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Theorem pntrlog2bndlem6 27800

Description: Lemma for pntrlog2bnd 27801. 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 13420	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 13038	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11226	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13450	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
7	6	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
8	7	simpld 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
9	5, 2, 8	ltled 11375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13117	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrlog2bnd.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27780	. . . . . . . . . . . 12 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7082	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	10, 13	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ)
15	14	recnd 11254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
16	15	abscld 15516	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
17	10	relogcld 26841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
18	16, 17	remulcld 11256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
19		2re 12332	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
21	2, 8	rplogcld 26847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
22	20, 21	rerpdivcld 13109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
23		fzfid 14029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
24	10	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
25		elfznn 13600	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
26	25	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
27	26	nnrpd 13076	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
28	24, 27	rpdivcld 13095	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29	12	ffvelcdmi 7082	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
30	28, 29	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
31	30	recnd 11254	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
32	31	abscld 15516	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
33	27	relogcld 26841	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
34	32, 33	remulcld 11256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
35	23, 34	fsumrecl 15810	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
36	22, 35	remulcld 11256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
37	18, 36	resubcld 11659	. . . . . 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 11254	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ)
39		fzfid 14029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ∈ Fin)
40		ssun2 4132	. . . . . . . . . . 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 27799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))))
48	40, 47	sseqtrrid 3981	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
49	48	sselda 3938	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
50	49, 34	syldan 603	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
51	39, 50	fsumrecl 15810	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
52	22, 51	remulcld 11256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
53	52	recnd 11254	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
54	2	recnd 11254	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
55	10	rpne0d 13083	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
56	38, 53, 54, 55	divdird 12046	. . . 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 11254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
58	36	recnd 11254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
59	57, 58, 53	subsubd 11614	. . . . . 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 11254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
61	35	recnd 11254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
62	51	recnd 11254	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 11687	. . . . . . . 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 14029	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ∈ Fin)
65		ssun1 4131	. . . . . . . . . . . . . . 15 ⊢ (1...(⌊‘(𝑥 / 𝐴))) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))
66	65, 47	sseqtrrid 3981	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ⊆ (1...(⌊‘𝑥)))
67	66	sselda 3938	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → 𝑛 ∈ (1...(⌊‘𝑥)))
68	67, 34	syldan 603	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
69	64, 68	fsumrecl 15810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
70	69	recnd 11254	. . . . . . . . . 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 13117	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
73	72	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ⁺)
74	2, 73	rerpdivcld 13109	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ)
75		reflcl 13849	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / 𝐴) ∈ ℝ → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
76	74, 75	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
77	76	ltp1d 12162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1))
78		fzdisj 13598	. . . . . . . . . . . 12 ⊢ ((⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1) → ((1...(⌊‘(𝑥 / 𝐴))) ∩ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅)
79	77, 78	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((1...(⌊‘(𝑥 / 𝐴))) ∩ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅)
80	34	recnd 11254	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
81	79, 47, 23, 80	fsumsplit 15817	. . . . . . . . . 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 11643	. . . . . . . . 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 7435	. . . . . . . 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 2802	. . . . . . 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 7435	. . . . . 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 2802	. . . . 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 7434	. . . 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 2802	. . 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 5206	. 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 13109	. . 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 13109	. . 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 27798	. . 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 13452	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
94	93	a1i 11	. . . 4 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ)
95		1red 11226	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
96	19	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
97	43	rpred 13078	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
98	72	relogcld 26841	. . . . . . 7 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
99	98, 95	readdcld 11255	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
100	97, 99	remulcld 11256	. . . . 5 ⊢ (𝜑 → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
101	96, 100	remulcld 11256	. . . 4 ⊢ (𝜑 → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
102	51, 21	rerpdivcld 13109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
103	97	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
104	73	relogcld 26841	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℝ)
105	104, 5	readdcld 11255	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℝ)
106	103, 105	remulcld 11256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
107	2, 106	remulcld 11256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
108		2rp 13039	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
109	108	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ⁺)
110	109	rpge0d 13082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
111	103, 2	remulcld 11256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℝ)
112	49, 25	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
113	112	nnrecred 12304	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
114	39, 113	fsumrecl 15810	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
115	111, 114	remulcld 11256	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
116	21	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ⁺)
117	50, 116	rerpdivcld 13109	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
118	103	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
119	2	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
120	118, 119	remulcld 11256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℝ)
121	120, 113	remulcld 11256	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) · (1 / 𝑛)) ∈ ℝ)
122	49, 32	syldan 603	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
123	119, 112	nndivred 12307	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
124	118, 123	remulcld 11256	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) ∈ ℝ)
125	49, 27	syldan 603	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
126	125	relogcld 26841	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
127	10	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
128	127	relogcld 26841	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ)
129	49, 31	syldan 603	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
130	129	absge0d 15524	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
131		elfzle2 13574	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) → 𝑛 ≤ (⌊‘𝑥))
132	131	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ (⌊‘𝑥))
133	112	nnzd 12634	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
134		flge 13858	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ 𝑥 ↔ 𝑛 ≤ (⌊‘𝑥)))
135	119, 133, 134	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛 ≤ 𝑥 ↔ 𝑛 ≤ (⌊‘𝑥)))
136	132, 135	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
137	125, 127	logled 26845	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛 ≤ 𝑥 ↔ (log‘𝑛) ≤ (log‘𝑥)))
138	136, 137	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ≤ (log‘𝑥))
139	126, 128, 122, 130, 138	lemul2ad 12172	. . . . . . . . . . . . . 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 13132	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
142	123	recnd 11254	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
143	49, 28	syldan 603	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
144	143	rpne0d 13083	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
145	129, 142, 144	absdivd 15535	. . . . . . . . . . . . . . . 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 13082	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
147	146	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ 𝑥)
148	119, 125, 147	divge0d 13118	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
149	123, 148	absidd 15500	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
150	149	oveq2d 7435	. . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
152		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛)))
153		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
154	152, 153	oveq12d 7437	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
155	154	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
156	155	breq1d 5121	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
157	44	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
158	156, 157, 143	rspcdva 3584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
159	151, 158	eqbrtrrd 5137	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵)
160	122, 118, 143	ledivmul2d 13132	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛)))
162	117, 122, 124, 141, 161	letrd 11384	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝐵 · (𝑥 / 𝑛)))
163	118	recnd 11254	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
164	54	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
165	112	nncnd 12266	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
166	112	nnne0d 12303	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≠ 0)
167	163, 164, 165, 166	divassd 12043	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = (𝐵 · (𝑥 / 𝑛)))
168	163, 164	mulcld 11246	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℂ)
169	168, 165, 166	divrecd 12011	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = ((𝐵 · 𝑥) · (1 / 𝑛)))
170	167, 169	eqtr3d 2802	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) = ((𝐵 · 𝑥) · (1 / 𝑛)))
171	162, 170	breqtrd 5139	. . . . . . . . . . 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 15876	. . . . . . . . . 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 11254	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
174	49, 80	syldan 603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
175	21	rpne0d 13083	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
176	39, 173, 174, 175	fsumdivc 15862	. . . . . . . . . 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 11254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
178	177, 54	mulcld 11246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℂ)
179	113	recnd 11254	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
180	39, 178, 179	fsummulc2 15860	. . . . . . . . . 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 5145	. . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ⁺)
183	182	rpge0d 13082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
184	103, 2, 183, 146	mulge0d 11808	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐵 · 𝑥))
185	26	nnrecred 12304	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
186	23, 185	fsumrecl 15810	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
187	17, 104	resubcld 11659	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ∈ ℝ)
188	17, 5	readdcld 11255	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
189	67, 185	syldan 603	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
190	64, 189	fsumrecl 15810	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
191		harmonicubnd 27227	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
192	2, 9, 191	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
193	10, 73	relogdivd 26844	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) = ((log‘𝑥) − (log‘𝐴)))
194	10, 73	rpdivcld 13095	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ⁺)
195		harmoniclbnd 27226	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝐴) ∈ ℝ⁺ → (log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
196	194, 195	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
197	193, 196	eqbrtrrd 5137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
198	186, 187, 188, 190, 192, 197	le2subd 11851	. . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → 𝑛 ∈ ℕ)
200	199	nnrecred 12304	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
201	64, 200	fsumrecl 15810	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
202	201	recnd 11254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℂ)
203	114	recnd 11254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
204	26	nncnd 12266	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
205	26	nnne0d 12303	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
206	204, 205	reccld 12001	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
207	79, 47, 23, 206	fsumsplit 15817	. . . . . . . . . . . . 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 11644	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))
209		1cnd 11219	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
210	104	recnd 11254	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℂ)
211	173, 209, 210	pnncand 11625	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = (1 + (log‘𝐴)))
212	209, 210, 211	comraddd 11441	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = ((log‘𝐴) + 1))
213	198, 208, 212	3brtr3d 5144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
214	114, 105, 111, 184, 213	lemul2ad 12172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ ((𝐵 · 𝑥) · ((log‘𝐴) + 1)))
215	105	recnd 11254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
216	177, 54, 215	mulassd 11249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝐵 · (𝑥 · ((log‘𝐴) + 1))))
217	177, 54, 215	mul12d 11436	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · (𝑥 · ((log‘𝐴) + 1))) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
218	216, 217	eqtrd 2800	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
219	214, 218	breqtrd 5139	. . . . . . . . 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 11384	. . . . . . . 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 12172	. . . . . . 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 12336	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
223	222, 173, 62, 175	div32d 12031	. . . . . . 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 11245	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
225	177, 224	mulcld 11246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℂ)
226	54, 222, 225	mul12d 11436	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))) = (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1)))))
227	221, 223, 226	3brtr4d 5145	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
229	52, 228, 10	ledivmuld 13131	. . . . . 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 260	. . . . 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 730	. . . 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 15600	. . 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 15704	. 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 2866	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 ifcif 4489 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 ℂcc 11115 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 +∞cpnf 11257 < clt 11260 ≤ cle 11261 − cmin 11458 / cdiv 11888 ℕcn 12250 2c2 12312 ℤcz 12608 ℝ⁺crp 13034 (,)cioo 13390 ...cfz 13553 ⌊cfl 13843 abscabs 15311 ≤𝑂(1)clo1 15564 Σcsu 15763 logclog 26772 Λcvma 27309 ψcchp 27310

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-xnn0 12595 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-o1 15567 df-lo1 15568 df-sum 15764 df-ef 16145 df-e 16146 df-sin 16147 df-cos 16148 df-tan 16149 df-pi 16150 df-dvds 16335 df-gcd 16577 df-prm 16754 df-pc 16921 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-pt 17521 df-prds 17524 df-xrs 17580 df-qtop 17585 df-imas 17586 df-xps 17588 df-mre 17662 df-mrc 17663 df-acs 17665 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-mulg 19180 df-cntz 19433 df-cmn 19898 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-fbas 21571 df-fg 21572 df-cnfld 21575 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-cld 23228 df-ntr 23229 df-cls 23230 df-nei 23307 df-lp 23345 df-perf 23346 df-cn 23436 df-cnp 23437 df-haus 23524 df-cmp 23596 df-tx 23772 df-hmeo 23965 df-fil 24056 df-fm 24148 df-flim 24149 df-flf 24150 df-xms 24530 df-ms 24531 df-tms 24532 df-cncf 25090 df-limc 26078 df-dv 26079 df-ulm 26593 df-log 26774 df-cxp 26775 df-atan 27085 df-em 27210 df-cht 27314 df-vma 27315 df-chp 27316 df-ppi 27317 df-mu 27318

This theorem is used by: pntrlog2bnd 27801

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