isConjRoot_of_algEquiv₂ — Mathlib

English

Two automorphisms s1,s2 yield IsConjRoot R (s1 x) (s2 x).

Русский

Два автоморфизма s1,s2 дают IsConjRoot R (s1 x) (s2 x).

LaTeX

$$IsConjRoot R (s1 x) (s2 x)$$

Lean4

/-- Let `s₁` and `s₂` be two `R`-algebra isomorphisms. Then `s₂ x` is a conjugate root of `s₁ x`.
-/
@[simp]
theorem isConjRoot_of_algEquiv₂ (x : A) (s₁ s₂ : A ≃ₐ[R] A) : IsConjRoot R (s₁ x) (s₂ x) :=
  isConjRoot_def.mpr <| (minpoly.algEquiv_eq s₂ x) ▸ (minpoly.algEquiv_eq s₁ x)

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