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Theorem hmeoco 12499
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 12488 . . 3  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
2 hmeocn 12488 . . 3  |-  ( G  e.  ( K Homeo L )  ->  G  e.  ( K  Cn  L
) )
3 cnco 12404 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( K  Cn  L ) )  -> 
( G  o.  F
)  e.  ( J  Cn  L ) )
41, 2, 3syl2an 287 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  ( G  o.  F )  e.  ( J  Cn  L ) )
5 cnvco 4724 . . 3  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
6 hmeocnvcn 12489 . . . 4  |-  ( G  e.  ( K Homeo L )  ->  `' G  e.  ( L  Cn  K
) )
7 hmeocnvcn 12489 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
8 cnco 12404 . . . 4  |-  ( ( `' G  e.  ( L  Cn  K )  /\  `' F  e.  ( K  Cn  J ) )  ->  ( `' F  o.  `' G )  e.  ( L  Cn  J ) )
96, 7, 8syl2anr 288 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  ( `' F  o.  `' G
)  e.  ( L  Cn  J ) )
105, 9eqeltrid 2226 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  G  e.  ( K Homeo L ) )  ->  `' ( G  o.  F )  e.  ( L  Cn  J
) )
11 ishmeo 12487 . 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 413 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 103    e. wcel 1480   `'ccnv 4538    o. ccom 4543  (class class class)co 5774    Cn ccn 12368   Homeochmeo 12483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12179  df-topon 12192  df-cn 12371  df-hmeo 12484
This theorem is referenced by: (None)
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