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Theorem pntrlog2bndlem5 27796

Description: Lemma for pntrlog2bnd 27799. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)

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

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

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

**Proof of Theorem pntrlog2bndlem5**
Step	Hyp	Ref	Expression
1		elioore 13418	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
2	1	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3		1rp 13036	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
4	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
5		1red 11224	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6		eliooord 13448	. . . . . . . . . . . . . . 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 11373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
10	2, 4, 9	rpgecld 13115	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
11		pntrlog2bnd.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27778	. . . . . . . . . . . 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 11252	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
16	15	abscld 15514	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
17	16	recnd 11252	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℂ)
18	10	relogcld 26839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
19	18	recnd 11252	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
20	17, 19	mulcld 11244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
21		2cnd 12334	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
22	2, 8	rplogcld 26845	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
23	22	rpne0d 13081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
24	21, 19, 23	divcld 12006	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
25		fzfid 14027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
26	10	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
27		elfznn 13598	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
28	27	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
29	28	nnrpd 13074	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
30	26, 29	rpdivcld 13093	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
31	12	ffvelcdmi 7082	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
32	30, 31	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
33	32	recnd 11252	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
34	33	abscld 15514	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
35	29	relogcld 26839	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
36		1red 11224	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
37	35, 36	readdcld 11253	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℝ)
38	34, 37	remulcld 11254	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
39	38	recnd 11252	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
40	25, 39	fsumcl 15807	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
41	24, 40	mulcld 11244	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℂ)
42	20, 41	subcld 11584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℂ)
43	34	recnd 11252	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
44	25, 43	fsumcl 15807	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
45	24, 44	mulcld 11244	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℂ)
46	2	recnd 11252	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
47	10	rpne0d 13081	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
48	42, 45, 46, 47	divdird 12044	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) / 𝑥) = (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)))
49	16, 18	remulcld 11254	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
50	49	recnd 11252	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℂ)
51	50, 41, 45	subsubd 11612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))))
52	24, 40, 44	subdid 11685	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))))
53	25, 39, 43	fsumsub 15862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))
54	37	recnd 11252	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℂ)
55		1cnd 11217	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
56	43, 54, 55	subdid 11685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1)))
57	35	recnd 11252	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
58	57, 55	pncand 11585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘𝑛) + 1) − 1) = (log‘𝑛))
59	58	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
60	43	mulridd 11241	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1) = (abs‘(𝑅‘(𝑥 / 𝑛))))
61	60	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))))
62	56, 59, 61	3eqtr3rd 2809	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
63	62	sumeq2dv 15777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
64	53, 63	eqtr3d 2802	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
65	64	oveq2d 7435	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
66	52, 65	eqtr3d 2802	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
67	66	oveq2d 7435	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
68	51, 67	eqtr3d 2802	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
69	68	oveq1d 7434	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) / 𝑥) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
70	48, 69	eqtr3d 2802	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) = ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
71	70	mpteq2dva 5206	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)))
72		2re 12330	. . . . . . . 8 ⊢ 2 ∈ ℝ
73	72	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
74	73, 22	rerpdivcld 13107	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
75	25, 38	fsumrecl 15808	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
76	74, 75	remulcld 11254	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℝ)
77	49, 76	resubcld 11657	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℝ)
78	77, 10	rerpdivcld 13107	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ∈ ℝ)
79	25, 34	fsumrecl 15808	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
80	74, 79	remulcld 11254	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
81	80, 10	rerpdivcld 13107	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
82		1red 11224	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
83		pntsval.1	. . . . . 6 ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
84		pntrlog2bnd.t	. . . . . 6 ⊢ 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0))
85	83, 11, 84	pntrlog2bndlem4 27795	. . . . 5 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1)
86	85	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1))
87	28	nnred 12263	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
88		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → 𝑎 ∈ ℝ)
89		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → 𝑎 ∈ ℝ⁺)
90	89	relogcld 26839	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → (log‘𝑎) ∈ ℝ)
91	88, 90	remulcld 11254	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ⁺) → (𝑎 · (log‘𝑎)) ∈ ℝ)
92		0red 11226	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℝ ∧ ¬ 𝑎 ∈ ℝ⁺) → 0 ∈ ℝ)
93	91, 92	ifclda 4525	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℝ → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) ∈ ℝ)
94	84, 93	fmpti 7111	. . . . . . . . . . . 12 ⊢ 𝑇:ℝ⟶ℝ
95	94	ffvelcdmi 7082	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝑇‘𝑛) ∈ ℝ)
96	87, 95	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇‘𝑛) ∈ ℝ)
97	87, 36	resubcld 11657	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
98	94	ffvelcdmi 7082	. . . . . . . . . . 11 ⊢ ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) ∈ ℝ)
99	97, 98	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇‘(𝑛 − 1)) ∈ ℝ)
100	96, 99	resubcld 11657	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ∈ ℝ)
101	34, 100	remulcld 11254	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
102	25, 101	fsumrecl 15808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
103	74, 102	remulcld 11254	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))))) ∈ ℝ)
104	49, 103	resubcld 11657	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) ∈ ℝ)
105	104, 10	rerpdivcld 13107	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥) ∈ ℝ)
106		2rp 13037	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ⁺
107	106	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ⁺)
108	107	rpge0d 13080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
109	73, 22, 108	divge0d 13116	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 / (log‘𝑥)))
110	33	absge0d 15522	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
111	29	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ⁺)
112	111	rpcnd 13078	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℂ)
113	57	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℂ)
114	112, 113	mulcld 11244	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘𝑛)) ∈ ℂ)
115		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 < 𝑛)
116		1re 11223	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ
117	111	rpred 13076	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ)
118		difrp 13072	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ⁺))
119	116, 117, 118	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ⁺))
120	115, 119	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 − 1) ∈ ℝ⁺)
121	120	relogcld 26839	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℝ)
122	121	recnd 11252	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℂ)
123	112, 122	mulcld 11244	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘(𝑛 − 1))) ∈ ℂ)
124	114, 123, 122	subsubd 11612	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · (log‘𝑛)) − ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1)))) = (((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
125		rpre 13041	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
126		eleq1 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑛 → (𝑎 ∈ ℝ⁺ ↔ 𝑛 ∈ ℝ⁺))
127		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑛 → 𝑎 = 𝑛)
128		fveq2 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛))
129	127, 128	oveq12d 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑛 → (𝑎 · (log‘𝑎)) = (𝑛 · (log‘𝑛)))
130	126, 129	ifbieq1d 4514	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑛 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
131		ovex 7452	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 · (log‘𝑛)) ∈ V
132		c0ex 11215	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ V
133	131, 132	ifex 4540	. . . . . . . . . . . . . . . . . 18 ⊢ if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0) ∈ V
134	130, 84, 133	fvmpt 6993	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → (𝑇‘𝑛) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
135	125, 134	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → (𝑇‘𝑛) = if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0))
136		iftrue 4495	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℝ⁺ → if(𝑛 ∈ ℝ⁺, (𝑛 · (log‘𝑛)), 0) = (𝑛 · (log‘𝑛)))
137	135, 136	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (𝑇‘𝑛) = (𝑛 · (log‘𝑛)))
138	111, 137	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘𝑛) = (𝑛 · (log‘𝑛)))
139		rpre 13041	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑛 − 1) ∈ ℝ)
140		eleq1 2853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝑛 − 1) → (𝑎 ∈ ℝ⁺ ↔ (𝑛 − 1) ∈ ℝ⁺))
141		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑛 − 1) → 𝑎 = (𝑛 − 1))
142		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑛 − 1) → (log‘𝑎) = (log‘(𝑛 − 1)))
143	141, 142	oveq12d 7437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (𝑛 − 1) → (𝑎 · (log‘𝑎)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
144	140, 143	ifbieq1d 4514	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (𝑛 − 1) → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
145		ovex 7452	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) · (log‘(𝑛 − 1))) ∈ V
146	145, 132	ifex 4540	. . . . . . . . . . . . . . . . . . 19 ⊢ if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) ∈ V
147	144, 84, 146	fvmpt 6993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
148	139, 147	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑇‘(𝑛 − 1)) = if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
149		iftrue 4495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → if((𝑛 − 1) ∈ ℝ⁺, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
150	148, 149	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → (𝑇‘(𝑛 − 1)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
151	120, 150	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
152		1cnd 11217	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℂ)
153	112, 152, 122	subdird 11686	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · (log‘(𝑛 − 1))) = ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))))
154	122	mullidd 11242	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · (log‘(𝑛 − 1))) = (log‘(𝑛 − 1)))
155	154	oveq2d 7435	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
156	151, 153, 155	3eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
157	138, 156	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · (log‘𝑛)) − ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1)))))
158	112, 113, 122	subdid 11685	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))))
159	158	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) = (((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
160	124, 157, 159	3eqtr4d 2810	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
161	111	relogcld 26839	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℝ)
162	161, 121	resubcld 11657	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℝ)
163	162	recnd 11252	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℂ)
164	112, 152, 163	subdird 11686	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − (1 · ((log‘𝑛) − (log‘(𝑛 − 1))))))
165	163	mullidd 11242	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
166	165	oveq2d 7435	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − (1 · ((log‘𝑛) − (log‘(𝑛 − 1))))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − ((log‘𝑛) − (log‘(𝑛 − 1)))))
167	117, 162	remulcld 11254	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℝ)
168	167	recnd 11252	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℂ)
169	168, 113, 122	subsub3d 11614	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − ((log‘𝑛) − (log‘(𝑛 − 1)))) = (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)))
170	164, 166, 169	3eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) = (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)))
171	112, 152	npcand 11588	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) + 1) = 𝑛)
172	171	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
173	172	oveq1d 7434	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
174		logdifbnd 27209	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ⁺ → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
175	120, 174	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
176	173, 175	eqbrtrrd 5137	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
177		1red 11224	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℝ)
178	162, 177, 120	lemuldiv2d 13126	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1 ↔ ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1))))
179	176, 178	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1)
180	170, 179	eqbrtrrd 5137	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1)
181	167, 121	readdcld 11253	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ∈ ℝ)
182	181, 161, 177	lesubadd2d 11828	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1 ↔ ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1)))
183	180, 182	mpbid 235	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
184	160, 183	eqbrtrd 5135	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
185		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑇‘𝑛) = (𝑇‘1))
186		id 23	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 1 → 𝑎 = 1)
187	186, 3	eqeltrdi 2873	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 1 → 𝑎 ∈ ℝ⁺)
188	187	iftrued 4497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 1 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = (𝑎 · (log‘𝑎)))
189		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 1 → (log‘𝑎) = (log‘1))
190		log1 26801	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (log‘1) = 0
191	189, 190	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 1 → (log‘𝑎) = 0)
192	186, 191	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 1 → (𝑎 · (log‘𝑎)) = (1 · 0))
193		ax-1cn 11173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℂ
194	193	mul01i 11415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 · 0) = 0
195	192, 194	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 1 → (𝑎 · (log‘𝑎)) = 0)
196	188, 195	eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 1 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = 0)
197	196, 84, 132	fvmpt 6993	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℝ → (𝑇‘1) = 0)
198	116, 197	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇‘1) = 0
199	185, 198	eqtrdi 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑇‘𝑛) = 0)
200		oveq1 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
201		1m1e0 12328	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 − 1) = 0
202	200, 201	eqtrdi 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝑛 − 1) = 0)
203	202	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = (𝑇‘0))
204		0re 11225	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
205		rpne0 13049	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ∈ ℝ⁺ → 𝑎 ≠ 0)
206	205	necon2bi 2990	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 0 → ¬ 𝑎 ∈ ℝ⁺)
207	206	iffalsed 4500	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 0 → if(𝑎 ∈ ℝ⁺, (𝑎 · (log‘𝑎)), 0) = 0)
208	207, 84, 132	fvmpt 6993	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℝ → (𝑇‘0) = 0)
209	204, 208	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇‘0) = 0
210	203, 209	eqtrdi 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = 0)
211	199, 210	oveq12d 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 1 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = (0 − 0))
212		0m0e0 12374	. . . . . . . . . . . . . . 15 ⊢ (0 − 0) = 0
213	211, 212	eqtrdi 2816	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = 0)
214	213	eqcoms 2773	. . . . . . . . . . . . 13 ⊢ (1 = 𝑛 → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = 0)
215	214	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) = 0)
216		0red 11226	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ∈ ℝ)
217	28	nnge1d 12299	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
218	87, 217	logge0d 26846	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
219	35	lep1d 12161	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ≤ ((log‘𝑛) + 1))
220	216, 35, 37, 218, 219	letrd 11382	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘𝑛) + 1))
221	220	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → 0 ≤ ((log‘𝑛) + 1))
222	215, 221	eqbrtrd 5135	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
223		elfzle1 13571	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 1 ≤ 𝑛)
224	223	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
225	36, 87	leloed 11368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 ≤ 𝑛 ↔ (1 < 𝑛 ∨ 1 = 𝑛)))
226	224, 225	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 < 𝑛 ∨ 1 = 𝑛))
227	184, 222, 226	mpjaodan 973	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
228	100, 37, 34, 110, 227	lemul2ad 12170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))
229	25, 101, 38, 228	fsumle 15874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))
230	102, 75, 74, 109, 229	lemul2ad 12170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1))))) ≤ ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))))
231	103, 76, 49, 230	lesub2dd 11846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ≤ (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))))
232	77, 104, 10, 231	lediv1dd 13134	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ≤ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥))
233	232	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ≤ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇‘𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥))
234	82, 86, 105, 78, 233	lo1le 15727	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥)) ∈ ≤𝑂(1))
235	106	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ⁺)
236		pntrlog2bndlem5.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
237	235, 236	rpmulcld 13092	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ⁺)
238	237	rpred 13076	. . . . . 6 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
239	238	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ)
240	5, 22	rerpdivcld 13107	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
241	5, 240	readdcld 11253	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
242		ioossre 13450	. . . . . 6 ⊢ (1(,)+∞) ⊆ ℝ
243		lo1const 15696	. . . . . 6 ⊢ (((1(,)+∞) ⊆ ℝ ∧ (2 · 𝐵) ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
244	242, 238, 243	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
245		lo1const 15696	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
246	242, 82, 245	sylancr 599	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
247		divlogrlim 26851	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0
248		rlimo1 15692	. . . . . . . 8 ⊢ ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
249	247, 248	mp1i 14	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
250	240, 249	o1lo1d 15614	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ ≤𝑂(1))
251	5, 240, 246, 250	lo1add 15702	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ ≤𝑂(1))
252	237	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ⁺)
253	252	rpge0d 13080	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 · 𝐵))
254	239, 241, 244, 251, 253	lo1mul 15703	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 · 𝐵) · (1 + (1 / (log‘𝑥))))) ∈ ≤𝑂(1))
255	239, 241	remulcld 11254	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
256	79, 10	rerpdivcld 13107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
257	18, 5	readdcld 11253	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
258	236	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ⁺)
259	258	rpred 13076	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
260	257, 259	remulcld 11254	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) · 𝐵) ∈ ℝ)
261	28	nnrecred 12302	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
262	25, 261	fsumrecl 15808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
263	262, 259	remulcld 11254	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ∈ ℝ)
264	34, 26	rerpdivcld 13107	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
265	259	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
266	261, 265	remulcld 11254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · 𝐵) ∈ ℝ)
267	30	rpcnd 13078	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
268	30	rpne0d 13081	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
269	33, 267, 268	absdivd 15533	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))))
270	2	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
271	270, 28	nndivred 12305	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
272	30	rpge0d 13080	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
273	271, 272	absidd 15498	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
274	273	oveq2d 7435	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
275	269, 274	eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
276	46	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
277	87	recnd 11252	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
278	47	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
279	28	nnne0d 12301	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
280	43, 276, 277, 278, 279	divdiv2d 12038	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥))
281	43, 277, 276, 278	div23d 12043	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
282	275, 280, 281	3eqtrd 2804	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
283		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (𝑅‘𝑦) = (𝑅‘(𝑥 / 𝑛)))
284		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
285	283, 284	oveq12d 7437	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 / 𝑛) → ((𝑅‘𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
286	285	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅‘𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
287	286	breq1d 5121	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
288		pntrlog2bndlem5.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
289	288	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (abs‘((𝑅‘𝑦) / 𝑦)) ≤ 𝐵)
290	287, 289, 30	rspcdva 3584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
291	282, 290	eqbrtrrd 5137	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵)
292	264, 265, 29	lemuldivd 13125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵 ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛)))
293	291, 292	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛))
294	265	recnd 11252	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
295	294, 277, 279	divrec2d 12010	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐵 / 𝑛) = ((1 / 𝑛) · 𝐵))
296	293, 295	breqtrd 5139	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ ((1 / 𝑛) · 𝐵))
297	25, 264, 266, 296	fsumle 15874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
298	25, 46, 43, 47	fsumdivc 15860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥))
299	258	rpcnd 13078	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
300	261	recnd 11252	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
301	25, 299, 300	fsummulc1 15859	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) = Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
302	297, 298, 301	3brtr4d 5145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵))
303	258	rpge0d 13080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
304		harmonicubnd 27225	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
305	2, 9, 304	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
306	262, 257, 259, 303, 305	lemul1ad 12169	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ≤ (((log‘𝑥) + 1) · 𝐵))
307	256, 263, 260, 302, 306	letrd 11382	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (((log‘𝑥) + 1) · 𝐵))
308	256, 260, 74, 109, 307	lemul2ad 12170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)) ≤ ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
309	24, 44, 46, 47	divassd 12041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) = ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)))
310	241	recnd 11252	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℂ)
311	21, 299, 310	mul32d 11435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵))
312		1cnd 11217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
313	19, 312, 19, 23	divdird 12044	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
314	19, 23	dividd 12004	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
315	314	oveq1d 7434	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
316	313, 315	eqtr2d 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
317	316	oveq2d 7435	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
318	19, 312	addcld 11243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
319	21, 19, 318, 23	div32d 12029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
320	317, 319	eqtr4d 2803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)))
321	320	oveq1d 7434	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵) = (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵))
322	24, 318, 299	mulassd 11247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
323	311, 321, 322	3eqtrd 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
324	308, 309, 323	3brtr4d 5145	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
325	324	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
326	82, 254, 255, 81, 325	lo1le 15727	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
327	78, 81, 234, 326	lo1add 15702	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥))) ∈ ≤𝑂(1))
328	71, 327	eqeltrrd 2866	1 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ⊆ wss 3906 ifcif 4489 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 +∞cpnf 11255 < clt 11258 ≤ cle 11259 − cmin 11456 / cdiv 11886 ℕcn 12248 2c2 12310 ℝ⁺crp 13032 (,)cioo 13388 ...cfz 13551 ⌊cfl 13841 abscabs 15309 ⇝_𝑟 crli 15560 𝑂(1)co1 15561 ≤𝑂(1)clo1 15562 Σcsu 15761 logclog 26770 Λcvma 27307 ψcchp 27308

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 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

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 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-fac 14328 df-bc 14357 df-hash 14385 df-shft 15128 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-o1 15565 df-lo1 15566 df-sum 15762 df-ef 16143 df-e 16144 df-sin 16145 df-cos 16146 df-tan 16147 df-pi 16148 df-dvds 16333 df-gcd 16575 df-prm 16752 df-pc 16919 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-cmp 23594 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-limc 26076 df-dv 26077 df-ulm 26591 df-log 26772 df-cxp 26773 df-atan 27083 df-em 27208 df-cht 27312 df-vma 27313 df-chp 27314 df-ppi 27315 df-mu 27316

This theorem is used by: pntrlog2bndlem6 27798

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