Theorem lgslem1 13970

Description: When a is coprime to the prime p, a ^ ( ( p - 1 ) / 2 ) is equivalent mod p to 1 or -u 1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.)

Assertion

Ref	Expression
lgslem1	|- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )

Proof of Theorem lgslem1

Step	Hyp	Ref	Expression
1		eldifi 3255	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	3ad2ant2 1019	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. Prime )
3		prmnn 12075	. . . . . . . 8 |- ( P e. Prime -> P e. NN )
4	2, 3	syl 14	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. NN )
5		simp1 997	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> A e. ZZ )
6		prmz 12076	. . . . . . . . . 10 |- ( P e. Prime -> P e. ZZ )
7	2, 6	syl 14	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. ZZ )
8	5, 7	gcdcomd 11940	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
9		simp3 999	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> -. P || A )
10		coprm 12109	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
11	2, 5, 10	syl2anc 411	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
12	9, 11	mpbid 147	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P gcd A ) = 1 )
13	8, 12	eqtrd 2208	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A gcd P ) = 1 )
14		eulerth 12198	. . . . . . 7 |- ( ( P e. NN /\ A e. ZZ /\ ( A gcd P ) = 1 ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
15	4, 5, 13, 14	syl3anc 1238	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
16		phiprm 12188	. . . . . . . . . 10 |- ( P e. Prime -> ( phi ` P ) = ( P - 1 ) )
17	2, 16	syl 14	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) = ( P - 1 ) )
18		nnm1nn0 9188	. . . . . . . . . 10 |- ( P e. NN -> ( P - 1 ) e. NN0 )
19	4, 18	syl 14	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P - 1 ) e. NN0 )
20	17, 19	eqeltrd 2252	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) e. NN0 )
21		zexpcl 10503	. . . . . . . 8 |- ( ( A e. ZZ /\ ( phi ` P ) e. NN0 ) -> ( A ^ ( phi ` P ) ) e. ZZ )
22	5, 20, 21	syl2anc 411	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) e. ZZ )
23		1zzd 9251	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 1 e. ZZ )
24		moddvds 11772	. . . . . . 7 |- ( ( P e. NN /\ ( A ^ ( phi ` P ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( phi ` P ) ) - 1 ) ) )
25	4, 22, 23, 24	syl3anc 1238	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( phi ` P ) ) - 1 ) ) )
26	15, 25	mpbid 147	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P || ( ( A ^ ( phi ` P ) ) - 1 ) )
27	19	nn0cnd 9202	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P - 1 ) e. CC )
28		2cnd 8963	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. CC )
29		2ap0 8983	. . . . . . . . . . . . 13 |- 2 # 0
30	29	a1i 9	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 # 0 )
31	27, 28, 30	divcanap1d 8720	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( P - 1 ) / 2 ) x. 2 ) = ( P - 1 ) )
32	17, 31	eqtr4d 2211	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) = ( ( ( P - 1 ) / 2 ) x. 2 ) )
33	32	oveq2d 5881	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( ( P - 1 ) / 2 ) x. 2 ) ) )
34	5	zcnd 9347	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> A e. CC )
35		2nn0 9164	. . . . . . . . . . 11 |- 2 e. NN0
36	35	a1i 9	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. NN0 )
37		oddprm 12224	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
38	37	3ad2ant2 1019	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
39	38	nnnn0d 9200	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
40	34, 36, 39	expmuld 10624	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( ( ( P - 1 ) / 2 ) x. 2 ) ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) )
41	33, 40	eqtrd 2208	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) )
42	41	oveq1d 5880	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - 1 ) )
43		sq1 10581	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
44	43	oveq2i 5876	. . . . . . 7 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - 1 )
45	42, 44	eqtr4di 2226	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) )
46		zexpcl 10503	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
47	5, 39, 46	syl2anc 411	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
48	47	zcnd 9347	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
49		ax-1cn 7879	. . . . . . 7 |- 1 e. CC
50		subsq 10594	. . . . . . 7 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) e. CC /\ 1 e. CC ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) x. ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
51	48, 49, 50	sylancl 413	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) x. ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
52	45, 51	eqtrd 2208	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) x. ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
53	26, 52	breqtrd 4024	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P || ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) x. ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
54	47	peano2zd 9349	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
55		peano2zm 9262	. . . . . 6 |- ( ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ -> ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) e. ZZ )
56	47, 55	syl 14	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) e. ZZ )
57		euclemma 12111	. . . . 5 |- ( ( P e. Prime /\ ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ /\ ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) e. ZZ ) -> ( P || ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) x. ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) <-> ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) \/ P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) ) )
58	2, 54, 56, 57	syl3anc 1238	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P || ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) x. ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) <-> ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) \/ P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) ) )
59	53, 58	mpbid 147	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) \/ P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
60		dvdsval3 11764	. . . . 5 |- ( ( P e. NN /\ ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ ) -> ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) <-> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 ) )
61	4, 54, 60	syl2anc 411	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) <-> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 ) )
62		2z 9252	. . . . . . 7 |- 2 e. ZZ
63	62	a1i 9	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. ZZ )
64		moddvds 11772	. . . . . 6 |- ( ( P e. NN /\ ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ /\ 2 e. ZZ ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 2 mod P ) <-> P || ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) ) )
65	4, 54, 63, 64	syl3anc 1238	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 2 mod P ) <-> P || ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) ) )
66		zq 9597	. . . . . . . 8 |- ( 2 e. ZZ -> 2 e. QQ )
67	62, 66	mp1i 10	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. QQ )
68		zq 9597	. . . . . . . 8 |- ( P e. ZZ -> P e. QQ )
69	7, 68	syl 14	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. QQ )
70		0le2 8980	. . . . . . . 8 |- 0 <_ 2
71	70	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 0 <_ 2 )
72		eldifsni 3718	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
73	72	3ad2ant2 1019	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P =/= 2 )
74		zapne 9298	. . . . . . . . . 10 |- ( ( P e. ZZ /\ 2 e. ZZ ) -> ( P # 2 <-> P =/= 2 ) )
75	7, 62, 74	sylancl 413	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P # 2 <-> P =/= 2 ) )
76	73, 75	mpbird 167	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P # 2 )
77		2re 8960	. . . . . . . . . 10 |- 2 e. RR
78	77	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. RR )
79	4	nnred 8903	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. RR )
80		prmuz2 12096	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
81	2, 80	syl 14	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. ( ZZ>= ` 2 ) )
82		eluzle 9511	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 2 ) -> 2 <_ P )
83	81, 82	syl 14	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 <_ P )
84	78, 79, 83	leltapd 8570	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( 2 < P <-> P # 2 ) )
85	76, 84	mpbird 167	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 < P )
86		modqid 10317	. . . . . . 7 |- ( ( ( 2 e. QQ /\ P e. QQ ) /\ ( 0 <_ 2 /\ 2 < P ) ) -> ( 2 mod P ) = 2 )
87	67, 69, 71, 85, 86	syl22anc 1239	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( 2 mod P ) = 2 )
88	87	eqeq2d 2187	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = ( 2 mod P ) <-> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) )
89		df-2 8949	. . . . . . . 8 |- 2 = ( 1 + 1 )
90	89	oveq2i 5876	. . . . . . 7 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - ( 1 + 1 ) )
91	49	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 1 e. CC )
92	48, 91, 91	pnpcan2d 8280	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - ( 1 + 1 ) ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) )
93	90, 92	eqtrid 2220	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) )
94	93	breq2d 4010	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P || ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) <-> P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
95	65, 88, 94	3bitr3rd 219	. . . 4 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) <-> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) )
96	61, 95	orbi12d 793	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P || ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) \/ P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) <-> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) ) )
97	59, 96	mpbid 147	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) )
98	54, 4	zmodcld 10313	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
99		elprg 3609	. . 3 |- ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } <-> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) ) )
100	98, 99	syl 14	. 2 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } <-> ( ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 0 \/ ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) = 2 ) ) )
101	97, 100	mpbird 167	1 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. { 0 , 2 } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2146   =/= wne 2345   \ cdif 3124   {csn 3589   {cpr 3590   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   < clt 7966   <_ cle 7967   - cmin 8102   # cap 8512   / cdiv 8601   NNcn 8890   2c2 8941   NN0cn0 9147   ZZcz 9224   ZZ>=cuz 9499   QQcq 9590   mod cmo 10290   ^cexp 10487   || cdvds 11760   gcd cgcd 11908   Primecprime 12072   phicphi 12174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-2o 6408  df-oadd 6411  df-er 6525  df-en 6731  df-dom 6732  df-fin 6733  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-ihash 10722  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-clim 11253  df-proddc 11525  df-dvds 11761  df-gcd 11909  df-prm 12073  df-phi 12176

This theorem is referenced by:  lgslem4  13973