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Theorem hmeoco 17479
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 17467 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
2 hmeocn 17467 . . 3  |-  ( G  e.  ( K  Homeo  L )  ->  G  e.  ( K  Cn  L
) )
3 cnco 17011 . . 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 4881 . . 3  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
6 hmeocnvcn 17468 . . . 4  |-  ( G  e.  ( K  Homeo  L )  ->  `' G  e.  ( L  Cn  K
) )
7 hmeocnvcn 17468 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
8 cnco 17011 . . . 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 2380 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  `' ( G  o.  F )  e.  ( L  Cn  J
) )
11 ishmeo 17466 . 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 1696   `'ccnv 4704    o. ccom 4709  (class class class)co 5874    Cn ccn 16970    Homeo chmeo 17460
This theorem is referenced by:  hmphtr  17490  xpstopnlem1  17516  tgpconcomp  17811  tsmsxplem1  17851  hmeogrpi  25639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462
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