Theorem xmstps 23206

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 23200	. 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
5	4	simplbi 501	1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   × cxp 5523   ↾ cres 5527  ‘cfv 6339  Basecbs 16586  distcds 16677  TopOpenctopn 16798  MetOpencmopn 20207  TopSpctps 21683  ∞MetSpcxms 23070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-res 5537  df-iota 6297  df-fv 6347  df-xms 23073

This theorem is referenced by:  mstps  23208  ressxms  23278  prdsxmslem2  23282  tmsxpsmopn  23290  minveclem4a  24182  rrhcn  31517  rrhf  31518  rrexttps  31526  sitmcl  31888