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Theorem selberg4lem1 26148
Description: Lemma for selberg4 26149. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Hypotheses
Ref Expression
selberg4lem1.1 (𝜑𝐴 ∈ ℝ+)
selberg4lem1.2 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴)
Assertion
Ref Expression
selberg4lem1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
Distinct variable groups:   𝑖,𝑚,𝑛,𝑥,𝑦,𝐴   𝜑,𝑚,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑖)

Proof of Theorem selberg4lem1
StepHypRef Expression
1 2cnd 11707 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
2 fzfid 13340 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
3 elfznn 12935 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
43adantl 485 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
5 vmacl 25707 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
64, 5syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
76, 4nndivred 11683 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
8 elioore 12760 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
98adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
10 1rp 12385 . . . . . . . . . . . . . . 15 1 ∈ ℝ+
1110a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
12 1red 10635 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
13 eliooord 12788 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
1413adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
1514simpld 498 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
1612, 9, 15ltled 10781 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
179, 11, 16rpgecld 12462 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
1817adantr 484 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
194nnrpd 12421 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
2018, 19rpdivcld 12440 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
2120relogcld 25218 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
227, 21remulcld 10664 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
232, 22fsumrecl 15087 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
249, 15rplogcld 25224 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2523, 24rerpdivcld 12454 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℝ)
2625recnd 10662 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
2717relogcld 25218 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
2827rehalfcld 11876 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ)
2928recnd 10662 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
301, 26, 29subdid 11089 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2))))
3127recnd 10662 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
32 2ne0 11733 . . . . . . . 8 2 ≠ 0
3332a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ≠ 0)
3431, 1, 33divcan2d 11411 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · ((log‘𝑥) / 2)) = (log‘𝑥))
3534oveq2d 7155 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (2 · ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))
3630, 35eqtrd 2836 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))
3736mpteq2dva 5128 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))))
38 2re 11703 . . . . 5 2 ∈ ℝ
3938a1i 11 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
4025, 28resubcld 11061 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ)
41 ioossre 12790 . . . . . 6 (1(,)+∞) ⊆ ℝ
42 2cn 11704 . . . . . 6 2 ∈ ℂ
43 o1const 14972 . . . . . 6 (((1(,)+∞) ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
4441, 42, 43mp2an 691 . . . . 5 (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1)
4544a1i 11 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 2) ∈ 𝑂(1))
46 vmalogdivsum2 26126 . . . . 5 (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
4746a1i 11 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
4839, 40, 45, 47o1mul2 14977 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (2 · ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
4937, 48eqeltrrd 2894 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
50 fzfid 13340 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
51 elfznn 12935 . . . . . . . . . . . 12 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
5251adantl 485 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
53 vmacl 25707 . . . . . . . . . . 11 (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
5452, 53syl 17 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑚) ∈ ℝ)
5552nnrpd 12421 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ+)
5655relogcld 25218 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
579adantr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
5857, 4nndivred 11683 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
5958adantr 484 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑥 / 𝑛) ∈ ℝ)
6059, 52nndivred 11683 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((𝑥 / 𝑛) / 𝑚) ∈ ℝ)
61 chpcl 25713 . . . . . . . . . . . 12 (((𝑥 / 𝑛) / 𝑚) ∈ ℝ → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
6260, 61syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (ψ‘((𝑥 / 𝑛) / 𝑚)) ∈ ℝ)
6356, 62readdcld 10663 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))) ∈ ℝ)
6454, 63remulcld 10664 . . . . . . . . 9 ((((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
6550, 64fsumrecl 15087 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℝ)
666, 65remulcld 10664 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
672, 66fsumrecl 15087 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℝ)
6817, 24rpmulcld 12439 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ+)
6967, 68rerpdivcld 12454 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
7069, 27resubcld 11061 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) ∈ ℝ)
7170recnd 10662 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) ∈ ℂ)
7223recnd 10662 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
7324rpne0d 12428 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
7472, 31, 73divcld 11409 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
751, 74mulcld 10654 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) ∈ ℂ)
7675, 31subcld 10990 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)) ∈ ℂ)
7769recnd 10662 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
7877, 75, 31nnncan2d 11025 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)))))
7967recnd 10662 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ)
809recnd 10662 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
8117rpne0d 12428 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
8279, 80, 31, 81, 73divdiv1d 11440 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))))
831, 72, 31, 73divassd 11444 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
8482, 83oveq12d 7157 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) − ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)))))
8567, 17rerpdivcld 12454 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℝ)
8685recnd 10662 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℂ)
871, 72mulcld 10654 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
8886, 87, 31, 73divsubdird 11448 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) − ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥))))
8981adantr 484 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
9066, 57, 89redivcld 11461 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℝ)
9190recnd 10662 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) ∈ ℂ)
9238a1i 11 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℝ)
9392, 22remulcld 10664 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
9493recnd 10662 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
952, 91, 94fsumsub 15139 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
966recnd 10662 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
9765, 57, 89redivcld 11461 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) ∈ ℝ)
9897recnd 10662 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) ∈ ℂ)
99 2cnd 11707 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
10021recnd 10662 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
1014nncnd 11645 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
1024nnne0d 11679 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
103100, 101, 102divcld 11409 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) / 𝑛) ∈ ℂ)
10499, 103mulcld 10654 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) ∈ ℂ)
10596, 98, 104subdid 11089 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥)) − ((Λ‘𝑛) · (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))
10665recnd 10662 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) ∈ ℂ)
10780adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
10896, 106, 107, 89divassd 11444 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) = ((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥)))
10996, 101, 100, 102div32d 11432 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = ((Λ‘𝑛) · ((log‘(𝑥 / 𝑛)) / 𝑛)))
110109oveq2d 7155 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (2 · ((Λ‘𝑛) · ((log‘(𝑥 / 𝑛)) / 𝑛))))
11199, 96, 103mul12d 10842 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((Λ‘𝑛) · ((log‘(𝑥 / 𝑛)) / 𝑛))) = ((Λ‘𝑛) · (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))
112110, 111eqtrd 2836 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = ((Λ‘𝑛) · (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))
113108, 112oveq12d 7157 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (((Λ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥)) − ((Λ‘𝑛) · (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))
114105, 113eqtr4d 2839 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) = ((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
115114sumeq2dv 15056 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
11666recnd 10662 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) ∈ ℂ)
1172, 80, 116, 81fsumdivc 15137 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥))
11822recnd 10662 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
1192, 1, 118fsummulc2 15135 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
120117, 119oveq12d 7157 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
12195, 115, 1203eqtr4rd 2847 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))))
122121oveq1d 7154 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) − (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)))
12388, 122eqtr3d 2838 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / 𝑥) / (log‘𝑥)) − ((2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)))
12478, 84, 1233eqtr2d 2842 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)))
125124mpteq2dva 5128 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))))
126 1red 10635 . . . . 5 (𝜑 → 1 ∈ ℝ)
127 selberg4lem1.1 . . . . . . . 8 (𝜑𝐴 ∈ ℝ+)
128127adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ+)
129128rpred 12423 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
1302, 7fsumrecl 15087 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
131130, 24rerpdivcld 12454 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ)
132127rpcnd 12425 . . . . . . 7 (𝜑𝐴 ∈ ℂ)
133 o1const 14972 . . . . . . 7 (((1(,)+∞) ⊆ ℝ ∧ 𝐴 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
13441, 132, 133sylancr 590 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 𝐴) ∈ 𝑂(1))
135 1cnd 10629 . . . . . . . 8 (𝜑 → 1 ∈ ℂ)
136 o1const 14972 . . . . . . . 8 (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
13741, 135, 136sylancr 590 . . . . . . 7 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
138131recnd 10662 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ)
139 1cnd 10629 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
140130recnd 10662 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
141140, 31, 31, 73divsubdird 11448 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))))
142140, 31subcld 10990 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
143142, 31, 73divrecd 11412 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
14431, 73dividd 11407 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
145144oveq2d 7155 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))
146141, 143, 1453eqtr3d 2844 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))
147146mpteq2dva 5128 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)))
148130, 27resubcld 11061 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
14912, 24rerpdivcld 12454 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
15017ex 416 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
151150ssrdv 3924 . . . . . . . . . . 11 (𝜑 → (1(,)+∞) ⊆ ℝ+)
152 vmadivsum 26070 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
153152a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
154151, 153o1res2 14916 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
155 divlogrlim 25230 . . . . . . . . . . 11 (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
156 rlimo1 14969 . . . . . . . . . . 11 ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
157155, 156mp1i 13 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
158148, 149, 154, 157o1mul2 14977 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
159147, 158eqeltrrd 2894 . . . . . . . 8 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈ 𝑂(1))
160138, 139, 159o1dif 14982 . . . . . . 7 (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)))
161137, 160mpbird 260 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1))
162129, 131, 134, 161o1mul2 14977 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) ∈ 𝑂(1))
163129, 131remulcld 10664 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ ℝ)
16421, 4nndivred 11683 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛)) / 𝑛) ∈ ℝ)
16592, 164remulcld 10664 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) ∈ ℝ)
16697, 165resubcld 11061 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) ∈ ℝ)
1676, 166remulcld 10664 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℝ)
1682, 167fsumrecl 15087 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℝ)
169168, 24rerpdivcld 12454 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)) ∈ ℝ)
170169recnd 10662 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥)) ∈ ℂ)
171168recnd 10662 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℂ)
172171abscld 14792 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ∈ ℝ)
173129, 130remulcld 10664 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) ∈ ℝ)
17498, 104subcld 10990 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) ∈ ℂ)
17596, 174mulcld 10654 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℂ)
176175abscld 14792 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ∈ ℝ)
1772, 176fsumrecl 15087 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ∈ ℝ)
178167recnd 10662 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℂ)
1792, 178fsumabs 15152 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))))
180129adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
181180, 7remulcld 10664 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
182174abscld 14792 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ∈ ℝ)
183180, 4nndivred 11683 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 / 𝑛) ∈ ℝ)
184 vmage0 25710 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
1854, 184syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
186106, 107, 101, 89, 102divdiv2d 11441 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) · 𝑛) / 𝑥))
187106, 101, 107, 89div23d 11446 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) · 𝑛) / 𝑥) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛))
188186, 187eqtrd 2836 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛))
18999, 103, 101mulassd 10657 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛) = (2 · (((log‘(𝑥 / 𝑛)) / 𝑛) · 𝑛)))
190100, 101, 102divcan1d 11410 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛)) / 𝑛) · 𝑛) = (log‘(𝑥 / 𝑛)))
191190oveq2d 7155 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((log‘(𝑥 / 𝑛)) / 𝑛) · 𝑛)) = (2 · (log‘(𝑥 / 𝑛))))
192189, 191eqtr2d 2837 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) = ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛))
193188, 192oveq12d 7157 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛)))) = (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛) − ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛)))
19498, 104, 101subdird 11090 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛) = (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) · 𝑛) − ((2 · ((log‘(𝑥 / 𝑛)) / 𝑛)) · 𝑛)))
195193, 194eqtr4d 2839 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛)))) = (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛))
196195fveq2d 6653 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) = (abs‘(((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛)))
197174, 101absmuld 14810 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))) · 𝑛)) = ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · (abs‘𝑛)))
1984nnred 11644 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
19919rpge0d 12427 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑛)
200198, 199absidd 14778 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑛) = 𝑛)
201200oveq2d 7155 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · (abs‘𝑛)) = ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛))
202196, 197, 2013eqtrd 2840 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) = ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛))
203 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑚 → (Λ‘𝑖) = (Λ‘𝑚))
204 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (log‘𝑖) = (log‘𝑚))
205 oveq2 7147 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 = 𝑚 → (𝑦 / 𝑖) = (𝑦 / 𝑚))
206205fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (ψ‘(𝑦 / 𝑖)) = (ψ‘(𝑦 / 𝑚)))
207204, 206oveq12d 7157 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 = 𝑚 → ((log‘𝑖) + (ψ‘(𝑦 / 𝑖))) = ((log‘𝑚) + (ψ‘(𝑦 / 𝑚))))
208203, 207oveq12d 7157 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑚 → ((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) = ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))))
209208cbvsumv 15049 . . . . . . . . . . . . . . . . . . . . . 22 Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) = Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚))))
210 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑥 / 𝑛) → (⌊‘𝑦) = (⌊‘(𝑥 / 𝑛)))
211210oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑥 / 𝑛) → (1...(⌊‘𝑦)) = (1...(⌊‘(𝑥 / 𝑛))))
212 fvoveq1 7162 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = (𝑥 / 𝑛) → (ψ‘(𝑦 / 𝑚)) = (ψ‘((𝑥 / 𝑛) / 𝑚)))
213212oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 = (𝑥 / 𝑛) → ((log‘𝑚) + (ψ‘(𝑦 / 𝑚))) = ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))
214213oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑥 / 𝑛) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) = ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))
215214adantr 484 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 = (𝑥 / 𝑛) ∧ 𝑚 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) = ((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))
216211, 215sumeq12dv 15059 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑥 / 𝑛) → Σ𝑚 ∈ (1...(⌊‘𝑦))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘(𝑦 / 𝑚)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))
217209, 216syl5eq 2848 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / 𝑛) → Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))))
218 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / 𝑛) → 𝑦 = (𝑥 / 𝑛))
219217, 218oveq12d 7157 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑥 / 𝑛) → (Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)))
220 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / 𝑛) → (log‘𝑦) = (log‘(𝑥 / 𝑛)))
221220oveq2d 7155 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑥 / 𝑛) → (2 · (log‘𝑦)) = (2 · (log‘(𝑥 / 𝑛))))
222219, 221oveq12d 7157 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑥 / 𝑛) → ((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦))) = ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛)))))
223222fveq2d 6653 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑥 / 𝑛) → (abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) = (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))))
224223breq1d 5043 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑥 / 𝑛) → ((abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴 ↔ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) ≤ 𝐴))
225 selberg4lem1.2 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴)
226225ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑦 / 𝑖)))) / 𝑦) − (2 · (log‘𝑦)))) ≤ 𝐴)
227101mulid2d 10652 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
228 fznnfl 13229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
2299, 228syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
230229simplbda 503 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛𝑥)
231227, 230eqbrtrd 5055 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
232 1red 10635 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
233232, 57, 19lemuldivd 12472 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
234231, 233mpbid 235 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
235 1re 10634 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
236 elicopnf 12827 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℝ → ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛))))
237235, 236ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝑥 / 𝑛) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)))
23858, 234, 237sylanbrc 586 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ (1[,)+∞))
239224, 226, 238rspcdva 3576 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / (𝑥 / 𝑛)) − (2 · (log‘(𝑥 / 𝑛))))) ≤ 𝐴)
240202, 239eqbrtrrd 5057 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛) ≤ 𝐴)
241182, 180, 19lemuldivd 12472 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) · 𝑛) ≤ 𝐴 ↔ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ≤ (𝐴 / 𝑛)))
242240, 241mpbid 235 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) ≤ (𝐴 / 𝑛))
243182, 183, 6, 185, 242lemul2ad 11573 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ ((Λ‘𝑛) · (𝐴 / 𝑛)))
24496, 174absmuld 14810 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) = ((abs‘(Λ‘𝑛)) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))))
2456, 185absidd 14778 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Λ‘𝑛)) = (Λ‘𝑛))
246245oveq1d 7154 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(Λ‘𝑛)) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) = ((Λ‘𝑛) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))))
247244, 246eqtrd 2836 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) = ((Λ‘𝑛) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))))
248132ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
249248, 96, 101, 102div12d 11445 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · ((Λ‘𝑛) / 𝑛)) = ((Λ‘𝑛) · (𝐴 / 𝑛)))
250243, 247, 2493brtr4d 5065 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ (𝐴 · ((Λ‘𝑛) / 𝑛)))
2512, 176, 181, 250fsumle 15150 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝐴 · ((Λ‘𝑛) / 𝑛)))
252132adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℂ)
2537recnd 10662 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
2542, 252, 253fsummulc2 15135 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(𝐴 · ((Λ‘𝑛) / 𝑛)))
255251, 254breqtrrd 5061 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ (𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)))
256172, 177, 173, 179, 255letrd 10790 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) ≤ (𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)))
257172, 173, 24, 256lediv1dd 12481 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥)) ≤ ((𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) / (log‘𝑥)))
258252, 140, 31, 73divassd 11444 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 · Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛)) / (log‘𝑥)) = (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))))
259257, 258breqtrd 5059 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥)) ≤ (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))))
260171, 31, 73absdivd 14811 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (abs‘(log‘𝑥))))
26124rpge0d 12427 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (log‘𝑥))
26227, 261absidd 14778 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(log‘𝑥)) = (log‘𝑥))
263262oveq2d 7155 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (abs‘(log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥)))
264260, 263eqtrd 2836 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) = ((abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛))))) / (log‘𝑥)))
265128rpge0d 12427 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴)
2666, 19, 185divge0d 12463 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
2672, 7, 266fsumge0 15146 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
268130, 24, 267divge0d 12463 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))
269129, 131, 265, 268mulge0d 11210 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))))
270163, 269absidd 14778 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))) = (𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))))
271259, 264, 2703brtr4d 5065 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) ≤ (abs‘(𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))))
272271adantrr 716 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) ≤ (abs‘(𝐴 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))))
273126, 162, 163, 170, 272o1le 15005 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚)))) / 𝑥) − (2 · ((log‘(𝑥 / 𝑛)) / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1))
274125, 273eqeltrd 2893 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥)) − ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥)))) ∈ 𝑂(1))
27571, 76, 274o1dif 14982 . 2 (𝜑 → ((𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1)))
27649, 275mpbird 260 1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) · ((log‘𝑚) + (ψ‘((𝑥 / 𝑛) / 𝑚))))) / (𝑥 · (log‘𝑥))) − (log‘𝑥))) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  wss 3884   class class class wbr 5033  cmpt 5113  cfv 6328  (class class class)co 7139  cc 10528  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   · cmul 10535  +∞cpnf 10665   < clt 10668  cle 10669  cmin 10863   / cdiv 11290  cn 11629  2c2 11684  +crp 12381  (,)cioo 12730  [,)cico 12732  ...cfz 12889  cfl 13159  abscabs 14589  𝑟 crli 14838  𝑂(1)co1 14839  Σcsu 15038  logclog 25150  Λcvma 25681  ψcchp 25682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609  ax-mulf 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13430  df-fac 13634  df-bc 13663  df-hash 13691  df-shft 14422  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-limsup 14824  df-clim 14841  df-rlim 14842  df-o1 14843  df-lo1 14844  df-sum 15039  df-ef 15417  df-e 15418  df-sin 15419  df-cos 15420  df-tan 15421  df-pi 15422  df-dvds 15604  df-gcd 15838  df-prm 16010  df-pc 16168  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-starv 16576  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-unif 16584  df-hom 16585  df-cco 16586  df-rest 16692  df-topn 16693  df-0g 16711  df-gsum 16712  df-topgen 16713  df-pt 16714  df-prds 16717  df-xrs 16771  df-qtop 16776  df-imas 16777  df-xps 16779  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-submnd 17953  df-mulg 18221  df-cntz 18443  df-cmn 18904  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-fbas 20092  df-fg 20093  df-cnfld 20096  df-top 21503  df-topon 21520  df-topsp 21542  df-bases 21555  df-cld 21628  df-ntr 21629  df-cls 21630  df-nei 21707  df-lp 21745  df-perf 21746  df-cn 21836  df-cnp 21837  df-haus 21924  df-cmp 21996  df-tx 22171  df-hmeo 22364  df-fil 22455  df-fm 22547  df-flim 22548  df-flf 22549  df-xms 22931  df-ms 22932  df-tms 22933  df-cncf 23487  df-limc 24473  df-dv 24474  df-ulm 24976  df-log 25152  df-cxp 25153  df-atan 25457  df-em 25582  df-cht 25686  df-vma 25687  df-chp 25688  df-ppi 25689
This theorem is referenced by:  selberg4  26149
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