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Theorem selberg2lem 27786

Description: Lemma for selberg2 27787. 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 13051	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2		chpcl 27360	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
3	1, 2	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
4	3	recnd 11261	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
5		rprege0 13058	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6		flge0nn0 13881	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
7	5, 6	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℕ₀)
8		nn0p1nn 12567	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
9	7, 8	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℕ)
10	9	nnrpd 13084	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ ℝ⁺)
11	10	relogcld 26860	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
12	11	recnd 11261	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13		relogcl 26812	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
14	13	recnd 11261	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
15	12, 14	subcld 11593	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
16	4, 15	mulcld 11253	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17		fzfid 14037	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
18		elfznn 13608	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
19	18	adantl 487	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
20	19	nnrpd 13084	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
21		1rp 13046	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
22		rpaddcl 13066	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑛 + 1) ∈ ℝ⁺)
23	21, 22	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 + 1) ∈ ℝ⁺)
24	23	relogcld 26860	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘(𝑛 + 1)) ∈ ℝ)
25		relogcl 26812	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
26	24, 25	resubcld 11666	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27		rpre 13051	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
28		chpcl 27360	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
29	27, 28	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℝ)
30	26, 29	remulcld 11263	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
31	30	recnd 11261	. . . . . . . 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 15819	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34		rpcnne0 13061	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35		divsubdir 11932	. . . . . 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 11253	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
38	4, 14	mulcld 11253	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
39	37, 38, 33	sub32d 11625	. . . . . . 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 11694	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
41	40	oveq1d 7428	. . . . . . 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 6878	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43		fvoveq1 7436	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
44	42, 43	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
45		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
46		fvoveq1 7436	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
47	45, 46	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
48		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (log‘𝑚) = (log‘1))
49		log1 26822	. . . . . . . . . . . 12 ⊢ (log‘1) = 0
50	48, 49	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (log‘𝑚) = 0)
51		oveq1 7420	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52		1m1e0 12337	. . . . . . . . . . . . . 14 ⊢ (1 − 1) = 0
53	51, 52	eqtrdi 2811	. . . . . . . . . . . . 13 ⊢ (𝑚 = 1 → (𝑚 − 1) = 0)
54	53	fveq2d 6882	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55		2pos 12369	. . . . . . . . . . . . 13 ⊢ 0 < 2
56		0re 11234	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
57		chpeq0 27444	. . . . . . . . . . . . . 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 2811	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
61	50, 60	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
62		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
63		fvoveq1 7436	. . . . . . . . . . 11 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
64	62, 63	jca 521	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
65		nnuz 12926	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
66	9, 65	eleqtrdi 2870	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
67		elfznn 13608	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
68	67	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
69	68	nnrpd 13084	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ⁺)
70	69	relogcld 26860	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
71	70	recnd 11261	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
72	68	nnred 12272	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73		peano2rem 11549	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
74	72, 73	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75		chpcl 27360	. . . . . . . . . . . 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 11261	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
78	44, 47, 61, 64, 66, 71, 77	fsumparts 15893	. . . . . . . . 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 12640	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℤ)
80		fzval3 13790	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
81	79, 80	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
82	81	eqcomd 2766	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83		nnm1nn0 12569	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
84	19, 83	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ₀)
85	84	nn0red 12590	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86		chpcl 27360	. . . . . . . . . . . . . . 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 11261	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89		vmacl 27354	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
90	19, 89	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
91	90	recnd 11261	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
92	19	nncnd 12273	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93		ax-1cn 11182	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
94		pncan 11487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
95	92, 93, 94	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
96		npcan 11490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
97	92, 93, 96	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
98	95, 97	eqtr4d 2798	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
99	98	fveq2d 6882	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
100		chpp1 27391	. . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
103	102	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
104	99, 101, 103	3eqtrd 2799	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
105	88, 91, 104	mvrladdd 11651	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106	105	oveq2d 7429	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
107	20	relogcld 26860	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108	107	recnd 11261	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
109	91, 108	mulcomd 11254	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110	106, 109	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
111	82, 110	sumeq12rdv 15793	. . . . . . . . 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 12591	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℂ)
113		pncan 11487	. . . . . . . . . . . . . . . . 17 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
114	112, 93, 113	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
115	114	fveq2d 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
116		chpfl 27386	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
117	1, 116	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118	115, 117	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119	118	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
120	12, 4	mulcomd 11254	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121	119, 120	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122		0cn 11222	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
123	122	mul01i 11424	. . . . . . . . . . . . 13 ⊢ (0 · 0) = 0
124	123	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (0 · 0) = 0)
125	121, 124	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
126	37	subid1d 11582	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127	125, 126	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
128	95	fveq2d 6882	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
129	128	oveq2d 7429	. . . . . . . . . . 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 15793	. . . . . . . . . 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 7431	. . . . . . . . 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 2803	. . . . . . . 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 7428	. . . . . . 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 2805	. . . . . 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 7428	. . . . 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 11915	. . . . . . 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 7428	. . . . 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 2804	. . . 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 5200	. . 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 7448	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142		ovexd 7448	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143		reex 11215	. . . . . . . 8 ⊢ ℝ ∈ V
144		rpssre 13050	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
145	143, 144	ssexi 5287	. . . . . . 7 ⊢ ℝ⁺ ∈ V
146	145	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ⁺ ∈ V)
147		ovexd 7448	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ V)
148	15	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149		eqidd 2761	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)))
150		eqidd 2761	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151	146, 147, 148, 149, 150	offval2 7698	. . . . 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 27716	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153		0red 11235	. . . . . . . 8 ⊢ (⊤ → 0 ∈ ℝ)
154		1red 11233	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℝ)
155		divrcnv 15941	. . . . . . . . 9 ⊢ (1 ∈ ℂ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
156	93, 155	mp1i 14	. . . . . . . 8 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)) ⇝_𝑟 0)
157		rpreccl 13070	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
158	157	rpred 13086	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ)
159	158	adantl 487	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ)
160	11, 13	resubcld 11666	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161	160	adantl 487	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162		rpaddcl 13066	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (𝑥 + 1) ∈ ℝ⁺)
163	21, 162	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 + 1) ∈ ℝ⁺)
164	163	relogcld 26860	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(𝑥 + 1)) ∈ ℝ)
165	164, 13	resubcld 11666	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
166	7	nn0red 12590	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ∈ ℝ)
167		1red 11233	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 1 ∈ ℝ)
168		flle 13860	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
169	1, 168	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (⌊‘𝑥) ≤ 𝑥)
170	166, 1, 167, 169	leadd1dd 11852	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
171	10, 163	logled 26864	. . . . . . . . . . . 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 11854	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174		logdifbnd 27230	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175	160, 165, 158, 173, 174	letrd 11391	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
176	175	ad2antrl 741	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
177		fllep1 13862	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
178	1, 177	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≤ ((⌊‘𝑥) + 1))
179		logleb 26840	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ ((⌊‘𝑥) + 1) ∈ ℝ⁺) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
180	10, 179	mpdan 700	. . . . . . . . . . 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 11828	. . . . . . . . . 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 741	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
185	153, 154, 156, 159, 161, 176, 184	rlimsqz2 15738	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝_𝑟 0)
186		rlimo1 15704	. . . . . . 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 15702	. . . . . 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 599	. . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ ℝ⁺ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
190	151, 189	eqeltrrd 2861	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191		nnrp 13054	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
192	191	ssriv 3935	. . . . . . . 8 ⊢ ℕ ⊆ ℝ⁺
193	192	a1i 11	. . . . . . 7 ⊢ (⊤ → ℕ ⊆ ℝ⁺)
194	193	sselda 3931	. . . . . 6 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ⁺)
195	194, 31	syl 18	. . . . 5 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196		chpo1ub 27716	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197	196	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198		rerpdivcl 13074	. . . . . . . . 9 ⊢ (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
199	29, 198	mpancom 701	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
200	199	adantl 487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
201	31	adantl 487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℝ⁺) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
202		rpreccl 13070	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ⁺)
203	202	rpred 13086	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (1 / 𝑛) ∈ ℝ)
204		chpge0 27362	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
205	27, 204	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑛))
206		logdifbnd 27230	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
207	26, 203, 29, 205, 206	lemul1ad 12178	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
208	27	lep1d 12170	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≤ (𝑛 + 1))
209		logleb 26840	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℝ⁺ ∧ (𝑛 + 1) ∈ ℝ⁺) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
210	23, 209	mpdan 700	. . . . . . . . . . . . 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 11828	. . . . . . . . . . . 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 11815	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
215	30, 214	absidd 15510	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216		rpregt0 13057	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217		divge0 12108	. . . . . . . . . . . 12 ⊢ ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
218	29, 205, 216, 217	syl21anc 851	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑛) / 𝑛))
219	199, 218	absidd 15510	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
220	29	recnd 11261	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → (ψ‘𝑛) ∈ ℂ)
221		rpcn 13053	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
222		rpne0 13059	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
223	220, 221, 222	divrec2d 12019	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224	219, 223	eqtrd 2795	. . . . . . . . 9 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225	207, 215, 224	3brtr4d 5137	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
226	225	ad2antrl 741	. . . . . . 7 ⊢ ((⊤ ∧ (𝑛 ∈ ℝ⁺ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
227	154, 197, 200, 201, 226	o1le 15740	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℝ⁺ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228	193, 227	o1res2 15650	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229	195, 228	o1fsum 15900	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230	141, 142, 190, 229	o1sub2 15713	. . 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 2865	. 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

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7413 ∘_f cof 7676 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 < clt 11267 ≤ cle 11268 − cmin 11465 / cdiv 11895 ℕcn 12257 2c2 12319 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12887 ℝ⁺crp 13042 ...cfz 13561 ..^cfzo 13709 ⌊cfl 13851 abscabs 15321 ⇝_𝑟 crli 15572 𝑂(1)co1 15573 Σcsu 15773 logclog 26791 Λcvma 27328 ψcchp 27329

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-o1 15577 df-lo1 15578 df-sum 15774 df-ef 16153 df-e 16154 df-sin 16155 df-cos 16156 df-pi 16158 df-dvds 16343 df-gcd 16585 df-prm 16762 df-pc 16929 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094 df-log 26793 df-cxp 26794 df-cht 27333 df-vma 27334 df-chp 27335 df-ppi 27336

This theorem is used by: selberg2 27787 selberg3lem2 27794

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