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Theorem tgplacthmeo 23607
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 = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯))
tgplacthmeo.2 𝑋 = (Baseβ€˜πΊ)
tgplacthmeo.3 + = (+gβ€˜πΊ)
tgplacthmeo.4 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
tgplacthmeo ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽Homeo𝐽))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐺   π‘₯,𝐽   π‘₯, +   π‘₯,𝑋
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem tgplacthmeo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 tgptmd 23583 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopMnd)
2 tgplacthmeo.1 . . . 4 𝐹 = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯))
3 tgplacthmeo.2 . . . 4 𝑋 = (Baseβ€˜πΊ)
4 tgplacthmeo.3 . . . 4 + = (+gβ€˜πΊ)
5 tgplacthmeo.4 . . . 4 𝐽 = (TopOpenβ€˜πΊ)
62, 3, 4, 5tmdlactcn 23606 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 581 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 23582 . . . . . 6 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
9 eqid 2733 . . . . . . 7 (𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯))) = (𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))
10 eqid 2733 . . . . . . 7 (invgβ€˜πΊ) = (invgβ€˜πΊ)
119, 3, 4, 10grplactcnv 18926 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄):𝑋–1-1-onto→𝑋 ∧ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜((invgβ€˜πΊ)β€˜π΄))))
128, 11sylan 581 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄):𝑋–1-1-onto→𝑋 ∧ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜((invgβ€˜πΊ)β€˜π΄))))
1312simprd 497 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜((invgβ€˜πΊ)β€˜π΄)))
149, 3grplactfval 18924 . . . . . . 7 (𝐴 ∈ 𝑋 β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)))
1514adantl 483 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)))
1615, 2eqtr4di 2791 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = 𝐹)
1716cnveqd 5876 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ◑𝐹)
183, 10grpinvcl 18872 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋)
198, 18sylan 581 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋)
209, 3grplactfval 18924 . . . . 5 (((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋 β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜((invgβ€˜πΊ)β€˜π΄)) = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)))
2119, 20syl 17 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜((invgβ€˜πΊ)β€˜π΄)) = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)))
2213, 17, 213eqtr3d 2781 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ◑𝐹 = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)))
23 eqid 2733 . . . . 5 (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯))
2423, 3, 4, 5tmdlactcn 23606 . . . 4 ((𝐺 ∈ TopMnd ∧ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) ∈ (𝐽 Cn 𝐽))
251, 19, 24syl2an2r 684 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) ∈ (𝐽 Cn 𝐽))
2622, 25eqeltrd 2834 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ◑𝐹 ∈ (𝐽 Cn 𝐽))
27 ishmeo 23263 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ ◑𝐹 ∈ (𝐽 Cn 𝐽)))
287, 26, 27sylanbrc 584 1 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5232  β—‘ccnv 5676  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  TopOpenctopn 17367  Grpcgrp 18819  invgcminusg 18820   Cn ccn 22728  Homeochmeo 23257  TopMndctmd 23574  TopGrpctgp 23575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-0g 17387  df-topgen 17389  df-plusf 18560  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cn 22731  df-cnp 22732  df-tx 23066  df-hmeo 23259  df-tmd 23576  df-tgp 23577
This theorem is referenced by:  subgntr  23611  opnsubg  23612  cldsubg  23615  tgpconncompeqg  23616  tgpconncomp  23617  snclseqg  23620  qustgpopn  23624  tsmsxplem1  23657
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