Prove2Me

`BookProof.ChapterA.Rpseudoreal_realCommutant_eq_quaternion` (M : System ℂ V) (hSchur : IsSchurFull M) {θ : AntiUnitary V} (hθ : ∀ x, θ (θ x) = -x) (hθc : CommutesAntiUnitary M θ)

Proved
BookProof.ChapterA.Rpseudoreal_realCommutant_eq_quaternion

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA2c.

BookProof.ChapterA.Rpseudoreal_realCommutant_eq_quaternion (M : System ℂ V) (hSchur : IsSchurFull M) {θ : AntiUnitary V} (hθ : ∀ x, θ (θ x) = -x) (hθc : CommutesAntiUnitary M θ) (S : V →L[ℝ] V) : RealCommutes M S ↔ ∃ q : Quaternion ℝ, S = qembed θ hθ q

Formalization note: Lean 4 identifier BookProof.ChapterA.Rpseudoreal_realCommutant_eq_quaternion.

Preamble
-- Generated from ChapterA2c.lean — theorem BookProof.ChapterA.Rpseudoreal_realCommutant_eq_quaternion
import Mathlib
import Definitions.Def_ChapterA2c
import Definitions.Def_ChapterA
import Definitions.Def_ChapterA1
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace Quaternion


variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
Formal statement
theorem BookProof.ChapterA.Rpseudoreal_realCommutant_eq_quaternion (M : System ℂ V) (hSchur : IsSchurFull M)
    {θ : AntiUnitary V} (hθ : ∀ x, θ (θ x) = -x) (hθc : CommutesAntiUnitary M θ)
    (S : V →L[ℝ] V) :
    RealCommutes M S ↔ ∃ q : Quaternion ℝ, S = qembed θ hθ q := by sorry

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