Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mudivsum Structured version   Visualization version   GIF version

Theorem mudivsum 25019
 Description: Asymptotic formula for Σ𝑛 ≤ 𝑥, μ(𝑛) / 𝑛 = 𝑂(1). Equation 10.2.1 of [Shapiro], p. 405. (Contributed by Mario Carneiro, 14-May-2016.)
Assertion
Ref Expression
mudivsum (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)
Distinct variable group:   𝑥,𝑛

Proof of Theorem mudivsum
Dummy variables 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1red 9934 . . 3 (⊤ → 1 ∈ ℝ)
2 reex 9906 . . . . . . 7 ℝ ∈ V
3 rpssre 11719 . . . . . . 7 + ⊆ ℝ
42, 3ssexi 4731 . . . . . 6 + ∈ V
54a1i 11 . . . . 5 (⊤ → ℝ+ ∈ V)
6 fzfid 12634 . . . . . . . 8 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
7 rpre 11715 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
8 elfznn 12241 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
9 nndivre 10933 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
107, 8, 9syl2an 493 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
1110recnd 9947 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
12 reflcl 12459 . . . . . . . . . . . 12 ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
1310, 12syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
1413recnd 9947 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
1511, 14subcld 10271 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
168adantl 481 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
17 mucl 24667 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
1816, 17syl 17 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
1918zcnd 11359 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
2015, 19mulcld 9939 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
216, 20fsumcl 14311 . . . . . . 7 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
22 rpcn 11717 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
23 rpne0 11724 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ≠ 0)
2421, 22, 23divcld 10680 . . . . . 6 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) ∈ ℂ)
2524adantl 481 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) ∈ ℂ)
26 ovex 6577 . . . . . 6 (1 / 𝑥) ∈ V
2726a1i 11 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ V)
28 eqidd 2611 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)))
29 eqidd 2611 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)))
305, 25, 27, 28, 29offval2 6812 . . . 4 (⊤ → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∘𝑓 + (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))))
313a1i 11 . . . . . 6 (⊤ → ℝ+ ⊆ ℝ)
3221adantr 480 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
3322adantr 480 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℂ)
3423adantr 480 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 𝑥 ≠ 0)
3532, 33, 34absdivd 14042 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / (abs‘𝑥)))
36 rprege0 11723 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
37 absid 13884 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
3836, 37syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (abs‘𝑥) = 𝑥)
3938adantr 480 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
4039oveq2d 6565 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / (abs‘𝑥)) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / 𝑥))
4135, 40eqtrd 2644 . . . . . . . 8 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / 𝑥))
4232abscld 14023 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
43 fzfid 12634 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ∈ Fin)
4420adantlr 747 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
4544abscld 14023 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
4643, 45fsumrecl 14312 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
477adantr 480 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ)
4843, 44fsumabs 14374 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))))
49 reflcl 12459 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
5047, 49syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℝ)
51 1red 9934 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
5215adantlr 747 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
53 elfznn 12241 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
5453ssriv 3572 . . . . . . . . . . . . . . . . . . . 20 (1...(⌊‘𝑥)) ⊆ ℕ
5554a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ⊆ ℕ)
5655sselda 3568 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
5756, 17syl 17 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
5857zcnd 11359 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
5952, 58absmuld 14041 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) = ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))))
6052abscld 14023 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
6158abscld 14023 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
6252absge0d 14031 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
6358absge0d 14031 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(μ‘𝑛)))
64 simpl 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ+)
658nnrpd 11746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
66 rpdivcl 11732 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ℝ+𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
6764, 65, 66syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
683, 67sseldi 3566 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
6968, 12syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
70 flle 12462 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
7168, 70syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
7269, 68, 71abssubge0d 14018 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))))
73 fracle1 12466 . . . . . . . . . . . . . . . . . . 19 ((𝑥 / 𝑛) ∈ ℝ → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
7468, 73syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
7572, 74eqbrtrd 4605 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ 1)
76 mule1 24674 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
7756, 76syl 17 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
7860, 51, 61, 51, 62, 63, 75, 77lemul12ad 10845 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))) ≤ (1 · 1))
79 1t1e1 11052 . . . . . . . . . . . . . . . 16 (1 · 1) = 1
8078, 79syl6breq 4624 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))) ≤ 1)
8159, 80eqbrtrd 4605 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 1)
8243, 45, 51, 81fsumle 14372 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))1)
83 1cnd 9935 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 1 ∈ ℂ)
84 fsumconst 14364 . . . . . . . . . . . . . . 15 (((1...(⌊‘𝑥)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = ((#‘(1...(⌊‘𝑥))) · 1))
8543, 83, 84syl2anc 691 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = ((#‘(1...(⌊‘𝑥))) · 1))
86 flge1nn 12484 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
877, 86sylan 487 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
8887nnnn0d 11228 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
89 hashfz1 12996 . . . . . . . . . . . . . . . 16 ((⌊‘𝑥) ∈ ℕ0 → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
9088, 89syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
9190oveq1d 6564 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((#‘(1...(⌊‘𝑥))) · 1) = ((⌊‘𝑥) · 1))
9250recnd 9947 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℂ)
9392mulid1d 9936 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((⌊‘𝑥) · 1) = (⌊‘𝑥))
9485, 91, 933eqtrd 2648 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = (⌊‘𝑥))
9582, 94breqtrd 4609 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (⌊‘𝑥))
96 flle 12462 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
9747, 96syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ≤ 𝑥)
9846, 50, 47, 95, 97letrd 10073 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 𝑥)
9942, 46, 47, 48, 98letrd 10073 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 𝑥)
10033mulid1d 9936 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (𝑥 · 1) = 𝑥)
10199, 100breqtrrd 4611 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (𝑥 · 1))
102 1red 9934 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 1 ∈ ℝ)
10342, 102, 64ledivmuld 11801 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / 𝑥) ≤ 1 ↔ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (𝑥 · 1)))
104101, 103mpbird 246 . . . . . . . 8 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / 𝑥) ≤ 1)
10541, 104eqbrtrd 4605 . . . . . . 7 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ≤ 1)
106105adantl 481 . . . . . 6 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ≤ 1)
10731, 25, 1, 1, 106elo1d 14115 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
108 ax-1cn 9873 . . . . . . 7 1 ∈ ℂ
109 divrcnv 14423 . . . . . . 7 (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
110108, 109ax-mp 5 . . . . . 6 (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0
111 rlimo1 14195 . . . . . 6 ((𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0 → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ∈ 𝑂(1))
112110, 111mp1i 13 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ∈ 𝑂(1))
113 o1add 14192 . . . . 5 (((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∘𝑓 + (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) ∈ 𝑂(1))
114107, 112, 113syl2anc 691 . . . 4 (⊤ → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∘𝑓 + (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) ∈ 𝑂(1))
11530, 114eqeltrrd 2689 . . 3 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
116 ovex 6577 . . . 4 ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)) ∈ V
117116a1i 11 . . 3 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)) ∈ V)
11818zred 11358 . . . . . . 7 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
119118, 16nndivred 10946 . . . . . 6 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
120119recnd 9947 . . . . 5 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
1216, 120fsumcl 14311 . . . 4 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
122121adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
123121adantr 480 . . . . . 6 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
124123abscld 14023 . . . . 5 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ ℝ)
125120adantlr 747 . . . . . . . . . 10 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
12643, 33, 125fsummulc2 14358 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · ((μ‘𝑛) / 𝑛)))
12714, 19mulcld 9939 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) ∈ ℂ)
128127adantlr 747 . . . . . . . . . . 11 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) ∈ ℂ)
12943, 44, 128fsumadd 14317 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))))
13011adantlr 747 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
13114adantlr 747 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
132130, 131npcand 10275 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) + (⌊‘(𝑥 / 𝑛))) = (𝑥 / 𝑛))
133132oveq1d 6564 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) + (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) = ((𝑥 / 𝑛) · (μ‘𝑛)))
13452, 131, 58adddird 9944 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) + (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) = ((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))))
13533adantr 480 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
13656nnrpd 11746 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
137 rpcnne0 11726 . . . . . . . . . . . . . 14 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
138136, 137syl 17 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
139 div23 10583 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (μ‘𝑛)) / 𝑛) = ((𝑥 / 𝑛) · (μ‘𝑛)))
140 divass 10582 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (μ‘𝑛)) / 𝑛) = (𝑥 · ((μ‘𝑛) / 𝑛)))
141139, 140eqtr3d 2646 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 / 𝑛) · (μ‘𝑛)) = (𝑥 · ((μ‘𝑛) / 𝑛)))
142135, 58, 138, 141syl3anc 1318 . . . . . . . . . . . 12 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) · (μ‘𝑛)) = (𝑥 · ((μ‘𝑛) / 𝑛)))
143133, 134, 1423eqtr3d 2652 . . . . . . . . . . 11 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))) = (𝑥 · ((μ‘𝑛) / 𝑛)))
144143sumeq2dv 14281 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · ((μ‘𝑛) / 𝑛)))
145 eqidd 2611 . . . . . . . . . . . . 13 (𝑘 = (𝑛 · 𝑚) → (μ‘𝑛) = (μ‘𝑛))
146 ssrab2 3650 . . . . . . . . . . . . . . . 16 {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ ℕ
147 simprr 792 . . . . . . . . . . . . . . . 16 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
148146, 147sseldi 3566 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ ℕ)
149148, 17syl 17 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (μ‘𝑛) ∈ ℤ)
150149zcnd 11359 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (μ‘𝑛) ∈ ℂ)
151145, 47, 150dvdsflsumcom 24714 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (μ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛))
1521503impb 1252 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (μ‘𝑛) ∈ ℂ)
153152mulid1d 9936 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → ((μ‘𝑛) · 1) = (μ‘𝑛))
1541532sumeq2dv 14283 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((μ‘𝑛) · 1) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (μ‘𝑛))
155 eqidd 2611 . . . . . . . . . . . . . 14 (𝑘 = 1 → 1 = 1)
156 nnuz 11599 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
15787, 156syl6eleq 2698 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ‘1))
158 eluzfz1 12219 . . . . . . . . . . . . . . 15 ((⌊‘𝑥) ∈ (ℤ‘1) → 1 ∈ (1...(⌊‘𝑥)))
159157, 158syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → 1 ∈ (1...(⌊‘𝑥)))
160 1cnd 9935 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
161155, 43, 55, 159, 160musumsum 24718 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((μ‘𝑛) · 1) = 1)
162154, 161eqtr3d 2646 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (μ‘𝑛) = 1)
163 fzfid 12634 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
164 fsumconst 14364 . . . . . . . . . . . . . . 15 (((1...(⌊‘(𝑥 / 𝑛))) ∈ Fin ∧ (μ‘𝑛) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((#‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)))
165163, 58, 164syl2anc 691 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((#‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)))
166 rprege0 11723 . . . . . . . . . . . . . . . 16 ((𝑥 / 𝑛) ∈ ℝ+ → ((𝑥 / 𝑛) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑛)))
167 flge0nn0 12483 . . . . . . . . . . . . . . . 16 (((𝑥 / 𝑛) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑛)) → (⌊‘(𝑥 / 𝑛)) ∈ ℕ0)
168 hashfz1 12996 . . . . . . . . . . . . . . . 16 ((⌊‘(𝑥 / 𝑛)) ∈ ℕ0 → (#‘(1...(⌊‘(𝑥 / 𝑛)))) = (⌊‘(𝑥 / 𝑛)))
16967, 166, 167, 1684syl 19 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (#‘(1...(⌊‘(𝑥 / 𝑛)))) = (⌊‘(𝑥 / 𝑛)))
170169oveq1d 6564 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((#‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)) = ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
171165, 170eqtrd 2644 . . . . . . . . . . . . 13 (((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
172171sumeq2dv 14281 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
173151, 162, 1723eqtr3rd 2653 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) = 1)
174173oveq2d 6565 . . . . . . . . . 10 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1))
175129, 144, 1743eqtr3d 2652 . . . . . . . . 9 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · ((μ‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1))
176126, 175eqtrd 2644 . . . . . . . 8 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1))
177176oveq1d 6564 . . . . . . 7 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1) / 𝑥))
178123, 33, 34divcan3d 10685 . . . . . . 7 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛))
179 rpcnne0 11726 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
180179adantr 480 . . . . . . . 8 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
181 divdir 10589 . . . . . . . 8 ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)))
18232, 83, 180, 181syl3anc 1318 . . . . . . 7 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)))
183177, 178, 1823eqtr3d 2652 . . . . . 6 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)))
184183fveq2d 6107 . . . . 5 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))))
185 eqle 10018 . . . . 5 (((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ ℝ ∧ (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)))) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))))
186124, 184, 185syl2anc 691 . . . 4 ((𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))))
187186adantl 481 . . 3 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))))
1881, 115, 117, 122, 187o1le 14231 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1))
189188trud 1484 1 (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  ...cfz 12197  ⌊cfl 12453  #chash 12979  abscabs 13822   ⇝𝑟 crli 14064  𝑂(1)co1 14065  Σcsu 14264   ∥ cdvds 14821  μcmu 24621 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-o1 14069  df-lo1 14070  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-mu 24627 This theorem is referenced by:  mulogsumlem  25020  mulog2sumlem3  25025  selberglem1  25034
 Copyright terms: Public domain W3C validator