Theorems · Theorem · commutative algebra
smul_ne_zero
∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {r : R} {m : M}
[Module.IsTorsionFree R M] [IsCancelMulZero R], r ≠ 0 → m ≠ 0 → r • m ≠ 0- Defined in
- Mathlib.Algebra.Module.Torsion.Free
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Module.IsTorsionFreestatement and proof · cited by 600
- IsCancelMulZerostatement and proof · cited by 177
Cited by6
Results whose statement or proof uses this declaration.
- MeromorphicNFAt.smul_analyticproof · cited by 2
- Polynomial.natDegree_smulproof · cited by 2
- InnerProductGeometry.angle_eq_angle_add_add_angle_add_of_mem_spanproof · cited by 1
- Mathlib.Meta.Positivity.smul_ne_zero_of_ne_zero_of_posproof · cited by 0
- Mathlib.Meta.Positivity.smul_ne_zero_of_pos_of_ne_zeroproof · cited by 0
- Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smulproof · cited by 0