Prove2Me

`BookProof.ChapterA.complex_irreducible_iff_no_Jinvariant_subsystem` [CompleteSpace V] (M : System ℂ V) : M.IsIrreducible ↔ ∀ Y : Submodule ℝ V, (rxSystem M).IsSubsystem Y → (∀ y ∈

Proved
BookProof.ChapterA.complex_irreducible_iff_no_Jinvariant_subsystem

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1d.

BookProof.ChapterA.complex_irreducible_iff_no_Jinvariant_subsystem [CompleteSpace V] (M : System ℂ V) : M.IsIrreducible ↔ ∀ Y : Submodule ℝ V, (rxSystem M).IsSubsystem Y → (∀ y ∈ Y, Jmap y ∈ Y) → Y = ⊥ ∨ Y = ⊤

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

Preamble
-- Generated from ChapterA1d.lean — theorem BookProof.ChapterA.complex_irreducible_iff_no_Jinvariant_subsystem
import Mathlib
import Definitions.Def_ChapterA1d
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace




attribute [local instance] InnerProductSpace.rclikeToReal

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
Formal statement
theorem BookProof.ChapterA.complex_irreducible_iff_no_Jinvariant_subsystem [CompleteSpace V] (M : System ℂ V) :
    M.IsIrreducible ↔
      ∀ Y : Submodule ℝ V, (rxSystem M).IsSubsystem Y →
        (∀ y ∈ Y, Jmap y ∈ Y) → Y = ⊥ ∨ Y = ⊤ := by sorry

View graph