Theorem addge0d

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

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	addge0d.3	|- ( ph -> 0 <_ A )
	addge0d.4	|- ( ph -> 0 <_ B )
Assertion	addge0d	|- ( ph -> 0 <_ ( A + B ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		addge0d.3	|- ( ph -> 0 <_ A )
4		addge0d.4	|- ( ph -> 0 <_ B )
5		addge0	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A + B ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> 0 <_ ( A + B ) )