Theorem xmstps 24477

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 2734	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2734	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2734	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isxms 24471	. 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 1537   ∈ wcel 2103   × cxp 5697   ↾ cres 5701  ‘cfv 6572  Basecbs 17253  distcds 17315  TopOpenctopn 17476  MetOpencmopn 21372  TopSpctps 22952  ∞MetSpcxms 24341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-xp 5705  df-res 5711  df-iota 6524  df-fv 6580  df-xms 24344

This theorem is referenced by:  mstps  24479  ressxms  24552  prdsxmslem2  24556  tmsxpsmopn  24564  minveclem4a  25476  rrhcn  33935  rrhf  33936  rrexttps  33944  sitmcl  34308