`BookProof.ChapterFreeFieldBornFiberStabilizer.bornFiber_card_mul_signStab_card` (x : ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)) : Nat.card ↥(bornMapSphere n ⁻¹' {bornMapSp

Proved

BookProof.ChapterFreeFieldBornFiberStabilizer.bornFiber_card_mul_signStab_card

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornFiberStabilizer.

BookProof.ChapterFreeFieldBornFiberStabilizer.bornFiber_card_mul_signStab_card (x : ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)) : Nat.card ↥(bornMapSphere n ⁻¹' {bornMapSphere n x}) * (signStab (x : EuclideanSpace ℝ (Fin n))).card = 2 ^ n

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornFiberStabilizer.bornFiber_card_mul_signStab_card.

Preamble

-- Generated from ChapterFreeFieldBornFiberStabilizer.lean — theorem BookProof.ChapterFreeFieldBornFiberStabilizer.bornFiber_card_mul_signStab_card
import Definitions.Def_ChapterFreeFieldBorn
import Definitions.Def_ChapterFreeFieldBornQuotient
import Definitions.Def_ChapterFreeFieldBornSignGauge
import Definitions.Def_ChapterFreeFieldBornFiberCardGeneral
import Definitions.Def_ChapterFreeFieldBornFiberBounds
import Mathlib
import Definitions.Def_ChapterFreeFieldBornFiberStabilizer
open BookProof.ChapterFreeFieldBornFiberStabilizer


open MeasureTheory
open BookProof.ChapterFreeFieldBorn
open BookProof.ChapterFreeFieldBornQuotient
open BookProof.ChapterFreeFieldBornSignGauge
open BookProof.ChapterFreeFieldBornFiberCardGeneral
open BookProof.ChapterFreeFieldBornFiberBounds


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterFreeFieldBornFiberStabilizer.bornFiber_card_mul_signStab_card
    (x : ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)) :
    Nat.card ↥(bornMapSphere n ⁻¹' {bornMapSphere n x}) *
        (signStab (x : EuclideanSpace ℝ (Fin n))).card = 2 ^ n := by sorry

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