Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : 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] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ 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 (R ⧸ Ideal.span {c}) n e₂
f : R[X]
T₁ : ConstructibleSetData (Localization.Away c)
T₂ : ConstructibleSetData (R ⧸ Ideal.span {c})
hT₁ : comap Polynomial.C '' (zeroLocus (Set.range e₁.val) \ zeroLocus {(mapRingHom ↑q₁) f}) = T₁.toSet
hT₁deg : ∀ x ∈ T₁, x.n ≤ e₁.degBound
hT₁span : ∀ x ∈ T₁, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₁ ^ e₁.powBound
hT₂ : comap Polynomial.C '' (zeroLocus (Set.range e₂.val) \ zeroLocus {(mapRingHom ↑q₂) f}) = T₂.toSet
hT₂deg : ∀ x ∈ T₂, x.n ≤ e₂.degBound
hT₂span : ∀ x ∈ T₂, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₂ ^ e₂.powBound
x✝ : Invertible (q₁ c) := { invOf := IsLocalization.Away.invSelf c, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
⊢ ∃ T,
    comap Polynomial.C '' (zeroLocus (Set.range e.val) \ zeroLocus {f}) = T.toSet ∧
      ∀ C ∈ T, C.n ≤ e.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ coeffSubmodule R₀ e ^ e.powBound

Tactic:
have he₁span :
      e₁.coeffSubmodule R₀ ^ e₁.powBound = ⅟(q₁ c ^ e₁.powBound) •
        (span R₀ ({c} ∪ ⋃ i, coeff(e i)) ^ e₁.powBound).map q₁.toLinearMap := by
    unfold coeffSubmodule
    rw [Submodule.map_pow, map_span, invOf_pow, ← smul_pow, ← span_smul]
    simp [Set.image_insert_eq, Set.smul_set_insert, Set.image_iUnion, Set.smul_set_iUnion, q₁, e₁]
    congr! with i
    change _ = IsLocalization.Away.invSelf c • _
    simp [← Set.range_comp, Set.smul_set_range]
    ext
    simp

Goal state after:
R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : 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] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ 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 (R ⧸ Ideal.span {c}) n e₂
f : R[X]
T₁ : ConstructibleSetData (Localization.Away c)
T₂ : ConstructibleSetData (R ⧸ Ideal.span {c})
hT₁ : comap Polynomial.C '' (zeroLocus (Set.range e₁.val) \ zeroLocus {(mapRingHom ↑q₁) f}) = T₁.toSet
hT₁deg : ∀ x ∈ T₁, x.n ≤ e₁.degBound
hT₁span : ∀ x ∈ T₁, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₁ ^ e₁.powBound
hT₂ : comap Polynomial.C '' (zeroLocus (Set.range e₂.val) \ zeroLocus {(mapRingHom ↑q₂) f}) = T₂.toSet
hT₂deg : ∀ x ∈ T₂, x.n ≤ e₂.degBound
hT₂span : ∀ x ∈ T₂, ∀ (i : Fin x.n), x.g i ∈ coeffSubmodule R₀ e₂ ^ e₂.powBound
x✝ : Invertible (q₁ c) := { invOf := IsLocalization.Away.invSelf c, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
he₁span :
  coeffSubmodule R₀ e₁ ^ e₁.powBound =
    ⅟(q₁ c ^ e₁.powBound) • Submodule.map q₁.toLinearMap (span R₀ ({c} ∪ ⋃ i, coeff(e.val i)) ^ e₁.powBound)
⊢ ∃ T,
    comap Polynomial.C '' (zeroLocus (Set.range e.val) \ zeroLocus {f}) = T.toSet ∧
      ∀ C ∈ T, C.n ≤ e.degBound ∧ ∀ (i : Fin C.n), C.g i ∈ coeffSubmodule R₀ e ^ e.powBound