`BookProof.ChapterA.rimaginary_symm_apply` (J : W ≃ₗᵢ[ℝ] W) (hJ : ∀ x, J (J x) = -x) (x : W) : J.symm x = -J x

Proved

BookProof.ChapterA.rimaginary_symm_apply

by leonardopedro · Oct 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterA1.

BookProof.ChapterA.rimaginary_symm_apply (J : W ≃ₗᵢ[ℝ] W) (hJ : ∀ x, J (J x) = -x) (x : W) : J.symm x = -J x

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

Preamble

-- Generated from ChapterA1.lean — theorem BookProof.ChapterA.rimaginary_symm_apply
import Mathlib
import Definitions.Def_ChapterA1
import Definitions.Def_ChapterA
open BookProof.ChapterA

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


open scoped ComplexConjugate InnerProductSpace RealInnerProductSpace

Formal statement

theorem BookProof.ChapterA.rimaginary_symm_apply (J : W ≃ₗᵢ[ℝ] W) (hJ : ∀ x, J (J x) = -x) (x : W) :
    J.symm x = -J x := by sorry

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