Theorems · Theorem · commutative algebra
Subring.closure_le
∀ {R : Type u} [inst : NonAssocRing R] {s : Set R} {t : Subring R}, Subring.closure s ≤ t ↔ s ⊆ ↑tA subring t includes closure s if and only if it includes s.
- Defined in
- Mathlib.Algebra.Ring.Subring.Basic
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 72 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.
Cites8
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
- Set.Subset.transproof · cited by 218
- sInf_leproof · cited by 110
- Subring.closurestatement · cited by 78
- Subring.subset_closureproof · cited by 18
Cited by15
Results whose statement or proof uses this declaration.
- Subring.giproof · cited by 6
- Subring.closure_inductionproof · cited by 4
- Subring.closure_monoproof · cited by 2
- Subring.closure_singleton_intCastproof · cited by 2
- Algebra.adjoin_intproof · cited by 2
- Subring.closure_le_centralizer_centralizerproof · cited by 1
- RingHom.eqOn_set_closureproof · cited by 1
- Subring.comap_map_eqproof · cited by 1
- Subring.closure_eq_of_leproof · cited by 1
- Subfield.subring_closure_leproof · cited by 0
- iInf_valuationSubring_supersetproof · cited by 0
- RingHom.closure_preimage_leproof · cited by 0