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Theorem grpinvhmeo 23973

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 23972	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4		tgpgrp 23965	. . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5		eqid 2729	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5, 2	grpinvcnv 18938	. . . 4 ⊢ (𝐺 ∈ Grp → ^◡𝐼 = 𝐼)
7	4, 6	syl 17	. . 3 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 = 𝐼)
8	7, 3	eqeltrd 2828	. 2 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 ∈ (𝐽 Cn 𝐽))
9		ishmeo 23646	. 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 2109 ^◡ccnv 5637 ‘cfv 6511 (class class class)co 7387 Basecbs 17179 TopOpenctopn 17384 Grpcgrp 18865 inv_gcminusg 18866 Cn ccn 23111 Homeochmeo 23640 TopGrpctgp 23958

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-map 8801 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-top 22781 df-topon 22798 df-cn 23114 df-hmeo 23642 df-tgp 23960

This theorem is referenced by: tgpconncomp 24000 tsmsxplem1 24040

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