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Theorem mullt0 8264
Description: The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
Assertion
Ref Expression
mullt0  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
0  <  ( A  x.  B ) )

Proof of Theorem mullt0
StepHypRef Expression
1 renegcl 8045 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
21adantr 274 . . . 4  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  -u A  e.  RR )
3 lt0neg1 8252 . . . . 5  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
43biimpa 294 . . . 4  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
0  <  -u A )
52, 4jca 304 . . 3  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
( -u A  e.  RR  /\  0  <  -u A
) )
6 renegcl 8045 . . . . 5  |-  ( B  e.  RR  ->  -u B  e.  RR )
76adantr 274 . . . 4  |-  ( ( B  e.  RR  /\  B  <  0 )  ->  -u B  e.  RR )
8 lt0neg1 8252 . . . . 5  |-  ( B  e.  RR  ->  ( B  <  0  <->  0  <  -u B ) )
98biimpa 294 . . . 4  |-  ( ( B  e.  RR  /\  B  <  0 )  -> 
0  <  -u B )
107, 9jca 304 . . 3  |-  ( ( B  e.  RR  /\  B  <  0 )  -> 
( -u B  e.  RR  /\  0  <  -u B
) )
11 mulgt0 7861 . . 3  |-  ( ( ( -u A  e.  RR  /\  0  <  -u A )  /\  ( -u B  e.  RR  /\  0  <  -u B ) )  ->  0  <  ( -u A  x.  -u B
) )
125, 10, 11syl2an 287 . 2  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
0  <  ( -u A  x.  -u B ) )
13 recn 7775 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
14 recn 7775 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
15 mul2neg 8182 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  -u B )  =  ( A  x.  B ) )
1613, 14, 15syl2an 287 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  x.  -u B )  =  ( A  x.  B ) )
1716ad2ant2r 501 . 2  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
( -u A  x.  -u B
)  =  ( A  x.  B ) )
1812, 17breqtrd 3960 1  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  B  <  0 ) )  -> 
0  <  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   class class class wbr 3935  (class class class)co 5780   CCcc 7640   RRcr 7641   0cc0 7642    x. cmul 7647    < clt 7822   -ucneg 7956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-distr 7746  ax-i2m1 7747  ax-0id 7750  ax-rnegex 7751  ax-cnre 7753  ax-pre-ltadd 7758  ax-pre-mulgt0 7759
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-iota 5094  df-fun 5131  df-fv 5137  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-pnf 7824  df-mnf 7825  df-ltxr 7827  df-sub 7957  df-neg 7958
This theorem is referenced by:  inelr  8368  apsqgt0  8385
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