feat(Data/Nat/Prime): add 2 theorems (#11620)

Mathlib/Data/Nat/Prime.lean

@@ -454,6 +454,18 @@ theorem exists_dvd_of_not_prime2 {n : ℕ} (n2 : 2 ≤ n) (np : ¬Prime n) :
     (not_prime_iff_minFac_lt n2).1 np⟩
 #align nat.exists_dvd_of_not_prime2 Nat.exists_dvd_of_not_prime2
 
+theorem not_prime_of_dvd_of_ne {m n : ℕ} (h1 : m ∣ n) (h2 : m ≠ 1) (h3 : m ≠ n) : ¬Prime n :=
+  fun h => Or.elim (h.eq_one_or_self_of_dvd m h1) h2 h3
+
+theorem not_prime_of_dvd_of_lt {m n : ℕ} (h1 : m ∣ n) (h2 : 2 ≤ m) (h3 : m < n) : ¬Prime n :=
+  not_prime_of_dvd_of_ne h1 (ne_of_gt h2) (ne_of_lt h3)
+
+theorem not_prime_iff_exists_dvd_ne {n : ℕ} (h : 2 ≤ n) : (¬Prime n) ↔ ∃ m, m ∣ n ∧ m ≠ 1 ∧ m ≠ n :=
+  ⟨exists_dvd_of_not_prime h, fun ⟨_, h1, h2, h3⟩ => not_prime_of_dvd_of_ne h1 h2 h3⟩
+
+theorem not_prime_iff_exists_dvd_lt {n : ℕ} (h : 2 ≤ n) : (¬Prime n) ↔ ∃ m, m ∣ n ∧ 2 ≤ m ∧ m < n :=
+  ⟨exists_dvd_of_not_prime2 h, fun ⟨_, h1, h2, h3⟩ => not_prime_of_dvd_of_lt h1 h2 h3⟩
+
 theorem exists_prime_and_dvd {n : ℕ} (hn : n ≠ 1) : ∃ p, Prime p ∧ p ∣ n :=
   ⟨minFac n, minFac_prime hn, minFac_dvd _⟩
 #align nat.exists_prime_and_dvd Nat.exists_prime_and_dvd