Theorem xmet0

Description: The distance function of a metric space is zero if its arguments are equal. Definition 14-1.1(a) of Gleason p. 223. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xmet0	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 𝐴 = 𝐴
2		xmeteq0	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐴 ) = 0 ↔ 𝐴 = 𝐴 ) )
3	2	3anidm23	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐴 ) = 0 ↔ 𝐴 = 𝐴 ) )
4	1 3	mpbiri	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )