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Theorem pntrsumo1 25254

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 10039	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		elicopnf 12269	. . . . . . . . . . 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 476	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ)
5		0red 10041	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 ∈ ℝ)
6		1red 10055	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℝ)
7		0lt1 10550	. . . . . . . . . . 11 ⊢ 0 < 1
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 1)
9	3	simprbi 480	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
10	5, 6, 4, 8, 9	ltletrd 10197	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 0 < 𝑥)
11	4, 10	elrpd 11869	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
12	11	ssriv 3607	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ⁺)
14		rpssre 11843	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
15	13, 14	syl6ss 3615	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
16	15	resmptd 5452	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
17		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
18		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
19	18	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
20	19, 18	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))
21	17, 20	jca 554	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((1 / 𝑚) = (1 / 𝑛) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(𝑛 − 1)) − (𝑛 − 1))))
22		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
23		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
24	23	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
25	24, 23	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
26	22, 25	jca 554	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((1 / 𝑚) = (1 / (𝑛 + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))))
27		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (1 / 𝑚) = (1 / 1))
28		1div1e1 10717	. . . . . . . . . . . . 13 ⊢ (1 / 1) = 1
29	27, 28	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (1 / 𝑚) = 1)
30		oveq1 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
31		1m1e0 11089	. . . . . . . . . . . . . . . . 17 ⊢ (1 − 1) = 0
32	30, 31	syl6eq 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
33	32	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
34		2pos 11112	. . . . . . . . . . . . . . . 16 ⊢ 0 < 2
35		0re 10040	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
36		chpeq0 24933	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
37	35, 36	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
38	34, 37	mpbir 221	. . . . . . . . . . . . . . 15 ⊢ (ψ‘0) = 0
39	33, 38	syl6eq 2672	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
40	39, 32	oveq12d 6668	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = (0 − 0))
41		0m0e0 11130	. . . . . . . . . . . . 13 ⊢ (0 − 0) = 0
42	40, 41	syl6eq 2672	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0)
43	29, 42	jca 554	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((1 / 𝑚) = 1 ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = 0))
44		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)))
45		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
46	45	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
47	46, 45	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)))
48	44, 47	jca 554	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((1 / 𝑚) = (1 / ((⌊‘𝑥) + 1)) ∧ ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) = ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))))
49	11	rprege0d 11879	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
50		flge0nn0 12621	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ₀)
52		nn0p1nn 11332	. . . . . . . . . . . . 13 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℕ)
54		nnuz 11723	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
55	53, 54	syl6eleq 2711	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
56		elfznn 12370	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
57	56	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
58	57	nnrecred 11066	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℝ)
59	58	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (1 / 𝑚) ∈ ℂ)
60	57	nnred 11035	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
61		peano2rem 10348	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
62	60, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
63		chpcl 24850	. . . . . . . . . . . . . 14 ⊢ ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
64	62, 63	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
65	64, 62	resubcld 10458	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℝ)
66	65	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑚 − 1)) − (𝑚 − 1)) ∈ ℂ)
67	21, 26, 43, 48, 55, 59, 66	fsumparts 14538	. . . . . . . . . 10 ⊢ (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)))))
68	4	flcld 12599	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℤ)
69		fzval3 12536	. . . . . . . . . . . . 13 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
70	68, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
71	70	eqcomd 2628	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
72		elfznn 12370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
73	72	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
74	73	nncnd 11036	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
75		ax-1cn 9994	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
76		pncan 10287	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
77	74, 75, 76	sylancl 694	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
78	73	nnred 11035	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
79	77, 78	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ)
80		chpcl 24850	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 + 1) − 1) ∈ ℝ → (ψ‘((𝑛 + 1) − 1)) ∈ ℝ)
81	79, 80	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℝ)
82	81	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) ∈ ℂ)
83	79	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℂ)
84		peano2rem 10348	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
85	78, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86		chpcl 24850	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
87	85, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
88	87	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89		1cnd 10056	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
90	74, 89	subcld 10392	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℂ)
91	82, 83, 88, 90	sub4d 10441	. . . . . . . . . . . . . 14 ⊢ ((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))))
92		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
93	73, 92	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
94		chpp1 24881	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
96		npcan 10290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
97	74, 75, 96	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
98	97, 77	eqtr4d 2659	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = ((𝑛 + 1) − 1))
99	98	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = (ψ‘((𝑛 + 1) − 1)))
100	97	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
101	100	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
102	95, 99, 101	3eqtr3d 2664	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
103	102	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))))
104		vmacl 24844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
105	73, 104	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
106	105	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
107	88, 106	pncan2d 10394	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
108	103, 107	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
109		peano2cn 10208	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈ ℂ)
110	74, 109	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
111	110, 74, 89	nnncan2d 10427	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = ((𝑛 + 1) − 𝑛))
112		pncan2 10288	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 𝑛) = 1)
113	74, 75, 112	sylancl 694	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
114	111, 113	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 1) − (𝑛 − 1)) = 1)
115	108, 114	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) − (((𝑛 + 1) − 1) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
116	91, 115	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1))) = ((Λ‘𝑛) − 1))
117	116	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
118		peano2rem 10348	. . . . . . . . . . . . . . 15 ⊢ ((Λ‘𝑛) ∈ ℝ → ((Λ‘𝑛) − 1) ∈ ℝ)
119	105, 118	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℝ)
120	119	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℂ)
121	73	nnne0d 11065	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
122	120, 74, 121	divrec2d 10805	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = ((1 / 𝑛) · ((Λ‘𝑛) − 1)))
123	117, 122	eqtr4d 2659	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · (((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) − ((ψ‘(𝑛 − 1)) − (𝑛 − 1)))) = (((Λ‘𝑛) − 1) / 𝑛))
124	71, 123	sumeq12rdv 14438	. . . . . . . . . 10 ⊢ (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))
125	51	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℂ)
126		pncan 10287	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
127	125, 75, 126	sylancl 694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1[,)+∞) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
128	127	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
129		chpfl 24876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
130	4, 129	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
131	128, 130	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
132	131	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)))
133		chpcl 24850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
134	4, 133	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℝ)
135	134	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → (ψ‘𝑥) ∈ ℂ)
136	53	nncnd 11036	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ∈ ℂ)
137		1cnd 10056	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ∈ ℂ)
138	135, 136, 137	subsub3d 10422	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
139	132, 138	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1)) = (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1)))
140	139	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))))
141	53	nnrecred 11066	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℝ)
142	141	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (1 / ((⌊‘𝑥) + 1)) ∈ ℂ)
143		peano2cn 10208	. . . . . . . . . . . . . . . 16 ⊢ ((ψ‘𝑥) ∈ ℂ → ((ψ‘𝑥) + 1) ∈ ℂ)
144	135, 143	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℂ)
145	142, 144, 136	subdid 10486	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · (((ψ‘𝑥) + 1) − ((⌊‘𝑥) + 1))) = (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) − ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1))))
146	53	nnne0d 11065	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1[,)+∞) → ((⌊‘𝑥) + 1) ≠ 0)
147	144, 136, 146	divrec2d 10805	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) = ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)))
148	147	eqcomd 2628	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
149	136, 146	recid2d 10797	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1)) = 1)
150	148, 149	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘𝑥) + 1)) − ((1 / ((⌊‘𝑥) + 1)) · ((⌊‘𝑥) + 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
151	140, 145, 150	3eqtrd 2660	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
152	75	mul01i 10226	. . . . . . . . . . . . . 14 ⊢ (1 · 0) = 0
153	152	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → (1 · 0) = 0)
154	151, 153	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0))
155		peano2re 10209	. . . . . . . . . . . . . . . . 17 ⊢ ((ψ‘𝑥) ∈ ℝ → ((ψ‘𝑥) + 1) ∈ ℝ)
156	134, 155	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1[,)+∞) → ((ψ‘𝑥) + 1) ∈ ℝ)
157	156, 53	nndivred 11069	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
158	157	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1[,)+∞) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
159		subcl 10280	. . . . . . . . . . . . . 14 ⊢ (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ ∧ 1 ∈ ℂ) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
160	158, 75, 159	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℂ)
161	160	subid1d 10381	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − 0) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
162	154, 161	eqtrd 2656	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) → (((1 / ((⌊‘𝑥) + 1)) · ((ψ‘(((⌊‘𝑥) + 1) − 1)) − (((⌊‘𝑥) + 1) − 1))) − (1 · 0)) = ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))
163		peano2nn 11032	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
164		nnmulcl 11043	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
165	163, 164	mpdan 702	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → (𝑛 · (𝑛 + 1)) ∈ ℕ)
166	73, 165	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℕ)
167	166	nnrecred 11066	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℝ)
168	167	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) ∈ ℂ)
169		nnrp 11842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
170		pntrval.r	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
171	170	pntrf 25252	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑅:ℝ⁺⟶ℝ
172	171	ffvelrni 6358	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℝ⁺ → (𝑅‘𝑛) ∈ ℝ)
173	169, 172	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑅‘𝑛) ∈ ℝ)
174	73, 173	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℝ)
175	174	recnd 10068	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘𝑛) ∈ ℂ)
176	168, 175	mulneg1d 10483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
177	74, 89	mulcld 10060	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) ∈ ℂ)
178	74, 110	mulcld 10060	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ∈ ℂ)
179	166	nnne0d 11065	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (𝑛 + 1)) ≠ 0)
180	110, 177, 178, 179	divsubdird 10840	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (((𝑛 + 1) / (𝑛 · (𝑛 + 1))) − ((𝑛 · 1) / (𝑛 · (𝑛 + 1)))))
181	74	mulid1d 10057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · 1) = 𝑛)
182	181	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = ((𝑛 + 1) − 𝑛))
183	182, 113	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − (𝑛 · 1)) = 1)
184	183	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − (𝑛 · 1)) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 · (𝑛 + 1))))
185	110	mulid1d 10057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 1) = (𝑛 + 1))
186	110, 74	mulcomd 10061	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) · 𝑛) = (𝑛 · (𝑛 + 1)))
187	185, 186	oveq12d 6668	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = ((𝑛 + 1) / (𝑛 · (𝑛 + 1))))
188	73, 163	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
189	188	nnne0d 11065	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
190	89, 74, 110, 121, 189	divcan5d 10827	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) · 1) / ((𝑛 + 1) · 𝑛)) = (1 / 𝑛))
191	187, 190	eqtr3d 2658	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) / (𝑛 · (𝑛 + 1))) = (1 / 𝑛))
192	89, 110, 74, 189, 121	divcan5d 10827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 · 1) / (𝑛 · (𝑛 + 1))) = (1 / (𝑛 + 1)))
193	191, 192	oveq12d 6668	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) / (𝑛 · (𝑛 + 1))) − ((𝑛 · 1) / (𝑛 · (𝑛 + 1)))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
194	180, 184, 193	3eqtr3d 2664	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 · (𝑛 + 1))) = ((1 / 𝑛) − (1 / (𝑛 + 1))))
195	194	negeqd 10275	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(1 / (𝑛 · (𝑛 + 1))) = -((1 / 𝑛) − (1 / (𝑛 + 1))))
196	73	nnrecred 11066	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
197	196	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
198	188	nnrecred 11066	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℝ)
199	198	recnd 10068	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / (𝑛 + 1)) ∈ ℂ)
200	197, 199	negsubdi2d 10408	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((1 / 𝑛) − (1 / (𝑛 + 1))) = ((1 / (𝑛 + 1)) − (1 / 𝑛)))
201	195, 200	eqtr2d 2657	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / (𝑛 + 1)) − (1 / 𝑛)) = -(1 / (𝑛 · (𝑛 + 1))))
202	73	nnrpd 11870	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
203	77, 202	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) ∈ ℝ⁺)
204	170	pntrval 25251	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 + 1) − 1) ∈ ℝ⁺ → (𝑅‘((𝑛 + 1) − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
205	203, 204	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)))
206	77	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘((𝑛 + 1) − 1)) = (𝑅‘𝑛))
207	205, 206	eqtr3d 2658	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1)) = (𝑅‘𝑛))
208	201, 207	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = (-(1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
209	175, 178, 179	divrec2d 10805	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = ((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
210	209	negeqd 10275	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -((1 / (𝑛 · (𝑛 + 1))) · (𝑅‘𝑛)))
211	176, 208, 210	3eqtr4d 2666	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((1 / (𝑛 + 1)) − (1 / 𝑛)) · ((ψ‘((𝑛 + 1) − 1)) − ((𝑛 + 1) − 1))) = -((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
212	71, 211	sumeq12rdv 14438	. . . . . . . . . . . 12 ⊢ (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))))
213		fzfid 12772	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (1[,)+∞) → (1...(⌊‘𝑥)) ∈ Fin)
214	173, 165	nndivred 11069	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
215	73, 214	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
216	215	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (1[,)+∞) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
217	213, 216	fsumneg 14519	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))-((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
218	212, 217	eqtrd 2656	. . . . . . . . . . 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))))
219	162, 218	oveq12d 6668	. . . . . . . . . 10 ⊢ (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)))))
220	67, 124, 219	3eqtr3d 2664	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) − -Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
221		fzfid 12772	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (1...(⌊‘𝑥)) ∈ Fin)
222	72	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
223	222, 214	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
224	221, 223	fsumrecl 14465	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℝ)
225	224	recnd 10068	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
226	4, 225	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))) ∈ ℂ)
227	160, 226	subnegd 10399	. . . . . . . . 9 ⊢ (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)))))
228	220, 227	eqtrd 2656	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = (((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) + Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))))
229	228	oveq1d 6665	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) = ((((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) + Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)))
230	160, 226	pncan2d 10394	. . . . . . 7 ⊢ (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))))
231	229, 230	eqtrd 2656	. . . . . 6 ⊢ (𝑥 ∈ (1[,)+∞) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
232	231	mpteq2ia 4740	. . . . 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))))
233		fzfid 12772	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
234	72	adantl 482	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
235	234, 104	syl 17	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
236	235, 118	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) − 1) ∈ ℝ)
237	236, 234	nndivred 11069	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
238	233, 237	fsumrecl 14465	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) ∈ ℝ)
239		rpre 11839	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
240	239	adantl 482	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
241	240, 133	syl 17	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℝ)
242	241, 155	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ)
243		rprege0 11847	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
244	243, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
245	244	adantl 482	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℕ₀)
246	245, 52	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
247	242, 246	nndivred 11069	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ)
248		peano2rem 10348	. . . . . . . 8 ⊢ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℝ → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℝ)
249	247, 248	syl 17	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1) ∈ ℝ)
250		reex 10027	. . . . . . . . . . . 12 ⊢ ℝ ∈ V
251	250, 14	ssexi 4803	. . . . . . . . . . 11 ⊢ ℝ⁺ ∈ V
252	251	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ∈ V)
253	235, 234	nndivred 11069	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
254	253	recnd 10068	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
255	233, 254	fsumcl 14464	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
256		relogcl 24322	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
257	256	adantl 482	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
258	257	recnd 10068	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
259	255, 258	subcld 10392	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
260	234	nnrecred 11066	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
261	233, 260	fsumrecl 14465	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
262	261, 257	resubcld 10458	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℝ)
263		eqidd 2623	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
264		eqidd 2623	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))))
265	252, 259, 262, 263, 264	offval2 6914	. . . . . . . . 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)))))
266	260	recnd 10068	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
267	233, 254, 266	fsumsub 14520	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
268	235	recnd 10068	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
269		1cnd 10056	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
270	234	nncnd 11036	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
271	234	nnne0d 11065	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
272	268, 269, 270, 271	divsubdird 10840	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) − 1) / 𝑛) = (((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
273	272	sumeq2dv 14433	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − (1 / 𝑛)))
274	261	recnd 10068	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
275	255, 274, 258	nnncan2d 10427	. . . . . . . . . . 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)))
276	267, 273, 275	3eqtr4rd 2667	. . . . . . . . . 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))
277	276	mpteq2dva 4744	. . . . . . . . 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)))
278	265, 277	eqtrd 2656	. . . . . . . 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)))
279		vmadivsum 25171	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
280	14	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
281	262	recnd 10068	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ∈ ℂ)
282		1red 10055	. . . . . . . . . 10 ⊢ (⊤ → 1 ∈ ℝ)
283		harmoniclbnd 24735	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
284	283	adantl 482	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛))
285	257, 261, 284	abssubge0d 14170	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
286	285	adantrr 753	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))
287	239	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
288		simprr 796	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
289		harmonicubnd 24736	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
290	287, 288, 289	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
291		1red 10055	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
292	261, 257, 291	lesubadd2d 10626	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
293	292	adantrr 753	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1 ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1)))
294	290, 293	mpbird 247	. . . . . . . . . . 11 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)) ≤ 1)
295	286, 294	eqbrtrd 4675	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ≤ 1)
296	280, 281, 282, 282, 295	elo1d 14267	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
297		o1sub 14346	. . . . . . . . 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))
298	279, 296, 297	sylancr 695	. . . . . . . 8 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘_𝑓 − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − (log‘𝑥)))) ∈ 𝑂(1))
299	278, 298	eqeltrrd 2702	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛)) ∈ 𝑂(1))
300	247	recnd 10068	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ∈ ℂ)
301		1cnd 10056	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
302	241	recnd 10068	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (ψ‘𝑥) ∈ ℂ)
303		rpcnne0 11850	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
304	303	adantl 482	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
305		divdir 10710	. . . . . . . . . . . 12 ⊢ (((ψ‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
306	302, 301, 304, 305	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) = (((ψ‘𝑥) / 𝑥) + (1 / 𝑥)))
307	306	mpteq2dva 4744	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
308		simpr 477	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
309	241, 308	rerpdivcld 11903	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
310		rpreccl 11857	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
311	310	adantl 482	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
312		eqidd 2623	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
313		eqidd 2623	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
314	252, 309, 311, 312, 313	offval2 6914	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_𝑓 + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))))
315		chpo1ub 25169	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
316		divrcnv 14584	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
317	75, 316	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0
318		rlimo1 14347	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
319	317, 318	mp1i 13	. . . . . . . . . . . 12 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ∈ 𝑂(1))
320		o1add 14344	. . . . . . . . . . . 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))
321	315, 319, 320	sylancr 695	. . . . . . . . . . 11 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_𝑓 + (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))) ∈ 𝑂(1))
322	314, 321	eqeltrrd 2702	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) + (1 / 𝑥))) ∈ 𝑂(1))
323	307, 322	eqeltrd 2701	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / 𝑥)) ∈ 𝑂(1))
324	242, 308	rerpdivcld 11903	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℝ)
325		chpge0 24852	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
326	240, 325	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (ψ‘𝑥))
327	241, 326	ge0p1rpd 11902	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) + 1) ∈ ℝ⁺)
328	327	rprege0d 11879	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)))
329	246	nnrpd 11870	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
330	329	rpregt0d 11878	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1)))
331		divge0 10892	. . . . . . . . . . . . 13 ⊢ (((((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 ≤ ((ψ‘𝑥) + 1)) ∧ (((⌊‘𝑥) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝑥) + 1))) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
332	328, 330, 331	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
333	247, 332	absidd 14161	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) = (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)))
334	324	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ∈ ℂ)
335	334	abscld 14175	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / 𝑥)) ∈ ℝ)
336		fllep1 12602	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
337	240, 336	syl 17	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≤ ((⌊‘𝑥) + 1))
338		rpregt0 11846	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
339	338	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
340	327	rpregt0d 11878	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) ∈ ℝ ∧ 0 < ((ψ‘𝑥) + 1)))
341		lediv2 10913	. . . . . . . . . . . . . 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)))
342	339, 330, 340, 341	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥)))
343	337, 342	mpbid 222	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (((ψ‘𝑥) + 1) / 𝑥))
344	324	leabsd 14153	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / 𝑥) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
345	247, 324, 335, 343, 344	letrd 10194	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
346	333, 345	eqbrtrd 4675	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
347	346	adantrr 753	. . . . . . . . 9 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ≤ (abs‘(((ψ‘𝑥) + 1) / 𝑥)))
348	282, 323, 324, 300, 347	o1le 14383	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1))) ∈ 𝑂(1))
349		o1const 14350	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
350	14, 75, 349	mp2an 708	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
351	350	a1i 11	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
352	300, 301, 348, 351	o1sub2 14356	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1)) ∈ 𝑂(1))
353	238, 249, 299, 352	o1sub2 14356	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
354	13, 353	o1res2 14294	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) − 1) / 𝑛) − ((((ψ‘𝑥) + 1) / ((⌊‘𝑥) + 1)) − 1))) ∈ 𝑂(1))
355	232, 354	syl5eqelr 2706	. . . 4 ⊢ (⊤ → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
356	16, 355	eqeltrd 2701	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ↾ (1[,)+∞)) ∈ 𝑂(1))
357		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1))))
358	357, 225	fmpti 6383	. . . . 5 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))):ℝ⟶ℂ
359	358	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))):ℝ⟶ℂ)
360		ssid 3624	. . . . 5 ⊢ ℝ ⊆ ℝ
361	360	a1i 11	. . . 4 ⊢ (⊤ → ℝ ⊆ ℝ)
362	359, 361, 282	o1resb 14297	. . 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)))
363	356, 362	mpbird 247	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1))
364	363	trud 1493	1 ⊢ (𝑥 ∈ ℝ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘𝑛) / (𝑛 · (𝑛 + 1)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 [,)cico 12177 ...cfz 12326 ..^cfzo 12465 ⌊cfl 12591 abscabs 13974 ⇝_𝑟 crli 14216 𝑂(1)co1 14217 Σcsu 14416 logclog 24301 Λcvma 24818 ψcchp 24819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-o1 14221 df-lo1 14222 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-em 24719 df-cht 24823 df-vma 24824 df-chp 24825 df-ppi 24826

This theorem is referenced by: pntrsumbnd 25255

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