Theorem msxms 24662

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 2765	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2765	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2765	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24657	. 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 2146   × cxp 5661   ↾ cres 5665  ‘cfv 6540  Basecbs 17291  distcds 17341  TopOpenctopn 17496  Metcmet 21558  ∞MetSpcxms 24525  MetSpcms 24526

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 2148  ax-9 2156  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-res 5675  df-iota 6496  df-fv 6548  df-ms 24529

This theorem is used by:  mstps  24663  imasf1oms  24698  ressms  24734  prdsms  24739  ngpxms  24809  ngptgp  24844  nlmvscnlem2  24893  nlmvscn  24895  nrginvrcn  24900  nghmcn  24953  cnfldxms  24984  nmhmcn  25330  ipcnlem2  25454  ipcn  25456  nglmle  25512  cmetcusp1  25563  dya2icoseg2  34733