`BookProof.MixedLinearEsa.hasDerivAt_phaseFun_line` {W θ : V → ℝ} {m : V} (hθd : ∀ x, HasDerivAt (fun t : ℝ => θ (x + t • m)) (-(W x)) 0) (x : V) : HasDerivAt (fun t : ℝ => phaseFu

Proved

BookProof.MixedLinearEsa.hasDerivAt_phaseFun_line

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMixedLinearEsa.

BookProof.MixedLinearEsa.hasDerivAt_phaseFun_line {W θ : V → ℝ} {m : V} (hθd : ∀ x, HasDerivAt (fun t : ℝ => θ (x + t • m)) (-(W x)) 0) (x : V) : HasDerivAt (fun t : ℝ => phaseFun θ (x + t • m)) (phaseFun θ x * (Complex.I * ((-(W x) : ℝ) : ℂ))) 0

Formalization note: Lean 4 identifier BookProof.MixedLinearEsa.hasDerivAt_phaseFun_line.

Preamble

-- Generated from ChapterMixedLinearEsa.lean — theorem BookProof.MixedLinearEsa.hasDerivAt_phaseFun_line
import Definitions.Def_ChapterStrichartzWave
import Definitions.Def_ChapterFourierMultiplierEsa
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterMixedLinearEsa
open BookProof.MixedLinearEsa



open MeasureTheory SchwartzMap FourierTransform ComplexInnerProductSpace LineDeriv
open BookProof.StrichartzWave BookProof.FourierMultiplierEsa BookProof.FarisLavine

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
  [MeasurableSpace V] [BorelSpace V]

Formal statement

theorem BookProof.MixedLinearEsa.hasDerivAt_phaseFun_line {W θ : V → ℝ} {m : V}
    (hθd : ∀ x, HasDerivAt (fun t : ℝ => θ (x + t • m)) (-(W x)) 0) (x : V) :
    HasDerivAt (fun t : ℝ => phaseFun θ (x + t • m))
      (phaseFun θ x * (Complex.I * ((-(W x) : ℝ) : ℂ))) 0 := by sorry

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