Theorem selberg3lem2 27469

Description: Lemma for selberg3 27470. 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 11174	. . . . . . . 8 ⊢ 1 ∈ ℝ
2		elicopnf 13406	. . . . . . . 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 3950	. . . . 5 ⊢ (1[,)+∞) ⊆ ℝ
6	5	a1i 11	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
8		fzfid 13938	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (1...(⌊‘𝑦)) ∈ Fin)
9		elfznn 13514	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘𝑦)) → 𝑚 ∈ ℕ)
10	9	adantl 481	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℕ)
11		vmacl 27028	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑚) ∈ ℝ)
13	10	nnrpd 12993	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ+)
14	13	relogcld 26532	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (log‘𝑚) ∈ ℝ)
15	12, 14	remulcld 11204	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
16	8, 15	fsumrecl 15700	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
17	4	adantl 481	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ)
18		chpcl 27034	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (ψ‘𝑦) ∈ ℝ)
20		1rp 12955	. . . . . . . . . . 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 13034	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ+)
25	24	relogcld 26532	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (log‘𝑦) ∈ ℝ)
26	19, 25	remulcld 11204	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
27	16, 26	resubcld 11606	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
28	27, 24	rerpdivcld 13026	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℝ)
29	28	recnd 11202	. . . 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 3952	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
32		selberg2lem 27461	. . . . . 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 15529	. . . 4 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1))
35		fzfid 13938	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
36		elfznn 13514	. . . . . . . . 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 12993	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
40	39	relogcld 26532	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
41	38, 40	remulcld 11204	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
42	35, 41	fsumrecl 15700	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
43		chpcl 27034	. . . . . . 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 13034	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
49	48	relogcld 26532	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
50	44, 49	remulcld 11204	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
51	42, 50	readdcld 11203	. . . 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 11202	. . . . . . 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 12997	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℂ)
56	54	rpne0d 13000	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≠ 0)
57	53, 55, 56	absdivd 15424	. . . . . 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 12999	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ 𝑦)
60	58, 59	absidd 15389	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘𝑦) = 𝑦)
61	60	oveq2d 7403	. . . . . 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 2764	. . . . 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 15405	. . . . . . 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 13026	. . . . . 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 11322	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≤ 𝑥)
70	66, 54, 69	rpgecld 13034	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ+)
71	70	relogcld 26532	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑥) ∈ ℝ)
72	67, 71	remulcld 11204	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
73	65, 72	readdcld 11203	. . . . . 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 15413	. . . . . . . 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 13017	. . . . . . 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 11202	. . . . . . . 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 11950	. . . . . . 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 5133	. . . . . 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 26532	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ∈ ℝ)
84	82, 83	remulcld 11204	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
85	81, 84	readdcld 11203	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
86	81	recnd 11202	. . . . . . . . 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 11202	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℂ)
89	86, 88	abs2dif2d 15427	. . . . . . . 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 27031	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
91	10, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑚))
92	10	nnred 12201	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ)
93	10	nnge1d 12234	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑚)
94	92, 93	logge0d 26539	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (log‘𝑚))
95	12, 14, 91, 94	mulge0d 11755	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
96	8, 15, 95	fsumge0 15761	. . . . . . . . . . 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 15389	. . . . . . . . 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 27036	. . . . . . . . . . . 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 26539	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (log‘𝑦))
102	82, 83, 100, 101	mulge0d 11755	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ ((ψ‘𝑦) · (log‘𝑦)))
103	87, 102	absidd 15389	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((ψ‘𝑦) · (log‘𝑦))) = ((ψ‘𝑦) · (log‘𝑦)))
104	98, 103	oveq12d 7405	. . . . . . . 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 5133	. . . . . . 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 13938	. . . . . . . . 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 12993	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
110	109	relogcld 26532	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
111	108, 110	remulcld 11204	. . . . . . . . 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 12201	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
114	107	nnge1d 12234	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑚)
115	113, 114	logge0d 26539	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑚))
116	108, 110, 112, 115	mulge0d 11755	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
117		flword2 13775	. . . . . . . . . . 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 13525	. . . . . . . . . 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 15762	. . . . . . . 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 27067	. . . . . . . . . 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 26536	. . . . . . . . . 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 12125	. . . . . . . 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 11797	. . . . . . 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 11331	. . . . . 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 11331	. . . . 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 5129	. . . 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 15508	. . 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 27468	. . 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 3126	. 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 3044  ∃wrex 3053   ⊆ wss 3914   class class class wbr 5107   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073  +∞cpnf 11205   < clt 11208   ≤ cle 11209   − cmin 11405   / cdiv 11835  ℕcn 12186  2c2 12241  ℤ_≥cuz 12793  ℝ⁺crp 12951  (,)cioo 13306  [,)cico 13308  ...cfz 13468  ⌊cfl 13752  abscabs 15200  𝑂(1)co1 15452  Σcsu 15652  logclog 26463  Λcvma 27002  ψcchp 27003

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-fi 9362  df-sup 9393  df-inf 9394  df-oi 9463  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-xnn0 12516  df-z 12530  df-dec 12650  df-uz 12794  df-q 12908  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-ioo 13310  df-ioc 13311  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-seq 13967  df-exp 14027  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15033  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-limsup 15437  df-clim 15454  df-rlim 15455  df-o1 15456  df-lo1 15457  df-sum 15653  df-ef 16033  df-e 16034  df-sin 16035  df-cos 16036  df-pi 16038  df-dvds 16223  df-gcd 16465  df-prm 16642  df-pc 16808  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-xrs 17465  df-qtop 17470  df-imas 17471  df-xps 17473  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-mulg 19000  df-cntz 19249  df-cmn 19712  df-psmet 21256  df-xmet 21257  df-met 21258  df-bl 21259  df-mopn 21260  df-fbas 21261  df-fg 21262  df-cnfld 21265  df-top 22781  df-topon 22798  df-topsp 22820  df-bases 22833  df-cld 22906  df-ntr 22907  df-cls 22908  df-nei 22985  df-lp 23023  df-perf 23024  df-cn 23114  df-cnp 23115  df-haus 23202  df-cmp 23274  df-tx 23449  df-hmeo 23642  df-fil 23733  df-fm 23825  df-flim 23826  df-flf 23827  df-xms 24208  df-ms 24209  df-tms 24210  df-cncf 24771  df-limc 25767  df-dv 25768  df-log 26465  df-cxp 26466  df-cht 27007  df-vma 27008  df-chp 27009  df-ppi 27010

This theorem is referenced by:  selberg3  27470  selberg4  27472