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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 22621 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
4 | tgpgrp 22614 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
5 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 5, 2 | grpinvcnv 18105 | . . . 4 ⊢ (𝐺 ∈ Grp → ◡𝐼 = 𝐼) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 = 𝐼) |
8 | 7, 3 | eqeltrd 2910 | . 2 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 ∈ (𝐽 Cn 𝐽)) |
9 | ishmeo 22295 | . 2 ⊢ (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ ◡𝐼 ∈ (𝐽 Cn 𝐽))) | |
10 | 3, 8, 9 | sylanbrc 583 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ◡ccnv 5547 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TopOpenctopn 16683 Grpcgrp 18041 invgcminusg 18042 Cn ccn 21760 Homeochmeo 22289 TopGrpctgp 22607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-top 21430 df-topon 21447 df-cn 21763 df-hmeo 22291 df-tgp 22609 |
This theorem is referenced by: tgpconncomp 22648 tsmsxplem1 22688 |
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