`BookProof.ChapterFreeFieldBornSignOrientation.flipMatrix_mem_specialOrthogonalGroup_iff` (b : Fin n → Bool) : flipMatrix b ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔ Even (flipCo

Proved

BookProof.ChapterFreeFieldBornSignOrientation.flipMatrix_mem_specialOrthogonalGroup_iff

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornSignOrientation.

BookProof.ChapterFreeFieldBornSignOrientation.flipMatrix_mem_specialOrthogonalGroup_iff (b : Fin n → Bool) : flipMatrix b ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔ Even (flipCount b)

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornSignOrientation.flipMatrix_mem_specialOrthogonalGroup_iff.

Preamble

-- Generated from ChapterFreeFieldBornSignOrientation.lean — theorem BookProof.ChapterFreeFieldBornSignOrientation.flipMatrix_mem_specialOrthogonalGroup_iff
import Definitions.Def_ChapterFreeFieldBornSignAction
import Definitions.Def_ChapterFreeFieldBornSignHom
import Definitions.Def_ChapterFreeFieldBornSignMatrix
import Mathlib
import Definitions.Def_ChapterFreeFieldBornSignOrientation
open BookProof.ChapterFreeFieldBornSignOrientation

variable {n : ℕ}


open BookProof.ChapterFreeFieldBornSignAction
open BookProof.ChapterFreeFieldBornSignHom
open BookProof.ChapterFreeFieldBornSignMatrix

Formal statement

theorem BookProof.ChapterFreeFieldBornSignOrientation.flipMatrix_mem_specialOrthogonalGroup_iff (b : Fin n → Bool) :
    flipMatrix b ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔ Even (flipCount b) := by sorry

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