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Theorem xmstps 23752
Description: An extended 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 2736 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2736 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2736 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isxms 23746 . 2 (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
54simplbi 498 1 (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106   × cxp 5629  cres 5633  cfv 6493  Basecbs 17037  distcds 17096  TopOpenctopn 17257  MetOpencmopn 20733  TopSpctps 22227  ∞MetSpcxms 23616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-res 5643  df-iota 6445  df-fv 6501  df-xms 23619
This theorem is referenced by:  mstps  23754  ressxms  23827  prdsxmslem2  23831  tmsxpsmopn  23839  minveclem4a  24740  rrhcn  32406  rrhf  32407  rrexttps  32415  sitmcl  32779
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