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Theorem selberg4 26137

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 11712	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
3		elioore 12769	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
4	3	adantl 484	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
5		eliooord 12797	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
6	5	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	simpld 497	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
8	4, 7	rplogcld 25212	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
9	2, 8	rerpdivcld 12463	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
10		fzfid 13342	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
11		elfznn 12937	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
12	11	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
13		vmacl 25695	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
14	12, 13	syl 17	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
15	4	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
16	15, 12	nndivred 11692	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑚) ∈ ℝ)
17		chpcl 25701	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
19	14, 18	remulcld 10671	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℝ)
20	12	nnrpd 12430	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
21	20	relogcld 25206	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
22	19, 21	remulcld 10671	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) ∈ ℝ)
23	10, 22	fsumrecl 15091	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) ∈ ℝ)
24	9, 23	remulcld 10671	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) ∈ ℝ)
25	10, 19	fsumrecl 15091	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℝ)
26	24, 25	resubcld 11068	. . . . . . . . 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 12394	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
28	27	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
29		1red 10642	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
30	29, 4, 7	ltled 10788	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
31	4, 28, 30	rpgecld 12471	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
32	26, 31	rerpdivcld 12463	. . . . . . . 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 10669	. . . . . . 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 25701	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
35	4, 34	syl 17	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
36	31	relogcld 25206	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
37	35, 36	remulcld 10671	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
38	37, 25	readdcld 10670	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℝ)
39	38, 31	rerpdivcld 12463	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℝ)
40	39	recnd 10669	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) ∈ ℂ)
41	2, 36	remulcld 10671	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
42	41	recnd 10669	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (log‘𝑥)) ∈ ℂ)
43	33, 40, 42	addsubassd 11017	. . . . . 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 10669	. . . . . . . . 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 10669	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) ∈ ℂ)
46	4	recnd 10669	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
47	31	rpne0d 12437	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
48	44, 45, 46, 47	divdird 11454	. . . . . . . 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 10669	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))) ∈ ℂ)
50	25	recnd 10669	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) ∈ ℂ)
51	37	recnd 10669	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
52	49, 50, 51	nppcan3d 11024	. . . . . . . . . 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 12937	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))) → 𝑛 ∈ ℕ)
54	53	ad2antll 727	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → 𝑛 ∈ ℕ)
55		vmacl 25695	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
56	54, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑛) ∈ ℝ)
57	14	adantrr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑚) ∈ ℝ)
58	20	adantrr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → 𝑚 ∈ ℝ⁺)
59	58	relogcld 25206	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (log‘𝑚) ∈ ℝ)
60	57, 59	remulcld 10671	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
61	56, 60	remulcld 10671	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
62	61	recnd 10669	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
63	4, 62	fsumfldivdiag 25767	. . . . . . . . . . . . 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 10669	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℂ)
65	18	recnd 10669	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℂ)
66	21	recnd 10669	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℂ)
67	64, 65, 66	mul32d 10850	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))))
68	64, 66	mulcld 10661	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
69	68, 65	mulcomd 10662	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))) = ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))))
70		chpval 25699	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
71	16, 70	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
72	71	oveq1d 7171	. . . . . . . . . . . . . . . 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 13342	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑚))) ∈ Fin)
74	56	anassrs 470	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℝ)
75	74	recnd 10669	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℂ)
76	73, 68, 75	fsummulc1 15140	. . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . . 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 15060	. . . . . . . . . . . . 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 13342	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
81		elfznn 12937	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
82	81	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
83	82, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
84	83	recnd 10669	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
85		elfznn 12937	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
86	85	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
87	86, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
88	86	nnrpd 12430	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
89	88	relogcld 25206	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
90	87, 89	remulcld 10671	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
91	90	recnd 10669	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
92	80, 84, 91	fsummulc2 15139	. . . . . . . . . . . . . 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 15060	. . . . . . . . . . . . 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 2866	. . . . . . . . . . . 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 7172	. . . . . . . . . . 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 7171	. . . . . . . . . 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 2856	. . . . . . . . 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 7171	. . . . . . . 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 2858	. . . . . . 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 7171	. . . . . 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 2858	. . . . 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 5161	. . . 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 11068	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
104		selberg3lem2 26134	. . . . . 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 415	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
107	106	ssrdv 3973	. . . . . 6 ⊢ (⊤ → (1(,)+∞) ⊆ ℝ⁺)
108		selberg2 26127	. . . . . . 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 14920	. . . . 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 14980	. . . 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 2914	. . 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 15091	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
114	83, 113	remulcld 10671	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
115	10, 114	fsumrecl 15091	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℝ)
116	9, 115	remulcld 10671	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) ∈ ℝ)
117	116, 37	readdcld 10670	. . . . . . 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 12463	. . . . . 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 11068	. . . . 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 10669	. . . 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 483	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
122	121, 82	nndivred 11692	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
123	122	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ)
124	123, 86	nndivred 11692	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ)
125		chpcl 25701	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 𝑛) / 𝑚) ∈ ℝ → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
126	124, 125	syl 17	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
127	87, 126	remulcld 10671	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
128	80, 127	fsumrecl 15091	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
129	83, 128	remulcld 10671	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
130	10, 129	fsumrecl 15091	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
131	9, 130	remulcld 10671	. . . . . . 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 11068	. . . . . 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 12463	. . . . 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 10669	. . . 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 10669	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))) ∈ ℂ)
136	131	recnd 10669	. . . . . . . . . . . 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 11036	. . . . . . . . . . 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 10842	. . . . . . . . . . . 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 7171	. . . . . . . . . . 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 10669	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℂ)
141	89	recnd 10669	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℂ)
142	126	recnd 10669	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
143	140, 141, 142	adddid 10665	. . . . . . . . . . . . . . . . . . 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 15060	. . . . . . . . . . . . . . . . . 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 10669	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
146	80, 91, 145	fsumadd 15096	. . . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . . 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 7172	. . . . . . . . . . . . . . . 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 10669	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
150	128	recnd 10669	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
151	84, 149, 150	adddid 10665	. . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . 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 15060	. . . . . . . . . . . . . 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 10669	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
155	129	recnd 10669	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
156	10, 154, 155	fsumadd 15096	. . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . 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 7172	. . . . . . . . . . . 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 10669	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
160	115	recnd 10669	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
161	130	recnd 10669	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
162	159, 160, 161	adddid 10665	. . . . . . . . . . . 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 2856	. . . . . . . . . . 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 2866	. . . . . . . . . 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 7171	. . . . . . . . 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 10669	. . . . . . . . . 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 10997	. . . . . . . . . 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 11455	. . . . . . . . 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 11716	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
170	89, 126	readdcld 10670	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
171	87, 170	remulcld 10671	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
172	80, 171	fsumrecl 15091	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
173	83, 172	remulcld 10671	. . . . . . . . . . . . . . 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 15091	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
175	174	recnd 10669	. . . . . . . . . . . . 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 10661	. . . . . . . . . . . 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 10669	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
178	8	rpne0d 12437	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
179	176, 177, 46, 178, 47	divdiv1d 11447	. . . . . . . . . . 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 10662	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) · 𝑥) = (𝑥 · (log‘𝑥)))
181	180	oveq2d 7172	. . . . . . . . . . 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 2856	. . . . . . . . . 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 11453	. . . . . . . . . . 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 7171	. . . . . . . . . 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 12448	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
186	185	rpcnd 12434	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
187	185	rpne0d 12437	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
188	169, 175, 186, 187	divassd 11451	. . . . . . . . . 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 2864	. . . . . . . . 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 2864	. . . . . . . 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 7171	. . . . . . 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 10669	. . . . . . . 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 11029	. . . . . . 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 12463	. . . . . . . . 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 10669	. . . . . . . 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 11096	. . . . . . 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 2866	. . . . . 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 5161	. . . . 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 11068	. . . . . 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 12799	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
201		2cnd 11716	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
202		o1const 14976	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
203	200, 201, 202	sylancr 589	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
204		selbergb 26125	. . . . . . 7 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝑐
205		simpl 485	. . . . . . . . 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 487	. . . . . . . . 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 26136	. . . . . . . 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 3281	. . . . . . 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 13	. . . . . 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 14981	. . . . 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 2913	. . . 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 14986	. . 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 234	. 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 1544	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 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 +∞cpnf 10672 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 2c2 11693 ℝ⁺crp 12390 (,)cioo 12739 [,)cico 12741 ...cfz 12893 ⌊cfl 13161 abscabs 14593 𝑂(1)co1 14843 Σcsu 15042 logclog 25138 Λcvma 25669 ψcchp 25670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-o1 14847 df-lo1 14848 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-dvds 15608 df-gcd 15844 df-prm 16016 df-pc 16174 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-ulm 24965 df-log 25140 df-cxp 25141 df-atan 25445 df-em 25570 df-cht 25674 df-vma 25675 df-chp 25676 df-ppi 25677 df-mu 25678

This theorem is referenced by: selberg4r 26146

W3C validator