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Theorem ltabs 10862
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 523 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < ( abs ` A ) )
3 simpll 518 . . . . . . 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 7770 . . . . . . 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 7884 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 <_ A )
8 absid 10846 . . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
94, 7, 8syl2anc 408 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> ( abs ` A ) = A )
102, 9breqtrd 3954 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < A )
114ltnrd 7878 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> -. A < A )
1210, 11pm2.65da 650 . . 3 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> -. 0 < A )
13 recn 7756 . . . . . . . 8 |- ( A e. RR -> A e. CC )
14 abscl 10826 . . . . . . . 8 |- ( A e. CC -> ( abs ` A ) e. RR )
1513, 14syl 14 . . . . . . 7 |- ( A e. RR -> ( abs ` A ) e. RR )
1615ad2antrr 479 . . . . . 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 8394 . . . . 5 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) # 0 )
19 abs00ap 10837 . . . . . 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 7769 . . . . 5 |- 0 e. RR
23 reaplt 8353 . . . . 5 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/ 0 < A ) ) )
243, 22, 23sylancl 409 . . . 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 1327 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A < 0 )
27 axltwlin 7835 . . . . 5 |- ( ( A e. RR /\ ( abs ` A ) e. RR /\ 0 e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2822, 27mp3an3 1304 . . . 4 |- ( ( A e. RR /\ ( abs ` A ) e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2915, 28mpdan 417 . . 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 787 1 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123   CCcc 7621   RRcr 7622   0cc0 7623   < clt 7803   <_ cle 7804   # cap 8346   abscabs 10772
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-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741  ax-arch 7742  ax-caucvg 7743
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  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-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-div 8436  df-inn 8724  df-2 8782  df-3 8783  df-4 8784  df-n0 8981  df-z 9058  df-uz 9330  df-rp 9445  df-seqfrec 10222  df-exp 10296  df-cj 10617  df-re 10618  df-im 10619  df-rsqrt 10773  df-abs 10774
This theorem is referenced by:  abslt  10863  absle  10864  maxabslemlub  10982
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