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	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to ADB\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		metf	$$⊢D\in \mathrm{Met}\left(X\right)\to D:X\times X⟶ℝ$$
2		fovcdm	$$⊢\left(D:X\times X⟶ℝ\wedge A\in X\wedge B\in X\right)\to ADB\in ℝ$$
3	1 2	syl3an1	$$⊢\left(D\in \mathrm{Met}\left(X\right)\wedge A\in X\wedge B\in X\right)\to ADB\in ℝ$$