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Theorem grpinvhmeo 23989

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 23988	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
4		tgpgrp 23981	. . . 4 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
5		eqid 2729	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	5, 2	grpinvcnv 18903	. . . 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 23662	. 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 5622 ‘cfv 6486 (class class class)co 7353 Basecbs 17138 TopOpenctopn 17343 Grpcgrp 18830 inv_gcminusg 18831 Cn ccn 23127 Homeochmeo 23656 TopGrpctgp 23974

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-map 8762 df-0g 17363 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-grp 18833 df-minusg 18834 df-top 22797 df-topon 22814 df-cn 23130 df-hmeo 23658 df-tgp 23976

This theorem is referenced by: tgpconncomp 24016 tsmsxplem1 24056

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