English
A non-unital k-algebra homomorphism from MonoidAlgebra k G to A is determined by its values on the basis elements single x 1.
Русский
Некоторые негодованный алгебра-гомоморфизм из MonoidAlgebra k G в A определяется значениями на базисных элементах single x 1.
LaTeX
$$$\\forall x, \\phi_1(\\mathrm{single}\\ x\\ 1) = \\phi_2(\\mathrm{single}\\ x\\ 1) \\Rightarrow \\phi_1 = \\phi_2$$$
Lean4
/-- A non_unital `k`-algebra homomorphism from `MonoidAlgebra k G` is uniquely defined by its
values on the functions `single a 1`. -/
@[to_additive (dont_translate := k) /-- A non_unital `k`-algebra homomorphism from `k[G]` is uniquely defined by its
values on the functions `single a 1`. -/
]
theorem nonUnitalAlgHom_ext [DistribMulAction k A] {φ₁ φ₂ : MonoidAlgebra k G →ₙₐ[k] A}
(h : ∀ x, φ₁ (single x 1) = φ₂ (single x 1)) : φ₁ = φ₂ :=
NonUnitalAlgHom.to_distribMulActionHom_injective <|
Finsupp.distribMulActionHom_ext' fun a => DistribMulActionHom.ext_ring (h a)
