Theorem xmstps 23514

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 2738	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2738	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2738	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isxms 23508	. 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
5	4	simplbi 497	1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   × cxp 5578   ↾ cres 5582  ‘cfv 6418  Basecbs 16840  distcds 16897  TopOpenctopn 17049  MetOpencmopn 20500  TopSpctps 21989  ∞MetSpcxms 23378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-res 5592  df-iota 6376  df-fv 6426  df-xms 23381

This theorem is referenced by:  mstps  23516  ressxms  23587  prdsxmslem2  23591  tmsxpsmopn  23599  minveclem4a  24499  rrhcn  31847  rrhf  31848  rrexttps  31856  sitmcl  32218