| Step | Hyp | Ref
| Expression |
| 1 | | rpssre 13042 |
. . . 4
⊢
ℝ+ ⊆ ℝ |
| 2 | | ax-1cn 11213 |
. . . 4
⊢ 1 ∈
ℂ |
| 3 | | o1const 15656 |
. . . 4
⊢
((ℝ+ ⊆ ℝ ∧ 1 ∈ ℂ) →
(𝑥 ∈
ℝ+ ↦ 1) ∈ 𝑂(1)) |
| 4 | 1, 2, 3 | mp2an 692 |
. . 3
⊢ (𝑥 ∈ ℝ+
↦ 1) ∈ 𝑂(1) |
| 5 | | 1cnd 11256 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → 1 ∈ ℂ) |
| 6 | | fzfid 14014 |
. . . . . 6
⊢ (𝑥 ∈ ℝ+
→ (1...(⌊‘𝑥)) ∈ Fin) |
| 7 | | elfznn 13593 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℕ) |
| 8 | 7 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
| 9 | | mucl 27184 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ →
(μ‘𝑛) ∈
ℤ) |
| 10 | 8, 9 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℤ) |
| 11 | 10 | zred 12722 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℝ) |
| 12 | 11, 8 | nndivred 12320 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℝ) |
| 13 | 7 | nnrpd 13075 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℝ+) |
| 14 | | rpdivcl 13060 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
ℝ+) → (𝑥 / 𝑛) ∈
ℝ+) |
| 15 | 13, 14 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ+) |
| 16 | 15 | relogcld 26665 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℝ) |
| 17 | 12, 16 | remulcld 11291 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛))) ∈
ℝ) |
| 18 | 17 | recnd 11289 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ+
∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛))) ∈
ℂ) |
| 19 | 6, 18 | fsumcl 15769 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
→ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ) |
| 20 | 19 | adantl 481 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ) |
| 21 | | mulogsumlem 27575 |
. . . . 5
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) |
| 22 | | sumex 15724 |
. . . . . . . 8
⊢
Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ V |
| 23 | 22 | a1i 11 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ V) |
| 24 | 21 | a1i 11 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)) |
| 25 | 23, 24 | o1mptrcl 15659 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
| 26 | 5, 20 | subcld 11620 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ ℝ+) → (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ) |
| 27 | | 1red 11262 |
. . . . . 6
⊢ (⊤
→ 1 ∈ ℝ) |
| 28 | | fz1ssnn 13595 |
. . . . . . . . . . . . . . . 16
⊢
(1...(⌊‘𝑥)) ⊆ ℕ |
| 29 | 28 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
(1...(⌊‘𝑥))
⊆ ℕ) |
| 30 | 29 | sselda 3983 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
| 31 | 30, 9 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℤ) |
| 32 | 31 | zred 12722 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℝ) |
| 33 | 32, 30 | nndivred 12320 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℝ) |
| 34 | 33 | recnd 11289 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑛) /
𝑛) ∈
ℂ) |
| 35 | | fzfid 14014 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin) |
| 36 | | elfznn 13593 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛))) → 𝑚 ∈
ℕ) |
| 37 | 36 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℕ) |
| 38 | 37 | nnrpd 13075 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℝ+) |
| 39 | 38 | rpcnne0d 13086 |
. . . . . . . . . . . 12
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) |
| 40 | | reccl 11929 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℂ ∧ 𝑚 ≠ 0) → (1 / 𝑚) ∈
ℂ) |
| 41 | 39, 40 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (1 / 𝑚) ∈
ℂ) |
| 42 | 35, 41 | fsumcl 15769 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) ∈ ℂ) |
| 43 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
𝑥 ∈
ℝ+) |
| 44 | 43, 13, 14 | syl2an 596 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑛) ∈
ℝ+) |
| 45 | 44 | relogcld 26665 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℝ) |
| 46 | 45 | recnd 11289 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (log‘(𝑥 /
𝑛)) ∈
ℂ) |
| 47 | 34, 42, 46 | subdid 11719 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · (Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = ((((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) |
| 48 | 47 | sumeq2dv 15738 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) |
| 49 | | fzfid 14014 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
(1...(⌊‘𝑥))
∈ Fin) |
| 50 | 34, 42 | mulcld 11281 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚)) ∈ ℂ) |
| 51 | 18 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) ·
(log‘(𝑥 / 𝑛))) ∈
ℂ) |
| 52 | 49, 50, 51 | fsumsub 15824 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑛 ∈
(1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) |
| 53 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑛 · 𝑚) → (1 / 𝑘) = (1 / (𝑛 · 𝑚))) |
| 54 | 53 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑛 · 𝑚) → ((μ‘𝑛) · (1 / 𝑘)) = ((μ‘𝑛) · (1 / (𝑛 · 𝑚)))) |
| 55 | | rpre 13043 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
| 56 | 55 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
𝑥 ∈
ℝ) |
| 57 | | ssrab2 4080 |
. . . . . . . . . . . . . . 15
⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ |
| 58 | | simprr 773 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧
(𝑘 ∈
(1...(⌊‘𝑥))
∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) |
| 59 | 57, 58 | sselid 3981 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧
(𝑘 ∈
(1...(⌊‘𝑥))
∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ ℕ) |
| 60 | 59, 9 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧
(𝑘 ∈
(1...(⌊‘𝑥))
∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℤ) |
| 61 | 60 | zcnd 12723 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧
(𝑘 ∈
(1...(⌊‘𝑥))
∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℂ) |
| 62 | | elfznn 13593 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(1...(⌊‘𝑥))
→ 𝑘 ∈
ℕ) |
| 63 | 62 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑘 ∈
(1...(⌊‘𝑥)))
→ 𝑘 ∈
ℕ) |
| 64 | 63 | nnrecred 12317 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑘 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑘) ∈
ℝ) |
| 65 | 64 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑘 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑘) ∈
ℂ) |
| 66 | 65 | adantrr 717 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧
(𝑘 ∈
(1...(⌊‘𝑥))
∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (1 / 𝑘) ∈ ℂ) |
| 67 | 61, 66 | mulcld 11281 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧
(𝑘 ∈
(1...(⌊‘𝑥))
∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → ((μ‘𝑛) · (1 / 𝑘)) ∈ ℂ) |
| 68 | 54, 56, 67 | dvdsflsumcom 27231 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑘 ∈
(1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · (1 / 𝑘)) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚)))) |
| 69 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑘 = 1 → (1 / 𝑘) = (1 / 1)) |
| 70 | | 1div1e1 11958 |
. . . . . . . . . . . 12
⊢ (1 / 1) =
1 |
| 71 | 69, 70 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝑘 = 1 → (1 / 𝑘) = 1) |
| 72 | | flge1nn 13861 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ) |
| 73 | 55, 72 | sylan 580 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
(⌊‘𝑥) ∈
ℕ) |
| 74 | | nnuz 12921 |
. . . . . . . . . . . . 13
⊢ ℕ =
(ℤ≥‘1) |
| 75 | 73, 74 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
(⌊‘𝑥) ∈
(ℤ≥‘1)) |
| 76 | | eluzfz1 13571 |
. . . . . . . . . . . 12
⊢
((⌊‘𝑥)
∈ (ℤ≥‘1) → 1 ∈
(1...(⌊‘𝑥))) |
| 77 | 75, 76 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) → 1
∈ (1...(⌊‘𝑥))) |
| 78 | 71, 49, 29, 77, 65 | musumsum 27235 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑘 ∈
(1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · (1 / 𝑘)) = 1) |
| 79 | 31 | zcnd 12723 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑛)
∈ ℂ) |
| 80 | 79 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
(μ‘𝑛) ∈
ℂ) |
| 81 | 30 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑛 ∈
ℕ) |
| 82 | 81 | nnrpd 13075 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑛 ∈
ℝ+) |
| 83 | 82 | rpcnne0d 13086 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) |
| 84 | | divdiv1 11978 |
. . . . . . . . . . . . . . 15
⊢
(((μ‘𝑛)
∈ ℂ ∧ (𝑛
∈ ℂ ∧ 𝑛 ≠
0) ∧ (𝑚 ∈ ℂ
∧ 𝑚 ≠ 0)) →
(((μ‘𝑛) / 𝑛) / 𝑚) = ((μ‘𝑛) / (𝑛 · 𝑚))) |
| 85 | 80, 83, 39, 84 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
(((μ‘𝑛) / 𝑛) / 𝑚) = ((μ‘𝑛) / (𝑛 · 𝑚))) |
| 86 | 34 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
((μ‘𝑛) / 𝑛) ∈
ℂ) |
| 87 | 37 | nncnd 12282 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ∈
ℂ) |
| 88 | 37 | nnne0d 12316 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → 𝑚 ≠ 0) |
| 89 | 86, 87, 88 | divrecd 12046 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
(((μ‘𝑛) / 𝑛) / 𝑚) = (((μ‘𝑛) / 𝑛) · (1 / 𝑚))) |
| 90 | | nnmulcl 12290 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑛 · 𝑚) ∈ ℕ) |
| 91 | 30, 36, 90 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (𝑛 · 𝑚) ∈ ℕ) |
| 92 | 91 | nncnd 12282 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (𝑛 · 𝑚) ∈ ℂ) |
| 93 | 91 | nnne0d 12316 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) → (𝑛 · 𝑚) ≠ 0) |
| 94 | 80, 92, 93 | divrecd 12046 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
((μ‘𝑛) / (𝑛 · 𝑚)) = ((μ‘𝑛) · (1 / (𝑛 · 𝑚)))) |
| 95 | 85, 89, 94 | 3eqtr3rd 2786 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))) →
((μ‘𝑛) · (1
/ (𝑛 · 𝑚))) = (((μ‘𝑛) / 𝑛) · (1 / 𝑚))) |
| 96 | 95 | sumeq2dv 15738 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((μ‘𝑛) / 𝑛) · (1 / 𝑚))) |
| 97 | 35, 34, 41 | fsummulc2 15820 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (((μ‘𝑛) /
𝑛) · Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((μ‘𝑛) / 𝑛) · (1 / 𝑚))) |
| 98 | 96, 97 | eqtr4d 2780 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = (((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚))) |
| 99 | 98 | sumeq2dv 15738 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑛 ∈
(1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · (1 / (𝑛 · 𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚))) |
| 100 | 68, 78, 99 | 3eqtr3rd 2786 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) = 1) |
| 101 | 100 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
(Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) |
| 102 | 48, 52, 101 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ+
∧ 1 ≤ 𝑥) →
Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) |
| 103 | 102 | adantl 481 |
. . . . . 6
⊢
((⊤ ∧ (𝑥
∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) |
| 104 | 25, 26, 27, 103 | o1eq 15606 |
. . . . 5
⊢ (⊤
→ ((𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ+
↦ (1 − Σ𝑛
∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))) |
| 105 | 21, 104 | mpbii 233 |
. . . 4
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ (1 − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)) |
| 106 | 5, 20, 105 | o1dif 15666 |
. . 3
⊢ (⊤
→ ((𝑥 ∈
ℝ+ ↦ 1) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ+ ↦
Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1))) |
| 107 | 4, 106 | mpbii 233 |
. 2
⊢ (⊤
→ (𝑥 ∈
ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)) |
| 108 | 107 | mptru 1547 |
1
⊢ (𝑥 ∈ ℝ+
↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1) |