Theorems · Theorem · commutative algebra
Subsemiring.subset_closure
∀ {R : Type u} [inst : NonAssocSemiring R] {s : Set R}, s ⊆ ↑(Subsemiring.closure s)The subsemiring generated by a set includes the set.
- Defined in
- Mathlib.Algebra.Ring.Subsemiring.Basic
- Cited by
- 13 results in Mathlib
- Foundations
- Depth 66 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.
Cites6
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
- NonAssocSemiringstatement and proof · cited by 805
- Subsemiringstatement and proof · cited by 456
- Subsemiring.closurestatement · cited by 53
- Subsemiring.mem_closureproof · cited by 2
Cited by13
Results whose statement or proof uses this declaration.
- Subsemiring.closure_leproof · cited by 12
- Algebra.gcproof · cited by 7
- Subsemiring.closure_monoproof · cited by 4
- Subsemiring.closure_inductionstatement and proof · cited by 3
- Algebra.adjoin_natproof · cited by 1
- Subsemiring.mem_closure_iff_exists_listproof · cited by 1
- IsHomogeneous.subsemiringClosureproof · cited by 1
- Subsemiring.closure_submonoid_closureproof · cited by 1
- Subsemiring.closure_induction₂statement and proof · cited by 0
- Subsemiring.mem_closure_of_memproof · cited by 0
- Subsemiring.notMem_of_notMem_closureproof · cited by 0
- MvPolynomial.aeval_rangeproof · cited by 0