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Theorem selbergr 26143
 Description: Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
Hypothesis
Ref Expression
pntrval.r 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
Assertion
Ref Expression
selbergr (𝑥 ∈ ℝ+ ↦ ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1)
Distinct variable groups:   𝑎,𝑑,𝑥   𝑅,𝑑,𝑥
Allowed substitution hint:   𝑅(𝑎)

Proof of Theorem selbergr
StepHypRef Expression
1 reex 10627 . . . . . . 7 ℝ ∈ V
2 rpssre 12395 . . . . . . 7 + ⊆ ℝ
31, 2ssexi 5225 . . . . . 6 + ∈ V
43a1i 11 . . . . 5 (⊤ → ℝ+ ∈ V)
5 ovexd 7190 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6 ovexd 7190 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)) ∈ V)
7 eqidd 2822 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))))
8 eqidd 2822 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))))
94, 5, 6, 7, 8offval2 7425 . . . 4 (⊤ → ((𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f − (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))))
109mptru 1540 . . 3 ((𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f − (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))))
11 pntrval.r . . . . . . . . . . . 12 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
1211pntrf 26138 . . . . . . . . . . 11 𝑅:ℝ+⟶ℝ
1312ffvelrni 6849 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑅𝑥) ∈ ℝ)
1413recnd 10668 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (𝑅𝑥) ∈ ℂ)
15 relogcl 25158 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1615recnd 10668 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
1714, 16mulcld 10660 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((𝑅𝑥) · (log‘𝑥)) ∈ ℂ)
18 fzfid 13340 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
19 elfznn 12935 . . . . . . . . . . . . 13 (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
2019adantl 484 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
21 vmacl 25694 . . . . . . . . . . . 12 (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
2220, 21syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
2322recnd 10668 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
24 rpre 12396 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
25 nndivre 11677 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
2624, 19, 25syl2an 597 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
27 chpcl 25700 . . . . . . . . . . . 12 ((𝑥 / 𝑑) ∈ ℝ → (ψ‘(𝑥 / 𝑑)) ∈ ℝ)
2826, 27syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℝ)
2928recnd 10668 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℂ)
3023, 29mulcld 10660 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
3118, 30fsumcl 15089 . . . . . . . 8 (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
3217, 31addcld 10659 . . . . . . 7 (𝑥 ∈ ℝ+ → (((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
33 rpcn 12398 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
34 rpne0 12404 . . . . . . 7 (𝑥 ∈ ℝ+𝑥 ≠ 0)
3532, 33, 34divcld 11415 . . . . . 6 (𝑥 ∈ ℝ+ → ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
3622, 20nndivred 11690 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℝ)
3736recnd 10668 . . . . . . 7 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℂ)
3818, 37fsumcl 15089 . . . . . 6 (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) ∈ ℂ)
3935, 38, 16nnncan2d 11031 . . . . 5 (𝑥 ∈ ℝ+ → ((((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))
40 chpcl 25700 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
4124, 40syl 17 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
4241recnd 10668 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
4342, 16mulcld 10660 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
4443, 31addcld 10659 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
4544, 33, 34divcld 11415 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
4645, 16, 16subsub4d 11027 . . . . . . 7 (𝑥 ∈ ℝ+ → ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((log‘𝑥) + (log‘𝑥))))
4711pntrval 26137 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑅𝑥) = ((ψ‘𝑥) − 𝑥))
4847oveq1d 7170 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((𝑅𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥)))
4942, 33, 16subdird 11096 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
5048, 49eqtrd 2856 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((𝑅𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
5150oveq1d 7170 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) = ((((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))))
5233, 16mulcld 10660 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 · (log‘𝑥)) ∈ ℂ)
5343, 31, 52addsubd 11017 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))))
5451, 53eqtr4d 2859 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))))
5554oveq1d 7170 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) / 𝑥))
56 rpcnne0 12406 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
57 divsubdir 11333 . . . . . . . . . 10 (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ ∧ (𝑥 · (log‘𝑥)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)))
5844, 52, 56, 57syl3anc 1367 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)))
5916, 33, 34divcan3d 11420 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥))
6059oveq2d 7171 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)))
6155, 58, 603eqtrd 2860 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)))
6261oveq1d 7170 . . . . . . 7 (𝑥 ∈ ℝ+ → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) = ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)))
63162timesd 11879 . . . . . . . 8 (𝑥 ∈ ℝ+ → (2 · (log‘𝑥)) = ((log‘𝑥) + (log‘𝑥)))
6463oveq2d 7171 . . . . . . 7 (𝑥 ∈ ℝ+ → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((log‘𝑥) + (log‘𝑥))))
6546, 62, 643eqtr4d 2866 . . . . . 6 (𝑥 ∈ ℝ+ → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))))
6665oveq1d 7170 . . . . 5 (𝑥 ∈ ℝ+ → ((((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))))
6733, 38mulcld 10660 . . . . . . 7 (𝑥 ∈ ℝ+ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ)
68 divsubdir 11333 . . . . . . 7 (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ ∧ (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) / 𝑥) = (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥)))
6932, 67, 56, 68syl3anc 1367 . . . . . 6 (𝑥 ∈ ℝ+ → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) / 𝑥) = (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥)))
7017, 31, 67addsubassd 11016 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = (((𝑅𝑥) · (log‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))))
7133adantr 483 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
7271, 37mulcld 10660 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) ∈ ℂ)
7318, 30, 72fsumsub 15142 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))(((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑))))
7426recnd 10668 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℂ)
7523, 29, 74subdid 11095 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − ((Λ‘𝑑) · (𝑥 / 𝑑))))
7619nnrpd 12428 . . . . . . . . . . . . . . 15 (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ+)
77 rpdivcl 12413 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑑 ∈ ℝ+) → (𝑥 / 𝑑) ∈ ℝ+)
7876, 77sylan2 594 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ+)
7911pntrval 26137 . . . . . . . . . . . . . 14 ((𝑥 / 𝑑) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑑)) = ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑)))
8078, 79syl 17 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑑)) = ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑)))
8180oveq2d 7171 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))))
8220nnrpd 12428 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ+)
83 rpcnne0 12406 . . . . . . . . . . . . . . 15 (𝑑 ∈ ℝ+ → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
8482, 83syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
85 div12 11319 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ (Λ‘𝑑) ∈ ℂ ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝑥 · ((Λ‘𝑑) / 𝑑)) = ((Λ‘𝑑) · (𝑥 / 𝑑)))
8671, 23, 84, 85syl3anc 1367 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) = ((Λ‘𝑑) · (𝑥 / 𝑑)))
8786oveq2d 7171 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − ((Λ‘𝑑) · (𝑥 / 𝑑))))
8875, 81, 873eqtr4d 2866 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
8988sumeq2dv 15059 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
9018, 33, 37fsummulc2 15138 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑)))
9190oveq2d 7171 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑))))
9273, 89, 913eqtr4rd 2867 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))))
9392oveq2d 7171 . . . . . . . 8 (𝑥 ∈ ℝ+ → (((𝑅𝑥) · (log‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))) = (((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))))
9470, 93eqtrd 2856 . . . . . . 7 (𝑥 ∈ ℝ+ → ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = (((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))))
9594oveq1d 7170 . . . . . 6 (𝑥 ∈ ℝ+ → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) / 𝑥) = ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
9638, 33, 34divcan3d 11420 . . . . . . 7 (𝑥 ∈ ℝ+ → ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))
9796oveq2d 7171 . . . . . 6 (𝑥 ∈ ℝ+ → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥)) = (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))
9869, 95, 973eqtr3rd 2865 . . . . 5 (𝑥 ∈ ℝ+ → (((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
9939, 66, 983eqtr3d 2864 . . . 4 (𝑥 ∈ ℝ+ → ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
10099mpteq2ia 5156 . . 3 (𝑥 ∈ ℝ+ ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
10110, 100eqtri 2844 . 2 ((𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f − (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
102 selberg2 26126 . . 3 (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
103 vmadivsum 26057 . . 3 (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)
104 o1sub 14971 . . 3 (((𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f − (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) ∈ 𝑂(1))
105102, 103, 104mp2an 690 . 2 ((𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘f − (𝑥 ∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) ∈ 𝑂(1)
106101, 105eqeltrri 2910 1 (𝑥 ∈ ℝ+ ↦ ((((𝑅𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 398   = wceq 1533  ⊤wtru 1534   ∈ wcel 2110   ≠ wne 3016  Vcvv 3494   ↦ cmpt 5145  ‘cfv 6354  (class class class)co 7155   ∘f cof 7406  ℂcc 10534  ℝcr 10535  0cc0 10536  1c1 10537   + caddc 10539   · cmul 10541   − cmin 10869   / cdiv 11296  ℕcn 11637  2c2 11691  ℝ+crp 12388  ...cfz 12891  ⌊cfl 13159  𝑂(1)co1 14842  Σcsu 15041  logclog 25137  Λcvma 25668  ψcchp 25669 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614  ax-addf 10615  ax-mulf 10616 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-iin 4921  df-disj 5031  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-supp 7830  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-ixp 8461  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-fsupp 8833  df-fi 8874  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-xnn0 11967  df-z 11981  df-dec 12098  df-uz 12243  df-q 12348  df-rp 12389  df-xneg 12506  df-xadd 12507  df-xmul 12508  df-ioo 12741  df-ioc 12742  df-ico 12743  df-icc 12744  df-fz 12892  df-fzo 13033  df-fl 13161  df-mod 13237  df-seq 13369  df-exp 13429  df-fac 13633  df-bc 13662  df-hash 13690  df-shft 14425  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-limsup 14827  df-clim 14844  df-rlim 14845  df-o1 14846  df-lo1 14847  df-sum 15042  df-ef 15420  df-e 15421  df-sin 15422  df-cos 15423  df-tan 15424  df-pi 15425  df-dvds 15607  df-gcd 15843  df-prm 16015  df-pc 16173  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-mulr 16578  df-starv 16579  df-sca 16580  df-vsca 16581  df-ip 16582  df-tset 16583  df-ple 16584  df-ds 16586  df-unif 16587  df-hom 16588  df-cco 16589  df-rest 16695  df-topn 16696  df-0g 16714  df-gsum 16715  df-topgen 16716  df-pt 16717  df-prds 16720  df-xrs 16774  df-qtop 16779  df-imas 16780  df-xps 16782  df-mre 16856  df-mrc 16857  df-acs 16859  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-submnd 17956  df-mulg 18224  df-cntz 18446  df-cmn 18907  df-psmet 20536  df-xmet 20537  df-met 20538  df-bl 20539  df-mopn 20540  df-fbas 20541  df-fg 20542  df-cnfld 20545  df-top 21501  df-topon 21518  df-topsp 21540  df-bases 21553  df-cld 21626  df-ntr 21627  df-cls 21628  df-nei 21705  df-lp 21743  df-perf 21744  df-cn 21834  df-cnp 21835  df-haus 21922  df-cmp 21994  df-tx 22169  df-hmeo 22362  df-fil 22453  df-fm 22545  df-flim 22546  df-flf 22547  df-xms 22929  df-ms 22930  df-tms 22931  df-cncf 23485  df-limc 24463  df-dv 24464  df-ulm 24964  df-log 25139  df-cxp 25140  df-atan 25444  df-em 25569  df-cht 25673  df-vma 25674  df-chp 25675  df-ppi 25676  df-mu 25677 This theorem is referenced by: (None)
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