Theorem met0

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 NM, 30-Aug-2006)

	Ref	Expression
Assertion	met0	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		metxmet	⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) )
2		xmet0	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )
3	1 2	sylan	⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )