Step |
Hyp |
Ref |
Expression |
1 |
|
gausslemma2dlem0.p |
⊢ ( 𝜑 → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
2 |
|
gausslemma2dlem0.m |
⊢ 𝑀 = ( ⌊ ‘ ( 𝑃 / 4 ) ) |
3 |
|
gausslemma2dlem0.h |
⊢ 𝐻 = ( ( 𝑃 − 1 ) / 2 ) |
4 |
|
gausslemma2dlem0.n |
⊢ 𝑁 = ( 𝐻 − 𝑀 ) |
5 |
|
2z |
⊢ 2 ∈ ℤ |
6 |
|
id |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
7 |
6
|
gausslemma2dlem0a |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → 𝑃 ∈ ℕ ) |
8 |
7
|
nnzd |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → 𝑃 ∈ ℤ ) |
9 |
1 8
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ℤ ) |
10 |
|
lgscl1 |
⊢ ( ( 2 ∈ ℤ ∧ 𝑃 ∈ ℤ ) → ( 2 /L 𝑃 ) ∈ { - 1 , 0 , 1 } ) |
11 |
5 9 10
|
sylancr |
⊢ ( 𝜑 → ( 2 /L 𝑃 ) ∈ { - 1 , 0 , 1 } ) |
12 |
|
ovex |
⊢ ( 2 /L 𝑃 ) ∈ V |
13 |
12
|
eltp |
⊢ ( ( 2 /L 𝑃 ) ∈ { - 1 , 0 , 1 } ↔ ( ( 2 /L 𝑃 ) = - 1 ∨ ( 2 /L 𝑃 ) = 0 ∨ ( 2 /L 𝑃 ) = 1 ) ) |
14 |
1 2 3 4
|
gausslemma2dlem0h |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
15 |
14
|
nn0zd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
16 |
|
m1expcl2 |
⊢ ( 𝑁 ∈ ℤ → ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } ) |
17 |
15 16
|
syl |
⊢ ( 𝜑 → ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } ) |
18 |
|
ovex |
⊢ ( - 1 ↑ 𝑁 ) ∈ V |
19 |
18
|
elpr |
⊢ ( ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } ↔ ( ( - 1 ↑ 𝑁 ) = - 1 ∨ ( - 1 ↑ 𝑁 ) = 1 ) ) |
20 |
|
eqcom |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 ↔ - 1 = ( - 1 ↑ 𝑁 ) ) |
21 |
20
|
biimpi |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → - 1 = ( - 1 ↑ 𝑁 ) ) |
22 |
21
|
2a1d |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
23 |
|
eldifi |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → 𝑃 ∈ ℙ ) |
24 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
25 |
24
|
nnred |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℝ ) |
26 |
|
prmgt1 |
⊢ ( 𝑃 ∈ ℙ → 1 < 𝑃 ) |
27 |
25 26
|
jca |
⊢ ( 𝑃 ∈ ℙ → ( 𝑃 ∈ ℝ ∧ 1 < 𝑃 ) ) |
28 |
23 27
|
syl |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → ( 𝑃 ∈ ℝ ∧ 1 < 𝑃 ) ) |
29 |
|
1mod |
⊢ ( ( 𝑃 ∈ ℝ ∧ 1 < 𝑃 ) → ( 1 mod 𝑃 ) = 1 ) |
30 |
1 28 29
|
3syl |
⊢ ( 𝜑 → ( 1 mod 𝑃 ) = 1 ) |
31 |
30
|
eqeq2d |
⊢ ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( 1 mod 𝑃 ) ↔ ( - 1 mod 𝑃 ) = 1 ) ) |
32 |
|
oddprmge3 |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → 𝑃 ∈ ( ℤ≥ ‘ 3 ) ) |
33 |
|
m1modge3gt1 |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 3 ) → 1 < ( - 1 mod 𝑃 ) ) |
34 |
|
breq2 |
⊢ ( ( - 1 mod 𝑃 ) = 1 → ( 1 < ( - 1 mod 𝑃 ) ↔ 1 < 1 ) ) |
35 |
|
1re |
⊢ 1 ∈ ℝ |
36 |
35
|
ltnri |
⊢ ¬ 1 < 1 |
37 |
36
|
pm2.21i |
⊢ ( 1 < 1 → - 1 = 1 ) |
38 |
34 37
|
syl6bi |
⊢ ( ( - 1 mod 𝑃 ) = 1 → ( 1 < ( - 1 mod 𝑃 ) → - 1 = 1 ) ) |
39 |
33 38
|
syl5com |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 3 ) → ( ( - 1 mod 𝑃 ) = 1 → - 1 = 1 ) ) |
40 |
1 32 39
|
3syl |
⊢ ( 𝜑 → ( ( - 1 mod 𝑃 ) = 1 → - 1 = 1 ) ) |
41 |
31 40
|
sylbid |
⊢ ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( 1 mod 𝑃 ) → - 1 = 1 ) ) |
42 |
|
oveq1 |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) = ( 1 mod 𝑃 ) ) |
43 |
42
|
eqeq2d |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ ( - 1 mod 𝑃 ) = ( 1 mod 𝑃 ) ) ) |
44 |
|
eqeq2 |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( - 1 = ( - 1 ↑ 𝑁 ) ↔ - 1 = 1 ) ) |
45 |
43 44
|
imbi12d |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ↔ ( ( - 1 mod 𝑃 ) = ( 1 mod 𝑃 ) → - 1 = 1 ) ) ) |
46 |
41 45
|
syl5ibr |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
47 |
22 46
|
jaoi |
⊢ ( ( ( - 1 ↑ 𝑁 ) = - 1 ∨ ( - 1 ↑ 𝑁 ) = 1 ) → ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
48 |
19 47
|
sylbi |
⊢ ( ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } → ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
49 |
17 48
|
mpcom |
⊢ ( 𝜑 → ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ) |
50 |
|
oveq1 |
⊢ ( ( 2 /L 𝑃 ) = - 1 → ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( - 1 mod 𝑃 ) ) |
51 |
50
|
eqeq1d |
⊢ ( ( 2 /L 𝑃 ) = - 1 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ) ) |
52 |
|
eqeq1 |
⊢ ( ( 2 /L 𝑃 ) = - 1 → ( ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ↔ - 1 = ( - 1 ↑ 𝑁 ) ) ) |
53 |
51 52
|
imbi12d |
⊢ ( ( 2 /L 𝑃 ) = - 1 → ( ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ↔ ( ( - 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → - 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
54 |
49 53
|
syl5ibr |
⊢ ( ( 2 /L 𝑃 ) = - 1 → ( 𝜑 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ) ) |
55 |
1
|
gausslemma2dlem0a |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
56 |
55
|
nnrpd |
⊢ ( 𝜑 → 𝑃 ∈ ℝ+ ) |
57 |
|
0mod |
⊢ ( 𝑃 ∈ ℝ+ → ( 0 mod 𝑃 ) = 0 ) |
58 |
56 57
|
syl |
⊢ ( 𝜑 → ( 0 mod 𝑃 ) = 0 ) |
59 |
58
|
eqeq1d |
⊢ ( 𝜑 → ( ( 0 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ) ) |
60 |
|
oveq1 |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) = ( - 1 mod 𝑃 ) ) |
61 |
60
|
eqeq2d |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 0 = ( - 1 mod 𝑃 ) ) ) |
62 |
61
|
adantr |
⊢ ( ( ( - 1 ↑ 𝑁 ) = - 1 ∧ 𝜑 ) → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 0 = ( - 1 mod 𝑃 ) ) ) |
63 |
|
negmod0 |
⊢ ( ( 1 ∈ ℝ ∧ 𝑃 ∈ ℝ+ ) → ( ( 1 mod 𝑃 ) = 0 ↔ ( - 1 mod 𝑃 ) = 0 ) ) |
64 |
|
eqcom |
⊢ ( ( - 1 mod 𝑃 ) = 0 ↔ 0 = ( - 1 mod 𝑃 ) ) |
65 |
63 64
|
bitrdi |
⊢ ( ( 1 ∈ ℝ ∧ 𝑃 ∈ ℝ+ ) → ( ( 1 mod 𝑃 ) = 0 ↔ 0 = ( - 1 mod 𝑃 ) ) ) |
66 |
35 56 65
|
sylancr |
⊢ ( 𝜑 → ( ( 1 mod 𝑃 ) = 0 ↔ 0 = ( - 1 mod 𝑃 ) ) ) |
67 |
30
|
eqeq1d |
⊢ ( 𝜑 → ( ( 1 mod 𝑃 ) = 0 ↔ 1 = 0 ) ) |
68 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
69 |
|
eqneqall |
⊢ ( 1 = 0 → ( 1 ≠ 0 → 0 = ( - 1 ↑ 𝑁 ) ) ) |
70 |
68 69
|
mpi |
⊢ ( 1 = 0 → 0 = ( - 1 ↑ 𝑁 ) ) |
71 |
67 70
|
syl6bi |
⊢ ( 𝜑 → ( ( 1 mod 𝑃 ) = 0 → 0 = ( - 1 ↑ 𝑁 ) ) ) |
72 |
66 71
|
sylbird |
⊢ ( 𝜑 → ( 0 = ( - 1 mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
73 |
72
|
adantl |
⊢ ( ( ( - 1 ↑ 𝑁 ) = - 1 ∧ 𝜑 ) → ( 0 = ( - 1 mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
74 |
62 73
|
sylbid |
⊢ ( ( ( - 1 ↑ 𝑁 ) = - 1 ∧ 𝜑 ) → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
75 |
74
|
ex |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( 𝜑 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) ) |
76 |
42
|
eqeq2d |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 0 = ( 1 mod 𝑃 ) ) ) |
77 |
76
|
adantr |
⊢ ( ( ( - 1 ↑ 𝑁 ) = 1 ∧ 𝜑 ) → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 0 = ( 1 mod 𝑃 ) ) ) |
78 |
|
eqcom |
⊢ ( 0 = ( 1 mod 𝑃 ) ↔ ( 1 mod 𝑃 ) = 0 ) |
79 |
78 67
|
syl5bb |
⊢ ( 𝜑 → ( 0 = ( 1 mod 𝑃 ) ↔ 1 = 0 ) ) |
80 |
79 70
|
syl6bi |
⊢ ( 𝜑 → ( 0 = ( 1 mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
81 |
80
|
adantl |
⊢ ( ( ( - 1 ↑ 𝑁 ) = 1 ∧ 𝜑 ) → ( 0 = ( 1 mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
82 |
77 81
|
sylbid |
⊢ ( ( ( - 1 ↑ 𝑁 ) = 1 ∧ 𝜑 ) → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
83 |
82
|
ex |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( 𝜑 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) ) |
84 |
75 83
|
jaoi |
⊢ ( ( ( - 1 ↑ 𝑁 ) = - 1 ∨ ( - 1 ↑ 𝑁 ) = 1 ) → ( 𝜑 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) ) |
85 |
19 84
|
sylbi |
⊢ ( ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } → ( 𝜑 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) ) |
86 |
17 85
|
mpcom |
⊢ ( 𝜑 → ( 0 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) |
87 |
59 86
|
sylbid |
⊢ ( 𝜑 → ( ( 0 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 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 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ ( 0 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ) ) |
90 |
|
eqeq1 |
⊢ ( ( 2 /L 𝑃 ) = 0 → ( ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ↔ 0 = ( - 1 ↑ 𝑁 ) ) ) |
91 |
89 90
|
imbi12d |
⊢ ( ( 2 /L 𝑃 ) = 0 → ( ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ↔ ( ( 0 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 0 = ( - 1 ↑ 𝑁 ) ) ) ) |
92 |
87 91
|
syl5ibr |
⊢ ( ( 2 /L 𝑃 ) = 0 → ( 𝜑 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ) ) |
93 |
30
|
eqeq1d |
⊢ ( 𝜑 → ( ( 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ) ) |
94 |
|
eqcom |
⊢ ( 1 = ( - 1 mod 𝑃 ) ↔ ( - 1 mod 𝑃 ) = 1 ) |
95 |
|
eqcom |
⊢ ( 1 = - 1 ↔ - 1 = 1 ) |
96 |
40 94 95
|
3imtr4g |
⊢ ( 𝜑 → ( 1 = ( - 1 mod 𝑃 ) → 1 = - 1 ) ) |
97 |
60
|
eqeq2d |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ 1 = ( - 1 mod 𝑃 ) ) ) |
98 |
|
eqeq2 |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( 1 = ( - 1 ↑ 𝑁 ) ↔ 1 = - 1 ) ) |
99 |
97 98
|
imbi12d |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ↔ ( 1 = ( - 1 mod 𝑃 ) → 1 = - 1 ) ) ) |
100 |
96 99
|
syl5ibr |
⊢ ( ( - 1 ↑ 𝑁 ) = - 1 → ( 𝜑 → ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
101 |
|
eqcom |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 ↔ 1 = ( - 1 ↑ 𝑁 ) ) |
102 |
101
|
biimpi |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → 1 = ( - 1 ↑ 𝑁 ) ) |
103 |
102
|
2a1d |
⊢ ( ( - 1 ↑ 𝑁 ) = 1 → ( 𝜑 → ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
104 |
100 103
|
jaoi |
⊢ ( ( ( - 1 ↑ 𝑁 ) = - 1 ∨ ( - 1 ↑ 𝑁 ) = 1 ) → ( 𝜑 → ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
105 |
19 104
|
sylbi |
⊢ ( ( - 1 ↑ 𝑁 ) ∈ { - 1 , 1 } → ( 𝜑 → ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
106 |
17 105
|
mpcom |
⊢ ( 𝜑 → ( 1 = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ) |
107 |
93 106
|
sylbid |
⊢ ( 𝜑 → ( ( 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 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 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ↔ ( 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) ) ) |
110 |
|
eqeq1 |
⊢ ( ( 2 /L 𝑃 ) = 1 → ( ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ↔ 1 = ( - 1 ↑ 𝑁 ) ) ) |
111 |
109 110
|
imbi12d |
⊢ ( ( 2 /L 𝑃 ) = 1 → ( ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ↔ ( ( 1 mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → 1 = ( - 1 ↑ 𝑁 ) ) ) ) |
112 |
107 111
|
syl5ibr |
⊢ ( ( 2 /L 𝑃 ) = 1 → ( 𝜑 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ) ) |
113 |
54 92 112
|
3jaoi |
⊢ ( ( ( 2 /L 𝑃 ) = - 1 ∨ ( 2 /L 𝑃 ) = 0 ∨ ( 2 /L 𝑃 ) = 1 ) → ( 𝜑 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ) ) |
114 |
13 113
|
sylbi |
⊢ ( ( 2 /L 𝑃 ) ∈ { - 1 , 0 , 1 } → ( 𝜑 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ) ) |
115 |
11 114
|
mpcom |
⊢ ( 𝜑 → ( ( ( 2 /L 𝑃 ) mod 𝑃 ) = ( ( - 1 ↑ 𝑁 ) mod 𝑃 ) → ( 2 /L 𝑃 ) = ( - 1 ↑ 𝑁 ) ) ) |