Theorem grpsubfval 16720

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 5870	. . . . . 6 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
3		grpsubval.b	. . . . . 6 |- B = ( Base ` G )
4	2, 3	syl6eqr 2505	. . . . 5 |- ( g = G -> ( Base ` g ) = B )
5		fveq2 5870	. . . . . . 7 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
6		grpsubval.p	. . . . . . 7 |- .+ = ( +g ` G )
7	5, 6	syl6eqr 2505	. . . . . 6 |- ( g = G -> ( +g ` g ) = .+ )
8		eqidd 2454	. . . . . 6 |- ( g = G -> x = x )
9		fveq2 5870	. . . . . . . 8 |- ( g = G -> ( invg ` g ) = ( invg ` G ) )
10		grpsubval.i	. . . . . . . 8 |- I = ( invg ` G )
11	9, 10	syl6eqr 2505	. . . . . . 7 |- ( g = G -> ( invg ` g ) = I )
12	11	fveq1d 5872	. . . . . 6 |- ( g = G -> ( ( invg ` g ) ` y ) = ( I ` y ) )
13	7, 8, 12	oveq123d 6316	. . . . 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 16687	. . . 4 |- -g = ( g e. _V |-> ( x e. ( Base ` g ) , y e. ( Base ` g ) |-> ( x ( +g ` g ) ( ( invg ` g ) ` y ) ) ) )
16		fvex 5880	. . . . . 6 |- ( Base ` G ) e. _V
17	3, 16	eqeltri 2527	. . . . 5 |- B e. _V
18	17, 17	mpt2ex 6875	. . . 4 |- ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) e. _V
19	14, 15, 18	fvmpt 5953	. . 3 |- ( G e. _V -> ( -g ` G ) = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
20	1, 19	syl5eq 2499	. 2 |- ( G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
21		fvprc 5864	. . . 4 |- ( -. G e. _V -> ( -g ` G ) = (/) )
22	1, 21	syl5eq 2499	. . 3 |- ( -. G e. _V -> .- = (/) )
23		fvprc 5864	. . . . . 6 |- ( -. G e. _V -> ( Base ` G ) = (/) )
24	3, 23	syl5eq 2499	. . . . 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 667	. . . 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 2503	. . 3 |- ( -. G e. _V -> ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) = (/) )
29	22, 28	eqtr4d 2490	. 2 |- ( -. G e. _V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
30	20, 29	pm2.61i 168	1 |- .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) )

Syntax hints:   -. wn 3   = wceq 1446   e. wcel 1889   _Vcvv 3047   (/)c0 3733   ` cfv 5585  (class class class)co 6295   |-> cmpt2 6297   Basecbs 15133   +g cplusg 15202   invgcminusg 16682   -gcsg 16683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-sbg 16687

This theorem is referenced by:  grpsubval  16721  grpsubf  16745  grpsubpropd  16768  grpsubpropd2  16769  tgpsubcn  21117  tngtopn  21670