rotation_rotation — Mathlib

English

Rotating the first of two vectors by θ scales their Kahler form by (-θ) in the complex coefficient: o.kahler(o.rotation(θ)x, y) = (-θ).toCircle o.kahler(x, y).

Русский

При повороте первого вектора на θ коэффициентом Kahler изменяется на (-θ): o.kahler(o.rotation(θ)x, y) = (-θ).toCircle o.kahler(x, y).

LaTeX

$$$ o.kahler (o.rotation(\theta) x) y = (-\theta).toCircle \cdot o.kahler(x, y) $$$

Lean4

/-- Rotating twice is equivalent to rotating by the sum of the angles. -/
@[simp]
theorem rotation_rotation (θ₁ θ₂ : Real.Angle) (x : V) : o.rotation θ₁ (o.rotation θ₂ x) = o.rotation (θ₁ + θ₂) x :=
  by
  simp only [o.rotation_apply, Real.Angle.cos_add, Real.Angle.sin_add, LinearIsometryEquiv.map_add, map_smul,
    rightAngleRotation_rightAngleRotation]
  module

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