`BookProof.ChapterSchurFiniteDim.schur_scalar_of_irreducible` [FiniteDimensional ℂ V] [Nontrivial V] (M : System ℂ V) (hirr : IsIrreducibleSystem M) (f : V →ₗ[ℂ] V) (hf : ∀ m ∈ M.o

Proved

BookProof.ChapterSchurFiniteDim.schur_scalar_of_irreducible

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSchurFiniteDim.

BookProof.ChapterSchurFiniteDim.schur_scalar_of_irreducible [FiniteDimensional ℂ V] [Nontrivial V] (M : System ℂ V) (hirr : IsIrreducibleSystem M) (f : V →ₗ[ℂ] V) (hf : ∀ m ∈ M.ops, ∀ x, f (m x) = m (f x)) : ∃ c : ℂ, ∀ x, f x = c • x

Formalization note: Lean 4 identifier BookProof.ChapterSchurFiniteDim.schur_scalar_of_irreducible.

Preamble

-- Generated from ChapterSchurFiniteDim.lean — theorem BookProof.ChapterSchurFiniteDim.schur_scalar_of_irreducible
import Mathlib
import Definitions.Def_ChapterSchurFiniteDim
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterSchurFiniteDim


open Module


open BookProof.ChapterA

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]

Formal statement

theorem BookProof.ChapterSchurFiniteDim.schur_scalar_of_irreducible [FiniteDimensional ℂ V] [Nontrivial V]
    (M : System ℂ V) (hirr : IsIrreducibleSystem M) (f : V →ₗ[ℂ] V)
    (hf : ∀ m ∈ M.ops, ∀ x, f (m x) = m (f x)) :
    ∃ c : ℂ, ∀ x, f x = c • x := by sorry

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