Theorem selberg2b 26681

Description: Convert eventual boundedness in selberg2 26680 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 10959	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 13159	. . . . . . 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 3931	. . 3 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
8		chpcl 26254	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
9	4, 8	syl 17	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (ψ‘𝑥) ∈ ℝ)
10		1rp 12716	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
12	3	simplbda 499	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
13	4, 11, 12	rpgecld 12793	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ+)
14	13	relogcld 25759	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (log‘𝑥) ∈ ℝ)
15	9, 14	remulcld 10989	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
16		fzfid 13674	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
17		elfznn 13267	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
18	17	adantl 481	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
19		vmacl 26248	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
21	4	adantr 480	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
22	21, 18	nndivred 12010	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
23		chpcl 26254	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
24	22, 23	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
25	20, 24	remulcld 10989	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
26	16, 25	fsumrecl 15427	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
27	15, 26	readdcld 10988	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
28	27, 13	rerpdivcld 12785	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
29		2re 12030	. . . . . . 7 ⊢ 2 ∈ ℝ
30	29	a1i 11	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 2 ∈ ℝ)
31	30, 14	remulcld 10989	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
32	28, 31	resubcld 11386	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
33	32	recnd 10987	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
34	13	ex 412	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ+))
35	34	ssrdv 3931	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
36		selberg2 26680	. . . . 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 15253	. . 3 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
39		chpcl 26254	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
40	39	ad2antrl 724	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (ψ‘𝑦) ∈ ℝ)
41		simprl 767	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ)
42	10	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ∈ ℝ+)
43		simprr 769	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ≤ 𝑦)
44	41, 42, 43	rpgecld 12793	. . . . . . 7 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ+)
45	44	relogcld 25759	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (log‘𝑦) ∈ ℝ)
46	40, 45	remulcld 10989	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
47		fzfid 13674	. . . . . 6 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
48		elfznn 13267	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑦)) → 𝑛 ∈ ℕ)
49	48	adantl 481	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
50	49, 19	syl 17	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
51	41	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
52	51, 49	nndivred 12010	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
53		chpcl 26254	. . . . . . . 8 ⊢ ((𝑦 / 𝑛) ∈ ℝ → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
54	52, 53	syl 17	. . . . . . 7 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
55	50, 54	remulcld 10989	. . . . . 6 ⊢ (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
56	47, 55	fsumrecl 15427	. . . . 5 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
57	46, 56	readdcld 10988	. . . 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 10989	. . . 4 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
60	57, 59	readdcld 10988	. . 3 ⊢ ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
61	32	adantr 480	. . . . . 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 10987	. . . . 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 15129	. . . 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 480	. . . . . . 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 10987	. . . . . 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 15129	. . . . 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 480	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℝ)
68	67	recnd 10987	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℂ)
69	68	abscld 15129	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ∈ ℝ)
70	66, 69	readdcld 10988	. . . 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 743	. . . 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 15151	. . . 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 775	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ)
74	73, 39	syl 17	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑦) ∈ ℝ)
75	13	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ+)
76	4	adantr 480	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ)
77		simprr 769	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
78	76, 73, 77	ltled 11106	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ≤ 𝑦)
79	73, 75, 78	rpgecld 12793	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ+)
80	79	relogcld 25759	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑦) ∈ ℝ)
81	74, 80	remulcld 10989	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
82	56	ad2ant2r 743	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
83	81, 82	readdcld 10988	. . . . 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 10989	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
86	76, 8	syl 17	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ∈ ℝ)
87	75	relogcld 25759	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ∈ ℝ)
88	86, 87	remulcld 10989	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
89	26	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
90	88, 89	readdcld 10988	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
91		chpge0 26256	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
92	76, 91	syl 17	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (ψ‘𝑥))
93	12	adantr 480	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ≤ 𝑥)
94	76, 93	logge0d 25766	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (log‘𝑥))
95	86, 87, 92, 94	mulge0d 11535	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ ((ψ‘𝑥) · (log‘𝑥)))
96		vmage0 26251	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
97	18, 96	syl 17	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
98		chpge0 26256	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ → 0 ≤ (ψ‘(𝑥 / 𝑛)))
99	22, 98	syl 17	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
100	20, 24, 97, 99	mulge0d 11535	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
101	16, 25, 100	fsumge0 15488	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
102	101	adantr 480	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
103	88, 89, 95, 102	addge0d 11534	. . . . . . . 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 12794	. . . . . . 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 15115	. . . . . 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 26287	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
108	76, 73, 78, 107	syl3anc 1369	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ≤ (ψ‘𝑦))
109	75, 79	logled 25763	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (𝑥 ≤ 𝑦 ↔ (log‘𝑥) ≤ (log‘𝑦)))
110	78, 109	mpbid 231	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ≤ (log‘𝑦))
111	86, 74, 87, 80, 92, 94, 108, 110	lemul12ad 11900	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ≤ ((ψ‘𝑦) · (log‘𝑦)))
112		fzfid 13674	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
113	48	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
114	113, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
115	76	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ∈ ℝ)
116	115, 113	nndivred 12010	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ∈ ℝ)
117	116, 23	syl 17	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
118	114, 117	remulcld 10989	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
119	112, 118	fsumrecl 15427	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
120	113, 96	syl 17	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑛))
121	116, 98	syl 17	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
122	114, 117, 120, 121	mulge0d 11535	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
123		flword2 13514	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ≤ 𝑦) → (⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)))
124	76, 73, 78, 123	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)))
125		fzss2 13278	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑦) ∈ (ℤ≥‘(⌊‘𝑥)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
126	124, 125	syl 17	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
127	112, 118, 122, 126	fsumless 15489	. . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
129	128, 113	nndivred 12010	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
130	129, 53	syl 17	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
131	114, 130	remulcld 10989	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
132	113	nnrpd 12752	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ+)
133	78	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ≤ 𝑦)
134	115, 128, 132, 133	lediv1dd 12812	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ≤ (𝑦 / 𝑛))
135		chpwordi 26287	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 𝑛) ∈ ℝ ∧ (𝑦 / 𝑛) ∈ ℝ ∧ (𝑥 / 𝑛) ≤ (𝑦 / 𝑛)) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
136	116, 129, 134, 135	syl3anc 1369	. . . . . . . . . . . 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 11898	. . . . . . . . . . 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 15492	. . . . . . . . . 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 11115	. . . . . . . . 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 11577	. . . . . . . 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 12813	. . . . . . 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 10987	. . . . . . . 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 11726	. . . . . . 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 5104	. . . . . 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 5100	. . . . 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 12717	. . . . . . . . 9 ⊢ 2 ∈ ℝ+
147		rpge0 12725	. . . . . . . . 9 ⊢ (2 ∈ ℝ+ → 0 ≤ 2)
148	146, 147	mp1i 13	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ 2)
149	84, 87, 148, 94	mulge0d 11535	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (2 · (log‘𝑥)))
150	67, 149	absidd 15115	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) = (2 · (log‘𝑥)))
151	87, 80, 84, 148, 110	lemul2ad 11898	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ≤ (2 · (log‘𝑦)))
152	150, 151	eqbrtrd 5100	. . . . 5 ⊢ (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ≤ (2 · (log‘𝑦)))
153	66, 69, 83, 85, 145, 152	le2addd 11577	. . . 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 11115	. . 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 15232	. 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 1548	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 205   ∧ wa 395  ⊤wtru 1542   ∈ wcel 2109  ∀wral 3065  ∃wrex 3066   ⊆ wss 3891   class class class wbr 5078   ↦ cmpt 5161  ‘cfv 6430  (class class class)co 7268  ℝcr 10854  0cc0 10855  1c1 10856   + caddc 10858   · cmul 10860  +∞cpnf 10990   < clt 10993   ≤ cle 10994   − cmin 11188   / cdiv 11615  ℕcn 11956  2c2 12011  ℤ_≥cuz 12564  ℝ⁺crp 12712  [,)cico 13063  ...cfz 13221  ⌊cfl 13491  abscabs 14926  𝑂(1)co1 15176  Σcsu 15378  logclog 25691  Λcvma 26222  ψcchp 26223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-inf2 9360  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932  ax-pre-sup 10933  ax-addf 10934  ax-mulf 10935

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-disj 5044  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-om 7701  df-1st 7817  df-2nd 7818  df-supp 7962  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-2o 8282  df-oadd 8285  df-er 8472  df-map 8591  df-pm 8592  df-ixp 8660  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-fsupp 9090  df-fi 9131  df-sup 9162  df-inf 9163  df-oi 9230  df-dju 9643  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-div 11616  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-7 12024  df-8 12025  df-9 12026  df-n0 12217  df-xnn0 12289  df-z 12303  df-dec 12420  df-uz 12565  df-q 12671  df-rp 12713  df-xneg 12830  df-xadd 12831  df-xmul 12832  df-ioo 13065  df-ioc 13066  df-ico 13067  df-icc 13068  df-fz 13222  df-fzo 13365  df-fl 13493  df-mod 13571  df-seq 13703  df-exp 13764  df-fac 13969  df-bc 13998  df-hash 14026  df-shft 14759  df-cj 14791  df-re 14792  df-im 14793  df-sqrt 14927  df-abs 14928  df-limsup 15161  df-clim 15178  df-rlim 15179  df-o1 15180  df-lo1 15181  df-sum 15379  df-ef 15758  df-e 15759  df-sin 15760  df-cos 15761  df-tan 15762  df-pi 15763  df-dvds 15945  df-gcd 16183  df-prm 16358  df-pc 16519  df-struct 16829  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-plusg 16956  df-mulr 16957  df-starv 16958  df-sca 16959  df-vsca 16960  df-ip 16961  df-tset 16962  df-ple 16963  df-ds 16965  df-unif 16966  df-hom 16967  df-cco 16968  df-rest 17114  df-topn 17115  df-0g 17133  df-gsum 17134  df-topgen 17135  df-pt 17136  df-prds 17139  df-xrs 17194  df-qtop 17199  df-imas 17200  df-xps 17202  df-mre 17276  df-mrc 17277  df-acs 17279  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-submnd 18412  df-mulg 18682  df-cntz 18904  df-cmn 19369  df-psmet 20570  df-xmet 20571  df-met 20572  df-bl 20573  df-mopn 20574  df-fbas 20575  df-fg 20576  df-cnfld 20579  df-top 22024  df-topon 22041  df-topsp 22063  df-bases 22077  df-cld 22151  df-ntr 22152  df-cls 22153  df-nei 22230  df-lp 22268  df-perf 22269  df-cn 22359  df-cnp 22360  df-haus 22447  df-cmp 22519  df-tx 22694  df-hmeo 22887  df-fil 22978  df-fm 23070  df-flim 23071  df-flf 23072  df-xms 23454  df-ms 23455  df-tms 23456  df-cncf 24022  df-limc 25011  df-dv 25012  df-ulm 25517  df-log 25693  df-cxp 25694  df-atan 25998  df-em 26123  df-cht 26227  df-vma 26228  df-chp 26229  df-ppi 26230  df-mu 26231

This theorem is referenced by:  chpdifbnd  26684