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Theorem hmeoco 23207
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 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)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))
41, 2, 3syl2an 596 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))
5 cnvco 5878 . . 3 (𝐺𝐹) = (𝐹𝐺)
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)) → (𝐹𝐺) ∈ (𝐿 Cn 𝐽))
96, 7, 8syl2anr 597 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐹𝐺) ∈ (𝐿 Cn 𝐽))
105, 9eqeltrid 2837 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐿 Cn 𝐽))
11 ishmeo 23194 . 2 ((𝐺𝐹) ∈ (𝐽Homeo𝐿) ↔ ((𝐺𝐹) ∈ (𝐽 Cn 𝐿) ∧ (𝐺𝐹) ∈ (𝐿 Cn 𝐽)))
124, 10, 11sylanbrc 583 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (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
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