Theorem xrge0addcld

Description: Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019)

	Ref	Expression
Hypotheses	xrge0addcld.a	|- ( ph -> A e. ( 0 [,] +oo ) )
	xrge0addcld.b	|- ( ph -> B e. ( 0 [,] +oo ) )
Assertion	xrge0addcld	|- ( ph -> ( A +e B ) e. ( 0 [,] +oo ) )

Proof

Step	Hyp	Ref	Expression
1		xrge0addcld.a	|- ( ph -> A e. ( 0 [,] +oo ) )
2		xrge0addcld.b	|- ( ph -> B e. ( 0 [,] +oo ) )
3		elxrge0	|- ( A e. ( 0 [,] +oo ) <-> ( A e. RR* /\ 0 <_ A ) )
4	1 3	sylib	|- ( ph -> ( A e. RR* /\ 0 <_ A ) )
5	4	simpld	|- ( ph -> A e. RR* )
6		elxrge0	|- ( B e. ( 0 [,] +oo ) <-> ( B e. RR* /\ 0 <_ B ) )
7	2 6	sylib	|- ( ph -> ( B e. RR* /\ 0 <_ B ) )
8	7	simpld	|- ( ph -> B e. RR* )
9	5 8	xaddcld	|- ( ph -> ( A +e B ) e. RR* )
10	4	simprd	|- ( ph -> 0 <_ A )
11	7	simprd	|- ( ph -> 0 <_ B )
12		xaddge0	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A +e B ) )
13	5 8 10 11 12	syl22anc	|- ( ph -> 0 <_ ( A +e B ) )
14		elxrge0	|- ( ( A +e B ) e. ( 0 [,] +oo ) <-> ( ( A +e B ) e. RR* /\ 0 <_ ( A +e B ) ) )
15	9 13 14	sylanbrc	|- ( ph -> ( A +e B ) e. ( 0 [,] +oo ) )