l3lab/ntp-mathlib huggingface dataset:
theorem mul_toSubmodule_le (S T : Subalgebra R A) :
    (Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by
  rw [Submodule.mul_le]
  intro y hy z hz
  show y * z ∈ S ⊔ T
  

Proof state:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S T : Subalgebra R A
y : A
hy : y ∈ toSubmodule S
z : A
hz : z ∈ toSubmodule T
⊢ y * z ∈ S ⊔ T

Next tactic:
exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) := by
  

Proof state:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A
⊢ toSubmodule S * toSubmodule S = toSubmodule S

Next tactic:
apply le_antisymm
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) := by
  apply le_antisymm
  · 

Proof state:
case a
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A
⊢ toSubmodule S * toSubmodule S ≤ toSubmodule S

Next tactic:
refine' (mul_toSubmodule_le _ _).trans_eq _
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) := by
  apply le_antisymm
  · refine' (mul_toSubmodule_le _ _).trans_eq _
    

Proof state:
case a
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A
⊢ toSubmodule (S ⊔ S) = toSubmodule S

Next tactic:
rw [sup_idem]
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) := by
  apply le_antisymm
  · refine' (mul_toSubmodule_le _ _).trans_eq _
    rw [sup_idem]
  · 

Proof state:
case a
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A
⊢ toSubmodule S ≤ toSubmodule S * toSubmodule S

Next tactic:
intro x hx1
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) := by
  apply le_antisymm
  · refine' (mul_toSubmodule_le _ _).trans_eq _
    rw [sup_idem]
  · intro x hx1
    

Proof state:
case a
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A
x : A
hx1 : x ∈ toSubmodule S
⊢ x ∈ toSubmodule S * toSubmodule S

Next tactic:
rw [← mul_one x]
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem mul_self (S : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule S)
    = (Subalgebra.toSubmodule S) := by
  apply le_antisymm
  · refine' (mul_toSubmodule_le _ _).trans_eq _
    rw [sup_idem]
  · intro x hx1
    rw [← mul_one x]
    

Proof state:
case a
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S : Subalgebra R A
x : A
hx1 : x ∈ toSubmodule S
⊢ x * 1 ∈ toSubmodule S * toSubmodule S

Next tactic:
exact Submodule.mul_mem_mul hx1 (show (1 : A) ∈ S from one_mem S)
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  

Proof state:
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
⊢ toSubmodule S * toSubmodule T = toSubmodule (S ⊔ T)

Next tactic:
refine' le_antisymm (mul_toSubmodule_le _ _) _
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  

Proof state:
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
⊢ toSubmodule (S ⊔ T) ≤ toSubmodule S * toSubmodule T

Next tactic:
rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  

Proof state:
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x : A
hx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)
⊢ x ∈ toSubmodule S * toSubmodule T

Next tactic:
refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · 

Proof state:
case refine'_1
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x : A
hx : x ∈ ↑S ∪ ↑T
⊢ x ∈ toSubmodule S * toSubmodule T

Next tactic:
cases' hx with hxS hxT
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · 

Proof state:
case refine'_1.inl
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x : A
hxS : x ∈ ↑S
⊢ x ∈ toSubmodule S * toSubmodule T

Next tactic:
rw [← mul_one x]
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      

Proof state:
case refine'_1.inl
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x : A
hxS : x ∈ ↑S
⊢ x * 1 ∈ toSubmodule S * toSubmodule T

Next tactic:
exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · 

Proof state:
case refine'_1.inr
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x : A
hxT : x ∈ ↑T
⊢ x ∈ toSubmodule S * toSubmodule T

Next tactic:
rw [← one_mul x]
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · rw [← one_mul x]
      

Proof state:
case refine'_1.inr
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x : A
hxT : x ∈ ↑T
⊢ 1 * x ∈ toSubmodule S * toSubmodule T

Next tactic:
exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · rw [← one_mul x]
      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT
  · 

Proof state:
case refine'_2
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x : A
hx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)
r : R
⊢ (algebraMap R A) r ∈ toSubmodule S * toSubmodule T

Next tactic:
rw [← one_mul (algebraMap _ _ _)]
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · rw [← one_mul x]
      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT
  · rw [← one_mul (algebraMap _ _ _)]
    

Proof state:
case refine'_2
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x : A
hx : x ∈ Algebra.adjoin R (↑S ∪ ↑T)
r : R
⊢ 1 * (algebraMap R A) r ∈ toSubmodule S * toSubmodule T

Next tactic:
exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · rw [← one_mul x]
      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT
  · rw [← one_mul (algebraMap _ _ _)]
    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)
  

Proof state:
case refine'_3
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x y : A
hx : x ∈ toSubmodule S * toSubmodule T
hy : y ∈ toSubmodule S * toSubmodule T
⊢ x * y ∈ toSubmodule S * toSubmodule T

Next tactic:
have := Submodule.mul_mem_mul hx hy
/-- When `A` is commutative, `Subalgebra.mul_toSubmodule_le` is strict. -/
theorem mul_toSubmodule {R : Type*} {A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
    (S T : Subalgebra R A) : (Subalgebra.toSubmodule S) * (Subalgebra.toSubmodule T)
        = Subalgebra.toSubmodule (S ⊔ T) := by
  refine' le_antisymm (mul_toSubmodule_le _ _) _
  rintro x (hx : x ∈ Algebra.adjoin R (S ∪ T : Set A))
  refine'
    Algebra.adjoin_induction hx (fun x hx => _) (fun r => _) (fun _ _ => Submodule.add_mem _)
      fun x y hx hy => _
  · cases' hx with hxS hxT
    · rw [← mul_one x]
      exact Submodule.mul_mem_mul hxS (show (1 : A) ∈ T from one_mem T)
    · rw [← one_mul x]
      exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) hxT
  · rw [← one_mul (algebraMap _ _ _)]
    exact Submodule.mul_mem_mul (show (1 : A) ∈ S from one_mem S) (algebraMap_mem T _)
  have := Submodule.mul_mem_mul hx hy
  

Proof state:
case refine'_3
R✝ : Type u_1
A✝ : Type u_2
inst✝⁵ : CommSemiring R✝
inst✝⁴ : Semiring A✝
inst✝³ : Algebra R✝ A✝
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S T : Subalgebra R A
x✝ : A
hx✝ : x✝ ∈ Algebra.adjoin R (↑S ∪ ↑T)
x y : A
hx : x ∈ toSubmodule S * toSubmodule T
hy : y ∈ toSubmodule S * toSubmodule T
this : x * y ∈ toSubmodule S * toSubmodule T * (toSubmodule S * toSubmodule T)
⊢ x * y ∈ toSubmodule S * toSubmodule T

Next tactic:
rwa [mul_assoc, mul_comm _ (Subalgebra.toSubmodule T), ← mul_assoc _ _ (Subalgebra.toSubmodule S),
    mul_self, mul_comm (Subalgebra.toSubmodule T), ← mul_assoc, mul_self] at this
lemma fermatLastTheoremFor_zero : FermatLastTheoremFor 0 :=
  fun _ _ _ _ _ _ ↦ by 

Proof state:
x✝⁵ x✝⁴ x✝³ : ℕ
x✝² : x✝⁵ ≠ 0
x✝¹ : x✝⁴ ≠ 0
x✝ : x✝³ ≠ 0
⊢ x✝⁵ ^ 0 + x✝⁴ ^ 0 ≠ x✝³ ^ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_one : ¬ FermatLastTheoremFor 1 :=
  fun h ↦ h 1 1 2 (by 

Proof state:
h : FermatLastTheoremFor 1
⊢ 1 ≠ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_one : ¬ FermatLastTheoremFor 1 :=
  fun h ↦ h 1 1 2 (by norm_num) (by 

Proof state:
h : FermatLastTheoremFor 1
⊢ 1 ≠ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_one : ¬ FermatLastTheoremFor 1 :=
  fun h ↦ h 1 1 2 (by norm_num) (by norm_num) (by 

Proof state:
h : FermatLastTheoremFor 1
⊢ 2 ≠ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_one : ¬ FermatLastTheoremFor 1 :=
  fun h ↦ h 1 1 2 (by norm_num) (by norm_num) (by norm_num) (by 

Proof state:
h : FermatLastTheoremFor 1
⊢ 1 ^ 1 + 1 ^ 1 = 2 ^ 1

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_two : ¬ FermatLastTheoremFor 2 :=
  fun h ↦ h 3 4 5 (by 

Proof state:
h : FermatLastTheoremFor 2
⊢ 3 ≠ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_two : ¬ FermatLastTheoremFor 2 :=
  fun h ↦ h 3 4 5 (by norm_num) (by 

Proof state:
h : FermatLastTheoremFor 2
⊢ 4 ≠ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_two : ¬ FermatLastTheoremFor 2 :=
  fun h ↦ h 3 4 5 (by norm_num) (by norm_num) (by 

Proof state:
h : FermatLastTheoremFor 2
⊢ 5 ≠ 0

Next tactic:
norm_num
lemma not_fermatLastTheoremFor_two : ¬ FermatLastTheoremFor 2 :=
  fun h ↦ h 3 4 5 (by norm_num) (by norm_num) (by norm_num) (by 

Proof state:
h : FermatLastTheoremFor 2
⊢ 3 ^ 2 + 4 ^ 2 = 5 ^ 2

Next tactic:
norm_num
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n : ℕ
hmn : m ∣ n
hm : FermatLastTheoremWith α m
⊢ FermatLastTheoremWith α n

Next tactic:
rintro a b c ha hb hc
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  rintro a b c ha hb hc
  

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n : ℕ
hmn : m ∣ n
hm : FermatLastTheoremWith α m
a b c : α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
⊢ a ^ n + b ^ n ≠ c ^ n

Next tactic:
obtain ⟨k, rfl⟩ := hmn
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  rintro a b c ha hb hc
  obtain ⟨k, rfl⟩ := hmn
  

Proof state:
case intro
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m : ℕ
hm : FermatLastTheoremWith α m
a b c : α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ a ^ (m * k) + b ^ (m * k) ≠ c ^ (m * k)

Next tactic:
simp_rw [pow_mul']
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  rintro a b c ha hb hc
  obtain ⟨k, rfl⟩ := hmn
  simp_rw [pow_mul']
  

Proof state:
case intro
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m : ℕ
hm : FermatLastTheoremWith α m
a b c : α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ (a ^ k) ^ m + (b ^ k) ^ m ≠ (c ^ k) ^ m

Next tactic:
refine hm _ _ _ ?_ ?_ ?_
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  rintro a b c ha hb hc
  obtain ⟨k, rfl⟩ := hmn
  simp_rw [pow_mul']
  refine hm _ _ _ ?_ ?_ ?_ <;> 

Proof state:
case intro.refine_1
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m : ℕ
hm : FermatLastTheoremWith α m
a b c : α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ a ^ k ≠ 0

Next tactic:
exact pow_ne_zero _ ‹_›
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  rintro a b c ha hb hc
  obtain ⟨k, rfl⟩ := hmn
  simp_rw [pow_mul']
  refine hm _ _ _ ?_ ?_ ?_ <;> 

Proof state:
case intro.refine_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m : ℕ
hm : FermatLastTheoremWith α m
a b c : α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ b ^ k ≠ 0

Next tactic:
exact pow_ne_zero _ ‹_›
lemma FermatLastTheoremWith.mono (hmn : m ∣ n) (hm : FermatLastTheoremWith α m) :
    FermatLastTheoremWith α n := by
  rintro a b c ha hb hc
  obtain ⟨k, rfl⟩ := hmn
  simp_rw [pow_mul']
  refine hm _ _ _ ?_ ?_ ?_ <;> 

Proof state:
case intro.refine_3
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m : ℕ
hm : FermatLastTheoremWith α m
a b c : α
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
k : ℕ
⊢ c ^ k ≠ 0

Next tactic:
exact pow_ne_zero _ ‹_›
lemma FermatLastTheoremFor.mono (hmn : m ∣ n) (hm : FermatLastTheoremFor m) :
    FermatLastTheoremFor n := by
  

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n : ℕ
hmn : m ∣ n
hm : FermatLastTheoremFor m
⊢ FermatLastTheoremFor n

Next tactic:
exact FermatLastTheoremWith.mono hmn hm
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
⊢ TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n]

Next tactic:
tfae_have 1 → 2
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · 

Proof state:
case tfae_1_to_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
⊢ FermatLastTheoremWith ℕ n → FermatLastTheoremWith ℤ n

Next tactic:
rintro h a b c ha hb hc habc
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    

Proof state:
case tfae_1_to_2
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
⊢ False

Next tactic:
obtain hn | hn := n.even_or_odd
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · 

Proof state:
case tfae_1_to_2.inl
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ False

Next tactic:
refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
        (Int.coe_nat_inj'.1 _)
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ Int.natAbs a ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ Int.natAbs b ≠ 0

Next tactic:
positivity
lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
    TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
  tfae_have 1 → 2
  · rintro h a b c ha hb hc habc
    obtain hn | hn := n.even_or_odd
    · refine' h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by 

Proof state:
α : Type u_1
inst✝¹ : Semiring α
inst✝ : NoZeroDivisors α
m n✝ n : ℕ
h : FermatLastTheoremWith ℕ n
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
hn : Even n
⊢ Int.natAbs c ≠ 0

Next tactic:
positivity