Theorem grpodivval 23851

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 23850	. . . 4 |- ( G e. GrpOp -> D = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )
5	4	oveqd 6193	. . 3 |- ( G e. GrpOp -> ( A D B ) = ( A ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) B ) )
6		oveq1 6183	. . . 4 |- ( x = A -> ( x G ( N ` y ) ) = ( A G ( N ` y ) ) )
7		fveq2 5775	. . . . 5 |- ( y = B -> ( N ` y ) = ( N ` B ) )
8	7	oveq2d 6192	. . . 4 |- ( y = B -> ( A G ( N ` y ) ) = ( A G ( N ` B ) ) )
9		eqid 2450	. . . 4 |- ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) = ( x e. X , y e. X |-> ( x G ( N ` y ) ) )
10		ovex 6201	. . . 4 |- ( A G ( N ` B ) ) e. _V
11	6, 8, 9, 10	ovmpt2 6312	. . 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 2510	. 2 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X ) ) -> ( A D B ) = ( A G ( N ` B ) ) )
13	12	3impb 1184	1 |- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A D B ) = ( A G ( N ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1757   ran crn 4925   ` cfv 5502  (class class class)co 6176   |-> cmpt2 6178   GrpOpcgr 23794   invcgn 23796   /g cgs 23797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-gdiv 23802

This theorem is referenced by:  grpodivinv  23852  grpoinvdiv  23853  grpodivdiv  23856  grpomuldivass  23857  grpodivid  23858  grponpcan  23860  grpopnpcan2  23861  grponnncan2  23862  ablodivdiv4  23899  nvmval  24143  rngosub  28878