Theorem grpinvfvalg 13824

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 13786	. . 3 |- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) )
3		fveq2 5690	. . . . 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 5690	. . . . . . . 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 6092	. . . . . 6 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
10		fveq2 5690	. . . . . . 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 6031	. . . 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 4208	. . 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 13389	. . . . . 6 |- Base Fn _V
18		funfvex 5707	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5478	. . . . . 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 5935	. . 3 |- ( G e. V -> ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) e. _V )
23	2, 15, 16, 22	fvmptd3 5793	. 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. ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4187   Fn wfn 5367   ` cfv 5372   iota_crio 6027  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   invgcminusg 13783

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-minusg 13786

This theorem is referenced by:  grpinvval  13825  grpinvfng  13826  grpsubval  13828  grpinvf  13829  grpinvpropdg  13857  opprnegg  14362