Theorems · Theorem · geometry

EuclideanGeometry.orthogonalProjection_eq_iff_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 : Nonempty ↥s] [inst_6 : s.direction.HasOrthogonalProjection] {p : P} {q : ↥s},
  (EuclideanGeometry.orthogonalProjection s) p = q ↔ p -ᵥ ↑q ∈ s.directionᗮ

The characteristic property of the orthogonal projection, for a point given in the relevant subspace. This form is typically more convenient to use than inter_eq_singleton_orthogonalProjection.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

1 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection