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Theorem abs2dif 11070
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 8139 . . . 4 |- ( A e. CC -> ( A - 0 ) = A )
21fveq2d 5500 . . 3 |- ( A e. CC -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
3 subid1 8139 . . . 4 |- ( B e. CC -> ( B - 0 ) = B )
43fveq2d 5500 . . 3 |- ( B e. CC -> ( abs ` ( B - 0 ) ) = ( abs ` B ) )
52, 4oveqan12d 5872 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) = ( ( abs ` A ) - ( abs ` B ) ) )
6 0cn 7912 . . . 4 |- 0 e. CC
7 abs3dif 11069 . . . 4 |- ( ( A e. CC /\ 0 e. CC /\ B e. CC ) -> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) )
86, 7mp3an2 1320 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - 0 ) ) <_ ( ( abs ` ( A - B ) ) + ( abs ` ( B - 0 ) ) ) )
9 subcl 8118 . . . . . . . 8 |- ( ( A e. CC /\ 0 e. CC ) -> ( A - 0 ) e. CC )
106, 9mpan2 423 . . . . . . 7 |- ( A e. CC -> ( A - 0 ) e. CC )
11 abscl 11015 . . . . . . 7 |- ( ( A - 0 ) e. CC -> ( abs ` ( A - 0 ) ) e. RR )
1210, 11syl 14 . . . . . 6 |- ( A e. CC -> ( abs ` ( A - 0 ) ) e. RR )
13 subcl 8118 . . . . . . . 8 |- ( ( B e. CC /\ 0 e. CC ) -> ( B - 0 ) e. CC )
146, 13mpan2 423 . . . . . . 7 |- ( B e. CC -> ( B - 0 ) e. CC )
15 abscl 11015 . . . . . . 7 |- ( ( B - 0 ) e. CC -> ( abs ` ( B - 0 ) ) e. RR )
1614, 15syl 14 . . . . . 6 |- ( B e. CC -> ( abs ` ( B - 0 ) ) e. RR )
1712, 16anim12i 336 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR ) )
18 subcl 8118 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
19 abscl 11015 . . . . . 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 975 . . . . 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 415 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) e. RR /\ ( abs ` ( B - 0 ) ) e. RR /\ ( abs ` ( A - B ) ) e. RR ) )
23 lesubadd 8353 . . . 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 166 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - 0 ) ) - ( abs ` ( B - 0 ) ) ) <_ ( abs ` ( A - B ) ) )
265, 25eqbrtrrd 4013 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 103   <-> wb 104   /\ w3a 973   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   + caddc 7777   <_ cle 7955   - cmin 8090   abscabs 10961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963
This theorem is referenced by:  abs2difabs  11072  caubnd2  11081  abs2difd  11161
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