Theorem selberg3lem2 27878

Description: Lemma for selberg3 27879. 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 11301	. . . . . . . 8 ⊢ 1 ∈ ℝ
2		elicopnf 13569	. . . . . . . 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 502	. . . . . 6 ⊢ (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ)
5	4	ssriv 3935	. . . . 5 ⊢ (1[,)+∞) ⊆ ℝ
6	5	a1i 11	. . . 4 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ)
7	1	a1i 11	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
8		fzfid 14109	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (1...(⌊‘𝑦)) ∈ Fin)
9		elfznn 13680	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘𝑦)) → 𝑚 ∈ ℕ)
10	9	adantl 487	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℕ)
11		vmacl 27438	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑚) ∈ ℝ)
13	10	nnrpd 13155	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ+)
14	13	relogcld 26944	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → (log‘𝑚) ∈ ℝ)
15	12, 14	remulcld 11332	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
16	8, 15	fsumrecl 15893	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
17	4	adantl 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ)
18		chpcl 27444	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
19	17, 18	syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (ψ‘𝑦) ∈ ℝ)
20		1rp 13117	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ+
21	20	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
22	3	simprbi 503	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (1[,)+∞) → 1 ≤ 𝑦)
23	22	adantl 487	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 1 ≤ 𝑦)
24	17, 21, 23	rpgecld 13196	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 𝑦 ∈ ℝ+)
25	24	relogcld 26944	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (log‘𝑦) ∈ ℝ)
26	19, 25	remulcld 11332	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
27	16, 26	resubcld 11737	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
28	27, 24	rerpdivcld 13188	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℝ)
29	28	recnd 11330	. . . 4 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) ∈ ℂ)
30	24	ex 418	. . . . . 6 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) → 𝑦 ∈ ℝ+))
31	30	ssrdv 3937	. . . . 5 ⊢ (⊤ → (1[,)+∞) ⊆ ℝ+)
32		selberg2lem 27870	. . . . . 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 15723	. . . 4 ⊢ (⊤ → (𝑦 ∈ (1[,)+∞) ↦ ((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ∈ 𝑂(1))
35		fzfid 14109	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
36		elfznn 13680	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
37	36	adantl 487	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
38	37, 11	syl 18	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
39	37	nnrpd 13155	. . . . . . . 8 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
40	39	relogcld 26944	. . . . . . 7 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
41	38, 40	remulcld 11332	. . . . . 6 ⊢ (((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
42	35, 41	fsumrecl 15893	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
43		chpcl 27444	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
44	43	ad2antrl 741	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (ψ‘𝑥) ∈ ℝ)
45		simprl 783	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
46	20	a1i 11	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ+)
47		simprr 785	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
48	45, 46, 47	rpgecld 13196	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
49	48	relogcld 26944	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
50	44, 49	remulcld 11332	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
51	42, 50	readdcld 11331	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑥) · (log‘𝑥))) ∈ ℝ)
52	27	adantr 486	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
53	52	recnd 11330	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) ∈ ℂ)
54	24	adantr 486	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ+)
55	54	rpcnd 13159	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℂ)
56	54	rpne0d 13162	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≠ 0)
57	53, 55, 56	absdivd 15618	. . . . . 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 486	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ∈ ℝ)
59	54	rpge0d 13161	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ 𝑦)
60	58, 59	absidd 15583	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘𝑦) = 𝑦)
61	60	oveq2d 7434	. . . . . 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 2796	. . . . 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 15599	. . . . . . 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 13188	. . . . . 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 760	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
66		simprll 791	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ)
67	66, 43	syl 18	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑥) ∈ ℝ)
68		simprr 785	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 < 𝑥)
69	58, 66, 68	ltled 11451	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑦 ≤ 𝑥)
70	66, 54, 69	rpgecld 13196	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 𝑥 ∈ ℝ+)
71	70	relogcld 26944	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑥) ∈ ℝ)
72	67, 71	remulcld 11332	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
73	65, 72	readdcld 11331	. . . . . 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 15607	. . . . . . . 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 486	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 1 ≤ 𝑦)
77	74, 54, 63, 75, 76	lediv2ad 13179	. . . . . . 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 11330	. . . . . . . 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 12078	. . . . . . 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 5131	. . . . . 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 486	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
82	58, 18	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ∈ ℝ)
83	54	relogcld 26944	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ∈ ℝ)
84	82, 83	remulcld 11332	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
85	81, 84	readdcld 11331	. . . . . . 7 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘𝑦) · (log‘𝑦))) ∈ ℝ)
86	81	recnd 11330	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
87	26	adantr 486	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
88	87	recnd 11330	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℂ)
89	86, 88	abs2dif2d 15621	. . . . . . . 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 27441	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
91	10, 90	syl 18	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑚))
92	10	nnred 12343	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 𝑚 ∈ ℝ)
93	10	nnge1d 12379	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 1 ≤ 𝑚)
94	92, 93	logge0d 26951	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ (log‘𝑚))
95	12, 14, 91, 94	mulge0d 11886	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
96	8, 15, 95	fsumge0 15955	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,)+∞)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
97	96	adantr 486	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)))
98	81, 97	absidd 15583	. . . . . . . . 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 27446	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 0 ≤ (ψ‘𝑦))
100	58, 99	syl 18	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (ψ‘𝑦))
101	58, 76	logge0d 26951	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ (log‘𝑦))
102	82, 83, 100, 101	mulge0d 11886	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → 0 ≤ ((ψ‘𝑦) · (log‘𝑦)))
103	87, 102	absidd 15583	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (abs‘((ψ‘𝑦) · (log‘𝑦))) = ((ψ‘𝑦) · (log‘𝑦)))
104	98, 103	oveq12d 7436	. . . . . . . 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 5131	. . . . . . 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 14109	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
107	36	adantl 487	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
108	107, 11	syl 18	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
109	107	nnrpd 13155	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
110	109	relogcld 26944	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
111	108, 110	remulcld 11332	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
112	107, 90	syl 18	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑚))
113	107	nnred 12343	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ)
114	107	nnge1d 12379	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑚)
115	113, 114	logge0d 26951	. . . . . . . . . 10 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑚))
116	108, 110, 112, 115	mulge0d 11886	. . . . . . . . 9 ⊢ ((((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑚) · (log‘𝑚)))
117		flword2 13946	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)))
118	58, 66, 69, 117	syl3anc 1398	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)))
119		fzss2 13691	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ (ℤ≥‘(⌊‘𝑦)) → (1...(⌊‘𝑦)) ⊆ (1...(⌊‘𝑥)))
120	118, 119	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (1...(⌊‘𝑦)) ⊆ (1...(⌊‘𝑥)))
121	106, 111, 116, 120	fsumless 15956	. . . . . . . 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 27477	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ≤ 𝑥) → (ψ‘𝑦) ≤ (ψ‘𝑥))
123	58, 66, 69, 122	syl3anc 1398	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (ψ‘𝑦) ≤ (ψ‘𝑥))
124	54, 70	logled 26948	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (𝑦 ≤ 𝑥 ↔ (log‘𝑦) ≤ (log‘𝑥)))
125	69, 124	mpbid 235	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑦 ∈ (1[,)+∞)) ∧ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ 𝑦 < 𝑥)) → (log‘𝑦) ≤ (log‘𝑥))
126	82, 67, 83, 71, 100, 101, 123, 125	lemul12ad 12252	. . . . . . . 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 11928	. . . . . . 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 11460	. . . . . 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 11460	. . . . 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 5127	. . . 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 15702	. . 3 ⊢ (⊤ → ∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐)
132	131	mptru 1577	. 2 ⊢ ∃𝑐 ∈ ℝ+ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐
133		simpl 488	. . . 4 ⊢ ((𝑐 ∈ ℝ+ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝑐) → 𝑐 ∈ ℝ+)
134		simpr 490	. . . 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 27877	. . 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 3156	. 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)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ⊤wtru 1571   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6537  (class class class)co 7418  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   · cmul 11198  +∞cpnf 11333   < clt 11336   ≤ cle 11337   − cmin 11534   / cdiv 11966  ℕcn 12328  2c2 12390  ℤ_≥cuz 12958  ℝ⁺crp 13113  (,)cioo 13469  [,)cico 13471  ...cfz 13632  ⌊cfl 13923  abscabs 15394  𝑂(1)co1 15646  Σcsu 15846  logclog 26875  Λcvma 27412  ψcchp 27413

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271  ax-addf 11272

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-fi 9396  df-sup 9427  df-inf 9428  df-oi 9497  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-xnn0 12673  df-z 12687  df-dec 12808  df-uz 12959  df-q 13069  df-rp 13114  df-xneg 13234  df-xadd 13235  df-xmul 13236  df-ioo 13473  df-ioc 13474  df-ico 13475  df-icc 13476  df-fz 13633  df-fzo 13782  df-fl 13925  df-mod 14003  df-seq 14138  df-exp 14198  df-fac 14411  df-bc 14440  df-hash 14468  df-shft 15213  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-limsup 15631  df-clim 15648  df-rlim 15649  df-o1 15650  df-lo1 15651  df-sum 15847  df-ef 16226  df-e 16227  df-sin 16228  df-cos 16229  df-pi 16231  df-dvds 16416  df-gcd 16658  df-prm 16840  df-pc 17008  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-starv 17436  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-unif 17444  df-hom 17445  df-cco 17446  df-rest 17586  df-topn 17587  df-0g 17605  df-gsum 17606  df-topgen 17607  df-pt 17608  df-prds 17611  df-xrs 17667  df-qtop 17672  df-imas 17673  df-xps 17675  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-submnd 18972  df-mulg 19271  df-cntz 19524  df-cmn 19989  df-psmet 21663  df-xmet 21664  df-met 21665  df-bl 21666  df-mopn 21667  df-fbas 21668  df-fg 21669  df-cnfld 21672  df-top 23205  df-topon 23222  df-topsp 23244  df-bases 23257  df-cld 23330  df-ntr 23331  df-cls 23332  df-nei 23409  df-lp 23447  df-perf 23448  df-cn 23538  df-cnp 23539  df-haus 23626  df-cmp 23698  df-tx 23874  df-hmeo 24067  df-fil 24158  df-fm 24250  df-flim 24251  df-flf 24252  df-xms 24632  df-ms 24633  df-tms 24634  df-cncf 25192  df-limc 26179  df-dv 26180  df-log 26877  df-cxp 26878  df-cht 27417  df-vma 27418  df-chp 27419  df-ppi 27420

This theorem is used by:  selberg3  27879  selberg4  27881