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 11290 |
. . . . . . 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 11839 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (𝐴 + 𝐵)) |
| 7 | | cxpaddle.5 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈
ℝ+) |
| 8 | 7 | rpred 13077 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 9 | 3, 6, 8 | recxpcld 26765 |
. . . . . 6
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) ∈ ℝ) |
| 10 | 9 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ∈ ℝ) |
| 11 | 10 | recnd 11289 |
. . . 4
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ∈ ℂ) |
| 12 | 11 | mullidd 11279 |
. . 3
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑𝑐𝐶)) = ((𝐴 + 𝐵)↑𝑐𝐶)) |
| 13 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℝ) |
| 14 | 3 | anim1i 615 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) |
| 15 | | elrp 13036 |
. . . . . . . 8
⊢ ((𝐴 + 𝐵) ∈ ℝ+ ↔ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) |
| 16 | 14, 15 | sylibr 234 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈
ℝ+) |
| 17 | 13, 16 | rerpdivcld 13108 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ∈ ℝ) |
| 18 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℝ) |
| 19 | 18, 16 | rerpdivcld 13108 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ∈ ℝ) |
| 20 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐴) |
| 21 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℝ) |
| 22 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 < (𝐴 + 𝐵)) |
| 23 | | divge0 12137 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤
𝐴) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐴 / (𝐴 + 𝐵))) |
| 24 | 13, 20, 21, 22, 23 | syl22anc 839 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐴 / (𝐴 + 𝐵))) |
| 25 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℝ) |
| 26 | 17, 24, 25 | recxpcld 26765 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) ∈
ℝ) |
| 27 | 5 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ 𝐵) |
| 28 | | divge0 12137 |
. . . . . . . 8
⊢ (((𝐵 ∈ ℝ ∧ 0 ≤
𝐵) ∧ ((𝐴 + 𝐵) ∈ ℝ ∧ 0 < (𝐴 + 𝐵))) → 0 ≤ (𝐵 / (𝐴 + 𝐵))) |
| 29 | 18, 27, 21, 22, 28 | syl22anc 839 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 0 ≤ (𝐵 / (𝐴 + 𝐵))) |
| 30 | 19, 29, 25 | recxpcld 26765 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶) ∈
ℝ) |
| 31 | 1, 2 | addge01d 11851 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) |
| 32 | 5, 31 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ≤ (𝐴 + 𝐵)) |
| 33 | 32 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ (𝐴 + 𝐵)) |
| 34 | 21 | recnd 11289 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ∈ ℂ) |
| 35 | 34 | mulridd 11278 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) · 1) = (𝐴 + 𝐵)) |
| 36 | 33, 35 | breqtrrd 5171 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ≤ ((𝐴 + 𝐵) · 1)) |
| 37 | | 1red 11262 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ∈ ℝ) |
| 38 | | ledivmul 12144 |
. . . . . . . . 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 1376 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐴 ≤ ((𝐴 + 𝐵) · 1))) |
| 40 | 36, 39 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ 1) |
| 41 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈
ℝ+) |
| 42 | | cxpaddle.6 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ≤ 1) |
| 43 | 42 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ≤ 1) |
| 44 | 17, 24, 40, 41, 43 | cxpaddlelem 26794 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 / (𝐴 + 𝐵)) ≤ ((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶)) |
| 45 | 2, 1 | addge02d 11852 |
. . . . . . . . . . 11
⊢ (𝜑 → (0 ≤ 𝐴 ↔ 𝐵 ≤ (𝐴 + 𝐵))) |
| 46 | 4, 45 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ≤ (𝐴 + 𝐵)) |
| 47 | 46 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ (𝐴 + 𝐵)) |
| 48 | 47, 35 | breqtrrd 5171 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ≤ ((𝐴 + 𝐵) · 1)) |
| 49 | | ledivmul 12144 |
. . . . . . . . 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 1376 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵)) ≤ 1 ↔ 𝐵 ≤ ((𝐴 + 𝐵) · 1))) |
| 51 | 48, 50 | mpbird 257 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ 1) |
| 52 | 19, 29, 51, 41, 43 | cxpaddlelem 26794 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵 / (𝐴 + 𝐵)) ≤ ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶)) |
| 53 | 17, 19, 26, 30, 44, 52 | le2addd 11882 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) ≤ (((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) + ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶))) |
| 54 | 13 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐴 ∈ ℂ) |
| 55 | 18 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐵 ∈ ℂ) |
| 56 | 16 | rpne0d 13082 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴 + 𝐵) ≠ 0) |
| 57 | 54, 55, 34, 56 | divdird 12081 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵)))) |
| 58 | 34, 56 | dividd 12041 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵) / (𝐴 + 𝐵)) = 1) |
| 59 | 57, 58 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵)) + (𝐵 / (𝐴 + 𝐵))) = 1) |
| 60 | 8 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 61 | 60 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 𝐶 ∈ ℂ) |
| 62 | 13, 20, 16, 61 | divcxpd 26764 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 / (𝐴 + 𝐵))↑𝑐𝐶) = ((𝐴↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
| 63 | 18, 27, 16, 61 | divcxpd 26764 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐵 / (𝐴 + 𝐵))↑𝑐𝐶) = ((𝐵↑𝑐𝐶) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
| 64 | 62, 63 | oveq12d 7449 |
. . . . . 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 26765 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴↑𝑐𝐶) ∈ ℝ) |
| 66 | 65 | recnd 11289 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑𝑐𝐶) ∈ ℂ) |
| 67 | 66 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐴↑𝑐𝐶) ∈ ℂ) |
| 68 | 2, 5, 8 | recxpcld 26765 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵↑𝑐𝐶) ∈ ℝ) |
| 69 | 68 | recnd 11289 |
. . . . . . . 8
⊢ (𝜑 → (𝐵↑𝑐𝐶) ∈ ℂ) |
| 70 | 69 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (𝐵↑𝑐𝐶) ∈ ℂ) |
| 71 | 16, 25 | rpcxpcld 26775 |
. . . . . . . 8
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ∈
ℝ+) |
| 72 | 71 | rpne0d 13082 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ≠ 0) |
| 73 | 67, 70, 11, 72 | divdird 12081 |
. . . . . 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 2780 |
. . . . 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 5174 |
. . . 4
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → 1 ≤ (((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) / ((𝐴 + 𝐵)↑𝑐𝐶))) |
| 76 | 65, 68 | readdcld 11290 |
. . . . . 6
⊢ (𝜑 → ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ∈ ℝ) |
| 77 | 76 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ∈ ℝ) |
| 78 | 37, 77, 71 | lemuldivd 13126 |
. . . 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 257 |
. . 3
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → (1 · ((𝐴 + 𝐵)↑𝑐𝐶)) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
| 80 | 12, 79 | eqbrtrrd 5167 |
. 2
⊢ ((𝜑 ∧ 0 < (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
| 81 | 7 | rpne0d 13082 |
. . . . . 6
⊢ (𝜑 → 𝐶 ≠ 0) |
| 82 | 60, 81 | 0cxpd 26752 |
. . . . 5
⊢ (𝜑 →
(0↑𝑐𝐶) = 0) |
| 83 | 1, 4, 8 | cxpge0d 26766 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (𝐴↑𝑐𝐶)) |
| 84 | 2, 5, 8 | cxpge0d 26766 |
. . . . . 6
⊢ (𝜑 → 0 ≤ (𝐵↑𝑐𝐶)) |
| 85 | 65, 68, 83, 84 | addge0d 11839 |
. . . . 5
⊢ (𝜑 → 0 ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
| 86 | 82, 85 | eqbrtrd 5165 |
. . . 4
⊢ (𝜑 →
(0↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
| 87 | | oveq1 7438 |
. . . . 5
⊢ (0 =
(𝐴 + 𝐵) → (0↑𝑐𝐶) = ((𝐴 + 𝐵)↑𝑐𝐶)) |
| 88 | 87 | breq1d 5153 |
. . . 4
⊢ (0 =
(𝐴 + 𝐵) → ((0↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)) ↔ ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)))) |
| 89 | 86, 88 | syl5ibcom 245 |
. . 3
⊢ (𝜑 → (0 = (𝐴 + 𝐵) → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶)))) |
| 90 | 89 | imp 406 |
. 2
⊢ ((𝜑 ∧ 0 = (𝐴 + 𝐵)) → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |
| 91 | | 0re 11263 |
. . . 4
⊢ 0 ∈
ℝ |
| 92 | | leloe 11347 |
. . . 4
⊢ ((0
∈ ℝ ∧ (𝐴 +
𝐵) ∈ ℝ) →
(0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵)))) |
| 93 | 91, 3, 92 | sylancr 587 |
. . 3
⊢ (𝜑 → (0 ≤ (𝐴 + 𝐵) ↔ (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵)))) |
| 94 | 6, 93 | mpbid 232 |
. 2
⊢ (𝜑 → (0 < (𝐴 + 𝐵) ∨ 0 = (𝐴 + 𝐵))) |
| 95 | 80, 90, 94 | mpjaodan 961 |
1
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑐𝐶) ≤ ((𝐴↑𝑐𝐶) + (𝐵↑𝑐𝐶))) |