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Theorem tgplacthmeo 23827
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 23803 . . 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 23826 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 578 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 23802 . . . . . 6 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
9 eqid 2730 . . . . . . 7 (𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯))) = (𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))
10 eqid 2730 . . . . . . 7 (invgβ€˜πΊ) = (invgβ€˜πΊ)
119, 3, 4, 10grplactcnv 18962 . . . . . 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 578 . . . . 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 494 . . . 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 18960 . . . . . . 7 (𝐴 ∈ 𝑋 β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)))
1514adantl 480 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)))
1615, 2eqtr4di 2788 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = 𝐹)
1716cnveqd 5874 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ◑𝐹)
183, 10grpinvcl 18908 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋)
198, 18sylan 578 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋)
209, 3grplactfval 18960 . . . . 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 2778 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ◑𝐹 = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)))
23 eqid 2730 . . . . 5 (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯))
2423, 3, 4, 5tmdlactcn 23826 . . . 4 ((𝐺 ∈ TopMnd ∧ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) ∈ (𝐽 Cn 𝐽))
251, 19, 24syl2an2r 681 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) ∈ (𝐽 Cn 𝐽))
2622, 25eqeltrd 2831 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ◑𝐹 ∈ (𝐽 Cn 𝐽))
27 ishmeo 23483 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ ◑𝐹 ∈ (𝐽 Cn 𝐽)))
287, 26, 27sylanbrc 581 1 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ↦ cmpt 5230  β—‘ccnv 5674  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  TopOpenctopn 17371  Grpcgrp 18855  invgcminusg 18856   Cn ccn 22948  Homeochmeo 23477  TopMndctmd 23794  TopGrpctgp 23795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-0g 17391  df-topgen 17393  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cn 22951  df-cnp 22952  df-tx 23286  df-hmeo 23479  df-tmd 23796  df-tgp 23797
This theorem is referenced by:  subgntr  23831  opnsubg  23832  cldsubg  23835  tgpconncompeqg  23836  tgpconncomp  23837  snclseqg  23840  qustgpopn  23844  tsmsxplem1  23877
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