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 e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || ( abs ` N ) ) )

Proof

Step	Hyp	Ref	Expression
1		absdvdsb	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || N ) )
2		zabscl	|- ( M e. ZZ -> ( abs ` M ) e. ZZ )
3		dvdsabsb	|- ( ( ( abs ` M ) e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( abs ` M ) || ( abs ` N ) ) )
4	2 3	sylan	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( abs ` M ) || N <-> ( abs ` M ) || ( abs ` N ) ) )
5	1 4	bitrd	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> ( abs ` M ) || ( abs ` N ) ) )