Theorems · Theorem · commutative algebra
Subring.mem_closure_iff
∀ {R : Type u} [inst : NonAssocRing R] {s : Set R} {x : R},
x ∈ Subring.closure s ↔ x ∈ AddSubgroup.closure ↑(Submonoid.closure s)- Defined in
- Mathlib.Algebra.Ring.Subring.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 74 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.
Cites27
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
- AddSubgroupstatement · cited by 3,232
- Submonoidstatement · cited by 3,086
- MulZeroClass.mul_zeroproof · cited by 2,091
- MulZeroClass.zero_mulproof · cited by 1,625
- neg_mulproof · cited by 654
- Subringstatement · cited by 602
- mul_negproof · cited by 590
- NonAssocRingstatement and proof · cited by 483
- mul_addproof · cited by 413
- add_mulproof · cited by 363
Cited by1
Results whose statement or proof uses this declaration.
- Subring.exists_list_of_mem_closureproof · cited by 1