Theorem eqglact 13892

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 2231	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
3		eqglact.3	. . . . . . 7 |- .+ = ( +g ` G )
4		eqger.r	. . . . . . 7 |- .~ = ( G ~QG Y )
5	1, 2, 3, 4	eqgval 13890	. . . . . 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 2350	. 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 6748	. . 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 2231	. . . . . . . . 9 |- ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) = ( g e. X |-> ( x e. X |-> ( g .+ x ) ) )
14	13, 1, 3, 2	grplactcnv 13765	. . . . . . . 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 13764	. . . . . . . . 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 4912	. . . . . . 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 13711	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
20	13, 1	grplactfval 13764	. . . . . . . 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 2272	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> `' ( x e. X |-> ( A .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
23	22	cnveqd 4912	. . . . 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 5081	. . 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 5208	. . 3 |- ( `' `' ( x e. X |-> ( A .+ x ) ) " Y ) = ( ( x e. X |-> ( A .+ x ) ) " Y )
27		eqid 2231	. . . . 5 |- ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) )
28	27	mptpreima 5237	. . . 4 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y }
29		df-rab 2520	. . . 4 |- { x e. X | ( ( ( invg ` G ) ` A ) .+ x ) e. Y } = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
30	28, 29	eqtri 2252	. . 3 |- ( `' ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) " Y ) = { x | ( x e. X /\ ( ( ( invg ` G ) ` A ) .+ x ) e. Y ) }
31	25, 26, 30	3eqtr3g 2287	. 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 2274	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 2202   {cab 2217   {crab 2515   C_ wss 3201   class class class wbr 4093   |-> cmpt 4155   `'ccnv 4730   "cima 4734   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   [cec 6743   Basecbs 13162   +g cplusg 13240   Grpcgrp 13663   invgcminusg 13664   ~_QG cqg 13836

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  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-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-ec 6747  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-eqg 13839

This theorem is referenced by:  eqgen  13894