Theorem msxms 23588

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

Step	Hyp	Ref	Expression
1		eqid 2739	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2739	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2739	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 23583	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 497	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2109   × cxp 5586   ↾ cres 5590  ‘cfv 6430  Basecbs 16893  distcds 16952  TopOpenctopn 17113  Metcmet 20564  ∞MetSpcxms 23451  MetSpcms 23452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-xp 5594  df-res 5600  df-iota 6388  df-fv 6438  df-ms 23455

This theorem is referenced by:  mstps  23589  imasf1oms  23627  ressms  23663  prdsms  23668  ngpxms  23738  ngptgp  23773  nlmvscnlem2  23830  nlmvscn  23832  nrginvrcn  23837  nghmcn  23890  cnfldxms  23921  nmhmcn  24264  ipcnlem2  24389  ipcn  24391  nglmle  24447  cmetcusp1  24498  dya2icoseg2  32224