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Theorem dvdsmod 13891
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 996 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  ZZ )
21zred 10955 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  RR )
3 simpl2 995 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  NN )
43nnrpd 11244 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  RR+ )
5 modval 11954 . . . 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 4452 . 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 994 . . . . . . . . 9  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  NN )
109nnzd 10954 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  e.  ZZ )
113nnzd 10954 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  N  e.  ZZ )
122, 3nndivred 10573 . . . . . . . . 9  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  /  N )  e.  RR )
1312flcld 11892 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( |_ `  ( K  /  N
) )  e.  ZZ )
14 dvdsmultr1 13868 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ  /\  ( |_ `  ( K  /  N ) )  e.  ZZ )  ->  ( P  ||  N  ->  P  ||  ( N  x.  ( |_ `  ( K  /  N ) ) ) ) )
1510, 11, 13, 14syl3anc 1223 . . . . . . 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 10961 . . . . . . . 8  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  ZZ )
1817zcnd 10956 . . . . . . 7  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( N  x.  ( |_ `  ( K  /  N ) ) )  e.  CC )
1918subid1d 9908 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( ( N  x.  ( |_ `  ( K  /  N
) ) )  - 
0 )  =  ( N  x.  ( |_
`  ( K  /  N ) ) ) )
2016, 19breqtrrd 4466 . . . . 5  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  P  ||  (
( N  x.  ( |_ `  ( K  /  N ) ) )  -  0 ) )
21 0zd 10865 . . . . . 6  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  0  e.  ZZ )
22 moddvds 13843 . . . . . 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 1223 . . . . 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 2474 . . 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 13843 . . . 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 1223 . . 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 13843 . . . 4  |-  ( ( P  e.  NN  /\  K  e.  ZZ  /\  0  e.  ZZ )  ->  (
( K  mod  P
)  =  ( 0  mod  P )  <->  P  ||  ( K  -  0 ) ) )
299, 1, 21, 28syl3anc 1223 . . 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 10956 . . . 4  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  K  e.  CC )
3231subid1d 9908 . . 3  |-  ( ( ( P  e.  NN  /\  N  e.  NN  /\  K  e.  ZZ )  /\  P  ||  N )  ->  ( K  - 
0 )  =  K )
3332breq2d 4452 . 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 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   RRcr 9480   0cc0 9481    x. cmul 9486    - cmin 9794    / cdiv 10195   NNcn 10525   ZZcz 10853   RR+crp 11209   |_cfl 11884    mod cmo 11952    || cdivides 13836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953  df-dvds 13837
This theorem is referenced by:  ppiublem1  23198  lgsdir2lem2  23320
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