MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  selberg3lem1 Structured version   Visualization version   GIF version

Theorem selberg3lem1 25537
Description: Introduce a log weighting on the summands of Σ𝑚 · 𝑛𝑥, Λ(𝑚)Λ(𝑛), the core of selberg2 25531 (written here as Σ𝑛𝑥, Λ(𝑛)ψ(𝑥 / 𝑛)). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Hypotheses
Ref Expression
selberg3lem1.1 (𝜑𝐴 ∈ ℝ+)
selberg3lem1.2 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)
Assertion
Ref Expression
selberg3lem1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))
Distinct variable groups:   𝑘,𝑛,𝑥,𝑦,𝐴   𝜑,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑘)

Proof of Theorem selberg3lem1
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 1red 10294 . 2 (𝜑 → 1 ∈ ℝ)
2 ioossre 12437 . . . 4 (1(,)+∞) ⊆ ℝ
3 selberg3lem1.1 . . . . 5 (𝜑𝐴 ∈ ℝ+)
43rpcnd 12072 . . . 4 (𝜑𝐴 ∈ ℂ)
5 o1const 14637 . . . 4 (((1(,)+∞) ⊆ ℝ ∧ 𝐴 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
62, 4, 5sylancr 581 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
7 fzfid 12980 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
8 elfznn 12577 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
98adantl 473 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
10 vmacl 25135 . . . . . . . . 9 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
119, 10syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
1211, 9nndivred 11326 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
137, 12fsumrecl 14752 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
14 elioore 12407 . . . . . . . . 9 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
15 eliooord 12435 . . . . . . . . . 10 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
1615simpld 488 . . . . . . . . 9 (𝑥 ∈ (1(,)+∞) → 1 < 𝑥)
1714, 16rplogcld 24666 . . . . . . . 8 (𝑥 ∈ (1(,)+∞) → (log‘𝑥) ∈ ℝ+)
18 rpdivcl 12054 . . . . . . . 8 ((𝐴 ∈ ℝ+ ∧ (log‘𝑥) ∈ ℝ+) → (𝐴 / (log‘𝑥)) ∈ ℝ+)
193, 17, 18syl2an 589 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 / (log‘𝑥)) ∈ ℝ+)
2019rpred 12070 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 / (log‘𝑥)) ∈ ℝ)
2113, 20remulcld 10324 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) ∈ ℝ)
2221recnd 10322 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) ∈ ℂ)
234adantr 472 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℂ)
2413recnd 10322 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
2517adantl 473 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2625rpcnd 12072 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2719rpcnd 12072 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 / (log‘𝑥)) ∈ ℂ)
2824, 26, 27subdird 10741 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − ((log‘𝑥) · (𝐴 / (log‘𝑥)))))
2925rpne0d 12075 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
3023, 26, 29divcan2d 11057 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) · (𝐴 / (log‘𝑥))) = 𝐴)
3130oveq2d 6858 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − ((log‘𝑥) · (𝐴 / (log‘𝑥)))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴))
3228, 31eqtrd 2799 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴))
3332mpteq2dva 4903 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴)))
3425rpred 12070 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
3513, 34resubcld 10712 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
3614adantl 473 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
37 0red 10297 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ∈ ℝ)
38 1red 10294 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
39 0lt1 10804 . . . . . . . . . . . 12 0 < 1
4039a1i 11 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 < 1)
4116adantl 473 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
4237, 38, 36, 40, 41lttrd 10452 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 < 𝑥)
4336, 42elrpd 12067 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
4443ex 401 . . . . . . . 8 (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
4544ssrdv 3767 . . . . . . 7 (𝜑 → (1(,)+∞) ⊆ ℝ+)
46 vmadivsum 25462 . . . . . . . 8 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
4746a1i 11 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
4845, 47o1res2 14581 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
492a1i 11 . . . . . . 7 (𝜑 → (1(,)+∞) ⊆ ℝ)
50 ere 15103 . . . . . . . 8 e ∈ ℝ
5150a1i 11 . . . . . . 7 (𝜑 → e ∈ ℝ)
523rpred 12070 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
5319adantrr 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / (log‘𝑥)) ∈ ℝ+)
5453rprege0d 12077 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → ((𝐴 / (log‘𝑥)) ∈ ℝ ∧ 0 ≤ (𝐴 / (log‘𝑥))))
55 absid 14323 . . . . . . . . 9 (((𝐴 / (log‘𝑥)) ∈ ℝ ∧ 0 ≤ (𝐴 / (log‘𝑥))) → (abs‘(𝐴 / (log‘𝑥))) = (𝐴 / (log‘𝑥)))
5654, 55syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (abs‘(𝐴 / (log‘𝑥))) = (𝐴 / (log‘𝑥)))
57 loge 24624 . . . . . . . . . . 11 (log‘e) = 1
58 simprr 789 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → e ≤ 𝑥)
59 epr 15220 . . . . . . . . . . . . 13 e ∈ ℝ+
6043adantrr 708 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 𝑥 ∈ ℝ+)
61 logleb 24640 . . . . . . . . . . . . 13 ((e ∈ ℝ+𝑥 ∈ ℝ+) → (e ≤ 𝑥 ↔ (log‘e) ≤ (log‘𝑥)))
6259, 60, 61sylancr 581 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (e ≤ 𝑥 ↔ (log‘e) ≤ (log‘𝑥)))
6358, 62mpbid 223 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (log‘e) ≤ (log‘𝑥))
6457, 63syl5eqbrr 4845 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 1 ≤ (log‘𝑥))
65 1rp 12032 . . . . . . . . . . . 12 1 ∈ ℝ+
66 rpregt0 12044 . . . . . . . . . . . 12 (1 ∈ ℝ+ → (1 ∈ ℝ ∧ 0 < 1))
6765, 66mp1i 13 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (1 ∈ ℝ ∧ 0 < 1))
6825adantrr 708 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (log‘𝑥) ∈ ℝ+)
6968rpregt0d 12076 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → ((log‘𝑥) ∈ ℝ ∧ 0 < (log‘𝑥)))
703adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 𝐴 ∈ ℝ+)
7170rpregt0d 12076 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
72 lediv2 11167 . . . . . . . . . . 11 (((1 ∈ ℝ ∧ 0 < 1) ∧ ((log‘𝑥) ∈ ℝ ∧ 0 < (log‘𝑥)) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ (log‘𝑥) ↔ (𝐴 / (log‘𝑥)) ≤ (𝐴 / 1)))
7367, 69, 71, 72syl3anc 1490 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (1 ≤ (log‘𝑥) ↔ (𝐴 / (log‘𝑥)) ≤ (𝐴 / 1)))
7464, 73mpbid 223 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / (log‘𝑥)) ≤ (𝐴 / 1))
754adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → 𝐴 ∈ ℂ)
7675div1d 11047 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / 1) = 𝐴)
7774, 76breqtrd 4835 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (𝐴 / (log‘𝑥)) ≤ 𝐴)
7856, 77eqbrtrd 4831 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ e ≤ 𝑥)) → (abs‘(𝐴 / (log‘𝑥))) ≤ 𝐴)
7949, 27, 51, 52, 78elo1d 14554 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 / (log‘𝑥))) ∈ 𝑂(1))
8035, 20, 48, 79o1mul2 14642 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (𝐴 / (log‘𝑥)))) ∈ 𝑂(1))
8133, 80eqeltrrd 2845 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) − 𝐴)) ∈ 𝑂(1))
8222, 23, 81o1dif 14647 . . 3 (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1)))
836, 82mpbird 248 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))) ∈ 𝑂(1))
84 2re 11346 . . . . . . 7 2 ∈ ℝ
85 rerpdivcl 12059 . . . . . . 7 ((2 ∈ ℝ ∧ (log‘𝑥) ∈ ℝ+) → (2 / (log‘𝑥)) ∈ ℝ)
8684, 25, 85sylancr 581 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 / (log‘𝑥)) ∈ ℝ)
87 nndivre 11313 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
8836, 8, 87syl2an 589 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
89 chpcl 25141 . . . . . . . . . 10 ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
9088, 89syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
9111, 90remulcld 10324 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
929nnrpd 12068 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
9392relogcld 24660 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
9491, 93remulcld 10324 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
957, 94fsumrecl 14752 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
9686, 95remulcld 10324 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℝ)
977, 91fsumrecl 14752 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
9896, 97resubcld 10712 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
9998, 43rerpdivcld 12101 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
10099recnd 10322 . 2 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
101100abscld 14462 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ℝ)
10222abscld 14462 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))) ∈ ℝ)
103 2cnd 11350 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
10495recnd 10322 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
105103, 104mulcld 10314 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
10697recnd 10322 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
107106, 26mulcld 10314 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) ∈ ℂ)
108105, 107subcld 10646 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) ∈ ℂ)
109108abscld 14462 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ∈ ℝ)
11042gt0ne0d 10846 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
111109, 36, 110redivcld 11107 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) ∈ ℝ)
11252adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
11313, 112remulcld 10324 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) ∈ ℝ)
11411recnd 10322 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
115 fzfid 12980 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
116 elfznn 12577 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
117116adantl 473 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
118 vmacl 25135 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
119117, 118syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
120117nnrpd 12068 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ+)
121120relogcld 24660 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
122119, 121remulcld 10324 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
123115, 122fsumrecl 14752 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
1248nnrpd 12068 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
125 rpdivcl 12054 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
12643, 124, 125syl2an 589 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
127126relogcld 24660 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
12890, 127remulcld 10324 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
129123, 128resubcld 10712 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
130129recnd 10322 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
131114, 130mulcld 10314 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
1327, 131fsumcl 14751 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
133132abscld 14462 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ∈ ℝ)
134131abscld 14462 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ∈ ℝ)
1357, 134fsumrecl 14752 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ∈ ℝ)
136112, 36remulcld 10324 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 · 𝑥) ∈ ℝ)
13713, 136remulcld 10324 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) ∈ ℝ)
1387, 131fsumabs 14819 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
13952ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
14036adantr 472 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
141139, 140remulcld 10324 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · 𝑥) ∈ ℝ)
14212, 141remulcld 10324 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) ∈ ℝ)
143130abscld 14462 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ∈ ℝ)
144141, 9nndivred 11326 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 · 𝑥) / 𝑛) ∈ ℝ)
145 vmage0 25138 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
1469, 145syl 17 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
14788recnd 10322 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
148126rpne0d 12075 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ≠ 0)
149130, 147, 148absdivd 14481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (abs‘(𝑥 / 𝑛))))
150126rpge0d 12074 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / 𝑛))
15188, 150absidd 14448 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 / 𝑛)) = (𝑥 / 𝑛))
152151oveq2d 6858 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (abs‘(𝑥 / 𝑛))) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)))
153149, 152eqtrd 2799 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) = ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)))
154 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑚 → (Λ‘𝑘) = (Λ‘𝑚))
155 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑚 → (log‘𝑘) = (log‘𝑚))
156154, 155oveq12d 6860 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → ((Λ‘𝑘) · (log‘𝑘)) = ((Λ‘𝑚) · (log‘𝑚)))
157156cbvsumv 14713 . . . . . . . . . . . . . . . . . . . . . 22 Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚))
158 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑥 / 𝑛) → (⌊‘𝑦) = (⌊‘(𝑥 / 𝑛)))
159158oveq2d 6858 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑥 / 𝑛) → (1...(⌊‘𝑦)) = (1...(⌊‘(𝑥 / 𝑛))))
160159sumeq1d 14718 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑥 / 𝑛) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · (log‘𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))
161157, 160syl5eq 2811 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / 𝑛) → Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)))
162 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑥 / 𝑛) → (ψ‘𝑦) = (ψ‘(𝑥 / 𝑛)))
163 fveq2 6375 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑥 / 𝑛) → (log‘𝑦) = (log‘(𝑥 / 𝑛)))
164162, 163oveq12d 6860 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / 𝑛) → ((ψ‘𝑦) · (log‘𝑦)) = ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))
165161, 164oveq12d 6860 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑥 / 𝑛) → (Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))
166 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
167165, 166oveq12d 6860 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑥 / 𝑛) → ((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛)))
168167fveq2d 6379 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑥 / 𝑛) → (abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) = (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))))
169168breq1d 4819 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑥 / 𝑛) → ((abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴 ↔ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) ≤ 𝐴))
170 selberg3lem1.2 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)
171170ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑘 ∈ (1...(⌊‘𝑦))((Λ‘𝑘) · (log‘𝑘)) − ((ψ‘𝑦) · (log‘𝑦))) / 𝑦)) ≤ 𝐴)
1729nncnd 11292 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
173172mulid2d 10312 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
174 fznnfl 12869 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
17536, 174syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
176175simplbda 493 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛𝑥)
177173, 176eqbrtrd 4831 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
178 1red 10294 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
179178, 140, 92lemuldivd 12119 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
180177, 179mpbid 223 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
181 1re 10293 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
182 elicopnf 12472 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℝ → ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛))))
183181, 182ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)))
18488, 180, 183sylanbrc 578 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ (1[,)+∞))
185169, 171, 184rspcdva 3467 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) / (𝑥 / 𝑛))) ≤ 𝐴)
186153, 185eqbrtrrd 4833 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)) ≤ 𝐴)
187143, 139, 126ledivmul2d 12124 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) / (𝑥 / 𝑛)) ≤ 𝐴 ↔ (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ≤ (𝐴 · (𝑥 / 𝑛))))
188186, 187mpbid 223 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ≤ (𝐴 · (𝑥 / 𝑛)))
18923adantr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
190140recnd 10322 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
1919nnne0d 11322 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
192189, 190, 172, 191divassd 11090 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 · 𝑥) / 𝑛) = (𝐴 · (𝑥 / 𝑛)))
193188, 192breqtrrd 4837 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) ≤ ((𝐴 · 𝑥) / 𝑛))
194143, 144, 11, 146, 193lemul2ad 11218 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ ((Λ‘𝑛) · ((𝐴 · 𝑥) / 𝑛)))
195114, 130absmuld 14480 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) = ((abs‘(Λ‘𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
19611, 146absidd 14448 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Λ‘𝑛)) = (Λ‘𝑛))
197196oveq1d 6857 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Λ‘𝑛)) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) = ((Λ‘𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
198195, 197eqtrd 2799 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) = ((Λ‘𝑛) · (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
199141recnd 10322 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · 𝑥) ∈ ℂ)
200114, 172, 199, 191div32d 11078 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) = ((Λ‘𝑛) · ((𝐴 · 𝑥) / 𝑛)))
201194, 198, 2003brtr4d 4841 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ (((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
2027, 134, 142, 201fsumle 14817 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
20336recnd 10322 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
20423, 203mulcld 10314 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 · 𝑥) ∈ ℂ)
205114, 172, 191divcld 11055 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
2067, 204, 205fsummulc1 14803 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
207202, 206breqtrrd 4837 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
208133, 135, 137, 138, 207letrd 10448 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
209123recnd 10322 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
21090recnd 10322 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
21193recnd 10322 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
212210, 211mulcld 10314 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)) ∈ ℂ)
213209, 212addcld 10313 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) ∈ ℂ)
214114, 213mulcld 10314 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) ∈ ℂ)
215114, 210mulcld 10314 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
21626adantr 472 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈ ℂ)
217215, 216mulcld 10314 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) ∈ ℂ)
2187, 214, 217fsumsub 14806 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
219210, 216mulcld 10314 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) ∈ ℂ)
220114, 213, 219subdid 10740 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))))
22143adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
222221, 92relogdivd 24663 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) = ((log‘𝑥) − (log‘𝑛)))
223222oveq2d 6858 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) · ((log‘𝑥) − (log‘𝑛))))
224210, 216, 211subdid 10740 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · ((log‘𝑥) − (log‘𝑛))) = (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
225223, 224eqtrd 2799 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))) = (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
226225oveq2d 6858 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
227209, 219, 212subsub3d 10676 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − (((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥))))
228226, 227eqtrd 2799 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥))))
229228oveq2d 6858 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) = ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) − ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))))
230114, 210, 216mulassd 10317 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) = ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥))))
231230oveq2d 6858 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑥)))))
232220, 229, 2313eqtr4d 2809 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
233232sumeq2dv 14720 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
234 fveq2 6375 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (Λ‘𝑛) = (Λ‘𝑚))
235 oveq2 6850 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑚 → (𝑥 / 𝑛) = (𝑥 / 𝑚))
236235fveq2d 6379 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑚 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑚)))
237234, 236oveq12d 6860 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))))
238 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑚 → (log‘𝑛) = (log‘𝑚))
239237, 238oveq12d 6860 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑚 → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)))
240239cbvsumv 14713 . . . . . . . . . . . . . . 15 Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚))
241 elfznn 12577 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))) → 𝑛 ∈ ℕ)
242241adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → 𝑛 ∈ ℕ)
243242, 10syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℝ)
244243recnd 10322 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))) → (Λ‘𝑛) ∈ ℂ)
245244anasss 458 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → (Λ‘𝑛) ∈ ℂ)
246 elfznn 12577 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
247246adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
248247, 118syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℝ)
249248recnd 10322 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑚) ∈ ℂ)
250247nnrpd 12068 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ+)
251250relogcld 24660 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℝ)
252251recnd 10322 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (log‘𝑚) ∈ ℂ)
253249, 252mulcld 10314 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
254253adantrr 708 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
255245, 254mulcld 10314 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ (𝑚 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚))))) → ((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
25636, 255fsumfldivdiag 25207 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
25736adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
258257, 247nndivred 11326 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑚) ∈ ℝ)
259 chpcl 25141 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
260258, 259syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
261260recnd 10322 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) ∈ ℂ)
262249, 261, 252mul32d 10500 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))))
263248, 251remulcld 10324 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℝ)
264263recnd 10322 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
265264, 261mulcomd 10315 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (log‘𝑚)) · (ψ‘(𝑥 / 𝑚))) = ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))))
266 chpval 25139 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
267258, 266syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛))
268267oveq1d 6857 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))) = (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
269 fzfid 12980 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑚))) ∈ Fin)
270269, 264, 244fsummulc1 14803 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))(Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
271268, 270eqtrd 2799 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑚)) · ((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
272262, 265, 2713eqtrd 2803 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
273272sumeq2dv 14720 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ (1...(⌊‘(𝑥 / 𝑚)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
274122recnd 10322 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · (log‘𝑚)) ∈ ℂ)
275115, 114, 274fsummulc2 14802 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
276275sumeq2dv 14720 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) · ((Λ‘𝑚) · (log‘𝑚))))
277256, 273, 2763eqtr4d 2809 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((Λ‘𝑚) · (ψ‘(𝑥 / 𝑚))) · (log‘𝑚)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))
278240, 277syl5eq 2811 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))))
279114, 210, 211mulassd 10317 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
280279sumeq2dv 14720 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))))
281278, 280oveq12d 6860 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
2821042timesd 11521 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))))
283114, 209mulcld 10314 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) ∈ ℂ)
284114, 212mulcld 10314 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛))) ∈ ℂ)
2857, 283, 284fsumadd 14757 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
286281, 282, 2853eqtr4d 2809 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
287114, 209, 212adddid 10318 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
288287sumeq2dv 14720 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚))) + ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
289286, 288eqtr4d 2802 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))))
29091recnd 10322 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
2917, 26, 290fsummulc1 14803 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))
292289, 291oveq12d 6860 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) + ((ψ‘(𝑥 / 𝑛)) · (log‘𝑛)))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))))
293218, 233, 2923eqtr4rd 2810 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛))))))
294293fveq2d 6379 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) = (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · (log‘𝑚)) − ((ψ‘(𝑥 / 𝑛)) · (log‘(𝑥 / 𝑛)))))))
29524, 23, 203mulassd 10317 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) · 𝑥) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 · 𝑥)))
296208, 294, 2953brtr4d 4841 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) · 𝑥))
297109, 113, 43ledivmul2d 12124 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) ↔ (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) · 𝑥)))
298296, 297mpbird 248 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴))
299111, 113, 25, 298lediv1dd 12128 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) / (log‘𝑥)) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)))
300109recnd 10322 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) ∈ ℂ)
301300, 203, 26, 110, 29divdiv1d 11086 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) / (log‘𝑥)) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (𝑥 · (log‘𝑥))))
302108, 26, 203, 29, 110divdiv32d 11080 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) / 𝑥) = ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / 𝑥) / (log‘𝑥)))
303105, 107, 26, 29divsubdird 11094 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) = (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) / (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) / (log‘𝑥))))
304103, 104, 26, 29div23d 11092 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) / (log‘𝑥)) = ((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))))
305106, 26, 29divcan4d 11061 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) / (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
306304, 305oveq12d 6860 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) / (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)) / (log‘𝑥))) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
307303, 306eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) = (((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
308307oveq1d 6857 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (log‘𝑥)) / 𝑥) = ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
309108, 203, 26, 110, 29divdiv1d 11086 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / 𝑥) / (log‘𝑥)) = (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥))))
310302, 308, 3093eqtr3d 2807 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) = (((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥))))
311310fveq2d 6379 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = (abs‘(((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥)))))
31243, 25rpmulcld 12086 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ+)
313312rpcnd 12072 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
314312rpne0d 12075 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
315108, 313, 314absdivd 14481 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥))) / (𝑥 · (log‘𝑥)))) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (abs‘(𝑥 · (log‘𝑥)))))
316312rpred 12070 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ)
317312rpge0d 12074 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝑥 · (log‘𝑥)))
318316, 317absidd 14448 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝑥 · (log‘𝑥))) = (𝑥 · (log‘𝑥)))
319318oveq2d 6858 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (abs‘(𝑥 · (log‘𝑥)))) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (𝑥 · (log‘𝑥))))
320311, 315, 3193eqtrd 2803 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = ((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / (𝑥 · (log‘𝑥))))
321301, 320eqtr4d 2802 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((abs‘((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑥)))) / 𝑥) / (log‘𝑥)) = (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)))
32224, 23, 26, 29divassd 11090 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝐴) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))))
323299, 321, 3223brtr3d 4840 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))))
32421leabsd 14440 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥))) ≤ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))))
325101, 21, 102, 323, 324letrd 10448 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))))
326325adantrr 708 . 2 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (𝐴 / (log‘𝑥)))))
3271, 83, 21, 100, 326o1le 14670 1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((((2 / (log‘𝑥)) · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) · (log‘𝑛))) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wss 3732   class class class wbr 4809  cmpt 4888  cfv 6068  (class class class)co 6842  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  +∞cpnf 10325   < clt 10328  cle 10329  cmin 10520   / cdiv 10938  cn 11274  2c2 11327  +crp 12028  (,)cioo 12377  [,)cico 12379  ...cfz 12533  cfl 12799  abscabs 14261  𝑂(1)co1 14504  Σcsu 14703  eceu 15077  logclog 24592  Λcvma 25109  ψcchp 25110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ioc 12382  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14094  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-limsup 14489  df-clim 14506  df-rlim 14507  df-o1 14508  df-lo1 14509  df-sum 14704  df-ef 15082  df-e 15083  df-sin 15084  df-cos 15085  df-pi 15087  df-dvds 15268  df-gcd 15500  df-prm 15668  df-pc 15823  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-starv 16231  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-unif 16239  df-hom 16240  df-cco 16241  df-rest 16351  df-topn 16352  df-0g 16370  df-gsum 16371  df-topgen 16372  df-pt 16373  df-prds 16376  df-xrs 16430  df-qtop 16435  df-imas 16436  df-xps 16438  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-submnd 17604  df-mulg 17810  df-cntz 18015  df-cmn 18461  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-fbas 20016  df-fg 20017  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-lp 21220  df-perf 21221  df-cn 21311  df-cnp 21312  df-haus 21399  df-cmp 21470  df-tx 21645  df-hmeo 21838  df-fil 21929  df-fm 22021  df-flim 22022  df-flf 22023  df-xms 22404  df-ms 22405  df-tms 22406  df-cncf 22960  df-limc 23921  df-dv 23922  df-log 24594  df-cxp 24595  df-cht 25114  df-vma 25115  df-chp 25116  df-ppi 25117
This theorem is referenced by:  selberg3lem2  25538
  Copyright terms: Public domain W3C validator