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Theorem pntrlog2bndlem5 25467
 Description: Lemma for pntrlog2bnd 25470. 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 12396 . . . . . . . . . . . . 13 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
21adantl 473 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3 1rp 12027 . . . . . . . . . . . . 13 1 ∈ ℝ+
43a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
5 1red 10245 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6 eliooord 12424 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
76adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
87simpld 477 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
95, 2, 8ltled 10375 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
102, 4, 9rpgecld 12102 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
11 pntrlog2bnd.r . . . . . . . . . . . . 13 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
1211pntrf 25449 . . . . . . . . . . . 12 𝑅:ℝ+⟶ℝ
1312ffvelrni 6519 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑅𝑥) ∈ ℝ)
1410, 13syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℝ)
1514recnd 10258 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℂ)
1615abscld 14372 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅𝑥)) ∈ ℝ)
1716recnd 10258 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅𝑥)) ∈ ℂ)
1810relogcld 24566 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
1918recnd 10258 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2017, 19mulcld 10250 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℂ)
21 2cnd 11283 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
222, 8rplogcld 24572 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2322rpne0d 12068 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
2421, 19, 23divcld 10991 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
25 fzfid 12964 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2610adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
27 elfznn 12561 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2827adantl 473 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2928nnrpd 12061 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
3026, 29rpdivcld 12080 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
3112ffvelrni 6519 . . . . . . . . . . . . 13 ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
3230, 31syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
3332recnd 10258 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
3433abscld 14372 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
3529relogcld 24566 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
36 1red 10245 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
3735, 36readdcld 10259 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℝ)
3834, 37remulcld 10260 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
3938recnd 10258 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
4025, 39fsumcl 14661 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℂ)
4124, 40mulcld 10250 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℂ)
4220, 41subcld 10582 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℂ)
4334recnd 10258 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
4425, 43fsumcl 14661 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
4524, 44mulcld 10250 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℂ)
462recnd 10258 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
4710rpne0d 12068 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
4842, 45, 46, 47divdird 11029 . . . 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 10260 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℝ)
5049recnd 10258 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℂ)
5150, 41, 45subsubd 10610 . . . . . 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 10676 . . . . . . . 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 14717 . . . . . . . . . 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 10258 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + 1) ∈ ℂ)
55 1cnd 10246 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
5643, 54, 55subdid 10676 . . . . . . . . . . . 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 10258 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
5857, 55pncand 10583 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘𝑛) + 1) − 1) = (log‘𝑛))
5958oveq2d 6827 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (((log‘𝑛) + 1) − 1)) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
6043mulid1d 10247 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · 1) = (abs‘(𝑅‘(𝑥 / 𝑛))))
6160oveq2d 6827 . . . . . . . . . . . 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 2801 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) − (abs‘(𝑅‘(𝑥 / 𝑛)))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
6362sumeq2dv 14630 . . . . . . . . . 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 2794 . . . . . . . . 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 6827 . . . . . . . 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 2794 . . . . . . 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 6827 . . . . . 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 2794 . . . . 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 6826 . . . 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 2794 . . 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 4894 . 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 11280 . . . . . . . 8 2 ∈ ℝ
7372a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
7473, 22rerpdivcld 12094 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
7525, 38fsumrecl 14662 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)) ∈ ℝ)
7674, 75remulcld 10260 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1))) ∈ ℝ)
7749, 76resubcld 10648 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) ∈ ℝ)
7877, 10rerpdivcld 12094 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((log‘𝑛) + 1)))) / 𝑥) ∈ ℝ)
7925, 34fsumrecl 14662 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
8074, 79remulcld 10260 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
8180, 10rerpdivcld 12094 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
82 1red 10245 . . . 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 25466 . . . . 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 11225 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
88 simpl 474 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → 𝑎 ∈ ℝ)
89 simpr 479 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → 𝑎 ∈ ℝ+)
9089relogcld 24566 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → (log‘𝑎) ∈ ℝ)
9188, 90remulcld 10260 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℝ ∧ 𝑎 ∈ ℝ+) → (𝑎 · (log‘𝑎)) ∈ ℝ)
92 0red 10231 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℝ ∧ ¬ 𝑎 ∈ ℝ+) → 0 ∈ ℝ)
9391, 92ifclda 4262 . . . . . . . . . . . . 13 (𝑎 ∈ ℝ → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) ∈ ℝ)
9484, 93fmpti 6544 . . . . . . . . . . . 12 𝑇:ℝ⟶ℝ
9594ffvelrni 6519 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (𝑇𝑛) ∈ ℝ)
9687, 95syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇𝑛) ∈ ℝ)
9787, 36resubcld 10648 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
9894ffvelrni 6519 . . . . . . . . . . 11 ((𝑛 − 1) ∈ ℝ → (𝑇‘(𝑛 − 1)) ∈ ℝ)
9997, 98syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑇‘(𝑛 − 1)) ∈ ℝ)
10096, 99resubcld 10648 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ∈ ℝ)
10134, 100remulcld 10260 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
10225, 101fsumrecl 14662 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))) ∈ ℝ)
10374, 102remulcld 10260 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1))))) ∈ ℝ)
10449, 103resubcld 10648 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · ((𝑇𝑛) − (𝑇‘(𝑛 − 1)))))) ∈ ℝ)
105104, 10rerpdivcld 12094 . . . 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 12028 . . . . . . . . . . 11 2 ∈ ℝ+
107106a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ+)
108107rpge0d 12067 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
10973, 22, 108divge0d 12103 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 / (log‘𝑥)))
11033absge0d 14380 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
11129adantr 472 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ+)
112111rpcnd 12065 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℂ)
11357adantr 472 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℂ)
114112, 113mulcld 10250 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘𝑛)) ∈ ℂ)
115 simpr 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 < 𝑛)
116 1re 10229 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
117111rpred 12063 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 𝑛 ∈ ℝ)
118 difrp 12059 . . . . . . . . . . . . . . . . . . 19 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ+))
119116, 117, 118sylancr 698 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 < 𝑛 ↔ (𝑛 − 1) ∈ ℝ+))
120115, 119mpbid 222 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 − 1) ∈ ℝ+)
121120relogcld 24566 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℝ)
122121recnd 10258 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘(𝑛 − 1)) ∈ ℂ)
123112, 122mulcld 10250 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · (log‘(𝑛 − 1))) ∈ ℂ)
124114, 123, 122subsubd 10610 . . . . . . . . . . . . 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 12030 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
126 eleq1 2825 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑛 → (𝑎 ∈ ℝ+𝑛 ∈ ℝ+))
127 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑛𝑎 = 𝑛)
128 fveq2 6350 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑛 → (log‘𝑎) = (log‘𝑛))
129127, 128oveq12d 6829 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑛 → (𝑎 · (log‘𝑎)) = (𝑛 · (log‘𝑛)))
130126, 129ifbieq1d 4251 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑛 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0))
131 ovex 6839 . . . . . . . . . . . . . . . . . . 19 (𝑛 · (log‘𝑛)) ∈ V
132 c0ex 10224 . . . . . . . . . . . . . . . . . . 19 0 ∈ V
133131, 132ifex 4298 . . . . . . . . . . . . . . . . . 18 if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0) ∈ V
134130, 84, 133fvmpt 6442 . . . . . . . . . . . . . . . . 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 4234 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℝ+ → if(𝑛 ∈ ℝ+, (𝑛 · (log‘𝑛)), 0) = (𝑛 · (log‘𝑛)))
137135, 136eqtrd 2792 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℝ+ → (𝑇𝑛) = (𝑛 · (log‘𝑛)))
138111, 137syl 17 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇𝑛) = (𝑛 · (log‘𝑛)))
139 rpre 12030 . . . . . . . . . . . . . . . . . 18 ((𝑛 − 1) ∈ ℝ+ → (𝑛 − 1) ∈ ℝ)
140 eleq1 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (𝑛 − 1) → (𝑎 ∈ ℝ+ ↔ (𝑛 − 1) ∈ ℝ+))
141 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑛 − 1) → 𝑎 = (𝑛 − 1))
142 fveq2 6350 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑛 − 1) → (log‘𝑎) = (log‘(𝑛 − 1)))
143141, 142oveq12d 6829 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (𝑛 − 1) → (𝑎 · (log‘𝑎)) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
144140, 143ifbieq1d 4251 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝑛 − 1) → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0))
145 ovex 6839 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 − 1) · (log‘(𝑛 − 1))) ∈ V
146145, 132ifex 4298 . . . . . . . . . . . . . . . . . . 19 if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) ∈ V
147144, 84, 146fvmpt 6442 . . . . . . . . . . . . . . . . . 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 4234 . . . . . . . . . . . . . . . . 17 ((𝑛 − 1) ∈ ℝ+ → if((𝑛 − 1) ∈ ℝ+, ((𝑛 − 1) · (log‘(𝑛 − 1))), 0) = ((𝑛 − 1) · (log‘(𝑛 − 1))))
150148, 149eqtrd 2792 . . . . . . . . . . . . . . . 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 10246 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℂ)
153112, 152, 122subdird 10677 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · (log‘(𝑛 − 1))) = ((𝑛 · (log‘(𝑛 − 1))) − (1 · (log‘(𝑛 − 1)))))
154122mulid2d 10248 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · (log‘(𝑛 − 1))) = (log‘(𝑛 − 1)))
155154oveq2d 6827 . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑇‘(𝑛 − 1)) = ((𝑛 · (log‘(𝑛 − 1))) − (log‘(𝑛 − 1))))
157138, 156oveq12d 6829 . . . . . . . . . . . . 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 10676 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((𝑛 · (log‘𝑛)) − (𝑛 · (log‘(𝑛 − 1)))))
159158oveq1d 6826 . . . . . . . . . . . . 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 2802 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))))
161111relogcld 24566 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘𝑛) ∈ ℝ)
162161, 121resubcld 10648 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℝ)
163162recnd 10258 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ∈ ℂ)
164112, 152, 163subdird 10677 . . . . . . . . . . . . . . 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 10248 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (1 · ((log‘𝑛) − (log‘(𝑛 − 1)))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
166165oveq2d 6827 . . . . . . . . . . . . . . 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 10260 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℝ)
168167recnd 10258 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) ∈ ℂ)
169168, 113, 122subsub3d 10612 . . . . . . . . . . . . . . 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 2796 . . . . . . . . . . . . . 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 10586 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) + 1) = 𝑛)
172171fveq2d 6354 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (log‘((𝑛 − 1) + 1)) = (log‘𝑛))
173172oveq1d 6826 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘((𝑛 − 1) + 1)) − (log‘(𝑛 − 1))) = ((log‘𝑛) − (log‘(𝑛 − 1))))
174 logdifbnd 24917 . . . . . . . . . . . . . . . . 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 4826 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((log‘𝑛) − (log‘(𝑛 − 1))) ≤ (1 / (𝑛 − 1)))
177 1red 10245 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → 1 ∈ ℝ)
178162, 177, 120lemuldiv2d 12113 . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 − 1) · ((log‘𝑛) − (log‘(𝑛 − 1)))) ≤ 1)
180170, 179eqbrtrrd 4826 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → (((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) − (log‘𝑛)) ≤ 1)
181167, 121readdcld 10259 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ∈ ℝ)
182181, 161, 177lesubadd2d 10816 . . . . . . . . . . . . 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 222 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑛 · ((log‘𝑛) − (log‘(𝑛 − 1)))) + (log‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
184160, 183eqbrtrd 4824 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 < 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
185 fveq2 6350 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑇𝑛) = (𝑇‘1))
186 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 1 → 𝑎 = 1)
187186, 3syl6eqel 2845 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 1 → 𝑎 ∈ ℝ+)
188187iftrued 4236 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 1 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = (𝑎 · (log‘𝑎)))
189 fveq2 6350 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 1 → (log‘𝑎) = (log‘1))
190 log1 24529 . . . . . . . . . . . . . . . . . . . . . . 23 (log‘1) = 0
191189, 190syl6eq 2808 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 1 → (log‘𝑎) = 0)
192186, 191oveq12d 6829 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 1 → (𝑎 · (log‘𝑎)) = (1 · 0))
193 ax-1cn 10184 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℂ
194193mul01i 10416 . . . . . . . . . . . . . . . . . . . . 21 (1 · 0) = 0
195192, 194syl6eq 2808 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 1 → (𝑎 · (log‘𝑎)) = 0)
196188, 195eqtrd 2792 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 1 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = 0)
197196, 84, 132fvmpt 6442 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℝ → (𝑇‘1) = 0)
198116, 197ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑇‘1) = 0
199185, 198syl6eq 2808 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑇𝑛) = 0)
200 oveq1 6818 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → (𝑛 − 1) = (1 − 1))
201 1m1e0 11279 . . . . . . . . . . . . . . . . . . 19 (1 − 1) = 0
202200, 201syl6eq 2808 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (𝑛 − 1) = 0)
203202fveq2d 6354 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = (𝑇‘0))
204 0re 10230 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
205 rpne0 12039 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ℝ+𝑎 ≠ 0)
206205necon2bi 2960 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 0 → ¬ 𝑎 ∈ ℝ+)
207206iffalsed 4239 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 0 → if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0) = 0)
208207, 84, 132fvmpt 6442 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → (𝑇‘0) = 0)
209204, 208ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑇‘0) = 0
210203, 209syl6eq 2808 . . . . . . . . . . . . . . . 16 (𝑛 = 1 → (𝑇‘(𝑛 − 1)) = 0)
211199, 210oveq12d 6829 . . . . . . . . . . . . . . 15 (𝑛 = 1 → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = (0 − 0))
212 0m0e0 11320 . . . . . . . . . . . . . . 15 (0 − 0) = 0
213211, 212syl6eq 2808 . . . . . . . . . . . . . 14 (𝑛 = 1 → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = 0)
214213eqcoms 2766 . . . . . . . . . . . . 13 (1 = 𝑛 → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = 0)
215214adantl 473 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) = 0)
216 0red 10231 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ∈ ℝ)
21728nnge1d 11253 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
21887, 217logge0d 24573 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (log‘𝑛))
21935lep1d 11145 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ≤ ((log‘𝑛) + 1))
220216, 35, 37, 218, 219letrd 10384 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((log‘𝑛) + 1))
221220adantr 472 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → 0 ≤ ((log‘𝑛) + 1))
222215, 221eqbrtrd 4824 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 1 = 𝑛) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
223 elfzle1 12535 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘𝑥)) → 1 ≤ 𝑛)
224223adantl 473 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑛)
22536, 87leloed 10370 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 ≤ 𝑛 ↔ (1 < 𝑛 ∨ 1 = 𝑛)))
226224, 225mpbid 222 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 < 𝑛 ∨ 1 = 𝑛))
227184, 222, 226mpjaodan 862 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑇𝑛) − (𝑇‘(𝑛 − 1))) ≤ ((log‘𝑛) + 1))
228100, 37, 34, 110, 227lemul2ad 11154 . . . . . . . . 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 14728 . . . . . . . 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 11154 . . . . . . 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 10834 . . . . . 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 12121 . . . . 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 755 . . . 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 14579 . . 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 12079 . . . . . . 7 (𝜑 → (2 · 𝐵) ∈ ℝ+)
238237rpred 12063 . . . . . 6 (𝜑 → (2 · 𝐵) ∈ ℝ)
239238adantr 472 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ)
2405, 22rerpdivcld 12094 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
2415, 240readdcld 10259 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
242 ioossre 12426 . . . . . 6 (1(,)+∞) ⊆ ℝ
243 lo1const 14548 . . . . . 6 (((1(,)+∞) ⊆ ℝ ∧ (2 · 𝐵) ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
244242, 238, 243sylancr 698 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · 𝐵)) ∈ ≤𝑂(1))
245 lo1const 14548 . . . . . . 7 (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℝ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
246242, 82, 245sylancr 698 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ ≤𝑂(1))
247 divlogrlim 24578 . . . . . . . 8 (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
248 rlimo1 14544 . . . . . . . 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 14467 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ ≤𝑂(1))
2515, 240, 246, 250lo1add 14554 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ ≤𝑂(1))
252237adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · 𝐵) ∈ ℝ+)
253252rpge0d 12067 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (2 · 𝐵))
254239, 241, 244, 251, 253lo1mul 14555 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 · 𝐵) · (1 + (1 / (log‘𝑥))))) ∈ ≤𝑂(1))
255239, 241remulcld 10260 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
25679, 10rerpdivcld 12094 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
25718, 5readdcld 10259 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
258236adantr 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ+)
259258rpred 12063 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
260257, 259remulcld 10260 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) · 𝐵) ∈ ℝ)
26128nnrecred 11256 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
26225, 261fsumrecl 14662 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
263262, 259remulcld 10260 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ∈ ℝ)
26434, 26rerpdivcld 12094 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ∈ ℝ)
265259adantr 472 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
266261, 265remulcld 10260 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 / 𝑛) · 𝐵) ∈ ℝ)
26730rpcnd 12065 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
26830rpne0d 12068 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
26933, 267, 268absdivd 14391 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))))
2702adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
271270, 28nndivred 11259 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
27230rpge0d 12067 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
273271, 272absidd 14358 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
274273oveq2d 6827 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
275269, 274eqtrd 2792 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
27646adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
27787recnd 10258 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
27847adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
27928nnne0d 11255 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
28043, 276, 277, 278, 279divdiv2d 11023 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) = (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥))
28143, 277, 276, 278div23d 11028 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · 𝑛) / 𝑥) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
282275, 280, 2813eqtrd 2796 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛))
283 pntrlog2bndlem5.2 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
284283ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
285 fveq2 6350 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑥 / 𝑛) → (𝑅𝑦) = (𝑅‘(𝑥 / 𝑛)))
286 id 22 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
287285, 286oveq12d 6829 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑥 / 𝑛) → ((𝑅𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
288287fveq2d 6354 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
289288breq1d 4812 . . . . . . . . . . . . . . 15 (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
290289rspcv 3443 . . . . . . . . . . . . . 14 ((𝑥 / 𝑛) ∈ ℝ+ → (∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵 → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
29130, 284, 290sylc 65 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
292282, 291eqbrtrrd 4826 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵)
293264, 265, 29lemuldivd 12112 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) · 𝑛) ≤ 𝐵 ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛)))
294292, 293mpbid 222 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (𝐵 / 𝑛))
295265recnd 10258 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
296295, 277, 279divrec2d 10995 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐵 / 𝑛) = ((1 / 𝑛) · 𝐵))
297294, 296breqtrd 4828 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ ((1 / 𝑛) · 𝐵))
29825, 264, 266, 297fsumle 14728 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
29925, 46, 43, 47fsumdivc 14715 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥))
300258rpcnd 12065 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
301261recnd 10258 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
30225, 300, 301fsummulc1 14714 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) = Σ𝑛 ∈ (1...(⌊‘𝑥))((1 / 𝑛) · 𝐵))
303298, 299, 3023brtr4d 4834 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵))
304258rpge0d 12067 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
305 harmonicubnd 24933 . . . . . . . . . 10 ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
3062, 9, 305syl2anc 696 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
307262, 257, 259, 304, 306lemul1ad 11153 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) · 𝐵) ≤ (((log‘𝑥) + 1) · 𝐵))
308256, 263, 260, 303, 307letrd 10384 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥) ≤ (((log‘𝑥) + 1) · 𝐵))
309256, 260, 74, 109, 308lemul2ad 11154 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛))) / 𝑥)) ≤ ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
31024, 44, 46, 47divassd 11026 . . . . . 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)))
311241recnd 10258 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℂ)
31221, 300, 311mul32d 10436 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵))
313 1cnd 10246 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
31419, 313, 19, 23divdird 11029 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
31519, 23dividd 10989 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
316315oveq1d 6826 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
317314, 316eqtr2d 2793 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
318317oveq2d 6827 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
31919, 313addcld 10249 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
32021, 19, 319, 23div32d 11014 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) = (2 · (((log‘𝑥) + 1) / (log‘𝑥))))
321318, 320eqtr4d 2795 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · ((log‘𝑥) + 1)))
322321oveq1d 6826 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · (1 + (1 / (log‘𝑥)))) · 𝐵) = (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵))
32324, 319, 300mulassd 10253 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · ((log‘𝑥) + 1)) · 𝐵) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
324312, 322, 3233eqtrd 2796 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))) = ((2 / (log‘𝑥)) · (((log‘𝑥) + 1) · 𝐵)))
325309, 310, 3243brtr4d 4834 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
326325adantrr 755 . . . 4 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((2 · 𝐵) · (1 + (1 / (log‘𝑥)))))
32782, 254, 255, 81, 326lo1le 14579 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘(𝑅‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
32878, 81, 234, 327lo1add 14554 . 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))
32971, 328eqeltrrd 2838 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 196   ∨ wo 382   ∧ wa 383   = wceq 1630   ∈ wcel 2137   ≠ wne 2930  ∀wral 3048   ⊆ wss 3713  ifcif 4228   class class class wbr 4802   ↦ cmpt 4879  ‘cfv 6047  (class class class)co 6811  ℂcc 10124  ℝcr 10125  0cc0 10126  1c1 10127   + caddc 10129   · cmul 10131  +∞cpnf 10261   < clt 10264   ≤ cle 10265   − cmin 10456   / cdiv 10874  ℕcn 11210  2c2 11260  ℝ+crp 12023  (,)cioo 12366  ...cfz 12517  ⌊cfl 12783  abscabs 14171   ⇝𝑟 crli 14413  𝑂(1)co1 14414  ≤𝑂(1)clo1 14415  Σcsu 14613  logclog 24498  Λcvma 25015  ψcchp 25016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204  ax-addf 10205  ax-mulf 10206 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-disj 4771  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-fi 8480  df-sup 8511  df-inf 8512  df-oi 8578  df-card 8953  df-cda 9180  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-xnn0 11554  df-z 11568  df-dec 11684  df-uz 11878  df-q 11980  df-rp 12024  df-xneg 12137  df-xadd 12138  df-xmul 12139  df-ioo 12370  df-ioc 12371  df-ico 12372  df-icc 12373  df-fz 12518  df-fzo 12658  df-fl 12785  df-mod 12861  df-seq 12994  df-exp 13053  df-fac 13253  df-bc 13282  df-hash 13310  df-shft 14004  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-limsup 14399  df-clim 14416  df-rlim 14417  df-o1 14418  df-lo1 14419  df-sum 14614  df-ef 14995  df-e 14996  df-sin 14997  df-cos 14998  df-pi 15000  df-dvds 15181  df-gcd 15417  df-prm 15586  df-pc 15742  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-mulr 16155  df-starv 16156  df-sca 16157  df-vsca 16158  df-ip 16159  df-tset 16160  df-ple 16161  df-ds 16164  df-unif 16165  df-hom 16166  df-cco 16167  df-rest 16283  df-topn 16284  df-0g 16302  df-gsum 16303  df-topgen 16304  df-pt 16305  df-prds 16308  df-xrs 16362  df-qtop 16367  df-imas 16368  df-xps 16370  df-mre 16446  df-mrc 16447  df-acs 16449  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-submnd 17535  df-mulg 17740  df-cntz 17948  df-cmn 18393  df-psmet 19938  df-xmet 19939  df-met 19940  df-bl 19941  df-mopn 19942  df-fbas 19943  df-fg 19944  df-cnfld 19947  df-top 20899  df-topon 20916  df-topsp 20937  df-bases 20950  df-cld 21023  df-ntr 21024  df-cls 21025  df-nei 21102  df-lp 21140  df-perf 21141  df-cn 21231  df-cnp 21232  df-haus 21319  df-cmp 21390  df-tx 21565  df-hmeo 21758  df-fil 21849  df-fm 21941  df-flim 21942  df-flf 21943  df-xms 22324  df-ms 22325  df-tms 22326  df-cncf 22880  df-limc 23827  df-dv 23828  df-log 24500  df-cxp 24501  df-em 24916  df-cht 25020  df-vma 25021  df-chp 25022  df-ppi 25023  df-mu 25024 This theorem is referenced by:  pntrlog2bndlem6  25469
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