Theorem wilth 24797

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 24796 for the forward implication. This is Metamath 100 proof #51. (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.

Step	Hyp	Ref	Expression
1		prmuz2 15408	. . 3 |- ( N e. Prime -> N e. ( ZZ>= ` 2 ) )
2		eqid 2622	. . . 4 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
3		eleq2 2690	. . . . . 6 |- ( z = x -> ( ( N - 1 ) e. z <-> ( N - 1 ) e. x ) )
4		oveq1 6657	. . . . . . . . . 10 |- ( n = y -> ( n ^ ( N - 2 ) ) = ( y ^ ( N - 2 ) ) )
5	4	oveq1d 6665	. . . . . . . . 9 |- ( n = y -> ( ( n ^ ( N - 2 ) ) mod N ) = ( ( y ^ ( N - 2 ) ) mod N ) )
6	5	eleq1d 2686	. . . . . . . 8 |- ( n = y -> ( ( ( n ^ ( N - 2 ) ) mod N ) e. z <-> ( ( y ^ ( N - 2 ) ) mod N ) e. z ) )
7	6	cbvralv 3171	. . . . . . 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 2690	. . . . . . . 8 |- ( z = x -> ( ( ( y ^ ( N - 2 ) ) mod N ) e. z <-> ( ( y ^ ( N - 2 ) ) mod N ) e. x ) )
9	8	raleqbi1dv 3146	. . . . . . 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 ) )
10	7, 9	syl5bb 272	. . . . . 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 ) )
11	3, 10	anbi12d 747	. . . . 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 ) ) )
12	11	cbvrabv 3199	. . . 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 ) }
13	2, 12	wilthlem3 24796	. . 3 |- ( N e. Prime -> N || ( ( ! ` ( N - 1 ) ) + 1 ) )
14	1, 13	jca 554	. 2 |- ( N e. Prime -> ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) )
15		simpl 473	. . 3 |- ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) -> N e. ( ZZ>= ` 2 ) )
16		elfzuz 12338	. . . . . . . . 9 |- ( n e. ( 2 ... ( N - 1 ) ) -> n e. ( ZZ>= ` 2 ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> n e. ( ZZ>= ` 2 ) )
18		eluz2nn 11726	. . . . . . . 8 |- ( n e. ( ZZ>= ` 2 ) -> n e. NN )
19	17, 18	syl 17	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> n e. NN )
20		elfzuz3 12339	. . . . . . . 8 |- ( n e. ( 2 ... ( N - 1 ) ) -> ( N - 1 ) e. ( ZZ>= ` n ) )
21	20	adantl 482	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` n ) )
22		dvdsfac 15048	. . . . . . 7 |- ( ( n e. NN /\ ( N - 1 ) e. ( ZZ>= ` n ) ) -> n || ( ! ` ( N - 1 ) ) )
23	19, 21, 22	syl2anc 693	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> n || ( ! ` ( N - 1 ) ) )
24		eluz2nn 11726	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
25	24	ad2antrr 762	. . . . . . . . 9 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> N e. NN )
26		nnm1nn0 11334	. . . . . . . . 9 |- ( N e. NN -> ( N - 1 ) e. NN0 )
27		faccl 13070	. . . . . . . . 9 |- ( ( N - 1 ) e. NN0 -> ( ! ` ( N - 1 ) ) e. NN )
28	25, 26, 27	3syl 18	. . . . . . . 8 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( ! ` ( N - 1 ) ) e. NN )
29	28	nnzd 11481	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( ! ` ( N - 1 ) ) e. ZZ )
30		eluz2b2 11761	. . . . . . . . 9 |- ( n e. ( ZZ>= ` 2 ) <-> ( n e. NN /\ 1 < n ) )
31	30	simprbi 480	. . . . . . . 8 |- ( n e. ( ZZ>= ` 2 ) -> 1 < n )
32	17, 31	syl 17	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> 1 < n )
33		ndvdsp1 15135	. . . . . . 7 |- ( ( ( ! ` ( N - 1 ) ) e. ZZ /\ n e. NN /\ 1 < n ) -> ( n || ( ! ` ( N - 1 ) ) -> -. n || ( ( ! ` ( N - 1 ) ) + 1 ) ) )
34	29, 19, 32, 33	syl3anc 1326	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( n || ( ! ` ( N - 1 ) ) -> -. n || ( ( ! ` ( N - 1 ) ) + 1 ) ) )
35	23, 34	mpd 15	. . . . 5 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> -. n || ( ( ! ` ( N - 1 ) ) + 1 ) )
36		simplr 792	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> N || ( ( ! ` ( N - 1 ) ) + 1 ) )
37	19	nnzd 11481	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> n e. ZZ )
38	25	nnzd 11481	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> N e. ZZ )
39	29	peano2zd 11485	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( ( ! ` ( N - 1 ) ) + 1 ) e. ZZ )
40		dvdstr 15018	. . . . . . 7 |- ( ( n e. ZZ /\ N e. ZZ /\ ( ( ! ` ( N - 1 ) ) + 1 ) e. ZZ ) -> ( ( n || N /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) -> n || ( ( ! ` ( N - 1 ) ) + 1 ) ) )
41	37, 38, 39, 40	syl3anc 1326	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( ( n || N /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) -> n || ( ( ! ` ( N - 1 ) ) + 1 ) ) )
42	36, 41	mpan2d 710	. . . . 5 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> ( n || N -> n || ( ( ! ` ( N - 1 ) ) + 1 ) ) )
43	35, 42	mtod 189	. . . 4 |- ( ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) /\ n e. ( 2 ... ( N - 1 ) ) ) -> -. n || N )
44	43	ralrimiva 2966	. . 3 |- ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) -> A. n e. ( 2 ... ( N - 1 ) ) -. n || N )
45		isprm3 15396	. . 3 |- ( N e. Prime <-> ( N e. ( ZZ>= ` 2 ) /\ A. n e. ( 2 ... ( N - 1 ) ) -. n || N ) )
46	15, 44, 45	sylanbrc 698	. 2 |- ( ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) -> N e. Prime )
47	14, 46	impbii 199	1 |- ( N e. Prime <-> ( N e. ( ZZ>= ` 2 ) /\ N || ( ( ! ` ( N - 1 ) ) + 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   {crab 2916   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   mod cmo 12668   ^cexp 12860   !cfa 13060   || cdvds 14983   Primecprime 15385  mulGrpcmgp 18489  ℂ_fldccnfld 19746

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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  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-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  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-phi 15471  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-cnfld 19747

This theorem is referenced by:  wilthimp  24798