ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abs3lem Unicode version
Theorem abs3lem 11255
Description: Lemma involving absolute value of differences. (Contributed by NM, 2-Oct-1999.)
Assertion
Ref Expression
abs3lem |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
Proof of Theorem abs3lem
StepHypRef Expression
1 simplll 533 . . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> A e. CC )
2 simpllr 534 . . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> B e. CC )
31, 2subcld 8330 . . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( A - B ) e. CC )
4 abscl 11195 . . . 4 |- ( ( A - B ) e. CC -> ( abs ` ( A - B ) ) e. RR )
53, 4syl 14 . . 3 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( A - B ) ) e. RR )
6 simplrl 535 . . . . . 6 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> C e. CC )
71, 6subcld 8330 . . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( A - C ) e. CC )
8 abscl 11195 . . . . 5 |- ( ( A - C ) e. CC -> ( abs ` ( A - C ) ) e. RR )
97, 8syl 14 . . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( A - C ) ) e. RR )
106, 2subcld 8330 . . . . 5 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( C - B ) e. CC )
11 abscl 11195 . . . . 5 |- ( ( C - B ) e. CC -> ( abs ` ( C - B ) ) e. RR )
1210, 11syl 14 . . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( C - B ) ) e. RR )
139, 12readdcld 8049 . . 3 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) e. RR )
14 simplrr 536 . . 3 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> D e. RR )
15 abs3dif 11249 . . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
161, 2, 6, 15syl3anc 1249 . . 3 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) )
17 simprl 529 . . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( A - C ) ) < ( D / 2 ) )
18 simprr 531 . . . 4 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( C - B ) ) < ( D / 2 ) )
199, 12, 14, 17, 18lt2halvesd 9230 . . 3 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) < D )
205, 13, 14, 16, 19lelttrd 8144 . 2 |- ( ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) /\ ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) ) -> ( abs ` ( A - B ) ) < D )
2120ex 115 1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. RR ) ) -> ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   + caddc 7875   < clt 8054   <_ cle 8055   - cmin 8190   / cdiv 8691   2c2 9033   abscabs 11141
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143
This theorem is referenced by:  cau3  11259  abs3lemi  11301  abs3lemd  11345  climuni  11436  2clim  11444  addcn2  11453  mulcn2  11455
  Copyright terms: Public domain W3C validator