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Theorem hmeoco 23682

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 23670	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		hmeocn 23670	. . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿))
3		cnco 23176	. . 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 5820	. . 3 ⊢ ^◡(𝐺 ∘ 𝐹) = (^◡𝐹 ∘ ^◡𝐺)
6		hmeocnvcn 23671	. . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ^◡𝐺 ∈ (𝐿 Cn 𝐾))
7		hmeocnvcn 23671	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
8		cnco 23176	. . . 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 2835	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽))
11		ishmeo 23669	. 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 2111 ^◡ccnv 5610 ∘ ccom 5615 (class class class)co 7341 Cn ccn 23134 Homeochmeo 23663

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5506 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-map 8747 df-top 22804 df-topon 22821 df-cn 23137 df-hmeo 23665

This theorem is referenced by: hmphtr 23693 xpstopnlem1 23719 tgpconncomp 24023 tsmsxplem1 24063

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