Theorem dvdsleabs

Description: The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in ApostolNT p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011) (Proof shortened by Fan Zheng, 3-Jul-2016)

	Ref	Expression
Assertion	dvdsleabs	|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> M <_ ( abs ` N ) ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsabsb	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> M || ( abs ` N ) ) )
2	1	3adant3	|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N <-> M || ( abs ` N ) ) )
3		nnabscl	|- ( ( N e. ZZ /\ N =/= 0 ) -> ( abs ` N ) e. NN )
4		dvdsle	|- ( ( M e. ZZ /\ ( abs ` N ) e. NN ) -> ( M || ( abs ` N ) -> M <_ ( abs ` N ) ) )
5	3 4	sylan2	|- ( ( M e. ZZ /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M || ( abs ` N ) -> M <_ ( abs ` N ) ) )
6	5	3impb	|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || ( abs ` N ) -> M <_ ( abs ` N ) ) )
7	2 6	sylbid	|- ( ( M e. ZZ /\ N e. ZZ /\ N =/= 0 ) -> ( M || N -> M <_ ( abs ` N ) ) )