Theorem vmalogdivsum 27871

Description: The sum Σ𝑛 ≤ 𝑥, Λ(𝑛)log𝑛 / 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
vmalogdivsum	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

Distinct variable group:   𝑥,𝑛

Proof of Theorem vmalogdivsum

Step	Hyp	Ref	Expression
1		elioore 13509	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 13127	. . . . . . . 8 ⊢ 1 ∈ ℝ+
4	3	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
5		1red 11312	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13539	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	adantl 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
8	7	simpld 500	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
9	5, 2, 8	ltled 11461	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13206	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
11	10	ex 418	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
12	11	ssrdv 3914	. . . 4 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ+)
13		vmadivsum 27814	. . . . 5 ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
14	13	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
15	12, 14	o1res2 15735	. . 3 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
16		fzfid 14119	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
17		elfznn 13690	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
18	17	adantl 487	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
19		vmacl 27450	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
20	18, 19	syl 18	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
21	20, 18	nndivred 12395	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
22	21	recnd 11340	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
23	16, 22	fsumcl 15904	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
24	10	relogcld 26956	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
25	24	recnd 11340	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
26	23, 25	subcld 11672	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
27	18	nnrpd 13165	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
28	27	relogcld 26956	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
29	21, 28	remulcld 11342	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℝ)
30	16, 29	fsumrecl 15905	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℝ)
31	2, 8	rplogcld 26962	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
32	30, 31	rerpdivcld 13198	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
33	24	rehalfcld 12596	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ)
34	32, 33	resubcld 11747	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ)
35	34	recnd 11340	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
36	33	recnd 11340	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
37	23, 36	subcld 11672	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) ∈ ℂ)
38	32	recnd 11340	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℂ)
39	37, 38, 36	nnncan2d 11707	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
40	23, 36, 36	subsub4d 11703	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (((log‘𝑥) / 2) + ((log‘𝑥) / 2))))
41	25	2halvesd 12595	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / 2) + ((log‘𝑥) / 2)) = (log‘𝑥))
42	41	oveq2d 7436	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (((log‘𝑥) / 2) + ((log‘𝑥) / 2))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
43	40, 42	eqtrd 2795	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
44	43	oveq1d 7435	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))))
45	23, 36, 38	sub32d 11704	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − ((log‘𝑥) / 2)))
46	10	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
47	46	relogcld 26956	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ)
48	21, 47	remulcld 11342	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℝ)
49	48	recnd 11340	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℂ)
50	29	recnd 11340	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℂ)
51	16, 49, 50	fsumsub 15959	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
52	46, 27	relogdivd 26959	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) = ((log‘𝑥) − (log‘𝑛)))
53	52	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = (((Λ‘𝑛) / 𝑛) · ((log‘𝑥) − (log‘𝑛))))
54	25	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈ ℂ)
55	28	recnd 11340	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
56	22, 54, 55	subdid 11775	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · ((log‘𝑥) − (log‘𝑛))) = ((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
57	53, 56	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = ((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
58	57	sumeq2dv 15874	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
59	20	recnd 11340	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
60	18	nncnd 12354	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
61	18	nnne0d 12391	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
62	59, 60, 61	divcld 12096	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
63	16, 25, 62	fsummulc1 15956	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)))
64	63	oveq1d 7435	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
65	51, 58, 64	3eqtr4d 2805	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
66	65	oveq1d 7435	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) / (log‘𝑥)))
67	23, 25	mulcld 11332	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℂ)
68	30	recnd 11340	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℂ)
69	31	rpne0d 13172	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
70	67, 68, 25, 69	divsubdird 12135	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
71	23, 25, 69	divcan4d 12102	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
72	71	oveq1d 7435	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
73	66, 70, 72	3eqtrd 2799	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
74	73	oveq1d 7435	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − ((log‘𝑥) / 2)))
75	45, 74	eqtr4d 2798	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))
76	39, 44, 75	3eqtr3d 2803	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))
77	76	mpteq2dva 5197	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))))
78		vmalogdivsum2 27870	. . . . 5 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
79	77, 78	eqeltrdi 2868	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
80	26, 35, 79	o1dif 15802	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)))
81	15, 80	mpbid 235	. 2 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
82	81	mptru 1576	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

This proof depends on syntax axioms:   ∧ wa 401  ⊤wtru 1570   ∈ wcel 2144   class class class wbr 5102   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420  ℂcc 11201  ℝcr 11202  1c1 11204   + caddc 11206   · cmul 11208  +∞cpnf 11343   < clt 11346   − cmin 11544   / cdiv 11976  ℕcn 12338  2c2 12400  ℝ⁺crp 13123  (,)cioo 13479  ...cfz 13642  ⌊cfl 13933  𝑂(1)co1 15658  Σcsu 15858  logclog 26887  Λcvma 27424

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281  ax-addf 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8180  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  df-er 8720  df-map 8852  df-pm 8853  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fsupp 9357  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9507  df-dju 9985  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-xnn0 12683  df-z 12697  df-dec 12818  df-uz 12969  df-q 13079  df-rp 13124  df-xneg 13244  df-xadd 13245  df-xmul 13246  df-ioo 13483  df-ioc 13484  df-ico 13485  df-icc 13486  df-fz 13643  df-fzo 13792  df-fl 13935  df-mod 14013  df-seq 14148  df-exp 14208  df-fac 14421  df-bc 14450  df-hash 14478  df-shft 15225  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-limsup 15643  df-clim 15660  df-rlim 15661  df-o1 15662  df-lo1 15663  df-sum 15859  df-ef 16238  df-e 16239  df-sin 16240  df-cos 16241  df-tan 16242  df-pi 16243  df-dvds 16428  df-gcd 16670  df-prm 16852  df-pc 17020  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-starv 17448  df-sca 17449  df-vsca 17450  df-ip 17451  df-tset 17452  df-ple 17453  df-ds 17455  df-unif 17456  df-hom 17457  df-cco 17458  df-rest 17598  df-topn 17599  df-0g 17617  df-gsum 17618  df-topgen 17619  df-pt 17620  df-prds 17623  df-xrs 17679  df-qtop 17684  df-imas 17685  df-xps 17687  df-mre 17761  df-mrc 17762  df-acs 17764  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-submnd 18984  df-mulg 19283  df-cntz 19536  df-cmn 20001  df-psmet 21675  df-xmet 21676  df-met 21677  df-bl 21678  df-mopn 21679  df-fbas 21680  df-fg 21681  df-cnfld 21684  df-top 23217  df-topon 23234  df-topsp 23256  df-bases 23269  df-cld 23342  df-ntr 23343  df-cls 23344  df-nei 23421  df-lp 23459  df-perf 23460  df-cn 23550  df-cnp 23551  df-haus 23638  df-cmp 23710  df-tx 23886  df-hmeo 24079  df-fil 24170  df-fm 24262  df-flim 24263  df-flf 24264  df-xms 24644  df-ms 24645  df-tms 24646  df-cncf 25204  df-limc 26191  df-dv 26192  df-ulm 26709  df-log 26889  df-cxp 26890  df-atan 27200  df-em 27325  df-cht 27429  df-vma 27430  df-chp 27431  df-ppi 27432

This theorem is used by:  selberg3r  27901