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