Theorem tgplacthmeo 24051

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 24027	. . 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 24050	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
7	1, 6	sylan 578	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
8		tgpgrp 24026	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
9		eqid 2725	. . . . . . 7 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
10		eqid 2725	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
11	9, 3, 4, 10	grplactcnv 19007	. . . . . 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 578	. . . . 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 494	. . . 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 19005	. . . . . . 7 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
15	14	adantl 480	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
16	15, 2	eqtr4di 2783	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
17	16	cnveqd 5878	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡𝐹)
18	3, 10	grpinvcl 18952	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
19	8, 18	sylan 578	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	9, 3	grplactfval 19005	. . . . 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 2773	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡𝐹 = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23		eqid 2725	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
24	23, 3, 4, 5	tmdlactcn 24050	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ ((invg‘𝐺)‘𝐴) ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
25	1, 19, 24	syl2an2r 683	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
26	22, 25	eqeltrd 2825	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → ◡𝐹 ∈ (𝐽 Cn 𝐽))
27		ishmeo 23707	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ ◡𝐹 ∈ (𝐽 Cn 𝐽)))
28	7, 26, 27	sylanbrc 581	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5232  ^◡ccnv 5677  –1-1-onto→wf1o 6548  ‘cfv 6549  (class class class)co 7419  Basecbs 17183  +_gcplusg 17236  TopOpenctopn 17406  Grpcgrp 18898  inv_gcminusg 18899   Cn ccn 23172  Homeochmeo 23701  TopMndctmd 24018  TopGrpctgp 24019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847  df-0g 17426  df-topgen 17428  df-plusf 18602  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cn 23175  df-cnp 23176  df-tx 23510  df-hmeo 23703  df-tmd 24020  df-tgp 24021

This theorem is referenced by:  subgntr  24055  opnsubg  24056  cldsubg  24059  tgpconncompeqg  24060  tgpconncomp  24061  snclseqg  24064  qustgpopn  24068  tsmsxplem1  24101