Theorems · Theorem · commutative algebra
Submodule.mem_span_mul_finite_of_mem_mul
∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] {P Q : Submodule R A}
{x : A}, x ∈ P * Q → ∃ T T', ↑T ⊆ ↑P ∧ ↑T' ⊆ ↑Q ∧ x ∈ Submodule.span R (↑T * ↑T')- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
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 · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Finsetstatement · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- Submodule.spanstatement and proof · cited by 1,504
- Set.mulstatement · cited by 297
- Submodule.span_eqproof · cited by 26
- Submodule.span_mul_spanproof · cited by 18
- Submodule.mem_span_mul_finite_of_mem_span_mulproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Submodule.fg_unitproof · cited by 3
- IsGCDMonoid.isPrincipal_of_exists_mul_ne_zero_isPrincipalproof · cited by 1
- FractionalIdeal.mem_span_mul_finite_of_mem_mulproof · cited by 0