Theorem grplactfval 13745

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 6035	. . 3 |- ( g = A -> ( g .+ a ) = ( A .+ a ) )
3	2	mpteq2dv 4185	. 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 13202	. . . . 5 |- Base Fn _V
7	5	basmex 13203	. . . . 5 |- ( A e. X -> G e. _V )
8		funfvex 5665	. . . . . 6 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
9	8	funfni 5439	. . . . 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 5891	. 2 |- ( A e. X -> ( a e. X |-> ( A .+ a ) ) e. _V )
13	1, 3, 4, 12	fvmptd3 5749	1 |- ( A e. X -> ( F ` A ) = ( a e. X |-> ( A .+ a ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   |-> cmpt 4155   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   Basecbs 13143

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-inn 9187  df-ndx 13146  df-slot 13147  df-base 13149

This theorem is referenced by:  grplactcnv  13746  eqglact  13873  eqgen  13875