Theorem pockthi 12288

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 12287 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.

Step	Hyp	Ref	Expression
1		pockthi.d	. 2 |- D e. NN
2		pockthi.p	. . . . . 6 |- P e. Prime
3		prmnn 12042	. . . . . 6 |- ( P e. Prime -> P e. NN )
4	2, 3	ax-mp 5	. . . . 5 |- P e. NN
5		pockthi.e	. . . . . 6 |- E e. NN
6	5	nnnn0i 9122	. . . . 5 |- E e. NN0
7		nnexpcl 10468	. . . . 5 |- ( ( P e. NN /\ E e. NN0 ) -> ( P ^ E ) e. NN )
8	4, 6, 7	mp2an 423	. . . 4 |- ( P ^ E ) e. NN
9	8	a1i 9	. . 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 )
12	11	a1i 9	. . 3 |- ( D e. NN -> D < ( P ^ E ) )
13		pockthi.n	. . . . 5 |- N = ( M + 1 )
14		pockthi.fac	. . . . . . 7 |- M = ( D x. ( P ^ E ) )
15	1	nncni 8867	. . . . . . . 8 |- D e. CC
16	8	nncni 8867	. . . . . . . 8 |- ( P ^ E ) e. CC
17	15, 16	mulcomi 7905	. . . . . . 7 |- ( D x. ( P ^ E ) ) = ( ( P ^ E ) x. D )
18	14, 17	eqtri 2186	. . . . . 6 |- M = ( ( P ^ E ) x. D )
19	18	oveq1i 5852	. . . . 5 |- ( M + 1 ) = ( ( ( P ^ E ) x. D ) + 1 )
20	13, 19	eqtri 2186	. . . 4 |- N = ( ( ( P ^ E ) x. D ) + 1 )
21	20	a1i 9	. . 3 |- ( D e. NN -> N = ( ( ( P ^ E ) x. D ) + 1 ) )
22		prmdvdsexpb 12081	. . . . . . 7 |- ( ( x e. Prime /\ P e. Prime /\ E e. NN ) -> ( x || ( P ^ E ) <-> x = P ) )
23	2, 5, 22	mp3an23 1319	. . . . . 6 |- ( x e. Prime -> ( x || ( P ^ E ) <-> x = P ) )
24		pockthi.m	. . . . . . . . . . . . 13 |- M = ( G x. P )
25		pockthi.g	. . . . . . . . . . . . . 14 |- G e. NN
26	25, 4	nnmulcli 8879	. . . . . . . . . . . . 13 |- ( G x. P ) e. NN
27	24, 26	eqeltri 2239	. . . . . . . . . . . 12 |- M e. NN
28	27	nncni 8867	. . . . . . . . . . 11 |- M e. CC
29		ax-1cn 7846	. . . . . . . . . . 11 |- 1 e. CC
30	28, 29, 13	mvrraddi 8115	. . . . . . . . . 10 |- ( N - 1 ) = M
31	30	oveq2i 5853	. . . . . . . . 9 |- ( A ^ ( N - 1 ) ) = ( A ^ M )
32	31	oveq1i 5852	. . . . . . . 8 |- ( ( A ^ ( N - 1 ) ) mod N ) = ( ( A ^ M ) mod N )
33		pockthi.mod	. . . . . . . . 9 |- ( ( A ^ M ) mod N ) = ( 1 mod N )
34		peano2nn 8869	. . . . . . . . . . . . 13 |- ( M e. NN -> ( M + 1 ) e. NN )
35	27, 34	ax-mp 5	. . . . . . . . . . . 12 |- ( M + 1 ) e. NN
36	13, 35	eqeltri 2239	. . . . . . . . . . 11 |- N e. NN
37		nnq 9571	. . . . . . . . . . 11 |- ( N e. NN -> N e. QQ )
38	36, 37	ax-mp 5	. . . . . . . . . 10 |- N e. QQ
39	27	nngt0i 8887	. . . . . . . . . . . 12 |- 0 < M
40	27	nnrei 8866	. . . . . . . . . . . . 13 |- M e. RR
41		1re 7898	. . . . . . . . . . . . 13 |- 1 e. RR
42		ltaddpos2 8351	. . . . . . . . . . . . 13 |- ( ( M e. RR /\ 1 e. RR ) -> ( 0 < M <-> 1 < ( M + 1 ) ) )
43	40, 41, 42	mp2an 423	. . . . . . . . . . . 12 |- ( 0 < M <-> 1 < ( M + 1 ) )
44	39, 43	mpbi 144	. . . . . . . . . . 11 |- 1 < ( M + 1 )
45	44, 13	breqtrri 4009	. . . . . . . . . 10 |- 1 < N
46		q1mod 10291	. . . . . . . . . 10 |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
47	38, 45, 46	mp2an 423	. . . . . . . . 9 |- ( 1 mod N ) = 1
48	33, 47	eqtri 2186	. . . . . . . 8 |- ( ( A ^ M ) mod N ) = 1
49	32, 48	eqtri 2186	. . . . . . 7 |- ( ( A ^ ( N - 1 ) ) mod N ) = 1
50		oveq2 5850	. . . . . . . . . . . 12 |- ( x = P -> ( ( N - 1 ) / x ) = ( ( N - 1 ) / P ) )
51	25	nncni 8867	. . . . . . . . . . . . . . 15 |- G e. CC
52	4	nncni 8867	. . . . . . . . . . . . . . 15 |- P e. CC
53	51, 52	mulcomi 7905	. . . . . . . . . . . . . 14 |- ( G x. P ) = ( P x. G )
54	30, 24, 53	3eqtrri 2191	. . . . . . . . . . . . 13 |- ( P x. G ) = ( N - 1 )
55	36	nncni 8867	. . . . . . . . . . . . . . 15 |- N e. CC
56	55, 29	subcli 8174	. . . . . . . . . . . . . 14 |- ( N - 1 ) e. CC
57	4	nnap0i 8888	. . . . . . . . . . . . . 14 |- P # 0
58	56, 52, 51, 57	divmulapi 8662	. . . . . . . . . . . . 13 |- ( ( ( N - 1 ) / P ) = G <-> ( P x. G ) = ( N - 1 ) )
59	54, 58	mpbir 145	. . . . . . . . . . . 12 |- ( ( N - 1 ) / P ) = G
60	50, 59	eqtrdi 2215	. . . . . . . . . . 11 |- ( x = P -> ( ( N - 1 ) / x ) = G )
61	60	oveq2d 5858	. . . . . . . . . 10 |- ( x = P -> ( A ^ ( ( N - 1 ) / x ) ) = ( A ^ G ) )
62	61	oveq1d 5857	. . . . . . . . 9 |- ( x = P -> ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ G ) - 1 ) )
63	62	oveq1d 5857	. . . . . . . 8 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ G ) - 1 ) gcd N ) )
64		pockthi.gcd	. . . . . . . 8 |- ( ( ( A ^ G ) - 1 ) gcd N ) = 1
65	63, 64	eqtrdi 2215	. . . . . . 7 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 )
66		pockthi.a	. . . . . . . . 9 |- A e. NN
67	66	nnzi 9212	. . . . . . . 8 |- A e. ZZ
68		oveq1 5849	. . . . . . . . . . . 12 |- ( y = A -> ( y ^ ( N - 1 ) ) = ( A ^ ( N - 1 ) ) )
69	68	oveq1d 5857	. . . . . . . . . . 11 |- ( y = A -> ( ( y ^ ( N - 1 ) ) mod N ) = ( ( A ^ ( N - 1 ) ) mod N ) )
70	69	eqeq1d 2174	. . . . . . . . . 10 |- ( y = A -> ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 <-> ( ( A ^ ( N - 1 ) ) mod N ) = 1 ) )
71		oveq1 5849	. . . . . . . . . . . . 13 |- ( y = A -> ( y ^ ( ( N - 1 ) / x ) ) = ( A ^ ( ( N - 1 ) / x ) ) )
72	71	oveq1d 5857	. . . . . . . . . . . 12 |- ( y = A -> ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) )
73	72	oveq1d 5857	. . . . . . . . . . 11 |- ( y = A -> ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) )
74	73	eqeq1d 2174	. . . . . . . . . 10 |- ( y = A -> ( ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 <-> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
75	70, 74	anbi12d 465	. . . . . . . . 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 ) ) )
76	75	rspcev 2830	. . . . . . . 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 ) )
77	67, 76	mpan 421	. . . . . . 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 ) )
78	49, 65, 77	sylancr 411	. . . . . 6 |- ( x = P -> E. y e. ZZ ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 /\ ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
79	23, 78	syl6bi 162	. . . . 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 ) ) )
80	79	rgen 2519	. . . 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 ) )
81	80	a1i 9	. . 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 ) ) )
82	9, 10, 12, 21, 81	pockthg 12287	. 2 |- ( D e. NN -> N e. Prime )
83	1, 82	ax-mp 5	1 |- N e. Prime

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   - cmin 8069   / cdiv 8568   NNcn 8857   NN0cn0 9114   ZZcz 9191   QQcq 9557   mod cmo 10257   ^cexp 10454   || cdvds 11727   gcd cgcd 11875   Primecprime 12039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-2o 6385  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-xnn0 9178  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-proddc 11492  df-dvds 11728  df-gcd 11876  df-prm 12040  df-odz 12142  df-phi 12143  df-pc 12217

This theorem is referenced by: (None)