Theorem dvdsr 16726

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
dvdsr.1	|- B = ( Base ` R )
dvdsr.2	|- .|| = ( ||r ` R )
dvdsr.3	|- .x. = ( .r ` R )

Assertion

Ref	Expression
dvdsr	|- ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) )

Distinct variable groups:   z, B   z, X   z, Y   z, R   z, .x.

Allowed substitution hint:   .|| ( z)

Proof of Theorem dvdsr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr.2	. . . 4 |- .|| = ( ||r ` R )
2	1	reldvdsr 16724	. . 3 |- Rel .||
3		brrelex12 4871	. . 3 |- ( ( Rel .|| /\ X .|| Y ) -> ( X e. _V /\ Y e. _V ) )
4	2, 3	mpan 670	. 2 |- ( X .|| Y -> ( X e. _V /\ Y e. _V ) )
5		elex 2976	. . 3 |- ( X e. B -> X e. _V )
6		id 22	. . . . 5 |- ( ( z .x. X ) = Y -> ( z .x. X ) = Y )
7		ovex 6111	. . . . 5 |- ( z .x. X ) e. _V
8	6, 7	syl6eqelr 2527	. . . 4 |- ( ( z .x. X ) = Y -> Y e. _V )
9	8	rexlimivw 2832	. . 3 |- ( E. z e. B ( z .x. X ) = Y -> Y e. _V )
10	5, 9	anim12i 566	. 2 |- ( ( X e. B /\ E. z e. B ( z .x. X ) = Y ) -> ( X e. _V /\ Y e. _V ) )
11		simpl 457	. . . . 5 |- ( ( x = X /\ y = Y ) -> x = X )
12	11	eleq1d 2504	. . . 4 |- ( ( x = X /\ y = Y ) -> ( x e. B <-> X e. B ) )
13	11	oveq2d 6102	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( z .x. x ) = ( z .x. X ) )
14		simpr 461	. . . . . 6 |- ( ( x = X /\ y = Y ) -> y = Y )
15	13, 14	eqeq12d 2452	. . . . 5 |- ( ( x = X /\ y = Y ) -> ( ( z .x. x ) = y <-> ( z .x. X ) = Y ) )
16	15	rexbidv 2731	. . . 4 |- ( ( x = X /\ y = Y ) -> ( E. z e. B ( z .x. x ) = y <-> E. z e. B ( z .x. X ) = Y ) )
17	12, 16	anbi12d 710	. . 3 |- ( ( x = X /\ y = Y ) -> ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
18		dvdsr.1	. . . 4 |- B = ( Base ` R )
19		dvdsr.3	. . . 4 |- .x. = ( .r ` R )
20	18, 1, 19	dvdsrval 16725	. . 3 |- .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) }
21	17, 20	brabga 4598	. 2 |- ( ( X e. _V /\ Y e. _V ) -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
22	4, 10, 21	pm5.21nii 353	1 |- ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   E.wrex 2711   _Vcvv 2967   class class class wbr 4287   Rel wrel 4840   ` cfv 5413  (class class class)co 6086   Basecbs 14166   .rcmulr 14231   ||_rcdsr 16718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-dvdsr 16721

This theorem is referenced by:  dvdsr2  16727  dvdsrmul  16728  dvdsrcl  16729  dvdsrcl2  16730  dvdsrtr  16732  dvdsrmul1  16733  opprunit  16741  crngunit  16742  subrgdvds  16857  rhmdvdsr  26237