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Theorem pntrlog2bndlem6 27967

Description: Lemma for pntrlog2bnd 27968. 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 13546	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 13164	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11348	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13576	. . . . . . . . . . . . . . 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 11497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrlog2bnd.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27947	. . . . . . . . . . . 12 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7115	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	10, 13	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ)
15	14	recnd 11376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
16	15	abscld 15650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
17	10	relogcld 27008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
18	16, 17	remulcld 11378	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
19		2re 12457	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
20	19	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
21	2, 8	rplogcld 27014	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
22	20, 21	rerpdivcld 13235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
23		fzfid 14156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
24	10	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
25		elfznn 13727	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
26	25	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
27	26	nnrpd 13202	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
28	24, 27	rpdivcld 13221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
29	12	ffvelcdmi 7115	. . . . . . . . . . . . 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 11376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
32	31	abscld 15650	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
33	27	relogcld 27008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
34	32, 33	remulcld 11378	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
35	23, 34	fsumrecl 15944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
36	22, 35	remulcld 11378	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
37	18, 36	resubcld 11783	. . . . . 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 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ)
39		fzfid 14156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ∈ Fin)
40		ssun2 4180	. . . . . . . . . . 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 27966	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))))
48	40, 47	sseqtrrid 3956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
49	48	sselda 3913	. . . . . . . . 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 15944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
52	22, 51	remulcld 11378	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
53	52	recnd 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
54	2	recnd 11376	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
55	10	rpne0d 13209	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
56	38, 53, 54, 55	divdird 12171	. . . 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 11376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
58	36	recnd 11376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
59	57, 58, 53	subsubd 11736	. . . . . 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 11376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
61	35	recnd 11376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
62	51	recnd 11376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 11811	. . . . . . . 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 14156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ∈ Fin)
65		ssun1 4179	. . . . . . . . . . . . . . 15 ⊢ (1...(⌊‘(𝑥 / 𝐴))) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))
66	65, 47	sseqtrrid 3956	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ⊆ (1...(⌊‘𝑥)))
67	66	sselda 3913	. . . . . . . . . . . . 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 15944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
70	69	recnd 11376	. . . . . . . . . 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 13243	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
73	72	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ⁺)
74	2, 73	rerpdivcld 13235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ)
75		reflcl 13976	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / 𝐴) ∈ ℝ → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
76	74, 75	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
77	76	ltp1d 12287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1))
78		fzdisj 13725	. . . . . . . . . . . 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 11376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
81	79, 47, 23, 80	fsumsplit 15951	. . . . . . . . . 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 11766	. . . . . . . . 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 7467	. . . . . . . 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 2800	. . . . . . 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 7467	. . . . . 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 2800	. . . . 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 7466	. . . 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 2800	. . 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 5204	. 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 13235	. . 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 13235	. . 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 27965	. . 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 13578	. . . . 5 ⊢ (1(,)+∞) ⊆ ℝ
94	93	a1i 11	. . . 4 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ)
95		1red 11348	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
96	19	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ)
97	43	rpred 13204	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
98	72	relogcld 27008	. . . . . . 7 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
99	98, 95	readdcld 11377	. . . . . 6 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
100	97, 99	remulcld 11378	. . . . 5 ⊢ (𝜑 → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
101	96, 100	remulcld 11378	. . . 4 ⊢ (𝜑 → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
102	51, 21	rerpdivcld 13235	. . . . . . . 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 27008	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℝ)
105	104, 5	readdcld 11377	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℝ)
106	103, 105	remulcld 11378	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
107	2, 106	remulcld 11378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
108		2rp 13165	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
109	108	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ⁺)
110	109	rpge0d 13208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
111	103, 2	remulcld 11378	. . . . . . . . . 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 12429	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
114	39, 113	fsumrecl 15944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
115	111, 114	remulcld 11378	. . . . . . . . 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 13235	. . . . . . . . . . 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 11378	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℝ)
121	120, 113	remulcld 11378	. . . . . . . . . . 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 12432	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
124	118, 123	remulcld 11378	. . . . . . . . . . . . 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 27008	. . . . . . . . . . . . . . 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 27008	. . . . . . . . . . . . . . 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 15658	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
131		elfzle2 13701	. . . . . . . . . . . . . . . . . 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 12759	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
134		flge 13985	. . . . . . . . . . . . . . . . . 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 27012	. . . . . . . . . . . . . . . 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 12297	. . . . . . . . . . . . . 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 13258	. . . . . . . . . . . . . 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 11376	. . . . . . . . . . . . . . . . 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 13209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
145	129, 142, 144	absdivd 15669	. . . . . . . . . . . . . . . 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 13208	. . . . . . . . . . . . . . . . . . . 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 13244	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
149	123, 148	absidd 15634	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
150	149	oveq2d 7467	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
152		fveq2 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛)))
153		id 23	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
154	152, 153	oveq12d 7469	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
155	154	fveq2d 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
156	155	breq1d 5118	. . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
159	151, 158	eqbrtrrd 5135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵)
160	122, 118, 143	ledivmul2d 13258	. . . . . . . . . . . . . 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 11506	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝐵 · (𝑥 / 𝑛)))
163	118	recnd 11376	. . . . . . . . . . . . . 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 12391	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
166	112	nnne0d 12428	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≠ 0)
167	163, 164, 165, 166	divassd 12168	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = (𝐵 · (𝑥 / 𝑛)))
168	163, 164	mulcld 11368	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℂ)
169	168, 165, 166	divrecd 12136	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = ((𝐵 · 𝑥) · (1 / 𝑛)))
170	167, 169	eqtr3d 2800	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) = ((𝐵 · 𝑥) · (1 / 𝑛)))
171	162, 170	breqtrd 5137	. . . . . . . . . . 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 16010	. . . . . . . . . 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 11376	. . . . . . . . . . 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 13209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
176	39, 173, 174, 175	fsumdivc 15996	. . . . . . . . . 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 11376	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
178	177, 54	mulcld 11368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℂ)
179	113	recnd 11376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
180	39, 178, 179	fsummulc2 15994	. . . . . . . . . 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 5143	. . . . . . . . 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 13208	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
184	103, 2, 183, 146	mulge0d 11933	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐵 · 𝑥))
185	26	nnrecred 12429	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
186	23, 185	fsumrecl 15944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
187	17, 104	resubcld 11783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ∈ ℝ)
188	17, 5	readdcld 11377	. . . . . . . . . . . . 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 15944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
191		harmonicubnd 27394	. . . . . . . . . . . . . 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 27011	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) = ((log‘𝑥) − (log‘𝐴)))
194	10, 73	rpdivcld 13221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ⁺)
195		harmoniclbnd 27393	. . . . . . . . . . . . . . 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 5135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
198	186, 187, 188, 190, 192, 197	le2subd 11976	. . . . . . . . . . . 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 12429	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
201	64, 200	fsumrecl 15944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
202	201	recnd 11376	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℂ)
203	114	recnd 11376	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
204	26	nncnd 12391	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
205	26	nnne0d 12428	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
206	204, 205	reccld 12126	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
207	79, 47, 23, 206	fsumsplit 15951	. . . . . . . . . . . . 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 11767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))
209		1cnd 11341	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
210	104	recnd 11376	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℂ)
211	173, 209, 210	pnncand 11747	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = (1 + (log‘𝐴)))
212	209, 210, 211	comraddd 11563	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = ((log‘𝐴) + 1))
213	198, 208, 212	3brtr3d 5142	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
214	114, 105, 111, 184, 213	lemul2ad 12297	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ ((𝐵 · 𝑥) · ((log‘𝐴) + 1)))
215	105	recnd 11376	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
216	177, 54, 215	mulassd 11371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝐵 · (𝑥 · ((log‘𝐴) + 1))))
217	177, 54, 215	mul12d 11558	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · (𝑥 · ((log‘𝐴) + 1))) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
218	216, 217	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
219	214, 218	breqtrd 5137	. . . . . . . . 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 11506	. . . . . . . 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 12297	. . . . . . 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 12461	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
223	222, 173, 62, 175	div32d 12156	. . . . . . 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 11367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
225	177, 224	mulcld 11368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℂ)
226	54, 222, 225	mul12d 11558	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))) = (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1)))))
227	221, 223, 226	3brtr4d 5143	. . . . . 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 13257	. . . . . 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 15734	. . 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 15838	. 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 2865	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 2147 ∀wral 3080 ⊆ wss 3900 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 ifcif 4485 class class class wbr 5108 ↦ cmpt 5192 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 +∞cpnf 11379 < clt 11382 ≤ cle 11383 − cmin 11580 / cdiv 12013 ℕcn 12375 2c2 12437 ℤcz 12733 ℝ⁺crp 13160 (,)cioo 13516 ...cfz 13679 ⌊cfl 13970 abscabs 15444 ≤𝑂(1)clo1 15698 Σcsu 15897 logclog 26939 Λcvma 27476 ψcchp 27477

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-o1 15701 df-lo1 15702 df-sum 15898 df-ef 16277 df-e 16278 df-sin 16279 df-cos 16280 df-tan 16281 df-pi 16282 df-dvds 16467 df-gcd 16709 df-prm 16891 df-pc 17059 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-ulm 26761 df-log 26941 df-cxp 26942 df-atan 27252 df-em 27377 df-cht 27481 df-vma 27482 df-chp 27483 df-ppi 27484 df-mu 27485

This theorem is used by: pntrlog2bnd 27968

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