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Metamath Proof Explorer

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 ) )