Theorems · Theorem · linear algebra

SameRay.sameRay_pos_smul_left

∀ {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] {S : Type u_5} [inst_5 : CommSemiring S] [inst_6 : PartialOrder S]
  [inst_7 : Algebra S R] [inst_8 : Module S M] [SMulPosMono S R] [IsScalarTower S R M] {a : S} (v : M),
  0 < a → SameRay R (a • v) v

A positive multiple of a vector is in the same ray as that vector.

Defined in

Mathlib.LinearAlgebra.Ray

Cited by

3 results in Mathlib

Foundations

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

Assumes

CommSemiringPartialOrderIsStrictOrderedRingAddCommMonoidModuleCommSemiringPartialOrderAlgebraModuleSMulPosMonoIsScalarTower