simp_rw [LinearMap.smul_def, Polynomial.cyclotomic_prime ℤ p, AddEquivClass.map_eq_zero_iff,
  map_sum, map_pow, aeval_X, LinearMap.coeFn_sum, sum_apply, ← relativeUnitsMapHom_apply,
  ← map_pow, ← Units.val_pow_eq_pow_val, ← map_pow, AlgEquiv.val_algHomUnitsEquiv_symm_apply,
  relativeUnitsMapHom_apply, Monoid.EndAdditive_apply,
  addMonoidEndRingEquivInt_apply, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
  LinearEquiv.coe_coe, addMonoidHomLequivInt_apply, AddMonoidHom.coe_toIntLinearMap,
  AddMonoidHom.coe_mk, ZeroHom.coe_mk, toMul_ofMul, relativeUnitsMap_mk]
case intro.intro.intro
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
this : Fact (↑p).Prime
x : (𝓞 K)ˣ
⊢ (Module.AEval.of ℤ (Additive (RelativeUnits k K))
        ((addMonoidEndRingEquivInt (Additive (RelativeUnits k K)))
          (Monoid.EndAdditive (relativeUnitsMap ↑((AlgEquiv.algHomUnitsEquiv k K).symm σ)))))
      ((aeval
            ((addMonoidEndRingEquivInt (Additive (RelativeUnits k K)))
              (Monoid.EndAdditive (relativeUnitsMap ↑((AlgEquiv.algHomUnitsEquiv k K).symm σ)))))
          (cyclotomic ↑p ℤ) •
        Additive.ofMul ↑x) =
    0
case intro.intro.intro
p : ℕ+
K : Type u_1
inst✝¹³ : Field K
inst✝¹² : NumberField K
k : Type u_2
inst✝¹¹ : Field k
inst✝¹⁰ : NumberField k
hp : (↑p).Prime
G : Type u_3
H : Type u_4
inst✝⁹ : AddCommGroup G
inst✝⁸ : CommGroup H
inst✝⁷ : Fintype H
hCard : Fintype.card H = ↑p
σ✝ : H
hσ✝ : Subgroup.zpowers σ✝ = ⊤
r : ℕ
inst✝⁶ : DistribMulAction H G
inst✝⁵ : Module.Free ℤ G
inst✝⁴ : Module.Finite ℤ G
hf : finrank ℤ G = r * (↑p - 1)
inst✝³ : Algebra k K
inst✝² : IsGalois k K
inst✝¹ : FiniteDimensional k K
inst✝ : IsUnramifiedAtInfinitePlaces k K
hKL : finrank k K = ↑p
σ : K ≃ₐ[k] K
hσ : ∀ (x : K ≃ₐ[k] K), x ∈ Subgroup.zpowers σ
this : Fact (↑p).Prime
x : (𝓞 K)ˣ
⊢ ∑ x_1 ∈ range ↑p, Additive.ofMul ↑((Units.map ↑((galRestrictHom (𝓞 k) k K (𝓞 K)) ↑(σ ^ x_1))) x) = 0