Prove2Me

`BookProof.ChapterAbelianCyclicCommutant.commutant_repSet_isCommutative_inc` {S R : H →L[ℂ] H} (hS : S ∈ (repSet pi).centralizer) (hR : R ∈ (repSet pi).centralizer) : S * R = R * S

Proved
BookProof.ChapterAbelianCyclicCommutant.commutant_repSet_isCommutative_inc

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicCommutant.

BookProof.ChapterAbelianCyclicCommutant.commutant_repSet_isCommutative_inc {S R : H →L[ℂ] H} (hS : S ∈ (repSet pi).centralizer) (hR : R ∈ (repSet pi).centralizer) : S * R = R * S

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

Replaces the retired node BookProof.ChapterAbelianCyclicCommutant.commutant_repSet_isCommutative, whose stored statement omitted the hcyc hypothesis binder (dropped include).

Preamble
-- Generated from ChapterAbelianCyclicCommutant.lean — theorem BookProof.ChapterAbelianCyclicCommutant.commutant_repSet_isCommutative_inc
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))


include hcyc in
Formal statement
theorem BookProof.ChapterAbelianCyclicCommutant.commutant_repSet_isCommutative_inc {S R : H →L[ℂ] H}
    (hS : S ∈ (repSet pi).centralizer) (hR : R ∈ (repSet pi).centralizer) :
    S * R = R * S := by sorry

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