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Theorem ltabs 11234
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 110 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ A < 0 ) -> A < 0 )
2 simpllr 534 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < ( abs ` A ) )
3 simpll 527 . . . . . . 7 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A e. RR )
43adantr 276 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A e. RR )
5 0red 8022 . . . . . . 7 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 e. RR )
6 simpr 110 . . . . . . 7 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 < A )
75, 4, 6ltled 8140 . . . . . 6 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> 0 <_ A )
8 absid 11218 . . . . . 6 |- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )
94, 7, 8syl2anc 411 . . . . 5 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> ( abs ` A ) = A )
102, 9breqtrd 4056 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> A < A )
114ltnrd 8133 . . . 4 |- ( ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) /\ 0 < A ) -> -. A < A )
1210, 11pm2.65da 662 . . 3 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> -. 0 < A )
13 recn 8007 . . . . . . . 8 |- ( A e. RR -> A e. CC )
14 abscl 11198 . . . . . . . 8 |- ( A e. CC -> ( abs ` A ) e. RR )
1513, 14syl 14 . . . . . . 7 |- ( A e. RR -> ( abs ` A ) e. RR )
1615ad2antrr 488 . . . . . 6 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) e. RR )
17 simpr 110 . . . . . 6 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> 0 < ( abs ` A ) )
1816, 17gt0ap0d 8650 . . . . 5 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( abs ` A ) # 0 )
19 abs00ap 11209 . . . . . 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 147 . . . 4 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A # 0 )
22 0re 8021 . . . . 5 |- 0 e. RR
23 reaplt 8609 . . . . 5 |- ( ( A e. RR /\ 0 e. RR ) -> ( A # 0 <-> ( A < 0 \/ 0 < A ) ) )
243, 22, 23sylancl 413 . . . 4 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( A # 0 <-> ( A < 0 \/ 0 < A ) ) )
2521, 24mpbid 147 . . 3 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> ( A < 0 \/ 0 < A ) )
2612, 25ecased 1360 . 2 |- ( ( ( A e. RR /\ A < ( abs ` A ) ) /\ 0 < ( abs ` A ) ) -> A < 0 )
27 axltwlin 8089 . . . . 5 |- ( ( A e. RR /\ ( abs ` A ) e. RR /\ 0 e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2822, 27mp3an3 1337 . . . 4 |- ( ( A e. RR /\ ( abs ` A ) e. RR ) -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
2915, 28mpdan 421 . . 3 |- ( A e. RR -> ( A < ( abs ` A ) -> ( A < 0 \/ 0 < ( abs ` A ) ) ) )
3029imp 124 . 2 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> ( A < 0 \/ 0 < ( abs ` A ) ) )
311, 26, 30mpjaodan 799 1 |- ( ( A e. RR /\ A < ( abs ` A ) ) -> A < 0 )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   class class class wbr 4030   ` cfv 5255   CCcc 7872   RRcr 7873   0cc0 7874   < clt 8056   <_ cle 8057   # cap 8602   abscabs 11144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-rp 9723  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146
This theorem is referenced by:  abslt  11235  absle  11236  maxabslemlub  11354
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