MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abslt Structured version   Unicode version

Theorem abslt 12120
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 732 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  RR )
21renegcld 9466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  e.  RR )
31recnd 9116 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  e.  CC )
4 abscl 12085 . . . . . . 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 733 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  B  e.  RR )
7 leabs 12106 . . . . . . . 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 12084 . . . . . . . 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 4238 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <_  ( abs `  A ) )
12 simpr 449 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( abs `  A
)  <  B )
132, 5, 6, 11, 12lelttrd 9230 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  -> 
-u A  <  B
)
14 leabs 12106 . . . . . . 7  |-  ( A  e.  RR  ->  A  <_  ( abs `  A
) )
1514ad2antrr 708 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <_  ( abs `  A ) )
161, 5, 6, 15, 12lelttrd 9230 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  A  <  B )
1713, 16jca 520 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( abs `  A
)  <  B )  ->  ( -u A  < 
B  /\  A  <  B ) )
1817ex 425 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  ->  (
-u A  <  B  /\  A  <  B ) ) )
19 absor 12107 . . . . 5  |-  ( A  e.  RR  ->  (
( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A ) )
2019adantr 453 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A ) )
21 breq1 4217 . . . . . . 7  |-  ( ( abs `  A )  =  A  ->  (
( abs `  A
)  <  B  <->  A  <  B ) )
2221biimprd 216 . . . . . 6  |-  ( ( abs `  A )  =  A  ->  ( A  <  B  ->  ( abs `  A )  < 
B ) )
23 breq1 4217 . . . . . . 7  |-  ( ( abs `  A )  =  -u A  ->  (
( abs `  A
)  <  B  <->  -u A  < 
B ) )
2423biimprd 216 . . . . . 6  |-  ( ( abs `  A )  =  -u A  ->  ( -u A  <  B  -> 
( abs `  A
)  <  B )
)
2522, 24jaoa 498 . . . . 5  |-  ( ( ( abs `  A
)  =  A  \/  ( abs `  A )  =  -u A )  -> 
( ( A  < 
B  /\  -u A  < 
B )  ->  ( abs `  A )  < 
B ) )
2625ancomsd 442 . . . 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 185 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs `  A
)  <  B  <->  ( -u A  <  B  /\  A  < 
B ) ) )
29 ltnegcon1 9531 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u A  < 
B  <->  -u B  <  A
) )
3029anbi1d 687 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( -u A  <  B  /\  A  < 
B )  <->  ( -u B  <  A  /\  A  < 
B ) ) )
3128, 30bitrd 246 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456   CCcc 8990   RRcr 8991    < clt 9122    <_ cle 9123   -ucneg 9294   abscabs 12041
This theorem is referenced by:  absdiflt  12123  abslti  12196  absltd  12234  tanregt0  20443  argregt0  20507  efopnlem2  20550  ftc1anclem1  26282  stoweidlem7  27734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043
  Copyright terms: Public domain W3C validator