Theorem psmetcl

Description: Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018)

	Ref	Expression
Assertion	psmetcl	$$⊢\left(D\in \mathrm{PsMet}\left(X\right)\wedge A\in X\wedge B\in X\right)\to ADB\in {ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		psmetf	$$⊢D\in \mathrm{PsMet}\left(X\right)\to D:X\times X⟶{ℝ}^{*}$$
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	$$⊢\left(D\in \mathrm{PsMet}\left(X\right)\wedge A\in X\wedge B\in X\right)\to ADB\in {ℝ}^{*}$$