Theorem xmstopn 15539

Description: The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypotheses

Ref	Expression
isms.j	⊢ 𝐽 = (TopOpen‘𝐾)
isms.x	⊢ 𝑋 = (Base‘𝐾)
isms.d	⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))

Assertion

Ref	Expression
xmstopn	⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn

Step	Hyp	Ref	Expression
1		isms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝐾)
2		isms.x	. . 3 ⊢ 𝑋 = (Base‘𝐾)
3		isms.d	. . 3 ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
4	1, 2, 3	isxms 15535	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
5	4	simprbi 275	1 ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209   × cxp 4770   ↾ cres 4774  ‘cfv 5375  Basecbs 13335  distcds 13423  TopOpenctopn 13577  MetOpencmopn 14861  TopSpctps 15114  ∞MetSpcxms 15420

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-res 4784  df-iota 5335  df-fv 5383  df-xms 15423

This theorem is referenced by: (None)