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Mirrors > Home > MPE Home > Th. List > hmeoco | Structured version Visualization version GIF version |
Description: The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmeoco | ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocn 21686 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | hmeocn 21686 | . . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿)) | |
3 | cnco 21193 | . . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿)) | |
4 | 1, 2, 3 | syl2an 495 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿)) |
5 | cnvco 5415 | . . 3 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
6 | hmeocnvcn 21687 | . . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ◡𝐺 ∈ (𝐿 Cn 𝐾)) | |
7 | hmeocnvcn 21687 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
8 | cnco 21193 | . . . 4 ⊢ ((◡𝐺 ∈ (𝐿 Cn 𝐾) ∧ ◡𝐹 ∈ (𝐾 Cn 𝐽)) → (◡𝐹 ∘ ◡𝐺) ∈ (𝐿 Cn 𝐽)) | |
9 | 6, 7, 8 | syl2anr 496 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (◡𝐹 ∘ ◡𝐺) ∈ (𝐿 Cn 𝐽)) |
10 | 5, 9 | syl5eqel 2807 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → ◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽)) |
11 | ishmeo 21685 | . 2 ⊢ ((𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿) ↔ ((𝐺 ∘ 𝐹) ∈ (𝐽 Cn 𝐿) ∧ ◡(𝐺 ∘ 𝐹) ∈ (𝐿 Cn 𝐽))) | |
12 | 4, 10, 11 | sylanbrc 701 | 1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺 ∘ 𝐹) ∈ (𝐽Homeo𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2103 ◡ccnv 5217 ∘ ccom 5222 (class class class)co 6765 Cn ccn 21151 Homeochmeo 21679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-ral 3019 df-rex 3020 df-rab 3023 df-v 3306 df-sbc 3542 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-op 4292 df-uni 4545 df-br 4761 df-opab 4821 df-mpt 4838 df-id 5128 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-fv 6009 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-map 7976 df-top 20822 df-topon 20839 df-cn 21154 df-hmeo 21681 |
This theorem is referenced by: hmphtr 21709 xpstopnlem1 21735 tgpconncomp 22038 tsmsxplem1 22078 |
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