Theorems · Theorem · ring theory
Algebra.mem_adjoin_iff
∀ {R : Type uR} {A : Type uA} [inst : CommRing R] [inst_1 : Ring A] [inst_2 : Algebra R A] {s : Set A} {x : A},
x ∈ Algebra.adjoin R s ↔ x ∈ Subring.closure (Set.range ⇑(algebraMap R A) ∪ s)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Algebra.algebraMapstatement and proof · cited by 4,706
- Set.rangestatement and proof · cited by 4,705
- Subalgebrastatement · cited by 1,353
- Subringstatement and proof · cited by 602
- Algebra.adjoinstatement · cited by 535
- Subring.closurestatement and proof · cited by 78
Cited by1
Results whose statement or proof uses this declaration.
- RingHom.IsIntegralElem.of_mem_closureproof · cited by 4