Theorems · Theorem · commutative algebra
Subring.add_mem
∀ {R : Type u} [inst : NonAssocRing R] (s : Subring R) {x y : R}, x ∈ s → y ∈ s → x + y ∈ sA subring is closed under addition.
- Defined in
- Mathlib.Algebra.Ring.Subring.Defs
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext
- Assumes
- NonAssocRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Subringstatement and proof · cited by 602
- NonAssocRingstatement and proof · cited by 483
- AddMemClass.add_memproof · cited by 229
Cited by6
Results whose statement or proof uses this declaration.
- RingHom.IsIntegralElem.addproof · cited by 3
- Algebra.adjoin_eq_ring_closureproof · cited by 2
- FreeCommRing.isSupported_addproof · cited by 2
- ValuationSubring.add_memproof · cited by 1
- Subring.comap_map_eqproof · cited by 1
- Polynomial.mem_closure_X_union_Cproof · cited by 1