| Step | Hyp | Ref
| Expression |
| 1 | | 2fveq3 6911 |
. . . . . 6
⊢ (𝑛 = (𝑑 · 𝑚) → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘(𝑑 · 𝑚)))) |
| 2 | | id 22 |
. . . . . 6
⊢ (𝑛 = (𝑑 · 𝑚) → 𝑛 = (𝑑 · 𝑚)) |
| 3 | 1, 2 | oveq12d 7449 |
. . . . 5
⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) |
| 4 | | oveq2 7439 |
. . . . . 6
⊢ (𝑛 = (𝑑 · 𝑚) → (𝐴 / 𝑛) = (𝐴 / (𝑑 · 𝑚))) |
| 5 | 4 | fveq2d 6910 |
. . . . 5
⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝐴 / 𝑛)) = (log‘(𝐴 / (𝑑 · 𝑚)))) |
| 6 | 3, 5 | oveq12d 7449 |
. . . 4
⊢ (𝑛 = (𝑑 · 𝑚) → (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛))) = (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚))))) |
| 7 | 6 | oveq2d 7447 |
. . 3
⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛)))) = ((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 8 | | dchrvmasum.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈
ℝ+) |
| 9 | 8 | rpred 13077 |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 10 | | elrabi 3687 |
. . . . . . 7
⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} → 𝑑 ∈ ℕ) |
| 11 | 10 | ad2antll 729 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝑑 ∈ ℕ) |
| 12 | | mucl 27184 |
. . . . . 6
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℤ) |
| 13 | 11, 12 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ) |
| 14 | 13 | zcnd 12723 |
. . . 4
⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ) |
| 15 | | rpvmasum.g |
. . . . . . . 8
⊢ 𝐺 = (DChr‘𝑁) |
| 16 | | rpvmasum.z |
. . . . . . . 8
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
| 17 | | rpvmasum.d |
. . . . . . . 8
⊢ 𝐷 = (Base‘𝐺) |
| 18 | | rpvmasum.l |
. . . . . . . 8
⊢ 𝐿 = (ℤRHom‘𝑍) |
| 19 | | dchrisum.b |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ 𝐷) |
| 20 | 19 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷) |
| 21 | | elfzelz 13564 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝐴))
→ 𝑛 ∈
ℤ) |
| 22 | 21 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℤ) |
| 23 | 15, 16, 17, 18, 20, 22 | dchrzrhcl 27289 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ) |
| 24 | | elfznn 13593 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝐴))
→ 𝑛 ∈
ℕ) |
| 25 | 24 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ) |
| 26 | 25 | nncnd 12282 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ) |
| 27 | 25 | nnne0d 12316 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0) |
| 28 | 23, 26, 27 | divcld 12043 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ) |
| 29 | 24 | nnrpd 13075 |
. . . . . . . . 9
⊢ (𝑛 ∈
(1...(⌊‘𝐴))
→ 𝑛 ∈
ℝ+) |
| 30 | | rpdivcl 13060 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ+
∧ 𝑛 ∈
ℝ+) → (𝐴 / 𝑛) ∈
ℝ+) |
| 31 | 8, 29, 30 | syl2an 596 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈
ℝ+) |
| 32 | 31 | relogcld 26665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑛)) ∈ ℝ) |
| 33 | 32 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘(𝐴 / 𝑛)) ∈ ℂ) |
| 34 | 28, 33 | mulcld 11281 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛))) ∈ ℂ) |
| 35 | 34 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛))) ∈ ℂ) |
| 36 | 14, 35 | mulcld 11281 |
. . 3
⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛)))) ∈ ℂ) |
| 37 | 7, 9, 36 | dvdsflsumcom 27231 |
. 2
⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛)))) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 38 | | 2fveq3 6911 |
. . . . . 6
⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1))) |
| 39 | | id 22 |
. . . . . 6
⊢ (𝑛 = 1 → 𝑛 = 1) |
| 40 | 38, 39 | oveq12d 7449 |
. . . . 5
⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1)) |
| 41 | | oveq2 7439 |
. . . . . 6
⊢ (𝑛 = 1 → (𝐴 / 𝑛) = (𝐴 / 1)) |
| 42 | 41 | fveq2d 6910 |
. . . . 5
⊢ (𝑛 = 1 → (log‘(𝐴 / 𝑛)) = (log‘(𝐴 / 1))) |
| 43 | 40, 42 | oveq12d 7449 |
. . . 4
⊢ (𝑛 = 1 → (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛))) = (((𝑋‘(𝐿‘1)) / 1) · (log‘(𝐴 / 1)))) |
| 44 | | fzfid 14014 |
. . . 4
⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin) |
| 45 | | fz1ssnn 13595 |
. . . . 5
⊢
(1...(⌊‘𝐴)) ⊆ ℕ |
| 46 | 45 | a1i 11 |
. . . 4
⊢ (𝜑 → (1...(⌊‘𝐴)) ⊆
ℕ) |
| 47 | | dchrvmasum2.2 |
. . . . . . 7
⊢ (𝜑 → 1 ≤ 𝐴) |
| 48 | | flge1nn 13861 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤
𝐴) →
(⌊‘𝐴) ∈
ℕ) |
| 49 | 9, 47, 48 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (⌊‘𝐴) ∈
ℕ) |
| 50 | | nnuz 12921 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
| 51 | 49, 50 | eleqtrdi 2851 |
. . . . 5
⊢ (𝜑 → (⌊‘𝐴) ∈
(ℤ≥‘1)) |
| 52 | | eluzfz1 13571 |
. . . . 5
⊢
((⌊‘𝐴)
∈ (ℤ≥‘1) → 1 ∈
(1...(⌊‘𝐴))) |
| 53 | 51, 52 | syl 17 |
. . . 4
⊢ (𝜑 → 1 ∈
(1...(⌊‘𝐴))) |
| 54 | 43, 44, 46, 53, 34 | musumsum 27235 |
. . 3
⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛)))) = (((𝑋‘(𝐿‘1)) / 1) · (log‘(𝐴 / 1)))) |
| 55 | 15, 16, 17, 18, 19 | dchrzrh1 27288 |
. . . . . 6
⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) |
| 56 | 55 | oveq1d 7446 |
. . . . 5
⊢ (𝜑 → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1)) |
| 57 | | 1div1e1 11958 |
. . . . 5
⊢ (1 / 1) =
1 |
| 58 | 56, 57 | eqtrdi 2793 |
. . . 4
⊢ (𝜑 → ((𝑋‘(𝐿‘1)) / 1) = 1) |
| 59 | 8 | rpcnd 13079 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 60 | 59 | div1d 12035 |
. . . . 5
⊢ (𝜑 → (𝐴 / 1) = 𝐴) |
| 61 | 60 | fveq2d 6910 |
. . . 4
⊢ (𝜑 → (log‘(𝐴 / 1)) = (log‘𝐴)) |
| 62 | 58, 61 | oveq12d 7449 |
. . 3
⊢ (𝜑 → (((𝑋‘(𝐿‘1)) / 1) · (log‘(𝐴 / 1))) = (1 ·
(log‘𝐴))) |
| 63 | 8 | relogcld 26665 |
. . . . 5
⊢ (𝜑 → (log‘𝐴) ∈
ℝ) |
| 64 | 63 | recnd 11289 |
. . . 4
⊢ (𝜑 → (log‘𝐴) ∈
ℂ) |
| 65 | 64 | mullidd 11279 |
. . 3
⊢ (𝜑 → (1 ·
(log‘𝐴)) =
(log‘𝐴)) |
| 66 | 54, 62, 65 | 3eqtrrd 2782 |
. 2
⊢ (𝜑 → (log‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑛)) / 𝑛) · (log‘(𝐴 / 𝑛))))) |
| 67 | | fzfid 14014 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin) |
| 68 | 19 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷) |
| 69 | | elfzelz 13564 |
. . . . . . . 8
⊢ (𝑑 ∈
(1...(⌊‘𝐴))
→ 𝑑 ∈
ℤ) |
| 70 | 69 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℤ) |
| 71 | 15, 16, 17, 18, 68, 70 | dchrzrhcl 27289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
| 72 | | fznnfl 13902 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℝ → (𝑑 ∈
(1...(⌊‘𝐴))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝐴))) |
| 73 | 9, 72 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴))) |
| 74 | 73 | simprbda 498 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ) |
| 75 | 74, 12 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (μ‘𝑑) ∈ ℤ) |
| 76 | 75 | zred 12722 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (μ‘𝑑) ∈ ℝ) |
| 77 | 76, 74 | nndivred 12320 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℝ) |
| 78 | 77 | recnd 11289 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℂ) |
| 79 | 71, 78 | mulcld 11281 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
| 80 | 19 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑋 ∈ 𝐷) |
| 81 | | elfzelz 13564 |
. . . . . . . 8
⊢ (𝑚 ∈
(1...(⌊‘(𝐴 /
𝑑))) → 𝑚 ∈
ℤ) |
| 82 | 81 | adantl 481 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℤ) |
| 83 | 15, 16, 17, 18, 80, 82 | dchrzrhcl 27289 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
| 84 | | elfznn 13593 |
. . . . . . . . . . . 12
⊢ (𝑑 ∈
(1...(⌊‘𝐴))
→ 𝑑 ∈
ℕ) |
| 85 | 84 | nnrpd 13075 |
. . . . . . . . . . 11
⊢ (𝑑 ∈
(1...(⌊‘𝐴))
→ 𝑑 ∈
ℝ+) |
| 86 | | rpdivcl 13060 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ+
∧ 𝑑 ∈
ℝ+) → (𝐴 / 𝑑) ∈
ℝ+) |
| 87 | 8, 85, 86 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑑) ∈
ℝ+) |
| 88 | | elfznn 13593 |
. . . . . . . . . . 11
⊢ (𝑚 ∈
(1...(⌊‘(𝐴 /
𝑑))) → 𝑚 ∈
ℕ) |
| 89 | 88 | nnrpd 13075 |
. . . . . . . . . 10
⊢ (𝑚 ∈
(1...(⌊‘(𝐴 /
𝑑))) → 𝑚 ∈
ℝ+) |
| 90 | | rpdivcl 13060 |
. . . . . . . . . 10
⊢ (((𝐴 / 𝑑) ∈ ℝ+ ∧ 𝑚 ∈ ℝ+)
→ ((𝐴 / 𝑑) / 𝑚) ∈
ℝ+) |
| 91 | 87, 89, 90 | syl2an 596 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝐴 / 𝑑) / 𝑚) ∈
ℝ+) |
| 92 | 91 | relogcld 26665 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℝ) |
| 93 | 88 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℕ) |
| 94 | 92, 93 | nndivred 12320 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚) ∈ ℝ) |
| 95 | 94 | recnd 11289 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚) ∈ ℂ) |
| 96 | 83, 95 | mulcld 11281 |
. . . . 5
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) ∈ ℂ) |
| 97 | 67, 79, 96 | fsummulc2 15820 |
. . . 4
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))) |
| 98 | 71 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
| 99 | 76 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (μ‘𝑑) ∈ ℝ) |
| 100 | 99 | recnd 11289 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (μ‘𝑑) ∈ ℂ) |
| 101 | 74 | nnrpd 13075 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ+) |
| 102 | 101 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ∈ ℝ+) |
| 103 | 102 | rpcnne0d 13086 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) |
| 104 | | div12 11944 |
. . . . . . . 8
⊢ (((𝑋‘(𝐿‘𝑑)) ∈ ℂ ∧ (μ‘𝑑) ∈ ℂ ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑))) |
| 105 | 98, 100, 103, 104 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑))) |
| 106 | 92 | recnd 11289 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℂ) |
| 107 | 93 | nnrpd 13075 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℝ+) |
| 108 | 107 | rpcnne0d 13086 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) |
| 109 | | div12 11944 |
. . . . . . . 8
⊢ (((𝑋‘(𝐿‘𝑚)) ∈ ℂ ∧ (log‘((𝐴 / 𝑑) / 𝑚)) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) = ((log‘((𝐴 / 𝑑) / 𝑚)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 110 | 83, 106, 108, 109 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)) = ((log‘((𝐴 / 𝑑) / 𝑚)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 111 | 105, 110 | oveq12d 7449 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = (((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((log‘((𝐴 / 𝑑) / 𝑚)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))) |
| 112 | 102 | rpcnd 13079 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ∈ ℂ) |
| 113 | 102 | rpne0d 13082 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ≠ 0) |
| 114 | 98, 112, 113 | divcld 12043 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ) |
| 115 | 93 | nncnd 12282 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℂ) |
| 116 | 93 | nnne0d 12316 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ≠ 0) |
| 117 | 83, 115, 116 | divcld 12043 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
| 118 | 114, 117 | mulcld 11281 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ) |
| 119 | 100, 106,
118 | mulassd 11284 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) · (log‘((𝐴 / 𝑑) / 𝑚))) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) = ((μ‘𝑑) · ((log‘((𝐴 / 𝑑) / 𝑚)) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))))) |
| 120 | 100, 114,
106, 117 | mul4d 11473 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((log‘((𝐴 / 𝑑) / 𝑚)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (((μ‘𝑑) · (log‘((𝐴 / 𝑑) / 𝑚))) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))) |
| 121 | 69 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ∈ ℤ) |
| 122 | 15, 16, 17, 18, 80, 121, 82 | dchrzrhmul 27290 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘(𝑑 · 𝑚))) = ((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚)))) |
| 123 | 122 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚))) |
| 124 | | divmuldiv 11967 |
. . . . . . . . . . . 12
⊢ ((((𝑋‘(𝐿‘𝑑)) ∈ ℂ ∧ (𝑋‘(𝐿‘𝑚)) ∈ ℂ) ∧ ((𝑑 ∈ ℂ ∧ 𝑑 ≠ 0) ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))) → (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚))) |
| 125 | 98, 83, 103, 108, 124 | syl22anc 839 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚))) |
| 126 | 123, 125 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) = (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
| 127 | 59 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝐴 ∈ ℂ) |
| 128 | | divdiv1 11978 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0) ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → ((𝐴 / 𝑑) / 𝑚) = (𝐴 / (𝑑 · 𝑚))) |
| 129 | 127, 103,
108, 128 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝐴 / 𝑑) / 𝑚) = (𝐴 / (𝑑 · 𝑚))) |
| 130 | 129 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝐴 / (𝑑 · 𝑚)) = ((𝐴 / 𝑑) / 𝑚)) |
| 131 | 130 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘(𝐴 / (𝑑 · 𝑚))) = (log‘((𝐴 / 𝑑) / 𝑚))) |
| 132 | 126, 131 | oveq12d 7449 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))) = ((((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) · (log‘((𝐴 / 𝑑) / 𝑚)))) |
| 133 | 118, 106 | mulcomd 11282 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) · (log‘((𝐴 / 𝑑) / 𝑚))) = ((log‘((𝐴 / 𝑑) / 𝑚)) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))) |
| 134 | 132, 133 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))) = ((log‘((𝐴 / 𝑑) / 𝑚)) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))) |
| 135 | 134 | oveq2d 7447 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚))))) = ((μ‘𝑑) · ((log‘((𝐴 / 𝑑) / 𝑚)) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))))) |
| 136 | 119, 120,
135 | 3eqtr4d 2787 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((log‘((𝐴 / 𝑑) / 𝑚)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) = ((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 137 | 111, 136 | eqtrd 2777 |
. . . . 5
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = ((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 138 | 137 | sumeq2dv 15738 |
. . . 4
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 139 | 97, 138 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 140 | 139 | sumeq2dv 15738 |
. 2
⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚))) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((μ‘𝑑) · (((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) · (log‘(𝐴 / (𝑑 · 𝑚)))))) |
| 141 | 37, 66, 140 | 3eqtr4d 2787 |
1
⊢ (𝜑 → (log‘𝐴) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘((𝐴 / 𝑑) / 𝑚)) / 𝑚)))) |