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Theorem mulltgt0 27669
Description: The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Assertion
Ref Expression
mulltgt0  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  0 )

Proof of Theorem mulltgt0
StepHypRef Expression
1 renegcl 9364 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
21ad2antrr 707 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  -u A  e.  RR )
3 lt0neg1 9534 . . . . . 6  |-  ( A  e.  RR  ->  ( A  <  0  <->  0  <  -u A ) )
43biimpa 471 . . . . 5  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
0  <  -u A )
54adantr 452 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  -u A )
6 simpr 448 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( B  e.  RR  /\  0  <  B ) )
7 mulgt0 9153 . . . 4  |-  ( ( ( -u A  e.  RR  /\  0  <  -u A )  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
0  <  ( -u A  x.  B ) )
82, 5, 6, 7syl21anc 1183 . . 3  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( -u A  x.  B ) )
9 recn 9080 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
109ad2antrr 707 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  e.  CC )
11 recn 9080 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
1211ad2antrl 709 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  e.  CC )
1310, 12mulneg1d 9486 . . 3  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( -u A  x.  B
)  =  -u ( A  x.  B )
)
148, 13breqtrd 4236 . 2  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  -u ( A  x.  B ) )
15 remulcl 9075 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
1615ad2ant2r 728 . . 3  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  x.  B
)  e.  RR )
1716lt0neg1d 9596 . 2  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( A  x.  B )  <  0  <->  0  <  -u ( A  x.  B ) ) )
1814, 17mpbird 224 1  |-  ( ( ( A  e.  RR  /\  A  <  0 )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    x. cmul 8995    < clt 9120   -ucneg 9292
This theorem is referenced by:  stoweidlem26  27751  stirlinglem5  27803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294
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