Theorems · Theorem · linear algebra
sameRay_or_sameRay_neg_iff_not_linearIndependent
∀ {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] {x y : M},
SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]Two vectors are in the same ray, or the first is in the same ray as the negation of the second, if and only if they are not linearly independent.
- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites37
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- 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
- Finset.sumproof · cited by 5,195
- Algebra.algebraMapproof · cited by 4,706
- Finset.univproof · cited by 3,473
- Nat.cast_oneproof · cited by 2,501
- IsStrictOrderedRingstatement and proof · cited by 2,490
- zero_addproof · cited by 2,366
- Nat.cast_zeroproof · cited by 1,870
Cited by2
Results whose statement or proof uses this declaration.
- Orientation.eq_or_eq_neg_of_isEmptyproof · cited by 6
- sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependentproof · cited by 1