Theorem lgslem1 15748

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 3329	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	3ad2ant2 1045	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. Prime )
3		prmnn 12700	. . . . . . . 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 1023	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> A e. ZZ )
6		prmz 12701	. . . . . . . . . 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 12563	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
9		simp3 1025	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> -. P || A )
10		coprm 12734	. . . . . . . . . 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 2264	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A gcd P ) = 1 )
14		eulerth 12823	. . . . . . 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 1273	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
16		phiprm 12813	. . . . . . . . . 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 9443	. . . . . . . . . 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 2308	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) e. NN0 )
21		zexpcl 10817	. . . . . . . 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 9506	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 1 e. ZZ )
24		moddvds 12378	. . . . . . 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 1273	. . . . . 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 9457	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P - 1 ) e. CC )
28		2cnd 9216	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. CC )
29		2ap0 9236	. . . . . . . . . . . . 13 |- 2 # 0
30	29	a1i 9	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 # 0 )
31	27, 28, 30	divcanap1d 8971	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( P - 1 ) / 2 ) x. 2 ) = ( P - 1 ) )
32	17, 31	eqtr4d 2267	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) = ( ( ( P - 1 ) / 2 ) x. 2 ) )
33	32	oveq2d 6034	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( ( P - 1 ) / 2 ) x. 2 ) ) )
34	5	zcnd 9603	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> A e. CC )
35		2nn0 9419	. . . . . . . . . . 11 |- 2 e. NN0
36	35	a1i 9	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. NN0 )
37		oddprm 12850	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
38	37	3ad2ant2 1045	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
39	38	nnnn0d 9455	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
40	34, 36, 39	expmuld 10939	. . . . . . . . 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 2264	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) )
42	41	oveq1d 6033	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - 1 ) )
43		sq1 10896	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
44	43	oveq2i 6029	. . . . . . 7 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - 1 )
45	42, 44	eqtr4di 2282	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) )
46		zexpcl 10817	. . . . . . . . 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 9603	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
49		ax-1cn 8125	. . . . . . 7 |- 1 e. CC
50		subsq 10909	. . . . . . 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 2264	. . . . 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 4114	. . . 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 9605	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
55		peano2zm 9517	. . . . . 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 12736	. . . . 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 1273	. . . 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 12370	. . . . 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 9507	. . . . . . 7 |- 2 e. ZZ
63	62	a1i 9	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. ZZ )
64		moddvds 12378	. . . . . 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 1273	. . . . 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 9860	. . . . . . . 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 9860	. . . . . . . 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 9233	. . . . . . . 8 |- 0 <_ 2
71	70	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 0 <_ 2 )
72		eldifsni 3802	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
73	72	3ad2ant2 1045	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P =/= 2 )
74		zapne 9554	. . . . . . . . . 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 9213	. . . . . . . . . 10 |- 2 e. RR
78	77	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. RR )
79	4	nnred 9156	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. RR )
80		prmuz2 12721	. . . . . . . . . . 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 9768	. . . . . . . . . 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 8819	. . . . . . . 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 10612	. . . . . . 7 |- ( ( ( 2 e. QQ /\ P e. QQ ) /\ ( 0 <_ 2 /\ 2 < P ) ) -> ( 2 mod P ) = 2 )
87	67, 69, 71, 85, 86	syl22anc 1274	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( 2 mod P ) = 2 )
88	87	eqeq2d 2243	. . . . 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 9202	. . . . . . . 8 |- 2 = ( 1 + 1 )
90	89	oveq2i 6029	. . . . . . 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 8528	. . . . . . 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 2276	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) )
94	93	breq2d 4100	. . . . 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 800	. . 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 10608	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
99		elprg 3689	. . 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 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   \ cdif 3197   {csn 3669   {cpr 3670   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   <_ cle 8215   - cmin 8350   # cap 8761   / cdiv 8852   NNcn 9143   2c2 9194   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   QQcq 9853   mod cmo 10585   ^cexp 10801   || cdvds 12366   gcd cgcd 12542   Primecprime 12697   phicphi 12799

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857  df-proddc 12130  df-dvds 12367  df-gcd 12543  df-prm 12698  df-phi 12801

This theorem is referenced by:  lgslem4  15751