Theorems · Theorem · geometry

EuclideanGeometry.orthogonalProjection_sup_of_orthogonalProjection_eq

∀ {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] {s₁ s₂ : AffineSubspace 𝕜 P}
  [inst_5 : Nonempty ↥s₁] [inst_6 : Nonempty ↥s₂] [inst_7 : s₁.direction.HasOrthogonalProjection]
  [inst_8 : s₂.direction.HasOrthogonalProjection] {p : P},
  ↑((EuclideanGeometry.orthogonalProjection s₁) p) = ↑((EuclideanGeometry.orthogonalProjection s₂) p) →
    ∀ [inst_9 : (s₁ ⊔ s₂).direction.HasOrthogonalProjection],
      ↑((EuclideanGeometry.orthogonalProjection (s₁ ⊔ s₂)) p) = ↑((EuclideanGeometry.orthogonalProjection s₁) p)

If the orthogonal projections of a point onto two subspaces are equal, so is the projection onto their supremum.

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

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptyNonemptySubmodule.HasOrthogonalProjectionSubmodule.HasOrthogonalProjectionSubmodule.HasOrthogonalProjection