Theorems · Theorem · geometry

AffineSubspace.vadd_mem_iff_mem_direction

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] {s : AffineSubspace k P} (v : V) {p : P}, p ∈ s → (v +ᵥ p ∈ s ↔ v ∈ s.direction)

Adding a vector to a point in a subspace produces a point in the subspace if and only if the vector is in the direction.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs

Cited by

9 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Quot.sound

Assumes

RingAddCommGroupModuleAddTorsor