Φ₀ S transform

theorem φ₀_S_transform (z : ℍ) :    φ₀ (ModularGroup.S • z) = φ₀ z - (12 * Complex.I) / (π * z) * φ₂' z                             - 36 / (π ^ 2 * z ^ 2) * φ₄' z := by  have hz : (z : ℂ) ≠ 0 := ne_zero z  have hπ : (π : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr Real.pi_ne_zero  have hI : Complex.I ≠ 0 := Complex.I_ne_zero  unfold φ₀ φ₂' φ₄'  rw [E₂_S_transform, E₄_S_transform, E₆_S_transform, Δ_S_transform]  -- Let A = E₂ z * E₄ z - E₆ z (the key expression)  set A := E₂ z * E₄ z - E₆ z with hA  -- The numerator E₂(S•z) * E₄(S•z) - E₆(S•z) simplifies to z⁶ * (A + 6E₄/(πIz))  have h_numer : (z : ℂ) ^ 2 * (E₂ z + 6 / (π * Complex.I * z)) * (z ^ 4 * E₄ z) -                 z ^ 6 * E₆ z = z ^ 6 * (A + 6 * E₄ z / (π * Complex.I * z)) := by    ring_nf; rw [hA]; ring  -- The main algebraic simplification  rw [h_numer]  -- Now we have: (z⁶ * (A + 6E₄/(πIz)))² / (z¹² * Δ z)  -- = z¹² * (A + 6E₄/(πIz))² / (z¹² * Δ z) = (A + 6E₄/(πIz))² / Δ z  have h_sq : (z ^ 6 * (A + 6 * E₄ z / (π * Complex.I * z))) ^ 2 =              z ^ 12 * (A + 6 * E₄ z / (π * Complex.I * z)) ^ 2 := by    rw [mul_pow, sq (z ^ 6 : ℂ), ← pow_add]  rw [h_sq]  -- Simplify z¹² * X / (z¹² * Δ z) = X / Δ z  have h_div : z ^ 12 * (A + 6 * E₄ z / (π * Complex.I * z)) ^ 2 / (z ^ 12 * Δ z) =               (A + 6 * E₄ z / (π * Complex.I * z)) ^ 2 / Δ z := by    rw [mul_comm (z ^ 12 : ℂ) (Δ z)]; field_simp  rw [h_div]  -- Expand (A + 6E₄/(πIz))² = A² + 12AE₄/(πIz) + 36E₄²/(π²I²z²)  -- Since I² = -1, we get: A² + 12AE₄/(πIz) - 36E₄²/(π²z²)  have hI2 : Complex.I ^ 2 = -1 := Complex.I_sq  -- Expand the square and simplify  have h_expand : (A + 6 * E₄ z / (π * Complex.I * z)) ^ 2 / Δ z =                  A ^ 2 / Δ z + 12 * A * E₄ z / (π * Complex.I * z * Δ z) +                  36 * (E₄ z) ^ 2 / (π ^ 2 * Complex.I ^ 2 * z ^ 2 * Δ z) := by    have hπIz : π * Complex.I * z ≠ 0 := mul_ne_zero (mul_ne_zero hπ hI) hz    field_simp; ring  rw [h_expand, hI2]  -- Transform 12/(πIz) to -12I/(πz) using I⁻¹ = -I  have h_I_factor : (12 : ℂ) / (π * Complex.I * z) = -12 * Complex.I / (π * z) := by    field_simp [Complex.inv_I]; simp [Complex.I_sq]  have h_final : A ^ 2 / Δ z + 12 * A * E₄ z / (π * Complex.I * z * Δ z) +       36 * (E₄ z) ^ 2 / (π ^ 2 * (-1) * z ^ 2 * Δ z) =       A ^ 2 / Δ z - 12 * Complex.I / (π * z) * (E₄ z * A / Δ z) -       36 / (π ^ 2 * z ^ 2) * ((E₄ z) ^ 2 / Δ z) := by    have h1 : 12 * A * E₄ z / (π * Complex.I * z * Δ z) =              12 / (π * Complex.I * z) * (E₄ z * A / Δ z) := by field_simp    rw [h1, h_I_factor]; ring  -- The goal now matches h_final applied to the expanded expression  rw [h_final]