Theorem selbergb 27475

Description: Convert eventual boundedness in selberg 27474 to boundedness on [1, +∞). (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 30-May-2016.)

Assertion

Ref	Expression
selbergb	⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐

Distinct variable group:   𝑛,𝑐,𝑥

Proof of Theorem selbergb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 11238	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 13448	. . . . . . 7 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
3	1, 2	mp1i 13	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
4	3	simprbda 498	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ)
5	4	ex 412	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ))
6	5	ssrdv 3984	. . 3 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
8		fzfid 13964	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
9		elfznn 13556	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
10	9	adantl 481	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
11		vmacl 27043	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
13	10	nnrpd 13040	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
14	13	relogcld 26550	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
15	4	adantr 480	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
16	15, 10	nndivred 12290	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
17		chpcl 27049	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
19	14, 18	readdcld 11267	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
20	12, 19	remulcld 11268	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
21	8, 20	fsumrecl 15706	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
22		1rp 13004	. . . . . . . 8 ⊢ 1 ∈ ℝ+
23	22	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
24	3	simplbda 499	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
25	4, 23, 24	rpgecld 13081	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ+)
26	21, 25	rerpdivcld 13073	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
27		2re 12310	. . . . . . 7 ⊢ 2 ∈ ℝ
28	27	a1i 11	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 2 ∈ ℝ)
29	25	relogcld 26550	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (log‘𝑥) ∈ ℝ)
30	28, 29	remulcld 11268	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
31	26, 30	resubcld 11666	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
32	31	recnd 11266	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
33	25	ex 412	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ+))
34	33	ssrdv 3984	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
35		selberg 27474	. . . . 5 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
36	35	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
37	34, 36	o1res2 15533	. . 3 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
38		fzfid 13964	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
39		elfznn 13556	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(⌊‘𝑦)) → 𝑛 ∈ ℕ)
40	39	adantl 481	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
41	40, 11	syl 17	. . . . . 6 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
42	40	nnrpd 13040	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ+)
43	42	relogcld 26550	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (log‘𝑛) ∈ ℝ)
44		simprl 770	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ)
45	44	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
46	45, 40	nndivred 12290	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
47		chpcl 27049	. . . . . . . 8 ⊢ ((𝑦 / 𝑛) ∈ ℝ → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
48	46, 47	syl 17	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
49	43, 48	readdcld 11267	. . . . . 6 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((log‘𝑛) + (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
50	41, 49	remulcld 11268	. . . . 5 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
51	38, 50	fsumrecl 15706	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
52	27	a1i 11	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 2 ∈ ℝ)
53	22	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ∈ ℝ+)
54		simprr 772	. . . . . . 7 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ≤ 𝑦)
55	44, 53, 54	rpgecld 13081	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ+)
56	55	relogcld 26550	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (log‘𝑦) ∈ ℝ)
57	52, 56	remulcld 11268	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
58	51, 57	readdcld 11267	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
59	31	adantr 480	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
60	59	recnd 11266	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
61	60	abscld 15409	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ ℝ)
62	26	adantr 480	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
63	30	adantr 480	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℝ)
64	62, 63	readdcld 11267	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) + (2 · (log‘𝑥))) ∈ ℝ)
65		fzfid 13964	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
66	39	adantl 481	. . . . . . . 8 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
67	66, 11	syl 17	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
68	66	nnrpd 13040	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ+)
69	68	relogcld 26550	. . . . . . . 8 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (log‘𝑛) ∈ ℝ)
70		simprll 778	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ)
71	70	adantr 480	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
72	71, 66	nndivred 12290	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
73	72, 47	syl 17	. . . . . . . 8 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
74	69, 73	readdcld 11267	. . . . . . 7 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((log‘𝑛) + (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
75	67, 74	remulcld 11268	. . . . . 6 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
76	65, 75	fsumrecl 15706	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
77	27	a1i 11	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℝ)
78	25	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ+)
79	4	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ)
80		simprr 772	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
81	79, 70, 80	ltled 11386	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ≤ 𝑦)
82	70, 78, 81	rpgecld 13081	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ+)
83	82	relogcld 26550	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑦) ∈ ℝ)
84	77, 83	remulcld 11268	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
85	76, 84	readdcld 11267	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
86	62	recnd 11266	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
87	63	recnd 11266	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℂ)
88	86, 87	abs2dif2d 15431	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))))
89	21	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
90		vmage0 27046	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
91	10, 90	syl 17	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
92	10	nnred 12251	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
93	10	nnge1d 12284	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
94	92, 93	logge0d 26557	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
95		chpge0 27051	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ → 0 ≤ (ψ‘(𝑥 / 𝑛)))
96	16, 95	syl 17	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
97	14, 18, 94, 96	addge0d 11814	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))))
98	12, 19, 91, 97	mulge0d 11815	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))))
99	8, 20, 98	fsumge0 15767	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))))
100	99	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))))
101	89, 78, 100	divge0d 13082	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥))
102	62, 101	absidd 15395	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥))
103	78	relogcld 26550	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ∈ ℝ)
104		2rp 13005	. . . . . . . . 9 ⊢ 2 ∈ ℝ+
105		rpge0 13013	. . . . . . . . 9 ⊢ (2 ∈ ℝ+ → 0 ≤ 2)
106	104, 105	mp1i 13	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ 2)
107	24	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ≤ 𝑥)
108	79, 107	logge0d 26557	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (log‘𝑥))
109	77, 103, 106, 108	mulge0d 11815	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (2 · (log‘𝑥)))
110	63, 109	absidd 15395	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) = (2 · (log‘𝑥)))
111	102, 110	oveq12d 7432	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) + (2 · (log‘𝑥))))
112	88, 111	breqtrd 5168	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) + (2 · (log‘𝑥))))
113	22	a1i 11	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ∈ ℝ+)
114	79	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ∈ ℝ)
115	114, 66	nndivred 12290	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ∈ ℝ)
116	115, 17	syl 17	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
117	69, 116	readdcld 11267	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
118	67, 117	remulcld 11268	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
119	65, 118	fsumrecl 15706	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
120	66, 90	syl 17	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑛))
121	66	nnred 12251	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ)
122	66	nnge1d 12284	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑛)
123	121, 122	logge0d 26557	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (log‘𝑛))
124	115, 95	syl 17	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
125	69, 116, 123, 124	addge0d 11814	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))))
126	67, 117, 120, 125	mulge0d 11815	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))))
127		flword2 13804	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)))
128	79, 70, 81, 127	syl3anc 1369	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)))
129		fzss2 13567	. . . . . . . . . 10 ⊢ ((⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
130	128, 129	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
131	65, 118, 126, 130	fsumless 15768	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))))
132	81	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ≤ 𝑦)
133	114, 71, 68, 132	lediv1dd 13100	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ≤ (𝑦 / 𝑛))
134		chpwordi 27082	. . . . . . . . . . . 12 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ (𝑦 / 𝑛) ∈ ℝ ∧ (𝑥 / 𝑛) ≤ (𝑦 / 𝑛)) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
135	115, 72, 133, 134	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
136	116, 73, 69, 135	leadd2dd 11853	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) ≤ ((log‘𝑛) + (ψ‘(𝑦 / 𝑛))))
137	117, 74, 67, 120, 136	lemul2ad 12178	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))))
138	65, 118, 75, 137	fsumle 15771	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))))
139	89, 119, 76, 131, 138	letrd 11395	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))))
140	89, 76, 113, 79, 100, 139, 107	lediv12ad 13101	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) / 1))
141	76	recnd 11266	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) ∈ ℂ)
142	141	div1d 12006	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) / 1) = Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))))
143	140, 142	breqtrd 5168	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))))
144	78, 82	logled 26554	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ≤ 𝑦 ↔ (log‘𝑥) ≤ (log‘𝑦)))
145	81, 144	mpbid 231	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ≤ (log‘𝑦))
146	103, 83, 77, 106, 145	lemul2ad 12178	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ≤ (2 · (log‘𝑦)))
147	62, 63, 76, 84, 143, 146	le2addd 11857	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) + (2 · (log‘𝑥))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))))
148	61, 64, 85, 112, 147	letrd 11395	. . 3 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))))
149	6, 7, 32, 37, 58, 148	o1bddrp 15512	. 2 ⊢ (⊤ → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐)
150	149	mptru 1541	1 ⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐

Syntax hints:   ↔ wb 205   ∧ wa 395  ⊤wtru 1535   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065   ⊆ wss 3944   class class class wbr 5142   ↦ cmpt 5225  ‘cfv 6542  (class class class)co 7414  ℝcr 11131  0cc0 11132  1c1 11133   + caddc 11135   · cmul 11137  +∞cpnf 11269   < clt 11272   ≤ cle 11273   − cmin 11468   / cdiv 11895  ℕcn 12236  2c2 12291  ℤ_≥cuz 12846  ℝ⁺crp 13000  [,)cico 13352  ...cfz 13510  ⌊cfl 13781  abscabs 15207  𝑂(1)co1 15456  Σcsu 15658  logclog 26481  Λcvma 27017  ψcchp 27018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210  ax-addf 11211

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9380  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9527  df-dju 9918  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12497  df-xnn0 12569  df-z 12583  df-dec 12702  df-uz 12847  df-q 12957  df-rp 13001  df-xneg 13118  df-xadd 13119  df-xmul 13120  df-ioo 13354  df-ioc 13355  df-ico 13356  df-icc 13357  df-fz 13511  df-fzo 13654  df-fl 13783  df-mod 13861  df-seq 13993  df-exp 14053  df-fac 14259  df-bc 14288  df-hash 14316  df-shft 15040  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-limsup 15441  df-clim 15458  df-rlim 15459  df-o1 15460  df-lo1 15461  df-sum 15659  df-ef 16037  df-e 16038  df-sin 16039  df-cos 16040  df-tan 16041  df-pi 16042  df-dvds 16225  df-gcd 16463  df-prm 16636  df-pc 16799  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-mulr 17240  df-starv 17241  df-sca 17242  df-vsca 17243  df-ip 17244  df-tset 17245  df-ple 17246  df-ds 17248  df-unif 17249  df-hom 17250  df-cco 17251  df-rest 17397  df-topn 17398  df-0g 17416  df-gsum 17417  df-topgen 17418  df-pt 17419  df-prds 17422  df-xrs 17477  df-qtop 17482  df-imas 17483  df-xps 17485  df-mre 17559  df-mrc 17560  df-acs 17562  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-submnd 18734  df-mulg 19017  df-cntz 19261  df-cmn 19730  df-psmet 21264  df-xmet 21265  df-met 21266  df-bl 21267  df-mopn 21268  df-fbas 21269  df-fg 21270  df-cnfld 21273  df-top 22789  df-topon 22806  df-topsp 22828  df-bases 22842  df-cld 22916  df-ntr 22917  df-cls 22918  df-nei 22995  df-lp 23033  df-perf 23034  df-cn 23124  df-cnp 23125  df-haus 23212  df-cmp 23284  df-tx 23459  df-hmeo 23652  df-fil 23743  df-fm 23835  df-flim 23836  df-flf 23837  df-xms 24219  df-ms 24220  df-tms 24221  df-cncf 24791  df-limc 25788  df-dv 25789  df-ulm 26306  df-log 26483  df-cxp 26484  df-atan 26792  df-em 26918  df-vma 27023  df-chp 27024  df-mu 27026

This theorem is referenced by:  selberg4  27487  selbergsb  27501