`BookProof.ChapterFreeFieldBorn.bornMap_mem_stdSimplex` {x : EuclideanSpace ℝ (Fin n)} (hx : x ∈ Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1) : bornMap x ∈ stdSimplex ℝ (Fin n)

Proved

BookProof.ChapterFreeFieldBorn.bornMap_mem_stdSimplex

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterFreeFieldBorn.

BookProof.ChapterFreeFieldBorn.bornMap_mem_stdSimplex {x : EuclideanSpace ℝ (Fin n)} (hx : x ∈ Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1) : bornMap x ∈ stdSimplex ℝ (Fin n)

Formalization note: Lean 4 identifier BookProof.ChapterFreeFieldBorn.bornMap_mem_stdSimplex.

Preamble

-- Generated from ChapterFreeFieldBorn.lean — theorem BookProof.ChapterFreeFieldBorn.bornMap_mem_stdSimplex
import Definitions.Def_ChapterFreeFieldGaussian
import Definitions.Def_ChapterFreeFieldSphere
import Definitions.Def_ChapterFreeFieldSphereSupport
import Mathlib
import Definitions.Def_ChapterFreeFieldBorn
open BookProof.ChapterFreeFieldBorn

variable {n : ℕ}


open MeasureTheory
open BookProof.ChapterFreeFieldGaussian BookProof.ChapterFreeFieldSphere
open BookProof.ChapterFreeFieldSphereSupport

Formal statement

theorem BookProof.ChapterFreeFieldBorn.bornMap_mem_stdSimplex {x : EuclideanSpace ℝ (Fin n)}
    (hx : x ∈ Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1) :
    bornMap x ∈ stdSimplex ℝ (Fin n) := by sorry

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