coprime_of_coprime — Mathlib

English

If gcd x y = 1, then gcd y z = 1.

Русский

Если gcd(x,y)=1, то gcd(y,z)=1.

LaTeX

$$$\gcd(y,z) = 1$$$

Lean4

theorem coprime_of_coprime (hc : Int.gcd x y = 1) : Int.gcd y z = 1 :=
  by
  by_contra H
  obtain ⟨p, hp, hpy, hpz⟩ := Nat.Prime.not_coprime_iff_dvd.mp H
  apply hp.not_dvd_one
  rw [← hc]
  apply Nat.dvd_gcd (Int.Prime.dvd_natAbs_of_coe_dvd_sq hp _ _) hpy
  rw [sq, eq_sub_of_add_eq h]
  rw [← Int.natCast_dvd] at hpy hpz
  exact dvd_sub (hpz.mul_right _) (hpy.mul_right _)

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