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Theorem hmeoco 17463
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  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  ( G  o.  F )  e.  ( J  Homeo  L )
)

Proof of Theorem hmeoco
StepHypRef Expression
1 hmeocn 17451 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
2 hmeocn 17451 . . 3  |-  ( G  e.  ( K  Homeo  L )  ->  G  e.  ( K  Cn  L
) )
3 cnco 16995 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )
41, 2, 3syl2an 463 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  ( G  o.  F )  e.  ( J  Cn  L ) )
5 cnvco 4865 . . 3  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
6 hmeocnvcn 17452 . . . 4  |-  ( G  e.  ( K  Homeo  L )  ->  `' G  e.  ( L  Cn  K
) )
7 hmeocnvcn 17452 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
8 cnco 16995 . . . 4  |-  ( ( `' G  e.  ( L  Cn  K )  /\  `' F  e.  ( K  Cn  J ) )  ->  ( `' F  o.  `' G )  e.  ( L  Cn  J ) )
96, 7, 8syl2anr 464 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  ( `' F  o.  `' G
)  e.  ( L  Cn  J ) )
105, 9syl5eqel 2367 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  `' ( G  o.  F )  e.  ( L  Cn  J
) )
11 ishmeo 17450 . 2  |-  ( ( G  o.  F )  e.  ( J  Homeo  L )  <->  ( ( G  o.  F )  e.  ( J  Cn  L
)  /\  `' ( G  o.  F )  e.  ( L  Cn  J
) ) )
124, 10, 11sylanbrc 645 1  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  ( G  o.  F )  e.  ( J  Homeo  L )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   `'ccnv 4688    o. ccom 4693  (class class class)co 5858    Cn ccn 16954    Homeo chmeo 17444
This theorem is referenced by:  hmphtr  17474  xpstopnlem1  17500  tgpconcomp  17795  tsmsxplem1  17835  hmeogrpi  24948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446
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