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Theorem abslt 12047
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 731 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  RR )
21renegcld 9398 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  e.  RR )
31recnd 9049 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  CC )
4 abscl 12012 . . . . . . 7  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
53, 4syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  e.  RR )
6 simplr 732 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  B  e.  RR )
7 leabs 12033 . . . . . . . 8  |-  ( -u A  e.  RR  ->  -u A  <_  ( abs `  -u A
) )
82, 7syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  -u A ) )
9 absneg 12011 . . . . . . . 8  |-  ( A  e.  CC  ->  ( abs `  -u A )  =  ( abs `  A
) )
103, 9syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  -u A
)  =  ( abs `  A ) )
118, 10breqtrd 4179 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  A ) )
12 simpr 448 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
132, 5, 6, 11, 12lelttrd 9162 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <  B
)
14 leabs 12033 . . . . . . 7  |-  ( A  e.  RR  ->  A  <_  ( abs `  A
) )
1514ad2antrr 707 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <_  ( abs `  A ) )
161, 5, 6, 15, 12lelttrd 9162 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <  B )
1713, 16jca 519 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( -u A  < 
B  /\  A  <  B ) )
1817ex 424 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  ->  (
-u A  <  B  /\  A  <  B ) ) )
19 absor 12034 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A ) )
2019adantr 452 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A ) )
21 breq1 4158 . . . . . . 7  |-  ( ( abs `  A )  =  A  ->  (
( abs `  A
)  <  B  <->  A  <  B ) )
2221biimprd 215 . . . . . 6  |-  ( ( abs `  A )  =  A  ->  ( A  <  B  ->  ( abs `  A )  < 
B ) )
23 breq1 4158 . . . . . . 7  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  <  B  <->  -u A  < 
B ) )
2423biimprd 215 . . . . . 6  |-  ( ( abs `  A )  =  -u A  ->  ( -u A  <  B  -> 
( abs `  A
)  <  B )
)
2522, 24jaoa 497 . . . . 5  |-  ( ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A )  -> 
( ( A  < 
B  /\  -u A  < 
B )  ->  ( abs `  A )  < 
B ) )
2625ancomsd 441 . . . 4  |-  ( ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A )  -> 
( ( -u A  <  B  /\  A  < 
B )  ->  ( abs `  A )  < 
B ) )
2720, 26syl 16 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  ->  ( abs `  A )  < 
B ) )
2818, 27impbid 184 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u A  <  B  /\  A  < 
B ) ) )
29 ltnegcon1 9463 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  < 
B  <->  -u B  <  A
) )
3029anbi1d 686 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  <->  ( -u B  <  A  /\  A  < 
B ) ) )
3128, 30bitrd 245 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 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396   CCcc 8923   RRcr 8924    < clt 9055    <_ cle 9056   -ucneg 9226   abscabs 11968
This theorem is referenced by:  absdiflt  12050  abslti  12123  absltd  12161  tanregt0  20310  argregt0  20374  efopnlem2  20417  stoweidlem7  27426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970
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