Proof of Theorem nn0rppwr
Step | Hyp | Ref
| Expression |
1 | | elnn0 12092 |
. 2
⊢ (𝑁 ∈ ℕ0
↔ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
2 | | elnn0 12092 |
. . . . 5
⊢ (𝐴 ∈ ℕ0
↔ (𝐴 ∈ ℕ
∨ 𝐴 =
0)) |
3 | | elnn0 12092 |
. . . . 5
⊢ (𝐵 ∈ ℕ0
↔ (𝐵 ∈ ℕ
∨ 𝐵 =
0)) |
4 | | rppwr 16121 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)) |
5 | 4 | 3expia 1123 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
6 | | simp1l 1199 |
. . . . . . . . . . 11
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0) |
7 | 6 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (0↑𝑁)) |
8 | | 0exp 13670 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ →
(0↑𝑁) =
0) |
9 | 8 | 3ad2ant2 1136 |
. . . . . . . . . 10
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0) |
10 | 7, 9 | eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 0) |
11 | 6 | oveq1d 7228 |
. . . . . . . . . . . 12
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵)) |
12 | | simp3 1140 |
. . . . . . . . . . . 12
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1) |
13 | | simp1r 1200 |
. . . . . . . . . . . . 13
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ) |
14 | | nnz 12199 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℤ) |
15 | | gcd0id 16078 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℤ → (0 gcd
𝐵) = (abs‘𝐵)) |
16 | 14, 15 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈ ℕ → (0 gcd
𝐵) = (abs‘𝐵)) |
17 | | nnre 11837 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℝ) |
18 | | 0red 10836 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ ℕ → 0 ∈
ℝ) |
19 | | nngt0 11861 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ ℕ → 0 <
𝐵) |
20 | 18, 17, 19 | ltled 10980 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 0 ≤
𝐵) |
21 | 17, 20 | absidd 14986 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈ ℕ →
(abs‘𝐵) = 𝐵) |
22 | 16, 21 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℕ → (0 gcd
𝐵) = 𝐵) |
23 | 13, 22 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0 gcd 𝐵) = 𝐵) |
24 | 11, 12, 23 | 3eqtr3rd 2786 |
. . . . . . . . . . 11
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 1) |
25 | 24 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (1↑𝑁)) |
26 | | nnz 12199 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℤ) |
27 | 26 | 3ad2ant2 1136 |
. . . . . . . . . . 11
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ) |
28 | | 1exp 13664 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℤ →
(1↑𝑁) =
1) |
29 | 27, 28 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1) |
30 | 25, 29 | eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 1) |
31 | 10, 30 | oveq12d 7231 |
. . . . . . . 8
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd 1)) |
32 | | 1z 12207 |
. . . . . . . . . 10
⊢ 1 ∈
ℤ |
33 | | gcd0id 16078 |
. . . . . . . . . 10
⊢ (1 ∈
ℤ → (0 gcd 1) = (abs‘1)) |
34 | 32, 33 | ax-mp 5 |
. . . . . . . . 9
⊢ (0 gcd 1)
= (abs‘1) |
35 | | abs1 14861 |
. . . . . . . . 9
⊢
(abs‘1) = 1 |
36 | 34, 35 | eqtri 2765 |
. . . . . . . 8
⊢ (0 gcd 1)
= 1 |
37 | 31, 36 | eqtrdi 2794 |
. . . . . . 7
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1) |
38 | 37 | 3exp 1121 |
. . . . . 6
⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
39 | | simp1r 1200 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0) |
40 | 39 | oveq2d 7229 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0)) |
41 | | simp3 1140 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1) |
42 | | simp1l 1199 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ) |
43 | 42 | nnnn0d 12150 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈
ℕ0) |
44 | | nn0gcdid0 16080 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℕ0
→ (𝐴 gcd 0) = 𝐴) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 0) = 𝐴) |
46 | 40, 41, 45 | 3eqtr3rd 2786 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 1) |
47 | 46 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (1↑𝑁)) |
48 | 26 | 3ad2ant2 1136 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ) |
49 | 48, 28 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1↑𝑁) = 1) |
50 | 47, 49 | eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = 1) |
51 | 39 | oveq1d 7228 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (0↑𝑁)) |
52 | 8 | 3ad2ant2 1136 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (0↑𝑁) = 0) |
53 | 51, 52 | eqtrd 2777 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = 0) |
54 | 50, 53 | oveq12d 7231 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 0)) |
55 | | 1nn0 12106 |
. . . . . . . . 9
⊢ 1 ∈
ℕ0 |
56 | | nn0gcdid0 16080 |
. . . . . . . . 9
⊢ (1 ∈
ℕ0 → (1 gcd 0) = 1) |
57 | 55, 56 | mp1i 13 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (1 gcd 0) =
1) |
58 | 54, 57 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1) |
59 | 58 | 3exp 1121 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
60 | | oveq12 7222 |
. . . . . . . . . 10
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0)) |
61 | | gcd0val 16056 |
. . . . . . . . . . . 12
⊢ (0 gcd 0)
= 0 |
62 | | 0ne1 11901 |
. . . . . . . . . . . 12
⊢ 0 ≠
1 |
63 | 61, 62 | eqnetri 3011 |
. . . . . . . . . . 11
⊢ (0 gcd 0)
≠ 1 |
64 | 63 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (0 gcd 0) ≠
1) |
65 | 60, 64 | eqnetrd 3008 |
. . . . . . . . 9
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) ≠ 1) |
66 | 65 | neneqd 2945 |
. . . . . . . 8
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ¬ (𝐴 gcd 𝐵) = 1) |
67 | 66 | pm2.21d 121 |
. . . . . . 7
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)) |
68 | 67 | a1d 25 |
. . . . . 6
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
69 | 5, 38, 59, 68 | ccase 1038 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
70 | 2, 3, 69 | syl2anb 601 |
. . . 4
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
71 | | oveq2 7221 |
. . . . . . . . . 10
⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) |
72 | 71 | 3ad2ant3 1137 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐴↑𝑁) = (𝐴↑0)) |
73 | | nn0cn 12100 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℂ) |
74 | 73 | 3ad2ant1 1135 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → 𝐴 ∈
ℂ) |
75 | 74 | exp0d 13710 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐴↑0) =
1) |
76 | 72, 75 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐴↑𝑁) = 1) |
77 | | oveq2 7221 |
. . . . . . . . . 10
⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0)) |
78 | 77 | 3ad2ant3 1137 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐵↑𝑁) = (𝐵↑0)) |
79 | | nn0cn 12100 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ ℕ0
→ 𝐵 ∈
ℂ) |
80 | 79 | 3ad2ant2 1136 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → 𝐵 ∈
ℂ) |
81 | 80 | exp0d 13710 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐵↑0) =
1) |
82 | 78, 81 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐵↑𝑁) = 1) |
83 | 76, 82 | oveq12d 7231 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1)) |
84 | | 1gcd 16093 |
. . . . . . . 8
⊢ (1 ∈
ℤ → (1 gcd 1) = 1) |
85 | 32, 84 | mp1i 13 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (1 gcd 1) = 1) |
86 | 83, 85 | eqtrd 2777 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1) |
87 | 86 | 3expia 1123 |
. . . . 5
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (𝑁 = 0 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)) |
88 | 87 | a1dd 50 |
. . . 4
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (𝑁 = 0 → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
89 | 70, 88 | jaod 859 |
. . 3
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) |
90 | 89 | 3impia 1119 |
. 2
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)) |
91 | 1, 90 | syl3an3b 1407 |
1
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)) |