`BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor` {b₁ b₂ : Fin n → Bool} (h₁ : flipMatrix b₁ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ) (h₂ : flipMatr

Proved

BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor

by leonardopedro · Oct 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornSignOrientationKernel.

BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor {b₁ b₂ : Fin n → Bool} (h₁ : flipMatrix b₁ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ) (h₂ : flipMatrix b₂ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ) : flipMatrix (fun k => xor (b₁ k) (b₂ k)) ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor.

Preamble

-- Generated from ChapterFreeFieldBornSignOrientationKernel.lean — theorem BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor
import Definitions.Def_ChapterFreeFieldBornSignHom
import Definitions.Def_ChapterFreeFieldBornSignMatrix
import Definitions.Def_ChapterFreeFieldBornSignOrientation
import Mathlib
import Definitions.Def_ChapterFreeFieldBornSignOrientationKernel
open BookProof.ChapterFreeFieldBornSignOrientationKernel

variable {n : ℕ}


open BookProof.ChapterFreeFieldBornSignHom
open BookProof.ChapterFreeFieldBornSignMatrix
open BookProof.ChapterFreeFieldBornSignOrientation

Formal statement

theorem BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor {b₁ b₂ : Fin n → Bool}
    (h₁ : flipMatrix b₁ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ)
    (h₂ : flipMatrix b₂ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ) :
    flipMatrix (fun k => xor (b₁ k) (b₂ k)) ∈
      Matrix.specialOrthogonalGroup (Fin n) ℝ := by sorry

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