Theorems · Theorem · ring theory
Algebra.adjoin_toSubsemiring
∀ (R : Type u) {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (s : Set A),
(Algebra.adjoin R s).toSubsemiring = Subsemiring.closure (Set.range ⇑(algebraMap R A) ∪ s)- Cited by
- 4 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
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.
- DFunLike.coestatement · cited by 62,936
- Setstatement and proof · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement · cited by 4,706
- Set.rangestatement · cited by 4,705
- Algebra.adjoinstatement and proof · cited by 535
- Subsemiringstatement · cited by 456
- Subalgebra.toSubsemiringstatement and proof · cited by 115
- Subsemiring.closurestatement · cited by 53
Cited by4
Results whose statement or proof uses this declaration.
- IntermediateField.LinearDisjoint.adjoin_rank_eq_rank_left_of_isAlgebraicproof · cited by 2
- Algebra.sSup_toSubsemiringproof · cited by 1
- Algebra.sup_toSubsemiringproof · cited by 0