![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpinvhmeo | Structured version Visualization version GIF version |
Description: The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgpinv.5 | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvhmeo | ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgpcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐺) | |
2 | tgpinv.5 | . . 3 ⊢ 𝐼 = (invg‘𝐺) | |
3 | 1, 2 | tgpinv 22297 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
4 | tgpgrp 22290 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
5 | eqid 2778 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 5, 2 | grpinvcnv 17870 | . . . 4 ⊢ (𝐺 ∈ Grp → ◡𝐼 = 𝐼) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 = 𝐼) |
8 | 7, 3 | eqeltrd 2859 | . 2 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 ∈ (𝐽 Cn 𝐽)) |
9 | ishmeo 21971 | . 2 ⊢ (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ ◡𝐼 ∈ (𝐽 Cn 𝐽))) | |
10 | 3, 8, 9 | sylanbrc 578 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ◡ccnv 5354 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 TopOpenctopn 16468 Grpcgrp 17809 invgcminusg 17810 Cn ccn 21436 Homeochmeo 21965 TopGrpctgp 22283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-top 21106 df-topon 21123 df-cn 21439 df-hmeo 21967 df-tgp 22285 |
This theorem is referenced by: tgpconncomp 22324 tsmsxplem1 22364 |
Copyright terms: Public domain | W3C validator |