Theorems · Theorem · ring theory
Subalgebra.mem_toSubring
∀ {R : Type u} {A : Type v} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] {S : Subalgebra R A} {x : A},
x ∈ S.toSubring ↔ x ∈ S- Defined in
- Mathlib.Algebra.Algebra.Subalgebra.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Subalgebrastatement and proof · cited by 1,353
- Subringstatement · cited by 602
- Subalgebra.toSubringstatement · cited by 30
Cited by3
Results whose statement or proof uses this declaration.
- Subalgebra.toSubring_injectiveproof · cited by 3
- ValuationSubring.le_ofPrimeproof · cited by 2
- Algebra.mem_adjoin_iffproof · cited by 1