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Metamath Proof Explorer

Theorem xmetgt0

Description: The distance function of an extended metric space is positive for unequal points. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmetgt0 ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≠ 𝐵 ↔ 0 < ( 𝐴 𝐷 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 xmetge0 ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 0 ≤ ( 𝐴 𝐷 𝐵 ) )
2 1 biantrud ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ↔ ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ∧ 0 ≤ ( 𝐴 𝐷 𝐵 ) ) ) )
3 xmetcl ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* )
4 0xr ⊢ 0 ∈ ℝ*
5 xrletri3 ⊢ ( ( ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ∧ 0 ≤ ( 𝐴 𝐷 𝐵 ) ) ) )
6 3 4 5 sylancl ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ∧ 0 ≤ ( 𝐴 𝐷 𝐵 ) ) ) )
7 2 6 bitr4d ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ↔ ( 𝐴 𝐷 𝐵 ) = 0 ) )
8 xrlenlt ⊢ ( ( ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ↔ ¬ 0 < ( 𝐴 𝐷 𝐵 ) ) )
9 3 4 8 sylancl ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) ≤ 0 ↔ ¬ 0 < ( 𝐴 𝐷 𝐵 ) ) )
10 xmeteq0 ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝐴 𝐷 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
11 7 9 10 3bitr3d ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ¬ 0 < ( 𝐴 𝐷 𝐵 ) ↔ 𝐴 = 𝐵 ) )
12 11 necon1abid ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≠ 𝐵 ↔ 0 < ( 𝐴 𝐷 𝐵 ) ) )