Metamath Proof Explorer

Theorem mulltgt0d

Description: Negative times positive is negative. (Contributed by SN, 26-Nov-2025)

Ref Expression
Hypotheses mullt0b1d.a 
|- ( ph -> A e. RR )
mullt0b1d.b 
|- ( ph -> B e. RR )
mullt0b1d.1 
|- ( ph -> A < 0 )
mulltgt0d.2 
|- ( ph -> 0 < B )
Assertion mulltgt0d 
|- ( ph -> ( A x. B ) < 0 )

Proof

Step Hyp Ref Expression
1 mullt0b1d.a 
 |-  ( ph -> A e. RR )
2 mullt0b1d.b 
 |-  ( ph -> B e. RR )
3 mullt0b1d.1 
 |-  ( ph -> A < 0 )
4 mulltgt0d.2 
 |-  ( ph -> 0 < B )
5 3 lt0ne0d 
 |-  ( ph -> A =/= 0 )
6 4 gt0ne0d 
 |-  ( ph -> B =/= 0 )
7 5 6 jca 
 |-  ( ph -> ( A =/= 0 /\ B =/= 0 ) )
8 neanior 
 |-  ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) )
9 7 8 sylib 
 |-  ( ph -> -. ( A = 0 \/ B = 0 ) )
10 1 2 sn-remul0ord 
 |-  ( ph -> ( ( A x. B ) = 0 <-> ( A = 0 \/ B = 0 ) ) )
11 9 10 mtbird 
 |-  ( ph -> -. ( A x. B ) = 0 )
12 0red 
 |-  ( ph -> 0 e. RR )
13 1 12 3 ltnsymd 
 |-  ( ph -> -. 0 < A )
14 1 2 4 mulgt0b2d 
 |-  ( ph -> ( 0 < A <-> 0 < ( A x. B ) ) )
15 13 14 mtbid 
 |-  ( ph -> -. 0 < ( A x. B ) )
16 ioran 
 |-  ( -. ( ( A x. B ) = 0 \/ 0 < ( A x. B ) ) <-> ( -. ( A x. B ) = 0 /\ -. 0 < ( A x. B ) ) )
17 11 15 16 sylanbrc 
 |-  ( ph -> -. ( ( A x. B ) = 0 \/ 0 < ( A x. B ) ) )
18 1 2 remulcld 
 |-  ( ph -> ( A x. B ) e. RR )
19 18 12 lttrid 
 |-  ( ph -> ( ( A x. B ) < 0 <-> -. ( ( A x. B ) = 0 \/ 0 < ( A x. B ) ) ) )
20 17 19 mpbird 
 |-  ( ph -> ( A x. B ) < 0 )