Theorem msxms 24369

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 2731	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2731	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2731	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isms 24364	. 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 2111   × cxp 5612   ↾ cres 5616  ‘cfv 6481  Basecbs 17120  distcds 17170  TopOpenctopn 17325  Metcmet 21277  ∞MetSpcxms 24232  MetSpcms 24233

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-res 5626  df-iota 6437  df-fv 6489  df-ms 24236

This theorem is referenced by:  mstps  24370  imasf1oms  24405  ressms  24441  prdsms  24446  ngpxms  24516  ngptgp  24551  nlmvscnlem2  24600  nlmvscn  24602  nrginvrcn  24607  nghmcn  24660  cnfldxms  24691  nmhmcn  25047  ipcnlem2  25171  ipcn  25173  nglmle  25229  cmetcusp1  25280  dya2icoseg2  34291