Proof of Theorem cxpaddlelem
Step | Hyp | Ref
| Expression |
1 | | cxpaddlelem.1 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) |
2 | | cxpaddlelem.2 |
. . . . . 6
⊢ (𝜑 → 0 ≤ 𝐴) |
3 | | 1re 9918 |
. . . . . . 7
⊢ 1 ∈
ℝ |
4 | | cxpaddlelem.4 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
5 | 4 | rpred 11748 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
6 | | resubcl 10224 |
. . . . . . 7
⊢ ((1
∈ ℝ ∧ 𝐵
∈ ℝ) → (1 − 𝐵) ∈ ℝ) |
7 | 3, 5, 6 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → (1 − 𝐵) ∈
ℝ) |
8 | 1, 2, 7 | recxpcld 24269 |
. . . . 5
⊢ (𝜑 → (𝐴↑𝑐(1 − 𝐵)) ∈
ℝ) |
9 | 8 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐(1 − 𝐵)) ∈
ℝ) |
10 | | 1red 9934 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → 1 ∈ ℝ) |
11 | | recxpcl 24221 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴 ∧ 𝐵 ∈ ℝ) → (𝐴↑𝑐𝐵) ∈ ℝ) |
12 | | cxpge0 24229 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴 ∧ 𝐵 ∈ ℝ) → 0 ≤ (𝐴↑𝑐𝐵)) |
13 | 11, 12 | jca 553 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤
𝐴 ∧ 𝐵 ∈ ℝ) → ((𝐴↑𝑐𝐵) ∈ ℝ ∧ 0 ≤ (𝐴↑𝑐𝐵))) |
14 | 1, 2, 5, 13 | syl3anc 1318 |
. . . . 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 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 𝐴 ≤ 1) |
18 | 1 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 𝐴 ∈ ℝ) |
19 | 2 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 0 ≤ 𝐴) |
20 | | 1red 9934 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 1 ∈
ℝ) |
21 | | 0le1 10430 |
. . . . . . . . 9
⊢ 0 ≤
1 |
22 | 21 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → 0 ≤ 1) |
23 | | difrp 11744 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℝ ∧ 1 ∈
ℝ) → (𝐵 < 1
↔ (1 − 𝐵) ∈
ℝ+)) |
24 | 5, 3, 23 | sylancl 693 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 < 1 ↔ (1 − 𝐵) ∈
ℝ+)) |
25 | 24 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐵 < 1 ↔ (1 − 𝐵) ∈
ℝ+)) |
26 | 25 | biimpa 500 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (1 − 𝐵) ∈
ℝ+) |
27 | 18, 19, 20, 22, 26 | cxple2d 24273 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴 ≤ 1 ↔ (𝐴↑𝑐(1 − 𝐵)) ≤
(1↑𝑐(1 − 𝐵)))) |
28 | 17, 27 | mpbid 221 |
. . . . . 6
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴↑𝑐(1 − 𝐵)) ≤
(1↑𝑐(1 − 𝐵))) |
29 | 7 | recnd 9947 |
. . . . . . . 8
⊢ (𝜑 → (1 − 𝐵) ∈
ℂ) |
30 | 29 | 1cxpd 24253 |
. . . . . . 7
⊢ (𝜑 →
(1↑𝑐(1 − 𝐵)) = 1) |
31 | 30 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) →
(1↑𝑐(1 − 𝐵)) = 1) |
32 | 28, 31 | breqtrd 4609 |
. . . . 5
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 < 1) → (𝐴↑𝑐(1 − 𝐵)) ≤ 1) |
33 | | simpr 476 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → 𝐵 = 1) |
34 | 33 | oveq2d 6565 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (1 − 𝐵) = (1 − 1)) |
35 | | 1m1e0 10966 |
. . . . . . . . 9
⊢ (1
− 1) = 0 |
36 | 34, 35 | syl6eq 2660 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (1 − 𝐵) = 0) |
37 | 36 | oveq2d 6565 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐(1 − 𝐵)) = (𝐴↑𝑐0)) |
38 | 1 | recnd 9947 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
39 | 38 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → 𝐴 ∈ ℂ) |
40 | 39 | cxp0d 24251 |
. . . . . . 7
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐0) =
1) |
41 | 37, 40 | eqtrd 2644 |
. . . . . 6
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐(1 − 𝐵)) = 1) |
42 | | 1le1 10534 |
. . . . . 6
⊢ 1 ≤
1 |
43 | 41, 42 | syl6eqbr 4622 |
. . . . 5
⊢ (((𝜑 ∧ 0 < 𝐴) ∧ 𝐵 = 1) → (𝐴↑𝑐(1 − 𝐵)) ≤ 1) |
44 | | cxpaddlelem.5 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ≤ 1) |
45 | | leloe 10003 |
. . . . . . . 8
⊢ ((𝐵 ∈ ℝ ∧ 1 ∈
ℝ) → (𝐵 ≤ 1
↔ (𝐵 < 1 ∨ 𝐵 = 1))) |
46 | 5, 3, 45 | sylancl 693 |
. . . . . . 7
⊢ (𝜑 → (𝐵 ≤ 1 ↔ (𝐵 < 1 ∨ 𝐵 = 1))) |
47 | 44, 46 | mpbid 221 |
. . . . . 6
⊢ (𝜑 → (𝐵 < 1 ∨ 𝐵 = 1)) |
48 | 47 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐵 < 1 ∨ 𝐵 = 1)) |
49 | 32, 43, 48 | mpjaodan 823 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐(1 − 𝐵)) ≤ 1) |
50 | | lemul1a 10756 |
. . . 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 1321 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑𝑐(1 − 𝐵)) · (𝐴↑𝑐𝐵)) ≤ (1 · (𝐴↑𝑐𝐵))) |
52 | | ax-1cn 9873 |
. . . . . . 7
⊢ 1 ∈
ℂ |
53 | 5 | recnd 9947 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) |
54 | | npcan 10169 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ 𝐵
∈ ℂ) → ((1 − 𝐵) + 𝐵) = 1) |
55 | 52, 53, 54 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → ((1 − 𝐵) + 𝐵) = 1) |
56 | 55 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → ((1 − 𝐵) + 𝐵) = 1) |
57 | 56 | oveq2d 6565 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐((1 − 𝐵) + 𝐵)) = (𝐴↑𝑐1)) |
58 | 38 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈ ℂ) |
59 | 1 | anim1i 590 |
. . . . . . 7
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
60 | | elrp 11710 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ+
↔ (𝐴 ∈ ℝ
∧ 0 < 𝐴)) |
61 | 59, 60 | sylibr 223 |
. . . . . 6
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ∈
ℝ+) |
62 | 61 | rpne0d 11753 |
. . . . 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 24263 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐((1 − 𝐵) + 𝐵)) = ((𝐴↑𝑐(1 − 𝐵)) · (𝐴↑𝑐𝐵))) |
66 | 38 | cxp1d 24252 |
. . . . 5
⊢ (𝜑 → (𝐴↑𝑐1) = 𝐴) |
67 | 66 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 0 < 𝐴) → (𝐴↑𝑐1) = 𝐴) |
68 | 57, 65, 67 | 3eqtr3d 2652 |
. . 3
⊢ ((𝜑 ∧ 0 < 𝐴) → ((𝐴↑𝑐(1 − 𝐵)) · (𝐴↑𝑐𝐵)) = 𝐴) |
69 | 38, 53 | cxpcld 24254 |
. . . . 5
⊢ (𝜑 → (𝐴↑𝑐𝐵) ∈ ℂ) |
70 | 69 | mulid2d 9937 |
. . . 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 4614 |
. 2
⊢ ((𝜑 ∧ 0 < 𝐴) → 𝐴 ≤ (𝐴↑𝑐𝐵)) |
73 | 1, 2, 5 | cxpge0d 24270 |
. . . 4
⊢ (𝜑 → 0 ≤ (𝐴↑𝑐𝐵)) |
74 | | breq1 4586 |
. . . 4
⊢ (0 =
𝐴 → (0 ≤ (𝐴↑𝑐𝐵) ↔ 𝐴 ≤ (𝐴↑𝑐𝐵))) |
75 | 73, 74 | syl5ibcom 234 |
. . 3
⊢ (𝜑 → (0 = 𝐴 → 𝐴 ≤ (𝐴↑𝑐𝐵))) |
76 | 75 | imp 444 |
. 2
⊢ ((𝜑 ∧ 0 = 𝐴) → 𝐴 ≤ (𝐴↑𝑐𝐵)) |
77 | | 0re 9919 |
. . . 4
⊢ 0 ∈
ℝ |
78 | | leloe 10003 |
. . . 4
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
79 | 77, 1, 78 | sylancr 694 |
. . 3
⊢ (𝜑 → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴))) |
80 | 2, 79 | mpbid 221 |
. 2
⊢ (𝜑 → (0 < 𝐴 ∨ 0 = 𝐴)) |
81 | 72, 76, 80 | mpjaodan 823 |
1
⊢ (𝜑 → 𝐴 ≤ (𝐴↑𝑐𝐵)) |