Proof of Theorem cxpaddlelem
| Step | Hyp | Ref
| Expression |
| 1 | | cxpaddlelem.1 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 2 | | cxpaddlelem.2 |
. . . . . 6
⊢ (𝜑 → 0 ≤ 𝐴) |
| 3 | | 1re 11261 |
. . . . . . 7
⊢ 1 ∈
ℝ |
| 4 | | cxpaddlelem.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
| 5 | 4 | rpred 13077 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 6 | | resubcl 11573 |
. . . . . . 7
⊢ ((1
∈ ℝ ∧ 𝐵
∈ ℝ) → (1 − 𝐵) ∈ ℝ) |
| 7 | 3, 5, 6 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (1 − 𝐵) ∈
ℝ) |
| 8 | 1, 2, 7 | recxpcld 26765 |
. . . . 5
⊢ (𝜑 → (𝐴↑𝑐(1 − 𝐵)) ∈
ℝ) |
| 9 | 8 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐(1 − 𝐵)) ∈
ℝ) |
| 10 | | 1red 11262 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 1 ∈ ℝ) |
| 11 | | recxpcl 26717 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴 ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) ∈ ℝ) |
| 12 | | cxpge0 26725 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴 ∧ 𝐵 ∈ ℝ) → 0 ≤ (𝐴↑𝑐𝐵)) |
| 13 | 11, 12 | jca 511 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴 ∧ 𝐵 ∈ ℝ) → ((𝐴↑𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑𝑐𝐵))) |
| 14 | 1, 2, 5, 13 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → ((𝐴↑𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑𝑐𝐵))) |
| 15 | 14 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑𝑐𝐵))) |
| 16 | | cxpaddlelem.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ≤ 1) |
| 17 | 16 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 𝐴 ≤ 1) |
| 18 | 1 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 𝐴 ∈ ℝ) |
| 19 | 2 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 0 ≤ 𝐴) |
| 20 | | 1red 11262 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 1 ∈
ℝ) |
| 21 | | 0le1 11786 |
. . . . . . . . 9
⊢ 0 ≤
1 |
| 22 | 21 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 0 ≤ 1) |
| 23 | | difrp 13073 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℝ ∧ 1 ∈
ℝ) → (𝐵 < 1
↔ (1 − 𝐵) ∈
ℝ+)) |
| 24 | 5, 3, 23 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 < 1 ↔ (1 − 𝐵) ∈
ℝ+)) |
| 25 | 24 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐵 < 1 ↔ (1 − 𝐵) ∈
ℝ+)) |
| 26 | 25 | biimpa 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (1 − 𝐵) ∈
ℝ+) |
| 27 | 18, 19, 20, 22, 26 | cxple2d 26769 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴 ≤ 1 ↔ (𝐴↑𝑐(1 − 𝐵)) ≤
(1↑𝑐(1 − 𝐵)))) |
| 28 | 17, 27 | mpbid 232 |
. . . . . 6
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴↑𝑐(1 − 𝐵)) ≤
(1↑𝑐(1 − 𝐵))) |
| 29 | 7 | recnd 11289 |
. . . . . . . 8
⊢ (𝜑 → (1 − 𝐵) ∈
ℂ) |
| 30 | 29 | 1cxpd 26749 |
. . . . . . 7
⊢ (𝜑 →
(1↑𝑐(1 − 𝐵)) = 1) |
| 31 | 30 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) →
(1↑𝑐(1 − 𝐵)) = 1) |
| 32 | 28, 31 | breqtrd 5169 |
. . . . 5
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴↑𝑐(1 − 𝐵)) ≤ 1) |
| 33 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → 𝐵 = 1) |
| 34 | 33 | oveq2d 7447 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (1 − 𝐵) = (1 − 1)) |
| 35 | | 1m1e0 12338 |
. . . . . . . . 9
⊢ (1
− 1) = 0 |
| 36 | 34, 35 | eqtrdi 2793 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (1 − 𝐵) = 0) |
| 37 | 36 | oveq2d 7447 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐(1 − 𝐵)) = (𝐴↑𝑐0)) |
| 38 | 1 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 39 | 38 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → 𝐴 ∈ ℂ) |
| 40 | 39 | cxp0d 26747 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐0) =
1) |
| 41 | 37, 40 | eqtrd 2777 |
. . . . . 6
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐(1 − 𝐵)) = 1) |
| 42 | | 1le1 11891 |
. . . . . 6
⊢ 1 ≤
1 |
| 43 | 41, 42 | eqbrtrdi 5182 |
. . . . 5
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐(1 − 𝐵)) ≤ 1) |
| 44 | | cxpaddlelem.5 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ≤ 1) |
| 45 | | leloe 11347 |
. . . . . . . 8
⊢ ((𝐵 ∈ ℝ ∧ 1 ∈
ℝ) → (𝐵 ≤ 1
↔ (𝐵 < 1 ∨ 𝐵 = 1))) |
| 46 | 5, 3, 45 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → (𝐵 ≤ 1 ↔ (𝐵 < 1 ∨ 𝐵 = 1))) |
| 47 | 44, 46 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (𝐵 < 1 ∨ 𝐵 = 1)) |
| 48 | 47 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐵 < 1 ∨ 𝐵 = 1)) |
| 49 | 32, 43, 48 | mpjaodan 961 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐(1 − 𝐵)) ≤ 1) |
| 50 | | lemul1a 12121 |
. . . 4
⊢ ((((𝐴↑𝑐(1
− 𝐵)) ∈ ℝ
∧ 1 ∈ ℝ ∧ ((𝐴↑𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑𝑐𝐵))) ∧ (𝐴↑𝑐(1 − 𝐵)) ≤ 1) → ((𝐴↑𝑐(1
− 𝐵)) · (𝐴↑𝑐𝐵)) ≤ (1 · (𝐴↑𝑐𝐵))) |
| 51 | 9, 10, 15, 49, 50 | syl31anc 1375 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑𝑐(1 − 𝐵)) · (𝐴↑𝑐𝐵)) ≤ (1 · (𝐴↑𝑐𝐵))) |
| 52 | | ax-1cn 11213 |
. . . . . . 7
⊢ 1 ∈
ℂ |
| 53 | 5 | recnd 11289 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 54 | | npcan 11517 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ 𝐵
∈ ℂ) → ((1 − 𝐵) + 𝐵) = 1) |
| 55 | 52, 53, 54 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → ((1 − 𝐵) + 𝐵) = 1) |
| 56 | 55 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → ((1 − 𝐵) + 𝐵) = 1) |
| 57 | 56 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐((1 − 𝐵) + 𝐵)) = (𝐴↑𝑐1)) |
| 58 | 38 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℂ) |
| 59 | 1 | anim1i 615 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
| 60 | | elrp 13036 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ+
↔ (𝐴 ∈ ℝ
∧ 0 < 𝐴)) |
| 61 | 59, 60 | sylibr 234 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈
ℝ+) |
| 62 | 61 | rpne0d 13082 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ≠ 0) |
| 63 | 29 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → (1 − 𝐵) ∈ ℂ) |
| 64 | 53 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐵 ∈ ℂ) |
| 65 | 58, 62, 63, 64 | cxpaddd 26759 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐((1 − 𝐵) + 𝐵)) = ((𝐴↑𝑐(1 − 𝐵)) · (𝐴↑𝑐𝐵))) |
| 66 | 38 | cxp1d 26748 |
. . . . 5
⊢ (𝜑 → (𝐴↑𝑐1) = 𝐴) |
| 67 | 66 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐1) = 𝐴) |
| 68 | 57, 65, 67 | 3eqtr3d 2785 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑𝑐(1 − 𝐵)) · (𝐴↑𝑐𝐵)) = 𝐴) |
| 69 | 38, 53 | cxpcld 26750 |
. . . . 5
⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℂ) |
| 70 | 69 | mullidd 11279 |
. . . 4
⊢ (𝜑 → (1 · (𝐴↑𝑐𝐵)) = (𝐴↑𝑐𝐵)) |
| 71 | 70 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → (1 · (𝐴↑𝑐𝐵)) = (𝐴↑𝑐𝐵)) |
| 72 | 51, 68, 71 | 3brtr3d 5174 |
. 2
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ≤ (𝐴↑𝑐𝐵)) |
| 73 | 1, 2, 5 | cxpge0d 26766 |
. . . 4
⊢ (𝜑 → 0 ≤ (𝐴↑𝑐𝐵)) |
| 74 | | breq1 5146 |
. . . 4
⊢ (0 =
𝐴 → (0 ≤ (𝐴↑𝑐𝐵) ↔ 𝐴 ≤ (𝐴↑𝑐𝐵))) |
| 75 | 73, 74 | syl5ibcom 245 |
. . 3
⊢ (𝜑 → (0 = 𝐴 → 𝐴 ≤ (𝐴↑𝑐𝐵))) |
| 76 | 75 | imp 406 |
. 2
⊢ ((𝜑 ∧ 0 = 𝐴) → 𝐴 ≤ (𝐴↑𝑐𝐵)) |
| 77 | | 0re 11263 |
. . . 4
⊢ 0 ∈
ℝ |
| 78 | | leloe 11347 |
. . . 4
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
| 79 | 77, 1, 78 | sylancr 587 |
. . 3
⊢ (𝜑 → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
| 80 | 2, 79 | mpbid 232 |
. 2
⊢ (𝜑 → (0 < 𝐴 ∨ 0 = 𝐴)) |
| 81 | 72, 76, 80 | mpjaodan 961 |
1
⊢ (𝜑 → 𝐴 ≤ (𝐴↑𝑐𝐵)) |