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Metamath Proof Explorer

Theorem mulltgt0

Description: The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017)

Ref Expression
Assertion mulltgt0 
|- ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> ( A x. B ) < 0 )

Proof

Step Hyp Ref Expression
1 renegcl 
 |-  ( A e. RR -> -u A e. RR )
2 1 ad2antrr 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> -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 4 adantr 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < -u A )
6 simpr 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> ( B e. RR /\ 0 < B ) )
7 mulgt0 
 |-  ( ( ( -u A e. RR /\ 0 < -u A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( -u A x. B ) )
8 2 5 6 7 syl21anc 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( -u A x. B ) )
9 recn 
 |-  ( A e. RR -> A e. CC )
10 9 ad2antrr 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
11 recn 
 |-  ( B e. RR -> B e. CC )
12 11 ad2antrl 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
13 10 12 mulneg1d 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> ( -u A x. B ) = -u ( A x. B ) )
14 8 13 breqtrd 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < -u ( A x. B ) )
15 remulcl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
16 15 ad2ant2r 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> ( A x. B ) e. RR )
17 16 lt0neg1d 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A x. B ) < 0 <-> 0 < -u ( A x. B ) ) )
18 14 17 mpbird 
 |-  ( ( ( A e. RR /\ A < 0 ) /\ ( B e. RR /\ 0 < B ) ) -> ( A x. B ) < 0 )