`BookProof.Complexification.irreducible_iff_no_conj_subsystem` [CompleteSpace W] (M : System ℝ W) : M.IsIrreducible ↔ ∀ X : Submodule ℂ (Cx W), (cxSystem M).IsSubsystem X → (∀ x ∈

Proved

BookProof.Complexification.irreducible_iff_no_conj_subsystem

by leonardopedro · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1b.

BookProof.Complexification.irreducible_iff_no_conj_subsystem [CompleteSpace W] (M : System ℝ W) : M.IsIrreducible ↔ ∀ X : Submodule ℂ (Cx W), (cxSystem M).IsSubsystem X → (∀ x ∈ X, Cx.cxConj x ∈ X) → X = ⊥ ∨ X = ⊤

Formalization note: Lean 4 identifier BookProof.Complexification.irreducible_iff_no_conj_subsystem.

Preamble

-- Generated from ChapterA1b.lean — theorem BookProof.Complexification.irreducible_iff_no_conj_subsystem
import Mathlib
import Definitions.Def_ChapterA1b
import Definitions.Def_ChapterA
import Definitions.Def_Complexification
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.Complexification


open scoped RealInnerProductSpace
open BookProof.ChapterA


variable {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W]

variable [CompleteSpace W]

Formal statement

theorem BookProof.Complexification.irreducible_iff_no_conj_subsystem [CompleteSpace W] (M : System ℝ W) :
    M.IsIrreducible ↔
      ∀ X : Submodule ℂ (Cx W), (cxSystem M).IsSubsystem X →
        (∀ x ∈ X, Cx.cxConj x ∈ X) → X = ⊥ ∨ X = ⊤ := by sorry

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