Prove2Me

`BookProof.ChapterA.cxSystem_reducible_of_commuting_rImaginary` [Nontrivial W] (M : System ℝ W) (h : HasCommutingRImaginary M) : ¬ (cxSystem M).IsIrreducible

Proved
BookProof.ChapterA.cxSystem_reducible_of_commuting_rImaginary

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1h.

BookProof.ChapterA.cxSystem_reducible_of_commuting_rImaginary [Nontrivial W] (M : System ℝ W) (h : HasCommutingRImaginary M) : ¬ (cxSystem M).IsIrreducible

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

Preamble
-- Generated from ChapterA1h.lean — theorem BookProof.ChapterA.cxSystem_reducible_of_commuting_rImaginary
import Definitions.Def_Complexification
import Mathlib
import Definitions.Def_ChapterA1h
import Definitions.Def_ChapterA
import Definitions.Def_ChapterA1
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterA.AntiUnitary


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace


open BookProof.Complexification

variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [CompleteSpace W]
Formal statement
theorem BookProof.ChapterA.cxSystem_reducible_of_commuting_rImaginary [Nontrivial W] (M : System ℝ W)
    (h : HasCommutingRImaginary M) : ¬ (cxSystem M).IsIrreducible := by sorry

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