MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wilth Unicode version

Theorem wilth 20807
Description: Wilson's theorem. A number is prime iff it is greater or equal to  2 and  ( N  - 
1 ) ! is congruent to  -u 1,  mod  N, or alternatively if  N divides  ( N  - 
1 ) !  + 
1. In this part of the proof we show the relatively simple reverse implication; see wilthlem3 20806 for the forward implication. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by Fan Zheng, 16-Jun-2016.)
Assertion
Ref Expression
wilth  |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  N  ||  ( ( ! `  ( N  -  1 ) )  +  1 ) ) )

Proof of Theorem wilth
Dummy variables  x  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmuz2 13052 . . 3  |-  ( N  e.  Prime  ->  N  e.  ( ZZ>= `  2 )
)
2 eqid 2404 . . . 4  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
3 eleq2 2465 . . . . . 6  |-  ( z  =  x  ->  (
( N  -  1 )  e.  z  <->  ( N  -  1 )  e.  x ) )
4 oveq1 6047 . . . . . . . . . 10  |-  ( n  =  y  ->  (
n ^ ( N  -  2 ) )  =  ( y ^
( N  -  2 ) ) )
54oveq1d 6055 . . . . . . . . 9  |-  ( n  =  y  ->  (
( n ^ ( N  -  2 ) )  mod  N )  =  ( ( y ^ ( N  - 
2 ) )  mod 
N ) )
65eleq1d 2470 . . . . . . . 8  |-  ( n  =  y  ->  (
( ( n ^
( N  -  2 ) )  mod  N
)  e.  z  <->  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  z ) )
76cbvralv 2892 . . . . . . 7  |-  ( A. n  e.  z  (
( n ^ ( N  -  2 ) )  mod  N )  e.  z  <->  A. y  e.  z  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  z )
8 eleq2 2465 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( y ^
( N  -  2 ) )  mod  N
)  e.  z  <->  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) )
98raleqbi1dv 2872 . . . . . . 7  |-  ( z  =  x  ->  ( A. y  e.  z 
( ( y ^
( N  -  2 ) )  mod  N
)  e.  z  <->  A. y  e.  x  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) )
107, 9syl5bb 249 . . . . . 6  |-  ( z  =  x  ->  ( A. n  e.  z 
( ( n ^
( N  -  2 ) )  mod  N
)  e.  z  <->  A. y  e.  x  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) )
113, 10anbi12d 692 . . . . 5  |-  ( z  =  x  ->  (
( ( N  - 
1 )  e.  z  /\  A. n  e.  z  ( ( n ^ ( N  - 
2 ) )  mod 
N )  e.  z )  <->  ( ( N  -  1 )  e.  x  /\  A. y  e.  x  ( (
y ^ ( N  -  2 ) )  mod  N )  e.  x ) ) )
1211cbvrabv 2915 . . . 4  |-  { z  e.  ~P ( 1 ... ( N  - 
1 ) )  |  ( ( N  - 
1 )  e.  z  /\  A. n  e.  z  ( ( n ^ ( N  - 
2 ) )  mod 
N )  e.  z ) }  =  {
x  e.  ~P (
1 ... ( N  - 
1 ) )  |  ( ( N  - 
1 )  e.  x  /\  A. y  e.  x  ( ( y ^
( N  -  2 ) )  mod  N
)  e.  x ) }
132, 12wilthlem3 20806 . . 3  |-  ( N  e.  Prime  ->  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )
141, 13jca 519 . 2  |-  ( N  e.  Prime  ->  ( N  e.  ( ZZ>= `  2
)  /\  N  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
15 simpl 444 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  N  e.  ( ZZ>= `  2 )
)
16 elfzuz 11011 . . . . . . . . 9  |-  ( n  e.  ( 2 ... ( N  -  1 ) )  ->  n  e.  ( ZZ>= `  2 )
)
1716adantl 453 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  e.  (
ZZ>= `  2 ) )
18 eluz2b2 10504 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  2
)  <->  ( n  e.  NN  /\  1  < 
n ) )
1918simplbi 447 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
2017, 19syl 16 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  e.  NN )
21 elfzuz3 11012 . . . . . . . 8  |-  ( n  e.  ( 2 ... ( N  -  1 ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  n
) )
2221adantl 453 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( N  - 
1 )  e.  (
ZZ>= `  n ) )
23 dvdsfac 12859 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( N  -  1
)  e.  ( ZZ>= `  n ) )  ->  n  ||  ( ! `  ( N  -  1
) ) )
2420, 22, 23syl2anc 643 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  ||  ( ! `  ( N  -  1 ) ) )
25 eluz2b2 10504 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  1  < 
N ) )
2625simplbi 447 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN )
2726ad2antrr 707 . . . . . . . . 9  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  N  e.  NN )
28 nnm1nn0 10217 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
29 faccl 11531 . . . . . . . . 9  |-  ( ( N  -  1 )  e.  NN0  ->  ( ! `
 ( N  - 
1 ) )  e.  NN )
3027, 28, 293syl 19 . . . . . . . 8  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ! `  ( N  -  1
) )  e.  NN )
3130nnzd 10330 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ! `  ( N  -  1
) )  e.  ZZ )
3218simprbi 451 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  2
)  ->  1  <  n )
3317, 32syl 16 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  1  <  n
)
34 ndvdsp1 12884 . . . . . . 7  |-  ( ( ( ! `  ( N  -  1 ) )  e.  ZZ  /\  n  e.  NN  /\  1  <  n )  ->  (
n  ||  ( ! `  ( N  -  1 ) )  ->  -.  n  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) ) )
3531, 20, 33, 34syl3anc 1184 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( n  ||  ( ! `  ( N  -  1 ) )  ->  -.  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
3624, 35mpd 15 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  -.  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) )
37 simplr 732 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  N  ||  (
( ! `  ( N  -  1 ) )  +  1 ) )
3820nnzd 10330 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  n  e.  ZZ )
3927nnzd 10330 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  N  e.  ZZ )
4031peano2zd 10334 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ( ! `
 ( N  - 
1 ) )  +  1 )  e.  ZZ )
41 dvdstr 12838 . . . . . . 7  |-  ( ( n  e.  ZZ  /\  N  e.  ZZ  /\  (
( ! `  ( N  -  1 ) )  +  1 )  e.  ZZ )  -> 
( ( n  ||  N  /\  N  ||  (
( ! `  ( N  -  1 ) )  +  1 ) )  ->  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
4238, 39, 40, 41syl3anc 1184 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( ( n 
||  N  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  n  ||  ( ( ! `  ( N  -  1
) )  +  1 ) ) )
4337, 42mpan2d 656 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  ( n  ||  N  ->  n  ||  (
( ! `  ( N  -  1 ) )  +  1 ) ) )
4436, 43mtod 170 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  2 )  /\  N  ||  ( ( ! `
 ( N  - 
1 ) )  +  1 ) )  /\  n  e.  ( 2 ... ( N  - 
1 ) ) )  ->  -.  n  ||  N
)
4544ralrimiva 2749 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  A. n  e.  ( 2 ... ( N  -  1 ) )  -.  n  ||  N )
46 isprm3 13043 . . 3  |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  A. n  e.  ( 2 ... ( N  -  1 ) )  -.  n  ||  N
) )
4715, 45, 46sylanbrc 646 . 2  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  N  ||  ( ( ! `  ( N  -  1
) )  +  1 ) )  ->  N  e.  Prime )
4814, 47impbii 181 1  |-  ( N  e.  Prime  <->  ( N  e.  ( ZZ>= `  2 )  /\  N  ||  ( ( ! `  ( N  -  1 ) )  +  1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   A.wral 2666   {crab 2670   ~Pcpw 3759   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   1c1 8947    + caddc 8949    < clt 9076    - cmin 9247   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    mod cmo 11205   ^cexp 11337   !cfa 11521    || cdivides 12807   Primecprime 13034  mulGrpcmgp 15603  ℂfldccnfld 16658
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-phi 13110  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-mulg 14770  df-subg 14896  df-cntz 15071  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-cnfld 16659
  Copyright terms: Public domain W3C validator