Theorem msxms 24828

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 2763	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2763	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2763	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24823	. 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 2147   × cxp 5656   ↾ cres 5660  ‘cfv 6566  Basecbs 17431  distcds 17481  TopOpenctopn 17642  Metcmet 21715  ∞MetSpcxms 24691  MetSpcms 24692

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-xp 5664  df-res 5670  df-iota 6520  df-fv 6574  df-ms 24695

This theorem is used by:  mstps  24829  imasf1oms  24864  ressms  24900  prdsms  24905  ngpxms  24975  ngptgp  25010  nlmvscnlem2  25059  nlmvscn  25061  nrginvrcn  25066  nghmcn  25119  cnfldxms  25150  nmhmcn  25496  ipcnlem2  25620  ipcn  25622  nglmle  25678  cmetcusp1  25729  dya2icoseg2  35021