`BookProof.ChapterAbelianCyclicCommutant.conjRep_mul` (A B : Lp ℂ 2 (repMeasure pi xi) →L[ℂ] Lp ℂ 2 (repMeasure pi xi)) : conjRep pi xi hcyc (A * B) = conjRep pi xi hcyc A * conjRe

Proved

BookProof.ChapterAbelianCyclicCommutant.conjRep_mul

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicCommutant.

BookProof.ChapterAbelianCyclicCommutant.conjRep_mul (A B : Lp ℂ 2 (repMeasure pi xi) →L[ℂ] Lp ℂ 2 (repMeasure pi xi)) : conjRep pi xi hcyc (A * B) = conjRep pi xi hcyc A * conjRep pi xi hcyc B

Formalization note: Lean 4 identifier BookProof.ChapterAbelianCyclicCommutant.conjRep_mul.

Preamble

-- Generated from ChapterAbelianCyclicCommutant.lean — theorem BookProof.ChapterAbelianCyclicCommutant.conjRep_mul
import Definitions.Def_ChapterLinftyMultiplication
import Definitions.Def_ChapterLinftyMaximalAbelian
import Definitions.Def_ChapterAbelianGelfandModel
import Definitions.Def_ChapterSpectralCommutant
import Definitions.Def_ChapterAbelianCyclicModel
import Mathlib
import Definitions.Def_ChapterAbelianCyclicCommutant
open BookProof.ChapterAbelianCyclicCommutant


noncomputable section

open MeasureTheory Complex WeakDual


open BookProof.ChapterLinftyMultiplication BookProof.ChapterLinftyMaximalAbelian
open BookProof.ChapterAbelianGelfandModel BookProof.ChapterSpectralCommutant
open BookProof.ChapterAbelianCyclicModel


variable {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
  [MeasurableSpace X] [BorelSpace X]
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (pi : C(X, ℂ) →⋆ₐ[ℂ] (H →L[ℂ] H)) (xi : H) (hcyc : DenseRange (repVec pi xi))

Formal statement

theorem BookProof.ChapterAbelianCyclicCommutant.conjRep_mul (A B : Lp ℂ 2 (repMeasure pi xi) →L[ℂ] Lp ℂ 2 (repMeasure pi xi)) :
    conjRep pi xi hcyc (A * B) = conjRep pi xi hcyc A * conjRep pi xi hcyc B := by sorry

View graph