Theorem divides 12300

Description: Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16094). As proven in dvdsval3 12302, M || N <-> ( N mod M ) = 0. See divides 12300 and dvdsval2 12301 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 4084	. . 3 |- ( M || N <-> <. M , N >. e. || )
2		df-dvds 12299	. . . 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 6009	. . . . 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 4359	. 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 4083   {copab 4144  (class class class)co 6001   x. cmul 8004   ZZcz 9446   || cdvds 12298

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 4202  ax-pow 4258  ax-pr 4293

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 3889  df-br 4084  df-opab 4146  df-iota 5278  df-fv 5326  df-ov 6004  df-dvds 12299

This theorem is referenced by:  dvdsval2  12301  dvds0lem  12312  dvds1lem  12313  dvds2lem  12314  0dvds  12322  dvdsle  12355  divconjdvds  12360  odd2np1  12384  even2n  12385  oddm1even  12386  opeo  12408  omeo  12409  m1exp1  12412  divalgb  12436  modremain  12440  zeqzmulgcd  12491  gcddiv  12540  dvdssqim  12545  coprmdvds2  12615  congr  12622  divgcdcoprm0  12623  cncongr2  12626  dvdsnprmd  12647  prmpwdvds  12878  lgsquadlem2  15757