Theorem addgt0i

Description: Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

	Ref	Expression
Hypotheses	lt2.1	$$⊢A\in ℝ$$
	lt2.2	$$⊢B\in ℝ$$
Assertion	addgt0i	$$⊢\left(0<A\wedge 0<B\right)\to 0<A+B$$

Proof

Step	Hyp	Ref	Expression
1		lt2.1	$$⊢A\in ℝ$$
2		lt2.2	$$⊢B\in ℝ$$
3		addgt0	$$⊢\left(\left(A\in ℝ\wedge B\in ℝ\right)\wedge \left(0<A\wedge 0<B\right)\right)\to 0<A+B$$
4	1 2 3	mpanl12	$$⊢\left(0<A\wedge 0<B\right)\to 0<A+B$$