Metamath Proof Explorer


Theorem xmetutop

Description: The topology induced by a uniform structure generated by an extended metric D is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018)

Ref Expression
Assertion xmetutop ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) → ( unifTop ‘ ( metUnif ‘ 𝐷 ) ) = ( MetOpen ‘ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 xmetpsmet ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) )
2 psmetutop ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( unifTop ‘ ( metUnif ‘ 𝐷 ) ) = ( topGen ‘ ran ( ball ‘ 𝐷 ) ) )
3 1 2 sylan2 ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) → ( unifTop ‘ ( metUnif ‘ 𝐷 ) ) = ( topGen ‘ ran ( ball ‘ 𝐷 ) ) )
4 eqid ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 )
5 4 mopnval ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( MetOpen ‘ 𝐷 ) = ( topGen ‘ ran ( ball ‘ 𝐷 ) ) )
6 5 adantl ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) → ( MetOpen ‘ 𝐷 ) = ( topGen ‘ ran ( ball ‘ 𝐷 ) ) )
7 3 6 eqtr4d ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) → ( unifTop ‘ ( metUnif ‘ 𝐷 ) ) = ( MetOpen ‘ 𝐷 ) )