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Theorem pntrlog2bndlem5 27625

Description: Lemma for pntrlog2bnd 27628. 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 13417	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 13038	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11262	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13446	. . . . . . . . . . . . . . 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 11409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13116	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrlog2bnd.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27607	. . . . . . . . . . . 12 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7103	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	10, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ)
15	14	recnd 11289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
16	15	abscld 15475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
17	16	recnd 11289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℂ)
18	10	relogcld 26665	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
19	18	recnd 11289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
20	17, 19	mulcld 11281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
21		2cnd 12344	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
22	2, 8	rplogcld 26671	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
23	22	rpne0d 13082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
24	21, 19, 23	divcld 12043	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
25		fzfid 14014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
26	10	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
27		elfznn 13593	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
28	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
29	28	nnrpd 13075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
30	26, 29	rpdivcld 13094	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
31	12	ffvelcdmi 7103	. . . . . . . . . . . . 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 11289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
34	33	abscld 15475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
35	29	relogcld 26665	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
36		1red 11262	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
37	35, 36	readdcld 11290	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℝ)
38	34, 37	remulcld 11291	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
39	38	recnd 11289	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
40	25, 39	fsumcl 15769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
41	24, 40	mulcld 11281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℂ)
42	20, 41	subcld 11620	. . . . 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 11289	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
44	25, 43	fsumcl 15769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
45	24, 44	mulcld 11281	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℂ)
46	2	recnd 11289	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
47	10	rpne0d 13082	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
48	42, 45, 46, 47	divdird 12081	. . . 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 11291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
50	49	recnd 11289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
51	50, 41, 45	subsubd 11648	. . . . . 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 11719	. . . . . . . 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 15824	. . . . . . . . . 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 11289	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℂ)
55		1cnd 11256	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
56	43, 54, 55	subdid 11719	. . . . . . . . . . . 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 11289	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
58	57, 55	pncand 11621	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘𝑛) + 1) − 1) = (log‘𝑛))
59	58	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
60	43	mulridd 11278	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1) = (abs‘(𝑅‘(𝑥 / 𝑛))))
61	60	oveq2d 7447	. . . . . . . . . . . 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 2786	. . . . . . . . . . 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 15738	. . . . . . . . . 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 2779	. . . . . . . . 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 7447	. . . . . . . 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 2779	. . . . . . 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 7447	. . . . . 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 2779	. . . . 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 7446	. . . 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 2779	. . 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 5242	. 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 12340	. . . . . . . 8 ⊢ 2 ∈ ℝ
73	72	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
74	73, 22	rerpdivcld 13108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
75	25, 38	fsumrecl 15770	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
76	74, 75	remulcld 11291	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℝ)
77	49, 76	resubcld 11691	. . . 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 13108	. . 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 15770	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
80	74, 79	remulcld 11291	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
81	80, 10	rerpdivcld 13108	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
82		1red 11262	. . . 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 27624	. . . . 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 12281	. . . . . . . . . . 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 26665	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → (log‘𝑎) ∈ ℝ)
91	88, 90	remulcld 11291	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → (𝑎 · (log‘𝑎)) ∈ ℝ)
92		0red 11264	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ ¬ 𝑎 ∈ ℝ⁺) → 0 ∈ ℝ)
93	91, 92	ifclda 4561	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) ∈ ℝ)
94	84, 93	fmpti 7132	. . . . . . . . . . . 12 ⊢ 𝑇:ℝ⟶ℝ
95	94	ffvelcdmi 7103	. . . . . . . . . . 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 11691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
98	94	ffvelcdmi 7103	. . . . . . . . . . 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 11691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ∈ ℝ)
101	34, 100	remulcld 11291	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
102	25, 101	fsumrecl 15770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
103	74, 102	remulcld 11291	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))))) ∈ ℝ)
104	49, 103	resubcld 11691	. . . . 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 13108	. . . 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 13039	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
107	106	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ⁺)
108	107	rpge0d 13081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
109	73, 22, 108	divge0d 13117	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 / (log‘𝑥)))
110	33	absge0d 15483	. . . . . . . . . 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 13079	. . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . 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 11261	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
117	111	rpred 13077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ)
118		difrp 13073	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ⁺))
119	116, 117, 118	sylancr 587	. . . . . . . . . . . . . . . . . 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 26665	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℝ)
122	121	recnd 11289	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℂ)
123	112, 122	mulcld 11281	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘(𝑛 − 1))) ∈ ℂ)
124	114, 123, 122	subsubd 11648	. . . . . . . . . . . . 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 13043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
126		eleq1 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑛 → (𝑎 ∈ ℝ⁺ ↔ 𝑛 ∈ ℝ⁺))
127		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑛 → 𝑎 = 𝑛)
128		fveq2 6906	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛))
129	127, 128	oveq12d 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑛 → (𝑎 · (log‘𝑎)) = (𝑛 · (log‘𝑛)))
130	126, 129	ifbieq1d 4550	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑛 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
131		ovex 7464	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 · (log‘𝑛)) ∈ V
132		c0ex 11255	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
133	131, 132	ifex 4576	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0) ∈ V
134	130, 84, 133	fvmpt 7016	. . . . . . . . . . . . . . . . 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 4531	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0) = (𝑛 · (log‘𝑛)))
137	135, 136	eqtrd 2777	. . . . . . . . . . . . . . 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 13043	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑛 − 1) ∈ ℝ)
140		eleq1 2829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝑛 − 1) → (𝑎 ∈ ℝ⁺ ↔ (𝑛 − 1) ∈ ℝ⁺))
141		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑛 − 1) → 𝑎 = (𝑛 − 1))
142		fveq2 6906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑛 − 1) → (log‘𝑎) = (log‘(𝑛 − 1)))
143	141, 142	oveq12d 7449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝑛 − 1) → (𝑎 · (log‘𝑎)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
144	140, 143	ifbieq1d 4550	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝑛 − 1) → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
145		ovex 7464	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) · (log‘(𝑛 − 1))) ∈ V
146	145, 132	ifex 4576	. . . . . . . . . . . . . . . . . . 19 ⊢ if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) ∈ V
147	144, 84, 146	fvmpt 7016	. . . . . . . . . . . . . . . . . 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 4531	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
150	148, 149	eqtrd 2777	. . . . . . . . . . . . . . . 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 11256	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℂ)
153	112, 152, 122	subdird 11720	. . . . . . . . . . . . . . 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 11279	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · (log‘(𝑛 − 1))) = (log‘(𝑛 − 1)))
155	154	oveq2d 7447	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
157	138, 156	oveq12d 7449	. . . . . . . . . . . . 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 11719	. . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . 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 2787	. . . . . . . . . . . 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 26665	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℝ)
162	161, 121	resubcld 11691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℝ)
163	162	recnd 11289	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℂ)
164	112, 152, 163	subdird 11720	. . . . . . . . . . . . . . 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 11279	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
166	165	oveq2d 7447	. . . . . . . . . . . . . . 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 11291	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℝ)
168	167	recnd 11289	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℂ)
169	168, 113, 122	subsub3d 11650	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 11624	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) + 1) = 𝑛)
172	171	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
173	172	oveq1d 7446	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
174		logdifbnd 27037	. . . . . . . . . . . . . . . . 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 5167	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
177		1red 11262	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℝ)
178	162, 177, 120	lemuldiv2d 13127	. . . . . . . . . . . . . . 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 5167	. . . . . . . . . . . . 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 11290	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ∈ ℝ)
182	181, 161, 177	lesubadd2d 11862	. . . . . . . . . . . . 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 5165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
185		fveq2 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑇‘𝑛) = (𝑇‘1))
186		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 1 → 𝑎 = 1)
187	186, 3	eqeltrdi 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 1 → 𝑎 ∈ ℝ⁺)
188	187	iftrued 4533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 1 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = (𝑎 · (log‘𝑎)))
189		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 1 → (log‘𝑎) = (log‘1))
190		log1 26627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (log‘1) = 0
191	189, 190	eqtrdi 2793	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 1 → (log‘𝑎) = 0)
192	186, 191	oveq12d 7449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 1 → (𝑎 · (log‘𝑎)) = (1 · 0))
193		ax-1cn 11213	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℂ
194	193	mul01i 11451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 · 0) = 0
195	192, 194	eqtrdi 2793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 1 → (𝑎 · (log‘𝑎)) = 0)
196	188, 195	eqtrd 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 1 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = 0)
197	196, 84, 132	fvmpt 7016	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → (𝑇‘1) = 0)
198	116, 197	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇‘1) = 0
199	185, 198	eqtrdi 2793	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑇‘𝑛) = 0)
200		oveq1 7438	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
201		1m1e0 12338	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 1) = 0
202	200, 201	eqtrdi 2793	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
203	202	fveq2d 6910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = (𝑇‘0))
204		0re 11263	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
205		rpne0 13051	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℝ⁺ → 𝑎 ≠ 0)
206	205	necon2bi 2971	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → ¬ 𝑎 ∈ ℝ⁺)
207	206	iffalsed 4536	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 0 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = 0)
208	207, 84, 132	fvmpt 7016	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℝ → (𝑇‘0) = 0)
209	204, 208	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇‘0) = 0
210	203, 209	eqtrdi 2793	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = 0)
211	199, 210	oveq12d 7449	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = (0 − 0))
212		0m0e0 12386	. . . . . . . . . . . . . . 15 ⊢ (0 − 0) = 0
213	211, 212	eqtrdi 2793	. . . . . . . . . . . . . 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 11264	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ∈ ℝ)
217	28	nnge1d 12314	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
218	87, 217	logge0d 26672	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
219	35	lep1d 12199	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ≤ ((log‘𝑛) + 1))
220	216, 35, 37, 218, 219	letrd 11418	. . . . . . . . . . . . 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 5165	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
223		elfzle1 13567	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 1 ≤ 𝑛)
224	223	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
225	36, 87	leloed 11404	. . . . . . . . . . . 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 12208	. . . . . . . . 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 15835	. . . . . . . 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 12208	. . . . . . 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 11880	. . . . . 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 13135	. . . . 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 717	. . . 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 15688	. . 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 13093	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ⁺)
238	237	rpred 13077	. . . . . 6 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
239	238	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ)
240	5, 22	rerpdivcld 13108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
241	5, 240	readdcld 11290	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
242		ioossre 13448	. . . . . 6 ⊢ (1(,)+∞) ⊆ ℝ
243		lo1const 15657	. . . . . 6 ⊢ (((1(,)+∞) ⊆ ℝ ∧ (2 · 𝐵) ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
244	242, 238, 243	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
245		lo1const 15657	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
246	242, 82, 245	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
247		divlogrlim 26677	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0
248		rlimo1 15653	. . . . . . . 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 15575	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ ≤𝑂(1))
251	5, 240, 246, 250	lo1add 15663	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ ≤𝑂(1))
252	237	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ⁺)
253	252	rpge0d 13081	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 · 𝐵))
254	239, 241, 244, 251, 253	lo1mul 15664	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 · 𝐵) · (1 + (1 / (log‘𝑥))))) ∈ ≤𝑂(1))
255	239, 241	remulcld 11291	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
256	79, 10	rerpdivcld 13108	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
257	18, 5	readdcld 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
258	236	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ⁺)
259	258	rpred 13077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
260	257, 259	remulcld 11291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) · 𝐵) ∈ ℝ)
261	28	nnrecred 12317	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
262	25, 261	fsumrecl 15770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
263	262, 259	remulcld 11291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ∈ ℝ)
264	34, 26	rerpdivcld 13108	. . . . . . . . . 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 11291	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · 𝐵) ∈ ℝ)
267	30	rpcnd 13079	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
268	30	rpne0d 13082	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
269	33, 267, 268	absdivd 15494	. . . . . . . . . . . . . . 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 12320	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
272	30	rpge0d 13081	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
273	271, 272	absidd 15461	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
274	273	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
275	269, 274	eqtrd 2777	. . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . 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 12316	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
280	43, 276, 277, 278, 279	divdiv2d 12075	. . . . . . . . . . . . . 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 12080	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
282	275, 280, 281	3eqtrd 2781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
283		fveq2 6906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛)))
284		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
285	283, 284	oveq12d 7449	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
286	285	fveq2d 6910	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
287	286	breq1d 5153	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
290	287, 289, 30	rspcdva 3623	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
291	282, 290	eqbrtrrd 5167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵)
292	264, 265, 29	lemuldivd 13126	. . . . . . . . . . . 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 11289	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
295	294, 277, 279	divrec2d 12047	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐵 / 𝑛) = ((1 / 𝑛) · 𝐵))
296	293, 295	breqtrd 5169	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ ((1 / 𝑛) · 𝐵))
297	25, 264, 266, 296	fsumle 15835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
298	25, 46, 43, 47	fsumdivc 15822	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥))
299	258	rpcnd 13079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
300	261	recnd 11289	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
301	25, 299, 300	fsummulc1 15821	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) = Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
302	297, 298, 301	3brtr4d 5175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵))
303	258	rpge0d 13081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
304		harmonicubnd 27053	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
305	2, 9, 304	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
306	262, 257, 259, 303, 305	lemul1ad 12207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ≤ (((log‘𝑥) + 1) · 𝐵))
307	256, 263, 260, 302, 306	letrd 11418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (((log‘𝑥) + 1) · 𝐵))
308	256, 260, 74, 109, 307	lemul2ad 12208	. . . . . 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 12078	. . . . . 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 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℂ)
311	21, 299, 310	mul32d 11471	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵))
312		1cnd 11256	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
313	19, 312, 19, 23	divdird 12081	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
314	19, 23	dividd 12041	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
315	314	oveq1d 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
316	313, 315	eqtr2d 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
317	316	oveq2d 7447	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
318	19, 312	addcld 11280	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
319	21, 19, 318, 23	div32d 12066	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
320	317, 319	eqtr4d 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)))
321	320	oveq1d 7446	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵) = (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵))
322	24, 318, 299	mulassd 11284	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
323	311, 321, 322	3eqtrd 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
324	308, 309, 323	3brtr4d 5175	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
325	324	adantrr 717	. . . 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 15688	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
327	78, 81, 234, 326	lo1add 15663	. 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 2842	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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℝ⁺crp 13034 (,)cioo 13387 ...cfz 13547 ⌊cfl 13830 abscabs 15273 ⇝_𝑟 crli 15521 𝑂(1)co1 15522 ≤𝑂(1)clo1 15523 Σcsu 15722 logclog 26596 Λcvma 27135 ψcchp 27136

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-o1 15526 df-lo1 15527 df-sum 15723 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-cxp 26599 df-atan 26910 df-em 27036 df-cht 27140 df-vma 27141 df-chp 27142 df-ppi 27143 df-mu 27144

This theorem is referenced by: pntrlog2bndlem6 27627

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