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Theorem hmeoco 23666

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 23654	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		hmeocn 23654	. . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿))
3		cnco 23160	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿))
5		cnvco 5852	. . 3 ⊢ ^◡(𝐺 ∘ 𝐹) = (^◡𝐹 ∘ ^◡𝐺)
6		hmeocnvcn 23655	. . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ^◡𝐺 ∈ (𝐿 Cn 𝐾))
7		hmeocnvcn 23655	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
8		cnco 23160	. . . 4 ⊢ ((^◡𝐺 ∈ (𝐿 Cn 𝐾) ∧ ^◡𝐹 ∈ (𝐾 Cn 𝐽)) → (^◡𝐹 ∘ ^◡𝐺) ∈ (𝐿 Cn 𝐽))
9	6, 7, 8	syl2anr 597	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (^◡𝐹 ∘ ^◡𝐺) ∈ (𝐿 Cn 𝐽))
10	5, 9	eqeltrid 2833	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽))
11		ishmeo 23653	. 2 ⊢ ((𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿) ↔ ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿) ∧ ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽)))
12	4, 10, 11	sylanbrc 583	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 ^◡ccnv 5640 ∘ ccom 5645 (class class class)co 7390 Cn ccn 23118 Homeochmeo 23647

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-map 8804 df-top 22788 df-topon 22805 df-cn 23121 df-hmeo 23649

This theorem is referenced by: hmphtr 23677 xpstopnlem1 23703 tgpconncomp 24007 tsmsxplem1 24047

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