Theorems · Theorem · commutative algebra
Submodule.mem_smul_iff_inv_mul_mem
∀ {R : Type u} [inst : CommSemiring R] {S : Type u_1} [inst_1 : DivisionSemiring S] [inst_2 : Algebra R S] {x : S}
{p : Submodule R S} {y : S}, x ≠ 0 → (y ∈ x • p ↔ x⁻¹ * y ∈ p)- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coeproof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- DivisionSemiringstatement and proof · cited by 216
- Submodule.pointwiseDistribMulActionstatement · cited by 105
- DistribSMul.toLinearMapproof · cited by 50
- inv_mul_cancel_left₀proof · cited by 47
- mul_inv_cancel_left₀proof · cited by 42
- DistribSMul.toLinearMap_applyproof · cited by 18
Cited by1
Results whose statement or proof uses this declaration.
- conductor_mul_differentIdealproof · cited by 1