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Theorem pntrlog2bndlem5 27561

Description: Lemma for pntrlog2bnd 27564. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)

**Hypotheses**
Ref	Expression
pntsval.1	⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
pntrlog2bnd.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntrlog2bnd.t	⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0))
pntrlog2bndlem5.1	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntrlog2bndlem5.2	⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)

**Assertion**
Ref	Expression
pntrlog2bndlem5	⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))

Distinct variable groups: 𝑖,𝑎,𝑛,𝑥,𝑦 𝐵,𝑛,𝑥,𝑦 𝜑,𝑛,𝑥 𝑆,𝑛,𝑥,𝑦 𝑅,𝑛,𝑥,𝑦 𝑇,𝑛 Allowed substitution hints: 𝜑(𝑦,𝑖,𝑎) 𝐵(𝑖,𝑎) 𝑅(𝑖,𝑎) 𝑆(𝑖,𝑎) 𝑇(𝑥,𝑦,𝑖,𝑎)

**Proof of Theorem pntrlog2bndlem5**
Step	Hyp	Ref	Expression
1		elioore 13322	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 12940	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13352	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
8	7	simpld 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
9	5, 2, 8	ltled 11288	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13019	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrlog2bnd.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27543	. . . . . . . . . . . 12 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7030	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	10, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ)
15	14	recnd 11167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
16	15	abscld 15395	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
17	16	recnd 11167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℂ)
18	10	relogcld 26603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
19	18	recnd 11167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
20	17, 19	mulcld 11159	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
21		2cnd 12253	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
22	2, 8	rplogcld 26609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
23	22	rpne0d 12985	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
24	21, 19, 23	divcld 11925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
25		fzfid 13929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
26	10	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
27		elfznn 13501	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
28	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
29	28	nnrpd 12978	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
30	26, 29	rpdivcld 12997	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
31	12	ffvelcdmi 7030	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
33	32	recnd 11167	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
34	33	abscld 15395	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
35	29	relogcld 26603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
36		1red 11139	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
37	35, 36	readdcld 11168	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℝ)
38	34, 37	remulcld 11169	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
39	38	recnd 11167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
40	25, 39	fsumcl 15689	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
41	24, 40	mulcld 11159	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℂ)
42	20, 41	subcld 11499	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℂ)
43	34	recnd 11167	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
44	25, 43	fsumcl 15689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
45	24, 44	mulcld 11159	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℂ)
46	2	recnd 11167	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
47	10	rpne0d 12985	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
48	42, 45, 46, 47	divdird 11963	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) / 𝑥) = (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)))
49	16, 18	remulcld 11169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
50	49	recnd 11167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
51	50, 41, 45	subsubd 11527	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))))
52	24, 40, 44	subdid 11600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))))
53	25, 39, 43	fsumsub 15744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))
54	37	recnd 11167	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℂ)
55		1cnd 11133	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
56	43, 54, 55	subdid 11600	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1)))
57	35	recnd 11167	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
58	57, 55	pncand 11500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘𝑛) + 1) − 1) = (log‘𝑛))
59	58	oveq2d 7377	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
60	43	mulridd 11156	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1) = (abs‘(𝑅‘(𝑥 / 𝑛))))
61	60	oveq2d 7377	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))))
62	56, 59, 61	3eqtr3rd 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
63	62	sumeq2dv 15658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
64	53, 63	eqtr3d 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
65	64	oveq2d 7377	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
66	52, 65	eqtr3d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
67	66	oveq2d 7377	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
68	51, 67	eqtr3d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
69	68	oveq1d 7376	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) / 𝑥) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
70	48, 69	eqtr3d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
71	70	mpteq2dva 5179	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)))
72		2re 12249	. . . . . . . 8 ⊢ 2 ∈ ℝ
73	72	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
74	73, 22	rerpdivcld 13011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
75	25, 38	fsumrecl 15690	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
76	74, 75	remulcld 11169	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℝ)
77	49, 76	resubcld 11572	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℝ)
78	77, 10	rerpdivcld 13011	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ∈ ℝ)
79	25, 34	fsumrecl 15690	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
80	74, 79	remulcld 11169	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
81	80, 10	rerpdivcld 13011	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
82		1red 11139	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
83		pntsval.1	. . . . . 6 ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
84		pntrlog2bnd.t	. . . . . 6 ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0))
85	83, 11, 84	pntrlog2bndlem4 27560	. . . . 5 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1)
86	85	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1))
87	28	nnred 12183	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
88		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → 𝑎 ∈ ℝ)
89		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → 𝑎 ∈ ℝ⁺)
90	89	relogcld 26603	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → (log‘𝑎) ∈ ℝ)
91	88, 90	remulcld 11169	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → (𝑎 · (log‘𝑎)) ∈ ℝ)
92		0red 11141	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ ¬ 𝑎 ∈ ℝ⁺) → 0 ∈ ℝ)
93	91, 92	ifclda 4503	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) ∈ ℝ)
94	84, 93	fmpti 7059	. . . . . . . . . . . 12 ⊢ 𝑇:ℝ⟶ℝ
95	94	ffvelcdmi 7030	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝑇‘𝑛) ∈ ℝ)
96	87, 95	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇‘𝑛) ∈ ℝ)
97	87, 36	resubcld 11572	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
98	94	ffvelcdmi 7030	. . . . . . . . . . 11 ⊢ ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) ∈ ℝ)
99	97, 98	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇‘(𝑛 − 1)) ∈ ℝ)
100	96, 99	resubcld 11572	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ∈ ℝ)
101	34, 100	remulcld 11169	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
102	25, 101	fsumrecl 15690	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
103	74, 102	remulcld 11169	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))))) ∈ ℝ)
104	49, 103	resubcld 11572	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) ∈ ℝ)
105	104, 10	rerpdivcld 13011	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥) ∈ ℝ)
106		2rp 12941	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
107	106	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ⁺)
108	107	rpge0d 12984	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
109	73, 22, 108	divge0d 13020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 / (log‘𝑥)))
110	33	absge0d 15403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
111	29	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ⁺)
112	111	rpcnd 12982	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℂ)
113	57	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℂ)
114	112, 113	mulcld 11159	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘𝑛)) ∈ ℂ)
115		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 < 𝑛)
116		1re 11138	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
117	111	rpred 12980	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ)
118		difrp 12976	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ⁺))
119	116, 117, 118	sylancr 588	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ⁺))
120	115, 119	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 − 1) ∈ ℝ⁺)
121	120	relogcld 26603	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℝ)
122	121	recnd 11167	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℂ)
123	112, 122	mulcld 11159	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘(𝑛 − 1))) ∈ ℂ)
124	114, 123, 122	subsubd 11527	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · (log‘𝑛)) − ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1)))) = (((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
125		rpre 12945	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
126		eleq1 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑛 → (𝑎 ∈ ℝ⁺ ↔ 𝑛 ∈ ℝ⁺))
127		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑛 → 𝑎 = 𝑛)
128		fveq2 6835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛))
129	127, 128	oveq12d 7379	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑛 → (𝑎 · (log‘𝑎)) = (𝑛 · (log‘𝑛)))
130	126, 129	ifbieq1d 4492	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑛 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
131		ovex 7394	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 · (log‘𝑛)) ∈ V
132		c0ex 11132	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
133	131, 132	ifex 4518	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0) ∈ V
134	130, 84, 133	fvmpt 6942	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → (𝑇‘𝑛) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
135	125, 134	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → (𝑇‘𝑛) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
136		iftrue 4473	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0) = (𝑛 · (log‘𝑛)))
137	135, 136	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (𝑇‘𝑛) = (𝑛 · (log‘𝑛)))
138	111, 137	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘𝑛) = (𝑛 · (log‘𝑛)))
139		rpre 12945	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑛 − 1) ∈ ℝ)
140		eleq1 2825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝑛 − 1) → (𝑎 ∈ ℝ⁺ ↔ (𝑛 − 1) ∈ ℝ⁺))
141		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑛 − 1) → 𝑎 = (𝑛 − 1))
142		fveq2 6835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑛 − 1) → (log‘𝑎) = (log‘(𝑛 − 1)))
143	141, 142	oveq12d 7379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝑛 − 1) → (𝑎 · (log‘𝑎)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
144	140, 143	ifbieq1d 4492	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝑛 − 1) → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
145		ovex 7394	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) · (log‘(𝑛 − 1))) ∈ V
146	145, 132	ifex 4518	. . . . . . . . . . . . . . . . . . 19 ⊢ if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) ∈ V
147	144, 84, 146	fvmpt 6942	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
148	139, 147	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑇‘(𝑛 − 1)) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
149		iftrue 4473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
150	148, 149	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑇‘(𝑛 − 1)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
151	120, 150	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
152		1cnd 11133	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℂ)
153	112, 152, 122	subdird 11601	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · (log‘(𝑛 − 1))) = ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))))
154	122	mullidd 11157	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · (log‘(𝑛 − 1))) = (log‘(𝑛 − 1)))
155	154	oveq2d 7377	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
156	151, 153, 155	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
157	138, 156	oveq12d 7379	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · (log‘𝑛)) − ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1)))))
158	112, 113, 122	subdid 11600	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))))
159	158	oveq1d 7376	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) = (((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
160	124, 157, 159	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
161	111	relogcld 26603	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℝ)
162	161, 121	resubcld 11572	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℝ)
163	162	recnd 11167	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℂ)
164	112, 152, 163	subdird 11601	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − (1 · ((log‘𝑛) − (log‘(𝑛 − 1))))))
165	163	mullidd 11157	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
166	165	oveq2d 7377	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − (1 · ((log‘𝑛) − (log‘(𝑛 − 1))))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − ((log‘𝑛) − (log‘(𝑛 − 1)))))
167	117, 162	remulcld 11169	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℝ)
168	167	recnd 11167	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℂ)
169	168, 113, 122	subsub3d 11529	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − ((log‘𝑛) − (log‘(𝑛 − 1)))) = (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)))
170	164, 166, 169	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) = (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)))
171	112, 152	npcand 11503	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) + 1) = 𝑛)
172	171	fveq2d 6839	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
173	172	oveq1d 7376	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
174		logdifbnd 26974	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
175	120, 174	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
176	173, 175	eqbrtrrd 5110	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
177		1red 11139	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℝ)
178	162, 177, 120	lemuldiv2d 13030	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1 ↔ ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1))))
179	176, 178	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1)
180	170, 179	eqbrtrrd 5110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1)
181	167, 121	readdcld 11168	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ∈ ℝ)
182	181, 161, 177	lesubadd2d 11743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1 ↔ ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1)))
183	180, 182	mpbid 232	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
184	160, 183	eqbrtrd 5108	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
185		fveq2 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑇‘𝑛) = (𝑇‘1))
186		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 1 → 𝑎 = 1)
187	186, 3	eqeltrdi 2845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 1 → 𝑎 ∈ ℝ⁺)
188	187	iftrued 4475	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 1 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = (𝑎 · (log‘𝑎)))
189		fveq2 6835	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 1 → (log‘𝑎) = (log‘1))
190		log1 26565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (log‘1) = 0
191	189, 190	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 1 → (log‘𝑎) = 0)
192	186, 191	oveq12d 7379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 1 → (𝑎 · (log‘𝑎)) = (1 · 0))
193		ax-1cn 11090	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℂ
194	193	mul01i 11330	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 · 0) = 0
195	192, 194	eqtrdi 2788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 1 → (𝑎 · (log‘𝑎)) = 0)
196	188, 195	eqtrd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 1 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = 0)
197	196, 84, 132	fvmpt 6942	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → (𝑇‘1) = 0)
198	116, 197	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇‘1) = 0
199	185, 198	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑇‘𝑛) = 0)
200		oveq1 7368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
201		1m1e0 12247	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 1) = 0
202	200, 201	eqtrdi 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
203	202	fveq2d 6839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = (𝑇‘0))
204		0re 11140	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
205		rpne0 12953	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℝ⁺ → 𝑎 ≠ 0)
206	205	necon2bi 2963	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → ¬ 𝑎 ∈ ℝ⁺)
207	206	iffalsed 4478	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 0 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = 0)
208	207, 84, 132	fvmpt 6942	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℝ → (𝑇‘0) = 0)
209	204, 208	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇‘0) = 0
210	203, 209	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = 0)
211	199, 210	oveq12d 7379	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = (0 − 0))
212		0m0e0 12290	. . . . . . . . . . . . . . 15 ⊢ (0 − 0) = 0
213	211, 212	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = 0)
214	213	eqcoms 2745	. . . . . . . . . . . . 13 ⊢ (1 = 𝑛 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = 0)
215	214	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = 0)
216		0red 11141	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ∈ ℝ)
217	28	nnge1d 12219	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
218	87, 217	logge0d 26610	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
219	35	lep1d 12081	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ≤ ((log‘𝑛) + 1))
220	216, 35, 37, 218, 219	letrd 11297	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘𝑛) + 1))
221	220	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → 0 ≤ ((log‘𝑛) + 1))
222	215, 221	eqbrtrd 5108	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
223		elfzle1 13475	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 1 ≤ 𝑛)
224	223	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
225	36, 87	leloed 11283	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 ≤ 𝑛 ↔ (1 < 𝑛 ∨ 1 = 𝑛)))
226	224, 225	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 < 𝑛 ∨ 1 = 𝑛))
227	184, 222, 226	mpjaodan 961	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
228	100, 37, 34, 110, 227	lemul2ad 12090	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))
229	25, 101, 38, 228	fsumle 15756	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))
230	102, 75, 74, 109, 229	lemul2ad 12090	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))))) ≤ ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))))
231	103, 76, 49, 230	lesub2dd 11761	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ≤ (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))))
232	77, 104, 10, 231	lediv1dd 13038	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ≤ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥))
233	232	adantrr 718	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ≤ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥))
234	82, 86, 105, 78, 233	lo1le 15608	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥)) ∈ ≤𝑂(1))
235	106	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ⁺)
236		pntrlog2bndlem5.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
237	235, 236	rpmulcld 12996	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ⁺)
238	237	rpred 12980	. . . . . 6 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
239	238	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ)
240	5, 22	rerpdivcld 13011	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
241	5, 240	readdcld 11168	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
242		ioossre 13354	. . . . . 6 ⊢ (1(,)+∞) ⊆ ℝ
243		lo1const 15577	. . . . . 6 ⊢ (((1(,)+∞) ⊆ ℝ ∧ (2 · 𝐵) ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
244	242, 238, 243	sylancr 588	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
245		lo1const 15577	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
246	242, 82, 245	sylancr 588	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
247		divlogrlim 26615	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0
248		rlimo1 15573	. . . . . . . 8 ⊢ ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
249	247, 248	mp1i 13	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
250	240, 249	o1lo1d 15495	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ ≤𝑂(1))
251	5, 240, 246, 250	lo1add 15583	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ ≤𝑂(1))
252	237	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ⁺)
253	252	rpge0d 12984	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 · 𝐵))
254	239, 241, 244, 251, 253	lo1mul 15584	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 · 𝐵) · (1 + (1 / (log‘𝑥))))) ∈ ≤𝑂(1))
255	239, 241	remulcld 11169	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
256	79, 10	rerpdivcld 13011	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
257	18, 5	readdcld 11168	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
258	236	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ⁺)
259	258	rpred 12980	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
260	257, 259	remulcld 11169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) · 𝐵) ∈ ℝ)
261	28	nnrecred 12222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
262	25, 261	fsumrecl 15690	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
263	262, 259	remulcld 11169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ∈ ℝ)
264	34, 26	rerpdivcld 13011	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
265	259	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
266	261, 265	remulcld 11169	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · 𝐵) ∈ ℝ)
267	30	rpcnd 12982	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
268	30	rpne0d 12985	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
269	33, 267, 268	absdivd 15414	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))))
270	2	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
271	270, 28	nndivred 12225	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
272	30	rpge0d 12984	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
273	271, 272	absidd 15379	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
274	273	oveq2d 7377	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
275	269, 274	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
276	46	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
277	87	recnd 11167	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
278	47	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
279	28	nnne0d 12221	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
280	43, 276, 277, 278, 279	divdiv2d 11957	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥))
281	43, 277, 276, 278	div23d 11962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
282	275, 280, 281	3eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
283		fveq2 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛)))
284		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
285	283, 284	oveq12d 7379	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
286	285	fveq2d 6839	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
287	286	breq1d 5096	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
288		pntrlog2bndlem5.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
289	288	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
290	287, 289, 30	rspcdva 3566	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
291	282, 290	eqbrtrrd 5110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵)
292	264, 265, 29	lemuldivd 13029	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵 ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛)))
293	291, 292	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛))
294	265	recnd 11167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
295	294, 277, 279	divrec2d 11929	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐵 / 𝑛) = ((1 / 𝑛) · 𝐵))
296	293, 295	breqtrd 5112	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ ((1 / 𝑛) · 𝐵))
297	25, 264, 266, 296	fsumle 15756	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
298	25, 46, 43, 47	fsumdivc 15742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥))
299	258	rpcnd 12982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
300	261	recnd 11167	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
301	25, 299, 300	fsummulc1 15741	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) = Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
302	297, 298, 301	3brtr4d 5118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵))
303	258	rpge0d 12984	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
304		harmonicubnd 26990	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
305	2, 9, 304	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
306	262, 257, 259, 303, 305	lemul1ad 12089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ≤ (((log‘𝑥) + 1) · 𝐵))
307	256, 263, 260, 302, 306	letrd 11297	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (((log‘𝑥) + 1) · 𝐵))
308	256, 260, 74, 109, 307	lemul2ad 12090	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)) ≤ ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
309	24, 44, 46, 47	divassd 11960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) = ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)))
310	241	recnd 11167	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℂ)
311	21, 299, 310	mul32d 11350	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵))
312		1cnd 11133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
313	19, 312, 19, 23	divdird 11963	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
314	19, 23	dividd 11923	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
315	314	oveq1d 7376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
316	313, 315	eqtr2d 2773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
317	316	oveq2d 7377	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
318	19, 312	addcld 11158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
319	21, 19, 318, 23	div32d 11948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
320	317, 319	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)))
321	320	oveq1d 7376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵) = (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵))
322	24, 318, 299	mulassd 11162	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
323	311, 321, 322	3eqtrd 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
324	308, 309, 323	3brtr4d 5118	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
325	324	adantrr 718	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
326	82, 254, 255, 81, 325	lo1le 15608	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
327	78, 81, 234, 326	lo1add 15583	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥))) ∈ ≤𝑂(1))
328	71, 327	eqeltrrd 2838	1 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ⊆ wss 3890 ifcif 4467 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 +∞cpnf 11170 < clt 11173 ≤ cle 11174 − cmin 11371 / cdiv 11801 ℕcn 12168 2c2 12230 ℝ⁺crp 12936 (,)cioo 13292 ...cfz 13455 ⌊cfl 13743 abscabs 15190 ⇝_𝑟 crli 15441 𝑂(1)co1 15442 ≤𝑂(1)clo1 15443 Σcsu 15642 logclog 26534 Λcvma 27072 ψcchp 27073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-xnn0 12505 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ioc 13297 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15023 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-limsup 15427 df-clim 15444 df-rlim 15445 df-o1 15446 df-lo1 15447 df-sum 15643 df-ef 16026 df-e 16027 df-sin 16028 df-cos 16029 df-tan 16030 df-pi 16031 df-dvds 16216 df-gcd 16458 df-prm 16635 df-pc 16802 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-pt 17401 df-prds 17404 df-xrs 17460 df-qtop 17465 df-imas 17466 df-xps 17468 df-mre 17542 df-mrc 17543 df-acs 17545 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-mulg 19038 df-cntz 19286 df-cmn 19751 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cld 22997 df-ntr 22998 df-cls 22999 df-nei 23076 df-lp 23114 df-perf 23115 df-cn 23205 df-cnp 23206 df-haus 23293 df-cmp 23365 df-tx 23540 df-hmeo 23733 df-fil 23824 df-fm 23916 df-flim 23917 df-flf 23918 df-xms 24298 df-ms 24299 df-tms 24300 df-cncf 24858 df-limc 25846 df-dv 25847 df-ulm 26358 df-log 26536 df-cxp 26537 df-atan 26847 df-em 26973 df-cht 27077 df-vma 27078 df-chp 27079 df-ppi 27080 df-mu 27081

This theorem is referenced by: pntrlog2bndlem6 27563

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