Theorem wilthlem1 23211

Description: The only elements that are equal to their own inverses in the multiplicative group of nonzero elements in ZZ / P ZZ are 1 and -u 1 == P - 1. (Note that from prmdiveq 14190, ( N ^ ( P - 2 ) ) mod P is the modular inverse of N in ZZ / P ZZ. (Contributed by Mario Carneiro, 24-Jan-2015.)

Assertion

Ref	Expression
wilthlem1	|- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )

Proof of Theorem wilthlem1

Step	Hyp	Ref	Expression
1		elfzelz 11694	. . . . . . . . . 10 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. ZZ )
2	1	adantl 466	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ZZ )
3		peano2zm 10910	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
4	2, 3	syl 16	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. ZZ )
5	4	zcnd 10972	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. CC )
6	2	peano2zd 10974	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. ZZ )
7	6	zcnd 10972	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. CC )
8	5, 7	mulcomd 9617	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
9	2	zcnd 10972	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. CC )
10		ax-1cn 9550	. . . . . . 7 |- 1 e. CC
11		subsq 12251	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
12	9, 10, 11	sylancl 662	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
13	9	sqvald 12283	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N ^ 2 ) = ( N x. N ) )
14		sq1 12238	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
15	14	a1i 11	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 ^ 2 ) = 1 )
16	13, 15	oveq12d 6296	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N x. N ) - 1 ) )
17	8, 12, 16	3eqtr2d 2488	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N x. N ) - 1 ) )
18	17	breq2d 4446	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> P || ( ( N x. N ) - 1 ) ) )
19		1e0p1 11009	. . . . . . . 8 |- 1 = ( 0 + 1 )
20	19	oveq1i 6288	. . . . . . 7 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
21		0z 10878	. . . . . . . 8 |- 0 e. ZZ
22		fzp1ss 11737	. . . . . . . 8 |- ( 0 e. ZZ -> ( ( 0 + 1 ) ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) ) )
23	21, 22	ax-mp 5	. . . . . . 7 |- ( ( 0 + 1 ) ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
24	20, 23	eqsstri 3517	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
25		simpr 461	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 1 ... ( P - 1 ) ) )
26	24, 25	sseldi 3485	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 0 ... ( P - 1 ) ) )
27	26	biantrurd 508	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N x. N ) - 1 ) <-> ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) ) )
28	18, 27	bitrd 253	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) ) )
29		simpl 457	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime )
30		euclemma 14123	. . . 4 |- ( ( P e. Prime /\ ( N - 1 ) e. ZZ /\ ( N + 1 ) e. ZZ ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
31	29, 4, 6, 30	syl3anc 1227	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
32		prmnn 14094	. . . . 5 |- ( P e. Prime -> P e. NN )
33		fzm1ndvds 13912	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
34	32, 33	sylan 471	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
35		eqid 2441	. . . . 5 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
36	35	prmdiveq 14190	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ -. P || N ) -> ( ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) <-> N = ( ( N ^ ( P - 2 ) ) mod P ) ) )
37	29, 2, 34, 36	syl3anc 1227	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N e. ( 0 ... ( P - 1 ) ) /\ P || ( ( N x. N ) - 1 ) ) <-> N = ( ( N ^ ( P - 2 ) ) mod P ) ) )
38	28, 31, 37	3bitr3rd 284	. 2 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
39	29, 32	syl 16	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. NN )
40		1zzd 10898	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 e. ZZ )
41		moddvds 13867	. . . . 5 |- ( ( P e. NN /\ N e. ZZ /\ 1 e. ZZ ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
42	39, 2, 40, 41	syl3anc 1227	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
43		elfznn 11720	. . . . . . . 8 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. NN )
44	43	adantl 466	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN )
45	44	nnred 10554	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. RR )
46	39	nnrpd 11261	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR+ )
47	44	nnnn0d 10855	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN0 )
48	47	nn0ge0d 10858	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ N )
49		elfzle2 11696	. . . . . . . 8 |- ( N e. ( 1 ... ( P - 1 ) ) -> N <_ ( P - 1 ) )
50	49	adantl 466	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N <_ ( P - 1 ) )
51		prmz 14095	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
52		zltlem1 10919	. . . . . . . 8 |- ( ( N e. ZZ /\ P e. ZZ ) -> ( N < P <-> N <_ ( P - 1 ) ) )
53	1, 51, 52	syl2anr 478	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N < P <-> N <_ ( P - 1 ) ) )
54	50, 53	mpbird 232	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N < P )
55		modid 11996	. . . . . 6 |- ( ( ( N e. RR /\ P e. RR+ ) /\ ( 0 <_ N /\ N < P ) ) -> ( N mod P ) = N )
56	45, 46, 48, 54, 55	syl22anc 1228	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N mod P ) = N )
57	39	nnred 10554	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR )
58		prmuz2 14109	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
59	29, 58	syl 16	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. ( ZZ>= ` 2 ) )
60		eluz2b2 11160	. . . . . . . 8 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
61	60	simprbi 464	. . . . . . 7 |- ( P e. ( ZZ>= ` 2 ) -> 1 < P )
62	59, 61	syl 16	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 < P )
63		1mod 12004	. . . . . 6 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
64	57, 62, 63	syl2anc 661	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 mod P ) = 1 )
65	56, 64	eqeq12d 2463	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> N = 1 ) )
66	42, 65	bitr3d 255	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N - 1 ) <-> N = 1 ) )
67	40	znegcld 10973	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. ZZ )
68		moddvds 13867	. . . . 5 |- ( ( P e. NN /\ N e. ZZ /\ -u 1 e. ZZ ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
69	39, 2, 67, 68	syl3anc 1227	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
70	39	nncnd 10555	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. CC )
71	70	mulid2d 9614	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 x. P ) = P )
72	71	oveq2d 6294	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( -u 1 + P ) )
73		neg1cn 10642	. . . . . . . . 9 |- -u 1 e. CC
74		addcom 9766	. . . . . . . . 9 |- ( ( -u 1 e. CC /\ P e. CC ) -> ( -u 1 + P ) = ( P + -u 1 ) )
75	73, 70, 74	sylancr 663	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + P ) = ( P + -u 1 ) )
76		negsub 9869	. . . . . . . . 9 |- ( ( P e. CC /\ 1 e. CC ) -> ( P + -u 1 ) = ( P - 1 ) )
77	70, 10, 76	sylancl 662	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P + -u 1 ) = ( P - 1 ) )
78	72, 75, 77	3eqtrd 2486	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( P - 1 ) )
79	78	oveq1d 6293	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( ( P - 1 ) mod P ) )
80		neg1rr 10643	. . . . . . . 8 |- -u 1 e. RR
81	80	a1i 11	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. RR )
82		modcyc 12007	. . . . . . 7 |- ( ( -u 1 e. RR /\ P e. RR+ /\ 1 e. ZZ ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
83	81, 46, 40, 82	syl3anc 1227	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
84		peano2rem 9888	. . . . . . . 8 |- ( P e. RR -> ( P - 1 ) e. RR )
85	57, 84	syl 16	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. RR )
86		nnm1nn0 10840	. . . . . . . . 9 |- ( P e. NN -> ( P - 1 ) e. NN0 )
87	39, 86	syl 16	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) e. NN0 )
88	87	nn0ge0d 10858	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ ( P - 1 ) )
89	57	ltm1d 10481	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) < P )
90		modid 11996	. . . . . . 7 |- ( ( ( ( P - 1 ) e. RR /\ P e. RR+ ) /\ ( 0 <_ ( P - 1 ) /\ ( P - 1 ) < P ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
91	85, 46, 88, 89, 90	syl22anc 1228	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
92	79, 83, 91	3eqtr3d 2490	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 mod P ) = ( P - 1 ) )
93	56, 92	eqeq12d 2463	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> N = ( P - 1 ) ) )
94		subneg 9870	. . . . . 6 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - -u 1 ) = ( N + 1 ) )
95	9, 10, 94	sylancl 662	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - -u 1 ) = ( N + 1 ) )
96	95	breq2d 4446	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N - -u 1 ) <-> P || ( N + 1 ) ) )
97	69, 93, 96	3bitr3rd 284	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( N + 1 ) <-> N = ( P - 1 ) ) )
98	66, 97	orbi12d 709	. 2 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P || ( N - 1 ) \/ P || ( N + 1 ) ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )
99	38, 98	bitrd 253	1 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N = ( ( N ^ ( P - 2 ) ) mod P ) <-> ( N = 1 \/ N = ( P - 1 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1381   e. wcel 1802   C_ wss 3459   class class class wbr 4434   ` cfv 5575  (class class class)co 6278   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493   + caddc 9495   x. cmul 9497   < clt 9628   <_ cle 9629   - cmin 9807   -ucneg 9808   NNcn 10539   2c2 10588   NN0cn0 10798   ZZcz 10867   ZZ>=cuz 11087   RR+crp 11226   ...cfz 11678   mod cmo 11972   ^cexp 12142   || cdvds 13860   Primecprime 14091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-2o 7130  df-oadd 7133  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-sup 7900  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-2 10597  df-3 10598  df-n0 10799  df-z 10868  df-uz 11088  df-rp 11227  df-fz 11679  df-fzo 11801  df-fl 11905  df-mod 11973  df-seq 12084  df-exp 12143  df-hash 12382  df-cj 12908  df-re 12909  df-im 12910  df-sqrt 13044  df-abs 13045  df-dvds 13861  df-gcd 14019  df-prm 14092  df-phi 14170

This theorem is referenced by:  wilthlem2  23212