pntrlog2bndlem6 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pntrlog2bndlem6		Structured version Visualization version GIF version

Theorem pntrlog2bndlem6 27503

Description: Lemma for pntrlog2bnd 27504. 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)
pntrlog2bndlem6.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
pntrlog2bndlem6.2	⊢ (𝜑 → 1 ≤ 𝐴)

**Assertion**
Ref	Expression
pntrlog2bndlem6	⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))

Distinct variable groups: 𝑖,𝑎,𝑛,𝑥,𝑦,𝐴 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 pntrlog2bndlem6**
Step	Hyp	Ref	Expression
1		elioore 13378	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 13002	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13407	. . . . . . . . . . . . . . 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 11384	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrlog2bnd.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27483	. . . . . . . . . . . 12 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7087	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	10, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ)
15	14	recnd 11264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
16	15	abscld 15407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
17	10	relogcld 26544	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
18	16, 17	remulcld 11266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
19		2re 12308	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
21	2, 8	rplogcld 26550	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
22	20, 21	rerpdivcld 13071	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
23		fzfid 13962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
24	10	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
25		elfznn 13554	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
26	25	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
27	26	nnrpd 13038	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
28	24, 27	rpdivcld 13057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29	12	ffvelcdmi 7087	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
31	30	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
32	31	abscld 15407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
33	27	relogcld 26544	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
34	32, 33	remulcld 11266	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
35	23, 34	fsumrecl 15704	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
36	22, 35	remulcld 11266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
37	18, 36	resubcld 11664	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℝ)
38	37	recnd 11264	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ)
39		fzfid 13962	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ∈ Fin)
40		ssun2 4169	. . . . . . . . . . 11 ⊢ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))
41		pntsval.1	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
42		pntrlog2bnd.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0))
43		pntrlog2bndlem5.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
44		pntrlog2bndlem5.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
45		pntrlog2bndlem6.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
46		pntrlog2bndlem6.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝐴)
47	41, 11, 42, 43, 44, 45, 46	pntrlog2bndlem6a 27502	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))))
48	40, 47	sseqtrrid 4031	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
49	48	sselda 3978	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
50	49, 34	syldan 590	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
51	39, 50	fsumrecl 15704	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
52	22, 51	remulcld 11266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
53	52	recnd 11264	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
54	2	recnd 11264	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
55	10	rpne0d 13045	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
56	38, 53, 54, 55	divdird 12050	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)))
57	18	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
58	36	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
59	57, 58, 53	subsubd 11621	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
60	22	recnd 11264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
61	35	recnd 11264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
62	51	recnd 11264	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 11692	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
64		fzfid 13962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ∈ Fin)
65		ssun1 4168	. . . . . . . . . . . . . . 15 ⊢ (1...(⌊‘(𝑥 / 𝐴))) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))
66	65, 47	sseqtrrid 4031	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ⊆ (1...(⌊‘𝑥)))
67	66	sselda 3978	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → 𝑛 ∈ (1...(⌊‘𝑥)))
68	67, 34	syldan 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
69	64, 68	fsumrecl 15704	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
70	69	recnd 11264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
71	3	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ ℝ⁺)
72	45, 71, 46	rpgecld 13079	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
73	72	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ⁺)
74	2, 73	rerpdivcld 13071	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ)
75		reflcl 13785	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / 𝐴) ∈ ℝ → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
76	74, 75	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
77	76	ltp1d 12166	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1))
78		fzdisj 13552	. . . . . . . . . . . 12 ⊢ ((⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1) → ((1...(⌊‘(𝑥 / 𝐴))) ∩ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅)
79	77, 78	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((1...(⌊‘(𝑥 / 𝐴))) ∩ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅)
80	34	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
81	79, 47, 23, 80	fsumsplit 15711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
82	70, 62, 81	mvrraddd 11648	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
83	82	oveq2d 7430	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
84	63, 83	eqtr3d 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
85	84	oveq2d 7430	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
86	59, 85	eqtr3d 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
87	86	oveq1d 7429	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
88	56, 87	eqtr3d 2769	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
89	88	mpteq2dva 5242	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)))
90	37, 10	rerpdivcld 13071	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ∈ ℝ)
91	52, 10	rerpdivcld 13071	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ∈ ℝ)
92	41, 11, 42, 43, 44	pntrlog2bndlem5 27501	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
93		ioossre 13409	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
94	93	a1i 11	. . . 4 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ)
95		1red 11237	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
96	19	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
97	43	rpred 13040	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
98	72	relogcld 26544	. . . . . . 7 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
99	98, 95	readdcld 11265	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
100	97, 99	remulcld 11266	. . . . 5 ⊢ (𝜑 → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
101	96, 100	remulcld 11266	. . . 4 ⊢ (𝜑 → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
102	51, 21	rerpdivcld 13071	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
103	97	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
104	73	relogcld 26544	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℝ)
105	104, 5	readdcld 11265	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℝ)
106	103, 105	remulcld 11266	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
107	2, 106	remulcld 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
108		2rp 13003	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
109	108	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ⁺)
110	109	rpge0d 13044	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
111	103, 2	remulcld 11266	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℝ)
112	49, 25	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
113	112	nnrecred 12285	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
114	39, 113	fsumrecl 15704	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
115	111, 114	remulcld 11266	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
116	21	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ⁺)
117	50, 116	rerpdivcld 13071	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
118	103	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
119	2	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
120	118, 119	remulcld 11266	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℝ)
121	120, 113	remulcld 11266	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) · (1 / 𝑛)) ∈ ℝ)
122	49, 32	syldan 590	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
123	119, 112	nndivred 12288	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
124	118, 123	remulcld 11266	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) ∈ ℝ)
125	49, 27	syldan 590	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
126	125	relogcld 26544	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
127	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
128	127	relogcld 26544	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ)
129	49, 31	syldan 590	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
130	129	absge0d 15415	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
131		elfzle2 13529	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) → 𝑛 ≤ (⌊‘𝑥))
132	131	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ (⌊‘𝑥))
133	112	nnzd 12607	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
134		flge 13794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝑥 ↔ 𝑛 ≤ (⌊‘𝑥)))
135	119, 133, 134	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛 ≤ 𝑥 ↔ 𝑛 ≤ (⌊‘𝑥)))
136	132, 135	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
137	125, 127	logled 26548	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛 ≤ 𝑥 ↔ (log‘𝑛) ≤ (log‘𝑥)))
138	136, 137	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ≤ (log‘𝑥))
139	126, 128, 122, 130, 138	lemul2ad 12176	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑥)))
140	50, 122, 116	ledivmul2d 13094	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛))) ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑥))))
141	139, 140	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
142	123	recnd 11264	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
143	49, 28	syldan 590	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
144	143	rpne0d 13045	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
145	129, 142, 144	absdivd 15426	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))))
146	10	rpge0d 13044	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
147	146	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ 𝑥)
148	119, 125, 147	divge0d 13080	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
149	123, 148	absidd 15393	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
150	149	oveq2d 7430	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
151	145, 150	eqtrd 2767	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
152		fveq2 6891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛)))
153		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
154	152, 153	oveq12d 7432	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
155	154	fveq2d 6895	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
156	155	breq1d 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
157	44	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
158	156, 157, 143	rspcdva 3608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
159	151, 158	eqbrtrrd 5166	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵)
160	122, 118, 143	ledivmul2d 13094	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵 ↔ (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛))))
161	159, 160	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛)))
162	117, 122, 124, 141, 161	letrd 11393	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝐵 · (𝑥 / 𝑛)))
163	118	recnd 11264	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
164	54	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
165	112	nncnd 12250	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
166	112	nnne0d 12284	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≠ 0)
167	163, 164, 165, 166	divassd 12047	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = (𝐵 · (𝑥 / 𝑛)))
168	163, 164	mulcld 11256	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℂ)
169	168, 165, 166	divrecd 12015	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = ((𝐵 · 𝑥) · (1 / 𝑛)))
170	167, 169	eqtr3d 2769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) = ((𝐵 · 𝑥) · (1 / 𝑛)))
171	162, 170	breqtrd 5168	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ ((𝐵 · 𝑥) · (1 / 𝑛)))
172	39, 117, 121, 171	fsumle 15769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((𝐵 · 𝑥) · (1 / 𝑛)))
173	17	recnd 11264	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
174	49, 80	syldan 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
175	21	rpne0d 13045	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
176	39, 173, 174, 175	fsumdivc 15756	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)))
177	103	recnd 11264	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
178	177, 54	mulcld 11256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℂ)
179	113	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
180	39, 178, 179	fsummulc2 15754	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((𝐵 · 𝑥) · (1 / 𝑛)))
181	172, 176, 180	3brtr4d 5174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)))
182	43	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ⁺)
183	182	rpge0d 13044	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
184	103, 2, 183, 146	mulge0d 11813	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐵 · 𝑥))
185	26	nnrecred 12285	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
186	23, 185	fsumrecl 15704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
187	17, 104	resubcld 11664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ∈ ℝ)
188	17, 5	readdcld 11265	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
189	67, 185	syldan 590	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
190	64, 189	fsumrecl 15704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
191		harmonicubnd 26929	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
192	2, 9, 191	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
193	10, 73	relogdivd 26547	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) = ((log‘𝑥) − (log‘𝐴)))
194	10, 73	rpdivcld 13057	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ⁺)
195		harmoniclbnd 26928	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝐴) ∈ ℝ⁺ → (log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
196	194, 195	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
197	193, 196	eqbrtrrd 5166	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
198	186, 187, 188, 190, 192, 197	le2subd 11856	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) ≤ (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))))
199	67, 25	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → 𝑛 ∈ ℕ)
200	199	nnrecred 12285	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
201	64, 200	fsumrecl 15704	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
202	201	recnd 11264	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℂ)
203	114	recnd 11264	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
204	26	nncnd 12250	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
205	26	nnne0d 12284	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
206	204, 205	reccld 12005	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
207	79, 47, 23, 206	fsumsplit 15711	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)))
208	202, 203, 207	mvrladdd 11649	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))
209		1cnd 11231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
210	104	recnd 11264	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℂ)
211	173, 209, 210	pnncand 11632	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = (1 + (log‘𝐴)))
212	209, 210, 211	comraddd 11450	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = ((log‘𝐴) + 1))
213	198, 208, 212	3brtr3d 5173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
214	114, 105, 111, 184, 213	lemul2ad 12176	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ ((𝐵 · 𝑥) · ((log‘𝐴) + 1)))
215	105	recnd 11264	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
216	177, 54, 215	mulassd 11259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝐵 · (𝑥 · ((log‘𝐴) + 1))))
217	177, 54, 215	mul12d 11445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · (𝑥 · ((log‘𝐴) + 1))) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
218	216, 217	eqtrd 2767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
219	214, 218	breqtrd 5168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
220	102, 115, 107, 181, 219	letrd 11393	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
221	102, 107, 20, 110, 220	lemul2ad 12176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥))) ≤ (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1)))))
222		2cnd 12312	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
223	222, 173, 62, 175	div32d 12035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = (2 · (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥))))
224	210, 209	addcld 11255	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
225	177, 224	mulcld 11256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℂ)
226	54, 222, 225	mul12d 11445	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))) = (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1)))))
227	221, 223, 226	3brtr4d 5174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ≤ (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))))
228	101	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
229	52, 228, 10	ledivmuld 13093	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))) ↔ ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ≤ (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1))))))
230	227, 229	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))))
231	230	adantrr 716	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))))
232	94, 91, 95, 101, 231	ello1d 15491	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)) ∈ ≤𝑂(1))
233	90, 91, 92, 232	lo1add 15595	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) ∈ ≤𝑂(1))
234	89, 233	eqeltrrd 2829	1 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 +∞cpnf 11267 < clt 11270 ≤ cle 11271 − cmin 11466 / cdiv 11893 ℕcn 12234 2c2 12289 ℤcz 12580 ℝ⁺crp 12998 (,)cioo 13348 ...cfz 13508 ⌊cfl 13779 abscabs 15205 ≤𝑂(1)clo1 15455 Σcsu 15656 logclog 26475 Λcvma 27011 ψcchp 27012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-shft 15038 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-o1 15458 df-lo1 15459 df-sum 15657 df-ef 16035 df-e 16036 df-sin 16037 df-cos 16038 df-tan 16039 df-pi 16040 df-dvds 16223 df-gcd 16461 df-prm 16634 df-pc 16797 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-pt 17417 df-prds 17420 df-xrs 17475 df-qtop 17480 df-imas 17481 df-xps 17483 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-submnd 18732 df-mulg 19015 df-cntz 19259 df-cmn 19728 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-fbas 21263 df-fg 21264 df-cnfld 21267 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-cld 22910 df-ntr 22911 df-cls 22912 df-nei 22989 df-lp 23027 df-perf 23028 df-cn 23118 df-cnp 23119 df-haus 23206 df-cmp 23278 df-tx 23453 df-hmeo 23646 df-fil 23737 df-fm 23829 df-flim 23830 df-flf 23831 df-xms 24213 df-ms 24214 df-tms 24215 df-cncf 24785 df-limc 25782 df-dv 25783 df-ulm 26300 df-log 26477 df-cxp 26478 df-atan 26786 df-em 26912 df-cht 27016 df-vma 27017 df-chp 27018 df-ppi 27019 df-mu 27020

This theorem is referenced by: pntrlog2bnd 27504

W3C validator