Theorems · Theorem · commutative algebra
Subring.subset_closure
∀ {R : Type u} [inst : NonAssocRing R] {s : Set R}, s ⊆ ↑(Subring.closure s)The subring generated by a set includes the set.
- Defined in
- Mathlib.Algebra.Ring.Subring.Basic
- Cited by
- 18 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NonAssocRing
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.
- Setstatement and proof · cited by 53,352
- SetLike.coestatement and proof · cited by 8,199
- Subringstatement and proof · cited by 602
- NonAssocRingstatement and proof · cited by 483
- Subring.closurestatement · cited by 78
- Subring.mem_closureproof · cited by 1
Cited by18
Results whose statement or proof uses this declaration.
- Subring.closure_leproof · cited by 14
- Subring.closure_inductionstatement and proof · cited by 4
- RingHom.IsIntegralElem.mulproof · cited by 4
- RingHom.IsIntegralElem.addproof · cited by 3
- RingHom.IsIntegralElem.negproof · cited by 3
- Subring.closure_monoproof · cited by 2
- Algebra.adjoin_eq_ring_closureproof · cited by 2
- Algebra.adjoin_intproof · cited by 2
- Subring.mem_closure_iffproof · cited by 1
- FreeCommRing.isSupported_ofproof · cited by 1
- Polynomial.mem_closure_X_union_Cproof · cited by 1
- Algebra.TensorProduct.closure_range_union_range_eq_topproof · cited by 1