Theorem: norm_Int_zeta_sub_one (FltRegular/NumberTheory/Cyclotomic/MoreLemmas.lean)

State before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
⊢ (Algebra.norm ℤ) (↑hζ.unit' - 1) = ↑↑p
Tactic:
letI := IsCyclotomicExtension.numberField {p} ℚ K
State after:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
this : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
⊢ (Algebra.norm ℤ) (↑hζ.unit' - 1) = ↑↑p

State before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
this : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
⊢ (Algebra.norm ℤ) (↑hζ.unit' - 1) = ↑↑p
Tactic:
haveI : Fact (Nat.Prime p) := hpri
State after:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
this✝ : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
this : Fact (↑p).Prime
⊢ (Algebra.norm ℤ) (↑hζ.unit' - 1) = ↑↑p

State before:
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
this✝ : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
this : Fact (↑p).Prime
⊢ (Algebra.norm ℤ) (↑hζ.unit' - 1) = ↑↑p
Tactic:
apply RingHom.injective_int (algebraMap ℤ ℚ)
State after:
case a
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
this✝ : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
this : Fact (↑p).Prime
⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (↑hζ.unit' - 1)) = (algebraMap ℤ ℚ) ↑↑p

State before:
case a
K : Type u_1
p : ℕ+
hpri : Fact p.Prime
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
hp : p ≠ 2
this✝ : NumberField K := IsCyclotomicExtension.numberField {p} ℚ K
this : Fact (↑p).Prime
⊢ (algebraMap ℤ ℚ) ((Algebra.norm ℤ) (↑hζ.unit' - 1)) = (algebraMap ℤ ℚ) ↑↑p
Tactic:
simp [Algebra.coe_norm_int, hζ.norm_sub_one_of_prime_ne_two' (cyclotomic.irreducible_rat p.2) hp]
State after:
no goals