-- Mathlib/RingTheory/RingHom/FiniteType.lean
theorem finiteType_isStableUnderBaseChange : IsStableUnderBaseChange @FiniteType

Proof state:
case h₁

⊢ RespectsIso @FiniteType

Tactic:
exact finiteType_respectsIso

Result:
no goals

Proof state:
case h₂

⊢ ∀ ⦃R S T : Type u_1⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
    [inst_4 : Algebra R T], (algebraMap R T).FiniteType → Algebra.TensorProduct.includeLeftRingHom.FiniteType

Tactic:
introv h

Result:
case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : (algebraMap R T).FiniteType
⊢ Algebra.TensorProduct.includeLeftRingHom.FiniteType

Proof state:
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
⊢ Algebra.TensorProduct.includeLeftRingHom.FiniteType

Tactic:
rw [RingHom.FiniteType]

Result:
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
⊢ Algebra.FiniteType S (S ⊗[R] T)

Proof state:
case h.e'_5.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
r✝ : R
x✝ : T
⊢ (algebraMap R T) r✝ * x✝ = (algebraMap R T) r✝ * x✝

Tactic:
rfl

Result:
no goals

Proof state:
case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ Algebra.FiniteType S (S ⊗[R] T)

Tactic:
infer_instance

Result:
no goals

Proof state:
case h.e'_5
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ inst✝ = (algebraMap R T).toAlgebra

Tactic:
ext

Result:
case h.e'_5.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
r✝ : R
x✝ : T
⊢ (let_fun I := inst✝;
    r✝ • x✝) =
    r✝ • x✝

Proof state:
case h.e'_5.h.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
r✝ : S
x✝ : S ⊗[R] T
⊢ (let_fun I := Algebra.TensorProduct.includeLeftRingHom.toAlgebra;
    r✝ • x✝) =
    r✝ • x✝

Tactic:
simp_rw [Algebra.smul_def]

Result:
case h.e'_5.h.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
r✝ : S
x✝ : S ⊗[R] T
⊢ (algebraMap S (S ⊗[R] T)) r✝ * x✝ = (algebraMap S (S ⊗[R] T)) r✝ * x✝

Proof state:
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : (algebraMap R T).FiniteType
⊢ Algebra.FiniteType R T

Tactic:
rw [RingHom.FiniteType] at h

Result:
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ Algebra.FiniteType R T

Proof state:
⊢ IsStableUnderBaseChange @FiniteType

Tactic:
apply IsStableUnderBaseChange.mk

Result:
case h₁

⊢ RespectsIso @FiniteType

case h₂

⊢ ∀ ⦃R S T : Type u_1⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
    [inst_4 : Algebra R T], (algebraMap R T).FiniteType → Algebra.TensorProduct.includeLeftRingHom.FiniteType

Proof state:
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
⊢ Algebra.FiniteType S (S ⊗[R] T)

Tactic:
convert this

Result:
case h.e'_5.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
⊢ Algebra.TensorProduct.includeLeftRingHom.toAlgebra = Algebra.TensorProduct.leftAlgebra

Proof state:
case h.e'_5.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
r✝ : R
x✝ : T
⊢ (let_fun I := inst✝;
    r✝ • x✝) =
    r✝ • x✝

Tactic:
simp_rw [Algebra.smul_def]

Result:
case h.e'_5.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
r✝ : R
x✝ : T
⊢ (algebraMap R T) r✝ * x✝ = (algebraMap R T) r✝ * x✝

Proof state:
case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : (algebraMap R T).FiniteType
⊢ Algebra.TensorProduct.includeLeftRingHom.FiniteType

Tactic:
replace h : Algebra.FiniteType R T := by
  rw [RingHom.FiniteType] at h; convert h; ext; simp_rw [Algebra.smul_def]; rfl

Result:
case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ Algebra.TensorProduct.includeLeftRingHom.FiniteType

Proof state:
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ Algebra.FiniteType R T

Tactic:
convert h

Result:
case h.e'_5
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ inst✝ = (algebraMap R T).toAlgebra

Proof state:
case h.e'_5.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
⊢ Algebra.TensorProduct.includeLeftRingHom.toAlgebra = Algebra.TensorProduct.leftAlgebra

Tactic:
ext

Result:
case h.e'_5.h.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
r✝ : S
x✝ : S ⊗[R] T
⊢ (let_fun I := Algebra.TensorProduct.includeLeftRingHom.toAlgebra;
    r✝ • x✝) =
    r✝ • x✝

Proof state:
case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ Algebra.TensorProduct.includeLeftRingHom.FiniteType

Tactic:
suffices Algebra.FiniteType S (S ⊗[R] T) by
  rw [RingHom.FiniteType]; convert this; ext; simp_rw [Algebra.smul_def]; rfl

Result:
case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
⊢ Algebra.FiniteType S (S ⊗[R] T)

Proof state:
case h.e'_5.h.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FiniteType R T
this : Algebra.FiniteType S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
r✝ : S
x✝ : S ⊗[R] T
⊢ (algebraMap S (S ⊗[R] T)) r✝ * x✝ = (algebraMap S (S ⊗[R] T)) r✝ * x✝

Tactic:
rfl

Result:
no goals