Theorem ltsubnn0

Description: Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018)

	Ref	Expression
Assertion	ltsubnn0	|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B < A -> ( A - B ) e. NN0 ) )

Proof

Step	Hyp	Ref	Expression
1		nn0re	|- ( B e. NN0 -> B e. RR )
2		nn0re	|- ( A e. NN0 -> A e. RR )
3		ltle	|- ( ( B e. RR /\ A e. RR ) -> ( B < A -> B <_ A ) )
4	1 2 3	syl2anr	|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B < A -> B <_ A ) )
5		nn0sub	|- ( ( B e. NN0 /\ A e. NN0 ) -> ( B <_ A <-> ( A - B ) e. NN0 ) )
6	5	ancoms	|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B <_ A <-> ( A - B ) e. NN0 ) )
7	4 6	sylibd	|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B < A -> ( A - B ) e. NN0 ) )