Theorem metcl

Description: Closure of the distance function of a metric space. Part of Property M1 of Kreyszig p. 3. (Contributed by NM, 30-Aug-2006)

	Ref	Expression
Assertion	metcl	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		metf	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ )
2		fovrn	⊢ ( ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )
3	1 2	syl3an1	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ )