`BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_eq_iff_posSupport_card_eq` {p q : ↥(stdSimplex ℝ (Fin n))} : Nat.card ↥(bornMapSphere n ⁻¹' {p}) = Nat.card ↥(bornMapSpher

Proved

BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_eq_iff_posSupport_card_eq

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornFiberBounds.

BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_eq_iff_posSupport_card_eq {p q : ↥(stdSimplex ℝ (Fin n))} : Nat.card ↥(bornMapSphere n ⁻¹' {p}) = Nat.card ↥(bornMapSphere n ⁻¹' {q}) ↔ (posSupport (p : Fin n → ℝ)).card = (posSupport (q : Fin n → ℝ)).card

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_eq_iff_posSupport_card_eq.

Preamble

-- Generated from ChapterFreeFieldBornFiberBounds.lean — theorem BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_eq_iff_posSupport_card_eq
import Definitions.Def_ChapterFreeFieldBorn
import Definitions.Def_ChapterFreeFieldBornSurj
import Definitions.Def_ChapterFreeFieldBornCont
import Definitions.Def_ChapterFreeFieldBornQuotient
import Definitions.Def_ChapterFreeFieldBornFiberCardGeneral
import Definitions.Def_ChapterFreeFieldBornFiberTwo
import Mathlib
import Definitions.Def_ChapterFreeFieldBornFiberBounds
open BookProof.ChapterFreeFieldBornFiberBounds


open MeasureTheory
open BookProof.ChapterFreeFieldBorn BookProof.ChapterFreeFieldBornSurj
open BookProof.ChapterFreeFieldBornCont BookProof.ChapterFreeFieldBornQuotient
open BookProof.ChapterFreeFieldBornFiberCardGeneral
open BookProof.ChapterFreeFieldBornFiberTwo


variable {n : ℕ}

Formal statement

theorem BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_eq_iff_posSupport_card_eq
    {p q : ↥(stdSimplex ℝ (Fin n))} :
    Nat.card ↥(bornMapSphere n ⁻¹' {p}) = Nat.card ↥(bornMapSphere n ⁻¹' {q}) ↔
      (posSupport (p : Fin n → ℝ)).card = (posSupport (q : Fin n → ℝ)).card := by sorry

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