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Theorem pntrsumo1 27530

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 11130	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		elicopnf 13359	. . . . . . . . . . 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 11133	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 ∈ ℝ)
6		1red 11131	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℝ)
7		0lt1 11657	. . . . . . . . . . 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 11291	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 𝑥)
11	4, 10	elrpd 12944	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
12	11	ssriv 3935	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ⁺)
14		rpssre 12911	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
15	13, 14	sstrdi 3944	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
16	15	resmptd 5997	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
17		chpcl 27088	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	4, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℝ)
19		peano2re 11304	. . . . . . . . . . 11 ⊢ ((ψ‘𝑥) ∈ ℝ → ((ψ‘𝑥) + 1) ∈ ℝ)
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℝ)
21	11	rprege0d 12954	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 13738	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ₀)
24		nn0p1nn 12438	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℕ)
26	20, 25	nndivred 12197	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
27	26	recnd 11158	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
28		ax-1cn 11082	. . . . . . . 8 ⊢ 1 ∈ ℂ
29		subcl 11377	. . . . . . . 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 13894	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (1...(⌊‘𝑥)) ∈ Fin)
32		elfznn 13467	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
33	32	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
34		nnrp 12915	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
35		pntrval.r	. . . . . . . . . . . . . . 15 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
36	35	pntrf 27528	. . . . . . . . . . . . . 14 ⊢ 𝑅:ℝ⁺⟶ℝ
37	36	ffvelcdmi 7026	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
38	34, 37	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
39		peano2nn 12155	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
40		nnmulcl 12167	. . . . . . . . . . . . 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 12197	. . . . . . . . . . 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 15655	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
45	44	recnd 11158	. . . . . . . 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 7364	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
48		fvoveq1 7379	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
49		oveq1 7363	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
50	48, 49	oveq12d 7374	. . . . . . . . . . 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 7364	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
53		fvoveq1 7379	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
54		oveq1 7363	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
55	53, 54	oveq12d 7374	. . . . . . . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
58		1div1e1 11830	. . . . . . . . . . . 12 ⊢ (1 / 1) = 1
59	57, 58	eqtrdi 2785	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
60		oveq1 7363	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
61		1m1e0 12215	. . . . . . . . . . . . . . . 16 ⊢ (1 − 1) = 0
62	60, 61	eqtrdi 2785	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
63	62	fveq2d 6836	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
64		2pos 12246	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
65		0re 11132	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
66		chpeq0 27173	. . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = (0 − 0))
71		0m0e0 12258	. . . . . . . . . . . 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 7364	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)))
75		fvoveq1 7379	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
76		oveq1 7363	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
77	75, 76	oveq12d 7374	. . . . . . . . . . 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 12788	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
80	25, 79	eleqtrdi 2844	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
81		elfznn 13467	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
82	81	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
83	82	nnrecred 12194	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℝ)
84	83	recnd 11158	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℂ)
85	82	nnred 12158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
86		peano2rem 11446	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
87	85, 86	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
88		chpcl 27088	. . . . . . . . . . . . 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 11563	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℝ)
91	90	recnd 11158	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℂ)
92	51, 56, 73, 78, 80, 84, 91	fsumparts 15727	. . . . . . . . 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 13716	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℤ)
94		fzval3 13648	. . . . . . . . . . . 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 12159	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
99		pncan 11384	. . . . . . . . . . . . . . . . . 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 12158	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
102	100, 101	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ)
103		chpcl 27088	. . . . . . . . . . . . . . . 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 11158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℂ)
106	102	recnd 11158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℂ)
107		peano2rem 11446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
108	101, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
109		chpcl 27088	. . . . . . . . . . . . . . . 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 11158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
112		1cnd 11125	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
113	98, 112	subcld 11490	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℂ)
114	105, 106, 111, 113	sub4d 11539	. . . . . . . . . . . . 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 27082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
116	97, 115	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
117	116	recnd 11158	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
118		nnm1nn0 12440	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
119	97, 118	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
120		chpp1 27119	. . . . . . . . . . . . . . . . 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 11387	. . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = (ψ‘((𝑛 + 1) − 1)))
126	123	fveq2d 6836	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
127	126	oveq2d 7372	. . . . . . . . . . . . . . . 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 11548	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
130		peano2cn 11303	. . . . . . . . . . . . . . . . 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 11525	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = ((𝑛 + 1) − 𝑛))
133		pncan2 11385	. . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . 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 7372	. . . . . . . . . . 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 11446	. . . . . . . . . . . . . 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 11158	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℂ)
142	97	nnne0d 12193	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
143	141, 98, 142	divrec2d 11919	. . . . . . . . . . 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 15628	. . . . . . . . 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 12462	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℂ)
147		pncan 11384	. . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
150		chpfl 27114	. . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)))
154	18	recnd 11158	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℂ)
155	25	nncnd 12159	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℂ)
156		1cnd 11125	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℂ)
157	154, 155, 156	subsub3d 11520	. . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))))
160	25	nnrecred 12194	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℝ)
161	160	recnd 11158	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℂ)
162		peano2cn 11303	. . . . . . . . . . . . . . 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 11591	. . . . . . . . . . . . 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 12193	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ≠ 0)
166	163, 155, 165	divrec2d 11919	. . . . . . . . . . . . . . 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 11911	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1)) = 1)
169	167, 168	oveq12d 7374	. . . . . . . . . . . . 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 11321	. . . . . . . . . . . . 13 ⊢ (1 · 0) = 0
172	171	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1 · 0) = 0)
173	170, 172	oveq12d 7374	. . . . . . . . . . 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 11479	. . . . . . . . . . 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 12194	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℝ)
178	177	recnd 11158	. . . . . . . . . . . . . 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 11158	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℂ)
181	178, 180	mulneg1d 11588	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
182	98, 112	mulcld 11150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) ∈ ℂ)
183	98, 131	mulcld 11150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℂ)
184	176	nnne0d 12193	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ≠ 0)
185	131, 182, 183, 184	divsubdird 11954	. . . . . . . . . . . . . . . . 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 11147	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) = 𝑛)
187	186	oveq2d 7372	. . . . . . . . . . . . . . . . . . 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 7371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 · (𝑛 + 1))))
190	131	mulridd 11147	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 1) = (𝑛 + 1))
191	131, 98	mulcomd 11151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 𝑛) = (𝑛 · (𝑛 + 1)))
192	190, 191	oveq12d 7374	. . . . . . . . . . . . . . . . . . 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 12193	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
195	112, 98, 131, 142, 194	divcan5d 11941	. . . . . . . . . . . . . . . . . . 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 11941	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 · 1) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 + 1)))
198	196, 197	oveq12d 7374	. . . . . . . . . . . . . . . . 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 11372	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(1 / (𝑛 · (𝑛 + 1))) = -((1 / 𝑛) − (1 / (𝑛 + 1))))
201	97	nnrecred 12194	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
202	201	recnd 11158	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
203	193	nnrecred 12194	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℝ)
204	203	recnd 11158	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℂ)
205	202, 204	negsubdi2d 11506	. . . . . . . . . . . . . . 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 12945	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
208	100, 207	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ⁺)
209	35	pntrval 27527	. . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . 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 11919	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = ((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
215	214	negeqd 11372	. . . . . . . . . . . . 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 15628	. . . . . . . . . . 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 13894	. . . . . . . . . . . 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 11158	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
221	218, 220	fsumneg 15708	. . . . . . . . . . 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 7374	. . . . . . . . 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 11497	. . . . . . . 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 11548	. . . . . 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 5191	. . . . 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 13894	. . . . . . . 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 12197	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
234	229, 233	fsumrecl 15655	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
235		rpre 12912	. . . . . . . . . . . 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 12919	. . . . . . . . . . . 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 12197	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
244		peano2rem 11446	. . . . . . . 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 11115	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
247	246, 14	ssexi 5265	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ V
248	247	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ∈ V)
249	231, 230	nndivred 12197	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
250	249	recnd 11158	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
251	229, 250	fsumcl 15654	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
252		relogcl 26538	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
253	252	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
254	253	recnd 11158	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
255	251, 254	subcld 11490	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
256	230	nnrecred 12194	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
257	229, 256	fsumrecl 15655	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
258	257, 253	resubcld 11563	. . . . . . . . . 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 7640	. . . . . . . . 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 11158	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
263	229, 250, 262	fsumsub 15709	. . . . . . . . . . 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 11158	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
265		1cnd 11125	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
266	230	nncnd 12159	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
267	230	nnne0d 12193	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
268	264, 265, 266, 267	divsubdird 11954	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = (((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
269	268	sumeq2dv 15623	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
270	257	recnd 11158	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
271	251, 270, 254	nnncan2d 11525	. . . . . . . . . . 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 5189	. . . . . . . . 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 27447	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
276	14	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
277	258	recnd 11158	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℂ)
278		1red 11131	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
279		harmoniclbnd 26973	. . . . . . . . . . . . . 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 15355	. . . . . . . . . . . 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 26974	. . . . . . . . . . . . 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 11131	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
288	257, 253, 287	lesubadd2d 11734	. . . . . . . . . . . . 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 5118	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ≤ 1)
292	276, 277, 278, 278, 291	elo1d 15457	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
293		o1sub 15537	. . . . . . . . 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 2835	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)) ∈ 𝑂(1))
296	243	recnd 11158	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
297		1cnd 11125	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
298	237	recnd 11158	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℂ)
299		rpcnne0 12922	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
300	299	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
301		divdir 11819	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
303	302	mpteq2dva 5189	. . . . . . . . . 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 12978	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
306		rpreccl 12931	. . . . . . . . . . . . 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 7640	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
311		chpo1ub 27445	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
312		divrcnv 15773	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
313	28, 312	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
314		rlimo1 15538	. . . . . . . . . . . . 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 15535	. . . . . . . . . . . 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 2835	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
319	303, 318	eqeltrd 2834	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) ∈ 𝑂(1))
320	238, 304	rerpdivcld 12978	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℝ)
321		chpge0 27090	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
322	236, 321	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (ψ‘𝑥))
323	237, 322	ge0p1rpd 12977	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ⁺)
324	323	rprege0d 12954	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)))
325	242	nnrpd 12945	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
326	325	rpregt0d 12953	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)))
327		divge0 12009	. . . . . . . . . . . . 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 15344	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
330	320	recnd 11158	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℂ)
331	330	abscld 15360	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / 𝑥)) ∈ ℝ)
332		fllep1 13719	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
333	236, 332	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≤ ((⌊‘𝑥) + 1))
334		rpregt0 12918	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
335	334	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
336	323	rpregt0d 12953	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1)))
337		lediv2 12030	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 15336	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
341	243, 320, 331, 339, 340	letrd 11288	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
342	329, 341	eqbrtrd 5118	. . . . . . . . . 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 15574	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ∈ 𝑂(1))
345		o1const 15541	. . . . . . . . . 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 15547	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) ∈ 𝑂(1))
349	234, 245, 295, 348	o1sub2 15547	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
350	13, 349	o1res2 15484	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
351	228, 350	eqeltrrid 2839	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
352	16, 351	eqeltrd 2834	. . 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 7055	. . . . 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 3955	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
357	355, 356, 278	o1resb 15487	. . 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 1548	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 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2930 Vcvv 3438 ⊆ wss 3899 class class class wbr 5096 ↦ cmpt 5177 ↾ cres 5624 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ∘_f cof 7618 ℂcc 11022 ℝcr 11023 0cc0 11024 1c1 11025 + caddc 11027 · cmul 11029 +∞cpnf 11161 < clt 11164 ≤ cle 11165 − cmin 11362 -cneg 11363 / cdiv 11792 ℕcn 12143 2c2 12198 ℕ₀cn0 12399 ℤcz 12486 ℤ_≥cuz 12749 ℝ⁺crp 12903 [,)cico 13261 ...cfz 13421 ..^cfzo 13568 ⌊cfl 13708 abscabs 15155 ⇝_𝑟 crli 15406 𝑂(1)co1 15407 Σcsu 15607 logclog 26517 Λcvma 27056 ψcchp 27057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-xnn0 12473 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-fac 14195 df-bc 14224 df-hash 14252 df-shft 14988 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-o1 15411 df-lo1 15412 df-sum 15608 df-ef 15988 df-e 15989 df-sin 15990 df-cos 15991 df-tan 15992 df-pi 15993 df-dvds 16178 df-gcd 16420 df-prm 16597 df-pc 16763 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-limc 25821 df-dv 25822 df-ulm 26340 df-log 26519 df-cxp 26520 df-atan 26831 df-em 26957 df-cht 27061 df-vma 27062 df-chp 27063 df-ppi 27064

This theorem is referenced by: pntrsumbnd 27531

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