Theorem xmstps 24772

Description: An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
xmstps	⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Proof of Theorem xmstps

Step	Hyp	Ref	Expression
1		eqid 2760	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2760	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2760	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isxms 24766	. 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
5	4	simplbi 502	1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   × cxp 5648   ↾ cres 5652  ‘cfv 6538  Basecbs 17387  distcds 17437  TopOpenctopn 17592  MetOpencmopn 21668  TopSpctps 23250  ∞MetSpcxms 24636

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-res 5662  df-iota 6494  df-fv 6546  df-xms 24639

This theorem is used by:  mstps  24774  ressxms  24844  prdsxmslem2  24848  tmsxpsmopn  24856  minveclem4a  25751  rrhcn  34629  rrhf  34630  rrexttps  34638  sitmcl  34983