Theorem grpsubfval 15592

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 5703	. . . . . 6 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
3		grpsubval.b	. . . . . 6 |- B = ( Base ` G )
4	2, 3	syl6eqr 2493	. . . . 5 |- ( g = G -> ( Base ` g ) = B )
5		fveq2 5703	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
6		grpsubval.p	. . . . . . 7 |- .+ = ( +g ` G )
7	5, 6	syl6eqr 2493	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
8		eqidd 2444	. . . . . 6 |- ( g = G -> x = x )
9		fveq2 5703	. . . . . . . 8 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
10		grpsubval.i	. . . . . . . 8 |- I = ( invg ` G )
11	9, 10	syl6eqr 2493	. . . . . . 7 |- ( g = G -> ( invg ` g ) = I )
12	11	fveq1d 5705	. . . . . 6 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
13	7, 8, 12	oveq123d 6124	. . . . 5 |- ( g = G -> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) = ( x .+ ( I ` y ) ) )
14	4, 4, 13	mpt2eq123dv 6160	. . . 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 15559	. . . 4 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
16		fvex 5713	. . . . . 6 |- ( Base ` G ) e. _V
17	3, 16	eqeltri 2513	. . . . 5 |- B e. _V
18	17, 17	mpt2ex 6662	. . . 4 |- ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V
19	14, 15, 18	fvmpt 5786	. . 3 |- ( G e. _V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
20	1, 19	syl5eq 2487	. 2 |- ( G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
21		fvprc 5697	. . . 4 |- ( -. G e. _V -> ( -g ` G ) = (/) )
22	1, 21	syl5eq 2487	. . 3 |- ( -. G e. _V -> .- = (/) )
23		fvprc 5697	. . . . . 6 |- ( -. G e. _V -> ( Base ` G ) = (/) )
24	3, 23	syl5eq 2487	. . . . 5 |- ( -. G e. _V -> B = (/) )
25		mpt2eq12 6158	. . . . 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 6168	. . . 4 |- ( x e. (/) , y e. (/) |-> ( x .+ ( I ` y ) ) ) = (/)
28	26, 27	syl6eq 2491	. . 3 |- ( -. G e. _V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = (/) )
29	22, 28	eqtr4d 2478	. 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 1369   e. wcel 1756   _Vcvv 2984   (/)c0 3649   ` cfv 5430  (class class class)co 6103   e. cmpt2 6105   Basecbs 14186   +g cplusg 14250   invgcminusg 15423   -gcsg 15425

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 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384

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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-sbg 15559

This theorem is referenced by:  grpsubval  15593  grpsubf  15617  grpsubpropd  15638  grpsubpropd2  15639  tgpsubcn  19673  tngtopn  20248