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`BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor_iff` (b₁ b₂ : Fin n → Bool) : flipMatrix (fun k => xor (b₁ k) (b₂ k)) ∈ Matrix.specialOrthogonalGroup

Proved
BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor_iff

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornSignOrientationKernel.

BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor_iff (b₁ b₂ : Fin n → Bool) : flipMatrix (fun k => xor (b₁ k) (b₂ k)) ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔ (flipMatrix b₁ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔ flipMatrix b₂ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ)

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

Preamble
-- Generated from ChapterFreeFieldBornSignOrientationKernel.lean — theorem BookProof.ChapterFreeFieldBornSignOrientationKernel.orientationPreserving_xor_iff
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_iff (b₁ b₂ : Fin n → Bool) :
    flipMatrix (fun k => xor (b₁ k) (b₂ k)) ∈
        Matrix.specialOrthogonalGroup (Fin n) ℝ ↔
      (flipMatrix b₁ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔
       flipMatrix b₂ ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ) := by sorry

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