Step | Hyp | Ref
| Expression |
1 | | elfvdm 6700 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
2 | | isxmet 23070 |
. . . . . . 7
⊢ (𝑋 ∈ dom ∞Met →
(𝐷 ∈
(∞Met‘𝑋) ↔
(𝐷:(𝑋 × 𝑋)⟶ℝ* ∧
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
3 | 1, 2 | syl 17 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))))) |
4 | 3 | ibi 270 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧
∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))) |
5 | | simpr 488 |
. . . . . 6
⊢ ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
6 | 5 | 2ralimi 3076 |
. . . . 5
⊢
(∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
7 | 4, 6 | simpl2im 507 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))) |
8 | | oveq1 7171 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (𝑥𝐷𝑦) = (𝐴𝐷𝑦)) |
9 | | oveq2 7172 |
. . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑧𝐷𝑥) = (𝑧𝐷𝐴)) |
10 | 9 | oveq1d 7179 |
. . . . . 6
⊢ (𝑥 = 𝐴 → ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝑦))) |
11 | 8, 10 | breq12d 5040 |
. . . . 5
⊢ (𝑥 = 𝐴 → ((𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) ↔ (𝐴𝐷𝑦) ≤ ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝑦)))) |
12 | | oveq2 7172 |
. . . . . 6
⊢ (𝑦 = 𝐵 → (𝐴𝐷𝑦) = (𝐴𝐷𝐵)) |
13 | | oveq2 7172 |
. . . . . . 7
⊢ (𝑦 = 𝐵 → (𝑧𝐷𝑦) = (𝑧𝐷𝐵)) |
14 | 13 | oveq2d 7180 |
. . . . . 6
⊢ (𝑦 = 𝐵 → ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝐵))) |
15 | 12, 14 | breq12d 5040 |
. . . . 5
⊢ (𝑦 = 𝐵 → ((𝐴𝐷𝑦) ≤ ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝑦)) ↔ (𝐴𝐷𝐵) ≤ ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝐵)))) |
16 | | oveq1 7171 |
. . . . . . 7
⊢ (𝑧 = 𝐶 → (𝑧𝐷𝐴) = (𝐶𝐷𝐴)) |
17 | | oveq1 7171 |
. . . . . . 7
⊢ (𝑧 = 𝐶 → (𝑧𝐷𝐵) = (𝐶𝐷𝐵)) |
18 | 16, 17 | oveq12d 7182 |
. . . . . 6
⊢ (𝑧 = 𝐶 → ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝐵)) = ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) |
19 | 18 | breq2d 5039 |
. . . . 5
⊢ (𝑧 = 𝐶 → ((𝐴𝐷𝐵) ≤ ((𝑧𝐷𝐴) +𝑒 (𝑧𝐷𝐵)) ↔ (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))) |
20 | 11, 15, 19 | rspc3v 3537 |
. . . 4
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))) |
21 | 7, 20 | syl5 34 |
. . 3
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))) |
22 | 21 | 3comr 1126 |
. 2
⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))) |
23 | 22 | impcom 411 |
1
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))) |