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Theorem grpinvhmeo 24460

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 24459	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4		tgpgrp 24452	. . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5		eqid 2763	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5, 2	grpinvcnv 19267	. . . 4 ⊢ (𝐺 ∈ Grp → ^◡𝐼 = 𝐼)
7	4, 6	syl 18	. . 3 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 = 𝐼)
8	7, 3	eqeltrd 2864	. 2 ⊢ (𝐺 ∈ TopGrp → ^◡𝐼 ∈ (𝐽 Cn 𝐽))
9		ishmeo 24133	. 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 2147 ^◡ccnv 5657 ‘cfv 6566 (class class class)co 7451 Basecbs 17431 TopOpenctopn 17642 Grpcgrp 19194 inv_gcminusg 19195 Cn ccn 23597 Homeochmeo 24127 TopGrpctgp 24445

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-map 8887 df-0g 17662 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-grp 19197 df-minusg 19198 df-top 23267 df-topon 23284 df-cn 23600 df-hmeo 24129 df-tgp 24447

This theorem is used by: tgpconncomp 24487 tsmsxplem1 24527

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