Theorem divides 11495

Description: Define the divides relation. M || N means M divides into N with no remainder. For example, 3 || 6 (ex-dvds 12942). As proven in dvdsval3 11497, M || N <-> ( N mod M ) = 0. See divides 11495 and dvdsval2 11496 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 3930	. . 3 |- ( M || N <-> <. M , N >. e. || )
2		df-dvds 11494	. . . 4 |- || = { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) }
3	2	eleq2i 2206	. . 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 183	. 2 |- ( M || N <-> <. M , N >. e. { <. x , y >. | ( ( x e. ZZ /\ y e. ZZ ) /\ E. n e. ZZ ( n x. x ) = y ) } )
5		oveq2 5782	. . . . 5 |- ( x = M -> ( n x. x ) = ( n x. M ) )
6	5	eqeq1d 2148	. . . 4 |- ( x = M -> ( ( n x. x ) = y <-> ( n x. M ) = y ) )
7	6	rexbidv 2438	. . 3 |- ( x = M -> ( E. n e. ZZ ( n x. x ) = y <-> E. n e. ZZ ( n x. M ) = y ) )
8		eqeq2 2149	. . . 4 |- ( y = N -> ( ( n x. M ) = y <-> ( n x. M ) = N ) )
9	8	rexbidv 2438	. . 3 |- ( y = N -> ( E. n e. ZZ ( n x. M ) = y <-> E. n e. ZZ ( n x. M ) = N ) )
10	7, 9	opelopab2 4192	. 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	syl5bb 191	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. n e. ZZ ( n x. M ) = N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   <.cop 3530   class class class wbr 3929   {copab 3988  (class class class)co 5774   x. cmul 7625   ZZcz 9054   || cdvds 11493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-iota 5088  df-fv 5131  df-ov 5777  df-dvds 11494

This theorem is referenced by:  dvdsval2  11496  dvds0lem  11503  dvds1lem  11504  dvds2lem  11505  0dvds  11513  dvdsle  11542  divconjdvds  11547  odd2np1  11570  even2n  11571  oddm1even  11572  opeo  11594  omeo  11595  m1exp1  11598  divalgb  11622  modremain  11626  zeqzmulgcd  11659  gcddiv  11707  dvdssqim  11712  coprmdvds2  11774  congr  11781  divgcdcoprm0  11782  cncongr2  11785  dvdsnprmd  11806