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Theorem msxms 21083
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 2457 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2457 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2457 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 21078 . 2  |-  ( M  e.  MetSp 
<->  ( M  e.  *MetSp  /\  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )  e.  ( Met `  ( Base `  M ) ) ) )
54simplbi 460 1  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    X. cxp 5006    |` cres 5010   ` cfv 5594   Basecbs 14644   distcds 14721   TopOpenctopn 14839   Metcme 18531   *MetSpcxme 20946   MetSpcmt 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-res 5020  df-iota 5557  df-fv 5602  df-ms 20950
This theorem is referenced by:  mstps  21084  imasf1oms  21119  ressms  21155  prdsms  21160  ngpxms  21247  ngptgp  21276  nlmvscnlem2  21320  nlmvscn  21322  nrginvrcn  21326  nghmcn  21378  cnfldxms  21410  nmhmcn  21729  ipcnlem2  21810  ipcn  21812  cmetcusp1OLD  21917  cmetcusp1  21918  dya2icoseg2  28422
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