Theorems · Theorem · ring theory

Algebra.adjoin_union_coe_submodule

∀ (R : Type uR) {A : Type uA} [inst : CommSemiring R] [inst_1 : CommSemiring A] [inst_2 : Algebra R A] (s t : Set A),
  Subalgebra.toSubmodule (Algebra.adjoin R (s ∪ t)) =
    Subalgebra.toSubmodule (Algebra.adjoin R s) * Subalgebra.toSubmodule (Algebra.adjoin R t)

Defined in: Mathlib.Algebra.Algebra.Subalgebra.Lattice
Cited by: 3 results in Mathlib
Foundations: Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring CommSemiring Algebra

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Cites19

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
Moduleproof · cited by 20,661
Semiringproof · cited by 13,802
AddCommMonoidproof · cited by 12,281
Algebrastatement and proof · cited by 11,388
CommSemiringstatement and proof · cited by 10,911
SetLike.coeproof · cited by 8,199
Submodulestatement and proof · cited by 7,192
Set.extproof · cited by 2,266
Submodule.spanproof · cited by 1,504
Subalgebrastatement · cited by 1,353

Cited by3

Results whose statement or proof uses this declaration.

fg_adjoin_of_finiteproof · cited by 9
RingHom.IsIntegralElem.of_mem_closureproof · cited by 4
Subalgebra.rank_sup_le_of_freeproof · cited by 2