Theorems · Theorem · ring theory
Subalgebra.mem_toSubmodule
∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (S : Subalgebra R A)
{x : A}, x ∈ Subalgebra.toSubmodule S ↔ x ∈ S- Defined in
- Mathlib.Algebra.Algebra.Subalgebra.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 30 from the axioms · uses propext, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement · cited by 7,192
- Subalgebrastatement and proof · cited by 1,353
- OrderEmbeddingstatement · cited by 619
- Subalgebra.toSubmodulestatement · cited by 141
Cited by5
Results whose statement or proof uses this declaration.
- Algebra.denominator_dvd_iffproof · cited by 3
- Submodule.span_range_natDegree_eq_adjoinproof · cited by 3
- ContinuousMap.adjoin_id_eq_span_one_addproof · cited by 1
- ContinuousMap.adjoin_id_eq_span_one_unionproof · cited by 0
- mem_adjoin_map_integralClosure_of_isStandardEtaleproof · cited by 0