Theorems · Theorem · linear algebra
sameRay_neg_smul_right_iff_of_ne
∀ {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},
v ≠ 0 → r ≠ 0 → (SameRay R (-v) (r • v) ↔ r < 0)- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearOrderstatement and proof · cited by 8,572
- IsStrictOrderedRingstatement and proof · cited by 2,490
- Module.IsTorsionFreestatement and proof · cited by 600
- SameRaystatement · cited by 149
- Ne.le_iff_ltproof · cited by 8
Cited by1
Results whose statement or proof uses this declaration.
- sameRay_neg_smul_left_iff_of_neproof · cited by 1