Theorem divides 13664

Description: Divides relation. M||N means M divides into N with no remainder. (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

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (x = M -> (n x. x) = (n x. M))
2	1	eqeq1d 1892	. . . 4 |- (x = M -> ((n x. x) = y <-> (n x. M) = y))
3	2	rexbidv 2124	. . 3 |- (x = M -> (E.n e. ZZ (n x. x) = y <-> E.n e. ZZ (n x. M) = y))
4		eqeq2 1893	. . . 4 |- (y = N -> ((n x. M) = y <-> (n x. M) = N))
5	4	rexbidv 2124	. . 3 |- (y = N -> (E.n e. ZZ (n x. M) = y <-> E.n e. ZZ (n x. M) = N))
6	3, 5	opelopab2 3569	. 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))
7		df-br 3339	. . 3 |- (M||N <-> <.M, N>. e. ||)
8		df-divides 13663	. . . 4 |- || = {<.x, y>. | ((x e. ZZ /\ y e. ZZ) /\ E.n e. ZZ (n x. x) = y)}
9	8	eleq2i 1961	. . 3 |- (<.M, N>. e. || <-> <.M, N>. e. {<.x, y>. | ((x e. ZZ /\ y e. ZZ) /\ E.n e. ZZ (n x. x) = y)})
10	7, 9	bitri 190	. 2 |- (M||N <-> <.M, N>. e. {<.x, y>. | ((x e. ZZ /\ y e. ZZ) /\ E.n e. ZZ (n x. x) = y)})
11	6, 10	syl5bb 591	1 |- ((M e. ZZ /\ N e. ZZ) -> (M||N <-> E.n e. ZZ (n x. M) = N))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  <.cop 3046   class class class wbr 3338  {copab 3395  (class class class)co 4884   x. cmul 6391  ZZcz 6451  ||cdivides 13662

This theorem is referenced by:  dvds0lem 13665  dvds1lem 13666  dvds2lem 13667  0dvds 13675  dvdsle 13693  divalglem4 13699  divalglem9 13704  divalgb 13707  dvdsgcd 13765

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-divides 13663

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