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Theorem tgplacthmeo 22641
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 22617 . . 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 22640 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 580 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 22616 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
9 eqid 2821 . . . . . . 7 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
10 eqid 2821 . . . . . . 7 (invg𝐺) = (invg𝐺)
119, 3, 4, 10grplactcnv 18142 . . . . . 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 580 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1312simprd 496 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
149, 3grplactfval 18140 . . . . . . 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 2874 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
1716cnveqd 5740 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
183, 10grpinvcl 18091 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
198, 18sylan 580 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
209, 3grplactfval 18140 . . . . 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 2864 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
23 eqid 2821 . . . . 5 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2423, 3, 4, 5tmdlactcn 22640 . . . 4 ((𝐺 ∈ TopMnd ∧ ((invg𝐺)‘𝐴) ∈ 𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
251, 19, 24syl2an2r 681 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
2622, 25eqeltrd 2913 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
27 ishmeo 22297 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐽)))
287, 26, 27sylanbrc 583 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cmpt 5138  ccnv 5548  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7145  Basecbs 16473  +gcplusg 16555  TopOpenctopn 16685  Grpcgrp 18043  invgcminusg 18044   Cn ccn 21762  Homeochmeo 22291  TopMndctmd 22608  TopGrpctgp 22609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7680  df-2nd 7681  df-map 8398  df-0g 16705  df-topgen 16707  df-plusf 17841  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-grp 18046  df-minusg 18047  df-top 21432  df-topon 21449  df-topsp 21471  df-bases 21484  df-cn 21765  df-cnp 21766  df-tx 22100  df-hmeo 22293  df-tmd 22610  df-tgp 22611
This theorem is referenced by:  subgntr  22644  opnsubg  22645  cldsubg  22648  tgpconncompeqg  22649  tgpconncomp  22650  snclseqg  22653  qustgpopn  22657  tsmsxplem1  22690
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