`BookProof.ChapterMaschkeFiniteGroup.avgProj_idempotent` [Fintype G] {ρ : Representation ℂ G V} {W : Submodule ℂ V} (hW : IsInvariant ρ W) (pi : V →ₗ[ℂ] V) (hpi_mem : ∀ x, pi x ∈ W

Proved

BookProof.ChapterMaschkeFiniteGroup.avgProj_idempotent

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterMaschkeFiniteGroup.

BookProof.ChapterMaschkeFiniteGroup.avgProj_idempotent [Fintype G] {ρ : Representation ℂ G V} {W : Submodule ℂ V} (hW : IsInvariant ρ W) (pi : V →ₗ[ℂ] V) (hpi_mem : ∀ x, pi x ∈ W) (hpi_id : ∀ x ∈ W, pi x = x) (x : V) : avgProj ρ pi (avgProj ρ pi x) = avgProj ρ pi x

Formalization note: Lean 4 identifier BookProof.ChapterMaschkeFiniteGroup.avgProj_idempotent.

Preamble

-- Generated from ChapterMaschkeFiniteGroup.lean — theorem BookProof.ChapterMaschkeFiniteGroup.avgProj_idempotent
import Mathlib
import Definitions.Def_ChapterMaschkeFiniteGroup
open BookProof.ChapterMaschkeFiniteGroup



variable {G : Type*} [Group G] {V : Type*} [AddCommGroup V] [Module ℂ V]

Formal statement

theorem BookProof.ChapterMaschkeFiniteGroup.avgProj_idempotent [Fintype G] {ρ : Representation ℂ G V} {W : Submodule ℂ V}
    (hW : IsInvariant ρ W) (pi : V →ₗ[ℂ] V) (hpi_mem : ∀ x, pi x ∈ W)
    (hpi_id : ∀ x ∈ W, pi x = x) (x : V) :
    avgProj ρ pi (avgProj ρ pi x) = avgProj ρ pi x := by sorry

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