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	$$⊢\left(A\in {\mathbb{N}}_0\wedge B\in {\mathbb{N}}_0\right)\to \left(B<A\to A-B\in {\mathbb{N}}_0\right)$$

Proof

Step	Hyp	Ref	Expression
1		nn0re	$$⊢B\in {\mathbb{N}}_0\to B\in ℝ$$
2		nn0re	$$⊢A\in {\mathbb{N}}_0\to A\in ℝ$$
3		ltle	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(B<A\to B\le A\right)$$
4	1 2 3	syl2anr	$$⊢\left(A\in {\mathbb{N}}_0\wedge B\in {\mathbb{N}}_0\right)\to \left(B<A\to B\le A\right)$$
5		nn0sub	$$⊢\left(B\in {\mathbb{N}}_0\wedge A\in {\mathbb{N}}_0\right)\to \left(B\le A\leftrightarrow A-B\in {\mathbb{N}}_0\right)$$
6	5	ancoms	$$⊢\left(A\in {\mathbb{N}}_0\wedge B\in {\mathbb{N}}_0\right)\to \left(B\le A\leftrightarrow A-B\in {\mathbb{N}}_0\right)$$
7	4 6	sylibd	$$⊢\left(A\in {\mathbb{N}}_0\wedge B\in {\mathbb{N}}_0\right)\to \left(B<A\to A-B\in {\mathbb{N}}_0\right)$$