Theorem ge0addcl

Description: The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Assertion	ge0addcl	|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A + B ) e. ( 0 [,) +oo ) )

Proof

Step	Hyp	Ref	Expression
1		elrege0	|- ( A e. ( 0 [,) +oo ) <-> ( A e. RR /\ 0 <_ A ) )
2		elrege0	|- ( B e. ( 0 [,) +oo ) <-> ( B e. RR /\ 0 <_ B ) )
3		readdcl	|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	3	ad2ant2r	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A + B ) e. RR )
5		addge0	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A + B ) )
6	5	an4s	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A + B ) )
7		elrege0	|- ( ( A + B ) e. ( 0 [,) +oo ) <-> ( ( A + B ) e. RR /\ 0 <_ ( A + B ) ) )
8	4 6 7	sylanbrc	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A + B ) e. ( 0 [,) +oo ) )
9	1 2 8	syl2anb	|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A + B ) e. ( 0 [,) +oo ) )