Theorem msxms 24404

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 2725	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2725	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2725	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24399	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 496	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2098   × cxp 5676   ↾ cres 5680  ‘cfv 6549  Basecbs 17183  distcds 17245  TopOpenctopn 17406  Metcmet 21282  ∞MetSpcxms 24267  MetSpcms 24268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-res 5690  df-iota 6501  df-fv 6557  df-ms 24271

This theorem is referenced by:  mstps  24405  imasf1oms  24443  ressms  24479  prdsms  24484  ngpxms  24554  ngptgp  24589  nlmvscnlem2  24646  nlmvscn  24648  nrginvrcn  24653  nghmcn  24706  cnfldxms  24737  nmhmcn  25091  ipcnlem2  25216  ipcn  25218  nglmle  25274  cmetcusp1  25325  dya2icoseg2  34029