-- Module: Mathlib.NumberTheory.Multiplicity
Goal:
R : Type u_1
inst : CommRing R
a b : R
p : ℕ
hp : Odd p
h1 :
  ∀ (i : ℕ),
    Eq (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
      ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
          ((a + (↑p : R) * b) ^ i) :
        HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
      ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
          (a ^ (i - 1) * ((↑p : R) * b) * (↑i : R) + a ^ i) :
        HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
s : R := HPow.hPow (α := R) (β := ℕ) (↑p : R) 2
simp_rw :
  Eq (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
    ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
      (∑ i ∈ range p, (a + (↑p : R) * b) ^ i * a ^ (p - 1 - i)))
    (∑ x ∈ range p,
      (Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
        ((a ^ (x - 1) * ((↑p : R) * b) * (↑x : R) + a ^ x) * a ^ (p - 1 - x)))
simp_rw₁ :
  Eq.{u_1 + 1} (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
    (∑ i ∈ range p,
      HPow.hPow (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
          ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
              (a + (↑p : R) * b) :
            HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
          i *
        HPow.hPow (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
          ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2}) a :
            HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
          (p - 1 - i))
    (∑ x ∈ range p,
      (Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
        ((a ^ (x - 1) * ((↑p : R) * b) * (↑x : R) + a ^ x) * a ^ (p - 1 - x)))
simp_rw₂ :
  Eq.{u_1 + 1} (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
    (∑ x ∈ range p,
      HMul.hMul (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2})) (β :=
        HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
        ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
            ((a + (↑p : R) * b) ^ x) :
          HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
        ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
            (a ^ (p - 1 - x)) :
          HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2})))
    (∑ x ∈ range p,
      (Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
        ((a ^ (x - 1) * ((↑p : R) * b) * (↑x : R) + a ^ x) * a ^ (p - 1 - x)))
simp_rw₃ :
  Eq.{u_1 + 1} (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
    (∑ x ∈ range p,
      HMul.hMul (α := HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2})) (β :=
        HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
        ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
            (a ^ (x - 1) * ((↑p : R) * b) * (↑x : R) + a ^ x) :
          HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}))
        ((Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
            (a ^ (p - 1 - x)) :
          HasQuotient.Quotient R (B := Ideal R) (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2})))
    (∑ x ∈ range p,
      (Ideal.Quotient.mk (span {HPow.hPow (α := R) (β := ℕ) (↑p : R) 2}) : (a : R) → R ⧸ span {(↑p : R) ^ 2})
        ((a ^ (x - 1) * ((↑p : R) * b) * (↑x : R) + a ^ x) * a ^ (p - 1 - x)))
⊢ Eq (α := HasQuotient.Quotient R (B := Ideal R) (span {s}))
    ((Ideal.Quotient.mk (span {s}) : (a : R) → R ⧸ span {s}) (∑ i ∈ range p, (a + (↑p : R) * b) ^ i * a ^ (p - 1 - i)))
    (∑ i ∈ range p,
      (Ideal.Quotient.mk (span {s}) : (a : R) → R ⧸ span {s})
        ((a ^ (i - 1) * ((↑p : R) * b) * (↑i : R) + a ^ i) * a ^ (p - 1 - i)))

Tactic:
simp_rw [s, RingHom.map_geom_sum₂, ← map_pow, h1, ← map_mul]

Premises used: map_pow, map_mul, RingHom.map_geom_sum₂