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Theorem pntrlog2bndlem1 26725
Description: The sum of selberg3r 26717 and selberg4r 26718. (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‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (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.
StepHypRef Expression
1 1red 10976 . . 3 (⊤ → 1 ∈ ℝ)
2 pntrlog2bnd.r . . . . 5 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
32selberg34r 26719 . . . 4 (𝑥 ∈ (1(,)+∞) ↦ ((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
4 elioore 13109 . . . . . . . . . . . 12 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
54adantl 482 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
6 1rp 12734 . . . . . . . . . . . 12 1 ∈ ℝ+
76a1i 11 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
8 1red 10976 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
9 eliooord 13138 . . . . . . . . . . . . . 14 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
109adantl 482 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
1110simpld 495 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
128, 5, 11ltled 11123 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
135, 7, 12rpgecld 12811 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
142pntrf 26711 . . . . . . . . . . 11 𝑅:ℝ+⟶ℝ
1514ffvelrni 6960 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑅𝑥) ∈ ℝ)
1613, 15syl 17 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℝ)
1713relogcld 25778 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
1816, 17remulcld 11005 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑅𝑥) · (log‘𝑥)) ∈ ℝ)
19 fzfid 13693 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2013adantr 481 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
21 elfznn 13285 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2221adantl 482 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2322nnrpd 12770 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
2420, 23rpdivcld 12789 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
2514ffvelrni 6960 . . . . . . . . . . . 12 ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
2624, 25syl 17 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
27 fzfid 13693 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
28 dvdsssfz1 16027 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦𝑛} ⊆ (1...𝑛))
2922, 28syl 17 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦𝑛} ⊆ (1...𝑛))
3027, 29ssfid 9042 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦𝑛} ∈ Fin)
31 ssrab2 4013 . . . . . . . . . . . . . . . 16 {𝑦 ∈ ℕ ∣ 𝑦𝑛} ⊆ ℕ
32 simpr 485 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛})
3331, 32sselid 3919 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → 𝑚 ∈ ℕ)
34 vmacl 26267 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
3533, 34syl 17 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → (Λ‘𝑚) ∈ ℝ)
36 dvdsdivcl 16025 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℕ ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → (𝑛 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛})
3722, 36sylan 580 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → (𝑛 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛})
3831, 37sselid 3919 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → (𝑛 / 𝑚) ∈ ℕ)
39 vmacl 26267 . . . . . . . . . . . . . . 15 ((𝑛 / 𝑚) ∈ ℕ → (Λ‘(𝑛 / 𝑚)) ∈ ℝ)
4038, 39syl 17 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → (Λ‘(𝑛 / 𝑚)) ∈ ℝ)
4135, 40remulcld 11005 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) ∈ ℝ)
4230, 41fsumrecl 15446 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) ∈ ℝ)
43 vmacl 26267 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
4422, 43syl 17 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
4523relogcld 25778 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
4644, 45remulcld 11005 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℝ)
4742, 46resubcld 11403 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))) ∈ ℝ)
4826, 47remulcld 11005 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℝ)
4919, 48fsumrecl 15446 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℝ)
505, 11rplogcld 25784 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
5149, 50rerpdivcld 12803 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)) ∈ ℝ)
5218, 51resubcld 11403 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) ∈ ℝ)
5352, 13rerpdivcld 12803 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥) ∈ ℝ)
5453recnd 11003 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥) ∈ ℂ)
5554lo1o12 15242 . . . 4 (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ ((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥))) ∈ ≤𝑂(1)))
563, 55mpbii 232 . . 3 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥))) ∈ ≤𝑂(1))
5754abscld 15148 . . 3 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) ∈ ℝ)
5816recnd 11003 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℂ)
5958abscld 15148 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅𝑥)) ∈ ℝ)
6059, 17remulcld 11005 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℝ)
6126recnd 11003 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
6261abscld 15148 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
6322nnred 11988 . . . . . . . . . 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)))))
6564pntsf 26721 . . . . . . . . . . 11 𝑆:ℝ⟶ℝ
6665ffvelrni 6960 . . . . . . . . . 10 (𝑛 ∈ ℝ → (𝑆𝑛) ∈ ℝ)
6763, 66syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆𝑛) ∈ ℝ)
68 1red 10976 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
6963, 68resubcld 11403 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
7065ffvelrni 6960 . . . . . . . . . 10 ((𝑛 − 1) ∈ ℝ → (𝑆‘(𝑛 − 1)) ∈ ℝ)
7169, 70syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) ∈ ℝ)
7267, 71resubcld 11403 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆𝑛) − (𝑆‘(𝑛 − 1))) ∈ ℝ)
7362, 72remulcld 11005 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) ∈ ℝ)
7419, 73fsumrecl 15446 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) ∈ ℝ)
7574, 50rerpdivcld 12803 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥)) ∈ ℝ)
7660, 75resubcld 11403 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) ∈ ℝ)
7776, 13rerpdivcld 12803 . . 3 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ∈ ℝ)
7817recnd 11003 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
7958, 78mulcld 10995 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((𝑅𝑥) · (log‘𝑥)) ∈ ℂ)
8049recnd 11003 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℂ)
8150rpne0d 12777 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
8280, 78, 81divcld 11751 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)) ∈ ℂ)
8379, 82subcld 11332 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) ∈ ℂ)
8483abscld 15148 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) ∈ ℝ)
8580abscld 15148 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ∈ ℝ)
8685, 50rerpdivcld 12803 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)) ∈ ℝ)
8760, 86resubcld 11403 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))) ∈ ℝ)
8848recnd 11003 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℂ)
8988abscld 15148 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ∈ ℝ)
9019, 89fsumrecl 15446 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ∈ ℝ)
9119, 88fsumabs 15513 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))))
9247recnd 11003 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))) ∈ ℂ)
9361, 92absmuld 15166 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))))
9492abscld 15148 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ∈ ℝ)
9561absge0d 15156 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
9642recnd 11003 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) ∈ ℂ)
9746recnd 11003 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
9896, 97abs2dif2d 15170 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ≤ ((abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) + (abs‘((Λ‘𝑛) · (log‘𝑛)))))
9971recnd 11003 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) ∈ ℂ)
10096, 97addcld 10994 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))) ∈ ℂ)
10199, 100pncan2d 11334 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))) − (𝑆‘(𝑛 − 1))) = (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))))
102 elfzuz 13252 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ (ℤ‘1))
103102adantl 482 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ (ℤ‘1))
104 elfznn 13285 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ)
105104adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
106 vmacl 26267 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ ℕ → (Λ‘𝑘) ∈ ℝ)
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (Λ‘𝑘) ∈ ℝ)
108105nnrpd 12770 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℝ+)
109108relogcld 25778 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (log‘𝑘) ∈ ℝ)
110107, 109remulcld 11005 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → ((Λ‘𝑘) · (log‘𝑘)) ∈ ℝ)
111 fzfid 13693 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (1...𝑘) ∈ Fin)
112 dvdsssfz1 16027 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘))
113105, 112syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘))
114111, 113ssfid 9042 . . . . . . . . . . . . . . . . . . . . 21 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ∈ Fin)
115 ssrab2 4013 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ ℕ
116 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
117115, 116sselid 3919 . . . . . . . . . . . . . . . . . . . . . . 23 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → 𝑚 ∈ ℕ)
118117, 34syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (Λ‘𝑚) ∈ ℝ)
119 dvdsdivcl 16025 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℕ ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (𝑘 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
120105, 119sylan 580 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (𝑘 / 𝑚) ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
121115, 120sselid 3919 . . . . . . . . . . . . . . . . . . . . . . 23 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (𝑘 / 𝑚) ∈ ℕ)
122 vmacl 26267 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 / 𝑚) ∈ ℕ → (Λ‘(𝑘 / 𝑚)) ∈ ℝ)
123121, 122syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (Λ‘(𝑘 / 𝑚)) ∈ ℝ)
124118, 123remulcld 11005 . . . . . . . . . . . . . . . . . . . . 21 (((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) ∈ ℝ)
125114, 124fsumrecl 15446 . . . . . . . . . . . . . . . . . . . 20 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) ∈ ℝ)
126110, 125readdcld 11004 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) ∈ ℝ)
127126recnd 11003 . . . . . . . . . . . . . . . . . 18 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...𝑛)) → (((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) ∈ ℂ)
128 fveq2 6774 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → (Λ‘𝑘) = (Λ‘𝑛))
129 fveq2 6774 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → (log‘𝑘) = (log‘𝑛))
130128, 129oveq12d 7293 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → ((Λ‘𝑘) · (log‘𝑘)) = ((Λ‘𝑛) · (log‘𝑛)))
131 breq2 5078 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → (𝑦𝑘𝑦𝑛))
132131rabbidv 3414 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑛 → {𝑦 ∈ ℕ ∣ 𝑦𝑘} = {𝑦 ∈ ℕ ∣ 𝑦𝑛})
133 fvoveq1 7298 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛 → (Λ‘(𝑘 / 𝑚)) = (Λ‘(𝑛 / 𝑚)))
134133oveq2d 7291 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑛 → ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) = ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
135134adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 = 𝑛𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) = ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
136132, 135sumeq12rdv 15419 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚))) = Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
137130, 136oveq12d 7293 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))))
138103, 127, 137fsumm1 15463 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...𝑛)(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = (Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) + (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))))
13964pntsval2 26724 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ → (𝑆𝑛) = Σ𝑘 ∈ (1...(⌊‘𝑛))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
14063, 139syl 17 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆𝑛) = Σ𝑘 ∈ (1...(⌊‘𝑛))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
14122nnzd 12425 . . . . . . . . . . . . . . . . . . . . 21 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
142 flid 13528 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ℤ → (⌊‘𝑛) = 𝑛)
143141, 142syl 17 . . . . . . . . . . . . . . . . . . . 20 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑛) = 𝑛)
144143oveq2d 7291 . . . . . . . . . . . . . . . . . . 19 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘𝑛)) = (1...𝑛))
145144sumeq1d 15413 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...(⌊‘𝑛))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = Σ𝑘 ∈ (1...𝑛)(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
146140, 145eqtrd 2778 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆𝑛) = Σ𝑘 ∈ (1...𝑛)(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
14764pntsval2 26724 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) ∈ ℝ → (𝑆‘(𝑛 − 1)) = Σ𝑘 ∈ (1...(⌊‘(𝑛 − 1)))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
14869, 147syl 17 . . . . . . . . . . . . . . . . . . 19 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) = Σ𝑘 ∈ (1...(⌊‘(𝑛 − 1)))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
149 1zzd 12351 . . . . . . . . . . . . . . . . . . . . . . 23 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℤ)
150141, 149zsubcld 12431 . . . . . . . . . . . . . . . . . . . . . 22 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℤ)
151 flid 13528 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 − 1) ∈ ℤ → (⌊‘(𝑛 − 1)) = (𝑛 − 1))
152150, 151syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑛 − 1)) = (𝑛 − 1))
153152oveq2d 7291 . . . . . . . . . . . . . . . . . . . 20 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑛 − 1))) = (1...(𝑛 − 1)))
154153sumeq1d 15413 . . . . . . . . . . . . . . . . . . 19 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...(⌊‘(𝑛 − 1)))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) = Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
155148, 154eqtrd 2778 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆‘(𝑛 − 1)) = Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))))
15696, 97addcomd 11177 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))) = (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))))
157155, 156oveq12d 7293 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))) = (Σ𝑘 ∈ (1...(𝑛 − 1))(((Λ‘𝑘) · (log‘𝑘)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑚) · (Λ‘(𝑘 / 𝑚)))) + (((Λ‘𝑛) · (log‘𝑛)) + Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))))
158138, 146, 1573eqtr4d 2788 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑆𝑛) = ((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))))
159158oveq1d 7290 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆𝑛) − (𝑆‘(𝑛 − 1))) = (((𝑆‘(𝑛 − 1)) + (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛)))) − (𝑆‘(𝑛 − 1))))
160 vmage0 26270 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → 0 ≤ (Λ‘𝑚))
16133, 160syl 17 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → 0 ≤ (Λ‘𝑚))
162 vmage0 26270 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 / 𝑚) ∈ ℕ → 0 ≤ (Λ‘(𝑛 / 𝑚)))
16338, 162syl 17 . . . . . . . . . . . . . . . . . . 19 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → 0 ≤ (Λ‘(𝑛 / 𝑚)))
16435, 40, 161, 163mulge0d 11552 . . . . . . . . . . . . . . . . . 18 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛}) → 0 ≤ ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
16530, 41, 164fsumge0 15507 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
16642, 165absidd 15134 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) = Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))))
167 vmage0 26270 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
16822, 167syl 17 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
16922nnge1d 12021 . . . . . . . . . . . . . . . . . . 19 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
17063, 169logge0d 25785 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
17144, 45, 168, 170mulge0d 11552 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · (log‘𝑛)))
17246, 171absidd 15134 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (log‘𝑛))) = ((Λ‘𝑛) · (log‘𝑛)))
173166, 172oveq12d 7293 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) + (abs‘((Λ‘𝑛) · (log‘𝑛)))) = (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) + ((Λ‘𝑛) · (log‘𝑛))))
174101, 159, 1733eqtr4d 2788 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑆𝑛) − (𝑆‘(𝑛 − 1))) = ((abs‘Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚)))) + (abs‘((Λ‘𝑛) · (log‘𝑛)))))
17598, 174breqtrrd 5102 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) ≤ ((𝑆𝑛) − (𝑆‘(𝑛 − 1))))
17694, 72, 62, 95, 175lemul2ad 11915 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (abs‘(Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))))
17793, 176eqbrtrd 5096 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))))
17819, 89, 73, 177fsumle 15511 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))))
17985, 90, 74, 91, 178letrd 11132 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))))
18085, 74, 50, 179lediv1dd 12830 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥)))
18186, 75, 60, 180lesub2dd 11592 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) ≤ (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))))
18258, 78absmuld 15166 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝑅𝑥) · (log‘𝑥))) = ((abs‘(𝑅𝑥)) · (abs‘(log‘𝑥))))
1835, 12logge0d 25785 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (log‘𝑥))
18417, 183absidd 15134 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(log‘𝑥)) = (log‘𝑥))
185184oveq2d 7291 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (abs‘(log‘𝑥))) = ((abs‘(𝑅𝑥)) · (log‘𝑥)))
186182, 185eqtrd 2778 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((𝑅𝑥) · (log‘𝑥))) = ((abs‘(𝑅𝑥)) · (log‘𝑥)))
18780, 78, 81absdivd 15167 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (abs‘(log‘𝑥))))
188184oveq2d 7291 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (abs‘(log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)))
189187, 188eqtrd 2778 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥)))
190186, 189oveq12d 7293 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((𝑅𝑥) · (log‘𝑥))) − (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) = (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))))
19179, 82abs2difd 15169 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘((𝑅𝑥) · (log‘𝑥))) − (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) ≤ (abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))))
192190, 191eqbrtrrd 5098 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛))))) / (log‘𝑥))) ≤ (abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))))
19376, 87, 84, 181, 192letrd 11132 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) ≤ (abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))))
19476, 84, 13, 193lediv1dd 12830 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ≤ ((abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / 𝑥))
19552recnd 11003 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) ∈ ℂ)
1965recnd 11003 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
19713rpne0d 12777 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
198195, 196, 197absdivd 15167 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) = ((abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / (abs‘𝑥)))
19913rpge0d 12776 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
2005, 199absidd 15134 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘𝑥) = 𝑥)
201200oveq2d 7291 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / (abs‘𝑥)) = ((abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / 𝑥))
202198, 201eqtrd 2778 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)) = ((abs‘(((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥)))) / 𝑥))
203194, 202breqtrrd 5102 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ≤ (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)))
204203adantrr 714 . . 3 ((⊤ ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥) ≤ (abs‘((((𝑅𝑥) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑅‘(𝑥 / 𝑛)) · (Σ𝑚 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑛} ((Λ‘𝑚) · (Λ‘(𝑛 / 𝑚))) − ((Λ‘𝑛) · (log‘𝑛)))) / (log‘𝑥))) / 𝑥)))
2051, 56, 57, 77, 204lo1le 15363 . 2 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1))
206205mptru 1546 1 (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑆𝑛) − (𝑆‘(𝑛 − 1)))) / (log‘𝑥))) / 𝑥)) ∈ ≤𝑂(1)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wtru 1540  wcel 2106  {crab 3068  wss 3887   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  cr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  +∞cpnf 11006   < clt 11009  cle 11010  cmin 11205   / cdiv 11632  cn 11973  cz 12319  cuz 12582  +crp 12730  (,)cioo 13079  ...cfz 13239  cfl 13510  abscabs 14945  𝑂(1)co1 15195  ≤𝑂(1)clo1 15196  Σcsu 15397  cdvds 15963  logclog 25710  Λcvma 26241  ψcchp 26242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-o1 15199  df-lo1 15200  df-sum 15398  df-ef 15777  df-e 15778  df-sin 15779  df-cos 15780  df-tan 15781  df-pi 15782  df-dvds 15964  df-gcd 16202  df-prm 16377  df-pc 16538  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031  df-ulm 25536  df-log 25712  df-cxp 25713  df-atan 26017  df-em 26142  df-cht 26246  df-vma 26247  df-chp 26248  df-ppi 26249  df-mu 26250
This theorem is referenced by:  pntrlog2bndlem4  26728
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