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Theorem xmstopn 20153
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 20149 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
54simprbi 464 1  |-  ( K  e.  *MetSp  ->  J  =  ( MetOpen `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    X. cxp 4941    |` cres 4945   ` cfv 5521   Basecbs 14287   distcds 14361   TopOpenctopn 14474   MetOpencmopn 17926   TopSpctps 18628   *MetSpcxme 20019
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
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-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-xp 4949  df-res 4955  df-iota 5484  df-fv 5529  df-xms 20022
This theorem is referenced by:  imasf1oxms  20191  ressxms  20227  prdsxmslem2  20231  tmsxpsmopn  20239  xmsuspOLD  20287  xmsusp  20288  cmetcusp1OLD  20990  cmetcusp1  20991  minveclem4a  21044  minveclem4  21046  qqhcn  26560  rrhcn  26566  rrexthaus  26576  dya2icoseg2  26832
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