Theorem grpinvfng 13832

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 13394	. . . . . . 7 |- Base Fn _V
3		elex 2833	. . . . . . 7 |- ( G e. V -> G e. _V )
4		funfvex 5710	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
5	4	funfni 5481	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
6	2, 3, 5	sylancr 418	. . . . . 6 |- ( G e. V -> ( Base ` G ) e. _V )
7	1, 6	eqeltrid 2325	. . . . 5 |- ( G e. V -> B e. _V )
8		riotaexg 6036	. . . . 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 2624	. . 3 |- ( G e. V -> A. x e. B ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) e. _V )
11		eqid 2238	. . . 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 5508	. . 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 2238	. . . 4 |- ( +g ` G ) = ( +g ` G )
15		eqid 2238	. . . 4 |- ( 0g ` G ) = ( 0g ` G )
16		grpinvfn.n	. . . 4 |- N = ( invg ` G )
17	1, 14, 15, 16	grpinvfvalg 13830	. . 3 |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y ( +g ` G ) x ) = ( 0g ` G ) ) ) )
18	17	fneq1d 5469	. 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 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   |-> cmpt 4190   Fn wfn 5370   ` cfv 5375   iota_crio 6031  (class class class)co 6079   Basecbs 13335   +g cplusg 13414   0gc0g 13593   invgcminusg 13789

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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-minusg 13792

This theorem is referenced by:  isgrpinv  13842  mulgval  13908  mulgfng  13910  invrfvald  14412