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Theorem msxms 22479
Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
msxms (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Proof of Theorem msxms
StepHypRef Expression
1 eqid 2771 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2771 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2771 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isms 22474 . 2 (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
54simplbi 485 1 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145   × cxp 5247  cres 5251  cfv 6031  Basecbs 16064  distcds 16158  TopOpenctopn 16290  Metcme 19947  ∞MetSpcxme 22342  MetSpcmt 22343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-ms 22346
This theorem is referenced by:  mstps  22480  imasf1oms  22515  ressms  22551  prdsms  22556  ngpxms  22625  ngptgp  22660  nlmvscnlem2  22709  nlmvscn  22711  nrginvrcn  22716  nghmcn  22769  cnfldxms  22800  nmhmcn  23139  ipcnlem2  23262  ipcn  23264  nglmle  23319  cmetcusp1  23368  dya2icoseg2  30680
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