Theorem nn0addge2

Description: A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005)

	Ref	Expression
Assertion	nn0addge2	$$⊢\left(A\in ℝ\wedge N\in {\mathbb{N}}_0\right)\to A\le N+A$$

Proof

Step	Hyp	Ref	Expression
1		nn0re	$$⊢N\in {\mathbb{N}}_0\to N\in ℝ$$
2		nn0ge0	$$⊢N\in {\mathbb{N}}_0\to 0\le N$$
3	1 2	jca	$$⊢N\in {\mathbb{N}}_0\to \left(N\in ℝ\wedge 0\le N\right)$$
4		addge02	$$⊢\left(A\in ℝ\wedge N\in ℝ\right)\to \left(0\le N\leftrightarrow A\le N+A\right)$$
5	4	biimp3a	$$⊢\left(A\in ℝ\wedge N\in ℝ\wedge 0\le N\right)\to A\le N+A$$
6	5	3expb	$$⊢\left(A\in ℝ\wedge \left(N\in ℝ\wedge 0\le N\right)\right)\to A\le N+A$$
7	3 6	sylan2	$$⊢\left(A\in ℝ\wedge N\in {\mathbb{N}}_0\right)\to A\le N+A$$