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Theorem msxms 18437
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
msxms  |-  ( M  e.  MetSp  ->  M  e.  *
MetSp )

Proof of Theorem msxms
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2404 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2404 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 18432 . 2  |-  ( M  e.  MetSp 
<->  ( M  e.  * MetSp  /\  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )  e.  ( Met `  ( Base `  M ) ) ) )
54simplbi 447 1  |-  ( M  e.  MetSp  ->  M  e.  *
MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721    X. cxp 4835    |` cres 4839   ` cfv 5413   Basecbs 13424   distcds 13493   TopOpenctopn 13604   Metcme 16642   *
MetSpcxme 18300   MetSpcmt 18301
This theorem is referenced by:  mstps  18438  imasf1oms  18473  ressms  18509  prdsms  18514  ngpxms  18601  ngptgp  18630  nlmvscnlem2  18674  nlmvscn  18676  nrginvrcn  18680  nghmcn  18732  cnfldxms  18764  nmhmcn  19081  ipcnlem2  19151  ipcn  19153  cmetcusp1OLD  19258  cmetcusp1  19259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-res 4849  df-iota 5377  df-fv 5421  df-ms 18304
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