Theorem xmetcl

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

Proof

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