`BookProof.ChapterH5.shift_pow_sub_pow_mem` (H : E →ₗ[K] E) (γ : K) (v : E) (m : ℕ) : (((H - γ • 1) ^ m) v) - ((H ^ m) v) ∈ krylovSpan H v m

Proved

BookProof.ChapterH5.shift_pow_sub_pow_mem

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterH5.

BookProof.ChapterH5.shift_pow_sub_pow_mem (H : E →ₗ[K] E) (γ : K) (v : E) (m : ℕ) : (((H - γ • 1) ^ m) v) - ((H ^ m) v) ∈ krylovSpan H v m

Formalization note: Lean 4 identifier BookProof.ChapterH5.shift_pow_sub_pow_mem.

Preamble

-- Generated from ChapterH5.lean — theorem BookProof.ChapterH5.shift_pow_sub_pow_mem
import Mathlib
import Definitions.Def_ChapterH5
open BookProof.ChapterH5


noncomputable section



variable {K E : Type*} [Field K] [AddCommGroup E] [Module K E]

variable {H : E →ₗ[K] E} {v : E}

Formal statement

theorem BookProof.ChapterH5.shift_pow_sub_pow_mem (H : E →ₗ[K] E) (γ : K) (v : E) (m : ℕ) :
    (((H - γ • 1) ^ m) v) - ((H ^ m) v) ∈ krylovSpan H v m := by sorry

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