`BookProof.ChapterA.conjugation_avg_antifixed` (θ : AntiUnitary V) (hθ : ∀ x, θ (θ x) = x) (x : V) : θ ((2⁻¹ : ℂ) • (x - θ x)) = -((2⁻¹ : ℂ) • (x - θ x))
ProvedBookProof.ChapterA.conjugation_avg_antifixedtheoremstimepiece
Prove the following Lean 4 theorem from ChapterA1.
BookProof.ChapterA.conjugation_avg_antifixed (θ : AntiUnitary V) (hθ : ∀ x, θ (θ x) = x) (x : V) : θ ((2⁻¹ : ℂ) • (x - θ x)) = -((2⁻¹ : ℂ) • (x - θ x))
Formalization note: Lean 4 identifier BookProof.ChapterA.conjugation_avg_antifixed.
Preamble
-- Generated from ChapterA1.lean — theorem BookProof.ChapterA.conjugation_avg_antifixed
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 RealInnerProductSpaceFormal statement
theorem BookProof.ChapterA.conjugation_avg_antifixed (θ : AntiUnitary V) (hθ : ∀ x, θ (θ x) = x) (x : V) :
θ ((2⁻¹ : ℂ) • (x - θ x)) = -((2⁻¹ : ℂ) • (x - θ x)) := by sorry