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Metamath Proof Explorer

Theorem mscl

Description: Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006) (Revised by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses mscl.x ⊢ 𝑋 = ( Base ‘ 𝑀 )
mscl.d ⊢ 𝐷 = ( dist ‘ 𝑀 )
Assertion mscl ( ( 𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 mscl.x ⊢ 𝑋 = ( Base ‘ 𝑀 )
2 mscl.d ⊢ 𝐷 = ( dist ‘ 𝑀 )
3 ovres ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
4 3 3adant1 ⊢ ( ( 𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) = ( 𝐴 𝐷 𝐵 ) )
5 1 2 msmet2 ⊢ ( 𝑀 ∈ MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( Met ‘ 𝑋 ) )
6 metcl ⊢ ( ( ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) ∈ ℝ )
7 5 6 syl3an1 ⊢ ( ( 𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 𝐵 ) ∈ ℝ )
8 4 7 eqeltrrd ⊢ ( ( 𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )