aeval_eq_smeval — Mathlib

English

In a commutative semiring with an algebra, evaluating polynomials equals smeval: aeval x p = p.smeval x.

Русский

В коммутативной полусфере с алгеброй: аeval x p = p.smeval x.

LaTeX

$$aeval\ x\ p = p.smeval\ x$$

Lean4

theorem aeval_eq_smeval {R : Type*} [CommSemiring R] {S : Type*} [Semiring S] [Algebra R S] (x : S) (p : R[X]) :
    aeval x p = p.smeval x :=
  by
  rw [aeval_def, eval₂_def, Algebra.algebraMap_eq_smul_one', smeval_def]
  simp only [Algebra.smul_mul_assoc, one_mul]
  exact rfl

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