Theorem xmstps 15451

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 2234	. . 3 ⊢ (TopOpen‘𝑀) = (TopOpen‘𝑀)
2		eqid 2234	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2234	. . 3 ⊢ ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))) = ((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀)))
4	1, 2, 3	isxms 15445	. 2 ⊢ (𝑀 ∈ ∞MetSp ↔ (𝑀 ∈ TopSp ∧ (TopOpen‘𝑀) = (MetOpen‘((dist‘𝑀) ↾ ((Base‘𝑀) × (Base‘𝑀))))))
5	4	simplbi 274	1 ⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205   × cxp 4752   ↾ cres 4756  ‘cfv 5357  Basecbs 13299  distcds 13386  TopOpenctopn 13540  MetOpencmopn 14818  TopSpctps 15024  ∞MetSpcxms 15330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-res 4766  df-iota 5317  df-fv 5365  df-xms 15333

This theorem is referenced by:  mstps  15453