![]() |
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 23580 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
4 | tgpgrp 23573 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
5 | eqid 2732 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 5, 2 | grpinvcnv 18887 | . . . 4 ⊢ (𝐺 ∈ Grp → ◡𝐼 = 𝐼) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 = 𝐼) |
8 | 7, 3 | eqeltrd 2833 | . 2 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 ∈ (𝐽 Cn 𝐽)) |
9 | ishmeo 23254 | . 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 1541 ∈ wcel 2106 ◡ccnv 5674 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 TopOpenctopn 17363 Grpcgrp 18815 invgcminusg 18816 Cn ccn 22719 Homeochmeo 23248 TopGrpctgp 23566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8818 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-top 22387 df-topon 22404 df-cn 22722 df-hmeo 23250 df-tgp 23568 |
This theorem is referenced by: tgpconncomp 23608 tsmsxplem1 23648 |
Copyright terms: Public domain | W3C validator |