`BookProof.ChapterAbelianCyclicModel.cyclic_representation_multiplication_model_inc` (hxi : ‖xi‖ = 1) : ∃ (mu : Measure X) (_ : IsProbabilityMeasure mu) (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H),...

Proved

BookProof.ChapterAbelianCyclicModel.cyclic_representation_multiplication_model_inc

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianCyclicModel.

BookProof.ChapterAbelianCyclicModel.cyclic_representation_multiplication_model_inc (hxi : ‖xi‖ = 1) : ∃ (mu : Measure X) (_ : IsProbabilityMeasure mu) (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H), (∀ (g : C(X, ℂ)) (u : Lp ℂ 2 mu), U (mulRep mu g u) = pi g (U u)) ∧ (∀ (g : C(X, ℂ)) (v : H), U.symm (pi g v) = mulRep mu g (U.symm v))

Formalization note: Lean 4 identifier BookProof.ChapterAbelianCyclicModel.cyclic_representation_multiplication_model_inc.

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

Preamble

-- Generated from ChapterAbelianCyclicModel.lean — theorem BookProof.ChapterAbelianCyclicModel.cyclic_representation_multiplication_model_inc
import Definitions.Def_ChapterAbelianGelfandModel
import Mathlib
import Definitions.Def_ChapterAbelianCyclicModel
open BookProof.ChapterAbelianCyclicModel


open MeasureTheory Complex WeakDual
open scoped ComplexOrder


open BookProof.ChapterAbelianGelfandModel


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)

variable (hcyc : DenseRange (repVec pi xi))

include hcyc in

Formal statement

theorem BookProof.ChapterAbelianCyclicModel.cyclic_representation_multiplication_model_inc (hxi : ‖xi‖ = 1) :
    ∃ (mu : Measure X) (_ : IsProbabilityMeasure mu) (U : Lp ℂ 2 mu ≃ₗᵢ[ℂ] H),
      (∀ (g : C(X, ℂ)) (u : Lp ℂ 2 mu), U (mulRep mu g u) = pi g (U u)) ∧
      (∀ (g : C(X, ℂ)) (v : H), U.symm (pi g v) = mulRep mu g (U.symm v)) := by sorry

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