`BookProof.ChapterA.conjugation_avg_fixed` (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_fixedtheoremstimepiece
Prove the following Lean 4 theorem from ChapterA1.
BookProof.ChapterA.conjugation_avg_fixed (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_fixed.
Preamble
-- Generated from ChapterA1.lean — theorem BookProof.ChapterA.conjugation_avg_fixed
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_fixed (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