Proof of Theorem nn0rppwr
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elnn0 12528 | . 2
⊢ (𝑁 ∈ ℕ0
↔ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) | 
| 2 |  | elnn0 12528 | . . . . 5
⊢ (𝐴 ∈ ℕ0
↔ (𝐴 ∈ ℕ
∨ 𝐴 =
0)) | 
| 3 |  | elnn0 12528 | . . . . 5
⊢ (𝐵 ∈ ℕ0
↔ (𝐵 ∈ ℕ
∨ 𝐵 =
0)) | 
| 4 |  | rppwr 16597 | . . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1)) | 
| 5 | 4 | 3expia 1122 | . . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) | 
| 6 |  | simp1l 1198 | . . . . . . . . . . 11
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 = 0) | 
| 7 | 6 | oveq1d 7446 | . . . . . . . . . 10
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (0↑𝑁)) | 
| 8 |  | 0exp 14138 | . . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ →
(0↑𝑁) =
0) | 
| 9 | 8 | 3ad2ant2 1135 | . . . . . . . . . 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 7446 | . . . . . . . . . . . 12
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (0 gcd 𝐵)) | 
| 12 |  | simp3 1139 | . . . . . . . . . . . 12
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1) | 
| 13 |  | simp1r 1199 | . . . . . . . . . . . . 13
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 ∈ ℕ) | 
| 14 |  | nnz 12634 | . . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℤ) | 
| 15 |  | gcd0id 16556 | . . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℤ → (0 gcd
𝐵) = (abs‘𝐵)) | 
| 16 | 14, 15 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝐵 ∈ ℕ → (0 gcd
𝐵) = (abs‘𝐵)) | 
| 17 |  | nnre 12273 | . . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℝ) | 
| 18 |  | 0red 11264 | . . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ ℕ → 0 ∈
ℝ) | 
| 19 |  | nngt0 12297 | . . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ ℕ → 0 <
𝐵) | 
| 20 | 18, 17, 19 | ltled 11409 | . . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ ℕ → 0 ≤
𝐵) | 
| 21 | 17, 20 | absidd 15461 | . . . . . . . . . . . . . 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 7446 | . . . . . . . . . 10
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (1↑𝑁)) | 
| 26 |  | nnz 12634 | . . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℤ) | 
| 27 | 26 | 3ad2ant2 1135 | . . . . . . . . . . 11
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝑁 ∈ ℤ) | 
| 28 |  | 1exp 14132 | . . . . . . . . . . 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 7449 | . . . . . . . 8
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (0 gcd 1)) | 
| 32 |  | 1z 12647 | . . . . . . . . . 10
⊢ 1 ∈
ℤ | 
| 33 |  | gcd0id 16556 | . . . . . . . . . 10
⊢ (1 ∈
ℤ → (0 gcd 1) = (abs‘1)) | 
| 34 | 32, 33 | ax-mp 5 | . . . . . . . . 9
⊢ (0 gcd 1)
= (abs‘1) | 
| 35 |  | abs1 15336 | . . . . . . . . 9
⊢
(abs‘1) = 1 | 
| 36 | 34, 35 | eqtri 2765 | . . . . . . . 8
⊢ (0 gcd 1)
= 1 | 
| 37 | 31, 36 | eqtrdi 2793 | . . . . . . 7
⊢ (((𝐴 = 0 ∧ 𝐵 ∈ ℕ) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1) | 
| 38 | 37 | 3exp 1120 | . . . . . 6
⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) | 
| 39 |  | simp1r 1199 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐵 = 0) | 
| 40 | 39 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = (𝐴 gcd 0)) | 
| 41 |  | simp3 1139 | . . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 gcd 𝐵) = 1) | 
| 42 |  | simp1l 1198 | . . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈ ℕ) | 
| 43 | 42 | nnnn0d 12587 | . . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → 𝐴 ∈
ℕ0) | 
| 44 |  | nn0gcdid0 16558 | . . . . . . . . . . . . 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 7446 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴↑𝑁) = (1↑𝑁)) | 
| 48 | 26 | 3ad2ant2 1135 | . . . . . . . . . . 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 7446 | . . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐵↑𝑁) = (0↑𝑁)) | 
| 52 | 8 | 3ad2ant2 1135 | . . . . . . . . . 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 7449 | . . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 = 0) ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 0)) | 
| 55 |  | 1nn0 12542 | . . . . . . . . 9
⊢ 1 ∈
ℕ0 | 
| 56 |  | nn0gcdid0 16558 | . . . . . . . . 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 1120 | . . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) | 
| 60 |  | oveq12 7440 | . . . . . . . . . 10
⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0)) | 
| 61 |  | gcd0val 16534 | . . . . . . . . . . . 12
⊢ (0 gcd 0)
= 0 | 
| 62 |  | 0ne1 12337 | . . . . . . . . . . . 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 598 | . . . 4
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (𝑁 ∈ ℕ → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) | 
| 71 |  | oveq2 7439 | . . . . . . . . . 10
⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) | 
| 72 | 71 | 3ad2ant3 1136 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐴↑𝑁) = (𝐴↑0)) | 
| 73 |  | nn0cn 12536 | . . . . . . . . . . 11
⊢ (𝐴 ∈ ℕ0
→ 𝐴 ∈
ℂ) | 
| 74 | 73 | 3ad2ant1 1134 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → 𝐴 ∈
ℂ) | 
| 75 | 74 | exp0d 14180 | . . . . . . . . 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 7439 | . . . . . . . . . 10
⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0)) | 
| 78 | 77 | 3ad2ant3 1136 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → (𝐵↑𝑁) = (𝐵↑0)) | 
| 79 |  | nn0cn 12536 | . . . . . . . . . . 11
⊢ (𝐵 ∈ ℕ0
→ 𝐵 ∈
ℂ) | 
| 80 | 79 | 3ad2ant2 1135 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → 𝐵 ∈
ℂ) | 
| 81 | 80 | exp0d 14180 | . . . . . . . . 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 7449 | . . . . . . 7
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0 ∧ 𝑁
= 0) → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = (1 gcd 1)) | 
| 84 |  | 1gcd 16570 | . . . . . . . 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 1122 | . . . . 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 860 | . . 3
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))) | 
| 90 | 89 | 3impia 1118 | . 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)) |