Theorem grpodivval 25371

Description: Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdiv.1	|- X = ran G
grpdiv.2	|- N = ( inv ` G )
grpdiv.3	|- D = ( /g ` G )

Assertion

Ref	Expression
grpodivval	|- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A D B ) = ( A G ( N ` B ) ) )

Proof of Theorem grpodivval

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

Step	Hyp	Ref	Expression
1		grpdiv.1	. . . . 5 |- X = ran G
2		grpdiv.2	. . . . 5 |- N = ( inv ` G )
3		grpdiv.3	. . . . 5 |- D = ( /g ` G )
4	1, 2, 3	grpodivfval 25370	. . . 4 |- ( G e. GrpOp -> D = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )
5	4	oveqd 6313	. . 3 |- ( G e. GrpOp -> ( A D B ) = ( A ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) B ) )
6		oveq1 6303	. . . 4 |- ( x = A -> ( x G ( N ` y ) ) = ( A G ( N ` y ) ) )
7		fveq2 5872	. . . . 5 |- ( y = B -> ( N ` y ) = ( N ` B ) )
8	7	oveq2d 6312	. . . 4 |- ( y = B -> ( A G ( N ` y ) ) = ( A G ( N ` B ) ) )
9		eqid 2457	. . . 4 |- ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) = ( x e. X , y e. X |-> ( x G ( N ` y ) ) )
10		ovex 6324	. . . 4 |- ( A G ( N ` B ) ) e. _V
11	6, 8, 9, 10	ovmpt2 6437	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) B ) = ( A G ( N ` B ) ) )
12	5, 11	sylan9eq 2518	. 2 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( A D B ) = ( A G ( N ` B ) ) )
13	12	3impb 1192	1 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A D B ) = ( A G ( N ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   ran crn 5009   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   GrpOpcgr 25314   invcgn 25316   /g cgs 25317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-gdiv 25322

This theorem is referenced by:  grpodivinv  25372  grpoinvdiv  25373  grpodivdiv  25376  grpomuldivass  25377  grpodivid  25378  grponpcan  25380  grpopnpcan2  25381  grponnncan2  25382  ablodivdiv4  25419  nvmval  25663  rngosub  30513