-- Module: Mathlib.Analysis.Complex.Poisson
Goal:
R φ θ r : ℝ
simp :
  (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) - 0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0))) *
              (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) /
            ((R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) +
          (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1))) *
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) /
            ((R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) -
        ((r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0))) *
              (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) /
            ((r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) +
          (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1))) *
              (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) /
            ((r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))))) =
      (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
              (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
            (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
              (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) /
          ((R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
              (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
            (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) +
        (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) +
              (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
            (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
              (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) /
          ((R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
              (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
            (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) ∧
    (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1))) *
              (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) /
            ((R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) -
          (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0))) *
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) /
            ((R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (R * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (r * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (R * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (r * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) -
        ((r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1))) *
              (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) /
            ((r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1))))) -
          (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0))) *
              (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                  0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) /
            ((r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) *
                (r * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) -
                  (R * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                    0 * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)))) +
              (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))) *
                (r * (0 + (Real.sin θ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos θ + (Real.sin θ * 0 - HMul.hMul (α := ℝ) 0 1)) -
                  (R * (0 + (Real.sin φ * 1 + HMul.hMul (α := ℝ) 0 0)) +
                    0 * (Real.cos φ + (Real.sin φ * 0 - HMul.hMul (α := ℝ) 0 1)))))) =
      0
⊢ (↑R : ℂ) * cexp ((↑θ : ℂ) * I) / ((↑R : ℂ) * cexp ((↑θ : ℂ) * I) - (↑r : ℂ) * cexp ((↑φ : ℂ) * I)) -
      (↑r : ℂ) * cexp ((↑θ : ℂ) * I) / ((↑r : ℂ) * cexp ((↑θ : ℂ) * I) - (↑R : ℂ) * cexp ((↑φ : ℂ) * I)) =
    (↑(((↑R : ℂ) * cexp ((↑θ : ℂ) * I) + (↑r : ℂ) * cexp ((↑φ : ℂ) * I)) /
            ((↑R : ℂ) * cexp ((↑θ : ℂ) * I) - (↑r : ℂ) * cexp ((↑φ : ℂ) * I))).re :
      ℂ)

Tactic:
simp only [Complex.ext_iff, exp_ofReal_mul_I, add_re, sub_re, mul_re, div_re, ofReal_re,
    I_re, I_im, add_im, sub_im, mul_im, div_im, ofReal_im, normSq_apply]

Premises used: Complex.mul_im, Complex.add_im, Complex.mul_re, Complex.sub_im, Complex.sub_re, Complex.div_im, Complex.add_re, Complex.I_re, Complex.ext_iff, Complex.I_im, Complex.div_re, Complex.ofReal_re, Complex.ofReal_im, Complex.normSq_apply, Complex.exp_ofReal_mul_I