-- Mathlib/FieldTheory/Finite/Basic.lean
theorem forall_pow_eq_one_iff (i : ℕ) : (∀ x : Kˣ, x ^ i = 1) ↔ q - 1 ∣ i

Proof state:
case intro.mpr.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
h : x ^ i = 1
hx : ∀ (x_1 : Kˣ), x_1 ∈ Submonoid.powers x
j : ℕ
⊢ (fun x_1 => x ^ x_1) j ^ i = 1

Tactic:
rw [← pow_mul, mul_comm, pow_mul, h, one_pow]

Result:
no goals

Proof state:
case intro
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ (∀ (x : Kˣ), x ^ i = 1) ↔ q - 1 ∣ i

Tactic:
rw [← Nat.card_eq_fintype_card, ← Nat.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hx,
  orderOf_dvd_iff_pow_eq_one]

Result:
case intro
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ (∀ (x : Kˣ), x ^ i = 1) ↔ x ^ i = 1

Proof state:
case intro.mpr
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ x ^ i = 1 → ∀ (x : Kˣ), x ^ i = 1

Tactic:
intro h y

Result:
case intro.mpr
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
h : x ^ i = 1
y : Kˣ
⊢ y ^ i = 1

Proof state:
case intro.mpr
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
h : x ^ i = 1
y : Kˣ
⊢ y ^ i = 1

Tactic:
simp_rw [← mem_powers_iff_mem_zpowers] at hx

Result:
case intro.mpr
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
h : x ^ i = 1
y : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Submonoid.powers x
⊢ y ^ i = 1

Proof state:
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
⊢ (∀ (x : Kˣ), x ^ i = 1) ↔ q - 1 ∣ i

Tactic:
obtain ⟨x, hx⟩ := IsCyclic.exists_generator (α := Kˣ)

Result:
case intro
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ (∀ (x : Kˣ), x ^ i = 1) ↔ q - 1 ∣ i

Proof state:
case intro.mpr
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
h : x ^ i = 1
y : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Submonoid.powers x
⊢ y ^ i = 1

Tactic:
rcases hx y with ⟨j, rfl⟩

Result:
case intro.mpr.intro
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
h : x ^ i = 1
hx : ∀ (x_1 : Kˣ), x_1 ∈ Submonoid.powers x
j : ℕ
⊢ (fun x_1 => x ^ x_1) j ^ i = 1

Proof state:
case intro.mp
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
h : ∀ (x : Kˣ), x ^ i = 1
⊢ x ^ i = 1

Tactic:
apply h

Result:
no goals

Proof state:
case intro
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ (∀ (x : Kˣ), x ^ i = 1) ↔ x ^ i = 1

Tactic:
constructor

Result:
case intro.mp
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ (∀ (x : Kˣ), x ^ i = 1) → x ^ i = 1

case intro.mpr
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ x ^ i = 1 → ∀ (x : Kˣ), x ^ i = 1

Proof state:
case intro.mp
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
⊢ (∀ (x : Kˣ), x ^ i = 1) → x ^ i = 1

Tactic:
intro h

Result:
case intro.mp
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
i : ℕ
x : Kˣ
hx : ∀ (x_1 : Kˣ), x_1 ∈ Subgroup.zpowers x
h : ∀ (x : Kˣ), x ^ i = 1
⊢ x ^ i = 1