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Theorem ltabs 10174
Description: A number which is less than its absolute value is negative. (Contributed by Jim Kingdon, 12-Aug-2021.)
Assertion
Ref Expression
ltabs  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <  0 )

Proof of Theorem ltabs
StepHypRef Expression
1 simpr 108 . 2  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  A  <  0 )  ->  A  <  0 )
2 simpllr 501 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  A  <  ( abs `  A
) )
3 simpll 496 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  A  e.  RR )
43adantr 270 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  A  e.  RR )
5 0red 7234 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  0  e.  RR )
6 simpr 108 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  0  <  A )
75, 4, 6ltled 7347 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  0  <_  A )
8 absid 10158 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( abs `  A
)  =  A )
94, 7, 8syl2anc 403 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  ( abs `  A )  =  A )
102, 9breqtrd 3829 . . . 4  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  A  <  A )
114ltnrd 7341 . . . 4  |-  ( ( ( ( A  e.  RR  /\  A  < 
( abs `  A
) )  /\  0  <  ( abs `  A
) )  /\  0  <  A )  ->  -.  A  <  A )
1210, 11pm2.65da 620 . . 3  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  -.  0  <  A )
13 recn 7220 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
14 abscl 10138 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
1513, 14syl 14 . . . . . . 7  |-  ( A  e.  RR  ->  ( abs `  A )  e.  RR )
1615ad2antrr 472 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  ( abs `  A )  e.  RR )
17 simpr 108 . . . . . 6  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  0  <  ( abs `  A
) )
1816, 17gt0ap0d 7847 . . . . 5  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  ( abs `  A ) #  0 )
19 abs00ap 10149 . . . . . 6  |-  ( A  e.  CC  ->  (
( abs `  A
) #  0  <->  A #  0
) )
203, 13, 193syl 17 . . . . 5  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  (
( abs `  A
) #  0  <->  A #  0
) )
2118, 20mpbid 145 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  A #  0 )
22 0re 7233 . . . . 5  |-  0  e.  RR
23 reaplt 7807 . . . . 5  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A #  0  <->  ( A  <  0  \/  0  <  A ) ) )
243, 22, 23sylancl 404 . . . 4  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  ( A #  0  <->  ( A  <  0  \/  0  < 
A ) ) )
2521, 24mpbid 145 . . 3  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  ( A  <  0  \/  0  <  A ) )
2612, 25ecased 1281 . 2  |-  ( ( ( A  e.  RR  /\  A  <  ( abs `  A ) )  /\  0  <  ( abs `  A
) )  ->  A  <  0 )
27 axltwlin 7299 . . . . 5  |-  ( ( A  e.  RR  /\  ( abs `  A )  e.  RR  /\  0  e.  RR )  ->  ( A  <  ( abs `  A
)  ->  ( A  <  0  \/  0  < 
( abs `  A
) ) ) )
2822, 27mp3an3 1258 . . . 4  |-  ( ( A  e.  RR  /\  ( abs `  A )  e.  RR )  -> 
( A  <  ( abs `  A )  -> 
( A  <  0  \/  0  <  ( abs `  A ) ) ) )
2915, 28mpdan 412 . . 3  |-  ( A  e.  RR  ->  ( A  <  ( abs `  A
)  ->  ( A  <  0  \/  0  < 
( abs `  A
) ) ) )
3029imp 122 . 2  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( A  <  0  \/  0  <  ( abs `  A
) ) )
311, 26, 30mpjaodan 745 1  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   class class class wbr 3805   ` cfv 4952   CCcc 7093   RRcr 7094   0cc0 7095    < clt 7267    <_ cle 7268   # cap 7800   abscabs 10084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-rp 8868  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086
This theorem is referenced by:  abslt  10175  absle  10176  maxabslemlub  10294
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