Proof of Theorem m1lgs
Step | Hyp | Ref
| Expression |
1 | | neg1z 9302 |
. . . . . . . . 9
⊢ -1 ∈
ℤ |
2 | | oddprm 12276 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ) |
3 | 2 | nnnn0d 9246 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℕ0) |
4 | | zexpcl 10552 |
. . . . . . . . 9
⊢ ((-1
∈ ℤ ∧ ((𝑃
− 1) / 2) ∈ ℕ0) → (-1↑((𝑃 − 1) / 2)) ∈
ℤ) |
5 | 1, 3, 4 | sylancr 414 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (-1↑((𝑃 −
1) / 2)) ∈ ℤ) |
6 | 5 | peano2zd 9395 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1↑((𝑃
− 1) / 2)) + 1) ∈ ℤ) |
7 | | eldifi 3271 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℙ) |
8 | | prmnn 12127 |
. . . . . . . 8
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
9 | 7, 8 | syl 14 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℕ) |
10 | 6, 9 | zmodcld 10362 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) ∈
ℕ0) |
11 | 10 | nn0cnd 9248 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) ∈ ℂ) |
12 | | 1cnd 7990 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 1 ∈ ℂ) |
13 | 11, 12, 12 | subaddd 8303 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃))) |
14 | | 2z 9298 |
. . . . . . . . 9
⊢ 2 ∈
ℤ |
15 | | zq 9643 |
. . . . . . . . 9
⊢ (2 ∈
ℤ → 2 ∈ ℚ) |
16 | 14, 15 | ax-mp 5 |
. . . . . . . 8
⊢ 2 ∈
ℚ |
17 | 16 | a1i 9 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℚ) |
18 | | prmz 12128 |
. . . . . . . 8
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
19 | | zq 9643 |
. . . . . . . 8
⊢ (𝑃 ∈ ℤ → 𝑃 ∈
ℚ) |
20 | 7, 18, 19 | 3syl 17 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℚ) |
21 | | 0le2 9026 |
. . . . . . . 8
⊢ 0 ≤
2 |
22 | 21 | a1i 9 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 0 ≤ 2) |
23 | | oddprmgt2 12151 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 < 𝑃) |
24 | | modqid 10366 |
. . . . . . 7
⊢ (((2
∈ ℚ ∧ 𝑃
∈ ℚ) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2) |
25 | 17, 20, 22, 23, 24 | syl22anc 1249 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 mod 𝑃) =
2) |
26 | | df-2 8995 |
. . . . . 6
⊢ 2 = (1 +
1) |
27 | 25, 26 | eqtrdi 2237 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 mod 𝑃) = (1 +
1)) |
28 | 27 | eqeq1d 2197 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) ↔ (1 +
1) = (((-1↑((𝑃 −
1) / 2)) + 1) mod 𝑃))) |
29 | | 2nn 9097 |
. . . . . . . 8
⊢ 2 ∈
ℕ |
30 | 2 | nnzd 9391 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℤ) |
31 | | dvdsdc 11822 |
. . . . . . . 8
⊢ ((2
∈ ℕ ∧ ((𝑃
− 1) / 2) ∈ ℤ) → DECID 2 ∥ ((𝑃 − 1) /
2)) |
32 | 29, 30, 31 | sylancr 414 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ DECID 2 ∥ ((𝑃 − 1) / 2)) |
33 | | eldifsni 3735 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ≠
2) |
34 | 33 | neneqd 2380 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ¬ 𝑃 =
2) |
35 | | prmuz2 12148 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
(ℤ≥‘2)) |
36 | 7, 35 | syl 14 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
(ℤ≥‘2)) |
37 | | 2prm 12144 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℙ |
38 | | dvdsprm 12154 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2)) |
39 | 36, 37, 38 | sylancl 413 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 ∥ 2 ↔
𝑃 = 2)) |
40 | 34, 39 | mtbird 674 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ¬ 𝑃 ∥
2) |
41 | 40 | adantr 276 |
. . . . . . . . 9
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ¬ 𝑃 ∥ 2) |
42 | | 1cnd 7990 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → 1 ∈ ℂ) |
43 | 2 | adantr 276 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((𝑃
− 1) / 2) ∈ ℕ) |
44 | | simpr 110 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ¬ 2 ∥ ((𝑃 − 1) / 2)) |
45 | | oexpneg 11899 |
. . . . . . . . . . . . . . . 16
⊢ ((1
∈ ℂ ∧ ((𝑃
− 1) / 2) ∈ ℕ ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) =
-(1↑((𝑃 − 1) /
2))) |
46 | 42, 43, 44, 45 | syl3anc 1248 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) /
2))) |
47 | 43 | nnzd 9391 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((𝑃
− 1) / 2) ∈ ℤ) |
48 | | 1exp 10566 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑃 − 1) / 2) ∈ ℤ
→ (1↑((𝑃 −
1) / 2)) = 1) |
49 | 47, 48 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (1↑((𝑃 − 1) / 2)) = 1) |
50 | 49 | negeqd 8169 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1) |
51 | 46, 50 | eqtrd 2221 |
. . . . . . . . . . . . . 14
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1) |
52 | 51 | oveq1d 5905 |
. . . . . . . . . . . . 13
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 +
1)) |
53 | | ax-1cn 7921 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
54 | | neg1cn 9041 |
. . . . . . . . . . . . . 14
⊢ -1 ∈
ℂ |
55 | | 1pneg1e0 9047 |
. . . . . . . . . . . . . 14
⊢ (1 + -1)
= 0 |
56 | 53, 54, 55 | addcomli 8119 |
. . . . . . . . . . . . 13
⊢ (-1 + 1)
= 0 |
57 | 52, 56 | eqtrdi 2237 |
. . . . . . . . . . . 12
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0) |
58 | 57 | oveq2d 5906 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 −
0)) |
59 | | 2cn 9007 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℂ |
60 | 59 | subid1i 8246 |
. . . . . . . . . . 11
⊢ (2
− 0) = 2 |
61 | 58, 60 | eqtrdi 2237 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) =
2) |
62 | 61 | breq2d 4029 |
. . . . . . . . 9
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → (𝑃
∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2)) |
63 | 41, 62 | mtbird 674 |
. . . . . . . 8
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ ¬ 2 ∥ ((𝑃
− 1) / 2)) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) +
1))) |
64 | 63 | ex 115 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (¬ 2 ∥ ((𝑃
− 1) / 2) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) +
1)))) |
65 | | condc 854 |
. . . . . . 7
⊢
(DECID 2 ∥ ((𝑃 − 1) / 2) → ((¬ 2 ∥
((𝑃 − 1) / 2) →
¬ 𝑃 ∥ (2 −
((-1↑((𝑃 − 1) /
2)) + 1))) → (𝑃
∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) → 2 ∥
((𝑃 − 1) /
2)))) |
66 | 32, 64, 65 | sylc 62 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 ∥ (2
− ((-1↑((𝑃
− 1) / 2)) + 1)) → 2 ∥ ((𝑃 − 1) / 2))) |
67 | 14 | a1i 9 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℤ) |
68 | | moddvds 11823 |
. . . . . . 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)))) |
69 | 9, 67, 6, 68 | syl3anc 1248 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) ↔
𝑃 ∥ (2 −
((-1↑((𝑃 − 1) /
2)) + 1)))) |
70 | | 4z 9300 |
. . . . . . . . 9
⊢ 4 ∈
ℤ |
71 | | 4ne0 9034 |
. . . . . . . . 9
⊢ 4 ≠
0 |
72 | | nnm1nn0 9234 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈
ℕ0) |
73 | 9, 72 | syl 14 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 − 1) ∈
ℕ0) |
74 | 73 | nn0zd 9390 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 − 1) ∈
ℤ) |
75 | | dvdsval2 11814 |
. . . . . . . . 9
⊢ ((4
∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4
∥ (𝑃 − 1)
↔ ((𝑃 − 1) / 4)
∈ ℤ)) |
76 | 70, 71, 74, 75 | mp3an12i 1351 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) ↔ ((𝑃
− 1) / 4) ∈ ℤ)) |
77 | 73 | nn0cnd 9248 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (𝑃 − 1) ∈
ℂ) |
78 | 59 | a1i 9 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℂ) |
79 | 29 | a1i 9 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 ∈ ℕ) |
80 | 79 | nnap0d 8982 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 2 # 0) |
81 | 77, 78, 78, 80, 80 | divdivap1d 8796 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((𝑃 − 1) / 2)
/ 2) = ((𝑃 − 1) / (2
· 2))) |
82 | | 2t2e4 9090 |
. . . . . . . . . . 11
⊢ (2
· 2) = 4 |
83 | 82 | oveq2i 5901 |
. . . . . . . . . 10
⊢ ((𝑃 − 1) / (2 · 2)) =
((𝑃 − 1) /
4) |
84 | 81, 83 | eqtrdi 2237 |
. . . . . . . . 9
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((𝑃 − 1) / 2)
/ 2) = ((𝑃 − 1) /
4)) |
85 | 84 | eleq1d 2257 |
. . . . . . . 8
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((((𝑃 − 1) /
2) / 2) ∈ ℤ ↔ ((𝑃 − 1) / 4) ∈
ℤ)) |
86 | 76, 85 | bitr4d 191 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) ↔ (((𝑃
− 1) / 2) / 2) ∈ ℤ)) |
87 | | 2ne0 9028 |
. . . . . . . 8
⊢ 2 ≠
0 |
88 | | dvdsval2 11814 |
. . . . . . . 8
⊢ ((2
∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2
∥ ((𝑃 − 1) / 2)
↔ (((𝑃 − 1) / 2)
/ 2) ∈ ℤ)) |
89 | 14, 87, 30, 88 | mp3an12i 1351 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 ∥ ((𝑃
− 1) / 2) ↔ (((𝑃
− 1) / 2) / 2) ∈ ℤ)) |
90 | 86, 89 | bitr4d 191 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) ↔ 2 ∥ ((𝑃 − 1) / 2))) |
91 | 66, 69, 90 | 3imtr4d 203 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) → 4
∥ (𝑃 −
1))) |
92 | 54 | a1i 9 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → -1 ∈ ℂ) |
93 | | neg1ap0 9045 |
. . . . . . . . . . . 12
⊢ -1 #
0 |
94 | 93 | a1i 9 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → -1 # 0) |
95 | 14 | a1i 9 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → 2 ∈ ℤ) |
96 | 86 | biimpa 296 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (((𝑃 − 1)
/ 2) / 2) ∈ ℤ) |
97 | | expmulzap 10583 |
. . . . . . . . . . 11
⊢ (((-1
∈ ℂ ∧ -1 # 0) ∧ (2 ∈ ℤ ∧ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
→ (-1↑(2 · (((𝑃 − 1) / 2) / 2))) =
((-1↑2)↑(((𝑃
− 1) / 2) / 2))) |
98 | 92, 94, 95, 96, 97 | syl22anc 1249 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) =
((-1↑2)↑(((𝑃
− 1) / 2) / 2))) |
99 | 2 | nncnd 8950 |
. . . . . . . . . . . . 13
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 − 1) / 2)
∈ ℂ) |
100 | 99, 78, 80 | divcanap2d 8766 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (2 · (((𝑃
− 1) / 2) / 2)) = ((𝑃
− 1) / 2)) |
101 | 100 | adantr 276 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (2 · (((𝑃
− 1) / 2) / 2)) = ((𝑃
− 1) / 2)) |
102 | 101 | oveq2d 5906 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) /
2))) |
103 | | neg1sqe1 10632 |
. . . . . . . . . . . 12
⊢
(-1↑2) = 1 |
104 | 103 | oveq1i 5900 |
. . . . . . . . . . 11
⊢
((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) /
2)) |
105 | | 1exp 10566 |
. . . . . . . . . . . 12
⊢ ((((𝑃 − 1) / 2) / 2) ∈
ℤ → (1↑(((𝑃
− 1) / 2) / 2)) = 1) |
106 | 96, 105 | syl 14 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (1↑(((𝑃
− 1) / 2) / 2)) = 1) |
107 | 104, 106 | eqtrid 2233 |
. . . . . . . . . 10
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1) |
108 | 98, 102, 107 | 3eqtr3d 2229 |
. . . . . . . . 9
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (-1↑((𝑃
− 1) / 2)) = 1) |
109 | 108 | oveq1d 5905 |
. . . . . . . 8
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → ((-1↑((𝑃
− 1) / 2)) + 1) = (1 + 1)) |
110 | 26, 109 | eqtr4id 2240 |
. . . . . . 7
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1)) |
111 | 110 | oveq1d 5905 |
. . . . . 6
⊢ ((𝑃 ∈ (ℙ ∖ {2})
∧ 4 ∥ (𝑃 −
1)) → (2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃)) |
112 | 111 | ex 115 |
. . . . 5
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (4 ∥ (𝑃
− 1) → (2 mod 𝑃)
= (((-1↑((𝑃 − 1)
/ 2)) + 1) mod 𝑃))) |
113 | 91, 112 | impbid 129 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((2 mod 𝑃) =
(((-1↑((𝑃 − 1) /
2)) + 1) mod 𝑃) ↔ 4
∥ (𝑃 −
1))) |
114 | 13, 28, 113 | 3bitr2d 216 |
. . 3
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (((((-1↑((𝑃
− 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ 4 ∥ (𝑃 − 1))) |
115 | | lgsval3 14802 |
. . . . 5
⊢ ((-1
∈ ℤ ∧ 𝑃
∈ (ℙ ∖ {2})) → (-1 /L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) |
116 | 1, 115 | mpan 424 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (-1 /L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) |
117 | 116 | eqeq1d 2197 |
. . 3
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1 /L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1)) |
118 | | 4nn 9099 |
. . . . 5
⊢ 4 ∈
ℕ |
119 | 118 | a1i 9 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 4 ∈ ℕ) |
120 | 7, 18 | syl 14 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℤ) |
121 | | 1zzd 9297 |
. . . 4
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 1 ∈ ℤ) |
122 | | moddvds 11823 |
. . . 4
⊢ ((4
∈ ℕ ∧ 𝑃
∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1))) |
123 | 119, 120,
121, 122 | syl3anc 1248 |
. . 3
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((𝑃 mod 4) = (1 mod
4) ↔ 4 ∥ (𝑃
− 1))) |
124 | 114, 117,
123 | 3bitr4d 220 |
. 2
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1 /L 𝑃) = 1 ↔ (𝑃 mod 4) = (1 mod 4))) |
125 | | 1z 9296 |
. . . . 5
⊢ 1 ∈
ℤ |
126 | | zq 9643 |
. . . . 5
⊢ (1 ∈
ℤ → 1 ∈ ℚ) |
127 | 125, 126 | ax-mp 5 |
. . . 4
⊢ 1 ∈
ℚ |
128 | | zq 9643 |
. . . . 5
⊢ (4 ∈
ℤ → 4 ∈ ℚ) |
129 | 70, 128 | ax-mp 5 |
. . . 4
⊢ 4 ∈
ℚ |
130 | | 0le1 8455 |
. . . 4
⊢ 0 ≤
1 |
131 | | 1lt4 9110 |
. . . 4
⊢ 1 <
4 |
132 | | modqid 10366 |
. . . 4
⊢ (((1
∈ ℚ ∧ 4 ∈ ℚ) ∧ (0 ≤ 1 ∧ 1 < 4)) →
(1 mod 4) = 1) |
133 | 127, 129,
130, 131, 132 | mp4an 427 |
. . 3
⊢ (1 mod 4)
= 1 |
134 | 133 | eqeq2i 2199 |
. 2
⊢ ((𝑃 mod 4) = (1 mod 4) ↔
(𝑃 mod 4) =
1) |
135 | 124, 134 | bitrdi 196 |
1
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ ((-1 /L 𝑃) = 1 ↔ (𝑃 mod 4) = 1)) |