English
In an ultrametric normed ring, the integer cast n ↦ n in R has norm ≤ 1 for any n ∈ ℕ.
Русский
В ультраметрическом нормированном кольце нормa целого числа n, вложенного в R, не превышает 1.
LaTeX
$$$\| n \| \le 1 \quad (n \in \mathbb{N}).$$$
Lean4
/-- The formula `Ring.inverse (x + t) = Ring.inverse (1 + x⁻¹ * t) * x⁻¹` holds for `t` sufficiently
small. -/
theorem inverse_add (x : Rˣ) : ∀ᶠ t in 𝓝 0, inverse ((x : R) + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹ :=
by
nontriviality R
rw [Metric.eventually_nhds_iff]
refine ⟨‖(↑x⁻¹ : R)‖⁻¹, by cancel_denoms, fun t ht ↦ ?_⟩
rw [dist_zero_right] at ht
rw [← x.val_add t ht, inverse_unit, Units.add, Units.copy_eq, mul_inv_rev, Units.val_mul, ← inverse_unit,
Units.val_oneSub, sub_neg_eq_add]
