Metamath Proof Explorer

Theorem absdvdsabsb

Description: Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024)

Ref Expression
Assertion absdvdsabsb ⊢ M ∈ ℤ ∧ N ∈ ℤ → M ∥ N ↔ M ∥ N

Proof

Step Hyp Ref Expression
1 absdvdsb ⊢ M ∈ ℤ ∧ N ∈ ℤ → M ∥ N ↔ M ∥ N
2 zabscl ⊢ M ∈ ℤ → M ∈ ℤ
3 dvdsabsb ⊢ M ∈ ℤ ∧ N ∈ ℤ → M ∥ N ↔ M ∥ N
4 2 3 sylan ⊢ M ∈ ℤ ∧ N ∈ ℤ → M ∥ N ↔ M ∥ N
5 1 4 bitrd ⊢ M ∈ ℤ ∧ N ∈ ℤ → M ∥ N ↔ M ∥ N