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Theorem selberg2lem 27582

Description: Lemma for selberg2 27583. 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 13038	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2		chpcl 27155	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
4	3	recnd 11294	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
5		rprege0 13045	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6		flge0nn0 13842	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
7	5, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
8		nn0p1nn 12565	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℕ)
10	9	nnrpd 13070	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
11	10	relogcld 26653	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
12	11	recnd 11294	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13		relogcl 26605	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
14	13	recnd 11294	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
15	12, 14	subcld 11623	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
16	4, 15	mulcld 11286	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17		fzfid 13995	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
18		elfznn 13586	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
19	18	adantl 480	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
20	19	nnrpd 13070	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
21		1rp 13034	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
22		rpaddcl 13052	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑛 + 1) ∈ ℝ⁺)
23	21, 22	mpan2 689	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 + 1) ∈ ℝ⁺)
24	23	relogcld 26653	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘(𝑛 + 1)) ∈ ℝ)
25		relogcl 26605	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
26	24, 25	resubcld 11694	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27		rpre 13038	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
28		chpcl 27155	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℝ)
30	26, 29	remulcld 11296	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
31	30	recnd 11294	. . . . . . . 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 15739	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34		rpcnne0 13048	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35		divsubdir 11961	. . . . . 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 1368	. . . . 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 11286	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
38	4, 14	mulcld 11286	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
39	37, 38, 33	sub32d 11655	. . . . . . 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 11722	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
41	40	oveq1d 7441	. . . . . . 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 6903	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43		fvoveq1 7449	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
44	42, 43	jca 510	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
45		fveq2 6903	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
46		fvoveq1 7449	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
47	45, 46	jca 510	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
48		fveq2 6903	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (log‘𝑚) = (log‘1))
49		log1 26615	. . . . . . . . . . . 12 ⊢ (log‘1) = 0
50	48, 49	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (log‘𝑚) = 0)
51		oveq1 7433	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52		1m1e0 12338	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
53	51, 52	eqtrdi 2782	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
54	53	fveq2d 6907	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55		2pos 12369	. . . . . . . . . . . . 13 ⊢ 0 < 2
56		0re 11268	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
57		chpeq0 27240	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
58	56, 57	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((ψ‘0) = 0 ↔ 0 < 2)
59	55, 58	mpbir 230	. . . . . . . . . . . 12 ⊢ (ψ‘0) = 0
60	54, 59	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
61	50, 60	jca 510	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
62		fveq2 6903	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
63		fvoveq1 7449	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
64	62, 63	jca 510	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
65		nnuz 12919	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
66	9, 65	eleqtrdi 2836	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
67		elfznn 13586	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
68	67	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
69	68	nnrpd 13070	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ⁺)
70	69	relogcld 26653	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
71	70	recnd 11294	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
72	68	nnred 12281	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73		peano2rem 11579	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75		chpcl 27155	. . . . . . . . . . . 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 11294	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
78	44, 47, 61, 64, 66, 71, 77	fsumparts 15812	. . . . . . . . 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 12638	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℤ)
80		fzval3 13757	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
82	81	eqcomd 2732	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83		nnm1nn0 12567	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
84	19, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
85	84	nn0red 12587	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86		chpcl 27155	. . . . . . . . . . . . . . 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 11294	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89		vmacl 27149	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
90	19, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
91	90	recnd 11294	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
92	19	nncnd 12282	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93		ax-1cn 11218	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
94		pncan 11518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
95	92, 93, 94	sylancl 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
96		npcan 11521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
97	92, 93, 96	sylancl 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
98	95, 97	eqtr4d 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
99	98	fveq2d 6907	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
100		chpp1 27186	. . . . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
103	102	oveq2d 7442	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
104	99, 101, 103	3eqtrd 2770	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
105	88, 91, 104	mvrladdd 11679	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106	105	oveq2d 7442	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
107	20	relogcld 26653	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108	107	recnd 11294	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
109	91, 108	mulcomd 11287	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110	106, 109	eqtr4d 2769	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
111	82, 110	sumeq12rdv 15713	. . . . . . . . 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 12588	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
113		pncan 11518	. . . . . . . . . . . . . . . . 17 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
114	112, 93, 113	sylancl 584	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
115	114	fveq2d 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
116		chpfl 27181	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
117	1, 116	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118	115, 117	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119	118	oveq2d 7442	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
120	12, 4	mulcomd 11287	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121	119, 120	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122		0cn 11258	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
123	122	mul01i 11456	. . . . . . . . . . . . 13 ⊢ (0 · 0) = 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (0 · 0) = 0)
125	121, 124	oveq12d 7444	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
126	37	subid1d 11612	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127	125, 126	eqtrd 2766	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
128	95	fveq2d 6907	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
129	128	oveq2d 7442	. . . . . . . . . . 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 15713	. . . . . . . . . 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 7444	. . . . . . . . 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 2774	. . . . . . . 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 7441	. . . . . . 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 2776	. . . . . 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 7441	. . . . 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 11944	. . . . . . 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 1368	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
138	137	oveq1d 7441	. . . . 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 2775	. . . 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 5258	. . 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 7461	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142		ovexd 7461	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143		reex 11251	. . . . . . . 8 ⊢ ℝ ∈ V
144		rpssre 13037	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
145	143, 144	ssexi 5329	. . . . . . 7 ⊢ ℝ⁺ ∈ V
146	145	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ⁺ ∈ V)
147		ovexd 7461	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ V)
148	15	adantl 480	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149		eqidd 2727	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
150		eqidd 2727	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151	146, 147, 148, 149, 150	offval2 7712	. . . . 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 27512	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153		0red 11269	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℝ)
154		1red 11267	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ)
155		divrcnv 15858	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
156	93, 155	mp1i 13	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
157		rpreccl 13056	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
158	157	rpred 13072	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
159	158	adantl 480	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
160	11, 13	resubcld 11694	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161	160	adantl 480	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162		rpaddcl 13052	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
163	21, 162	mpan2 689	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 + 1) ∈ ℝ⁺)
164	163	relogcld 26653	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(𝑥 + 1)) ∈ ℝ)
165	164, 13	resubcld 11694	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
166	7	nn0red 12587	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
167		1red 11267	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
168		flle 13821	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
169	1, 168	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
170	166, 1, 167, 169	leadd1dd 11880	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
171	10, 163	logled 26657	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
172	170, 171	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
173	11, 164, 13, 172	lesub1dd 11882	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174		logdifbnd 27025	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175	160, 165, 158, 173, 174	letrd 11423	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
176	175	ad2antrl 726	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
177		fllep1 13823	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
178	1, 177	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≤ ((⌊‘𝑥) + 1))
179		logleb 26633	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((⌊‘𝑥) + 1) ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
180	10, 179	mpdan 685	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
181	178, 180	mpbid 231	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
182	11, 13	subge0d 11856	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
183	181, 182	mpbird 256	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
184	183	ad2antrl 726	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
185	153, 154, 156, 159, 161, 176, 184	rlimsqz2 15657	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝_𝑟 0)
186		rlimo1 15621	. . . . . . 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 15619	. . . . . 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 585	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
190	151, 189	eqeltrrd 2827	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191		nnrp 13041	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
192	191	ssriv 3983	. . . . . . . 8 ⊢ ℕ ⊆ ℝ⁺
193	192	a1i 11	. . . . . . 7 ⊢ (⊤ → ℕ ⊆ ℝ⁺)
194	193	sselda 3979	. . . . . 6 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
195	194, 31	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196		chpo1ub 27512	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197	196	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198		rerpdivcl 13060	. . . . . . . . 9 ⊢ (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
199	29, 198	mpancom 686	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
200	199	adantl 480	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
201	31	adantl 480	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
202		rpreccl 13056	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ⁺)
203	202	rpred 13072	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
204		chpge0 27157	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
205	27, 204	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑛))
206		logdifbnd 27025	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
207	26, 203, 29, 205, 206	lemul1ad 12207	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
208	27	lep1d 12199	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≤ (𝑛 + 1))
209		logleb 26633	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ⁺ ∧ (𝑛 + 1) ∈ ℝ⁺) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
210	23, 209	mpdan 685	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
211	208, 210	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
212	24, 25	subge0d 11856	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
213	211, 212	mpbird 256	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
214	26, 29, 213, 205	mulge0d 11843	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
215	30, 214	absidd 15429	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216		rpregt0 13044	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217		divge0 12137	. . . . . . . . . . . 12 ⊢ ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
218	29, 205, 216, 217	syl21anc 836	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑛) / 𝑛))
219	199, 218	absidd 15429	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
220	29	recnd 11294	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℂ)
221		rpcn 13040	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
222		rpne0 13046	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
223	220, 221, 222	divrec2d 12047	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224	219, 223	eqtrd 2766	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225	207, 215, 224	3brtr4d 5187	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
226	225	ad2antrl 726	. . . . . . 7 ⊢ ((⊤ ∧ (𝑛 ∈ ℝ⁺ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
227	154, 197, 200, 201, 226	o1le 15659	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228	193, 227	o1res2 15567	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229	195, 228	o1fsum 15819	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230	141, 142, 190, 229	o1sub2 15630	. . 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 2831	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
232	231	mptru 1541	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 205 ∧ wa 394 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 ⊆ wss 3947 class class class wbr 5155 ↦ cmpt 5238 ‘cfv 6556 (class class class)co 7426 ∘_f cof 7690 ℂcc 11158 ℝcr 11159 0cc0 11160 1c1 11161 + caddc 11163 · cmul 11165 < clt 11300 ≤ cle 11301 − cmin 11496 / cdiv 11923 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12612 ℤ_≥cuz 12876 ℝ⁺crp 13030 ...cfz 13540 ..^cfzo 13683 ⌊cfl 13812 abscabs 15241 ⇝_𝑟 crli 15489 𝑂(1)co1 15490 Σcsu 15692 logclog 26584 Λcvma 27123 ψcchp 27124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 ax-addf 11239

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-supp 8177 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-oadd 8502 df-er 8736 df-map 8859 df-pm 8860 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9408 df-fi 9456 df-sup 9487 df-inf 9488 df-oi 9555 df-dju 9946 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12599 df-z 12613 df-dec 12732 df-uz 12877 df-q 12987 df-rp 13031 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13384 df-ioc 13385 df-ico 13386 df-icc 13387 df-fz 13541 df-fzo 13684 df-fl 13814 df-mod 13892 df-seq 14024 df-exp 14084 df-fac 14293 df-bc 14322 df-hash 14350 df-shft 15074 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-limsup 15475 df-clim 15492 df-rlim 15493 df-o1 15494 df-lo1 15495 df-sum 15693 df-ef 16071 df-e 16072 df-sin 16073 df-cos 16074 df-pi 16076 df-dvds 16259 df-gcd 16497 df-prm 16675 df-pc 16841 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-starv 17283 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-unif 17291 df-hom 17292 df-cco 17293 df-rest 17439 df-topn 17440 df-0g 17458 df-gsum 17459 df-topgen 17460 df-pt 17461 df-prds 17464 df-xrs 17519 df-qtop 17524 df-imas 17525 df-xps 17527 df-mre 17601 df-mrc 17602 df-acs 17604 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-submnd 18776 df-mulg 19064 df-cntz 19313 df-cmn 19782 df-psmet 21337 df-xmet 21338 df-met 21339 df-bl 21340 df-mopn 21341 df-fbas 21342 df-fg 21343 df-cnfld 21346 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cld 23017 df-ntr 23018 df-cls 23019 df-nei 23096 df-lp 23134 df-perf 23135 df-cn 23225 df-cnp 23226 df-haus 23313 df-tx 23560 df-hmeo 23753 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-xms 24320 df-ms 24321 df-tms 24322 df-cncf 24892 df-limc 25889 df-dv 25890 df-log 26586 df-cxp 26587 df-cht 27128 df-vma 27129 df-chp 27130 df-ppi 27131

This theorem is referenced by: selberg2 27583 selberg3lem2 27590

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