Prove2Me

`BookProof.EnergyBandDecomposition.norm_mul_energy_sub_scalar_le` (hε : 0 < ε) (hx : x ∈ band E ε k) (z : ℂ) : ‖(E x : ℂ) * z - ((k : ℝ) * ε : ℝ) * z‖ ≤ ε * ‖z‖

Proved
BookProof.EnergyBandDecomposition.norm_mul_energy_sub_scalar_le

by leonardopedro · Oct 8, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterEnergyBandDecomposition.

BookProof.EnergyBandDecomposition.norm_mul_energy_sub_scalar_le (hε : 0 < ε) (hx : x ∈ band E ε k) (z : ℂ) : ‖(E x : ℂ) * z - ((k : ℝ) * ε : ℝ) * z‖ ≤ ε * ‖z‖

Formalization note: Lean 4 identifier BookProof.EnergyBandDecomposition.norm_mul_energy_sub_scalar_le.

Preamble
-- Generated from ChapterEnergyBandDecomposition.lean — theorem BookProof.EnergyBandDecomposition.norm_mul_energy_sub_scalar_le
import Mathlib
import Definitions.Def_ChapterEnergyBandDecomposition
open BookProof.EnergyBandDecomposition



open MeasureTheory

variable {X : Type*} {E : X → ℝ} {ε : ℝ} {k : ℤ} {x : X}
Formal statement
theorem BookProof.EnergyBandDecomposition.norm_mul_energy_sub_scalar_le (hε : 0 < ε) (hx : x ∈ band E ε k) (z : ℂ) :
    ‖(E x : ℂ) * z - ((k : ℝ) * ε : ℝ) * z‖ ≤ ε * ‖z‖ := by sorry

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