Theorem vmalogdivsum2 27526

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 13933	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 13505	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
3	2	adantl 482	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
4	3	nnrpd 12982	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ+)
5	4	relogcld 26612	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (log‘𝑘) ∈ ℝ)
6	5, 3	nndivred 12229	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) ∈ ℝ)
7	1, 6	fsumrecl 15694	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℝ)
8	7	recnd 11171	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℂ)
9		elioore 13326	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
10	9	adantl 482	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
11		1rp 12944	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ+
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
13		1red 11143	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
14		eliooord 13356	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
15	14	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
16	15	simpld 495	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
17	13, 10, 16	ltled 11292	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
18	10, 12, 17	rpgecld 13023	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
19	18	relogcld 26612	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
20	19	resqcld 14085	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℝ)
21	20	rehalfcld 12422	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℝ)
22	21	recnd 11171	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℂ)
23	19	recnd 11171	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
24	10, 16	rplogcld 26618	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
25	24	rpne0d 12989	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
26	8, 22, 23, 25	divsubdird 11968	. . . . . 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 11576	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℝ)
28	27	recnd 11171	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℂ)
29	28, 23, 25	divrecd 11932	. . . . . 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 11171	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℂ)
31		2cnd 12257	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
32		2ne0 12283	. . . . . . . . . 10 ⊢ 2 ≠ 0
33	32	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ≠ 0)
34	30, 31, 23, 33, 25	divdiv32d 11954	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((((log‘𝑥)↑2) / (log‘𝑥)) / 2))
35	23	sqvald 14103	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) = ((log‘𝑥) · (log‘𝑥)))
36	35	oveq1d 7378	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)))
37	23, 23, 25	divcan3d 11934	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)) = (log‘𝑥))
38	36, 37	eqtrd 2775	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (log‘𝑥))
39	38	oveq1d 7378	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / (log‘𝑥)) / 2) = ((log‘𝑥) / 2))
40	34, 39	eqtrd 2775	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((log‘𝑥) / 2))
41	40	oveq2d 7379	. . . . . 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 2784	. . . . 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 5172	. . . 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 12995	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
45	18	ex 413	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
46	45	ssrdv 3928	. . . . . 6 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ+)
47		eqid 2740	. . . . . . . . 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 27521	. . . . . . . 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 1146	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟
50		rlimdmo1 15578	. . . . . . 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 15523	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
53		divlogrlim 26624	. . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
54		rlimo1 15577	. . . . . 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 15585	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
57	43, 56	eqeltrd 2840	. . 3 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
58	8, 23, 25	divcld 11929	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) ∈ ℂ)
59	23	halfcld 12420	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
60	58, 59	subcld 11503	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
61		elfznn 13505	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
62	61	adantl 482	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
63		vmacl 27106	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
65	64, 62	nndivred 12229	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
66	18	adantr 481	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
67	62	nnrpd 12982	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
68	66, 67	rpdivcld 13001	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
69	68	relogcld 26612	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
70	65, 69	remulcld 11173	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
71	1, 70	fsumrecl 15694	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
72	71	recnd 11171	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
73	24	rpcnd 12986	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
74	72, 73, 25	divcld 11929	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
75	73	halfcld 12420	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
76	74, 75	subcld 11503	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
77	58, 74, 59	nnncan2d 11538	. . . . . . 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 11968	. . . . . . 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 13933	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
80	64	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℝ)
81	62	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℕ)
82		elfznn 13505	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
83	82	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
84	81, 83	nnmulcld 12228	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ∈ ℕ)
85	80, 84	nndivred 12229	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
86	79, 85	fsumrecl 15694	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
87	86	recnd 11171	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℂ)
88	70	recnd 11171	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
89	1, 87, 88	fsumsub 15748	. . . . . . . . 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 11171	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
91	62	nncnd 12188	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
92	62	nnne0d 12225	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
93	90, 91, 92	divcld 11929	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
94	83	nnrecred 12226	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
95	79, 94	fsumrecl 15694	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
96	95	recnd 11171	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
97	69	recnd 11171	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
98	93, 96, 97	subdid 11604	. . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℂ)
100	91	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℂ)
101	83	nncnd 12188	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
102	92	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ≠ 0)
103	83	nnne0d 12225	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≠ 0)
104	99, 100, 101, 102, 103	divdiv1d 11960	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
105	99, 100, 102	divcld 11929	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
106	105, 101, 103	divrecd 11932	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
107	104, 106	eqtr3d 2777	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
108	107	sumeq2dv 15662	. . . . . . . . . . . . 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 11922	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℂ)
110	79, 93, 109	fsummulc2 15744	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 7378	. . . . . . . . . . 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 2778	. . . . . . . . . 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 15662	. . . . . . . . 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 27204	. . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} (Λ‘𝑛) / 𝑘) = ((log‘𝑘) / 𝑘))
118		fzfid 13933	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
119		dvdsssfz1 16285	. . . . . . . . . . . . . . . 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 9176	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ∈ Fin)
122	3	nncnd 12188	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℂ)
123		ssrab2 4018	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ
124		simprr 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
125	123, 124	sselid 3920	. . . . . . . . . . . . . . . . 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 11171	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → (Λ‘𝑛) ∈ ℂ)
128	127	anassrs 468	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (Λ‘𝑛) ∈ ℂ)
129	3	nnne0d 12225	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ≠ 0)
130	121, 122, 128, 129	fsumdivc 15746	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑛) / 𝑘))
132	131	sumeq2dv 15662	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑛) / 𝑘))
133		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑛 · 𝑚) → ((Λ‘𝑛) / 𝑘) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
134	2	ad2antrl 734	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑘 ∈ ℕ)
135	134	nncnd 12188	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑘 ∈ ℂ)
136	134	nnne0d 12225	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → 𝑘 ≠ 0)
137	127, 135, 136	divcld 11929	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})) → ((Λ‘𝑛) / 𝑘) ∈ ℂ)
138	133, 10, 137	dvdsflsumcom 27176	. . . . . . . . . . 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 2775	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)))
140	139	oveq1d 7378	. . . . . . . . 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 2786	. . . . . . . 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 7378	. . . . . . 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 2781	. . . . . 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 5172	. . . . 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 11143	. . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ)
146	1, 65	fsumrecl 15694	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
147	146, 24	rerpdivcld 13015	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ)
148		ioossre 13358	. . . . . . . . . . 11 ⊢ (1(,)+∞) ⊆ ℝ
149		ax-1cn 11094	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
150		o1const 15580	. . . . . . . . . . 11 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
151	148, 149, 150	mp2an 698	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)
152	151	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
153	147	recnd 11171	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ)
154	12	rpcnd 12986	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
155	146	recnd 11171	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
156	155, 23, 23, 25	divsubdird 11968	. . . . . . . . . . . . 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 11503	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
158	157, 23, 25	divrecd 11932	. . . . . . . . . . . . 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 11927	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
160	159	oveq2d 7379	. . . . . . . . . . . . 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 2784	. . . . . . . . . . . 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 5172	. . . . . . . . . . 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 11576	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
164		vmadivsum 27470	. . . . . . . . . . . . . 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 15523	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
167	163, 44, 166, 55	o1mul2 15585	. . . . . . . . . . 11 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
168	162, 167	eqeltrd 2840	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈ 𝑂(1))
169	153, 154, 168	o1dif 15590	. . . . . . . . 9 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)))
170	152, 169	mpbird 258	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1))
171	147, 170	o1lo1d 15499	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ ≤𝑂(1))
172	95, 69	resubcld 11576	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
173	65, 172	remulcld 11173	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
174	1, 173	fsumrecl 15694	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
175	174, 24	rerpdivcld 13015	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ∈ ℝ)
176		1red 11143	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
177		vmage0 27109	. . . . . . . . . . . . . 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 13024	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
180	68	rpred 12984	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
181	91	mullidd 11161	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
182		fznnfl 13819	. . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
185	181, 184	eqbrtrd 5101	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
186	10	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
187	176, 186, 67	lemuldivd 13033	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
188	185, 187	mpbid 233	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
189		harmonicubnd 26998	. . . . . . . . . . . . . 14 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
190	180, 188, 189	syl2anc 590	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
191	95, 69, 176	lesubadd2d 11747	. . . . . . . . . . . . 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 258	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ≤ 1)
193	172, 176, 65, 179, 192	lemul2ad 12094	. . . . . . . . . . 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 11160	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 1) = ((Λ‘𝑛) / 𝑛))
195	193, 194	breqtrd 5105	. . . . . . . . . 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 15760	. . . . . . . . 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 13042	. . . . . . . 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 723	. . . . . . 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 15612	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ ≤𝑂(1))
200		0red 11145	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ)
201		harmoniclbnd 26997	. . . . . . . . . . . 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 11738	. . . . . . . . . . 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 258	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))
205	65, 172, 179, 204	mulge0d 11725	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
206	1, 173, 205	fsumge0 15756	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
207	174, 24, 206	divge0d 13024	. . . . . . 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 15498	. . . . . 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 258	. . . . 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 2840	. . . 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 15590	. . 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 233	. 2 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
213	212	mptru 1554	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119   ≠ wne 2935  {crab 3392   ⊆ wss 3890   class class class wbr 5079   ↦ cmpt 5160  dom cdm 5625  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  ℂcc 11034  ℝcr 11035  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041  +∞cpnf 11174   < clt 11177   ≤ cle 11178   − cmin 11375   / cdiv 11805  ℕcn 12172  2c2 12234  ℝ⁺crp 12940  (,)cioo 13296  ...cfz 13459  ⌊cfl 13747  ↑cexp 14021  abscabs 15194   ⇝_𝑟 crli 15445  𝑂(1)co1 15446  ≤𝑂(1)clo1 15447  Σcsu 15646  eceu 16025   ∥ cdvds 16219  logclog 26543  Λcvma 27080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114  ax-addf 11115

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-iin 4931  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-om 7814  df-1st 7938  df-2nd 7939  df-supp 8108  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-er 8640  df-map 8772  df-pm 8773  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9272  df-fi 9321  df-sup 9352  df-inf 9353  df-oi 9422  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-xnn0 12509  df-z 12523  df-dec 12643  df-uz 12787  df-q 12897  df-rp 12941  df-xneg 13061  df-xadd 13062  df-xmul 13063  df-ioo 13300  df-ioc 13301  df-ico 13302  df-icc 13303  df-fz 13460  df-fzo 13607  df-fl 13749  df-mod 13827  df-seq 13962  df-exp 14022  df-fac 14234  df-bc 14263  df-hash 14291  df-shft 15027  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-limsup 15431  df-clim 15448  df-rlim 15449  df-o1 15450  df-lo1 15451  df-sum 15647  df-ef 16030  df-e 16031  df-sin 16032  df-cos 16033  df-tan 16034  df-pi 16035  df-dvds 16220  df-gcd 16462  df-prm 16639  df-pc 16806  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-starv 17233  df-sca 17234  df-vsca 17235  df-ip 17236  df-tset 17237  df-ple 17238  df-ds 17240  df-unif 17241  df-hom 17242  df-cco 17243  df-rest 17383  df-topn 17384  df-0g 17402  df-gsum 17403  df-topgen 17404  df-pt 17405  df-prds 17408  df-xrs 17464  df-qtop 17469  df-imas 17470  df-xps 17472  df-mre 17546  df-mrc 17547  df-acs 17549  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-submnd 18750  df-mulg 19042  df-cntz 19290  df-cmn 19755  df-psmet 21346  df-xmet 21347  df-met 21348  df-bl 21349  df-mopn 21350  df-fbas 21351  df-fg 21352  df-cnfld 21355  df-top 22884  df-topon 22901  df-topsp 22923  df-bases 22936  df-cld 23009  df-ntr 23010  df-cls 23011  df-nei 23088  df-lp 23126  df-perf 23127  df-cn 23217  df-cnp 23218  df-haus 23305  df-cmp 23377  df-tx 23552  df-hmeo 23745  df-fil 23836  df-fm 23928  df-flim 23929  df-flf 23930  df-xms 24310  df-ms 24311  df-tms 24312  df-cncf 24870  df-limc 25858  df-dv 25859  df-ulm 26367  df-log 26545  df-cxp 26546  df-atan 26856  df-em 26981  df-cht 27085  df-vma 27086  df-chp 27087  df-ppi 27088

This theorem is referenced by:  vmalogdivsum  27527  2vmadivsumlem  27528  selberg4lem1  27548