rotation — Mathlib

English

For any x ∈ ℂ_ℤ and any integer n, x^2 ≠ (n : ℂ)^2.

Русский

Для любого x ∈ ℂ_ℤ и любого целого n верно, что x^2 ≠ (n)^2.

LaTeX

$$$\forall x \in \mathbb{C}_{\mathbb{Z}}, \forall n \in \mathbb{Z}, x^2 \neq (n)^2$$$

Lean4

/-- An element of the unit circle defines a `LinearIsometryEquiv` from `ℂ` to itself, by
rotation. -/
def rotation : Circle →* ℂ ≃ₗᵢ[ℝ] ℂ
    where
  toFun
    a :=
    { DistribMulAction.toLinearEquiv ℝ ℂ a with
      norm_map' x := show ‖a * x‖ = ‖x‖ by rw [norm_mul, Circle.norm_coe, one_mul] }
  map_one' := LinearIsometryEquiv.ext <| by simp
  map_mul' a b := LinearIsometryEquiv.ext <| mul_smul a b

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