Theorem addgtge0d

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

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	addgtge0d.3	|- ( ph -> 0 < A )
	addgtge0d.4	|- ( ph -> 0 <_ B )
Assertion	addgtge0d	|- ( ph -> 0 < ( A + B ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		addgtge0d.3	|- ( ph -> 0 < A )
4		addgtge0d.4	|- ( ph -> 0 <_ B )
5		addgtge0	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) -> 0 < ( A + B ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> 0 < ( A + B ) )