Theorem grpinvfvalg 13906

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 13868	. . 3 |- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) )
3		fveq2 5698	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpinvval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2289	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5698	. . . . . . . 8 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpinvval.p	. . . . . . . 8 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2289	. . . . . . 7 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 6105	. . . . . 6 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
10		fveq2 5698	. . . . . . 7 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
11		grpinvval.o	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2289	. . . . . 6 |- ( g = G -> ( 0g ` g ) = .0. )
13	9, 12	eqeq12d 2253	. . . . 5 |- ( g = G -> ( ( y ( +g ` g ) x ) = ( 0g ` g ) <-> ( y .+ x ) = .0. ) )
14	5, 13	riotaeqbidv 6044	. . . 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 4214	. . 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 2833	. . 3 |- ( G e. V -> G e. _V )
17		basfn 13469	. . . . . 6 |- Base Fn _V
18		funfvex 5715	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5486	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 418	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2325	. . . 4 |- ( G e. V -> B e. _V )
22	21	mptexd 5947	. . 3 |- ( G e. V -> ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) e. _V )
23	2, 15, 16, 22	fvmptd3 5802	. 2 |- ( G e. V -> ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
24	1, 23	eqtrid 2283	1 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4193   Fn wfn 5375   ` cfv 5380   iota_crio 6040  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   0gc0g 13669   invgcminusg 13865

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416  df-minusg 13868

This theorem is used by:  grpinvval  13907  grpinvfng  13908  grpsubval  13910  grpinvf  13911  grpinvpropdg  13939  opprnegg  14498