ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dvdsmod Unicode version

Theorem dvdsmod 12004
Description: Any number  K whose mod base  N is divisible by a divisor  P of the base is also divisible by 
P. This means that primes will also be relatively prime to the base when reduced  mod 
N for any base. (Contributed by Mario Carneiro, 13-Mar-2014.)
Assertion
Ref Expression
dvdsmod  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  K )
)

Proof of Theorem dvdsmod
StepHypRef Expression
1 simpl3 1004 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  ZZ )
2 zq 9691 . . . . 5  |-  ( K  e.  ZZ  ->  K  e.  QQ )
31, 2syl 14 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  QQ )
4 simpl2 1003 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  NN )
5 nnq 9698 . . . . 5  |-  ( N  e.  NN  ->  N  e.  QQ )
64, 5syl 14 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  QQ )
74nngt0d 9026 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  <  N
)
8 modqval 10395 . . . 4  |-  ( ( K  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( K  mod  N )  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N
) ) ) ) )
93, 6, 7, 8syl3anc 1249 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  mod  N )  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
109breq2d 4041 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) ) )
11 simpl1 1002 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  NN )
1211nnzd 9438 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  ZZ )
134nnzd 9438 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  ZZ )
14 znq 9689 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  N  e.  NN )  ->  ( K  /  N
)  e.  QQ )
151, 4, 14syl2anc 411 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  /  N )  e.  QQ )
1615flqcld 10346 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( |_ `  ( K  /  N
) )  e.  ZZ )
17 simpr 110 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  N
)
1812, 13, 16, 17dvdsmultr1d 11975 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N
) ) ) )
1913, 16zmulcld 9445 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  ZZ )
2019zcnd 9440 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  CC )
2120subid1d 8319 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  - 
0 )  =  ( N  x.  ( |_
`  ( K  /  N ) ) ) )
2218, 21breqtrrd 4057 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  (
( N  x.  ( |_ `  ( K  /  N ) ) )  -  0 ) )
23 0zd 9329 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  e.  ZZ )
24 moddvds 11942 . . . . . 6  |-  ( ( P  e.  NN  /\  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  ZZ  /\  0  e.  ZZ )  ->  (
( ( N  x.  ( |_ `  ( K  /  N ) ) )  mod  P )  =  ( 0  mod 
P )  <->  P  ||  (
( N  x.  ( |_ `  ( K  /  N ) ) )  -  0 ) ) )
2511, 19, 23, 24syl3anc 1249 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( ( N  x.  ( |_
`  ( K  /  N ) ) )  mod  P )  =  ( 0  mod  P
)  <->  P  ||  ( ( N  x.  ( |_
`  ( K  /  N ) ) )  -  0 ) ) )
2622, 25mpbird 167 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  mod 
P )  =  ( 0  mod  P ) )
2726eqeq2d 2205 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( K  mod  P )  =  ( ( N  x.  ( |_ `  ( K  /  N ) ) )  mod  P )  <-> 
( K  mod  P
)  =  ( 0  mod  P ) ) )
28 moddvds 11942 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  ZZ  /\  ( N  x.  ( |_ `  ( K  /  N
) ) )  e.  ZZ )  ->  (
( K  mod  P
)  =  ( ( N  x.  ( |_
`  ( K  /  N ) ) )  mod  P )  <->  P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) ) )
2911, 1, 19, 28syl3anc 1249 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( K  mod  P )  =  ( ( N  x.  ( |_ `  ( K  /  N ) ) )  mod  P )  <-> 
P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) ) )
30 moddvds 11942 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  ZZ  /\  0  e.  ZZ )  ->  (
( K  mod  P
)  =  ( 0  mod  P )  <->  P  ||  ( K  -  0 ) ) )
3111, 1, 23, 30syl3anc 1249 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( K  mod  P )  =  ( 0  mod  P
)  <->  P  ||  ( K  -  0 ) ) )
3227, 29, 313bitr3d 218 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N
) ) ) )  <-> 
P  ||  ( K  -  0 ) ) )
331zcnd 9440 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  CC )
3433subid1d 8319 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  - 
0 )  =  K )
3534breq2d 4041 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  0
)  <->  P  ||  K ) )
3610, 32, 353bitrd 214 1  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  K )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   0cc0 7872    x. cmul 7877    < clt 8054    - cmin 8190    / cdiv 8691   NNcn 8982   ZZcz 9317   QQcq 9684   |_cfl 10337    mod cmo 10393    || cdvds 11930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-q 9685  df-rp 9720  df-fl 10339  df-mod 10394  df-dvds 11931
This theorem is referenced by:  lgsdir2lem2  15145
  Copyright terms: Public domain W3C validator