not_wbtw_of_injective — Mathlib

English

Equivalent reformulations of the SameRay condition with vsub expressions characterize Wbtw.

Русский

Эквивалентные переформулировки условия SameRay через vsub, характеризующие Wbtw.

LaTeX

$$$Wbtw_R x y z \iff SameRay_R (instVSub y x) (instVSub z y)$$$

Lean4

/-- If `T` is an affine independent family of points,
then any 3 distinct points form a triangle. -/
theorem not_wbtw_of_injective {ι} (i j k : ι) (h : Function.Injective ![i, j, k]) {T : ι → P}
    (hT : AffineIndependent R T) : ¬Wbtw R (T i) (T j) (T k) :=
  by
  replace hT := hT.comp_embedding ⟨_, h⟩
  rw [affineIndependent_iff_not_collinear] at hT
  contrapose! hT
  simp [Set.range_comp, Set.image_insert_eq, hT.symm.collinear]

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