Theorem grpinvfng 13841

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 13355	. . . . . . 7 |- Base Fn _V
3		elex 2827	. . . . . . 7 |- ( G e. V -> G e. _V )
4		funfvex 5692	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
5	4	funfni 5463	. . . . . . 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 2321	. . . . 5 |- ( G e. V -> B e. _V )
8		riotaexg 6015	. . . . 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 2618	. . 3 |- ( G e. V -> A. x e. B ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) e. _V )
11		eqid 2234	. . . 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 5490	. . 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 2234	. . . 4 |- ( +g ` G ) = ( +g ` G )
15		eqid 2234	. . . 4 |- ( 0g ` G ) = ( 0g ` G )
16		grpinvfn.n	. . . 4 |- N = ( invg ` G )
17	1, 14, 15, 16	grpinvfvalg 13839	. . 3 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) )
18	17	fneq1d 5451	. 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 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   |-> cmpt 4176   Fn wfn 5352   ` cfv 5357   iota_crio 6010  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   invgcminusg 13798

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-minusg 13801

This theorem is referenced by:  isgrpinv  13851  mulgval  13923  mulgfng  13925  invrfvald  14352