`BookProof.ChapterCompactCompleteReducibility.avgOp_apply_eq_self` {ρ : G →* (V ≃L[ℂ] V)} (hρ : Continuous fun g => (ρ g : V →L[ℂ] V)) {W : Submodule ℂ V} {T : V →L[ℂ] V} (hTid...

Proved

BookProof.ChapterCompactCompleteReducibility.avgOp_apply_eq_self

by leonardopedro · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterCompactCompleteReducibility.

BookProof.ChapterCompactCompleteReducibility.avgOp_apply_eq_self {ρ : G →* (V ≃L[ℂ] V)} (hρ : Continuous fun g => (ρ g : V →L[ℂ] V)) {W : Submodule ℂ V} {T : V →L[ℂ] V} (hTid : ∀ w ∈ W, T w = w) (hW : ∀ g : G, ∀ x ∈ W, ρ g x ∈ W) {w : V} (hw : w ∈ W) : avgOp μ ρ T w = w

Formalization note: Lean 4 identifier BookProof.ChapterCompactCompleteReducibility.avgOp_apply_eq_self.

Preamble

-- Generated from ChapterCompactCompleteReducibility.lean — theorem BookProof.ChapterCompactCompleteReducibility.avgOp_apply_eq_self
import Mathlib
import Definitions.Def_ChapterCompactCompleteReducibility
open BookProof.ChapterCompactCompleteReducibility



open MeasureTheory

variable {G : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
  [MeasurableSpace G] [BorelSpace G] {μ : Measure G} [IsProbabilityMeasure μ]
  [μ.IsMulLeftInvariant]
variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℂ V] [FiniteDimensional ℂ V]

Formal statement

theorem BookProof.ChapterCompactCompleteReducibility.avgOp_apply_eq_self {ρ : G →* (V ≃L[ℂ] V)} (hρ : Continuous fun g => (ρ g : V →L[ℂ] V))
    {W : Submodule ℂ V} {T : V →L[ℂ] V} (hTid : ∀ w ∈ W, T w = w)
    (hW : ∀ g : G, ∀ x ∈ W, ρ g x ∈ W) {w : V} (hw : w ∈ W) :
    avgOp μ ρ T w = w := by sorry

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