Theorem addge0d

Description: Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	addge0d.3	$$⊢\phi \to 0\le A$$
	addge0d.4	$$⊢\phi \to 0\le B$$
Assertion	addge0d	$$⊢\phi \to 0\le A+B$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		addge0d.3	$$⊢\phi \to 0\le A$$
4		addge0d.4	$$⊢\phi \to 0\le B$$
5		addge0	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0\le A\wedge 0\le B\right)\right)\to 0\le A+B$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to 0\le A+B$$