Theorem grpinvfvalg 13570

Description: The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.)

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
grpinvfvalg	|- ( G e. V -> 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)   V( x, y)   .0. ( y)

Proof of Theorem grpinvfvalg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvval.n	. 2 |- N = ( invg ` G )
2		df-minusg 13532	. . 3 |- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) )
3		fveq2 5626	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpinvval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2280	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5626	. . . . . . . 8 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpinvval.p	. . . . . . . 8 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2280	. . . . . . 7 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 6017	. . . . . 6 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
10		fveq2 5626	. . . . . . 7 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
11		grpinvval.o	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2280	. . . . . 6 |- ( g = G -> ( 0g ` g ) = .0. )
13	9, 12	eqeq12d 2244	. . . . 5 |- ( g = G -> ( ( y ( +g ` g ) x ) = ( 0g ` g ) <-> ( y .+ x ) = .0. ) )
14	5, 13	riotaeqbidv 5956	. . . 4 |- ( g = G -> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) = ( iota_ y e. B ( y .+ x ) = .0. ) )
15	5, 14	mpteq12dv 4165	. . 3 |- ( 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. ) ) )
16		elex 2811	. . 3 |- ( G e. V -> G e. _V )
17		basfn 13086	. . . . . 6 |- Base Fn _V
18		funfvex 5643	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5422	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 414	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2316	. . . 4 |- ( G e. V -> B e. _V )
22	21	mptexd 5865	. . 3 |- ( G e. V -> ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) e. _V )
23	2, 15, 16, 22	fvmptd3 5727	. 2 |- ( G e. V -> ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
24	1, 23	eqtrid 2274	1 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   |-> cmpt 4144   Fn wfn 5312   ` cfv 5317   iota_crio 5952  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   invgcminusg 13529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-minusg 13532

This theorem is referenced by:  grpinvval  13571  grpinvfng  13572  grpsubval  13574  grpinvf  13575  grpinvpropdg  13603  opprnegg  14041