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Theorem pntrlog2bndlem2 27557

Description: Lemma for pntrlog2bnd 27563. 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))
pntrlog2bndlem2.1	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntrlog2bndlem2.2	⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (ψ‘𝑦) ≤ (𝐴 · 𝑦))

**Assertion**
Ref	Expression
pntrlog2bndlem2	⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))

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

Proof of Theorem pntrlog2bndlem2 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 11145	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		elioore 13303	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
4		chpcl 27102	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
6	5	recnd 11172	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℂ)
7		fzfid 13908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
8	3	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
9		elfznn 13481	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
10	9	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
11	10	peano2nnd 12174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
12	8, 11	nndivred 12211	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ)
13		chpcl 27102	. . . . . . . . 9 ⊢ ((𝑥 / (𝑛 + 1)) ∈ ℝ → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
15	14, 12	readdcld 11173	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
16	7, 15	fsumrecl 15669	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
17	16	recnd 11172	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
18	3	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
19		eliooord 13333	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
20	19	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
21	20	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
22	3, 21	rplogcld 26606	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
23	22	rpcnd 12963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
24	18, 23	mulcld 11164	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
25		1rp 12921	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
27		1red 11145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
28	27, 3, 21	ltled 11293	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
29	3, 26, 28	rpgecld 13000	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
30	29	rpne0d 12966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
31	22	rpne0d 12966	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
32	18, 23, 30, 31	mulne0d 11801	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
33	6, 17, 24, 32	divdird 11967	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) = (((ψ‘𝑥) / (𝑥 · (log‘𝑥))) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))))
34	33	mpteq2dva 5193	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / (𝑥 · (log‘𝑥))) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))))
35	29, 22	rpmulcld 12977	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
36	5, 35	rerpdivcld 12992	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) ∈ ℝ)
37	16, 35	rerpdivcld 12992	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
38	6, 18, 23, 30, 31	divdiv1d 11960	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = ((ψ‘𝑥) / (𝑥 · (log‘𝑥))))
39	5, 29	rerpdivcld 12992	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	39	recnd 11172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
41	40, 23, 31	divrecd 11932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
42	38, 41	eqtr3d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
43	42	mpteq2dva 5193	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))))
44	22	rprecred 12972	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
45	29	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
46	45	ssrdv 3941	. . . . . . 7 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ⁺)
47		chpo1ub 27459	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
48	47	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
49	46, 48	o1res2 15498	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
50		divlogrlim 26612	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0
51		rlimo1 15552	. . . . . . 7 ⊢ ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
52	50, 51	mp1i 13	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
53	39, 44, 49, 52	o1mul2 15560	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
54	43, 53	eqeltrd 2837	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
55		pntrlog2bndlem2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
56	55	rpred 12961	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	56, 1	readdcld 11173	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ)
58	57	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℝ)
59	27, 44	readdcld 11173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
60		ioossre 13335	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
61	57	recnd 11172	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℂ)
62		o1const 15555	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ (𝐴 + 1) ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 + 1)) ∈ 𝑂(1))
63	60, 61, 62	sylancr 588	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 + 1)) ∈ 𝑂(1))
64		1cnd 11139	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
65		o1const 15555	. . . . . . . 8 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
66	60, 64, 65	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
67	27, 44, 66, 52	o1add2 15559	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ 𝑂(1))
68	58, 59, 63, 67	o1mul2 15560	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ 𝑂(1))
69	58, 59	remulcld 11174	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
70	37	recnd 11172	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
71		chpge0 27104	. . . . . . . . . . . 12 ⊢ ((𝑥 / (𝑛 + 1)) ∈ ℝ → 0 ≤ (ψ‘(𝑥 / (𝑛 + 1))))
72	12, 71	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (ψ‘(𝑥 / (𝑛 + 1))))
73	10	nnrpd 12959	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
74	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ⁺)
75	73, 74	rpaddcld 12976	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℝ⁺)
76	29	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
77	76	rpge0d 12965	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑥)
78	8, 75, 77	divge0d 13001	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / (𝑛 + 1)))
79	14, 12, 72, 78	addge0d 11725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
80	7, 15, 79	fsumge0 15730	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
81	16, 35, 80	divge0d 13001	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
82	37, 81	absidd 15358	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
83	69	recnd 11172	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℂ)
84	83	abscld 15374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ ℝ)
85	16, 29	rerpdivcld 12992	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ∈ ℝ)
86	29	relogcld 26600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
87	86, 27	readdcld 11173	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
88	58, 87	remulcld 11174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · ((log‘𝑥) + 1)) ∈ ℝ)
89	58, 3	remulcld 11174	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
90	10	nnrecred 12208	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
91	7, 90	fsumrecl 15669	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
92	89, 91	remulcld 11174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
93	89, 87	remulcld 11174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) ∈ ℝ)
94	56	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
95		1red 11145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
96	94, 95	readdcld 11173	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℝ)
97	96, 8	remulcld 11174	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
98	97, 90	remulcld 11174	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) · (1 / 𝑛)) ∈ ℝ)
99	97, 11	nndivred 12211	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ∈ ℝ)
100	97, 10	nndivred 12211	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) ∈ ℝ)
101	94, 12	remulcld 11174	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · (𝑥 / (𝑛 + 1))) ∈ ℝ)
102		fveq2 6842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → (ψ‘𝑦) = (ψ‘(𝑥 / (𝑛 + 1))))
103		oveq2 7376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → (𝐴 · 𝑦) = (𝐴 · (𝑥 / (𝑛 + 1))))
104	102, 103	breq12d 5113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → ((ψ‘𝑦) ≤ (𝐴 · 𝑦) ↔ (ψ‘(𝑥 / (𝑛 + 1))) ≤ (𝐴 · (𝑥 / (𝑛 + 1)))))
105		pntrlog2bndlem2.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
106	105	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
107	76, 75	rpdivcld 12978	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ⁺)
108	104, 106, 107	rspcdva 3579	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (𝐴 · (𝑥 / (𝑛 + 1))))
109	14, 101, 12, 108	leadd1dd 11763	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
110	61	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℂ)
111	18	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
112	10	nncnd 12173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
113		1cnd 11139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
114	112, 113	addcld 11163	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
115	11	nnne0d 12207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
116	110, 111, 114, 115	divassd 11964	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 + 1) · (𝑥 / (𝑛 + 1))))
117	94	recnd 11172	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
118	111, 114, 115	divcld 11929	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℂ)
119	117, 113, 118	adddird 11169	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · (𝑥 / (𝑛 + 1))) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (1 · (𝑥 / (𝑛 + 1)))))
120	118	mullidd 11162	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · (𝑥 / (𝑛 + 1))) = (𝑥 / (𝑛 + 1)))
121	120	oveq2d 7384	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 · (𝑥 / (𝑛 + 1))) + (1 · (𝑥 / (𝑛 + 1)))) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
122	116, 119, 121	3eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
123	109, 122	breqtrrd 5128	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / (𝑛 + 1)))
124	56	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
125	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ⁺)
126	125	rpge0d 12965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴)
127	26	rpge0d 12965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 1)
128	124, 27, 126, 127	addge0d 11725	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐴 + 1))
129	29	rpge0d 12965	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
130	58, 3, 128, 129	mulge0d 11726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ ((𝐴 + 1) · 𝑥))
131	130	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝐴 + 1) · 𝑥))
132	10	nnred 12172	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
133	132	lep1d 12085	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ (𝑛 + 1))
134	73, 75, 97, 131, 133	lediv2ad 12983	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
135	15, 99, 100, 123, 134	letrd 11302	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
136	97	recnd 11172	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
137	10	nnne0d 12207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
138	136, 112, 137	divrecd 11932	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) = (((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
139	135, 138	breqtrd 5126	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
140	7, 15, 98, 139	fsumle 15734	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
141	89	recnd 11172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
142	112, 137	reccld 11922	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
143	7, 141, 142	fsummulc2 15719	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
144	140, 143	breqtrrd 5128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
145		harmonicubnd 26988	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
146	3, 28, 145	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
147	91, 87, 89, 130, 146	lemul2ad 12094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) ≤ (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)))
148	16, 92, 93, 144, 147	letrd 11302	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)))
149	61	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℂ)
150	87	recnd 11172	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
151	149, 18, 150	mul32d 11355	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) = (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥))
152	148, 151	breqtrd 5126	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥))
153	16, 88, 29	ledivmul2d 13015	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ≤ ((𝐴 + 1) · ((log‘𝑥) + 1)) ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥)))
154	152, 153	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ≤ ((𝐴 + 1) · ((log‘𝑥) + 1)))
155	85, 88, 22, 154	lediv1dd 13019	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) / (log‘𝑥)) ≤ (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)))
156	17, 18, 23, 30, 31	divdiv1d 11960	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
157		1cnd 11139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
158	23, 157	addcld 11163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
159	149, 158, 23, 31	divassd 11964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
160	23, 157, 23, 31	divdird 11967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
161	23, 31	dividd 11927	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
162	161	oveq1d 7383	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
163	160, 162	eqtr2d 2773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
164	163	oveq2d 7384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
165	159, 164	eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))))
166	155, 156, 165	3brtr3d 5131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ≤ ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))))
167	69	leabsd 15350	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
168	37, 69, 84, 166, 167	letrd 11302	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
169	82, 168	eqbrtrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
170	169	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
171	1, 68, 69, 70, 170	o1le 15588	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
172	36, 37, 54, 171	o1add2 15559	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / (𝑥 · (log‘𝑥))) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))) ∈ 𝑂(1))
173	34, 172	eqeltrd 2837	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
174	5, 16	readdcld 11173	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
175	174, 35	rerpdivcld 12992	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
176		pntrlog2bnd.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
177	176	pntrf 27542	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
178	177	ffvelcdmi 7037	. . . . . . . . . 10 ⊢ ((𝑥 / (𝑛 + 1)) ∈ ℝ⁺ → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
179	107, 178	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
180	179	recnd 11172	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
181	76, 73	rpdivcld 12978	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
182	177	ffvelcdmi 7037	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
183	181, 182	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
184	183	recnd 11172	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
185	180, 184	subcld 11504	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
186	185	abscld 15374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
187	132, 186	remulcld 11174	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
188	7, 187	fsumrecl 15669	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
189	188, 35	rerpdivcld 12992	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
190	189	recnd 11172	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
191	73	rpge0d 12965	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑛)
192	185	absge0d 15382	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))))
193	132, 186, 191, 192	mulge0d 11726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
194	7, 187, 193	fsumge0 15730	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
195	188, 35, 194	divge0d 13001	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))))
196	189, 195	absidd 15358	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))))
197	6, 17	addcld 11163	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
198	197, 24, 32	divcld 11929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
199	198	abscld 15374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))) ∈ ℝ)
200	8, 10	nndivred 12211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
201		chpcl 27102	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
202	200, 201	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
203	202, 200	readdcld 11173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℝ)
204	203, 15	resubcld 11577	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
205	132, 204	remulcld 11174	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℝ)
206	176	pntrval 27541	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / (𝑛 + 1)) ∈ ℝ⁺ → (𝑅‘(𝑥 / (𝑛 + 1))) = ((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))))
207	107, 206	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) = ((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))))
208	176	pntrval 27541	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑛)) = ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛)))
209	181, 208	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) = ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛)))
210	207, 209	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) = (((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛))))
211	14	recnd 11172	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
212	202	recnd 11172	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
213	111, 112, 137	divcld 11929	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
214	211, 118, 212, 213	sub4d 11553	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛))) = (((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))))
215	210, 214	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) = (((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))))
216	215	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) = (abs‘(((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))))
217	211, 212	subcld 11504	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
218	118, 213	subcld 11504	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)) ∈ ℂ)
219	217, 218	abs2dif2d 15396	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))) ≤ ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))))
220	216, 219	eqbrtrd 5122	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ≤ ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))))
221	73, 75, 8, 77, 133	lediv2ad 12983	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ≤ (𝑥 / 𝑛))
222		chpwordi 27135	. . . . . . . . . . . . . 14 ⊢ (((𝑥 / (𝑛 + 1)) ∈ ℝ ∧ (𝑥 / 𝑛) ∈ ℝ ∧ (𝑥 / (𝑛 + 1)) ≤ (𝑥 / 𝑛)) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (ψ‘(𝑥 / 𝑛)))
223	12, 200, 221, 222	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (ψ‘(𝑥 / 𝑛)))
224	14, 202, 223	abssuble0d 15370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) = ((ψ‘(𝑥 / 𝑛)) − (ψ‘(𝑥 / (𝑛 + 1)))))
225	12, 200, 221	abssuble0d 15370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))) = ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1))))
226	224, 225	oveq12d 7386	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))) = (((ψ‘(𝑥 / 𝑛)) − (ψ‘(𝑥 / (𝑛 + 1)))) + ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1)))))
227	212, 213, 211, 118	addsub4d 11551	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (((ψ‘(𝑥 / 𝑛)) − (ψ‘(𝑥 / (𝑛 + 1)))) + ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1)))))
228	226, 227	eqtr4d 2775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))) = (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
229	220, 228	breqtrd 5126	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ≤ (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
230	186, 204, 132, 191, 229	lemul2ad 12094	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ≤ (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
231	7, 187, 205, 230	fsumle 15734	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
232	204	recnd 11172	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
233	112, 232	mulcld 11164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℂ)
234	7, 233	fsumcl 15668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℂ)
235	6, 17	negdi2d 11518	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (-(ψ‘𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
236	29	rprege0d 12968	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
237		flge0nn0 13752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
238		nn0p1nn 12452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
239	236, 237, 238	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℕ)
240	3, 239	nndivred 12211	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℝ)
241		2re 12231	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
242	241	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
243		flltp1 13732	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → 𝑥 < ((⌊‘𝑥) + 1))
244	3, 243	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 < ((⌊‘𝑥) + 1))
245	239	nncnd 12173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℂ)
246	245	mulridd 11161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · 1) = ((⌊‘𝑥) + 1))
247	244, 246	breqtrrd 5128	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 < (((⌊‘𝑥) + 1) · 1))
248	239	nnrpd 12959	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
249	3, 27, 248	ltdivmuld 13012	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑥 / ((⌊‘𝑥) + 1)) < 1 ↔ 𝑥 < (((⌊‘𝑥) + 1) · 1)))
250	247, 249	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 1)
251		1lt2 12323	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 2
252	251	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 2)
253	240, 27, 242, 250, 252	lttrd 11306	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 2)
254		chpeq0 27187	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 / ((⌊‘𝑥) + 1)) ∈ ℝ → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0 ↔ (𝑥 / ((⌊‘𝑥) + 1)) < 2))
255	240, 254	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0 ↔ (𝑥 / ((⌊‘𝑥) + 1)) < 2))
256	253, 255	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0)
257	256	oveq1d 7383	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (0 + (𝑥 / ((⌊‘𝑥) + 1))))
258	240	recnd 11172	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℂ)
259	258	addlidd 11346	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (0 + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
260	257, 259	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
261	260	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))))
262	239	nnne0d 12207	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ≠ 0)
263	18, 245, 262	divcan2d 11931	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))) = 𝑥)
264	261, 263	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = 𝑥)
265	18	div1d 11921	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 1) = 𝑥)
266	265	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / 1)) = (ψ‘𝑥))
267	266, 265	oveq12d 7386	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / 1)) + (𝑥 / 1)) = ((ψ‘𝑥) + 𝑥))
268	267	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = (1 · ((ψ‘𝑥) + 𝑥)))
269	5, 3	readdcld 11173	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℝ)
270	269	recnd 11172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℂ)
271	270	mullidd 11162	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘𝑥) + 𝑥)) = ((ψ‘𝑥) + 𝑥))
272	268, 271	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = ((ψ‘𝑥) + 𝑥))
273	264, 272	oveq12d 7386	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) = (𝑥 − ((ψ‘𝑥) + 𝑥)))
274	270, 18	negsubdi2d 11520	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = (𝑥 − ((ψ‘𝑥) + 𝑥)))
275	6, 18	pncand 11505	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + 𝑥) − 𝑥) = (ψ‘𝑥))
276	275	negeqd 11386	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = -(ψ‘𝑥))
277	273, 274, 276	3eqtr2d 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) = -(ψ‘𝑥))
278	3	flcld 13730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘𝑥) ∈ ℤ)
279		fzval3 13662	. . . . . . . . . . . . . 14 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
280	278, 279	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
281	280	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
282	112, 113	pncan2d 11506	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
283	282	oveq1d 7383	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (1 · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
284	15	recnd 11172	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
285	284	mullidd 11162	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
286	283, 285	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
287	281, 286	sumeq12rdv 15642	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
288	277, 287	oveq12d 7386	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = (-(ψ‘𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
289		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑥 / 𝑚) = (𝑥 / 𝑛))
290	289	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 𝑛)))
291	290, 289	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))
292	291	ancli 548	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = 𝑛 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛))))
293		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 + 1) → (𝑥 / 𝑚) = (𝑥 / (𝑛 + 1)))
294	293	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / (𝑛 + 1))))
295	294, 293	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
296	295	ancli 548	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 = (𝑛 + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
297		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑥 / 𝑚) = (𝑥 / 1))
298	297	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 1)))
299	298, 297	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))
300	299	ancli 548	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑚 = 1 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1))))
301		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑥 / 𝑚) = (𝑥 / ((⌊‘𝑥) + 1)))
302	301	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / ((⌊‘𝑥) + 1))))
303	302, 301	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))))
304	303	ancli 548	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 = ((⌊‘𝑥) + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))))
305		nnuz 12802	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
306	239, 305	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
307		elfznn 13481	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
308	307	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
309	308	nncnd 12173	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℂ)
310	3	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑥 ∈ ℝ)
311	310, 308	nndivred 12211	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑥 / 𝑚) ∈ ℝ)
312		chpcl 27102	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
313	311, 312	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
314	313, 311	readdcld 11173	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℝ)
315	314	recnd 11172	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℂ)
316	292, 296, 300, 304, 306, 309, 315	fsumparts 15741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))) = (((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
317	212, 213	addcld 11163	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℂ)
318	211, 118	addcld 11163	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
319	317, 318	negsubdi2d 11520	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛))))
320	319	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · -(((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = (𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))))
321	112, 232	mulneg2d 11603	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · -(((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = -(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
322	320, 321	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))) = -(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
323	281, 322	sumeq12rdv 15642	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
324	316, 323	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
325	235, 288, 324	3eqtr2d 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
326	7, 233	fsumneg 15722	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
327	325, 326	eqtr2d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = -((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
328	234, 197, 327	neg11d 11516	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
329	231, 328	breqtrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ≤ ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
330	188, 174, 35, 329	lediv1dd 13019	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ≤ (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))))
331	175	leabsd 15350	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
332	189, 175, 199, 330, 331	letrd 11302	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
333	196, 332	eqbrtrd 5122	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
334	333	adantrr 718	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
335	1, 173, 175, 190, 334	o1le 15588	1 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3903 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 +∞cpnf 11175 < clt 11178 ≤ cle 11179 − cmin 11376 -cneg 11377 / cdiv 11806 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ℝ⁺crp 12917 (,)cioo 13273 ...cfz 13435 ..^cfzo 13582 ⌊cfl 13722 abscabs 15169 ⇝_𝑟 crli 15420 𝑂(1)co1 15421 Σcsu 15621 logclog 26531 Λcvma 27070 ψcchp 27071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-o1 15425 df-lo1 15426 df-sum 15622 df-ef 16002 df-e 16003 df-sin 16004 df-cos 16005 df-pi 16007 df-dvds 16192 df-gcd 16434 df-prm 16611 df-pc 16777 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-log 26533 df-cxp 26534 df-em 26971 df-cht 27075 df-vma 27076 df-chp 27077 df-ppi 27078

This theorem is referenced by: pntrlog2bndlem3 27558

W3C validator