Theorems · Theorem · linear algebra
SameRay.exists_nonneg_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 → ∃ r, 0 ≤ r ∧ r • x = yIf a vector v₂ is on the same ray as a nonzero vector v₁, then it is equal to c • v₁ for
some nonnegative c.
- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 47 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- le_rflproof · cited by 1,558
- le_of_ltproof · cited by 1,175
- eq_or_neproof · cited by 1,117
- zero_smulproof · cited by 716
- SameRaystatement and proof · cited by 149
- SameRay.exists_pos_leftproof · cited by 5
Cited by5
Results whose statement or proof uses this declaration.
- exists_nonneg_left_iff_sameRayproof · cited by 4
- norm_injOn_ray_leftproof · cited by 1
- SameRay.exists_nonneg_rightproof · cited by 1
- Orientation.nonneg_inner_and_areaForm_eq_zero_iff_sameRayproof · cited by 1
- Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smulproof · cited by 0