Theorem xmstps 24450

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 2726	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2726	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2726	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isxms 24444	. 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
5	4	simplbi 496	1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   × cxp 5680   ↾ cres 5684  ‘cfv 6554  Basecbs 17213  distcds 17275  TopOpenctopn 17436  MetOpencmopn 21333  TopSpctps 22925  ∞MetSpcxms 24314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-res 5694  df-iota 6506  df-fv 6562  df-xms 24317

This theorem is referenced by:  mstps  24452  ressxms  24525  prdsxmslem2  24529  tmsxpsmopn  24537  minveclem4a  25449  rrhcn  33812  rrhf  33813  rrexttps  33821  sitmcl  34185