Theorems · Theorem · linear algebra
SameRay.symm
∀ {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] {x y : M}, SameRay R x y → SameRay R y xSameRay is symmetric.
- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses propext
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- PartialOrderstatement and proof · cited by 6,410
- IsStrictOrderedRingstatement and proof · cited by 2,490
- SameRaystatement and proof · cited by 149
Cited by9
Results whose statement or proof uses this declaration.
- SameRay.sameRay_commproof · cited by 11
- SameRay.sameRay_nonneg_smul_leftproof · cited by 7
- AffineSubspace.wSameSide_commproof · cited by 5
- EuclideanGeometry.angle_eq_iff_oangle_eq_or_wbtwproof · cited by 2
- SameRay.exists_nonneg_rightproof · cited by 1
- abs_lt_norm_sub_of_not_sameRayproof · cited by 1
- SameRay.nonneg_smul_leftproof · cited by 1
- SameRay.add_rightproof · cited by 0
- SameRay.exists_pos_rightproof · cited by 0