smul_eval — Mathlib

English

For any g in G, the action by g leaves the product prodXSubSMul G R x unchanged: g • prodXSubSMul G R x = prodXSubSMul G R x.

Русский

Действие элемента g сохраняет произведение: g • prodXSubSMul G R x = prodXSubSMul G R x.

LaTeX

$$$g \cdot \bigl(prodXSubSMul\ G\ R\ x\bigr) = prodXSubSMul\ G\ R\ x$.$$

Lean4

theorem smul_eval [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) : (g • f).eval x = g • f.eval (g⁻¹ • x) := by
  rw [← smul_eval_smul, smul_inv_smul]

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