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Theorem abs3lem 12829
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 757 . . . . 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 758 . . . . 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 9722 . . . 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 12770 . . . 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 759 . . . . . 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 9722 . . . . 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 12770 . . . . 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 9722 . . . . 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 12770 . . . . 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 9416 . . 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 760 . . 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 12822 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( abs `  ( A  -  B ) )  <_ 
( ( abs `  ( A  -  C )
)  +  ( abs `  ( C  -  B
) ) ) )
161, 2, 6, 15syl3anc 1218 . . 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 755 . . . 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 756 . . . 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 10575 . . 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 9532 . 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 1756   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   CCcc 9283   RRcr 9284    + caddc 9288    < clt 9421    <_ cle 9422    - cmin 9598    / cdiv 9996   2c2 10374   abscabs 12726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728
This theorem is referenced by:  cau3  12846  abs3lemd  12950  rlimuni  13031  climuni  13033  2clim  13053  addcn2  13074  mulcn2  13076  ulmcaulem  21862  ulmcau  21863
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