Theorem eqglact 13089

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 2177	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
3		eqglact.3	. . . . . . 7 |- .+ = ( +g ` G )
4		eqger.r	. . . . . . 7 |- .~ = ( G ~QG Y )
5	1, 2, 3, 4	eqgval 13087	. . . . . 6 |- ( ( G e. Grp /\ Y C_ X ) -> ( A .~ x <-> ( A e. X /\ x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
6		3anass 982	. . . . . 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 923	. . . 4 |- ( ( ( G e. Grp /\ Y C_ X ) /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
9	8	3impa 1194	. . 3 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
10	9	abbidv 2295	. 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 6540	. . 3 |- ( A e. X -> [ A ] .~ = { x | A .~ x } )
12	11	3ad2ant3 1020	. 2 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = { x | A .~ x } )
13		eqid 2177	. . . . . . . . 9 |- ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) = ( g e. X |-> ( x e. X |-> ( g .+ x ) ) )
14	13, 1, 3, 2	grplactcnv 12977	. . . . . . . 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 12976	. . . . . . . . 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 4805	. . . . . . 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 12926	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
20	13, 1	grplactfval 12976	. . . . . . . 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 2218	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> `' ( x e. X |-> ( A .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
23	22	cnveqd 4805	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> `' `' ( x e. X |-> ( A .+ x ) ) = `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
24	23	3adant2 1016	. . . 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 4971	. . 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 5095	. . 3 |- ( `' `' ( x e. X |-> ( A .+ x ) ) " Y ) = ( ( x e. X |-> ( A .+ x ) ) " Y )
27		eqid 2177	. . . . 5 |- ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) )
28	27	mptpreima 5124	. . . 4 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y }
29		df-rab 2464	. . . 4 |- { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y } = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
30	28, 29	eqtri 2198	. . 3 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
31	25, 26, 30	3eqtr3g 2233	. 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 2220	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 978   = wceq 1353   e. wcel 2148   {cab 2163   {crab 2459   C_ wss 3131   class class class wbr 4005   |-> cmpt 4066   `'ccnv 4627   "cima 4631   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877   [cec 6535   Basecbs 12464   +g cplusg 12538   Grpcgrp 12882   invgcminusg 12883   ~_QG cqg 13034

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-ec 6539  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-eqg 13037

This theorem is referenced by:  eqgen  13091