Theorem addgt0d

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

	Ref	Expression
Hypotheses	leidd.1	$$⊢\phi \to A\in ℝ$$
	ltnegd.2	$$⊢\phi \to B\in ℝ$$
	addgt0d.3	$$⊢\phi \to 0<A$$
	addgt0d.4	$$⊢\phi \to 0<B$$
Assertion	addgt0d	$$⊢\phi \to 0<A+B$$

Proof

Step	Hyp	Ref	Expression
1		leidd.1	$$⊢\phi \to A\in ℝ$$
2		ltnegd.2	$$⊢\phi \to B\in ℝ$$
3		addgt0d.3	$$⊢\phi \to 0<A$$
4		addgt0d.4	$$⊢\phi \to 0<B$$
5		0red	$$⊢\phi \to 0\in ℝ$$
6	5 1 3	ltled	$$⊢\phi \to 0\le A$$
7	1 2 6 4	addgegt0d	$$⊢\phi \to 0<A+B$$