Theorem msxms 24437

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 2739	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2739	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2739	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24432	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 497	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2119   × cxp 5616   ↾ cres 5620  ‘cfv 6485  Basecbs 17170  distcds 17220  TopOpenctopn 17375  Metcmet 21333  ∞MetSpcxms 24300  MetSpcms 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-res 5630  df-iota 6441  df-fv 6493  df-ms 24304

This theorem is referenced by:  mstps  24438  imasf1oms  24473  ressms  24509  prdsms  24514  ngpxms  24584  ngptgp  24619  nlmvscnlem2  24668  nlmvscn  24670  nrginvrcn  24675  nghmcn  24728  cnfldxms  24759  nmhmcn  25105  ipcnlem2  25229  ipcn  25231  nglmle  25287  cmetcusp1  25338  dya2icoseg2  34462