Theorem pockthi 15611

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 15610 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 15388	. . . . . 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 11300	. . . . 5 |- E e. NN0
7		nnexpcl 12873	. . . . 5 |- ( ( P e. NN /\ E e. NN0 ) -> ( P ^ E ) e. NN )
8	4, 6, 7	mp2an 708	. . . 4 |- ( P ^ E ) e. NN
9	8	a1i 11	. . 3 |- ( D e. NN -> ( P ^ E ) e. NN )
10		id 22	. . 3 |- ( D e. NN -> D e. NN )
11		pockthi.gt	. . . 4 |- D < ( P ^ E )
12	11	a1i 11	. . 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 11030	. . . . . . . 8 |- D e. CC
16	8	nncni 11030	. . . . . . . 8 |- ( P ^ E ) e. CC
17	15, 16	mulcomi 10046	. . . . . . 7 |- ( D x. ( P ^ E ) ) = ( ( P ^ E ) x. D )
18	14, 17	eqtri 2644	. . . . . 6 |- M = ( ( P ^ E ) x. D )
19	18	oveq1i 6660	. . . . 5 |- ( M + 1 ) = ( ( ( P ^ E ) x. D ) + 1 )
20	13, 19	eqtri 2644	. . . 4 |- N = ( ( ( P ^ E ) x. D ) + 1 )
21	20	a1i 11	. . 3 |- ( D e. NN -> N = ( ( ( P ^ E ) x. D ) + 1 ) )
22		prmdvdsexpb 15428	. . . . . . 7 |- ( ( x e. Prime /\ P e. Prime /\ E e. NN ) -> ( x || ( P ^ E ) <-> x = P ) )
23	2, 5, 22	mp3an23 1416	. . . . . 6 |- ( x e. Prime -> ( x || ( P ^ E ) <-> x = P ) )
24	13	eqcomi 2631	. . . . . . . . . . 11 |- ( M + 1 ) = N
25		pockthi.m	. . . . . . . . . . . . . . . 16 |- M = ( G x. P )
26		pockthi.g	. . . . . . . . . . . . . . . . 17 |- G e. NN
27	26, 4	nnmulcli 11044	. . . . . . . . . . . . . . . 16 |- ( G x. P ) e. NN
28	25, 27	eqeltri 2697	. . . . . . . . . . . . . . 15 |- M e. NN
29		peano2nn 11032	. . . . . . . . . . . . . . 15 |- ( M e. NN -> ( M + 1 ) e. NN )
30	28, 29	ax-mp 5	. . . . . . . . . . . . . 14 |- ( M + 1 ) e. NN
31	13, 30	eqeltri 2697	. . . . . . . . . . . . 13 |- N e. NN
32	31	nncni 11030	. . . . . . . . . . . 12 |- N e. CC
33		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
34	28	nncni 11030	. . . . . . . . . . . 12 |- M e. CC
35	32, 33, 34	subadd2i 10369	. . . . . . . . . . 11 |- ( ( N - 1 ) = M <-> ( M + 1 ) = N )
36	24, 35	mpbir 221	. . . . . . . . . 10 |- ( N - 1 ) = M
37	36	oveq2i 6661	. . . . . . . . 9 |- ( A ^ ( N - 1 ) ) = ( A ^ M )
38	37	oveq1i 6660	. . . . . . . 8 |- ( ( A ^ ( N - 1 ) ) mod N ) = ( ( A ^ M ) mod N )
39		pockthi.mod	. . . . . . . . 9 |- ( ( A ^ M ) mod N ) = ( 1 mod N )
40	31	nnrei 11029	. . . . . . . . . 10 |- N e. RR
41	28	nngt0i 11054	. . . . . . . . . . . 12 |- 0 < M
42	28	nnrei 11029	. . . . . . . . . . . . 13 |- M e. RR
43		1re 10039	. . . . . . . . . . . . 13 |- 1 e. RR
44		ltaddpos2 10519	. . . . . . . . . . . . 13 |- ( ( M e. RR /\ 1 e. RR ) -> ( 0 < M <-> 1 < ( M + 1 ) ) )
45	42, 43, 44	mp2an 708	. . . . . . . . . . . 12 |- ( 0 < M <-> 1 < ( M + 1 ) )
46	41, 45	mpbi 220	. . . . . . . . . . 11 |- 1 < ( M + 1 )
47	46, 13	breqtrri 4680	. . . . . . . . . 10 |- 1 < N
48		1mod 12702	. . . . . . . . . 10 |- ( ( N e. RR /\ 1 < N ) -> ( 1 mod N ) = 1 )
49	40, 47, 48	mp2an 708	. . . . . . . . 9 |- ( 1 mod N ) = 1
50	39, 49	eqtri 2644	. . . . . . . 8 |- ( ( A ^ M ) mod N ) = 1
51	38, 50	eqtri 2644	. . . . . . 7 |- ( ( A ^ ( N - 1 ) ) mod N ) = 1
52		oveq2 6658	. . . . . . . . . . . 12 |- ( x = P -> ( ( N - 1 ) / x ) = ( ( N - 1 ) / P ) )
53	26	nncni 11030	. . . . . . . . . . . . . . 15 |- G e. CC
54	4	nncni 11030	. . . . . . . . . . . . . . 15 |- P e. CC
55	53, 54	mulcomi 10046	. . . . . . . . . . . . . 14 |- ( G x. P ) = ( P x. G )
56	36, 25, 55	3eqtrri 2649	. . . . . . . . . . . . 13 |- ( P x. G ) = ( N - 1 )
57	32, 33	subcli 10357	. . . . . . . . . . . . . 14 |- ( N - 1 ) e. CC
58	4	nnne0i 11055	. . . . . . . . . . . . . 14 |- P =/= 0
59	57, 54, 53, 58	divmuli 10779	. . . . . . . . . . . . 13 |- ( ( ( N - 1 ) / P ) = G <-> ( P x. G ) = ( N - 1 ) )
60	56, 59	mpbir 221	. . . . . . . . . . . 12 |- ( ( N - 1 ) / P ) = G
61	52, 60	syl6eq 2672	. . . . . . . . . . 11 |- ( x = P -> ( ( N - 1 ) / x ) = G )
62	61	oveq2d 6666	. . . . . . . . . 10 |- ( x = P -> ( A ^ ( ( N - 1 ) / x ) ) = ( A ^ G ) )
63	62	oveq1d 6665	. . . . . . . . 9 |- ( x = P -> ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ G ) - 1 ) )
64	63	oveq1d 6665	. . . . . . . 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
66	64, 65	syl6eq 2672	. . . . . . 7 |- ( x = P -> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 )
67		pockthi.a	. . . . . . . . 9 |- A e. NN
68	67	nnzi 11401	. . . . . . . 8 |- A e. ZZ
69		oveq1 6657	. . . . . . . . . . . 12 |- ( y = A -> ( y ^ ( N - 1 ) ) = ( A ^ ( N - 1 ) ) )
70	69	oveq1d 6665	. . . . . . . . . . 11 |- ( y = A -> ( ( y ^ ( N - 1 ) ) mod N ) = ( ( A ^ ( N - 1 ) ) mod N ) )
71	70	eqeq1d 2624	. . . . . . . . . 10 |- ( y = A -> ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 <-> ( ( A ^ ( N - 1 ) ) mod N ) = 1 ) )
72		oveq1 6657	. . . . . . . . . . . . 13 |- ( y = A -> ( y ^ ( ( N - 1 ) / x ) ) = ( A ^ ( ( N - 1 ) / x ) ) )
73	72	oveq1d 6665	. . . . . . . . . . . 12 |- ( y = A -> ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) = ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) )
74	73	oveq1d 6665	. . . . . . . . . . 11 |- ( y = A -> ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) )
75	74	eqeq1d 2624	. . . . . . . . . 10 |- ( y = A -> ( ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 <-> ( ( ( A ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
76	71, 75	anbi12d 747	. . . . . . . . 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 ) ) )
77	76	rspcev 3309	. . . . . . . 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 ) )
78	68, 77	mpan 706	. . . . . . 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 ) )
79	51, 66, 78	sylancr 695	. . . . . 6 |- ( x = P -> E. y e. ZZ ( ( ( y ^ ( N - 1 ) ) mod N ) = 1 /\ ( ( ( y ^ ( ( N - 1 ) / x ) ) - 1 ) gcd N ) = 1 ) )
80	23, 79	syl6bi 243	. . . . 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 ) ) )
81	80	rgen 2922	. . . 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 ) )
82	81	a1i 11	. . 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 ) ) )
83	9, 10, 12, 21, 82	pockthg 15610	. 2 |- ( D e. NN -> N e. Prime )
84	1, 83	ax-mp 5	1 |- N e. Prime

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   mod cmo 12668   ^cexp 12860   || cdvds 14983   gcd cgcd 15216   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-odz 15470  df-phi 15471  df-pc 15542

This theorem is referenced by:  1259prm  15843  2503prm  15847  4001prm  15852