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Theorem ltabs 11055
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 530 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < ( abs ` A ) )
3 simpll 525 . . . . . . 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 7925 . . . . . . 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 8042 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 <_ A )
8 absid 11039 . . . . . 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 4016 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < A )
114ltnrd 8035 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> -. A < A )
1210, 11pm2.65da 657 . . 3 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> -. 0 < A )
13 recn 7911 . . . . . . . 8 |- ( A e. RR -> A e. CC )
14 abscl 11019 . . . . . . . 8 |- ( A e. CC -> ( abs ` A ) e. RR )
1513, 14syl 14 . . . . . . 7 |- ( A e. RR -> ( abs ` A ) e. RR )
1615ad2antrr 486 . . . . . 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 8552 . . . . 5 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) # 0 )
19 abs00ap 11030 . . . . . 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 7924 . . . . 5 |- 0 e. RR
23 reaplt 8511 . . . . 5 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/ 0 < A ) ) )
243, 22, 23sylancl 411 . . . 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 1345 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A < 0 )
27 axltwlin 7991 . . . . 5 |- ( ( A e. RR /\ ( abs ` A ) e. RR /\ 0 e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2822, 27mp3an3 1322 . . . 4 |- ( ( A e. RR /\ ( abs ` A ) e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2915, 28mpdan 419 . . 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 794 1 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 704   = wceq 1349   e. wcel 2142   class class class wbr 3990   ` cfv 5200   CCcc 7776   RRcr 7777   0cc0 7778   < clt 7958   <_ cle 7959   # cap 8504   abscabs 10965
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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-mulrcl 7877  ax-addcom 7878  ax-mulcom 7879  ax-addass 7880  ax-mulass 7881  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-1rid 7885  ax-0id 7886  ax-rnegex 7887  ax-precex 7888  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-apti 7893  ax-pre-ltadd 7894  ax-pre-mulgt0 7895  ax-pre-mulext 7896  ax-arch 7897  ax-caucvg 7898
This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-reap 8498  df-ap 8505  df-div 8594  df-inn 8883  df-2 8941  df-3 8942  df-4 8943  df-n0 9140  df-z 9217  df-uz 9492  df-rp 9615  df-seqfrec 10406  df-exp 10480  df-cj 10810  df-re 10811  df-im 10812  df-rsqrt 10966  df-abs 10967
This theorem is referenced by:  abslt  11056  absle  11057  maxabslemlub  11175
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