Theorem divides 12574

Description: Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16866). As proven in dvdsval3 12576, M || N <-> ( N mod M ) = 0. See divides 12574 and dvdsval2 12575 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.

Step	Hyp	Ref	Expression
1		df-br 4131	. . 3 |- ( M || N <-> <. M , N >. e. || )
2		df-dvds 12573	. . . 4 |- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
3	2	eleq2i 2305	. . 3 |- ( <. M , N >. e. || <-> <. M , N >. e. { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) } )
4	1, 3	bitri 184	. 2 |- ( M || N <-> <. M , N >. e. { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) } )
5		oveq2 6093	. . . . 5 |- ( x = M -> ( n x. x ) = ( n x. M ) )
6	5	eqeq1d 2247	. . . 4 |- ( x = M -> ( ( n x. x ) = y <-> ( n x. M ) = y ) )
7	6	rexbidv 2551	. . 3 |- ( x = M -> ( E. n e. ZZ ( n x. x ) = y <-> E. n e. ZZ ( n x. M ) = y ) )
8		eqeq2 2248	. . . 4 |- ( y = N -> ( ( n x. M ) = y <-> ( n x. M ) = N ) )
9	8	rexbidv 2551	. . 3 |- ( y = N -> ( E. n e. ZZ ( n x. M ) = y <-> E. n e. ZZ ( n x. M ) = N ) )
10	7, 9	opelopab2 4413	. 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 ) )
11	4, 10	bitrid 192	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. n e. ZZ ( n x. M ) = N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   <.cop 3712   class class class wbr 4130   {copab 4191  (class class class)co 6085   x. cmul 8185   ZZcz 9649   || cdvds 12572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-iota 5337  df-fv 5385  df-ov 6088  df-dvds 12573

This theorem is used by:  dvdsval2  12575  dvds0lem  12586  dvds1lem  12587  dvds2lem  12588  0dvds  12596  dvdsle  12629  divconjdvds  12634  odd2np1  12658  even2n  12659  oddm1even  12660  opeo  12682  omeo  12683  m1exp1  12686  divalgb  12710  modremain  12714  zeqzmulgcd  12765  gcddiv  12814  dvdssqim  12819  coprmdvds2  12889  congr  12896  divgcdcoprm0  12897  cncongr2  12900  dvdsnprmd  12921  prmpwdvds  13156  lgsquadlem2  16319