norm_cast_le — Mathlib

English

A subalgebra of a seminormed ring is itself a seminormed ring under the restricted norm.

Русский

Подалгебра полуправила нормированного кольца образует самóе сепримедованное кольцо с ограниченной нормой.

LaTeX

$$$\text{If } S \text{ is a Subalgebra of a seminormed ring } E, \text{ then } S \text{ is a SeminormedRing with restricted norm.}$$$

Lean4

theorem norm_cast_le : ∀ n : ℕ, ‖(n : α)‖ ≤ n * ‖(1 : α)‖
  | 0 => by simp
  | n + 1 => by
    rw [n.cast_succ, n.cast_succ, add_mul, one_mul]
    exact norm_add_le_of_le (Nat.norm_cast_le n) le_rfl

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