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Theorem xmstopn 21120
Description: The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
isms.j  |-  J  =  ( TopOpen `  K )
isms.x  |-  X  =  ( Base `  K
)
isms.d  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
xmstopn  |-  ( K  e.  *MetSp  ->  J  =  ( MetOpen `  D
) )

Proof of Theorem xmstopn
StepHypRef Expression
1 isms.j . . 3  |-  J  =  ( TopOpen `  K )
2 isms.x . . 3  |-  X  =  ( Base `  K
)
3 isms.d . . 3  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
41, 2, 3isxms 21116 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
54simprbi 462 1  |-  ( K  e.  *MetSp  ->  J  =  ( MetOpen `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    X. cxp 4986    |` cres 4990   ` cfv 5570   Basecbs 14716   distcds 14793   TopOpenctopn 14911   MetOpencmopn 18603   TopSpctps 19564   *MetSpcxme 20986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-res 5000  df-iota 5534  df-fv 5578  df-xms 20989
This theorem is referenced by:  imasf1oxms  21158  ressxms  21194  prdsxmslem2  21198  tmsxpsmopn  21206  xmsuspOLD  21254  xmsusp  21255  cmetcusp1OLD  21957  cmetcusp1  21958  minveclem4a  22011  minveclem4  22013  qqhcn  28206  rrhcn  28212  rrexthaus  28222  dya2icoseg2  28486
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