selberg3lem1 - Metamath Proof Explorer

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Theorem selberg3lem1 27619

Description: Introduce a log weighting on the summands of Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 27613 (written here as Σ𝑛 ≤ 𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)

**Hypotheses**
Ref	Expression
selberg3lem1.1	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
selberg3lem1.2	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)

**Assertion**
Ref	Expression
selberg3lem1	⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))

Distinct variable groups: 𝑘,𝑛,𝑥,𝑦,𝐴 𝜑,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑦,𝑘)

Proof of Theorem selberg3lem1 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 11291	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		ioossre 13468	. . . 4 ⊢ (1(,)+∞) ⊆ ℝ
3		selberg3lem1.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
4	3	rpcnd 13101	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		o1const 15666	. . . 4 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 𝐴 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
6	2, 4, 5	sylancr 586	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
7		fzfid 14024	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
8		elfznn 13613	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
9	8	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
10		vmacl 27179	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
11	9, 10	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
12	11, 9	nndivred 12347	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
13	7, 12	fsumrecl 15782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
14		elioore 13437	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
15		eliooord 13466	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
16	15	simpld 494	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → 1 < 𝑥)
17	14, 16	rplogcld 26689	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ⁺)
18		rpdivcl 13082	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ (log‘𝑥) ∈ ℝ⁺) → (𝐴 / (log‘𝑥)) ∈ ℝ⁺)
19	3, 17, 18	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 / (log‘𝑥)) ∈ ℝ⁺)
20	19	rpred 13099	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 / (log‘𝑥)) ∈ ℝ)
21	13, 20	remulcld 11320	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) ∈ ℝ)
22	21	recnd 11318	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) ∈ ℂ)
23	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℂ)
24	13	recnd 11318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
25	17	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
26	25	rpcnd 13101	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
27	19	rpcnd 13101	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 / (log‘𝑥)) ∈ ℂ)
28	24, 26, 27	subdird 11747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − ((log‘𝑥) · (𝐴 / (log‘𝑥)))))
29	25	rpne0d 13104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
30	23, 26, 29	divcan2d 12072	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) · (𝐴 / (log‘𝑥))) = 𝐴)
31	30	oveq2d 7464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − ((log‘𝑥) · (𝐴 / (log‘𝑥)))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴))
32	28, 31	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴))
33	32	mpteq2dva 5266	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴)))
34	25	rpred 13099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
35	13, 34	resubcld 11718	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
36	14	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
37		0red 11293	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ∈ ℝ)
38		1red 11291	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
39		0lt1 11812	. . . . . . . . . . . 12 ⊢ 0 < 1
40	39	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 < 1)
41	16	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
42	37, 38, 36, 40, 41	lttrd 11451	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 < 𝑥)
43	36, 42	elrpd 13096	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
44	43	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
45	44	ssrdv 4014	. . . . . . 7 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ⁺)
46		vmadivsum 27544	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
47	46	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
48	45, 47	o1res2 15609	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
49	2	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ)
50		ere 16137	. . . . . . . 8 ⊢ e ∈ ℝ
51	50	a1i 11	. . . . . . 7 ⊢ (𝜑 → e ∈ ℝ)
52	3	rpred 13099	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
53	19	adantrr 716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / (log‘𝑥)) ∈ ℝ⁺)
54	53	rprege0d 13106	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → ((𝐴 / (log‘𝑥)) ∈ ℝ ∧ 0 ≤ (𝐴 / (log‘𝑥))))
55		absid 15345	. . . . . . . . 9 ⊢ (((𝐴 / (log‘𝑥)) ∈ ℝ ∧ 0 ≤ (𝐴 / (log‘𝑥))) → (abs‘(𝐴 / (log‘𝑥))) = (𝐴 / (log‘𝑥)))
56	54, 55	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (abs‘(𝐴 / (log‘𝑥))) = (𝐴 / (log‘𝑥)))
57		loge 26646	. . . . . . . . . . 11 ⊢ (log‘e) = 1
58		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → e ≤ 𝑥)
59		epr 16256	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ⁺
60	43	adantrr 716	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
61		logleb 26663	. . . . . . . . . . . . 13 ⊢ ((e ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (e ≤ 𝑥 ↔ (log‘e) ≤ (log‘𝑥)))
62	59, 60, 61	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (e ≤ 𝑥 ↔ (log‘e) ≤ (log‘𝑥)))
63	58, 62	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (log‘e) ≤ (log‘𝑥))
64	57, 63	eqbrtrrid 5202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 1 ≤ (log‘𝑥))
65		1rp 13061	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
66		rpregt0 13071	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ⁺ → (1 ∈ ℝ ∧ 0 < 1))
67	65, 66	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (1 ∈ ℝ ∧ 0 < 1))
68	25	adantrr 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (log‘𝑥) ∈ ℝ⁺)
69	68	rpregt0d 13105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → ((log‘𝑥) ∈ ℝ ∧ 0 < (log‘𝑥)))
70	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 𝐴 ∈ ℝ⁺)
71	70	rpregt0d 13105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
72		lediv2 12185	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((log‘𝑥) ∈ ℝ ∧ 0 < (log‘𝑥)) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ (log‘𝑥) ↔ (𝐴 / (log‘𝑥)) ≤ (𝐴 / 1)))
73	67, 69, 71, 72	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (1 ≤ (log‘𝑥) ↔ (𝐴 / (log‘𝑥)) ≤ (𝐴 / 1)))
74	64, 73	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / (log‘𝑥)) ≤ (𝐴 / 1))
75	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 𝐴 ∈ ℂ)
76	75	div1d 12062	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / 1) = 𝐴)
77	74, 76	breqtrd 5192	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / (log‘𝑥)) ≤ 𝐴)
78	56, 77	eqbrtrd 5188	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (abs‘(𝐴 / (log‘𝑥))) ≤ 𝐴)
79	49, 27, 51, 52, 78	elo1d 15582	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 / (log‘𝑥))) ∈ 𝑂(1))
80	35, 20, 48, 79	o1mul2 15671	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥)))) ∈ 𝑂(1))
81	33, 80	eqeltrrd 2845	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴)) ∈ 𝑂(1))
82	22, 23, 81	o1dif 15676	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1)))
83	6, 82	mpbird 257	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))) ∈ 𝑂(1))
84		2re 12367	. . . . . . 7 ⊢ 2 ∈ ℝ
85		rerpdivcl 13087	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ⁺) → (2 / (log‘𝑥)) ∈ ℝ)
86	84, 25, 85	sylancr 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
87		nndivre 12334	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
88	36, 8, 87	syl2an 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
89		chpcl 27185	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
90	88, 89	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
91	11, 90	remulcld 11320	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
92	9	nnrpd 13097	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
93	92	relogcld 26683	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
94	91, 93	remulcld 11320	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
95	7, 94	fsumrecl 15782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
96	86, 95	remulcld 11320	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
97	7, 91	fsumrecl 15782	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
98	96, 97	resubcld 11718	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
99	98, 43	rerpdivcld 13130	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
100	99	recnd 11318	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
101	100	abscld 15485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ℝ)
102	22	abscld 15485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))) ∈ ℝ)
103		2cnd 12371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
104	95	recnd 11318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
105	103, 104	mulcld 11310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
106	97	recnd 11318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
107	106, 26	mulcld 11310	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) ∈ ℂ)
108	105, 107	subcld 11647	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) ∈ ℂ)
109	108	abscld 15485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ∈ ℝ)
110	42	gt0ne0d 11854	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
111	109, 36, 110	redivcld 12122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) ∈ ℝ)
112	52	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
113	13, 112	remulcld 11320	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) ∈ ℝ)
114	11	recnd 11318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
115		fzfid 14024	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
116		elfznn 13613	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
117	116	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
118		vmacl 27179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
119	117, 118	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
120	117	nnrpd 13097	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
121	120	relogcld 26683	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
122	119, 121	remulcld 11320	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
123	115, 122	fsumrecl 15782	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
124	8	nnrpd 13097	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
125		rpdivcl 13082	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
126	43, 124, 125	syl2an 595	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
127	126	relogcld 26683	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
128	90, 127	remulcld 11320	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
129	123, 128	resubcld 11718	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
130	129	recnd 11318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
131	114, 130	mulcld 11310	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
132	7, 131	fsumcl 15781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
133	132	abscld 15485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ∈ ℝ)
134	131	abscld 15485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ∈ ℝ)
135	7, 134	fsumrecl 15782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ∈ ℝ)
136	112, 36	remulcld 11320	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 · 𝑥) ∈ ℝ)
137	13, 136	remulcld 11320	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) ∈ ℝ)
138	7, 131	fsumabs 15849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
139	52	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
140	36	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
141	139, 140	remulcld 11320	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · 𝑥) ∈ ℝ)
142	12, 141	remulcld 11320	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) ∈ ℝ)
143	130	abscld 15485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ∈ ℝ)
144	141, 9	nndivred 12347	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 · 𝑥) / 𝑛) ∈ ℝ)
145		vmage0 27182	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
146	9, 145	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
147	88	recnd 11318	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
148	126	rpne0d 13104	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
149	130, 147, 148	absdivd 15504	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (abs‘(𝑥 / 𝑛))))
150	126	rpge0d 13103	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
151	88, 150	absidd 15471	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
152	151	oveq2d 7464	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)))
153	149, 152	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)))
154		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑚 → (Λ‘𝑘) = (Λ‘𝑚))
155		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝑚 → (log‘𝑘) = (log‘𝑚))
156	154, 155	oveq12d 7466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑚 → ((Λ‘𝑘) · (log‘𝑘)) = ((Λ‘𝑚) · (log‘𝑚)))
157	156	cbvsumv 15744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))
158		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑥 / 𝑛) → (⌊‘𝑦) = (⌊‘(𝑥 / 𝑛)))
159	158	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑥 / 𝑛) → (1...(⌊‘𝑦)) = (1...(⌊‘(𝑥 / 𝑛))))
160	159	sumeq1d 15748	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑥 / 𝑛) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))
161	157, 160	eqtrid 2792	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / 𝑛) → Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))
162		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑥 / 𝑛) → (ψ‘𝑦) = (ψ‘(𝑥 / 𝑛)))
163		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑥 / 𝑛) → (log‘𝑦) = (log‘(𝑥 / 𝑛)))
164	162, 163	oveq12d 7466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / 𝑛) → ((ψ‘𝑦) · (log‘𝑦)) = ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))
165	161, 164	oveq12d 7466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 / 𝑛) → (Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))
166		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
167	165, 166	oveq12d 7466	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → ((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛)))
168	167	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) = (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))))
169	168	breq1d 5176	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴 ↔ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) ≤ 𝐴))
170		selberg3lem1.2	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)
171	170	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)
172	9	nncnd 12309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
173	172	mullidd 11308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
174		fznnfl 13913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
175	36, 174	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
176	175	simplbda 499	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
177	173, 176	eqbrtrd 5188	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
178		1red 11291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
179	178, 140, 92	lemuldivd 13148	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
180	177, 179	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
181		1re 11290	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
182		elicopnf 13505	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℝ → ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛))))
183	181, 182	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)))
184	88, 180, 183	sylanbrc 582	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ (1[,)+∞))
185	169, 171, 184	rspcdva 3636	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) ≤ 𝐴)
186	153, 185	eqbrtrrd 5190	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)) ≤ 𝐴)
187	143, 139, 126	ledivmul2d 13153	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)) ≤ 𝐴 ↔ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ≤ (𝐴 · (𝑥 / 𝑛))))
188	186, 187	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ≤ (𝐴 · (𝑥 / 𝑛)))
189	23	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
190	140	recnd 11318	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
191	9	nnne0d 12343	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
192	189, 190, 172, 191	divassd 12105	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 · 𝑥) / 𝑛) = (𝐴 · (𝑥 / 𝑛)))
193	188, 192	breqtrrd 5194	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ≤ ((𝐴 · 𝑥) / 𝑛))
194	143, 144, 11, 146, 193	lemul2ad 12235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ ((Λ‘𝑛) · ((𝐴 · 𝑥) / 𝑛)))
195	114, 130	absmuld 15503	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) = ((abs‘(Λ‘𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
196	11, 146	absidd 15471	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Λ‘𝑛)) = (Λ‘𝑛))
197	196	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Λ‘𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) = ((Λ‘𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
198	195, 197	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) = ((Λ‘𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
199	141	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · 𝑥) ∈ ℂ)
200	114, 172, 199, 191	div32d 12093	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) = ((Λ‘𝑛) · ((𝐴 · 𝑥) / 𝑛)))
201	194, 198, 200	3brtr4d 5198	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ (((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
202	7, 134, 142, 201	fsumle 15847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
203	36	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
204	23, 203	mulcld 11310	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 · 𝑥) ∈ ℂ)
205	114, 172, 191	divcld 12070	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
206	7, 204, 205	fsummulc1 15833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
207	202, 206	breqtrrd 5194	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
208	133, 135, 137, 138, 207	letrd 11447	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
209	123	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
210	90	recnd 11318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
211	93	recnd 11318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
212	210, 211	mulcld 11310	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)) ∈ ℂ)
213	209, 212	addcld 11309	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) ∈ ℂ)
214	114, 213	mulcld 11310	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) ∈ ℂ)
215	114, 210	mulcld 11310	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
216	26	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈ ℂ)
217	215, 216	mulcld 11310	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) ∈ ℂ)
218	7, 214, 217	fsumsub 15836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
219	210, 216	mulcld 11310	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) ∈ ℂ)
220	114, 213, 219	subdid 11746	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))))
221	43	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
222	221, 92	relogdivd 26686	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) = ((log‘𝑥) − (log‘𝑛)))
223	222	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) · ((log‘𝑥) − (log‘𝑛))))
224	210, 216, 211	subdid 11746	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · ((log‘𝑥) − (log‘𝑛))) = (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
225	223, 224	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))) = (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
226	225	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
227	209, 219, 212	subsub3d 11677	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥))))
228	226, 227	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥))))
229	228	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) = ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))))
230	114, 210, 216	mulassd 11313	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) = ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥))))
231	230	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))))
232	220, 229, 231	3eqtr4d 2790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
233	232	sumeq2dv 15750	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
234		fveq2 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (Λ‘𝑛) = (Λ‘𝑚))
235		oveq2 7456	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (𝑥 / 𝑛) = (𝑥 / 𝑚))
236	235	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑚)))
237	234, 236	oveq12d 7466	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))))
238		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → (log‘𝑛) = (log‘𝑚))
239	237, 238	oveq12d 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)))
240	239	cbvsumv 15744	. . . . . . . . . . . . . . 15 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))
241		elfznn 13613	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))) → 𝑛 ∈ ℕ)
242	241	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → 𝑛 ∈ ℕ)
243	242, 10	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℝ)
244	243	recnd 11318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℂ)
245	244	anasss 466	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑛) ∈ ℂ)
246		elfznn 13613	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
247	246	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
248	247, 118	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
249	248	recnd 11318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℂ)
250	247	nnrpd 13097	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
251	250	relogcld 26683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
252	251	recnd 11318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℂ)
253	249, 252	mulcld 11310	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
254	253	adantrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
255	245, 254	mulcld 11310	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
256	36, 255	fsumfldivdiag 27251	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
257	36	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
258	257, 247	nndivred 12347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑚) ∈ ℝ)
259		chpcl 27185	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
260	258, 259	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
261	260	recnd 11318	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℂ)
262	249, 261, 252	mul32d 11500	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))))
263	248, 251	remulcld 11320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
264	263	recnd 11318	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
265	264, 261	mulcomd 11311	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))) = ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))))
266		chpval 27183	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
267	258, 266	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
268	267	oveq1d 7463	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
269		fzfid 14024	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑚))) ∈ Fin)
270	269, 264, 244	fsummulc1 15833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
271	268, 270	eqtrd 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
272	262, 265, 271	3eqtrd 2784	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
273	272	sumeq2dv 15750	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
274	122	recnd 11318	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
275	115, 114, 274	fsummulc2 15832	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
276	275	sumeq2dv 15750	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
277	256, 273, 276	3eqtr4d 2790	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))
278	240, 277	eqtrid 2792	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))
279	114, 210, 211	mulassd 11313	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
280	279	sumeq2dv 15750	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
281	278, 280	oveq12d 7466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
282	104	2timesd 12536	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))))
283	114, 209	mulcld 11310	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
284	114, 212	mulcld 11310	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) ∈ ℂ)
285	7, 283, 284	fsumadd 15788	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
286	281, 282, 285	3eqtr4d 2790	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
287	114, 209, 212	adddid 11314	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
288	287	sumeq2dv 15750	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
289	286, 288	eqtr4d 2783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
290	91	recnd 11318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
291	7, 26, 290	fsummulc1 15833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))
292	289, 291	oveq12d 7466	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
293	218, 233, 292	3eqtr4rd 2791	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))))
294	293	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
295	24, 23, 203	mulassd 11313	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) · 𝑥) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
296	208, 294, 295	3brtr4d 5198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) · 𝑥))
297	109, 113, 43	ledivmul2d 13153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) ↔ (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) · 𝑥)))
298	296, 297	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴))
299	111, 113, 25, 298	lediv1dd 13157	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) / (log‘𝑥)) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)))
300	109	recnd 11318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ∈ ℂ)
301	300, 203, 26, 110, 29	divdiv1d 12101	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) / (log‘𝑥)) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (𝑥 · (log‘𝑥))))
302	108, 26, 203, 29, 110	divdiv32d 12095	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) / 𝑥) = ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / 𝑥) / (log‘𝑥)))
303	105, 107, 26, 29	divsubdird 12109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) = (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) / (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) / (log‘𝑥))))
304	103, 104, 26, 29	div23d 12107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) / (log‘𝑥)) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))))
305	106, 26, 29	divcan4d 12076	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) / (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
306	304, 305	oveq12d 7466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) / (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) / (log‘𝑥))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
307	303, 306	eqtrd 2780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
308	307	oveq1d 7463	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) / 𝑥) = ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
309	108, 203, 26, 110, 29	divdiv1d 12101	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / 𝑥) / (log‘𝑥)) = (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥))))
310	302, 308, 309	3eqtr3d 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) = (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥))))
311	310	fveq2d 6924	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = (abs‘(((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥)))))
312	43, 25	rpmulcld 13115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
313	312	rpcnd 13101	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
314	312	rpne0d 13104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
315	108, 313, 314	absdivd 15504	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥)))) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (abs‘(𝑥 · (log‘𝑥)))))
316	312	rpred 13099	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ)
317	312	rpge0d 13103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝑥 · (log‘𝑥)))
318	316, 317	absidd 15471	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑥 · (log‘𝑥))) = (𝑥 · (log‘𝑥)))
319	318	oveq2d 7464	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (abs‘(𝑥 · (log‘𝑥)))) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (𝑥 · (log‘𝑥))))
320	311, 315, 319	3eqtrd 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (𝑥 · (log‘𝑥))))
321	301, 320	eqtr4d 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) / (log‘𝑥)) = (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)))
322	24, 23, 26, 29	divassd 12105	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))))
323	299, 321, 322	3brtr3d 5197	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))))
324	21	leabsd 15463	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) ≤ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))))
325	101, 21, 102, 323, 324	letrd 11447	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))))
326	325	adantrr 716	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))))
327	1, 83, 21, 100, 326	o1le 15701	1 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ⊆ wss 3976 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 +∞cpnf 11321 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℝ⁺crp 13057 (,)cioo 13407 [,)cico 13409 ...cfz 13567 ⌊cfl 13841 abscabs 15283 𝑂(1)co1 15532 Σcsu 15734 eceu 16110 logclog 26614 Λcvma 27153 ψcchp 27154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-o1 15536 df-lo1 15537 df-sum 15735 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 df-pc 16884 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-cxp 26617 df-cht 27158 df-vma 27159 df-chp 27160 df-ppi 27161

This theorem is referenced by: selberg3lem2 27620

W3C validator