Proof of Theorem fmtnoprmfac1lem
| Step | Hyp | Ref
| Expression |
| 1 | | eldifi 4131 |
. . . . . . 7
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℙ) |
| 2 | | prmnn 16711 |
. . . . . . 7
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
| 3 | 1, 2 | syl 17 |
. . . . . 6
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℕ) |
| 4 | 3 | ad2antlr 727 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ 𝑃 ∥
(FermatNo‘𝑁)) →
𝑃 ∈
ℕ) |
| 5 | | nnnn0 12533 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
| 6 | | fmtno 47516 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ0
→ (FermatNo‘𝑁) =
((2↑(2↑𝑁)) +
1)) |
| 7 | 5, 6 | syl 17 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ →
(FermatNo‘𝑁) =
((2↑(2↑𝑁)) +
1)) |
| 8 | 7 | breq2d 5155 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → (𝑃 ∥ (FermatNo‘𝑁) ↔ 𝑃 ∥ ((2↑(2↑𝑁)) + 1))) |
| 9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (𝑃 ∥
(FermatNo‘𝑁) ↔
𝑃 ∥
((2↑(2↑𝑁)) +
1))) |
| 10 | 9 | biimpa 476 |
. . . . 5
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ 𝑃 ∥
(FermatNo‘𝑁)) →
𝑃 ∥
((2↑(2↑𝑁)) +
1)) |
| 11 | | dvdsmod0 16296 |
. . . . 5
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ∥ ((2↑(2↑𝑁)) + 1)) →
(((2↑(2↑𝑁)) + 1)
mod 𝑃) =
0) |
| 12 | 4, 10, 11 | syl2anc 584 |
. . . 4
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ 𝑃 ∥
(FermatNo‘𝑁)) →
(((2↑(2↑𝑁)) + 1)
mod 𝑃) =
0) |
| 13 | 12 | ex 412 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (𝑃 ∥
(FermatNo‘𝑁) →
(((2↑(2↑𝑁)) + 1)
mod 𝑃) =
0)) |
| 14 | | 2nn 12339 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ |
| 15 | 14 | a1i 11 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 2 ∈
ℕ) |
| 16 | | 2nn0 12543 |
. . . . . . . . . . 11
⊢ 2 ∈
ℕ0 |
| 17 | 16 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → 2 ∈
ℕ0) |
| 18 | 17, 5 | nn0expcld 14285 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ →
(2↑𝑁) ∈
ℕ0) |
| 19 | 15, 18 | nnexpcld 14284 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ →
(2↑(2↑𝑁)) ∈
ℕ) |
| 20 | 19 | nnzd 12640 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ →
(2↑(2↑𝑁)) ∈
ℤ) |
| 21 | 20 | adantr 480 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (2↑(2↑𝑁))
∈ ℤ) |
| 22 | | 1zzd 12648 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 1 ∈ ℤ) |
| 23 | 3 | adantl 481 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 𝑃 ∈
ℕ) |
| 24 | | summodnegmod 16324 |
. . . . . 6
⊢
(((2↑(2↑𝑁)) ∈ ℤ ∧ 1 ∈ ℤ
∧ 𝑃 ∈ ℕ)
→ ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 ↔ ((2↑(2↑𝑁)) mod 𝑃) = (-1 mod 𝑃))) |
| 25 | 21, 22, 23, 24 | syl3anc 1373 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 ↔ ((2↑(2↑𝑁)) mod 𝑃) = (-1 mod 𝑃))) |
| 26 | | neg1z 12653 |
. . . . . . . . . 10
⊢ -1 ∈
ℤ |
| 27 | 21, 26 | jctir 520 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((2↑(2↑𝑁))
∈ ℤ ∧ -1 ∈ ℤ)) |
| 28 | 27 | adantr 480 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑𝑁))
mod 𝑃) = (-1 mod 𝑃)) → ((2↑(2↑𝑁)) ∈ ℤ ∧ -1
∈ ℤ)) |
| 29 | 2 | nnrpd 13075 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℝ+) |
| 30 | 1, 29 | syl 17 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℝ+) |
| 31 | 17, 30 | anim12i 613 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (2 ∈ ℕ0 ∧ 𝑃 ∈
ℝ+)) |
| 32 | 31 | adantr 480 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑𝑁))
mod 𝑃) = (-1 mod 𝑃)) → (2 ∈
ℕ0 ∧ 𝑃
∈ ℝ+)) |
| 33 | | simpr 484 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑𝑁))
mod 𝑃) = (-1 mod 𝑃)) → ((2↑(2↑𝑁)) mod 𝑃) = (-1 mod 𝑃)) |
| 34 | | modexp 14277 |
. . . . . . . 8
⊢
((((2↑(2↑𝑁)) ∈ ℤ ∧ -1 ∈ ℤ)
∧ (2 ∈ ℕ0 ∧ 𝑃 ∈ ℝ+) ∧
((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃)) →
(((2↑(2↑𝑁))↑2) mod 𝑃) = ((-1↑2) mod 𝑃)) |
| 35 | 28, 32, 33, 34 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑𝑁))
mod 𝑃) = (-1 mod 𝑃)) →
(((2↑(2↑𝑁))↑2) mod 𝑃) = ((-1↑2) mod 𝑃)) |
| 36 | 35 | ex 412 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (((2↑(2↑𝑁)) mod 𝑃) = (-1 mod 𝑃) → (((2↑(2↑𝑁))↑2) mod 𝑃) = ((-1↑2) mod 𝑃))) |
| 37 | | 2cnd 12344 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → 2 ∈
ℂ) |
| 38 | 37, 18, 17 | 3jca 1129 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → (2 ∈
ℂ ∧ (2↑𝑁)
∈ ℕ0 ∧ 2 ∈
ℕ0)) |
| 39 | 38 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (2 ∈ ℂ ∧ (2↑𝑁) ∈ ℕ0 ∧ 2 ∈
ℕ0)) |
| 40 | | expmul 14148 |
. . . . . . . . . . 11
⊢ ((2
∈ ℂ ∧ (2↑𝑁) ∈ ℕ0 ∧ 2 ∈
ℕ0) → (2↑((2↑𝑁) · 2)) = ((2↑(2↑𝑁))↑2)) |
| 41 | 39, 40 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (2↑((2↑𝑁)
· 2)) = ((2↑(2↑𝑁))↑2)) |
| 42 | | 2cnd 12344 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 2 ∈ ℂ) |
| 43 | 5 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 𝑁 ∈
ℕ0) |
| 44 | 42, 43 | expp1d 14187 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (2↑(𝑁 + 1)) =
((2↑𝑁) ·
2)) |
| 45 | 44 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((2↑𝑁) ·
2) = (2↑(𝑁 +
1))) |
| 46 | 45 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (2↑((2↑𝑁)
· 2)) = (2↑(2↑(𝑁 + 1)))) |
| 47 | 41, 46 | eqtr3d 2779 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((2↑(2↑𝑁))↑2) = (2↑(2↑(𝑁 + 1)))) |
| 48 | 47 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (((2↑(2↑𝑁))↑2) mod 𝑃) = ((2↑(2↑(𝑁 + 1))) mod 𝑃)) |
| 49 | | neg1sqe1 14235 |
. . . . . . . . . . 11
⊢
(-1↑2) = 1 |
| 50 | 49 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (-1↑2) = 1) |
| 51 | 50 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((-1↑2) mod 𝑃)
= (1 mod 𝑃)) |
| 52 | 3 | nnred 12281 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℝ) |
| 53 | | prmgt1 16734 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ ℙ → 1 <
𝑃) |
| 54 | 1, 53 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 1 < 𝑃) |
| 55 | | 1mod 13943 |
. . . . . . . . . . 11
⊢ ((𝑃 ∈ ℝ ∧ 1 <
𝑃) → (1 mod 𝑃) = 1) |
| 56 | 52, 54, 55 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ (1 mod 𝑃) =
1) |
| 57 | 56 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (1 mod 𝑃) =
1) |
| 58 | 51, 57 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((-1↑2) mod 𝑃)
= 1) |
| 59 | 48, 58 | eqeq12d 2753 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((((2↑(2↑𝑁))↑2) mod 𝑃) = ((-1↑2) mod 𝑃) ↔ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) |
| 60 | | simpll 767 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
(((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0) → (𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖
{2}))) |
| 61 | 20 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
(2↑(2↑𝑁)) ∈
ℤ) |
| 62 | | 1zzd 12648 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 1 ∈
ℤ) |
| 63 | 2 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈
ℕ) |
| 64 | 61, 62, 63 | 3jca 1129 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) →
((2↑(2↑𝑁)) ∈
ℤ ∧ 1 ∈ ℤ ∧ 𝑃 ∈ ℕ)) |
| 65 | 1, 64 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((2↑(2↑𝑁))
∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑃 ∈ ℕ)) |
| 66 | 65 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) →
((2↑(2↑𝑁)) ∈
ℤ ∧ 1 ∈ ℤ ∧ 𝑃 ∈ ℕ)) |
| 67 | 66, 24 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) →
((((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0 ↔
((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃))) |
| 68 | | m1modnnsub1 13958 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃 ∈ ℕ → (-1 mod
𝑃) = (𝑃 − 1)) |
| 69 | 23, 68 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (-1 mod 𝑃) = (𝑃 − 1)) |
| 70 | | eldifsni 4790 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ≠
2) |
| 71 | 70 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 𝑃 ≠
2) |
| 72 | 71 | necomd 2996 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 2 ≠ 𝑃) |
| 73 | 3 | nncnd 12282 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑃 ∈ (ℙ ∖ {2})
→ 𝑃 ∈
ℂ) |
| 74 | 73 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 𝑃 ∈
ℂ) |
| 75 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ 1 ∈ ℂ) |
| 76 | 74, 75, 75 | subadd2d 11639 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((𝑃 − 1) = 1
↔ (1 + 1) = 𝑃)) |
| 77 | | 1p1e2 12391 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (1 + 1) =
2 |
| 78 | 77 | eqeq1i 2742 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((1 + 1)
= 𝑃 ↔ 2 = 𝑃) |
| 79 | 76, 78 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((𝑃 − 1) = 1
↔ 2 = 𝑃)) |
| 80 | 79 | necon3bid 2985 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((𝑃 − 1) ≠
1 ↔ 2 ≠ 𝑃)) |
| 81 | 72, 80 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (𝑃 − 1) ≠
1) |
| 82 | 69, 81 | eqnetrd 3008 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (-1 mod 𝑃) ≠
1) |
| 83 | 82 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) → (-1
mod 𝑃) ≠
1) |
| 84 | 83 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃)) → (-1 mod 𝑃) ≠ 1) |
| 85 | | eqeq1 2741 |
. . . . . . . . . . . . . . . 16
⊢
(((2↑(2↑𝑁)) mod 𝑃) = (-1 mod 𝑃) → (((2↑(2↑𝑁)) mod 𝑃) = 1 ↔ (-1 mod 𝑃) = 1)) |
| 86 | 85 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃)) →
(((2↑(2↑𝑁)) mod
𝑃) = 1 ↔ (-1 mod 𝑃) = 1)) |
| 87 | 86 | necon3bid 2985 |
. . . . . . . . . . . . . 14
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃)) →
(((2↑(2↑𝑁)) mod
𝑃) ≠ 1 ↔ (-1 mod
𝑃) ≠
1)) |
| 88 | 84, 87 | mpbird 257 |
. . . . . . . . . . . . 13
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃)) → ((2↑(2↑𝑁)) mod 𝑃) ≠ 1) |
| 89 | 88 | ex 412 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) →
(((2↑(2↑𝑁)) mod
𝑃) = (-1 mod 𝑃) → ((2↑(2↑𝑁)) mod 𝑃) ≠ 1)) |
| 90 | 67, 89 | sylbid 240 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) →
((((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0 →
((2↑(2↑𝑁)) mod
𝑃) ≠
1)) |
| 91 | 90 | imp 406 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
(((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0) →
((2↑(2↑𝑁)) mod
𝑃) ≠ 1) |
| 92 | | simplr 769 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
(((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0) →
((2↑(2↑(𝑁 + 1)))
mod 𝑃) =
1) |
| 93 | | odz2prm2pw 47550 |
. . . . . . . . . 10
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ (((2↑(2↑𝑁))
mod 𝑃) ≠ 1 ∧
((2↑(2↑(𝑁 + 1)))
mod 𝑃) = 1)) →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) |
| 94 | 60, 91, 92, 93 | syl12anc 837 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) ∧
(((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0) →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) |
| 95 | 94 | ex 412 |
. . . . . . . 8
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
∧ ((2↑(2↑(𝑁 +
1))) mod 𝑃) = 1) →
((((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0 →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))) |
| 96 | 95 | ex 412 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (((2↑(2↑(𝑁
+ 1))) mod 𝑃) = 1 →
((((2↑(2↑𝑁)) + 1)
mod 𝑃) = 0 →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))))) |
| 97 | 59, 96 | sylbid 240 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((((2↑(2↑𝑁))↑2) mod 𝑃) = ((-1↑2) mod 𝑃) → ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))))) |
| 98 | 36, 97 | syld 47 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (((2↑(2↑𝑁)) mod 𝑃) = (-1 mod 𝑃) → ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))))) |
| 99 | 25, 98 | sylbid 240 |
. . . 4
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 → ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))))) |
| 100 | 99 | pm2.43d 53 |
. . 3
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ ((((2↑(2↑𝑁)) + 1) mod 𝑃) = 0 →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))) |
| 101 | 13, 100 | syld 47 |
. 2
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}))
→ (𝑃 ∥
(FermatNo‘𝑁) →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))) |
| 102 | 101 | 3impia 1118 |
1
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})
∧ 𝑃 ∥
(FermatNo‘𝑁)) →
((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) |