Theorem vmalogdivsum2 27474

Description: The sum Σ𝑛 ≤ 𝑥, Λ(𝑛)log(𝑥 / 𝑛) / 𝑛 is asymptotic to log↑2(𝑥) / 2 + 𝑂(log𝑥). Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)

Assertion

Ref	Expression
vmalogdivsum2	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

Distinct variable group:   𝑥,𝑛

Proof of Theorem vmalogdivsum2

Dummy variables 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13877	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 13450	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
3	2	adantl 481	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
4	3	nnrpd 12929	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ+)
5	4	relogcld 26557	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (log‘𝑘) ∈ ℝ)
6	5, 3	nndivred 12176	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) ∈ ℝ)
7	1, 6	fsumrecl 15638	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℝ)
8	7	recnd 11137	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℂ)
9		elioore 13272	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
10	9	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
11		1rp 12891	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
13		1red 11110	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
14		eliooord 13302	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
15	14	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
16	15	simpld 494	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
17	13, 10, 16	ltled 11258	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
18	10, 12, 17	rpgecld 12970	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
19	18	relogcld 26557	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
20	19	resqcld 14029	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℝ)
21	20	rehalfcld 12365	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℝ)
22	21	recnd 11137	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℂ)
23	19	recnd 11137	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
24	10, 16	rplogcld 26563	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
25	24	rpne0d 12936	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
26	8, 22, 23, 25	divsubdird 11933	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) / (log‘𝑥)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((((log‘𝑥)↑2) / 2) / (log‘𝑥))))
27	7, 21	resubcld 11542	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℝ)
28	27	recnd 11137	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℂ)
29	28, 23, 25	divrecd 11897	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) / (log‘𝑥)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥))))
30	20	recnd 11137	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℂ)
31		2cnd 12200	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
32		2ne0 12226	. . . . . . . . . 10 ⊢ 2 ≠ 0
33	32	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ≠ 0)
34	30, 31, 23, 33, 25	divdiv32d 11919	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((((log‘𝑥)↑2) / (log‘𝑥)) / 2))
35	23	sqvald 14047	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) = ((log‘𝑥) · (log‘𝑥)))
36	35	oveq1d 7361	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)))
37	23, 23, 25	divcan3d 11899	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)) = (log‘𝑥))
38	36, 37	eqtrd 2766	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (log‘𝑥))
39	38	oveq1d 7361	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / (log‘𝑥)) / 2) = ((log‘𝑥) / 2))
40	34, 39	eqtrd 2766	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((log‘𝑥) / 2))
41	40	oveq2d 7362	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((((log‘𝑥)↑2) / 2) / (log‘𝑥))) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)))
42	26, 29, 41	3eqtr3rd 2775	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥))))
43	42	mpteq2dva 5184	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥)))))
44	24	rprecred 12942	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
45	18	ex 412	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
46	45	ssrdv 3940	. . . . . 6 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ+)
47		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)))
48	47	logdivsum 27469	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))):ℝ+⟶ℝ ∧ (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟 ∧ (((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ⇝𝑟 1 ∧ 1 ∈ ℝ+ ∧ e ≤ 1) → (abs‘(((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)))‘1) − 1)) ≤ ((log‘1) / 1)))
49	48	simp2i 1140	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟
50		rlimdmo1 15522	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟 → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
51	49, 50	mp1i 13	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
52	46, 51	o1res2 15467	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
53		divlogrlim 26569	. . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
54		rlimo1 15521	. . . . . 6 ⊢ ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
55	53, 54	mp1i 13	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
56	27, 44, 52, 55	o1mul2 15529	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
57	43, 56	eqeltrd 2831	. . 3 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
58	8, 23, 25	divcld 11894	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) ∈ ℂ)
59	23	halfcld 12363	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
60	58, 59	subcld 11469	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
61		elfznn 13450	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
62	61	adantl 481	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
63		vmacl 27053	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
65	64, 62	nndivred 12176	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
66	18	adantr 480	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
67	62	nnrpd 12929	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
68	66, 67	rpdivcld 12948	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
69	68	relogcld 26557	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
70	65, 69	remulcld 11139	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
71	1, 70	fsumrecl 15638	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
72	71	recnd 11137	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
73	24	rpcnd 12933	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
74	72, 73, 25	divcld 11894	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
75	73	halfcld 12363	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
76	74, 75	subcld 11469	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
77	58, 74, 59	nnncan2d 11504	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
78	8, 72, 23, 25	divsubdird 11933	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
79		fzfid 13877	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
80	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℝ)
81	62	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℕ)
82		elfznn 13450	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
83	82	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
84	81, 83	nnmulcld 12175	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ∈ ℕ)
85	80, 84	nndivred 12176	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
86	79, 85	fsumrecl 15638	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
87	86	recnd 11137	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℂ)
88	70	recnd 11137	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
89	1, 87, 88	fsumsub 15692	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
90	64	recnd 11137	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
91	62	nncnd 12138	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
92	62	nnne0d 12172	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
93	90, 91, 92	divcld 11894	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
94	83	nnrecred 12173	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
95	79, 94	fsumrecl 15638	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
96	95	recnd 11137	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
97	69	recnd 11137	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
98	93, 96, 97	subdid 11570	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
99	90	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℂ)
100	91	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℂ)
101	83	nncnd 12138	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
102	92	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ≠ 0)
103	83	nnne0d 12172	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≠ 0)
104	99, 100, 101, 102, 103	divdiv1d 11925	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
105	99, 100, 102	divcld 11894	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
106	105, 101, 103	divrecd 11897	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
107	104, 106	eqtr3d 2768	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
108	107	sumeq2dv 15606	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
109	101, 103	reccld 11887	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℂ)
110	79, 93, 109	fsummulc2 15688	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
111	108, 110	eqtr4d 2769	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
112	111	oveq1d 7361	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
113	98, 112	eqtr4d 2769	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
114	113	sumeq2dv 15606	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
115		vmasum 27152	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (Λ‘𝑛) = (log‘𝑘))
116	3, 115	syl 17	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (Λ‘𝑛) = (log‘𝑘))
117	116	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (Λ‘𝑛) / 𝑘) = ((log‘𝑘) / 𝑘))
118		fzfid 13877	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
119		dvdsssfz1 16226	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ (1...𝑘))
120	3, 119	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ (1...𝑘))
121	118, 120	ssfid 9153	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ∈ Fin)
122	3	nncnd 12138	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℂ)
123		ssrab2 4030	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ
124		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
125	123, 124	sselid 3932	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ ℕ)
126	125, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (Λ‘𝑛) ∈ ℝ)
127	126	recnd 11137	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (Λ‘𝑛) ∈ ℂ)
128	127	anassrs 467	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (Λ‘𝑛) ∈ ℂ)
129	3	nnne0d 12172	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ≠ 0)
130	121, 122, 128, 129	fsumdivc 15690	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (Λ‘𝑛) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑛) / 𝑘))
131	117, 130	eqtr3d 2768	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑛) / 𝑘))
132	131	sumeq2dv 15606	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑛) / 𝑘))
133		oveq2 7354	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 · 𝑚) → ((Λ‘𝑛) / 𝑘) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
134	2	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑘 ∈ ℕ)
135	134	nncnd 12138	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑘 ∈ ℂ)
136	134	nnne0d 12172	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑘 ≠ 0)
137	127, 135, 136	divcld 11894	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → ((Λ‘𝑛) / 𝑘) ∈ ℂ)
138	133, 10, 137	dvdsflsumcom 27123	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑛) / 𝑘) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)))
139	132, 138	eqtrd 2766	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)))
140	139	oveq1d 7361	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
141	89, 114, 140	3eqtr4rd 2777	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
142	141	oveq1d 7361	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
143	77, 78, 142	3eqtr2d 2772	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
144	143	mpteq2dva 5184	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))))
145		1red 11110	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ)
146	1, 65	fsumrecl 15638	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
147	146, 24	rerpdivcld 12962	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ)
148		ioossre 13304	. . . . . . . . . . 11 ⊢ (1(,)+∞) ⊆ ℝ
149		ax-1cn 11061	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
150		o1const 15524	. . . . . . . . . . 11 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
151	148, 149, 150	mp2an 692	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)
152	151	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
153	147	recnd 11137	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ)
154	12	rpcnd 12933	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
155	146	recnd 11137	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
156	155, 23, 23, 25	divsubdird 11933	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))))
157	155, 23	subcld 11469	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
158	157, 23, 25	divrecd 11897	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
159	23, 25	dividd 11892	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
160	159	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))
161	156, 158, 160	3eqtr3rd 2775	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
162	161	mpteq2dva 5184	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))))
163	146, 19	resubcld 11542	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
164		vmadivsum 27418	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
165	164	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
166	46, 165	o1res2 15467	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
167	163, 44, 166, 55	o1mul2 15529	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
168	162, 167	eqeltrd 2831	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈ 𝑂(1))
169	153, 154, 168	o1dif 15534	. . . . . . . . 9 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)))
170	152, 169	mpbird 257	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1))
171	147, 170	o1lo1d 15443	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ ≤𝑂(1))
172	95, 69	resubcld 11542	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
173	65, 172	remulcld 11139	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
174	1, 173	fsumrecl 15638	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
175	174, 24	rerpdivcld 12962	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ∈ ℝ)
176		1red 11110	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
177		vmage0 27056	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
178	62, 177	syl 17	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
179	64, 67, 178	divge0d 12971	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
180	68	rpred 12931	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
181	91	mullidd 11127	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
182		fznnfl 13763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
183	10, 182	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
184	183	simplbda 499	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
185	181, 184	eqbrtrd 5113	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
186	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
187	176, 186, 67	lemuldivd 12980	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
188	185, 187	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
189		harmonicubnd 26945	. . . . . . . . . . . . . 14 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
190	180, 188, 189	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
191	95, 69, 176	lesubadd2d 11713	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ≤ 1 ↔ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1)))
192	190, 191	mpbird 257	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ≤ 1)
193	172, 176, 65, 179, 192	lemul2ad 12059	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ (((Λ‘𝑛) / 𝑛) · 1))
194	93	mulridd 11126	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 1) = ((Λ‘𝑛) / 𝑛))
195	193, 194	breqtrd 5117	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) / 𝑛))
196	1, 173, 65, 195	fsumle 15703	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
197	174, 146, 24, 196	lediv1dd 12989	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))
198	197	adantrr 717	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))
199	145, 171, 147, 175, 198	lo1le 15556	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ ≤𝑂(1))
200		0red 11112	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ)
201		harmoniclbnd 26944	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ+ → (log‘(𝑥 / 𝑛)) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚))
202	68, 201	syl 17	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚))
203	95, 69	subge0d 11704	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ↔ (log‘(𝑥 / 𝑛)) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
204	202, 203	mpbird 257	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))
205	65, 172, 179, 204	mulge0d 11691	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
206	1, 173, 205	fsumge0 15699	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
207	174, 24, 206	divge0d 12971	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
208	175, 200, 207	o1lo12 15442	. . . . . 6 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ ≤𝑂(1)))
209	199, 208	mpbird 257	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1))
210	144, 209	eqeltrd 2831	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
211	60, 76, 210	o1dif 15534	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)))
212	57, 211	mpbid 232	. 2 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
213	212	mptru 1548	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ⊤wtru 1542   ∈ wcel 2111   ≠ wne 2928  {crab 3395   ⊆ wss 3902   class class class wbr 5091   ↦ cmpt 5172  dom cdm 5616  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℂcc 11001  ℝcr 11002  0cc0 11003  1c1 11004   + caddc 11006   · cmul 11008  +∞cpnf 11140   < clt 11143   ≤ cle 11144   − cmin 11341   / cdiv 11771  ℕcn 12122  2c2 12177  ℝ⁺crp 12887  (,)cioo 13242  ...cfz 13404  ⌊cfl 13691  ↑cexp 13965  abscabs 15138   ⇝_𝑟 crli 15389  𝑂(1)co1 15390  ≤𝑂(1)clo1 15391  Σcsu 15590  eceu 15966   ∥ cdvds 16160  logclog 26488  Λcvma 27027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080  ax-pre-sup 11081  ax-addf 11082

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-fi 9295  df-sup 9326  df-inf 9327  df-oi 9396  df-dju 9791  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-div 11772  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-xnn0 12452  df-z 12466  df-dec 12586  df-uz 12730  df-q 12844  df-rp 12888  df-xneg 13008  df-xadd 13009  df-xmul 13010  df-ioo 13246  df-ioc 13247  df-ico 13248  df-icc 13249  df-fz 13405  df-fzo 13552  df-fl 13693  df-mod 13771  df-seq 13906  df-exp 13966  df-fac 14178  df-bc 14207  df-hash 14235  df-shft 14971  df-cj 15003  df-re 15004  df-im 15005  df-sqrt 15139  df-abs 15140  df-limsup 15375  df-clim 15392  df-rlim 15393  df-o1 15394  df-lo1 15395  df-sum 15591  df-ef 15971  df-e 15972  df-sin 15973  df-cos 15974  df-tan 15975  df-pi 15976  df-dvds 16161  df-gcd 16403  df-prm 16580  df-pc 16746  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-plusg 17171  df-mulr 17172  df-starv 17173  df-sca 17174  df-vsca 17175  df-ip 17176  df-tset 17177  df-ple 17178  df-ds 17180  df-unif 17181  df-hom 17182  df-cco 17183  df-rest 17323  df-topn 17324  df-0g 17342  df-gsum 17343  df-topgen 17344  df-pt 17345  df-prds 17348  df-xrs 17403  df-qtop 17408  df-imas 17409  df-xps 17411  df-mre 17485  df-mrc 17486  df-acs 17488  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-submnd 18689  df-mulg 18978  df-cntz 19227  df-cmn 19692  df-psmet 21281  df-xmet 21282  df-met 21283  df-bl 21284  df-mopn 21285  df-fbas 21286  df-fg 21287  df-cnfld 21290  df-top 22807  df-topon 22824  df-topsp 22846  df-bases 22859  df-cld 22932  df-ntr 22933  df-cls 22934  df-nei 23011  df-lp 23049  df-perf 23050  df-cn 23140  df-cnp 23141  df-haus 23228  df-cmp 23300  df-tx 23475  df-hmeo 23668  df-fil 23759  df-fm 23851  df-flim 23852  df-flf 23853  df-xms 24233  df-ms 24234  df-tms 24235  df-cncf 24796  df-limc 25792  df-dv 25793  df-ulm 26311  df-log 26490  df-cxp 26491  df-atan 26802  df-em 26928  df-cht 27032  df-vma 27033  df-chp 27034  df-ppi 27035

This theorem is referenced by:  vmalogdivsum  27475  2vmadivsumlem  27476  selberg4lem1  27496