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Theorem divalglem0 12944
Description: Lemma for divalg 12954. (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 12894 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  D )
3 dvdsabsb 12900 . . . . . . . 8  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
43anidms 628 . . . . . . 7  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
52, 4mpbid 203 . . . . . 6  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
61, 5ax-mp 5 . . . . 5  |-  D  ||  ( abs `  D )
7 nn0abscl 12148 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
81, 7ax-mp 5 . . . . . . 7  |-  ( abs `  D )  e.  NN0
98nn0zi 10337 . . . . . 6  |-  ( abs `  D )  e.  ZZ
10 dvdsmultr2 12916 . . . . . 6  |-  ( ( D  e.  ZZ  /\  K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  ->  ( D  ||  ( abs `  D
)  ->  D  ||  ( K  x.  ( abs `  D ) ) ) )
111, 9, 10mp3an13 1271 . . . . 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 454 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  D  ||  ( K  x.  ( abs `  D
) ) )
14 divalglem0.1 . . . . 5  |-  N  e.  ZZ
15 zsubcl 10350 . . . . 5  |-  ( ( N  e.  ZZ  /\  R  e.  ZZ )  ->  ( N  -  R
)  e.  ZZ )
1614, 15mpan 653 . . . 4  |-  ( R  e.  ZZ  ->  ( N  -  R )  e.  ZZ )
17 zmulcl 10355 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  -> 
( K  x.  ( abs `  D ) )  e.  ZZ )
189, 17mpan2 654 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  ZZ )
19 dvds2add 12912 . . . . 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 1267 . . . 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 465 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( D  ||  ( N  -  R
)  /\  D  ||  ( K  x.  ( abs `  D ) ) )  ->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
2213, 21mpan2d 657 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) ) )
23 zcn 10318 . . . 4  |-  ( R  e.  ZZ  ->  R  e.  CC )
2418zcnd 10407 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  CC )
25 zcn 10318 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
2614, 25ax-mp 5 . . . . 5  |-  N  e.  CC
27 subsub 9362 . . . . 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 1267 . . . 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 465 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  ( R  -  ( K  x.  ( abs `  D
) ) ) )  =  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) )
3029breq2d 4249 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) )  <->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
3122, 30sylibrd 227 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 178    /\ wa 360    = wceq 1653    e. wcel 1727   class class class wbr 4237   ` cfv 5483  (class class class)co 6110   CCcc 9019    + caddc 9024    x. cmul 9026    - cmin 9322   NN0cn0 10252   ZZcz 10313   abscabs 12070    || cdivides 12883
This theorem is referenced by:  divalglem5  12948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-sup 7475  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-rp 10644  df-seq 11355  df-exp 11414  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-dvds 12884
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