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Theorem mul2lt0bi 26040
Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
Hypotheses
Ref Expression
mul2lt0.1  |-  ( ph  ->  A  e.  RR )
mul2lt0.2  |-  ( ph  ->  B  e.  RR )
Assertion
Ref Expression
mul2lt0bi  |-  ( ph  ->  ( ( A  x.  B )  <  0  <->  ( ( A  <  0  /\  0  <  B )  \/  ( 0  < 
A  /\  B  <  0 ) ) ) )

Proof of Theorem mul2lt0bi
StepHypRef Expression
1 mul2lt0.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
2 mul2lt0.2 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
31, 2remulcld 9412 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  RR )
4 0re 9384 . . . . . . . . 9  |-  0  e.  RR
54a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
63, 5ltnled 9519 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  B )  <  0  <->  -.  0  <_  ( A  x.  B ) ) )
71adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  ->  A  e.  RR )
82adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  ->  B  e.  RR )
9 simprl 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  -> 
0  <_  A )
10 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  -> 
0  <_  B )
117, 8, 9, 10mulge0d 9914 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  -> 
0  <_  ( A  x.  B ) )
1211ex 434 . . . . . . . 8  |-  ( ph  ->  ( ( 0  <_  A  /\  0  <_  B
)  ->  0  <_  ( A  x.  B ) ) )
1312con3d 133 . . . . . . 7  |-  ( ph  ->  ( -.  0  <_ 
( A  x.  B
)  ->  -.  (
0  <_  A  /\  0  <_  B ) ) )
146, 13sylbid 215 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  <  0  ->  -.  ( 0  <_  A  /\  0  <_  B
) ) )
15 ianor 488 . . . . . 6  |-  ( -.  ( 0  <_  A  /\  0  <_  B )  <-> 
( -.  0  <_  A  \/  -.  0  <_  B ) )
1614, 15syl6ib 226 . . . . 5  |-  ( ph  ->  ( ( A  x.  B )  <  0  ->  ( -.  0  <_  A  \/  -.  0  <_  B ) ) )
171, 5ltnled 9519 . . . . . 6  |-  ( ph  ->  ( A  <  0  <->  -.  0  <_  A )
)
182, 5ltnled 9519 . . . . . 6  |-  ( ph  ->  ( B  <  0  <->  -.  0  <_  B )
)
1917, 18orbi12d 709 . . . . 5  |-  ( ph  ->  ( ( A  <  0  \/  B  <  0 )  <->  ( -.  0  <_  A  \/  -.  0  <_  B ) ) )
2016, 19sylibrd 234 . . . 4  |-  ( ph  ->  ( ( A  x.  B )  <  0  ->  ( A  <  0  \/  B  <  0
) ) )
2120imp 429 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  <  0  \/  B  <  0 ) )
22 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  A  <  0 )
231adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
242adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
25 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  x.  B )  <  0
)
2623, 24, 25mul2lt0llt0 26038 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  0  <  B )
2722, 26jca 532 . . . . 5  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  ( A  <  0  /\  0  <  B ) )
2827ex 434 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  <  0  ->  ( A  <  0  /\  0  < 
B ) ) )
2923, 24, 25mul2lt0rlt0 26036 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  B  <  0 )  ->  0  <  A )
30 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  B  <  0 )  ->  B  <  0 )
3129, 30jca 532 . . . . 5  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  B  <  0 )  ->  (
0  <  A  /\  B  <  0 ) )
3231ex 434 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  <  0  ->  ( 0  <  A  /\  B  <  0 ) ) )
3328, 32orim12d 834 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( A  <  0  \/  B  <  0 )  ->  (
( A  <  0  /\  0  <  B )  \/  ( 0  < 
A  /\  B  <  0 ) ) ) )
3421, 33mpd 15 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( A  <  0  /\  0  <  B )  \/  (
0  <  A  /\  B  <  0 ) ) )
351adantr 465 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  A  e.  RR )
364a1i 11 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
0  e.  RR )
372adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  RR )
38 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
0  <  B )
3937, 38elrpd 11023 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  RR+ )
40 simprl 755 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  A  <  0 )
4135, 36, 39, 40ltmul1dd 11076 . . . 4  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  ( 0  x.  B ) )
4237recnd 9410 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  CC )
4342mul02d 9565 . . . 4  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( 0  x.  B
)  =  0 )
4441, 43breqtrd 4314 . . 3  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  0 )
452adantr 465 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  B  e.  RR )
464a1i 11 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
0  e.  RR )
471adantr 465 . . . . . 6  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  RR )
48 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
0  <  A )
4947, 48elrpd 11023 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  RR+ )
50 simprr 756 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  B  <  0 )
5145, 46, 49, 50ltmul2dd 11077 . . . 4  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  B
)  <  ( A  x.  0 ) )
5249rpcnd 11027 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  CC )
5352mul01d 9566 . . . 4  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  0 )  =  0 )
5451, 53breqtrd 4314 . . 3  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  B
)  <  0 )
5544, 54jaodan 783 . 2  |-  ( (
ph  /\  ( ( A  <  0  /\  0  <  B )  \/  (
0  <  A  /\  B  <  0 ) ) )  ->  ( A  x.  B )  <  0
)
5634, 55impbida 828 1  |-  ( ph  ->  ( ( A  x.  B )  <  0  <->  ( ( A  <  0  /\  0  <  B )  \/  ( 0  < 
A  /\  B  <  0 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    e. wcel 1756   class class class wbr 4290  (class class class)co 6089   RRcr 9279   0cc0 9280    x. cmul 9285    < clt 9416    <_ cle 9417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-rp 10990
This theorem is referenced by:  ztprmneprm  30736
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