hmeoco - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hmeoco		Structured version Visualization version GIF version

Theorem hmeoco 23707

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 23695	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		hmeocn 23695	. . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿))
3		cnco 23201	. . 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 5831	. . 3 ⊢ ^◡(𝐺 ∘ 𝐹) = (^◡𝐹 ∘ ^◡𝐺)
6		hmeocnvcn 23696	. . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ^◡𝐺 ∈ (𝐿 Cn 𝐾))
7		hmeocnvcn 23696	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
8		cnco 23201	. . . 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 2837	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → ^◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽))
11		ishmeo 23694	. 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 2113 ^◡ccnv 5620 ∘ ccom 5625 (class class class)co 7355 Cn ccn 23159 Homeochmeo 23688

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-fv 6497 df-ov 7358 df-oprab 7359 df-mpo 7360 df-map 8761 df-top 22829 df-topon 22846 df-cn 23162 df-hmeo 23690

This theorem is referenced by: hmphtr 23718 xpstopnlem1 23744 tgpconncomp 24048 tsmsxplem1 24088

W3C validator