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Theorem msxms 23057
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 2824 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2824 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2824 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isms 23052 . 2 (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
54simplbi 501 1 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115   × cxp 5540  cres 5544  cfv 6343  Basecbs 16479  distcds 16570  TopOpenctopn 16691  Metcmet 20524  ∞MetSpcxms 22920  MetSpcms 22921
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-res 5554  df-iota 6302  df-fv 6351  df-ms 22924
This theorem is referenced by:  mstps  23058  imasf1oms  23093  ressms  23129  prdsms  23134  ngpxms  23203  ngptgp  23238  nlmvscnlem2  23287  nlmvscn  23289  nrginvrcn  23294  nghmcn  23347  cnfldxms  23378  nmhmcn  23721  ipcnlem2  23844  ipcn  23846  nglmle  23902  cmetcusp1  23953  dya2icoseg2  31561
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