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 ∈ 𝑋 ↦ (𝐴 + 𝑥))
tgplacthmeo.2	⊢ 𝑋 = (Base‘𝐺)
tgplacthmeo.3	⊢ + = (+g‘𝐺)
tgplacthmeo.4	⊢ 𝐽 = (TopOpen‘𝐺)

Assertion

Ref	Expression
tgplacthmeo	⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))

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 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tgplacthmeo.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 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 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
7	1, 6	sylan 488	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
8		tgpgrp 21882	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
9		eqid 2622	. . . . . . 7 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
10		eqid 2622	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 3, 4, 10	grplactcnv 17518	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴))))
12	8, 11	sylan 488	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴))))
13	12	simprd 479	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)))
14	9, 3	grplactfval 17516	. . . . . . 7 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
15	14	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
16	15, 2	syl6eqr 2674	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
17	16	cnveqd 5298	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡𝐹)
18	3, 10	grpinvcl 17467	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
19	8, 18	sylan 488	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	9, 3	grplactfval 17516	. . . . 5 ⊢ (((invg‘𝐺)‘𝐴) ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
21	19, 20	syl 17	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
22	13, 17, 21	3eqtr3d 2664	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡𝐹 = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
24	23, 3, 4, 5	tmdlactcn 21906	. . . . 5 ⊢ ((𝐺 ∈ TopMnd ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
25	1, 24	sylan 488	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
26	19, 25	syldan 487	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
27	22, 26	eqeltrd 2701	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡𝐹 ∈ (𝐽 Cn 𝐽))
28		ishmeo 21562	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ ◡𝐹 ∈ (𝐽 Cn 𝐽)))
29	7, 27, 28	sylanbrc 698	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729  ^◡ccnv 5113  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  TopOpenctopn 16082  Grpcgrp 17422  inv_gcminusg 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