Theorem addgt0d

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

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	addgt0d.3	|- ( ph -> 0 < A )
	addgt0d.4	|- ( ph -> 0 < B )
Assertion	addgt0d	|- ( ph -> 0 < ( A + B ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		addgt0d.3	|- ( ph -> 0 < A )
4		addgt0d.4	|- ( ph -> 0 < B )
5		0red	|- ( ph -> 0 e. RR )
6	5 1 3	ltled	|- ( ph -> 0 <_ A )
7	1 2 6 4	addgegt0d	|- ( ph -> 0 < ( A + B ) )