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Theorem msxms 18000
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 2283 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2283 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2283 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 17995 . 2  |-  ( M  e.  MetSp 
<->  ( M  e.  * MetSp  /\  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )  e.  ( Met `  ( Base `  M ) ) ) )
54simplbi 446 1  |-  ( M  e.  MetSp  ->  M  e.  *
MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   distcds 13217   TopOpenctopn 13326   Metcme 16370   *
MetSpcxme 17882   MetSpcmt 17883
This theorem is referenced by:  mstps  18001  imasf1oms  18036  ressms  18072  prdsms  18077  ngpxms  18123  ngptgp  18152  nlmvscnlem2  18196  nlmvscn  18198  nrginvrcn  18202  nghmcn  18254  cnfldxms  18286  nmhmcn  18601  ipcnlem2  18671  ipcn  18673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263  df-ms 17886
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