Theorem msxms 24681

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 2760	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2760	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2760	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24676	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 502	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   × cxp 5653   ↾ cres 5657  ‘cfv 6533  Basecbs 17302  distcds 17352  TopOpenctopn 17507  Metcmet 21572  ∞MetSpcxms 24544  MetSpcms 24545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-res 5667  df-iota 6489  df-fv 6541  df-ms 24548

This theorem is used by:  mstps  24682  imasf1oms  24717  ressms  24753  prdsms  24758  ngpxms  24828  ngptgp  24863  nlmvscnlem2  24912  nlmvscn  24914  nrginvrcn  24919  nghmcn  24972  cnfldxms  25003  nmhmcn  25349  ipcnlem2  25473  ipcn  25475  nglmle  25531  cmetcusp1  25582  dya2icoseg2  34790