isCoprime_of_isMaximal — Mathlib

English

If I and J are maximal ideals with I ≠ J, then I and J are coprime.

Русский

Если I и J — максимальныеideалы и I ≠ J, то I и J взаимно просты.

LaTeX

$$$$ \text{If } I,J \text{ are maximal and } I \neq J, \text{ then } IsCoprime(I,J). $$$$

Lean4

theorem isCoprime_of_isMaximal [I.IsMaximal] [J.IsMaximal] (ne : I ≠ J) : IsCoprime I J :=
  by
  rw [isCoprime_iff_codisjoint, isMaximal_def] at *
  exact IsCoatom.codisjoint_of_ne ‹_› ‹_› ne

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