Theorem grpodivval 24918

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 24917	. . . 4 |- ( G e. GrpOp -> D = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )
5	4	oveqd 6299	. . 3 |- ( G e. GrpOp -> ( A D B ) = ( A ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) B ) )
6		oveq1 6289	. . . 4 |- ( x = A -> ( x G ( N ` y ) ) = ( A G ( N ` y ) ) )
7		fveq2 5864	. . . . 5 |- ( y = B -> ( N ` y ) = ( N ` B ) )
8	7	oveq2d 6298	. . . 4 |- ( y = B -> ( A G ( N ` y ) ) = ( A G ( N ` B ) ) )
9		eqid 2467	. . . 4 |- ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) = ( x e. X , y e. X |-> ( x G ( N ` y ) ) )
10		ovex 6307	. . . 4 |- ( A G ( N ` B ) ) e. _V
11	6, 8, 9, 10	ovmpt2 6420	. . 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 2528	. 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 1379   e. wcel 1767   ran crn 5000   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   GrpOpcgr 24861   invcgn 24863   /g cgs 24864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-gdiv 24869

This theorem is referenced by:  grpodivinv  24919  grpoinvdiv  24920  grpodivdiv  24923  grpomuldivass  24924  grpodivid  24925  grponpcan  24927  grpopnpcan2  24928  grponnncan2  24929  ablodivdiv4  24966  nvmval  25210  rngosub  29952