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Theorem msxms 18052
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 2316 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2316 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2316 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 18047 . 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 1701    X. cxp 4724    |` cres 4728   ` cfv 5292   Basecbs 13195   distcds 13264   TopOpenctopn 13375   Metcme 16419   *
MetSpcxme 17934   MetSpcmt 17935
This theorem is referenced by:  mstps  18053  imasf1oms  18088  ressms  18124  prdsms  18129  ngpxms  18175  ngptgp  18204  nlmvscnlem2  18248  nlmvscn  18250  nrginvrcn  18254  nghmcn  18306  cnfldxms  18338  nmhmcn  18654  ipcnlem2  18724  ipcn  18726  cmetcusp1  23511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-res 4738  df-iota 5256  df-fv 5300  df-ms 17938
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