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Theorem ltabs 10969
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 7862 . . . . . . 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 7977 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 <_ A )
8 absid 10953 . . . . . 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 3990 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < A )
114ltnrd 7971 . . . 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 7848 . . . . . . . 8 |- ( A e. RR -> A e. CC )
14 abscl 10933 . . . . . . . 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 8487 . . . . 5 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) # 0 )
19 abs00ap 10944 . . . . . 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 7861 . . . . 5 |- 0 e. RR
23 reaplt 8446 . . . . 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 1331 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A < 0 )
27 axltwlin 7928 . . . . 5 |- ( ( A e. RR /\ ( abs ` A ) e. RR /\ 0 e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2822, 27mp3an3 1308 . . . 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 1335   e. wcel 2128   class class class wbr 3965   ` cfv 5167   CCcc 7713   RRcr 7714   0cc0 7715   < clt 7895   <_ cle 7896   # cap 8439   abscabs 10879
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-frec 6332  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-rp 9543  df-seqfrec 10327  df-exp 10401  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881
This theorem is referenced by:  abslt  10970  absle  10971  maxabslemlub  11089
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