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Theorem divalglem7 12645
Description: Lemma for divalg 12649. (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 5907 . . . . 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 2382 . . . 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 285 . . 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 307 . 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 2487 . . 3  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( K  =/=  0  <->  if ( K  e.  ZZ ,  K ,  0 )  =/=  0 ) )
6 oveq1 5907 . . . . . 6  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( K  x.  ( abs `  D ) )  =  ( if ( K  e.  ZZ ,  K ,  0 )  x.  ( abs `  D
) ) )
76oveq2d 5916 . . . . 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 2382 . . . 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 285 . . 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 311 . 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 11856 . . . 4  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
1411, 12, 13mp2an 653 . . 3  |-  ( abs `  D )  e.  NN
15 0z 10082 . . . . 5  |-  0  e.  ZZ
16 0le0 9872 . . . . 5  |-  0  <_  0
1714nngt0i 9824 . . . . 5  |-  0  <  ( abs `  D
)
1814nnzi 10094 . . . . . 6  |-  ( abs `  D )  e.  ZZ
19 elfzm11 10900 . . . . . 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 653 . . . . 5  |-  ( 0  e.  ( 0 ... ( ( abs `  D
)  -  1 ) )  <->  ( 0  e.  ZZ  /\  0  <_ 
0  /\  0  <  ( abs `  D ) ) )
2115, 16, 17, 20mpbir3an 1134 . . . 4  |-  0  e.  ( 0 ... (
( abs `  D
)  -  1 ) )
2221elimel 3651 . . 3  |-  if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  e.  ( 0 ... (
( abs `  D
)  -  1 ) )
2315elimel 3651 . . 3  |-  if ( K  e.  ZZ ,  K ,  0 )  e.  ZZ
2414, 22, 23divalglem6 12644 . 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 3641 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   ifcif 3599   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    < clt 8912    <_ cle 8913    - cmin 9082   NNcn 9791   ZZcz 10071   ...cfz 10829   abscabs 11766
This theorem is referenced by:  divalglem8  12646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768
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