θ : ℝ
k : ℤ
⊢ -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π ↔ (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π
θ : ℝ
k : ℤ
⊢ toReal ↑θ = θ - 2 * ↑k * π ↔ θ ∈ Set.Ioc ((2 * ↑k - 1) * π) ((2 * ↑k + 1) * π)
rw [← sub_zero (θ : Angle), ← zsmul_zero k, ← coe_two_pi, ← coe_zsmul, ← coe_sub, zsmul_eq_mul, ←
  mul_assoc, mul_comm (k : ℝ), toReal_coe_eq_self_iff, Set.mem_Ioc]
no goals
θ : ℝ
k : ℤ
⊢ -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π ↔ (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π
exact ⟨fun h => ⟨by linarith, by linarith⟩, fun h => ⟨by linarith, by linarith⟩⟩
no goals
θ : ℝ
k : ℤ
h : -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π
⊢ (2 * ↑k - 1) * π < θ
linarith
no goals
θ : ℝ
k : ℤ
h : -π < θ - 2 * ↑k * π ∧ θ - 2 * ↑k * π ≤ π
⊢ θ ≤ (2 * ↑k + 1) * π
linarith
no goals
θ : ℝ
k : ℤ
h : (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π
⊢ -π < θ - 2 * ↑k * π
linarith
no goals
θ : ℝ
k : ℤ
h : (2 * ↑k - 1) * π < θ ∧ θ ≤ (2 * ↑k + 1) * π
⊢ θ - 2 * ↑k * π ≤ π
linarith