Theorems · Theorem · commutative algebra
Submodule.mem_span_mul_finite_of_mem_span_mul
∀ {R : Type u_1} {A : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Mul A] [inst_3 : Module R A]
{S S' : Set A} {x : A}, x ∈ Submodule.span R (S * S') → ∃ T T', ↑T ⊆ S ∧ ↑T' ⊆ S' ∧ x ∈ Submodule.span R (↑T * ↑T')- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- Finsetstatement and proof · cited by 13,712
- AddCommMonoidstatement and proof · cited by 12,281
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement · cited by 7,192
- Submodule.spanstatement and proof · cited by 1,504
- Set.mulstatement · cited by 297
- Submodule.span_monoproof · cited by 85
- Submodule.mem_span_finite_of_mem_spanproof · cited by 6
- Finset.subset_mulproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.mem_span_mul_finite_of_mem_mulproof · cited by 3