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Theorem hmeoco 22831

Description: The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
hmeoco	⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿))

**Proof of Theorem hmeoco**
Step	Hyp	Ref	Expression
1		hmeocn 22819	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		hmeocn 22819	. . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿))
3		cnco 22325	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))
5		cnvco 5783	. . 3 ⊢ ^◡(𝐺 ∘ 𝐹) = (^◡𝐹 ∘ ^◡𝐺)
6		hmeocnvcn 22820	. . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ^◡𝐺 ∈ (𝐿 Cn 𝐾))
7		hmeocnvcn 22820	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
8		cnco 22325	. . . 4 ⊢ ((^◡𝐺 ∈ (𝐿 Cn 𝐾) ∧ ^◡𝐹 ∈ (𝐾 Cn 𝐽)) → (^◡𝐹 ∘ ^◡𝐺) ∈ (𝐿 Cn 𝐽))
9	6, 7, 8	syl2anr 596	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (^◡𝐹 ∘ ^◡𝐺) ∈ (𝐿 Cn 𝐽))
10	5, 9	eqeltrid 2843	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽))
11		ishmeo 22818	. 2 ⊢ ((𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿) ↔ ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿) ∧ ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽)))
12	4, 10, 11	sylanbrc 582	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ^◡ccnv 5579 ∘ ccom 5584 (class class class)co 7255 Cn ccn 22283 Homeochmeo 22812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-top 21951 df-topon 21968 df-cn 22286 df-hmeo 22814

This theorem is referenced by: hmphtr 22842 xpstopnlem1 22868 tgpconncomp 23172 tsmsxplem1 23212

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