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Theorem mul2lt0bi 11402
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 9670 . . . . . . . 8  |-  ( ph  ->  ( A  x.  B
)  e.  RR )
4 0red 9643 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
53, 4ltnled 9781 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  B )  <  0  <->  -.  0  <_  ( A  x.  B ) ) )
61adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  ->  A  e.  RR )
72adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  ->  B  e.  RR )
8 simprl 762 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  -> 
0  <_  A )
9 simprr 764 . . . . . . . . . 10  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  -> 
0  <_  B )
106, 7, 8, 9mulge0d 10189 . . . . . . . . 9  |-  ( (
ph  /\  ( 0  <_  A  /\  0  <_  B ) )  -> 
0  <_  ( A  x.  B ) )
1110ex 435 . . . . . . . 8  |-  ( ph  ->  ( ( 0  <_  A  /\  0  <_  B
)  ->  0  <_  ( A  x.  B ) ) )
1211con3d 138 . . . . . . 7  |-  ( ph  ->  ( -.  0  <_ 
( A  x.  B
)  ->  -.  (
0  <_  A  /\  0  <_  B ) ) )
135, 12sylbid 218 . . . . . 6  |-  ( ph  ->  ( ( A  x.  B )  <  0  ->  -.  ( 0  <_  A  /\  0  <_  B
) ) )
14 ianor 490 . . . . . 6  |-  ( -.  ( 0  <_  A  /\  0  <_  B )  <-> 
( -.  0  <_  A  \/  -.  0  <_  B ) )
1513, 14syl6ib 229 . . . . 5  |-  ( ph  ->  ( ( A  x.  B )  <  0  ->  ( -.  0  <_  A  \/  -.  0  <_  B ) ) )
161, 4ltnled 9781 . . . . . 6  |-  ( ph  ->  ( A  <  0  <->  -.  0  <_  A )
)
172, 4ltnled 9781 . . . . . 6  |-  ( ph  ->  ( B  <  0  <->  -.  0  <_  B )
)
1816, 17orbi12d 714 . . . . 5  |-  ( ph  ->  ( ( A  <  0  \/  B  <  0 )  <->  ( -.  0  <_  A  \/  -.  0  <_  B ) ) )
1915, 18sylibrd 237 . . . 4  |-  ( ph  ->  ( ( A  x.  B )  <  0  ->  ( A  <  0  \/  B  <  0
) ) )
2019imp 430 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  <  0  \/  B  <  0 ) )
21 simpr 462 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  A  <  0 )
221adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  A  e.  RR )
232adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  B  e.  RR )
24 simpr 462 . . . . . . 7  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  x.  B )  <  0
)
2522, 23, 24mul2lt0llt0 11400 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  0  <  B )
2621, 25jca 534 . . . . 5  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  A  <  0 )  ->  ( A  <  0  /\  0  <  B ) )
2726ex 435 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( A  <  0  ->  ( A  <  0  /\  0  < 
B ) ) )
2822, 23, 24mul2lt0rlt0 11398 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  B  <  0 )  ->  0  <  A )
29 simpr 462 . . . . . 6  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  B  <  0 )  ->  B  <  0 )
3028, 29jca 534 . . . . 5  |-  ( ( ( ph  /\  ( A  x.  B )  <  0 )  /\  B  <  0 )  ->  (
0  <  A  /\  B  <  0 ) )
3130ex 435 . . . 4  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( B  <  0  ->  ( 0  <  A  /\  B  <  0 ) ) )
3227, 31orim12d 846 . . 3  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( A  <  0  \/  B  <  0 )  ->  (
( A  <  0  /\  0  <  B )  \/  ( 0  < 
A  /\  B  <  0 ) ) ) )
3320, 32mpd 15 . 2  |-  ( (
ph  /\  ( A  x.  B )  <  0
)  ->  ( ( A  <  0  /\  0  <  B )  \/  (
0  <  A  /\  B  <  0 ) ) )
341adantr 466 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  A  e.  RR )
35 0red 9643 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
0  e.  RR )
362adantr 466 . . . . . 6  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  RR )
37 simprr 764 . . . . . 6  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
0  <  B )
3836, 37elrpd 11338 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  RR+ )
39 simprl 762 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  A  <  0 )
4034, 35, 38, 39ltmul1dd 11393 . . . 4  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  ( 0  x.  B ) )
4136recnd 9668 . . . . 5  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  ->  B  e.  CC )
4241mul02d 9830 . . . 4  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( 0  x.  B
)  =  0 )
4340, 42breqtrd 4450 . . 3  |-  ( (
ph  /\  ( A  <  0  /\  0  < 
B ) )  -> 
( A  x.  B
)  <  0 )
442adantr 466 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  B  e.  RR )
45 0red 9643 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
0  e.  RR )
461adantr 466 . . . . . 6  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  RR )
47 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
0  <  A )
4846, 47elrpd 11338 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  RR+ )
49 simprr 764 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  B  <  0 )
5044, 45, 48, 49ltmul2dd 11394 . . . 4  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  B
)  <  ( A  x.  0 ) )
5146recnd 9668 . . . . 5  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  ->  A  e.  CC )
5251mul01d 9831 . . . 4  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  0 )  =  0 )
5350, 52breqtrd 4450 . . 3  |-  ( (
ph  /\  ( 0  <  A  /\  B  <  0 ) )  -> 
( A  x.  B
)  <  0 )
5443, 53jaodan 792 . 2  |-  ( (
ph  /\  ( ( A  <  0  /\  0  <  B )  \/  (
0  <  A  /\  B  <  0 ) ) )  ->  ( A  x.  B )  <  0
)
5533, 54impbida 840 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 187    \/ wo 369    /\ wa 370    e. wcel 1870   class class class wbr 4426  (class class class)co 6305   RRcr 9537   0cc0 9538    x. cmul 9543    < clt 9674    <_ cle 9675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-rp 11303
This theorem is referenced by:  ztprmneprm  38888
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