Theorem wilthlem1 23478

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 14337, ( 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 11627	. . . . . . . . . 10 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. ZZ )
2	1	adantl 464	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ZZ )
3		peano2zm 10842	. . . . . . . . 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 10903	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - 1 ) e. CC )
6	2	peano2zd 10905	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. ZZ )
7	6	zcnd 10903	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N + 1 ) e. CC )
8	5, 7	mulcomd 9546	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
9	2	zcnd 10903	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. CC )
10		ax-1cn 9479	. . . . . . 7 |- 1 e. CC
11		subsq 12197	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
12	9, 10, 11	sylancl 660	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N + 1 ) x. ( N - 1 ) ) )
13	9	sqvald 12228	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N ^ 2 ) = ( N x. N ) )
14		sq1 12184	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
15	14	a1i 11	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 ^ 2 ) = 1 )
16	13, 15	oveq12d 6232	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N ^ 2 ) - ( 1 ^ 2 ) ) = ( ( N x. N ) - 1 ) )
17	8, 12, 16	3eqtr2d 2439	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N - 1 ) x. ( N + 1 ) ) = ( ( N x. N ) - 1 ) )
18	17	breq2d 4392	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> P || ( ( N x. N ) - 1 ) ) )
19		1e0p1 10941	. . . . . . . 8 |- 1 = ( 0 + 1 )
20	19	oveq1i 6224	. . . . . . 7 |- ( 1 ... ( P - 1 ) ) = ( ( 0 + 1 ) ... ( P - 1 ) )
21		0z 10810	. . . . . . . 8 |- 0 e. ZZ
22		fzp1ss 11671	. . . . . . . 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 3460	. . . . . 6 |- ( 1 ... ( P - 1 ) ) C_ ( 0 ... ( P - 1 ) )
25		simpr 459	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 1 ... ( P - 1 ) ) )
26	24, 25	sseldi 3428	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. ( 0 ... ( P - 1 ) ) )
27	26	biantrurd 506	. . . 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 455	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. Prime )
30		euclemma 14270	. . . 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 1226	. . 3 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P || ( ( N - 1 ) x. ( N + 1 ) ) <-> ( P || ( N - 1 ) \/ P || ( N + 1 ) ) ) )
32		prmnn 14241	. . . . 5 |- ( P e. Prime -> P e. NN )
33		fzm1ndvds 14059	. . . . 5 |- ( ( P e. NN /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
34	32, 33	sylan 469	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -. P || N )
35		eqid 2392	. . . . 5 |- ( ( N ^ ( P - 2 ) ) mod P ) = ( ( N ^ ( P - 2 ) ) mod P )
36	35	prmdiveq 14337	. . . 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 1226	. . 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 10830	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 1 e. ZZ )
41		moddvds 14014	. . . . 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 1226	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( 1 mod P ) <-> P || ( N - 1 ) ) )
43		elfznn 11653	. . . . . . . 8 |- ( N e. ( 1 ... ( P - 1 ) ) -> N e. NN )
44	43	adantl 464	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN )
45	44	nnred 10485	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. RR )
46	39	nnrpd 11193	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR+ )
47	44	nnnn0d 10787	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N e. NN0 )
48	47	nn0ge0d 10790	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ N )
49		elfzle2 11629	. . . . . . . 8 |- ( N e. ( 1 ... ( P - 1 ) ) -> N <_ ( P - 1 ) )
50	49	adantl 464	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> N <_ ( P - 1 ) )
51		prmz 14242	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
52		zltlem1 10851	. . . . . . . 8 |- ( ( N e. ZZ /\ P e. ZZ ) -> ( N < P <-> N <_ ( P - 1 ) ) )
53	1, 51, 52	syl2anr 476	. . . . . . 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 11940	. . . . . 6 |- ( ( ( N e. RR /\ P e. RR+ ) /\ ( 0 <_ N /\ N < P ) ) -> ( N mod P ) = N )
56	45, 46, 48, 54, 55	syl22anc 1227	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N mod P ) = N )
57	39	nnred 10485	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. RR )
58		prmuz2 14256	. . . . . . . 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 11091	. . . . . . . 8 |- ( P e. ( ZZ>= ` 2 ) <-> ( P e. NN /\ 1 < P ) )
61	60	simprbi 462	. . . . . . 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 11948	. . . . . 6 |- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 )
64	57, 62, 63	syl2anc 659	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 mod P ) = 1 )
65	56, 64	eqeq12d 2414	. . . 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 10904	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. ZZ )
68		moddvds 14014	. . . . 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 1226	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> P || ( N - -u 1 ) ) )
70	39	nncnd 10486	. . . . . . . . . 10 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> P e. CC )
71	70	mulid2d 9543	. . . . . . . . 9 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( 1 x. P ) = P )
72	71	oveq2d 6230	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( -u 1 + P ) )
73		neg1cn 10574	. . . . . . . . 9 |- -u 1 e. CC
74		addcom 9695	. . . . . . . . 9 |- ( ( -u 1 e. CC /\ P e. CC ) -> ( -u 1 + P ) = ( P + -u 1 ) )
75	73, 70, 74	sylancr 661	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + P ) = ( P + -u 1 ) )
76		negsub 9798	. . . . . . . . 9 |- ( ( P e. CC /\ 1 e. CC ) -> ( P + -u 1 ) = ( P - 1 ) )
77	70, 10, 76	sylancl 660	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P + -u 1 ) = ( P - 1 ) )
78	72, 75, 77	3eqtrd 2437	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 + ( 1 x. P ) ) = ( P - 1 ) )
79	78	oveq1d 6229	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( ( P - 1 ) mod P ) )
80		neg1rr 10575	. . . . . . . 8 |- -u 1 e. RR
81	80	a1i 11	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> -u 1 e. RR )
82		modcyc 11951	. . . . . . 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 1226	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( -u 1 + ( 1 x. P ) ) mod P ) = ( -u 1 mod P ) )
84		peano2rem 9817	. . . . . . . 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 10772	. . . . . . . . 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 10790	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> 0 <_ ( P - 1 ) )
89	57	ltm1d 10412	. . . . . . 7 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( P - 1 ) < P )
90		modid 11940	. . . . . . 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 1227	. . . . . 6 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( P - 1 ) mod P ) = ( P - 1 ) )
92	79, 83, 91	3eqtr3d 2441	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( -u 1 mod P ) = ( P - 1 ) )
93	56, 92	eqeq12d 2414	. . . 4 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( ( N mod P ) = ( -u 1 mod P ) <-> N = ( P - 1 ) ) )
94		subneg 9799	. . . . . 6 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - -u 1 ) = ( N + 1 ) )
95	9, 10, 94	sylancl 660	. . . . 5 |- ( ( P e. Prime /\ N e. ( 1 ... ( P - 1 ) ) ) -> ( N - -u 1 ) = ( N + 1 ) )
96	95	breq2d 4392	. . . 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 707	. 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 366   /\ wa 367   = wceq 1399   e. wcel 1836   C_ wss 3402   class class class wbr 4380   ` cfv 5509  (class class class)co 6214   CCcc 9419   RRcr 9420   0cc0 9421   1c1 9422   + caddc 9424   x. cmul 9426   < clt 9557   <_ cle 9558   - cmin 9736   -ucneg 9737   NNcn 10470   2c2 10520   NN0cn0 10730   ZZcz 10799   ZZ>=cuz 11019   RR+crp 11157   ...cfz 11611   mod cmo 11915   ^cexp 12088   || cdvds 14007   Primecprime 14238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498  ax-pre-sup 9499

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-recs 6978  df-rdg 7012  df-1o 7066  df-2o 7067  df-oadd 7070  df-er 7247  df-map 7358  df-en 7454  df-dom 7455  df-sdom 7456  df-fin 7457  df-sup 7834  df-card 8251  df-cda 8479  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-div 10142  df-nn 10471  df-2 10529  df-3 10530  df-n0 10731  df-z 10800  df-uz 11020  df-rp 11158  df-fz 11612  df-fzo 11736  df-fl 11847  df-mod 11916  df-seq 12030  df-exp 12089  df-hash 12327  df-cj 12953  df-re 12954  df-im 12955  df-sqrt 13089  df-abs 13090  df-dvds 14008  df-gcd 14166  df-prm 14239  df-phi 14317

This theorem is referenced by:  wilthlem2  23479