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Theorem selberg2 25531
Description: Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Assertion
Ref Expression
selberg2 (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Distinct variable group:   𝑥,𝑛

Proof of Theorem selberg2
StepHypRef Expression
1 reex 10280 . . . . . . 7 ℝ ∈ V
2 rpssre 12035 . . . . . . 7 + ⊆ ℝ
31, 2ssexi 4964 . . . . . 6 + ∈ V
43a1i 11 . . . . 5 (⊤ → ℝ+ ∈ V)
5 ovexd 6876 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6 ovexd 6876 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ V)
7 eqidd 2766 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))))
8 eqidd 2766 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
94, 5, 6, 7, 8offval2 7112 . . . 4 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))))
109mptru 1660 . . 3 ((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
11 fzfid 12980 . . . . . . . 8 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
12 elfznn 12577 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1312adantl 473 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
14 vmacl 25135 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
1513, 14syl 17 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
1615recnd 10322 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
1713nnrpd 12068 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
18 relogcl 24613 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
1917, 18syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
2019recnd 10322 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
21 rpre 12036 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
22 nndivre 11313 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
2321, 12, 22syl2an 589 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
24 chpcl 25141 . . . . . . . . . . . 12 ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
2523, 24syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
2625recnd 10322 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
2720, 26addcld 10313 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
2816, 27mulcld 10314 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ)
2911, 28fsumcl 14749 . . . . . . 7 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ)
30 rpcn 12040 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
31 rpne0 12046 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ≠ 0)
3229, 30, 31divcld 11055 . . . . . 6 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
33 2cn 11347 . . . . . . 7 2 ∈ ℂ
34 relogcl 24613 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
3534recnd 10322 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
36 mulcl 10273 . . . . . . 7 ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
3733, 35, 36sylancr 581 . . . . . 6 (𝑥 ∈ ℝ+ → (2 · (log‘𝑥)) ∈ ℂ)
3816, 20mulcld 10314 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
3911, 38fsumcl 14749 . . . . . . . 8 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
40 chpcl 25141 . . . . . . . . . . 11 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
4121, 40syl 17 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
4241recnd 10322 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
4342, 35mulcld 10314 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
4439, 43subcld 10646 . . . . . . 7 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ)
4544, 30, 31divcld 11055 . . . . . 6 (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ ℂ)
4632, 37, 45sub32d 10678 . . . . 5 (𝑥 ∈ ℝ+ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) − (2 · (log‘𝑥))))
47 rpcnne0 12048 . . . . . . . 8 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
48 divsubdir 10975 . . . . . . . 8 ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
4929, 44, 47, 48syl3anc 1490 . . . . . . 7 (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
5016, 20, 26adddid 10318 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
5150sumeq2dv 14718 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
5216, 26mulcld 10314 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
5311, 38, 52fsumadd 14755 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
5451, 53eqtrd 2799 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
5554oveq1d 6857 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))))
5611, 52fsumcl 14749 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
5739, 56, 43pnncand 10685 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((ψ‘𝑥) · (log‘𝑥))))
5856, 43addcomd 10492 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((ψ‘𝑥) · (log‘𝑥))) = (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
5955, 57, 583eqtrd 2803 . . . . . . . 8 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
6059oveq1d 6857 . . . . . . 7 (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
6149, 60eqtr3d 2801 . . . . . 6 (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
6261oveq1d 6857 . . . . 5 (𝑥 ∈ ℝ+ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) − (2 · (log‘𝑥))) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
6346, 62eqtrd 2799 . . . 4 (𝑥 ∈ ℝ+ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
6463mpteq2ia 4899 . . 3 (𝑥 ∈ ℝ+ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
6510, 64eqtri 2787 . 2 ((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
66 selberg 25528 . . 3 (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
67 selberg2lem 25530 . . 3 (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
68 o1sub 14631 . . 3 (((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) ∈ 𝑂(1))
6966, 67, 68mp2an 683 . 2 ((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) ∈ 𝑂(1)
7065, 69eqeltrri 2841 1 (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wtru 1653  wcel 2155  wne 2937  Vcvv 3350  cmpt 4888  cfv 6068  (class class class)co 6842  𝑓 cof 7093  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  cmin 10520   / cdiv 10938  cn 11274  2c2 11327  +crp 12028  ...cfz 12533  cfl 12799  𝑂(1)co1 14502  Σcsu 14701  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-disj 4778  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 14092  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-limsup 14487  df-clim 14504  df-rlim 14505  df-o1 14506  df-lo1 14507  df-sum 14702  df-ef 15080  df-e 15081  df-sin 15082  df-cos 15083  df-pi 15085  df-dvds 15266  df-gcd 15498  df-prm 15666  df-pc 15821  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-rest 16349  df-topn 16350  df-0g 16368  df-gsum 16369  df-topgen 16370  df-pt 16371  df-prds 16374  df-xrs 16428  df-qtop 16433  df-imas 16434  df-xps 16436  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-mulg 17808  df-cntz 18013  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-em 25010  df-cht 25114  df-vma 25115  df-chp 25116  df-ppi 25117  df-mu 25118
This theorem is referenced by:  selberg2b  25532  selberg3  25539  selberg4  25541  selbergr  25548
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