Theorem xmstps 24619

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 24613	. 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
5	4	simplbi 501	1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2143   × cxp 5659   ↾ cres 5663  ‘cfv 6536  Basecbs 17273  distcds 17323  TopOpenctopn 17478  MetOpencmopn 21521  TopSpctps 23098  ∞MetSpcxms 24483

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-res 5673  df-iota 6492  df-fv 6544  df-xms 24486

This theorem is used by:  mstps  24621  ressxms  24691  prdsxmslem2  24695  tmsxpsmopn  24703  minveclem4a  25598  rrhcn  34396  rrhf  34397  rrexttps  34405  sitmcl  34750