MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeoco Structured version   Unicode version

Theorem hmeoco 17804
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 17792 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
2 hmeocn 17792 . . 3  |-  ( G  e.  ( K  Homeo  L )  ->  G  e.  ( K  Cn  L
) )
3 cnco 17330 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )
41, 2, 3syl2an 464 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  ( G  o.  F )  e.  ( J  Cn  L ) )
5 cnvco 5056 . . 3  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
6 hmeocnvcn 17793 . . . 4  |-  ( G  e.  ( K  Homeo  L )  ->  `' G  e.  ( L  Cn  K
) )
7 hmeocnvcn 17793 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
8 cnco 17330 . . . 4  |-  ( ( `' G  e.  ( L  Cn  K )  /\  `' F  e.  ( K  Cn  J ) )  ->  ( `' F  o.  `' G )  e.  ( L  Cn  J ) )
96, 7, 8syl2anr 465 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  ( `' F  o.  `' G
)  e.  ( L  Cn  J ) )
105, 9syl5eqel 2520 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  G  e.  ( K  Homeo  L ) )  ->  `' ( G  o.  F )  e.  ( L  Cn  J
) )
11 ishmeo 17791 . 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 646 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 359    e. wcel 1725   `'ccnv 4877    o. ccom 4882  (class class class)co 6081    Cn ccn 17288    Homeo chmeo 17785
This theorem is referenced by:  hmphtr  17815  xpstopnlem1  17841  tgpconcomp  18142  tsmsxplem1  18182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291  df-hmeo 17787
  Copyright terms: Public domain W3C validator