Theorems · Theorem · linear algebra
eq_zero_of_sameRay_self_neg
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : PartialOrder R] [inst_2 : IsStrictOrderedRing R] {M : Type u_2}
[inst_3 : AddCommGroup M] [inst_4 : Module R M] {x : M} [IsDomain R] [Module.IsTorsionFree R M],
SameRay R x (-x) → x = 0If a vector is in the same ray as its negation, that vector is zero.
- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- PartialOrderstatement and proof · cited by 6,410
- IsStrictOrderedRingstatement and proof · cited by 2,490
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement and proof · cited by 600
- SameRaystatement and proof · cited by 149
- neg_lt_zeroproof · cited by 36
- neg_one_smulproof · cited by 23
- zero_lt_one'proof · cited by 16
- eq_zero_of_sameRay_neg_smul_rightproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Module.Ray.ne_neg_selfproof · cited by 3
- Orientation.volumeForm_zero_negproof · cited by 1