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

Theorem msxms 22069
 Description: A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
msxms (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)

Proof of Theorem msxms
StepHypRef Expression
1 eqid 2610 . . 3 (TopOpen‘𝑀) = (TopOpen‘𝑀)
2 eqid 2610 . . 3 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2610 . . 3 ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
41, 2, 3isms 22064 . 2 (𝑀 ∈ MetSp ↔ (𝑀 ∈ ∞MetSp ∧ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) ∈ (Met‘(Base‘𝑀))))
54simplbi 475 1 (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   × cxp 5036   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  distcds 15777  TopOpenctopn 15905  Metcme 19553  ∞MetSpcxme 21932  MetSpcmt 21933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-ms 21936 This theorem is referenced by:  mstps  22070  imasf1oms  22105  ressms  22141  prdsms  22146  ngpxms  22215  ngptgp  22250  nlmvscnlem2  22299  nlmvscn  22301  nrginvrcn  22306  nghmcn  22359  cnfldxms  22390  nmhmcn  22728  ipcnlem2  22851  ipcn  22853  nglmle  22908  cmetcusp1  22957  dya2icoseg2  29667
 Copyright terms: Public domain W3C validator