Prove2Me

`BookProof.ChapterA.realification_splits` (M : System ℂ V) (h : M.IsIrreducible) (Y : Submodule ℝ V) (hY : (rxSystem M).IsSubsystem Y) : Y = ⊥ ∨ Y = ⊤ ∨ (Y ⊓ JY Y = ⊥ ∧ (Y ⊔ JY Y).

Proved
BookProof.ChapterA.realification_splits

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1e.

BookProof.ChapterA.realification_splits (M : System ℂ V) (h : M.IsIrreducible) (Y : Submodule ℝ V) (hY : (rxSystem M).IsSubsystem Y) : Y = ⊥ ∨ Y = ⊤ ∨ (Y ⊓ JY Y = ⊥ ∧ (Y ⊔ JY Y).topologicalClosure = ⊤)

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

Preamble
-- Generated from ChapterA1e.lean — theorem BookProof.ChapterA.realification_splits
import Mathlib
import Definitions.Def_ChapterA1e
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] [CompleteSpace V]
Formal statement
theorem BookProof.ChapterA.realification_splits (M : System ℂ V) (h : M.IsIrreducible)
    (Y : Submodule ℝ V) (hY : (rxSystem M).IsSubsystem Y) :
    Y = ⊥ ∨ Y = ⊤ ∨ (Y ⊓ JY Y = ⊥ ∧ (Y ⊔ JY Y).topologicalClosure = ⊤) := by sorry

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