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Theorem pntrlog2bndlem6 25471
Description: Lemma for pntrlog2bnd 25472. 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)
pntrlog2bndlem6.1 (𝜑𝐴 ∈ ℝ)
pntrlog2bndlem6.2 (𝜑 → 1 ≤ 𝐴)
Assertion
Ref Expression
pntrlog2bndlem6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
Distinct variable groups:   𝑖,𝑎,𝑛,𝑥,𝑦,𝐴   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 pntrlog2bndlem6
StepHypRef Expression
1 elioore 12398 . . . . . . . . . . . . 13 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
21adantl 473 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3 1rp 12029 . . . . . . . . . . . . 13 1 ∈ ℝ+
43a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
5 1red 10247 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6 eliooord 12426 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
76adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
87simpld 477 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
95, 2, 8ltled 10377 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
102, 4, 9rpgecld 12104 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
11 pntrlog2bnd.r . . . . . . . . . . . . 13 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
1211pntrf 25451 . . . . . . . . . . . 12 𝑅:ℝ+⟶ℝ
1312ffvelrni 6521 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑅𝑥) ∈ ℝ)
1410, 13syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℝ)
1514recnd 10260 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑅𝑥) ∈ ℂ)
1615abscld 14374 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝑅𝑥)) ∈ ℝ)
1710relogcld 24568 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
1816, 17remulcld 10262 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℝ)
19 2re 11282 . . . . . . . . . 10 2 ∈ ℝ
2019a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
212, 8rplogcld 24574 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2220, 21rerpdivcld 12096 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
23 fzfid 12966 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2410adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
25 elfznn 12563 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
2625adantl 473 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2726nnrpd 12063 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
2824, 27rpdivcld 12082 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
2912ffvelrni 6521 . . . . . . . . . . . . 13 ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
3028, 29syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
3130recnd 10260 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
3231abscld 14374 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
3327relogcld 24568 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
3432, 33remulcld 10262 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
3523, 34fsumrecl 14664 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
3622, 35remulcld 10262 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
3718, 36resubcld 10650 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℝ)
3837recnd 10260 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) ∈ ℂ)
39 fzfid 12966 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ∈ Fin)
40 ssun2 3920 . . . . . . . . . . 11 (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))
41 pntsval.1 . . . . . . . . . . . 12 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
42 pntrlog2bnd.t . . . . . . . . . . . 12 𝑇 = (𝑎 ∈ ℝ ↦ if(𝑎 ∈ ℝ+, (𝑎 · (log‘𝑎)), 0))
43 pntrlog2bndlem5.1 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ+)
44 pntrlog2bndlem5.2 . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
45 pntrlog2bndlem6.1 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ)
46 pntrlog2bndlem6.2 . . . . . . . . . . . 12 (𝜑 → 1 ≤ 𝐴)
4741, 11, 42, 43, 44, 45, 46pntrlog2bndlem6a 25470 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))))
4840, 47syl5sseqr 3795 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑥)))
4948sselda 3744 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ (1...(⌊‘𝑥)))
5049, 34syldan 488 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
5139, 50fsumrecl 14664 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
5222, 51remulcld 10262 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
5352recnd 10260 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
542recnd 10260 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
5510rpne0d 12070 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
5638, 53, 54, 55divdird 11031 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)))
5718recnd 10260 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘(𝑅𝑥)) · (log‘𝑥)) ∈ ℂ)
5836recnd 10260 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
5957, 58, 53subsubd 10612 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) = ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
6022recnd 10260 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℂ)
6135recnd 10260 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
6251recnd 10260 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
6360, 61, 62subdid 10678 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
643a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑 → 1 ∈ ℝ+)
6545, 64, 46rpgecld 12104 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ ℝ+)
6665adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ+)
672, 66rerpdivcld 12096 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ)
68 reflcl 12791 . . . . . . . . . . . . . . 15 ((𝑥 / 𝐴) ∈ ℝ → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
6967, 68syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) ∈ ℝ)
7069ltp1d 11146 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1))
71 fzdisj 12561 . . . . . . . . . . . . 13 ((⌊‘(𝑥 / 𝐴)) < ((⌊‘(𝑥 / 𝐴)) + 1) → ((1...(⌊‘(𝑥 / 𝐴))) ∩ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅)
7270, 71syl 17 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((1...(⌊‘(𝑥 / 𝐴))) ∩ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) = ∅)
7334recnd 10260 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
7472, 47, 23, 73fsumsplit 14670 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
7574oveq1d 6828 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = ((Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
76 fzfid 12966 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ∈ Fin)
77 ssun1 3919 . . . . . . . . . . . . . . . 16 (1...(⌊‘(𝑥 / 𝐴))) ⊆ ((1...(⌊‘(𝑥 / 𝐴))) ∪ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)))
7877, 47syl5sseqr 3795 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘(𝑥 / 𝐴))) ⊆ (1...(⌊‘𝑥)))
7978sselda 3744 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → 𝑛 ∈ (1...(⌊‘𝑥)))
8079, 34syldan 488 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
8176, 80fsumrecl 14664 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
8281recnd 10260 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
8382, 62pncand 10585 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
8475, 83eqtrd 2794 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))
8584oveq2d 6829 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · (Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) − Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
8663, 85eqtr3d 2796 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))
8786oveq2d 6829 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘(𝑅𝑥)) · (log‘𝑥)) − (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))))) = (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
8859, 87eqtr3d 2796 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) = (((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))))
8988oveq1d 6828 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) + ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) = ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
9056, 89eqtr3d 2796 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)) = ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥))
9190mpteq2dva 4896 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) = (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)))
9237, 10rerpdivcld 12096 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) ∈ ℝ)
9352, 10rerpdivcld 12096 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ∈ ℝ)
9441, 11, 42, 43, 44pntrlog2bndlem5 25469 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
95 ioossre 12428 . . . . 5 (1(,)+∞) ⊆ ℝ
9695a1i 11 . . . 4 (𝜑 → (1(,)+∞) ⊆ ℝ)
97 1red 10247 . . . 4 (𝜑 → 1 ∈ ℝ)
9819a1i 11 . . . . 5 (𝜑 → 2 ∈ ℝ)
9943rpred 12065 . . . . . 6 (𝜑𝐵 ∈ ℝ)
10065relogcld 24568 . . . . . . 7 (𝜑 → (log‘𝐴) ∈ ℝ)
101100, 97readdcld 10261 . . . . . 6 (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
10299, 101remulcld 10262 . . . . 5 (𝜑 → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
10398, 102remulcld 10262 . . . 4 (𝜑 → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
10451, 21rerpdivcld 12096 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
10599adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ)
10666relogcld 24568 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℝ)
107106, 5readdcld 10261 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℝ)
108105, 107remulcld 10262 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℝ)
1092, 108remulcld 10262 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
110 2rp 12030 . . . . . . . . . 10 2 ∈ ℝ+
111110a1i 11 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ+)
112111rpge0d 12069 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 2)
113105, 2remulcld 10262 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℝ)
11449, 25syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
115114nnrecred 11258 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
11639, 115fsumrecl 14664 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
117113, 116remulcld 10262 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
11821adantr 472 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ+)
11950, 118rerpdivcld 12096 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
120105adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℝ)
1212adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
122120, 121remulcld 10262 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℝ)
123122, 115remulcld 10262 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) · (1 / 𝑛)) ∈ ℝ)
12449, 32syldan 488 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ∈ ℝ)
125121, 114nndivred 11261 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
126120, 125remulcld 10262 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) ∈ ℝ)
12749, 27syldan 488 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
128127relogcld 24568 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
12910adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
130129relogcld 24568 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ)
13149, 31syldan 488 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
132131absge0d 14382 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
133 elfzle2 12538 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥)) → 𝑛 ≤ (⌊‘𝑥))
134133adantl 473 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≤ (⌊‘𝑥))
135114nnzd 11673 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
136 flge 12800 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛𝑥𝑛 ≤ (⌊‘𝑥)))
137121, 135, 136syl2anc 696 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛𝑥𝑛 ≤ (⌊‘𝑥)))
138134, 137mpbird 247 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛𝑥)
139127, 129logled 24572 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑛𝑥 ↔ (log‘𝑛) ≤ (log‘𝑥)))
140138, 139mpbid 222 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (log‘𝑛) ≤ (log‘𝑥))
141128, 130, 124, 132, 140lemul2ad 11156 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑥)))
14250, 124, 118ledivmul2d 12119 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛))) ↔ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ≤ ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑥))))
143141, 142mpbird 247 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (abs‘(𝑅‘(𝑥 / 𝑛))))
144125recnd 10260 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
14549, 28syldan 488 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
146145rpne0d 12070 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
147131, 144, 146absdivd 14393 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))))
14810rpge0d 12069 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
149148adantr 472 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ 𝑥)
150121, 127, 149divge0d 12105 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
151125, 150absidd 14360 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
152151oveq2d 6829 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
153147, 152eqtrd 2794 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) = ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)))
154 fveq2 6352 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑥 / 𝑛) → (𝑅𝑦) = (𝑅‘(𝑥 / 𝑛)))
155 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
156154, 155oveq12d 6831 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑥 / 𝑛) → ((𝑅𝑦) / 𝑦) = ((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛)))
157156fveq2d 6356 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑥 / 𝑛) → (abs‘((𝑅𝑦) / 𝑦)) = (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))))
158157breq1d 4814 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑥 / 𝑛) → ((abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵 ↔ (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵))
15944ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ+ (abs‘((𝑅𝑦) / 𝑦)) ≤ 𝐵)
160158, 159, 145rspcdva 3455 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / 𝑛)) / (𝑥 / 𝑛))) ≤ 𝐵)
161153, 160eqbrtrrd 4828 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵)
162124, 120, 145ledivmul2d 12119 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) / (𝑥 / 𝑛)) ≤ 𝐵 ↔ (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛))))
163161, 162mpbid 222 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (abs‘(𝑅‘(𝑥 / 𝑛))) ≤ (𝐵 · (𝑥 / 𝑛)))
164119, 124, 126, 143, 163letrd 10386 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝐵 · (𝑥 / 𝑛)))
165120recnd 10260 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝐵 ∈ ℂ)
16654adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
167114nncnd 11228 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
168114nnne0d 11257 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → 𝑛 ≠ 0)
169165, 166, 167, 168divassd 11028 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = (𝐵 · (𝑥 / 𝑛)))
170165, 166mulcld 10252 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · 𝑥) ∈ ℂ)
171170, 167, 168divrecd 10996 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((𝐵 · 𝑥) / 𝑛) = ((𝐵 · 𝑥) · (1 / 𝑛)))
172169, 171eqtr3d 2796 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (𝐵 · (𝑥 / 𝑛)) = ((𝐵 · 𝑥) · (1 / 𝑛)))
173164, 172breqtrd 4830 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ ((𝐵 · 𝑥) · (1 / 𝑛)))
17439, 119, 123, 173fsumle 14730 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((𝐵 · 𝑥) · (1 / 𝑛)))
17517recnd 10260 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
17649, 73syldan 488 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → ((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
17721rpne0d 12070 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
17839, 175, 176, 177fsumdivc 14717 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)))
179105recnd 10260 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℂ)
180179, 54mulcld 10252 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐵 · 𝑥) ∈ ℂ)
181115recnd 10260 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
18239, 180, 181fsummulc2 14715 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((𝐵 · 𝑥) · (1 / 𝑛)))
183174, 178, 1823brtr4d 4836 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)))
18443adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐵 ∈ ℝ+)
185184rpge0d 12069 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐵)
186105, 2, 185, 148mulge0d 10796 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐵 · 𝑥))
18726nnrecred 11258 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
18823, 187fsumrecl 14664 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
18917, 106resubcld 10650 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ∈ ℝ)
19017, 5readdcld 10261 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
19179, 187syldan 488 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
19276, 191fsumrecl 14664 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
193 harmonicubnd 24935 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
1942, 9, 193syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
19510, 66relogdivd 24571 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) = ((log‘𝑥) − (log‘𝐴)))
19610, 66rpdivcld 12082 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / 𝐴) ∈ ℝ+)
197 harmoniclbnd 24934 . . . . . . . . . . . . . . 15 ((𝑥 / 𝐴) ∈ ℝ+ → (log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
198196, 197syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘(𝑥 / 𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
199195, 198eqbrtrrd 4828 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) − (log‘𝐴)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛))
200188, 189, 190, 192, 194, 199le2subd 10839 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) ≤ (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))))
20126nncnd 11228 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
20226nnne0d 11257 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
203201, 202reccld 10986 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
20472, 47, 23, 203fsumsplit 14670 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)))
205204oveq1d 6828 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = ((Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)))
20679, 25syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → 𝑛 ∈ ℕ)
207206nnrecred 11258 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))) → (1 / 𝑛) ∈ ℝ)
20876, 207fsumrecl 14664 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℝ)
209208recnd 10260 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) ∈ ℂ)
210116recnd 10260 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ∈ ℂ)
211209, 210pncan2d 10586 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))
212205, 211eqtrd 2794 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛))
213 1cnd 10248 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
214106recnd 10260 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝐴) ∈ ℂ)
215175, 213, 214pnncand 10623 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = (1 + (log‘𝐴)))
216213, 214addcomd 10430 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (log‘𝐴)) = ((log‘𝐴) + 1))
217215, 216eqtrd 2794 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) − ((log‘𝑥) − (log‘𝐴))) = ((log‘𝐴) + 1))
218200, 212, 2173brtr3d 4835 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
219116, 107, 113, 186, 218lemul2ad 11156 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ ((𝐵 · 𝑥) · ((log‘𝐴) + 1)))
220107recnd 10260 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
221179, 54, 220mulassd 10255 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝐵 · (𝑥 · ((log‘𝐴) + 1))))
222179, 54, 220mul12d 10437 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐵 · (𝑥 · ((log‘𝐴) + 1))) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
223221, 222eqtrd 2794 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · ((log‘𝐴) + 1)) = (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
224219, 223breqtrd 4830 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐵 · 𝑥) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))(1 / 𝑛)) ≤ (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
225104, 117, 109, 183, 224letrd 10386 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥)) ≤ (𝑥 · (𝐵 · ((log‘𝐴) + 1))))
226104, 109, 20, 112, 225lemul2ad 11156 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥))) ≤ (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1)))))
227 2cnd 11285 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
228227, 175, 62, 177div32d 11016 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) = (2 · (Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)) / (log‘𝑥))))
229214, 213addcld 10251 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝐴) + 1) ∈ ℂ)
230179, 229mulcld 10252 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐵 · ((log‘𝐴) + 1)) ∈ ℂ)
23154, 227, 230mul12d 10437 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))) = (2 · (𝑥 · (𝐵 · ((log‘𝐴) + 1)))))
232226, 228, 2313brtr4d 4836 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ≤ (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1)))))
233103adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (𝐵 · ((log‘𝐴) + 1))) ∈ ℝ)
23452, 233, 10ledivmuld 12118 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))) ↔ ((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) ≤ (𝑥 · (2 · (𝐵 · ((log‘𝐴) + 1))))))
235232, 234mpbird 247 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))))
236235adantrr 755 . . . 4 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥) ≤ (2 · (𝐵 · ((log‘𝐴) + 1))))
23796, 93, 97, 103, 236ello1d 14453 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥)) ∈ ≤𝑂(1))
23892, 93, 94, 237lo1add 14556 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥) + (((2 / (log‘𝑥)) · Σ𝑛 ∈ (((⌊‘(𝑥 / 𝐴)) + 1)...(⌊‘𝑥))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛))) / 𝑥))) ∈ ≤𝑂(1))
23991, 238eqeltrrd 2840 1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((abs‘(𝑅𝑥)) · (log‘𝑥)) − ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝐴)))((abs‘(𝑅‘(𝑥 / 𝑛))) · (log‘𝑛)))) / 𝑥)) ∈ ≤𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cun 3713  cin 3714  wss 3715  c0 4058  ifcif 4230   class class class wbr 4804  cmpt 4881  cfv 6049  (class class class)co 6813  cc 10126  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133  +∞cpnf 10263   < clt 10266  cle 10267  cmin 10458   / cdiv 10876  cn 11212  2c2 11262  cz 11569  +crp 12025  (,)cioo 12368  ...cfz 12519  cfl 12785  abscabs 14173  ≤𝑂(1)clo1 14417  Σcsu 14615  logclog 24500  Λcvma 25017  ψcchp 25018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207  ax-mulf 10208
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-fi 8482  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-xnn0 11556  df-z 11570  df-dec 11686  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ioo 12372  df-ioc 12373  df-ico 12374  df-icc 12375  df-fz 12520  df-fzo 12660  df-fl 12787  df-mod 12863  df-seq 12996  df-exp 13055  df-fac 13255  df-bc 13284  df-hash 13312  df-shft 14006  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-limsup 14401  df-clim 14418  df-rlim 14419  df-o1 14420  df-lo1 14421  df-sum 14616  df-ef 14997  df-e 14998  df-sin 14999  df-cos 15000  df-pi 15002  df-dvds 15183  df-gcd 15419  df-prm 15588  df-pc 15744  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-starv 16158  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-unif 16167  df-hom 16168  df-cco 16169  df-rest 16285  df-topn 16286  df-0g 16304  df-gsum 16305  df-topgen 16306  df-pt 16307  df-prds 16310  df-xrs 16364  df-qtop 16369  df-imas 16370  df-xps 16372  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-mulg 17742  df-cntz 17950  df-cmn 18395  df-psmet 19940  df-xmet 19941  df-met 19942  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-cnfld 19949  df-top 20901  df-topon 20918  df-topsp 20939  df-bases 20952  df-cld 21025  df-ntr 21026  df-cls 21027  df-nei 21104  df-lp 21142  df-perf 21143  df-cn 21233  df-cnp 21234  df-haus 21321  df-cmp 21392  df-tx 21567  df-hmeo 21760  df-fil 21851  df-fm 21943  df-flim 21944  df-flf 21945  df-xms 22326  df-ms 22327  df-tms 22328  df-cncf 22882  df-limc 23829  df-dv 23830  df-log 24502  df-cxp 24503  df-em 24918  df-cht 25022  df-vma 25023  df-chp 25024  df-ppi 25025  df-mu 25026
This theorem is referenced by:  pntrlog2bnd  25472
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