Theorem selberg2b 27681

Description: Convert eventual boundedness in selberg2 27680 to boundedness on any interval [𝐴, +∞). (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 25-May-2016.)

Assertion

Ref	Expression
selberg2b	⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐

Distinct variable group:   𝑛,𝑐,𝑥

Proof of Theorem selberg2b

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 11207	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 13471	. . . . . . 7 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
3	1, 2	mp1i 14	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
4	3	simprbda 503	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ)
5	4	ex 417	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ))
6	5	ssrdv 3951	. . 3 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
8		chpcl 27253	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
9	4, 8	syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (ψ‘𝑥) ∈ ℝ)
10		1rp 13019	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
12	3	simplbda 504	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
13	4, 11, 12	rpgecld 13098	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ+)
14	13	relogcld 26753	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (log‘𝑥) ∈ ℝ)
15	9, 14	remulcld 11238	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
16		fzfid 14008	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
17		elfznn 13580	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
18	17	adantl 486	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
19		vmacl 27247	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
20	18, 19	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
21	4	adantr 485	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
22	21, 18	nndivred 12289	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
23		chpcl 27253	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
24	22, 23	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
25	20, 24	remulcld 11238	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
26	16, 25	fsumrecl 15784	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
27	15, 26	readdcld 11237	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
28	27, 13	rerpdivcld 13090	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
29		2re 12314	. . . . . . 7 ⊢ 2 ∈ ℝ
30	29	a1i 11	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 2 ∈ ℝ)
31	30, 14	remulcld 11238	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
32	28, 31	resubcld 11641	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
33	32	recnd 11236	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
34	13	ex 417	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ+))
35	34	ssrdv 3951	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
36		selberg2 27680	. . . . 5 ⊢ (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
37	36	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
38	35, 37	o1res2 15613	. . 3 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
39		chpcl 27253	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
40	39	ad2antrl 740	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (ψ‘𝑦) ∈ ℝ)
41		simprl 782	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ)
42	10	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ∈ ℝ+)
43		simprr 784	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ≤ 𝑦)
44	41, 42, 43	rpgecld 13098	. . . . . . 7 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ+)
45	44	relogcld 26753	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (log‘𝑦) ∈ ℝ)
46	40, 45	remulcld 11238	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
47		fzfid 14008	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
48		elfznn 13580	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑦)) → 𝑛 ∈ ℕ)
49	48	adantl 486	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
50	49, 19	syl 18	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
51	41	adantr 485	. . . . . . . . 9 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
52	51, 49	nndivred 12289	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
53		chpcl 27253	. . . . . . . 8 ⊢ ((𝑦 / 𝑛) ∈ ℝ → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
54	52, 53	syl 18	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
55	50, 54	remulcld 11238	. . . . . 6 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
56	47, 55	fsumrecl 15784	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
57	46, 56	readdcld 11237	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
58	29	a1i 11	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 2 ∈ ℝ)
59	58, 45	remulcld 11238	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
60	57, 59	readdcld 11237	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
61	32	adantr 485	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
62	61	recnd 11236	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
63	62	abscld 15489	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ ℝ)
64	28	adantr 485	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
65	64	recnd 11236	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
66	65	abscld 15489	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ℝ)
67	31	adantr 485	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℝ)
68	67	recnd 11236	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℂ)
69	68	abscld 15489	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ∈ ℝ)
70	66, 69	readdcld 11237	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))) ∈ ℝ)
71	60	ad2ant2r 759	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
72	65, 68	abs2dif2d 15511	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ ((abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))))
73		simprll 790	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ)
74	73, 39	syl 18	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑦) ∈ ℝ)
75	13	adantr 485	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ+)
76	4	adantr 485	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ)
77		simprr 784	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
78	76, 73, 77	ltled 11357	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ≤ 𝑦)
79	73, 75, 78	rpgecld 13098	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ+)
80	79	relogcld 26753	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑦) ∈ ℝ)
81	74, 80	remulcld 11238	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
82	56	ad2ant2r 759	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
83	81, 82	readdcld 11237	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
84	29	a1i 11	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℝ)
85	84, 80	remulcld 11238	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
86	76, 8	syl 18	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ∈ ℝ)
87	75	relogcld 26753	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ∈ ℝ)
88	86, 87	remulcld 11238	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
89	26	adantr 485	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
90	88, 89	readdcld 11237	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
91		chpge0 27255	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
92	76, 91	syl 18	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (ψ‘𝑥))
93	12	adantr 485	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ≤ 𝑥)
94	76, 93	logge0d 26760	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (log‘𝑥))
95	86, 87, 92, 94	mulge0d 11790	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ ((ψ‘𝑥) · (log‘𝑥)))
96		vmage0 27250	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
97	18, 96	syl 18	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
98		chpge0 27255	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ → 0 ≤ (ψ‘(𝑥 / 𝑛)))
99	22, 98	syl 18	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
100	20, 24, 97, 99	mulge0d 11790	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
101	16, 25, 100	fsumge0 15846	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
102	101	adantr 485	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
103	88, 89, 95, 102	addge0d 11789	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
104	90, 75, 103	divge0d 13099	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
105	64, 104	absidd 15473	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
106	10	a1i 11	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ∈ ℝ+)
107		chpwordi 27286	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
108	76, 73, 78, 107	syl3anc 1396	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ≤ (ψ‘𝑦))
109	75, 79	logled 26757	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ≤ 𝑦 ↔ (log‘𝑥) ≤ (log‘𝑦)))
110	78, 109	mpbid 235	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ≤ (log‘𝑦))
111	86, 74, 87, 80, 92, 94, 108, 110	lemul12ad 12156	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ≤ ((ψ‘𝑦) · (log‘𝑦)))
112		fzfid 14008	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
113	48	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
114	113, 19	syl 18	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
115	76	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ∈ ℝ)
116	115, 113	nndivred 12289	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ∈ ℝ)
117	116, 23	syl 18	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
118	114, 117	remulcld 11238	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
119	112, 118	fsumrecl 15784	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
120	113, 96	syl 18	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑛))
121	116, 98	syl 18	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
122	114, 117, 120, 121	mulge0d 11790	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
123		flword2 13845	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)))
124	76, 73, 78, 123	syl3anc 1396	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)))
125		fzss2 13591	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
126	124, 125	syl 18	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
127	112, 118, 122, 126	fsumless 15847	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
128	73	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
129	128, 113	nndivred 12289	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
130	129, 53	syl 18	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
131	114, 130	remulcld 11238	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
132	113	nnrpd 13057	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ+)
133	78	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ≤ 𝑦)
134	115, 128, 132, 133	lediv1dd 13117	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ≤ (𝑦 / 𝑛))
135		chpwordi 27286	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ (𝑦 / 𝑛) ∈ ℝ ∧ (𝑥 / 𝑛) ≤ (𝑦 / 𝑛)) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
136	116, 129, 134, 135	syl3anc 1396	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
137	117, 130, 114, 120, 136	lemul2ad 12154	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))))
138	112, 118, 131, 137	fsumle 15850	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))))
139	89, 119, 82, 127, 138	letrd 11366	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))))
140	88, 89, 81, 82, 111, 139	le2addd 11832	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ≤ (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
141	90, 83, 106, 76, 103, 140, 93	lediv12ad 13118	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) / 1))
142	83	recnd 11236	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℂ)
143	142	div1d 11982	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) / 1) = (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
144	141, 143	breqtrd 5141	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ≤ (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
145	105, 144	eqbrtrd 5137	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
146		2rp 13020	. . . . . . . . 9 ⊢ 2 ∈ ℝ+
147		rpge0 13029	. . . . . . . . 9 ⊢ (2 ∈ ℝ+ → 0 ≤ 2)
148	146, 147	mp1i 14	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ 2)
149	84, 87, 148, 94	mulge0d 11790	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (2 · (log‘𝑥)))
150	67, 149	absidd 15473	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) = (2 · (log‘𝑥)))
151	87, 80, 84, 148, 110	lemul2ad 12154	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ≤ (2 · (log‘𝑦)))
152	150, 151	eqbrtrd 5137	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ≤ (2 · (log‘𝑦)))
153	66, 69, 83, 85, 145, 152	le2addd 11832	. . . 4 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))) ≤ ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))))
154	63, 70, 71, 72, 153	letrd 11366	. . 3 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))))
155	6, 7, 33, 38, 60, 154	o1bddrp 15592	. 2 ⊢ (⊤ → ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐)
156	155	mptru 1574	1 ⊢ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐

Syntax hints:   ↔ wb 209   ∧ wa 400  ⊤wtru 1568   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913   class class class wbr 5113   ↦ cmpt 5196  ‘cfv 6537  (class class class)co 7411  ℝcr 11098  0cc0 11099  1c1 11100   + caddc 11102   · cmul 11104  +∞cpnf 11239   < clt 11242   ≤ cle 11243   − cmin 11440   / cdiv 11870  ℕcn 12232  2c2 12294  ℤ_≥cuz 12861  ℝ⁺crp 13015  [,)cico 13373  ...cfz 13534  ⌊cfl 13822  abscabs 15284  𝑂(1)co1 15536  Σcsu 15736  logclog 26684  Λcvma 27221  ψcchp 27222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-addf 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-xnn0 12577  df-z 12591  df-dec 12711  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ioc 13376  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13682  df-fl 13824  df-mod 13902  df-seq 14037  df-exp 14097  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15103  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-limsup 15521  df-clim 15538  df-rlim 15539  df-o1 15540  df-lo1 15541  df-sum 15737  df-ef 16120  df-e 16121  df-sin 16122  df-cos 16123  df-tan 16124  df-pi 16125  df-dvds 16310  df-gcd 16552  df-prm 16729  df-pc 16896  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-rest 17474  df-topn 17475  df-0g 17493  df-gsum 17494  df-topgen 17495  df-pt 17496  df-prds 17499  df-xrs 17555  df-qtop 17560  df-imas 17561  df-xps 17563  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-submnd 18841  df-mulg 19133  df-cntz 19386  df-cmn 19851  df-psmet 21482  df-xmet 21483  df-met 21484  df-bl 21485  df-mopn 21486  df-fbas 21487  df-fg 21488  df-cnfld 21491  df-top 23019  df-topon 23036  df-topsp 23058  df-bases 23071  df-cld 23144  df-ntr 23145  df-cls 23146  df-nei 23223  df-lp 23261  df-perf 23262  df-cn 23352  df-cnp 23353  df-haus 23440  df-cmp 23512  df-tx 23687  df-hmeo 23880  df-fil 23971  df-fm 24063  df-flim 24064  df-flf 24065  df-xms 24445  df-ms 24446  df-tms 24447  df-cncf 25005  df-limc 25993  df-dv 25994  df-ulm 26505  df-log 26686  df-cxp 26687  df-atan 26997  df-em 27122  df-cht 27226  df-vma 27227  df-chp 27228  df-ppi 27229  df-mu 27230

This theorem is referenced by:  chpdifbnd  27684