rotation_trans — Mathlib

English

Rotating the second vector by θ scales Kahler form by θ.toCircle: o.kahler(x, o.rotation(θ)y) = θ.toCircle o.kahler(x, y).

Русский

При повороте второго вектора на θ Kahler форма масштабируется на θ: o.kahler(x, o.rotation(θ)y) = θ.toCircle o.kahler(x, y).

LaTeX

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

Lean4

/-- Rotating twice is equivalent to rotating by the sum of the angles. -/
@[simp]
theorem rotation_trans (θ₁ θ₂ : Real.Angle) : (o.rotation θ₁).trans (o.rotation θ₂) = o.rotation (θ₂ + θ₁) :=
  LinearIsometryEquiv.ext fun _ => by rw [← rotation_rotation, LinearIsometryEquiv.trans_apply]

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