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Theorem abs2dif 11110
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 8175 . . . 4  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
21fveq2d 5519 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( A  - 
0 ) )  =  ( abs `  A
) )
3 subid1 8175 . . . 4  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
43fveq2d 5519 . . 3  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
52, 4oveqan12d 5893 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  -  ( abs `  ( B  -  0 ) ) )  =  ( ( abs `  A
)  -  ( abs `  B ) ) )
6 0cn 7948 . . . 4  |-  0  e.  CC
7 abs3dif 11109 . . . 4  |-  ( ( A  e.  CC  /\  0  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  - 
0 ) )  <_ 
( ( abs `  ( A  -  B )
)  +  ( abs `  ( B  -  0 ) ) ) )
86, 7mp3an2 1325 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  0 ) )  <_  ( ( abs `  ( A  -  B ) )  +  ( abs `  ( B  -  0 ) ) ) )
9 subcl 8154 . . . . . . . 8  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  ( A  -  0 )  e.  CC )
106, 9mpan2 425 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  -  0 )  e.  CC )
11 abscl 11055 . . . . . . 7  |-  ( ( A  -  0 )  e.  CC  ->  ( abs `  ( A  - 
0 ) )  e.  RR )
1210, 11syl 14 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( A  - 
0 ) )  e.  RR )
13 subcl 8154 . . . . . . . 8  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B  -  0 )  e.  CC )
146, 13mpan2 425 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  -  0 )  e.  CC )
15 abscl 11055 . . . . . . 7  |-  ( ( B  -  0 )  e.  CC  ->  ( abs `  ( B  - 
0 ) )  e.  RR )
1614, 15syl 14 . . . . . 6  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  e.  RR )
1712, 16anim12i 338 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR ) )
18 subcl 8154 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
19 abscl 11055 . . . . . 6  |-  ( ( A  -  B )  e.  CC  ->  ( abs `  ( A  -  B ) )  e.  RR )
2018, 19syl 14 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  B )
)  e.  RR )
21 df-3an 980 . . . . 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 417 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  e.  RR  /\  ( abs `  ( B  -  0 ) )  e.  RR  /\  ( abs `  ( A  -  B ) )  e.  RR ) )
23 lesubadd 8389 . . . 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 14 . . 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 167 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  0 ) )  -  ( abs `  ( B  -  0 ) ) )  <_ 
( abs `  ( A  -  B )
) )
265, 25eqbrtrrd 4027 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    e. wcel 2148   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810    + caddc 7813    <_ cle 7991    - cmin 8126   abscabs 11001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003
This theorem is referenced by:  abs2difabs  11112  caubnd2  11121  abs2difd  11201
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