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Theorem pntrlog2bndlem2 26431

Description: Lemma for pntrlog2bnd 26437. 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 10817	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		elioore 12948	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
3	2	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
4		chpcl 25978	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
6	5	recnd 10844	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℂ)
7		fzfid 13529	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
8	3	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
9		elfznn 13124	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
10	9	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
11	10	peano2nnd 11830	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
12	8, 11	nndivred 11867	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ)
13		chpcl 25978	. . . . . . . . 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 10845	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
16	7, 15	fsumrecl 15281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
17	16	recnd 10844	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
18	3	recnd 10844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
19		eliooord 12977	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
20	19	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
21	20	simpld 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
22	3, 21	rplogcld 25489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
23	22	rpcnd 12613	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
24	18, 23	mulcld 10836	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
25		1rp 12573	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
27		1red 10817	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
28	27, 3, 21	ltled 10963	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
29	3, 26, 28	rpgecld 12650	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
30	29	rpne0d 12616	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
31	22	rpne0d 12616	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
32	18, 23, 30, 31	mulne0d 11467	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
33	6, 17, 24, 32	divdird 11629	. . . 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 5139	. . 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 12627	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
36	5, 35	rerpdivcld 12642	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) ∈ ℝ)
37	16, 35	rerpdivcld 12642	. . . 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 11622	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = ((ψ‘𝑥) / (𝑥 · (log‘𝑥))))
39	5, 29	rerpdivcld 12642	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	39	recnd 10844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
41	40, 23, 31	divrecd 11594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
42	38, 41	eqtr3d 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
43	42	mpteq2dva 5139	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))))
44	22	rprecred 12622	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
45	29	ex 416	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
46	45	ssrdv 3897	. . . . . . 7 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ⁺)
47		chpo1ub 26333	. . . . . . . 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 15107	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
50		divlogrlim 25495	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0
51		rlimo1 15161	. . . . . . 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 15169	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
54	43, 53	eqeltrd 2834	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
55		pntrlog2bndlem2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
56	55	rpred 12611	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	56, 1	readdcld 10845	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ)
58	57	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℝ)
59	27, 44	readdcld 10845	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
60		ioossre 12979	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
61	57	recnd 10844	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℂ)
62		o1const 15164	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ (𝐴 + 1) ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 + 1)) ∈ 𝑂(1))
63	60, 61, 62	sylancr 590	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 + 1)) ∈ 𝑂(1))
64		1cnd 10811	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
65		o1const 15164	. . . . . . . 8 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
66	60, 64, 65	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
67	27, 44, 66, 52	o1add2 15168	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ 𝑂(1))
68	58, 59, 63, 67	o1mul2 15169	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ 𝑂(1))
69	58, 59	remulcld 10846	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
70	37	recnd 10844	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
71		chpge0 25980	. . . . . . . . . . . 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 12609	. . . . . . . . . . . . 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 12626	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℝ⁺)
76	29	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
77	76	rpge0d 12615	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑥)
78	8, 75, 77	divge0d 12651	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / (𝑛 + 1)))
79	14, 12, 72, 78	addge0d 11391	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
80	7, 15, 79	fsumge0 15340	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
81	16, 35, 80	divge0d 12651	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
82	37, 81	absidd 14969	. . . . . . 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 10844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℂ)
84	83	abscld 14983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ ℝ)
85	16, 29	rerpdivcld 12642	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ∈ ℝ)
86	29	relogcld 25483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
87	86, 27	readdcld 10845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
88	58, 87	remulcld 10846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · ((log‘𝑥) + 1)) ∈ ℝ)
89	58, 3	remulcld 10846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
90	10	nnrecred 11864	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
91	7, 90	fsumrecl 15281	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
92	89, 91	remulcld 10846	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
93	89, 87	remulcld 10846	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) ∈ ℝ)
94	56	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
95		1red 10817	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
96	94, 95	readdcld 10845	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℝ)
97	96, 8	remulcld 10846	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
98	97, 90	remulcld 10846	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) · (1 / 𝑛)) ∈ ℝ)
99	97, 11	nndivred 11867	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ∈ ℝ)
100	97, 10	nndivred 11867	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) ∈ ℝ)
101	94, 12	remulcld 10846	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · (𝑥 / (𝑛 + 1))) ∈ ℝ)
102		fveq2 6706	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → (ψ‘𝑦) = (ψ‘(𝑥 / (𝑛 + 1))))
103		oveq2 7210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → (𝐴 · 𝑦) = (𝐴 · (𝑥 / (𝑛 + 1))))
104	102, 103	breq12d 5056	. . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
107	76, 75	rpdivcld 12628	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ⁺)
108	104, 106, 107	rspcdva 3532	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (𝐴 · (𝑥 / (𝑛 + 1))))
109	14, 101, 12, 108	leadd1dd 11429	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
110	61	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℂ)
111	18	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
112	10	nncnd 11829	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
113		1cnd 10811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
114	112, 113	addcld 10835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
115	11	nnne0d 11863	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
116	110, 111, 114, 115	divassd 11626	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 + 1) · (𝑥 / (𝑛 + 1))))
117	94	recnd 10844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
118	111, 114, 115	divcld 11591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℂ)
119	117, 113, 118	adddird 10841	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · (𝑥 / (𝑛 + 1))) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (1 · (𝑥 / (𝑛 + 1)))))
120	118	mulid2d 10834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · (𝑥 / (𝑛 + 1))) = (𝑥 / (𝑛 + 1)))
121	120	oveq2d 7218	. . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
123	109, 122	breqtrrd 5071	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / (𝑛 + 1)))
124	56	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
125	55	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ⁺)
126	125	rpge0d 12615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴)
127	26	rpge0d 12615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 1)
128	124, 27, 126, 127	addge0d 11391	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐴 + 1))
129	29	rpge0d 12615	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
130	58, 3, 128, 129	mulge0d 11392	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ ((𝐴 + 1) · 𝑥))
131	130	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝐴 + 1) · 𝑥))
132	10	nnred 11828	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
133	132	lep1d 11746	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ (𝑛 + 1))
134	73, 75, 97, 131, 133	lediv2ad 12633	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
135	15, 99, 100, 123, 134	letrd 10972	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
136	97	recnd 10844	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
137	10	nnne0d 11863	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
138	136, 112, 137	divrecd 11594	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) = (((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
139	135, 138	breqtrd 5069	. . . . . . . . . . . . . . 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 15344	. . . . . . . . . . . . . 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 10844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
142	112, 137	reccld 11584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
143	7, 141, 142	fsummulc2 15329	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
144	140, 143	breqtrrd 5071	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
145		harmonicubnd 25864	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
146	3, 28, 145	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
147	91, 87, 89, 130, 146	lemul2ad 11755	. . . . . . . . . . . . 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 10972	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)))
149	61	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℂ)
150	87	recnd 10844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
151	149, 18, 150	mul32d 11025	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) = (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥))
152	148, 151	breqtrd 5069	. . . . . . . . . . 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 12665	. . . . . . . . . . 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 260	. . . . . . . . . 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 12669	. . . . . . . . 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 11622	. . . . . . . . 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 10811	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
158	23, 157	addcld 10835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
159	149, 158, 23, 31	divassd 11626	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
160	23, 157, 23, 31	divdird 11629	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
161	23, 31	dividd 11589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
162	161	oveq1d 7217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
163	160, 162	eqtr2d 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
164	163	oveq2d 7218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
165	159, 164	eqtr4d 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))))
166	155, 156, 165	3brtr3d 5074	. . . . . . . 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 14961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
168	37, 69, 84, 166, 167	letrd 10972	. . . . . . 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 5065	. . . . . 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 717	. . . . 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 15199	. . . 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 15168	. . 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 2834	. 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 10845	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
175	174, 35	rerpdivcld 12642	. 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 26416	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
178	177	ffvelrni 6892	. . . . . . . . . 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 10844	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
181	76, 73	rpdivcld 12628	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
182	177	ffvelrni 6892	. . . . . . . . . 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 10844	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
185	180, 184	subcld 11172	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
186	185	abscld 14983	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
187	132, 186	remulcld 10846	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
188	7, 187	fsumrecl 15281	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
189	188, 35	rerpdivcld 12642	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
190	189	recnd 10844	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
191	73	rpge0d 12615	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑛)
192	185	absge0d 14991	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))))
193	132, 186, 191, 192	mulge0d 11392	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
194	7, 187, 193	fsumge0 15340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
195	188, 35, 194	divge0d 12651	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))))
196	189, 195	absidd 14969	. . . 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 10835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
198	197, 24, 32	divcld 11591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
199	198	abscld 14983	. . . . 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 11867	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
201		chpcl 25978	. . . . . . . . . . . 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 10845	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℝ)
204	203, 15	resubcld 11243	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
205	132, 204	remulcld 10846	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℝ)
206	176	pntrval 26415	. . . . . . . . . . . . . . 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 26415	. . . . . . . . . . . . . . 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 7220	. . . . . . . . . . . . 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 10844	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
212	202	recnd 10844	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
213	111, 112, 137	divcld 11591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
214	211, 118, 212, 213	sub4d 11221	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 6710	. . . . . . . . . . 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 11172	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
218	118, 213	subcld 11172	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)) ∈ ℂ)
219	217, 218	abs2dif2d 15005	. . . . . . . . . . 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 5065	. . . . . . . . . 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 12633	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ≤ (𝑥 / 𝑛))
222		chpwordi 26011	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (ψ‘(𝑥 / 𝑛)))
224	14, 202, 223	abssuble0d 14979	. . . . . . . . . . . 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 14979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))) = ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1))))
226	224, 225	oveq12d 7220	. . . . . . . . . . 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 11219	. . . . . . . . . . 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 2777	. . . . . . . . . 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 5069	. . . . . . . . 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 11755	. . . . . . . 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 15344	. . . . . . 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 10844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
233	112, 232	mulcld 10836	. . . . . . . . 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 15280	. . . . . . . 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 11186	. . . . . . . . . 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 12618	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
237		flge0nn0 13378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
238		nn0p1nn 12112	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
239	236, 237, 238	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℕ)
240	3, 239	nndivred 11867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℝ)
241		2re 11887	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
242	241	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
243		flltp1 13358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → 𝑥 < ((⌊‘𝑥) + 1))
244	3, 243	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 < ((⌊‘𝑥) + 1))
245	239	nncnd 11829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℂ)
246	245	mulid1d 10833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · 1) = ((⌊‘𝑥) + 1))
247	244, 246	breqtrrd 5071	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 < (((⌊‘𝑥) + 1) · 1))
248	239	nnrpd 12609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
249	3, 27, 248	ltdivmuld 12662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑥 / ((⌊‘𝑥) + 1)) < 1 ↔ 𝑥 < (((⌊‘𝑥) + 1) · 1)))
250	247, 249	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 1)
251		1lt2 11984	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 2
252	251	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 2)
253	240, 27, 242, 250, 252	lttrd 10976	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 2)
254		chpeq0 26061	. . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0)
257	256	oveq1d 7217	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (0 + (𝑥 / ((⌊‘𝑥) + 1))))
258	240	recnd 10844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℂ)
259	258	addid2d 11016	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (0 + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
260	257, 259	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
261	260	oveq2d 7218	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))))
262	239	nnne0d 11863	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ≠ 0)
263	18, 245, 262	divcan2d 11593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))) = 𝑥)
264	261, 263	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = 𝑥)
265	18	div1d 11583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 1) = 𝑥)
266	265	fveq2d 6710	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / 1)) = (ψ‘𝑥))
267	266, 265	oveq12d 7220	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / 1)) + (𝑥 / 1)) = ((ψ‘𝑥) + 𝑥))
268	267	oveq2d 7218	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = (1 · ((ψ‘𝑥) + 𝑥)))
269	5, 3	readdcld 10845	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℝ)
270	269	recnd 10844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℂ)
271	270	mulid2d 10834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘𝑥) + 𝑥)) = ((ψ‘𝑥) + 𝑥))
272	268, 271	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = ((ψ‘𝑥) + 𝑥))
273	264, 272	oveq12d 7220	. . . . . . . . . . . 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 11188	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = (𝑥 − ((ψ‘𝑥) + 𝑥)))
275	6, 18	pncand 11173	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + 𝑥) − 𝑥) = (ψ‘𝑥))
276	275	negeqd 11055	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = -(ψ‘𝑥))
277	273, 274, 276	3eqtr2d 2780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) = -(ψ‘𝑥))
278	3	flcld 13356	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘𝑥) ∈ ℤ)
279		fzval3 13294	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
282	112, 113	pncan2d 11174	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
283	282	oveq1d 7217	. . . . . . . . . . . . 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 10844	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
285	284	mulid2d 10834	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 15254	. . . . . . . . . . 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 7220	. . . . . . . . . 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 7210	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑥 / 𝑚) = (𝑥 / 𝑛))
290	289	fveq2d 6710	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 𝑛)))
291	290, 289	oveq12d 7220	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))
292	291	ancli 552	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = 𝑛 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛))))
293		oveq2 7210	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 + 1) → (𝑥 / 𝑚) = (𝑥 / (𝑛 + 1)))
294	293	fveq2d 6710	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / (𝑛 + 1))))
295	294, 293	oveq12d 7220	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
296	295	ancli 552	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 = (𝑛 + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
297		oveq2 7210	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑥 / 𝑚) = (𝑥 / 1))
298	297	fveq2d 6710	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 1)))
299	298, 297	oveq12d 7220	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))
300	299	ancli 552	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑚 = 1 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1))))
301		oveq2 7210	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑥 / 𝑚) = (𝑥 / ((⌊‘𝑥) + 1)))
302	301	fveq2d 6710	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / ((⌊‘𝑥) + 1))))
303	302, 301	oveq12d 7220	. . . . . . . . . . . . 13 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))))
304	303	ancli 552	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 = ((⌊‘𝑥) + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))))
305		nnuz 12460	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
306	239, 305	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
307		elfznn 13124	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
308	307	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
309	308	nncnd 11829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℂ)
310	3	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑥 ∈ ℝ)
311	310, 308	nndivred 11867	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑥 / 𝑚) ∈ ℝ)
312		chpcl 25978	. . . . . . . . . . . . . . 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 10845	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℝ)
315	314	recnd 10844	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℂ)
316	292, 296, 300, 304, 306, 309, 315	fsumparts 15351	. . . . . . . . . . 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 10835	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℂ)
318	211, 118	addcld 10835	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
319	317, 318	negsubdi2d 11188	. . . . . . . . . . . . . 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 7218	. . . . . . . . . . . . 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 11269	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 15254	. . . . . . . . . . 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 2776	. . . . . . . . . 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 2780	. . . . . . . . 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 15332	. . . . . . . . 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 2775	. . . . . . . 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 11184	. . . . . . 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 5069	. . . . . 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 12669	. . . . 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 14961	. . . . 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 10972	. . . 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 5065	. . 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 717	. 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 15199	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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ⊆ wss 3857 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 · cmul 10717 +∞cpnf 10847 < clt 10850 ≤ cle 10851 − cmin 11045 -cneg 11046 / cdiv 11472 ℕcn 11813 2c2 11868 ℕ₀cn0 12073 ℤcz 12159 ℤ_≥cuz 12421 ℝ⁺crp 12569 (,)cioo 12918 ...cfz 13078 ..^cfzo 13221 ⌊cfl 13348 abscabs 14780 ⇝_𝑟 crli 15029 𝑂(1)co1 15030 Σcsu 15232 logclog 25415 Λcvma 25946 ψcchp 25947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-er 8380 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-xnn0 12146 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-o1 15034 df-lo1 15035 df-sum 15233 df-ef 15610 df-e 15611 df-sin 15612 df-cos 15613 df-pi 15615 df-dvds 15797 df-gcd 16035 df-prm 16210 df-pc 16371 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-mulg 18461 df-cntz 18683 df-cmn 19144 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-limc 24735 df-dv 24736 df-log 25417 df-cxp 25418 df-em 25847 df-cht 25951 df-vma 25952 df-chp 25953 df-ppi 25954

This theorem is referenced by: pntrlog2bndlem3 26432

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