Theorem grpsubfval 16219

Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
grpsubval.b	|- B = ( Base ` G )
grpsubval.p	|- .+ = ( +g ` G )
grpsubval.i	|- I = ( invg ` G )
grpsubval.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubfval	|- .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )

Distinct variable groups:   x, y, B   x, G, y   x, I, y   x, .+ , y

Allowed substitution hints:   .- ( x, y)

Proof of Theorem grpsubfval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsubval.m	. . 3 |- .- = ( -g ` G )
2		fveq2 5872	. . . . . 6 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
3		grpsubval.b	. . . . . 6 |- B = ( Base ` G )
4	2, 3	syl6eqr 2516	. . . . 5 |- ( g = G -> ( Base ` g ) = B )
5		fveq2 5872	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
6		grpsubval.p	. . . . . . 7 |- .+ = ( +g ` G )
7	5, 6	syl6eqr 2516	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
8		eqidd 2458	. . . . . 6 |- ( g = G -> x = x )
9		fveq2 5872	. . . . . . . 8 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
10		grpsubval.i	. . . . . . . 8 |- I = ( invg ` G )
11	9, 10	syl6eqr 2516	. . . . . . 7 |- ( g = G -> ( invg ` g ) = I )
12	11	fveq1d 5874	. . . . . 6 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
13	7, 8, 12	oveq123d 6317	. . . . 5 |- ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
14	4, 4, 13	mpt2eq123dv 6358	. . . 4 |- ( g = G -> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
15		df-sbg 16186	. . . 4 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
16		fvex 5882	. . . . . 6 |- ( Base ` G ) e. _V
17	3, 16	eqeltri 2541	. . . . 5 |- B e. _V
18	17, 17	mpt2ex 6876	. . . 4 |- ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V
19	14, 15, 18	fvmpt 5956	. . 3 |- ( G e. _V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
20	1, 19	syl5eq 2510	. 2 |- ( G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
21		fvprc 5866	. . . 4 |- ( -. G e. _V -> ( -g ` G ) = (/) )
22	1, 21	syl5eq 2510	. . 3 |- ( -. G e. _V -> .- = (/) )
23		fvprc 5866	. . . . . 6 |- ( -. G e. _V -> ( Base ` G ) = (/) )
24	3, 23	syl5eq 2510	. . . . 5 |- ( -. G e. _V -> B = (/) )
25		mpt2eq12 6356	. . . . 5 |- ( ( B = (/) /\ B = (/) ) -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = ( x e. (/) , y e. (/) |-> ( x .+ ( I ` y ) ) ) )
26	24, 24, 25	syl2anc 661	. . . 4 |- ( -. G e. _V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = ( x e. (/) , y e. (/) |-> ( x .+ ( I ` y ) ) ) )
27		mpt20 6366	. . . 4 |- ( x e. (/) , y e. (/) |-> ( x .+ ( I ` y ) ) ) = (/)
28	26, 27	syl6eq 2514	. . 3 |- ( -. G e. _V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = (/) )
29	22, 28	eqtr4d 2501	. 2 |- ( -. G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
30	20, 29	pm2.61i 164	1 |- .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )

Syntax hints:   -. wn 3   = wceq 1395   e. wcel 1819   _Vcvv 3109   (/)c0 3793   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   Basecbs 14644   +g cplusg 14712   invgcminusg 16181   -gcsg 16182

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-sbg 16186

This theorem is referenced by:  grpsubval  16220  grpsubf  16244  grpsubpropd  16267  grpsubpropd2  16268  tgpsubcn  20715  tngtopn  21290