Theorem msxms 23960

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 2733	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2733	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2733	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 23955	. 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 2107   × cxp 5675   ↾ cres 5679  ‘cfv 6544  Basecbs 17144  distcds 17206  TopOpenctopn 17367  Metcmet 20930  ∞MetSpcxms 23823  MetSpcms 23824

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689  df-iota 6496  df-fv 6552  df-ms 23827

This theorem is referenced by:  mstps  23961  imasf1oms  23999  ressms  24035  prdsms  24040  ngpxms  24110  ngptgp  24145  nlmvscnlem2  24202  nlmvscn  24204  nrginvrcn  24209  nghmcn  24262  cnfldxms  24293  nmhmcn  24636  ipcnlem2  24761  ipcn  24763  nglmle  24819  cmetcusp1  24870  dya2icoseg2  33277