1d 2096 . . . 4  |-  ( x  =  M  ->  (
( n  x.  x
)  =  y  <->  ( n  x.  M )  =  y ) )
76rexbidv 2381 . . 3  |-  ( x  =  M  ->  ( E. n  e.  ZZ  ( n  x.  x
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  y ) )
8 eqeq2 2097 . . . 4  |-  ( y  =  N  ->  (
( n  x.  M
)  =  y  <->  ( n  x.  M )  =  N ) )
98rexbidv 2381 . . 3  |-  ( y  =  N  ->  ( E. n  e.  ZZ  ( n  x.  M
)  =  y  <->  E. n  e.  ZZ  ( n  x.  M )  =  N ) )
107, 9opelopab2 4088 . 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 190 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   <.cop 3444   class class class wbr 3837   {copab 3890  (class class class)co 5634    x. cmul 7334   ZZcz 8720    || cdvds 10889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-iota 4967  df-fv 5010  df-ov 5637  df-dvds 10890
This theorem is referenced by:  dvdsval2  10892  dvds0lem  10899  dvds1lem  10900  dvds2lem  10901  0dvds  10909  dvdsle  10938  divconjdvds  10943  odd2np1  10966  even2n  10967  oddm1even  10968  opeo  10990  omeo  10991  m1exp1  10994  divalgb  11018  modremain  11022  zeqzmulgcd  11055  gcddiv  11101  dvdssqim  11106  coprmdvds2  11168  congr  11175  divgcdcoprm0  11176  cncongr2  11179  dvdsnprmd  11200
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