Theorem msxms 24620

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 2763	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2763	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2763	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24615	. 2 ⊢ (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
5	4	simplbi 501	1 ⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2143   × cxp 5659   ↾ cres 5663  ‘cfv 6536  Basecbs 17273  distcds 17323  TopOpenctopn 17478  Metcmet 21517  ∞MetSpcxms 24483  MetSpcms 24484

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-res 5673  df-iota 6492  df-fv 6544  df-ms 24487

This theorem is used by:  mstps  24621  imasf1oms  24656  ressms  24692  prdsms  24697  ngpxms  24767  ngptgp  24802  nlmvscnlem2  24851  nlmvscn  24853  nrginvrcn  24858  nghmcn  24911  cnfldxms  24942  nmhmcn  25288  ipcnlem2  25412  ipcn  25414  nglmle  25470  cmetcusp1  25521  dya2icoseg2  34677