Theorem eqglact 13295

Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	|- X = ( Base ` G )
eqger.r	|- .~ = ( G ~QG Y )
eqglact.3	|- .+ = ( +g ` G )

Assertion

Ref	Expression
eqglact	|- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = ( ( x e. X |-> ( A .+ x ) ) " Y ) )

Distinct variable groups:   x, .+   x, .~   x, G   x, X   x, A   x, Y

Proof of Theorem eqglact

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqger.x	. . . . . . 7 |- X = ( Base ` G )
2		eqid 2193	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
3		eqglact.3	. . . . . . 7 |- .+ = ( +g ` G )
4		eqger.r	. . . . . . 7 |- .~ = ( G ~QG Y )
5	1, 2, 3, 4	eqgval 13293	. . . . . 6 |- ( ( G e. Grp /\ Y C_ X ) -> ( A .~ x <-> ( A e. X /\ x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
6		3anass 984	. . . . . 6 |- ( ( A e. X /\ x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) <-> ( A e. X /\ ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
7	5, 6	bitrdi 196	. . . . 5 |- ( ( G e. Grp /\ Y C_ X ) -> ( A .~ x <-> ( A e. X /\ ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) ) )
8	7	baibd 924	. . . 4 |- ( ( ( G e. Grp /\ Y C_ X ) /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
9	8	3impa 1196	. . 3 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
10	9	abbidv 2311	. 2 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> { x | A .~ x } = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) } )
11		dfec2 6590	. . 3 |- ( A e. X -> [ A ] .~ = { x | A .~ x } )
12	11	3ad2ant3 1022	. 2 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = { x | A .~ x } )
13		eqid 2193	. . . . . . . . 9 |- ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) = ( g e. X |-> ( x e. X |-> ( g .+ x ) ) )
14	13, 1, 3, 2	grplactcnv 13174	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) : X -1-1-onto-> X /\ `' ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` ( ( invg ` G ) ` A ) ) ) )
15	14	simprd 114	. . . . . . 7 |- ( ( G e. Grp /\ A e. X ) -> `' ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` ( ( invg ` G ) ` A ) ) )
16	13, 1	grplactfval 13173	. . . . . . . . 9 |- ( A e. X -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( x e. X |-> ( A .+ x ) ) )
17	16	adantl 277	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( x e. X |-> ( A .+ x ) ) )
18	17	cnveqd 4838	. . . . . . 7 |- ( ( G e. Grp /\ A e. X ) -> `' ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = `' ( x e. X |-> ( A .+ x ) ) )
19	1, 2	grpinvcl 13120	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
20	13, 1	grplactfval 13173	. . . . . . . 8 |- ( ( ( invg ` G ) ` A ) e. X -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` ( ( invg ` G ) ` A ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
21	19, 20	syl 14	. . . . . . 7 |- ( ( G e. Grp /\ A e. X ) -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` ( ( invg ` G ) ` A ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
22	15, 18, 21	3eqtr3d 2234	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> `' ( x e. X |-> ( A .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
23	22	cnveqd 4838	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> `' `' ( x e. X |-> ( A .+ x ) ) = `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
24	23	3adant2 1018	. . . 4 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> `' `' ( x e. X |-> ( A .+ x ) ) = `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
25	24	imaeq1d 5004	. . 3 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> ( `' `' ( x e. X |-> ( A .+ x ) ) " Y ) = ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) )
26		imacnvcnv 5130	. . 3 |- ( `' `' ( x e. X |-> ( A .+ x ) ) " Y ) = ( ( x e. X |-> ( A .+ x ) ) " Y )
27		eqid 2193	. . . . 5 |- ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) )
28	27	mptpreima 5159	. . . 4 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y }
29		df-rab 2481	. . . 4 |- { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y } = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
30	28, 29	eqtri 2214	. . 3 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
31	25, 26, 30	3eqtr3g 2249	. 2 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> ( ( x e. X |-> ( A .+ x ) ) " Y ) = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) } )
32	10, 12, 31	3eqtr4d 2236	1 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = ( ( x e. X |-> ( A .+ x ) ) " Y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   C_ wss 3153   class class class wbr 4029   |-> cmpt 4090   `'ccnv 4658   "cima 4662   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   [cec 6585   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   invgcminusg 13073   ~_QG cqg 13239

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-ec 6589  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-eqg 13242

This theorem is referenced by:  eqgen  13297