Theorem psmet0

Description: The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018)

	Ref	Expression
Assertion	psmet0	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		elfvex	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝑋 ∈ V )
2		ispsmet	⊢ ( 𝑋 ∈ V → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +e ( 𝑐 𝐷 𝑏 ) ) ) ) ) )
3	1 2	syl	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +e ( 𝑐 𝐷 𝑏 ) ) ) ) ) )
4	3	ibi	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +e ( 𝑐 𝐷 𝑏 ) ) ) ) )
5	4	simprd	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ∀ 𝑎 ∈ 𝑋 ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +e ( 𝑐 𝐷 𝑏 ) ) ) )
6	5	r19.21bi	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) → ( ( 𝑎 𝐷 𝑎 ) = 0 ∧ ∀ 𝑏 ∈ 𝑋 ∀ 𝑐 ∈ 𝑋 ( 𝑎 𝐷 𝑏 ) ≤ ( ( 𝑐 𝐷 𝑎 ) +e ( 𝑐 𝐷 𝑏 ) ) ) )
7	6	simpld	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ 𝑋 ) → ( 𝑎 𝐷 𝑎 ) = 0 )
8	7	ralrimiva	⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ∀ 𝑎 ∈ 𝑋 ( 𝑎 𝐷 𝑎 ) = 0 )
9		id	⊢ ( 𝑎 = 𝐴 → 𝑎 = 𝐴 )
10	9 9	oveq12d	⊢ ( 𝑎 = 𝐴 → ( 𝑎 𝐷 𝑎 ) = ( 𝐴 𝐷 𝐴 ) )
11	10	eqeq1d	⊢ ( 𝑎 = 𝐴 → ( ( 𝑎 𝐷 𝑎 ) = 0 ↔ ( 𝐴 𝐷 𝐴 ) = 0 ) )
12	11	rspcv	⊢ ( 𝐴 ∈ 𝑋 → ( ∀ 𝑎 ∈ 𝑋 ( 𝑎 𝐷 𝑎 ) = 0 → ( 𝐴 𝐷 𝐴 ) = 0 ) )
13	8 12	mpan9	⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐴 ) = 0 )