Theorem eqglact 13561

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 2205	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
3		eqglact.3	. . . . . . 7 |- .+ = ( +g ` G )
4		eqger.r	. . . . . . 7 |- .~ = ( G ~QG Y )
5	1, 2, 3, 4	eqgval 13559	. . . . . 6 |- ( ( G e. Grp /\ Y C_ X ) -> ( A .~ x <-> ( A e. X /\ x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
6		3anass 985	. . . . . 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 925	. . . 4 |- ( ( ( G e. Grp /\ Y C_ X ) /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
9	8	3impa 1197	. . 3 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
10	9	abbidv 2323	. 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 6623	. . 3 |- ( A e. X -> [ A ] .~ = { x | A .~ x } )
12	11	3ad2ant3 1023	. 2 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = { x | A .~ x } )
13		eqid 2205	. . . . . . . . 9 |- ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) = ( g e. X |-> ( x e. X |-> ( g .+ x ) ) )
14	13, 1, 3, 2	grplactcnv 13434	. . . . . . . 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 13433	. . . . . . . . 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 4854	. . . . . . 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 13380	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
20	13, 1	grplactfval 13433	. . . . . . . 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 2246	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> `' ( x e. X |-> ( A .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
23	22	cnveqd 4854	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> `' `' ( x e. X |-> ( A .+ x ) ) = `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
24	23	3adant2 1019	. . . 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 5021	. . 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 5147	. . 3 |- ( `' `' ( x e. X |-> ( A .+ x ) ) " Y ) = ( ( x e. X |-> ( A .+ x ) ) " Y )
27		eqid 2205	. . . . 5 |- ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) )
28	27	mptpreima 5176	. . . 4 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y }
29		df-rab 2493	. . . 4 |- { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y } = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
30	28, 29	eqtri 2226	. . 3 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
31	25, 26, 30	3eqtr3g 2261	. 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 2248	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 981   = wceq 1373   e. wcel 2176   {cab 2191   {crab 2488   C_ wss 3166   class class class wbr 4044   |-> cmpt 4105   `'ccnv 4674   "cima 4678   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   [cec 6618   Basecbs 12832   +g cplusg 12909   Grpcgrp 13332   invgcminusg 13333   ~_QG cqg 13505

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-ec 6622  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-eqg 13508

This theorem is referenced by:  eqgen  13563