`BookProof.MixedLinearEsa.hasDerivAt_gaugeFun_line` (b m : V) (hm : m ≠ 0) (x : V) : HasDerivAt (fun t : ℝ => gaugeFun b m (x + t • m)) (gaugeFun b m x * (Complex.I * ((-(inner ℝ x

Proved

BookProof.MixedLinearEsa.hasDerivAt_gaugeFun_line

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMixedLinearEsa.

BookProof.MixedLinearEsa.hasDerivAt_gaugeFun_line (b m : V) (hm : m ≠ 0) (x : V) : HasDerivAt (fun t : ℝ => gaugeFun b m (x + t • m)) (gaugeFun b m x * (Complex.I * ((-(inner ℝ x b : ℝ) : ℝ) : ℂ))) 0

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

Preamble

-- Generated from ChapterMixedLinearEsa.lean — theorem BookProof.MixedLinearEsa.hasDerivAt_gaugeFun_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]


omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in

Formal statement

theorem BookProof.MixedLinearEsa.hasDerivAt_gaugeFun_line (b m : V) (hm : m ≠ 0) (x : V) :
    HasDerivAt (fun t : ℝ => gaugeFun b m (x + t • m))
      (gaugeFun b m x * (Complex.I * ((-(inner ℝ x b : ℝ) : ℝ) : ℂ))) 0 := by sorry

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