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Theorem selberg4 27725

Description: The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form Σ𝑖𝑗𝑘 ≤ 𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘); we eliminate one of the nested sums by using the definition of ψ(𝑥) = Σ𝑘 ≤ 𝑥, Λ(𝑘). This statement can thus equivalently be written ψ(𝑥)log↑2(𝑥) = 2Σ𝑖𝑗𝑘 ≤ 𝑥, Λ(𝑖)Λ(𝑗)Λ(𝑘) + 𝑂(𝑥log𝑥). Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)

**Assertion**
Ref	Expression
selberg4	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)

Distinct variable group: 𝑚,𝑛,𝑥

Proof of Theorem selberg4 Dummy variables 𝑖 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 12310	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
3		elioore 13397	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
4	3	adantl 486	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
5		eliooord 13427	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
6	5	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	simpld 499	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
8	4, 7	rplogcld 26794	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
9	2, 8	rerpdivcld 13086	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
10		fzfid 14005	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
11		elfznn 13577	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
12	11	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
13		vmacl 27282	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
14	12, 13	syl 18	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
15	4	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
16	15, 12	nndivred 12285	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑚) ∈ ℝ)
17		chpcl 27288	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
18	16, 17	syl 18	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
19	14, 18	remulcld 11234	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℝ)
20	12	nnrpd 13053	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
21	20	relogcld 26788	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
22	19, 21	remulcld 11234	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) ∈ ℝ)
23	10, 22	fsumrecl 15781	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) ∈ ℝ)
24	9, 23	remulcld 11234	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) ∈ ℝ)
25	10, 19	fsumrecl 15781	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℝ)
26	24, 25	resubcld 11637	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℝ)
27		1rp 13015	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
28	27	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
29		1red 11204	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
30	29, 4, 7	ltled 11353	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
31	4, 28, 30	rpgecld 13094	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
32	26, 31	rerpdivcld 13086	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℝ)
33	32	recnd 11232	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℂ)
34		chpcl 27288	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
35	4, 34	syl 18	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
36	31	relogcld 26788	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
37	35, 36	remulcld 11234	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
38	37, 25	readdcld 11233	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℝ)
39	38, 31	rerpdivcld 13086	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℝ)
40	39	recnd 11232	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℂ)
41	2, 36	remulcld 11234	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
42	41	recnd 11232	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℂ)
43	33, 40, 42	addsubassd 11584	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥)) − (2 · (log‘𝑥))) = (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥)))))
44	26	recnd 11232	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℂ)
45	38	recnd 11232	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℂ)
46	4	recnd 11232	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
47	31	rpne0d 13060	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
48	44, 45, 46, 47	divdird 12024	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) + (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))))) / 𝑥) = (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥)))
49	24	recnd 11232	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) ∈ ℂ)
50	25	recnd 11232	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℂ)
51	37	recnd 11232	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
52	49, 50, 51	nppcan3d 11591	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) + (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))))) = (((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) + ((ψ‘𝑥) · (log‘𝑥))))
53		elfznn 13577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))) → 𝑛 ∈ ℕ)
54	53	ad2antll 741	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → 𝑛 ∈ ℕ)
55		vmacl 27282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
56	54, 55	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑛) ∈ ℝ)
57	14	adantrr 729	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑚) ∈ ℝ)
58	20	adantrr 729	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → 𝑚 ∈ ℝ⁺)
59	58	relogcld 26788	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (log‘𝑚) ∈ ℝ)
60	57, 59	remulcld 11234	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
61	56, 60	remulcld 11234	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
62	61	recnd 11232	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
63	4, 62	fsumfldivdiag 27354	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 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))))
64	14	recnd 11232	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℂ)
65	18	recnd 11232	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℂ)
66	21	recnd 11232	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℂ)
67	64, 65, 66	mul32d 11415	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))))
68	64, 66	mulcld 11224	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
69	68, 65	mulcomd 11225	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))) = ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))))
70		chpval 27286	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
71	16, 70	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
72	71	oveq1d 7425	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
73		fzfid 14005	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑚))) ∈ Fin)
74	56	anassrs 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℝ)
75	74	recnd 11232	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℂ)
76	73, 68, 75	fsummulc1 15832	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 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))))
77	72, 76	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
78	67, 69, 77	3eqtrd 2802	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
79	78	sumeq2dv 15749	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 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))))
80		fzfid 14005	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
81		elfznn 13577	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
82	81	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
83	82, 55	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
84	83	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
85		elfznn 13577	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
86	85	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
87	86, 13	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
88	86	nnrpd 13053	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
89	88	relogcld 26788	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
90	87, 89	remulcld 11234	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
91	90	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
92	80, 84, 91	fsummulc2 15831	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 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))))
93	92	sumeq2dv 15749	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 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))))
94	63, 79, 93	3eqtr4d 2808	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 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))))
95	94	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))))
96	95	oveq1d 7425	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) + ((ψ‘𝑥) · (log‘𝑥))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))))
97	52, 96	eqtrd 2798	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) + (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))))
98	97	oveq1d 7425	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) + (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))))) / 𝑥) = ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
99	48, 98	eqtr3d 2800	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥)) = ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
100	99	oveq1d 7425	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥)) − (2 · (log‘𝑥))) = (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))))
101	43, 100	eqtr3d 2800	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥)))) = (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))))
102	101	mpteq2dva 5204	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥))))) = (𝑥 ∈ (1(,)+∞) ↦ (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥)))))
103	39, 41	resubcld 11637	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
104		selberg3lem2 27722	. . . . . 6 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥)) ∈ 𝑂(1)
105	104	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥)) ∈ 𝑂(1))
106	31	ex 417	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
107	106	ssrdv 3943	. . . . . 6 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ⁺)
108		selberg2 27715	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
109	108	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
110	107, 109	o1res2 15610	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
111	32, 103, 105, 110	o1add2 15671	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) + (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥))))) ∈ 𝑂(1))
112	102, 111	eqeltrrd 2864	. . 3 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
113	80, 90	fsumrecl 15781	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
114	83, 113	remulcld 11234	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
115	10, 114	fsumrecl 15781	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
116	9, 115	remulcld 11234	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) ∈ ℝ)
117	116, 37	readdcld 11233	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) ∈ ℝ)
118	117, 31	rerpdivcld 13086	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ ℝ)
119	118, 41	resubcld 11637	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
120	119	recnd 11232	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
121	4	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
122	121, 82	nndivred 12285	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
123	122	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ)
124	123, 86	nndivred 12285	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ)
125		chpcl 27288	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 𝑛) / 𝑚) ∈ ℝ → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
126	124, 125	syl 18	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
127	87, 126	remulcld 11234	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
128	80, 127	fsumrecl 15781	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
129	83, 128	remulcld 11234	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
130	10, 129	fsumrecl 15781	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
131	9, 130	remulcld 11234	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
132	37, 131	resubcld 11637	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) ∈ ℝ)
133	132, 31	rerpdivcld 13086	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥) ∈ ℝ)
134	133	recnd 11232	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥) ∈ ℂ)
135	116	recnd 11232	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) ∈ ℂ)
136	131	recnd 11232	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ)
137	51, 135, 136	pnncand 11603	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))) − (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))))
138	135, 51	addcomd 11407	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) = (((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))))
139	138	oveq1d 7425	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) − (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))) = ((((ψ‘𝑥) · (log‘𝑥)) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))) − (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))))
140	87	recnd 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℂ)
141	89	recnd 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℂ)
142	126	recnd 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
143	140, 141, 142	adddid 11228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) = (((Λ‘𝑚) · (log‘𝑚)) + ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))
144	143	sumeq2dv 15749	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) · (log‘𝑚)) + ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))
145	127	recnd 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
146	80, 91, 145	fsumadd 15787	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) · (log‘𝑚)) + ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))
147	144, 146	eqtrd 2798	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))
148	147	oveq2d 7426	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))
149	113	recnd 11232	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
150	128	recnd 11232	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
151	84, 149, 150	adddid 11228	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))) = (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))
152	148, 151	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) = (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))
153	152	sumeq2dv 15749	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))
154	114	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
155	129	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
156	10, 154, 155	fsumadd 15787	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))
157	153, 156	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))
158	157	oveq2d 7426	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) = ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))))
159	9	recnd 11232	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
160	115	recnd 11232	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
161	130	recnd 11232	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
162	159, 160, 161	adddid 11228	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))))
163	158, 162	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))))
164	137, 139, 163	3eqtr4d 2808	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) − (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))))
165	164	oveq1d 7425	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) − (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))) / 𝑥) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥))
166	117	recnd 11232	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ)
167	51, 136	subcld 11564	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) ∈ ℂ)
168	166, 167, 46, 47	divsubdird 12025	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) − (((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))))) / 𝑥) = (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)))
169		2cnd 12314	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
170	89, 126	readdcld 11233	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
171	87, 170	remulcld 11234	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
172	80, 171	fsumrecl 15781	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
173	83, 172	remulcld 11234	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
174	10, 173	fsumrecl 15781	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
175	174	recnd 11232	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ)
176	169, 175	mulcld 11224	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) ∈ ℂ)
177	36	recnd 11232	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
178	8	rpne0d 13060	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
179	176, 177, 46, 178, 47	divdiv1d 12017	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (log‘𝑥)) / 𝑥) = ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / ((log‘𝑥) · 𝑥)))
180	177, 46	mulcomd 11225	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) · 𝑥) = (𝑥 · (log‘𝑥)))
181	180	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / ((log‘𝑥) · 𝑥)) = ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (𝑥 · (log‘𝑥))))
182	179, 181	eqtrd 2798	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (log‘𝑥)) / 𝑥) = ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (𝑥 · (log‘𝑥))))
183	169, 175, 177, 178	div23d 12023	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (log‘𝑥)) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))))
184	183	oveq1d 7425	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (log‘𝑥)) / 𝑥) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥))
185	31, 8	rpmulcld 13071	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
186	185	rpcnd 13057	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
187	185	rpne0d 13060	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
188	169, 175, 186, 187	divassd 12021	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / (𝑥 · (log‘𝑥))) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥)))))
189	182, 184, 188	3eqtr3d 2806	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥)))))
190	165, 168, 189	3eqtr3d 2806	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥)))))
191	190	oveq1d 7425	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) − (2 · (log‘𝑥))) = ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥)))) − (2 · (log‘𝑥))))
192	118	recnd 11232	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ ℂ)
193	192, 42, 134	sub32d 11596	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) = ((((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) − (2 · (log‘𝑥))))
194	174, 185	rerpdivcld 13086	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
195	194	recnd 11232	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
196	169, 195, 177	subdid 11665	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) = ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥)))) − (2 · (log‘𝑥))))
197	191, 193, 196	3eqtr4d 2808	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) = (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))))
198	197	mpteq2dva 5204	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)))))
199	194, 36	resubcld 11637	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) ∈ ℝ)
200		ioossre 13429	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
201		2cnd 12314	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
202		o1const 15667	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
203	200, 201, 202	sylancr 598	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
204		selbergb 27713	. . . . . . 7 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐
205		simpl 487	. . . . . . . . 9 ⊢ ((𝑐 ∈ ℝ⁺ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐) → 𝑐 ∈ ℝ⁺)
206		simpr 489	. . . . . . . . 9 ⊢ ((𝑐 ∈ ℝ⁺ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐) → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐)
207	205, 206	selberg4lem1 27724	. . . . . . . 8 ⊢ ((𝑐 ∈ ℝ⁺ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐) → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
208	207	rexlimiva 3158	. . . . . . 7 ⊢ (∃𝑐 ∈ ℝ⁺ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
209	204, 208	mp1i 14	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
210	2, 199, 203, 209	o1mul2 15672	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)))) ∈ 𝑂(1))
211	198, 210	eqeltrd 2863	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥))) − ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥))) ∈ 𝑂(1))
212	120, 134, 211	o1dif 15677	. . 3 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) + ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)))
213	112, 212	mpbid 235	. 2 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1))
214	213	mptru 1577	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 400 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ⊆ wss 3905 class class class wbr 5109 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 1c1 11096 + caddc 11098 · cmul 11100 +∞cpnf 11235 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 ℕcn 12228 2c2 12290 ℝ⁺crp 13011 (,)cioo 13367 [,)cico 13369 ...cfz 13530 ⌊cfl 13819 abscabs 15281 𝑂(1)co1 15533 Σcsu 15733 logclog 26719 Λcvma 27256 ψcchp 27257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5077 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-o1 15537 df-lo1 15538 df-sum 15734 df-ef 16116 df-e 16117 df-sin 16118 df-cos 16119 df-tan 16120 df-pi 16121 df-dvds 16306 df-gcd 16548 df-prm 16725 df-pc 16892 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026 df-ulm 26540 df-log 26721 df-cxp 26722 df-atan 27032 df-em 27157 df-cht 27261 df-vma 27262 df-chp 27263 df-ppi 27264 df-mu 27265

This theorem is referenced by: selberg4r 27734

W3C validator