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Theorem divalglem0 12901
Description: Lemma for divalg 12911. (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 12851 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  D )
3 dvdsabsb 12857 . . . . . . . 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 12105 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
81, 7ax-mp 8 . . . . . . 7  |-  ( abs `  D )  e.  NN0
98nn0zi 10295 . . . . . 6  |-  ( abs `  D )  e.  ZZ
10 dvdsmultr2 12873 . . . . . 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 10308 . . . . 5  |-  ( ( N  e.  ZZ  /\  R  e.  ZZ )  ->  ( N  -  R
)  e.  ZZ )
1614, 15mpan 652 . . . 4  |-  ( R  e.  ZZ  ->  ( N  -  R )  e.  ZZ )
17 zmulcl 10313 . . . . 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 12869 . . . . 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 10276 . . . 4  |-  ( R  e.  ZZ  ->  R  e.  CC )
2418zcnd 10365 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  CC )
25 zcn 10276 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
2614, 25ax-mp 8 . . . . 5  |-  N  e.  CC
27 subsub 9320 . . . . 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 4216 . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   CCcc 8977    + caddc 8982    x. cmul 8984    - cmin 9280   NN0cn0 10210   ZZcz 10271   abscabs 12027    || cdivides 12840
This theorem is referenced by:  divalglem5  12905
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 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
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 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-dvds 12841
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