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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 = (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 24027 . . 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 24050 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 578 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 24026 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
9 eqid 2725 . . . . . . 7 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
10 eqid 2725 . . . . . . 7 (invg𝐺) = (invg𝐺)
119, 3, 4, 10grplactcnv 19007 . . . . . 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 578 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1312simprd 494 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
149, 3grplactfval 19005 . . . . . . 7 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1514adantl 480 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1615, 2eqtr4di 2783 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
1716cnveqd 5878 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
183, 10grpinvcl 18952 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
198, 18sylan 578 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
209, 3grplactfval 19005 . . . . 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 2773 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
23 eqid 2725 . . . . 5 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2423, 3, 4, 5tmdlactcn 24050 . . . 4 ((𝐺 ∈ TopMnd ∧ ((invg𝐺)‘𝐴) ∈ 𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
251, 19, 24syl2an2r 683 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
2622, 25eqeltrd 2825 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
27 ishmeo 23707 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐽)))
287, 26, 27sylanbrc 581 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cmpt 5232  ccnv 5677  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  TopOpenctopn 17406  Grpcgrp 18898  invgcminusg 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
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