Theorems · Definition · linear algebra

rayOfNeZero

(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] → (v : M) → v ≠ 0 → Module.Ray R M

The ray given by a nonzero vector.

Defined in: Mathlib.LinearAlgebra.Ray
Cited by: 22 results in Mathlib
Foundations: Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring PartialOrder IsStrictOrderedRing AddCommMonoid Module

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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
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
Equiv.symmproof · cited by 3,681
IsStrictOrderedRingstatement and proof · cited by 2,490
Module.Raystatement · cited by 30
RayVector.equivproof · cited by 6

Cited by23

Results whose statement or proof uses this declaration.

Module.Basis.orientationproof · cited by 28
Module.Ray.indstatement and proof · cited by 10
ray_eq_iffstatement · cited by 8
rayOfNeZero.congr_simpstatement and proof · cited by 4
smul_rayOfNeZerostatement · cited by 3
Module.Basis.map_orientation_eq_det_inv_smulproof · cited by 3
Module.Ray.ne_neg_selfproof · cited by 3
Module.Basis.orientation_mapproof · cited by 3
Orientation.map_applystatement · cited by 2
Orientation.map_of_isEmptyproof · cited by 2
neg_rayOfNeZerostatement · cited by 2
Module.Ray.units_smul_of_posproof · cited by 2