`BookProof.ChapterWeylSl2.Sl2Rep.E_pow_F` {w : V} {lam : ℂ} (hE : R.E w = 0) (h : R.H w = lam • w) (k : ℕ) : R.E ((R.F ^ (k + 1)) w) = ((k + 1 : ℂ) * (lam - k)) • (R.F ^ k) w

Proved

BookProof.ChapterWeylSl2.Sl2Rep.E_pow_F

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterWeylSl2.

BookProof.ChapterWeylSl2.Sl2Rep.E_pow_F {w : V} {lam : ℂ} (hE : R.E w = 0) (h : R.H w = lam • w) (k : ℕ) : R.E ((R.F ^ (k + 1)) w) = ((k + 1 : ℂ) * (lam - k)) • (R.F ^ k) w

Formalization note: Lean 4 identifier BookProof.ChapterWeylSl2.Sl2Rep.E_pow_F.

Preamble

-- Generated from ChapterWeylSl2.lean — theorem BookProof.ChapterWeylSl2.Sl2Rep.E_pow_F
import Mathlib
import Definitions.Def_ChapterWeylSl2
import Definitions.Def_ChapterDoubleSlit
import Definitions.Def_ChapterGleasonPureMixed
open BookProof.ChapterDoubleSlit
open BookProof.ChapterGleasonPureMixed
open BookProof.ChapterWeylSl2
open BookProof.ChapterWeylSl2



universe u

variable {V : Type u} [AddCommGroup V] [Module ℂ V]

variable (R : Sl2Rep V)
variable {R}
variable {R : Sl2Rep V}

Formal statement

theorem BookProof.ChapterWeylSl2.Sl2Rep.E_pow_F {w : V} {lam : ℂ} (hE : R.E w = 0) (h : R.H w = lam • w) (k : ℕ) :
    R.E ((R.F ^ (k + 1)) w) = ((k + 1 : ℂ) * (lam - k)) • (R.F ^ k) w := by sorry

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