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Theorem abslt 11814
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 730 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  RR )
21renegcld 9226 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  e.  RR )
31recnd 8877 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  CC )
4 abscl 11779 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
53, 4syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  e.  RR )
6 simplr 731 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  B  e.  RR )
7 leabs 11800 . . . . . . . 8  |-  ( -u A  e.  RR  ->  -u A  <_  ( abs `  -u A
) )
82, 7syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  -u A ) )
9 absneg 11778 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A
) )
103, 9syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  -u A
)  =  ( abs `  A ) )
118, 10breqtrd 4063 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  A ) )
12 simpr 447 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
132, 5, 6, 11, 12lelttrd 8990 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <  B
)
14 leabs 11800 . . . . . . 7  |-  ( A  e.  RR  ->  A  <_  ( abs `  A
) )
1514ad2antrr 706 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <_  ( abs `  A ) )
161, 5, 6, 15, 12lelttrd 8990 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <  B )
1713, 16jca 518 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( -u A  < 
B  /\  A  <  B ) )
1817ex 423 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  ->  (
-u A  <  B  /\  A  <  B ) ) )
19 absor 11801 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A ) )
2019adantr 451 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A ) )
21 breq1 4042 . . . . . . 7  |-  ( ( abs `  A )  =  A  ->  (
( abs `  A
)  <  B  <->  A  <  B ) )
2221biimprd 214 . . . . . 6  |-  ( ( abs `  A )  =  A  ->  ( A  <  B  ->  ( abs `  A )  < 
B ) )
23 breq1 4042 . . . . . . 7  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  <  B  <->  -u A  < 
B ) )
2423biimprd 214 . . . . . 6  |-  ( ( abs `  A )  =  -u A  ->  ( -u A  <  B  -> 
( abs `  A
)  <  B )
)
2522, 24jaoa 496 . . . . 5  |-  ( ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A )  -> 
( ( A  < 
B  /\  -u A  < 
B )  ->  ( abs `  A )  < 
B ) )
2625ancomsd 440 . . . 4  |-  ( ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A )  -> 
( ( -u A  <  B  /\  A  < 
B )  ->  ( abs `  A )  < 
B ) )
2720, 26syl 15 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  ->  ( abs `  A )  < 
B ) )
2818, 27impbid 183 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u A  <  B  /\  A  < 
B ) ) )
29 ltnegcon1 9291 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  < 
B  <->  -u B  <  A
) )
3029anbi1d 685 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  <->  ( -u B  <  A  /\  A  < 
B ) ) )
3128, 30bitrd 244 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    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   CCcc 8751   RRcr 8752    < clt 8883    <_ cle 8884   -ucneg 9054   abscabs 11735
This theorem is referenced by:  absdiflt  11817  abslti  11890  absltd  11928  tanregt0  19917  argregt0  19980  efopnlem2  20020  stoweidlem7  27859  stoweidlem13  27865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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