-- Mathlib/NumberTheory/NumberField/Embeddings.lean
lemma ComplexEmbedding.exists_comp_symm_eq_of_comp_eq [IsGalois k K] (φ ψ : K →+* ℂ)
    (h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)) :
    ∃ σ : K ≃ₐ[k] K, φ.comp σ.symm = ψ

Proof state:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this : Algebra K ℂ := φ.toAlgebra
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Tactic:
have : IsScalarTower k K ℂ := IsScalarTower.of_algebraMap_eq' rfl

Result:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Proof state:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Tactic:
letI := (φ.comp (algebraMap k K)).toAlgebra

Result:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Proof state:
case h.a
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
x : K
⊢ (φ.comp ↑(ψ'.restrictNormal' K).symm.symm) x = ψ x

Tactic:
exact AlgHom.restrictNormal_commutes ψ' K x

Result:
no goals

Proof state:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Tactic:
let ψ' : K →ₐ[k] ℂ := { ψ with commutes' := fun r ↦ (RingHom.congr_fun h r).symm }

Result:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Proof state:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Tactic:
letI := φ.toAlgebra

Result:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this : Algebra K ℂ := φ.toAlgebra
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Proof state:
case h
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
⊢ φ.comp ↑(ψ'.restrictNormal' K).symm.symm = ψ

Tactic:
ext1 x

Result:
case h.a
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
x : K
⊢ (φ.comp ↑(ψ'.restrictNormal' K).symm.symm) x = ψ x

Proof state:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
⊢ ∃ σ, φ.comp ↑σ.symm = ψ

Tactic:
use (AlgHom.restrictNormal' ψ' K).symm

Result:
case h
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
⊢ φ.comp ↑(ψ'.restrictNormal' K).symm.symm = ψ