Theorem nndivdvdsd

Description: A positive integer divides a natural number if and only if the quotient is a positive integer, a deduction version of nndivdvds . (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	nndivdvdsd.1	|- ( ph -> M e. NN )
	nndivdvdsd.2	|- ( ph -> N e. NN )
Assertion	nndivdvdsd	|- ( ph -> ( M || N <-> ( N / M ) e. NN ) )

Proof

Step	Hyp	Ref	Expression
1		nndivdvdsd.1	|- ( ph -> M e. NN )
2		nndivdvdsd.2	|- ( ph -> N e. NN )
3		nndivdvds	|- ( ( N e. NN /\ M e. NN ) -> ( M || N <-> ( N / M ) e. NN ) )
4	2 1 3	syl2anc	|- ( ph -> ( M || N <-> ( N / M ) e. NN ) )