Theorems · Theorem · geometry

Affine.Simplex.wOppSide_affineSpan_faceOpposite_iff

∀ {R : Type u_1} {V : Type u_2} {P : Type u_4} [inst : Field R] [inst_1 : LinearOrder R]
  [inst_2 : IsStrictOrderedRing R] [inst_3 : AddCommGroup V] [inst_4 : Module R V] [inst_5 : AddTorsor V P] {n : ℕ}
  [inst_6 : NeZero n] (s : Affine.Simplex R P n) {w₁ w₂ : Fin (n + 1) → R},
  ∑ j, w₁ j = 1 →
    ∑ j, w₂ j = 1 →
      ∀ {i : Fin (n + 1)},
        (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide
            ((Finset.affineCombination R Finset.univ s.points) w₁)
            ((Finset.affineCombination R Finset.univ s.points) w₂) ↔
          SignType.sign (w₁ i) = -SignType.sign (w₂ i) ∨ w₁ i = 0 ∨ w₂ i = 0

Defined in

Mathlib.Analysis.Convex.Side

Cited by

1 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRingAddCommGroupModuleAddTorsorNeZero