Theorems · Theorem · commutative algebra
smul_ne_zero_iff
∀ {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 • m ≠ 0 ↔ r ≠ 0 ∧ m ≠ 0- Defined in
- Mathlib.Algebra.Module.Torsion.Free
- Cited by
- 8 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 by8
Results whose statement or proof uses this declaration.
- EuclideanGeometry.angle_eq_pi_iff_sbtwproof · cited by 6
- Submodule.span_singleton_eq_span_singletonproof · cited by 5
- EuclideanGeometry.inner_pos_or_eq_of_dist_le_radiusproof · cited by 3
- MeromorphicAt.eventually_eq_zero_or_eventually_ne_zeroproof · cited by 3
- EuclideanGeometry.angle_eq_zero_iff_ne_and_wbtwproof · cited by 3
- sbtw_iff_left_ne_and_right_mem_image_Ioiproof · cited by 2
- Function.support_smulproof · cited by 1
- Polynomial.Sequence.linearIndependentproof · cited by 1