This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ge0xaddcl

Description: The nonnegative reals are closed under addition. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Assertion ge0xaddcl 
|- ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) -> ( A +e B ) e. ( 0 [,] +oo ) )

Proof

Step Hyp Ref Expression
1 elxrge0 
 |-  ( A e. ( 0 [,] +oo ) <-> ( A e. RR* /\ 0 <_ A ) )
2 elxrge0 
 |-  ( B e. ( 0 [,] +oo ) <-> ( B e. RR* /\ 0 <_ B ) )
3 xaddcl 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
4 3 ad2ant2r 
 |-  ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) ) -> ( A +e B ) e. RR* )
5 xaddge0 
 |-  ( ( ( A e. RR* /\ B e. RR* ) /\ ( 0 <_ A /\ 0 <_ B ) ) -> 0 <_ ( A +e B ) )
6 5 an4s 
 |-  ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) ) -> 0 <_ ( A +e B ) )
7 elxrge0 
 |-  ( ( A +e B ) e. ( 0 [,] +oo ) <-> ( ( A +e B ) e. RR* /\ 0 <_ ( A +e B ) ) )
8 4 6 7 sylanbrc 
 |-  ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) ) -> ( A +e B ) e. ( 0 [,] +oo ) )
9 1 2 8 syl2anb 
 |-  ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) -> ( A +e B ) e. ( 0 [,] +oo ) )