Theorem psmetf

Description: The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018)

	Ref	Expression
Assertion	psmetf	$$⊢D\in \mathrm{PsMet}\left(X\right)\to D:X\times X⟶{ℝ}^{*}$$

Proof

Step	Hyp	Ref	Expression
1		elfvex	$$⊢D\in \mathrm{PsMet}\left(X\right)\to X\in \mathrm{V}$$
2		ispsmet	$$⊢X\in \mathrm{V}\to \left(D\in \mathrm{PsMet}\left(X\right)\leftrightarrow \left(D:X\times X⟶{ℝ}^{*}\wedge \forall a\in X\left(aDa=0\wedge \forall b\in X\forall c\in XaDb\le \left(cDa\right){+}_{𝑒}\left(cDb\right)\right)\right)\right)$$
3	1 2	syl	$$⊢D\in \mathrm{PsMet}\left(X\right)\to \left(D\in \mathrm{PsMet}\left(X\right)\leftrightarrow \left(D:X\times X⟶{ℝ}^{*}\wedge \forall a\in X\left(aDa=0\wedge \forall b\in X\forall c\in XaDb\le \left(cDa\right){+}_{𝑒}\left(cDb\right)\right)\right)\right)$$
4	3	ibi	$$⊢D\in \mathrm{PsMet}\left(X\right)\to \left(D:X\times X⟶{ℝ}^{*}\wedge \forall a\in X\left(aDa=0\wedge \forall b\in X\forall c\in XaDb\le \left(cDa\right){+}_{𝑒}\left(cDb\right)\right)\right)$$
5	4	simpld	$$⊢D\in \mathrm{PsMet}\left(X\right)\to D:X\times X⟶{ℝ}^{*}$$