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Theorem msxms 20009
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 2438 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2438 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2438 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 20004 . 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 1756    X. cxp 4833    |` cres 4837   ` cfv 5413   Basecbs 14166   distcds 14239   TopOpenctopn 14352   Metcme 17782   *MetSpcxme 19872   MetSpcmt 19873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-res 4847  df-iota 5376  df-fv 5421  df-ms 19876
This theorem is referenced by:  mstps  20010  imasf1oms  20045  ressms  20081  prdsms  20086  ngpxms  20173  ngptgp  20202  nlmvscnlem2  20246  nlmvscn  20248  nrginvrcn  20252  nghmcn  20304  cnfldxms  20336  nmhmcn  20655  ipcnlem2  20736  ipcn  20738  cmetcusp1OLD  20843  cmetcusp1  20844  dya2icoseg2  26662
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