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Theorem xmstps 22068
 Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
xmstps (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Proof of Theorem xmstps
StepHypRef Expression
1 eqid 2610 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2610 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2610 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isxms 22062 . 2 (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
54simplbi 475 1 (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   × cxp 5036   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  distcds 15777  TopOpenctopn 15905  MetOpencmopn 19557  TopSpctps 20519  ∞MetSpcxme 21932 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-xms 21935 This theorem is referenced by:  mstps  22070  ressxms  22140  prdsxmslem2  22144  tmsxpsmopn  22152  minveclem4a  23009  rrhcn  29369  rrhf  29370  rrexttps  29378  sitmcl  29740
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