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Theorem tgplacthmeo 21812
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 = (𝑥𝑋 ↦ (𝐴 + 𝑥))
tgplacthmeo.2 𝑋 = (Base‘𝐺)
tgplacthmeo.3 + = (+g𝐺)
tgplacthmeo.4 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
tgplacthmeo ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 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.
StepHypRef Expression
1 tgptmd 21788 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tgplacthmeo.1 . . . 4 𝐹 = (𝑥𝑋 ↦ (𝐴 + 𝑥))
3 tgplacthmeo.2 . . . 4 𝑋 = (Base‘𝐺)
4 tgplacthmeo.3 . . . 4 + = (+g𝐺)
5 tgplacthmeo.4 . . . 4 𝐽 = (TopOpen‘𝐺)
62, 3, 4, 5tmdlactcn 21811 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 488 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 21787 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
9 eqid 2626 . . . . . . 7 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
10 eqid 2626 . . . . . . 7 (invg𝐺) = (invg𝐺)
119, 3, 4, 10grplactcnv 17434 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
128, 11sylan 488 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1312simprd 479 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
149, 3grplactfval 17432 . . . . . . 7 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1514adantl 482 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1615, 2syl6eqr 2678 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
1716cnveqd 5263 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
183, 10grpinvcl 17383 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
198, 18sylan 488 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
209, 3grplactfval 17432 . . . . 5 (((invg𝐺)‘𝐴) ∈ 𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2119, 20syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2213, 17, 213eqtr3d 2668 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
23 eqid 2626 . . . . . 6 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2423, 3, 4, 5tmdlactcn 21811 . . . . 5 ((𝐺 ∈ TopMnd ∧ ((invg𝐺)‘𝐴) ∈ 𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
251, 24sylan 488 . . . 4 ((𝐺 ∈ TopGrp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
2619, 25syldan 487 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
2722, 26eqeltrd 2704 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
28 ishmeo 21467 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐽)))
297, 27, 28sylanbrc 697 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  cmpt 4678  ccnv 5078  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  TopOpenctopn 15998  Grpcgrp 17338  invgcminusg 17339   Cn ccn 20933  Homeochmeo 21461  TopMndctmd 21779  TopGrpctgp 21780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-0g 16018  df-topgen 16020  df-plusf 17157  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-top 20616  df-bases 20617  df-topon 20618  df-topsp 20619  df-cn 20936  df-cnp 20937  df-tx 21270  df-hmeo 21463  df-tmd 21781  df-tgp 21782
This theorem is referenced by:  subgntr  21815  opnsubg  21816  cldsubg  21819  tgpconncompeqg  21820  tgpconncomp  21821  snclseqg  21824  qustgpopn  21828  tsmsxplem1  21861
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