ringHom_of_comp_eq — Mathlib

English

If H : Transcendental S B and f, g satisfy a composition equality, then Transcendental R A holds.

Русский

Если H: Transcendental(S, B) и f, g удовлетворяют композиционному равенству, то Transcendental(R, A).

LaTeX

$$$[H : Transcendental S B] \Rightarrow Transcendental R A$ при равенстве композиции.$$

Lean4

theorem ringHom_of_comp_eq [H : Algebra.Transcendental R A] (hf : Function.Surjective f) (hg : Function.Injective g)
    (h : RingHom.comp (algebraMap S B) f = RingHom.comp g (algebraMap R A)) : Algebra.Transcendental S B :=
  by
  rw [Algebra.transcendental_iff_not_isAlgebraic] at H ⊢
  exact fun halg ↦ H (halg.of_ringHom_of_comp_eq f g hf hg h)

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