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 |
|
1mod |
|- ( ( P e. RR /\ 1 < P ) -> ( 1 mod P ) = 1 ) |
29 |
1 23 27 28
|
4syl |
|- ( ph -> ( 1 mod P ) = 1 ) |
30 |
29
|
eqeq2d |
|- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) <-> ( -u 1 mod P ) = 1 ) ) |
31 |
|
oddprmge3 |
|- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) ) |
32 |
|
m1modge3gt1 |
|- ( P e. ( ZZ>= ` 3 ) -> 1 < ( -u 1 mod P ) ) |
33 |
|
breq2 |
|- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) <-> 1 < 1 ) ) |
34 |
|
1re |
|- 1 e. RR |
35 |
34
|
ltnri |
|- -. 1 < 1 |
36 |
35
|
pm2.21i |
|- ( 1 < 1 -> -u 1 = 1 ) |
37 |
33 36
|
biimtrdi |
|- ( ( -u 1 mod P ) = 1 -> ( 1 < ( -u 1 mod P ) -> -u 1 = 1 ) ) |
38 |
32 37
|
syl5com |
|- ( P e. ( ZZ>= ` 3 ) -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) ) |
39 |
1 31 38
|
3syl |
|- ( ph -> ( ( -u 1 mod P ) = 1 -> -u 1 = 1 ) ) |
40 |
30 39
|
sylbid |
|- ( ph -> ( ( -u 1 mod P ) = ( 1 mod P ) -> -u 1 = 1 ) ) |
41 |
|
oveq1 |
|- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 ^ N ) mod P ) = ( 1 mod P ) ) |
42 |
41
|
eqeq2d |
|- ( ( -u 1 ^ N ) = 1 -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( -u 1 mod P ) = ( 1 mod P ) ) ) |
43 |
|
eqeq2 |
|- ( ( -u 1 ^ N ) = 1 -> ( -u 1 = ( -u 1 ^ N ) <-> -u 1 = 1 ) ) |
44 |
42 43
|
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 ) ) ) |
45 |
40 44
|
imbitrrid |
|- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
46 |
22 45
|
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 ) ) ) ) |
47 |
19 46
|
sylbi |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) ) |
48 |
17 47
|
mpcom |
|- ( ph -> ( ( -u 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> -u 1 = ( -u 1 ^ N ) ) ) |
49 |
|
oveq1 |
|- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) mod P ) = ( -u 1 mod P ) ) |
50 |
49
|
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 ) ) ) |
51 |
|
eqeq1 |
|- ( ( 2 /L P ) = -u 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> -u 1 = ( -u 1 ^ N ) ) ) |
52 |
50 51
|
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 ) ) ) ) |
53 |
48 52
|
imbitrrid |
|- ( ( 2 /L P ) = -u 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
54 |
1
|
gausslemma2dlem0a |
|- ( ph -> P e. NN ) |
55 |
54
|
nnrpd |
|- ( ph -> P e. RR+ ) |
56 |
|
0mod |
|- ( P e. RR+ -> ( 0 mod P ) = 0 ) |
57 |
55 56
|
syl |
|- ( ph -> ( 0 mod P ) = 0 ) |
58 |
57
|
eqeq1d |
|- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( ( -u 1 ^ N ) mod P ) ) ) |
59 |
|
oveq1 |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ( -u 1 ^ N ) mod P ) = ( -u 1 mod P ) ) |
60 |
59
|
eqeq2d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) ) |
61 |
60
|
adantr |
|- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( -u 1 mod P ) ) ) |
62 |
|
negmod0 |
|- ( ( 1 e. RR /\ P e. RR+ ) -> ( ( 1 mod P ) = 0 <-> ( -u 1 mod P ) = 0 ) ) |
63 |
|
eqcom |
|- ( ( -u 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) |
64 |
62 63
|
bitrdi |
|- ( ( 1 e. RR /\ P e. RR+ ) -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) ) |
65 |
34 55 64
|
sylancr |
|- ( ph -> ( ( 1 mod P ) = 0 <-> 0 = ( -u 1 mod P ) ) ) |
66 |
29
|
eqeq1d |
|- ( ph -> ( ( 1 mod P ) = 0 <-> 1 = 0 ) ) |
67 |
|
ax-1ne0 |
|- 1 =/= 0 |
68 |
|
eqneqall |
|- ( 1 = 0 -> ( 1 =/= 0 -> 0 = ( -u 1 ^ N ) ) ) |
69 |
67 68
|
mpi |
|- ( 1 = 0 -> 0 = ( -u 1 ^ N ) ) |
70 |
66 69
|
biimtrdi |
|- ( ph -> ( ( 1 mod P ) = 0 -> 0 = ( -u 1 ^ N ) ) ) |
71 |
65 70
|
sylbird |
|- ( ph -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
72 |
71
|
adantl |
|- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( -u 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
73 |
61 72
|
sylbid |
|- ( ( ( -u 1 ^ N ) = -u 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
74 |
73
|
ex |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
75 |
41
|
eqeq2d |
|- ( ( -u 1 ^ N ) = 1 -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) ) |
76 |
75
|
adantr |
|- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) <-> 0 = ( 1 mod P ) ) ) |
77 |
|
eqcom |
|- ( 0 = ( 1 mod P ) <-> ( 1 mod P ) = 0 ) |
78 |
77 66
|
bitrid |
|- ( ph -> ( 0 = ( 1 mod P ) <-> 1 = 0 ) ) |
79 |
78 69
|
biimtrdi |
|- ( ph -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
80 |
79
|
adantl |
|- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( 1 mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
81 |
76 80
|
sylbid |
|- ( ( ( -u 1 ^ N ) = 1 /\ ph ) -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
82 |
81
|
ex |
|- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
83 |
74 82
|
jaoi |
|- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
84 |
19 83
|
sylbi |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) ) |
85 |
17 84
|
mpcom |
|- ( ph -> ( 0 = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
86 |
58 85
|
sylbid |
|- ( ph -> ( ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 0 = ( -u 1 ^ N ) ) ) |
87 |
|
oveq1 |
|- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) mod P ) = ( 0 mod P ) ) |
88 |
87
|
eqeq1d |
|- ( ( 2 /L P ) = 0 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 0 mod P ) = ( ( -u 1 ^ N ) mod P ) ) ) |
89 |
|
eqeq1 |
|- ( ( 2 /L P ) = 0 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 0 = ( -u 1 ^ N ) ) ) |
90 |
88 89
|
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 ) ) ) ) |
91 |
86 90
|
imbitrrid |
|- ( ( 2 /L P ) = 0 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
92 |
29
|
eqeq1d |
|- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( ( -u 1 ^ N ) mod P ) ) ) |
93 |
|
eqcom |
|- ( 1 = ( -u 1 mod P ) <-> ( -u 1 mod P ) = 1 ) |
94 |
|
eqcom |
|- ( 1 = -u 1 <-> -u 1 = 1 ) |
95 |
39 93 94
|
3imtr4g |
|- ( ph -> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) ) |
96 |
59
|
eqeq2d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( ( -u 1 ^ N ) mod P ) <-> 1 = ( -u 1 mod P ) ) ) |
97 |
|
eqeq2 |
|- ( ( -u 1 ^ N ) = -u 1 -> ( 1 = ( -u 1 ^ N ) <-> 1 = -u 1 ) ) |
98 |
96 97
|
imbi12d |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) <-> ( 1 = ( -u 1 mod P ) -> 1 = -u 1 ) ) ) |
99 |
95 98
|
imbitrrid |
|- ( ( -u 1 ^ N ) = -u 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
100 |
|
eqcom |
|- ( ( -u 1 ^ N ) = 1 <-> 1 = ( -u 1 ^ N ) ) |
101 |
100
|
biimpi |
|- ( ( -u 1 ^ N ) = 1 -> 1 = ( -u 1 ^ N ) ) |
102 |
101
|
2a1d |
|- ( ( -u 1 ^ N ) = 1 -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
103 |
99 102
|
jaoi |
|- ( ( ( -u 1 ^ N ) = -u 1 \/ ( -u 1 ^ N ) = 1 ) -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
104 |
19 103
|
sylbi |
|- ( ( -u 1 ^ N ) e. { -u 1 , 1 } -> ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) ) |
105 |
17 104
|
mpcom |
|- ( ph -> ( 1 = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) |
106 |
92 105
|
sylbid |
|- ( ph -> ( ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) -> 1 = ( -u 1 ^ N ) ) ) |
107 |
|
oveq1 |
|- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) mod P ) = ( 1 mod P ) ) |
108 |
107
|
eqeq1d |
|- ( ( 2 /L P ) = 1 -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) <-> ( 1 mod P ) = ( ( -u 1 ^ N ) mod P ) ) ) |
109 |
|
eqeq1 |
|- ( ( 2 /L P ) = 1 -> ( ( 2 /L P ) = ( -u 1 ^ N ) <-> 1 = ( -u 1 ^ N ) ) ) |
110 |
108 109
|
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 ) ) ) ) |
111 |
106 110
|
imbitrrid |
|- ( ( 2 /L P ) = 1 -> ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) ) |
112 |
53 91 111
|
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 ) ) ) ) |
113 |
13 112
|
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 ) ) ) ) |
114 |
11 113
|
mpcom |
|- ( ph -> ( ( ( 2 /L P ) mod P ) = ( ( -u 1 ^ N ) mod P ) -> ( 2 /L P ) = ( -u 1 ^ N ) ) ) |