Theorem grpinvfng 13563

Description: Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Hypotheses

Ref	Expression
grpinvfn.b	|- B = ( Base ` G )
grpinvfn.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvfng	|- ( G e. V -> N Fn B )

Proof of Theorem grpinvfng

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinvfn.b	. . . . . 6 |- B = ( Base ` G )
2		basfn 13077	. . . . . . 7 |- Base Fn _V
3		elex 2811	. . . . . . 7 |- ( G e. V -> G e. _V )
4		funfvex 5640	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
5	4	funfni 5419	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
6	2, 3, 5	sylancr 414	. . . . . 6 |- ( G e. V -> ( Base ` G ) e. _V )
7	1, 6	eqeltrid 2316	. . . . 5 |- ( G e. V -> B e. _V )
8		riotaexg 5951	. . . . 5 |- ( B e. _V -> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) e. _V )
9	7, 8	syl 14	. . . 4 |- ( G e. V -> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) e. _V )
10	9	ralrimivw 2604	. . 3 |- ( G e. V -> A. x e. B ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) e. _V )
11		eqid 2229	. . . 4 |- ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) = ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
12	11	fnmpt 5446	. . 3 |- ( A. x e. B ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) e. _V -> ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) Fn B )
13	10, 12	syl 14	. 2 |- ( G e. V -> ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) Fn B )
14		eqid 2229	. . . 4 |- ( +g ` G ) = ( +g ` G )
15		eqid 2229	. . . 4 |- ( 0g ` G ) = ( 0g ` G )
16		grpinvfn.n	. . . 4 |- N = ( invg ` G )
17	1, 14, 15, 16	grpinvfvalg 13561	. . 3 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) )
18	17	fneq1d 5407	. 2 |- ( G e. V -> ( N Fn B <-> ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) Fn B ) )
19	13, 18	mpbird 167	1 |- ( G e. V -> N Fn B )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   |-> cmpt 4144   Fn wfn 5309   ` cfv 5314   iota_crio 5946  (class class class)co 5994   Basecbs 13018   +g cplusg 13096   0gc0g 13275   invgcminusg 13520

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 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

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 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-inn 9099  df-ndx 13021  df-slot 13022  df-base 13024  df-minusg 13523

This theorem is referenced by:  isgrpinv  13573  mulgval  13645  mulgfng  13647  invrfvald  14071