Theorems · Theorem · linear algebra
exists_nonneg_left_iff_sameRay
∀ {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}, x ≠ 0 → ((∃ r, 0 ≤ r ∧ r • x = y) ↔ SameRay R x y)- Defined in
- Mathlib.LinearAlgebra.Ray
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 48 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- SameRaystatement and proof · cited by 149
- SameRay.sameRay_nonneg_smul_rightproof · cited by 6
- SameRay.exists_nonneg_leftproof · cited by 5
Cited by4
Results whose statement or proof uses this declaration.
- Collinear.oangle_sign_of_sameRay_vsubproof · cited by 2
- Orientation.abs_oangle_sub_left_toReal_lt_pi_div_twoproof · cited by 1
- isConnected_setOfPred_sameRayproof · cited by 1
- exists_nonneg_right_iff_sameRayproof · cited by 0