Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mul2lt0bi Structured version   Unicode version

Theorem mul2lt0bi 27265
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 9624 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  RR )
4 0re 9596 . . . . . . . . 9  |-  0  e.  RR
54a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
63, 5ltnled 9731 . . . . . . 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 10129 . . . . . . . . 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 9731 . . . . . 6  |-  ( ph  ->  ( A  <  0  <->  -.  0  <_  A )
)
182, 5ltnled 9731 . . . . . 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 27263 . . . . . 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 27261 . . . . . 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 836 . . 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 11254 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  RR+ )
40 simprl 755 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  A  <  0 )
4135, 36, 39, 40ltmul1dd 11307 . . . 4  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  ( 0  x.  B ) )
4237recnd 9622 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  CC )
4342mul02d 9777 . . . 4  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( 0  x.  B
)  =  0 )
4441, 43breqtrd 4471 . . 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 11254 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  RR+ )
50 simprr 756 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  B  <  0 )
5145, 46, 49, 50ltmul2dd 11308 . . . 4  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  B
)  <  ( A  x.  0 ) )
5249rpcnd 11258 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  CC )
5352mul01d 9778 . . . 4  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  0 )  =  0 )
5451, 53breqtrd 4471 . . 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 830 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 1767   class class class wbr 4447  (class class class)co 6284   RRcr 9491   0cc0 9492    x. cmul 9497    < clt 9628    <_ cle 9629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-rp 11221
This theorem is referenced by:  ztprmneprm  32032
  Copyright terms: Public domain W3C validator