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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 22696 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
4 | tgpgrp 22689 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
5 | eqid 2824 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 5, 2 | grpinvcnv 18170 | . . . 4 ⊢ (𝐺 ∈ Grp → ◡𝐼 = 𝐼) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 = 𝐼) |
8 | 7, 3 | eqeltrd 2916 | . 2 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 ∈ (𝐽 Cn 𝐽)) |
9 | ishmeo 22370 | . 2 ⊢ (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ ◡𝐼 ∈ (𝐽 Cn 𝐽))) | |
10 | 3, 8, 9 | sylanbrc 585 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ◡ccnv 5557 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 TopOpenctopn 16698 Grpcgrp 18106 invgcminusg 18107 Cn ccn 21835 Homeochmeo 22364 TopGrpctgp 22682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-top 21505 df-topon 21522 df-cn 21838 df-hmeo 22366 df-tgp 22684 |
This theorem is referenced by: tgpconncomp 22724 tsmsxplem1 22764 |
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