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Theorem divalglem7 14069
Description: Lemma for divalg 14073. (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 6303 . . . . 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 2526 . . . 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 294 . . 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 316 . 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 2738 . . 3  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( K  =/=  0  <->  if ( K  e.  ZZ ,  K ,  0 )  =/=  0 ) )
6 oveq1 6303 . . . . . 6  |-  ( K  =  if ( K  e.  ZZ ,  K ,  0 )  -> 
( K  x.  ( abs `  D ) )  =  ( if ( K  e.  ZZ ,  K ,  0 )  x.  ( abs `  D
) ) )
76oveq2d 6312 . . . . 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 2526 . . . 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 294 . . 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 320 . 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 13170 . . . 4  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  -> 
( abs `  D
)  e.  NN )
1411, 12, 13mp2an 672 . . 3  |-  ( abs `  D )  e.  NN
15 0z 10896 . . . . 5  |-  0  e.  ZZ
16 0le0 10646 . . . . 5  |-  0  <_  0
1714nngt0i 10590 . . . . 5  |-  0  <  ( abs `  D
)
1814nnzi 10909 . . . . . 6  |-  ( abs `  D )  e.  ZZ
19 elfzm11 11775 . . . . . 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 672 . . . . 5  |-  ( 0  e.  ( 0 ... ( ( abs `  D
)  -  1 ) )  <->  ( 0  e.  ZZ  /\  0  <_ 
0  /\  0  <  ( abs `  D ) ) )
2115, 16, 17, 20mpbir3an 1178 . . . 4  |-  0  e.  ( 0 ... (
( abs `  D
)  -  1 ) )
2221elimel 4007 . . 3  |-  if ( X  e.  ( 0 ... ( ( abs `  D )  -  1 ) ) ,  X ,  0 )  e.  ( 0 ... (
( abs `  D
)  -  1 ) )
2315elimel 4007 . . 3  |-  if ( K  e.  ZZ ,  K ,  0 )  e.  ZZ
2414, 22, 23divalglem6 14068 . 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 3997 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   ifcif 3944   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   ZZcz 10885   ...cfz 11697   abscabs 13079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081
This theorem is referenced by:  divalglem8  14070
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