`BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_expApply` (A : E →L[ℂ] E) (x : E) (t : ℝ) : HasDerivAt (fun s : ℝ => (NormedSpace.exp (s • A)) x) (A (NormedSpace.exp (t • A) x

Proved

BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_expApply

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeSemigroup.

BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_expApply (A : E →L[ℂ] E) (x : E) (t : ℝ) : HasDerivAt (fun s : ℝ => (NormedSpace.exp (s • A)) x) (A (NormedSpace.exp (t • A) x)) t

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_expApply.

Preamble

-- Generated from ChapterNumericalRangeSemigroup.lean — theorem BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_expApply
import Mathlib
import Definitions.Def_ChapterNumericalRangeSemigroup
open BookProof.ChapterNumericalRangeSemigroup


open scoped InnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E]

Formal statement

theorem BookProof.ChapterNumericalRangeSemigroup.hasDerivAt_expApply (A : E →L[ℂ] E) (x : E) (t : ℝ) :
    HasDerivAt (fun s : ℝ => (NormedSpace.exp (s • A)) x) (A (NormedSpace.exp (t • A) x)) t := by sorry

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