| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nnz 12636 | . . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) | 
| 2 |  | gcddvds 16541 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛)) | 
| 3 | 2 | simpld 494 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧) | 
| 4 | 1, 3 | sylan2 593 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧) | 
| 5 |  | gcdcl 16544 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈
ℕ0) | 
| 6 | 1, 5 | sylan2 593 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈
ℕ0) | 
| 7 | 6 | nn0zd 12641 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ) | 
| 8 |  | simpl 482 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈
ℤ) | 
| 9 | 1 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℤ) | 
| 10 |  | nnne0 12301 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | 
| 11 | 10 | neneqd 2944 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → ¬
𝑛 = 0) | 
| 12 | 11 | intnand 488 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ¬
(𝑧 = 0 ∧ 𝑛 = 0)) | 
| 13 | 12 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬
(𝑧 = 0 ∧ 𝑛 = 0)) | 
| 14 |  | gcdn0cl 16540 | . . . . . . . . . . . 12
⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬
(𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ) | 
| 15 | 8, 9, 13, 14 | syl21anc 837 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ) | 
| 16 |  | nnne0 12301 | . . . . . . . . . . 11
⊢ ((𝑧 gcd 𝑛) ∈ ℕ → (𝑧 gcd 𝑛) ≠ 0) | 
| 17 | 15, 16 | syl 17 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0) | 
| 18 |  | dvdsval2 16294 | . . . . . . . . . 10
⊢ (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)) | 
| 19 | 7, 17, 8, 18 | syl3anc 1372 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)) | 
| 20 | 4, 19 | mpbid 232 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ) | 
| 21 | 20 | 3adant3 1132 | . . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ) | 
| 22 | 2 | simprd 495 | . . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛) | 
| 23 | 1, 22 | sylan2 593 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛) | 
| 24 |  | dvdsval2 16294 | . . . . . . . . . . . 12
⊢ (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)) | 
| 25 | 7, 17, 9, 24 | syl3anc 1372 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)) | 
| 26 | 23, 25 | mpbid 232 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ) | 
| 27 |  | nnre 12274 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) | 
| 28 | 27 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℝ) | 
| 29 | 6 | nn0red 12590 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ) | 
| 30 |  | nngt0 12298 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 0 <
𝑛) | 
| 31 | 30 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 <
𝑛) | 
| 32 |  | nngt0 12298 | . . . . . . . . . . . 12
⊢ ((𝑧 gcd 𝑛) ∈ ℕ → 0 < (𝑧 gcd 𝑛)) | 
| 33 | 15, 32 | syl 17 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 <
(𝑧 gcd 𝑛)) | 
| 34 | 28, 29, 31, 33 | divgt0d 12204 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 <
(𝑛 / (𝑧 gcd 𝑛))) | 
| 35 | 26, 34 | jca 511 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛)))) | 
| 36 | 35 | 3adant3 1132 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛)))) | 
| 37 |  | elnnz 12625 | . . . . . . . 8
⊢ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛)))) | 
| 38 | 36, 37 | sylibr 234 | . . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ) | 
| 39 | 21, 38 | opelxpd 5723 | . . . . . 6
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 ∈ (ℤ ×
ℕ)) | 
| 40 | 20, 26 | gcdcld 16546 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈
ℕ0) | 
| 41 | 40 | nn0cnd 12591 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ) | 
| 42 |  | 1cnd 11257 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈
ℂ) | 
| 43 | 6 | nn0cnd 12591 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ) | 
| 44 | 43 | mulridd 11279 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛)) | 
| 45 |  | zcn 12620 | . . . . . . . . . . . 12
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℂ) | 
| 46 | 45 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈
ℂ) | 
| 47 | 46, 43, 17 | divcan2d 12046 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧) | 
| 48 |  | nncn 12275 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) | 
| 49 | 48 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℂ) | 
| 50 | 49, 43, 17 | divcan2d 12046 | . . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛) | 
| 51 | 47, 50 | oveq12d 7450 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = (𝑧 gcd 𝑛)) | 
| 52 |  | mulgcd 16586 | . . . . . . . . . 10
⊢ (((𝑧 gcd 𝑛) ∈ ℕ0 ∧ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))) | 
| 53 | 6, 20, 26, 52 | syl3anc 1372 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))) | 
| 54 | 44, 51, 53 | 3eqtr2rd 2783 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · 1)) | 
| 55 | 41, 42, 43, 17, 54 | mulcanad 11899 | . . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1) | 
| 56 | 55 | 3adant3 1132 | . . . . . 6
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1) | 
| 57 | 10 | adantl 481 | . . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0) | 
| 58 | 46, 49, 43, 57, 17 | divcan7d 12072 | . . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛)) | 
| 59 | 58 | eqeq2d 2747 | . . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛))) | 
| 60 | 59 | biimp3ar 1471 | . . . . . 6
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))) | 
| 61 |  | ovex 7465 | . . . . . . . . . . 11
⊢ (𝑧 / (𝑧 gcd 𝑛)) ∈ V | 
| 62 |  | ovex 7465 | . . . . . . . . . . 11
⊢ (𝑛 / (𝑧 gcd 𝑛)) ∈ V | 
| 63 | 61, 62 | op1std 8025 | . . . . . . . . . 10
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (1st ‘𝑥) = (𝑧 / (𝑧 gcd 𝑛))) | 
| 64 | 61, 62 | op2ndd 8026 | . . . . . . . . . 10
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (2nd ‘𝑥) = (𝑛 / (𝑧 gcd 𝑛))) | 
| 65 | 63, 64 | oveq12d 7450 | . . . . . . . . 9
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → ((1st ‘𝑥) gcd (2nd
‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) | 
| 66 | 65 | eqeq1d 2738 | . . . . . . . 8
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ↔ ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)) | 
| 67 | 63, 64 | oveq12d 7450 | . . . . . . . . 9
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → ((1st ‘𝑥) / (2nd ‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))) | 
| 68 | 67 | eqeq2d 2747 | . . . . . . . 8
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))) | 
| 69 | 66, 68 | anbi12d 632 | . . . . . . 7
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ↔ (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))) | 
| 70 | 69 | rspcev 3621 | . . . . . 6
⊢
((〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 ∈ (ℤ × ℕ)
∧ (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))) → ∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | 
| 71 | 39, 56, 60, 70 | syl12anc 836 | . . . . 5
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | 
| 72 |  | elxp6 8049 | . . . . . . 7
⊢ (𝑥 ∈ (ℤ ×
ℕ) ↔ (𝑥 =
〈(1st ‘𝑥), (2nd ‘𝑥)〉 ∧ ((1st ‘𝑥) ∈ ℤ ∧
(2nd ‘𝑥)
∈ ℕ))) | 
| 73 |  | elxp6 8049 | . . . . . . 7
⊢ (𝑦 ∈ (ℤ ×
ℕ) ↔ (𝑦 =
〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st ‘𝑦) ∈ ℤ ∧
(2nd ‘𝑦)
∈ ℕ))) | 
| 74 |  | simprl 770 | . . . . . . . . . . . 12
⊢ ((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st
‘𝑥) ∈
ℤ) | 
| 75 | 74 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → (1st
‘𝑥) ∈
ℤ) | 
| 76 |  | simprr 772 | . . . . . . . . . . . 12
⊢ ((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd
‘𝑥) ∈
ℕ) | 
| 77 | 76 | ad2antrr 726 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → (2nd
‘𝑥) ∈
ℕ) | 
| 78 |  | simprll 778 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → ((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1) | 
| 79 |  | simprl 770 | . . . . . . . . . . . 12
⊢ ((𝑦 = 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∧
((1st ‘𝑦)
∈ ℤ ∧ (2nd ‘𝑦) ∈ ℕ)) → (1st
‘𝑦) ∈
ℤ) | 
| 80 | 79 | ad2antlr 727 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → (1st
‘𝑦) ∈
ℤ) | 
| 81 |  | simprr 772 | . . . . . . . . . . . 12
⊢ ((𝑦 = 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∧
((1st ‘𝑦)
∈ ℤ ∧ (2nd ‘𝑦) ∈ ℕ)) → (2nd
‘𝑦) ∈
ℕ) | 
| 82 | 81 | ad2antlr 727 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → (2nd
‘𝑦) ∈
ℕ) | 
| 83 |  | simprrl 780 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → ((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1) | 
| 84 |  | simprlr 779 | . . . . . . . . . . . 12
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) | 
| 85 |  | simprrr 781 | . . . . . . . . . . . 12
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))) | 
| 86 | 84, 85 | eqtr3d 2778 | . . . . . . . . . . 11
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → ((1st
‘𝑥) / (2nd
‘𝑥)) =
((1st ‘𝑦)
/ (2nd ‘𝑦))) | 
| 87 |  | qredeq 16695 | . . . . . . . . . . 11
⊢
((((1st ‘𝑥) ∈ ℤ ∧ (2nd
‘𝑥) ∈ ℕ
∧ ((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1) ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ ∧ ((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1) ∧ ((1st ‘𝑥) / (2nd ‘𝑥)) = ((1st ‘𝑦) / (2nd ‘𝑦))) → ((1st
‘𝑥) = (1st
‘𝑦) ∧
(2nd ‘𝑥) =
(2nd ‘𝑦))) | 
| 88 | 75, 77, 78, 80, 82, 83, 86, 87 | syl331anc 1396 | . . . . . . . . . 10
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → ((1st
‘𝑥) = (1st
‘𝑦) ∧
(2nd ‘𝑥) =
(2nd ‘𝑦))) | 
| 89 |  | fvex 6918 | . . . . . . . . . . 11
⊢
(1st ‘𝑥) ∈ V | 
| 90 |  | fvex 6918 | . . . . . . . . . . 11
⊢
(2nd ‘𝑥) ∈ V | 
| 91 | 89, 90 | opth 5480 | . . . . . . . . . 10
⊢
(〈(1st ‘𝑥), (2nd ‘𝑥)〉 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ↔ ((1st
‘𝑥) = (1st
‘𝑦) ∧
(2nd ‘𝑥) =
(2nd ‘𝑦))) | 
| 92 | 88, 91 | sylibr 234 | . . . . . . . . 9
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) →
〈(1st ‘𝑥), (2nd ‘𝑥)〉 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | 
| 93 |  | simplll 774 | . . . . . . . . 9
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) | 
| 94 |  | simplrl 776 | . . . . . . . . 9
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → 𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉) | 
| 95 | 92, 93, 94 | 3eqtr4d 2786 | . . . . . . . 8
⊢ ((((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) ∧ ((((1st
‘𝑥) gcd
(2nd ‘𝑥))
= 1 ∧ 𝐴 =
((1st ‘𝑥)
/ (2nd ‘𝑥))) ∧ (((1st ‘𝑦) gcd (2nd
‘𝑦)) = 1 ∧ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦))))) → 𝑥 = 𝑦) | 
| 96 | 95 | ex 412 | . . . . . . 7
⊢ (((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st
‘𝑦) ∈ ℤ
∧ (2nd ‘𝑦) ∈ ℕ))) →
(((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦)) | 
| 97 | 72, 73, 96 | syl2anb 598 | . . . . . 6
⊢ ((𝑥 ∈ (ℤ ×
ℕ) ∧ 𝑦 ∈
(ℤ × ℕ)) → (((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦)) | 
| 98 | 97 | rgen2 3198 | . . . . 5
⊢
∀𝑥 ∈
(ℤ × ℕ)∀𝑦 ∈ (ℤ ×
ℕ)(((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦) | 
| 99 | 71, 98 | jctir 520 | . . . 4
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ ×
ℕ)∀𝑦 ∈
(ℤ × ℕ)(((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦))) | 
| 100 | 99 | 3expia 1121 | . . 3
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = (𝑧 / 𝑛) → (∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ ×
ℕ)∀𝑦 ∈
(ℤ × ℕ)(((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦)))) | 
| 101 | 100 | rexlimivv 3200 | . 2
⊢
(∃𝑧 ∈
ℤ ∃𝑛 ∈
ℕ 𝐴 = (𝑧 / 𝑛) → (∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ ×
ℕ)∀𝑦 ∈
(ℤ × ℕ)(((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦))) | 
| 102 |  | elq 12993 | . 2
⊢ (𝐴 ∈ ℚ ↔
∃𝑧 ∈ ℤ
∃𝑛 ∈ ℕ
𝐴 = (𝑧 / 𝑛)) | 
| 103 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦)) | 
| 104 |  | fveq2 6905 | . . . . . 6
⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦)) | 
| 105 | 103, 104 | oveq12d 7450 | . . . . 5
⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) gcd (2nd
‘𝑥)) =
((1st ‘𝑦)
gcd (2nd ‘𝑦))) | 
| 106 | 105 | eqeq1d 2738 | . . . 4
⊢ (𝑥 = 𝑦 → (((1st ‘𝑥) gcd (2nd
‘𝑥)) = 1 ↔
((1st ‘𝑦)
gcd (2nd ‘𝑦)) = 1)) | 
| 107 | 103, 104 | oveq12d 7450 | . . . . 5
⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) / (2nd ‘𝑥)) = ((1st
‘𝑦) / (2nd
‘𝑦))) | 
| 108 | 107 | eqeq2d 2747 | . . . 4
⊢ (𝑥 = 𝑦 → (𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦)))) | 
| 109 | 106, 108 | anbi12d 632 | . . 3
⊢ (𝑥 = 𝑦 → ((((1st ‘𝑥) gcd (2nd
‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ↔ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦))))) | 
| 110 | 109 | reu4 3736 | . 2
⊢
(∃!𝑥 ∈
(ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ↔ (∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ ×
ℕ)∀𝑦 ∈
(ℤ × ℕ)(((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))) ∧ (((1st
‘𝑦) gcd
(2nd ‘𝑦))
= 1 ∧ 𝐴 =
((1st ‘𝑦)
/ (2nd ‘𝑦)))) → 𝑥 = 𝑦))) | 
| 111 | 101, 102,
110 | 3imtr4i 292 | 1
⊢ (𝐴 ∈ ℚ →
∃!𝑥 ∈ (ℤ
× ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) |