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Theorem hmeoco 19458
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 19446 . . 3  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
2 hmeocn 19446 . . 3  |-  ( G  e.  ( K Homeo L )  ->  G  e.  ( K  Cn  L
) )
3 cnco 18983 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )
41, 2, 3syl2an 477 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  ( G  o.  F )  e.  ( J  Cn  L ) )
5 cnvco 5120 . . 3  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
6 hmeocnvcn 19447 . . . 4  |-  ( G  e.  ( K Homeo L )  ->  `' G  e.  ( L  Cn  K
) )
7 hmeocnvcn 19447 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
8 cnco 18983 . . . 4  |-  ( ( `' G  e.  ( L  Cn  K )  /\  `' F  e.  ( K  Cn  J ) )  ->  ( `' F  o.  `' G )  e.  ( L  Cn  J ) )
96, 7, 8syl2anr 478 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  ( `' F  o.  `' G
)  e.  ( L  Cn  J ) )
105, 9syl5eqel 2541 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  `' ( G  o.  F )  e.  ( L  Cn  J
) )
11 ishmeo 19445 . 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 664 1  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  ( G  o.  F )  e.  ( J Homeo L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   `'ccnv 4934    o. ccom 4939  (class class class)co 6187    Cn ccn 18941   Homeochmeo 19439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-map 7313  df-top 18616  df-topon 18619  df-cn 18944  df-hmeo 19441
This theorem is referenced by:  hmphtr  19469  xpstopnlem1  19495  tgpconcomp  19796  tsmsxplem1  19840
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