awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this : NeZero n₁
⊢ NeZero n₂

Tactic:
have : NeZero n₂ := NeZero.of_gt hn₂
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
⊢ IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)

Tactic:
have hζ := zeta_spec (n₁ * n₂) ℚ K
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
rw rw₁ : n₂ * n₁ = n₂ * n₁
⊢ n₁ * n₂ = n₂ * n₁

Tactic:
rw [mul_comm]

Premises used: mul_comm
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
⊢ IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁

Tactic:
have hζ₁ := hζ.pow (NeZero.pos _) (a := n₂) (b := n₁) (by rw [mul_comm])
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
⊢ IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯

Tactic:
have := hζ₁.intermediateField_adjoin_isCyclotomicExtension ℚ
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
⊢ IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁

Tactic:
have hζ₁' : IsPrimitiveRoot (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁ :=
      IsPrimitiveRoot.coe_submonoidClass_iff.mp hζ₁
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₁ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₁] [hK : IsCyclotomicExtension {n₁} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
⊢ (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))

Tactic:
replace hK₁ := @hK₁ ℚ⟮ζ ^ n₂⟯ _ _ _ _ (of_intermediateField _)
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
⊢ IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂

Tactic:
have hζ₂ := hζ.pow (NeZero.pos _) (a := n₁) (b := n₂) rfl
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
⊢ IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯

Tactic:
have := hζ₂.intermediateField_adjoin_isCyclotomicExtension ℚ
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
⊢ IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂

Tactic:
have hζ₂' : IsPrimitiveRoot (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂ :=
      IsPrimitiveRoot.coe_submonoidClass_iff.mp hζ₂
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
hK₂ :
  ∀ (K : Type u) [inst : Field K] [inst_2 : CharZero K] [hn : NeZero n₂] [hK : IsCyclotomicExtension {n₂} ℚ K]
    (this : NumberField K), (NumberField.discr K).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
⊢ (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))

Tactic:
replace hK₂ := @hK₂ ℚ⟮ζ ^ n₁⟯ _ _ _ _ (of_intermediateField _)
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
⊢ IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯

Tactic:
have : IsGalois ℚ ℚ⟮ζ ^ n₂⟯ := isGalois {n₁} ℚ _
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
⊢ IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤

Tactic:
have : IsCyclotomicExtension {n₁ * n₂} ℚ (⊤ : IntermediateField ℚ K) :=
          hK.equiv _ _ _ topEquiv.symm
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this_6 : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
this : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤
rw : IsCyclotomicExtension.{0, u} {n₁.lcm n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)
⊢ IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)

Tactic:
rw [← Nat.Coprime.lcm_eq_mul h]

Premises used: Nat.Coprime.lcm_eq_mul
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this_6 : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
this : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤
⊢ IsCyclotomicExtension.{0, u} {n₁.lcm n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)

Tactic:
isCyclotomicExtension_lcm_sup ℚ K n₁ n₂ ℚ⟮ζ ^ n₂⟯ ℚ⟮ζ ^ n₁⟯

Premises used: Set.Mem, Rat, IntermediateField.isCyclotomicExtension_lcm_sup, HPow.hPow, IntermediateField.adjoin, RingHom.codRestrict, Rat.castHom, Singleton.singleton
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this_6 : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
this : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤
⊢ IsCyclotomicExtension.{0, u} {n₁.lcm n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)

Tactic:
exact isCyclotomicExtension_lcm_sup ℚ K n₁ n₂ ℚ⟮ζ ^ n₂⟯ ℚ⟮ζ ^ n₁⟯

Premises used: Rat, IntermediateField.isCyclotomicExtension_lcm_sup, HPow.hPow, IntermediateField.adjoin, Singleton.singleton
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this_6 : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
this : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤
⊢ IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)

Tactic:
have : IsCyclotomicExtension {n₁ * n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯) := by
        rw [← Nat.Coprime.lcm_eq_mul h]
        exact isCyclotomicExtension_lcm_sup ℚ K n₁ n₂ ℚ⟮ζ ^ n₂⟯ ℚ⟮ζ ^ n₁⟯
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this_6 : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
this_7 : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤
this : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)
⊢ Eq (α := IntermediateField ℚ K) (ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯) ⊤

Tactic:
isCyclotomicExtension_eq {n₁ * n₂} ℚ K _ _

Premises used: HMul.hMul, Rat, IntermediateField.isCyclotomicExtension_eq, Singleton.singleton
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this_6 : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
this_7 : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥⊤
this : IsCyclotomicExtension.{0, u} {n₁ * n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯)
⊢ Eq (α := IntermediateField ℚ K) (ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯) ⊤

Tactic:
exact isCyclotomicExtension_eq {n₁ * n₂} ℚ K _ _

Premises used: HMul.hMul, Rat, IntermediateField.isCyclotomicExtension_eq, Singleton.singleton
-- Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
Goal:
n₁ n₂ : ℕ
hn₁ : 1 < n₁
hn₂ : 1 < n₂
h : n₁.Coprime n₂
K : Type u
inst : Field K
inst_1 : CharZero K
hn : NeZero (n₁ * n₂)
hK : IsCyclotomicExtension {n₁ * n₂} ℚ K
this_1 : NumberField K
this_2 : NeZero n₁
this_3 : NeZero n₂
ζ : K := zeta (n₁ * n₂) ℚ K
hζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n₂)
hζ₁ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₂) n₁
this_4 : IsCyclotomicExtension.{0, u} {n₁} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₂⟯
hζ₁' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₂⟯) (AdjoinSimple.gen ℚ (ζ ^ n₂)) n₁
hK₁ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₂⟯).natAbs = n₁ ^ φ n₁ / ∏ p ∈ n₁.primeFactors, p ^ (φ n₁ / (p - 1))
hζ₂ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K ^ n₁) n₂
this_5 : IsCyclotomicExtension.{0, u} {n₂} ℚ ↥ℚ⟮zeta (n₁ * n₂) ℚ K ^ n₁⟯
hζ₂' : IsPrimitiveRoot (M := ↥ℚ⟮ζ ^ n₁⟯) (AdjoinSimple.gen ℚ (ζ ^ n₁)) n₂
hK₂ : (NumberField.discr.{u} ↥ℚ⟮ζ ^ n₁⟯).natAbs = n₂ ^ φ n₂ / ∏ p ∈ n₂.primeFactors, p ^ (φ n₂ / (p - 1))
this : IsGalois.{0, u} ℚ ↥ℚ⟮ζ ^ n₂⟯
⊢ Eq (α := IntermediateField ℚ K) (ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯) ⊤

Tactic:
have h_top : ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯ = ⊤ := by
      have : IsCyclotomicExtension {n₁ * n₂} ℚ (⊤ : IntermediateField ℚ K) :=
          hK.equiv _ _ _ topEquiv.symm
      have : IsCyclotomicExtension {n₁ * n₂} ℚ ↥(ℚ⟮ζ ^ n₂⟯ ⊔ ℚ⟮ζ ^ n₁⟯) := by
        rw [← Nat.Coprime.lcm_eq_mul h]
        exact isCyclotomicExtension_lcm_sup ℚ K n₁ n₂ ℚ⟮ζ ^ n₂⟯ ℚ⟮ζ ^ n₁⟯
      exact isCyclotomicExtension_eq {n₁ * n₂} ℚ K _ _