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Theorem msxms 22991
Description: A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
msxms (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Proof of Theorem msxms
StepHypRef Expression
1 eqid 2818 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2818 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2818 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isms 22986 . 2 (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
54simplbi 498 1 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105   × cxp 5546  cres 5550  cfv 6348  Basecbs 16471  distcds 16562  TopOpenctopn 16683  Metcmet 20459  ∞MetSpcxms 22854  MetSpcms 22855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560  df-iota 6307  df-fv 6356  df-ms 22858
This theorem is referenced by:  mstps  22992  imasf1oms  23027  ressms  23063  prdsms  23068  ngpxms  23137  ngptgp  23172  nlmvscnlem2  23221  nlmvscn  23223  nrginvrcn  23228  nghmcn  23281  cnfldxms  23312  nmhmcn  23651  ipcnlem2  23774  ipcn  23776  nglmle  23832  cmetcusp1  23883  dya2icoseg2  31435
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