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Theorem mudivsum 26583

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 10907	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
2		reex 10893	. . . . . . 7 ⊢ ℝ ∈ V
3		rpssre 12666	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
4	2, 3	ssexi 5241	. . . . . 6 ⊢ ℝ⁺ ∈ V
5	4	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
6		fzfid 13621	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
7		rpre 12667	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
8		elfznn 13214	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
9		nndivre 11944	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
10	7, 8, 9	syl2an 595	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
11	10	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
12		reflcl 13444	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
13	10, 12	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
14	13	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
15	11, 14	subcld 11262	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
16	8	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
17		mucl 26195	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
19	18	zcnd 12356	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
20	15, 19	mulcld 10926	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
21	6, 20	fsumcl 15373	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
22		rpcn 12669	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
23		rpne0 12675	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
24	21, 22, 23	divcld 11681	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) ∈ ℂ)
25	24	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) ∈ ℂ)
26		ovexd 7290	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ V)
27		eqidd 2739	. . . . 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 2739	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
29	5, 25, 26, 27, 28	offval2 7531	. . . 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 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
32	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℂ)
33	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ≠ 0)
34	31, 32, 33	absdivd 15095	. . . . . . . . 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 12674	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
36		absid 14936	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘𝑥) = 𝑥)
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
39	38	oveq2d 7271	. . . . . . . . 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 2778	. . . . . . . 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 15076	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
42		fzfid 13621	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ∈ Fin)
43	20	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) ∈ ℂ)
44	43	abscld 15076	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
45	42, 44	fsumrecl 15374	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ∈ ℝ)
46	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ)
47	42, 43	fsumabs 15441	. . . . . . . . . . 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 13444	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
49	46, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℝ)
50		1red 10907	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
51	15	adantlr 711	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
52		fz1ssnn 13216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...(⌊‘𝑥)) ⊆ ℕ
53	52	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (1...(⌊‘𝑥)) ⊆ ℕ)
54	53	sselda 3917	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
55	54, 17	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
56	55	zcnd 12356	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
57	51, 56	absmuld 15094	. . . . . . . . . . . . . . 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 15076	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
59	56	abscld 15076	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
60	51	absge0d 15084	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
61	56	absge0d 15084	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(μ‘𝑛)))
62		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 𝑥 ∈ ℝ⁺)
63	8	nnrpd 12699	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
64		rpdivcl 12684	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
65	62, 63, 64	syl2an 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
66	3, 65	sselid 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
67	66, 12	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
68		flle 13447	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
69	66, 68	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
70	67, 66, 69	abssubge0d 15071	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))))
71		fracle1 13451	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 / 𝑛) ∈ ℝ → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
72	66, 71	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
73	70, 72	eqbrtrd 5092	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ 1)
74		mule1 26202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
75	54, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
76	58, 50, 59, 50, 60, 61, 73, 75	lemul12ad 11847	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))) ≤ (1 · 1))
77		1t1e1 12065	. . . . . . . . . . . . . . . 16 ⊢ (1 · 1) = 1
78	76, 77	breqtrdi 5111	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) · (abs‘(μ‘𝑛))) ≤ 1)
79	57, 78	eqbrtrd 5092	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 1)
80	42, 44, 50, 79	fsumle 15439	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))1)
81		1cnd 10901	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ ℂ)
82		fsumconst 15430	. . . . . . . . . . . . . . 15 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = ((♯‘(1...(⌊‘𝑥))) · 1))
83	42, 81, 82	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = ((♯‘(1...(⌊‘𝑥))) · 1))
84		flge1nn 13469	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
85	7, 84	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
86	85	nnnn0d 12223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
87		hashfz1 13988	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
88	86, 87	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
89	88	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((♯‘(1...(⌊‘𝑥))) · 1) = ((⌊‘𝑥) · 1))
90	49	recnd 10934	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℂ)
91	90	mulid1d 10923	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((⌊‘𝑥) · 1) = (⌊‘𝑥))
92	83, 89, 91	3eqtrd 2782	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))1 = (⌊‘𝑥))
93	80, 92	breqtrd 5096	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (⌊‘𝑥))
94		flle 13447	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
95	46, 94	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ≤ 𝑥)
96	45, 49, 46, 93, 95	letrd 11062	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 𝑥)
97	41, 45, 46, 47, 96	letrd 11062	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ 𝑥)
98	32	mulid1d 10923	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 · 1) = 𝑥)
99	97, 98	breqtrrd 5098	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) ≤ (𝑥 · 1))
100		1red 10907	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ ℝ)
101	41, 100, 62	ledivmuld 12754	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛))) / 𝑥) ≤ 1)
103	40, 102	eqbrtrd 5092	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ≤ 1)
104	103	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ≤ 1)
105	30, 25, 1, 1, 104	elo1d 15173	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
106		ax-1cn 10860	. . . . . . 7 ⊢ 1 ∈ ℂ
107		divrcnv 15492	. . . . . . 7 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
108	106, 107	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
109		rlimo1 15254	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
110	108, 109	mp1i 13	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
111		o1add 15251	. . . . 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 583	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
113	29, 112	eqeltrrd 2840	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
114		ovexd 7290	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) · (μ‘𝑛)) / 𝑥) + (1 / 𝑥)) ∈ V)
115	18	zred 12355	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
116	115, 16	nndivred 11957	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
117	116	recnd 10934	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
118	6, 117	fsumcl 15373	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
119	118	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
120	118	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ ℂ)
121	120	abscld 15076	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ ℝ)
122	117	adantlr 711	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
123	42, 32, 122	fsummulc2 15424	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑥 · ((μ‘𝑛) / 𝑛)))
124	14, 19	mulcld 10926	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) ∈ ℂ)
125	124	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) ∈ ℂ)
126	42, 43, 125	fsumadd 15380	. . . . . . . . . 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 711	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
128	14	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
129	127, 128	npcand 11266	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) + (⌊‘(𝑥 / 𝑛))) = (𝑥 / 𝑛))
130	129	oveq1d 7270	. . . . . . . . . . . 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 10931	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
133	54	nnrpd 12699	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
134		rpcnne0 12677	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
135	133, 134	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
136		div23 11582	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (μ‘𝑛)) / 𝑛) = ((𝑥 / 𝑛) · (μ‘𝑛)))
137		divass 11581	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (μ‘𝑛)) / 𝑛) = (𝑥 · ((μ‘𝑛) / 𝑛)))
138	136, 137	eqtr3d 2780	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ (μ‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 / 𝑛) · (μ‘𝑛)) = (𝑥 · ((μ‘𝑛) / 𝑛)))
139	132, 56, 135, 138	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) · (μ‘𝑛)) = (𝑥 · ((μ‘𝑛) / 𝑛)))
140	130, 131, 139	3eqtr3d 2786	. . . . . . . . . . 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 15343	. . . . . . . . . 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 2739	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑛 · 𝑚) → (μ‘𝑛) = (μ‘𝑛))
143		ssrab2 4009	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ
144		simprr 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
145	143, 144	sselid 3915	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ ℕ)
146	145, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℤ)
147	146	zcnd 12356	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (μ‘𝑛) ∈ ℂ)
148	142, 46, 147	dvdsflsumcom 26242	. . . . . . . . . . . 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 1113	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (μ‘𝑛) ∈ ℂ)
150	149	mulid1d 10923	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → ((μ‘𝑛) · 1) = (μ‘𝑛))
151	150	2sumeq2dv 15345	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → 1 = 1)
153		nnuz 12550	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
154	85, 153	eleqtrdi 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
155		eluzfz1 13192	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝑥)))
156	154, 155	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → 1 ∈ (1...(⌊‘𝑥)))
157		1cnd 10901	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
158	152, 42, 53, 156, 157	musumsum 26246	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((μ‘𝑛) · 1) = 1)
159	151, 158	eqtr3d 2780	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (μ‘𝑛) = 1)
160		fzfid 13621	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
161		fsumconst 15430	. . . . . . . . . . . . . . 15 ⊢ (((1...(⌊‘(𝑥 / 𝑛))) ∈ Fin ∧ (μ‘𝑛) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((♯‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)))
162	160, 56, 161	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((♯‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)))
163		rprege0 12674	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → ((𝑥 / 𝑛) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑛)))
164		flge0nn0 13468	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ 0 ≤ (𝑥 / 𝑛)) → (⌊‘(𝑥 / 𝑛)) ∈ ℕ₀)
165		hashfz1 13988	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(𝑥 / 𝑛)) ∈ ℕ₀ → (♯‘(1...(⌊‘(𝑥 / 𝑛)))) = (⌊‘(𝑥 / 𝑛)))
166	65, 163, 164, 165	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (♯‘(1...(⌊‘(𝑥 / 𝑛)))) = (⌊‘(𝑥 / 𝑛)))
167	166	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((♯‘(1...(⌊‘(𝑥 / 𝑛)))) · (μ‘𝑛)) = ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
168	162, 167	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(μ‘𝑛) = ((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)))
169	168	sumeq2dv 15343	. . . . . . . . . . . 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 2787	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))((⌊‘(𝑥 / 𝑛)) · (μ‘𝑛)) = 1)
171	170	oveq2d 7271	. . . . . . . . . 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 2786	. . . . . . . . 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 2778	. . . . . . . 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 7270	. . . . . . 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 11686	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → ((𝑥 · Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛))
176		rpcnne0 12677	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
177	176	adantr 480	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
178		divdir 11588	. . . . . . . 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 1369	. . . . . . 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 2786	. . . . . 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 6760	. . . . 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 11008	. . . 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 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))))
184	1, 113, 114, 119, 183	o1le 15292	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1))
185	184	mptru 1546	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1085 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 {crab 3067 Vcvv 3422 ⊆ wss 3883 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 Fincfn 8691 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ...cfz 13168 ⌊cfl 13438 ♯chash 13972 abscabs 14873 ⇝_𝑟 crli 15122 𝑂(1)co1 15123 Σcsu 15325 ∥ cdvds 15891 μcmu 26149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-ico 13014 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-o1 15127 df-lo1 15128 df-sum 15326 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-mu 26155

This theorem is referenced by: mulogsumlem 26584 mulog2sumlem3 26589 selberglem1 26598

W3C validator