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Theorem mul2lt0rlt0 27388
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 )
mul2lt0.3  |-  ( ph  ->  ( A  x.  B
)  <  0 )
Assertion
Ref Expression
mul2lt0rlt0  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )

Proof of Theorem mul2lt0rlt0
StepHypRef Expression
1 mul2lt0.3 . . . . 5  |-  ( ph  ->  ( A  x.  B
)  <  0 )
21adantr 465 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  ( A  x.  B )  <  0 )
3 mul2lt0.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
4 mul2lt0.2 . . . . . . 7  |-  ( ph  ->  B  e.  RR )
53, 4remulcld 9636 . . . . . 6  |-  ( ph  ->  ( A  x.  B
)  e.  RR )
65adantr 465 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  ( A  x.  B )  e.  RR )
7 0re 9608 . . . . . 6  |-  0  e.  RR
87a1i 11 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  0  e.  RR )
9 negelrp 27387 . . . . . . 7  |-  ( B  e.  RR  ->  ( -u B  e.  RR+  <->  B  <  0 ) )
104, 9syl 16 . . . . . 6  |-  ( ph  ->  ( -u B  e.  RR+ 
<->  B  <  0 ) )
1110biimpar 485 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  -u B  e.  RR+ )
126, 8, 11ltdiv1d 11309 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  <  0  <->  ( ( A  x.  B )  /  -u B )  < 
( 0  /  -u B
) ) )
132, 12mpbid 210 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  /  -u B
)  <  ( 0  /  -u B ) )
143recnd 9634 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  <  0 )  ->  A  e.  CC )
164recnd 9634 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
1716adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  <  0 )  ->  B  e.  CC )
1815, 17mulcld 9628 . . . . . 6  |-  ( (
ph  /\  B  <  0 )  ->  ( A  x.  B )  e.  CC )
194adantr 465 . . . . . . 7  |-  ( (
ph  /\  B  <  0 )  ->  B  e.  RR )
20 simpr 461 . . . . . . 7  |-  ( (
ph  /\  B  <  0 )  ->  B  <  0 )
2119, 20ltned 9732 . . . . . 6  |-  ( (
ph  /\  B  <  0 )  ->  B  =/=  0 )
2218, 17, 21divneg2d 10346 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  -u (
( A  x.  B
)  /  B )  =  ( ( A  x.  B )  /  -u B ) )
2315, 17, 21divcan4d 10338 . . . . . 6  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
2423negeqd 9826 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  -u (
( A  x.  B
)  /  B )  =  -u A )
2522, 24eqtr3d 2510 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  /  -u B
)  =  -u A
)
2617negcld 9929 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  -u B  e.  CC )
2717, 21negne0d 9940 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  -u B  =/=  0 )
28 div0 10247 . . . . 5  |-  ( (
-u B  e.  CC  /\  -u B  =/=  0
)  ->  ( 0  /  -u B )  =  0 )
2926, 27, 28syl2anc 661 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  (
0  /  -u B
)  =  0 )
3025, 29breq12d 4466 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  (
( ( A  x.  B )  /  -u B
)  <  ( 0  /  -u B )  <->  -u A  <  0 ) )
3113, 30mpbid 210 . 2  |-  ( (
ph  /\  B  <  0 )  ->  -u A  <  0 )
323adantr 465 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  A  e.  RR )
3332lt0neg2d 10135 . 2  |-  ( (
ph  /\  B  <  0 )  ->  (
0  <  A  <->  -u A  <  0 ) )
3431, 33mpbird 232 1  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509    < clt 9640   -ucneg 9818    / cdiv 10218   RR+crp 11232
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-rp 11233
This theorem is referenced by:  mul2lt0llt0  27390  mul2lt0bi  27392  sgnmul  28306  signsply0  28333
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