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Theorem ndvdsadd 12607
Description: Corollary of the division algorithm. If an integer  D greater than  1 divides  N, then it does not divide any of  N  +  1,  N  +  2...  N  +  ( D  -  1 ). (Contributed by Paul Chapman, 31-Mar-2011.)
Assertion
Ref Expression
ndvdsadd  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( D  ||  N  ->  -.  D  ||  ( N  +  K )
) )

Proof of Theorem ndvdsadd
StepHypRef Expression
1 nnre 9753 . . . . . . . . 9  |-  ( K  e.  NN  ->  K  e.  RR )
2 nnre 9753 . . . . . . . . 9  |-  ( D  e.  NN  ->  D  e.  RR )
3 posdif 9267 . . . . . . . . 9  |-  ( ( K  e.  RR  /\  D  e.  RR )  ->  ( K  <  D  <->  0  <  ( D  -  K ) ) )
41, 2, 3syl2anr 464 . . . . . . . 8  |-  ( ( D  e.  NN  /\  K  e.  NN )  ->  ( K  <  D  <->  0  <  ( D  -  K ) ) )
54pm5.32i 618 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  K  e.  NN )  /\  K  <  D
)  <->  ( ( D  e.  NN  /\  K  e.  NN )  /\  0  <  ( D  -  K
) ) )
6 nnz 10045 . . . . . . . . 9  |-  ( D  e.  NN  ->  D  e.  ZZ )
7 nnz 10045 . . . . . . . . 9  |-  ( K  e.  NN  ->  K  e.  ZZ )
8 zsubcl 10061 . . . . . . . . 9  |-  ( ( D  e.  ZZ  /\  K  e.  ZZ )  ->  ( D  -  K
)  e.  ZZ )
96, 7, 8syl2an 463 . . . . . . . 8  |-  ( ( D  e.  NN  /\  K  e.  NN )  ->  ( D  -  K
)  e.  ZZ )
10 elnnz 10034 . . . . . . . . 9  |-  ( ( D  -  K )  e.  NN  <->  ( ( D  -  K )  e.  ZZ  /\  0  < 
( D  -  K
) ) )
1110biimpri 197 . . . . . . . 8  |-  ( ( ( D  -  K
)  e.  ZZ  /\  0  <  ( D  -  K ) )  -> 
( D  -  K
)  e.  NN )
129, 11sylan 457 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  K  e.  NN )  /\  0  <  ( D  -  K )
)  ->  ( D  -  K )  e.  NN )
135, 12sylbi 187 . . . . . 6  |-  ( ( ( D  e.  NN  /\  K  e.  NN )  /\  K  <  D
)  ->  ( D  -  K )  e.  NN )
1413anasss 628 . . . . 5  |-  ( ( D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  -  K )  e.  NN )
15 nngt0 9775 . . . . . . . 8  |-  ( K  e.  NN  ->  0  <  K )
16 ltsubpos 9266 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  D  e.  RR )  ->  ( 0  <  K  <->  ( D  -  K )  <  D ) )
171, 2, 16syl2an 463 . . . . . . . . . 10  |-  ( ( K  e.  NN  /\  D  e.  NN )  ->  ( 0  <  K  <->  ( D  -  K )  <  D ) )
1817biimpd 198 . . . . . . . . 9  |-  ( ( K  e.  NN  /\  D  e.  NN )  ->  ( 0  <  K  ->  ( D  -  K
)  <  D )
)
1918expcom 424 . . . . . . . 8  |-  ( D  e.  NN  ->  ( K  e.  NN  ->  ( 0  <  K  -> 
( D  -  K
)  <  D )
) )
2015, 19mpdi 38 . . . . . . 7  |-  ( D  e.  NN  ->  ( K  e.  NN  ->  ( D  -  K )  <  D ) )
2120imp 418 . . . . . 6  |-  ( ( D  e.  NN  /\  K  e.  NN )  ->  ( D  -  K
)  <  D )
2221adantrr 697 . . . . 5  |-  ( ( D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( D  -  K )  <  D
)
2314, 22jca 518 . . . 4  |-  ( ( D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  ->  ( ( D  -  K )  e.  NN  /\  ( D  -  K )  < 
D ) )
24233adant1 973 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( ( D  -  K )  e.  NN  /\  ( D  -  K
)  <  D )
)
25 ndvdssub 12606 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  (
( D  -  K
)  e.  NN  /\  ( D  -  K
)  <  D )
)  ->  ( D  ||  N  ->  -.  D  ||  ( N  -  ( D  -  K )
) ) )
2624, 25syld3an3 1227 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( D  ||  N  ->  -.  D  ||  ( N  -  ( D  -  K ) ) ) )
27 zaddcl 10059 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  +  K
)  e.  ZZ )
287, 27sylan2 460 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  K  e.  NN )  ->  ( N  +  K
)  e.  ZZ )
29 dvdssubr 12570 . . . . . . . 8  |-  ( ( D  e.  ZZ  /\  ( N  +  K
)  e.  ZZ )  ->  ( D  ||  ( N  +  K
)  <->  D  ||  ( ( N  +  K )  -  D ) ) )
306, 28, 29syl2an 463 . . . . . . 7  |-  ( ( D  e.  NN  /\  ( N  e.  ZZ  /\  K  e.  NN ) )  ->  ( D  ||  ( N  +  K
)  <->  D  ||  ( ( N  +  K )  -  D ) ) )
3130an12s 776 . . . . . 6  |-  ( ( N  e.  ZZ  /\  ( D  e.  NN  /\  K  e.  NN ) )  ->  ( D  ||  ( N  +  K
)  <->  D  ||  ( ( N  +  K )  -  D ) ) )
32313impb 1147 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  K  e.  NN )  ->  ( D  ||  ( N  +  K )  <->  D  ||  (
( N  +  K
)  -  D ) ) )
33 zcn 10029 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  e.  CC )
34 nncn 9754 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  CC )
35 nncn 9754 . . . . . . 7  |-  ( K  e.  NN  ->  K  e.  CC )
36 subsub3 9079 . . . . . . 7  |-  ( ( N  e.  CC  /\  D  e.  CC  /\  K  e.  CC )  ->  ( N  -  ( D  -  K ) )  =  ( ( N  +  K )  -  D
) )
3733, 34, 35, 36syl3an 1224 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  K  e.  NN )  ->  ( N  -  ( D  -  K ) )  =  ( ( N  +  K )  -  D
) )
3837breq2d 4035 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  K  e.  NN )  ->  ( D  ||  ( N  -  ( D  -  K
) )  <->  D  ||  (
( N  +  K
)  -  D ) ) )
3932, 38bitr4d 247 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  K  e.  NN )  ->  ( D  ||  ( N  +  K )  <->  D  ||  ( N  -  ( D  -  K ) ) ) )
4039notbid 285 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  K  e.  NN )  ->  ( -.  D  ||  ( N  +  K )  <->  -.  D  ||  ( N  -  ( D  -  K )
) ) )
41403adant3r 1179 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( -.  D  ||  ( N  +  K
)  <->  -.  D  ||  ( N  -  ( D  -  K ) ) ) )
4226, 41sylibrd 225 1  |-  ( ( N  e.  ZZ  /\  D  e.  NN  /\  ( K  e.  NN  /\  K  <  D ) )  -> 
( D  ||  N  ->  -.  D  ||  ( N  +  K )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    < clt 8867    - cmin 9037   NNcn 9746   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  ndvdsp1  12608  ndvdsi  12609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532
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