Metamath Proof Explorer

Theorem dvdssqnn

Description: Two positive integers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014) (Revised by Mario Carneiro, 19-Apr-2014) (Proof shortened by AV, 16-Sep-2026)

Ref Expression
Assertion dvdssqnn ⊢ M ∈ ℕ ∧ N ∈ ℕ → M ∥ N ↔ M 2 ∥ N 2

Proof

Step Hyp Ref Expression
1 2nn ⊢ 2 ∈ ℕ
2 dvdsexpnn ⊢ M ∈ ℕ ∧ N ∈ ℕ ∧ 2 ∈ ℕ → M ∥ N ↔ M 2 ∥ N 2
3 1 2 mp3an3 ⊢ M ∈ ℕ ∧ N ∈ ℕ → M ∥ N ↔ M 2 ∥ N 2