![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23195 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | hmeocn 23195 | . . 3 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿)) | |
3 | cnco 22701 | . . 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 5878 | . . 3 ⊢ ◡(𝐺 ∘ 𝐹) = (◡𝐹 ∘ ◡𝐺) | |
6 | hmeocnvcn 23196 | . . . 4 ⊢ (𝐺 ∈ (𝐾Homeo𝐿) → ◡𝐺 ∈ (𝐿 Cn 𝐾)) | |
7 | hmeocnvcn 23196 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
8 | cnco 22701 | . . . 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 23194 | . 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 396 ∈ wcel 2106 ◡ccnv 5669 ∘ ccom 5674 (class class class)co 7394 Cn ccn 22659 Homeochmeo 23188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-map 8807 df-top 22327 df-topon 22344 df-cn 22662 df-hmeo 23190 |
This theorem is referenced by: hmphtr 23218 xpstopnlem1 23244 tgpconncomp 23548 tsmsxplem1 23588 |
Copyright terms: Public domain | W3C validator |