smeval_C_mul — Mathlib

English

smeval(C r · f) = r · smeval(f): scaling by a constant commutes with smeval.

Русский

smeval(C r · f) = r · smeval(f): масштабирование константой commuting with smeval.

LaTeX

$$$$ (C r \cdot f).smeval x = r \cdot (f.smeval x) $$$$

Lean4

@[simp]
theorem smeval_C_mul (r : R) : (C r * f).smeval x = r • (f.smeval x) := by
  induction f using LaurentPolynomial.induction_on' with
  | add p q hp hq => rw [mul_add, smeval_add, smeval_add, smul_add, hp, hq]
  | C_mul_T n s => rw [← mul_assoc, ← map_mul, smeval_C_mul_T_n, smeval_C_mul_T_n, mul_smul]

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