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Theorem abs3lem 10540
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 500 . . . . 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 501 . . . . 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 7791 . . . 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 10480 . . . 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 502 . . . . . 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 7791 . . . . 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 10480 . . . . 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 7791 . . . . 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 10480 . . . . 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 7515 . . 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 503 . . 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 10534 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( abs `  ( A  -  B ) )  <_ 
( ( abs `  ( A  -  C )
)  +  ( abs `  ( C  -  B
) ) ) )
161, 2, 6, 15syl3anc 1174 . . 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 498 . . . 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 499 . . . 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 8661 . . 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 7606 . 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 113 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 102    e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7346   RRcr 7347    + caddc 7351    < clt 7520    <_ cle 7521    - cmin 7651    / cdiv 8137   2c2 8471   abscabs 10426
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-rp 9133  df-iseq 9849  df-seq3 9850  df-exp 9951  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428
This theorem is referenced by:  cau3  10544  abs3lemi  10586  abs3lemd  10630  climuni  10677  2clim  10685  addcn2  10695  mulcn2  10697
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