-- Mathlib/FieldTheory/Finite/Basic.lean
theorem card_cast_subgroup_card_ne_zero [Ring K] [NoZeroDivisors K] [Nontrivial K]
    (G : Subgroup Kˣ) [Fintype G] : (Fintype.card G : K) ≠ 0

Proof state:
case inl
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
nzero : ↑(Fintype.card ↥G) = 0
p : ℕ
char_p : CharP K p
hd : p ∣ n
pprime : Nat.Prime p
fact_pprime : Fact (Nat.Prime p)
x : ↥G
hx : orderOf x = p
u : K := ↑↑x
hu : orderOf u = p
h : u = 1
⊢ False

Tactic:
apply pprime.one_lt.ne

Result:
case inl
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
nzero : ↑(Fintype.card ↥G) = 0
p : ℕ
char_p : CharP K p
hd : p ∣ n
pprime : Nat.Prime p
fact_pprime : Fact (Nat.Prime p)
x : ↥G
hx : orderOf x = p
u : K := ↑↑x
hu : orderOf u = p
h : u = 1
⊢ 1 = p

Proof state:
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
nzero : ↑(Fintype.card ↥G) = 0
p : ℕ
char_p : CharP K p
hd : p ∣ n
⊢ False

Tactic:
cases CharP.char_is_prime_or_zero K p with
| inr pzero =>
  exact (Fintype.card_pos).ne' <| Nat.eq_zero_of_zero_dvd <| pzero ▸ hd
| inl pprime =>
  have fact_pprime := Fact.mk pprime
  have ⟨x, hx⟩ := exists_prime_orderOf_dvd_card p hd
  let u := ((x : Kˣ) : K)
  have hu : orderOf u = p := by rwa [orderOf_units, Subgroup.orderOf_coe]
  have h : u = 1 := by
    rw [← sub_left_inj, sub_self 1]
    apply pow_eq_zero (n := p)
    rw [sub_pow_char_of_commute, one_pow, ← hu, pow_orderOf_eq_one, sub_self]
    exact Commute.one_right u
  apply pprime.one_lt.ne
  rw [← hu, h, orderOf_one]

Result:
no goals

Proof state:
case h
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
nzero : ↑(Fintype.card ↥G) = 0
p : ℕ
char_p : CharP K p
hd : p ∣ n
pprime : Nat.Prime p
fact_pprime : Fact (Nat.Prime p)
x : ↥G
hx : orderOf x = p
u : K := ↑↑x
hu : orderOf u = p
⊢ Commute u 1

Tactic:
exact Commute.one_right u

Result:
no goals

Proof state:
case inl
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
nzero : ↑(Fintype.card ↥G) = 0
p : ℕ
char_p : CharP K p
hd : p ∣ n
pprime : Nat.Prime p
fact_pprime : Fact (Nat.Prime p)
x : ↥G
hx : orderOf x = p
u : K := ↑↑x
hu : orderOf u = p
h : u = 1
⊢ 1 = p

Tactic:
rw [← hu, h, orderOf_one]

Result:
no goals

Proof state:
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
⊢ ↑(Fintype.card ↥G) ≠ 0

Tactic:
let n := Fintype.card G

Result:
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
⊢ ↑(Fintype.card ↥G) ≠ 0

Proof state:
K : Type u_1
inst✝³ : Ring K
inst✝² : NoZeroDivisors K
inst✝¹ : Nontrivial K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
n : ℕ := Fintype.card ↥G
nzero : ↑(Fintype.card ↥G) = 0
p : ℕ
char_p : CharP K p
hd : p ∣ n
pprime : Nat.Prime p
fact_pprime : Fact (Nat.Prime p)
x : ↥G
hx : orderOf x = p
u : K := ↑↑x
⊢ orderOf u = p

Tactic:
rwa [orderOf_units, Subgroup.orderOf_coe]

Result:
no goals