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Theorem selberg4 27945

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 12457	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
3		elioore 13546	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
4	3	adantl 487	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
5		eliooord 13576	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
6	5	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	simpld 500	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
8	4, 7	rplogcld 27014	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
9	2, 8	rerpdivcld 13235	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
10		fzfid 14156	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
11		elfznn 13727	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
12	11	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
13		vmacl 27502	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
14	12, 13	syl 18	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
15	4	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
16	15, 12	nndivred 12432	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑚) ∈ ℝ)
17		chpcl 27508	. . . . . . . . . . . . . . 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 11378	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℝ)
20	12	nnrpd 13202	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
21	20	relogcld 27008	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
22	19, 21	remulcld 11378	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) ∈ ℝ)
23	10, 22	fsumrecl 15944	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) ∈ ℝ)
24	9, 23	remulcld 11378	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) ∈ ℝ)
25	10, 19	fsumrecl 15944	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℝ)
26	24, 25	resubcld 11783	. . . . . . . . 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 13164	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
28	27	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
29		1red 11348	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
30	29, 4, 7	ltled 11497	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
31	4, 28, 30	rpgecld 13243	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
32	26, 31	rerpdivcld 13235	. . . . . . . 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 11376	. . . . . . 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 27508	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
35	4, 34	syl 18	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
36	31	relogcld 27008	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
37	35, 36	remulcld 11378	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
38	37, 25	readdcld 11377	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℝ)
39	38, 31	rerpdivcld 13235	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℝ)
40	39	recnd 11376	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℂ)
41	2, 36	remulcld 11378	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
42	41	recnd 11376	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℂ)
43	33, 40, 42	addsubassd 11728	. . . . . 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 11376	. . . . . . . . 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 11376	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℂ)
46	4	recnd 11376	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
47	31	rpne0d 13209	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
48	44, 45, 46, 47	divdird 12171	. . . . . . . 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 11376	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) ∈ ℂ)
50	25	recnd 11376	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℂ)
51	37	recnd 11376	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
52	49, 50, 51	nppcan3d 11735	. . . . . . . . . 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 13727	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))) → 𝑛 ∈ ℕ)
54	53	ad2antll 742	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → 𝑛 ∈ ℕ)
55		vmacl 27502	. . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑚) ∈ ℝ)
58	20	adantrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → 𝑚 ∈ ℝ⁺)
59	58	relogcld 27008	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (log‘𝑚) ∈ ℝ)
60	57, 59	remulcld 11378	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
61	56, 60	remulcld 11378	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
62	61	recnd 11376	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
63	4, 62	fsumfldivdiag 27574	. . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℂ)
65	18	recnd 11376	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℂ)
66	21	recnd 11376	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℂ)
67	64, 65, 66	mul32d 11559	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))))
68	64, 66	mulcld 11368	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
69	68, 65	mulcomd 11369	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))) = ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))))
70		chpval 27506	. . . . . . . . . . . . . . . . . 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 7466	. . . . . . . . . . . . . . . 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 14156	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑚))) ∈ Fin)
74	56	anassrs 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℝ)
75	74	recnd 11376	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℂ)
76	73, 68, 75	fsummulc1 15995	. . . . . . . . . . . . . . . 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 15913	. . . . . . . . . . . . 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 14156	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
81		elfznn 13727	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
82	81	adantl 487	. . . . . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
85		elfznn 13727	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
86	85	adantl 487	. . . . . . . . . . . . . . . . . 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 13202	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
89	88	relogcld 27008	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
90	87, 89	remulcld 11378	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
91	90	recnd 11376	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
92	80, 84, 91	fsummulc2 15994	. . . . . . . . . . . . . 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 15913	. . . . . . . . . . . . 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 7467	. . . . . . . . . . 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 7466	. . . . . . . . . 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 7466	. . . . . . . 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 7466	. . . . . 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 11783	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
104		selberg3lem2 27942	. . . . . 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 418	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
107	106	ssrdv 3919	. . . . . 6 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ⁺)
108		selberg2 27935	. . . . . . 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 15774	. . . . 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 15835	. . . 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 2865	. . 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 15944	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
114	83, 113	remulcld 11378	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
115	10, 114	fsumrecl 15944	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
116	9, 115	remulcld 11378	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) ∈ ℝ)
117	116, 37	readdcld 11377	. . . . . . 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 13235	. . . . . 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 11783	. . . . 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 11376	. . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
122	121, 82	nndivred 12432	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
123	122	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ)
124	123, 86	nndivred 12432	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ)
125		chpcl 27508	. . . . . . . . . . . . 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 11378	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
128	80, 127	fsumrecl 15944	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
129	83, 128	remulcld 11378	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
130	10, 129	fsumrecl 15944	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
131	9, 130	remulcld 11378	. . . . . . 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 11783	. . . . . 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 13235	. . . . 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 11376	. . . 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 11376	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) ∈ ℂ)
136	131	recnd 11376	. . . . . . . . . . . 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 11747	. . . . . . . . . . 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 11551	. . . . . . . . . . . 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 7466	. . . . . . . . . . 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 11376	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℂ)
141	89	recnd 11376	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℂ)
142	126	recnd 11376	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
143	140, 141, 142	adddid 11372	. . . . . . . . . . . . . . . . . . 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 15913	. . . . . . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
146	80, 91, 145	fsumadd 15950	. . . . . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
150	128	recnd 11376	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
151	84, 149, 150	adddid 11372	. . . . . . . . . . . . . . . 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 15913	. . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
155	129	recnd 11376	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
156	10, 154, 155	fsumadd 15950	. . . . . . . . . . . . . 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 7467	. . . . . . . . . . . 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 11376	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
160	115	recnd 11376	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
161	130	recnd 11376	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
162	159, 160, 161	adddid 11372	. . . . . . . . . . . 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 7466	. . . . . . . . 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 11376	. . . . . . . . . 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 11708	. . . . . . . . . 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 12172	. . . . . . . . 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 12461	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
170	89, 126	readdcld 11377	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
171	87, 170	remulcld 11378	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
172	80, 171	fsumrecl 15944	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
173	83, 172	remulcld 11378	. . . . . . . . . . . . . . 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 15944	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
175	174	recnd 11376	. . . . . . . . . . . . 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 11368	. . . . . . . . . . . 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 11376	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
178	8	rpne0d 13209	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
179	176, 177, 46, 178, 47	divdiv1d 12164	. . . . . . . . . . 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 11369	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) · 𝑥) = (𝑥 · (log‘𝑥)))
181	180	oveq2d 7467	. . . . . . . . . . 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 12170	. . . . . . . . . . 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 7466	. . . . . . . . . 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 13220	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
186	185	rpcnd 13206	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
187	185	rpne0d 13209	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
188	169, 175, 186, 187	divassd 12168	. . . . . . . . . 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 7466	. . . . . . 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 11376	. . . . . . . 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 11740	. . . . . . 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 13235	. . . . . . . . 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 11376	. . . . . . . 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 11811	. . . . . . 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 11783	. . . . . 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 13578	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
201		2cnd 12461	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
202		o1const 15831	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
203	200, 201, 202	sylancr 599	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
204		selbergb 27933	. . . . . . 7 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐
205		simpl 488	. . . . . . . . 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 490	. . . . . . . . 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 27944	. . . . . . . 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 3159	. . . . . . 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 15836	. . . . 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 2864	. . . 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 15841	. . 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

This proof depends on syntax axioms: ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 ⊆ wss 3900 class class class wbr 5108 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 1c1 11240 + caddc 11242 · cmul 11244 +∞cpnf 11379 < clt 11382 ≤ cle 11383 − cmin 11580 / cdiv 12013 ℕcn 12375 2c2 12437 ℝ⁺crp 13160 (,)cioo 13516 [,)cico 13518 ...cfz 13679 ⌊cfl 13970 abscabs 15444 𝑂(1)co1 15697 Σcsu 15897 logclog 26939 Λcvma 27476 ψcchp 27477

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-o1 15701 df-lo1 15702 df-sum 15898 df-ef 16277 df-e 16278 df-sin 16279 df-cos 16280 df-tan 16281 df-pi 16282 df-dvds 16467 df-gcd 16709 df-prm 16891 df-pc 17059 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-ulm 26761 df-log 26941 df-cxp 26942 df-atan 27252 df-em 27377 df-cht 27481 df-vma 27482 df-chp 27483 df-ppi 27484 df-mu 27485

This theorem is used by: selberg4r 27954

W3C validator