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 ) )