Theorem xmstps 24799

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 2763	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2763	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2763	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isxms 24793	. 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 1570   ∈ wcel 2147   × cxp 5653   ↾ cres 5657  ‘cfv 6555  Basecbs 17413  distcds 17463  TopOpenctopn 17618  MetOpencmopn 21695  TopSpctps 23277  ∞MetSpcxms 24663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-res 5667  df-iota 6509  df-fv 6563  df-xms 24666

This theorem is used by:  mstps  24801  ressxms  24871  prdsxmslem2  24875  tmsxpsmopn  24883  minveclem4a  25778  rrhcn  34696  rrhf  34697  rrexttps  34705  sitmcl  35050