-- Module: Mathlib.RingTheory.WittVector.StructurePolynomial
Goal:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
Φ : MvPolynomial idx ℚ
n : ℕ
this :
  HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ) (β := MvPolynomial (idx × ℕ) ℚ)
      ((bind₁ fun (k : ℕ) ↦
              (bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ k) :
                  (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
                Φ :
            (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ)
          (xInTermsOfW p ℚ n) :
        MvPolynomial (idx × ℕ) ℚ)
      ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) n) : MvPolynomial (idx × ℕ) ℚ) =
    (bind₁ fun (k : ℕ) ↦
          (bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ k) :
              (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
            Φ :
        (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ)
      (HSub.hSub (α := MvPolynomial ℕ ℚ) (X n)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial ℕ ℚ)
            ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial ℕ ℚ)
            (HPow.hPow (α := MvPolynomial ℕ ℚ) (xInTermsOfW p ℚ i) (p ^ (n - i)))))
rw :
  (bind₁ fun (k : ℕ) ↦
          (bind₁ fun (i : idx) ↦ (rename (Prod.mk i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ k) :
              (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
            Φ :
        (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ)
      (HSub.hSub (α := MvPolynomial ℕ ℚ) (X n)
        (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial ℕ ℚ)
            ((C : (a : ℚ) → MvPolynomial ℕ ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial ℕ ℚ)
            (HPow.hPow (α := MvPolynomial ℕ ℚ) (xInTermsOfW p ℚ i) (p ^ (n - i))))) =
    HSub.hSub (α := MvPolynomial (idx × ℕ) ℚ) (β := MvPolynomial (idx × ℕ) ℚ)
      ((bind₁ fun (b : idx) ↦
              (rename fun (i : ℕ) ↦ (b, i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ n) :
            (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
          Φ :
        MvPolynomial (idx × ℕ) ℚ)
      (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
            ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p Φ i ^ p ^ (n - i)) :
        MvPolynomial (idx × ℕ) ℚ)
⊢ HMul.hMul (β := MvPolynomial (idx × ℕ) ℚ) (wittStructureRat p Φ n)
      ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) n) : MvPolynomial (idx × ℕ) ℚ) =
    HSub.hSub (α := MvPolynomial (idx × ℕ) ℚ) (β := MvPolynomial (idx × ℕ) ℚ)
      ((bind₁ fun (b : idx) ↦
              (rename fun (i : ℕ) ↦ (b, i) : (a : MvPolynomial ℕ ℚ) → MvPolynomial (idx × ℕ) ℚ) (W_ ℚ n) :
            (a : MvPolynomial idx ℚ) → MvPolynomial (idx × ℕ) ℚ)
          Φ :
        MvPolynomial (idx × ℕ) ℚ)
      (∑ i ∈ Finset.range n,
          HMul.hMul (α := MvPolynomial (idx × ℕ) ℚ)
            ((C : (a : ℚ) → MvPolynomial (idx × ℕ) ℚ) (HPow.hPow (α := ℚ) (↑p : ℚ) i) : MvPolynomial (idx × ℕ) ℚ)
            (wittStructureRat p Φ i ^ p ^ (n - i)) :
        MvPolynomial (idx × ℕ) ℚ)

Tactic:
rw [wittStructureRat, this]

Premises used: wittStructureRat