ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abs3lem Unicode version

Theorem abs3lem 10834
Description: Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
Assertion
Ref Expression
abs3lem  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  -> 
( ( ( abs `  ( A  -  C
) )  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) )  -> 
( abs `  ( A  -  B )
)  <  D )
)

Proof of Theorem abs3lem
StepHypRef Expression
1 simplll 505 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  A  e.  CC )
2 simpllr 506 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  B  e.  CC )
31, 2subcld 8037 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( A  -  B )  e.  CC )
4 abscl 10774 . . . 4  |-  ( ( A  -  B )  e.  CC  ->  ( abs `  ( A  -  B ) )  e.  RR )
53, 4syl 14 . . 3  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( A  -  B )
)  e.  RR )
6 simplrl 507 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  C  e.  CC )
71, 6subcld 8037 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( A  -  C )  e.  CC )
8 abscl 10774 . . . . 5  |-  ( ( A  -  C )  e.  CC  ->  ( abs `  ( A  -  C ) )  e.  RR )
97, 8syl 14 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( A  -  C )
)  e.  RR )
106, 2subcld 8037 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( C  -  B )  e.  CC )
11 abscl 10774 . . . . 5  |-  ( ( C  -  B )  e.  CC  ->  ( abs `  ( C  -  B ) )  e.  RR )
1210, 11syl 14 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( C  -  B )
)  e.  RR )
139, 12readdcld 7759 . . 3  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( ( abs `  ( A  -  C
) )  +  ( abs `  ( C  -  B ) ) )  e.  RR )
14 simplrr 508 . . 3  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  D  e.  RR )
15 abs3dif 10828 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( abs `  ( A  -  B ) )  <_ 
( ( abs `  ( A  -  C )
)  +  ( abs `  ( C  -  B
) ) ) )
161, 2, 6, 15syl3anc 1199 . . 3  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( A  -  B )
)  <_  ( ( abs `  ( A  -  C ) )  +  ( abs `  ( C  -  B )
) ) )
17 simprl 503 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( A  -  C )
)  <  ( D  /  2 ) )
18 simprr 504 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( C  -  B )
)  <  ( D  /  2 ) )
199, 12, 14, 17, 18lt2halvesd 8921 . . 3  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( ( abs `  ( A  -  C
) )  +  ( abs `  ( C  -  B ) ) )  <  D )
205, 13, 14, 16, 19lelttrd 7851 . 2  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  /\  ( ( abs `  ( A  -  C )
)  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) ) )  ->  ( abs `  ( A  -  B )
)  <  D )
2120ex 114 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  RR ) )  -> 
( ( ( abs `  ( A  -  C
) )  <  ( D  /  2 )  /\  ( abs `  ( C  -  B ) )  <  ( D  / 
2 ) )  -> 
( abs `  ( A  -  B )
)  <  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1463   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   CCcc 7582   RRcr 7583    + caddc 7587    < clt 7764    <_ cle 7765    - cmin 7897    / cdiv 8395   2c2 8731   abscabs 10720
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-rp 9394  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722
This theorem is referenced by:  cau3  10838  abs3lemi  10880  abs3lemd  10924  climuni  11013  2clim  11021  addcn2  11030  mulcn2  11032
  Copyright terms: Public domain W3C validator