Theorem msxms 24393

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 2735	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2735	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2735	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24388	. 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 2108   × cxp 5652   ↾ cres 5656  ‘cfv 6531  Basecbs 17228  distcds 17280  TopOpenctopn 17435  Metcmet 21301  ∞MetSpcxms 24256  MetSpcms 24257

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-res 5666  df-iota 6484  df-fv 6539  df-ms 24260

This theorem is referenced by:  mstps  24394  imasf1oms  24429  ressms  24465  prdsms  24470  ngpxms  24540  ngptgp  24575  nlmvscnlem2  24624  nlmvscn  24626  nrginvrcn  24631  nghmcn  24684  cnfldxms  24715  nmhmcn  25071  ipcnlem2  25196  ipcn  25198  nglmle  25254  cmetcusp1  25305  dya2icoseg2  34310