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Theorem hmeoco 22473
 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)) → (𝐺𝐹) ∈ (𝐽Homeo𝐿))

Proof of Theorem hmeoco
StepHypRef Expression
1 hmeocn 22461 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2 hmeocn 22461 . . 3 (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐾 Cn 𝐿))
3 cnco 21967 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))
41, 2, 3syl2an 599 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))
5 cnvco 5726 . . 3 (𝐺𝐹) = (𝐹𝐺)
6 hmeocnvcn 22462 . . . 4 (𝐺 ∈ (𝐾Homeo𝐿) → 𝐺 ∈ (𝐿 Cn 𝐾))
7 hmeocnvcn 22462 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
8 cnco 21967 . . . 4 ((𝐺 ∈ (𝐿 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)) → (𝐹𝐺) ∈ (𝐿 Cn 𝐽))
96, 7, 8syl2anr 600 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐹𝐺) ∈ (𝐿 Cn 𝐽))
105, 9eqeltrid 2857 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐿 Cn 𝐽))
11 ishmeo 22460 . 2 ((𝐺𝐹) ∈ (𝐽Homeo𝐿) ↔ ((𝐺𝐹) ∈ (𝐽 Cn 𝐿) ∧ (𝐺𝐹) ∈ (𝐿 Cn 𝐽)))
124, 10, 11sylanbrc 587 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐽Homeo𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2112  ◡ccnv 5524   ∘ ccom 5529  (class class class)co 7151   Cn ccn 21925  Homeochmeo 22454 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-top 21595  df-topon 21612  df-cn 21928  df-hmeo 22456 This theorem is referenced by:  hmphtr  22484  xpstopnlem1  22510  tgpconncomp  22814  tsmsxplem1  22854
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