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Theorem mudivsum 27731

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.

Step	Hyp	Ref	Expression
1		1red 11227	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
2		reex 11209	. . . . . . 7 ⊢ ℝ ∈ V
3		rpssre 13042	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
4	2, 3	ssexi 5298	. . . . . 6 ⊢ ℝ⁺ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
6		fzfid 14029	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
7		rpre 13043	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
8		elfznn 13600	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
9		nndivre 12295	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
10	7, 8, 9	syl2an 608	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
11	10	recnd 11255	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
12		reflcl 13849	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
13	10, 12	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
14	13	recnd 11255	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
15	11, 14	subcld 11587	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
16	8	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
17		mucl 27342	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
18	16, 17	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
19	18	zcnd 12719	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
20	15, 19	mulcld 11247	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
21	6, 20	fsumcl 15810	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
22		rpcn 13045	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
23		rpne0 13051	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
24	21, 22, 23	divcld 12009	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) ∈ ℂ)
25	24	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) ∈ ℂ)
26		ovexd 7458	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
27		eqidd 2767	. . . . 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)))
28		eqidd 2767	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
29	5, 25, 26, 27, 28	offval2 7707	. . . 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))))
30	3	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
31	21	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
32	22	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℂ)
33	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ≠ 0)
34	31, 32, 33	absdivd 15535	. . . . . . . . 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)))
35		rprege0 13050	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
36		absid 15373	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘𝑥) = 𝑥)
38	37	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
39	38	oveq2d 7439	. . . . . . . . 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))
40	34, 39	eqtrd 2801	. . . . . . . 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))
41	31	abscld 15516	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
42		fzfid 14029	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ∈ Fin)
43	20	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
44	43	abscld 15516	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
45	42, 44	fsumrecl 15811	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
46	7	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ)
47	42, 43	fsumabs 15879	. . . . . . . . . . 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))))
48		reflcl 13849	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
49	46, 48	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℝ)
50		1red 11227	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
51	15	adantlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
52		fz1ssnn 13602	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...(⌊‘𝑥)) ⊆ ℕ
53	52	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ⊆ ℕ)
54	53	sselda 3940	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
55	54, 17	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
56	55	zcnd 12719	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
57	51, 56	absmuld 15534	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) = ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))))
58	51	abscld 15516	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
59	56	abscld 15516	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
60	51	absge0d 15524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
61	56	absge0d 15524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(μ‘𝑛)))
62		simpl 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ⁺)
63	8	nnrpd 13076	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
64		rpdivcl 13061	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
65	62, 63, 64	syl2an 608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
66	3, 65	sselid 3938	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
67	66, 12	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
68		flle 13852	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
69	66, 68	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
70	67, 66, 69	abssubge0d 15511	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))))
71		fracle1 13856	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 / 𝑛) ∈ ℝ → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
72	66, 71	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
73	70, 72	eqbrtrd 5138	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ 1)
74		mule1 27349	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
75	54, 74	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
76	58, 50, 59, 50, 60, 61, 73, 75	lemul12ad 12175	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))) ≤ (1 · 1))
77		1t1e1 12420	. . . . . . . . . . . . . . . 16 ⊢ (1 · 1) = 1
78	76, 77	breqtrdi 5157	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))) ≤ 1)
79	57, 78	eqbrtrd 5138	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 1)
80	42, 44, 50, 79	fsumle 15877	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))1)
81		1cnd 11220	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ ℂ)
82		fsumconst 15867	. . . . . . . . . . . . . . 15 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = ((♯‘(1...(⌊‘𝑥))) · 1))
83	42, 81, 82	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = ((♯‘(1...(⌊‘𝑥))) · 1))
84		flge1nn 13874	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
85	7, 84	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
86	85	nnnn0d 12583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
87		hashfz1 14402	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
88	86, 87	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
89	88	oveq1d 7438	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((♯‘(1...(⌊‘𝑥))) · 1) = ((⌊‘𝑥) · 1))
90	49	recnd 11255	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℂ)
91	90	mulridd 11244	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((⌊‘𝑥) · 1) = (⌊‘𝑥))
92	83, 89, 91	3eqtrd 2805	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = (⌊‘𝑥))
93	80, 92	breqtrd 5142	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (⌊‘𝑥))
94		flle 13852	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
95	46, 94	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ≤ 𝑥)
96	45, 49, 46, 93, 95	letrd 11385	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 𝑥)
97	41, 45, 46, 47, 96	letrd 11385	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 𝑥)
98	32	mulridd 11244	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 · 1) = 𝑥)
99	97, 98	breqtrrd 5144	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (𝑥 · 1))
100		1red 11227	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ ℝ)
101	41, 100, 62	ledivmuld 13131	. . . . . . . . 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)))
102	99, 101	mpbird 260	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / 𝑥) ≤ 1)
103	40, 102	eqbrtrd 5138	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ≤ 1)
104	103	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ≤ 1)
105	30, 25, 1, 1, 104	elo1d 15613	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
106		ax-1cn 11176	. . . . . . 7 ⊢ 1 ∈ ℂ
107		divrcnv 15932	. . . . . . 7 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
108	106, 107	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
109		rlimo1 15694	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
110	108, 109	mp1i 14	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
111		o1add 15691	. . . . 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))
112	105, 110, 111	syl2anc 596	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
113	29, 112	eqeltrrd 2867	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
114		ovexd 7458	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)) ∈ V)
115	18	zred 12718	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
116	115, 16	nndivred 12308	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
117	116	recnd 11255	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
118	6, 117	fsumcl 15810	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
119	118	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
120	118	adantr 486	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
121	120	abscld 15516	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ ℝ)
122	117	adantlr 728	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
123	42, 32, 122	fsummulc2 15861	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · ((μ‘𝑛) / 𝑛)))
124	14, 19	mulcld 11247	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) ∈ ℂ)
125	124	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) ∈ ℂ)
126	42, 43, 125	fsumadd 15817	. . . . . . . . . 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))))
127	11	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
128	14	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
129	127, 128	npcand 11591	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) + (⌊‘(𝑥 / 𝑛))) = (𝑥 / 𝑛))
130	129	oveq1d 7438	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) + (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) = ((𝑥 / 𝑛) · (μ‘𝑛)))
131	51, 128, 56	adddird 11252	. . . . . . . . . . . 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))))
132	32	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
133	54	nnrpd 13076	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
134		rpcnne0 13053	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
135	133, 134	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
136		div23 11909	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (μ‘𝑛)) / 𝑛) = ((𝑥 / 𝑛) · (μ‘𝑛)))
137		divass 11908	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (μ‘𝑛)) / 𝑛) = (𝑥 · ((μ‘𝑛) / 𝑛)))
138	136, 137	eqtr3d 2803	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 / 𝑛) · (μ‘𝑛)) = (𝑥 · ((μ‘𝑛) / 𝑛)))
139	132, 56, 135, 138	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) · (μ‘𝑛)) = (𝑥 · ((μ‘𝑛) / 𝑛)))
140	130, 131, 139	3eqtr3d 2809	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛))) = (𝑥 · ((μ‘𝑛) / 𝑛)))
141	140	sumeq2dv 15779	. . . . . . . . . 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)))
142		eqidd 2767	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 · 𝑚) → (μ‘𝑛) = (μ‘𝑛))
143		ssrab2 4037	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ
144		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
145	143, 144	sselid 3938	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ ℕ)
146	145, 17	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℤ)
147	146	zcnd 12719	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℂ)
148	142, 46, 147	dvdsflsumcom 27389	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (μ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛))
149	147	3impb 1132	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (μ‘𝑛) ∈ ℂ)
150	149	mulridd 11244	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → ((μ‘𝑛) · 1) = (μ‘𝑛))
151	150	2sumeq2dv 15782	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · 1) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (μ‘𝑛))
152		eqidd 2767	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → 1 = 1)
153		nnuz 12919	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
154	85, 153	eleqtrdi 2876	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
155		eluzfz1 13577	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝑥)))
156	154, 155	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ (1...(⌊‘𝑥)))
157		1cnd 11220	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
158	152, 42, 53, 156, 157	musumsum 27393	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · 1) = 1)
159	151, 158	eqtr3d 2803	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (μ‘𝑛) = 1)
160		fzfid 14029	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
161		fsumconst 15867	. . . . . . . . . . . . . . 15 ⊢ (((1...(⌊‘(𝑥 / 𝑛))) ∈ Fin ∧ (μ‘𝑛) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((♯‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)))
162	160, 56, 161	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((♯‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)))
163		rprege0 13050	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → ((𝑥 / 𝑛) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑛)))
164		flge0nn0 13873	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑛)) → (⌊‘(𝑥 / 𝑛)) ∈ ℕ₀)
165		hashfz1 14402	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(𝑥 / 𝑛)) ∈ ℕ₀ → (♯‘(1...(⌊‘(𝑥 / 𝑛)))) = (⌊‘(𝑥 / 𝑛)))
166	65, 163, 164, 165	4syl 20	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (♯‘(1...(⌊‘(𝑥 / 𝑛)))) = (⌊‘(𝑥 / 𝑛)))
167	166	oveq1d 7438	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((♯‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)) = ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
168	162, 167	eqtrd 2801	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
169	168	sumeq2dv 15779	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
170	148, 159, 169	3eqtr3rd 2810	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) = 1)
171	170	oveq2d 7439	. . . . . . . . . 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))
172	126, 141, 171	3eqtr3d 2809	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · ((μ‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1))
173	123, 172	eqtrd 2801	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1))
174	173	oveq1d 7438	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) + 1) / 𝑥))
175	120, 32, 33	divcan3d 12014	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛))
176		rpcnne0 13053	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
177	176	adantr 486	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
178		divdir 11915	. . . . . . . 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)))
179	31, 81, 177, 178	syl3anc 1398	. . . . . . 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)))
180	174, 175, 179	3eqtr3d 2809	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)))
181	180	fveq2d 6892	. . . . 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))))
182	121, 181	eqled 11331	. . . 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))))
183	182	adantl 487	. . 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))))
184	1, 113, 114, 119, 183	o1le 15730	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1))
185	184	mptru 1577	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2146 ≠ wne 2961 {crab 3419 Vcvv 3458 ⊆ wss 3908 class class class wbr 5114 ↦ cmpt 5197 ‘cfv 6543 (class class class)co 7423 ∘_f cof 7685 Fincfn 8952 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 ≤ cle 11262 − cmin 11459 / cdiv 11889 ℕcn 12251 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ℝ⁺crp 13034 ...cfz 13553 ⌊cfl 13843 ♯chash 14386 abscabs 15311 ⇝_𝑟 crli 15562 𝑂(1)co1 15563 Σcsu 15763 ∥ cdvds 16335 μcmu 27296

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-disj 5082 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-q 12991 df-rp 13035 df-ico 13396 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566 df-o1 15567 df-lo1 15568 df-sum 15764 df-dvds 16336 df-gcd 16578 df-prm 16755 df-pc 16922 df-mu 27302

This theorem is used by: mulogsumlem 27732 mulog2sumlem3 27737 selberglem1 27746

W3C validator