Theorems · Theorem · ring theory
Algebra.adjoin_range_eq_range_freeAlgebra_lift
∀ (R : Type u_1) (X : Type u_2) [inst : CommSemiring R] {A : Type u_3} [inst_1 : Semiring A] [inst_2 : Algebra R A]
(f : X → A), Algebra.adjoin R (Set.range f) = ((FreeAlgebra.lift R) f).rangeNoncommutative version of Algebra.adjoin_range_eq_range_aeval.
- Defined in
- Mathlib.Algebra.FreeAlgebra
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 77 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.
Cites16
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
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Equivstatement · cited by 8,337
- Set.rangestatement and proof · cited by 4,705
- AlgHomstatement · cited by 3,236
- Subalgebrastatement · cited by 1,353
- Algebra.adjoinstatement and proof · cited by 535
- AlgHom.rangestatement · cited by 169
- Subalgebra.mapproof · cited by 90
- FreeAlgebrastatement · cited by 48
Cited by2
Results whose statement or proof uses this declaration.
- Algebra.adjoin_eq_range_freeAlgebra_liftproof · cited by 2
- Algebra.FiniteType.iff_quotient_freeAlgebraproof · cited by 1