Theorems · Theorem · linear algebra
SameRay.exists_pos_left
∀ {R : Type u_1} [inst : Field R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] {M : Type u_2}
[inst_3 : AddCommGroup M] [inst_4 : Module R M] {x y : M}, SameRay R x y → x ≠ 0 → y ≠ 0 → ∃ r, 0 < r ∧ r • x = y- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 46 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
- AddCommGroupstatement and proof · cited by 12,871
- LinearOrderstatement and proof · cited by 8,572
- Fieldstatement and proof · cited by 7,404
- IsStrictOrderedRingstatement and proof · cited by 2,490
- LT.lt.ne'proof · cited by 1,417
- mul_posproof · cited by 374
- SemigroupAction.mul_smulproof · cited by 291
- SameRaystatement and proof · cited by 149
- inv_posproof · cited by 124
- inv_smul_smul₀proof · cited by 80
- SameRay.exists_posproof · cited by 3
Cited by5
Results whose statement or proof uses this declaration.
- SameRay.exists_nonneg_leftproof · cited by 5
- exists_pos_left_iff_sameRayproof · cited by 3
- EuclideanGeometry.angle_eq_iff_oangle_eq_or_wbtwproof · cited by 2
- Orientation.angle_eq_iff_oangle_eq_or_sameRayproof · cited by 1
- SameRay.exists_pos_rightproof · cited by 0