Step | Hyp | Ref
| Expression |
1 | | oveq2 7240 |
. . . . . . . 8
⊢ (𝑘 = 1 → (𝐴↑𝑘) = (𝐴↑1)) |
2 | 1 | oveq1d 7247 |
. . . . . . 7
⊢ (𝑘 = 1 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑1) gcd 𝐵)) |
3 | 2 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑘 = 1 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑1) gcd 𝐵) = 1)) |
4 | 3 | imbi2d 344 |
. . . . 5
⊢ (𝑘 = 1 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1))) |
5 | | oveq2 7240 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (𝐴↑𝑘) = (𝐴↑𝑛)) |
6 | 5 | oveq1d 7247 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵)) |
7 | 6 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑘 = 𝑛 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑𝑛) gcd 𝐵) = 1)) |
8 | 7 | imbi2d 344 |
. . . . 5
⊢ (𝑘 = 𝑛 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = 1))) |
9 | | oveq2 7240 |
. . . . . . . 8
⊢ (𝑘 = (𝑛 + 1) → (𝐴↑𝑘) = (𝐴↑(𝑛 + 1))) |
10 | 9 | oveq1d 7247 |
. . . . . . 7
⊢ (𝑘 = (𝑛 + 1) → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑(𝑛 + 1)) gcd 𝐵)) |
11 | 10 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑘 = (𝑛 + 1) → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)) |
12 | 11 | imbi2d 344 |
. . . . 5
⊢ (𝑘 = (𝑛 + 1) → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))) |
13 | | oveq2 7240 |
. . . . . . . 8
⊢ (𝑘 = 𝑁 → (𝐴↑𝑘) = (𝐴↑𝑁)) |
14 | 13 | oveq1d 7247 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → ((𝐴↑𝑘) gcd 𝐵) = ((𝐴↑𝑁) gcd 𝐵)) |
15 | 14 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (((𝐴↑𝑘) gcd 𝐵) = 1 ↔ ((𝐴↑𝑁) gcd 𝐵) = 1)) |
16 | 15 | imbi2d 344 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑘) gcd 𝐵) = 1) ↔ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd 𝐵) = 1))) |
17 | | nncn 11863 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℂ) |
18 | 17 | exp1d 13739 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℕ → (𝐴↑1) = 𝐴) |
19 | 18 | oveq1d 7247 |
. . . . . . . 8
⊢ (𝐴 ∈ ℕ → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵)) |
20 | 19 | adantr 484 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴↑1) gcd 𝐵) = (𝐴 gcd 𝐵)) |
21 | 20 | eqeq1d 2740 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((𝐴↑1) gcd 𝐵) = 1 ↔ (𝐴 gcd 𝐵) = 1)) |
22 | 21 | biimpar 481 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑1) gcd 𝐵) = 1) |
23 | | df-3an 1091 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ↔ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈
ℕ)) |
24 | | simpl1 1193 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ) |
25 | 24 | nncnd 11871 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℂ) |
26 | | simpl3 1195 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ) |
27 | 26 | nnnn0d 12175 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝑛 ∈ ℕ0) |
28 | 25, 27 | expp1d 13745 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = ((𝐴↑𝑛) · 𝐴)) |
29 | | simp1 1138 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈
ℕ) |
30 | | nnnn0 12122 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
31 | 30 | 3ad2ant3 1137 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℕ0) |
32 | 29, 31 | nnexpcld 13840 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℕ) |
33 | 32 | nnzd 12306 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ∈ ℤ) |
34 | 33 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℤ) |
35 | 34 | zcnd 12308 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℂ) |
36 | 35, 25 | mulcomd 10879 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) · 𝐴) = (𝐴 · (𝐴↑𝑛))) |
37 | 28, 36 | eqtrd 2778 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) = (𝐴 · (𝐴↑𝑛))) |
38 | 37 | oveq2d 7248 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴 · (𝐴↑𝑛)))) |
39 | | simpl2 1194 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ) |
40 | 32 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑛) ∈ ℕ) |
41 | | nnz 12224 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℤ) |
42 | 41 | 3ad2ant1 1135 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐴 ∈
ℤ) |
43 | | nnz 12224 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℤ) |
44 | 43 | 3ad2ant2 1136 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐵 ∈
ℤ) |
45 | 42, 44 | gcdcomd 16101 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴)) |
46 | 45 | eqeq1d 2740 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 ↔ (𝐵 gcd 𝐴) = 1)) |
47 | 46 | biimpa 480 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd 𝐴) = 1) |
48 | | rpmulgcd 16146 |
. . . . . . . . . . . . . 14
⊢ (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ (𝐴↑𝑛) ∈ ℕ) ∧ (𝐵 gcd 𝐴) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛))) |
49 | 39, 24, 40, 47, 48 | syl31anc 1375 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴 · (𝐴↑𝑛))) = (𝐵 gcd (𝐴↑𝑛))) |
50 | 38, 49 | eqtrd 2778 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐵 gcd (𝐴↑(𝑛 + 1))) = (𝐵 gcd (𝐴↑𝑛))) |
51 | | peano2nn 11867 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) |
52 | 51 | 3ad2ant3 1137 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈
ℕ) |
53 | 52 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈ ℕ) |
54 | 53 | nnnn0d 12175 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝑛 + 1) ∈
ℕ0) |
55 | 24, 54 | nnexpcld 13840 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℕ) |
56 | 55 | nnzd 12306 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑(𝑛 + 1)) ∈ ℤ) |
57 | 44 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℤ) |
58 | 56, 57 | gcdcomd 16101 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = (𝐵 gcd (𝐴↑(𝑛 + 1)))) |
59 | 34, 57 | gcdcomd 16101 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = (𝐵 gcd (𝐴↑𝑛))) |
60 | 50, 58, 59 | 3eqtr4d 2788 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = ((𝐴↑𝑛) gcd 𝐵)) |
61 | 60 | eqeq1d 2740 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1 ↔ ((𝐴↑𝑛) gcd 𝐵) = 1)) |
62 | 61 | biimprd 251 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)) |
63 | 23, 62 | sylanbr 585 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)) |
64 | 63 | an32s 652 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑛 ∈ ℕ) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1)) |
65 | 64 | expcom 417 |
. . . . . 6
⊢ (𝑛 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → (((𝐴↑𝑛) gcd 𝐵) = 1 → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))) |
66 | 65 | a2d 29 |
. . . . 5
⊢ (𝑛 ∈ ℕ → ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑛) gcd 𝐵) = 1) → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑(𝑛 + 1)) gcd 𝐵) = 1))) |
67 | 4, 8, 12, 16, 22, 66 | nnind 11873 |
. . . 4
⊢ (𝑁 ∈ ℕ → (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd 𝐵) = 1)) |
68 | 67 | expd 419 |
. . 3
⊢ (𝑁 ∈ ℕ → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1))) |
69 | 68 | com12 32 |
. 2
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1))) |
70 | 69 | 3impia 1119 |
1
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd 𝐵) = 1)) |