Theorem msxms 23656

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 2736	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2736	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2736	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 23651	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 499	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2104   × cxp 5598   ↾ cres 5602  ‘cfv 6458  Basecbs 16961  distcds 17020  TopOpenctopn 17181  Metcmet 20632  ∞MetSpcxms 23519  MetSpcms 23520

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-xp 5606  df-res 5612  df-iota 6410  df-fv 6466  df-ms 23523

This theorem is referenced by:  mstps  23657  imasf1oms  23695  ressms  23731  prdsms  23736  ngpxms  23806  ngptgp  23841  nlmvscnlem2  23898  nlmvscn  23900  nrginvrcn  23905  nghmcn  23958  cnfldxms  23989  nmhmcn  24332  ipcnlem2  24457  ipcn  24459  nglmle  24515  cmetcusp1  24566  dya2icoseg2  32294