MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  divalglem0 Unicode version

Theorem divalglem0 12555
Description: Lemma for divalg 12565. (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 12505 . . . . . . 7  |-  ( D  e.  ZZ  ->  D  ||  D )
3 dvdsabsb 12511 . . . . . . . 8  |-  ( ( D  e.  ZZ  /\  D  e.  ZZ )  ->  ( D  ||  D  <->  D 
||  ( abs `  D
) ) )
43anidms 629 . . . . . . 7  |-  ( D  e.  ZZ  ->  ( D  ||  D  <->  D  ||  ( abs `  D ) ) )
52, 4mpbid 203 . . . . . 6  |-  ( D  e.  ZZ  ->  D  ||  ( abs `  D
) )
61, 5ax-mp 10 . . . . 5  |-  D  ||  ( abs `  D )
7 nn0abscl 11763 . . . . . . . 8  |-  ( D  e.  ZZ  ->  ( abs `  D )  e. 
NN0 )
81, 7ax-mp 10 . . . . . . 7  |-  ( abs `  D )  e.  NN0
98nn0zi 10016 . . . . . 6  |-  ( abs `  D )  e.  ZZ
10 dvdsmultr2 12527 . . . . . 6  |-  ( ( D  e.  ZZ  /\  K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  ->  ( D  ||  ( abs `  D
)  ->  D  ||  ( K  x.  ( abs `  D ) ) ) )
111, 9, 10mp3an13 1273 . . . . 5  |-  ( K  e.  ZZ  ->  ( D  ||  ( abs `  D
)  ->  D  ||  ( K  x.  ( abs `  D ) ) ) )
126, 11mpi 18 . . . 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 10029 . . . . 5  |-  ( ( N  e.  ZZ  /\  R  e.  ZZ )  ->  ( N  -  R
)  e.  ZZ )
1614, 15mpan 654 . . . 4  |-  ( R  e.  ZZ  ->  ( N  -  R )  e.  ZZ )
17 zmulcl 10034 . . . . 5  |-  ( ( K  e.  ZZ  /\  ( abs `  D )  e.  ZZ )  -> 
( K  x.  ( abs `  D ) )  e.  ZZ )
189, 17mpan2 655 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  ZZ )
19 dvds2add 12523 . . . . 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 1269 . . . 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 658 . 2  |-  ( ( R  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  ||  ( N  -  R )  ->  D  ||  ( ( N  -  R )  +  ( K  x.  ( abs `  D ) ) ) ) )
23 zcn 9997 . . . 4  |-  ( R  e.  ZZ  ->  R  e.  CC )
2418zcnd 10086 . . . 4  |-  ( K  e.  ZZ  ->  ( K  x.  ( abs `  D ) )  e.  CC )
25 zcn 9997 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
2614, 25ax-mp 10 . . . . 5  |-  N  e.  CC
27 subsub 9045 . . . . 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 1269 . . . 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 4009 . 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   CCcc 8703    + caddc 8708    x. cmul 8710    - cmin 9005   NN0cn0 9933   ZZcz 9992   abscabs 11685    || cdivides 12494
This theorem is referenced by:  divalglem5  12559
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-divides 12495
  Copyright terms: Public domain W3C validator