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Theorem ltabs 11029
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 109 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ A < 0 ) -> A < 0 )
2 simpllr 524 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < ( abs ` A ) )
3 simpll 519 . . . . . . 7 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A e. RR )
43adantr 274 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A e. RR )
5 0red 7900 . . . . . . 7 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 e. RR )
6 simpr 109 . . . . . . 7 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 < A )
75, 4, 6ltled 8017 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 <_ A )
8 absid 11013 . . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
94, 7, 8syl2anc 409 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> ( abs ` A ) = A )
102, 9breqtrd 4008 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < A )
114ltnrd 8010 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> -. A < A )
1210, 11pm2.65da 651 . . 3 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> -. 0 < A )
13 recn 7886 . . . . . . . 8 |- ( A e. RR -> A e. CC )
14 abscl 10993 . . . . . . . 8 |- ( A e. CC -> ( abs ` A ) e. RR )
1513, 14syl 14 . . . . . . 7 |- ( A e. RR -> ( abs ` A ) e. RR )
1615ad2antrr 480 . . . . . 6 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) e. RR )
17 simpr 109 . . . . . 6 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> 0 < ( abs ` A ) )
1816, 17gt0ap0d 8527 . . . . 5 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) # 0 )
19 abs00ap 11004 . . . . . 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 146 . . . 4 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A # 0 )
22 0re 7899 . . . . 5 |- 0 e. RR
23 reaplt 8486 . . . . 5 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/ 0 < A ) ) )
243, 22, 23sylancl 410 . . . 4 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( A # 0 <-> ( A < 0 \/ 0 < A ) ) )
2521, 24mpbid 146 . . 3 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( A < 0 \/ 0 < A ) )
2612, 25ecased 1339 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A < 0 )
27 axltwlin 7966 . . . . 5 |- ( ( A e. RR /\ ( abs ` A ) e. RR /\ 0 e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2822, 27mp3an3 1316 . . . 4 |- ( ( A e. RR /\ ( abs ` A ) e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2915, 28mpdan 418 . . 3 |- ( A e. RR -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
3029imp 123 . 2 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( A < 0 \/ 0 < ( abs ` A ) ) )
311, 26, 30mpjaodan 788 1 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188   CCcc 7751   RRcr 7752   0cc0 7753   < clt 7933   <_ cle 7934   # cap 8479   abscabs 10939
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941
This theorem is referenced by:  abslt  11030  absle  11031  maxabslemlub  11149
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