Theorem grpinvfval 17460

Description: The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Hypotheses

Ref	Expression
grpinvval.b	|- B = ( Base ` G )
grpinvval.p	|- .+ = ( +g ` G )
grpinvval.o	|- .0. = ( 0g ` G )
grpinvval.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvfval	|- N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )

Distinct variable groups:   x, y, B   x, G, y   x, .0.   x, .+

Allowed substitution hints:   .+ ( y)   N( x, y)   .0. ( y)

Proof of Theorem grpinvfval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvval.n	. 2 |- N = ( invg ` G )
2		fveq2 6191	. . . . . 6 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
3		grpinvval.b	. . . . . 6 |- B = ( Base ` G )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( g = G -> ( Base ` g ) = B )
5		fveq2 6191	. . . . . . . . 9 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
6		grpinvval.p	. . . . . . . . 9 |- .+ = ( +g ` G )
7	5, 6	syl6eqr 2674	. . . . . . . 8 |- ( g = G -> ( +g ` g ) = .+ )
8	7	oveqd 6667	. . . . . . 7 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
9		fveq2 6191	. . . . . . . 8 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
10		grpinvval.o	. . . . . . . 8 |- .0. = ( 0g ` G )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( g = G -> ( 0g ` g ) = .0. )
12	8, 11	eqeq12d 2637	. . . . . 6 |- ( g = G -> ( ( y ( +g ` g ) x ) = ( 0g ` g ) <-> ( y .+ x ) = .0. ) )
13	4, 12	riotaeqbidv 6614	. . . . 5 |- ( g = G -> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) = ( iota_ y e. B ( y .+ x ) = .0. ) )
14	4, 13	mpteq12dv 4733	. . . 4 |- ( g = G -> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
15		df-minusg 17426	. . . 4 |- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) )
16		fvex 6201	. . . . . 6 |- ( Base ` G ) e. _V
17	3, 16	eqeltri 2697	. . . . 5 |- B e. _V
18	17	mptex 6486	. . . 4 |- ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) e. _V
19	14, 15, 18	fvmpt 6282	. . 3 |- ( G e. _V -> ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
20		fvprc 6185	. . . . 5 |- ( -. G e. _V -> ( invg ` G ) = (/) )
21		mpt0 6021	. . . . 5 |- ( x e. (/) |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) = (/)
22	20, 21	syl6eqr 2674	. . . 4 |- ( -. G e. _V -> ( invg ` G ) = ( x e. (/) |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
23		fvprc 6185	. . . . . 6 |- ( -. G e. _V -> ( Base ` G ) = (/) )
24	3, 23	syl5eq 2668	. . . . 5 |- ( -. G e. _V -> B = (/) )
25	24	mpteq1d 4738	. . . 4 |- ( -. G e. _V -> ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) = ( x e. (/) |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
26	22, 25	eqtr4d 2659	. . 3 |- ( -. G e. _V -> ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
27	19, 26	pm2.61i 176	. 2 |- ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
28	1, 27	eqtri 2644	1 |- N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   invgcminusg 17423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-minusg 17426

This theorem is referenced by:  grpinvval  17461  grpinvfn  17462  grpinvf  17466  grpinvpropd  17490  opprneg  18635