Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsmulf1o.1 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
dvdsmulf1o.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
3 |
|
dvdsmulf1o.3 |
⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) = 1 ) |
4 |
|
dvdsmulf1o.x |
⊢ 𝑋 = { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀 } |
5 |
|
dvdsmulf1o.y |
⊢ 𝑌 = { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁 } |
6 |
|
dvdsmulf1o.z |
⊢ 𝑍 = { 𝑥 ∈ ℕ ∣ 𝑥 ∥ ( 𝑀 · 𝑁 ) } |
7 |
|
ax-mulf |
⊢ · : ( ℂ × ℂ ) ⟶ ℂ |
8 |
|
ffn |
⊢ ( · : ( ℂ × ℂ ) ⟶ ℂ → · Fn ( ℂ × ℂ ) ) |
9 |
7 8
|
ax-mp |
⊢ · Fn ( ℂ × ℂ ) |
10 |
4
|
ssrab3 |
⊢ 𝑋 ⊆ ℕ |
11 |
|
nnsscn |
⊢ ℕ ⊆ ℂ |
12 |
10 11
|
sstri |
⊢ 𝑋 ⊆ ℂ |
13 |
5
|
ssrab3 |
⊢ 𝑌 ⊆ ℕ |
14 |
13 11
|
sstri |
⊢ 𝑌 ⊆ ℂ |
15 |
|
xpss12 |
⊢ ( ( 𝑋 ⊆ ℂ ∧ 𝑌 ⊆ ℂ ) → ( 𝑋 × 𝑌 ) ⊆ ( ℂ × ℂ ) ) |
16 |
12 14 15
|
mp2an |
⊢ ( 𝑋 × 𝑌 ) ⊆ ( ℂ × ℂ ) |
17 |
|
fnssres |
⊢ ( ( · Fn ( ℂ × ℂ ) ∧ ( 𝑋 × 𝑌 ) ⊆ ( ℂ × ℂ ) ) → ( · ↾ ( 𝑋 × 𝑌 ) ) Fn ( 𝑋 × 𝑌 ) ) |
18 |
9 16 17
|
mp2an |
⊢ ( · ↾ ( 𝑋 × 𝑌 ) ) Fn ( 𝑋 × 𝑌 ) |
19 |
18
|
a1i |
⊢ ( 𝜑 → ( · ↾ ( 𝑋 × 𝑌 ) ) Fn ( 𝑋 × 𝑌 ) ) |
20 |
|
ovres |
⊢ ( ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) → ( 𝑖 ( · ↾ ( 𝑋 × 𝑌 ) ) 𝑗 ) = ( 𝑖 · 𝑗 ) ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 ( · ↾ ( 𝑋 × 𝑌 ) ) 𝑗 ) = ( 𝑖 · 𝑗 ) ) |
22 |
|
breq1 |
⊢ ( 𝑥 = 𝑖 → ( 𝑥 ∥ 𝑀 ↔ 𝑖 ∥ 𝑀 ) ) |
23 |
22 4
|
elrab2 |
⊢ ( 𝑖 ∈ 𝑋 ↔ ( 𝑖 ∈ ℕ ∧ 𝑖 ∥ 𝑀 ) ) |
24 |
23
|
simplbi |
⊢ ( 𝑖 ∈ 𝑋 → 𝑖 ∈ ℕ ) |
25 |
24
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑖 ∈ ℕ ) |
26 |
|
breq1 |
⊢ ( 𝑥 = 𝑗 → ( 𝑥 ∥ 𝑁 ↔ 𝑗 ∥ 𝑁 ) ) |
27 |
26 5
|
elrab2 |
⊢ ( 𝑗 ∈ 𝑌 ↔ ( 𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑁 ) ) |
28 |
27
|
simplbi |
⊢ ( 𝑗 ∈ 𝑌 → 𝑗 ∈ ℕ ) |
29 |
28
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑗 ∈ ℕ ) |
30 |
25 29
|
nnmulcld |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 · 𝑗 ) ∈ ℕ ) |
31 |
27
|
simprbi |
⊢ ( 𝑗 ∈ 𝑌 → 𝑗 ∥ 𝑁 ) |
32 |
31
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑗 ∥ 𝑁 ) |
33 |
23
|
simprbi |
⊢ ( 𝑖 ∈ 𝑋 → 𝑖 ∥ 𝑀 ) |
34 |
33
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑖 ∥ 𝑀 ) |
35 |
29
|
nnzd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑗 ∈ ℤ ) |
36 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑁 ∈ ℕ ) |
37 |
36
|
nnzd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑁 ∈ ℤ ) |
38 |
25
|
nnzd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑖 ∈ ℤ ) |
39 |
|
dvdscmul |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( 𝑗 ∥ 𝑁 → ( 𝑖 · 𝑗 ) ∥ ( 𝑖 · 𝑁 ) ) ) |
40 |
35 37 38 39
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑗 ∥ 𝑁 → ( 𝑖 · 𝑗 ) ∥ ( 𝑖 · 𝑁 ) ) ) |
41 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑀 ∈ ℕ ) |
42 |
41
|
nnzd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → 𝑀 ∈ ℤ ) |
43 |
|
dvdsmulc |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑖 ∥ 𝑀 → ( 𝑖 · 𝑁 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
44 |
38 42 37 43
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 ∥ 𝑀 → ( 𝑖 · 𝑁 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
45 |
30
|
nnzd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 · 𝑗 ) ∈ ℤ ) |
46 |
38 37
|
zmulcld |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 · 𝑁 ) ∈ ℤ ) |
47 |
42 37
|
zmulcld |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑀 · 𝑁 ) ∈ ℤ ) |
48 |
|
dvdstr |
⊢ ( ( ( 𝑖 · 𝑗 ) ∈ ℤ ∧ ( 𝑖 · 𝑁 ) ∈ ℤ ∧ ( 𝑀 · 𝑁 ) ∈ ℤ ) → ( ( ( 𝑖 · 𝑗 ) ∥ ( 𝑖 · 𝑁 ) ∧ ( 𝑖 · 𝑁 ) ∥ ( 𝑀 · 𝑁 ) ) → ( 𝑖 · 𝑗 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
49 |
45 46 47 48
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( ( ( 𝑖 · 𝑗 ) ∥ ( 𝑖 · 𝑁 ) ∧ ( 𝑖 · 𝑁 ) ∥ ( 𝑀 · 𝑁 ) ) → ( 𝑖 · 𝑗 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
50 |
40 44 49
|
syl2and |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( ( 𝑗 ∥ 𝑁 ∧ 𝑖 ∥ 𝑀 ) → ( 𝑖 · 𝑗 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
51 |
32 34 50
|
mp2and |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 · 𝑗 ) ∥ ( 𝑀 · 𝑁 ) ) |
52 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑖 · 𝑗 ) → ( 𝑥 ∥ ( 𝑀 · 𝑁 ) ↔ ( 𝑖 · 𝑗 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
53 |
52 6
|
elrab2 |
⊢ ( ( 𝑖 · 𝑗 ) ∈ 𝑍 ↔ ( ( 𝑖 · 𝑗 ) ∈ ℕ ∧ ( 𝑖 · 𝑗 ) ∥ ( 𝑀 · 𝑁 ) ) ) |
54 |
30 51 53
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 · 𝑗 ) ∈ 𝑍 ) |
55 |
21 54
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ( 𝑖 ( · ↾ ( 𝑋 × 𝑌 ) ) 𝑗 ) ∈ 𝑍 ) |
56 |
55
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑋 ∀ 𝑗 ∈ 𝑌 ( 𝑖 ( · ↾ ( 𝑋 × 𝑌 ) ) 𝑗 ) ∈ 𝑍 ) |
57 |
|
ffnov |
⊢ ( ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) ⟶ 𝑍 ↔ ( ( · ↾ ( 𝑋 × 𝑌 ) ) Fn ( 𝑋 × 𝑌 ) ∧ ∀ 𝑖 ∈ 𝑋 ∀ 𝑗 ∈ 𝑌 ( 𝑖 ( · ↾ ( 𝑋 × 𝑌 ) ) 𝑗 ) ∈ 𝑍 ) ) |
58 |
19 56 57
|
sylanbrc |
⊢ ( 𝜑 → ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) ⟶ 𝑍 ) |
59 |
25
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∈ ℕ ) |
60 |
59
|
nnnn0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∈ ℕ0 ) |
61 |
|
simprll |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∈ 𝑋 ) |
62 |
|
breq1 |
⊢ ( 𝑥 = 𝑚 → ( 𝑥 ∥ 𝑀 ↔ 𝑚 ∥ 𝑀 ) ) |
63 |
62 4
|
elrab2 |
⊢ ( 𝑚 ∈ 𝑋 ↔ ( 𝑚 ∈ ℕ ∧ 𝑚 ∥ 𝑀 ) ) |
64 |
63
|
simplbi |
⊢ ( 𝑚 ∈ 𝑋 → 𝑚 ∈ ℕ ) |
65 |
61 64
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∈ ℕ ) |
66 |
65
|
nnnn0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∈ ℕ0 ) |
67 |
59
|
nnzd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∈ ℤ ) |
68 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑗 ∈ ℕ ) |
69 |
68
|
nnzd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑗 ∈ ℤ ) |
70 |
|
dvdsmul1 |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ) → 𝑖 ∥ ( 𝑖 · 𝑗 ) ) |
71 |
67 69 70
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∥ ( 𝑖 · 𝑗 ) ) |
72 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) |
73 |
12 61
|
sselid |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∈ ℂ ) |
74 |
|
simprlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑛 ∈ 𝑌 ) |
75 |
|
breq1 |
⊢ ( 𝑥 = 𝑛 → ( 𝑥 ∥ 𝑁 ↔ 𝑛 ∥ 𝑁 ) ) |
76 |
75 5
|
elrab2 |
⊢ ( 𝑛 ∈ 𝑌 ↔ ( 𝑛 ∈ ℕ ∧ 𝑛 ∥ 𝑁 ) ) |
77 |
76
|
simplbi |
⊢ ( 𝑛 ∈ 𝑌 → 𝑛 ∈ ℕ ) |
78 |
74 77
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑛 ∈ ℕ ) |
79 |
78
|
nncnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑛 ∈ ℂ ) |
80 |
73 79
|
mulcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑚 · 𝑛 ) = ( 𝑛 · 𝑚 ) ) |
81 |
72 80
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 · 𝑗 ) = ( 𝑛 · 𝑚 ) ) |
82 |
71 81
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∥ ( 𝑛 · 𝑚 ) ) |
83 |
78
|
nnzd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑛 ∈ ℤ ) |
84 |
37
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑁 ∈ ℤ ) |
85 |
67 84
|
gcdcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 gcd 𝑁 ) = ( 𝑁 gcd 𝑖 ) ) |
86 |
42
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑀 ∈ ℤ ) |
87 |
2
|
nnzd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
88 |
1
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
89 |
87 88
|
gcdcomd |
⊢ ( 𝜑 → ( 𝑁 gcd 𝑀 ) = ( 𝑀 gcd 𝑁 ) ) |
90 |
89 3
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 gcd 𝑀 ) = 1 ) |
91 |
90
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑁 gcd 𝑀 ) = 1 ) |
92 |
34
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∥ 𝑀 ) |
93 |
|
rpdvds |
⊢ ( ( ( 𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( ( 𝑁 gcd 𝑀 ) = 1 ∧ 𝑖 ∥ 𝑀 ) ) → ( 𝑁 gcd 𝑖 ) = 1 ) |
94 |
84 67 86 91 92 93
|
syl32anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑁 gcd 𝑖 ) = 1 ) |
95 |
85 94
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 gcd 𝑁 ) = 1 ) |
96 |
76
|
simprbi |
⊢ ( 𝑛 ∈ 𝑌 → 𝑛 ∥ 𝑁 ) |
97 |
74 96
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑛 ∥ 𝑁 ) |
98 |
|
rpdvds |
⊢ ( ( ( 𝑖 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( ( 𝑖 gcd 𝑁 ) = 1 ∧ 𝑛 ∥ 𝑁 ) ) → ( 𝑖 gcd 𝑛 ) = 1 ) |
99 |
67 83 84 95 97 98
|
syl32anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 gcd 𝑛 ) = 1 ) |
100 |
65
|
nnzd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∈ ℤ ) |
101 |
|
coprmdvds |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ ) → ( ( 𝑖 ∥ ( 𝑛 · 𝑚 ) ∧ ( 𝑖 gcd 𝑛 ) = 1 ) → 𝑖 ∥ 𝑚 ) ) |
102 |
67 83 100 101
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( ( 𝑖 ∥ ( 𝑛 · 𝑚 ) ∧ ( 𝑖 gcd 𝑛 ) = 1 ) → 𝑖 ∥ 𝑚 ) ) |
103 |
82 99 102
|
mp2and |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∥ 𝑚 ) |
104 |
|
dvdsmul1 |
⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ ) → 𝑚 ∥ ( 𝑚 · 𝑛 ) ) |
105 |
100 83 104
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∥ ( 𝑚 · 𝑛 ) ) |
106 |
59
|
nncnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ∈ ℂ ) |
107 |
68
|
nncnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑗 ∈ ℂ ) |
108 |
106 107
|
mulcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 · 𝑗 ) = ( 𝑗 · 𝑖 ) ) |
109 |
72 108
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑚 · 𝑛 ) = ( 𝑗 · 𝑖 ) ) |
110 |
105 109
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∥ ( 𝑗 · 𝑖 ) ) |
111 |
100 84
|
gcdcomd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑚 gcd 𝑁 ) = ( 𝑁 gcd 𝑚 ) ) |
112 |
63
|
simprbi |
⊢ ( 𝑚 ∈ 𝑋 → 𝑚 ∥ 𝑀 ) |
113 |
61 112
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∥ 𝑀 ) |
114 |
|
rpdvds |
⊢ ( ( ( 𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( ( 𝑁 gcd 𝑀 ) = 1 ∧ 𝑚 ∥ 𝑀 ) ) → ( 𝑁 gcd 𝑚 ) = 1 ) |
115 |
84 100 86 91 113 114
|
syl32anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑁 gcd 𝑚 ) = 1 ) |
116 |
111 115
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑚 gcd 𝑁 ) = 1 ) |
117 |
32
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑗 ∥ 𝑁 ) |
118 |
|
rpdvds |
⊢ ( ( ( 𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( ( 𝑚 gcd 𝑁 ) = 1 ∧ 𝑗 ∥ 𝑁 ) ) → ( 𝑚 gcd 𝑗 ) = 1 ) |
119 |
100 69 84 116 117 118
|
syl32anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑚 gcd 𝑗 ) = 1 ) |
120 |
|
coprmdvds |
⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ) → ( ( 𝑚 ∥ ( 𝑗 · 𝑖 ) ∧ ( 𝑚 gcd 𝑗 ) = 1 ) → 𝑚 ∥ 𝑖 ) ) |
121 |
100 69 67 120
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( ( 𝑚 ∥ ( 𝑗 · 𝑖 ) ∧ ( 𝑚 gcd 𝑗 ) = 1 ) → 𝑚 ∥ 𝑖 ) ) |
122 |
110 119 121
|
mp2and |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑚 ∥ 𝑖 ) |
123 |
|
dvdseq |
⊢ ( ( ( 𝑖 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ) ∧ ( 𝑖 ∥ 𝑚 ∧ 𝑚 ∥ 𝑖 ) ) → 𝑖 = 𝑚 ) |
124 |
60 66 103 122 123
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 = 𝑚 ) |
125 |
59
|
nnne0d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑖 ≠ 0 ) |
126 |
124
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 · 𝑛 ) = ( 𝑚 · 𝑛 ) ) |
127 |
72 126
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → ( 𝑖 · 𝑗 ) = ( 𝑖 · 𝑛 ) ) |
128 |
107 79 106 125 127
|
mulcanad |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 𝑗 = 𝑛 ) |
129 |
124 128
|
opeq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ∧ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) → 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) |
130 |
129
|
expr |
⊢ ( ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) ∧ ( 𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑌 ) ) → ( ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) → 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) ) |
131 |
130
|
ralrimivva |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑌 ) ) → ∀ 𝑚 ∈ 𝑋 ∀ 𝑛 ∈ 𝑌 ( ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) → 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) ) |
132 |
131
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑋 ∀ 𝑗 ∈ 𝑌 ∀ 𝑚 ∈ 𝑋 ∀ 𝑛 ∈ 𝑌 ( ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) → 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) ) |
133 |
|
fvres |
⊢ ( 𝑢 ∈ ( 𝑋 × 𝑌 ) → ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( · ‘ 𝑢 ) ) |
134 |
|
fvres |
⊢ ( 𝑣 ∈ ( 𝑋 × 𝑌 ) → ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑣 ) = ( · ‘ 𝑣 ) ) |
135 |
133 134
|
eqeqan12d |
⊢ ( ( 𝑢 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑣 ∈ ( 𝑋 × 𝑌 ) ) → ( ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑣 ) ↔ ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) ) ) |
136 |
135
|
imbi1d |
⊢ ( ( 𝑢 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑣 ∈ ( 𝑋 × 𝑌 ) ) → ( ( ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ↔ ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) ) |
137 |
136
|
ralbidva |
⊢ ( 𝑢 ∈ ( 𝑋 × 𝑌 ) → ( ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ↔ ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) ) |
138 |
137
|
ralbiia |
⊢ ( ∀ 𝑢 ∈ ( 𝑋 × 𝑌 ) ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ↔ ∀ 𝑢 ∈ ( 𝑋 × 𝑌 ) ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
139 |
|
fveq2 |
⊢ ( 𝑣 = 〈 𝑚 , 𝑛 〉 → ( · ‘ 𝑣 ) = ( · ‘ 〈 𝑚 , 𝑛 〉 ) ) |
140 |
|
df-ov |
⊢ ( 𝑚 · 𝑛 ) = ( · ‘ 〈 𝑚 , 𝑛 〉 ) |
141 |
139 140
|
eqtr4di |
⊢ ( 𝑣 = 〈 𝑚 , 𝑛 〉 → ( · ‘ 𝑣 ) = ( 𝑚 · 𝑛 ) ) |
142 |
141
|
eqeq2d |
⊢ ( 𝑣 = 〈 𝑚 , 𝑛 〉 → ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) ↔ ( · ‘ 𝑢 ) = ( 𝑚 · 𝑛 ) ) ) |
143 |
|
eqeq2 |
⊢ ( 𝑣 = 〈 𝑚 , 𝑛 〉 → ( 𝑢 = 𝑣 ↔ 𝑢 = 〈 𝑚 , 𝑛 〉 ) ) |
144 |
142 143
|
imbi12d |
⊢ ( 𝑣 = 〈 𝑚 , 𝑛 〉 → ( ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) → 𝑢 = 𝑣 ) ↔ ( ( · ‘ 𝑢 ) = ( 𝑚 · 𝑛 ) → 𝑢 = 〈 𝑚 , 𝑛 〉 ) ) ) |
145 |
144
|
ralxp |
⊢ ( ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) → 𝑢 = 𝑣 ) ↔ ∀ 𝑚 ∈ 𝑋 ∀ 𝑛 ∈ 𝑌 ( ( · ‘ 𝑢 ) = ( 𝑚 · 𝑛 ) → 𝑢 = 〈 𝑚 , 𝑛 〉 ) ) |
146 |
|
fveq2 |
⊢ ( 𝑢 = 〈 𝑖 , 𝑗 〉 → ( · ‘ 𝑢 ) = ( · ‘ 〈 𝑖 , 𝑗 〉 ) ) |
147 |
|
df-ov |
⊢ ( 𝑖 · 𝑗 ) = ( · ‘ 〈 𝑖 , 𝑗 〉 ) |
148 |
146 147
|
eqtr4di |
⊢ ( 𝑢 = 〈 𝑖 , 𝑗 〉 → ( · ‘ 𝑢 ) = ( 𝑖 · 𝑗 ) ) |
149 |
148
|
eqeq1d |
⊢ ( 𝑢 = 〈 𝑖 , 𝑗 〉 → ( ( · ‘ 𝑢 ) = ( 𝑚 · 𝑛 ) ↔ ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) ) ) |
150 |
|
eqeq1 |
⊢ ( 𝑢 = 〈 𝑖 , 𝑗 〉 → ( 𝑢 = 〈 𝑚 , 𝑛 〉 ↔ 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) ) |
151 |
149 150
|
imbi12d |
⊢ ( 𝑢 = 〈 𝑖 , 𝑗 〉 → ( ( ( · ‘ 𝑢 ) = ( 𝑚 · 𝑛 ) → 𝑢 = 〈 𝑚 , 𝑛 〉 ) ↔ ( ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) → 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) ) ) |
152 |
151
|
2ralbidv |
⊢ ( 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 〉 ) ) ) |
153 |
145 152
|
syl5bb |
⊢ ( 𝑢 = 〈 𝑖 , 𝑗 〉 → ( ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( · ‘ 𝑢 ) = ( · ‘ 𝑣 ) → 𝑢 = 𝑣 ) ↔ ∀ 𝑚 ∈ 𝑋 ∀ 𝑛 ∈ 𝑌 ( ( 𝑖 · 𝑗 ) = ( 𝑚 · 𝑛 ) → 〈 𝑖 , 𝑗 〉 = 〈 𝑚 , 𝑛 〉 ) ) ) |
154 |
153
|
ralxp |
⊢ ( ∀ 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 〉 ) ) |
155 |
138 154
|
bitri |
⊢ ( ∀ 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 〉 ) ) |
156 |
132 155
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑢 ∈ ( 𝑋 × 𝑌 ) ∀ 𝑣 ∈ ( 𝑋 × 𝑌 ) ( ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
157 |
|
dff13 |
⊢ ( ( · ↾ ( 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 ) ) ) |
158 |
58 156 157
|
sylanbrc |
⊢ ( 𝜑 → ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –1-1→ 𝑍 ) |
159 |
|
breq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 ∥ ( 𝑀 · 𝑁 ) ↔ 𝑤 ∥ ( 𝑀 · 𝑁 ) ) ) |
160 |
159 6
|
elrab2 |
⊢ ( 𝑤 ∈ 𝑍 ↔ ( 𝑤 ∈ ℕ ∧ 𝑤 ∥ ( 𝑀 · 𝑁 ) ) ) |
161 |
160
|
simplbi |
⊢ ( 𝑤 ∈ 𝑍 → 𝑤 ∈ ℕ ) |
162 |
161
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑤 ∈ ℕ ) |
163 |
162
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑤 ∈ ℤ ) |
164 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑀 ∈ ℕ ) |
165 |
164
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑀 ∈ ℤ ) |
166 |
164
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑀 ≠ 0 ) |
167 |
|
simpr |
⊢ ( ( 𝑤 = 0 ∧ 𝑀 = 0 ) → 𝑀 = 0 ) |
168 |
167
|
necon3ai |
⊢ ( 𝑀 ≠ 0 → ¬ ( 𝑤 = 0 ∧ 𝑀 = 0 ) ) |
169 |
166 168
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ¬ ( 𝑤 = 0 ∧ 𝑀 = 0 ) ) |
170 |
|
gcdn0cl |
⊢ ( ( ( 𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ¬ ( 𝑤 = 0 ∧ 𝑀 = 0 ) ) → ( 𝑤 gcd 𝑀 ) ∈ ℕ ) |
171 |
163 165 169 170
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd 𝑀 ) ∈ ℕ ) |
172 |
|
gcddvds |
⊢ ( ( 𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( ( 𝑤 gcd 𝑀 ) ∥ 𝑤 ∧ ( 𝑤 gcd 𝑀 ) ∥ 𝑀 ) ) |
173 |
163 165 172
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( ( 𝑤 gcd 𝑀 ) ∥ 𝑤 ∧ ( 𝑤 gcd 𝑀 ) ∥ 𝑀 ) ) |
174 |
173
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd 𝑀 ) ∥ 𝑀 ) |
175 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑤 gcd 𝑀 ) → ( 𝑥 ∥ 𝑀 ↔ ( 𝑤 gcd 𝑀 ) ∥ 𝑀 ) ) |
176 |
175 4
|
elrab2 |
⊢ ( ( 𝑤 gcd 𝑀 ) ∈ 𝑋 ↔ ( ( 𝑤 gcd 𝑀 ) ∈ ℕ ∧ ( 𝑤 gcd 𝑀 ) ∥ 𝑀 ) ) |
177 |
171 174 176
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd 𝑀 ) ∈ 𝑋 ) |
178 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑁 ∈ ℕ ) |
179 |
178
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑁 ∈ ℤ ) |
180 |
178
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑁 ≠ 0 ) |
181 |
|
simpr |
⊢ ( ( 𝑤 = 0 ∧ 𝑁 = 0 ) → 𝑁 = 0 ) |
182 |
181
|
necon3ai |
⊢ ( 𝑁 ≠ 0 → ¬ ( 𝑤 = 0 ∧ 𝑁 = 0 ) ) |
183 |
180 182
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ¬ ( 𝑤 = 0 ∧ 𝑁 = 0 ) ) |
184 |
|
gcdn0cl |
⊢ ( ( ( 𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑤 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑤 gcd 𝑁 ) ∈ ℕ ) |
185 |
163 179 183 184
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd 𝑁 ) ∈ ℕ ) |
186 |
|
gcddvds |
⊢ ( ( 𝑤 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑤 gcd 𝑁 ) ∥ 𝑤 ∧ ( 𝑤 gcd 𝑁 ) ∥ 𝑁 ) ) |
187 |
163 179 186
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( ( 𝑤 gcd 𝑁 ) ∥ 𝑤 ∧ ( 𝑤 gcd 𝑁 ) ∥ 𝑁 ) ) |
188 |
187
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd 𝑁 ) ∥ 𝑁 ) |
189 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑤 gcd 𝑁 ) → ( 𝑥 ∥ 𝑁 ↔ ( 𝑤 gcd 𝑁 ) ∥ 𝑁 ) ) |
190 |
189 5
|
elrab2 |
⊢ ( ( 𝑤 gcd 𝑁 ) ∈ 𝑌 ↔ ( ( 𝑤 gcd 𝑁 ) ∈ ℕ ∧ ( 𝑤 gcd 𝑁 ) ∥ 𝑁 ) ) |
191 |
185 188 190
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd 𝑁 ) ∈ 𝑌 ) |
192 |
177 191
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ∈ ( 𝑋 × 𝑌 ) ) |
193 |
192
|
fvresd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) = ( · ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) ) |
194 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑀 gcd 𝑁 ) = 1 ) |
195 |
|
rpmulgcd2 |
⊢ ( ( ( 𝑤 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝑀 gcd 𝑁 ) = 1 ) → ( 𝑤 gcd ( 𝑀 · 𝑁 ) ) = ( ( 𝑤 gcd 𝑀 ) · ( 𝑤 gcd 𝑁 ) ) ) |
196 |
163 165 179 194 195
|
syl31anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd ( 𝑀 · 𝑁 ) ) = ( ( 𝑤 gcd 𝑀 ) · ( 𝑤 gcd 𝑁 ) ) ) |
197 |
|
df-ov |
⊢ ( ( 𝑤 gcd 𝑀 ) · ( 𝑤 gcd 𝑁 ) ) = ( · ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) |
198 |
196 197
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd ( 𝑀 · 𝑁 ) ) = ( · ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) ) |
199 |
160
|
simprbi |
⊢ ( 𝑤 ∈ 𝑍 → 𝑤 ∥ ( 𝑀 · 𝑁 ) ) |
200 |
199
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑤 ∥ ( 𝑀 · 𝑁 ) ) |
201 |
1 2
|
nnmulcld |
⊢ ( 𝜑 → ( 𝑀 · 𝑁 ) ∈ ℕ ) |
202 |
|
gcdeq |
⊢ ( ( 𝑤 ∈ ℕ ∧ ( 𝑀 · 𝑁 ) ∈ ℕ ) → ( ( 𝑤 gcd ( 𝑀 · 𝑁 ) ) = 𝑤 ↔ 𝑤 ∥ ( 𝑀 · 𝑁 ) ) ) |
203 |
161 201 202
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( ( 𝑤 gcd ( 𝑀 · 𝑁 ) ) = 𝑤 ↔ 𝑤 ∥ ( 𝑀 · 𝑁 ) ) ) |
204 |
200 203
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ( 𝑤 gcd ( 𝑀 · 𝑁 ) ) = 𝑤 ) |
205 |
193 198 204
|
3eqtr2rd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → 𝑤 = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) ) |
206 |
|
fveq2 |
⊢ ( 𝑢 = 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 → ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) ) |
207 |
206
|
rspceeqv |
⊢ ( ( 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ∈ ( 𝑋 × 𝑌 ) ∧ 𝑤 = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 〈 ( 𝑤 gcd 𝑀 ) , ( 𝑤 gcd 𝑁 ) 〉 ) ) → ∃ 𝑢 ∈ ( 𝑋 × 𝑌 ) 𝑤 = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) ) |
208 |
192 205 207
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝑍 ) → ∃ 𝑢 ∈ ( 𝑋 × 𝑌 ) 𝑤 = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) ) |
209 |
208
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ 𝑍 ∃ 𝑢 ∈ ( 𝑋 × 𝑌 ) 𝑤 = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) ) |
210 |
|
dffo3 |
⊢ ( ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –onto→ 𝑍 ↔ ( ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) ⟶ 𝑍 ∧ ∀ 𝑤 ∈ 𝑍 ∃ 𝑢 ∈ ( 𝑋 × 𝑌 ) 𝑤 = ( ( · ↾ ( 𝑋 × 𝑌 ) ) ‘ 𝑢 ) ) ) |
211 |
58 209 210
|
sylanbrc |
⊢ ( 𝜑 → ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –onto→ 𝑍 ) |
212 |
|
df-f1o |
⊢ ( ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –1-1-onto→ 𝑍 ↔ ( ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –1-1→ 𝑍 ∧ ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –onto→ 𝑍 ) ) |
213 |
158 211 212
|
sylanbrc |
⊢ ( 𝜑 → ( · ↾ ( 𝑋 × 𝑌 ) ) : ( 𝑋 × 𝑌 ) –1-1-onto→ 𝑍 ) |