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Theorem divalglem7 12911
Description: Lemma for divalg 12915. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
divalglem7.1  |-  D  e.  ZZ
divalglem7.2  |-  D  =/=  0
Assertion
Ref Expression
divalglem7  |-  ( ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) )  /\  K  e.  ZZ )  ->  ( K  =/=  0  ->  -.  ( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ) )

Proof of Theorem divalglem7
StepHypRef Expression
1 oveq1 6080 . . . . 5  |-  ( X  =  if ( X  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ,  X , 
0 )  ->  ( X  +  ( K  x.  ( abs `  D
) ) )  =  ( if ( X  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ,  X , 
0 )  +  ( K  x.  ( abs `  D ) ) ) )
21eleq1d 2501 . . . 4  |-  ( X  =  if ( X  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ,  X , 
0 )  ->  (
( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D
)  -  1 ) )  <->  ( if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ) )
32notbid 286 . . 3  |-  ( X  =  if ( X  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ,  X , 
0 )  ->  ( -.  ( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D
)  -  1 ) )  <->  -.  ( if ( X  e.  (
0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ) )
43imbi2d 308 . 2  |-  ( X  =  if ( X  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ,  X , 
0 )  ->  (
( K  =/=  0  ->  -.  ( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) )  <->  ( K  =/=  0  ->  -.  ( if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( K  x.  ( abs `  D
) ) )  e.  ( 0 ... (
( abs `  D
)  -  1 ) ) ) ) )
5 neeq1 2606 . . 3  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( K  =/=  0  <->  if ( K  e.  ZZ ,  K ,  0 )  =/=  0 ) )
6 oveq1 6080 . . . . . 6  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( K  x.  ( abs `  D ) )  =  ( if ( K  e.  ZZ ,  K ,  0 )  x.  ( abs `  D
) ) )
76oveq2d 6089 . . . . 5  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( if ( X  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ,  X , 
0 )  +  ( K  x.  ( abs `  D ) ) )  =  ( if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( if ( K  e.  ZZ ,  K ,  0 )  x.  ( abs `  D
) ) ) )
87eleq1d 2501 . . . 4  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( ( if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D )  -  1 ) )  <->  ( if ( X  e.  (
0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( if ( K  e.  ZZ ,  K ,  0 )  x.  ( abs `  D
) ) )  e.  ( 0 ... (
( abs `  D
)  -  1 ) ) ) )
98notbid 286 . . 3  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( -.  ( if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( K  x.  ( abs `  D
) ) )  e.  ( 0 ... (
( abs `  D
)  -  1 ) )  <->  -.  ( if ( X  e.  (
0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( if ( K  e.  ZZ ,  K ,  0 )  x.  ( abs `  D
) ) )  e.  ( 0 ... (
( abs `  D
)  -  1 ) ) ) )
105, 9imbi12d 312 . 2  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( ( K  =/=  0  ->  -.  ( if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  +  ( K  x.  ( abs `  D
) ) )  e.  ( 0 ... (
( abs `  D
)  -  1 ) ) )  <->  ( if ( K  e.  ZZ ,  K ,  0 )  =/=  0  ->  -.  ( if ( X  e.  ( 0 ... (
( abs `  D
)  -  1 ) ) ,  X , 
0 )  +  ( if ( K  e.  ZZ ,  K , 
0 )  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ) ) )
11 divalglem7.1 . . . 4  |-  D  e.  ZZ
12 divalglem7.2 . . . 4  |-  D  =/=  0
13 nnabscl 12121 . . . 4  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
1411, 12, 13mp2an 654 . . 3  |-  ( abs `  D )  e.  NN
15 0z 10285 . . . . 5  |-  0  e.  ZZ
16 0le0 10073 . . . . 5  |-  0  <_  0
1714nngt0i 10025 . . . . 5  |-  0  <  ( abs `  D
)
1814nnzi 10297 . . . . . 6  |-  ( abs `  D )  e.  ZZ
19 elfzm11 11108 . . . . . 6  |-  ( ( 0  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  -> 
( 0  e.  ( 0 ... ( ( abs `  D )  -  1 ) )  <-> 
( 0  e.  ZZ  /\  0  <_  0  /\  0  <  ( abs `  D
) ) ) )
2015, 18, 19mp2an 654 . . . . 5  |-  ( 0  e.  ( 0 ... ( ( abs `  D
)  -  1 ) )  <->  ( 0  e.  ZZ  /\  0  <_ 
0  /\  0  <  ( abs `  D ) ) )
2115, 16, 17, 20mpbir3an 1136 . . . 4  |-  0  e.  ( 0 ... (
( abs `  D
)  -  1 ) )
2221elimel 3783 . . 3  |-  if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  e.  ( 0 ... (
( abs `  D
)  -  1 ) )
2315elimel 3783 . . 3  |-  if ( K  e.  ZZ ,  K ,  0 )  e.  ZZ
2414, 22, 23divalglem6 12910 . 2  |-  ( if ( K  e.  ZZ ,  K ,  0 )  =/=  0  ->  -.  ( if ( X  e.  ( 0 ... (
( abs `  D
)  -  1 ) ) ,  X , 
0 )  +  ( if ( K  e.  ZZ ,  K , 
0 )  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) )
254, 10, 24dedth2h 3773 1  |-  ( ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) )  /\  K  e.  ZZ )  ->  ( K  =/=  0  ->  -.  ( X  +  ( K  x.  ( abs `  D ) ) )  e.  ( 0 ... ( ( abs `  D
)  -  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283   NNcn 9992   ZZcz 10274   ...cfz 11035   abscabs 12031
This theorem is referenced by:  divalglem8  12912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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