Theorem selberg3lem2 27602

Description: Lemma for selberg3 27603. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)

Assertion

Ref	Expression
selberg3lem2	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1)

Distinct variable group:   𝑥,𝑛

Proof of Theorem selberg3lem2

Dummy variables 𝑚 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1re 11261	. . . . . . . 8 ⊢ 1 ∈ ℝ
2		elicopnf 13485	. . . . . . . 8 ⊢ (1 ∈ ℝ → (𝑦 ∈ (1[,)+∞) ↔ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)))
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ (1[,)+∞) ↔ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦))
4	3	simplbi 497	. . . . . 6 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ)
5	4	ssriv 3987	. . . . 5 ⊢ (1[,)+∞) ⊆ ℝ
6	5	a1i 11	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
8		fzfid 14014	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (1...(⌊‘𝑦)) ∈ Fin)
9		elfznn 13593	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘𝑦)) → 𝑚 ∈ ℕ)
10	9	adantl 481	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℕ)
11		vmacl 27161	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑚) ∈ ℝ)
13	10	nnrpd 13075	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ+)
14	13	relogcld 26665	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (log‘𝑚) ∈ ℝ)
15	12, 14	remulcld 11291	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
16	8, 15	fsumrecl 15770	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
17	4	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ)
18		chpcl 27167	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (ψ‘𝑦) ∈ ℝ)
20		1rp 13038	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
21	20	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
22	3	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦)
23	22	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦)
24	17, 21, 23	rpgecld 13116	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ+)
25	24	relogcld 26665	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (log‘𝑦) ∈ ℝ)
26	19, 25	remulcld 11291	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
27	16, 26	resubcld 11691	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
28	27, 24	rerpdivcld 13108	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℝ)
29	28	recnd 11289	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℂ)
30	24	ex 412	. . . . . 6 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ+))
31	30	ssrdv 3989	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
32		selberg2lem 27594	. . . . . 6 ⊢ (𝑦 ∈ ℝ+ ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (𝑦 ∈ ℝ+ ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1))
34	31, 33	o1res2 15599	. . . 4 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1))
35		fzfid 14014	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
36		elfznn 13593	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
37	36	adantl 481	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
38	37, 11	syl 17	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
39	37	nnrpd 13075	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
40	39	relogcld 26665	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
41	38, 40	remulcld 11291	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
42	35, 41	fsumrecl 15770	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
43		chpcl 27167	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
44	43	ad2antrl 728	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (ψ‘𝑥) ∈ ℝ)
45		simprl 771	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
46	20	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ+)
47		simprr 773	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
48	45, 46, 47	rpgecld 13116	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
49	48	relogcld 26665	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
50	44, 49	remulcld 11291	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
51	42, 50	readdcld 11290	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))) ∈ ℝ)
52	27	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
53	52	recnd 11289	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℂ)
54	24	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ+)
55	54	rpcnd 13079	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℂ)
56	54	rpne0d 13082	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≠ 0)
57	53, 55, 56	absdivd 15494	. . . . . 6 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / (abs‘𝑦)))
58	17	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ)
59	54	rpge0d 13081	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ 𝑦)
60	58, 59	absidd 15461	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘𝑦) = 𝑦)
61	60	oveq2d 7447	. . . . . 6 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / (abs‘𝑦)) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦))
62	57, 61	eqtrd 2777	. . . . 5 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦))
63	53	abscld 15475	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ∈ ℝ)
64	63, 54	rerpdivcld 13108	. . . . . 6 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ∈ ℝ)
65	42	ad2ant2r 747	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
66		simprll 779	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ)
67	66, 43	syl 17	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑥) ∈ ℝ)
68		simprr 773	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 < 𝑥)
69	58, 66, 68	ltled 11409	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≤ 𝑥)
70	66, 54, 69	rpgecld 13116	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ+)
71	70	relogcld 26665	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑥) ∈ ℝ)
72	67, 71	remulcld 11291	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
73	65, 72	readdcld 11290	. . . . . 6 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))) ∈ ℝ)
74	20	a1i 11	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 1 ∈ ℝ+)
75	53	absge0d 15483	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))))
76	23	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 1 ≤ 𝑦)
77	74, 54, 63, 75, 76	lediv2ad 13099	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ≤ ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 1))
78	63	recnd 11289	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ∈ ℂ)
79	78	div1d 12035	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 1) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))))
80	77, 79	breqtrd 5169	. . . . . 6 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))))
81	16	adantr 480	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
82	58, 18	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ∈ ℝ)
83	54	relogcld 26665	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ∈ ℝ)
84	82, 83	remulcld 11291	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
85	81, 84	readdcld 11290	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
86	81	recnd 11289	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
87	26	adantr 480	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
88	87	recnd 11289	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℂ)
89	86, 88	abs2dif2d 15497	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ≤ ((abs‘Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) + (abs‘((ψ‘𝑦) · (log‘𝑦)))))
90		vmage0 27164	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
91	10, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑚))
92	10	nnred 12281	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ)
93	10	nnge1d 12314	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑚)
94	92, 93	logge0d 26672	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (log‘𝑚))
95	12, 14, 91, 94	mulge0d 11840	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
96	8, 15, 95	fsumge0 15831	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
97	96	adantr 480	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
98	81, 97	absidd 15461	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
99		chpge0 27169	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 0 ≤ (ψ‘𝑦))
100	58, 99	syl 17	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (ψ‘𝑦))
101	58, 76	logge0d 26672	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (log‘𝑦))
102	82, 83, 100, 101	mulge0d 11840	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ ((ψ‘𝑦) · (log‘𝑦)))
103	87, 102	absidd 15461	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((ψ‘𝑦) · (log‘𝑦))) = ((ψ‘𝑦) · (log‘𝑦)))
104	98, 103	oveq12d 7449	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))) + (abs‘((ψ‘𝑦) · (log‘𝑦)))) = (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))))
105	89, 104	breqtrd 5169	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))))
106		fzfid 14014	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
107	36	adantl 481	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
108	107, 11	syl 17	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
109	107	nnrpd 13075	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
110	109	relogcld 26665	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
111	108, 110	remulcld 11291	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
112	107, 90	syl 17	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑚))
113	107	nnred 12281	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
114	107	nnge1d 12314	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑚)
115	113, 114	logge0d 26672	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑚))
116	108, 110, 112, 115	mulge0d 11840	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
117		flword2 13853	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)))
118	58, 66, 69, 117	syl3anc 1373	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)))
119		fzss2 13604	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)) → (1...(⌊‘𝑦)) ⊆ (1...(⌊‘𝑥)))
120	118, 119	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑦)) ⊆ (1...(⌊‘𝑥)))
121	106, 111, 116, 120	fsumless 15832	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ≤ Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)))
122		chpwordi 27200	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (ψ‘𝑦) ≤ (ψ‘𝑥))
123	58, 66, 69, 122	syl3anc 1373	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ≤ (ψ‘𝑥))
124	54, 70	logled 26669	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (𝑦 ≤ 𝑥 ↔ (log‘𝑦) ≤ (log‘𝑥)))
125	69, 124	mpbid 232	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ≤ (log‘𝑥))
126	82, 67, 83, 71, 100, 101, 123, 125	lemul12ad 12210	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ≤ ((ψ‘𝑥) · (log‘𝑥)))
127	81, 84, 65, 72, 121, 126	le2addd 11882	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))))
128	63, 85, 73, 105, 127	letrd 11418	. . . . . 6 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))))
129	64, 63, 73, 80, 128	letrd 11418	. . . . 5 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦)))) / 𝑦) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))))
130	62, 129	eqbrtrd 5165	. . . 4 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))))
131	6, 7, 29, 34, 51, 130	o1bddrp 15578	. . 3 ⊢ (⊤ → ∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐)
132	131	mptru 1547	. 2 ⊢ ∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐
133		simpl 482	. . . 4 ⊢ ((𝑐 ∈ ℝ+ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → 𝑐 ∈ ℝ+)
134		simpr 484	. . . 4 ⊢ ((𝑐 ∈ ℝ+ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐)
135	133, 134	selberg3lem1 27601	. . 3 ⊢ ((𝑐 ∈ ℝ+ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))
136	135	rexlimiva 3147	. 2 ⊢ (∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))
137	132, 136	ax-mp 5	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1)

Syntax hints:   ↔ wb 206   ∧ wa 395  ⊤wtru 1541   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  +∞cpnf 11292   < clt 11295   ≤ cle 11296   − cmin 11492   / cdiv 11920  ℕcn 12266  2c2 12321  ℤ_≥cuz 12878  ℝ⁺crp 13034  (,)cioo 13387  [,)cico 13389  ...cfz 13547  ⌊cfl 13830  abscabs 15273  𝑂(1)co1 15522  Σcsu 15722  logclog 26596  Λcvma 27135  ψcchp 27136

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  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 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-o1 15526  df-lo1 15527  df-sum 15723  df-ef 16103  df-e 16104  df-sin 16105  df-cos 16106  df-pi 16108  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-log 26598  df-cxp 26599  df-cht 27140  df-vma 27141  df-chp 27142  df-ppi 27143

This theorem is referenced by:  selberg3  27603  selberg4  27605