`BookProof.ChapterSchurFiniteDimensional.Rreal_commutant_eq_real_scalars_finiteDimensional` [FiniteDimensional ℂ V] [Nontrivial V] (M : System ℂ V) (hirr : M.IsIrreducible) {θ : An

Proved

BookProof.ChapterSchurFiniteDimensional.Rreal_commutant_eq_real_scalars_finiteDimensional

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSchurFiniteDimensional.

BookProof.ChapterSchurFiniteDimensional.Rreal_commutant_eq_real_scalars_finiteDimensional [FiniteDimensional ℂ V] [Nontrivial V] (M : System ℂ V) (hirr : M.IsIrreducible) {θ : AntiUnitary V} (hθ : IsConjugation M θ) (S : V →L[ℂ] V) : (M.Commutes S ∧ CommutesConj θ S) ↔ ∃ r : ℝ, S = ((r : ℂ)) • (1 : V →L[ℂ] V)

Formalization note: Lean 4 identifier BookProof.ChapterSchurFiniteDimensional.Rreal_commutant_eq_real_scalars_finiteDimensional.

Preamble

-- Generated from ChapterSchurFiniteDimensional.lean — theorem BookProof.ChapterSchurFiniteDimensional.Rreal_commutant_eq_real_scalars_finiteDimensional
import Mathlib
import Definitions.Def_ChapterSchurFiniteDimensional
import Definitions.Def_ChapterA
import Definitions.Def_ChapterA1
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterSchurFiniteDimensional


open scoped ComplexConjugate InnerProductSpace


open BookProof.ChapterA BookProof.ChapterA

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

Formal statement

theorem BookProof.ChapterSchurFiniteDimensional.Rreal_commutant_eq_real_scalars_finiteDimensional [FiniteDimensional ℂ V] [Nontrivial V]
    (M : System ℂ V) (hirr : M.IsIrreducible) {θ : AntiUnitary V} (hθ : IsConjugation M θ)
    (S : V →L[ℂ] V) :
    (M.Commutes S ∧ CommutesConj θ S) ↔ ∃ r : ℝ, S = ((r : ℂ)) • (1 : V →L[ℂ] V) := by sorry

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