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Theorem divalglem0 12840
Description: Lemma for divalg 12850. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
divalglem0.1  |-  N  e.  ZZ
divalglem0.2  |-  D  e.  ZZ
Assertion
Ref Expression
divalglem0  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) ) ) )

Proof of Theorem divalglem0
StepHypRef Expression
1 divalglem0.2 . . . . . 6  |-  D  e.  ZZ
2 iddvds 12790 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  D )
3 dvdsabsb 12796 . . . . . . . 8  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
43anidms 627 . . . . . . 7  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
52, 4mpbid 202 . . . . . 6  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
61, 5ax-mp 8 . . . . 5  |-  D  ||  ( abs `  D )
7 nn0abscl 12044 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
81, 7ax-mp 8 . . . . . . 7  |-  ( abs `  D )  e.  NN0
98nn0zi 10238 . . . . . 6  |-  ( abs `  D )  e.  ZZ
10 dvdsmultr2 12812 . . . . . 6  |-  ( ( D  e.  ZZ  /\  K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  ->  ( D  ||  ( abs `  D
)  ->  D  ||  ( K  x.  ( abs `  D ) ) ) )
111, 9, 10mp3an13 1270 . . . . 5  |-  ( K  e.  ZZ  ->  ( D  ||  ( abs `  D
)  ->  D  ||  ( K  x.  ( abs `  D ) ) ) )
126, 11mpi 17 . . . 4  |-  ( K  e.  ZZ  ->  D  ||  ( K  x.  ( abs `  D ) ) )
1312adantl 453 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  D  ||  ( K  x.  ( abs `  D
) ) )
14 divalglem0.1 . . . . 5  |-  N  e.  ZZ
15 zsubcl 10251 . . . . 5  |-  ( ( N  e.  ZZ  /\  R  e.  ZZ )  ->  ( N  -  R
)  e.  ZZ )
1614, 15mpan 652 . . . 4  |-  ( R  e.  ZZ  ->  ( N  -  R )  e.  ZZ )
17 zmulcl 10256 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  -> 
( K  x.  ( abs `  D ) )  e.  ZZ )
189, 17mpan2 653 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  ZZ )
19 dvds2add 12808 . . . . 5  |-  ( ( D  e.  ZZ  /\  ( N  -  R
)  e.  ZZ  /\  ( K  x.  ( abs `  D ) )  e.  ZZ )  -> 
( ( D  ||  ( N  -  R
)  /\  D  ||  ( K  x.  ( abs `  D ) ) )  ->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
201, 19mp3an1 1266 . . . 4  |-  ( ( ( N  -  R
)  e.  ZZ  /\  ( K  x.  ( abs `  D ) )  e.  ZZ )  -> 
( ( D  ||  ( N  -  R
)  /\  D  ||  ( K  x.  ( abs `  D ) ) )  ->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
2116, 18, 20syl2an 464 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( D  ||  ( N  -  R
)  /\  D  ||  ( K  x.  ( abs `  D ) ) )  ->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
2213, 21mpan2d 656 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) ) )
23 zcn 10219 . . . 4  |-  ( R  e.  ZZ  ->  R  e.  CC )
2418zcnd 10308 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  CC )
25 zcn 10219 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
2614, 25ax-mp 8 . . . . 5  |-  N  e.  CC
27 subsub 9263 . . . . 5  |-  ( ( N  e.  CC  /\  R  e.  CC  /\  ( K  x.  ( abs `  D ) )  e.  CC )  ->  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) )  =  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) )
2826, 27mp3an1 1266 . . . 4  |-  ( ( R  e.  CC  /\  ( K  x.  ( abs `  D ) )  e.  CC )  -> 
( N  -  ( R  -  ( K  x.  ( abs `  D
) ) ) )  =  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) )
2923, 24, 28syl2an 464 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  ( R  -  ( K  x.  ( abs `  D
) ) ) )  =  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) )
3029breq2d 4165 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) )  <->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
3122, 30sylibrd 226 1  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921    + caddc 8926    x. cmul 8928    - cmin 9223   NN0cn0 10153   ZZcz 10214   abscabs 11966    || cdivides 12779
This theorem is referenced by:  divalglem5  12844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780
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