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Theorem grpinvhmeo 22260
Description: The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
tgpcn.j 𝐽 = (TopOpen‘𝐺)
tgpinv.5 𝐼 = (invg𝐺)
Assertion
Ref Expression
grpinvhmeo (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))

Proof of Theorem grpinvhmeo
StepHypRef Expression
1 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
2 tgpinv.5 . . 3 𝐼 = (invg𝐺)
31, 2tgpinv 22259 . 2 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4 tgpgrp 22252 . . . 4 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5 eqid 2825 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
65, 2grpinvcnv 17837 . . . 4 (𝐺 ∈ Grp → 𝐼 = 𝐼)
74, 6syl 17 . . 3 (𝐺 ∈ TopGrp → 𝐼 = 𝐼)
87, 3eqeltrd 2906 . 2 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
9 ishmeo 21933 . 2 (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
103, 8, 9sylanbrc 580 1 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  ccnv 5341  cfv 6123  (class class class)co 6905  Basecbs 16222  TopOpenctopn 16435  Grpcgrp 17776  invgcminusg 17777   Cn ccn 21399  Homeochmeo 21927  TopGrpctgp 22245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-top 21069  df-topon 21086  df-cn 21402  df-hmeo 21929  df-tgp 22247
This theorem is referenced by:  tgpconncomp  22286  tsmsxplem1  22326
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