Theorems · Theorem · commutative algebra
Subring.closure_mono
∀ {R : Type u} [inst : NonAssocRing R] ⦃s t : Set R⦄, s ⊆ t → Subring.closure s ≤ Subring.closure tSubring closure of a set is monotone in its argument: if s ⊆ t,
then closure s ≤ closure t.
- Defined in
- Mathlib.Algebra.Ring.Subring.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 73 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.
Cites7
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
- Subringstatement · cited by 602
- NonAssocRingstatement and proof · cited by 483
- Set.Subset.transproof · cited by 218
- Subring.closurestatement · cited by 78
- Subring.subset_closureproof · cited by 18
- Subring.closure_leproof · cited by 14
Cited by2
Results whose statement or proof uses this declaration.
- RingHom.IsIntegralElem.of_mem_closureproof · cited by 4
- FreeCommRing.isSupported_upwardsproof · cited by 1