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Theorems · Theorem · linear algebra

sameRay_smul_right_iff

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] {M : Type u_2}
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [Module.IsTorsionFree R M] {v : M} {r : R},
  SameRay R v (r • v) ↔ 0 ≤ r ∨ v = 0
Defined in
Mathlib.LinearAlgebra.Ray
Cited by
2 results in Mathlib
Foundations
Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommRingLinearOrderIsStrictOrderedRingAddCommGroupModuleModule.IsTorsionFree
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

sameRay_neg_smul_right_iff · cited by 1sameRay_neg_smul_right_iffsameRay_smul_left_iff · cited by 0sameRay_smul_left_iffModule · cited by 20661ModuleCommRing · cited by 17173CommRingAddCommGroup · cited by 12871AddCommGroupLinearOrder · cited by 8572LinearOrderIsStrictOrderedRing · cited by 2490IsStrictOrderedRingModule.IsTorsionFree · cited by 600Module.IsTorsionFreenot_le · cited by 328not_leSameRay · cited by 149SameRaySameRay.sameRay_nonneg_smul_right · cited by 6SameRay.sameRay_nonneg_sm…SameRay.zero_left · cited by 6SameRay.zero_lefteq_zero_of_sameRay_neg_smul_right · cited by 2eq_zero_of_sameRay_neg_sm…sameRay_smul_right_iffCITED BYCITES

Cites11

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Cited by2

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