Theorem addge02

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005)

	Ref	Expression
Assertion	addge02	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(0\le B\leftrightarrow A\le B+A\right)$$

Proof

Step	Hyp	Ref	Expression
1		addge01	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(0\le B\leftrightarrow A\le A+B\right)$$
2		recn	$$⊢A\in ℝ\to A\in ℂ$$
3		recn	$$⊢B\in ℝ\to B\in ℂ$$
4		addcom	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+B=B+A$$
5	2 3 4	syl2an	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to A+B=B+A$$
6	5	breq2d	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A\le A+B\leftrightarrow A\le B+A\right)$$
7	1 6	bitrd	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(0\le B\leftrightarrow A\le B+A\right)$$