Theorem msxms 24782

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 24777	. 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 2144   × cxp 5649   ↾ cres 5653  ‘cfv 6543  Basecbs 17396  distcds 17446  TopOpenctopn 17601  Metcmet 21673  ∞MetSpcxms 24645  MetSpcms 24646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5657  df-res 5663  df-iota 6497  df-fv 6551  df-ms 24649

This theorem is used by:  mstps  24783  imasf1oms  24818  ressms  24854  prdsms  24859  ngpxms  24929  ngptgp  24964  nlmvscnlem2  25013  nlmvscn  25015  nrginvrcn  25020  nghmcn  25073  cnfldxms  25104  nmhmcn  25450  ipcnlem2  25574  ipcn  25576  nglmle  25632  cmetcusp1  25683  dya2icoseg2  34960