Proof of Theorem cxpaddle
Step | Hyp | Ref
| Expression |
1 | | cxpaddle.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | | cxpaddle.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℝ) |
3 | 1, 2 | readdcld 10748 |
. . . . . . 7
⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | | cxpaddle.2 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ 𝐴) |
5 | | cxpaddle.4 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ 𝐵) |
6 | 1, 2, 4, 5 | addge0d 11294 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
7 | | cxpaddle.5 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈
ℝ+) |
8 | 7 | rpred 12514 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 3, 6, 8 | recxpcld 25466 |
. . . . . 6
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) ∈ ℝ) |
10 | 9 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ∈ ℝ) |
11 | 10 | recnd 10747 |
. . . 4
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ∈ ℂ) |
12 | 11 | mulid2d 10737 |
. . 3
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑𝑐𝐶)) = ((𝐴 + 𝐵)↑𝑐𝐶)) |
13 | 1 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℝ) |
14 | 3 | anim1i 618 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) |
15 | | elrp 12474 |
. . . . . . . 8
⊢ ((𝐴 + 𝐵) ∈ ℝ+ ↔ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) |
16 | 14, 15 | sylibr 237 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈
ℝ+) |
17 | 13, 16 | rerpdivcld 12545 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ∈ ℝ) |
18 | 2 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℝ) |
19 | 18, 16 | rerpdivcld 12545 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ∈ ℝ) |
20 | 4 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐴) |
21 | 3 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℝ) |
22 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 < (𝐴 + 𝐵)) |
23 | | divge0 11587 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐴 / (𝐴 + 𝐵))) |
24 | 13, 20, 21, 22, 23 | syl22anc 838 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐴 / (𝐴 + 𝐵))) |
25 | 8 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℝ) |
26 | 17, 24, 25 | recxpcld 25466 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) ∈
ℝ) |
27 | 5 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐵) |
28 | | divge0 11587 |
. . . . . . . 8
⊢ (((𝐵 ∈ ℝ ∧ 0 ≤
𝐵) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐵 / (𝐴 + 𝐵))) |
29 | 18, 27, 21, 22, 28 | syl22anc 838 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 / (𝐴 + 𝐵))) |
30 | 19, 29, 25 | recxpcld 25466 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶) ∈
ℝ) |
31 | 1, 2 | addge01d 11306 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) |
32 | 5, 31 | mpbid 235 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ≤ (𝐴 + 𝐵)) |
33 | 32 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ (𝐴 + 𝐵)) |
34 | 21 | recnd 10747 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℂ) |
35 | 34 | mulid1d 10736 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵)) |
36 | 33, 35 | breqtrrd 5058 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ ((𝐴 + 𝐵) · 1)) |
37 | | 1red 10720 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ∈ ℝ) |
38 | | ledivmul 11594 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 1 ∈
ℝ ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 <
(𝐴 + 𝐵))) → ((𝐴 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐴 ≤ ((𝐴 + 𝐵) · 1))) |
39 | 13, 37, 21, 22, 38 | syl112anc 1375 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐴 ≤ ((𝐴 + 𝐵) · 1))) |
40 | 36, 39 | mpbird 260 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ 1) |
41 | 7 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈
ℝ+) |
42 | | cxpaddle.6 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ≤ 1) |
43 | 42 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ≤ 1) |
44 | 17, 24, 40, 41, 43 | cxpaddlelem 25492 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ ((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶)) |
45 | 2, 1 | addge02d 11307 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵))) |
46 | 4, 45 | mpbid 235 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ≤ (𝐴 + 𝐵)) |
47 | 46 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ (𝐴 + 𝐵)) |
48 | 47, 35 | breqtrrd 5058 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ ((𝐴 + 𝐵) · 1)) |
49 | | ledivmul 11594 |
. . . . . . . . 9
⊢ ((𝐵 ∈ ℝ ∧ 1 ∈
ℝ ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 <
(𝐴 + 𝐵))) → ((𝐵 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐵 ≤ ((𝐴 + 𝐵) · 1))) |
50 | 18, 37, 21, 22, 49 | syl112anc 1375 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐵 ≤ ((𝐴 + 𝐵) · 1))) |
51 | 48, 50 | mpbird 260 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ 1) |
52 | 19, 29, 51, 41, 43 | cxpaddlelem 25492 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶)) |
53 | 17, 19, 26, 30, 44, 52 | le2addd 11337 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) ≤ (((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶))) |
54 | 13 | recnd 10747 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℂ) |
55 | 18 | recnd 10747 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℂ) |
56 | 16 | rpne0d 12519 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ≠ 0) |
57 | 54, 55, 34, 56 | divdird 11532 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵)))) |
58 | 34, 56 | dividd 11492 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = 1) |
59 | 57, 58 | eqtr3d 2775 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) = 1) |
60 | 8 | recnd 10747 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℂ) |
61 | 60 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℂ) |
62 | 13, 20, 16, 61 | divcxpd 25465 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) = ((𝐴↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
63 | 18, 27, 16, 61 | divcxpd 25465 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶) = ((𝐵↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
64 | 62, 63 | oveq12d 7188 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶)) = (((𝐴↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶)) + ((𝐵↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶)))) |
65 | 1, 4, 8 | recxpcld 25466 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴↑𝑐𝐶) ∈ ℝ) |
66 | 65 | recnd 10747 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑𝑐𝐶) ∈ ℂ) |
67 | 66 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴↑𝑐𝐶) ∈ ℂ) |
68 | 2, 5, 8 | recxpcld 25466 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵↑𝑐𝐶) ∈ ℝ) |
69 | 68 | recnd 10747 |
. . . . . . . 8
⊢ (𝜑 → (𝐵↑𝑐𝐶) ∈ ℂ) |
70 | 69 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵↑𝑐𝐶) ∈ ℂ) |
71 | 16, 25 | rpcxpcld 25475 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ∈
ℝ+) |
72 | 71 | rpne0d 12519 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ≠ 0) |
73 | 67, 70, 11, 72 | divdird 11532 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) / ((𝐴 + 𝐵)↑𝑐𝐶)) = (((𝐴↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶)) + ((𝐵↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶)))) |
74 | 64, 73 | eqtr4d 2776 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶)) = (((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
75 | 53, 59, 74 | 3brtr3d 5061 |
. . . 4
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ≤ (((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
76 | 65, 68 | readdcld 10748 |
. . . . . 6
⊢ (𝜑 → ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ∈ ℝ) |
77 | 76 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ∈ ℝ) |
78 | 37, 77, 71 | lemuldivd 12563 |
. . . 4
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((1 · ((𝐴 + 𝐵)↑𝑐𝐶)) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ↔ 1 ≤ (((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) / ((𝐴 + 𝐵)↑𝑐𝐶)))) |
79 | 75, 78 | mpbird 260 |
. . 3
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑𝑐𝐶)) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
80 | 12, 79 | eqbrtrrd 5054 |
. 2
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
81 | 7 | rpne0d 12519 |
. . . . . 6
⊢ (𝜑 → 𝐶 ≠ 0) |
82 | 60, 81 | 0cxpd 25453 |
. . . . 5
⊢ (𝜑 →
(0↑𝑐𝐶) = 0) |
83 | 1, 4, 8 | cxpge0d 25467 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (𝐴↑𝑐𝐶)) |
84 | 2, 5, 8 | cxpge0d 25467 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (𝐵↑𝑐𝐶)) |
85 | 65, 68, 83, 84 | addge0d 11294 |
. . . . 5
⊢ (𝜑 → 0 ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
86 | 82, 85 | eqbrtrd 5052 |
. . . 4
⊢ (𝜑 →
(0↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
87 | | oveq1 7177 |
. . . . 5
⊢ (0 =
(𝐴 + 𝐵) → (0↑𝑐𝐶) = ((𝐴 + 𝐵)↑𝑐𝐶)) |
88 | 87 | breq1d 5040 |
. . . 4
⊢ (0 =
(𝐴 + 𝐵) → ((0↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ↔ ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)))) |
89 | 86, 88 | syl5ibcom 248 |
. . 3
⊢ (𝜑 → (0 = (𝐴 + 𝐵) → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)))) |
90 | 89 | imp 410 |
. 2
⊢ ((𝜑 ∧ 0 = (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
91 | | 0re 10721 |
. . . 4
⊢ 0 ∈
ℝ |
92 | | leloe 10805 |
. . . 4
⊢ ((0
∈ ℝ ∧ (𝐴 +
𝐵) ∈ ℝ) →
(0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵)))) |
93 | 91, 3, 92 | sylancr 590 |
. . 3
⊢ (𝜑 → (0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵)))) |
94 | 6, 93 | mpbid 235 |
. 2
⊢ (𝜑 → (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵))) |
95 | 80, 90, 94 | mpjaodan 958 |
1
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |