Theorems · Theorem · ring theory

AddMonoidAlgebra.bialgHom_ext

∀ {R : Type u_1} {A : Type u_3} {B : Type u_4} {M : Type u_8} [inst : CommSemiring R] [inst_1 : Semiring A]
  [inst_2 : Semiring B] [inst_3 : Bialgebra R A] [inst_4 : Bialgebra R B] [inst_5 : AddMonoid M]
  ⦃φ₁ φ₂ : AddMonoidAlgebra A M →ₐc[R] B⦄,
  (∀ (m : M), φ₁ (AddMonoidAlgebra.single m 1) = φ₂ (AddMonoidAlgebra.single m 1)) →
    (↑φ₁).comp AddMonoidAlgebra.singleZeroAlgHom = (↑φ₂).comp AddMonoidAlgebra.singleZeroAlgHom → φ₁ = φ₂

A R-bialgebra homomorphism from A[M] is uniquely defined by its values on the functions single m 1 and single 1 a. See note [partially-applied ext lemmas]. Note that the first assumption isn't written as an equality of AddMonoidHoms because of doesn't multiplicativise.

Defined in

Mathlib.RingTheory.Bialgebra.MonoidAlgebra

Cited by

3 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringSemiringSemiringBialgebraBialgebraAddMonoid