| Step | Hyp | Ref
| Expression |
| 1 | | dvrelogpow2b.5 |
. . . 4
⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁)) |
| 2 | 1 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁))) |
| 3 | 2 | oveq2d 7447 |
. 2
⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁)))) |
| 4 | | reelprrecn 11247 |
. . . . 5
⊢ ℝ
∈ {ℝ, ℂ} |
| 5 | 4 | a1i 11 |
. . . 4
⊢ (𝜑 → ℝ ∈ {ℝ,
ℂ}) |
| 6 | | cnelprrecn 11248 |
. . . . 5
⊢ ℂ
∈ {ℝ, ℂ} |
| 7 | 6 | a1i 11 |
. . . 4
⊢ (𝜑 → ℂ ∈ {ℝ,
ℂ}) |
| 8 | | elioore 13417 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ) |
| 9 | 8 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℝ) |
| 10 | 9 | recnd 11289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ) |
| 11 | | dvrelogpow2b.1 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 12 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ) |
| 13 | | dvrelogpow2b.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 14 | 13 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ) |
| 15 | | dvrelogpow2b.3 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝐴) |
| 16 | 15 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < 𝐴) |
| 17 | | dvrelogpow2b.4 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 18 | 17 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵) |
| 19 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴(,)𝐵)) |
| 20 | 12, 14, 16, 18, 19 | 0nonelalab 42068 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥) |
| 21 | 20 | necomd 2996 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0) |
| 22 | 10, 21 | logcld 26612 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘𝑥) ∈ ℂ) |
| 23 | | 2cnd 12344 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ ℂ) |
| 24 | | 0ne2 12473 |
. . . . . . . . 9
⊢ 0 ≠
2 |
| 25 | 24 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 2) |
| 26 | 25 | necomd 2996 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0) |
| 27 | 23, 26 | logcld 26612 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈
ℂ) |
| 28 | | 0red 11264 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ) |
| 29 | | 1lt2 12437 |
. . . . . . . . . 10
⊢ 1 <
2 |
| 30 | 29 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 < 2) |
| 31 | | 2rp 13039 |
. . . . . . . . . 10
⊢ 2 ∈
ℝ+ |
| 32 | | loggt0b 26674 |
. . . . . . . . . 10
⊢ (2 ∈
ℝ+ → (0 < (log‘2) ↔ 1 <
2)) |
| 33 | 31, 32 | ax-mp 5 |
. . . . . . . . 9
⊢ (0 <
(log‘2) ↔ 1 < 2) |
| 34 | 30, 33 | sylibr 234 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 <
(log‘2)) |
| 35 | 28, 34 | ltned 11397 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠
(log‘2)) |
| 36 | 35 | necomd 2996 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ≠
0) |
| 37 | 22, 27, 36 | divcld 12043 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥) / (log‘2)) ∈
ℂ) |
| 38 | | 1red 11262 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℝ) |
| 39 | 38, 30 | ltned 11397 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ≠ 2) |
| 40 | 39 | necomd 2996 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 1) |
| 41 | 26, 40 | nelprd 4657 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 2 ∈ {0,
1}) |
| 42 | 23, 41 | eldifd 3962 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈ (ℂ ∖ {0,
1})) |
| 43 | | necom 2994 |
. . . . . . . . . . . 12
⊢ (0 ≠
𝑥 ↔ 𝑥 ≠ 0) |
| 44 | 43 | imbi2i 336 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≠ 𝑥) ↔ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0)) |
| 45 | 20, 44 | mpbi 230 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ≠ 0) |
| 46 | 45 | neneqd 2945 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 = 0) |
| 47 | | velsn 4642 |
. . . . . . . . 9
⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0) |
| 48 | 46, 47 | sylnibr 329 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ¬ 𝑥 ∈ {0}) |
| 49 | 10, 48 | eldifd 3962 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (ℂ ∖
{0})) |
| 50 | | logbval 26809 |
. . . . . . 7
⊢ ((2
∈ (ℂ ∖ {0, 1}) ∧ 𝑥 ∈ (ℂ ∖ {0})) → (2
logb 𝑥) =
((log‘𝑥) /
(log‘2))) |
| 51 | 42, 49, 50 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 logb 𝑥) = ((log‘𝑥) /
(log‘2))) |
| 52 | 51 | eleq1d 2826 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 logb 𝑥) ∈ ℂ ↔
((log‘𝑥) /
(log‘2)) ∈ ℂ)) |
| 53 | 37, 52 | mpbird 257 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (2 logb 𝑥) ∈
ℂ) |
| 54 | 31 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ∈
ℝ+) |
| 55 | 54 | relogcld 26665 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈
ℝ) |
| 56 | 9, 55 | remulcld 11291 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈
ℝ) |
| 57 | 54 | rpne0d 13082 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 2 ≠ 0) |
| 58 | 23, 57 | logcld 26612 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (log‘2) ∈
ℂ) |
| 59 | 10, 58, 21, 36 | mulne0d 11915 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ≠
0) |
| 60 | 38, 56, 59 | redivcld 12095 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (1 / (𝑥 · (log‘2))) ∈
ℝ) |
| 61 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) |
| 62 | | dvrelogpow2b.8 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 63 | 62 | nnnn0d 12587 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 64 | 63 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈
ℕ0) |
| 65 | 61, 64 | expcld 14186 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑁) ∈ ℂ) |
| 66 | 62 | nncnd 12282 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 67 | 66 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑁 ∈ ℂ) |
| 68 | | nnm1nn0 12567 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
| 69 | 62, 68 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑁 − 1) ∈
ℕ0) |
| 70 | 69 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 − 1) ∈
ℕ0) |
| 71 | 61, 70 | expcld 14186 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦↑(𝑁 − 1)) ∈
ℂ) |
| 72 | 67, 71 | mulcld 11281 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑁 · (𝑦↑(𝑁 − 1))) ∈
ℂ) |
| 73 | 11 | rexrd 11311 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 74 | 13 | rexrd 11311 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 75 | | 0red 11264 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℝ) |
| 76 | 75, 11, 15 | ltled 11409 |
. . . . 5
⊢ (𝜑 → 0 ≤ 𝐴) |
| 77 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥)) |
| 78 | | eqid 2737 |
. . . . 5
⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2)))) |
| 79 | 73, 74, 76, 17, 77, 78 | dvrelog2b 42067 |
. . . 4
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))) |
| 80 | | dvexp 25991 |
. . . . 5
⊢ (𝑁 ∈ ℕ → (ℂ
D (𝑦 ∈ ℂ ↦
(𝑦↑𝑁))) = (𝑦 ∈ ℂ ↦ (𝑁 · (𝑦↑(𝑁 − 1))))) |
| 81 | 62, 80 | syl 17 |
. . . 4
⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝑁))) = (𝑦 ∈ ℂ ↦ (𝑁 · (𝑦↑(𝑁 − 1))))) |
| 82 | | oveq1 7438 |
. . . 4
⊢ (𝑦 = (2 logb 𝑥) → (𝑦↑𝑁) = ((2 logb 𝑥)↑𝑁)) |
| 83 | | oveq1 7438 |
. . . . 5
⊢ (𝑦 = (2 logb 𝑥) → (𝑦↑(𝑁 − 1)) = ((2 logb 𝑥)↑(𝑁 − 1))) |
| 84 | 83 | oveq2d 7447 |
. . . 4
⊢ (𝑦 = (2 logb 𝑥) → (𝑁 · (𝑦↑(𝑁 − 1))) = (𝑁 · ((2 logb 𝑥)↑(𝑁 − 1)))) |
| 85 | 5, 7, 53, 60, 65, 72, 79, 81, 82, 84 | dvmptco 26010 |
. . 3
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 ·
(log‘2)))))) |
| 86 | | dvrelogpow2b.6 |
. . . . . 6
⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥))) |
| 87 | 86 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))) |
| 88 | | dvrelogpow2b.7 |
. . . . . . . . . . . . . 14
⊢ 𝐶 = (𝑁 / ((log‘2)↑𝑁)) |
| 89 | 88 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐶 = (𝑁 / ((log‘2)↑𝑁))) |
| 90 | 89 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥))) |
| 91 | 66 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℂ) |
| 92 | 63 | nn0zd 12639 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 93 | 92 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 ∈ ℤ) |
| 94 | 27, 36, 93 | expclzd 14191 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ∈
ℂ) |
| 95 | 69 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈
ℕ0) |
| 96 | 22, 95 | expcld 14186 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘𝑥)↑(𝑁 − 1)) ∈
ℂ) |
| 97 | 27, 36, 93 | expne0d 14192 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) ≠ 0) |
| 98 | 91, 94, 96, 10, 97, 21 | divmuldivd 12084 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑𝑁) · 𝑥))) |
| 99 | 94, 10 | mulcomd 11282 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2)↑𝑁))) |
| 100 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 1 ∈
ℂ) |
| 101 | 100, 66 | pncan3d 11623 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁) |
| 102 | 101 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑁 = (1 + (𝑁 − 1))) |
| 103 | 102 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑁 = (1 + (𝑁 − 1))) |
| 104 | 103 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2)↑(1 +
(𝑁 −
1)))) |
| 105 | | 1nn0 12542 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 1 ∈
ℕ0 |
| 106 | 105 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈
ℕ0) |
| 107 | 27, 95, 106 | expaddd 14188 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(1 + (𝑁 − 1))) =
(((log‘2)↑1) · ((log‘2)↑(𝑁 − 1)))) |
| 108 | 104, 107 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = (((log‘2)↑1)
· ((log‘2)↑(𝑁 − 1)))) |
| 109 | 27 | exp1d 14181 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑1) =
(log‘2)) |
| 110 | 109 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑1) ·
((log‘2)↑(𝑁
− 1))) = ((log‘2) · ((log‘2)↑(𝑁 − 1)))) |
| 111 | 108, 110 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑𝑁) = ((log‘2) ·
((log‘2)↑(𝑁
− 1)))) |
| 112 | 111 | oveq2d 7447 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2)↑𝑁)) = (𝑥 · ((log‘2) ·
((log‘2)↑(𝑁
− 1))))) |
| 113 | 99, 112 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (𝑥 · ((log‘2) ·
((log‘2)↑(𝑁
− 1))))) |
| 114 | 27, 95 | expcld 14186 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ∈
ℂ) |
| 115 | 10, 27, 114 | mulassd 11284 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) ·
((log‘2)↑(𝑁
− 1))) = (𝑥 ·
((log‘2) · ((log‘2)↑(𝑁 − 1))))) |
| 116 | 115 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · ((log‘2) ·
((log‘2)↑(𝑁
− 1)))) = ((𝑥
· (log‘2)) · ((log‘2)↑(𝑁 − 1)))) |
| 117 | 113, 116 | eqtrd 2777 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = ((𝑥 · (log‘2)) ·
((log‘2)↑(𝑁
− 1)))) |
| 118 | 10, 27 | mulcld 11281 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑥 · (log‘2)) ∈
ℂ) |
| 119 | 118, 114 | mulcomd 11282 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑥 · (log‘2)) ·
((log‘2)↑(𝑁
− 1))) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))) |
| 120 | 117, 119 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘2)↑𝑁) · 𝑥) = (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))) |
| 121 | 120 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑𝑁) · 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 ·
(log‘2))))) |
| 122 | 98, 121 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 / ((log‘2)↑𝑁)) · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 ·
(log‘2))))) |
| 123 | 90, 122 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 ·
(log‘2))))) |
| 124 | 91, 96 | mulcld 11281 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((log‘𝑥)↑(𝑁 − 1))) ∈
ℂ) |
| 125 | | 1zzd 12648 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 1 ∈ ℤ) |
| 126 | 93, 125 | zsubcld 12727 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 − 1) ∈ ℤ) |
| 127 | 27, 36, 126 | expne0d 14192 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((log‘2)↑(𝑁 − 1)) ≠
0) |
| 128 | 124, 114,
118, 127, 59 | divdiv1d 12074 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 · (log‘2))) =
((𝑁 ·
((log‘𝑥)↑(𝑁 − 1))) /
(((log‘2)↑(𝑁
− 1)) · (𝑥
· (log‘2))))) |
| 129 | 128 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / (((log‘2)↑(𝑁 − 1)) · (𝑥 · (log‘2)))) =
(((𝑁 ·
((log‘𝑥)↑(𝑁 − 1))) /
((log‘2)↑(𝑁
− 1))) / (𝑥 ·
(log‘2)))) |
| 130 | 123, 129 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 ·
(log‘2)))) |
| 131 | 91, 96, 114, 127 | divassd 12078 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))))) |
| 132 | 131 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((𝑁 · ((log‘𝑥)↑(𝑁 − 1))) / ((log‘2)↑(𝑁 − 1))) / (𝑥 · (log‘2))) =
((𝑁 ·
(((log‘𝑥)↑(𝑁 − 1)) /
((log‘2)↑(𝑁
− 1)))) / (𝑥 ·
(log‘2)))) |
| 133 | 130, 132 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) / (𝑥 ·
(log‘2)))) |
| 134 | 22, 27, 36, 95 | expdivd 14200 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥) / (log‘2))↑(𝑁 − 1)) = (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) |
| 135 | 134 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1))) =
(((log‘𝑥) /
(log‘2))↑(𝑁
− 1))) |
| 136 | 135 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))) |
| 137 | 136 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · (((log‘𝑥)↑(𝑁 − 1)) / ((log‘2)↑(𝑁 − 1)))) / (𝑥 · (log‘2))) =
((𝑁 ·
(((log‘𝑥) /
(log‘2))↑(𝑁
− 1))) / (𝑥 ·
(log‘2)))) |
| 138 | 133, 137 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2)))) |
| 139 | 51 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 logb 𝑥)↑(𝑁 − 1)) = (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) |
| 140 | 139 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) = (𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1)))) |
| 141 | 140 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 ·
(log‘2)))) |
| 142 | 141 | eqcomd 2743 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · (((log‘𝑥) / (log‘2))↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · ((2 logb
𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2)))) |
| 143 | 138, 142 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2)))) |
| 144 | 53, 95 | expcld 14186 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((2 logb 𝑥)↑(𝑁 − 1)) ∈
ℂ) |
| 145 | 91, 144 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) ∈
ℂ) |
| 146 | 145, 118,
59 | divrecd 12046 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) / (𝑥 · (log‘2))) = ((𝑁 · ((2 logb
𝑥)↑(𝑁 − 1))) · (1 / (𝑥 ·
(log‘2))))) |
| 147 | 143, 146 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)) = ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 ·
(log‘2))))) |
| 148 | 147 | mpteq2dva 5242 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 ·
(log‘2)))))) |
| 149 | 87, 148 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 ·
(log‘2)))))) |
| 150 | 149 | eqcomd 2743 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((𝑁 · ((2 logb 𝑥)↑(𝑁 − 1))) · (1 / (𝑥 · (log‘2))))) =
𝐺) |
| 151 | 85, 150 | eqtrd 2777 |
. 2
⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁))) = 𝐺) |
| 152 | 3, 151 | eqtrd 2777 |
1
⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |