English
If α and β are nonunital seminormed rings, then their product α × β carries a natural nonunital seminormed ring structure, using the sup norm ∥(x,y)∥ = max(∥x∥, ∥y∥).
Русский
Если α и β — ненулевые полууглеродные кольца с нормой, то их произведение α × β естественным образом образует ненулевое полнормированное кольцо, используя суп Norm: ∥(x,y)∥ = max(∥x∥, ∥y∥).
LaTeX
$$$\alpha,\beta$ — ненулевые ненульевыe нормированные кольца; \\ (\alpha \times \beta, (x,y) \mapsto (xy_1, y_2) ) ext{ with } \| (x,y) \| = \max( \|x\|, \|y\| )$ is a NonUnitalSeminormedRing.$$
Lean4
/-- Non-unital seminormed ring structure on the product of two non-unital seminormed rings,
using the sup norm. -/
instance nonUnitalSeminormedRing [NonUnitalSeminormedRing β] : NonUnitalSeminormedRing (α × β) :=
{ seminormedAddCommGroup, instNonUnitalRing with
norm_mul_le x
y :=
calc
‖x * y‖ = ‖(x.1 * y.1, x.2 * y.2)‖ := rfl
_ = max ‖x.1 * y.1‖ ‖x.2 * y.2‖ := rfl
_ ≤ max (‖x.1‖ * ‖y.1‖) (‖x.2‖ * ‖y.2‖) := (max_le_max (norm_mul_le x.1 y.1) (norm_mul_le x.2 y.2))
_ = max (‖x.1‖ * ‖y.1‖) (‖y.2‖ * ‖x.2‖) := by simp [mul_comm]
_ ≤ max ‖x.1‖ ‖x.2‖ * max ‖y.2‖ ‖y.1‖ := by apply max_mul_mul_le_max_mul_max <;> simp [norm_nonneg]
_ = max ‖x.1‖ ‖x.2‖ * max ‖y.1‖ ‖y.2‖ := by simp [max_comm]
_ = ‖x‖ * ‖y‖ := rfl }
