`BookProof.ChapterNumericalRangeSemigroup.norm_cexp_le_of_re_le` {ω t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z.re ≤ ω) : ‖Complex.exp ((t : ℂ) * z)‖ ≤ Real.exp (ω * t)

Proved

BookProof.ChapterNumericalRangeSemigroup.norm_cexp_le_of_re_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeSemigroup.

BookProof.ChapterNumericalRangeSemigroup.norm_cexp_le_of_re_le {ω t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z.re ≤ ω) : ‖Complex.exp ((t : ℂ) * z)‖ ≤ Real.exp (ω * t)

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

Preamble

-- Generated from ChapterNumericalRangeSemigroup.lean — theorem BookProof.ChapterNumericalRangeSemigroup.norm_cexp_le_of_re_le
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.norm_cexp_le_of_re_le {ω t : ℝ} (ht : 0 ≤ t) {z : ℂ} (hz : z.re ≤ ω) :
    ‖Complex.exp ((t : ℂ) * z)‖ ≤ Real.exp (ω * t) := by sorry

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