-- Mathlib/NumberTheory/Cyclotomic/Rat.lean
theorem zeta_sub_one_prime_of_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hodd : p ≠ 2) :
    Prime (hζ.toInteger - 1)

Proof state:
case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger - 1 = 0
⊢ False

Tactic:
apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow₀ hp.out.one_lt (by simp))

Result:
case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger - 1 = 0
⊢ ζ ^ 1 = 1

Proof state:
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Prime (hζ.toInteger - 1)

Tactic:
refine Ideal.prime_of_irreducible_absNorm_span (fun h ↦ ?_) ?_

Result:
case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger - 1 = 0
⊢ False

case refine_2
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))

Proof state:
case h.e'_3.a
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) ↑↑p

Tactic:
simp only [PNat.pow_coe, id.map_eq_id, RingHomCompTriple.comp_eq, RingHom.coe_coe,
  Subalgebra.coe_val, algebraMap_int_eq, map_natCast]

Result:
case h.e'_3.a
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = ↑↑p

Proof state:
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger - 1 = 0
⊢ k + 1 ≠ 0

Tactic:
simp

Result:
no goals

Proof state:
case h.e'_3.a
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = ↑↑p

Tactic:
exact hζ.norm_sub_one_of_prime_ne_two (Polynomial.cyclotomic.irreducible_rat (PNat.pos _)) hodd

Result:
no goals

Proof state:
case refine_2
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))

Tactic:
convert Nat.prime_iff_prime_int.1 hp.out

Result:
case h.e'_3
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑↑p

Proof state:
case refine_2
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Irreducible (Ideal.absNorm (Ideal.span {hζ.toInteger - 1}))

Tactic:
rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
  ← Int.prime_iff_natAbs_prime]

Result:
case refine_2
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))

Proof state:
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
⊢ Prime (hζ.toInteger - 1)

Tactic:
letI := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K

Result:
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ Prime (hζ.toInteger - 1)

Proof state:
case h.e'_3.a
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) ↑↑p

Tactic:
rw [← Algebra.norm_localization (Sₘ := K) ℤ (nonZeroDivisors ℤ)]

Result:
case h.e'_3.a
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) ↑↑p

Proof state:
case h.e'_3
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑↑p

Tactic:
apply RingHom.injective_int (algebraMap ℤ ℚ)

Result:
case h.e'_3.a
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (hζ.toInteger - 1)) = (algebraMap ℤ ℚ) ↑↑p

Proof state:
case refine_1
p : ℕ+
k : ℕ
K : Type u
inst✝² : Field K
ζ : K
hp : Fact (Nat.Prime ↑p)
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hodd : p ≠ 2
this : NumberField K := numberField {p ^ (k + 1)} ℚ K
h : hζ.toInteger = 1
⊢ ζ ^ 1 = 1

Tactic:
simpa using congrArg (algebraMap _ K) h

Result:
no goals