distspace

Description: A set X together with a (distance) function D which is a pseudometric is adistance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set X equipped with adistance D , which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021) (Revised by AV, 5-Jul-2022)

		Ref	Expression
	Assertion	distspace	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to ((D : X \times X ⟶ ℝ^{*} \land A D A = 0) \land (0 \leq A D B \land A D B = B D A))$

Proof

Step	Hyp	Ref	Expression
1		psmetf	$⊢ D \in PsMet (X) \to D : X \times X ⟶ ℝ^{*}$
2	1	3ad2ant1	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to D : X \times X ⟶ ℝ^{*}$
3		psmet0	$⊢ (D \in PsMet (X) \land A \in X) \to A D A = 0$
4	3	3adant3	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to A D A = 0$
5	2 4	jca	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to (D : X \times X ⟶ ℝ^{*} \land A D A = 0)$
6		psmetge0	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to 0 \leq A D B$
7		psmetsym	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to A D B = B D A$
8	5 6 7	jca32	$⊢ (D \in PsMet (X) \land A \in X \land B \in X) \to ((D : X \times X ⟶ ℝ^{*} \land A D A = 0) \land (0 \leq A D B \land A D B = B D A))$

Metamath Proof Explorer

Theorem distspace

Proof