Theorem msxms 24773

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 2761	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2761	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2761	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24768	. 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 5649   ↾ cres 5653  ‘cfv 6538  Basecbs 17387  distcds 17437  TopOpenctopn 17592  Metcmet 21664  ∞MetSpcxms 24636  MetSpcms 24637

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  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 5657  df-res 5663  df-iota 6494  df-fv 6546  df-ms 24640

This theorem is used by:  mstps  24774  imasf1oms  24809  ressms  24845  prdsms  24850  ngpxms  24920  ngptgp  24955  nlmvscnlem2  25004  nlmvscn  25006  nrginvrcn  25011  nghmcn  25064  cnfldxms  25095  nmhmcn  25441  ipcnlem2  25565  ipcn  25567  nglmle  25623  cmetcusp1  25674  dya2icoseg2  34910