Theorem grplactfval 13898

Description: The left group action of element A of group G. (Contributed by Paul Chapman, 18-Mar-2008.)

Hypotheses

Ref	Expression
grplact.1	|- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )
grplact.2	|- X = ( Base ` G )

Assertion

Ref	Expression
grplactfval	|- ( A e. X -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )

Distinct variable groups:   g, a, A   G, a, g   .+ , a, g   X, a, g

Allowed substitution hints:   F( g, a)

Proof of Theorem grplactfval

Step	Hyp	Ref	Expression
1		grplact.1	. 2 |- F = ( g e. X |-> ( a e. X |-> ( g .+ a ) ) )
2		oveq1 6065	. . 3 |- ( g = A -> ( g .+ a ) = ( A .+ a ) )
3	2	mpteq2dv 4206	. 2 |- ( g = A -> ( a e. X |-> ( g .+ a ) ) = ( a e. X |-> ( A .+ a ) ) )
4		id 19	. 2 |- ( A e. X -> A e. X )
5		grplact.2	. . . 4 |- X = ( Base ` G )
6		basfn 13355	. . . . 5 |- Base Fn _V
7	5	basmex 13356	. . . . 5 |- ( A e. X -> G e. _V )
8		funfvex 5692	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
9	8	funfni 5463	. . . . 5 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
10	6, 7, 9	sylancr 414	. . . 4 |- ( A e. X -> ( Base ` G ) e. _V )
11	5, 10	eqeltrid 2321	. . 3 |- ( A e. X -> X e. _V )
12	11	mptexd 5918	. 2 |- ( A e. X -> ( a e. X |-> ( A .+ a ) ) e. _V )
13	1, 3, 4, 12	fvmptd3 5776	1 |- ( A e. X -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   _Vcvv 2815   |-> cmpt 4176   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296

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-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302

This theorem is referenced by:  grplactcnv  13899  eqglact  14026  eqgen  14028