`BookProof.ChapterA.rImagCx_commutes` {M : System ℝ W} {J : W ≃ₗᵢ[ℝ] W} (hJ : IsRImaginary M J) {m : W →L[ℝ] W} (hm : m ∈ M.ops) (x : Cx W) : rImagCx J (Cx.cxMap m x) = Cx.cxMap m

Proved

BookProof.ChapterA.rImagCx_commutes

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1h.

BookProof.ChapterA.rImagCx_commutes {M : System ℝ W} {J : W ≃ₗᵢ[ℝ] W} (hJ : IsRImaginary M J) {m : W →L[ℝ] W} (hm : m ∈ M.ops) (x : Cx W) : rImagCx J (Cx.cxMap m x) = Cx.cxMap m (rImagCx J x)

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

Preamble

-- Generated from ChapterA1h.lean — theorem BookProof.ChapterA.rImagCx_commutes
import Definitions.Def_Complexification
import Mathlib
import Definitions.Def_ChapterA1h
import Definitions.Def_ChapterA
import Definitions.Def_ChapterA1
import Definitions.Def_ChapterQgOuterFockCoreFL
open BookProof.ChapterA
open BookProof.ChapterA
open BookProof.ChapterA.AntiUnitary
open BookProof.QgOuterFockCoreFL
open BookProof.QgOuterFockCoreFL.CoreData


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace


open BookProof.Complexification

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

Formal statement

theorem BookProof.ChapterA.rImagCx_commutes {M : System ℝ W} {J : W ≃ₗᵢ[ℝ] W} (hJ : IsRImaginary M J)
    {m : W →L[ℝ] W} (hm : m ∈ M.ops) (x : Cx W) :
    rImagCx J (Cx.cxMap m x) = Cx.cxMap m (rImagCx J x) := by sorry

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