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 ) ) ) |