Theorem addgtge0d

Description: Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021)

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	addgtge0d.3	$$⊢\phi \to 0<A$$
	addgtge0d.4	$$⊢\phi \to 0\le B$$
Assertion	addgtge0d	$$⊢\phi \to 0<A+B$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		addgtge0d.3	$$⊢\phi \to 0<A$$
4		addgtge0d.4	$$⊢\phi \to 0\le B$$
5		addgtge0	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0<A\wedge 0\le B\right)\right)\to 0<A+B$$
6	1 2 3 4 5	syl22anc	$$⊢\phi \to 0<A+B$$