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Theorem dvdsmod 12869
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 962 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  ZZ )
21zred 10339 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  RR )
3 simpl2 961 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  NN )
43nnrpd 10611 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  RR+ )
5 modval 11215 . . . 4  |-  ( ( K  e.  RR  /\  N  e.  RR+ )  -> 
( K  mod  N
)  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
62, 4, 5syl2anc 643 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  mod  N )  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
76breq2d 4192 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) ) )
8 simpr 448 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  N
)
9 simpl1 960 . . . . . . . . 9  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  NN )
109nnzd 10338 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  ZZ )
113nnzd 10338 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  ZZ )
122, 3nndivred 10012 . . . . . . . . 9  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  /  N )  e.  RR )
1312flcld 11170 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( |_ `  ( K  /  N
) )  e.  ZZ )
14 dvdsmultr1 12847 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ  /\  ( |_ `  ( K  /  N ) )  e.  ZZ )  ->  ( P  ||  N  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
1510, 11, 13, 14syl3anc 1184 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  N  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N
) ) ) ) )
168, 15mpd 15 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N
) ) ) )
1711, 13zmulcld 10345 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  ZZ )
1817zcnd 10340 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  CC )
1918subid1d 9364 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  - 
0 )  =  ( N  x.  ( |_
`  ( K  /  N ) ) ) )
2016, 19breqtrrd 4206 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  (
( N  x.  ( |_ `  ( K  /  N ) ) )  -  0 ) )
21 0z 10257 . . . . . . 7  |-  0  e.  ZZ
2221a1i 11 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  e.  ZZ )
23 moddvds 12822 . . . . . 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 ) ) )
249, 17, 22, 23syl3anc 1184 . . . . 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 ) ) )
2520, 24mpbird 224 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  mod 
P )  =  ( 0  mod  P ) )
2625eqeq2d 2423 . . 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 ) ) )
27 moddvds 12822 . . . 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 ) ) ) ) ) )
289, 1, 17, 27syl3anc 1184 . . 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 ) ) ) ) ) )
29 moddvds 12822 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  ZZ  /\  0  e.  ZZ )  ->  (
( K  mod  P
)  =  ( 0  mod  P )  <->  P  ||  ( K  -  0 ) ) )
309, 1, 22, 29syl3anc 1184 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( K  mod  P )  =  ( 0  mod  P
)  <->  P  ||  ( K  -  0 ) ) )
3126, 28, 303bitr3d 275 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N
) ) ) )  <-> 
P  ||  ( K  -  0 ) ) )
321zcnd 10340 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  CC )
3332subid1d 9364 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  - 
0 )  =  K )
3433breq2d 4192 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  0
)  <->  P  ||  K ) )
357, 31, 343bitrd 271 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    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954    x. cmul 8959    - cmin 9255    / cdiv 9641   NNcn 9964   ZZcz 10246   RR+crp 10576   |_cfl 11164    mod cmo 11213    || cdivides 12815
This theorem is referenced by:  ppiublem1  20947  lgsdir2lem2  21069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fl 11165  df-mod 11214  df-dvds 12816
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