Step |
Hyp |
Ref |
Expression |
1 |
|
gausslemma2dlem0.p |
|- ( ph -> P e. ( Prime \ { 2 } ) ) |
2 |
|
gausslemma2dlem0.m |
|- M = ( |_ ` ( P / 4 ) ) |
3 |
|
gausslemma2dlem0.h |
|- H = ( ( P - 1 ) / 2 ) |
4 |
|
gausslemma2dlem0.n |
|- N = ( H - M ) |
5 |
|
2z |
|- 2 e. ZZ |
6 |
|
id |
|- ( P e. ( Prime \ { 2 } ) -> P e. ( Prime \ { 2 } ) ) |
7 |
6
|
gausslemma2dlem0a |
|- ( P e. ( Prime \ { 2 } ) -> P e. NN ) |
8 |
7
|
nnzd |
|- ( P e. ( Prime \ { 2 } ) -> P e. ZZ ) |
9 |
1 8
|
syl |
|- ( ph -> P e. ZZ ) |
10 |
|
lgscl1 |
|- ( ( 2 e. ZZ /\ P e. ZZ ) -> ( 2 /L P ) e. { -u 1 , 0 , 1 } ) |
11 |
5 9 10
|
sylancr |
|- ( ph -> ( 2 /L P ) e. { -u 1 , 0 , 1 } ) |
12 |
|
ovex |
|- ( 2 /L P ) e. _V |
13 |
12
|
eltp |
|- ( ( 2 /L P ) e. { -u 1 , 0 , 1 } <-> ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) ) |
14 |
1 2 3 4
|
gausslemma2dlem0h |
|- ( ph -> N e. NN0 ) |
15 |
14
|
nn0zd |
|- ( ph -> N e. ZZ ) |
16 |
|
m1expcl2 |
|- ( N e. ZZ -> ( -u 1 ^ N ) e. { -u 1 , 1 } ) |
17 |
15 16
|
syl |
|- ( ph -> ( -u 1 ^ N ) e. { -u 1 , 1 } ) |
18 |
|
ovex |
|- ( -u 1 ^ N ) e. _V |
19 |
18
|
elpr |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } <-> ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) ) |
20 |
|
eqcom |
|- ( ( -u 1 ^ N ) = -u 1 <-> -u 1 = ( -u 1 ^ N ) ) |
21 |
20
|
biimpi |
|- ( ( -u 1 ^ N ) = -u 1 -> -u 1 = ( -u 1 ^ N ) ) |
22 |
21
|
2a1d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
23 |
|
eldifi |
|- ( P e. ( Prime \ { 2 } ) -> P e. Prime ) |
24 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
25 |
24
|
nnred |
|- ( P e. Prime -> P e. RR ) |
26 |
|
prmgt1 |
|- ( P e. Prime -> 1 < P ) |
27 |
25 26
|
jca |
|- ( P e. Prime -> ( P e. RR /\ 1 < P ) ) |
28 |
23 27
|
syl |
|- ( P e. ( Prime \ { 2 } ) -> ( P e. RR /\ 1 < P ) ) |
29 |
|
1mod |
|- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 ) |
30 |
1 28 29
|
3syl |
|- ( ph -> ( 1 mod P ) = 1 ) |
31 |
30
|
eqeq2d |
|- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) ) |
32 |
|
oddprmge3 |
|- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) ) |
33 |
|
m1modge3gt1 |
|- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) ) |
34 |
|
breq2 |
|- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) ) |
35 |
|
1re |
|- 1 e. RR |
36 |
35
|
ltnri |
|- -. 1 < 1 |
37 |
36
|
pm2.21i |
|- ( 1 < 1 -> -u 1 = 1 ) |
38 |
34 37
|
syl6bi |
|- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) -> -u 1 = 1 ) ) |
39 |
33 38
|
syl5com |
|- ( P e. ( ZZ>= ` 3 ) -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) ) |
40 |
1 32 39
|
3syl |
|- ( ph -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) ) |
41 |
31 40
|
sylbid |
|- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) -> -u 1 = 1 ) ) |
42 |
|
oveq1 |
|- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) ) |
43 |
42
|
eqeq2d |
|- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) ) |
44 |
|
eqeq2 |
|- ( ( -u 1 ^ N ) = 1 -> ( -u 1 = ( -u 1 ^ N ) <-> -u 1 = 1 ) ) |
45 |
43 44
|
imbi12d |
|- ( ( -u 1 ^ N ) = 1 -> ( ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) <-> ( ( -u 1 mod P ) = ( 1 mod P ) -> -u 1 = 1 ) ) ) |
46 |
41 45
|
syl5ibr |
|- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
47 |
22 46
|
jaoi |
|- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
48 |
19 47
|
sylbi |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
49 |
17 48
|
mpcom |
|- ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) |
50 |
|
oveq1 |
|- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) ) |
51 |
50
|
eqeq1d |
|- ( ( 2 /L P ) = -u 1 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) ) ) |
52 |
|
eqeq1 |
|- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> -u 1 = ( -u 1 ^ N ) ) ) |
53 |
51 52
|
imbi12d |
|- ( ( 2 /L P ) = -u 1 -> ( ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) <-> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
54 |
49 53
|
syl5ibr |
|- ( ( 2 /L P ) = -u 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
55 |
1
|
gausslemma2dlem0a |
|- ( ph -> P e. NN ) |
56 |
55
|
nnrpd |
|- ( ph -> P e. RR+ ) |
57 |
|
0mod |
|- ( P e. RR+ -> ( 0 mod P ) = 0 ) |
58 |
56 57
|
syl |
|- ( ph -> ( 0 mod P ) = 0 ) |
59 |
58
|
eqeq1d |
|- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) ) |
60 |
|
oveq1 |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) ) |
61 |
60
|
eqeq2d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) ) |
62 |
61
|
adantr |
|- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) ) |
63 |
|
negmod0 |
|- ( ( 1 e. RR /\ P e. RR+ ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) ) |
64 |
|
eqcom |
|- ( ( -u 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) |
65 |
63 64
|
bitrdi |
|- ( ( 1 e. RR /\ P e. RR+ ) -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) ) |
66 |
35 56 65
|
sylancr |
|- ( ph -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) ) |
67 |
30
|
eqeq1d |
|- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) ) |
68 |
|
ax-1ne0 |
|- 1 =/= 0 |
69 |
|
eqneqall |
|- ( 1 = 0 -> ( 1 =/= 0 -> 0 = ( -u 1 ^ N ) ) ) |
70 |
68 69
|
mpi |
|- ( 1 = 0 -> 0 = ( -u 1 ^ N ) ) |
71 |
67 70
|
syl6bi |
|- ( ph -> ( ( 1 mod P ) = 0 -> 0 = ( -u 1 ^ N ) ) ) |
72 |
66 71
|
sylbird |
|- ( ph -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
73 |
72
|
adantl |
|- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
74 |
62 73
|
sylbid |
|- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
75 |
74
|
ex |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
76 |
42
|
eqeq2d |
|- ( ( -u 1 ^ N ) = 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) ) |
77 |
76
|
adantr |
|- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) ) |
78 |
|
eqcom |
|- ( 0 = ( 1 mod P ) <-> ( 1 mod P ) = 0 ) |
79 |
78 67
|
syl5bb |
|- ( ph -> ( 0 = ( 1 mod P ) <-> 1 = 0 ) ) |
80 |
79 70
|
syl6bi |
|- ( ph -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
81 |
80
|
adantl |
|- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
82 |
77 81
|
sylbid |
|- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
83 |
82
|
ex |
|- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
84 |
75 83
|
jaoi |
|- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
85 |
19 84
|
sylbi |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
86 |
17 85
|
mpcom |
|- ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
87 |
59 86
|
sylbid |
|- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
88 |
|
oveq1 |
|- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) ) |
89 |
88
|
eqeq1d |
|- ( ( 2 /L P ) = 0 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) ) ) |
90 |
|
eqeq1 |
|- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 0 = ( -u 1 ^ N ) ) ) |
91 |
89 90
|
imbi12d |
|- ( ( 2 /L P ) = 0 -> ( ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) <-> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
92 |
87 91
|
syl5ibr |
|- ( ( 2 /L P ) = 0 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
93 |
30
|
eqeq1d |
|- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) ) |
94 |
|
eqcom |
|- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 ) |
95 |
|
eqcom |
|- ( 1 = -u 1 <-> -u 1 = 1 ) |
96 |
40 94 95
|
3imtr4g |
|- ( ph -> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) ) |
97 |
60
|
eqeq2d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) ) |
98 |
|
eqeq2 |
|- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( -u 1 ^ N ) <-> 1 = -u 1 ) ) |
99 |
97 98
|
imbi12d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) <-> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) ) ) |
100 |
96 99
|
syl5ibr |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
101 |
|
eqcom |
|- ( ( -u 1 ^ N ) = 1 <-> 1 = ( -u 1 ^ N ) ) |
102 |
101
|
biimpi |
|- ( ( -u 1 ^ N ) = 1 -> 1 = ( -u 1 ^ N ) ) |
103 |
102
|
2a1d |
|- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
104 |
100 103
|
jaoi |
|- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
105 |
19 104
|
sylbi |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
106 |
17 105
|
mpcom |
|- ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) |
107 |
93 106
|
sylbid |
|- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) |
108 |
|
oveq1 |
|- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) ) |
109 |
108
|
eqeq1d |
|- ( ( 2 /L P ) = 1 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) ) ) |
110 |
|
eqeq1 |
|- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 1 = ( -u 1 ^ N ) ) ) |
111 |
109 110
|
imbi12d |
|- ( ( 2 /L P ) = 1 -> ( ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) <-> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
112 |
107 111
|
syl5ibr |
|- ( ( 2 /L P ) = 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
113 |
54 92 112
|
3jaoi |
|- ( ( ( 2 /L P ) = -u 1 \/ ( 2 /L P ) = 0 \/ ( 2 /L P ) = 1 ) -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
114 |
13 113
|
sylbi |
|- ( ( 2 /L P ) e. { -u 1 , 0 , 1 } -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
115 |
11 114
|
mpcom |
|- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) |