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Theorem msxms 20145
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 2451 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2451 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2451 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 20140 . 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 1758    X. cxp 4936    |` cres 4940   ` cfv 5516   Basecbs 14276   distcds 14349   TopOpenctopn 14462   Metcme 17911   *MetSpcxme 20008   MetSpcmt 20009
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 1952  ax-ext 2430
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 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-xp 4944  df-res 4950  df-iota 5479  df-fv 5524  df-ms 20012
This theorem is referenced by:  mstps  20146  imasf1oms  20181  ressms  20217  prdsms  20222  ngpxms  20309  ngptgp  20338  nlmvscnlem2  20382  nlmvscn  20384  nrginvrcn  20388  nghmcn  20440  cnfldxms  20472  nmhmcn  20791  ipcnlem2  20872  ipcn  20874  cmetcusp1OLD  20979  cmetcusp1  20980  dya2icoseg2  26827
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