Theorems · Theorem · linear algebra

SameRay.exists_pos

∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : PartialOrder R] [inst_2 : IsStrictOrderedRing R] {M : Type u_2}
  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] {x y : M},
  SameRay R x y → x ≠ 0 → y ≠ 0 → ∃ r₁ r₂, 0 < r₁ ∧ 0 < r₂ ∧ r₁ • x = r₂ • y

If x and y are nonzero vectors on the same ray, then there exist positive numbers r₁ r₂ such that r₁ • x = r₂ • y.

Defined in

Mathlib.LinearAlgebra.Ray

Cited by

3 results in Mathlib

Foundations

Depth 13 from the axioms · uses no axioms

Assumes

CommSemiringPartialOrderIsStrictOrderedRingAddCommMonoidModule