Theorems · Theorem · commutative algebra
smul_eq_zero_iff_left
∀ {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], m ≠ 0 → (r • m = 0 ↔ r = 0)- Defined in
- Mathlib.Algebra.Module.Torsion.Free
- Cited by
- 10 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 by10
Results whose statement or proof uses this declaration.
- smul_left_injectiveproof · cited by 25
- LieAlgebra.IsKilling.apply_coroot_eq_cast'proof · cited by 4
- RootPairing.pairing_smul_root_eq_of_not_linearIndependentproof · cited by 3
- ContinuousAlgEquiv.eq_continuousLinearEquivConjContinuousAlgEquivproof · cited by 2
- IsSl2Triple.HasPrimitiveVectorWith.exists_natproof · cited by 2
- isIntegral_of_smul_mem_submoduleproof · cited by 1
- Polynomial.Sequence.linearIndependentproof · cited by 1
- EuclideanGeometry.Sphere.ncard_inter_orthRadius_eq_two_of_dist_lt_radiusproof · cited by 1
- MvPolynomial.nonTorsionWeight_ofproof · cited by 1
- Module.stabilizer_units_eq_bot_of_ne_zeroproof · cited by 0