l3lab/ntp-mathlib huggingface dataset:
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
⊢ adjoin S ↑(adjoin R s) = adjoin S s

Next tactic:
apply le_antisymm (adjoin_le _)
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  apply le_antisymm (adjoin_le _)
  · 

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
⊢ adjoin S s ≤ adjoin S ↑(adjoin R s)

Next tactic:
exact adjoin_mono subset_adjoin
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  apply le_antisymm (adjoin_le _)
  · exact adjoin_mono subset_adjoin
  · 

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
⊢ ↑(adjoin R s) ⊆ ↑(adjoin S s)

Next tactic:
change adjoin R s ≤ (adjoin S s).restrictScalars R
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  apply le_antisymm (adjoin_le _)
  · exact adjoin_mono subset_adjoin
  · change adjoin R s ≤ (adjoin S s).restrictScalars R
    

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
⊢ adjoin R s ≤ Subalgebra.restrictScalars R (adjoin S s)

Next tactic:
refine' adjoin_le _
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  apply le_antisymm (adjoin_le _)
  · exact adjoin_mono subset_adjoin
  · change adjoin R s ≤ (adjoin S s).restrictScalars R
    refine' adjoin_le _
    -- porting note: unclear why this was broken
    

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
⊢ s ⊆ ↑(Subalgebra.restrictScalars R (adjoin S s))

Next tactic:
have : (Subalgebra.restrictScalars R (adjoin S s) : Set A) = adjoin S s := rfl
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  apply le_antisymm (adjoin_le _)
  · exact adjoin_mono subset_adjoin
  · change adjoin R s ≤ (adjoin S s).restrictScalars R
    refine' adjoin_le _
    -- porting note: unclear why this was broken
    have : (Subalgebra.restrictScalars R (adjoin S s) : Set A) = adjoin S s := rfl
    

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
this : ↑(Subalgebra.restrictScalars R (adjoin S s)) = ↑(adjoin S s)
⊢ s ⊆ ↑(Subalgebra.restrictScalars R (adjoin S s))

Next tactic:
rw [this]
theorem adjoin_adjoin_of_tower (s : Set A) : adjoin S (adjoin R s : Set A) = adjoin S s := by
  apply le_antisymm (adjoin_le _)
  · exact adjoin_mono subset_adjoin
  · change adjoin R s ≤ (adjoin S s).restrictScalars R
    refine' adjoin_le _
    -- porting note: unclear why this was broken
    have : (Subalgebra.restrictScalars R (adjoin S s) : Set A) = adjoin S s := rfl
    rw [this]
    

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
this : ↑(Subalgebra.restrictScalars R (adjoin S s)) = ↑(adjoin S s)
⊢ s ⊆ ↑(adjoin S s)

Next tactic:
exact subset_adjoin
theorem adjoin_union_eq_adjoin_adjoin :
    adjoin R (s ∪ t) = (adjoin (adjoin R s) t).restrictScalars R := by
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s t : Set A
⊢ adjoin R (s ∪ t) = Subalgebra.restrictScalars R (adjoin (↥(adjoin R s)) t)

Next tactic:
simpa using adjoin_algebraMap_image_union_eq_adjoin_adjoin R s t
theorem adjoin_union_coe_submodule :
    Subalgebra.toSubmodule (adjoin R (s ∪ t)) =
      Subalgebra.toSubmodule (adjoin R s) * Subalgebra.toSubmodule (adjoin R t) := by
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s t : Set A
⊢ Subalgebra.toSubmodule (adjoin R (s ∪ t)) = Subalgebra.toSubmodule (adjoin R s) * Subalgebra.toSubmodule (adjoin R t)

Next tactic:
rw [adjoin_eq_span, adjoin_eq_span, adjoin_eq_span, span_mul_span]
theorem adjoin_union_coe_submodule :
    Subalgebra.toSubmodule (adjoin R (s ∪ t)) =
      Subalgebra.toSubmodule (adjoin R s) * Subalgebra.toSubmodule (adjoin R t) := by
  rw [adjoin_eq_span, adjoin_eq_span, adjoin_eq_span, span_mul_span]
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s t : Set A
⊢ span R ↑(Submonoid.closure (s ∪ t)) = span R (↑(Submonoid.closure s) * ↑(Submonoid.closure t))

Next tactic:
congr 1 with z
theorem adjoin_union_coe_submodule :
    Subalgebra.toSubmodule (adjoin R (s ∪ t)) =
      Subalgebra.toSubmodule (adjoin R s) * Subalgebra.toSubmodule (adjoin R t) := by
  rw [adjoin_eq_span, adjoin_eq_span, adjoin_eq_span, span_mul_span]
  congr 1 with z; 

Proof state:
case e_s.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s t : Set A
z : A
⊢ z ∈ ↑(Submonoid.closure (s ∪ t)) ↔ z ∈ ↑(Submonoid.closure s) * ↑(Submonoid.closure t)

Next tactic:
simp [Submonoid.closure_union, Submonoid.mem_sup, Set.mem_mul]
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' := by
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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'
x : B
hx : x ∈ adjoin R s
hr : (algebraMap A B) r ∈ B'
⊢ ∃ n₀, ∀ n ≥ n₀, r ^ n • x ∈ B'

Next tactic:
change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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'
x : B
hr : (algebraMap A B) r ∈ B'
hx : x ∈ Subalgebra.toSubmodule (adjoin R s)
⊢ ∃ n₀, ∀ n ≥ n₀, r ^ n • x ∈ B'

Next tactic:
rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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'
x : B
hr : (algebraMap A B) r ∈ B'
hx : ∃ l, (Finsupp.total (↑↑(Submonoid.closure s)) B R Subtype.val) l = x
⊢ ∃ n₀, ∀ n ≥ n₀, r ^ n • x ∈ B'

Next tactic:
rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  

Proof state:
case intro
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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
⊢ ∃ n₀, ∀ n ≥ n₀, (r ^ n • Finsupp.sum l fun i c => c • ↑i) ∈ B'

Next tactic:
choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  

Proof state:
case intro
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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)
⊢ ∃ n₀, ∀ n ≥ n₀, (r ^ n • Finsupp.sum l fun i c => c • ↑i) ∈ B'

Next tactic:
use l.support.sup n₁
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁, (r ^ n • Finsupp.sum l fun i c => c • ↑i) ∈ B'

Next tactic:
intro n hn
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁
⊢ (r ^ n • Finsupp.sum l fun i c => c • ↑i) ∈ B'

Next tactic:
rw [Finsupp.smul_sum]
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁
⊢ (Finsupp.sum l fun a b => r ^ n • b • ↑a) ∈ B'

Next tactic:
refine' B'.toSubmodule.sum_mem _
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  refine' B'.toSubmodule.sum_mem _
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁
⊢ ∀ c ∈ l.support, (fun a b => r ^ n • b • ↑a) c (l c) ∈ Subalgebra.toSubmodule B'

Next tactic:
intro a ha
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  refine' B'.toSubmodule.sum_mem _
  intro a ha
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
⊢ (fun a b => r ^ n • b • ↑a) a (l a) ∈ Subalgebra.toSubmodule B'

Next tactic:
have : n ≥ n₁ a := le_trans (Finset.le_sup ha) hn
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  refine' B'.toSubmodule.sum_mem _
  intro a ha
  have : n ≥ n₁ a := le_trans (Finset.le_sup ha) hn
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support 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'

Next tactic:
dsimp only
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  refine' B'.toSubmodule.sum_mem _
  intro a ha
  have : n ≥ n₁ a := le_trans (Finset.le_sup ha) hn
  dsimp only
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ r ^ n • l a • ↑a ∈ Subalgebra.toSubmodule B'

Next 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]
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  refine' B'.toSubmodule.sum_mem _
  intro a ha
  have : n ≥ n₁ a := le_trans (Finset.le_sup ha) hn
  dsimp only
  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]
  

Proof state:
case h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support 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'

Next tactic:
apply Subalgebra.mul_mem _ (Subalgebra.pow_mem _ hr _) _
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' := by
  change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
  rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
  choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
  use l.support.sup n₁
  intro n hn
  rw [Finsupp.smul_sum]
  refine' B'.toSubmodule.sum_mem _
  intro a ha
  have : n ≥ n₁ a := le_trans (Finset.le_sup ha) hn
  dsimp only
  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]
  apply Subalgebra.mul_mem _ (Subalgebra.pow_mem _ hr _) _
  

Proof state:
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring A
inst✝⁴ : Algebra R A
s✝ t : Set 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 ≥ Finset.sup l.support n₁
a : ↑↑(Submonoid.closure s)
ha : a ∈ l.support
this : n ≥ n₁ a
⊢ l a • r ^ n₁ a • ↑a ∈ B'

Next tactic:
refine' Subalgebra.smul_mem _ _ _