ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abs2difabs Unicode version
Theorem abs2difabs 9994
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 9992 . . . 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 9937 . . . . 5 |- ( A e. CC -> ( abs ` A ) e. RR )
43recnd 7147 . . . 4 |- ( A e. CC -> ( abs ` A ) e. CC )
5 abscl 9937 . . . . 5 |- ( B e. CC -> ( abs ` B ) e. RR )
65recnd 7147 . . . 4 |- ( B e. CC -> ( abs ` B ) e. CC )
7 negsubdi2 7367 . . . 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 9987 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( abs ` ( A - B ) ) = ( abs ` ( B - A ) ) )
102, 8, 93brtr4d 3815 . 2 |- ( ( A e. CC /\ B e. CC ) -> -u ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
11 abs2dif 9992 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` A ) - ( abs ` B ) ) <_ ( abs ` ( A - B ) ) )
12 resubcl 7372 . . . . 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 7307 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
15 abscl 9937 . . . . 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 9975 . . . 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 7570 . . . . 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 452 . . 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 885 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 1284   e. wcel 1433   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   <_ cle 7154   - cmin 7279   -ucneg 7280   abscabs 9883
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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885
This theorem is referenced by:  abs2difabsd  10085  abscn2  10153
  Copyright terms: Public domain W3C validator