`BookProof.Complexification.Cx.complexify_realPart_of_invariant` {X : Submodule ℂ (Cx W)} (hX : ∀ x ∈ X, cxConj x ∈ X) : complexify (realPart X) = X

Proved

BookProof.Complexification.Cx.complexify_realPart_of_invariant

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1b.

BookProof.Complexification.Cx.complexify_realPart_of_invariant {X : Submodule ℂ (Cx W)} (hX : ∀ x ∈ X, cxConj x ∈ X) : complexify (realPart X) = X

Formalization note: Lean 4 identifier BookProof.Complexification.Cx.complexify_realPart_of_invariant.

Preamble

-- Generated from ChapterA1b.lean — theorem BookProof.Complexification.Cx.complexify_realPart_of_invariant
import Definitions.Def_ChapterA
import Mathlib
import Definitions.Def_ChapterA1b
import Definitions.Def_ChapterQgOuterFockCoreFL
import Definitions.Def_Complexification
open BookProof.QgOuterFockCoreFL
open BookProof.QgOuterFockCoreFL.CoreData
open BookProof.Complexification
open BookProof.Complexification


open scoped RealInnerProductSpace
open BookProof.ChapterA


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

Formal statement

theorem BookProof.Complexification.Cx.complexify_realPart_of_invariant {X : Submodule ℂ (Cx W)}
    (hX : ∀ x ∈ X, cxConj x ∈ X) : complexify (realPart X) = X := by sorry

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