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