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Theorem hmeoco 21698
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 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)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))
41, 2, 3syl2an 495 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐽 Cn 𝐿))
5 cnvco 5415 . . 3 (𝐺𝐹) = (𝐹𝐺)
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)) → (𝐹𝐺) ∈ (𝐿 Cn 𝐽))
96, 7, 8syl2anr 496 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐹𝐺) ∈ (𝐿 Cn 𝐽))
105, 9syl5eqel 2807 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (𝐿 Cn 𝐽))
11 ishmeo 21685 . 2 ((𝐺𝐹) ∈ (𝐽Homeo𝐿) ↔ ((𝐺𝐹) ∈ (𝐽 Cn 𝐿) ∧ (𝐺𝐹) ∈ (𝐿 Cn 𝐽)))
124, 10, 11sylanbrc 701 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐺 ∈ (𝐾Homeo𝐿)) → (𝐺𝐹) ∈ (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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