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Theorem hmeoco 23596

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 23584	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		hmeocn 23584	. . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿))
3		cnco 23090	. . 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 5885	. . 3 ⊢ ^◡(𝐺 ∘ 𝐹) = (^◡𝐹 ∘ ^◡𝐺)
6		hmeocnvcn 23585	. . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ^◡𝐺 ∈ (𝐿 Cn 𝐾))
7		hmeocnvcn 23585	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
8		cnco 23090	. . . 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 2836	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽))
11		ishmeo 23583	. 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 2105 ^◡ccnv 5675 ∘ ccom 5680 (class class class)co 7412 Cn ccn 23048 Homeochmeo 23577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8828 df-top 22716 df-topon 22733 df-cn 23051 df-hmeo 23579

This theorem is referenced by: hmphtr 23607 xpstopnlem1 23633 tgpconncomp 23937 tsmsxplem1 23977

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