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Theorem tgplacthmeo 21954
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 21930 . . 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 21953 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 487 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 21929 . . . . . 6 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
9 eqid 2651 . . . . . . 7 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
10 eqid 2651 . . . . . . 7 (invg𝐺) = (invg𝐺)
119, 3, 4, 10grplactcnv 17565 . . . . . 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 487 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1312simprd 478 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
149, 3grplactfval 17563 . . . . . . 7 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1514adantl 481 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1615, 2syl6eqr 2703 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
1716cnveqd 5330 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = 𝐹)
183, 10grpinvcl 17514 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
198, 18sylan 487 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
209, 3grplactfval 17563 . . . . 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 2693 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
23 eqid 2651 . . . . . 6 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2423, 3, 4, 5tmdlactcn 21953 . . . . 5 ((𝐺 ∈ TopMnd ∧ ((invg𝐺)‘𝐴) ∈ 𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
251, 24sylan 487 . . . 4 ((𝐺 ∈ TopGrp ∧ ((invg𝐺)‘𝐴) ∈ 𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
2619, 25syldan 486 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) ∈ (𝐽 Cn 𝐽))
2722, 26eqeltrd 2730 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
28 ishmeo 21610 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐽)))
297, 27, 28sylanbrc 699 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cmpt 4762  ccnv 5142  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  Grpcgrp 17469  invgcminusg 17470   Cn ccn 21076  Homeochmeo 21604  TopMndctmd 21921  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-0g 16149  df-topgen 16151  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cn 21079  df-cnp 21080  df-tx 21413  df-hmeo 21606  df-tmd 21923  df-tgp 21924
This theorem is referenced by:  subgntr  21957  opnsubg  21958  cldsubg  21961  tgpconncompeqg  21962  tgpconncomp  21963  snclseqg  21966  qustgpopn  21970  tsmsxplem1  22003
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