Theorem selberg3lem2 26134

Description: Lemma for selberg3 26135. 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 10641	. . . . . . . 8 ⊢ 1 ∈ ℝ
2		elicopnf 12834	. . . . . . . 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 500	. . . . . 6 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ)
5	4	ssriv 3971	. . . . 5 ⊢ (1[,)+∞) ⊆ ℝ
6	5	a1i 11	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
8		fzfid 13342	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (1...(⌊‘𝑦)) ∈ Fin)
9		elfznn 12937	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘𝑦)) → 𝑚 ∈ ℕ)
10	9	adantl 484	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℕ)
11		vmacl 25695	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑚) ∈ ℝ)
13	10	nnrpd 12430	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ+)
14	13	relogcld 25206	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (log‘𝑚) ∈ ℝ)
15	12, 14	remulcld 10671	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
16	8, 15	fsumrecl 15091	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
17	4	adantl 484	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ)
18		chpcl 25701	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
19	17, 18	syl 17	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (ψ‘𝑦) ∈ ℝ)
20		1rp 12394	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
21	20	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
22	3	simprbi 499	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦)
23	22	adantl 484	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦)
24	17, 21, 23	rpgecld 12471	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ+)
25	24	relogcld 25206	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (log‘𝑦) ∈ ℝ)
26	19, 25	remulcld 10671	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
27	16, 26	resubcld 11068	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
28	27, 24	rerpdivcld 12463	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℝ)
29	28	recnd 10669	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℂ)
30	24	ex 415	. . . . . 6 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ+))
31	30	ssrdv 3973	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
32		selberg2lem 26126	. . . . . 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 14920	. . . 4 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1))
35		fzfid 13342	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
36		elfznn 12937	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
37	36	adantl 484	. . . . . . . 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 12430	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
40	39	relogcld 25206	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
41	38, 40	remulcld 10671	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
42	35, 41	fsumrecl 15091	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
43		chpcl 25701	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
44	43	ad2antrl 726	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (ψ‘𝑥) ∈ ℝ)
45		simprl 769	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
46	20	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ+)
47		simprr 771	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
48	45, 46, 47	rpgecld 12471	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
49	48	relogcld 25206	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
50	44, 49	remulcld 10671	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
51	42, 50	readdcld 10670	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))) ∈ ℝ)
52	27	adantr 483	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
53	52	recnd 10669	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℂ)
54	24	adantr 483	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ+)
55	54	rpcnd 12434	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℂ)
56	54	rpne0d 12437	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≠ 0)
57	53, 55, 56	absdivd 14815	. . . . . 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 483	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ)
59	54	rpge0d 12436	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ 𝑦)
60	58, 59	absidd 14782	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘𝑦) = 𝑦)
61	60	oveq2d 7172	. . . . . 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 2856	. . . . 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 14796	. . . . . . 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 12463	. . . . . 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 745	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
66		simprll 777	. . . . . . . . 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 771	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 < 𝑥)
69	58, 66, 68	ltled 10788	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≤ 𝑥)
70	66, 54, 69	rpgecld 12471	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ+)
71	70	relogcld 25206	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑥) ∈ ℝ)
72	67, 71	remulcld 10671	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
73	65, 72	readdcld 10670	. . . . . 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 14804	. . . . . . . 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 483	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 1 ≤ 𝑦)
77	74, 54, 63, 75, 76	lediv2ad 12454	. . . . . . 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 10669	. . . . . . . 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 11408	. . . . . . 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 5092	. . . . . 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 483	. . . . . . . 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 25206	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ∈ ℝ)
84	82, 83	remulcld 10671	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
85	81, 84	readdcld 10670	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
86	81	recnd 10669	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
87	26	adantr 483	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
88	87	recnd 10669	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℂ)
89	86, 88	abs2dif2d 14818	. . . . . . . 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 25698	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
91	10, 90	syl 17	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑚))
92	10	nnred 11653	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ)
93	10	nnge1d 11686	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑚)
94	92, 93	logge0d 25213	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (log‘𝑚))
95	12, 14, 91, 94	mulge0d 11217	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
96	8, 15, 95	fsumge0 15150	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
97	96	adantr 483	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
98	81, 97	absidd 14782	. . . . . . . . 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 25703	. . . . . . . . . . . 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 25213	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (log‘𝑦))
102	82, 83, 100, 101	mulge0d 11217	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ ((ψ‘𝑦) · (log‘𝑦)))
103	87, 102	absidd 14782	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((ψ‘𝑦) · (log‘𝑦))) = ((ψ‘𝑦) · (log‘𝑦)))
104	98, 103	oveq12d 7174	. . . . . . . 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 5092	. . . . . . 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 13342	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
107	36	adantl 484	. . . . . . . . . . 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 12430	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
110	109	relogcld 25206	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
111	108, 110	remulcld 10671	. . . . . . . . 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 11653	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
114	107	nnge1d 11686	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑚)
115	113, 114	logge0d 25213	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑚))
116	108, 110, 112, 115	mulge0d 11217	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
117		flword2 13184	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)))
118	58, 66, 69, 117	syl3anc 1367	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)))
119		fzss2 12948	. . . . . . . . . 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 15151	. . . . . . . 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 25734	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (ψ‘𝑦) ≤ (ψ‘𝑥))
123	58, 66, 69, 122	syl3anc 1367	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ≤ (ψ‘𝑥))
124	54, 70	logled 25210	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (𝑦 ≤ 𝑥 ↔ (log‘𝑦) ≤ (log‘𝑥)))
125	69, 124	mpbid 234	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ≤ (log‘𝑥))
126	82, 67, 83, 71, 100, 101, 123, 125	lemul12ad 11582	. . . . . . . 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 11259	. . . . . . 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 10797	. . . . . 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 10797	. . . . 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 5088	. . . 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 14899	. . 3 ⊢ (⊤ → ∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐)
132	131	mptru 1544	. 2 ⊢ ∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐
133		simpl 485	. . . 4 ⊢ ((𝑐 ∈ ℝ+ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → 𝑐 ∈ ℝ+)
134		simpr 487	. . . 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 26133	. . 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 3281	. 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 208   ∧ wa 398  ⊤wtru 1538   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936   class class class wbr 5066   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156  ℝcr 10536  0cc0 10537  1c1 10538   + caddc 10540   · cmul 10542  +∞cpnf 10672   < clt 10675   ≤ cle 10676   − cmin 10870   / cdiv 11297  ℕcn 11638  2c2 11693  ℤ_≥cuz 12244  ℝ⁺crp 12390  (,)cioo 12739  [,)cico 12741  ...cfz 12893  ⌊cfl 13161  abscabs 14593  𝑂(1)co1 14843  Σcsu 15042  logclog 25138  Λcvma 25669  ψcchp 25670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-xnn0 11969  df-z 11983  df-dec 12100  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ioo 12743  df-ioc 12744  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-fl 13163  df-mod 13239  df-seq 13371  df-exp 13431  df-fac 13635  df-bc 13664  df-hash 13692  df-shft 14426  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-limsup 14828  df-clim 14845  df-rlim 14846  df-o1 14847  df-lo1 14848  df-sum 15043  df-ef 15421  df-e 15422  df-sin 15423  df-cos 15424  df-pi 15426  df-dvds 15608  df-gcd 15844  df-prm 16016  df-pc 16174  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-pt 16718  df-prds 16721  df-xrs 16775  df-qtop 16780  df-imas 16781  df-xps 16783  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-mulg 18225  df-cntz 18447  df-cmn 18908  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-fbas 20542  df-fg 20543  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-lp 21744  df-perf 21745  df-cn 21835  df-cnp 21836  df-haus 21923  df-cmp 21995  df-tx 22170  df-hmeo 22363  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-xms 22930  df-ms 22931  df-tms 22932  df-cncf 23486  df-limc 24464  df-dv 24465  df-log 25140  df-cxp 25141  df-cht 25674  df-vma 25675  df-chp 25676  df-ppi 25677

This theorem is referenced by:  selberg3  26135  selberg4  26137