Theorem msxms 24396

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 2734	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2734	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2734	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24391	. 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 2113   × cxp 5620   ↾ cres 5624  ‘cfv 6490  Basecbs 17134  distcds 17184  TopOpenctopn 17339  Metcmet 21293  ∞MetSpcxms 24259  MetSpcms 24260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-xp 5628  df-res 5634  df-iota 6446  df-fv 6498  df-ms 24263

This theorem is referenced by:  mstps  24397  imasf1oms  24432  ressms  24468  prdsms  24473  ngpxms  24543  ngptgp  24578  nlmvscnlem2  24627  nlmvscn  24629  nrginvrcn  24634  nghmcn  24687  cnfldxms  24718  nmhmcn  25074  ipcnlem2  25198  ipcn  25200  nglmle  25256  cmetcusp1  25307  dya2icoseg2  34384