Theorems · Definition · ring theory

AlgHom.comp

{R : Type u} →
  {A : Type v} →
    {B : Type w} →
      {C : Type u₁} →
        [inst : CommSemiring R] →
          [inst_1 : Semiring A] →
            [inst_2 : Semiring B] →
              [inst_3 : Semiring C] →
                [inst_4 : Algebra R A] →
                  [inst_5 : Algebra R B] → [inst_6 : Algebra R C] → (B →ₐ[R] C) → (A →ₐ[R] B) → A →ₐ[R] C

If φ₁ and φ₂ are R-algebra homomorphisms with the domain of φ₁ equal to the codomain of φ₂, then φ₁.comp φ₂ is the algebra homomorphism x ↦ φ₁ (φ₂ x).

Defined in

Mathlib.Algebra.Algebra.Hom

Cited by

501 results in Mathlib

Foundations

Depth 23 from the axioms, rests on 179 definitions · uses propext, Quot.sound

Assumes

CommSemiringSemiringSemiringSemiringAlgebraAlgebraAlgebra