`BookProof.ChapterPvmFibreInducedSystem.induced_system_of_isHilbertSum` {ι : Type*} [Countable ι] (P : Pvm X H) (μ : Measure X) (V : ∀ _ : ι, Lp ℂ 2 μ →ₗᵢ[ℂ] H) (hsum : IsHilbertSu

Proved

BookProof.ChapterPvmFibreInducedSystem.induced_system_of_isHilbertSum

by leonardopedro · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPvmFibreInducedSystem.

BookProof.ChapterPvmFibreInducedSystem.induced_system_of_isHilbertSum {ι : Type*} [Countable ι] (P : Pvm X H) (μ : Measure X) (V : ∀ _ : ι, Lp ℂ 2 μ →ₗᵢ[ℂ] H) (hsum : IsHilbertSum ℂ (fun _ : ι => Lp ℂ 2 μ) V) (hint : ∀ (i : ι) (E : Set X) (hE : MeasurableSet E) (f : Lp ℂ 2 μ), V i (proj μ hE f) = P.p E (V i f)) : ∃ W : H ≃ₗᵢ[ℂ] Lp (Fibre ι) 2 μ, ∀ (E : Set X) (hE : MeasurableSet E) (v : H), W (P.p E v) = proj μ hE (W v)

Formalization note: Lean 4 identifier BookProof.ChapterPvmFibreInducedSystem.induced_system_of_isHilbertSum.

Preamble

-- Generated from ChapterPvmFibreInducedSystem.lean — theorem BookProof.ChapterPvmFibreInducedSystem.induced_system_of_isHilbertSum
import Definitions.Def_ChapterPvmMeasure
import Definitions.Def_ChapterPvmCyclicUnitary
import Definitions.Def_ChapterPvmCyclicDecomposition
import Definitions.Def_ChapterPvmInducedSystem
import Definitions.Def_ChapterMackeyQuasiInvariant
import Definitions.Def_ChapterL2FibreSum
import Definitions.Def_ChapterHilbertSumIntertwine
import Mathlib
import Definitions.Def_ChapterPvmFibreInducedSystem
import Definitions.Def_ChapterElectroweakFieldStrength
open BookProof.ChapterElectroweakFieldStrength
open BookProof.ChapterPvmFibreInducedSystem


open MeasureTheory
open scoped InnerProductSpace


open BookProof.ChapterPvmMeasure BookProof.ChapterPvmCyclicUnitary
open BookProof.ChapterPvmCyclicDecomposition BookProof.ChapterPvmInducedSystem
open BookProof.ChapterMackeyQuasiInvariant BookProof.ChapterL2FibreSum
open BookProof.ChapterHilbertSumIntertwine

variable {X : Type*} [MeasurableSpace X]
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

Formal statement

theorem BookProof.ChapterPvmFibreInducedSystem.induced_system_of_isHilbertSum {ι : Type*} [Countable ι]
    (P : Pvm X H) (μ : Measure X) (V : ∀ _ : ι, Lp ℂ 2 μ →ₗᵢ[ℂ] H)
    (hsum : IsHilbertSum ℂ (fun _ : ι => Lp ℂ 2 μ) V)
    (hint : ∀ (i : ι) (E : Set X) (hE : MeasurableSet E) (f : Lp ℂ 2 μ),
      V i (proj μ hE f) = P.p E (V i f)) :
    ∃ W : H ≃ₗᵢ[ℂ] Lp (Fibre ι) 2 μ,
      ∀ (E : Set X) (hE : MeasurableSet E) (v : H), W (P.p E v) = proj μ hE (W v) := by sorry

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