Theorems · Theorem · commutative algebra
Submodule.span_mul_span
∀ (R : Type u) [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] (S T : Set A),
Submodule.span R S * Submodule.span R T = Submodule.span R (S * T)- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 18 results in Mathlib
- Foundations
- Depth 72 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.
Cites10
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
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement and proof · cited by 7,192
- Submodule.spanstatement and proof · cited by 1,504
- Set.mulstatement · cited by 297
- LinearMap.mulproof · cited by 61
- Submodule.mul_eq_map₂proof · cited by 6
- Submodule.map₂_span_spanproof · cited by 6
Cited by18
Results whose statement or proof uses this declaration.
- FractionalIdeal.spanSingleton_mul_spanSingletonproof · cited by 11
- Submodule.mem_span_mul_finite_of_mem_mulproof · cited by 3
- Submodule.fg_unitproof · cited by 3
- Algebra.adjoin_union_coe_submoduleproof · cited by 3
- Submodule.prod_spanproof · cited by 2
- Subalgebra.rank_sup_le_of_freeproof · cited by 2
- exists_subalgebra_of_fgproof · cited by 1
- PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domainproof · cited by 1
- Submodule.ker_unitsToPicproof · cited by 1
- MvPolynomial.restrictSupport_addproof · cited by 1
- coeff_minpolyDiv_sub_pow_mem_spanproof · cited by 1
- FractionalIdeal.isNoetherian_spanSingleton_inv_to_map_mulproof · cited by 1