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Theorem xmstps 23206
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 2738 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2738 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2738 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isxms 23200 . 2 (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
54simplbi 501 1 (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114   × cxp 5523  cres 5527  cfv 6339  Basecbs 16586  distcds 16677  TopOpenctopn 16798  MetOpencmopn 20207  TopSpctps 21683  ∞MetSpcxms 23070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-res 5537  df-iota 6297  df-fv 6347  df-xms 23073
This theorem is referenced by:  mstps  23208  ressxms  23278  prdsxmslem2  23282  tmsxpsmopn  23290  minveclem4a  24182  rrhcn  31517  rrhf  31518  rrexttps  31526  sitmcl  31888
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