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Theorem selberg2lem 27695

Description: Lemma for selberg2 27696. 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 13026	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2		chpcl 27269	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
3	1, 2	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
4	3	recnd 11238	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
5		rprege0 13033	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6		flge0nn0 13855	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
7	5, 6	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
8		nn0p1nn 12544	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
9	7, 8	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℕ)
10	9	nnrpd 13059	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
11	10	relogcld 26769	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
12	11	recnd 11238	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13		relogcl 26721	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
14	13	recnd 11238	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
15	12, 14	subcld 11570	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
16	4, 15	mulcld 11230	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17		fzfid 14011	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
18		elfznn 13583	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
19	18	adantl 486	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
20	19	nnrpd 13059	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
21		1rp 13021	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
22		rpaddcl 13041	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑛 + 1) ∈ ℝ⁺)
23	21, 22	mpan2 703	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 + 1) ∈ ℝ⁺)
24	23	relogcld 26769	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘(𝑛 + 1)) ∈ ℝ)
25		relogcl 26721	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
26	24, 25	resubcld 11643	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27		rpre 13026	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
28		chpcl 27269	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
29	27, 28	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℝ)
30	26, 29	remulcld 11240	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
31	30	recnd 11238	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
32	20, 31	syl 18	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
33	17, 32	fsumcl 15786	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34		rpcnne0 13036	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35		divsubdir 11909	. . . . . 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 1398	. . . . 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 11230	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
38	4, 14	mulcld 11230	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
39	37, 38, 33	sub32d 11602	. . . . . . 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 11671	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
41	40	oveq1d 7427	. . . . . . 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 6883	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43		fvoveq1 7435	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
44	42, 43	jca 520	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
45		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
46		fvoveq1 7435	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
47	45, 46	jca 520	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
48		fveq2 6883	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (log‘𝑚) = (log‘1))
49		log1 26731	. . . . . . . . . . . 12 ⊢ (log‘1) = 0
50	48, 49	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (log‘𝑚) = 0)
51		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52		1m1e0 12314	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
53	51, 52	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
54	53	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55		2pos 12346	. . . . . . . . . . . . 13 ⊢ 0 < 2
56		0re 11211	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
57		chpeq0 27353	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
58	56, 57	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
59	55, 58	mpbir 234	. . . . . . . . . . . 12 ⊢ (ψ‘0) = 0
60	54, 59	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
61	50, 60	jca 520	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
62		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
63		fvoveq1 7435	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
64	62, 63	jca 520	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
65		nnuz 12902	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
66	9, 65	eleqtrdi 2873	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
67		elfznn 13583	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
68	67	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
69	68	nnrpd 13059	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ⁺)
70	69	relogcld 26769	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
71	70	recnd 11238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
72	68	nnred 12249	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73		peano2rem 11526	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
74	72, 73	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75		chpcl 27269	. . . . . . . . . . . 12 ⊢ ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
76	74, 75	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
77	76	recnd 11238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
78	44, 47, 61, 64, 66, 71, 77	fsumparts 15860	. . . . . . . . 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 12617	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℤ)
80		fzval3 13765	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
81	79, 80	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
82	81	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83		nnm1nn0 12546	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
84	19, 83	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
85	84	nn0red 12567	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86		chpcl 27269	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
87	85, 86	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
88	87	recnd 11238	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89		vmacl 27263	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
90	19, 89	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
91	90	recnd 11238	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
92	19	nncnd 12250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93		ax-1cn 11159	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
94		pncan 11464	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
95	92, 93, 94	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
96		npcan 11467	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
97	92, 93, 96	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
98	95, 97	eqtr4d 2801	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
99	98	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
100		chpp1 27300	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 1) ∈ ℕ₀ → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
101	84, 100	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
102	97	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
103	102	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
104	99, 101, 103	3eqtrd 2802	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
105	88, 91, 104	mvrladdd 11628	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106	105	oveq2d 7428	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
107	20	relogcld 26769	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108	107	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
109	91, 108	mulcomd 11231	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110	106, 109	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
111	82, 110	sumeq12rdv 15760	. . . . . . . . 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 12568	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
113		pncan 11464	. . . . . . . . . . . . . . . . 17 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
114	112, 93, 113	sylancl 597	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
115	114	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
116		chpfl 27295	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
117	1, 116	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118	115, 117	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119	118	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
120	12, 4	mulcomd 11231	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121	119, 120	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122		0cn 11199	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
123	122	mul01i 11401	. . . . . . . . . . . . 13 ⊢ (0 · 0) = 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (0 · 0) = 0)
125	121, 124	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
126	37	subid1d 11559	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127	125, 126	eqtrd 2798	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
128	95	fveq2d 6887	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
129	128	oveq2d 7428	. . . . . . . . . . 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 15760	. . . . . . . . . 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 7430	. . . . . . . . 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 2806	. . . . . . . 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 7427	. . . . . . 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 2808	. . . . . 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 7427	. . . . 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 11892	. . . . . . 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 1398	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
138	137	oveq1d 7427	. . . . 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 2807	. . . 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 5207	. . 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 7447	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142		ovexd 7447	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143		reex 11192	. . . . . . . 8 ⊢ ℝ ∈ V
144		rpssre 13025	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
145	143, 144	ssexi 5294	. . . . . . 7 ⊢ ℝ⁺ ∈ V
146	145	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ⁺ ∈ V)
147		ovexd 7447	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ V)
148	15	adantl 486	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149		eqidd 2764	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
150		eqidd 2764	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151	146, 147, 148, 149, 150	offval2 7696	. . . . 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 27625	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153		0red 11212	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℝ)
154		1red 11210	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ)
155		divrcnv 15908	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
156	93, 155	mp1i 14	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
157		rpreccl 13045	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
158	157	rpred 13061	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
159	158	adantl 486	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
160	11, 13	resubcld 11643	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161	160	adantl 486	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162		rpaddcl 13041	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
163	21, 162	mpan2 703	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 + 1) ∈ ℝ⁺)
164	163	relogcld 26769	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(𝑥 + 1)) ∈ ℝ)
165	164, 13	resubcld 11643	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
166	7	nn0red 12567	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
167		1red 11210	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
168		flle 13834	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
169	1, 168	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
170	166, 1, 167, 169	leadd1dd 11829	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
171	10, 163	logled 26773	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
172	170, 171	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
173	11, 164, 13, 172	lesub1dd 11831	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174		logdifbnd 27139	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175	160, 165, 158, 173, 174	letrd 11368	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
176	175	ad2antrl 740	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
177		fllep1 13836	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
178	1, 177	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≤ ((⌊‘𝑥) + 1))
179		logleb 26749	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((⌊‘𝑥) + 1) ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
180	10, 179	mpdan 699	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
181	178, 180	mpbid 235	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
182	11, 13	subge0d 11805	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
183	181, 182	mpbird 260	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
184	183	ad2antrl 740	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
185	153, 154, 156, 159, 161, 176, 184	rlimsqz2 15704	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝_𝑟 0)
186		rlimo1 15670	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝_𝑟 0 → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
187	185, 186	syl 18	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
188		o1mul 15668	. . . . . 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 598	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
190	151, 189	eqeltrrd 2864	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191		nnrp 13029	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
192	191	ssriv 3942	. . . . . . . 8 ⊢ ℕ ⊆ ℝ⁺
193	192	a1i 11	. . . . . . 7 ⊢ (⊤ → ℕ ⊆ ℝ⁺)
194	193	sselda 3938	. . . . . 6 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
195	194, 31	syl 18	. . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196		chpo1ub 27625	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197	196	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198		rerpdivcl 13049	. . . . . . . . 9 ⊢ (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
199	29, 198	mpancom 700	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
200	199	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
201	31	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
202		rpreccl 13045	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ⁺)
203	202	rpred 13061	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
204		chpge0 27271	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
205	27, 204	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑛))
206		logdifbnd 27139	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
207	26, 203, 29, 205, 206	lemul1ad 12155	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
208	27	lep1d 12147	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≤ (𝑛 + 1))
209		logleb 26749	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ⁺ ∧ (𝑛 + 1) ∈ ℝ⁺) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
210	23, 209	mpdan 699	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
211	208, 210	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
212	24, 25	subge0d 11805	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
213	211, 212	mpbird 260	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
214	26, 29, 213, 205	mulge0d 11792	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
215	30, 214	absidd 15476	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216		rpregt0 13032	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217		divge0 12085	. . . . . . . . . . . 12 ⊢ ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
218	29, 205, 216, 217	syl21anc 850	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑛) / 𝑛))
219	199, 218	absidd 15476	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
220	29	recnd 11238	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℂ)
221		rpcn 13028	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
222		rpne0 13034	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
223	220, 221, 222	divrec2d 11996	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224	219, 223	eqtrd 2798	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225	207, 215, 224	3brtr4d 5144	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
226	225	ad2antrl 740	. . . . . . 7 ⊢ ((⊤ ∧ (𝑛 ∈ ℝ⁺ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
227	154, 197, 200, 201, 226	o1le 15706	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228	193, 227	o1res2 15616	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229	195, 228	o1fsum 15867	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230	141, 142, 190, 229	o1sub2 15679	. . 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 2868	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
232	231	mptru 1577	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 209 ∧ wa 400 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ⊆ wss 3906 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6538 (class class class)co 7412 ∘_f cof 7674 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 < clt 11244 ≤ cle 11245 − cmin 11442 / cdiv 11872 ℕcn 12234 2c2 12296 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ℝ⁺crp 13017 ...cfz 13536 ..^cfzo 13684 ⌊cfl 13825 abscabs 15287 ⇝_𝑟 crli 15538 𝑂(1)co1 15539 Σcsu 15739 logclog 26700 Λcvma 27237 ψcchp 27238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-xnn0 12579 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-o1 15543 df-lo1 15544 df-sum 15740 df-ef 16122 df-e 16123 df-sin 16124 df-cos 16125 df-pi 16127 df-dvds 16312 df-gcd 16554 df-prm 16731 df-pc 16898 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-limc 26006 df-dv 26007 df-log 26702 df-cxp 26703 df-cht 27242 df-vma 27243 df-chp 27244 df-ppi 27245

This theorem is referenced by: selberg2 27696 selberg3lem2 27703

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