selberg4r - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > selberg4r		Structured version Visualization version GIF version

Theorem selberg4r 27514

Description: Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))

**Assertion**
Ref	Expression
selberg4r	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)

Distinct variable groups: 𝑚,𝑎,𝑛,𝑥 𝑅,𝑚,𝑛,𝑥 Allowed substitution hint: 𝑅(𝑎)

**Proof of Theorem selberg4r**
Step	Hyp	Ref	Expression
1		elioore 13312	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 12931	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11151	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13342	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
8	7	simpld 494	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
9	5, 2, 8	ltled 11298	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13010	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrval.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrval 27506	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
13	10, 12	syl 17	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
14	13	oveq1d 7384	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥)))
15		chpcl 27067	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
16	2, 15	syl 17	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
17	16	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℂ)
18	2	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
19	10	relogcld 26565	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
20	19	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
21	17, 18, 20	subdird 11611	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
22	14, 21	eqtrd 2764	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
23	10	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑥 ∈ ℝ⁺)
24		elfznn 13490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
25	24	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
26	25	nnrpd 12969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
27	26	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℝ⁺)
28	23, 27	rpdivcld 12988	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29		elfznn 13490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
30	29	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
31	30	nnrpd 12969	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
32	28, 31	rpdivcld 12988	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ⁺)
33	11	pntrval 27506	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 / 𝑛) / 𝑚) ∈ ℝ⁺ → (𝑅‘((𝑥 / 𝑛) / 𝑚)) = ((ψ‘((𝑥 / 𝑛) / 𝑚)) − ((𝑥 / 𝑛) / 𝑚)))
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑅‘((𝑥 / 𝑛) / 𝑚)) = ((ψ‘((𝑥 / 𝑛) / 𝑚)) − ((𝑥 / 𝑛) / 𝑚)))
35	34	oveq2d 7385	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚))) = ((Λ‘𝑚) · ((ψ‘((𝑥 / 𝑛) / 𝑚)) − ((𝑥 / 𝑛) / 𝑚))))
36		vmacl 27061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
37	30, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
38	37	recnd 11178	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℂ)
39	2	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
40	39, 25	nndivred 12216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
41	40	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ)
42	41, 30	nndivred 12216	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ)
43		chpcl 27067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 / 𝑛) / 𝑚) ∈ ℝ → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
44	42, 43	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
45	44	recnd 11178	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
46	42	recnd 11178	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℂ)
47	38, 45, 46	subdid 11610	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((ψ‘((𝑥 / 𝑛) / 𝑚)) − ((𝑥 / 𝑛) / 𝑚))) = (((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))
48	35, 47	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚))) = (((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))
49	48	sumeq2dv 15644	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))
50		fzfid 13914	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
51	37, 44	remulcld 11180	. . . . . . . . . . . . . . . . 17 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
52	51	recnd 11178	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
53	38, 46	mulcld 11170	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
54	50, 52, 53	fsumsub 15730	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))
55	49, 54	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))
56	55	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))
57		vmacl 27061	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
58	25, 57	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
59	58	recnd 11178	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
60	50, 51	fsumrecl 15676	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
61	60	recnd 11178	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
62	50, 53	fsumcl 15675	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
63	59, 61, 62	subdid 11610	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))) − Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) = (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))
64	56, 63	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))) = (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))
65	64	sumeq2dv 15644	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))
66		fzfid 13914	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
67	58, 60	remulcld 11180	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
68	67	recnd 11178	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
69	59, 62	mulcld 11170	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
70	66, 68, 69	fsumsub 15730	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))
71	65, 70	eqtrd 2764	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))
72	71	oveq2d 7385	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚))))) = ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))))
73		2re 12236	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
74	73	a1i 11	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
75	2, 8	rplogcld 26571	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
76	74, 75	rerpdivcld 13002	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
77	76	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
78	66, 67	fsumrecl 15676	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
79	78	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
80	66, 69	fsumcl 15675	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
81	77, 79, 80	subdid 11610	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) = (((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) · ((𝑥 / 𝑛) / 𝑚))))))
82	72, 81	eqtrd 2764	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚))))) = (((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) · ((𝑥 / 𝑛) / 𝑚))))))
83	22, 82	oveq12d 7387	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) = ((((ψ‘𝑥) · (log‘𝑥)) − (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) · ((𝑥 / 𝑛) / 𝑚)))))))
84	16, 19	remulcld 11180	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
85	84	recnd 11178	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
86	18, 20	mulcld 11170	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
87	76, 78	remulcld 11180	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
88	87	recnd 11178	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ)
89	77, 80	mulcld 11170	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
90	85, 86, 88, 89	sub4d 11558	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) − (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) · ((𝑥 / 𝑛) / 𝑚)))))) = ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) − ((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))))
91	83, 90	eqtrd 2764	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) = ((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) − ((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))))
92	91	oveq1d 7384	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) − ((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))) / 𝑥))
93	84, 87	resubcld 11582	. . . . . . . 8 ⊢ ((⊤ ∧ 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)
94	93	recnd 11178	. . . . . . 7 ⊢ ((⊤ ∧ 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)
95	2, 19	remulcld 11180	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ)
96	37, 42	remulcld 11180	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
97	50, 96	fsumrecl 15676	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
98	58, 97	remulcld 11180	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
99	66, 98	fsumrecl 15676	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
100	76, 99	remulcld 11180	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
101	95, 100	resubcld 11582	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
102	101	recnd 11178	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ)
103	10	rpne0d 12976	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
104	94, 102, 18, 103	divsubdird 11973	. . . . . 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)))))) − ((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))) / 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 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) / 𝑥)))
105	95	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
106	99	recnd 11178	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
107	77, 106	mulcld 11170	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
108	105, 107, 18, 103	divsubdird 11973	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) / 𝑥) = (((𝑥 · (log‘𝑥)) / 𝑥) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) / 𝑥)))
109	20, 18, 103	divcan3d 11939	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥))
110	77, 106, 18, 103	divassd 11969	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) / 𝑥) = ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥)))
111	98	recnd 11178	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) ∈ ℂ)
112	66, 18, 111, 103	fsumdivc 15728	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥))
113	41	recnd 11178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℂ)
114	30	nncnd 12178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
115	30	nnne0d 12212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≠ 0)
116	113, 38, 114, 115	div12d 11970	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) · ((Λ‘𝑚) / 𝑚)) = ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))
117	18	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
118	117	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑥 ∈ ℂ)
119	25	nncnd 12178	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
120	119	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℂ)
121	37, 30	nndivred 12216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) / 𝑚) ∈ ℝ)
122	121	recnd 11178	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) / 𝑚) ∈ ℂ)
123	25	nnne0d 12212	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
124	123	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ≠ 0)
125	118, 120, 122, 124	div32d 11957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) · ((Λ‘𝑚) / 𝑚)) = (𝑥 · (((Λ‘𝑚) / 𝑚) / 𝑛)))
126	116, 125	eqtr3d 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) = (𝑥 · (((Λ‘𝑚) / 𝑚) / 𝑛)))
127	126	oveq1d 7384	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥) = ((𝑥 · (((Λ‘𝑚) / 𝑚) / 𝑛)) / 𝑥))
128	25	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℕ)
129	121, 128	nndivred 12216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑚) / 𝑚) / 𝑛) ∈ ℝ)
130	129	recnd 11178	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑚) / 𝑚) / 𝑛) ∈ ℂ)
131	103	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
132	131	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑥 ≠ 0)
133	130, 118, 132	divcan3d 11939	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 · (((Λ‘𝑚) / 𝑚) / 𝑛)) / 𝑥) = (((Λ‘𝑚) / 𝑚) / 𝑛))
134	127, 133	eqtrd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥) = (((Λ‘𝑚) / 𝑚) / 𝑛))
135	134	sumeq2dv 15644	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) / 𝑚) / 𝑛))
136	96	recnd 11178	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
137	50, 117, 136, 131	fsumdivc 15728	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥))
138	50, 119, 122, 123	fsumdivc 15728	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) / 𝑛) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑚) / 𝑚) / 𝑛))
139	135, 137, 138	3eqtr4d 2774	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) / 𝑛))
140	139	oveq2d 7385	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥)) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) / 𝑛)))
141	97	recnd 11178	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) ∈ ℂ)
142	59, 141, 117, 131	divassd 11969	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)) / 𝑥)))
143	50, 121	fsumrecl 15676	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) ∈ ℝ)
144	143	recnd 11178	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) ∈ ℂ)
145	59, 119, 144, 123	div32d 11957	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) / 𝑛)))
146	140, 142, 145	3eqtr4d 2774	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥) = (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)))
147	146	sumeq2dv 15644	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)))
148	112, 147	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)))
149	148	oveq2d 7385	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))) / 𝑥)) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))))
150	110, 149	eqtrd 2764	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) / 𝑥) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))))
151	109, 150	oveq12d 7387	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑥 · (log‘𝑥)) / 𝑥) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))) / 𝑥)) = ((log‘𝑥) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)))))
152	108, 151	eqtrd 2764	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) / 𝑥) = ((log‘𝑥) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)))))
153	152	oveq2d 7385	. . . . . . 7 ⊢ ((⊤ ∧ 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) − (((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (ψ‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥) − ((log‘𝑥) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))))))
154	94, 18, 103	divcld 11934	. . . . . . . 8 ⊢ ((⊤ ∧ 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)
155	58, 25	nndivred 12216	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
156	155, 143	remulcld 11180	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℝ)
157	66, 156	fsumrecl 15676	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℝ)
158	76, 157	remulcld 11180	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) ∈ ℝ)
159	158	recnd 11178	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) ∈ ℂ)
160	154, 20, 159	subsub2d 11538	. . . . . . 7 ⊢ ((⊤ ∧ 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) − ((log‘𝑥) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))))) = (((((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) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥))))
161	153, 160	eqtrd 2764	. . . . . 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)))))) / 𝑥) − (((𝑥 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚))))) / 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) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥))))
162	104, 161	eqtrd 2764	. . . . 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 · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((𝑥 / 𝑛) / 𝑚)))))) / 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) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥))))
163	92, 162	eqtrd 2764	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 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) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥))))
164	163	mpteq2dva 5195	. . 3 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) = (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) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥)))))
165	93, 10	rerpdivcld 13002	. . . 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) ∈ ℝ)
166	158, 19	resubcld 11582	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥)) ∈ ℝ)
167		selberg4 27505	. . . . 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)) ∈ 𝑂(1)
168	167	a1i 11	. . . 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)) ∈ 𝑂(1))
169		2cnd 12240	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
170	157, 75	rerpdivcld 13002	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) ∈ ℝ)
171	170	recnd 11178	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) ∈ ℂ)
172	19	rehalfcld 12405	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ)
173	172	recnd 11178	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
174	169, 171, 173	subdid 11610	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2))))
175	157	recnd 11178	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℂ)
176	75	rpne0d 12976	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
177	169, 20, 175, 176	div32d 11957	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥))))
178	177	eqcomd 2735	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))))
179		2ne0 12266	. . . . . . . . . 10 ⊢ 2 ≠ 0
180	179	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ≠ 0)
181	20, 169, 180	divcan2d 11936	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
182	178, 181	oveq12d 7387	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥)))
183	174, 182	eqtrd 2764	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥)))
184	183	mpteq2dva 5195	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥))))
185	170, 172	resubcld 11582	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ)
186		ioossre 13344	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
187		2cnd 12240	. . . . . . 7 ⊢ (⊤ → 2 ∈ ℂ)
188		o1const 15562	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
189	186, 187, 188	sylancr 587	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
190		2vmadivsum 27485	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
191	190	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
192	74, 185, 189, 191	o1mul2 15567	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
193	184, 192	eqeltrrd 2829	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥))) ∈ 𝑂(1))
194	165, 166, 168, 193	o1add2 15566	. . 3 ⊢ (⊤ → (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) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))) − (log‘𝑥)))) ∈ 𝑂(1))
195	164, 194	eqeltrd 2828	. 2 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1))
196	195	mptru 1547	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (𝑅‘((𝑥 / 𝑛) / 𝑚)))))) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3911 class class class wbr 5102 ↦ cmpt 5183 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 +∞cpnf 11181 < clt 11184 − cmin 11381 / cdiv 11811 ℕcn 12162 2c2 12217 ℝ⁺crp 12927 (,)cioo 13282 ...cfz 13444 ⌊cfl 13728 𝑂(1)co1 15428 Σcsu 15628 logclog 26496 Λcvma 27035 ψcchp 27036

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-o1 15432 df-lo1 15433 df-sum 15629 df-ef 16009 df-e 16010 df-sin 16011 df-cos 16012 df-tan 16013 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801 df-ulm 26319 df-log 26498 df-cxp 26499 df-atan 26810 df-em 26936 df-cht 27040 df-vma 27041 df-chp 27042 df-ppi 27043 df-mu 27044

This theorem is referenced by: selberg34r 27515

W3C validator