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Theorem tgplacthmeo 23477
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 23453 . . 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 23476 . . 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 23452 . . . . . 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 18858 . . . . . 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 18856 . . . . . . 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 5835 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ β—‘((𝑔 ∈ 𝑋 ↦ (π‘₯ ∈ 𝑋 ↦ (𝑔 + π‘₯)))β€˜π΄) = ◑𝐹)
183, 10grpinvcl 18806 . . . . . 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 18856 . . . . 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 23476 . . . 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 23133 . 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 5192  β—‘ccnv 5636  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  TopOpenctopn 17311  Grpcgrp 18756  invgcminusg 18757   Cn ccn 22598  Homeochmeo 23127  TopMndctmd 23444  TopGrpctgp 23445
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-0g 17331  df-topgen 17333  df-plusf 18504  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cn 22601  df-cnp 22602  df-tx 22936  df-hmeo 23129  df-tmd 23446  df-tgp 23447
This theorem is referenced by:  subgntr  23481  opnsubg  23482  cldsubg  23485  tgpconncompeqg  23486  tgpconncomp  23487  snclseqg  23490  qustgpopn  23494  tsmsxplem1  23527
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