-- Mathlib/RingTheory/Adjoin/Basic.lean
theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B] [Algebra A B]
    [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
    (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B'

Proof state:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ (l.sum fun a b => r ^ n • b • ↑a) ∈ B'

Tactic:
refine B'.toSubmodule.sum_mem ?_

Result:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ ∀ c ∈ l.support, (fun a b => r ^ n • b • ↑a) c (l c) ∈ Subalgebra.toSubmodule B'

Proof state:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ (fun a b => r ^ n • b • ↑a) a (l a) ∈ Subalgebra.toSubmodule B'

Tactic:
dsimp only

Result:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ r ^ n • l a • ↑a ∈ Subalgebra.toSubmodule B'

Proof state:
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ r ^ n₁ a • ↑a ∈ B'.toSubmonoid

Tactic:
rw [← Submonoid.closure_eq B'.toSubmonoid]

Result:
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ r ^ n₁ a • ↑a ∈ Submonoid.closure ↑B'.toSubmonoid

Proof state:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ (r ^ n • l.sum fun i c => c • ↑i) ∈ B'

Tactic:
rw [Finsupp.smul_sum]

Result:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ (l.sum fun a b => r ^ n • b • ↑a) ∈ B'

Proof state:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ r ^ n • l a • ↑a ∈ Subalgebra.toSubmodule B'

Tactic:
rw [← tsub_add_cancel_of_le this, pow_add, ← smul_smul, ←
  IsScalarTower.algebraMap_smul A (l a) (a : B), smul_smul (r ^ n₁ a), mul_comm, ← smul_smul,
  smul_def, map_pow, IsScalarTower.algebraMap_smul]

Result:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ (algebraMap A B) r ^ (n - n₁ a) * l a • r ^ n₁ a • ↑a ∈ Subalgebra.toSubmodule B'

Proof state:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
⊢ ∀ c ∈ l.support, (fun a b => r ^ n • b • ↑a) c (l c) ∈ Subalgebra.toSubmodule B'

Tactic:
intro a ha

Result:
case h
R : Type uR
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
inst✝³ : CommSemiring B
inst✝² : Algebra R B
inst✝¹ : Algebra A B
inst✝ : IsScalarTower R A B
r : A
s : Set B
B' : Subalgebra R B
hs : r • s ⊆ ↑B'
hr : (algebraMap A B) r ∈ B'
l : ↑↑(Submonoid.closure s) →₀ R
n₁ : ↥(Submonoid.closure s) → ℕ
n₂ : ∀ (x : ↥(Submonoid.closure s)), r ^ n₁ x • ↑x ∈ Submonoid.closure (r • s)
n : ℕ
hn : n ≥ l.support.sup n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
⊢ (fun a b => r ^ n • b • ↑a) a (l a) ∈ Subalgebra.toSubmodule B'