-- Mathlib/RingTheory/Algebraic/Integral.lean
theorem restrictScalars [Algebra.IsAlgebraic R S]
    {a : A} (h : IsAlgebraic S a) : IsAlgebraic R a

Proof state:
case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p✝ : S[X]
hp : p✝ ≠ 0
eval0 : (aeval a) p✝ = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p✝.coeff p✝.support, IsIntegral R (r • z)
p : (↥(integralClosure R S).toSubring)[X] := (r • p✝).toSubring (integralClosure R S).toSubring ⋯
⊢ p ≠ 0

Tactic:
have : FaithfulSMul R S := (faithfulSMul_iff_algebraMap_injective R S).mpr hRS

Result:
case refine_1
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p✝ : S[X]
hp : p✝ ≠ 0
eval0 : (aeval a) p✝ = 0
hRS : Function.Injective ⇑(algebraMap R S)
this✝ : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p✝.coeff p✝.support, IsIntegral R (r • z)
p : (↥(integralClosure R S).toSubring)[X] := (r • p✝).toSubring (integralClosure R S).toSubring ⋯
this : FaithfulSMul R S
⊢ p ≠ 0

Proof state:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p✝ : S[X]
hp : p✝ ≠ 0
eval0 : (aeval a) p✝ = 0
hRS : Function.Injective ⇑(algebraMap R S)
this✝ : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p✝.coeff p✝.support, IsIntegral R (r • z)
p : (↥(integralClosure R S).toSubring)[X] := (r • p✝).toSubring (integralClosure R S).toSubring ⋯
this : IsAlgebraic (↥(integralClosure R S)) a
⊢ IsAlgebraic R a

Tactic:
exact restrictScalars_of_isIntegral _ this

Result:
no goals

Proof state:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
⊢ IsAlgebraic R a

case neg
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : ¬Function.Injective ⇑(algebraMap R S)
⊢ IsAlgebraic R a

Tactic:
on_goal 2 => exact (Algebra.isAlgebraic_of_not_injective
  fun h ↦ hRS <| .of_comp (IsScalarTower.algebraMap_eq R S A ▸ h)).1 _

Result:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
⊢ IsAlgebraic R a

Proof state:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p.coeff p.support, IsIntegral R (r • z)
⊢ IsAlgebraic R a

Tactic:
let p := (r • p).toSubring (integralClosure R S).toSubring fun s hs ↦ by
  obtain ⟨n, hn, rfl⟩ := mem_coeffs_iff.mp hs
  exact int _ (Finset.mem_image_of_mem _ <| support_smul _ _ hn)

Result:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p✝ : S[X]
hp : p✝ ≠ 0
eval0 : (aeval a) p✝ = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p✝.coeff p✝.support, IsIntegral R (r • z)
p : (↥(integralClosure R S).toSubring)[X] := (r • p✝).toSubring (integralClosure R S).toSubring ⋯
⊢ IsAlgebraic R a

Proof state:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
⊢ IsAlgebraic R a

Tactic:
have := hRS.noZeroDivisors _ (map_zero _) (map_mul _)

Result:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
⊢ IsAlgebraic R a

Proof state:
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p.coeff p.support, IsIntegral R (r • z)
s : S
hs : s ∈ ↑(r • p).coeffs
⊢ s ∈ ↑(integralClosure R S).toSubring

Tactic:
obtain ⟨n, hn, rfl⟩ := mem_coeffs_iff.mp hs

Result:
case intro.intro
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p.coeff p.support, IsIntegral R (r • z)
n : ℕ
hn : n ∈ (r • p).support
hs : (r • p).coeff n ∈ ↑(r • p).coeffs
⊢ (r • p).coeff n ∈ ↑(integralClosure R S).toSubring

Proof state:
case neg
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : ¬Function.Injective ⇑(algebraMap R S)
⊢ IsAlgebraic R a

Tactic:
exact (Algebra.isAlgebraic_of_not_injective
fun h ↦ hRS <| .of_comp (IsScalarTower.algebraMap_eq R S A ▸ h)).1 _

Result:
no goals