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Theorem abslt 10891
Description: Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005.) (Revised by Mario Carneiro, 29-May-2016.)
Assertion
Ref Expression
abslt  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u B  <  A  /\  A  < 
B ) ) )

Proof of Theorem abslt
StepHypRef Expression
1 simpll 519 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  RR )
21renegcld 8165 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  e.  RR )
31recnd 7817 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  CC )
4 abscl 10854 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
53, 4syl 14 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  e.  RR )
6 simplr 520 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  B  e.  RR )
7 leabs 10877 . . . . . . 7  |-  ( -u A  e.  RR  ->  -u A  <_  ( abs `  -u A
) )
82, 7syl 14 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  -u A ) )
9 absneg 10853 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A
) )
103, 9syl 14 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  -u A
)  =  ( abs `  A ) )
118, 10breqtrd 3961 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  A ) )
12 simpr 109 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
132, 5, 6, 11, 12lelttrd 7910 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <  B
)
14 leabs 10877 . . . . . 6  |-  ( A  e.  RR  ->  A  <_  ( abs `  A
) )
1514ad2antrr 480 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <_  ( abs `  A ) )
161, 5, 6, 15, 12lelttrd 7910 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <  B )
1713, 16jca 304 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( -u A  < 
B  /\  A  <  B ) )
18 simpll 519 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  e.  RR )
19 simpl 108 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  e.  RR )
2019recnd 7817 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  e.  CC )
2120, 9syl 14 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  -u A )  =  ( abs `  A
) )
2219renegcld 8165 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  -u A  e.  RR )
23 0red 7790 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  0  e.  RR )
24 ltabs 10890 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <  0 )
2519, 23, 24ltled 7904 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <_  0 )
2619le0neg1d 8302 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( A  <_  0  <->  0  <_  -u A ) )
2725, 26mpbid 146 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  0  <_ 
-u A )
28 absid 10874 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  0  <_  -u A )  ->  ( abs `  -u A
)  =  -u A
)
2922, 27, 28syl2anc 409 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  -u A )  = 
-u A )
3021, 29eqtr3d 2175 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  A )  = 
-u A )
3118, 30sylan 281 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  ( abs `  A
)  =  -u A
)
32 simplrl 525 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  -u A  <  B
)
3331, 32eqbrtrd 3957 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  ( abs `  A
)  <  B )
34 simpr 109 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
35 simprr 522 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  <  B )
36 simplr 520 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  B  e.  RR )
3718recnd 7817 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  e.  CC )
3837, 4syl 14 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( abs `  A
)  e.  RR )
39 axltwlin 7855 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( abs `  A )  e.  RR )  ->  ( A  <  B  ->  ( A  <  ( abs `  A
)  \/  ( abs `  A )  <  B
) ) )
4018, 36, 38, 39syl3anc 1217 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( A  <  B  ->  ( A  <  ( abs `  A )  \/  ( abs `  A
)  <  B )
) )
4135, 40mpd 13 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( A  <  ( abs `  A )  \/  ( abs `  A
)  <  B )
)
4233, 34, 41mpjaodan 788 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( abs `  A
)  <  B )
4317, 42impbida 586 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u A  <  B  /\  A  < 
B ) ) )
44 ltnegcon1 8248 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  < 
B  <->  -u B  <  A
) )
4544anbi1d 461 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  <->  ( -u B  <  A  /\  A  < 
B ) ) )
4643, 45bitrd 187 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u B  <  A  /\  A  < 
B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   class class class wbr 3936   ` cfv 5130   CCcc 7641   RRcr 7642   0cc0 7643    < clt 7823    <_ cle 7824   -ucneg 7957   abscabs 10800
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-rp 9470  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802
This theorem is referenced by:  absdiflt  10895  abslti  10941  absltd  10977
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