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Theorem tgplacthmeo 23606
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 23582 . . 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 23605 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
71, 6sylan 580 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐽))
8 tgpgrp 23581 . . . . . 6 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
9 eqid 2732 . . . . . . 7 (𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯))) = (𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))
10 eqid 2732 . . . . . . 7 (invgβ€˜πΊ) = (invgβ€˜πΊ)
119, 3, 4, 10grplactcnv 18925 . . . . . 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 580 . . . . 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 496 . . . 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 18923 . . . . . . 7 (𝐴 ∈ 𝑋 β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)))
1514adantl 482 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = (π‘₯ ∈ 𝑋 ↦ (𝐴 + π‘₯)))
1615, 2eqtr4di 2790 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = 𝐹)
1716cnveqd 5875 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ◑𝐹)
183, 10grpinvcl 18871 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋)
198, 18sylan 580 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋)
209, 3grplactfval 18923 . . . . 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 2780 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ◑𝐹 = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)))
23 eqid 2732 . . . . 5 (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) = (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯))
2423, 3, 4, 5tmdlactcn 23605 . . . 4 ((𝐺 ∈ TopMnd ∧ ((invgβ€˜πΊ)β€˜π΄) ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) ∈ (𝐽 Cn 𝐽))
251, 19, 24syl2an2r 683 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘₯ ∈ 𝑋 ↦ (((invgβ€˜πΊ)β€˜π΄) + π‘₯)) ∈ (𝐽 Cn 𝐽))
2622, 25eqeltrd 2833 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ◑𝐹 ∈ (𝐽 Cn 𝐽))
27 ishmeo 23262 . 2 (𝐹 ∈ (𝐽Homeo𝐽) ↔ (𝐹 ∈ (𝐽 Cn 𝐽) ∧ ◑𝐹 ∈ (𝐽 Cn 𝐽)))
287, 26, 27sylanbrc 583 1 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐹 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5231  β—‘ccnv 5675  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  TopOpenctopn 17366  Grpcgrp 18818  invgcminusg 18819   Cn ccn 22727  Homeochmeo 23256  TopMndctmd 23573  TopGrpctgp 23574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-0g 17386  df-topgen 17388  df-plusf 18559  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-top 22395  df-topon 22412  df-topsp 22434  df-bases 22448  df-cn 22730  df-cnp 22731  df-tx 23065  df-hmeo 23258  df-tmd 23575  df-tgp 23576
This theorem is referenced by:  subgntr  23610  opnsubg  23611  cldsubg  23614  tgpconncompeqg  23615  tgpconncomp  23616  snclseqg  23619  qustgpopn  23623  tsmsxplem1  23656
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