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Theorem msxms 18485
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 2437 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2437 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2437 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 18480 . 2  |-  ( M  e.  MetSp 
<->  ( M  e.  * MetSp  /\  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )  e.  ( Met `  ( Base `  M ) ) ) )
54simplbi 448 1  |-  ( M  e.  MetSp  ->  M  e.  *
MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    X. cxp 4877    |` cres 4881   ` cfv 5455   Basecbs 13470   distcds 13539   TopOpenctopn 13650   Metcme 16688   *
MetSpcxme 18348   MetSpcmt 18349
This theorem is referenced by:  mstps  18486  imasf1oms  18521  ressms  18557  prdsms  18562  ngpxms  18649  ngptgp  18678  nlmvscnlem2  18722  nlmvscn  18724  nrginvrcn  18728  nghmcn  18780  cnfldxms  18812  nmhmcn  19129  ipcnlem2  19199  ipcn  19201  cmetcusp1OLD  19306  cmetcusp1  19307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-xp 4885  df-res 4891  df-iota 5419  df-fv 5463  df-ms 18352
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