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Theorem pockthi 12970
Description: Pocklington's theorem, which gives a sufficient criterion for a number  N to be prime. This is the preferred method for verifying large primes, being much more efficient to compute than trial division. This form has been optimized for application to specific large primes; see pockthg 12969 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)
Hypotheses
Ref Expression
pockthi.p  |-  P  e. 
Prime
pockthi.g  |-  G  e.  NN
pockthi.m  |-  M  =  ( G  x.  P
)
pockthi.n  |-  N  =  ( M  +  1 )
pockthi.d  |-  D  e.  NN
pockthi.e  |-  E  e.  NN
pockthi.a  |-  A  e.  NN
pockthi.fac  |-  M  =  ( D  x.  ( P ^ E ) )
pockthi.gt  |-  D  < 
( P ^ E
)
pockthi.mod  |-  ( ( A ^ M )  mod  N )  =  ( 1  mod  N
)
pockthi.gcd  |-  ( ( ( A ^ G
)  -  1 )  gcd  N )  =  1
Assertion
Ref Expression
pockthi  |-  N  e. 
Prime

Proof of Theorem pockthi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pockthi.d . 2  |-  D  e.  NN
2 pockthi.p . . . . . 6  |-  P  e. 
Prime
3 prmnn 12777 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
42, 3ax-mp 8 . . . . 5  |-  P  e.  NN
5 pockthi.e . . . . . 6  |-  E  e.  NN
65nnnn0i 9989 . . . . 5  |-  E  e. 
NN0
7 nnexpcl 11132 . . . . 5  |-  ( ( P  e.  NN  /\  E  e.  NN0 )  -> 
( P ^ E
)  e.  NN )
84, 6, 7mp2an 653 . . . 4  |-  ( P ^ E )  e.  NN
98a1i 10 . . 3  |-  ( D  e.  NN  ->  ( P ^ E )  e.  NN )
10 id 19 . . 3  |-  ( D  e.  NN  ->  D  e.  NN )
11 pockthi.gt . . . 4  |-  D  < 
( P ^ E
)
1211a1i 10 . . 3  |-  ( D  e.  NN  ->  D  <  ( P ^ E
) )
13 pockthi.n . . . . 5  |-  N  =  ( M  +  1 )
14 pockthi.fac . . . . . . 7  |-  M  =  ( D  x.  ( P ^ E ) )
151nncni 9772 . . . . . . . 8  |-  D  e.  CC
168nncni 9772 . . . . . . . 8  |-  ( P ^ E )  e.  CC
1715, 16mulcomi 8859 . . . . . . 7  |-  ( D  x.  ( P ^ E ) )  =  ( ( P ^ E )  x.  D
)
1814, 17eqtri 2316 . . . . . 6  |-  M  =  ( ( P ^ E )  x.  D
)
1918oveq1i 5884 . . . . 5  |-  ( M  +  1 )  =  ( ( ( P ^ E )  x.  D )  +  1 )
2013, 19eqtri 2316 . . . 4  |-  N  =  ( ( ( P ^ E )  x.  D )  +  1 )
2120a1i 10 . . 3  |-  ( D  e.  NN  ->  N  =  ( ( ( P ^ E )  x.  D )  +  1 ) )
22 prmdvdsexpb 12810 . . . . . . 7  |-  ( ( x  e.  Prime  /\  P  e.  Prime  /\  E  e.  NN )  ->  ( x 
||  ( P ^ E )  <->  x  =  P ) )
232, 5, 22mp3an23 1269 . . . . . 6  |-  ( x  e.  Prime  ->  ( x 
||  ( P ^ E )  <->  x  =  P ) )
2413eqcomi 2300 . . . . . . . . . . 11  |-  ( M  +  1 )  =  N
25 pockthi.m . . . . . . . . . . . . . . . 16  |-  M  =  ( G  x.  P
)
26 pockthi.g . . . . . . . . . . . . . . . . 17  |-  G  e.  NN
2726, 4nnmulcli 9786 . . . . . . . . . . . . . . . 16  |-  ( G  x.  P )  e.  NN
2825, 27eqeltri 2366 . . . . . . . . . . . . . . 15  |-  M  e.  NN
29 peano2nn 9774 . . . . . . . . . . . . . . 15  |-  ( M  e.  NN  ->  ( M  +  1 )  e.  NN )
3028, 29ax-mp 8 . . . . . . . . . . . . . 14  |-  ( M  +  1 )  e.  NN
3113, 30eqeltri 2366 . . . . . . . . . . . . 13  |-  N  e.  NN
3231nncni 9772 . . . . . . . . . . . 12  |-  N  e.  CC
33 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
3428nncni 9772 . . . . . . . . . . . 12  |-  M  e.  CC
3532, 33, 34subadd2i 9150 . . . . . . . . . . 11  |-  ( ( N  -  1 )  =  M  <->  ( M  +  1 )  =  N )
3624, 35mpbir 200 . . . . . . . . . 10  |-  ( N  -  1 )  =  M
3736oveq2i 5885 . . . . . . . . 9  |-  ( A ^ ( N  - 
1 ) )  =  ( A ^ M
)
3837oveq1i 5884 . . . . . . . 8  |-  ( ( A ^ ( N  -  1 ) )  mod  N )  =  ( ( A ^ M )  mod  N
)
39 pockthi.mod . . . . . . . . 9  |-  ( ( A ^ M )  mod  N )  =  ( 1  mod  N
)
4031nnrei 9771 . . . . . . . . . 10  |-  N  e.  RR
4128nngt0i 9795 . . . . . . . . . . . 12  |-  0  <  M
4228nnrei 9771 . . . . . . . . . . . . 13  |-  M  e.  RR
43 1re 8853 . . . . . . . . . . . . 13  |-  1  e.  RR
44 ltaddpos2 9281 . . . . . . . . . . . . 13  |-  ( ( M  e.  RR  /\  1  e.  RR )  ->  ( 0  <  M  <->  1  <  ( M  + 
1 ) ) )
4542, 43, 44mp2an 653 . . . . . . . . . . . 12  |-  ( 0  <  M  <->  1  <  ( M  +  1 ) )
4641, 45mpbi 199 . . . . . . . . . . 11  |-  1  <  ( M  +  1 )
4746, 13breqtrri 4064 . . . . . . . . . 10  |-  1  <  N
48 1mod 11012 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  1  <  N )  -> 
( 1  mod  N
)  =  1 )
4940, 47, 48mp2an 653 . . . . . . . . 9  |-  ( 1  mod  N )  =  1
5039, 49eqtri 2316 . . . . . . . 8  |-  ( ( A ^ M )  mod  N )  =  1
5138, 50eqtri 2316 . . . . . . 7  |-  ( ( A ^ ( N  -  1 ) )  mod  N )  =  1
52 oveq2 5882 . . . . . . . . . . . 12  |-  ( x  =  P  ->  (
( N  -  1 )  /  x )  =  ( ( N  -  1 )  /  P ) )
5326nncni 9772 . . . . . . . . . . . . . . 15  |-  G  e.  CC
544nncni 9772 . . . . . . . . . . . . . . 15  |-  P  e.  CC
5553, 54mulcomi 8859 . . . . . . . . . . . . . 14  |-  ( G  x.  P )  =  ( P  x.  G
)
5636, 25, 553eqtrri 2321 . . . . . . . . . . . . 13  |-  ( P  x.  G )  =  ( N  -  1 )
5732, 33subcli 9138 . . . . . . . . . . . . . 14  |-  ( N  -  1 )  e.  CC
584nnne0i 9796 . . . . . . . . . . . . . 14  |-  P  =/=  0
5957, 54, 53, 58divmuli 9530 . . . . . . . . . . . . 13  |-  ( ( ( N  -  1 )  /  P )  =  G  <->  ( P  x.  G )  =  ( N  -  1 ) )
6056, 59mpbir 200 . . . . . . . . . . . 12  |-  ( ( N  -  1 )  /  P )  =  G
6152, 60syl6eq 2344 . . . . . . . . . . 11  |-  ( x  =  P  ->  (
( N  -  1 )  /  x )  =  G )
6261oveq2d 5890 . . . . . . . . . 10  |-  ( x  =  P  ->  ( A ^ ( ( N  -  1 )  /  x ) )  =  ( A ^ G
) )
6362oveq1d 5889 . . . . . . . . 9  |-  ( x  =  P  ->  (
( A ^ (
( N  -  1 )  /  x ) )  -  1 )  =  ( ( A ^ G )  - 
1 ) )
6463oveq1d 5889 . . . . . . . 8  |-  ( x  =  P  ->  (
( ( A ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  ( ( ( A ^ G )  -  1 )  gcd 
N ) )
65 pockthi.gcd . . . . . . . 8  |-  ( ( ( A ^ G
)  -  1 )  gcd  N )  =  1
6664, 65syl6eq 2344 . . . . . . 7  |-  ( x  =  P  ->  (
( ( A ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  1 )
67 pockthi.a . . . . . . . . 9  |-  A  e.  NN
6867nnzi 10063 . . . . . . . 8  |-  A  e.  ZZ
69 oveq1 5881 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
y ^ ( N  -  1 ) )  =  ( A ^
( N  -  1 ) ) )
7069oveq1d 5889 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( y ^ ( N  -  1 ) )  mod  N )  =  ( ( A ^ ( N  - 
1 ) )  mod 
N ) )
7170eqeq1d 2304 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( ( y ^
( N  -  1 ) )  mod  N
)  =  1  <->  (
( A ^ ( N  -  1 ) )  mod  N )  =  1 ) )
72 oveq1 5881 . . . . . . . . . . . . 13  |-  ( y  =  A  ->  (
y ^ ( ( N  -  1 )  /  x ) )  =  ( A ^
( ( N  - 
1 )  /  x
) ) )
7372oveq1d 5889 . . . . . . . . . . . 12  |-  ( y  =  A  ->  (
( y ^ (
( N  -  1 )  /  x ) )  -  1 )  =  ( ( A ^ ( ( N  -  1 )  /  x ) )  - 
1 ) )
7473oveq1d 5889 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
( ( y ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  ( ( ( A ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N ) )
7574eqeq1d 2304 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( ( ( y ^ ( ( N  -  1 )  /  x ) )  - 
1 )  gcd  N
)  =  1  <->  (
( ( A ^
( ( N  - 
1 )  /  x
) )  -  1 )  gcd  N )  =  1 ) )
7671, 75anbi12d 691 . . . . . . . . 9  |-  ( y  =  A  ->  (
( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 )  <->  ( ( ( A ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( A ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) ) )
7776rspcev 2897 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( ( ( A ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( A ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( y ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) )
7868, 77mpan 651 . . . . . . 7  |-  ( ( ( ( A ^
( N  -  1 ) )  mod  N
)  =  1  /\  ( ( ( A ^ ( ( N  -  1 )  /  x ) )  - 
1 )  gcd  N
)  =  1 )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) )
7951, 66, 78sylancr 644 . . . . . 6  |-  ( x  =  P  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( y ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) )
8023, 79syl6bi 219 . . . . 5  |-  ( x  e.  Prime  ->  ( x 
||  ( P ^ E )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  -  1 ) )  mod  N )  =  1  /\  ( ( ( y ^ (
( N  -  1 )  /  x ) )  -  1 )  gcd  N )  =  1 ) ) )
8180rgen 2621 . . . 4  |-  A. x  e.  Prime  ( x  ||  ( P ^ E )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) )
8281a1i 10 . . 3  |-  ( D  e.  NN  ->  A. x  e.  Prime  ( x  ||  ( P ^ E )  ->  E. y  e.  ZZ  ( ( ( y ^ ( N  - 
1 ) )  mod 
N )  =  1  /\  ( ( ( y ^ ( ( N  -  1 )  /  x ) )  -  1 )  gcd 
N )  =  1 ) ) )
839, 10, 12, 21, 82pockthg 12969 . 2  |-  ( D  e.  NN  ->  N  e.  Prime )
841, 83ax-mp 8 1  |-  N  e. 
Prime
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040    mod cmo 10989   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Primecprime 12774
This theorem is referenced by:  1259prm  13150  2503prm  13154  4001prm  13159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-odz 12849  df-phi 12850  df-pc 12906
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