Proof of Theorem m1lgs
Step | Hyp | Ref
| Expression |
1 | | neg1z 11413 |
. . . . . . . . 9
⊢ -1 ∈
ℤ |
2 | | oddprm 15515 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ) |
3 | 2 | nnnn0d 11351 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ0) |
4 | | zexpcl 12875 |
. . . . . . . . 9
⊢ ((-1
∈ ℤ ∧ ((𝑃
− 1) / 2) ∈ ℕ0) → (-1↑((𝑃 − 1) / 2)) ∈
ℤ) |
5 | 1, 3, 4 | sylancr 695 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (-1↑((𝑃 −
1) / 2)) ∈ ℤ) |
6 | 5 | peano2zd 11485 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1↑((𝑃
− 1) / 2)) + 1) ∈ ℤ) |
7 | | eldifi 3732 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℙ) |
8 | | prmnn 15388 |
. . . . . . . 8
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
9 | 7, 8 | syl 17 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℕ) |
10 | 6, 9 | zmodcld 12691 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) ∈
ℕ0) |
11 | 10 | nn0cnd 11353 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) ∈ ℂ) |
12 | | 1cnd 10056 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 1 ∈ ℂ) |
13 | 11, 12, 12 | subaddd 10410 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃))) |
14 | | 2re 11090 |
. . . . . . . 8
⊢ 2 ∈
ℝ |
15 | 14 | a1i 11 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℝ) |
16 | 9 | nnrpd 11870 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℝ+) |
17 | | 0le2 11111 |
. . . . . . . 8
⊢ 0 ≤
2 |
18 | 17 | a1i 11 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 0 ≤ 2) |
19 | | eldifsni 4320 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ≠
2) |
20 | 9 | nnred 11035 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℝ) |
21 | | prmuz2 15408 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
(ℤ≥‘2)) |
22 | 7, 21 | syl 17 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
(ℤ≥‘2)) |
23 | | eluzle 11700 |
. . . . . . . . . 10
⊢ (𝑃 ∈
(ℤ≥‘2) → 2 ≤ 𝑃) |
24 | 22, 23 | syl 17 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ≤ 𝑃) |
25 | 15, 20, 24 | leltned 10190 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 < 𝑃 ↔
𝑃 ≠ 2)) |
26 | 19, 25 | mpbird 247 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 < 𝑃) |
27 | | modid 12695 |
. . . . . . 7
⊢ (((2
∈ ℝ ∧ 𝑃
∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2) |
28 | 15, 16, 18, 26, 27 | syl22anc 1327 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 mod 𝑃) =
2) |
29 | | df-2 11079 |
. . . . . 6
⊢ 2 = (1 +
1) |
30 | 28, 29 | syl6eq 2672 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 mod 𝑃) = (1 +
1)) |
31 | 30 | eqeq1d 2624 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) ↔ (1 +
1) = (((-1↑((𝑃 −
1) / 2)) + 1) mod 𝑃))) |
32 | 19 | neneqd 2799 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ¬ 𝑃 =
2) |
33 | | 2prm 15405 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℙ |
34 | | dvdsprm 15415 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2)) |
35 | 22, 33, 34 | sylancl 694 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 ∥ 2 ↔
𝑃 = 2)) |
36 | 32, 35 | mtbird 315 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ¬ 𝑃 ∥
2) |
37 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ¬ 𝑃 ∥ 2) |
38 | | 1cnd 10056 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → 1 ∈ ℂ) |
39 | 2 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((𝑃
− 1) / 2) ∈ ℕ) |
40 | | simpr 477 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ¬ 2 ∥ ((𝑃 − 1) / 2)) |
41 | | oexpneg 15069 |
. . . . . . . . . . . . . . . 16
⊢ ((1
∈ ℂ ∧ ((𝑃
− 1) / 2) ∈ ℕ ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) =
-(1↑((𝑃 − 1) /
2))) |
42 | 38, 39, 40, 41 | syl3anc 1326 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) /
2))) |
43 | 39 | nnzd 11481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((𝑃
− 1) / 2) ∈ ℤ) |
44 | | 1exp 12889 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑃 − 1) / 2) ∈ ℤ
→ (1↑((𝑃 −
1) / 2)) = 1) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (1↑((𝑃 − 1) / 2)) = 1) |
46 | 45 | negeqd 10275 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1) |
47 | 42, 46 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1) |
48 | 47 | oveq1d 6665 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 +
1)) |
49 | | ax-1cn 9994 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
50 | | neg1cn 11124 |
. . . . . . . . . . . . . 14
⊢ -1 ∈
ℂ |
51 | | 1pneg1e0 11129 |
. . . . . . . . . . . . . 14
⊢ (1 + -1)
= 0 |
52 | 49, 50, 51 | addcomli 10228 |
. . . . . . . . . . . . 13
⊢ (-1 + 1)
= 0 |
53 | 48, 52 | syl6eq 2672 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0) |
54 | 53 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 −
0)) |
55 | | 2cn 11091 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℂ |
56 | 55 | subid1i 10353 |
. . . . . . . . . . 11
⊢ (2
− 0) = 2 |
57 | 54, 56 | syl6eq 2672 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) =
2) |
58 | 57 | breq2d 4665 |
. . . . . . . . 9
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (𝑃
∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2)) |
59 | 37, 58 | mtbird 315 |
. . . . . . . 8
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) +
1))) |
60 | 59 | ex 450 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (¬ 2 ∥ ((𝑃
− 1) / 2) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) +
1)))) |
61 | 60 | con4d 114 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 ∥ (2
− ((-1↑((𝑃
− 1) / 2)) + 1)) → 2 ∥ ((𝑃 − 1) / 2))) |
62 | | 2z 11409 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
63 | 62 | a1i 11 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℤ) |
64 | | moddvds 14991 |
. . . . . . 7
⊢ ((𝑃 ∈ ℕ ∧ 2 ∈
ℤ ∧ ((-1↑((𝑃
− 1) / 2)) + 1) ∈ ℤ) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) +
1)))) |
65 | 9, 63, 6, 64 | syl3anc 1326 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) ↔
𝑃 ∥ (2 −
((-1↑((𝑃 − 1) /
2)) + 1)))) |
66 | | 4z 11411 |
. . . . . . . . . 10
⊢ 4 ∈
ℤ |
67 | 66 | a1i 11 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 4 ∈ ℤ) |
68 | | 4ne0 11117 |
. . . . . . . . . 10
⊢ 4 ≠
0 |
69 | 68 | a1i 11 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 4 ≠ 0) |
70 | | nnm1nn0 11334 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈
ℕ0) |
71 | 9, 70 | syl 17 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 − 1) ∈
ℕ0) |
72 | 71 | nn0zd 11480 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 − 1) ∈
ℤ) |
73 | | dvdsval2 14986 |
. . . . . . . . 9
⊢ ((4
∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4
∥ (𝑃 − 1)
↔ ((𝑃 − 1) / 4)
∈ ℤ)) |
74 | 67, 69, 72, 73 | syl3anc 1326 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) ↔ ((𝑃
− 1) / 4) ∈ ℤ)) |
75 | 71 | nn0cnd 11353 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 − 1) ∈
ℂ) |
76 | 55 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℂ) |
77 | | 2ne0 11113 |
. . . . . . . . . . . 12
⊢ 2 ≠
0 |
78 | 77 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ≠ 0) |
79 | 75, 76, 76, 78, 78 | divdiv1d 10832 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((𝑃 − 1) / 2)
/ 2) = ((𝑃 − 1) / (2
· 2))) |
80 | | 2t2e4 11177 |
. . . . . . . . . . 11
⊢ (2
· 2) = 4 |
81 | 80 | oveq2i 6661 |
. . . . . . . . . 10
⊢ ((𝑃 − 1) / (2 · 2)) =
((𝑃 − 1) /
4) |
82 | 79, 81 | syl6eq 2672 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((𝑃 − 1) / 2)
/ 2) = ((𝑃 − 1) /
4)) |
83 | 82 | eleq1d 2686 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((((𝑃 − 1) /
2) / 2) ∈ ℤ ↔ ((𝑃 − 1) / 4) ∈
ℤ)) |
84 | 74, 83 | bitr4d 271 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) ↔ (((𝑃
− 1) / 2) / 2) ∈ ℤ)) |
85 | 2 | nnzd 11481 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℤ) |
86 | | dvdsval2 14986 |
. . . . . . . 8
⊢ ((2
∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2
∥ ((𝑃 − 1) / 2)
↔ (((𝑃 − 1) / 2)
/ 2) ∈ ℤ)) |
87 | 63, 78, 85, 86 | syl3anc 1326 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 ∥ ((𝑃
− 1) / 2) ↔ (((𝑃
− 1) / 2) / 2) ∈ ℤ)) |
88 | 84, 87 | bitr4d 271 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) ↔ 2 ∥ ((𝑃 − 1) / 2))) |
89 | 61, 65, 88 | 3imtr4d 283 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) → 4
∥ (𝑃 −
1))) |
90 | 50 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → -1 ∈ ℂ) |
91 | | neg1ne0 11126 |
. . . . . . . . . . . 12
⊢ -1 ≠
0 |
92 | 91 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → -1 ≠ 0) |
93 | 62 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → 2 ∈ ℤ) |
94 | 84 | biimpa 501 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (((𝑃 − 1)
/ 2) / 2) ∈ ℤ) |
95 | | expmulz 12906 |
. . . . . . . . . . 11
⊢ (((-1
∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ (((𝑃 − 1) / 2) / 2) ∈
ℤ)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) =
((-1↑2)↑(((𝑃
− 1) / 2) / 2))) |
96 | 90, 92, 93, 94, 95 | syl22anc 1327 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) =
((-1↑2)↑(((𝑃
− 1) / 2) / 2))) |
97 | 2 | nncnd 11036 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℂ) |
98 | 97, 76, 78 | divcan2d 10803 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 · (((𝑃
− 1) / 2) / 2)) = ((𝑃
− 1) / 2)) |
99 | 98 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (2 · (((𝑃
− 1) / 2) / 2)) = ((𝑃
− 1) / 2)) |
100 | 99 | oveq2d 6666 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) /
2))) |
101 | | neg1sqe1 12959 |
. . . . . . . . . . . 12
⊢
(-1↑2) = 1 |
102 | 101 | oveq1i 6660 |
. . . . . . . . . . 11
⊢
((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) /
2)) |
103 | | 1exp 12889 |
. . . . . . . . . . . 12
⊢ ((((𝑃 − 1) / 2) / 2) ∈
ℤ → (1↑(((𝑃
− 1) / 2) / 2)) = 1) |
104 | 94, 103 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (1↑(((𝑃
− 1) / 2) / 2)) = 1) |
105 | 102, 104 | syl5eq 2668 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1) |
106 | 96, 100, 105 | 3eqtr3d 2664 |
. . . . . . . . 9
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (-1↑((𝑃
− 1) / 2)) = 1) |
107 | 106 | oveq1d 6665 |
. . . . . . . 8
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → ((-1↑((𝑃
− 1) / 2)) + 1) = (1 + 1)) |
108 | 107, 29 | syl6reqr 2675 |
. . . . . . 7
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1)) |
109 | 108 | oveq1d 6665 |
. . . . . 6
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃)) |
110 | 109 | ex 450 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) → (2 mod 𝑃)
= (((-1↑((𝑃 − 1)
/ 2)) + 1) mod 𝑃))) |
111 | 89, 110 | impbid 202 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) ↔ 4
∥ (𝑃 −
1))) |
112 | 13, 31, 111 | 3bitr2d 296 |
. . 3
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ 4 ∥ (𝑃 − 1))) |
113 | | lgsval3 25040 |
. . . . 5
⊢ ((-1
∈ ℤ ∧ 𝑃
∈ (ℙ ∖ {2})) → (-1 /L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) |
114 | 1, 113 | mpan 706 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (-1 /L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) |
115 | 114 | eqeq1d 2624 |
. . 3
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1 /L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1)) |
116 | | 4nn 11187 |
. . . . 5
⊢ 4 ∈
ℕ |
117 | 116 | a1i 11 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 4 ∈ ℕ) |
118 | | prmz 15389 |
. . . . 5
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
119 | 7, 118 | syl 17 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℤ) |
120 | | 1zzd 11408 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 1 ∈ ℤ) |
121 | | moddvds 14991 |
. . . 4
⊢ ((4
∈ ℕ ∧ 𝑃
∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1))) |
122 | 117, 119,
120, 121 | syl3anc 1326 |
. . 3
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 mod 4) = (1 mod
4) ↔ 4 ∥ (𝑃
− 1))) |
123 | 112, 115,
122 | 3bitr4d 300 |
. 2
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1 /L 𝑃) = 1 ↔ (𝑃 mod 4) = (1 mod 4))) |
124 | | 1re 10039 |
. . . 4
⊢ 1 ∈
ℝ |
125 | | nnrp 11842 |
. . . . 5
⊢ (4 ∈
ℕ → 4 ∈ ℝ+) |
126 | 116, 125 | ax-mp 5 |
. . . 4
⊢ 4 ∈
ℝ+ |
127 | | 0le1 10551 |
. . . 4
⊢ 0 ≤
1 |
128 | | 1lt4 11199 |
. . . 4
⊢ 1 <
4 |
129 | | modid 12695 |
. . . 4
⊢ (((1
∈ ℝ ∧ 4 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 <
4)) → (1 mod 4) = 1) |
130 | 124, 126,
127, 128, 129 | mp4an 709 |
. . 3
⊢ (1 mod 4)
= 1 |
131 | 130 | eqeq2i 2634 |
. 2
⊢ ((𝑃 mod 4) = (1 mod 4) ↔
(𝑃 mod 4) =
1) |
132 | 123, 131 | syl6bb 276 |
1
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1 /L 𝑃) = 1 ↔ (𝑃 mod 4) = 1)) |