Theorem msxms 24464

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 2737	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2737	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2737	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24459	. 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 5683   ↾ cres 5687  ‘cfv 6561  Basecbs 17247  distcds 17306  TopOpenctopn 17466  Metcmet 21350  ∞MetSpcxms 24327  MetSpcms 24328

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-res 5697  df-iota 6514  df-fv 6569  df-ms 24331

This theorem is referenced by:  mstps  24465  imasf1oms  24503  ressms  24539  prdsms  24544  ngpxms  24614  ngptgp  24649  nlmvscnlem2  24706  nlmvscn  24708  nrginvrcn  24713  nghmcn  24766  cnfldxms  24797  nmhmcn  25153  ipcnlem2  25278  ipcn  25280  nglmle  25336  cmetcusp1  25387  dya2icoseg2  34280