Theorem divides 12355

Description: Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 16348). As proven in dvdsval3 12357, M || N <-> ( N mod M ) = 0. See divides 12355 and dvdsval2 12356 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 4089	. . 3 |- ( M || N <-> <. M , N >. e. || )
2		df-dvds 12354	. . . 4 |- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
3	2	eleq2i 2298	. . 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 6026	. . . . 5 |- ( x = M -> ( n x. x ) = ( n x. M ) )
6	5	eqeq1d 2240	. . . 4 |- ( x = M -> ( ( n x. x ) = y <-> ( n x. M ) = y ) )
7	6	rexbidv 2533	. . 3 |- ( x = M -> ( E. n e. ZZ ( n x. x ) = y <-> E. n e. ZZ ( n x. M ) = y ) )
8		eqeq2 2241	. . . 4 |- ( y = N -> ( ( n x. M ) = y <-> ( n x. M ) = N ) )
9	8	rexbidv 2533	. . 3 |- ( y = N -> ( E. n e. ZZ ( n x. M ) = y <-> E. n e. ZZ ( n x. M ) = N ) )
10	7, 9	opelopab2 4365	. 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 1397   e. wcel 2202   E.wrex 2511   <.cop 3672   class class class wbr 4088   {copab 4149  (class class class)co 6018   x. cmul 8037   ZZcz 9479   || cdvds 12353

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-iota 5286  df-fv 5334  df-ov 6021  df-dvds 12354

This theorem is referenced by:  dvdsval2  12356  dvds0lem  12367  dvds1lem  12368  dvds2lem  12369  0dvds  12377  dvdsle  12410  divconjdvds  12415  odd2np1  12439  even2n  12440  oddm1even  12441  opeo  12463  omeo  12464  m1exp1  12467  divalgb  12491  modremain  12495  zeqzmulgcd  12546  gcddiv  12595  dvdssqim  12600  coprmdvds2  12670  congr  12677  divgcdcoprm0  12678  cncongr2  12681  dvdsnprmd  12702  prmpwdvds  12933  lgsquadlem2  15813