Theorem grplactfval 13689

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 6025	. . 3 |- ( g = A -> ( g .+ a ) = ( A .+ a ) )
3	2	mpteq2dv 4180	. 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 13146	. . . . 5 |- Base Fn _V
7	5	basmex 13147	. . . . 5 |- ( A e. X -> G e. _V )
8		funfvex 5656	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
9	8	funfni 5432	. . . . 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 2318	. . 3 |- ( A e. X -> X e. _V )
12	11	mptexd 5881	. 2 |- ( A e. X -> ( a e. X |-> ( A .+ a ) ) e. _V )
13	1, 3, 4, 12	fvmptd3 5740	1 |- ( A e. X -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   |-> cmpt 4150   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   Basecbs 13087

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-inn 9144  df-ndx 13090  df-slot 13091  df-base 13093

This theorem is referenced by:  grplactcnv  13690  eqglact  13817  eqgen  13819