Theorem divides 12286

Description: Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16024). As proven in dvdsval3 12288, M || N <-> ( N mod M ) = 0. See divides 12286 and dvdsval2 12287 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 4083	. . 3 |- ( M || N <-> <. M , N >. e. || )
2		df-dvds 12285	. . . 4 |- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
3	2	eleq2i 2296	. . 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 6002	. . . . 5 |- ( x = M -> ( n x. x ) = ( n x. M ) )
6	5	eqeq1d 2238	. . . 4 |- ( x = M -> ( ( n x. x ) = y <-> ( n x. M ) = y ) )
7	6	rexbidv 2531	. . 3 |- ( x = M -> ( E. n e. ZZ ( n x. x ) = y <-> E. n e. ZZ ( n x. M ) = y ) )
8		eqeq2 2239	. . . 4 |- ( y = N -> ( ( n x. M ) = y <-> ( n x. M ) = N ) )
9	8	rexbidv 2531	. . 3 |- ( y = N -> ( E. n e. ZZ ( n x. M ) = y <-> E. n e. ZZ ( n x. M ) = N ) )
10	7, 9	opelopab2 4358	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   <.cop 3669   class class class wbr 4082   {copab 4143  (class class class)co 5994   x. cmul 7992   ZZcz 9434   || cdvds 12284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-iota 5274  df-fv 5322  df-ov 5997  df-dvds 12285

This theorem is referenced by:  dvdsval2  12287  dvds0lem  12298  dvds1lem  12299  dvds2lem  12300  0dvds  12308  dvdsle  12341  divconjdvds  12346  odd2np1  12370  even2n  12371  oddm1even  12372  opeo  12394  omeo  12395  m1exp1  12398  divalgb  12422  modremain  12426  zeqzmulgcd  12477  gcddiv  12526  dvdssqim  12531  coprmdvds2  12601  congr  12608  divgcdcoprm0  12609  cncongr2  12612  dvdsnprmd  12633  prmpwdvds  12864  lgsquadlem2  15742