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Theorem grpinvhmeo 21803
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 21802 . 2 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4 tgpgrp 21795 . . . 4 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5 eqid 2621 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
65, 2grpinvcnv 17407 . . . 4 (𝐺 ∈ Grp → 𝐼 = 𝐼)
74, 6syl 17 . . 3 (𝐺 ∈ TopGrp → 𝐼 = 𝐼)
87, 3eqeltrd 2698 . 2 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
9 ishmeo 21475 . 2 (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
103, 8, 9sylanbrc 697 1 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  ccnv 5075  cfv 5849  (class class class)co 6607  Basecbs 15784  TopOpenctopn 16006  Grpcgrp 17346  invgcminusg 17347   Cn ccn 20941  Homeochmeo 21469  TopGrpctgp 21788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-map 7807  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-top 20621  df-topon 20638  df-cn 20944  df-hmeo 21471  df-tgp 21790
This theorem is referenced by:  tgpconncomp  21829  tsmsxplem1  21869
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