Theorem pntrlog2bndlem1 27636

Description: The sum of selberg3r 27628 and selberg4r 27629. (Contributed by Mario Carneiro, 31-May-2016.)

Hypotheses

Ref	Expression
pntsval.1	⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
pntrlog2bnd.r	⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))

Assertion

Ref	Expression
pntrlog2bndlem1	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1)

Distinct variable groups:   𝑖,𝑎,𝑛,𝑥   𝑆,𝑛,𝑥   𝑅,𝑛,𝑥

Allowed substitution hints:   𝑅(𝑖,𝑎)   𝑆(𝑖,𝑎)

Proof of Theorem pntrlog2bndlem1

Dummy variables 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 11260	. . 3 ⊢ (⊤ → 1 ∈ ℝ)
2		pntrlog2bnd.r	. . . . 5 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
3	2	selberg34r 27630	. . . 4 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
4		elioore 13414	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
5	4	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
6		1rp 13036	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ+
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
8		1red 11260	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
9		eliooord 13443	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (1(,)+∞) → (1 < 𝑥 ∧ 𝑥 < +∞))
10	9	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥 ∧ 𝑥 < +∞))
11	10	simpld 494	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
12	8, 5, 11	ltled 11407	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
13	5, 7, 12	rpgecld 13114	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
14	2	pntrf 27622	. . . . . . . . . . 11 ⊢ 𝑅:ℝ+⟶ℝ
15	14	ffvelcdmi 7103	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (𝑅‘𝑥) ∈ ℝ)
16	13, 15	syl 17	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℝ)
17	13	relogcld 26680	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
18	16, 17	remulcld 11289	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℝ)
19		fzfid 14011	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
20	13	adantr 480	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
21		elfznn 13590	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
22	21	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
23	22	nnrpd 13073	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
24	20, 23	rpdivcld 13092	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
25	14	ffvelcdmi 7103	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
27		fzfid 14011	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
28		dvdsssfz1 16352	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ (1...𝑛))
29	22, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ (1...𝑛))
30	27, 29	ssfid 9299	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ∈ Fin)
31		ssrab2 4090	. . . . . . . . . . . . . . . 16 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
32		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
33	31, 32	sselid 3993	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 𝑚 ∈ ℕ)
34		vmacl 27176	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
35	33, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (Λ‘𝑚) ∈ ℝ)
36		dvdsdivcl 16350	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
37	22, 36	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
38	31, 37	sselid 3993	. . . . . . . . . . . . . . 15 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑚) ∈ ℕ)
39		vmacl 27176	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 / 𝑚) ∈ ℕ → (Λ‘(𝑛 / 𝑚)) ∈ ℝ)
40	38, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (Λ‘(𝑛 / 𝑚)) ∈ ℝ)
41	35, 40	remulcld 11289	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) ∈ ℝ)
42	30, 41	fsumrecl 15767	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) ∈ ℝ)
43		vmacl 27176	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
44	22, 43	syl 17	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
45	23	relogcld 26680	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
46	44, 45	remulcld 11289	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℝ)
47	42, 46	resubcld 11689	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))) ∈ ℝ)
48	26, 47	remulcld 11289	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℝ)
49	19, 48	fsumrecl 15767	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℝ)
50	5, 11	rplogcld 26686	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
51	49, 50	rerpdivcld 13106	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)) ∈ ℝ)
52	18, 51	resubcld 11689	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) ∈ ℝ)
53	52, 13	rerpdivcld 13106	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥) ∈ ℝ)
54	53	recnd 11287	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥) ∈ ℂ)
55	54	lo1o12 15566	. . . 4 ⊢ (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ ((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥))) ∈ ≤𝑂(1)))
56	3, 55	mpbii 233	. . 3 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥))) ∈ ≤𝑂(1))
57	54	abscld 15472	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ ℝ)
58	16	recnd 11287	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅‘𝑥) ∈ ℂ)
59	58	abscld 15472	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅‘𝑥)) ∈ ℝ)
60	59, 17	remulcld 11289	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (log‘𝑥)) ∈ ℝ)
61	26	recnd 11287	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
62	61	abscld 15472	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
63	22	nnred 12279	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
64		pntsval.1	. . . . . . . . . . . 12 ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
65	64	pntsf 27632	. . . . . . . . . . 11 ⊢ 𝑆:ℝ⟶ℝ
66	65	ffvelcdmi 7103	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ → (𝑆‘𝑛) ∈ ℝ)
67	63, 66	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘𝑛) ∈ ℝ)
68		1red 11260	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
69	63, 68	resubcld 11689	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
70	65	ffvelcdmi 7103	. . . . . . . . . 10 ⊢ ((𝑛 − 1) ∈ ℝ → (𝑆‘(𝑛 − 1)) ∈ ℝ)
71	69, 70	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) ∈ ℝ)
72	67, 71	resubcld 11689	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1))) ∈ ℝ)
73	62, 72	remulcld 11289	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) ∈ ℝ)
74	19, 73	fsumrecl 15767	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) ∈ ℝ)
75	74, 50	rerpdivcld 13106	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥)) ∈ ℝ)
76	60, 75	resubcld 11689	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) ∈ ℝ)
77	76, 13	rerpdivcld 13106	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ∈ ℝ)
78	17	recnd 11287	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
79	58, 78	mulcld 11279	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℂ)
80	49	recnd 11287	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℂ)
81	50	rpne0d 13080	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
82	80, 78, 81	divcld 12041	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)) ∈ ℂ)
83	79, 82	subcld 11618	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) ∈ ℂ)
84	83	abscld 15472	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) ∈ ℝ)
85	80	abscld 15472	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ∈ ℝ)
86	85, 50	rerpdivcld 13106	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)) ∈ ℝ)
87	60, 86	resubcld 11689	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))) ∈ ℝ)
88	48	recnd 11287	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℂ)
89	88	abscld 15472	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ∈ ℝ)
90	19, 89	fsumrecl 15767	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ∈ ℝ)
91	19, 88	fsumabs 15834	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))))
92	47	recnd 11287	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))) ∈ ℂ)
93	61, 92	absmuld 15490	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))))
94	92	abscld 15472	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℝ)
95	61	absge0d 15480	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
96	42	recnd 11287	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) ∈ ℂ)
97	46	recnd 11287	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
98	96, 97	abs2dif2d 15494	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ≤ ((abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) + (abs‘((Λ‘𝑛) · (log‘𝑛)))))
99	71	recnd 11287	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) ∈ ℂ)
100	96, 97	addcld 11278	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))) ∈ ℂ)
101	99, 100	pncan2d 11620	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))) − (𝑆‘(𝑛 − 1))) = (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))))
102		elfzuz 13557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ (ℤ≥‘1))
103	102	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ (ℤ≥‘1))
104		elfznn 13590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
105	104	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
106		vmacl 27176	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ ℕ → (Λ‘𝑘) ∈ ℝ)
107	105, 106	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (Λ‘𝑘) ∈ ℝ)
108	105	nnrpd 13073	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℝ+)
109	108	relogcld 26680	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (log‘𝑘) ∈ ℝ)
110	107, 109	remulcld 11289	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → ((Λ‘𝑘) · (log‘𝑘)) ∈ ℝ)
111		fzfid 14011	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (1...𝑘) ∈ Fin)
112		dvdsssfz1 16352	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ (1...𝑘))
113	105, 112	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ (1...𝑘))
114	111, 113	ssfid 9299	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ∈ Fin)
115		ssrab2 4090	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ⊆ ℕ
116		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
117	115, 116	sselid 3993	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → 𝑚 ∈ ℕ)
118	117, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (Λ‘𝑚) ∈ ℝ)
119		dvdsdivcl 16350	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (𝑘 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
120	105, 119	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (𝑘 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘})
121	115, 120	sselid 3993	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (𝑘 / 𝑚) ∈ ℕ)
122		vmacl 27176	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 / 𝑚) ∈ ℕ → (Λ‘(𝑘 / 𝑚)) ∈ ℝ)
123	121, 122	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → (Λ‘(𝑘 / 𝑚)) ∈ ℝ)
124	118, 123	remulcld 11289	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘}) → ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) ∈ ℝ)
125	114, 124	fsumrecl 15767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) ∈ ℝ)
126	110, 125	readdcld 11288	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) ∈ ℝ)
127	126	recnd 11287	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) ∈ ℂ)
128		fveq2 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑛 → (Λ‘𝑘) = (Λ‘𝑛))
129		fveq2 6907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑛 → (log‘𝑘) = (log‘𝑛))
130	128, 129	oveq12d 7449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → ((Λ‘𝑘) · (log‘𝑘)) = ((Λ‘𝑛) · (log‘𝑛)))
131		breq2 5152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑛 → (𝑦 ∥ 𝑘 ↔ 𝑦 ∥ 𝑛))
132	131	rabbidv 3441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑛 → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} = {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
133		fvoveq1 7454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑛 → (Λ‘(𝑘 / 𝑚)) = (Λ‘(𝑛 / 𝑚)))
134	133	oveq2d 7447	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑛 → ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) = ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
135	134	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 = 𝑛 ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) = ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
136	132, 135	sumeq12rdv 15740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) = Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
137	130, 136	oveq12d 7449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))))
138	103, 127, 137	fsumm1 15784	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...𝑛)(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = (Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) + (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))))
139	64	pntsval2 27635	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ → (𝑆‘𝑛) = Σ𝑘 ∈ (1...(⌊‘𝑛))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
140	63, 139	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘𝑛) = Σ𝑘 ∈ (1...(⌊‘𝑛))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
141	22	nnzd 12638	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
142		flid 13845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ℤ → (⌊‘𝑛) = 𝑛)
143	141, 142	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑛) = 𝑛)
144	143	oveq2d 7447	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘𝑛)) = (1...𝑛))
145	144	sumeq1d 15733	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...(⌊‘𝑛))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = Σ𝑘 ∈ (1...𝑛)(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
146	140, 145	eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘𝑛) = Σ𝑘 ∈ (1...𝑛)(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
147	64	pntsval2 27635	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 − 1) ∈ ℝ → (𝑆‘(𝑛 − 1)) = Σ𝑘 ∈ (1...(⌊‘(𝑛 − 1)))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
148	69, 147	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) = Σ𝑘 ∈ (1...(⌊‘(𝑛 − 1)))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
149		1zzd 12646	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℤ)
150	141, 149	zsubcld 12725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℤ)
151		flid 13845	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 − 1) ∈ ℤ → (⌊‘(𝑛 − 1)) = (𝑛 − 1))
152	150, 151	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑛 − 1)) = (𝑛 − 1))
153	152	oveq2d 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑛 − 1))) = (1...(𝑛 − 1)))
154	153	sumeq1d 15733	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...(⌊‘(𝑛 − 1)))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
155	148, 154	eqtrd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) = Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
156	96, 97	addcomd 11461	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))) = (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))))
157	155, 156	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))) = (Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) + (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))))
158	138, 146, 157	3eqtr4d 2785	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘𝑛) = ((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))))
159	158	oveq1d 7446	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1))) = (((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))) − (𝑆‘(𝑛 − 1))))
160		vmage0 27179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
161	33, 160	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 0 ≤ (Λ‘𝑚))
162		vmage0 27179	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 / 𝑚) ∈ ℕ → 0 ≤ (Λ‘(𝑛 / 𝑚)))
163	38, 162	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 0 ≤ (Λ‘(𝑛 / 𝑚)))
164	35, 40, 161, 163	mulge0d 11838	. . . . . . . . . . . . . . . . . 18 ⊢ ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 0 ≤ ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
165	30, 41, 164	fsumge0 15828	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
166	42, 165	absidd 15458	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) = Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
167		vmage0 27179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
168	22, 167	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
169	22	nnge1d 12312	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
170	63, 169	logge0d 26687	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
171	44, 45, 168, 170	mulge0d 11838	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · (log‘𝑛)))
172	46, 171	absidd 15458	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (log‘𝑛))) = ((Λ‘𝑛) · (log‘𝑛)))
173	166, 172	oveq12d 7449	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) + (abs‘((Λ‘𝑛) · (log‘𝑛)))) = (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))))
174	101, 159, 173	3eqtr4d 2785	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1))) = ((abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) + (abs‘((Λ‘𝑛) · (log‘𝑛)))))
175	98, 174	breqtrrd 5176	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ≤ ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1))))
176	94, 72, 62, 95, 175	lemul2ad 12206	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))))
177	93, 176	eqbrtrd 5170	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))))
178	19, 89, 73, 177	fsumle 15832	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))))
179	85, 90, 74, 91, 178	letrd 11416	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))))
180	85, 74, 50, 179	lediv1dd 13133	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥)))
181	86, 75, 60, 180	lesub2dd 11878	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) ≤ (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))))
182	58, 78	absmuld 15490	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝑅‘𝑥) · (log‘𝑥))) = ((abs‘(𝑅‘𝑥)) · (abs‘(log‘𝑥))))
183	5, 12	logge0d 26687	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (log‘𝑥))
184	17, 183	absidd 15458	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(log‘𝑥)) = (log‘𝑥))
185	184	oveq2d 7447	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅‘𝑥)) · (abs‘(log‘𝑥))) = ((abs‘(𝑅‘𝑥)) · (log‘𝑥)))
186	182, 185	eqtrd 2775	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝑅‘𝑥) · (log‘𝑥))) = ((abs‘(𝑅‘𝑥)) · (log‘𝑥)))
187	80, 78, 81	absdivd 15491	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (abs‘(log‘𝑥))))
188	184	oveq2d 7447	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (abs‘(log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)))
189	187, 188	eqtrd 2775	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)))
190	186, 189	oveq12d 7449	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((𝑅‘𝑥) · (log‘𝑥))) − (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) = (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))))
191	79, 82	abs2difd 15493	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((𝑅‘𝑥) · (log‘𝑥))) − (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) ≤ (abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))))
192	190, 191	eqbrtrrd 5172	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))) ≤ (abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))))
193	76, 87, 84, 181, 192	letrd 11416	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) ≤ (abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))))
194	76, 84, 13, 193	lediv1dd 13133	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ≤ ((abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / 𝑥))
195	52	recnd 11287	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) ∈ ℂ)
196	5	recnd 11287	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
197	13	rpne0d 13080	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
198	195, 196, 197	absdivd 15491	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) = ((abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / (abs‘𝑥)))
199	13	rpge0d 13079	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
200	5, 199	absidd 15458	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘𝑥) = 𝑥)
201	200	oveq2d 7447	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / (abs‘𝑥)) = ((abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / 𝑥))
202	198, 201	eqtrd 2775	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) = ((abs‘(((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / 𝑥))
203	194, 202	breqtrrd 5176	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ≤ (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)))
204	203	adantrr 717	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ≤ (abs‘((((𝑅‘𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)))
205	1, 56, 57, 77, 204	lo1le 15685	. 2 ⊢ (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1))
206	205	mptru 1544	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅‘𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆‘𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1)

Syntax hints:   ∧ wa 395   = wceq 1537  ⊤wtru 1538   ∈ wcel 2106  {crab 3433   ⊆ wss 3963   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290   < clt 11293   ≤ cle 11294   − cmin 11490   / cdiv 11918  ℕcn 12264  ℤcz 12611  ℤ_≥cuz 12876  ℝ⁺crp 13032  (,)cioo 13384  ...cfz 13544  ⌊cfl 13827  abscabs 15270  𝑂(1)co1 15519  ≤𝑂(1)clo1 15520  Σcsu 15719   ∥ cdvds 16287  logclog 26611  Λcvma 27150  ψcchp 27151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-o1 15523  df-lo1 15524  df-sum 15720  df-ef 16100  df-e 16101  df-sin 16102  df-cos 16103  df-tan 16104  df-pi 16105  df-dvds 16288  df-gcd 16529  df-prm 16706  df-pc 16871  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-ulm 26435  df-log 26613  df-cxp 26614  df-atan 26925  df-em 27051  df-cht 27155  df-vma 27156  df-chp 27157  df-ppi 27158  df-mu 27159

This theorem is referenced by:  pntrlog2bndlem4  27639