pntrlog2bndlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pntrlog2bndlem2		Structured version Visualization version GIF version

Theorem pntrlog2bndlem2 27081

Description: Lemma for pntrlog2bnd 27087. 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 11215	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		elioore 13354	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
3	2	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
4		chpcl 26628	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
6	5	recnd 11242	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℂ)
7		fzfid 13938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
8	3	adantr 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
9		elfznn 13530	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
10	9	adantl 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
11	10	peano2nnd 12229	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
12	8, 11	nndivred 12266	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ)
13		chpcl 26628	. . . . . . . . 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 11243	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
16	7, 15	fsumrecl 15680	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
17	16	recnd 11242	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
18	3	recnd 11242	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
19		eliooord 13383	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
20	19	adantl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
21	20	simpld 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
22	3, 21	rplogcld 26137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
23	22	rpcnd 13018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
24	18, 23	mulcld 11234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
25		1rp 12978	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ⁺)
27		1red 11215	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
28	27, 3, 21	ltled 11362	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
29	3, 26, 28	rpgecld 13055	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ⁺)
30	29	rpne0d 13021	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
31	22	rpne0d 13021	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
32	18, 23, 30, 31	mulne0d 11866	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
33	6, 17, 24, 32	divdird 12028	. . . 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 5249	. . 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 13032	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ⁺)
36	5, 35	rerpdivcld 13047	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) ∈ ℝ)
37	16, 35	rerpdivcld 13047	. . . 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 12021	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = ((ψ‘𝑥) / (𝑥 · (log‘𝑥))))
39	5, 29	rerpdivcld 13047	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	39	recnd 11242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
41	40, 23, 31	divrecd 11993	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
42	38, 41	eqtr3d 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
43	42	mpteq2dva 5249	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))))
44	22	rprecred 13027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
45	29	ex 414	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ⁺))
46	45	ssrdv 3989	. . . . . . 7 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ⁺)
47		chpo1ub 26983	. . . . . . . 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 15507	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
50		divlogrlim 26143	. . . . . . 7 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝_𝑟 0
51		rlimo1 15561	. . . . . . 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 15569	. . . . 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 13016	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
57	56, 1	readdcld 11243	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ)
58	57	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℝ)
59	27, 44	readdcld 11243	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
60		ioossre 13385	. . . . . . 7 ⊢ (1(,)+∞) ⊆ ℝ
61	57	recnd 11242	. . . . . . 7 ⊢ (𝜑 → (𝐴 + 1) ∈ ℂ)
62		o1const 15564	. . . . . . 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 11209	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
65		o1const 15564	. . . . . . . 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 15568	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ 𝑂(1))
68	58, 59, 63, 67	o1mul2 15569	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ 𝑂(1))
69	58, 59	remulcld 11244	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
70	37	recnd 11242	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
71		chpge0 26630	. . . . . . . . . . . 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 13014	. . . . . . . . . . . . 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 13031	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℝ⁺)
76	29	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
77	76	rpge0d 13020	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑥)
78	8, 75, 77	divge0d 13056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / (𝑛 + 1)))
79	14, 12, 72, 78	addge0d 11790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
80	7, 15, 79	fsumge0 15741	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
81	16, 35, 80	divge0d 13056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
82	37, 81	absidd 15369	. . . . . . 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 11242	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℂ)
84	83	abscld 15383	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ ℝ)
85	16, 29	rerpdivcld 13047	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ∈ ℝ)
86	29	relogcld 26131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
87	86, 27	readdcld 11243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
88	58, 87	remulcld 11244	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · ((log‘𝑥) + 1)) ∈ ℝ)
89	58, 3	remulcld 11244	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
90	10	nnrecred 12263	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
91	7, 90	fsumrecl 15680	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
92	89, 91	remulcld 11244	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
93	89, 87	remulcld 11244	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) ∈ ℝ)
94	56	ad2antrr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
95		1red 11215	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
96	94, 95	readdcld 11243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℝ)
97	96, 8	remulcld 11244	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
98	97, 90	remulcld 11244	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) · (1 / 𝑛)) ∈ ℝ)
99	97, 11	nndivred 12266	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ∈ ℝ)
100	97, 10	nndivred 12266	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) ∈ ℝ)
101	94, 12	remulcld 11244	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · (𝑥 / (𝑛 + 1))) ∈ ℝ)
102		fveq2 6892	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → (ψ‘𝑦) = (ψ‘(𝑥 / (𝑛 + 1))))
103		oveq2 7417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 / (𝑛 + 1)) → (𝐴 · 𝑦) = (𝐴 · (𝑥 / (𝑛 + 1))))
104	102, 103	breq12d 5162	. . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ⁺ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
107	76, 75	rpdivcld 13033	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ⁺)
108	104, 106, 107	rspcdva 3614	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (𝐴 · (𝑥 / (𝑛 + 1))))
109	14, 101, 12, 108	leadd1dd 11828	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
110	61	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℂ)
111	18	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
112	10	nncnd 12228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
113		1cnd 11209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
114	112, 113	addcld 11233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
115	11	nnne0d 12262	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
116	110, 111, 114, 115	divassd 12025	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 + 1) · (𝑥 / (𝑛 + 1))))
117	94	recnd 11242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
118	111, 114, 115	divcld 11990	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℂ)
119	117, 113, 118	adddird 11239	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · (𝑥 / (𝑛 + 1))) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (1 · (𝑥 / (𝑛 + 1)))))
120	118	mullidd 11232	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · (𝑥 / (𝑛 + 1))) = (𝑥 / (𝑛 + 1)))
121	120	oveq2d 7425	. . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
123	109, 122	breqtrrd 5177	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / (𝑛 + 1)))
124	56	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
125	55	adantr 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ⁺)
126	125	rpge0d 13020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴)
127	26	rpge0d 13020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 1)
128	124, 27, 126, 127	addge0d 11790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐴 + 1))
129	29	rpge0d 13020	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
130	58, 3, 128, 129	mulge0d 11791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ ((𝐴 + 1) · 𝑥))
131	130	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝐴 + 1) · 𝑥))
132	10	nnred 12227	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
133	132	lep1d 12145	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ (𝑛 + 1))
134	73, 75, 97, 131, 133	lediv2ad 13038	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
135	15, 99, 100, 123, 134	letrd 11371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
136	97	recnd 11242	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
137	10	nnne0d 12262	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
138	136, 112, 137	divrecd 11993	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) = (((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
139	135, 138	breqtrd 5175	. . . . . . . . . . . . . . 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 15745	. . . . . . . . . . . . . 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 11242	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
142	112, 137	reccld 11983	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
143	7, 141, 142	fsummulc2 15730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
144	140, 143	breqtrrd 5177	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
145		harmonicubnd 26514	. . . . . . . . . . . . . . 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 12154	. . . . . . . . . . . . 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 11371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)))
149	61	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℂ)
150	87	recnd 11242	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
151	149, 18, 150	mul32d 11424	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) = (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥))
152	148, 151	breqtrd 5175	. . . . . . . . . . 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 13070	. . . . . . . . . . 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 13074	. . . . . . . . 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 12021	. . . . . . . . 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 11209	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
158	23, 157	addcld 11233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
159	149, 158, 23, 31	divassd 12025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
160	23, 157, 23, 31	divdird 12028	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
161	23, 31	dividd 11988	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
162	161	oveq1d 7424	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
163	160, 162	eqtr2d 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
164	163	oveq2d 7425	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
165	159, 164	eqtr4d 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))))
166	155, 156, 165	3brtr3d 5180	. . . . . . . 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 15361	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
168	37, 69, 84, 166, 167	letrd 11371	. . . . . . 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 5171	. . . . . 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 716	. . . . 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 15599	. . . 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 15568	. . 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 11243	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
175	174, 35	rerpdivcld 13047	. 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 27066	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
178	177	ffvelcdmi 7086	. . . . . . . . . 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 11242	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
181	76, 73	rpdivcld 13033	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
182	177	ffvelcdmi 7086	. . . . . . . . . 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 11242	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
185	180, 184	subcld 11571	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
186	185	abscld 15383	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
187	132, 186	remulcld 11244	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
188	7, 187	fsumrecl 15680	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
189	188, 35	rerpdivcld 13047	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
190	189	recnd 11242	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
191	73	rpge0d 13020	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑛)
192	185	absge0d 15391	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))))
193	132, 186, 191, 192	mulge0d 11791	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
194	7, 187, 193	fsumge0 15741	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
195	188, 35, 194	divge0d 13056	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))))
196	189, 195	absidd 15369	. . . 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 11233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
198	197, 24, 32	divcld 11990	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
199	198	abscld 15383	. . . . 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 12266	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
201		chpcl 26628	. . . . . . . . . . . 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 11243	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℝ)
204	203, 15	resubcld 11642	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
205	132, 204	remulcld 11244	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℝ)
206	176	pntrval 27065	. . . . . . . . . . . . . . 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 27065	. . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . 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 11242	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
212	202	recnd 11242	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
213	111, 112, 137	divcld 11990	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
214	211, 118, 212, 213	sub4d 11620	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 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 6896	. . . . . . . . . . 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 11571	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
218	118, 213	subcld 11571	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)) ∈ ℂ)
219	217, 218	abs2dif2d 15405	. . . . . . . . . . 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 5171	. . . . . . . . . 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 13038	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ≤ (𝑥 / 𝑛))
222		chpwordi 26661	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (ψ‘(𝑥 / 𝑛)))
224	14, 202, 223	abssuble0d 15379	. . . . . . . . . . . 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 15379	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))) = ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1))))
226	224, 225	oveq12d 7427	. . . . . . . . . . 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 11618	. . . . . . . . . . 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 2776	. . . . . . . . . 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 5175	. . . . . . . . 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 12154	. . . . . . . 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 15745	. . . . . . 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 11242	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
233	112, 232	mulcld 11234	. . . . . . . . 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 15679	. . . . . . . 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 11585	. . . . . . . . . 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 13023	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
237		flge0nn0 13785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
238		nn0p1nn 12511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
239	236, 237, 238	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℕ)
240	3, 239	nndivred 12266	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℝ)
241		2re 12286	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
242	241	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
243		flltp1 13765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ ℝ → 𝑥 < ((⌊‘𝑥) + 1))
244	3, 243	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 < ((⌊‘𝑥) + 1))
245	239	nncnd 12228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℂ)
246	245	mulridd 11231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · 1) = ((⌊‘𝑥) + 1))
247	244, 246	breqtrrd 5177	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 < (((⌊‘𝑥) + 1) · 1))
248	239	nnrpd 13014	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
249	3, 27, 248	ltdivmuld 13067	. . . . . . . . . . . . . . . . . . . 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 12383	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 2
252	251	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 2)
253	240, 27, 242, 250, 252	lttrd 11375	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 2)
254		chpeq0 26711	. . . . . . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (0 + (𝑥 / ((⌊‘𝑥) + 1))))
258	240	recnd 11242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℂ)
259	258	addlidd 11415	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (0 + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
260	257, 259	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
261	260	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))))
262	239	nnne0d 12262	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ≠ 0)
263	18, 245, 262	divcan2d 11992	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))) = 𝑥)
264	261, 263	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = 𝑥)
265	18	div1d 11982	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑥 / 1) = 𝑥)
266	265	fveq2d 6896	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / 1)) = (ψ‘𝑥))
267	266, 265	oveq12d 7427	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / 1)) + (𝑥 / 1)) = ((ψ‘𝑥) + 𝑥))
268	267	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = (1 · ((ψ‘𝑥) + 𝑥)))
269	5, 3	readdcld 11243	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℝ)
270	269	recnd 11242	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℂ)
271	270	mullidd 11232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘𝑥) + 𝑥)) = ((ψ‘𝑥) + 𝑥))
272	268, 271	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = ((ψ‘𝑥) + 𝑥))
273	264, 272	oveq12d 7427	. . . . . . . . . . . 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 11587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = (𝑥 − ((ψ‘𝑥) + 𝑥)))
275	6, 18	pncand 11572	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + 𝑥) − 𝑥) = (ψ‘𝑥))
276	275	negeqd 11454	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = -(ψ‘𝑥))
277	273, 274, 276	3eqtr2d 2779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) = -(ψ‘𝑥))
278	3	flcld 13763	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (⌊‘𝑥) ∈ ℤ)
279		fzval3 13701	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
282	112, 113	pncan2d 11573	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
283	282	oveq1d 7424	. . . . . . . . . . . . 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 11242	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
285	284	mullidd 11232	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 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 15653	. . . . . . . . . . 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 7427	. . . . . . . . . 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 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑥 / 𝑚) = (𝑥 / 𝑛))
290	289	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 𝑛)))
291	290, 289	oveq12d 7427	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))
292	291	ancli 550	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 = 𝑛 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛))))
293		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑛 + 1) → (𝑥 / 𝑚) = (𝑥 / (𝑛 + 1)))
294	293	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / (𝑛 + 1))))
295	294, 293	oveq12d 7427	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
296	295	ancli 550	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 = (𝑛 + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
297		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑥 / 𝑚) = (𝑥 / 1))
298	297	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 1)))
299	298, 297	oveq12d 7427	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))
300	299	ancli 550	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑚 = 1 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1))))
301		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑥 / 𝑚) = (𝑥 / ((⌊‘𝑥) + 1)))
302	301	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / ((⌊‘𝑥) + 1))))
303	302, 301	oveq12d 7427	. . . . . . . . . . . . 13 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))))
304	303	ancli 550	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 = ((⌊‘𝑥) + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))))
305		nnuz 12865	. . . . . . . . . . . . 13 ⊢ ℕ = (ℤ_≥‘1)
306	239, 305	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
307		elfznn 13530	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
308	307	adantl 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
309	308	nncnd 12228	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℂ)
310	3	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑥 ∈ ℝ)
311	310, 308	nndivred 12266	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑥 / 𝑚) ∈ ℝ)
312		chpcl 26628	. . . . . . . . . . . . . . 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 11243	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℝ)
315	314	recnd 11242	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℂ)
316	292, 296, 300, 304, 306, 309, 315	fsumparts 15752	. . . . . . . . . . 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 11233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℂ)
318	211, 118	addcld 11233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
319	317, 318	negsubdi2d 11587	. . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . 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 11668	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 15653	. . . . . . . . . . 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 2775	. . . . . . . . . 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 2779	. . . . . . . . 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 15733	. . . . . . . . 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 2774	. . . . . . . 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 11583	. . . . . . 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 5175	. . . . . 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 13074	. . . . 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 15361	. . . . 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 11371	. . . 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 5171	. . 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 716	. 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 15599	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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3949 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 +∞cpnf 11245 < clt 11248 ≤ cle 11249 − cmin 11444 -cneg 11445 / cdiv 11871 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 (,)cioo 13324 ...cfz 13484 ..^cfzo 13627 ⌊cfl 13755 abscabs 15181 ⇝_𝑟 crli 15429 𝑂(1)co1 15430 Σcsu 15632 logclog 26063 Λcvma 26596 ψcchp 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-o1 15434 df-lo1 15435 df-sum 15633 df-ef 16011 df-e 16012 df-sin 16013 df-cos 16014 df-pi 16016 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-cxp 26066 df-em 26497 df-cht 26601 df-vma 26602 df-chp 26603 df-ppi 26604

This theorem is referenced by: pntrlog2bndlem3 27082

W3C validator