Theorem selberg3lem2 27476

Description: Lemma for selberg3 27477. 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 11181	. . . . . . . 8 ⊢ 1 ∈ ℝ
2		elicopnf 13413	. . . . . . . 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 3953	. . . . 5 ⊢ (1[,)+∞) ⊆ ℝ
6	5	a1i 11	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
8		fzfid 13945	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (1...(⌊‘𝑦)) ∈ Fin)
9		elfznn 13521	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘𝑦)) → 𝑚 ∈ ℕ)
10	9	adantl 481	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℕ)
11		vmacl 27035	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑚) ∈ ℝ)
13	10	nnrpd 13000	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ+)
14	13	relogcld 26539	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (log‘𝑚) ∈ ℝ)
15	12, 14	remulcld 11211	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
16	8, 15	fsumrecl 15707	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
17	4	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ)
18		chpcl 27041	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (ψ‘𝑦) ∈ ℝ)
20		1rp 12962	. . . . . . . . . . 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 13041	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ+)
25	24	relogcld 26539	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (log‘𝑦) ∈ ℝ)
26	19, 25	remulcld 11211	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
27	16, 26	resubcld 11613	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
28	27, 24	rerpdivcld 13033	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℝ)
29	28	recnd 11209	. . . 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 3955	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
32		selberg2lem 27468	. . . . . 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 15536	. . . 4 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1))
35		fzfid 13945	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
36		elfznn 13521	. . . . . . . . 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 13000	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
40	39	relogcld 26539	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
41	38, 40	remulcld 11211	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
42	35, 41	fsumrecl 15707	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
43		chpcl 27041	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
44	43	ad2antrl 728	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (ψ‘𝑥) ∈ ℝ)
45		simprl 770	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
46	20	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ+)
47		simprr 772	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
48	45, 46, 47	rpgecld 13041	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
49	48	relogcld 26539	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
50	44, 49	remulcld 11211	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
51	42, 50	readdcld 11210	. . . 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 11209	. . . . . . 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 13004	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℂ)
56	54	rpne0d 13007	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≠ 0)
57	53, 55, 56	absdivd 15431	. . . . . 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 13006	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ 𝑦)
60	58, 59	absidd 15396	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘𝑦) = 𝑦)
61	60	oveq2d 7406	. . . . . 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 2765	. . . . 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 15412	. . . . . . 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 13033	. . . . . 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 778	. . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 < 𝑥)
69	58, 66, 68	ltled 11329	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≤ 𝑥)
70	66, 54, 69	rpgecld 13041	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ+)
71	70	relogcld 26539	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑥) ∈ ℝ)
72	67, 71	remulcld 11211	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
73	65, 72	readdcld 11210	. . . . . 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 15420	. . . . . . . 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 13024	. . . . . . 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 11209	. . . . . . . 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 11957	. . . . . . 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 5136	. . . . . 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 26539	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ∈ ℝ)
84	82, 83	remulcld 11211	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
85	81, 84	readdcld 11210	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
86	81	recnd 11209	. . . . . . . . 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 11209	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℂ)
89	86, 88	abs2dif2d 15434	. . . . . . . 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 27038	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
91	10, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑚))
92	10	nnred 12208	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ)
93	10	nnge1d 12241	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑚)
94	92, 93	logge0d 26546	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (log‘𝑚))
95	12, 14, 91, 94	mulge0d 11762	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
96	8, 15, 95	fsumge0 15768	. . . . . . . . . . 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 15396	. . . . . . . . 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 27043	. . . . . . . . . . . 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 26546	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (log‘𝑦))
102	82, 83, 100, 101	mulge0d 11762	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ ((ψ‘𝑦) · (log‘𝑦)))
103	87, 102	absidd 15396	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((ψ‘𝑦) · (log‘𝑦))) = ((ψ‘𝑦) · (log‘𝑦)))
104	98, 103	oveq12d 7408	. . . . . . . 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 5136	. . . . . . 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 13945	. . . . . . . . 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 13000	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
110	109	relogcld 26539	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
111	108, 110	remulcld 11211	. . . . . . . . 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 12208	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
114	107	nnge1d 12241	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑚)
115	113, 114	logge0d 26546	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑚))
116	108, 110, 112, 115	mulge0d 11762	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
117		flword2 13782	. . . . . . . . . . 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 13532	. . . . . . . . . 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 15769	. . . . . . . 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 27074	. . . . . . . . . 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 26543	. . . . . . . . . 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 12132	. . . . . . . 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 11804	. . . . . . 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 11338	. . . . . 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 11338	. . . . 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 5132	. . . 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 15515	. . 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 27475	. . 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 3127	. 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 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917   class class class wbr 5110   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080  +∞cpnf 11212   < clt 11215   ≤ cle 11216   − cmin 11412   / cdiv 11842  ℕcn 12193  2c2 12248  ℤ_≥cuz 12800  ℝ⁺crp 12958  (,)cioo 13313  [,)cico 13315  ...cfz 13475  ⌊cfl 13759  abscabs 15207  𝑂(1)co1 15459  Σcsu 15659  logclog 26470  Λcvma 27009  ψcchp 27010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-ioc 13318  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-mod 13839  df-seq 13974  df-exp 14034  df-fac 14246  df-bc 14275  df-hash 14303  df-shft 15040  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-limsup 15444  df-clim 15461  df-rlim 15462  df-o1 15463  df-lo1 15464  df-sum 15660  df-ef 16040  df-e 16041  df-sin 16042  df-cos 16043  df-pi 16045  df-dvds 16230  df-gcd 16472  df-prm 16649  df-pc 16815  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-rest 17392  df-topn 17393  df-0g 17411  df-gsum 17412  df-topgen 17413  df-pt 17414  df-prds 17417  df-xrs 17472  df-qtop 17477  df-imas 17478  df-xps 17480  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-mulg 19007  df-cntz 19256  df-cmn 19719  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-fbas 21268  df-fg 21269  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cld 22913  df-ntr 22914  df-cls 22915  df-nei 22992  df-lp 23030  df-perf 23031  df-cn 23121  df-cnp 23122  df-haus 23209  df-cmp 23281  df-tx 23456  df-hmeo 23649  df-fil 23740  df-fm 23832  df-flim 23833  df-flf 23834  df-xms 24215  df-ms 24216  df-tms 24217  df-cncf 24778  df-limc 25774  df-dv 25775  df-log 26472  df-cxp 26473  df-cht 27014  df-vma 27015  df-chp 27016  df-ppi 27017

This theorem is referenced by:  selberg3  27477  selberg4  27479