Theorem m1lgs 25113

Description: The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime P iff P == 1 (mod 4). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.)

Assertion

Ref	Expression
m1lgs	|- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 /L P ) = 1 <-> ( P mod 4 ) = 1 ) )

Proof of Theorem m1lgs

Step	Hyp	Ref	Expression
1		neg1z 11413	. . . . . . . . 9 |- -u 1 e. ZZ
2		oddprm 15515	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
3	2	nnnn0d 11351	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN0 )
4		zexpcl 12875	. . . . . . . . 9 |- ( ( -u 1 e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( -u 1 ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
5	1, 3, 4	sylancr 695	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( -u 1 ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
6	5	peano2zd 11485	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ )
7		eldifi 3732	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
8		prmnn 15388	. . . . . . . 8 |- ( P e. Prime -> P e. NN )
9	7, 8	syl 17	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. NN )
10	6, 9	zmodcld 12691	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. NN0 )
11	10	nn0cnd 11353	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) e. CC )
12		1cnd 10056	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> 1 e. CC )
13	11, 12, 12	subaddd 10410	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = 1 <-> ( 1 + 1 ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) ) )
14		2re 11090	. . . . . . . 8 |- 2 e. RR
15	14	a1i 11	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> 2 e. RR )
16	9	nnrpd 11870	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> P e. RR+ )
17		0le2 11111	. . . . . . . 8 |- 0 <_ 2
18	17	a1i 11	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> 0 <_ 2 )
19		eldifsni 4320	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> P =/= 2 )
20	9	nnred 11035	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. RR )
21		prmuz2 15408	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
22	7, 21	syl 17	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 2 ) )
23		eluzle 11700	. . . . . . . . . 10 |- ( P e. ( ZZ>= ` 2 ) -> 2 <_ P )
24	22, 23	syl 17	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> 2 <_ P )
25	15, 20, 24	leltned 10190	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( 2 < P <-> P =/= 2 ) )
26	19, 25	mpbird 247	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> 2 < P )
27		modid 12695	. . . . . . 7 |- ( ( ( 2 e. RR /\ P e. RR+ ) /\ ( 0 <_ 2 /\ 2 < P ) ) -> ( 2 mod P ) = 2 )
28	15, 16, 18, 26, 27	syl22anc 1327	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( 2 mod P ) = 2 )
29		df-2 11079	. . . . . 6 |- 2 = ( 1 + 1 )
30	28, 29	syl6eq 2672	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> ( 2 mod P ) = ( 1 + 1 ) )
31	30	eqeq1d 2624	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) <-> ( 1 + 1 ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) ) )
32	19	neneqd 2799	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> -. P = 2 )
33		2prm 15405	. . . . . . . . . . . 12 |- 2 e. Prime
34		dvdsprm 15415	. . . . . . . . . . . 12 |- ( ( P e. ( ZZ>= ` 2 ) /\ 2 e. Prime ) -> ( P || 2 <-> P = 2 ) )
35	22, 33, 34	sylancl 694	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( P || 2 <-> P = 2 ) )
36	32, 35	mtbird 315	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> -. P || 2 )
37	36	adantr 481	. . . . . . . . 9 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> -. P || 2 )
38		1cnd 10056	. . . . . . . . . . . . . . . 16 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> 1 e. CC )
39	2	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( ( P - 1 ) / 2 ) e. NN )
40		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> -. 2 || ( ( P - 1 ) / 2 ) )
41		oexpneg 15069	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. CC /\ ( ( P - 1 ) / 2 ) e. NN /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( -u 1 ^ ( ( P - 1 ) / 2 ) ) = -u ( 1 ^ ( ( P - 1 ) / 2 ) ) )
42	38, 39, 40, 41	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( -u 1 ^ ( ( P - 1 ) / 2 ) ) = -u ( 1 ^ ( ( P - 1 ) / 2 ) ) )
43	39	nnzd 11481	. . . . . . . . . . . . . . . . 17 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( ( P - 1 ) / 2 ) e. ZZ )
44		1exp 12889	. . . . . . . . . . . . . . . . 17 |- ( ( ( P - 1 ) / 2 ) e. ZZ -> ( 1 ^ ( ( P - 1 ) / 2 ) ) = 1 )
45	43, 44	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( 1 ^ ( ( P - 1 ) / 2 ) ) = 1 )
46	45	negeqd 10275	. . . . . . . . . . . . . . 15 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> -u ( 1 ^ ( ( P - 1 ) / 2 ) ) = -u 1 )
47	42, 46	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( -u 1 ^ ( ( P - 1 ) / 2 ) ) = -u 1 )
48	47	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) = ( -u 1 + 1 ) )
49		ax-1cn 9994	. . . . . . . . . . . . . 14 |- 1 e. CC
50		neg1cn 11124	. . . . . . . . . . . . . 14 |- -u 1 e. CC
51		1pneg1e0 11129	. . . . . . . . . . . . . 14 |- ( 1 + -u 1 ) = 0
52	49, 50, 51	addcomli 10228	. . . . . . . . . . . . 13 |- ( -u 1 + 1 ) = 0
53	48, 52	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) = 0 )
54	53	oveq2d 6666	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) = ( 2 - 0 ) )
55		2cn 11091	. . . . . . . . . . . 12 |- 2 e. CC
56	55	subid1i 10353	. . . . . . . . . . 11 |- ( 2 - 0 ) = 2
57	54, 56	syl6eq 2672	. . . . . . . . . 10 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) = 2 )
58	57	breq2d 4665	. . . . . . . . 9 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> ( P || ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) <-> P || 2 ) )
59	37, 58	mtbird 315	. . . . . . . 8 |- ( ( P e. ( Prime \ { 2 } ) /\ -. 2 || ( ( P - 1 ) / 2 ) ) -> -. P || ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) )
60	59	ex 450	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( -. 2 || ( ( P - 1 ) / 2 ) -> -. P || ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) ) )
61	60	con4d 114	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( P || ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) -> 2 || ( ( P - 1 ) / 2 ) ) )
62		2z 11409	. . . . . . . 8 |- 2 e. ZZ
63	62	a1i 11	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> 2 e. ZZ )
64		moddvds 14991	. . . . . . 7 |- ( ( P e. NN /\ 2 e. ZZ /\ ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) e. ZZ ) -> ( ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) <-> P || ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) ) )
65	9, 63, 6, 64	syl3anc 1326	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) <-> P || ( 2 - ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) ) ) )
66		4z 11411	. . . . . . . . . 10 |- 4 e. ZZ
67	66	a1i 11	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> 4 e. ZZ )
68		4ne0 11117	. . . . . . . . . 10 |- 4 =/= 0
69	68	a1i 11	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> 4 =/= 0 )
70		nnm1nn0 11334	. . . . . . . . . . 11 |- ( P e. NN -> ( P - 1 ) e. NN0 )
71	9, 70	syl 17	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> ( P - 1 ) e. NN0 )
72	71	nn0zd 11480	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( P - 1 ) e. ZZ )
73		dvdsval2 14986	. . . . . . . . 9 |- ( ( 4 e. ZZ /\ 4 =/= 0 /\ ( P - 1 ) e. ZZ ) -> ( 4 || ( P - 1 ) <-> ( ( P - 1 ) / 4 ) e. ZZ ) )
74	67, 69, 72, 73	syl3anc 1326	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( 4 || ( P - 1 ) <-> ( ( P - 1 ) / 4 ) e. ZZ ) )
75	71	nn0cnd 11353	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> ( P - 1 ) e. CC )
76	55	a1i 11	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> 2 e. CC )
77		2ne0 11113	. . . . . . . . . . . 12 |- 2 =/= 0
78	77	a1i 11	. . . . . . . . . . 11 |- ( P e. ( Prime \ { 2 } ) -> 2 =/= 0 )
79	75, 76, 76, 78, 78	divdiv1d 10832	. . . . . . . . . 10 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( P - 1 ) / 2 ) / 2 ) = ( ( P - 1 ) / ( 2 x. 2 ) ) )
80		2t2e4 11177	. . . . . . . . . . 11 |- ( 2 x. 2 ) = 4
81	80	oveq2i 6661	. . . . . . . . . 10 |- ( ( P - 1 ) / ( 2 x. 2 ) ) = ( ( P - 1 ) / 4 )
82	79, 81	syl6eq 2672	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( P - 1 ) / 2 ) / 2 ) = ( ( P - 1 ) / 4 ) )
83	82	eleq1d 2686	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ <-> ( ( P - 1 ) / 4 ) e. ZZ ) )
84	74, 83	bitr4d 271	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( 4 || ( P - 1 ) <-> ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ ) )
85	2	nnzd 11481	. . . . . . . 8 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. ZZ )
86		dvdsval2 14986	. . . . . . . 8 |- ( ( 2 e. ZZ /\ 2 =/= 0 /\ ( ( P - 1 ) / 2 ) e. ZZ ) -> ( 2 || ( ( P - 1 ) / 2 ) <-> ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ ) )
87	63, 78, 85, 86	syl3anc 1326	. . . . . . 7 |- ( P e. ( Prime \ { 2 } ) -> ( 2 || ( ( P - 1 ) / 2 ) <-> ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ ) )
88	84, 87	bitr4d 271	. . . . . 6 |- ( P e. ( Prime \ { 2 } ) -> ( 4 || ( P - 1 ) <-> 2 || ( ( P - 1 ) / 2 ) ) )
89	61, 65, 88	3imtr4d 283	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) -> 4 || ( P - 1 ) ) )
90	50	a1i 11	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> -u 1 e. CC )
91		neg1ne0 11126	. . . . . . . . . . . 12 |- -u 1 =/= 0
92	91	a1i 11	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> -u 1 =/= 0 )
93	62	a1i 11	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> 2 e. ZZ )
94	84	biimpa 501	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ )
95		expmulz 12906	. . . . . . . . . . 11 |- ( ( ( -u 1 e. CC /\ -u 1 =/= 0 ) /\ ( 2 e. ZZ /\ ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ ) ) -> ( -u 1 ^ ( 2 x. ( ( ( P - 1 ) / 2 ) / 2 ) ) ) = ( ( -u 1 ^ 2 ) ^ ( ( ( P - 1 ) / 2 ) / 2 ) ) )
96	90, 92, 93, 94, 95	syl22anc 1327	. . . . . . . . . 10 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( -u 1 ^ ( 2 x. ( ( ( P - 1 ) / 2 ) / 2 ) ) ) = ( ( -u 1 ^ 2 ) ^ ( ( ( P - 1 ) / 2 ) / 2 ) ) )
97	2	nncnd 11036	. . . . . . . . . . . . 13 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. CC )
98	97, 76, 78	divcan2d 10803	. . . . . . . . . . . 12 |- ( P e. ( Prime \ { 2 } ) -> ( 2 x. ( ( ( P - 1 ) / 2 ) / 2 ) ) = ( ( P - 1 ) / 2 ) )
99	98	adantr 481	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( 2 x. ( ( ( P - 1 ) / 2 ) / 2 ) ) = ( ( P - 1 ) / 2 ) )
100	99	oveq2d 6666	. . . . . . . . . 10 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( -u 1 ^ ( 2 x. ( ( ( P - 1 ) / 2 ) / 2 ) ) ) = ( -u 1 ^ ( ( P - 1 ) / 2 ) ) )
101		neg1sqe1 12959	. . . . . . . . . . . 12 |- ( -u 1 ^ 2 ) = 1
102	101	oveq1i 6660	. . . . . . . . . . 11 |- ( ( -u 1 ^ 2 ) ^ ( ( ( P - 1 ) / 2 ) / 2 ) ) = ( 1 ^ ( ( ( P - 1 ) / 2 ) / 2 ) )
103		1exp 12889	. . . . . . . . . . . 12 |- ( ( ( ( P - 1 ) / 2 ) / 2 ) e. ZZ -> ( 1 ^ ( ( ( P - 1 ) / 2 ) / 2 ) ) = 1 )
104	94, 103	syl 17	. . . . . . . . . . 11 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( 1 ^ ( ( ( P - 1 ) / 2 ) / 2 ) ) = 1 )
105	102, 104	syl5eq 2668	. . . . . . . . . 10 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( ( -u 1 ^ 2 ) ^ ( ( ( P - 1 ) / 2 ) / 2 ) ) = 1 )
106	96, 100, 105	3eqtr3d 2664	. . . . . . . . 9 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( -u 1 ^ ( ( P - 1 ) / 2 ) ) = 1 )
107	106	oveq1d 6665	. . . . . . . 8 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) = ( 1 + 1 ) )
108	107, 29	syl6reqr 2675	. . . . . . 7 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> 2 = ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) )
109	108	oveq1d 6665	. . . . . 6 |- ( ( P e. ( Prime \ { 2 } ) /\ 4 || ( P - 1 ) ) -> ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) )
110	109	ex 450	. . . . 5 |- ( P e. ( Prime \ { 2 } ) -> ( 4 || ( P - 1 ) -> ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) ) )
111	89, 110	impbid 202	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> ( ( 2 mod P ) = ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) <-> 4 || ( P - 1 ) ) )
112	13, 31, 111	3bitr2d 296	. . 3 |- ( P e. ( Prime \ { 2 } ) -> ( ( ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = 1 <-> 4 || ( P - 1 ) ) )
113		lgsval3 25040	. . . . 5 |- ( ( -u 1 e. ZZ /\ P e. ( Prime \ { 2 } ) ) -> ( -u 1 /L P ) = ( ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
114	1, 113	mpan 706	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> ( -u 1 /L P ) = ( ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) )
115	114	eqeq1d 2624	. . 3 |- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 /L P ) = 1 <-> ( ( ( ( -u 1 ^ ( ( P - 1 ) / 2 ) ) + 1 ) mod P ) - 1 ) = 1 ) )
116		4nn 11187	. . . . 5 |- 4 e. NN
117	116	a1i 11	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> 4 e. NN )
118		prmz 15389	. . . . 5 |- ( P e. Prime -> P e. ZZ )
119	7, 118	syl 17	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> P e. ZZ )
120		1zzd 11408	. . . 4 |- ( P e. ( Prime \ { 2 } ) -> 1 e. ZZ )
121		moddvds 14991	. . . 4 |- ( ( 4 e. NN /\ P e. ZZ /\ 1 e. ZZ ) -> ( ( P mod 4 ) = ( 1 mod 4 ) <-> 4 || ( P - 1 ) ) )
122	117, 119, 120, 121	syl3anc 1326	. . 3 |- ( P e. ( Prime \ { 2 } ) -> ( ( P mod 4 ) = ( 1 mod 4 ) <-> 4 || ( P - 1 ) ) )
123	112, 115, 122	3bitr4d 300	. 2 |- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 /L P ) = 1 <-> ( P mod 4 ) = ( 1 mod 4 ) ) )
124		1re 10039	. . . 4 |- 1 e. RR
125		nnrp 11842	. . . . 5 |- ( 4 e. NN -> 4 e. RR+ )
126	116, 125	ax-mp 5	. . . 4 |- 4 e. RR+
127		0le1 10551	. . . 4 |- 0 <_ 1
128		1lt4 11199	. . . 4 |- 1 < 4
129		modid 12695	. . . 4 |- ( ( ( 1 e. RR /\ 4 e. RR+ ) /\ ( 0 <_ 1 /\ 1 < 4 ) ) -> ( 1 mod 4 ) = 1 )
130	124, 126, 127, 128, 129	mp4an 709	. . 3 |- ( 1 mod 4 ) = 1
131	130	eqeq2i 2634	. 2 |- ( ( P mod 4 ) = ( 1 mod 4 ) <-> ( P mod 4 ) = 1 )
132	123, 131	syl6bb 276	1 |- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 /L P ) = 1 <-> ( P mod 4 ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   4c4 11072   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   mod cmo 12668   ^cexp 12860   || cdvds 14983   Primecprime 15385   /Lclgs 25019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-lgs 25020

This theorem is referenced by:  2sqlem11  25154  2sqblem  25156