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Theorem pntrlog2bndlem5 25843
Description: Lemma for pntrlog2bnd 25846. Bound on the difference between the Selberg function and its approximation, inside a sum. (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))
pntrlog2bnd.t 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))
pntrlog2bndlem5.1 (𝜑𝐵 ∈ ℝ+)
pntrlog2bndlem5.2 (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
Assertion
Ref Expression
pntrlog2bndlem5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
Distinct variable groups:   𝑖,𝑎,𝑛,𝑥,𝑦   𝐵,𝑛,𝑥,𝑦   𝜑,𝑛,𝑥   𝑆,𝑛,𝑥,𝑦   𝑅,𝑛,𝑥,𝑦   𝑇,𝑛
Allowed substitution hints:   𝜑(𝑦,𝑖,𝑎)   𝐵(𝑖,𝑎)   𝑅(𝑖,𝑎)   𝑆(𝑖,𝑎)   𝑇(𝑥,𝑦,𝑖,𝑎)

Proof of Theorem pntrlog2bndlem5
StepHypRef Expression
1 elioore 12622 . . . . . . . . . . . . 13 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
21adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3 1rp 12247 . . . . . . . . . . . . 13 1 ∈ ℝ+
43a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
5 1red 10495 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6 eliooord 12650 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
76adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
87simpld 495 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
95, 2, 8ltled 10641 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
102, 4, 9rpgecld 12324 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
11 pntrlog2bnd.r . . . . . . . . . . . . 13 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
1211pntrf 25825 . . . . . . . . . . . 12 𝑅:ℝ+⟶ℝ
1312ffvelrni 6722 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑅𝑥) ∈ ℝ)
1410, 13syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℝ)
1514recnd 10522 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℂ)
1615abscld 14634 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅𝑥)) ∈ ℝ)
1716recnd 10522 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅𝑥)) ∈ ℂ)
1810relogcld 24891 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
1918recnd 10522 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2017, 19mulcld 10514 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℂ)
21 2cnd 11569 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
222, 8rplogcld 24897 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2322rpne0d 12290 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
2421, 19, 23divcld 11270 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
25 fzfid 13195 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2610adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
27 elfznn 12790 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2827adantl 482 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2928nnrpd 12283 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
3026, 29rpdivcld 12302 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
3112ffvelrni 6722 . . . . . . . . . . . . 13 ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
3230, 31syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
3332recnd 10522 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
3433abscld 14634 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
3529relogcld 24891 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
36 1red 10495 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
3735, 36readdcld 10523 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℝ)
3834, 37remulcld 10524 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
3938recnd 10522 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
4025, 39fsumcl 14927 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
4124, 40mulcld 10514 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℂ)
4220, 41subcld 10851 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℂ)
4334recnd 10522 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
4425, 43fsumcl 14927 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
4524, 44mulcld 10514 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℂ)
462recnd 10522 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
4710rpne0d 12290 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
4842, 45, 46, 47divdird 11308 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) / 𝑥) = (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)))
4916, 18remulcld 10524 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℝ)
5049recnd 10522 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℂ)
5150, 41, 45subsubd 10879 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))) = ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))))
5224, 40, 44subdid 10950 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))))
5325, 39, 43fsumsub 14980 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))
5437recnd 10522 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℂ)
55 1cnd 10489 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
5643, 54, 55subdid 10950 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1)))
5735recnd 10522 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
5857, 55pncand 10852 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘𝑛) + 1) − 1) = (log‘𝑛))
5958oveq2d 7039 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
6043mulid1d 10511 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1) = (abs‘(𝑅‘(𝑥 / 𝑛))))
6160oveq2d 7039 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))))
6256, 59, 613eqtr3rd 2842 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
6362sumeq2dv 14897 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
6453, 63eqtr3d 2835 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
6564oveq2d 7039 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
6652, 65eqtr3d 2835 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
6766oveq2d 7039 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))))) = (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
6851, 67eqtr3d 2835 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) = (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
6968oveq1d 7038 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))))) / 𝑥) = ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
7048, 69eqtr3d 2835 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) = ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
7170mpteq2dva 5062 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)))
72 2re 11565 . . . . . . . 8 2 ∈ ℝ
7372a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
7473, 22rerpdivcld 12316 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
7525, 38fsumrecl 14928 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
7674, 75remulcld 10524 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℝ)
7749, 76resubcld 10922 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℝ)
7877, 10rerpdivcld 12316 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ∈ ℝ)
7925, 34fsumrecl 14928 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
8074, 79remulcld 10524 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
8180, 10rerpdivcld 12316 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
82 1red 10495 . . . 4 (𝜑 → 1 ∈ ℝ)
83 pntsval.1 . . . . . 6 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
84 pntrlog2bnd.t . . . . . 6 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))
8583, 11, 84pntrlog2bndlem4 25842 . . . . 5 (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1)
8685a1i 11 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥)) ∈ ≤𝑂(1))
8728nnred 11507 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
88 simpl 483 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → 𝑎 ∈ ℝ)
89 simpr 485 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → 𝑎 ∈ ℝ+)
9089relogcld 24891 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → (log‘𝑎) ∈ ℝ)
9188, 90remulcld 10524 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → (𝑎 · (log‘𝑎)) ∈ ℝ)
92 0red 10497 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℝ ∧ ¬ 𝑎 ∈ ℝ+) → 0 ∈ ℝ)
9391, 92ifclda 4421 . . . . . . . . . . . . 13 (𝑎 ∈ ℝ → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) ∈ ℝ)
9484, 93fmpti 6746 . . . . . . . . . . . 12 𝑇:ℝ⟶ℝ
9594ffvelrni 6722 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝑇𝑛) ∈ ℝ)
9687, 95syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇𝑛) ∈ ℝ)
9787, 36resubcld 10922 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
9894ffvelrni 6722 . . . . . . . . . . 11 ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) ∈ ℝ)
9997, 98syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇‘(𝑛 − 1)) ∈ ℝ)
10096, 99resubcld 10922 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ∈ ℝ)
10134, 100remulcld 10524 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
10225, 101fsumrecl 14928 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
10374, 102remulcld 10524 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1))))) ∈ ℝ)
10449, 103resubcld 10922 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) ∈ ℝ)
105104, 10rerpdivcld 12316 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥) ∈ ℝ)
106 2rp 12248 . . . . . . . . . . 11 2 ∈ ℝ+
107106a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ+)
108107rpge0d 12289 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
10973, 22, 108divge0d 12325 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 / (log‘𝑥)))
11033absge0d 14642 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
11129adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ+)
112111rpcnd 12287 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℂ)
11357adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℂ)
114112, 113mulcld 10514 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘𝑛)) ∈ ℂ)
115 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 < 𝑛)
116 1re 10494 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
117111rpred 12285 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ)
118 difrp 12281 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ+))
119116, 117, 118sylancr 587 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ+))
120115, 119mpbid 233 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 − 1) ∈ ℝ+)
121120relogcld 24891 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℝ)
122121recnd 10522 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℂ)
123112, 122mulcld 10514 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘(𝑛 − 1))) ∈ ℂ)
124114, 123, 122subsubd 10879 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · (log‘𝑛)) − ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1)))) = (((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
125 rpre 12251 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
126 eleq1 2872 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑛 → (𝑎 ∈ ℝ+𝑛 ∈ ℝ+))
127 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑛𝑎 = 𝑛)
128 fveq2 6545 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛))
129127, 128oveq12d 7041 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑛 → (𝑎 · (log‘𝑎)) = (𝑛 · (log‘𝑛)))
130126, 129ifbieq1d 4410 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑛 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0))
131 ovex 7055 . . . . . . . . . . . . . . . . . . 19 (𝑛 · (log‘𝑛)) ∈ V
132 c0ex 10488 . . . . . . . . . . . . . . . . . . 19 0 ∈ V
133131, 132ifex 4435 . . . . . . . . . . . . . . . . . 18 if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0) ∈ V
134130, 84, 133fvmpt 6642 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ → (𝑇𝑛) = if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0))
135125, 134syl 17 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ+ → (𝑇𝑛) = if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0))
136 iftrue 4393 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ+ → if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0) = (𝑛 · (log‘𝑛)))
137135, 136eqtrd 2833 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℝ+ → (𝑇𝑛) = (𝑛 · (log‘𝑛)))
138111, 137syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇𝑛) = (𝑛 · (log‘𝑛)))
139 rpre 12251 . . . . . . . . . . . . . . . . . 18 ((𝑛 − 1) ∈ ℝ+ → (𝑛 − 1) ∈ ℝ)
140 eleq1 2872 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (𝑛 − 1) → (𝑎 ∈ ℝ+ ↔ (𝑛 − 1) ∈ ℝ+))
141 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑛 − 1) → 𝑎 = (𝑛 − 1))
142 fveq2 6545 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑛 − 1) → (log‘𝑎) = (log‘(𝑛 − 1)))
143141, 142oveq12d 7041 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (𝑛 − 1) → (𝑎 · (log‘𝑎)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
144140, 143ifbieq1d 4410 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝑛 − 1) → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
145 ovex 7055 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) · (log‘(𝑛 − 1))) ∈ V
146145, 132ifex 4435 . . . . . . . . . . . . . . . . . . 19 if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) ∈ V
147144, 84, 146fvmpt 6642 . . . . . . . . . . . . . . . . . 18 ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) = if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
148139, 147syl 17 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℝ+ → (𝑇‘(𝑛 − 1)) = if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
149 iftrue 4393 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℝ+ → if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
150148, 149eqtrd 2833 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ℝ+ → (𝑇‘(𝑛 − 1)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
151120, 150syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
152 1cnd 10489 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℂ)
153112, 152, 122subdird 10951 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · (log‘(𝑛 − 1))) = ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))))
154122mulid2d 10512 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · (log‘(𝑛 − 1))) = (log‘(𝑛 − 1)))
155154oveq2d 7039 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
156151, 153, 1553eqtrd 2837 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
157138, 156oveq12d 7041 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · (log‘𝑛)) − ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1)))))
158112, 113, 122subdid 10950 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))))
159158oveq1d 7038 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) = (((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
160124, 157, 1593eqtr4d 2843 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
161111relogcld 24891 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℝ)
162161, 121resubcld 10922 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℝ)
163162recnd 10522 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℂ)
164112, 152, 163subdird 10951 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − (1 · ((log‘𝑛) − (log‘(𝑛 − 1))))))
165163mulid2d 10512 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
166165oveq2d 7039 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − (1 · ((log‘𝑛) − (log‘(𝑛 − 1))))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − ((log‘𝑛) − (log‘(𝑛 − 1)))))
167117, 162remulcld 10524 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℝ)
168167recnd 10522 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℂ)
169168, 113, 122subsub3d 10881 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) − ((log‘𝑛) − (log‘(𝑛 − 1)))) = (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)))
170164, 166, 1693eqtrd 2837 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) = (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)))
171112, 152npcand 10855 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) + 1) = 𝑛)
172171fveq2d 6549 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
173172oveq1d 7038 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
174 logdifbnd 25257 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℝ+ → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
175120, 174syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
176173, 175eqbrtrrd 4992 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
177 1red 10495 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℝ)
178162, 177, 120lemuldiv2d 12335 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1 ↔ ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1))))
179176, 178mpbird 258 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1)
180170, 179eqbrtrrd 4992 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1)
181167, 121readdcld 10523 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ∈ ℝ)
182181, 161, 177lesubadd2d 11093 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1 ↔ ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1)))
183180, 182mpbid 233 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
184160, 183eqbrtrd 4990 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
185 fveq2 6545 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑇𝑛) = (𝑇‘1))
186 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 1 → 𝑎 = 1)
187186, 3syl6eqel 2893 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 1 → 𝑎 ∈ ℝ+)
188187iftrued 4395 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 1 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = (𝑎 · (log‘𝑎)))
189 fveq2 6545 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 1 → (log‘𝑎) = (log‘1))
190 log1 24854 . . . . . . . . . . . . . . . . . . . . . . 23 (log‘1) = 0
191189, 190syl6eq 2849 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 1 → (log‘𝑎) = 0)
192186, 191oveq12d 7041 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 1 → (𝑎 · (log‘𝑎)) = (1 · 0))
193 ax-1cn 10448 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℂ
194193mul01i 10683 . . . . . . . . . . . . . . . . . . . . 21 (1 · 0) = 0
195192, 194syl6eq 2849 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 1 → (𝑎 · (log‘𝑎)) = 0)
196188, 195eqtrd 2833 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 1 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = 0)
197196, 84, 132fvmpt 6642 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℝ → (𝑇‘1) = 0)
198116, 197ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑇‘1) = 0
199185, 198syl6eq 2849 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑇𝑛) = 0)
200 oveq1 7030 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
201 1m1e0 11563 . . . . . . . . . . . . . . . . . . 19 (1 − 1) = 0
202200, 201syl6eq 2849 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 − 1) = 0)
203202fveq2d 6549 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = (𝑇‘0))
204 0re 10496 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
205 rpne0 12259 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ℝ+𝑎 ≠ 0)
206205necon2bi 3016 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 0 → ¬ 𝑎 ∈ ℝ+)
207206iffalsed 4398 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 0 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = 0)
208207, 84, 132fvmpt 6642 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → (𝑇‘0) = 0)
209204, 208ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑇‘0) = 0
210203, 209syl6eq 2849 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = 0)
211199, 210oveq12d 7041 . . . . . . . . . . . . . . 15 (𝑛 = 1 → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = (0 − 0))
212 0m0e0 11611 . . . . . . . . . . . . . . 15 (0 − 0) = 0
213211, 212syl6eq 2849 . . . . . . . . . . . . . 14 (𝑛 = 1 → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = 0)
214213eqcoms 2805 . . . . . . . . . . . . 13 (1 = 𝑛 → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = 0)
215214adantl 482 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = 0)
216 0red 10497 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ∈ ℝ)
21728nnge1d 11539 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
21887, 217logge0d 24898 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
21935lep1d 11425 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ≤ ((log‘𝑛) + 1))
220216, 35, 37, 218, 219letrd 10650 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘𝑛) + 1))
221220adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → 0 ≤ ((log‘𝑛) + 1))
222215, 221eqbrtrd 4990 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
223 elfzle1 12764 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘𝑥)) → 1 ≤ 𝑛)
224223adantl 482 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
22536, 87leloed 10636 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 ≤ 𝑛 ↔ (1 < 𝑛 ∨ 1 = 𝑛)))
226224, 225mpbid 233 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 < 𝑛 ∨ 1 = 𝑛))
227184, 222, 226mpjaodan 953 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
228100, 37, 34, 110, 227lemul2ad 11434 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))
22925, 101, 38, 228fsumle 14991 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))
230102, 75, 74, 109, 229lemul2ad 11434 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1))))) ≤ ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))))
231103, 76, 49, 230lesub2dd 11111 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ≤ (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))))
23277, 104, 10, 231lediv1dd 12343 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ≤ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥))
233232adantrr 713 . . . 4 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ≤ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) / 𝑥))
23482, 86, 105, 78, 233lo1le 14846 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥)) ∈ ≤𝑂(1))
235106a1i 11 . . . . . . . 8 (𝜑 → 2 ∈ ℝ+)
236 pntrlog2bndlem5.1 . . . . . . . 8 (𝜑𝐵 ∈ ℝ+)
237235, 236rpmulcld 12301 . . . . . . 7 (𝜑 → (2 · 𝐵) ∈ ℝ+)
238237rpred 12285 . . . . . 6 (𝜑 → (2 · 𝐵) ∈ ℝ)
239238adantr 481 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ)
2405, 22rerpdivcld 12316 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
2415, 240readdcld 10523 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
242 ioossre 12652 . . . . . 6 (1(,)+∞) ⊆ ℝ
243 lo1const 14815 . . . . . 6 (((1(,)+∞) ⊆ ℝ ∧ (2 · 𝐵) ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
244242, 238, 243sylancr 587 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
245 lo1const 14815 . . . . . . 7 (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
246242, 82, 245sylancr 587 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
247 divlogrlim 24903 . . . . . . . 8 (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
248 rlimo1 14811 . . . . . . . 8 ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
249247, 248mp1i 13 . . . . . . 7 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
250240, 249o1lo1d 14734 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ ≤𝑂(1))
2515, 240, 246, 250lo1add 14821 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ ≤𝑂(1))
252237adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ+)
253252rpge0d 12289 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 · 𝐵))
254239, 241, 244, 251, 253lo1mul 14822 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 · 𝐵) · (1 + (1 / (log‘𝑥))))) ∈ ≤𝑂(1))
255239, 241remulcld 10524 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
25679, 10rerpdivcld 12316 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
25718, 5readdcld 10523 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
258236adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ+)
259258rpred 12285 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
260257, 259remulcld 10524 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) · 𝐵) ∈ ℝ)
26128nnrecred 11542 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
26225, 261fsumrecl 14928 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
263262, 259remulcld 10524 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ∈ ℝ)
26434, 26rerpdivcld 12316 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
265259adantr 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
266261, 265remulcld 10524 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · 𝐵) ∈ ℝ)
26730rpcnd 12287 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
26830rpne0d 12290 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
26933, 267, 268absdivd 14653 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))))
2702adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
271270, 28nndivred 11545 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
27230rpge0d 12289 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
273271, 272absidd 14620 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
274273oveq2d 7039 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
275269, 274eqtrd 2833 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
27646adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
27787recnd 10522 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
27847adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
27928nnne0d 11541 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
28043, 276, 277, 278, 279divdiv2d 11302 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥))
28143, 277, 276, 278div23d 11307 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
282275, 280, 2813eqtrd 2837 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
283 fveq2 6545 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑥 / 𝑛) → (𝑅𝑦) = (𝑅‘(𝑥 / 𝑛)))
284 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
285283, 284oveq12d 7041 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑥 / 𝑛) → ((𝑅𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
286285fveq2d 6549 . . . . . . . . . . . . . . 15 (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
287286breq1d 4978 . . . . . . . . . . . . . 14 (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
288 pntrlog2bndlem5.2 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
289288ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
290287, 289, 30rspcdva 3567 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
291282, 290eqbrtrrd 4992 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵)
292264, 265, 29lemuldivd 12334 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵 ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛)))
293291, 292mpbid 233 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛))
294265recnd 10522 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
295294, 277, 279divrec2d 11274 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐵 / 𝑛) = ((1 / 𝑛) · 𝐵))
296293, 295breqtrd 4994 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ ((1 / 𝑛) · 𝐵))
29725, 264, 266, 296fsumle 14991 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
29825, 46, 43, 47fsumdivc 14978 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥))
299258rpcnd 12287 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
300261recnd 10522 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
30125, 299, 300fsummulc1 14977 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) = Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
302297, 298, 3013brtr4d 5000 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵))
303258rpge0d 12289 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
304 harmonicubnd 25273 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
3052, 9, 304syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
306262, 257, 259, 303, 305lemul1ad 11433 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ≤ (((log‘𝑥) + 1) · 𝐵))
307256, 263, 260, 302, 306letrd 10650 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (((log‘𝑥) + 1) · 𝐵))
308256, 260, 74, 109, 307lemul2ad 11434 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)) ≤ ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
30924, 44, 46, 47divassd 11305 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) = ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)))
310241recnd 10522 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℂ)
31121, 299, 310mul32d 10703 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵))
312 1cnd 10489 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
31319, 312, 19, 23divdird 11308 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
31419, 23dividd 11268 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
315314oveq1d 7038 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
316313, 315eqtr2d 2834 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
317316oveq2d 7039 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
31819, 312addcld 10513 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
31921, 19, 318, 23div32d 11293 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
320317, 319eqtr4d 2836 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)))
321320oveq1d 7038 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵) = (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵))
32224, 318, 299mulassd 10517 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
323311, 321, 3223eqtrd 2837 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
324308, 309, 3233brtr4d 5000 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
325324adantrr 713 . . . 4 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
32682, 254, 255, 81, 325lo1le 14846 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
32778, 81, 234, 326lo1add 14821 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥))) ∈ ≤𝑂(1))
32871, 327eqeltrrd 2886 1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 842   = wceq 1525  wcel 2083  wne 2986  wral 3107  wss 3865  ifcif 4387   class class class wbr 4968  cmpt 5047  cfv 6232  (class class class)co 7023  cc 10388  cr 10389  0cc0 10390  1c1 10391   + caddc 10393   · cmul 10395  +∞cpnf 10525   < clt 10528  cle 10529  cmin 10723   / cdiv 11151  cn 11492  2c2 11546  +crp 12243  (,)cioo 12592  ...cfz 12746  cfl 13014  abscabs 14431  𝑟 crli 14680  𝑂(1)co1 14681  ≤𝑂(1)clo1 14682  Σcsu 14880  logclog 24823  Λcvma 25355  ψcchp 25356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-addf 10469  ax-mulf 10470
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-disj 4937  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-supp 7689  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-2o 7961  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-fsupp 8687  df-fi 8728  df-sup 8759  df-inf 8760  df-oi 8827  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-xnn0 11822  df-z 11836  df-dec 11953  df-uz 12098  df-q 12202  df-rp 12244  df-xneg 12361  df-xadd 12362  df-xmul 12363  df-ioo 12596  df-ioc 12597  df-ico 12598  df-icc 12599  df-fz 12747  df-fzo 12888  df-fl 13016  df-mod 13092  df-seq 13224  df-exp 13284  df-fac 13488  df-bc 13517  df-hash 13545  df-shft 14264  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-limsup 14666  df-clim 14683  df-rlim 14684  df-o1 14685  df-lo1 14686  df-sum 14881  df-ef 15258  df-e 15259  df-sin 15260  df-cos 15261  df-tan 15262  df-pi 15263  df-dvds 15445  df-gcd 15681  df-prm 15849  df-pc 16007  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-mulr 16412  df-starv 16413  df-sca 16414  df-vsca 16415  df-ip 16416  df-tset 16417  df-ple 16418  df-ds 16420  df-unif 16421  df-hom 16422  df-cco 16423  df-rest 16529  df-topn 16530  df-0g 16548  df-gsum 16549  df-topgen 16550  df-pt 16551  df-prds 16554  df-xrs 16608  df-qtop 16613  df-imas 16614  df-xps 16616  df-mre 16690  df-mrc 16691  df-acs 16693  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-submnd 17779  df-mulg 17986  df-cntz 18192  df-cmn 18639  df-psmet 20223  df-xmet 20224  df-met 20225  df-bl 20226  df-mopn 20227  df-fbas 20228  df-fg 20229  df-cnfld 20232  df-top 21190  df-topon 21207  df-topsp 21229  df-bases 21242  df-cld 21315  df-ntr 21316  df-cls 21317  df-nei 21394  df-lp 21432  df-perf 21433  df-cn 21523  df-cnp 21524  df-haus 21611  df-cmp 21683  df-tx 21858  df-hmeo 22051  df-fil 22142  df-fm 22234  df-flim 22235  df-flf 22236  df-xms 22617  df-ms 22618  df-tms 22619  df-cncf 23173  df-limc 24151  df-dv 24152  df-ulm 24652  df-log 24825  df-cxp 24826  df-atan 25130  df-em 25256  df-cht 25360  df-vma 25361  df-chp 25362  df-ppi 25363  df-mu 25364
This theorem is referenced by:  pntrlog2bndlem6  25845
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