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Theorem selberg2lem 27478

Description: Lemma for selberg2 27479. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)

**Assertion**
Ref	Expression
selberg2lem	⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)

Distinct variable group: 𝑥,𝑛

Proof of Theorem selberg2lem Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 12921	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2		chpcl 27051	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
4	3	recnd 11162	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
5		rprege0 12928	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6		flge0nn0 13743	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
8		nn0p1nn 12442	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℕ)
10	9	nnrpd 12954	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
11	10	relogcld 26549	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
12	11	recnd 11162	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13		relogcl 26501	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
14	13	recnd 11162	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
15	12, 14	subcld 11494	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
16	4, 15	mulcld 11154	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17		fzfid 13899	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
18		elfznn 13475	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
20	19	nnrpd 12954	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
21		1rp 12916	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
22		rpaddcl 12936	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑛 + 1) ∈ ℝ⁺)
23	21, 22	mpan2 691	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 + 1) ∈ ℝ⁺)
24	23	relogcld 26549	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘(𝑛 + 1)) ∈ ℝ)
25		relogcl 26501	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
26	24, 25	resubcld 11567	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27		rpre 12921	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
28		chpcl 27051	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℝ)
30	26, 29	remulcld 11164	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
31	30	recnd 11162	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
32	20, 31	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
33	17, 32	fsumcl 15659	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34		rpcnne0 12931	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35		divsubdir 11837	. . . . . 6 ⊢ ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ ∧ Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
36	16, 33, 34, 35	syl3anc 1373	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
37	4, 12	mulcld 11154	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
38	4, 14	mulcld 11154	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
39	37, 38, 33	sub32d 11526	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
40	4, 12, 14	subdid 11595	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
41	40	oveq1d 7368	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
42		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43		fvoveq1 7376	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
44	42, 43	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
45		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
46		fvoveq1 7376	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
47	45, 46	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
48		fveq2 6826	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (log‘𝑚) = (log‘1))
49		log1 26511	. . . . . . . . . . . 12 ⊢ (log‘1) = 0
50	48, 49	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (log‘𝑚) = 0)
51		oveq1 7360	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52		1m1e0 12219	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
53	51, 52	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
54	53	fveq2d 6830	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55		2pos 12250	. . . . . . . . . . . . 13 ⊢ 0 < 2
56		0re 11136	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
57		chpeq0 27136	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
58	56, 57	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
59	55, 58	mpbir 231	. . . . . . . . . . . 12 ⊢ (ψ‘0) = 0
60	54, 59	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
61	50, 60	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
62		fveq2 6826	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
63		fvoveq1 7376	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
64	62, 63	jca 511	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
65		nnuz 12797	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
66	9, 65	eleqtrdi 2838	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
67		elfznn 13475	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
68	67	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
69	68	nnrpd 12954	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ⁺)
70	69	relogcld 26549	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
71	70	recnd 11162	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
72	68	nnred 12162	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73		peano2rem 11450	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75		chpcl 27051	. . . . . . . . . . . 12 ⊢ ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
76	74, 75	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
77	76	recnd 11162	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
78	44, 47, 61, 64, 66, 71, 77	fsumparts 15732	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))))
79	7	nn0zd 12516	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℤ)
80		fzval3 13656	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
82	81	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83		nnm1nn0 12444	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
84	19, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
85	84	nn0red 12465	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86		chpcl 27051	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
87	85, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
88	87	recnd 11162	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89		vmacl 27045	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
90	19, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
91	90	recnd 11162	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
92	19	nncnd 12163	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93		ax-1cn 11086	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
94		pncan 11388	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
95	92, 93, 94	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
96		npcan 11391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
97	92, 93, 96	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
98	95, 97	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
99	98	fveq2d 6830	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
100		chpp1 27082	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
101	84, 100	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
102	97	fveq2d 6830	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
103	102	oveq2d 7369	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
104	99, 101, 103	3eqtrd 2768	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
105	88, 91, 104	mvrladdd 11552	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106	105	oveq2d 7369	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
107	20	relogcld 26549	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108	107	recnd 11162	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
109	91, 108	mulcomd 11155	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110	106, 109	eqtr4d 2767	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
111	82, 110	sumeq12rdv 15633	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
112	7	nn0cnd 12466	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
113		pncan 11388	. . . . . . . . . . . . . . . . 17 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
114	112, 93, 113	sylancl 586	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
115	114	fveq2d 6830	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
116		chpfl 27077	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
117	1, 116	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118	115, 117	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119	118	oveq2d 7369	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
120	12, 4	mulcomd 11155	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121	119, 120	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122		0cn 11126	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
123	122	mul01i 11325	. . . . . . . . . . . . 13 ⊢ (0 · 0) = 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (0 · 0) = 0)
125	121, 124	oveq12d 7371	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
126	37	subid1d 11483	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127	125, 126	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
128	95	fveq2d 6830	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
129	128	oveq2d 7369	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
130	82, 129	sumeq12rdv 15633	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
131	127, 130	oveq12d 7371	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
132	78, 111, 131	3eqtr3d 2772	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
133	132	oveq1d 7368	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
134	39, 41, 133	3eqtr4d 2774	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))))
135	134	oveq1d 7368	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
136		div23 11817	. . . . . . 7 ⊢ (((ψ‘𝑥) ∈ ℂ ∧ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
137	4, 15, 34, 136	syl3anc 1373	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
138	137	oveq1d 7368	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
139	36, 135, 138	3eqtr3rd 2773	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
140	139	mpteq2ia 5190	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
141		ovexd 7388	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142		ovexd 7388	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143		reex 11119	. . . . . . . 8 ⊢ ℝ ∈ V
144		rpssre 12920	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
145	143, 144	ssexi 5264	. . . . . . 7 ⊢ ℝ⁺ ∈ V
146	145	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ⁺ ∈ V)
147		ovexd 7388	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ V)
148	15	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149		eqidd 2730	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
150		eqidd 2730	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151	146, 147, 148, 149, 150	offval2 7637	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))))
152		chpo1ub 27408	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153		0red 11137	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℝ)
154		1red 11135	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ)
155		divrcnv 15778	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
156	93, 155	mp1i 13	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
157		rpreccl 12940	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
158	157	rpred 12956	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
159	158	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
160	11, 13	resubcld 11567	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161	160	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162		rpaddcl 12936	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
163	21, 162	mpan2 691	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 + 1) ∈ ℝ⁺)
164	163	relogcld 26549	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(𝑥 + 1)) ∈ ℝ)
165	164, 13	resubcld 11567	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
166	7	nn0red 12465	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
167		1red 11135	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
168		flle 13722	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
169	1, 168	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
170	166, 1, 167, 169	leadd1dd 11753	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
171	10, 163	logled 26553	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
172	170, 171	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
173	11, 164, 13, 172	lesub1dd 11755	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174		logdifbnd 26921	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175	160, 165, 158, 173, 174	letrd 11292	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
176	175	ad2antrl 728	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
177		fllep1 13724	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
178	1, 177	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≤ ((⌊‘𝑥) + 1))
179		logleb 26529	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((⌊‘𝑥) + 1) ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
180	10, 179	mpdan 687	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
181	178, 180	mpbid 232	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
182	11, 13	subge0d 11729	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
183	181, 182	mpbird 257	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
184	183	ad2antrl 728	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
185	153, 154, 156, 159, 161, 176, 184	rlimsqz2 15577	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝_𝑟 0)
186		rlimo1 15543	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
187	185, 186	syl 17	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
188		o1mul 15541	. . . . . 6 ⊢ (((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
189	152, 187, 188	sylancr 587	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
190	151, 189	eqeltrrd 2829	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191		nnrp 12924	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
192	191	ssriv 3941	. . . . . . . 8 ⊢ ℕ ⊆ ℝ⁺
193	192	a1i 11	. . . . . . 7 ⊢ (⊤ → ℕ ⊆ ℝ⁺)
194	193	sselda 3937	. . . . . 6 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
195	194, 31	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196		chpo1ub 27408	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197	196	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198		rerpdivcl 12944	. . . . . . . . 9 ⊢ (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
199	29, 198	mpancom 688	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
200	199	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
201	31	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
202		rpreccl 12940	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ⁺)
203	202	rpred 12956	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
204		chpge0 27053	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
205	27, 204	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑛))
206		logdifbnd 26921	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
207	26, 203, 29, 205, 206	lemul1ad 12083	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
208	27	lep1d 12075	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≤ (𝑛 + 1))
209		logleb 26529	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ⁺ ∧ (𝑛 + 1) ∈ ℝ⁺) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
210	23, 209	mpdan 687	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
211	208, 210	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
212	24, 25	subge0d 11729	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
213	211, 212	mpbird 257	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
214	26, 29, 213, 205	mulge0d 11716	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
215	30, 214	absidd 15349	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216		rpregt0 12927	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217		divge0 12013	. . . . . . . . . . . 12 ⊢ ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
218	29, 205, 216, 217	syl21anc 837	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑛) / 𝑛))
219	199, 218	absidd 15349	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
220	29	recnd 11162	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℂ)
221		rpcn 12923	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
222		rpne0 12929	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
223	220, 221, 222	divrec2d 11923	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224	219, 223	eqtrd 2764	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225	207, 215, 224	3brtr4d 5127	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
226	225	ad2antrl 728	. . . . . . 7 ⊢ ((⊤ ∧ (𝑛 ∈ ℝ⁺ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
227	154, 197, 200, 201, 226	o1le 15579	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228	193, 227	o1res2 15489	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229	195, 228	o1fsum 15739	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230	141, 142, 190, 229	o1sub2 15552	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) ∈ 𝑂(1))
231	140, 230	eqeltrrid 2833	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
232	231	mptru 1547	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 Vcvv 3438 ⊆ wss 3905 class class class wbr 5095 ↦ cmpt 5176 ‘cfv 6486 (class class class)co 7353 ∘_f cof 7615 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12147 2c2 12202 ℕ₀cn0 12403 ℤcz 12490 ℤ_≥cuz 12754 ℝ⁺crp 12912 ...cfz 13429 ..^cfzo 13576 ⌊cfl 13713 abscabs 15160 ⇝_𝑟 crli 15411 𝑂(1)co1 15412 Σcsu 15612 logclog 26480 Λcvma 27019 ψcchp 27020

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 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 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-xnn0 12477 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13271 df-ioc 13272 df-ico 13273 df-icc 13274 df-fz 13430 df-fzo 13577 df-fl 13715 df-mod 13793 df-seq 13928 df-exp 13988 df-fac 14200 df-bc 14229 df-hash 14257 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15397 df-clim 15414 df-rlim 15415 df-o1 15416 df-lo1 15417 df-sum 15613 df-ef 15993 df-e 15994 df-sin 15995 df-cos 15996 df-pi 15998 df-dvds 16183 df-gcd 16425 df-prm 16602 df-pc 16768 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-gsum 17365 df-topgen 17366 df-pt 17367 df-prds 17370 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-mulg 18966 df-cntz 19215 df-cmn 19680 df-psmet 21272 df-xmet 21273 df-met 21274 df-bl 21275 df-mopn 21276 df-fbas 21277 df-fg 21278 df-cnfld 21281 df-top 22798 df-topon 22815 df-topsp 22837 df-bases 22850 df-cld 22923 df-ntr 22924 df-cls 22925 df-nei 23002 df-lp 23040 df-perf 23041 df-cn 23131 df-cnp 23132 df-haus 23219 df-tx 23466 df-hmeo 23659 df-fil 23750 df-fm 23842 df-flim 23843 df-flf 23844 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24788 df-limc 25784 df-dv 25785 df-log 26482 df-cxp 26483 df-cht 27024 df-vma 27025 df-chp 27026 df-ppi 27027

This theorem is referenced by: selberg2 27479 selberg3lem2 27486

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