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Metamath Proof Explorer

Theorem ge0xmulcl

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

Ref Expression
Assertion ge0xmulcl 
|- ( ( 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 xmulcl 
 |-  ( ( 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 xmulge0 
 |-  ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) ) -> 0 <_ ( A *e B ) )
6 elxrge0 
 |-  ( ( A *e B ) e. ( 0 [,] +oo ) <-> ( ( A *e B ) e. RR* /\ 0 <_ ( A *e B ) ) )
7 4 5 6 sylanbrc 
 |-  ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) ) -> ( A *e B ) e. ( 0 [,] +oo ) )
8 1 2 7 syl2anb 
 |-  ( ( A e. ( 0 [,] +oo ) /\ B e. ( 0 [,] +oo ) ) -> ( A *e B ) e. ( 0 [,] +oo ) )