smoothingSeminorm_apply_of_map_mul_eq_mul

English

If hμ1 and hx: μ(xy) = μ x μ y, then smoothingSeminorm μ hμ1 hna (xy) = μ x · μ y under the mapped seminorm.

Русский

Если hμ1 и hx: μ(xy) = μ x μ y, тогда smoothingSeminorm μ hμ1 hna (xy) = sm μ x · sm μ y.

LaTeX

$$$\operatorname{smearingSeminorm} μ hμ1 hna (instHMul.hMul x y) = instHMul.hMul (\operatorname{smearingSeminorm} μ hμ1 hna x) (\operatorname{smearingSeminorm} μ hμ1 hna y)$$$

Lean4

/-- If `μ 1 ≤ 1`, `μ` is nonarchimedean, and `∀ y : R, μ (x * y) = μ x * μ y`, then
  `smoothingSeminorm μ x = μ x`. -/
theorem smoothingSeminorm_apply_of_map_mul_eq_mul (hμ1 : μ 1 ≤ 1) (hna : IsNonarchimedean μ) {x : R}
    (hx : ∀ y : R, μ (x * y) = μ x * μ y) : smoothingSeminorm μ hμ1 hna x = μ x :=
  smoothingFun_apply_of_map_mul_eq_mul μ hμ1 hx

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