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Metamath Proof Explorer

Theorem xmeteq0

Description: The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmeteq0 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 elfvdm ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 ∈ dom ∞Met )
2 isxmet ⊢ ( 𝑋 ∈ dom ∞Met → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) ) )
3 1 2 syl ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ↔ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) ) )
4 3 ibi ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) ) )
5 simpl ⊢ ( ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) → ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) )
6 5 2ralimi ⊢ ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ∧ ∀ 𝑧 ∈ 𝑋 ( 𝑥 𝐷 𝑦 ) ≤ ( ( 𝑧 𝐷 𝑥 ) +𝑒 ( 𝑧 𝐷 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) )
7 4 6 simpl2im ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) )
8 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐷 𝑦 ) = ( 𝐴 𝐷 𝑦 ) )
9 8 eqeq1d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ ( 𝐴 𝐷 𝑦 ) = 0 ) )
10 eqeq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝐴 = 𝑦 ) )
11 9 10 bibi12d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) ↔ ( ( 𝐴 𝐷 𝑦 ) = 0 ↔ 𝐴 = 𝑦 ) ) )
12 oveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐷 𝑦 ) = ( 𝐴 𝐷 𝐵 ) )
13 12 eqeq1d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐷 𝑦 ) = 0 ↔ ( 𝐴 𝐷 𝐵 ) = 0 ) )
14 eqeq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 = 𝑦 ↔ 𝐴 = 𝐵 ) )
15 13 14 bibi12d ⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐷 𝑦 ) = 0 ↔ 𝐴 = 𝑦 ) ↔ ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) ) )
16 11 15 rspc2v ⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 𝐷 𝑦 ) = 0 ↔ 𝑥 = 𝑦 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) ) )
17 7 16 syl5com ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) ) )
18 17 3impib ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )