Theorem grpinvfvalg 12745

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 12712	. . 3 |- invg = ( g e. _V |-> ( x e. ( Base ` g ) |-> ( iota_ y e. ( Base ` g ) ( y ( +g ` g ) x ) = ( 0g ` g ) ) ) )
3		fveq2 5496	. . . . 5 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		grpinvval.b	. . . . 5 |- B = ( Base ` G )
5	3, 4	eqtr4di 2221	. . . 4 |- ( g = G -> ( Base ` g ) = B )
6		fveq2 5496	. . . . . . . 8 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
7		grpinvval.p	. . . . . . . 8 |- .+ = ( +g ` G )
8	6, 7	eqtr4di 2221	. . . . . . 7 |- ( g = G -> ( +g ` g ) = .+ )
9	8	oveqd 5870	. . . . . 6 |- ( g = G -> ( y ( +g ` g ) x ) = ( y .+ x ) )
10		fveq2 5496	. . . . . . 7 |- ( g = G -> ( 0g ` g ) = ( 0g ` G ) )
11		grpinvval.o	. . . . . . 7 |- .0. = ( 0g ` G )
12	10, 11	eqtr4di 2221	. . . . . 6 |- ( g = G -> ( 0g ` g ) = .0. )
13	9, 12	eqeq12d 2185	. . . . 5 |- ( g = G -> ( ( y ( +g ` g ) x ) = ( 0g ` g ) <-> ( y .+ x ) = .0. ) )
14	5, 13	riotaeqbidv 5812	. . . 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 4071	. . 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 2741	. . 3 |- ( G e. V -> G e. _V )
17		basfn 12473	. . . . . 6 |- Base Fn _V
18		funfvex 5513	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
19	18	funfni 5298	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
20	17, 16, 19	sylancr 412	. . . . 5 |- ( G e. V -> ( Base ` G ) e. _V )
21	4, 20	eqeltrid 2257	. . . 4 |- ( G e. V -> B e. _V )
22	21	mptexd 5723	. . 3 |- ( G e. V -> ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) e. _V )
23	2, 15, 16, 22	fvmptd3 5589	. 2 |- ( G e. V -> ( invg ` G ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
24	1, 23	eqtrid 2215	1 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   |-> cmpt 4050   Fn wfn 5193   ` cfv 5198   iota_crio 5808  (class class class)co 5853   Basecbs 12416   +g cplusg 12480   0gc0g 12596   invgcminusg 12709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-inn 8879  df-ndx 12419  df-slot 12420  df-base 12422  df-minusg 12712

This theorem is referenced by:  grpinvval  12746  grpinvfng  12747  grpsubval  12749  grpinvf  12750