Step |
Hyp |
Ref |
Expression |
1 |
|
olc |
⊢ ( ( 𝑄 mod 8 ) = 7 → ( ( 𝑄 mod 8 ) = 1 ∨ ( 𝑄 mod 8 ) = 7 ) ) |
2 |
|
ovex |
⊢ ( 𝑄 mod 8 ) ∈ V |
3 |
|
elprg |
⊢ ( ( 𝑄 mod 8 ) ∈ V → ( ( 𝑄 mod 8 ) ∈ { 1 , 7 } ↔ ( ( 𝑄 mod 8 ) = 1 ∨ ( 𝑄 mod 8 ) = 7 ) ) ) |
4 |
2 3
|
mp1i |
⊢ ( ( 𝑄 mod 8 ) = 7 → ( ( 𝑄 mod 8 ) ∈ { 1 , 7 } ↔ ( ( 𝑄 mod 8 ) = 1 ∨ ( 𝑄 mod 8 ) = 7 ) ) ) |
5 |
1 4
|
mpbird |
⊢ ( ( 𝑄 mod 8 ) = 7 → ( 𝑄 mod 8 ) ∈ { 1 , 7 } ) |
6 |
|
2lgs |
⊢ ( 𝑄 ∈ ℙ → ( ( 2 /L 𝑄 ) = 1 ↔ ( 𝑄 mod 8 ) ∈ { 1 , 7 } ) ) |
7 |
6
|
ad2antlr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 2 /L 𝑄 ) = 1 ↔ ( 𝑄 mod 8 ) ∈ { 1 , 7 } ) ) |
8 |
|
2z |
⊢ 2 ∈ ℤ |
9 |
|
simpr |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → 𝑄 ∈ ℙ ) |
10 |
9
|
adantr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 𝑄 ∈ ℙ ) |
11 |
|
2re |
⊢ 2 ∈ ℝ |
12 |
11
|
a1i |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 2 ∈ ℝ ) |
13 |
|
2m1e1 |
⊢ ( 2 − 1 ) = 1 |
14 |
11
|
a1i |
⊢ ( 𝑃 ∈ ℙ → 2 ∈ ℝ ) |
15 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
16 |
15
|
nnred |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℝ ) |
17 |
|
1lt2 |
⊢ 1 < 2 |
18 |
17
|
a1i |
⊢ ( 𝑃 ∈ ℙ → 1 < 2 ) |
19 |
|
prmgt1 |
⊢ ( 𝑃 ∈ ℙ → 1 < 𝑃 ) |
20 |
14 16 18 19
|
mulgt1d |
⊢ ( 𝑃 ∈ ℙ → 1 < ( 2 · 𝑃 ) ) |
21 |
13 20
|
eqbrtrid |
⊢ ( 𝑃 ∈ ℙ → ( 2 − 1 ) < ( 2 · 𝑃 ) ) |
22 |
|
1red |
⊢ ( 𝑃 ∈ ℙ → 1 ∈ ℝ ) |
23 |
|
2nn |
⊢ 2 ∈ ℕ |
24 |
23
|
a1i |
⊢ ( 𝑃 ∈ ℙ → 2 ∈ ℕ ) |
25 |
24 15
|
nnmulcld |
⊢ ( 𝑃 ∈ ℙ → ( 2 · 𝑃 ) ∈ ℕ ) |
26 |
25
|
nnred |
⊢ ( 𝑃 ∈ ℙ → ( 2 · 𝑃 ) ∈ ℝ ) |
27 |
14 22 26
|
ltsubaddd |
⊢ ( 𝑃 ∈ ℙ → ( ( 2 − 1 ) < ( 2 · 𝑃 ) ↔ 2 < ( ( 2 · 𝑃 ) + 1 ) ) ) |
28 |
21 27
|
mpbid |
⊢ ( 𝑃 ∈ ℙ → 2 < ( ( 2 · 𝑃 ) + 1 ) ) |
29 |
28
|
ad2antrr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 2 < ( ( 2 · 𝑃 ) + 1 ) ) |
30 |
|
breq2 |
⊢ ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) → ( 2 < 𝑄 ↔ 2 < ( ( 2 · 𝑃 ) + 1 ) ) ) |
31 |
30
|
adantl |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( 2 < 𝑄 ↔ 2 < ( ( 2 · 𝑃 ) + 1 ) ) ) |
32 |
29 31
|
mpbird |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 2 < 𝑄 ) |
33 |
12 32
|
gtned |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 𝑄 ≠ 2 ) |
34 |
|
eldifsn |
⊢ ( 𝑄 ∈ ( ℙ ∖ { 2 } ) ↔ ( 𝑄 ∈ ℙ ∧ 𝑄 ≠ 2 ) ) |
35 |
10 33 34
|
sylanbrc |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 𝑄 ∈ ( ℙ ∖ { 2 } ) ) |
36 |
|
lgsqrmodndvds |
⊢ ( ( 2 ∈ ℤ ∧ 𝑄 ∈ ( ℙ ∖ { 2 } ) ) → ( ( 2 /L 𝑄 ) = 1 → ∃ 𝑚 ∈ ℤ ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) ) ) |
37 |
8 35 36
|
sylancr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 2 /L 𝑄 ) = 1 → ∃ 𝑚 ∈ ℤ ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) ) ) |
38 |
|
prmnn |
⊢ ( 𝑄 ∈ ℙ → 𝑄 ∈ ℕ ) |
39 |
38
|
nncnd |
⊢ ( 𝑄 ∈ ℙ → 𝑄 ∈ ℂ ) |
40 |
39
|
adantl |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → 𝑄 ∈ ℂ ) |
41 |
|
1cnd |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → 1 ∈ ℂ ) |
42 |
|
2cnd |
⊢ ( 𝑃 ∈ ℙ → 2 ∈ ℂ ) |
43 |
15
|
nncnd |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℂ ) |
44 |
42 43
|
mulcld |
⊢ ( 𝑃 ∈ ℙ → ( 2 · 𝑃 ) ∈ ℂ ) |
45 |
44
|
adantr |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 2 · 𝑃 ) ∈ ℂ ) |
46 |
40 41 45
|
subadd2d |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( ( 𝑄 − 1 ) = ( 2 · 𝑃 ) ↔ ( ( 2 · 𝑃 ) + 1 ) = 𝑄 ) ) |
47 |
|
prmz |
⊢ ( 𝑄 ∈ ℙ → 𝑄 ∈ ℤ ) |
48 |
|
peano2zm |
⊢ ( 𝑄 ∈ ℤ → ( 𝑄 − 1 ) ∈ ℤ ) |
49 |
47 48
|
syl |
⊢ ( 𝑄 ∈ ℙ → ( 𝑄 − 1 ) ∈ ℤ ) |
50 |
49
|
zcnd |
⊢ ( 𝑄 ∈ ℙ → ( 𝑄 − 1 ) ∈ ℂ ) |
51 |
50
|
adantl |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 𝑄 − 1 ) ∈ ℂ ) |
52 |
43
|
adantr |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → 𝑃 ∈ ℂ ) |
53 |
|
2cnne0 |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) |
54 |
53
|
a1i |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
55 |
|
divmul2 |
⊢ ( ( ( 𝑄 − 1 ) ∈ ℂ ∧ 𝑃 ∈ ℂ ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( ( 𝑄 − 1 ) / 2 ) = 𝑃 ↔ ( 𝑄 − 1 ) = ( 2 · 𝑃 ) ) ) |
56 |
51 52 54 55
|
syl3anc |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( ( ( 𝑄 − 1 ) / 2 ) = 𝑃 ↔ ( 𝑄 − 1 ) = ( 2 · 𝑃 ) ) ) |
57 |
|
eqcom |
⊢ ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ↔ ( ( 2 · 𝑃 ) + 1 ) = 𝑄 ) |
58 |
57
|
a1i |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ↔ ( ( 2 · 𝑃 ) + 1 ) = 𝑄 ) ) |
59 |
46 56 58
|
3bitr4rd |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ↔ ( ( 𝑄 − 1 ) / 2 ) = 𝑃 ) ) |
60 |
59
|
biimpa |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 𝑄 − 1 ) / 2 ) = 𝑃 ) |
61 |
|
oveq2 |
⊢ ( ( ( 𝑄 − 1 ) / 2 ) = 𝑃 → ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) ) |
62 |
|
zsqcl |
⊢ ( 𝑚 ∈ ℤ → ( 𝑚 ↑ 2 ) ∈ ℤ ) |
63 |
62
|
ad2antlr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) → ( 𝑚 ↑ 2 ) ∈ ℤ ) |
64 |
8
|
a1i |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) → 2 ∈ ℤ ) |
65 |
|
oveq1 |
⊢ ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) → ( 𝑄 − 1 ) = ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) ) |
66 |
65
|
adantl |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( 𝑄 − 1 ) = ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) ) |
67 |
66
|
oveq1d |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 𝑄 − 1 ) / 2 ) = ( ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) / 2 ) ) |
68 |
|
pncan1 |
⊢ ( ( 2 · 𝑃 ) ∈ ℂ → ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) = ( 2 · 𝑃 ) ) |
69 |
44 68
|
syl |
⊢ ( 𝑃 ∈ ℙ → ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) = ( 2 · 𝑃 ) ) |
70 |
69
|
oveq1d |
⊢ ( 𝑃 ∈ ℙ → ( ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) / 2 ) = ( ( 2 · 𝑃 ) / 2 ) ) |
71 |
|
2ne0 |
⊢ 2 ≠ 0 |
72 |
71
|
a1i |
⊢ ( 𝑃 ∈ ℙ → 2 ≠ 0 ) |
73 |
43 42 72
|
divcan3d |
⊢ ( 𝑃 ∈ ℙ → ( ( 2 · 𝑃 ) / 2 ) = 𝑃 ) |
74 |
70 73
|
eqtrd |
⊢ ( 𝑃 ∈ ℙ → ( ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) / 2 ) = 𝑃 ) |
75 |
74
|
ad2antrr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( ( ( 2 · 𝑃 ) + 1 ) − 1 ) / 2 ) = 𝑃 ) |
76 |
67 75
|
eqtrd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 𝑄 − 1 ) / 2 ) = 𝑃 ) |
77 |
15
|
nnnn0d |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ0 ) |
78 |
77
|
ad2antrr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 𝑃 ∈ ℕ0 ) |
79 |
76 78
|
eqeltrd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 𝑄 − 1 ) / 2 ) ∈ ℕ0 ) |
80 |
38
|
nnrpd |
⊢ ( 𝑄 ∈ ℙ → 𝑄 ∈ ℝ+ ) |
81 |
80
|
ad2antlr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → 𝑄 ∈ ℝ+ ) |
82 |
79 81
|
jca |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( ( 𝑄 − 1 ) / 2 ) ∈ ℕ0 ∧ 𝑄 ∈ ℝ+ ) ) |
83 |
82
|
ad2antrr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) → ( ( ( 𝑄 − 1 ) / 2 ) ∈ ℕ0 ∧ 𝑄 ∈ ℝ+ ) ) |
84 |
|
simpr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) → ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) |
85 |
|
modexp |
⊢ ( ( ( ( 𝑚 ↑ 2 ) ∈ ℤ ∧ 2 ∈ ℤ ) ∧ ( ( ( 𝑄 − 1 ) / 2 ) ∈ ℕ0 ∧ 𝑄 ∈ ℝ+ ) ∧ ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) ) |
86 |
63 64 83 84 85
|
syl211anc |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) ) |
87 |
86
|
ex |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) ) ) |
88 |
87
|
adantr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) ) ) |
89 |
|
2cnd |
⊢ ( 𝑄 ∈ ℙ → 2 ∈ ℂ ) |
90 |
71
|
a1i |
⊢ ( 𝑄 ∈ ℙ → 2 ≠ 0 ) |
91 |
50 89 90
|
divcan2d |
⊢ ( 𝑄 ∈ ℙ → ( 2 · ( ( 𝑄 − 1 ) / 2 ) ) = ( 𝑄 − 1 ) ) |
92 |
91
|
eqcomd |
⊢ ( 𝑄 ∈ ℙ → ( 𝑄 − 1 ) = ( 2 · ( ( 𝑄 − 1 ) / 2 ) ) ) |
93 |
92
|
oveq2d |
⊢ ( 𝑄 ∈ ℙ → ( 𝑚 ↑ ( 𝑄 − 1 ) ) = ( 𝑚 ↑ ( 2 · ( ( 𝑄 − 1 ) / 2 ) ) ) ) |
94 |
93
|
ad3antlr |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( 𝑚 ↑ ( 𝑄 − 1 ) ) = ( 𝑚 ↑ ( 2 · ( ( 𝑄 − 1 ) / 2 ) ) ) ) |
95 |
|
zcn |
⊢ ( 𝑚 ∈ ℤ → 𝑚 ∈ ℂ ) |
96 |
95
|
adantl |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → 𝑚 ∈ ℂ ) |
97 |
79
|
adantr |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( 𝑄 − 1 ) / 2 ) ∈ ℕ0 ) |
98 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
99 |
98
|
a1i |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → 2 ∈ ℕ0 ) |
100 |
96 97 99
|
expmuld |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( 𝑚 ↑ ( 2 · ( ( 𝑄 − 1 ) / 2 ) ) ) = ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) ) |
101 |
94 100
|
eqtr2d |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 𝑚 ↑ ( 𝑄 − 1 ) ) ) |
102 |
101
|
oveq1d |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 𝑚 ↑ ( 𝑄 − 1 ) ) mod 𝑄 ) ) |
103 |
102
|
adantr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 𝑚 ↑ ( 𝑄 − 1 ) ) mod 𝑄 ) ) |
104 |
|
vfermltl |
⊢ ( ( 𝑄 ∈ ℙ ∧ 𝑚 ∈ ℤ ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( 𝑚 ↑ ( 𝑄 − 1 ) ) mod 𝑄 ) = 1 ) |
105 |
104
|
ad5ant245 |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( 𝑚 ↑ ( 𝑄 − 1 ) ) mod 𝑄 ) = 1 ) |
106 |
103 105
|
eqtrd |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = 1 ) |
107 |
|
oveq1 |
⊢ ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) |
108 |
106 107
|
eqeqan12d |
⊢ ( ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) ∧ ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) ) → ( ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) ↔ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) ) |
109 |
|
id |
⊢ ( 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) → 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) |
110 |
109
|
eqcomd |
⊢ ( 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) → ( ( 2 ↑ 𝑃 ) mod 𝑄 ) = 1 ) |
111 |
38
|
nnred |
⊢ ( 𝑄 ∈ ℙ → 𝑄 ∈ ℝ ) |
112 |
|
prmgt1 |
⊢ ( 𝑄 ∈ ℙ → 1 < 𝑄 ) |
113 |
|
1mod |
⊢ ( ( 𝑄 ∈ ℝ ∧ 1 < 𝑄 ) → ( 1 mod 𝑄 ) = 1 ) |
114 |
111 112 113
|
syl2anc |
⊢ ( 𝑄 ∈ ℙ → ( 1 mod 𝑄 ) = 1 ) |
115 |
114
|
eqcomd |
⊢ ( 𝑄 ∈ ℙ → 1 = ( 1 mod 𝑄 ) ) |
116 |
115
|
ad3antlr |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → 1 = ( 1 mod 𝑄 ) ) |
117 |
110 116
|
sylan9eqr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) → ( ( 2 ↑ 𝑃 ) mod 𝑄 ) = ( 1 mod 𝑄 ) ) |
118 |
38
|
ad4antlr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) → 𝑄 ∈ ℕ ) |
119 |
|
zexpcl |
⊢ ( ( 2 ∈ ℤ ∧ 𝑃 ∈ ℕ0 ) → ( 2 ↑ 𝑃 ) ∈ ℤ ) |
120 |
8 77 119
|
sylancr |
⊢ ( 𝑃 ∈ ℙ → ( 2 ↑ 𝑃 ) ∈ ℤ ) |
121 |
120
|
ad4antr |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) → ( 2 ↑ 𝑃 ) ∈ ℤ ) |
122 |
|
1zzd |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) → 1 ∈ ℤ ) |
123 |
|
moddvds |
⊢ ( ( 𝑄 ∈ ℕ ∧ ( 2 ↑ 𝑃 ) ∈ ℤ ∧ 1 ∈ ℤ ) → ( ( ( 2 ↑ 𝑃 ) mod 𝑄 ) = ( 1 mod 𝑄 ) ↔ 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
124 |
118 121 122 123
|
syl3anc |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) → ( ( ( 2 ↑ 𝑃 ) mod 𝑄 ) = ( 1 mod 𝑄 ) ↔ 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
125 |
117 124
|
mpbid |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) |
126 |
125
|
ex |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
127 |
126
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) ∧ ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) ) → ( 1 = ( ( 2 ↑ 𝑃 ) mod 𝑄 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
128 |
108 127
|
sylbid |
⊢ ( ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) ∧ ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) ) → ( ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
129 |
128
|
ex |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → ( ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
130 |
129
|
com23 |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( ( ( 𝑚 ↑ 2 ) ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) = ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) mod 𝑄 ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
131 |
88 130
|
syld |
⊢ ( ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
132 |
131
|
ex |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ¬ 𝑄 ∥ 𝑚 → ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) ) |
133 |
132
|
com23 |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) → ( ¬ 𝑄 ∥ 𝑚 → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) ) |
134 |
133
|
impd |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
135 |
134
|
com23 |
⊢ ( ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ∧ 𝑚 ∈ ℤ ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → ( ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
136 |
135
|
ex |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( 𝑚 ∈ ℤ → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → ( ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) ) |
137 |
136
|
com23 |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 2 ↑ ( ( 𝑄 − 1 ) / 2 ) ) = ( 2 ↑ 𝑃 ) → ( 𝑚 ∈ ℤ → ( ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) ) |
138 |
61 137
|
syl5 |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( ( 𝑄 − 1 ) / 2 ) = 𝑃 → ( 𝑚 ∈ ℤ → ( ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) ) |
139 |
60 138
|
mpd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( 𝑚 ∈ ℤ → ( ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
140 |
139
|
rexlimdv |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ∃ 𝑚 ∈ ℤ ( ( ( 𝑚 ↑ 2 ) mod 𝑄 ) = ( 2 mod 𝑄 ) ∧ ¬ 𝑄 ∥ 𝑚 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
141 |
37 140
|
syld |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 2 /L 𝑄 ) = 1 → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
142 |
7 141
|
sylbird |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 𝑄 mod 8 ) ∈ { 1 , 7 } → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
143 |
5 142
|
syl5 |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) → ( ( 𝑄 mod 8 ) = 7 → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) |
144 |
143
|
ex |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) → ( ( 𝑄 mod 8 ) = 7 → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
145 |
144
|
com23 |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ) → ( ( 𝑄 mod 8 ) = 7 → ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) |
146 |
145
|
ex |
⊢ ( 𝑃 ∈ ℙ → ( 𝑄 ∈ ℙ → ( ( 𝑄 mod 8 ) = 7 → ( 𝑄 = ( ( 2 · 𝑃 ) + 1 ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) ) ) ) |
147 |
146
|
3imp2 |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑄 ∈ ℙ ∧ ( 𝑄 mod 8 ) = 7 ∧ 𝑄 = ( ( 2 · 𝑃 ) + 1 ) ) ) → 𝑄 ∥ ( ( 2 ↑ 𝑃 ) − 1 ) ) |