MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  msxms Structured version   Visualization version   GIF version

Theorem msxms 24480
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
StepHypRef Expression
1 eqid 2735 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2735 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2735 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isms 24475 . 2 (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
54simplbi 497 1 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   × cxp 5687  cres 5691  cfv 6563  Basecbs 17245  distcds 17307  TopOpenctopn 17468  Metcmet 21368  ∞MetSpcxms 24343  MetSpcms 24344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-iota 6516  df-fv 6571  df-ms 24347
This theorem is referenced by:  mstps  24481  imasf1oms  24519  ressms  24555  prdsms  24560  ngpxms  24630  ngptgp  24665  nlmvscnlem2  24722  nlmvscn  24724  nrginvrcn  24729  nghmcn  24782  cnfldxms  24813  nmhmcn  25167  ipcnlem2  25292  ipcn  25294  nglmle  25350  cmetcusp1  25401  dya2icoseg2  34260
  Copyright terms: Public domain W3C validator