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Theorem xmstps 20026
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
xmstps  |-  ( M  e.  *MetSp  ->  M  e.  TopSp )

Proof of Theorem xmstps
StepHypRef Expression
1 eqid 2441 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2441 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2441 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isxms 20020 . 2  |-  ( M  e.  *MetSp  <->  ( M  e.  TopSp  /\  ( TopOpen `  M )  =  (
MetOpen `  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) ) ) ) )
54simplbi 460 1  |-  ( M  e.  *MetSp  ->  M  e.  TopSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    X. cxp 4836    |` cres 4840   ` cfv 5416   Basecbs 14172   distcds 14245   TopOpenctopn 14358   MetOpencmopn 17804   TopSpctps 18499   *MetSpcxme 19890
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 2422
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-xp 4844  df-res 4850  df-iota 5379  df-fv 5424  df-xms 19893
This theorem is referenced by:  mstps  20028  ressxms  20098  prdsxmslem2  20102  tmsxpsmopn  20110  minveclem4a  20915  rrhcn  26424  rrhf  26425  rrexttps  26433
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