Proof of Theorem fpprel2
Step | Hyp | Ref
| Expression |
1 | | 2nn 11903 |
. . . . 5
⊢ 2 ∈
ℕ |
2 | | fpprel 44853 |
. . . . 5
⊢ (2 ∈
ℕ → (𝑋 ∈ (
FPPr ‘2) ↔ (𝑋
∈ (ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1))) |
3 | 1, 2 | mp1i 13 |
. . . 4
⊢ (𝑋 ∈ ( FPPr ‘2) →
(𝑋 ∈ ( FPPr ‘2)
↔ (𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1))) |
4 | | eluz4eluz2 12481 |
. . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 𝑋 ∈
(ℤ≥‘2)) |
5 | 4 | 3ad2ant1 1135 |
. . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1) → 𝑋 ∈
(ℤ≥‘2)) |
6 | 5 | adantl 485 |
. . . . . . 7
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → 𝑋 ∈
(ℤ≥‘2)) |
7 | | fppr2odd 44856 |
. . . . . . . 8
⊢ (𝑋 ∈ ( FPPr ‘2) →
𝑋 ∈ Odd
) |
8 | 7 | adantr 484 |
. . . . . . 7
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → 𝑋 ∈ Odd ) |
9 | | simpr2 1197 |
. . . . . . 7
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → 𝑋 ∉ ℙ) |
10 | 6, 8, 9 | 3jca 1130 |
. . . . . 6
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → (𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ)) |
11 | | fpprwppr 44864 |
. . . . . . 7
⊢ (𝑋 ∈ ( FPPr ‘2) →
((2↑𝑋) mod 𝑋) = (2 mod 𝑋)) |
12 | | 2re 11904 |
. . . . . . . . . 10
⊢ 2 ∈
ℝ |
13 | 12 | a1i 11 |
. . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 2 ∈ ℝ) |
14 | | eluz4nn 12482 |
. . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘4) → 𝑋 ∈ ℕ) |
15 | 14 | nnrpd 12626 |
. . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 𝑋 ∈
ℝ+) |
16 | | 0le2 11932 |
. . . . . . . . . 10
⊢ 0 ≤
2 |
17 | 16 | a1i 11 |
. . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 0 ≤ 2) |
18 | | eluz2 12444 |
. . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 𝑋 ∈ ℤ ∧ 4 ≤
𝑋)) |
19 | | 4z 12211 |
. . . . . . . . . . . . . 14
⊢ 4 ∈
ℤ |
20 | | zlem1lt 12229 |
. . . . . . . . . . . . . 14
⊢ ((4
∈ ℤ ∧ 𝑋
∈ ℤ) → (4 ≤ 𝑋 ↔ (4 − 1) < 𝑋)) |
21 | 19, 20 | mpan 690 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℤ → (4 ≤
𝑋 ↔ (4 − 1) <
𝑋)) |
22 | | 4m1e3 11959 |
. . . . . . . . . . . . . . 15
⊢ (4
− 1) = 3 |
23 | 22 | breq1i 5060 |
. . . . . . . . . . . . . 14
⊢ ((4
− 1) < 𝑋 ↔ 3
< 𝑋) |
24 | 12 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 2 ∈
ℝ) |
25 | | 3re 11910 |
. . . . . . . . . . . . . . . . 17
⊢ 3 ∈
ℝ |
26 | 25 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 3 ∈
ℝ) |
27 | | zre 12180 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑋 ∈ ℤ → 𝑋 ∈
ℝ) |
28 | 27 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 𝑋 ∈ ℝ) |
29 | | 2lt3 12002 |
. . . . . . . . . . . . . . . . 17
⊢ 2 <
3 |
30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 2 <
3) |
31 | | simpr 488 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 3 < 𝑋) |
32 | 24, 26, 28, 30, 31 | lttrd 10993 |
. . . . . . . . . . . . . . 15
⊢ ((𝑋 ∈ ℤ ∧ 3 <
𝑋) → 2 < 𝑋) |
33 | 32 | ex 416 |
. . . . . . . . . . . . . 14
⊢ (𝑋 ∈ ℤ → (3 <
𝑋 → 2 < 𝑋)) |
34 | 23, 33 | syl5bi 245 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℤ → ((4
− 1) < 𝑋 → 2
< 𝑋)) |
35 | 21, 34 | sylbid 243 |
. . . . . . . . . . . 12
⊢ (𝑋 ∈ ℤ → (4 ≤
𝑋 → 2 < 𝑋)) |
36 | 35 | a1i 11 |
. . . . . . . . . . 11
⊢ (4 ∈
ℤ → (𝑋 ∈
ℤ → (4 ≤ 𝑋
→ 2 < 𝑋))) |
37 | 36 | 3imp 1113 |
. . . . . . . . . 10
⊢ ((4
∈ ℤ ∧ 𝑋
∈ ℤ ∧ 4 ≤ 𝑋) → 2 < 𝑋) |
38 | 18, 37 | sylbi 220 |
. . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘4) → 2 < 𝑋) |
39 | | modid 13469 |
. . . . . . . . 9
⊢ (((2
∈ ℝ ∧ 𝑋
∈ ℝ+) ∧ (0 ≤ 2 ∧ 2 < 𝑋)) → (2 mod 𝑋) = 2) |
40 | 13, 15, 17, 38, 39 | syl22anc 839 |
. . . . . . . 8
⊢ (𝑋 ∈
(ℤ≥‘4) → (2 mod 𝑋) = 2) |
41 | 40 | 3ad2ant1 1135 |
. . . . . . 7
⊢ ((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1) → (2 mod 𝑋) = 2) |
42 | 11, 41 | sylan9eq 2798 |
. . . . . 6
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → ((2↑𝑋) mod 𝑋) = 2) |
43 | 10, 42 | jca 515 |
. . . . 5
⊢ ((𝑋 ∈ ( FPPr ‘2) ∧
(𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1)) → ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2)) |
44 | 43 | ex 416 |
. . . 4
⊢ (𝑋 ∈ ( FPPr ‘2) →
((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((2↑(𝑋 − 1)) mod 𝑋) = 1) → ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2))) |
45 | 3, 44 | sylbid 243 |
. . 3
⊢ (𝑋 ∈ ( FPPr ‘2) →
(𝑋 ∈ ( FPPr ‘2)
→ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2))) |
46 | 45 | pm2.43i 52 |
. 2
⊢ (𝑋 ∈ ( FPPr ‘2) →
((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2)) |
47 | | ge2nprmge4 16258 |
. . . . . 6
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
(ℤ≥‘4)) |
48 | 47 | 3adant2 1133 |
. . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
(ℤ≥‘4)) |
49 | | simp3 1140 |
. . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 𝑋 ∉ ℙ) |
50 | 48, 49 | jca 515 |
. . . 4
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (𝑋 ∈ (ℤ≥‘4)
∧ 𝑋 ∉
ℙ)) |
51 | 50 | adantr 484 |
. . 3
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (𝑋 ∈ (ℤ≥‘4)
∧ 𝑋 ∉
ℙ)) |
52 | 1 | a1i 11 |
. . . 4
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → 2 ∈
ℕ) |
53 | 12 | a1i 11 |
. . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 2 ∈
ℝ) |
54 | | eluz2nn 12480 |
. . . . . . . . . 10
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈ ℕ) |
55 | 54 | nnrpd 12626 |
. . . . . . . . 9
⊢ (𝑋 ∈
(ℤ≥‘2) → 𝑋 ∈
ℝ+) |
56 | 55 | 3ad2ant1 1135 |
. . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
ℝ+) |
57 | 16 | a1i 11 |
. . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 0 ≤
2) |
58 | 48, 38 | syl 17 |
. . . . . . . 8
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 2 < 𝑋) |
59 | 53, 56, 57, 58, 39 | syl22anc 839 |
. . . . . . 7
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (2 mod 𝑋) = 2) |
60 | 59 | eqcomd 2743 |
. . . . . 6
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → 2 = (2 mod 𝑋)) |
61 | 60 | eqeq2d 2748 |
. . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (((2↑𝑋) mod 𝑋) = 2 ↔ ((2↑𝑋) mod 𝑋) = (2 mod 𝑋))) |
62 | 61 | biimpa 480 |
. . . 4
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → ((2↑𝑋) mod 𝑋) = (2 mod 𝑋)) |
63 | 52, 62 | jca 515 |
. . 3
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (2 ∈ ℕ ∧
((2↑𝑋) mod 𝑋) = (2 mod 𝑋))) |
64 | | gcd2odd1 44793 |
. . . . . 6
⊢ (𝑋 ∈ Odd → (𝑋 gcd 2) = 1) |
65 | 64 | 3ad2ant2 1136 |
. . . . 5
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) → (𝑋 gcd 2) = 1) |
66 | 65 | adantr 484 |
. . . 4
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (𝑋 gcd 2) = 1) |
67 | | fpprwpprb 44865 |
. . . 4
⊢ ((𝑋 gcd 2) = 1 → (𝑋 ∈ ( FPPr ‘2) ↔
((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ) ∧ (2 ∈ ℕ
∧ ((2↑𝑋) mod 𝑋) = (2 mod 𝑋))))) |
68 | 66, 67 | syl 17 |
. . 3
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → (𝑋 ∈ ( FPPr ‘2) ↔ ((𝑋 ∈
(ℤ≥‘4) ∧ 𝑋 ∉ ℙ) ∧ (2 ∈ ℕ
∧ ((2↑𝑋) mod 𝑋) = (2 mod 𝑋))))) |
69 | 51, 63, 68 | mpbir2and 713 |
. 2
⊢ (((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2) → 𝑋 ∈ ( FPPr ‘2)) |
70 | 46, 69 | impbii 212 |
1
⊢ (𝑋 ∈ ( FPPr ‘2) ↔
((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∈ Odd ∧ 𝑋 ∉ ℙ) ∧ ((2↑𝑋) mod 𝑋) = 2)) |