`BookProof.ChapterFreeFieldBornSignRepresentation.flipRepresentation_specialOrthogonal_iff` (b : Multiplicative (Fin n → Bool)) : (flipRepresentation n b : Matrix (Fin n) (Fin n) ℝ

Proved

BookProof.ChapterFreeFieldBornSignRepresentation.flipRepresentation_specialOrthogonal_iff

by leonardopedro · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornSignRepresentation.

BookProof.ChapterFreeFieldBornSignRepresentation.flipRepresentation_specialOrthogonal_iff (b : Multiplicative (Fin n → Bool)) : (flipRepresentation n b : Matrix (Fin n) (Fin n) ℝ) ∈ Matrix.specialOrthogonalGroup (Fin n) ℝ ↔ b.toAdd ∈ orientationPreservingSigns n

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornSignRepresentation.flipRepresentation_specialOrthogonal_iff.

Preamble

-- Generated from ChapterFreeFieldBornSignRepresentation.lean — theorem BookProof.ChapterFreeFieldBornSignRepresentation.flipRepresentation_specialOrthogonal_iff
import Definitions.Def_ChapterFreeFieldBornSignAction
import Definitions.Def_ChapterFreeFieldBornSignHom
import Definitions.Def_ChapterFreeFieldBornSignMatrix
import Definitions.Def_ChapterFreeFieldBornSignOrientation
import Definitions.Def_ChapterFreeFieldBornSignOrientationSubgroup
import Mathlib
import Definitions.Def_ChapterFreeFieldBornSignRepresentation
open BookProof.ChapterFreeFieldBornSignRepresentation


open BookProof.ChapterFreeFieldBornSignAction
open BookProof.ChapterFreeFieldBornSignHom
open BookProof.ChapterFreeFieldBornSignMatrix
open BookProof.ChapterFreeFieldBornSignOrientation
open BookProof.ChapterFreeFieldBornSignOrientationSubgroup


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterFreeFieldBornSignRepresentation.flipRepresentation_specialOrthogonal_iff
    (b : Multiplicative (Fin n → Bool)) :
    (flipRepresentation n b : Matrix (Fin n) (Fin n) ℝ) ∈
        Matrix.specialOrthogonalGroup (Fin n) ℝ ↔
      b.toAdd ∈ orientationPreservingSigns n := by sorry

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