Theorem msxms 23951

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 2732	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2732	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2732	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 23946	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 498	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2106   × cxp 5673   ↾ cres 5677  ‘cfv 6540  Basecbs 17140  distcds 17202  TopOpenctopn 17363  Metcmet 20922  ∞MetSpcxms 23814  MetSpcms 23815

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-res 5687  df-iota 6492  df-fv 6548  df-ms 23818

This theorem is referenced by:  mstps  23952  imasf1oms  23990  ressms  24026  prdsms  24031  ngpxms  24101  ngptgp  24136  nlmvscnlem2  24193  nlmvscn  24195  nrginvrcn  24200  nghmcn  24253  cnfldxms  24284  nmhmcn  24627  ipcnlem2  24752  ipcn  24754  nglmle  24810  cmetcusp1  24861  dya2icoseg2  33265