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Theorem abs2dif 13374
Description: Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
Assertion
Ref Expression
abs2dif  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) )

Proof of Theorem abs2dif
StepHypRef Expression
1 subid1 9893 . . . 4  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
21fveq2d 5885 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A  - 
0 ) )  =  ( abs `  A
) )
3 subid1 9893 . . . 4  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
43fveq2d 5885 . . 3  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
52, 4oveqan12d 6324 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  -  ( abs `  ( B  -  0 ) ) )  =  ( ( abs `  A
)  -  ( abs `  B ) ) )
6 0cn 9634 . . . 4  |-  0  e.  CC
7 abs3dif 13373 . . . 4  |-  ( ( A  e.  CC  /\  0  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  - 
0 ) )  <_ 
( ( abs `  ( A  -  B )
)  +  ( abs `  ( B  -  0 ) ) ) )
86, 7mp3an2 1348 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  0 ) )  <_  ( ( abs `  ( A  -  B ) )  +  ( abs `  ( B  -  0 ) ) ) )
9 subcl 9873 . . . . . . . 8  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( A  -  0 )  e.  CC )
106, 9mpan2 675 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  -  0 )  e.  CC )
11 abscl 13320 . . . . . . 7  |-  ( ( A  -  0 )  e.  CC  ->  ( abs `  ( A  - 
0 ) )  e.  RR )
1210, 11syl 17 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( A  - 
0 ) )  e.  RR )
13 subcl 9873 . . . . . . . 8  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B  -  0 )  e.  CC )
146, 13mpan2 675 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  -  0 )  e.  CC )
15 abscl 13320 . . . . . . 7  |-  ( ( B  -  0 )  e.  CC  ->  ( abs `  ( B  - 
0 ) )  e.  RR )
1614, 15syl 17 . . . . . 6  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  e.  RR )
1712, 16anim12i 568 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR ) )
18 subcl 9873 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
19 abscl 13320 . . . . . 6  |-  ( ( A  -  B )  e.  CC  ->  ( abs `  ( A  -  B ) )  e.  RR )
2018, 19syl 17 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  B )
)  e.  RR )
21 df-3an 984 . . . . 5  |-  ( ( ( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR  /\  ( abs `  ( A  -  B ) )  e.  RR )  <->  ( (
( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR )  /\  ( abs `  ( A  -  B ) )  e.  RR ) )
2217, 20, 21sylanbrc 668 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR  /\  ( abs `  ( A  -  B ) )  e.  RR ) )
23 lesubadd 10085 . . . 4  |-  ( ( ( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR  /\  ( abs `  ( A  -  B ) )  e.  RR )  ->  (
( ( abs `  ( A  -  0 ) )  -  ( abs `  ( B  -  0 ) ) )  <_ 
( abs `  ( A  -  B )
)  <->  ( abs `  ( A  -  0 ) )  <_  ( ( abs `  ( A  -  B ) )  +  ( abs `  ( B  -  0 ) ) ) ) )
2422, 23syl 17 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  ( A  -  0 ) )  -  ( abs `  ( B  - 
0 ) ) )  <_  ( abs `  ( A  -  B )
)  <->  ( abs `  ( A  -  0 ) )  <_  ( ( abs `  ( A  -  B ) )  +  ( abs `  ( B  -  0 ) ) ) ) )
258, 24mpbird 235 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  -  ( abs `  ( B  -  0 ) ) )  <_ 
( abs `  ( A  -  B )
) )
265, 25eqbrtrrd 4448 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538    + caddc 9541    <_ cle 9675    - cmin 9859   abscabs 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278
This theorem is referenced by:  abs2difabs  13376  absrdbnd  13383  caubnd2  13399  abs2difd  13497  abelthlem2  23252  logfacbnd3  24014  log2sumbnd  24245  abs2difi  30114
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