awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp_1 :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hp : Membership.mem (γ := Set R[X]) (insert X {p : R[X] | p.degree ≤ 0}) p
⊢ p = X ∨ p.degree ≤ 0

Tactic:
push _ ∈ _ at hp
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp_1 :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hp : p = X ∨ p.degree ≤ 0
refine_1 :
  ∀ (hy : p = X),
    φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) p)
refine_1₁ :
  ∀ (hy : p.degree ≤ 0),
    φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) p)
⊢ φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) p)

Tactic:
rcases hp with hy | hy
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = X
refine_1 : φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) X)
⊢ φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) p)

Tactic:
rw [hy]
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = X
⊢ φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) X)

Tactic:
φ.isIntegralElem_localization_at_leadingCoeff ((Ideal.Quotient.mk P) X)
        (pX.map (Ideal.Quotient.mk P')) ?_ M ?_

Premises used: Ideal.Quotient.mk, RingHom.isIntegralElem_localization_at_leadingCoeff, Polynomial.map, Polynomial.X, DFunLike.coe
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = X
refine_1 : eval₂ φ ((Ideal.Quotient.mk P : (a : R[X]) → R[X] ⧸ P) X) (map (Ideal.Quotient.mk P') pX) = 0
refine_1₁ : (map (Ideal.Quotient.mk P') pX).leadingCoeff ∈ M
⊢ φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) X)

Tactic:
refine φ.isIntegralElem_localization_at_leadingCoeff ((Ideal.Quotient.mk P) X)
        (pX.map (Ideal.Quotient.mk P')) ?_ M ?_

Premises used: Ideal.Quotient.mk, RingHom.isIntegralElem_localization_at_leadingCoeff, Polynomial.map, Polynomial.X, DFunLike.coe
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = X
refine_1 : eval₂ (φ.comp (Ideal.Quotient.mk P')) ((Ideal.Quotient.mk P : (a : R[X]) → R[X] ⧸ P) X) pX = 0
refine_1₁ : eval₂ (RingHom.comp (Ideal.Quotient.mk P) C) ((Ideal.Quotient.mk P : (a : R[X]) → R[X] ⧸ P) X) pX = 0
refine_1₂ : Eq (α := R[X] ⧸ P) ((Ideal.Quotient.mk P : (a : R[X]) → R[X] ⧸ P) (eval₂ C X pX) : R[X] ⧸ P) (0 : R[X] ⧸ P)
refine_1₃ : eval₂ C X pX ∈ P
refine_1₄ refine_1₅ refine_1₆ refine_1₇ : pX ∈ P
⊢ eval₂ φ ((Ideal.Quotient.mk P : (a : R[X]) → R[X] ⧸ P) X) (map (Ideal.Quotient.mk P') pX) = 0

Tactic:
rwa [eval₂_map, hφ', ← hom_eval₂, Quotient.eq_zero_iff_mem, eval₂_C_X]

Premises used: Polynomial.hom_eval₂, Polynomial.eval₂_C_X, Polynomial.eval₂_map, Ideal.Quotient.eq_zero_iff_mem
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = X
h :
  (fun (x : ℕ) ↦ HPow.hPow (α := R ⧸ P') (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) x)
      1 =
    (map (Ideal.Quotient.mk P') pX).leadingCoeff
⊢ (map (Ideal.Quotient.mk P') pX).leadingCoeff ∈ M

Tactic:
use 1
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = X
⊢ (fun (x : ℕ) ↦ HPow.hPow (α := R ⧸ P') (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) x)
      1 =
    (map (Ideal.Quotient.mk P') pX).leadingCoeff

Tactic:
simp only [P', pow_one]

Premises used: pow_one
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p.degree ≤ 0
⊢ p = (C : (a : R) → R[X]) (p.coeff 0)

Tactic:
rw [degree_le_zero_iff] at hy

Premises used: Polynomial.degree_le_zero_iff
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = (C : (a : R) → R[X]) (p.coeff 0)
refine_1 :
  φ'.IsIntegralElem
    ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ)
      ((C : (a : R) → R[X]) (p.coeff 0) : R[X]))
⊢ φ'.IsIntegralElem ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ) p)

Tactic:
rw [hy]
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = (C : (a : R) → R[X]) (p.coeff 0)
⊢ φ'.IsIntegralElem
    ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ)
      ((C : (a : R) → R[X]) (p.coeff 0) : R[X]))

Tactic:
⟨X - C (algebraMap _ _ ((Ideal.Quotient.mk P') (p.coeff 0))), monic_X_sub_C _, ?_⟩

Premises used: Polynomial.monic_X_sub_C, Polynomial.C, Algebra.algebraMap, Ideal.Quotient.mk, HSub.hSub, RingHom.IsIntegralElem, Polynomial.coeff, And.intro, Exists.intro, OfNat.ofNat, Polynomial.X, DFunLike.coe
-- Module: Mathlib.RingTheory.Jacobson.Ring
Goal:
R : Type u_1
inst : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst_1 : CommRing Rₘ
inst_2 : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst_3 : Algebra.{u_1, u_3} (HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P)) Rₘ
inst_4 :
  IsLocalization.Away (R := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P))
    (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)) Rₘ
inst_5 : Algebra.{u_1, u_4} (R[X] ⧸ P) Sₘ
inst_6 :
  IsLocalization
    (Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
      (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))))
    Sₘ
P' : Ideal R := comap (F := R →+* R[X]) C P
M : Submonoid (R ⧸ P') := Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX))
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (F := HasQuotient.Quotient R (B := Ideal R) (comap (F := R →+* R[X]) C P) →+* R[X] ⧸ P)
    (quotientMap P C ⋯) (Submonoid.powers (leadingCoeff (map (Ideal.Quotient.mk (comap (F := R →+* R[X]) C P)) pX)))
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = RingHom.comp (Ideal.Quotient.mk P) C
p_1 : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp :
  HMul.hMul (β := Sₘ) p_1 ((algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) q : Sₘ) =
    (algebraMap (R[X] ⧸ P) Sₘ : (a : R[X] ⧸ P) → Sₘ) p'
p : R[X]
hy : p = (C : (a : R) → R[X]) (p.coeff 0)
refine_1 :
  eval₂ φ'
      ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ)
        ((C : (a : R) → R[X]) (p.coeff 0) : R[X]))
      (HSub.hSub (α := Rₘ[X]) (β := Rₘ[X]) X
        ((C : (a : Rₘ) → Rₘ[X])
            ((algebraMap (R ⧸ P') Rₘ : (a : R ⧸ P') → Rₘ)
              ((Ideal.Quotient.mk P' : (a : R) → R ⧸ P') (p.coeff 0) : R ⧸ P')) :
          Rₘ[X])) =
    0
⊢ φ'.IsIntegralElem
    ((RingHom.comp (algebraMap (R[X] ⧸ P) Sₘ) (Ideal.Quotient.mk P) : (a : R[X]) → Sₘ)
      ((C : (a : R) → R[X]) (p.coeff 0) : R[X]))

Tactic:
refine ⟨X - C (algebraMap _ _ ((Ideal.Quotient.mk P') (p.coeff 0))), monic_X_sub_C _, ?_⟩