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Theorem mul2lt0rlt0 9558
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 mul2lt0.1 . . . . . 6  |-  ( ph  ->  A  e.  RR )
2 mul2lt0.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
31, 2remulcld 7808 . . . . 5  |-  ( ph  ->  ( A  x.  B
)  e.  RR )
43adantr 274 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  ( A  x.  B )  e.  RR )
5 0red 7779 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  0  e.  RR )
62adantr 274 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  B  e.  RR )
7 simpr 109 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  B  <  0 )
86, 7negelrpd 9488 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  -u B  e.  RR+ )
9 mul2lt0.3 . . . . 5  |-  ( ph  ->  ( A  x.  B
)  <  0 )
109adantr 274 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  ( A  x.  B )  <  0 )
114, 5, 8, 10ltdiv1dd 9553 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  /  -u B
)  <  ( 0  /  -u B ) )
121recnd 7806 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
1312adantr 274 . . . . . 6  |-  ( (
ph  /\  B  <  0 )  ->  A  e.  CC )
142recnd 7806 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
1514adantr 274 . . . . . 6  |-  ( (
ph  /\  B  <  0 )  ->  B  e.  CC )
1613, 15mulcld 7798 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  ( A  x.  B )  e.  CC )
176, 7lt0ap0d 8423 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  B #  0 )
1816, 15, 17divneg2apd 8576 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  -u (
( A  x.  B
)  /  B )  =  ( ( A  x.  B )  /  -u B ) )
1913, 15, 17divcanap4d 8568 . . . . 5  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  /  B )  =  A )
2019negeqd 7969 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  -u (
( A  x.  B
)  /  B )  =  -u A )
2118, 20eqtr3d 2174 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  (
( A  x.  B
)  /  -u B
)  =  -u A
)
2215negcld 8072 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  -u B  e.  CC )
2315, 17negap0d 8405 . . . 4  |-  ( (
ph  /\  B  <  0 )  ->  -u B #  0 )
2422, 23div0apd 8559 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  (
0  /  -u B
)  =  0 )
2511, 21, 243brtr3d 3959 . 2  |-  ( (
ph  /\  B  <  0 )  ->  -u A  <  0 )
261adantr 274 . . 3  |-  ( (
ph  /\  B  <  0 )  ->  A  e.  RR )
2726lt0neg2d 8290 . 2  |-  ( (
ph  /\  B  <  0 )  ->  (
0  <  A  <->  -u A  <  0 ) )
2825, 27mpbird 166 1  |-  ( (
ph  /\  B  <  0 )  ->  0  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632    x. cmul 7637    < clt 7812   -ucneg 7946    / cdiv 8444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-rp 9454
This theorem is referenced by:  mul2lt0llt0  9560
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