-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R₀ : Type u_1
inst : CommRing R₀
n : ℕ
R : Type u_6
inst_1 : CommRing R
inst_2 : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}) := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective :
  Surjective (α := R) (β := HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})) (⇑q₂ : (a : R) → R ⧸ Ideal.span {c})
e₁ : InductionObj (Localization.Away c) n :=
  {
    val :=
      HSMul.hSMul (α := (Localization.Away c)[X]) (β := Fin n → (Localization.Away c)[X])
        ((Polynomial.C : (a : Localization.Away c) → (Localization.Away c)[X]) (IsLocalization.Away.invSelf c))
        ((⇑(mapRingHom (↑q₁ : R →+* Localization.Away c)) : (a : R[X]) → (Localization.Away c)[X]) ∘ e.val) }
e₂ : InductionObj (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})) n :=
  { val := (⇑(mapRingHom (↑q₂ : R →+* R ⧸ Ideal.span {c})) : (a : R[X]) → (R ⧸ Ideal.span {c})[X]) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
degBound_e₂_lt : e₂.degBound < e.degBound
H₁ : Statement R₀ (Localization.Away c) n e₁
H₂ : Statement R₀ (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c})) n e₂
f : R[X]
T₁ : ConstructibleSetData (Localization.Away c)
T₂ : ConstructibleSetData (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))
hT₁ :
  comap Polynomial.C ''
      (zeroLocus (Set.range e₁.val) \
        zeroLocus
          (singleton (α := (Localization.Away c)[X])
            ((mapRingHom (↑q₁ : R →+* Localization.Away c) : (a : R[X]) → (Localization.Away c)[X]) f :
              (Localization.Away c)[X]))) =
    T₁.toSet
hT₁deg : ∀ (x : BasicConstructibleSetData (Localization.Away c)) (x_1 : x ∈ T₁), x.n ≤ e₁.degBound
hT₁span :
  ∀ (x : BasicConstructibleSetData (Localization.Away c)) (x_1 : x ∈ T₁) (i : Fin x.n),
    x.g i ∈ HPow.hPow (α := Submodule R₀ (Localization.Away c)) (coeffSubmodule R₀ e₁) e₁.powBound
hT₂ :
  comap Polynomial.C ''
      (zeroLocus (Set.range e₂.val) \
        zeroLocus
          (singleton (α := (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))[X])
            ((mapRingHom (↑q₂ : R →+* R ⧸ Ideal.span {c}) : (a : R[X]) → (R ⧸ Ideal.span {c})[X]) f :
              (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))[X]))) =
    T₂.toSet
hT₂deg :
  ∀ (x : BasicConstructibleSetData (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))) (x_1 : x ∈ T₂),
    x.n ≤ e₂.degBound
hT₂span :
  ∀ (x : BasicConstructibleSetData (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))) (x_1 : x ∈ T₂)
    (i : Fin x.n),
    x.g i ∈
      HPow.hPow (α := Submodule R₀ (HasQuotient.Quotient R (B := Ideal R) (Ideal.span {c}))) (coeffSubmodule R₀ e₂)
        e₂.powBound
x : Invertible (α := Localization.Away c) ((q₁ : (a : R) → Localization.Away c) c) :=
  { invOf := IsLocalization.Away.invSelf c, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
simp :
  HPow.hPow (α := Submodule R₀ (Localization.Away c))
      (span R₀
        (insert 1
          (⋃ (i : Fin n),
            coeff(HMul.hMul (α := (Localization.Away c)[X])
                ((Polynomial.C : (a : Localization.Away c) → (Localization.Away c)[X]) (IsLocalization.Away.invSelf c))
                (Polynomial.map (algebraMap R (Localization.Away c)) (e.val i))))))
      {
          val :=
            HSMul.hSMul (α := (Localization.Away c)[X]) (β := Fin n → (Localization.Away c)[X])
              ((Polynomial.C : (a : Localization.Away c) → (Localization.Away c)[X]) (IsLocalization.Away.invSelf c))
              (Polynomial.map (algebraMap R (Localization.Away c)) ∘ e.val) :
          InductionObj (Localization.Away c) n }.powBound =
    HPow.hPow (α := Submodule R₀ (Localization.Away c))
      (span R₀
        (insert 1
          (⋃ (i : Fin n),
            HSMul.hSMul (α := Localization.Away c) (β := Set (Localization.Away c))
              (@Invertible.invOf _ MulOne.toMul MulOne.toOne
                ((algebraMap R (Localization.Away c) : (a : R) → Localization.Away c) c) _)
              ((fun (a : R) ↦ (algebraMap R (Localization.Away c) : (a : R) → Localization.Away c) a) ''
                coeff(e.val i)))))
      {
          val :=
            HSMul.hSMul (α := (Localization.Away c)[X]) (β := Fin n → (Localization.Away c)[X])
              ((Polynomial.C : (a : Localization.Away c) → (Localization.Away c)[X]) (IsLocalization.Away.invSelf c))
              (Polynomial.map (algebraMap R (Localization.Away c)) ∘ e.val) :
          InductionObj (Localization.Away c) n }.powBound
⊢ HPow.hPow (α := Submodule R₀ (Localization.Away c)) (span R₀ ({1} ∪ ⋃ (i : Fin n), coeff(e₁.val i))) e₁.powBound =
    HPow.hPow (α := Submodule R₀ (Localization.Away c))
      (span R₀
        (HSMul.hSMul (α := Localization.Away c) (β := Set (Localization.Away c))
          (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((q₁ : (a : R) → Localization.Away c) c) _)
          (Set.image (α := R) (⇑q₁.toLinearMap : (a : R) → Localization.Away c) ({c} ∪ ⋃ (i : Fin n), coeff(e.val i)))))
      e₁.powBound

Tactic:
simp [Set.image_insert_eq, Set.smul_set_insert, Set.image_iUnion, Set.smul_set_iUnion, q₁, e₁]

Premises used: Set.smul_set_insert, Set.image_iUnion, Set.smul_set_iUnion, Set.image_insert_eq