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 ) → ( 𝑀𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 absdvdsb ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁 ↔ ( abs ‘ 𝑀 ) ∥ 𝑁 ) )
2 zabscl ( 𝑀 ∈ ℤ → ( abs ‘ 𝑀 ) ∈ ℤ )
3 dvdsabsb ( ( ( abs ‘ 𝑀 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) )
4 2 3 sylan ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) )
5 1 4 bitrd ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) )