This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem isms2

Description: Express the predicate " <. X , D >. is a metric space" with underlying set X and distance function D . (Contributed by NM, 27-Aug-2006) (Revised by Mario Carneiro, 24-Aug-2015)

Ref Expression
Hypotheses isms.j ⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
isms.x ⊢ 𝑋 = ( Base ‘ 𝐾 )
isms.d ⊢ 𝐷 = ( ( dist ‘ 𝐾 ) ↾ ( 𝑋 × 𝑋 ) )
Assertion isms2 ( 𝐾 ∈ MetSp ↔ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 isms.j ⊢ 𝐽 = ( TopOpen ‘ 𝐾 )
2 isms.x ⊢ 𝑋 = ( Base ‘ 𝐾 )
3 isms.d ⊢ 𝐷 = ( ( dist ‘ 𝐾 ) ↾ ( 𝑋 × 𝑋 ) )
4 1 2 3 isxms2 ⊢ ( 𝐾 ∈ ∞MetSp ↔ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )
5 4 anbi1i ⊢ ( ( 𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) ↔ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
6 1 2 3 isms ⊢ ( 𝐾 ∈ MetSp ↔ ( 𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
7 metxmet ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
8 7 pm4.71ri ⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ↔ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
9 8 anbi1i ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) ↔ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )
10 an32 ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) ↔ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
11 9 10 bitri ⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) ↔ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) ∧ 𝐷 ∈ ( Met ‘ 𝑋 ) ) )
12 5 6 11 3bitr4i ⊢ ( 𝐾 ∈ MetSp ↔ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐽 = ( MetOpen ‘ 𝐷 ) ) )