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Theorem abs3lem 12810
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 750 . . . . 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 751 . . . . 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 9707 . . . 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 12751 . . . 4  |-  ( ( A  -  B )  e.  CC  ->  ( abs `  ( A  -  B ) )  e.  RR )
53, 4syl 16 . . 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 752 . . . . . 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 9707 . . . . 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 12751 . . . . 5  |-  ( ( A  -  C )  e.  CC  ->  ( abs `  ( A  -  C ) )  e.  RR )
97, 8syl 16 . . . 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 9707 . . . . 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 12751 . . . . 5  |-  ( ( C  -  B )  e.  CC  ->  ( abs `  ( C  -  B ) )  e.  RR )
1210, 11syl 16 . . . 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 9401 . . 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 753 . . 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 12803 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( abs `  ( A  -  B ) )  <_ 
( ( abs `  ( A  -  C )
)  +  ( abs `  ( C  -  B
) ) ) )
161, 2, 6, 15syl3anc 1211 . . 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 748 . . . 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 749 . . . 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 10560 . . 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 9517 . 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 434 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1755   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269    + caddc 9273    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   2c2 10359   abscabs 12707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-seq 11791  df-exp 11850  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709
This theorem is referenced by:  cau3  12827  abs3lemd  12931  rlimuni  13012  climuni  13014  2clim  13034  addcn2  13055  mulcn2  13057  ulmcaulem  21744  ulmcau  21745
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