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Theorem divalglem0 13906
Description: Lemma for divalg 13916. (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 13854 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  D )
3 dvdsabsb 13860 . . . . . . . 8  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
43anidms 645 . . . . . . 7  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
52, 4mpbid 210 . . . . . 6  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
61, 5ax-mp 5 . . . . 5  |-  D  ||  ( abs `  D )
7 nn0abscl 13104 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
81, 7ax-mp 5 . . . . . . 7  |-  ( abs `  D )  e.  NN0
98nn0zi 10885 . . . . . 6  |-  ( abs `  D )  e.  ZZ
10 dvdsmultr2 13876 . . . . . 6  |-  ( ( D  e.  ZZ  /\  K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  ->  ( D  ||  ( abs `  D
)  ->  D  ||  ( K  x.  ( abs `  D ) ) ) )
111, 9, 10mp3an13 1315 . . . . 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 466 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  D  ||  ( K  x.  ( abs `  D
) ) )
14 divalglem0.1 . . . . 5  |-  N  e.  ZZ
15 zsubcl 10901 . . . . 5  |-  ( ( N  e.  ZZ  /\  R  e.  ZZ )  ->  ( N  -  R
)  e.  ZZ )
1614, 15mpan 670 . . . 4  |-  ( R  e.  ZZ  ->  ( N  -  R )  e.  ZZ )
17 zmulcl 10907 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  -> 
( K  x.  ( abs `  D ) )  e.  ZZ )
189, 17mpan2 671 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  ZZ )
19 dvds2add 13872 . . . . 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 1311 . . . 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 477 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( ( D  ||  ( N  -  R
)  /\  D  ||  ( K  x.  ( abs `  D ) ) )  ->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
2213, 21mpan2d 674 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) ) )
23 zcn 10865 . . . 4  |-  ( R  e.  ZZ  ->  R  e.  CC )
2418zcnd 10963 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  CC )
25 zcn 10865 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
2614, 25ax-mp 5 . . . . 5  |-  N  e.  CC
27 subsub 9845 . . . . 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 1311 . . . 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 477 . . 3  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  -  ( R  -  ( K  x.  ( abs `  D
) ) ) )  =  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) )
3029breq2d 4459 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) )  <->  D  ||  (
( N  -  R
)  +  ( K  x.  ( abs `  D
) ) ) ) )
3122, 30sylibrd 234 1  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( N  -  ( R  -  ( K  x.  ( abs `  D ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486    + caddc 9491    x. cmul 9493    - cmin 9801   NN0cn0 10791   ZZcz 10860   abscabs 13026    || cdivides 13843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844
This theorem is referenced by:  divalglem5  13910
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