Proof of Theorem metdcnlem
Step | Hyp | Ref
| Expression |
1 | | xmetdcn2.2 |
. . . . 5
⊢ 𝐶 =
(dist‘ℝ*𝑠) |
2 | 1 | xrsxmet 23564 |
. . . 4
⊢ 𝐶 ∈
(∞Met‘ℝ*) |
3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐶 ∈
(∞Met‘ℝ*)) |
4 | | metdcn.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
5 | | metdcn.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
6 | | metdcn.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑋) |
7 | | xmetcl 23087 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈
ℝ*) |
8 | 4, 5, 6, 7 | syl3anc 1372 |
. . 3
⊢ (𝜑 → (𝐴𝐷𝐵) ∈
ℝ*) |
9 | | metdcn.y |
. . . 4
⊢ (𝜑 → 𝑌 ∈ 𝑋) |
10 | | metdcn.z |
. . . 4
⊢ (𝜑 → 𝑍 ∈ 𝑋) |
11 | | xmetcl 23087 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝑌𝐷𝑍) ∈
ℝ*) |
12 | 4, 9, 10, 11 | syl3anc 1372 |
. . 3
⊢ (𝜑 → (𝑌𝐷𝑍) ∈
ℝ*) |
13 | | xmetcl 23087 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑌𝐷𝐵) ∈
ℝ*) |
14 | 4, 9, 6, 13 | syl3anc 1372 |
. . . . 5
⊢ (𝜑 → (𝑌𝐷𝐵) ∈
ℝ*) |
15 | | metdcn.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈
ℝ+) |
16 | 15 | rphalfcld 12529 |
. . . . . 6
⊢ (𝜑 → (𝑅 / 2) ∈
ℝ+) |
17 | 16 | rpred 12517 |
. . . . 5
⊢ (𝜑 → (𝑅 / 2) ∈ ℝ) |
18 | | xmetcl 23087 |
. . . . . . 7
⊢ ((𝐶 ∈
(∞Met‘ℝ*) ∧ (𝐴𝐷𝐵) ∈ ℝ* ∧ (𝑌𝐷𝐵) ∈ ℝ*) → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ∈
ℝ*) |
19 | 3, 8, 14, 18 | syl3anc 1372 |
. . . . . 6
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ∈
ℝ*) |
20 | 16 | rpxrd 12518 |
. . . . . 6
⊢ (𝜑 → (𝑅 / 2) ∈
ℝ*) |
21 | | xmetcl 23087 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐴𝐷𝑌) ∈
ℝ*) |
22 | 4, 5, 9, 21 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝐷𝑌) ∈
ℝ*) |
23 | 1 | xmetrtri2 23112 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ≤ (𝐴𝐷𝑌)) |
24 | 4, 5, 9, 6, 23 | syl13anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ≤ (𝐴𝐷𝑌)) |
25 | | metdcn.4 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝐷𝑌) < (𝑅 / 2)) |
26 | 19, 22, 20, 24, 25 | xrlelttrd 12639 |
. . . . . 6
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) < (𝑅 / 2)) |
27 | 19, 20, 26 | xrltled 12629 |
. . . . 5
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ≤ (𝑅 / 2)) |
28 | | xmetlecl 23102 |
. . . . 5
⊢ ((𝐶 ∈
(∞Met‘ℝ*) ∧ ((𝐴𝐷𝐵) ∈ ℝ* ∧ (𝑌𝐷𝐵) ∈ ℝ*) ∧ ((𝑅 / 2) ∈ ℝ ∧
((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ≤ (𝑅 / 2))) → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ∈ ℝ) |
29 | 3, 8, 14, 17, 27, 28 | syl122anc 1380 |
. . . 4
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) ∈ ℝ) |
30 | | xmetcl 23087 |
. . . . . . 7
⊢ ((𝐶 ∈
(∞Met‘ℝ*) ∧ (𝑌𝐷𝐵) ∈ ℝ* ∧ (𝑌𝐷𝑍) ∈ ℝ*) → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ∈
ℝ*) |
31 | 3, 14, 12, 30 | syl3anc 1372 |
. . . . . 6
⊢ (𝜑 → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ∈
ℝ*) |
32 | | xmetcl 23087 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝐵𝐷𝑍) ∈
ℝ*) |
33 | 4, 6, 10, 32 | syl3anc 1372 |
. . . . . . 7
⊢ (𝜑 → (𝐵𝐷𝑍) ∈
ℝ*) |
34 | | xmetsym 23103 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑌𝐷𝐵) = (𝐵𝐷𝑌)) |
35 | 4, 9, 6, 34 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝜑 → (𝑌𝐷𝐵) = (𝐵𝐷𝑌)) |
36 | | xmetsym 23103 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋) → (𝑌𝐷𝑍) = (𝑍𝐷𝑌)) |
37 | 4, 9, 10, 36 | syl3anc 1372 |
. . . . . . . . 9
⊢ (𝜑 → (𝑌𝐷𝑍) = (𝑍𝐷𝑌)) |
38 | 35, 37 | oveq12d 7191 |
. . . . . . . 8
⊢ (𝜑 → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) = ((𝐵𝐷𝑌)𝐶(𝑍𝐷𝑌))) |
39 | 1 | xmetrtri2 23112 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝑍 ∈ 𝑋 ∧ 𝑌 ∈ 𝑋)) → ((𝐵𝐷𝑌)𝐶(𝑍𝐷𝑌)) ≤ (𝐵𝐷𝑍)) |
40 | 4, 6, 10, 9, 39 | syl13anc 1373 |
. . . . . . . 8
⊢ (𝜑 → ((𝐵𝐷𝑌)𝐶(𝑍𝐷𝑌)) ≤ (𝐵𝐷𝑍)) |
41 | 38, 40 | eqbrtrd 5053 |
. . . . . . 7
⊢ (𝜑 → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (𝐵𝐷𝑍)) |
42 | | metdcn.5 |
. . . . . . 7
⊢ (𝜑 → (𝐵𝐷𝑍) < (𝑅 / 2)) |
43 | 31, 33, 20, 41, 42 | xrlelttrd 12639 |
. . . . . 6
⊢ (𝜑 → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) < (𝑅 / 2)) |
44 | 31, 20, 43 | xrltled 12629 |
. . . . 5
⊢ (𝜑 → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (𝑅 / 2)) |
45 | | xmetlecl 23102 |
. . . . 5
⊢ ((𝐶 ∈
(∞Met‘ℝ*) ∧ ((𝑌𝐷𝐵) ∈ ℝ* ∧ (𝑌𝐷𝑍) ∈ ℝ*) ∧ ((𝑅 / 2) ∈ ℝ ∧
((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (𝑅 / 2))) → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ∈ ℝ) |
46 | 3, 14, 12, 17, 44, 45 | syl122anc 1380 |
. . . 4
⊢ (𝜑 → ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)) ∈ ℝ) |
47 | 29, 46 | readdcld 10751 |
. . 3
⊢ (𝜑 → (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) + ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍))) ∈ ℝ) |
48 | | xmettri 23107 |
. . . . 5
⊢ ((𝐶 ∈
(∞Met‘ℝ*) ∧ ((𝐴𝐷𝐵) ∈ ℝ* ∧ (𝑌𝐷𝑍) ∈ ℝ* ∧ (𝑌𝐷𝐵) ∈ ℝ*)) →
((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) +𝑒 ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)))) |
49 | 3, 8, 12, 14, 48 | syl13anc 1373 |
. . . 4
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) +𝑒 ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)))) |
50 | 29, 46 | rexaddd 12713 |
. . . 4
⊢ (𝜑 → (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) +𝑒 ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍))) = (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) + ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)))) |
51 | 49, 50 | breqtrd 5057 |
. . 3
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) + ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍)))) |
52 | | xmetlecl 23102 |
. . 3
⊢ ((𝐶 ∈
(∞Met‘ℝ*) ∧ ((𝐴𝐷𝐵) ∈ ℝ* ∧ (𝑌𝐷𝑍) ∈ ℝ*) ∧
((((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) + ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍))) ∈ ℝ ∧ ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) ≤ (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) + ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍))))) → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) ∈ ℝ) |
53 | 3, 8, 12, 47, 51, 52 | syl122anc 1380 |
. 2
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) ∈ ℝ) |
54 | 15 | rpred 12517 |
. 2
⊢ (𝜑 → 𝑅 ∈ ℝ) |
55 | 29, 46, 54, 26, 43 | lt2halvesd 11967 |
. 2
⊢ (𝜑 → (((𝐴𝐷𝐵)𝐶(𝑌𝐷𝐵)) + ((𝑌𝐷𝐵)𝐶(𝑌𝐷𝑍))) < 𝑅) |
56 | 53, 47, 54, 51, 55 | lelttrd 10879 |
1
⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) < 𝑅) |