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Metamath Proof Explorer

Theorem mullt0

Description: The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009)

Ref Expression
Assertion mullt0 
|- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )

Proof

Step Hyp Ref Expression
1 renegcl 
 |-  ( A e. RR -> -u A e. RR )
2 1 adantr 
 |-  ( ( A e. RR /\ A < 0 ) -> -u A e. RR )
3 lt0neg1 
 |-  ( A e. RR -> ( A < 0 <-> 0 < -u A ) )
4 3 biimpa 
 |-  ( ( A e. RR /\ A < 0 ) -> 0 < -u A )
5 2 4 jca 
 |-  ( ( A e. RR /\ A < 0 ) -> ( -u A e. RR /\ 0 < -u A ) )
6 renegcl 
 |-  ( B e. RR -> -u B e. RR )
7 6 adantr 
 |-  ( ( B e. RR /\ B < 0 ) -> -u B e. RR )
8 lt0neg1 
 |-  ( B e. RR -> ( B < 0 <-> 0 < -u B ) )
9 8 biimpa 
 |-  ( ( B e. RR /\ B < 0 ) -> 0 < -u B )
10 7 9 jca 
 |-  ( ( B e. RR /\ B < 0 ) -> ( -u B e. RR /\ 0 < -u B ) )
11 mulgt0 
 |-  ( ( ( -u A e. RR /\ 0 < -u A ) /\ ( -u B e. RR /\ 0 < -u B ) ) -> 0 < ( -u A x. -u B ) )
12 5 10 11 syl2an 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( -u A x. -u B ) )
13 recn 
 |-  ( A e. RR -> A e. CC )
14 recn 
 |-  ( B e. RR -> B e. CC )
15 mul2neg 
 |-  ( ( A e. CC /\ B e. CC ) -> ( -u A x. -u B ) = ( A x. B ) )
16 13 14 15 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A x. -u B ) = ( A x. B ) )
17 16 ad2ant2r 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> ( -u A x. -u B ) = ( A x. B ) )
18 12 17 breqtrd 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ B < 0 ) ) -> 0 < ( A x. B ) )