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Theorem pntrsumo1 25667

Description: A bound on a sum over 𝑅. Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))

**Assertion**
Ref	Expression
pntrsumo1	⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)

Distinct variable groups: 𝑛,𝑎,𝑥 𝑅,𝑛,𝑥 Allowed substitution hint: 𝑅(𝑎)

Proof of Theorem pntrsumo1 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 10356	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		elicopnf 12558	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
4	3	simplbi 493	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ)
5		0red 10360	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 ∈ ℝ)
6		1red 10357	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℝ)
7		0lt1 10874	. . . . . . . . . . 11 ⊢ 0 < 1
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 1)
9	3	simprbi 492	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
10	5, 6, 4, 8, 9	ltletrd 10516	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 𝑥)
11	4, 10	elrpd 12153	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
12	11	ssriv 3831	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ⁺)
14		rpssre 12119	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
15	13, 14	syl6ss 3839	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
16	15	resmptd 5689	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
17		chpcl 25263	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	4, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℝ)
19		peano2re 10528	. . . . . . . . . . 11 ⊢ ((ψ‘𝑥) ∈ ℝ → ((ψ‘𝑥) + 1) ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℝ)
21	11	rprege0d 12163	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 12916	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ₀)
24		nn0p1nn 11659	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℕ)
26	20, 25	nndivred 11405	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
27	26	recnd 10385	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
28		ax-1cn 10310	. . . . . . . 8 ⊢ 1 ∈ ℂ
29		subcl 10600	. . . . . . . 8 ⊢ (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
30	27, 28, 29	sylancl 582	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
31		fzfid 13067	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (1...(⌊‘𝑥)) ∈ Fin)
32		elfznn 12663	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
33	32	adantl 475	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
34		nnrp 12125	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
35		pntrval.r	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
36	35	pntrf 25665	. . . . . . . . . . . . . 14 ⊢ 𝑅:ℝ⁺⟶ℝ
37	36	ffvelrni 6607	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
38	34, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
39		peano2nn 11364	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40		nnmulcl 11375	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
41	39, 40	mpdan 680	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
42	38, 41	nndivred 11405	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
43	33, 42	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
44	31, 43	fsumrecl 14842	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
45	44	recnd 10385	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
46	4, 45	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
47		oveq2 6913	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
48		fvoveq1 6928	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
49		oveq1 6912	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
50	48, 49	oveq12d 6923	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))
51	47, 50	jca 509	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1 / 𝑚) = (1 / 𝑛) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1))))
52		oveq2 6913	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
53		fvoveq1 6928	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
54		oveq1 6912	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
55	53, 54	oveq12d 6923	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
56	52, 55	jca 509	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((1 / 𝑚) = (1 / (𝑛 + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))))
57		oveq2 6913	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
58		1div1e1 11042	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
59	57, 58	syl6eq 2877	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
60		oveq1 6912	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
61		1m1e0 11423	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
62	60, 61	syl6eq 2877	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
63	62	fveq2d 6437	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
64		2pos 11461	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
65		0re 10358	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
66		chpeq0 25346	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
67	65, 66	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
68	64, 67	mpbir 223	. . . . . . . . . . . . . 14 ⊢ (ψ‘0) = 0
69	63, 68	syl6eq 2877	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
70	69, 62	oveq12d 6923	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = (0 − 0))
71		0m0e0 11478	. . . . . . . . . . . 12 ⊢ (0 − 0) = 0
72	70, 71	syl6eq 2877	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0)
73	59, 72	jca 509	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((1 / 𝑚) = 1 ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0))
74		oveq2 6913	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)))
75		fvoveq1 6928	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
76		oveq1 6912	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
77	75, 76	oveq12d 6923	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)))
78	74, 77	jca 509	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))))
79		nnuz 12005	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
80	25, 79	syl6eleq 2916	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
81		elfznn 12663	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
82	81	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
83	82	nnrecred 11402	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℝ)
84	83	recnd 10385	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℂ)
85	82	nnred 11367	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
86		peano2rem 10669	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
87	85, 86	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
88		chpcl 25263	. . . . . . . . . . . . 13 ⊢ ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
89	87, 88	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
90	89, 87	resubcld 10782	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℝ)
91	90	recnd 10385	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℂ)
92	51, 56, 73, 78, 80, 84, 91	fsumparts 14912	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = ((((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))))
93	4	flcld 12894	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℤ)
94		fzval3 12832	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
95	93, 94	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
96	95	eqcomd 2831	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
97	32	adantl 475	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
98	97	nncnd 11368	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
99		pncan 10607	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
100	98, 28, 99	sylancl 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
101	97	nnred 11367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
102	100, 101	eqeltrd 2906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ)
103		chpcl 25263	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) − 1) ∈ ℝ → (ψ‘((𝑛 + 1) − 1)) ∈ ℝ)
104	102, 103	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℝ)
105	104	recnd 10385	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℂ)
106	102	recnd 10385	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℂ)
107		peano2rem 10669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
108	101, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
109		chpcl 25263	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
110	108, 109	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
111	110	recnd 10385	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
112		1cnd 10351	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
113	98, 112	subcld 10713	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℂ)
114	105, 106, 111, 113	sub4d 10762	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1))) = (((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) − (((𝑛 + 1) − 1) − (𝑛 − 1))))
115		vmacl 25257	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
116	97, 115	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
117	116	recnd 10385	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
118		nnm1nn0 11661	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
119	97, 118	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
120		chpp1 25294	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
121	119, 120	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
122		npcan 10611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
123	98, 28, 122	sylancl 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
124	123, 100	eqtr4d 2864	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = ((𝑛 + 1) − 1))
125	124	fveq2d 6437	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = (ψ‘((𝑛 + 1) − 1)))
126	123	fveq2d 6437	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
127	126	oveq2d 6921	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
128	121, 125, 127	3eqtr3d 2869	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
129	111, 117, 128	mvrladdd 10767	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
130		peano2cn 10527	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
131	98, 130	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
132	131, 98, 112	nnncan2d 10748	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = ((𝑛 + 1) − 𝑛))
133		pncan2 10608	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 𝑛) = 1)
134	98, 28, 133	sylancl 582	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
135	132, 134	eqtrd 2861	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = 1)
136	129, 135	oveq12d 6923	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) − (((𝑛 + 1) − 1) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
137	114, 136	eqtrd 2861	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
138	137	oveq2d 6921	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
139		peano2rem 10669	. . . . . . . . . . . . . 14 ⊢ ((Λ‘𝑛) ∈ ℝ → ((Λ‘𝑛) − 1) ∈ ℝ)
140	116, 139	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℝ)
141	140	recnd 10385	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℂ)
142	97	nnne0d 11401	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
143	141, 98, 142	divrec2d 11131	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
144	138, 143	eqtr4d 2864	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = (((Λ‘𝑛) − 1) / 𝑛))
145	96, 144	sumeq12rdv 14815	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛))
146	23	nn0cnd 11680	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℂ)
147		pncan 10607	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
148	146, 28, 147	sylancl 582	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (1[,)+∞) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
149	148	fveq2d 6437	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
150		chpfl 25289	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
151	4, 150	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
152	149, 151	eqtrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
153	152	oveq1d 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)))
154	18	recnd 10385	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℂ)
155	25	nncnd 11368	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℂ)
156		1cnd 10351	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℂ)
157	154, 155, 156	subsub3d 10743	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
158	153, 157	eqtrd 2861	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
159	158	oveq2d 6921	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))))
160	25	nnrecred 11402	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℝ)
161	160	recnd 10385	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℂ)
162		peano2cn 10527	. . . . . . . . . . . . . . 15 ⊢ ((ψ‘𝑥) ∈ ℂ → ((ψ‘𝑥) + 1) ∈ ℂ)
163	154, 162	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℂ)
164	161, 163, 155	subdid 10810	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))) = (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) − ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1))))
165	25	nnne0d 11401	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ≠ 0)
166	163, 155, 165	divrec2d 11131	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) = ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)))
167	166	eqcomd 2831	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
168	155, 165	recid2d 11123	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1)) = 1)
169	167, 168	oveq12d 6923	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) − ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
170	159, 164, 169	3eqtrd 2865	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
171	28	mul01i 10545	. . . . . . . . . . . . 13 ⊢ (1 · 0) = 0
172	171	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1 · 0) = 0)
173	170, 172	oveq12d 6923	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0))
174	30	subid1d 10702	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
175	173, 174	eqtrd 2861	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
176	97, 41	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
177	176	nnrecred 11402	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℝ)
178	177	recnd 10385	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℂ)
179	97, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℝ)
180	179	recnd 10385	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℂ)
181	178, 180	mulneg1d 10807	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
182	98, 112	mulcld 10377	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) ∈ ℂ)
183	98, 131	mulcld 10377	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℂ)
184	176	nnne0d 11401	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ≠ 0)
185	131, 182, 183, 184	divsubdird 11166	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (((𝑛 + 1) / (𝑛 · (𝑛 + 1))) − ((𝑛 · 1) / (𝑛 · (𝑛 + 1)))))
186	98	mulid1d 10374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) = 𝑛)
187	186	oveq2d 6921	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = ((𝑛 + 1) − 𝑛))
188	187, 134	eqtrd 2861	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = 1)
189	188	oveq1d 6920	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 · (𝑛 + 1))))
190	131	mulid1d 10374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 1) = (𝑛 + 1))
191	131, 98	mulcomd 10378	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 𝑛) = (𝑛 · (𝑛 + 1)))
192	190, 191	oveq12d 6923	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = ((𝑛 + 1) / (𝑛 · (𝑛 + 1))))
193	97, 39	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
194	193	nnne0d 11401	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
195	112, 98, 131, 142, 194	divcan5d 11153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = (1 / 𝑛))
196	192, 195	eqtr3d 2863	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) / (𝑛 · (𝑛 + 1))) = (1 / 𝑛))
197	112, 131, 98, 194, 142	divcan5d 11153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 · 1) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 + 1)))
198	196, 197	oveq12d 6923	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) / (𝑛 · (𝑛 + 1))) − ((𝑛 · 1) / (𝑛 · (𝑛 + 1)))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
199	185, 189, 198	3eqtr3d 2869	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
200	199	negeqd 10595	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(1 / (𝑛 · (𝑛 + 1))) = -((1 / 𝑛) − (1 / (𝑛 + 1))))
201	97	nnrecred 11402	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
202	201	recnd 10385	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
203	193	nnrecred 11402	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℝ)
204	203	recnd 10385	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℂ)
205	202, 204	negsubdi2d 10729	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((1 / 𝑛) − (1 / (𝑛 + 1))) = ((1 / (𝑛 + 1)) − (1 / 𝑛)))
206	200, 205	eqtr2d 2862	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / (𝑛 + 1)) − (1 / 𝑛)) = -(1 / (𝑛 · (𝑛 + 1))))
207	97	nnrpd 12154	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
208	100, 207	eqeltrd 2906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ⁺)
209	35	pntrval 25664	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) − 1) ∈ ℝ⁺ → (𝑅‘((𝑛 + 1) − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
210	208, 209	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
211	100	fveq2d 6437	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = (𝑅‘𝑛))
212	210, 211	eqtr3d 2863	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) = (𝑅‘𝑛))
213	206, 212	oveq12d 6923	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
214	180, 183, 184	divrec2d 11131	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = ((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
215	214	negeqd 10595	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
216	181, 213, 215	3eqtr4d 2871	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
217	96, 216	sumeq12rdv 14815	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
218		fzfid 13067	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) ∈ Fin)
219	97, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
220	219	recnd 10385	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
221	218, 220	fsumneg 14893	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))-((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
222	217, 221	eqtrd 2861	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
223	175, 222	oveq12d 6923	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → ((((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
224	92, 145, 223	3eqtr3d 2869	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
225	30, 46	subnegd 10720	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) + Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
226	224, 225	eqtrd 2861	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) + Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
227	30, 46, 226	mvrladdd 10767	. . . . . 6 ⊢ (𝑥 ∈ (1[,)+∞) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
228	227	mpteq2ia 4963	. . . . 5 ⊢ (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
229		fzfid 13067	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
230	32	adantl 475	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
231	230, 115	syl 17	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
232	231, 139	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℝ)
233	232, 230	nndivred 11405	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
234	229, 233	fsumrecl 14842	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
235		rpre 12120	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
236	235	adantl 475	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
237	236, 17	syl 17	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℝ)
238	237, 19	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ)
239		rprege0 12129	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
240	239, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
241	240	adantl 475	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℕ₀)
242	241, 24	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
243	238, 242	nndivred 11405	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
244		peano2rem 10669	. . . . . . . 8 ⊢ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℝ)
245	243, 244	syl 17	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℝ)
246		reex 10343	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
247	246, 14	ssexi 5028	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ V
248	247	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ∈ V)
249	231, 230	nndivred 11405	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
250	249	recnd 10385	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
251	229, 250	fsumcl 14841	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
252		relogcl 24721	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
253	252	adantl 475	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
254	253	recnd 10385	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
255	251, 254	subcld 10713	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
256	230	nnrecred 11402	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
257	229, 256	fsumrecl 14842	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
258	257, 253	resubcld 10782	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℝ)
259		eqidd 2826	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
260		eqidd 2826	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))))
261	248, 255, 258, 259, 260	offval2 7174	. . . . . . . . 9 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_𝑓 − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))))
262	256	recnd 10385	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
263	229, 250, 262	fsumsub 14894	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
264	231	recnd 10385	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
265		1cnd 10351	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
266	230	nncnd 11368	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
267	230	nnne0d 11401	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
268	264, 265, 266, 267	divsubdird 11166	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = (((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
269	268	sumeq2dv 14810	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
270	257	recnd 10385	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
271	251, 270, 254	nnncan2d 10748	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
272	263, 269, 271	3eqtr4rd 2872	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛))
273	272	mpteq2dva 4967	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)))
274	261, 273	eqtrd 2861	. . . . . . . 8 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_𝑓 − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)))
275		vmadivsum 25584	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
276	14	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
277	258	recnd 10385	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℂ)
278		1red 10357	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
279		harmoniclbnd 25148	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
280	279	adantl 475	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
281	253, 257, 280	abssubge0d 14547	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
282	281	adantrr 710	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
283	235	ad2antrl 721	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
284		simprr 791	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
285		harmonicubnd 25149	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
286	283, 284, 285	syl2anc 581	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
287		1red 10357	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
288	257, 253, 287	lesubadd2d 10951	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
289	288	adantrr 710	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
290	286, 289	mpbird 249	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1)
291	282, 290	eqbrtrd 4895	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ≤ 1)
292	276, 277, 278, 278, 291	elo1d 14644	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
293		o1sub 14723	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_𝑓 − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) ∈ 𝑂(1))
294	275, 292, 293	sylancr 583	. . . . . . . 8 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_𝑓 − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) ∈ 𝑂(1))
295	274, 294	eqeltrrd 2907	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)) ∈ 𝑂(1))
296	243	recnd 10385	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
297		1cnd 10351	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
298	237	recnd 10385	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℂ)
299		rpcnne0 12132	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
300	299	adantl 475	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
301		divdir 11035	. . . . . . . . . . . 12 ⊢ (((ψ‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
302	298, 297, 300, 301	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
303	302	mpteq2dva 4967	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
304		simpr 479	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
305	237, 304	rerpdivcld 12187	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
306		rpreccl 12140	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
307	306	adantl 475	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
308		eqidd 2826	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
309		eqidd 2826	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
310	248, 305, 307, 308, 309	offval2 7174	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_𝑓 + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
311		chpo1ub 25582	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
312		divrcnv 14958	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
313	28, 312	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
314		rlimo1 14724	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
315	313, 314	mp1i 13	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
316		o1add 14721	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_𝑓 + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
317	311, 315, 316	sylancr 583	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_𝑓 + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
318	310, 317	eqeltrrd 2907	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
319	303, 318	eqeltrd 2906	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) ∈ 𝑂(1))
320	238, 304	rerpdivcld 12187	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℝ)
321		chpge0 25265	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
322	236, 321	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (ψ‘𝑥))
323	237, 322	ge0p1rpd 12186	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ⁺)
324	323	rprege0d 12163	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)))
325	242	nnrpd 12154	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
326	325	rpregt0d 12162	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)))
327		divge0 11222	. . . . . . . . . . . . 13 ⊢ (((((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)) ∧ (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1))) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
328	324, 326, 327	syl2anc 581	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
329	243, 328	absidd 14538	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
330	320	recnd 10385	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℂ)
331	330	abscld 14552	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / 𝑥)) ∈ ℝ)
332		fllep1 12897	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
333	236, 332	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≤ ((⌊‘𝑥) + 1))
334		rpregt0 12128	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
335	334	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
336	323	rpregt0d 12162	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1)))
337		lediv2 11243	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ ∧ 0 < 𝑥) ∧ (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)) ∧ (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1))) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥)))
338	335, 326, 336, 337	syl3anc 1496	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥)))
339	333, 338	mpbid 224	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥))
340	320	leabsd 14530	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
341	243, 320, 331, 339, 340	letrd 10513	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
342	329, 341	eqbrtrd 4895	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
343	342	adantrr 710	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
344	278, 319, 320, 296, 343	o1le 14760	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ∈ 𝑂(1))
345		o1const 14727	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
346	14, 28, 345	mp2an 685	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
347	346	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
348	296, 297, 344, 347	o1sub2 14733	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) ∈ 𝑂(1))
349	234, 245, 295, 348	o1sub2 14733	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
350	13, 349	o1res2 14671	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
351	228, 350	syl5eqelr 2911	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
352	16, 351	eqeltrd 2906	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) ∈ 𝑂(1))
353		eqid 2825	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
354	353, 45	fmpti 6631	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))):ℝ⟶ℂ
355	354	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))):ℝ⟶ℂ)
356		ssidd 3849	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
357	355, 356, 278	o1resb 14674	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1) ↔ ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) ∈ 𝑂(1)))
358	352, 357	mpbird 249	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
359	358	mptru 1666	1 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1658 ⊤wtru 1659 ∈ wcel 2166 ≠ wne 2999 Vcvv 3414 ⊆ wss 3798 class class class wbr 4873 ↦ cmpt 4952 ↾ cres 5344 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ∘_𝑓 cof 7155 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 + caddc 10255 · cmul 10257 +∞cpnf 10388 < clt 10391 ≤ cle 10392 − cmin 10585 -cneg 10586 / cdiv 11009 ℕcn 11350 2c2 11406 ℕ₀cn0 11618 ℤcz 11704 ℤ_≥cuz 11968 ℝ⁺crp 12112 [,)cico 12465 ...cfz 12619 ..^cfzo 12760 ⌊cfl 12886 abscabs 14351 ⇝_𝑟 crli 14593 𝑂(1)co1 14594 Σcsu 14793 logclog 24700 Λcvma 25231 ψcchp 25232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-xnn0 11691 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ioc 12468 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-fac 13354 df-bc 13383 df-hash 13411 df-shft 14184 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-o1 14598 df-lo1 14599 df-sum 14794 df-ef 15170 df-e 15171 df-sin 15172 df-cos 15173 df-pi 15175 df-dvds 15358 df-gcd 15590 df-prm 15758 df-pc 15913 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-pt 16458 df-prds 16461 df-xrs 16515 df-qtop 16520 df-imas 16521 df-xps 16523 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-lp 21311 df-perf 21312 df-cn 21402 df-cnp 21403 df-haus 21490 df-cmp 21561 df-tx 21736 df-hmeo 21929 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-tms 22497 df-cncf 23051 df-limc 24029 df-dv 24030 df-log 24702 df-cxp 24703 df-em 25132 df-cht 25236 df-vma 25237 df-chp 25238 df-ppi 25239

This theorem is referenced by: pntrsumbnd 25668

W3C validator