theorem odd_sq_dvd_geom_sum₂_sub (hp : Odd p) :
    (p : R) ^ 2 ∣ (∑ i in range p, (a + p * b) ^ i * a ^ (p - 1 - i)) - p * a ^ (p - 1) := by
  have h1 : ∀ (i : ℕ),
      (p : R) ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * i + a ^ i) := by
    intro i
    calc
      ↑p ^ 2 ∣ (↑p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]
      _ ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) := by
        simp only [sq_dvd_add_pow_sub_sub (↑p * b) a i, ← sub_sub]
  simp_rw [← mem_span_singleton, ← Ideal.Quotient.eq] at *
  let s : R := (p : R)^2
  calc
    (Ideal.Quotient.mk (span {s})) (∑ i in range p, (a + (p : R) * b) ^ i * a ^ (p - 1 - i)) =
        ∑ i : ℕ in Finset.range p,
        mk (span {s}) ((a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) * a ^ (p - 1 - i)) :=
      by simp_rw [RingHom.map_geom_sum₂, ← map_pow, h1, ← _root_.map_mul]
    _ =
        mk (span {s})
            (∑ x : ℕ in Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +
          mk (span {s}) (∑ x : ℕ in Finset.range p, a ^ (x + (p - 1 - x))) := by
      ring_nf
      simp only [← pow_add, map_add, Finset.sum_add_distrib, ← map_sum]
      congr
      simp [pow_add a, mul_assoc]
    _ =
        mk (span {s})
            (∑ x : ℕ in Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +
          mk (span {s}) (∑ _x : ℕ in Finset.range p, a ^ (p - 1)) := by
      rw [add_right_inj]
      have : ∀ (x : ℕ), (hx : x ∈ range p) → a ^ (x + (p - 1 - x)) = a ^ (p - 1) := by
        intro x hx
        rw [← Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]
        exact Nat.le_sub_one_of_lt (Finset.mem_range.mp hx)
      rw [Finset.sum_congr rfl this]
    _ =
        mk (span {s})
            (∑ x : ℕ in Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +
          mk (span {s}) (↑p * a ^ (p - 1)) :=
      by simp only [add_right_inj, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
    _ =
        mk (span {s}) (↑p * b * ∑ x : ℕ in Finset.range p, a ^ (p - 2) * x) +
          mk (span {s}) (↑p * a ^ (p - 1)) := by
      simp only [Finset.mul_sum, ← mul_assoc, ← pow_add]
      rw [Finset.sum_congr rfl]
      rintro (⟨⟩ | ⟨x⟩) hx
      · rw [Nat.cast_zero, mul_zero, mul_zero]
      · have : x.succ - 1 + (p - 1 - x.succ) = p - 2 := by
          rw [← Nat.add_sub_assoc (Nat.le_sub_one_of_lt (Finset.mem_range.mp hx))]
          exact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)
        rw [this]
        ring1
    _ = mk (span {s}) (↑p * a ^ (p - 1)) := by
      have : Finset.sum (range p) (fun (x : ℕ) ↦ (x : R)) =
          ((Finset.sum (range p) (fun (x : ℕ) ↦ (x : ℕ)))) := by simp only [Nat.cast_sum]
      simp only [add_left_eq_self, ← Finset.mul_sum, this]
      norm_cast
      simp only [Finset.sum_range_id]
      norm_cast
      simp only [Nat.cast_mul, _root_.map_mul,
          Nat.mul_div_assoc p (even_iff_two_dvd.mp (Nat.Odd.sub_odd hp odd_one))]
      ring_nf
      rw [mul_assoc, mul_assoc]
      refine' mul_eq_zero_of_left _ _
      refine' Ideal.Quotient.eq_zero_iff_mem.mpr _
      

Proof state:
R : Type u_1
n : ℕ
inst✝ : CommRing R
a b x y : R
p : ℕ
hp : Odd p
h1 :
  ∀ (i : ℕ),
    (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =
      (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)
s : R := ↑p ^ 2
this : ∑ x in range p, ↑x = ↑(∑ x in range p, x)
⊢ (fun x => x ^ 2) ↑p ∈ span {↑p ^ 2}

Next tactic:
simp [mem_span_singleton]