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Theorem dvdsmod 10407
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 944 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  ZZ )
2 zq 8792 . . . . 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 943 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  NN )
5 nnq 8799 . . . . 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 8149 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  <  N
)
8 modqval 9406 . . . 4  |-  ( ( K  e.  QQ  /\  N  e.  QQ  /\  0  <  N )  ->  ( K  mod  N )  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N
) ) ) ) )
93, 6, 7, 8syl3anc 1170 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  mod  N )  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
109breq2d 3805 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) ) )
11 simpl1 942 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  NN )
1211nnzd 8549 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  ZZ )
134nnzd 8549 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  ZZ )
14 znq 8790 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  N  e.  NN )  ->  ( K  /  N
)  e.  QQ )
151, 4, 14syl2anc 403 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  /  N )  e.  QQ )
1615flqcld 9359 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( |_ `  ( K  /  N
) )  e.  ZZ )
17 simpr 108 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  N
)
1812, 13, 16, 17dvdsmultr1d 10379 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N
) ) ) )
1913, 16zmulcld 8556 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  ZZ )
2019zcnd 8551 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  CC )
2120subid1d 7475 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  - 
0 )  =  ( N  x.  ( |_
`  ( K  /  N ) ) ) )
2218, 21breqtrrd 3819 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  (
( N  x.  ( |_ `  ( K  /  N ) ) )  -  0 ) )
23 0zd 8444 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  e.  ZZ )
24 moddvds 10349 . . . . . 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 1170 . . . . 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 165 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  mod 
P )  =  ( 0  mod  P ) )
2726eqeq2d 2093 . . 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 10349 . . . 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 1170 . . 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 10349 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  ZZ  /\  0  e.  ZZ )  ->  (
( K  mod  P
)  =  ( 0  mod  P )  <->  P  ||  ( K  -  0 ) ) )
3111, 1, 23, 30syl3anc 1170 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( K  mod  P )  =  ( 0  mod  P
)  <->  P  ||  ( K  -  0 ) ) )
3227, 29, 313bitr3d 216 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N
) ) ) )  <-> 
P  ||  ( K  -  0 ) ) )
331zcnd 8551 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  CC )
3433subid1d 7475 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  - 
0 )  =  K )
3534breq2d 3805 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  0
)  <->  P  ||  K ) )
3610, 32, 353bitrd 212 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 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   0cc0 7043    x. cmul 7048    < clt 7215    - cmin 7346    / cdiv 7827   NNcn 8106   ZZcz 8432   QQcq 8785   |_cfl 9350    mod cmo 9404    || cdvds 10340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-n0 8356  df-z 8433  df-q 8786  df-rp 8816  df-fl 9352  df-mod 9405  df-dvds 10341
This theorem is referenced by: (None)
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