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Theorem abslt 10159
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 496 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  RR )
21renegcld 7587 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  e.  RR )
31recnd 7245 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  CC )
4 abscl 10122 . . . . . 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 497 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  B  e.  RR )
7 leabs 10145 . . . . . . 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 10121 . . . . . . 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 3830 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  A ) )
12 simpr 108 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
132, 5, 6, 11, 12lelttrd 7337 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <  B
)
14 leabs 10145 . . . . . 6  |-  ( A  e.  RR  ->  A  <_  ( abs `  A
) )
1514ad2antrr 472 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <_  ( abs `  A ) )
161, 5, 6, 15, 12lelttrd 7337 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <  B )
1713, 16jca 300 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( -u A  < 
B  /\  A  <  B ) )
18 simpll 496 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  e.  RR )
19 simpl 107 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  e.  RR )
2019recnd 7245 . . . . . . . 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 7587 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  -u A  e.  RR )
23 0red 7218 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  0  e.  RR )
24 ltabs 10158 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <  0 )
2519, 23, 24ltled 7331 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  A  <_  0 )
2619le0neg1d 7721 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( A  <_  0  <->  0  <_  -u A ) )
2725, 26mpbid 145 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  0  <_ 
-u A )
28 absid 10142 . . . . . . . 8  |-  ( (
-u A  e.  RR  /\  0  <_  -u A )  ->  ( abs `  -u A
)  =  -u A
)
2922, 27, 28syl2anc 403 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  -u A )  = 
-u A )
3021, 29eqtr3d 2117 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <  ( abs `  A
) )  ->  ( abs `  A )  = 
-u A )
3118, 30sylan 277 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  ( abs `  A
)  =  -u A
)
32 simplrl 502 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  -u A  <  B
)
3331, 32eqbrtrd 3826 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  A  <  ( abs `  A ) )  ->  ( abs `  A
)  <  B )
34 simpr 108 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  <  B ) )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
35 simprr 499 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  A  <  B )
36 simplr 497 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  ->  B  e.  RR )
3718recnd 7245 . . . . . . 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 7283 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( abs `  A )  e.  RR )  ->  ( A  <  B  ->  ( A  <  ( abs `  A
)  \/  ( abs `  A )  <  B
) ) )
4018, 36, 38, 39syl3anc 1170 . . . . 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 745 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( -u A  <  B  /\  A  < 
B ) )  -> 
( abs `  A
)  <  B )
4317, 42impbida 561 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u A  <  B  /\  A  < 
B ) ) )
44 ltnegcon1 7670 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  < 
B  <->  -u B  <  A
) )
4544anbi1d 453 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  <->  ( -u B  <  A  /\  A  < 
B ) ) )
4643, 45bitrd 186 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 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   class class class wbr 3806   ` cfv 4953   CCcc 7077   RRcr 7078   0cc0 7079    < clt 7251    <_ cle 7252   -ucneg 7383   abscabs 10068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192  ax-arch 7193  ax-caucvg 7194
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-if 3370  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-ilim 4153  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864  df-inn 8143  df-2 8201  df-3 8202  df-4 8203  df-n0 8392  df-z 8469  df-uz 8737  df-rp 8852  df-iseq 9558  df-iexp 9609  df-cj 9914  df-re 9915  df-im 9916  df-rsqrt 10069  df-abs 10070
This theorem is referenced by:  absdiflt  10163  abslti  10209  absltd  10245
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