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Theorem pntrsumo1 27545

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 11243	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		elicopnf 13467	. . . . . . . . . . 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 497	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ)
5		0red 11246	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 ∈ ℝ)
6		1red 11244	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℝ)
7		0lt1 11767	. . . . . . . . . . 11 ⊢ 0 < 1
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 1)
9	3	simprbi 496	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
10	5, 6, 4, 8, 9	ltletrd 11403	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 𝑥)
11	4, 10	elrpd 13056	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
12	11	ssriv 3967	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ⁺)
14		rpssre 13024	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
15	13, 14	sstrdi 3976	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
16	15	resmptd 6038	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
17		chpcl 27103	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	4, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℝ)
19		peano2re 11416	. . . . . . . . . . 11 ⊢ ((ψ‘𝑥) ∈ ℝ → ((ψ‘𝑥) + 1) ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℝ)
21	11	rprege0d 13066	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 13842	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ₀)
24		nn0p1nn 12548	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℕ)
26	20, 25	nndivred 12302	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
27	26	recnd 11271	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
28		ax-1cn 11195	. . . . . . . 8 ⊢ 1 ∈ ℂ
29		subcl 11489	. . . . . . . 8 ⊢ (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
30	27, 28, 29	sylancl 586	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
31		fzfid 13996	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (1...(⌊‘𝑥)) ∈ Fin)
32		elfznn 13575	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
33	32	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
34		nnrp 13028	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
35		pntrval.r	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
36	35	pntrf 27543	. . . . . . . . . . . . . 14 ⊢ 𝑅:ℝ⁺⟶ℝ
37	36	ffvelcdmi 7083	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
38	34, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
39		peano2nn 12260	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40		nnmulcl 12272	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
41	39, 40	mpdan 687	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
42	38, 41	nndivred 12302	. . . . . . . . . . 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 15752	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
45	44	recnd 11271	. . . . . . . 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 7421	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
48		fvoveq1 7436	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
49		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
50	48, 49	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))
51	47, 50	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((1 / 𝑚) = (1 / 𝑛) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1))))
52		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
53		fvoveq1 7436	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
54		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
55	53, 54	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
56	52, 55	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((1 / 𝑚) = (1 / (𝑛 + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))))
57		oveq2 7421	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
58		1div1e1 11940	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
59	57, 58	eqtrdi 2785	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
60		oveq1 7420	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
61		1m1e0 12320	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
62	60, 61	eqtrdi 2785	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
63	62	fveq2d 6890	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
64		2pos 12351	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
65		0re 11245	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
66		chpeq0 27188	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
67	65, 66	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
68	64, 67	mpbir 231	. . . . . . . . . . . . . 14 ⊢ (ψ‘0) = 0
69	63, 68	eqtrdi 2785	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
70	69, 62	oveq12d 7431	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = (0 − 0))
71		0m0e0 12368	. . . . . . . . . . . 12 ⊢ (0 − 0) = 0
72	70, 71	eqtrdi 2785	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0)
73	59, 72	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((1 / 𝑚) = 1 ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0))
74		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)))
75		fvoveq1 7436	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
76		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
77	75, 76	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)))
78	74, 77	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))))
79		nnuz 12903	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
80	25, 79	eleqtrdi 2843	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
81		elfznn 13575	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
82	81	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
83	82	nnrecred 12299	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℝ)
84	83	recnd 11271	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℂ)
85	82	nnred 12263	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
86		peano2rem 11558	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
87	85, 86	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
88		chpcl 27103	. . . . . . . . . . . . 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 11673	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℝ)
91	90	recnd 11271	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℂ)
92	51, 56, 73, 78, 80, 84, 91	fsumparts 15824	. . . . . . . . 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 13820	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℤ)
94		fzval3 13755	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
95	93, 94	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
96	95	eqcomd 2740	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
97	32	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
98	97	nncnd 12264	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
99		pncan 11496	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
100	98, 28, 99	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
101	97	nnred 12263	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
102	100, 101	eqeltrd 2833	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ)
103		chpcl 27103	. . . . . . . . . . . . . . . 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 11271	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℂ)
106	102	recnd 11271	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℂ)
107		peano2rem 11558	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
108	101, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
109		chpcl 27103	. . . . . . . . . . . . . . . 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 11271	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
112		1cnd 11238	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
113	98, 112	subcld 11602	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℂ)
114	105, 106, 111, 113	sub4d 11651	. . . . . . . . . . . . 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 27097	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
116	97, 115	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
117	116	recnd 11271	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
118		nnm1nn0 12550	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
119	97, 118	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
120		chpp1 27134	. . . . . . . . . . . . . . . . 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 11499	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
123	98, 28, 122	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
124	123, 100	eqtr4d 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = ((𝑛 + 1) − 1))
125	124	fveq2d 6890	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = (ψ‘((𝑛 + 1) − 1)))
126	123	fveq2d 6890	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
127	126	oveq2d 7429	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
128	121, 125, 127	3eqtr3d 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
129	111, 117, 128	mvrladdd 11658	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
130		peano2cn 11415	. . . . . . . . . . . . . . . . 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 11637	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = ((𝑛 + 1) − 𝑛))
133		pncan2 11497	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 𝑛) = 1)
134	98, 28, 133	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
135	132, 134	eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = 1)
136	129, 135	oveq12d 7431	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
138	137	oveq2d 7429	. . . . . . . . . . 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 11558	. . . . . . . . . . . . . 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 11271	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℂ)
142	97	nnne0d 12298	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
143	141, 98, 142	divrec2d 12029	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
144	138, 143	eqtr4d 2772	. . . . . . . . . 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 15725	. . . . . . . . 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 12572	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℂ)
147		pncan 11496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
148	146, 28, 147	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (1[,)+∞) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
149	148	fveq2d 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
150		chpfl 27129	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
151	4, 150	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
152	149, 151	eqtrd 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
153	152	oveq1d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)))
154	18	recnd 11271	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℂ)
155	25	nncnd 12264	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℂ)
156		1cnd 11238	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℂ)
157	154, 155, 156	subsub3d 11632	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
158	153, 157	eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
159	158	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))))
160	25	nnrecred 12299	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℝ)
161	160	recnd 11271	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℂ)
162		peano2cn 11415	. . . . . . . . . . . . . . 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 11701	. . . . . . . . . . . . 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 12298	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ≠ 0)
166	163, 155, 165	divrec2d 12029	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) = ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)))
167	166	eqcomd 2740	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
168	155, 165	recid2d 12021	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1)) = 1)
169	167, 168	oveq12d 7431	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
171	28	mul01i 11433	. . . . . . . . . . . . 13 ⊢ (1 · 0) = 0
172	171	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1 · 0) = 0)
173	170, 172	oveq12d 7431	. . . . . . . . . . 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 11591	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
175	173, 174	eqtrd 2769	. . . . . . . . . 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 12299	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℝ)
178	177	recnd 11271	. . . . . . . . . . . . . 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 11271	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℂ)
181	178, 180	mulneg1d 11698	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
182	98, 112	mulcld 11263	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) ∈ ℂ)
183	98, 131	mulcld 11263	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℂ)
184	176	nnne0d 12298	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ≠ 0)
185	131, 182, 183, 184	divsubdird 12064	. . . . . . . . . . . . . . . . 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	mulridd 11260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) = 𝑛)
187	186	oveq2d 7429	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = ((𝑛 + 1) − 𝑛))
188	187, 134	eqtrd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = 1)
189	188	oveq1d 7428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 · (𝑛 + 1))))
190	131	mulridd 11260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 1) = (𝑛 + 1))
191	131, 98	mulcomd 11264	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 𝑛) = (𝑛 · (𝑛 + 1)))
192	190, 191	oveq12d 7431	. . . . . . . . . . . . . . . . . . 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 12298	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
195	112, 98, 131, 142, 194	divcan5d 12051	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = (1 / 𝑛))
196	192, 195	eqtr3d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) / (𝑛 · (𝑛 + 1))) = (1 / 𝑛))
197	112, 131, 98, 194, 142	divcan5d 12051	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 · 1) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 + 1)))
198	196, 197	oveq12d 7431	. . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
200	199	negeqd 11484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(1 / (𝑛 · (𝑛 + 1))) = -((1 / 𝑛) − (1 / (𝑛 + 1))))
201	97	nnrecred 12299	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
202	201	recnd 11271	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
203	193	nnrecred 12299	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℝ)
204	203	recnd 11271	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℂ)
205	202, 204	negsubdi2d 11618	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((1 / 𝑛) − (1 / (𝑛 + 1))) = ((1 / (𝑛 + 1)) − (1 / 𝑛)))
206	200, 205	eqtr2d 2770	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / (𝑛 + 1)) − (1 / 𝑛)) = -(1 / (𝑛 · (𝑛 + 1))))
207	97	nnrpd 13057	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
208	100, 207	eqeltrd 2833	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ⁺)
209	35	pntrval 27542	. . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = (𝑅‘𝑛))
212	210, 211	eqtr3d 2771	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) = (𝑅‘𝑛))
213	206, 212	oveq12d 7431	. . . . . . . . . . . . 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 12029	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = ((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
215	214	negeqd 11484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
216	181, 213, 215	3eqtr4d 2779	. . . . . . . . . . . 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 15725	. . . . . . . . . . 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 13996	. . . . . . . . . . . 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 11271	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
221	218, 220	fsumneg 15805	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))-((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
222	217, 221	eqtrd 2769	. . . . . . . . . 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 7431	. . . . . . . . 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 2777	. . . . . . . 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 11609	. . . . . . . 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 2769	. . . . . . 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 11658	. . . . . 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 5225	. . . . 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 13996	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
230	32	adantl 481	. . . . . . . . . . 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 12302	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
234	229, 233	fsumrecl 15752	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
235		rpre 13025	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
236	235	adantl 481	. . . . . . . . . . 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 13032	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
240	239, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
241	240	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℕ₀)
242	241, 24	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
243	238, 242	nndivred 12302	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
244		peano2rem 11558	. . . . . . . 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 11228	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
247	246, 14	ssexi 5302	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ V
248	247	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ∈ V)
249	231, 230	nndivred 12302	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
250	249	recnd 11271	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
251	229, 250	fsumcl 15751	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
252		relogcl 26553	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
253	252	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
254	253	recnd 11271	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
255	251, 254	subcld 11602	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
256	230	nnrecred 12299	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
257	229, 256	fsumrecl 15752	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
258	257, 253	resubcld 11673	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℝ)
259		eqidd 2735	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
260		eqidd 2735	. . . . . . . . . 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 7699	. . . . . . . . 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 11271	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
263	229, 250, 262	fsumsub 15806	. . . . . . . . . . 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 11271	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
265		1cnd 11238	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
266	230	nncnd 12264	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
267	230	nnne0d 12298	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
268	264, 265, 266, 267	divsubdird 12064	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = (((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
269	268	sumeq2dv 15720	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
270	257	recnd 11271	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
271	251, 270, 254	nnncan2d 11637	. . . . . . . . . . 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 2780	. . . . . . . . . 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 5222	. . . . . . . . 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 2769	. . . . . . . 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 27462	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
276	14	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
277	258	recnd 11271	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℂ)
278		1red 11244	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
279		harmoniclbnd 26988	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
280	279	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
281	253, 257, 280	abssubge0d 15452	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
282	281	adantrr 717	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
283	235	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
284		simprr 772	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
285		harmonicubnd 26989	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
286	283, 284, 285	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
287		1red 11244	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
288	257, 253, 287	lesubadd2d 11844	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
289	288	adantrr 717	. . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1)
291	282, 290	eqbrtrd 5145	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ≤ 1)
292	276, 277, 278, 278, 291	elo1d 15554	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
293		o1sub 15634	. . . . . . . . 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 587	. . . . . . . 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 2834	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)) ∈ 𝑂(1))
296	243	recnd 11271	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
297		1cnd 11238	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
298	237	recnd 11271	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℂ)
299		rpcnne0 13035	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
300	299	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
301		divdir 11929	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
303	302	mpteq2dva 5222	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
304		simpr 484	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
305	237, 304	rerpdivcld 13090	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
306		rpreccl 13043	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
307	306	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
308		eqidd 2735	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
309		eqidd 2735	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
310	248, 305, 307, 308, 309	offval2 7699	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
311		chpo1ub 27460	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
312		divrcnv 15870	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
313	28, 312	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
314		rlimo1 15635	. . . . . . . . . . . . 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 15632	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
318	310, 317	eqeltrrd 2834	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
319	303, 318	eqeltrd 2833	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) ∈ 𝑂(1))
320	238, 304	rerpdivcld 13090	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℝ)
321		chpge0 27105	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
322	236, 321	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (ψ‘𝑥))
323	237, 322	ge0p1rpd 13089	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ⁺)
324	323	rprege0d 13066	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)))
325	242	nnrpd 13057	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
326	325	rpregt0d 13065	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)))
327		divge0 12119	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
329	243, 328	absidd 15443	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
330	320	recnd 11271	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℂ)
331	330	abscld 15457	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / 𝑥)) ∈ ℝ)
332		fllep1 13823	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
333	236, 332	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≤ ((⌊‘𝑥) + 1))
334		rpregt0 13031	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
335	334	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
336	323	rpregt0d 13065	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1)))
337		lediv2 12140	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥)))
339	333, 338	mpbid 232	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥))
340	320	leabsd 15435	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
341	243, 320, 331, 339, 340	letrd 11400	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
342	329, 341	eqbrtrd 5145	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
343	342	adantrr 717	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
344	278, 319, 320, 296, 343	o1le 15671	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ∈ 𝑂(1))
345		o1const 15638	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
346	14, 28, 345	mp2an 692	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
347	346	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
348	296, 297, 344, 347	o1sub2 15644	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) ∈ 𝑂(1))
349	234, 245, 295, 348	o1sub2 15644	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
350	13, 349	o1res2 15581	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
351	228, 350	eqeltrrid 2838	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
352	16, 351	eqeltrd 2833	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) ∈ 𝑂(1))
353		eqid 2734	. . . . . 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 7112	. . . . 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 3987	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
357	355, 356, 278	o1resb 15584	. . 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 257	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
359	358	mptru 1546	1 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ≠ wne 2931 Vcvv 3463 ⊆ wss 3931 class class class wbr 5123 ↦ cmpt 5205 ↾ cres 5667 ⟶wf 6537 ‘cfv 6541 (class class class)co 7413 ∘_f cof 7677 ℂcc 11135 ℝcr 11136 0cc0 11137 1c1 11138 + caddc 11140 · cmul 11142 +∞cpnf 11274 < clt 11277 ≤ cle 11278 − cmin 11474 -cneg 11475 / cdiv 11902 ℕcn 12248 2c2 12303 ℕ₀cn0 12509 ℤcz 12596 ℤ_≥cuz 12860 ℝ⁺crp 13016 [,)cico 13371 ...cfz 13529 ..^cfzo 13676 ⌊cfl 13812 abscabs 15255 ⇝_𝑟 crli 15503 𝑂(1)co1 15504 Σcsu 15704 logclog 26532 Λcvma 27071 ψcchp 27072

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-inf2 9663 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215 ax-addf 11216

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-supp 8168 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8727 df-map 8850 df-pm 8851 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-fsupp 9384 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9532 df-dju 9923 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-xnn0 12583 df-z 12597 df-dec 12717 df-uz 12861 df-q 12973 df-rp 13017 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13373 df-ioc 13374 df-ico 13375 df-icc 13376 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14295 df-bc 14324 df-hash 14352 df-shft 15088 df-cj 15120 df-re 15121 df-im 15122 df-sqrt 15256 df-abs 15257 df-limsup 15489 df-clim 15506 df-rlim 15507 df-o1 15508 df-lo1 15509 df-sum 15705 df-ef 16085 df-e 16086 df-sin 16087 df-cos 16088 df-tan 16089 df-pi 16090 df-dvds 16273 df-gcd 16514 df-prm 16691 df-pc 16857 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17230 df-ress 17253 df-plusg 17286 df-mulr 17287 df-starv 17288 df-sca 17289 df-vsca 17290 df-ip 17291 df-tset 17292 df-ple 17293 df-ds 17295 df-unif 17296 df-hom 17297 df-cco 17298 df-rest 17438 df-topn 17439 df-0g 17457 df-gsum 17458 df-topgen 17459 df-pt 17460 df-prds 17463 df-xrs 17518 df-qtop 17523 df-imas 17524 df-xps 17526 df-mre 17600 df-mrc 17601 df-acs 17603 df-mgm 18622 df-sgrp 18701 df-mnd 18717 df-submnd 18766 df-mulg 19055 df-cntz 19304 df-cmn 19768 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-top 22848 df-topon 22865 df-topsp 22887 df-bases 22900 df-cld 22973 df-ntr 22974 df-cls 22975 df-nei 23052 df-lp 23090 df-perf 23091 df-cn 23181 df-cnp 23182 df-haus 23269 df-cmp 23341 df-tx 23516 df-hmeo 23709 df-fil 23800 df-fm 23892 df-flim 23893 df-flf 23894 df-xms 24275 df-ms 24276 df-tms 24277 df-cncf 24840 df-limc 25837 df-dv 25838 df-ulm 26356 df-log 26534 df-cxp 26535 df-atan 26846 df-em 26972 df-cht 27076 df-vma 27077 df-chp 27078 df-ppi 27079

This theorem is referenced by: pntrsumbnd 27546

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