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Theorem grpinvhmeo 24094

Description: The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)

**Hypotheses**
Ref	Expression
tgpcn.j	⊢ 𝐽 = (TopOpen‘𝐺)
tgpinv.5	⊢ 𝐼 = (inv_g‘𝐺)

**Assertion**
Ref	Expression
grpinvhmeo	⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))

**Proof of Theorem grpinvhmeo**
Step	Hyp	Ref	Expression
1		tgpcn.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐺)
2		tgpinv.5	. . 3 ⊢ 𝐼 = (inv_g‘𝐺)
3	1, 2	tgpinv 24093	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4		tgpgrp 24086	. . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5		eqid 2737	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5, 2	grpinvcnv 19024	. . . 4 ⊢ (𝐺 ∈ Grp → ^◡𝐼 = 𝐼)
7	4, 6	syl 17	. . 3 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 = 𝐼)
8	7, 3	eqeltrd 2841	. 2 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 ∈ (𝐽 Cn 𝐽))
9		ishmeo 23767	. 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 1540 ∈ wcel 2108 ^◡ccnv 5684 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TopOpenctopn 17466 Grpcgrp 18951 inv_gcminusg 18952 Cn ccn 23232 Homeochmeo 23761 TopGrpctgp 24079

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-top 22900 df-topon 22917 df-cn 23235 df-hmeo 23763 df-tgp 24081

This theorem is referenced by: tgpconncomp 24121 tsmsxplem1 24161

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