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[Merged by Bors] - chore(Data/Nat/GCD/Basic): automated extraction from #40597 #42712

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[Merged by Bors] - chore(Data/Nat/GCD/Basic): automated extraction from #40597 #42712

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11 changes: 11 additions & 0 deletions Mathlib/Data/Nat/GCD/Basic.lean
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Expand Up @@ -262,6 +262,17 @@ lemma div_mul_div (hkm : m ∣ k) (hkn : n ∣ m) : (k / m) * (m / n) = k / n :=
lemma div_dvd_div_left (hkm : m ∣ k) (hkn : n ∣ m) : k / m ∣ k / n :=
⟨_, (div_mul_div hkm hkn).symm⟩

/-- `k / m ∣ k / n` iff `n ∣ m`, when both `m` and `n` divide `k`. -/
lemma div_dvd_div_iff_left (hk : 0 < k) (hmk : m ∣ k) (hnk : n ∣ k) :
k / m ∣ k / n ↔ n ∣ m := by
grind [Nat.div_div_self, div_dvd_div_left, div_dvd_of_dvd]

/-- `m / k ∣ n / k` iff `m ∣ n`, when `k` divides both `m` and `n`. -/
lemma div_dvd_div_iff_right (hkm : k ∣ m) (hkn : k ∣ n) : m / k ∣ n / k ↔ m ∣ n := by
rcases k.eq_zero_or_pos with rfl | hk
· simp_all
· rw [← Nat.mul_dvd_mul_iff_left hk, Nat.mul_div_cancel' hkm, Nat.mul_div_cancel' hkn]

lemma div_lcm_eq_div_gcd (hkm : m ∣ k) (hkn : n ∣ k) : (k / m).lcm (k / n) = k / (m.gcd n) := by
rw [Nat.lcm_eq_iff]
refine ⟨div_dvd_div_left hkm (Nat.gcd_dvd_left m n),
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