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 21699 | . 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽)) |
4 | tgpgrp 21692 | . . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp) | |
5 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
6 | 5, 2 | grpinvcnv 17306 | . . . 4 ⊢ (𝐺 ∈ Grp → ◡𝐼 = 𝐼) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 = 𝐼) |
8 | 7, 3 | eqeltrd 2688 | . 2 ⊢ (𝐺 ∈ TopGrp → ◡𝐼 ∈ (𝐽 Cn 𝐽)) |
9 | ishmeo 21372 | . 2 ⊢ (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ ◡𝐼 ∈ (𝐽 Cn 𝐽))) | |
10 | 3, 8, 9 | sylanbrc 695 | 1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ◡ccnv 5037 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TopOpenctopn 15905 Grpcgrp 17245 invgcminusg 17246 Cn ccn 20838 Homeochmeo 21366 TopGrpctgp 21685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-top 20521 df-topon 20523 df-cn 20841 df-hmeo 21368 df-tgp 21687 |
This theorem is referenced by: tgpconcomp 21726 tsmsxplem1 21766 |
Copyright terms: Public domain | W3C validator |