Theorem eqglact 13978

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 2234	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
3		eqglact.3	. . . . . . 7 |- .+ = ( +g ` G )
4		eqger.r	. . . . . . 7 |- .~ = ( G ~QG Y )
5	1, 2, 3, 4	eqgval 13976	. . . . . 6 |- ( ( G e. Grp /\ Y C_ X ) -> ( A .~ x <-> ( A e. X /\ x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
6		3anass 1009	. . . . . 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 931	. . . 4 |- ( ( ( G e. Grp /\ Y C_ X ) /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
9	8	3impa 1221	. . 3 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> ( A .~ x <-> ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) ) )
10	9	abbidv 2354	. 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 6783	. . 3 |- ( A e. X -> [ A ] .~ = { x | A .~ x } )
12	11	3ad2ant3 1047	. 2 |- ( ( G e. Grp /\ Y C_ X /\ A e. X ) -> [ A ] .~ = { x | A .~ x } )
13		eqid 2234	. . . . . . . . 9 |- ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) = ( g e. X |-> ( x e. X |-> ( g .+ x ) ) )
14	13, 1, 3, 2	grplactcnv 13857	. . . . . . . 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 13856	. . . . . . . . 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 4936	. . . . . . 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 13803	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
20	13, 1	grplactfval 13856	. . . . . . . 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 2275	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> `' ( x e. X |-> ( A .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
23	22	cnveqd 4936	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> `' `' ( x e. X |-> ( A .+ x ) ) = `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
24	23	3adant2 1043	. . . 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 5105	. . 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 5232	. . 3 |- ( `' `' ( x e. X |-> ( A .+ x ) ) " Y ) = ( ( x e. X |-> ( A .+ x ) ) " Y )
27		eqid 2234	. . . . 5 |- ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) )
28	27	mptpreima 5261	. . . 4 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y }
29		df-rab 2531	. . . 4 |- { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y } = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
30	28, 29	eqtri 2255	. . 3 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
31	25, 26, 30	3eqtr3g 2290	. 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 2277	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 1005   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   `'ccnv 4753   "cima 4757   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   [cec 6778   Basecbs 13296   +g cplusg 13374   Grpcgrp 13755   invgcminusg 13756   ~_QG cqg 13922

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 619  ax-in2 620  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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-ec 6782  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-eqg 13925

This theorem is referenced by:  eqgen  13980