AlexKontorovich/PrimeNumberTheoremAnd
Source indexedlemma · leanprover/lean4:v4.32.0
Complex.CartanBound.intervalIntegrable_phi_div
PrimeNumberTheoremAnd.Mathlib.Analysis.Complex.CartanBound · PrimeNumberTheoremAnd/Mathlib/Analysis/Complex/CartanBound.lean:465 to 475
Mathematical statement
Exact Lean statement
lemma intervalIntegrable_phi_div {a R : ℝ} (ha : 0 < a) (hR : 0 ≤ R) :
IntervalIntegrable (fun r : ℝ => φ (r / a)) volume R (2 * R)Complete declaration
Lean source
Full Lean sourceLean 4
lemma intervalIntegrable_phi_div {a R : ℝ} (ha : 0 < a) (hR : 0 ≤ R) : IntervalIntegrable (fun r : ℝ => φ (r / a)) volume R (2 * R) := by have ha0 : a ≠ 0 := ne_of_gt ha have hRa_nonneg : 0 ≤ R / a := by exact div_nonneg hR (le_of_lt ha) have hφ : IntervalIntegrable φ volume (R / a) (2 * (R / a)) := intervalIntegrable_phi_dyadic (A := (R / a)) hRa_nonneg have := (hφ.comp_mul_right (c := (a⁻¹ : ℝ))) have hupper : a * (R * (a⁻¹ * 2)) = (2 * R) := by field_simp [ha0] simpa [div_eq_mul_inv, ha0, hupper, mul_assoc, mul_left_comm, mul_comm] using this