Theorem lgslem1 15865

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 3340	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
2	1	3ad2ant2 1046	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. Prime )
3		prmnn 12803	. . . . . . . 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 1024	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> A e. ZZ )
6		prmz 12804	. . . . . . . . . 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 12666	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A gcd P ) = ( P gcd A ) )
9		simp3 1026	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> -. P || A )
10		coprm 12837	. . . . . . . . . 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 2265	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A gcd P ) = 1 )
14		eulerth 12926	. . . . . . 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 1274	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) mod P ) = ( 1 mod P ) )
16		phiprm 12916	. . . . . . . . . 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 9536	. . . . . . . . . 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 2309	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) e. NN0 )
21		zexpcl 10915	. . . . . . . 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 9603	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 1 e. ZZ )
24		moddvds 12481	. . . . . . 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 1274	. . . . . 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 9554	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( P - 1 ) e. CC )
28		2cnd 9309	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. CC )
29		2ap0 9329	. . . . . . . . . . . . 13 |- 2 # 0
30	29	a1i 9	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 # 0 )
31	27, 28, 30	divcanap1d 9064	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( P - 1 ) / 2 ) x. 2 ) = ( P - 1 ) )
32	17, 31	eqtr4d 2268	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( phi ` P ) = ( ( ( P - 1 ) / 2 ) x. 2 ) )
33	32	oveq2d 6065	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( A ^ ( ( ( P - 1 ) / 2 ) x. 2 ) ) )
34	5	zcnd 9700	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> A e. CC )
35		2nn0 9512	. . . . . . . . . . 11 |- 2 e. NN0
36	35	a1i 9	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. NN0 )
37		oddprm 12953	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
38	37	3ad2ant2 1046	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P - 1 ) / 2 ) e. NN )
39	38	nnnn0d 9552	. . . . . . . . . 10 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( P - 1 ) / 2 ) e. NN0 )
40	34, 36, 39	expmuld 11037	. . . . . . . . 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 2265	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( phi ` P ) ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) )
42	41	oveq1d 6064	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - 1 ) )
43		sq1 10994	. . . . . . . 8 |- ( 1 ^ 2 ) = 1
44	43	oveq2i 6060	. . . . . . 7 |- ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - 1 )
45	42, 44	eqtr4di 2283	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( phi ` P ) ) - 1 ) = ( ( ( A ^ ( ( P - 1 ) / 2 ) ) ^ 2 ) - ( 1 ^ 2 ) ) )
46		zexpcl 10915	. . . . . . . . 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 9700	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. CC )
49		ax-1cn 8219	. . . . . . 7 |- 1 e. CC
50		subsq 11007	. . . . . . 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 2265	. . . . 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 4134	. . . 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 9702	. . . . 5 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
55		peano2zm 9614	. . . . . 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 12839	. . . . 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 1274	. . . 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 12473	. . . . 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 9604	. . . . . . 7 |- 2 e. ZZ
63	62	a1i 9	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. ZZ )
64		moddvds 12481	. . . . . 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 1274	. . . . 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 9957	. . . . . . . 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 9957	. . . . . . . 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 9326	. . . . . . . 8 |- 0 <_ 2
71	70	a1i 9	. . . . . . 7 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 0 <_ 2 )
72		eldifsni 3821	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
73	72	3ad2ant2 1046	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P =/= 2 )
74		zapne 9651	. . . . . . . . . 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 9306	. . . . . . . . . 10 |- 2 e. RR
78	77	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> 2 e. RR )
79	4	nnred 9249	. . . . . . . . 9 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> P e. RR )
80		prmuz2 12824	. . . . . . . . . . 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 9865	. . . . . . . . . 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 8912	. . . . . . . 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 10710	. . . . . . 7 |- ( ( ( 2 e. QQ /\ P e. QQ ) /\ ( 0 <_ 2 /\ 2 < P ) ) -> ( 2 mod P ) = 2 )
87	67, 69, 71, 85, 86	syl22anc 1275	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( 2 mod P ) = 2 )
88	87	eqeq2d 2244	. . . . 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 9295	. . . . . . . 8 |- 2 = ( 1 + 1 )
90	89	oveq2i 6060	. . . . . . 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 8621	. . . . . . 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 2277	. . . . . 6 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) - 2 ) = ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) )
94	93	breq2d 4120	. . . . 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 801	. . 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 10706	. . 3 |- ( ( A e. ZZ /\ P e. ( Prime \ { 2 } ) /\ -. P || A ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
99		elprg 3708	. . 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   \ cdif 3207   {csn 3688   {cpr 3689   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   CCcc 8124   RRcr 8125   0cc0 8126   1c1 8127   + caddc 8129   x. cmul 8131   < clt 8307   <_ cle 8308   - cmin 8443   # cap 8854   / cdiv 8945   NNcn 9236   2c2 9287   NN0cn0 9495   ZZcz 9576   ZZ>=cuz 9852   QQcq 9950   mod cmo 10683   ^cexp 10899   || cdvds 12469   gcd cgcd 12645   Primecprime 12800   phicphi 12902

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-ihash 11137  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-clim 11960  df-proddc 12233  df-dvds 12470  df-gcd 12646  df-prm 12801  df-phi 12904

This theorem is referenced by:  lgslem4  15868