Step | Hyp | Ref
| Expression |
1 | | nnz 11276 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
2 | | gcddvds 15063 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛)) |
3 | 2 | simpld 474 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧) |
4 | 1, 3 | sylan2 490 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧) |
5 | | gcdcl 15066 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈
ℕ0) |
6 | 1, 5 | sylan2 490 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈
ℕ0) |
7 | 6 | nn0zd 11356 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ) |
8 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈
ℤ) |
9 | 1 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℤ) |
10 | | nnne0 10930 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) |
11 | 10 | neneqd 2787 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → ¬
𝑛 = 0) |
12 | 11 | intnand 953 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ¬
(𝑧 = 0 ∧ 𝑛 = 0)) |
13 | 12 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬
(𝑧 = 0 ∧ 𝑛 = 0)) |
14 | | gcdn0cl 15062 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬
(𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ) |
15 | 8, 9, 13, 14 | syl21anc 1317 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ) |
16 | | nnne0 10930 |
. . . . . . . . . . 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 14824 |
. . . . . . . . . 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 1318 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)) |
20 | 4, 19 | mpbid 221 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ) |
21 | 20 | 3adant3 1074 |
. . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ) |
22 | 2 | simprd 478 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛) |
23 | 1, 22 | sylan2 490 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛) |
24 | | dvdsval2 14824 |
. . . . . . . . . . . 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 1318 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)) |
26 | 23, 25 | mpbid 221 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ) |
27 | | nnre 10904 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
28 | 27 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℝ) |
29 | 6 | nn0red 11229 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ) |
30 | | nngt0 10926 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 0 <
𝑛) |
31 | 30 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 <
𝑛) |
32 | | nngt0 10926 |
. . . . . . . . . . . 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 10838 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 <
(𝑛 / (𝑧 gcd 𝑛))) |
35 | 26, 34 | jca 553 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛)))) |
36 | 35 | 3adant3 1074 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛)))) |
37 | | elnnz 11264 |
. . . . . . . 8
⊢ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛)))) |
38 | 36, 37 | sylibr 223 |
. . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ) |
39 | | opelxpi 5072 |
. . . . . . 7
⊢ (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ) → 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 ∈ (ℤ ×
ℕ)) |
40 | 21, 38, 39 | syl2anc 691 |
. . . . . 6
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 ∈ (ℤ ×
ℕ)) |
41 | 20, 26 | gcdcld 15068 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈
ℕ0) |
42 | 41 | nn0cnd 11230 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ) |
43 | | 1cnd 9935 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈
ℂ) |
44 | 6 | nn0cnd 11230 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ) |
45 | 44 | mulid1d 9936 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛)) |
46 | | zcn 11259 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℂ) |
47 | 46 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈
ℂ) |
48 | 47, 44, 17 | divcan2d 10682 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧) |
49 | | nncn 10905 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
50 | 49 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
ℂ) |
51 | 50, 44, 17 | divcan2d 10682 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛) |
52 | 48, 51 | oveq12d 6567 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = (𝑧 gcd 𝑛)) |
53 | | mulgcd 15103 |
. . . . . . . . . 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))))) |
54 | 6, 20, 26, 53 | syl3anc 1318 |
. . . . . . . . 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))))) |
55 | 45, 52, 54 | 3eqtr2rd 2651 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · 1)) |
56 | 42, 43, 44, 17, 55 | mulcanad 10541 |
. . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1) |
57 | 56 | 3adant3 1074 |
. . . . . 6
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1) |
58 | 10 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0) |
59 | 47, 50, 44, 58, 17 | divcan7d 10708 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛)) |
60 | 59 | eqeq2d 2620 |
. . . . . . 7
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛))) |
61 | 60 | biimp3ar 1425 |
. . . . . 6
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))) |
62 | | ovex 6577 |
. . . . . . . . . . 11
⊢ (𝑧 / (𝑧 gcd 𝑛)) ∈ V |
63 | | ovex 6577 |
. . . . . . . . . . 11
⊢ (𝑛 / (𝑧 gcd 𝑛)) ∈ V |
64 | 62, 63 | op1std 7069 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (1st ‘𝑥) = (𝑧 / (𝑧 gcd 𝑛))) |
65 | 62, 63 | op2ndd 7070 |
. . . . . . . . . 10
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (2nd ‘𝑥) = (𝑛 / (𝑧 gcd 𝑛))) |
66 | 64, 65 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → ((1st ‘𝑥) gcd (2nd
‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) |
67 | 66 | eqeq1d 2612 |
. . . . . . . 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)) |
68 | 64, 65 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → ((1st ‘𝑥) / (2nd ‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))) |
69 | 68 | eqeq2d 2620 |
. . . . . . . 8
⊢ (𝑥 = 〈(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))〉 → (𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))) |
70 | 67, 69 | anbi12d 743 |
. . . . . . 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)))))) |
71 | 70 | rspcev 3282 |
. . . . . 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)))) |
72 | 40, 57, 61, 71 | syl12anc 1316 |
. . . . 5
⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ ×
ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) |
73 | | elxp6 7091 |
. . . . . . 7
⊢ (𝑥 ∈ (ℤ ×
ℕ) ↔ (𝑥 =
〈(1st ‘𝑥), (2nd ‘𝑥)〉 ∧ ((1st ‘𝑥) ∈ ℤ ∧
(2nd ‘𝑥)
∈ ℕ))) |
74 | | elxp6 7091 |
. . . . . . 7
⊢ (𝑦 ∈ (ℤ ×
ℕ) ↔ (𝑦 =
〈(1st ‘𝑦), (2nd ‘𝑦)〉 ∧ ((1st ‘𝑦) ∈ ℤ ∧
(2nd ‘𝑦)
∈ ℕ))) |
75 | | simprl 790 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) → (1st
‘𝑥) ∈
ℤ) |
76 | 75 | ad2antrr 758 |
. . . . . . . . . . 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) |
77 | | simprr 792 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 〈(1st
‘𝑥), (2nd
‘𝑥)〉 ∧
((1st ‘𝑥)
∈ ℤ ∧ (2nd ‘𝑥) ∈ ℕ)) → (2nd
‘𝑥) ∈
ℕ) |
78 | 77 | ad2antrr 758 |
. . . . . . . . . . 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) |
79 | | simprll 798 |
. . . . . . . . . . 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) |
80 | | simprl 790 |
. . . . . . . . . . . 12
⊢ ((𝑦 = 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∧
((1st ‘𝑦)
∈ ℤ ∧ (2nd ‘𝑦) ∈ ℕ)) → (1st
‘𝑦) ∈
ℤ) |
81 | 80 | ad2antlr 759 |
. . . . . . . . . . 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) |
82 | | simprr 792 |
. . . . . . . . . . . 12
⊢ ((𝑦 = 〈(1st
‘𝑦), (2nd
‘𝑦)〉 ∧
((1st ‘𝑦)
∈ ℤ ∧ (2nd ‘𝑦) ∈ ℕ)) → (2nd
‘𝑦) ∈
ℕ) |
83 | 82 | ad2antlr 759 |
. . . . . . . . . . 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) |
84 | | simprrl 800 |
. . . . . . . . . . 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) |
85 | | simprlr 799 |
. . . . . . . . . . . 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))) |
86 | | simprrr 801 |
. . . . . . . . . . . 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))) |
87 | 85, 86 | eqtr3d 2646 |
. . . . . . . . . . 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))) |
88 | | qredeq 15209 |
. . . . . . . . . . 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))) |
89 | 76, 78, 79, 81, 83, 84, 87, 88 | syl331anc 1343 |
. . . . . . . . . 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))) |
90 | | fvex 6113 |
. . . . . . . . . . 11
⊢
(1st ‘𝑥) ∈ V |
91 | | fvex 6113 |
. . . . . . . . . . 11
⊢
(2nd ‘𝑥) ∈ V |
92 | 90, 91 | opth 4871 |
. . . . . . . . . 10
⊢
(〈(1st ‘𝑥), (2nd ‘𝑥)〉 = 〈(1st ‘𝑦), (2nd ‘𝑦)〉 ↔ ((1st
‘𝑥) = (1st
‘𝑦) ∧
(2nd ‘𝑥) =
(2nd ‘𝑦))) |
93 | 89, 92 | sylibr 223 |
. . . . . . . . 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)〉) |
94 | | simplll 794 |
. . . . . . . . 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)〉) |
95 | | simplrl 796 |
. . . . . . . . 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)〉) |
96 | 93, 94, 95 | 3eqtr4d 2654 |
. . . . . . . 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) |
97 | 96 | ex 449 |
. . . . . . 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)) |
98 | 73, 74, 97 | syl2anb 495 |
. . . . . 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)) |
99 | 98 | rgen2a 2960 |
. . . . 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) |
100 | 72, 99 | jctir 559 |
. . . 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))) |
101 | 100 | 3expia 1259 |
. . 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)))) |
102 | 101 | rexlimivv 3018 |
. 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))) |
103 | | elq 11666 |
. 2
⊢ (𝐴 ∈ ℚ ↔
∃𝑧 ∈ ℤ
∃𝑛 ∈ ℕ
𝐴 = (𝑧 / 𝑛)) |
104 | | fveq2 6103 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (1st ‘𝑥) = (1st ‘𝑦)) |
105 | | fveq2 6103 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (2nd ‘𝑥) = (2nd ‘𝑦)) |
106 | 104, 105 | oveq12d 6567 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) gcd (2nd
‘𝑥)) =
((1st ‘𝑦)
gcd (2nd ‘𝑦))) |
107 | 106 | eqeq1d 2612 |
. . . 4
⊢ (𝑥 = 𝑦 → (((1st ‘𝑥) gcd (2nd
‘𝑥)) = 1 ↔
((1st ‘𝑦)
gcd (2nd ‘𝑦)) = 1)) |
108 | 104, 105 | oveq12d 6567 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((1st ‘𝑥) / (2nd ‘𝑥)) = ((1st
‘𝑦) / (2nd
‘𝑦))) |
109 | 108 | eqeq2d 2620 |
. . . 4
⊢ (𝑥 = 𝑦 → (𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑦) / (2nd ‘𝑦)))) |
110 | 107, 109 | anbi12d 743 |
. . 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))))) |
111 | 110 | reu4 3367 |
. 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))) |
112 | 102, 103,
111 | 3imtr4i 280 |
1
⊢ (𝐴 ∈ ℚ →
∃!𝑥 ∈ (ℤ
× ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) |