Theorem wilthlem1 23070

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 14171, ( 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 11684	. . . . . . . . . 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 10902	. . . . . . . . 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 10963	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. CC )
6	2	peano2zd 10965	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. ZZ )
7	6	zcnd 10963	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. CC )
8	5, 7	mulcomd 9613	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
9	2	zcnd 10963	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. CC )
10		ax-1cn 9546	. . . . . . 7 |- 1 e. CC
11		subsq 12239	. . . . . . 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 12271	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N ^ 2 ) = ( N x. N ) )
14		sq1 12226	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
15	14	a1i 11	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 ^ 2 ) = 1 )
16	13, 15	oveq12d 6300	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N x. N ) - 1 ) )
17	8, 12, 16	3eqtr2d 2514	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N x. N ) - 1 ) )
18	17	breq2d 4459	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> P || ( ( N x. N ) - 1 ) ) )
19		1e0p1 11000	. . . . . . . 8 |- 1 = ( 0 + 1 )
20	19	oveq1i 6292	. . . . . . 7 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
21		0z 10871	. . . . . . . 8 |- 0 e. ZZ
22		fzp1ss 11727	. . . . . . . 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 3534	. . . . . 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 3502	. . . . 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 14104	. . . 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 1228	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
32		prmnn 14075	. . . . 5 |- ( P e. Prime -> P e. NN )
33		fzm1ndvds 13893	. . . . 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 2467	. . . . 5 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
36	35	prmdiveq 14171	. . . 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 1228	. . 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 10891	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 e. ZZ )
41		moddvds 13850	. . . . 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 1228	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
43		elfznn 11710	. . . . . . . 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 10547	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. RR )
46	39	nnrpd 11251	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR+ )
47	44	nnnn0d 10848	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN0 )
48	47	nn0ge0d 10851	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ N )
49		elfzle2 11686	. . . . . . . 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 14076	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
52		zltlem1 10911	. . . . . . . 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 11984	. . . . . 6 |- ( ( ( N e. RR /\ P e. RR+ ) /\ ( 0 <_ N /\ N < P ) ) -> ( N mod P ) = N )
56	45, 46, 48, 54, 55	syl22anc 1229	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N mod P ) = N )
57	39	nnred 10547	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR )
58		prmuz2 14090	. . . . . . . 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 11150	. . . . . . . 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 11992	. . . . . 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 2489	. . . 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 10964	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. ZZ )
68		moddvds 13850	. . . . 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 1228	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
70	39	nncnd 10548	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. CC )
71	70	mulid2d 9610	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 x. P ) = P )
72	71	oveq2d 6298	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( -u 1 + P ) )
73		neg1cn 10635	. . . . . . . . 9 |- -u 1 e. CC
74		addcom 9761	. . . . . . . . 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 9863	. . . . . . . . 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 2512	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( P - 1 ) )
79	78	oveq1d 6297	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( ( P - 1 ) mod P ) )
80		neg1rr 10636	. . . . . . . 8 |- -u 1 e. RR
81	80	a1i 11	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. RR )
82		modcyc 11995	. . . . . . 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 1228	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
84		peano2rem 9882	. . . . . . . 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 10833	. . . . . . . . 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 10851	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ ( P - 1 ) )
89	57	ltm1d 10474	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) < P )
90		modid 11984	. . . . . . 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 1229	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
92	79, 83, 91	3eqtr3d 2516	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 mod P ) = ( P - 1 ) )
93	56, 92	eqeq12d 2489	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> N = ( P - 1 ) ) )
94		subneg 9864	. . . . . 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 4459	. . . 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 1379   e. wcel 1767   C_ wss 3476   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   + caddc 9491   x. cmul 9493   < clt 9624   <_ cle 9625   - cmin 9801   -ucneg 9802   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   ...cfz 11668   mod cmo 11960   ^cexp 12130   || cdivides 13843   Primecprime 14072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-dvds 13844  df-gcd 14000  df-prm 14073  df-phi 14151

This theorem is referenced by:  wilthlem2  23071