Theorem grpinvfvalg 13755

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 13717	. . 3 |- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) )
3		fveq2 5670	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpinvval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2283	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5670	. . . . . . . 8 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpinvval.p	. . . . . . . 8 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2283	. . . . . . 7 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 6067	. . . . . 6 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
10		fveq2 5670	. . . . . . 7 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
11		grpinvval.o	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2283	. . . . . 6 |- ( g = G -> ( 0g ` g ) = .0. )
13	9, 12	eqeq12d 2247	. . . . 5 |- ( g = G -> ( ( y ( +g ` g ) x ) = ( 0g ` g ) <-> ( y .+ x ) = .0. ) )
14	5, 13	riotaeqbidv 6006	. . . 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 4192	. . 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 2825	. . 3 |- ( G e. V -> G e. _V )
17		basfn 13271	. . . . . 6 |- Base Fn _V
18		funfvex 5687	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5458	. . . . . 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 2319	. . . 4 |- ( G e. V -> B e. _V )
22	21	mptexd 5913	. . 3 |- ( G e. V -> ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) e. _V )
23	2, 15, 16, 22	fvmptd3 5771	. 2 |- ( G e. V -> ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
24	1, 23	eqtrid 2277	1 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352   iota_crio 6002  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   0gc0g 13469   invgcminusg 13714

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-minusg 13717

This theorem is referenced by:  grpinvval  13756  grpinvfng  13757  grpsubval  13759  grpinvf  13760  grpinvpropdg  13788  opprnegg  14227