Theorems · Theorem · geometry

AffineSubspace.linear_equivMapOfInjective

∀ {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [inst : NormedField 𝕜]
  [inst_1 : SeminormedAddCommGroup V₁] [inst_2 : NormedSpace 𝕜 V₁] [inst_3 : PseudoMetricSpace P₁]
  [inst_4 : NormedAddTorsor V₁ P₁] [inst_5 : SeminormedAddCommGroup V₂] [inst_6 : NormedSpace 𝕜 V₂]
  [inst_7 : PseudoMetricSpace P₂] [inst_8 : NormedAddTorsor V₂ P₂] (E : AffineSubspace 𝕜 P₁) [inst_9 : Nonempty ↥E]
  (φ : P₁ →ᵃ[𝕜] P₂) (hφ : Function.Injective ⇑φ),
  (E.equivMapOfInjective φ hφ).linear =
    (Submodule.equivMapOfInjective φ.linear ⋯ E.direction).trans
      (LinearEquiv.ofEq (Submodule.map φ.linear E.direction) (AffineSubspace.map φ E).direction ⋯)

Defined in

Mathlib.Analysis.Normed.Affine.Isometry

Cited by

0 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldSeminormedAddCommGroupNormedSpacePseudoMetricSpaceNormedAddTorsorSeminormedAddCommGroupNormedSpacePseudoMetricSpaceNormedAddTorsorNonempty