| Step | Hyp | Ref
| Expression |
| 1 | | elnn1uz2 12967 |
. . 3
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
| 2 | | 5prm 17146 |
. . . . . . 7
⊢ 5 ∈
ℙ |
| 3 | | dvdsprime 16724 |
. . . . . . 7
⊢ ((5
∈ ℙ ∧ 𝑀
∈ ℕ) → (𝑀
∥ 5 ↔ (𝑀 = 5
∨ 𝑀 =
1))) |
| 4 | 2, 3 | mpan 690 |
. . . . . 6
⊢ (𝑀 ∈ ℕ → (𝑀 ∥ 5 ↔ (𝑀 = 5 ∨ 𝑀 = 1))) |
| 5 | | 1nn0 12542 |
. . . . . . . . 9
⊢ 1 ∈
ℕ0 |
| 6 | 5 | a1i 11 |
. . . . . . . 8
⊢ (𝑀 = 5 → 1 ∈
ℕ0) |
| 7 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → 𝑀 = 5) |
| 8 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑘 = 1 → (𝑘 · 4) = (1 ·
4)) |
| 9 | 8 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝑘 = 1 → ((𝑘 · 4) + 1) = ((1 · 4) +
1)) |
| 10 | 9 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → ((𝑘 · 4) + 1) = ((1 · 4) +
1)) |
| 11 | 7, 10 | eqeq12d 2753 |
. . . . . . . 8
⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → (𝑀 = ((𝑘 · 4) + 1) ↔ 5 = ((1 · 4)
+ 1))) |
| 12 | | df-5 12332 |
. . . . . . . . . 10
⊢ 5 = (4 +
1) |
| 13 | | 4cn 12351 |
. . . . . . . . . . . . 13
⊢ 4 ∈
ℂ |
| 14 | 13 | mullidi 11266 |
. . . . . . . . . . . 12
⊢ (1
· 4) = 4 |
| 15 | 14 | eqcomi 2746 |
. . . . . . . . . . 11
⊢ 4 = (1
· 4) |
| 16 | 15 | oveq1i 7441 |
. . . . . . . . . 10
⊢ (4 + 1) =
((1 · 4) + 1) |
| 17 | 12, 16 | eqtri 2765 |
. . . . . . . . 9
⊢ 5 = ((1
· 4) + 1) |
| 18 | 17 | a1i 11 |
. . . . . . . 8
⊢ (𝑀 = 5 → 5 = ((1 · 4)
+ 1)) |
| 19 | 6, 11, 18 | rspcedvd 3624 |
. . . . . . 7
⊢ (𝑀 = 5 → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · 4) + 1)) |
| 20 | | 0nn0 12541 |
. . . . . . . . 9
⊢ 0 ∈
ℕ0 |
| 21 | 20 | a1i 11 |
. . . . . . . 8
⊢ (𝑀 = 1 → 0 ∈
ℕ0) |
| 22 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → 𝑀 = 1) |
| 23 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑘 = 0 → (𝑘 · 4) = (0 ·
4)) |
| 24 | 23 | oveq1d 7446 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → ((𝑘 · 4) + 1) = ((0 · 4) +
1)) |
| 25 | 24 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → ((𝑘 · 4) + 1) = ((0 · 4) +
1)) |
| 26 | 22, 25 | eqeq12d 2753 |
. . . . . . . 8
⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → (𝑀 = ((𝑘 · 4) + 1) ↔ 1 = ((0 · 4)
+ 1))) |
| 27 | 13 | mul02i 11450 |
. . . . . . . . . . . 12
⊢ (0
· 4) = 0 |
| 28 | 27 | oveq1i 7441 |
. . . . . . . . . . 11
⊢ ((0
· 4) + 1) = (0 + 1) |
| 29 | | 0p1e1 12388 |
. . . . . . . . . . 11
⊢ (0 + 1) =
1 |
| 30 | 28, 29 | eqtri 2765 |
. . . . . . . . . 10
⊢ ((0
· 4) + 1) = 1 |
| 31 | 30 | eqcomi 2746 |
. . . . . . . . 9
⊢ 1 = ((0
· 4) + 1) |
| 32 | 31 | a1i 11 |
. . . . . . . 8
⊢ (𝑀 = 1 → 1 = ((0 · 4)
+ 1)) |
| 33 | 21, 26, 32 | rspcedvd 3624 |
. . . . . . 7
⊢ (𝑀 = 1 → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · 4) + 1)) |
| 34 | 19, 33 | jaoi 858 |
. . . . . 6
⊢ ((𝑀 = 5 ∨ 𝑀 = 1) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · 4) + 1)) |
| 35 | 4, 34 | biimtrdi 253 |
. . . . 5
⊢ (𝑀 ∈ ℕ → (𝑀 ∥ 5 → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · 4) + 1))) |
| 36 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑁 = 1 →
(FermatNo‘𝑁) =
(FermatNo‘1)) |
| 37 | | fmtno1 47528 |
. . . . . . . 8
⊢
(FermatNo‘1) = 5 |
| 38 | 36, 37 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝑁 = 1 →
(FermatNo‘𝑁) =
5) |
| 39 | 38 | breq2d 5155 |
. . . . . 6
⊢ (𝑁 = 1 → (𝑀 ∥ (FermatNo‘𝑁) ↔ 𝑀 ∥ 5)) |
| 40 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑁 = 1 → (𝑁 + 1) = (1 + 1)) |
| 41 | | 1p1e2 12391 |
. . . . . . . . . . . . 13
⊢ (1 + 1) =
2 |
| 42 | 40, 41 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑁 = 1 → (𝑁 + 1) = 2) |
| 43 | 42 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑁 = 1 → (2↑(𝑁 + 1)) =
(2↑2)) |
| 44 | | sq2 14236 |
. . . . . . . . . . 11
⊢
(2↑2) = 4 |
| 45 | 43, 44 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝑁 = 1 → (2↑(𝑁 + 1)) = 4) |
| 46 | 45 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑁 = 1 → (𝑘 · (2↑(𝑁 + 1))) = (𝑘 · 4)) |
| 47 | 46 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑁 = 1 → ((𝑘 · (2↑(𝑁 + 1))) + 1) = ((𝑘 · 4) + 1)) |
| 48 | 47 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑁 = 1 → (𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ 𝑀 = ((𝑘 · 4) + 1))) |
| 49 | 48 | rexbidv 3179 |
. . . . . 6
⊢ (𝑁 = 1 → (∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · 4) + 1))) |
| 50 | 39, 49 | imbi12d 344 |
. . . . 5
⊢ (𝑁 = 1 → ((𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) ↔ (𝑀 ∥ 5 → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · 4) + 1)))) |
| 51 | 35, 50 | imbitrrid 246 |
. . . 4
⊢ (𝑁 = 1 → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))) |
| 52 | | fmtnofac2 47556 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑛 ∈ ℕ0 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) |
| 53 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℕ0) |
| 54 | | 2nn0 12543 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℕ0 |
| 55 | 54 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
→ 2 ∈ ℕ0) |
| 56 | 53, 55 | nn0mulcld 12592 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
→ (𝑛 · 2)
∈ ℕ0) |
| 57 | 56 | adantl 481 |
. . . . . . . . 9
⊢ (((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 2) ∈
ℕ0) |
| 58 | 57 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → (𝑛 · 2) ∈
ℕ0) |
| 59 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) |
| 60 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑛 · 2) → (𝑘 · (2↑(𝑁 + 1))) = ((𝑛 · 2) · (2↑(𝑁 + 1)))) |
| 61 | 60 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝑘 = (𝑛 · 2) → ((𝑘 · (2↑(𝑁 + 1))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1)) |
| 62 | 59, 61 | eqeqan12d 2751 |
. . . . . . . 8
⊢
(((((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) ∧ 𝑘 = (𝑛 · 2)) → (𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ ((𝑛 · (2↑(𝑁 + 2))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1))) |
| 63 | | eluzge2nn0 12929 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈
ℕ0) |
| 64 | 63 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℂ) |
| 65 | | add1p1 12517 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
| 66 | 64, 65 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘2) → ((𝑁 + 1) + 1) = (𝑁 + 2)) |
| 67 | 66 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑁 + 2) = ((𝑁 + 1) + 1)) |
| 68 | 67 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → (2↑(𝑁 + 2)) = (2↑((𝑁 + 1) + 1))) |
| 69 | | 2cnd 12344 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → 2 ∈ ℂ) |
| 70 | | peano2nn0 12566 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
| 71 | 63, 70 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑁 + 1) ∈
ℕ0) |
| 72 | 69, 71 | expp1d 14187 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → (2↑((𝑁 + 1) + 1)) = ((2↑(𝑁 + 1)) · 2)) |
| 73 | 54 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑁 ∈
(ℤ≥‘2) → 2 ∈
ℕ0) |
| 74 | 73, 71 | nn0expcld 14285 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘2) → (2↑(𝑁 + 1)) ∈
ℕ0) |
| 75 | 74 | nn0cnd 12589 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → (2↑(𝑁 + 1)) ∈ ℂ) |
| 76 | 75, 69 | mulcomd 11282 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → ((2↑(𝑁 + 1)) · 2) = (2 ·
(2↑(𝑁 +
1)))) |
| 77 | 68, 72, 76 | 3eqtrd 2781 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘2) → (2↑(𝑁 + 2)) = (2 · (2↑(𝑁 + 1)))) |
| 78 | 77 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) →
(2↑(𝑁 + 2)) = (2
· (2↑(𝑁 +
1)))) |
| 79 | 78 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) → (𝑛 · (2↑(𝑁 + 2))) = (𝑛 · (2 · (2↑(𝑁 + 1))))) |
| 80 | | nn0cn 12536 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℂ) |
| 81 | 80 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℂ) |
| 82 | | 2cnd 12344 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) → 2 ∈
ℂ) |
| 83 | 75 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) →
(2↑(𝑁 + 1)) ∈
ℂ) |
| 84 | 81, 82, 83 | mulassd 11284 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) → ((𝑛 · 2) ·
(2↑(𝑁 + 1))) = (𝑛 · (2 ·
(2↑(𝑁 +
1))))) |
| 85 | 79, 84 | eqtr4d 2780 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑛 ∈ ℕ0) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1)))) |
| 86 | 85 | 3ad2antl1 1186 |
. . . . . . . . . 10
⊢ (((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1)))) |
| 87 | 86 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1)))) |
| 88 | 87 | oveq1d 7446 |
. . . . . . . 8
⊢ ((((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → ((𝑛 · (2↑(𝑁 + 2))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1)) |
| 89 | 58, 62, 88 | rspcedvd 3624 |
. . . . . . 7
⊢ ((((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) |
| 90 | 89 | rexlimdva2 3157 |
. . . . . 6
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → (∃𝑛 ∈ ℕ0 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1) → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))) |
| 91 | 52, 90 | mpd 15 |
. . . . 5
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) |
| 92 | 91 | 3exp 1120 |
. . . 4
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))) |
| 93 | 51, 92 | jaoi 858 |
. . 3
⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))
→ (𝑀 ∈ ℕ
→ (𝑀 ∥
(FermatNo‘𝑁) →
∃𝑘 ∈
ℕ0 𝑀 =
((𝑘 · (2↑(𝑁 + 1))) + 1)))) |
| 94 | 1, 93 | sylbi 217 |
. 2
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))) |
| 95 | 94 | 3imp 1111 |
1
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0
𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) |