Theorem msxms 24342

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 2729	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2729	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2729	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24337	. 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 2109   × cxp 5636   ↾ cres 5640  ‘cfv 6511  Basecbs 17179  distcds 17229  TopOpenctopn 17384  Metcmet 21250  ∞MetSpcxms 24205  MetSpcms 24206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-res 5650  df-iota 6464  df-fv 6519  df-ms 24209

This theorem is referenced by:  mstps  24343  imasf1oms  24378  ressms  24414  prdsms  24419  ngpxms  24489  ngptgp  24524  nlmvscnlem2  24573  nlmvscn  24575  nrginvrcn  24580  nghmcn  24633  cnfldxms  24664  nmhmcn  25020  ipcnlem2  25144  ipcn  25146  nglmle  25202  cmetcusp1  25253  dya2icoseg2  34269