Theorems · Theorem · geometry

EuclideanGeometry.dist_orthogonalProjection_eq_dist_iff_eq_of_mem

∀ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup V]
  [inst_2 : InnerProductSpace 𝕜 V] [inst_3 : MetricSpace P] [inst_4 : NormedAddTorsor V P] {s : AffineSubspace 𝕜 P}
  [inst_5 : s.direction.HasOrthogonalProjection] {p₁ p₂ : P} (hp₂ : p₂ ∈ s),
  dist p₁ ↑((EuclideanGeometry.orthogonalProjection s) p₁) = dist p₁ p₂ ↔
    ↑((EuclideanGeometry.orthogonalProjection s) p₁) = p₂

If the distance from p₁ to its orthogonal projection equals its distance to a point in s, the orthogonal projection is that point.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

2 results in Mathlib

Foundations

Depth 182 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorSubmodule.HasOrthogonalProjection