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Theorem abs2difabs 10132
Description: Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
Assertion
Ref Expression
abs2difabs  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  (
( abs `  A
)  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B )
) )

Proof of Theorem abs2difabs
StepHypRef Expression
1 abs2dif 10130 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( abs `  B
)  -  ( abs `  A ) )  <_ 
( abs `  ( B  -  A )
) )
21ancoms 264 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  B
)  -  ( abs `  A ) )  <_ 
( abs `  ( B  -  A )
) )
3 abscl 10075 . . . . 5  |-  ( A  e.  CC  ->  ( abs `  A )  e.  RR )
43recnd 7209 . . . 4  |-  ( A  e.  CC  ->  ( abs `  A )  e.  CC )
5 abscl 10075 . . . . 5  |-  ( B  e.  CC  ->  ( abs `  B )  e.  RR )
65recnd 7209 . . . 4  |-  ( B  e.  CC  ->  ( abs `  B )  e.  CC )
7 negsubdi2 7434 . . . 4  |-  ( ( ( abs `  A
)  e.  CC  /\  ( abs `  B )  e.  CC )  ->  -u ( ( abs `  A
)  -  ( abs `  B ) )  =  ( ( abs `  B
)  -  ( abs `  A ) ) )
84, 6, 7syl2an 283 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( ( abs `  A )  -  ( abs `  B ) )  =  ( ( abs `  B )  -  ( abs `  A ) ) )
9 abssub 10125 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  B )
)  =  ( abs `  ( B  -  A
) ) )
102, 8, 93brtr4d 3823 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B )
) )
11 abs2dif 10130 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) )
12 resubcl 7439 . . . . 5  |-  ( ( ( abs `  A
)  e.  RR  /\  ( abs `  B )  e.  RR )  -> 
( ( abs `  A
)  -  ( abs `  B ) )  e.  RR )
133, 5, 12syl2an 283 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A
)  -  ( abs `  B ) )  e.  RR )
14 subcl 7374 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
15 abscl 10075 . . . . 5  |-  ( ( A  -  B )  e.  CC  ->  ( abs `  ( A  -  B ) )  e.  RR )
1614, 15syl 14 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  ( A  -  B )
)  e.  RR )
17 absle 10113 . . . 4  |-  ( ( ( ( abs `  A
)  -  ( abs `  B ) )  e.  RR  /\  ( abs `  ( A  -  B
) )  e.  RR )  ->  ( ( abs `  ( ( abs `  A
)  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B )
)  <->  ( -u ( abs `  ( A  -  B ) )  <_ 
( ( abs `  A
)  -  ( abs `  B ) )  /\  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) ) ) )
1813, 16, 17syl2anc 403 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  (
( abs `  A
)  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B )
)  <->  ( -u ( abs `  ( A  -  B ) )  <_ 
( ( abs `  A
)  -  ( abs `  B ) )  /\  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) ) ) )
19 lenegcon1 7637 . . . . 5  |-  ( ( ( ( abs `  A
)  -  ( abs `  B ) )  e.  RR  /\  ( abs `  ( A  -  B
) )  e.  RR )  ->  ( -u (
( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
)  <->  -u ( abs `  ( A  -  B )
)  <_  ( ( abs `  A )  -  ( abs `  B ) ) ) )
2013, 16, 19syl2anc 403 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u ( ( abs `  A )  -  ( abs `  B
) )  <_  ( abs `  ( A  -  B ) )  <->  -u ( abs `  ( A  -  B
) )  <_  (
( abs `  A
)  -  ( abs `  B ) ) ) )
2120anbi1d 453 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( -u (
( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
)  /\  ( ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B
) ) )  <->  ( -u ( abs `  ( A  -  B ) )  <_ 
( ( abs `  A
)  -  ( abs `  B ) )  /\  ( ( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
) ) ) )
2218, 21bitr4d 189 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  (
( abs `  A
)  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B )
)  <->  ( -u (
( abs `  A
)  -  ( abs `  B ) )  <_ 
( abs `  ( A  -  B )
)  /\  ( ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B
) ) ) ) )
2310, 11, 22mpbir2and 886 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( abs `  (
( abs `  A
)  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   CCcc 7041   RRcr 7042    <_ cle 7216    - cmin 7346   -ucneg 7347   abscabs 10021
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-rp 8816  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023
This theorem is referenced by:  abs2difabsd  10223  abscn2  10291
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