Theorems · Theorem · commutative algebra
Submodule.mul_mem_mul
∀ {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 : Submodule R A} {m n : A}, m ∈ M → n ∈ N → m * n ∈ M * N- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 27 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_mem_smulproof · cited by 32
Cited by27
Results whose statement or proof uses this declaration.
- Submodule.mul_eq_map₂proof · cited by 6
- FractionalIdeal.mul_mem_mulproof · cited by 6
- Submodule.pow_induction_on_left'statement and proof · cited by 5
- MvPolynomial.IsHomogeneous.mulproof · cited by 5
- Submodule.mul_induction_on'statement and proof · cited by 4
- FractionalIdeal.div_spanSingletonproof · cited by 4
- CliffordAlgebra.ι_mul_ι_mem_evenOdd_zeroproof · cited by 4
- MvPolynomial.IsWeightedHomogeneous.mulproof · cited by 3
- CliffordAlgebra.evenOdd_inductionproof · cited by 2
- Submodule.pow_induction_on_right'statement and proof · cited by 2
- Submodule.mulMap_rangeproof · cited by 2
- Subalgebra.isIdempotentElem_toSubmoduleproof · cited by 2