`BookProof.ChapterA.Rreal_commutant_eq_real_scalars` (M : System ℂ V) (hSchur : IsSchurFull M) {θ : AntiUnitary V} (_hθ : IsConjugation M θ) (S : V →L[ℂ] V) : (M.Commutes S ∧ Commu

Proved

BookProof.ChapterA.Rreal_commutant_eq_real_scalars

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA2b.

BookProof.ChapterA.Rreal_commutant_eq_real_scalars (M : System ℂ V) (hSchur : IsSchurFull M) {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.ChapterA.Rreal_commutant_eq_real_scalars.

Preamble

-- Generated from ChapterA2b.lean — theorem BookProof.ChapterA.Rreal_commutant_eq_real_scalars
import Mathlib
import Definitions.Def_ChapterA2b
import Definitions.Def_ChapterA
import Definitions.Def_ChapterA1
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace


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

Formal statement

theorem BookProof.ChapterA.Rreal_commutant_eq_real_scalars (M : System ℂ V) (hSchur : IsSchurFull M)
    {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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