Theorem 2vmadivsumlem 27779

Description: Lemma for 2vmadivsum 27780. (Contributed by Mario Carneiro, 30-May-2016.)

Hypotheses

Ref	Expression
2vmadivsum.1	⊢ (𝜑 → 𝐴 ∈ ℝ+)
2vmadivsum.2	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝐴)

Assertion

Ref	Expression
2vmadivsumlem	⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))

Distinct variable groups:   𝑖,𝑚,𝑛,𝑥,𝑦,𝐴   𝜑,𝑚,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑦, 𝑖)

Proof of Theorem 2vmadivsumlem

Step	Hyp	Ref	Expression
1		vmalogdivsum2 27777	. . 3 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
3		fzfid 14040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
4		elfznn 13611	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
5	4	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
6		vmacl 27357	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
8	7, 5	nndivred 12317	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
9		fzfid 14040	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
10		elfznn 13611	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
11	10	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
12		vmacl 27357	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
13	11, 12	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
14	13, 11	nndivred 12317	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) / 𝑚) ∈ ℝ)
15	9, 14	fsumrecl 15823	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) ∈ ℝ)
16	8, 15	remulcld 11266	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℝ)
17	3, 16	fsumrecl 15823	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℝ)
18		elioore 13431	. . . . . . . 8 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
19	18	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
20		eliooord 13461	. . . . . . . . 9 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
21	20	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
22	21	simpld 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
23	19, 22	rplogcld 26869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
24	17, 23	rerpdivcld 13120	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) ∈ ℝ)
25		1rp 13049	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
26	25	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
27		1red 11236	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
28	27, 19, 22	ltled 11385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
29	19, 26, 28	rpgecld 13128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
30	29	relogcld 26863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
31	30	rehalfcld 12518	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ)
32	24, 31	resubcld 11669	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ)
33	32	recnd 11264	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
34	29	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
35	5	nnrpd 13087	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
36	34, 35	rpdivcld 13106	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
37	36	relogcld 26863	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
38	8, 37	remulcld 11266	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
39	3, 38	fsumrecl 15823	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
40	39, 23	rerpdivcld 13120	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℝ)
41	40, 31	resubcld 11669	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ)
42	41	recnd 11264	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
43	17	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℂ)
44	39	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
45	30	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
46	23	rpne0d 13094	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
47	43, 44, 45, 46	divsubdird 12057	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
48	8	recnd 11264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
49	15	recnd 11264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) ∈ ℂ)
50	37	recnd 11264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
51	48, 49, 50	subdid 11697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
52	51	sumeq2dv 15792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
53	16	recnd 11264	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) ∈ ℂ)
54	38	recnd 11264	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
55	3, 53, 54	fsumsub 15877	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
56	52, 55	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
57	56	oveq1d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
58	24	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) ∈ ℂ)
59	40	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
60	31	recnd 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
61	58, 59, 60	nnncan2d 11631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
62	47, 57, 61	3eqtr4d 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))))
63	62	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))))
64		1red 11236	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
65	3, 8	fsumrecl 15823	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
66	65, 23	rerpdivcld 13120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ)
67		2vmadivsum.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
68	67	rpred 13089	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
69	68	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
70		ioossre 13463	. . . . . . . 8 ⊢ (1(,)+∞) ⊆ ℝ
71		1cnd 11229	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℂ)
72		o1const 15710	. . . . . . . 8 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
73	70, 71, 72	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
74	66	recnd 11264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ)
75		1cnd 11229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
76	65	recnd 11264	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
77	76, 45, 45, 46	divsubdird 12057	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))))
78	76, 45	subcld 11596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
79	78, 45, 46	divrecd 12021	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
80	45, 46	dividd 12016	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
81	80	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))
82	77, 79, 81	3eqtr3d 2803	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))
83	82	mpteq2dva 5198	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)))
84	65, 30	resubcld 11669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
85	27, 23	rerpdivcld 13120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
86	29	ex 418	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
87	86	ssrdv 3937	. . . . . . . . . . 11 ⊢ (𝜑 → (1(,)+∞) ⊆ ℝ+)
88		vmadivsum 27721	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
89	88	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
90	87, 89	o1res2 15653	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
91		divlogrlim 26875	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
92		rlimo1 15707	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
93	91, 92	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
94	84, 85, 90, 93	o1mul2 15715	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
95	83, 94	eqeltrrd 2861	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈ 𝑂(1))
96	74, 75, 95	o1dif 15720	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)))
97	73, 96	mpbird 260	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1))
98	68	recnd 11264	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
99		o1const 15710	. . . . . . 7 ⊢ (((1(,)+∞) ⊆ ℝ ∧ 𝐴 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
100	70, 98, 99	sylancr 599	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
101	66, 69, 97, 100	o1mul2 15715	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴)) ∈ 𝑂(1))
102	66, 69	remulcld 11266	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴) ∈ ℝ)
103	15, 37	resubcld 11669	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
104	8, 103	remulcld 11266	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
105	3, 104	fsumrecl 15823	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
106	105	recnd 11264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
107	106, 45, 46	divcld 12018	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ∈ ℂ)
108	106	abscld 15529	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ ℝ)
109	65, 69	remulcld 11266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) ∈ ℝ)
110	104	recnd 11264	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℂ)
111	110	abscld 15529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ ℝ)
112	3, 111	fsumrecl 15823	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ∈ ℝ)
113	3, 110	fsumabs 15891	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))))
114	69	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
115	8, 114	remulcld 11266	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝐴) ∈ ℝ)
116	103	recnd 11264	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℂ)
117	48, 116	absmuld 15547	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) = ((abs‘((Λ‘𝑛) / 𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))))
118		vmage0 27360	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
119	5, 118	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
120	7, 35, 119	divge0d 13129	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
121	8, 120	absidd 15513	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) / 𝑛)) = ((Λ‘𝑛) / 𝑛))
122	121	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((Λ‘𝑛) / 𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) = (((Λ‘𝑛) / 𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))))
123	117, 122	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) = (((Λ‘𝑛) / 𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))))
124	116	abscld 15529	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
125		fveq2 6879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑚 → (Λ‘𝑖) = (Λ‘𝑚))
126		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑚 → 𝑖 = 𝑚)
127	125, 126	oveq12d 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → ((Λ‘𝑖) / 𝑖) = ((Λ‘𝑚) / 𝑚))
128	127	cbvsumv 15786	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) / 𝑚)
129		fveq2 6879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 / 𝑛) → (⌊‘𝑦) = (⌊‘(𝑥 / 𝑛)))
130	129	oveq2d 7430	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 / 𝑛) → (1...(⌊‘𝑦)) = (1...(⌊‘(𝑥 / 𝑛))))
131	130	sumeq1d 15790	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑥 / 𝑛) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) / 𝑚) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))
132	128, 131	eqtrid 2807	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚))
133		fveq2 6879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝑥 / 𝑛) → (log‘𝑦) = (log‘(𝑥 / 𝑛)))
134	132, 133	oveq12d 7432	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (𝑥 / 𝑛) → (Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦)) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))
135	134	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑛) → (abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) = (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))))
136	135	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑛) → ((abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝐴 ↔ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ 𝐴))
137		2vmadivsum.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝐴)
138	137	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝐴)
139	36	rpred 13089	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
140	5	nncnd 12276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
141	140	mullidd 11254	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
142		fznnfl 13926	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
143	19, 142	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
144	143	simplbda 505	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
145	141, 144	eqbrtrd 5127	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
146		1red 11236	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
147	19	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
148	146, 147, 35	lemuldivd 13138	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
149	145, 148	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
150		1re 11235	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
151		elicopnf 13501	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℝ → ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛))))
152	150, 151	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)))
153	139, 149, 152	sylanbrc 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ (1[,)+∞))
154	136, 138, 153	rspcdva 3577	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ 𝐴)
155	124, 114, 8, 120, 154	lemul2ad 12182	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ≤ (((Λ‘𝑛) / 𝑛) · 𝐴))
156	123, 155	eqbrtrd 5127	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ≤ (((Λ‘𝑛) / 𝑛) · 𝐴))
157	3, 111, 115, 156	fsumle 15889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝐴))
158	98	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℂ)
159	3, 158, 48	fsummulc1 15874	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝐴))
160	157, 159	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴))
161	108, 112, 109, 113, 160	letrd 11394	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴))
162	108, 109, 23, 161	lediv1dd 13147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) / (log‘𝑥)) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)))
163	106, 45, 46	absdivd 15548	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) / (abs‘(log‘𝑥))))
164	23	rpge0d 13093	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (log‘𝑥))
165	30, 164	absidd 15513	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(log‘𝑥)) = (log‘𝑥))
166	165	oveq2d 7430	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) / (abs‘(log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) / (log‘𝑥)))
167	163, 166	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛))))) / (log‘𝑥)))
168	3, 8, 120	fsumge0 15885	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
169	65, 23, 168	divge0d 13129	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))
170	67	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ+)
171	170	rpge0d 13093	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴)
172	66, 69, 169, 171	mulge0d 11818	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴))
173	102, 172	absidd 15513	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴))
174	76, 158, 45, 46	div23d 12055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴))
175	173, 174	eqtr4d 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)))
176	162, 167, 175	3brtr4d 5137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴)))
177	176	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) · 𝐴)))
178	64, 101, 102, 107, 177	o1le 15743	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1))
179	63, 178	eqeltrrd 2861	. . 3 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
180	33, 42, 179	o1dif 15720	. 2 ⊢ (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)))
181	2, 180	mpbird 260	1 ⊢ (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7414  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   · cmul 11132  +∞cpnf 11267   < clt 11270   ≤ cle 11271   − cmin 11468   / cdiv 11898  ℕcn 12260  2c2 12322  ℝ⁺crp 13045  (,)cioo 13401  [,)cico 13403  ...cfz 13564  ⌊cfl 13854  abscabs 15324   ⇝_𝑟 crli 15575  𝑂(1)co1 15576  Σcsu 15776  logclog 26794  Λcvma 27331

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 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205  ax-addf 11206

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  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 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-fi 9384  df-sup 9415  df-inf 9416  df-oi 9485  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-xnn0 12605  df-z 12619  df-dec 12740  df-uz 12891  df-q 13001  df-rp 13046  df-xneg 13166  df-xadd 13167  df-xmul 13168  df-ioo 13405  df-ioc 13406  df-ico 13407  df-icc 13408  df-fz 13565  df-fzo 13713  df-fl 13856  df-mod 13934  df-seq 14069  df-exp 14129  df-fac 14341  df-bc 14370  df-hash 14398  df-shft 15143  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-limsup 15561  df-clim 15578  df-rlim 15579  df-o1 15580  df-lo1 15581  df-sum 15777  df-ef 16156  df-e 16157  df-sin 16158  df-cos 16159  df-tan 16160  df-pi 16161  df-dvds 16346  df-gcd 16588  df-prm 16765  df-pc 16932  df-struct 17242  df-sets 17259  df-slot 17277  df-ndx 17289  df-base 17305  df-ress 17326  df-plusg 17358  df-mulr 17359  df-starv 17360  df-sca 17361  df-vsca 17362  df-ip 17363  df-tset 17364  df-ple 17365  df-ds 17367  df-unif 17368  df-hom 17369  df-cco 17370  df-rest 17510  df-topn 17511  df-0g 17529  df-gsum 17530  df-topgen 17531  df-pt 17532  df-prds 17535  df-xrs 17591  df-qtop 17596  df-imas 17597  df-xps 17599  df-mre 17673  df-mrc 17674  df-acs 17676  df-mgm 18733  df-sgrp 18824  df-mnd 18840  df-submnd 18895  df-mulg 19194  df-cntz 19447  df-cmn 19912  df-psmet 21580  df-xmet 21581  df-met 21582  df-bl 21583  df-mopn 21584  df-fbas 21585  df-fg 21586  df-cnfld 21589  df-top 23122  df-topon 23139  df-topsp 23161  df-bases 23174  df-cld 23247  df-ntr 23248  df-cls 23249  df-nei 23326  df-lp 23364  df-perf 23365  df-cn 23455  df-cnp 23456  df-haus 23543  df-cmp 23615  df-tx 23791  df-hmeo 23984  df-fil 24075  df-fm 24167  df-flim 24168  df-flf 24169  df-xms 24549  df-ms 24550  df-tms 24551  df-cncf 25109  df-limc 26096  df-dv 26097  df-ulm 26616  df-log 26796  df-cxp 26797  df-atan 27107  df-em 27232  df-cht 27336  df-vma 27337  df-chp 27338  df-ppi 27339

This theorem is used by:  2vmadivsum  27780