MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xmstps Structured version   Unicode version

Theorem xmstps 21081
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 2457 . . 3  |-  ( TopOpen `  M )  =  (
TopOpen `  M )
2 eqid 2457 . . 3  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2457 . . 3  |-  ( (
dist `  M )  |`  ( ( Base `  M
)  X.  ( Base `  M ) ) )  =  ( ( dist `  M )  |`  (
( Base `  M )  X.  ( Base `  M
) ) )
41, 2, 3isxms 21075 . 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 1395    e. wcel 1819    X. cxp 5006    |` cres 5010   ` cfv 5594   Basecbs 14643   distcds 14720   TopOpenctopn 14838   MetOpencmopn 18534   TopSpctps 19523   *MetSpcxme 20945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-res 5020  df-iota 5557  df-fv 5602  df-xms 20948
This theorem is referenced by:  mstps  21083  ressxms  21153  prdsxmslem2  21157  tmsxpsmopn  21165  minveclem4a  21970  rrhcn  28131  rrhf  28132  rrexttps  28140
  Copyright terms: Public domain W3C validator