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Theorem dvdsmod 13588
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 993 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  ZZ )
21zred 10745 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  RR )
3 simpl2 992 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  NN )
43nnrpd 11024 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  RR+ )
5 modval 11708 . . . 4  |-  ( ( K  e.  RR  /\  N  e.  RR+ )  -> 
( K  mod  N
)  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
62, 4, 5syl2anc 661 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  mod  N )  =  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
76breq2d 4302 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) ) )
8 simpr 461 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  N
)
9 simpl1 991 . . . . . . . . 9  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  NN )
109nnzd 10744 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  ZZ )
113nnzd 10744 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  ZZ )
122, 3nndivred 10368 . . . . . . . . 9  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  /  N )  e.  RR )
1312flcld 11646 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( |_ `  ( K  /  N
) )  e.  ZZ )
14 dvdsmultr1 13565 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ  /\  ( |_ `  ( K  /  N ) )  e.  ZZ )  ->  ( P  ||  N  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
1510, 11, 13, 14syl3anc 1218 . . . . . . 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 10751 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  ZZ )
1817zcnd 10746 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  CC )
1918subid1d 9706 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  - 
0 )  =  ( N  x.  ( |_
`  ( K  /  N ) ) ) )
2016, 19breqtrrd 4316 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  (
( N  x.  ( |_ `  ( K  /  N ) ) )  -  0 ) )
21 0zd 10656 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  e.  ZZ )
22 moddvds 13540 . . . . . 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 ) ) )
239, 17, 21, 22syl3anc 1218 . . . . 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 ) ) )
2420, 23mpbird 232 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  mod 
P )  =  ( 0  mod  P ) )
2524eqeq2d 2452 . . 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 ) ) )
26 moddvds 13540 . . . 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 ) ) ) ) ) )
279, 1, 17, 26syl3anc 1218 . . 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 ) ) ) ) ) )
28 moddvds 13540 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  ZZ  /\  0  e.  ZZ )  ->  (
( K  mod  P
)  =  ( 0  mod  P )  <->  P  ||  ( K  -  0 ) ) )
299, 1, 21, 28syl3anc 1218 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( K  mod  P )  =  ( 0  mod  P
)  <->  P  ||  ( K  -  0 ) ) )
3025, 27, 293bitr3d 283 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  ( N  x.  ( |_ `  ( K  /  N
) ) ) )  <-> 
P  ||  ( K  -  0 ) ) )
311zcnd 10746 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  CC )
3231subid1d 9706 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  - 
0 )  =  K )
3332breq2d 4302 . 2  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  -  0
)  <->  P  ||  K ) )
347, 30, 333bitrd 279 1  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( P  ||  ( K  mod  N )  <-> 
P  ||  K )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   RRcr 9279   0cc0 9280    x. cmul 9285    - cmin 9593    / cdiv 9991   NNcn 10320   ZZcz 10644   RR+crp 10989   |_cfl 11638    mod cmo 11706    || cdivides 13533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fl 11640  df-mod 11707  df-dvds 13534
This theorem is referenced by:  ppiublem1  22539  lgsdir2lem2  22661
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