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Theorem msxms 19871
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 2433 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2433 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2433 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isms 19866 . 2  |-  ( M  e.  MetSp 
<->  ( M  e.  *MetSp  /\  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )  e.  ( Met `  ( Base `  M ) ) ) )
54simplbi 457 1  |-  ( M  e.  MetSp  ->  M  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1755    X. cxp 4825    |` cres 4829   ` cfv 5406   Basecbs 14157   distcds 14230   TopOpenctopn 14343   Metcme 17646   *MetSpcxme 19734   MetSpcmt 19735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-xp 4833  df-res 4839  df-iota 5369  df-fv 5414  df-ms 19738
This theorem is referenced by:  mstps  19872  imasf1oms  19907  ressms  19943  prdsms  19948  ngpxms  20035  ngptgp  20064  nlmvscnlem2  20108  nlmvscn  20110  nrginvrcn  20114  nghmcn  20166  cnfldxms  20198  nmhmcn  20517  ipcnlem2  20598  ipcn  20600  cmetcusp1OLD  20705  cmetcusp1  20706  dya2icoseg2  26547
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