aeval_algHom_apply — Mathlib

English

The equality φ_t = φ_{t'} holds if and only if t = t'.

Русский

Сохранение равенства φ_t = φ_{t'} эквивалентно t = t'.

LaTeX

$$$\forall t,t' \in R,\; \operatorname{algEquivAevalXAddC}(t) = \operatorname{algEquivAevalXAddC}(t') \iff t = t'.$$$

Lean4

theorem aeval_algHom_apply {F : Type*} [FunLike F A B] [AlgHomClass F R A B] (f : F) (x : A) (p : R[X]) :
    aeval (f x) p = f (aeval x p) :=
  by
  refine
    Polynomial.induction_on p (by simp [AlgHomClass.commutes]) (fun p q hp hq => ?_) (by simp [AlgHomClass.commutes])
  rw [map_add, hp, hq, ← map_add, ← map_add]

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