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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 ) )