`BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_bounds` {p : ↥(stdSimplex ℝ (Fin n))} : 2 ≤ Nat.card ↥(bornMapSphere n ⁻¹' {p}) ∧ Nat.card ↥(bornMapSphere n ⁻¹' {p}) ≤ 2

Proved

BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_bounds

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBornFiberBounds.

BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_bounds {p : ↥(stdSimplex ℝ (Fin n))} : 2 ≤ Nat.card ↥(bornMapSphere n ⁻¹' {p}) ∧ Nat.card ↥(bornMapSphere n ⁻¹' {p}) ≤ 2 ^ n

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

Preamble

-- Generated from ChapterFreeFieldBornFiberBounds.lean — theorem BookProof.ChapterFreeFieldBornFiberBounds.bornFiber_card_bounds
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_bounds {p : ↥(stdSimplex ℝ (Fin n))} :
    2 ≤ Nat.card ↥(bornMapSphere n ⁻¹' {p}) ∧
      Nat.card ↥(bornMapSphere n ⁻¹' {p}) ≤ 2 ^ n := by sorry

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