Theorem pockthi 13137

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 13136 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 12888	. . . . . 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 9571	. . . . 5 |- E e. NN0
7		nnexpcl 10989	. . . . 5 |- ( ( P e. NN /\ E e. NN0 ) -> ( P ^ E ) e. NN )
8	4, 6, 7	mp2an 430	. . . 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 9314	. . . . . . . 8 |- D e. CC
16	8	nncni 9314	. . . . . . . 8 |- ( P ^ E ) e. CC
17	15, 16	mulcomi 8332	. . . . . . 7 |- ( D x. ( P ^ E ) ) = ( ( P ^ E ) x. D )
18	14, 17	eqtri 2259	. . . . . 6 |- M = ( ( P ^ E ) x. D )
19	18	oveq1i 6095	. . . . 5 |- ( M + 1 ) = ( ( ( P ^ E ) x. D ) + 1 )
20	13, 19	eqtri 2259	. . . 4 |- N = ( ( ( P ^ E ) x. D ) + 1 )
21	20	a1i 9	. . 3 |- ( D e. NN -> N = ( ( ( P ^ E ) x. D ) + 1 ) )
22		prmdvdsexpb 12927	. . . . . . 7 |- ( ( x e. Prime /\ P e. Prime /\ E e. NN ) -> ( x || ( P ^ E ) <-> x = P ) )
23	2, 5, 22	mp3an23 1370	. . . . . 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 9326	. . . . . . . . . . . . 13 |- ( G x. P ) e. NN
27	24, 26	eqeltri 2311	. . . . . . . . . . . 12 |- M e. NN
28	27	nncni 9314	. . . . . . . . . . 11 |- M e. CC
29		ax-1cn 8272	. . . . . . . . . . 11 |- 1 e. CC
30	28, 29, 13	mvrraddi 8543	. . . . . . . . . 10 |- ( N - 1 ) = M
31	30	oveq2i 6096	. . . . . . . . 9 |- ( A ^ ( N - 1 ) ) = ( A ^ M )
32	31	oveq1i 6095	. . . . . . . 8 |- ( ( A ^ ( N - 1 ) ) mod N ) = ( ( A ^ M ) mod N )
33		pockthi.mod	. . . . . . . . 9 |- ( ( A ^ M ) mod N ) = ( 1 mod N )
34		peano2nn 9316	. . . . . . . . . . . . 13 |- ( M e. NN -> ( M + 1 ) e. NN )
35	27, 34	ax-mp 5	. . . . . . . . . . . 12 |- ( M + 1 ) e. NN
36	13, 35	eqeltri 2311	. . . . . . . . . . 11 |- N e. NN
37		nnq 10033	. . . . . . . . . . 11 |- ( N e. NN -> N e. QQ )
38	36, 37	ax-mp 5	. . . . . . . . . 10 |- N e. QQ
39	27	nngt0i 9334	. . . . . . . . . . . 12 |- 0 < M
40	27	nnrei 9313	. . . . . . . . . . . . 13 |- M e. RR
41		1re 8325	. . . . . . . . . . . . 13 |- 1 e. RR
42		ltaddpos2 8781	. . . . . . . . . . . . 13 |- ( ( M e. RR /\ 1 e. RR ) -> ( 0 < M <-> 1 < ( M + 1 ) ) )
43	40, 41, 42	mp2an 430	. . . . . . . . . . . 12 |- ( 0 < M <-> 1 < ( M + 1 ) )
44	39, 43	mpbi 145	. . . . . . . . . . 11 |- 1 < ( M + 1 )
45	44, 13	breqtrri 4157	. . . . . . . . . 10 |- 1 < N
46		q1mod 10793	. . . . . . . . . 10 |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
47	38, 45, 46	mp2an 430	. . . . . . . . 9 |- ( 1 mod N ) = 1
48	33, 47	eqtri 2259	. . . . . . . 8 |- ( ( A ^ M ) mod N ) = 1
49	32, 48	eqtri 2259	. . . . . . 7 |- ( ( A ^ ( N - 1 ) ) mod N ) = 1
50		oveq2 6093	. . . . . . . . . . . 12 |- ( x = P -> ( ( N - 1 ) / x ) = ( ( N - 1 ) / P ) )
51	25	nncni 9314	. . . . . . . . . . . . . . 15 |- G e. CC
52	4	nncni 9314	. . . . . . . . . . . . . . 15 |- P e. CC
53	51, 52	mulcomi 8332	. . . . . . . . . . . . . 14 |- ( G x. P ) = ( P x. G )
54	30, 24, 53	3eqtrri 2264	. . . . . . . . . . . . 13 |- ( P x. G ) = ( N - 1 )
55	36	nncni 9314	. . . . . . . . . . . . . . 15 |- N e. CC
56	55, 29	subcli 8602	. . . . . . . . . . . . . 14 |- ( N - 1 ) e. CC
57	4	nnap0i 9335	. . . . . . . . . . . . . 14 |- P # 0
58	56, 52, 51, 57	divmulapi 9096	. . . . . . . . . . . . 13 |- ( ( ( N - 1 ) / P ) = G <-> ( P x. G ) = ( N - 1 ) )
59	54, 58	mpbir 146	. . . . . . . . . . . 12 |- ( ( N - 1 ) / P ) = G
60	50, 59	eqtrdi 2287	. . . . . . . . . . 11 |- ( x = P -> ( ( N - 1 ) / x ) = G )
61	60	oveq2d 6101	. . . . . . . . . 10 |- ( x = P -> ( A ^ ( ( N - 1 ) / x ) ) = ( A ^ G ) )
62	61	oveq1d 6100	. . . . . . . . 9 |- ( x = P -> ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ G ) - 1 ) )
63	62	oveq1d 6100	. . . . . . . 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 2287	. . . . . . 7 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 )
66		pockthi.a	. . . . . . . . 9 |- A e. NN
67	66	nnzi 9665	. . . . . . . 8 |- A e. ZZ
68		oveq1 6092	. . . . . . . . . . . 12 |- ( y = A -> ( y ^ ( N - 1 ) ) = ( A ^ ( N - 1 ) ) )
69	68	oveq1d 6100	. . . . . . . . . . 11 |- ( y = A -> ( ( y ^ ( N - 1 ) ) mod N ) = ( ( A ^ ( N - 1 ) ) mod N ) )
70	69	eqeq1d 2247	. . . . . . . . . 10 |- ( y = A -> ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 <-> ( ( A ^ ( N - 1 ) ) mod N ) = 1 ) )
71		oveq1 6092	. . . . . . . . . . . . 13 |- ( y = A -> ( y ^ ( ( N - 1 ) / x ) ) = ( A ^ ( ( N - 1 ) / x ) ) )
72	71	oveq1d 6100	. . . . . . . . . . . 12 |- ( y = A -> ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) )
73	72	oveq1d 6100	. . . . . . . . . . 11 |- ( y = A -> ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) )
74	73	eqeq1d 2247	. . . . . . . . . 10 |- ( y = A -> ( ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 <-> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
75	70, 74	anbi12d 477	. . . . . . . . 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 2929	. . . . . . . 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 428	. . . . . . 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 418	. . . . . 6 |- ( x = P -> E. y e. ZZ ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 /\ ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
79	23, 78	biimtrdi 163	. . . . 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 2603	. . . 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 13136	. 2 |- ( D e. NN -> N e. Prime )
83	1, 82	ax-mp 5	1 |- N e. Prime

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   / cdiv 9002   NNcn 9304   NN0cn0 9563   ZZcz 9644   QQcq 10019   mod cmo 10759   ^cexp 10975   || cdvds 12554   gcd cgcd 12730   Primecprime 12885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-xnn0 9631  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-proddc 12318  df-dvds 12555  df-gcd 12731  df-prm 12886  df-odz 12988  df-phi 12989  df-pc 13064

This theorem is used by: (None)