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Theorem abs2dif 11617
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 8366 . . . 4 |- ( A e. CC -> ( A - 0 ) = A )
21fveq2d 5631 . . 3 |- ( A e. CC -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
3 subid1 8366 . . . 4 |- ( B e. CC -> ( B - 0 ) = B )
43fveq2d 5631 . . 3 |- ( B e. CC -> ( abs ` ( B - 0 ) ) = ( abs ` B ) )
52, 4oveqan12d 6020 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) = ( ( abs ` A ) - ( abs ` B ) ) )
6 0cn 8138 . . . 4 |- 0 e. CC
7 abs3dif 11616 . . . 4 |- ( ( A e. CC /\ 0 e. CC /\ B e. CC ) -> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) )
86, 7mp3an2 1359 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) )
9 subcl 8345 . . . . . . . 8 |- ( ( A e. CC /\ 0 e. CC ) -> ( A - 0 ) e. CC )
106, 9mpan2 425 . . . . . . 7 |- ( A e. CC -> ( A - 0 ) e. CC )
11 abscl 11562 . . . . . . 7 |- ( ( A - 0 ) e. CC -> ( abs ` ( A - 0 ) ) e. RR )
1210, 11syl 14 . . . . . 6 |- ( A e. CC -> ( abs ` ( A - 0 ) ) e. RR )
13 subcl 8345 . . . . . . . 8 |- ( ( B e. CC /\ 0 e. CC ) -> ( B - 0 ) e. CC )
146, 13mpan2 425 . . . . . . 7 |- ( B e. CC -> ( B - 0 ) e. CC )
15 abscl 11562 . . . . . . 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 8345 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
19 abscl 11562 . . . . . 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 1004 . . . . 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 8581 . . . 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 4107 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 1002   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   + caddc 8002   <_ cle 8182   - cmin 8317   abscabs 11508
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510
This theorem is referenced by:  abs2difabs  11619  caubnd2  11628  abs2difd  11708
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