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Metamath Proof Explorer

Theorem ge0mulcl

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

Ref Expression
Assertion ge0mulcl 
|- ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A x. 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 remulcl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
4 3 ad2ant2r 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A x. B ) e. RR )
5 mulge0 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> 0 <_ ( A x. B ) )
6 elrege0 
 |-  ( ( A x. B ) e. ( 0 [,) +oo ) <-> ( ( A x. B ) e. RR /\ 0 <_ ( A x. B ) ) )
7 4 5 6 sylanbrc 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A x. B ) e. ( 0 [,) +oo ) )
8 1 2 7 syl2anb 
 |-  ( ( A e. ( 0 [,) +oo ) /\ B e. ( 0 [,) +oo ) ) -> ( A x. B ) e. ( 0 [,) +oo ) )