Theorem msxms 23595

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 2738	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2738	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2738	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 23590	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 498	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2106   × cxp 5583   ↾ cres 5587  ‘cfv 6427  Basecbs 16900  distcds 16959  TopOpenctopn 17120  Metcmet 20571  ∞MetSpcxms 23458  MetSpcms 23459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-opab 5137  df-xp 5591  df-res 5597  df-iota 6385  df-fv 6435  df-ms 23462

This theorem is referenced by:  mstps  23596  imasf1oms  23634  ressms  23670  prdsms  23675  ngpxms  23745  ngptgp  23780  nlmvscnlem2  23837  nlmvscn  23839  nrginvrcn  23844  nghmcn  23897  cnfldxms  23928  nmhmcn  24271  ipcnlem2  24396  ipcn  24398  nglmle  24454  cmetcusp1  24505  dya2icoseg2  32231