isConj_complexConj — Mathlib

English

For any φ: K → ℂ, IsConj φ (complexConj K) holds; i.e., φ is conjugate to its image under complexConj.

Русский

Для любого отображения φ: K → ℂ выполняется IsConj φ (complexConj K).

LaTeX

$$$IsConj(φ, complexConj\\,K).$$$

Lean4

/-- The complex conjugation is the conjugation of any complex embedding of a CM-field.
-/
theorem isConj_complexConj (φ : K →+* ℂ) : IsConj φ (complexConj K) :=
  by
  obtain ⟨σ, hσ⟩ := exists_isConj _ φ
  have := (exists_isConj K (Classical.choice (inferInstance : Nonempty (K →+* ℂ)))).choose_spec
  rwa [isConj_eq_isConj K hσ this] at hσ

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