Step | Hyp | Ref
| Expression |
1 | | ax-mulf 10611 |
. . . . . . 7
⊢ ·
:(ℂ × ℂ)⟶ℂ |
2 | | ffn 6508 |
. . . . . . 7
⊢ (
· :(ℂ × ℂ)⟶ℂ → · Fn (ℂ
× ℂ)) |
3 | 1, 2 | ax-mp 5 |
. . . . . 6
⊢ ·
Fn (ℂ × ℂ) |
4 | | dvdsmulf1o.x |
. . . . . . . . 9
⊢ 𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀} |
5 | 4 | ssrab3 4056 |
. . . . . . . 8
⊢ 𝑋 ⊆
ℕ |
6 | | nnsscn 11637 |
. . . . . . . 8
⊢ ℕ
⊆ ℂ |
7 | 5, 6 | sstri 3975 |
. . . . . . 7
⊢ 𝑋 ⊆
ℂ |
8 | | dvdsmulf1o.y |
. . . . . . . . 9
⊢ 𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} |
9 | 8 | ssrab3 4056 |
. . . . . . . 8
⊢ 𝑌 ⊆
ℕ |
10 | 9, 6 | sstri 3975 |
. . . . . . 7
⊢ 𝑌 ⊆
ℂ |
11 | | xpss12 5564 |
. . . . . . 7
⊢ ((𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ) → (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ)) |
12 | 7, 10, 11 | mp2an 690 |
. . . . . 6
⊢ (𝑋 × 𝑌) ⊆ (ℂ ×
ℂ) |
13 | | fnssres 6464 |
. . . . . 6
⊢ ((
· Fn (ℂ × ℂ) ∧ (𝑋 × 𝑌) ⊆ (ℂ × ℂ)) →
( · ↾ (𝑋
× 𝑌)) Fn (𝑋 × 𝑌)) |
14 | 3, 12, 13 | mp2an 690 |
. . . . 5
⊢ (
· ↾ (𝑋 ×
𝑌)) Fn (𝑋 × 𝑌) |
15 | 14 | a1i 11 |
. . . 4
⊢ (𝜑 → ( · ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)) |
16 | | ovres 7308 |
. . . . . . 7
⊢ ((𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌) → (𝑖( · ↾ (𝑋 × 𝑌))𝑗) = (𝑖 · 𝑗)) |
17 | 16 | adantl 484 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖( · ↾ (𝑋 × 𝑌))𝑗) = (𝑖 · 𝑗)) |
18 | | breq1 5061 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑖 → (𝑥 ∥ 𝑀 ↔ 𝑖 ∥ 𝑀)) |
19 | 18, 4 | elrab2 3682 |
. . . . . . . . . 10
⊢ (𝑖 ∈ 𝑋 ↔ (𝑖 ∈ ℕ ∧ 𝑖 ∥ 𝑀)) |
20 | 19 | simplbi 500 |
. . . . . . . . 9
⊢ (𝑖 ∈ 𝑋 → 𝑖 ∈ ℕ) |
21 | 20 | ad2antrl 726 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑖 ∈ ℕ) |
22 | | breq1 5061 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑗 → (𝑥 ∥ 𝑁 ↔ 𝑗 ∥ 𝑁)) |
23 | 22, 8 | elrab2 3682 |
. . . . . . . . . 10
⊢ (𝑗 ∈ 𝑌 ↔ (𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑁)) |
24 | 23 | simplbi 500 |
. . . . . . . . 9
⊢ (𝑗 ∈ 𝑌 → 𝑗 ∈ ℕ) |
25 | 24 | ad2antll 727 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑗 ∈ ℕ) |
26 | 21, 25 | nnmulcld 11684 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∈ ℕ) |
27 | 23 | simprbi 499 |
. . . . . . . . 9
⊢ (𝑗 ∈ 𝑌 → 𝑗 ∥ 𝑁) |
28 | 27 | ad2antll 727 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑗 ∥ 𝑁) |
29 | 19 | simprbi 499 |
. . . . . . . . 9
⊢ (𝑖 ∈ 𝑋 → 𝑖 ∥ 𝑀) |
30 | 29 | ad2antrl 726 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑖 ∥ 𝑀) |
31 | 25 | nnzd 12080 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑗 ∈ ℤ) |
32 | | dvdsmulf1o.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℕ) |
33 | 32 | adantr 483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑁 ∈ ℕ) |
34 | 33 | nnzd 12080 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑁 ∈ ℤ) |
35 | 21 | nnzd 12080 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑖 ∈ ℤ) |
36 | | dvdscmul 15630 |
. . . . . . . . . 10
⊢ ((𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑗 ∥ 𝑁 → (𝑖 · 𝑗) ∥ (𝑖 · 𝑁))) |
37 | 31, 34, 35, 36 | syl3anc 1367 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑗 ∥ 𝑁 → (𝑖 · 𝑗) ∥ (𝑖 · 𝑁))) |
38 | | dvdsmulf1o.1 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℕ) |
39 | 38 | adantr 483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑀 ∈ ℕ) |
40 | 39 | nnzd 12080 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → 𝑀 ∈ ℤ) |
41 | | dvdsmulc 15631 |
. . . . . . . . . 10
⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 ∥ 𝑀 → (𝑖 · 𝑁) ∥ (𝑀 · 𝑁))) |
42 | 35, 40, 34, 41 | syl3anc 1367 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 ∥ 𝑀 → (𝑖 · 𝑁) ∥ (𝑀 · 𝑁))) |
43 | 26 | nnzd 12080 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∈ ℤ) |
44 | 35, 34 | zmulcld 12087 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑁) ∈ ℤ) |
45 | 40, 34 | zmulcld 12087 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑀 · 𝑁) ∈ ℤ) |
46 | | dvdstr 15640 |
. . . . . . . . . 10
⊢ (((𝑖 · 𝑗) ∈ ℤ ∧ (𝑖 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → (((𝑖 · 𝑗) ∥ (𝑖 · 𝑁) ∧ (𝑖 · 𝑁) ∥ (𝑀 · 𝑁)) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁))) |
47 | 43, 44, 45, 46 | syl3anc 1367 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (((𝑖 · 𝑗) ∥ (𝑖 · 𝑁) ∧ (𝑖 · 𝑁) ∥ (𝑀 · 𝑁)) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁))) |
48 | 37, 42, 47 | syl2and 609 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → ((𝑗 ∥ 𝑁 ∧ 𝑖 ∥ 𝑀) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁))) |
49 | 28, 30, 48 | mp2and 697 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∥ (𝑀 · 𝑁)) |
50 | | breq1 5061 |
. . . . . . . 8
⊢ (𝑥 = (𝑖 · 𝑗) → (𝑥 ∥ (𝑀 · 𝑁) ↔ (𝑖 · 𝑗) ∥ (𝑀 · 𝑁))) |
51 | | dvdsmulf1o.z |
. . . . . . . 8
⊢ 𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)} |
52 | 50, 51 | elrab2 3682 |
. . . . . . 7
⊢ ((𝑖 · 𝑗) ∈ 𝑍 ↔ ((𝑖 · 𝑗) ∈ ℕ ∧ (𝑖 · 𝑗) ∥ (𝑀 · 𝑁))) |
53 | 26, 49, 52 | sylanbrc 585 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖 · 𝑗) ∈ 𝑍) |
54 | 17, 53 | eqeltrd 2913 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → (𝑖( · ↾ (𝑋 × 𝑌))𝑗) ∈ 𝑍) |
55 | 54 | ralrimivva 3191 |
. . . 4
⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 (𝑖( · ↾ (𝑋 × 𝑌))𝑗) ∈ 𝑍) |
56 | | ffnov 7272 |
. . . 4
⊢ ((
· ↾ (𝑋 ×
𝑌)):(𝑋 × 𝑌)⟶𝑍 ↔ (( · ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 (𝑖( · ↾ (𝑋 × 𝑌))𝑗) ∈ 𝑍)) |
57 | 15, 55, 56 | sylanbrc 585 |
. . 3
⊢ (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍) |
58 | 21 | adantr 483 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℕ) |
59 | 58 | nnnn0d 11949 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℕ0) |
60 | | simprll 777 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ 𝑋) |
61 | | breq1 5061 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑚 → (𝑥 ∥ 𝑀 ↔ 𝑚 ∥ 𝑀)) |
62 | 61, 4 | elrab2 3682 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑋 ↔ (𝑚 ∈ ℕ ∧ 𝑚 ∥ 𝑀)) |
63 | 62 | simplbi 500 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ 𝑋 → 𝑚 ∈ ℕ) |
64 | 60, 63 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℕ) |
65 | 64 | nnnn0d 11949 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℕ0) |
66 | 58 | nnzd 12080 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℤ) |
67 | 25 | adantr 483 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∈ ℕ) |
68 | 67 | nnzd 12080 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∈ ℤ) |
69 | | dvdsmul1 15625 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑖 ∥ (𝑖 · 𝑗)) |
70 | 66, 68, 69 | syl2anc 586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ (𝑖 · 𝑗)) |
71 | | simprr 771 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑚 · 𝑛)) |
72 | 7, 60 | sseldi 3964 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℂ) |
73 | | simprlr 778 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ 𝑌) |
74 | | breq1 5061 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑛 → (𝑥 ∥ 𝑁 ↔ 𝑛 ∥ 𝑁)) |
75 | 74, 8 | elrab2 3682 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑌 ↔ (𝑛 ∈ ℕ ∧ 𝑛 ∥ 𝑁)) |
76 | 75 | simplbi 500 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑌 → 𝑛 ∈ ℕ) |
77 | 73, 76 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ ℕ) |
78 | 77 | nncnd 11648 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ ℂ) |
79 | 72, 78 | mulcomd 10656 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 · 𝑛) = (𝑛 · 𝑚)) |
80 | 71, 79 | eqtrd 2856 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑛 · 𝑚)) |
81 | 70, 80 | breqtrd 5084 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ (𝑛 · 𝑚)) |
82 | 77 | nnzd 12080 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∈ ℤ) |
83 | 34 | adantr 483 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑁 ∈ ℤ) |
84 | | gcdcom 15856 |
. . . . . . . . . . . . 13
⊢ ((𝑖 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑖 gcd 𝑁) = (𝑁 gcd 𝑖)) |
85 | 66, 83, 84 | syl2anc 586 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 gcd 𝑁) = (𝑁 gcd 𝑖)) |
86 | 40 | adantr 483 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑀 ∈ ℤ) |
87 | 32 | nnzd 12080 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℤ) |
88 | 38 | nnzd 12080 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℤ) |
89 | | gcdcom 15856 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
90 | 87, 88, 89 | syl2anc 586 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
91 | | dvdsmulf1o.3 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
92 | 90, 91 | eqtrd 2856 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
93 | 92 | ad2antrr 724 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑁 gcd 𝑀) = 1) |
94 | 30 | adantr 483 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ 𝑀) |
95 | | rpdvds 15998 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑁 gcd 𝑀) = 1 ∧ 𝑖 ∥ 𝑀)) → (𝑁 gcd 𝑖) = 1) |
96 | 83, 66, 86, 93, 94, 95 | syl32anc 1374 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑁 gcd 𝑖) = 1) |
97 | 85, 96 | eqtrd 2856 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 gcd 𝑁) = 1) |
98 | 75 | simprbi 499 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ 𝑌 → 𝑛 ∥ 𝑁) |
99 | 73, 98 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑛 ∥ 𝑁) |
100 | | rpdvds 15998 |
. . . . . . . . . . 11
⊢ (((𝑖 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑖 gcd 𝑁) = 1 ∧ 𝑛 ∥ 𝑁)) → (𝑖 gcd 𝑛) = 1) |
101 | 66, 82, 83, 97, 99, 100 | syl32anc 1374 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 gcd 𝑛) = 1) |
102 | 64 | nnzd 12080 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∈ ℤ) |
103 | | coprmdvds 15991 |
. . . . . . . . . . 11
⊢ ((𝑖 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((𝑖 ∥ (𝑛 · 𝑚) ∧ (𝑖 gcd 𝑛) = 1) → 𝑖 ∥ 𝑚)) |
104 | 66, 82, 102, 103 | syl3anc 1367 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → ((𝑖 ∥ (𝑛 · 𝑚) ∧ (𝑖 gcd 𝑛) = 1) → 𝑖 ∥ 𝑚)) |
105 | 81, 101, 104 | mp2and 697 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∥ 𝑚) |
106 | | dvdsmul1 15625 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → 𝑚 ∥ (𝑚 · 𝑛)) |
107 | 102, 82, 106 | syl2anc 586 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ (𝑚 · 𝑛)) |
108 | 58 | nncnd 11648 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ∈ ℂ) |
109 | 67 | nncnd 11648 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∈ ℂ) |
110 | 108, 109 | mulcomd 10656 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑗 · 𝑖)) |
111 | 71, 110 | eqtr3d 2858 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 · 𝑛) = (𝑗 · 𝑖)) |
112 | 107, 111 | breqtrd 5084 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ (𝑗 · 𝑖)) |
113 | | gcdcom 15856 |
. . . . . . . . . . . . 13
⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 gcd 𝑁) = (𝑁 gcd 𝑚)) |
114 | 102, 83, 113 | syl2anc 586 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 gcd 𝑁) = (𝑁 gcd 𝑚)) |
115 | 62 | simprbi 499 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑋 → 𝑚 ∥ 𝑀) |
116 | 60, 115 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ 𝑀) |
117 | | rpdvds 15998 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ((𝑁 gcd 𝑀) = 1 ∧ 𝑚 ∥ 𝑀)) → (𝑁 gcd 𝑚) = 1) |
118 | 83, 102, 86, 93, 116, 117 | syl32anc 1374 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑁 gcd 𝑚) = 1) |
119 | 114, 118 | eqtrd 2856 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 gcd 𝑁) = 1) |
120 | 28 | adantr 483 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 ∥ 𝑁) |
121 | | rpdvds 15998 |
. . . . . . . . . . 11
⊢ (((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑚 gcd 𝑁) = 1 ∧ 𝑗 ∥ 𝑁)) → (𝑚 gcd 𝑗) = 1) |
122 | 102, 68, 83, 119, 120, 121 | syl32anc 1374 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑚 gcd 𝑗) = 1) |
123 | | coprmdvds 15991 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((𝑚 ∥ (𝑗 · 𝑖) ∧ (𝑚 gcd 𝑗) = 1) → 𝑚 ∥ 𝑖)) |
124 | 102, 68, 66, 123 | syl3anc 1367 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → ((𝑚 ∥ (𝑗 · 𝑖) ∧ (𝑚 gcd 𝑗) = 1) → 𝑚 ∥ 𝑖)) |
125 | 112, 122,
124 | mp2and 697 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑚 ∥ 𝑖) |
126 | | dvdseq 15658 |
. . . . . . . . 9
⊢ (((𝑖 ∈ ℕ0
∧ 𝑚 ∈
ℕ0) ∧ (𝑖 ∥ 𝑚 ∧ 𝑚 ∥ 𝑖)) → 𝑖 = 𝑚) |
127 | 59, 65, 105, 125, 126 | syl22anc 836 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 = 𝑚) |
128 | 58 | nnne0d 11681 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑖 ≠ 0) |
129 | 127 | oveq1d 7165 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑛) = (𝑚 · 𝑛)) |
130 | 71, 129 | eqtr4d 2859 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → (𝑖 · 𝑗) = (𝑖 · 𝑛)) |
131 | 109, 78, 108, 128, 130 | mulcanad 11269 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 𝑗 = 𝑛) |
132 | 127, 131 | opeq12d 4804 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌) ∧ (𝑖 · 𝑗) = (𝑚 · 𝑛))) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉) |
133 | 132 | expr 459 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌)) → ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉)) |
134 | 133 | ralrimivva 3191 |
. . . . 5
⊢ ((𝜑 ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌)) → ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉)) |
135 | 134 | ralrimivva 3191 |
. . . 4
⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉)) |
136 | | fvres 6683 |
. . . . . . . . 9
⊢ (𝑢 ∈ (𝑋 × 𝑌) → (( · ↾ (𝑋 × 𝑌))‘𝑢) = ( · ‘𝑢)) |
137 | | fvres 6683 |
. . . . . . . . 9
⊢ (𝑣 ∈ (𝑋 × 𝑌) → (( · ↾ (𝑋 × 𝑌))‘𝑣) = ( · ‘𝑣)) |
138 | 136, 137 | eqeqan12d 2838 |
. . . . . . . 8
⊢ ((𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) → ((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) ↔ ( · ‘𝑢) = ( · ‘𝑣))) |
139 | 138 | imbi1d 344 |
. . . . . . 7
⊢ ((𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (𝑋 × 𝑌)) → (((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ (( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣))) |
140 | 139 | ralbidva 3196 |
. . . . . 6
⊢ (𝑢 ∈ (𝑋 × 𝑌) → (∀𝑣 ∈ (𝑋 × 𝑌)((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑣 ∈ (𝑋 × 𝑌)(( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣))) |
141 | 140 | ralbiia 3164 |
. . . . 5
⊢
(∀𝑢 ∈
(𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)(( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣)) |
142 | | fveq2 6664 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈𝑚, 𝑛〉 → ( · ‘𝑣) = ( ·
‘〈𝑚, 𝑛〉)) |
143 | | df-ov 7153 |
. . . . . . . . . . 11
⊢ (𝑚 · 𝑛) = ( · ‘〈𝑚, 𝑛〉) |
144 | 142, 143 | syl6eqr 2874 |
. . . . . . . . . 10
⊢ (𝑣 = 〈𝑚, 𝑛〉 → ( · ‘𝑣) = (𝑚 · 𝑛)) |
145 | 144 | eqeq2d 2832 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑚, 𝑛〉 → (( · ‘𝑢) = ( · ‘𝑣) ↔ ( · ‘𝑢) = (𝑚 · 𝑛))) |
146 | | eqeq2 2833 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑚, 𝑛〉 → (𝑢 = 𝑣 ↔ 𝑢 = 〈𝑚, 𝑛〉)) |
147 | 145, 146 | imbi12d 347 |
. . . . . . . 8
⊢ (𝑣 = 〈𝑚, 𝑛〉 → ((( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣) ↔ (( · ‘𝑢) = (𝑚 · 𝑛) → 𝑢 = 〈𝑚, 𝑛〉))) |
148 | 147 | ralxp 5706 |
. . . . . . 7
⊢
(∀𝑣 ∈
(𝑋 × 𝑌)(( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 (( · ‘𝑢) = (𝑚 · 𝑛) → 𝑢 = 〈𝑚, 𝑛〉)) |
149 | | fveq2 6664 |
. . . . . . . . . . 11
⊢ (𝑢 = 〈𝑖, 𝑗〉 → ( · ‘𝑢) = ( ·
‘〈𝑖, 𝑗〉)) |
150 | | df-ov 7153 |
. . . . . . . . . . 11
⊢ (𝑖 · 𝑗) = ( · ‘〈𝑖, 𝑗〉) |
151 | 149, 150 | syl6eqr 2874 |
. . . . . . . . . 10
⊢ (𝑢 = 〈𝑖, 𝑗〉 → ( · ‘𝑢) = (𝑖 · 𝑗)) |
152 | 151 | eqeq1d 2823 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑖, 𝑗〉 → (( · ‘𝑢) = (𝑚 · 𝑛) ↔ (𝑖 · 𝑗) = (𝑚 · 𝑛))) |
153 | | eqeq1 2825 |
. . . . . . . . 9
⊢ (𝑢 = 〈𝑖, 𝑗〉 → (𝑢 = 〈𝑚, 𝑛〉 ↔ 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉)) |
154 | 152, 153 | imbi12d 347 |
. . . . . . . 8
⊢ (𝑢 = 〈𝑖, 𝑗〉 → ((( · ‘𝑢) = (𝑚 · 𝑛) → 𝑢 = 〈𝑚, 𝑛〉) ↔ ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉))) |
155 | 154 | 2ralbidv 3199 |
. . . . . . 7
⊢ (𝑢 = 〈𝑖, 𝑗〉 → (∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 (( · ‘𝑢) = (𝑚 · 𝑛) → 𝑢 = 〈𝑚, 𝑛〉) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉))) |
156 | 148, 155 | syl5bb 285 |
. . . . . 6
⊢ (𝑢 = 〈𝑖, 𝑗〉 → (∀𝑣 ∈ (𝑋 × 𝑌)(( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉))) |
157 | 156 | ralxp 5706 |
. . . . 5
⊢
(∀𝑢 ∈
(𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)(( · ‘𝑢) = ( · ‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉)) |
158 | 141, 157 | bitri 277 |
. . . 4
⊢
(∀𝑢 ∈
(𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣) ↔ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑌 ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑌 ((𝑖 · 𝑗) = (𝑚 · 𝑛) → 〈𝑖, 𝑗〉 = 〈𝑚, 𝑛〉)) |
159 | 135, 158 | sylibr 236 |
. . 3
⊢ (𝜑 → ∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣)) |
160 | | dff13 7007 |
. . 3
⊢ ((
· ↾ (𝑋 ×
𝑌)):(𝑋 × 𝑌)–1-1→𝑍 ↔ (( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑢 ∈ (𝑋 × 𝑌)∀𝑣 ∈ (𝑋 × 𝑌)((( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘𝑣) → 𝑢 = 𝑣))) |
161 | 57, 159, 160 | sylanbrc 585 |
. 2
⊢ (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1→𝑍) |
162 | | breq1 5061 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (𝑥 ∥ (𝑀 · 𝑁) ↔ 𝑤 ∥ (𝑀 · 𝑁))) |
163 | 162, 51 | elrab2 3682 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ 𝑍 ↔ (𝑤 ∈ ℕ ∧ 𝑤 ∥ (𝑀 · 𝑁))) |
164 | 163 | simplbi 500 |
. . . . . . . . . 10
⊢ (𝑤 ∈ 𝑍 → 𝑤 ∈ ℕ) |
165 | 164 | adantl 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ ℕ) |
166 | 165 | nnzd 12080 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ ℤ) |
167 | 38 | adantr 483 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑀 ∈ ℕ) |
168 | 167 | nnzd 12080 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑀 ∈ ℤ) |
169 | 167 | nnne0d 11681 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑀 ≠ 0) |
170 | | simpr 487 |
. . . . . . . . . 10
⊢ ((𝑤 = 0 ∧ 𝑀 = 0) → 𝑀 = 0) |
171 | 170 | necon3ai 3041 |
. . . . . . . . 9
⊢ (𝑀 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑀 = 0)) |
172 | 169, 171 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ¬ (𝑤 = 0 ∧ 𝑀 = 0)) |
173 | | gcdn0cl 15845 |
. . . . . . . 8
⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ ¬
(𝑤 = 0 ∧ 𝑀 = 0)) → (𝑤 gcd 𝑀) ∈ ℕ) |
174 | 166, 168,
172, 173 | syl21anc 835 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∈ ℕ) |
175 | | gcddvds 15846 |
. . . . . . . . 9
⊢ ((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀)) |
176 | 166, 168,
175 | syl2anc 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd 𝑀) ∥ 𝑤 ∧ (𝑤 gcd 𝑀) ∥ 𝑀)) |
177 | 176 | simprd 498 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∥ 𝑀) |
178 | | breq1 5061 |
. . . . . . . 8
⊢ (𝑥 = (𝑤 gcd 𝑀) → (𝑥 ∥ 𝑀 ↔ (𝑤 gcd 𝑀) ∥ 𝑀)) |
179 | 178, 4 | elrab2 3682 |
. . . . . . 7
⊢ ((𝑤 gcd 𝑀) ∈ 𝑋 ↔ ((𝑤 gcd 𝑀) ∈ ℕ ∧ (𝑤 gcd 𝑀) ∥ 𝑀)) |
180 | 174, 177,
179 | sylanbrc 585 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑀) ∈ 𝑋) |
181 | 32 | adantr 483 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑁 ∈ ℕ) |
182 | 181 | nnzd 12080 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑁 ∈ ℤ) |
183 | 181 | nnne0d 11681 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑁 ≠ 0) |
184 | | simpr 487 |
. . . . . . . . . 10
⊢ ((𝑤 = 0 ∧ 𝑁 = 0) → 𝑁 = 0) |
185 | 184 | necon3ai 3041 |
. . . . . . . . 9
⊢ (𝑁 ≠ 0 → ¬ (𝑤 = 0 ∧ 𝑁 = 0)) |
186 | 183, 185 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ¬ (𝑤 = 0 ∧ 𝑁 = 0)) |
187 | | gcdn0cl 15845 |
. . . . . . . 8
⊢ (((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑤 = 0 ∧ 𝑁 = 0)) → (𝑤 gcd 𝑁) ∈ ℕ) |
188 | 166, 182,
186, 187 | syl21anc 835 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∈ ℕ) |
189 | | gcddvds 15846 |
. . . . . . . . 9
⊢ ((𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁)) |
190 | 166, 182,
189 | syl2anc 586 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd 𝑁) ∥ 𝑤 ∧ (𝑤 gcd 𝑁) ∥ 𝑁)) |
191 | 190 | simprd 498 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∥ 𝑁) |
192 | | breq1 5061 |
. . . . . . . 8
⊢ (𝑥 = (𝑤 gcd 𝑁) → (𝑥 ∥ 𝑁 ↔ (𝑤 gcd 𝑁) ∥ 𝑁)) |
193 | 192, 8 | elrab2 3682 |
. . . . . . 7
⊢ ((𝑤 gcd 𝑁) ∈ 𝑌 ↔ ((𝑤 gcd 𝑁) ∈ ℕ ∧ (𝑤 gcd 𝑁) ∥ 𝑁)) |
194 | 188, 191,
193 | sylanbrc 585 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd 𝑁) ∈ 𝑌) |
195 | 180, 194 | opelxpd 5587 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉 ∈ (𝑋 × 𝑌)) |
196 | 195 | fvresd 6684 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (( · ↾ (𝑋 × 𝑌))‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉) = ( · ‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉)) |
197 | 91 | adantr 483 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑀 gcd 𝑁) = 1) |
198 | | rpmulgcd2 15994 |
. . . . . . . 8
⊢ (((𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 gcd 𝑁) = 1) → (𝑤 gcd (𝑀 · 𝑁)) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁))) |
199 | 166, 168,
182, 197, 198 | syl31anc 1369 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd (𝑀 · 𝑁)) = ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁))) |
200 | | df-ov 7153 |
. . . . . . 7
⊢ ((𝑤 gcd 𝑀) · (𝑤 gcd 𝑁)) = ( · ‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉) |
201 | 199, 200 | syl6eq 2872 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd (𝑀 · 𝑁)) = ( · ‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉)) |
202 | 163 | simprbi 499 |
. . . . . . . 8
⊢ (𝑤 ∈ 𝑍 → 𝑤 ∥ (𝑀 · 𝑁)) |
203 | 202 | adantl 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∥ (𝑀 · 𝑁)) |
204 | 38, 32 | nnmulcld 11684 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 · 𝑁) ∈ ℕ) |
205 | | gcdeq 15897 |
. . . . . . . 8
⊢ ((𝑤 ∈ ℕ ∧ (𝑀 · 𝑁) ∈ ℕ) → ((𝑤 gcd (𝑀 · 𝑁)) = 𝑤 ↔ 𝑤 ∥ (𝑀 · 𝑁))) |
206 | 164, 204,
205 | syl2anr 598 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ((𝑤 gcd (𝑀 · 𝑁)) = 𝑤 ↔ 𝑤 ∥ (𝑀 · 𝑁))) |
207 | 203, 206 | mpbird 259 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝑤 gcd (𝑀 · 𝑁)) = 𝑤) |
208 | 196, 201,
207 | 3eqtr2rd 2863 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 = (( · ↾ (𝑋 × 𝑌))‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉)) |
209 | | fveq2 6664 |
. . . . . 6
⊢ (𝑢 = 〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉 → (( · ↾ (𝑋 × 𝑌))‘𝑢) = (( · ↾ (𝑋 × 𝑌))‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉)) |
210 | 209 | rspceeqv 3637 |
. . . . 5
⊢
((〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉 ∈ (𝑋 × 𝑌) ∧ 𝑤 = (( · ↾ (𝑋 × 𝑌))‘〈(𝑤 gcd 𝑀), (𝑤 gcd 𝑁)〉)) → ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (( · ↾ (𝑋 × 𝑌))‘𝑢)) |
211 | 195, 208,
210 | syl2anc 586 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (( · ↾ (𝑋 × 𝑌))‘𝑢)) |
212 | 211 | ralrimiva 3182 |
. . 3
⊢ (𝜑 → ∀𝑤 ∈ 𝑍 ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (( · ↾ (𝑋 × 𝑌))‘𝑢)) |
213 | | dffo3 6862 |
. . 3
⊢ ((
· ↾ (𝑋 ×
𝑌)):(𝑋 × 𝑌)–onto→𝑍 ↔ (( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑤 ∈ 𝑍 ∃𝑢 ∈ (𝑋 × 𝑌)𝑤 = (( · ↾ (𝑋 × 𝑌))‘𝑢))) |
214 | 57, 212, 213 | sylanbrc 585 |
. 2
⊢ (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍) |
215 | | df-f1o 6356 |
. 2
⊢ ((
· ↾ (𝑋 ×
𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍 ↔ (( · ↾
(𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1→𝑍 ∧ ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑍)) |
216 | 161, 214,
215 | sylanbrc 585 |
1
⊢ (𝜑 → ( · ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍) |