Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpinvhmeo Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator