Theorems · Theorem · commutative algebra
Subsemiring.coe_closure_eq
∀ {R : Type u} [inst : NonAssocSemiring R] (s : Set R),
↑(Subsemiring.closure s) = ↑(AddSubmonoid.closure ↑(Submonoid.closure s))The elements of the subsemiring closure of M are exactly the elements of the additive closure
of a multiplicative submonoid M.
- Defined in
- Mathlib.Algebra.Ring.Subsemiring.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NonAssocSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Submonoidstatement · cited by 3,086
- AddSubmonoidstatement · cited by 1,178
- NonAssocSemiringstatement and proof · cited by 805
- Subsemiringstatement · cited by 456
- AddSubmonoid.closurestatement · cited by 224
- Submonoid.closurestatement and proof · cited by 167
- Subsemiring.closurestatement and proof · cited by 53
- Subsemiring.toAddSubmonoidproof · cited by 20
- Submonoid.subsemiringClosure_eq_closureproof · cited by 1
- Subsemiring.closure_submonoid_closureproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Subsemiring.mem_closure_iffproof · cited by 2