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Theorem grpinvhmeo 24272

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 24271	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4		tgpgrp 24264	. . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5		eqid 2765	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5, 2	grpinvcnv 19096	. . . 4 ⊢ (𝐺 ∈ Grp → ^◡𝐼 = 𝐼)
7	4, 6	syl 18	. . 3 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 = 𝐼)
8	7, 3	eqeltrd 2865	. 2 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 ∈ (𝐽 Cn 𝐽))
9		ishmeo 23945	. 2 ⊢ (𝐼 ∈ (𝐽Homeo𝐽) ↔ (𝐼 ∈ (𝐽 Cn 𝐽) ∧ ^◡𝐼 ∈ (𝐽 Cn 𝐽)))
10	3, 8, 9	sylanbrc 595	1 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ^◡ccnv 5662 ‘cfv 6540 (class class class)co 7416 Basecbs 17286 TopOpenctopn 17491 Grpcgrp 19023 inv_gcminusg 19024 Cn ccn 23410 Homeochmeo 23939 TopGrpctgp 24257

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8828 df-0g 17511 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-grp 19026 df-minusg 19027 df-top 23080 df-topon 23097 df-cn 23413 df-hmeo 23941 df-tgp 24259

This theorem is used by: tgpconncomp 24299 tsmsxplem1 24339

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