`BookProof.WeylCauchyRiemann.dbar_mul` {a b : ℂ → ℂ} {z : ℂ} (ha : DifferentiableAt ℝ a z) (hb : DifferentiableAt ℝ b z) : dbar (fun w => a w * b w) z = dbar a z * b z + a z * dbar

Proved

BookProof.WeylCauchyRiemann.dbar_mul

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylCauchyRiemann.

BookProof.WeylCauchyRiemann.dbar_mul {a b : ℂ → ℂ} {z : ℂ} (ha : DifferentiableAt ℝ a z) (hb : DifferentiableAt ℝ b z) : dbar (fun w => a w * b w) z = dbar a z * b z + a z * dbar b z

Formalization note: Lean 4 identifier BookProof.WeylCauchyRiemann.dbar_mul.

Preamble

-- Generated from ChapterWeylCauchyRiemann.lean — theorem BookProof.WeylCauchyRiemann.dbar_mul
import Definitions.Def_ChapterRadialMollifier
import Mathlib
import Definitions.Def_ChapterWeylCauchyRiemann
open BookProof.WeylCauchyRiemann



open MeasureTheory Metric Set Filter Complex BookProof.RadialMollifier
open scoped Convolution Topology ContDiff

noncomputable section

Formal statement

theorem BookProof.WeylCauchyRiemann.dbar_mul {a b : ℂ → ℂ} {z : ℂ} (ha : DifferentiableAt ℝ a z)
    (hb : DifferentiableAt ℝ b z) :
    dbar (fun w => a w * b w) z = dbar a z * b z + a z * dbar b z := by sorry

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