Theorems · Theorem · commutative algebra
Submodule.mul_le
∀ {R : Type u} [inst : Semiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Module R A]
[inst_3 : IsScalarTower R A A] {M N P : Submodule R A}, M * N ≤ P ↔ ∀ m ∈ M, ∀ n ∈ N, m * n ∈ P- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 70 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
- Submodulestatement and proof · cited by 7,192
- IsScalarTowerstatement and proof · cited by 3,896
- Submodule.smul_leproof · cited by 20
Cited by13
Results whose statement or proof uses this declaration.
- Submodule.mul_eq_map₂proof · cited by 6
- Submodule.le_div_iff_mul_leproof · cited by 3
- MvPolynomial.weightedHomogeneousSubmodule_mulproof · cited by 2
- Submodule.le_traceDual_iff_map_le_oneproof · cited by 2
- Submodule.map_op_mulproof · cited by 2
- Submodule.mulMap_rangeproof · cited by 2
- Subalgebra.mul_toSubmodule_leproof · cited by 2
- FractionalIdeal.isPrincipal.of_finite_maximals_of_invproof · cited by 1
- Submodule.mul_one_div_le_oneproof · cited by 1
- FractionalIdeal.mul_leproof · cited by 1
- MvPolynomial.coeffsIn_mulproof · cited by 0
- MvPolynomial.degreesLE_addproof · cited by 0