Theorems · Theorem · ring theory
NonUnitalSubsemiring.coe_closure_eq
∀ {R : Type u} [inst : NonUnitalNonAssocSemiring R] (s : Set R),
↑(NonUnitalSubsemiring.closure s) = ↑(AddSubmonoid.closure ↑(Subsemigroup.closure s))The elements of the non-unital subsemiring closure of M are exactly the elements of the
additive closure of a multiplicative subsemigroup M.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NonUnitalNonAssocSemiring
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
- AddSubmonoidstatement · cited by 1,178
- NonUnitalNonAssocSemiringstatement and proof · cited by 1,081
- Subsemigroupstatement · cited by 323
- AddSubmonoid.closurestatement · cited by 224
- NonUnitalSubsemiringstatement · cited by 201
- NonUnitalSubsemiring.toAddSubmonoidproof · cited by 56
- Subsemigroup.closurestatement and proof · cited by 43
- NonUnitalSubsemiring.closurestatement and proof · cited by 31
- Subsemigroup.nonUnitalSubsemiringClosure_eq_closureproof · cited by 2
- NonUnitalSubsemiring.closure_subsemigroup_closureproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- NonUnitalSubsemiring.mem_closure_iffproof · cited by 1