Theorem tgplacthmeo 21907

Description: The left group action of element A in a topological group G is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
tgplacthmeo.1	|- F = ( x e. X |-> ( A .+ x ) )
tgplacthmeo.2	|- X = ( Base ` G )
tgplacthmeo.3	|- .+ = ( +g ` G )
tgplacthmeo.4	|- J = ( TopOpen ` G )

Assertion

Ref	Expression
tgplacthmeo	|- ( ( G e. TopGrp /\ A e. X ) -> F e. ( J Homeo J ) )

Distinct variable groups:   x, A   x, G   x, J   x, .+   x, X

Allowed substitution hint:   F( x)

Proof of Theorem tgplacthmeo

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgptmd 21883	. . 3 |- ( G e. TopGrp -> G e. TopMnd )
2		tgplacthmeo.1	. . . 4 |- F = ( x e. X |-> ( A .+ x ) )
3		tgplacthmeo.2	. . . 4 |- X = ( Base ` G )
4		tgplacthmeo.3	. . . 4 |- .+ = ( +g ` G )
5		tgplacthmeo.4	. . . 4 |- J = ( TopOpen ` G )
6	2, 3, 4, 5	tmdlactcn 21906	. . 3 |- ( ( G e. TopMnd /\ A e. X ) -> F e. ( J Cn J ) )
7	1, 6	sylan 488	. 2 |- ( ( G e. TopGrp /\ A e. X ) -> F e. ( J Cn J ) )
8		tgpgrp 21882	. . . . . 6 |- ( G e. TopGrp -> G e. Grp )
9		eqid 2622	. . . . . . 7 |- ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) = ( g e. X |-> ( x e. X |-> ( g .+ x ) ) )
10		eqid 2622	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
11	9, 3, 4, 10	grplactcnv 17518	. . . . . 6 |- ( ( 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 ) ) ) )
12	8, 11	sylan 488	. . . . 5 |- ( ( G e. TopGrp /\ 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 ) ) ) )
13	12	simprd 479	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> `' ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` ( ( invg ` G ) ` A ) ) )
14	9, 3	grplactfval 17516	. . . . . . 7 |- ( A e. X -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( x e. X |-> ( A .+ x ) ) )
15	14	adantl 482	. . . . . 6 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = ( x e. X |-> ( A .+ x ) ) )
16	15, 2	syl6eqr 2674	. . . . 5 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = F )
17	16	cnveqd 5298	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> `' ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` A ) = `' F )
18	3, 10	grpinvcl 17467	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
19	8, 18	sylan 488	. . . . 5 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
20	9, 3	grplactfval 17516	. . . . 5 |- ( ( ( 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 17	. . . 4 |- ( ( G e. TopGrp /\ A e. X ) -> ( ( g e. X |-> ( x e. X |-> ( g .+ x ) ) ) ` ( ( invg ` G ) ` A ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
22	13, 17, 21	3eqtr3d 2664	. . 3 |- ( ( G e. TopGrp /\ A e. X ) -> `' F = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) )
23		eqid 2622	. . . . . 6 |- ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) = ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) )
24	23, 3, 4, 5	tmdlactcn 21906	. . . . 5 |- ( ( G e. TopMnd /\ ( ( invg ` G ) ` A ) e. X ) -> ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) e. ( J Cn J ) )
25	1, 24	sylan 488	. . . 4 |- ( ( G e. TopGrp /\ ( ( invg ` G ) ` A ) e. X ) -> ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) e. ( J Cn J ) )
26	19, 25	syldan 487	. . 3 |- ( ( G e. TopGrp /\ A e. X ) -> ( x e. X |-> ( ( ( invg ` G ) ` A ) .+ x ) ) e. ( J Cn J ) )
27	22, 26	eqeltrd 2701	. 2 |- ( ( G e. TopGrp /\ A e. X ) -> `' F e. ( J Cn J ) )
28		ishmeo 21562	. 2 |- ( F e. ( J Homeo J ) <-> ( F e. ( J Cn J ) /\ `' F e. ( J Cn J ) ) )
29	7, 27, 28	sylanbrc 698	1 |- ( ( G e. TopGrp /\ A e. X ) -> F e. ( J Homeo J ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   `'ccnv 5113   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   Grpcgrp 17422   invgcminusg 17423   Cn ccn 21028   Homeochmeo 21556  TopMndctmd 21874   TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-0g 16102  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-tmd 21876  df-tgp 21877

This theorem is referenced by:  subgntr  21910  opnsubg  21911  cldsubg  21914  tgpconncompeqg  21915  tgpconncomp  21916  snclseqg  21919  qustgpopn  21923  tsmsxplem1  21956