Theorem msxms 23061

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 2798	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2798	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2798	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 23056	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 501	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2111   × cxp 5517   ↾ cres 5521  ‘cfv 6324  Basecbs 16475  distcds 16566  TopOpenctopn 16687  Metcmet 20077  ∞MetSpcxms 22924  MetSpcms 22925

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

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 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-res 5531  df-iota 6283  df-fv 6332  df-ms 22928

This theorem is referenced by:  mstps  23062  imasf1oms  23097  ressms  23133  prdsms  23138  ngpxms  23207  ngptgp  23242  nlmvscnlem2  23291  nlmvscn  23293  nrginvrcn  23298  nghmcn  23351  cnfldxms  23382  nmhmcn  23725  ipcnlem2  23848  ipcn  23850  nglmle  23906  cmetcusp1  23957  dya2icoseg2  31646