Step |
Hyp |
Ref |
Expression |
1 |
|
qqhval2.0 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
qqhval2.1 |
⊢ / = ( /r ‘ 𝑅 ) |
3 |
|
qqhval2.2 |
⊢ 𝐿 = ( ℤRHom ‘ 𝑅 ) |
4 |
|
drngring |
⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring ) |
5 |
3
|
zrhrhm |
⊢ ( 𝑅 ∈ Ring → 𝐿 ∈ ( ℤring RingHom 𝑅 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝑅 ∈ DivRing → 𝐿 ∈ ( ℤring RingHom 𝑅 ) ) |
7 |
6
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝐿 ∈ ( ℤring RingHom 𝑅 ) ) |
8 |
|
simpr1 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑋 ∈ ℤ ) |
9 |
|
simpr2 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑌 ∈ ℤ ) |
10 |
8 9
|
gcdcld |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 gcd 𝑌 ) ∈ ℕ0 ) |
11 |
10
|
nn0zd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 gcd 𝑌 ) ∈ ℤ ) |
12 |
|
simpr3 |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑌 ≠ 0 ) |
13 |
|
gcdeq0 |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ) → ( ( 𝑋 gcd 𝑌 ) = 0 ↔ ( 𝑋 = 0 ∧ 𝑌 = 0 ) ) ) |
14 |
13
|
simplbda |
⊢ ( ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ) ∧ ( 𝑋 gcd 𝑌 ) = 0 ) → 𝑌 = 0 ) |
15 |
14
|
ex |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ) → ( ( 𝑋 gcd 𝑌 ) = 0 → 𝑌 = 0 ) ) |
16 |
15
|
necon3d |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ) → ( 𝑌 ≠ 0 → ( 𝑋 gcd 𝑌 ) ≠ 0 ) ) |
17 |
16
|
imp |
⊢ ( ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ) ∧ 𝑌 ≠ 0 ) → ( 𝑋 gcd 𝑌 ) ≠ 0 ) |
18 |
8 9 12 17
|
syl21anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 gcd 𝑌 ) ≠ 0 ) |
19 |
|
gcddvds |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ) → ( ( 𝑋 gcd 𝑌 ) ∥ 𝑋 ∧ ( 𝑋 gcd 𝑌 ) ∥ 𝑌 ) ) |
20 |
8 9 19
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝑋 gcd 𝑌 ) ∥ 𝑋 ∧ ( 𝑋 gcd 𝑌 ) ∥ 𝑌 ) ) |
21 |
20
|
simpld |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 gcd 𝑌 ) ∥ 𝑋 ) |
22 |
|
dvdsval2 |
⊢ ( ( ( 𝑋 gcd 𝑌 ) ∈ ℤ ∧ ( 𝑋 gcd 𝑌 ) ≠ 0 ∧ 𝑋 ∈ ℤ ) → ( ( 𝑋 gcd 𝑌 ) ∥ 𝑋 ↔ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) ) |
23 |
22
|
biimpa |
⊢ ( ( ( ( 𝑋 gcd 𝑌 ) ∈ ℤ ∧ ( 𝑋 gcd 𝑌 ) ≠ 0 ∧ 𝑋 ∈ ℤ ) ∧ ( 𝑋 gcd 𝑌 ) ∥ 𝑋 ) → ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) |
24 |
11 18 8 21 23
|
syl31anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) |
25 |
20
|
simprd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 gcd 𝑌 ) ∥ 𝑌 ) |
26 |
|
dvdsval2 |
⊢ ( ( ( 𝑋 gcd 𝑌 ) ∈ ℤ ∧ ( 𝑋 gcd 𝑌 ) ≠ 0 ∧ 𝑌 ∈ ℤ ) → ( ( 𝑋 gcd 𝑌 ) ∥ 𝑌 ↔ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) ) |
27 |
26
|
biimpa |
⊢ ( ( ( ( 𝑋 gcd 𝑌 ) ∈ ℤ ∧ ( 𝑋 gcd 𝑌 ) ≠ 0 ∧ 𝑌 ∈ ℤ ) ∧ ( 𝑋 gcd 𝑌 ) ∥ 𝑌 ) → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) |
28 |
11 18 9 25 27
|
syl31anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) |
29 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
30 |
29 1
|
rhmf |
⊢ ( 𝐿 ∈ ( ℤring RingHom 𝑅 ) → 𝐿 : ℤ ⟶ 𝐵 ) |
31 |
7 30
|
syl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝐿 : ℤ ⟶ 𝐵 ) |
32 |
31 28
|
ffvelrnd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ 𝐵 ) |
33 |
31
|
ffnd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝐿 Fn ℤ ) |
34 |
9
|
zcnd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑌 ∈ ℂ ) |
35 |
11
|
zcnd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑋 gcd 𝑌 ) ∈ ℂ ) |
36 |
34 35 12 18
|
divne0d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ≠ 0 ) |
37 |
|
ovex |
⊢ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ V |
38 |
37
|
elsn |
⊢ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ { 0 } ↔ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) = 0 ) |
39 |
38
|
necon3bbii |
⊢ ( ¬ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ { 0 } ↔ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ≠ 0 ) |
40 |
36 39
|
sylibr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ¬ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ { 0 } ) |
41 |
4
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑅 ∈ Ring ) |
42 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( chr ‘ 𝑅 ) = 0 ) |
43 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
44 |
1 3 43
|
zrhker |
⊢ ( 𝑅 ∈ Ring → ( ( chr ‘ 𝑅 ) = 0 ↔ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) = { 0 } ) ) |
45 |
44
|
biimpa |
⊢ ( ( 𝑅 ∈ Ring ∧ ( chr ‘ 𝑅 ) = 0 ) → ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) = { 0 } ) |
46 |
41 42 45
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) = { 0 } ) |
47 |
40 46
|
neleqtrrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ¬ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) |
48 |
|
elpreima |
⊢ ( 𝐿 Fn ℤ → ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ↔ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ∧ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ { ( 0g ‘ 𝑅 ) } ) ) ) |
49 |
48
|
baibd |
⊢ ( ( 𝐿 Fn ℤ ∧ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) → ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ↔ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ { ( 0g ‘ 𝑅 ) } ) ) |
50 |
49
|
biimprd |
⊢ ( ( 𝐿 Fn ℤ ∧ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) → ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ { ( 0g ‘ 𝑅 ) } → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) ) |
51 |
50
|
con3dimp |
⊢ ( ( ( 𝐿 Fn ℤ ∧ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) ∧ ¬ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) → ¬ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ { ( 0g ‘ 𝑅 ) } ) |
52 |
|
fvex |
⊢ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ V |
53 |
52
|
elsn |
⊢ ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ { ( 0g ‘ 𝑅 ) } ↔ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) = ( 0g ‘ 𝑅 ) ) |
54 |
53
|
necon3bbii |
⊢ ( ¬ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ { ( 0g ‘ 𝑅 ) } ↔ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ≠ ( 0g ‘ 𝑅 ) ) |
55 |
51 54
|
sylib |
⊢ ( ( ( 𝐿 Fn ℤ ∧ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) ∧ ¬ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) → ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ≠ ( 0g ‘ 𝑅 ) ) |
56 |
33 28 47 55
|
syl21anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ≠ ( 0g ‘ 𝑅 ) ) |
57 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
58 |
1 57 43
|
drngunit |
⊢ ( 𝑅 ∈ DivRing → ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ ( Unit ‘ 𝑅 ) ↔ ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ 𝐵 ∧ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ≠ ( 0g ‘ 𝑅 ) ) ) ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ ( Unit ‘ 𝑅 ) ↔ ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ 𝐵 ∧ ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ≠ ( 0g ‘ 𝑅 ) ) ) ) |
60 |
32 56 59
|
mpbir2and |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ ( Unit ‘ 𝑅 ) ) |
61 |
31 11
|
ffvelrnd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ 𝐵 ) |
62 |
|
ovex |
⊢ ( 𝑋 gcd 𝑌 ) ∈ V |
63 |
62
|
elsn |
⊢ ( ( 𝑋 gcd 𝑌 ) ∈ { 0 } ↔ ( 𝑋 gcd 𝑌 ) = 0 ) |
64 |
63
|
necon3bbii |
⊢ ( ¬ ( 𝑋 gcd 𝑌 ) ∈ { 0 } ↔ ( 𝑋 gcd 𝑌 ) ≠ 0 ) |
65 |
18 64
|
sylibr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ¬ ( 𝑋 gcd 𝑌 ) ∈ { 0 } ) |
66 |
65 46
|
neleqtrrd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ¬ ( 𝑋 gcd 𝑌 ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) |
67 |
|
elpreima |
⊢ ( 𝐿 Fn ℤ → ( ( 𝑋 gcd 𝑌 ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ↔ ( ( 𝑋 gcd 𝑌 ) ∈ ℤ ∧ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ { ( 0g ‘ 𝑅 ) } ) ) ) |
68 |
67
|
baibd |
⊢ ( ( 𝐿 Fn ℤ ∧ ( 𝑋 gcd 𝑌 ) ∈ ℤ ) → ( ( 𝑋 gcd 𝑌 ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ↔ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ { ( 0g ‘ 𝑅 ) } ) ) |
69 |
68
|
biimprd |
⊢ ( ( 𝐿 Fn ℤ ∧ ( 𝑋 gcd 𝑌 ) ∈ ℤ ) → ( ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ { ( 0g ‘ 𝑅 ) } → ( 𝑋 gcd 𝑌 ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) ) |
70 |
69
|
con3dimp |
⊢ ( ( ( 𝐿 Fn ℤ ∧ ( 𝑋 gcd 𝑌 ) ∈ ℤ ) ∧ ¬ ( 𝑋 gcd 𝑌 ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) → ¬ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ { ( 0g ‘ 𝑅 ) } ) |
71 |
|
fvex |
⊢ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ V |
72 |
71
|
elsn |
⊢ ( ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ { ( 0g ‘ 𝑅 ) } ↔ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) = ( 0g ‘ 𝑅 ) ) |
73 |
72
|
necon3bbii |
⊢ ( ¬ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ { ( 0g ‘ 𝑅 ) } ↔ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ≠ ( 0g ‘ 𝑅 ) ) |
74 |
70 73
|
sylib |
⊢ ( ( ( 𝐿 Fn ℤ ∧ ( 𝑋 gcd 𝑌 ) ∈ ℤ ) ∧ ¬ ( 𝑋 gcd 𝑌 ) ∈ ( ◡ 𝐿 “ { ( 0g ‘ 𝑅 ) } ) ) → ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ≠ ( 0g ‘ 𝑅 ) ) |
75 |
33 11 66 74
|
syl21anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ≠ ( 0g ‘ 𝑅 ) ) |
76 |
1 57 43
|
drngunit |
⊢ ( 𝑅 ∈ DivRing → ( ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ ( Unit ‘ 𝑅 ) ↔ ( ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ 𝐵 ∧ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ≠ ( 0g ‘ 𝑅 ) ) ) ) |
77 |
76
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ ( Unit ‘ 𝑅 ) ↔ ( ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ 𝐵 ∧ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ≠ ( 0g ‘ 𝑅 ) ) ) ) |
78 |
61 75 77
|
mpbir2and |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ ( Unit ‘ 𝑅 ) ) |
79 |
|
zringmulr |
⊢ · = ( .r ‘ ℤring ) |
80 |
57 29 2 79
|
rhmdvd |
⊢ ( ( 𝐿 ∈ ( ℤring RingHom 𝑅 ) ∧ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ∧ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ∧ ( 𝑋 gcd 𝑌 ) ∈ ℤ ) ∧ ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ ( Unit ‘ 𝑅 ) ∧ ( 𝐿 ‘ ( 𝑋 gcd 𝑌 ) ) ∈ ( Unit ‘ 𝑅 ) ) ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) ) ) |
81 |
7 24 28 11 60 78 80
|
syl132anc |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) ) ) |
82 |
|
divnumden |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℕ ) → ( ( numer ‘ ( 𝑋 / 𝑌 ) ) = ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∧ ( denom ‘ ( 𝑋 / 𝑌 ) ) = ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) |
83 |
8 82
|
sylan |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( ( numer ‘ ( 𝑋 / 𝑌 ) ) = ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∧ ( denom ‘ ( 𝑋 / 𝑌 ) ) = ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) |
84 |
83
|
simpld |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( numer ‘ ( 𝑋 / 𝑌 ) ) = ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) |
85 |
84
|
eqcomd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) = ( numer ‘ ( 𝑋 / 𝑌 ) ) ) |
86 |
85
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) = ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) ) |
87 |
83
|
simprd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( denom ‘ ( 𝑋 / 𝑌 ) ) = ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) |
88 |
87
|
eqcomd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) = ( denom ‘ ( 𝑋 / 𝑌 ) ) ) |
89 |
88
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) = ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) |
90 |
86 89
|
oveq12d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ 𝑌 ∈ ℕ ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) ) |
91 |
24
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) |
92 |
91
|
zcnd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℂ ) |
93 |
92
|
mulm1d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( - 1 · ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) = - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) |
94 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
95 |
94
|
a1i |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → - 1 ∈ ℂ ) |
96 |
95 92
|
mulcomd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( - 1 · ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) = ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) |
97 |
93 96
|
eqtr3d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) = ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) |
98 |
97
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝐿 ‘ - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) = ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) ) |
99 |
28
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ) |
100 |
99
|
zcnd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℂ ) |
101 |
100
|
mulm1d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( - 1 · ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) = - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) |
102 |
95 100
|
mulcomd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( - 1 · ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) = ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) |
103 |
101 102
|
eqtr3d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) = ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) |
104 |
103
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝐿 ‘ - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) = ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) ) |
105 |
98 104
|
oveq12d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( ( 𝐿 ‘ - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) / ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) ) ) |
106 |
8
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → 𝑋 ∈ ℤ ) |
107 |
9
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → 𝑌 ∈ ℤ ) |
108 |
|
simpr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → - 𝑌 ∈ ℕ ) |
109 |
|
divnumden2 |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ - 𝑌 ∈ ℕ ) → ( ( numer ‘ ( 𝑋 / 𝑌 ) ) = - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∧ ( denom ‘ ( 𝑋 / 𝑌 ) ) = - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) |
110 |
106 107 108 109
|
syl3anc |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( ( numer ‘ ( 𝑋 / 𝑌 ) ) = - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∧ ( denom ‘ ( 𝑋 / 𝑌 ) ) = - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) |
111 |
110
|
simpld |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( numer ‘ ( 𝑋 / 𝑌 ) ) = - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) |
112 |
111
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) = ( 𝐿 ‘ - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) ) |
113 |
110
|
simprd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( denom ‘ ( 𝑋 / 𝑌 ) ) = - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) |
114 |
113
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) = ( 𝐿 ‘ - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) |
115 |
112 114
|
oveq12d |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) = ( ( 𝐿 ‘ - ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ - ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) ) |
116 |
7
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → 𝐿 ∈ ( ℤring RingHom 𝑅 ) ) |
117 |
|
1zzd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → 1 ∈ ℤ ) |
118 |
117
|
znegcld |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → - 1 ∈ ℤ ) |
119 |
60
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ ( Unit ‘ 𝑅 ) ) |
120 |
|
neg1z |
⊢ - 1 ∈ ℤ |
121 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
122 |
121
|
absnegi |
⊢ ( abs ‘ - 1 ) = ( abs ‘ 1 ) |
123 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
124 |
122 123
|
eqtri |
⊢ ( abs ‘ - 1 ) = 1 |
125 |
|
zringunit |
⊢ ( - 1 ∈ ( Unit ‘ ℤring ) ↔ ( - 1 ∈ ℤ ∧ ( abs ‘ - 1 ) = 1 ) ) |
126 |
120 124 125
|
mpbir2an |
⊢ - 1 ∈ ( Unit ‘ ℤring ) |
127 |
126
|
a1i |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → - 1 ∈ ( Unit ‘ ℤring ) ) |
128 |
|
elrhmunit |
⊢ ( ( 𝐿 ∈ ( ℤring RingHom 𝑅 ) ∧ - 1 ∈ ( Unit ‘ ℤring ) ) → ( 𝐿 ‘ - 1 ) ∈ ( Unit ‘ 𝑅 ) ) |
129 |
116 127 128
|
syl2anc |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( 𝐿 ‘ - 1 ) ∈ ( Unit ‘ 𝑅 ) ) |
130 |
57 29 2 79
|
rhmdvd |
⊢ ( ( 𝐿 ∈ ( ℤring RingHom 𝑅 ) ∧ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ∧ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ∈ ℤ ∧ - 1 ∈ ℤ ) ∧ ( ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ∈ ( Unit ‘ 𝑅 ) ∧ ( 𝐿 ‘ - 1 ) ∈ ( Unit ‘ 𝑅 ) ) ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) / ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) ) ) |
131 |
116 91 99 118 119 129 130
|
syl132anc |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) / ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · - 1 ) ) ) ) |
132 |
105 115 131
|
3eqtr4rd |
⊢ ( ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) ∧ - 𝑌 ∈ ℕ ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) ) |
133 |
|
simp3 |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → 𝑌 ≠ 0 ) |
134 |
133
|
neneqd |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → ¬ 𝑌 = 0 ) |
135 |
|
simp2 |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ ℤ ) |
136 |
|
elz |
⊢ ( 𝑌 ∈ ℤ ↔ ( 𝑌 ∈ ℝ ∧ ( 𝑌 = 0 ∨ 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) ) |
137 |
135 136
|
sylib |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → ( 𝑌 ∈ ℝ ∧ ( 𝑌 = 0 ∨ 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) ) |
138 |
137
|
simprd |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → ( 𝑌 = 0 ∨ 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) |
139 |
|
3orass |
⊢ ( ( 𝑌 = 0 ∨ 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ↔ ( 𝑌 = 0 ∨ ( 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) ) |
140 |
138 139
|
sylib |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → ( 𝑌 = 0 ∨ ( 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) ) |
141 |
|
orel1 |
⊢ ( ¬ 𝑌 = 0 → ( ( 𝑌 = 0 ∨ ( 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) → ( 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) ) |
142 |
134 140 141
|
sylc |
⊢ ( ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) → ( 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) |
143 |
142
|
adantl |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝑌 ∈ ℕ ∨ - 𝑌 ∈ ℕ ) ) |
144 |
90 132 143
|
mpjaodan |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) ) |
145 |
8
|
zcnd |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → 𝑋 ∈ ℂ ) |
146 |
145 35 18
|
divcan1d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) = 𝑋 ) |
147 |
146
|
fveq2d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) = ( 𝐿 ‘ 𝑋 ) ) |
148 |
34 35 18
|
divcan1d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) = 𝑌 ) |
149 |
148
|
fveq2d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) = ( 𝐿 ‘ 𝑌 ) ) |
150 |
147 149
|
oveq12d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( ( 𝑋 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) / ( 𝐿 ‘ ( ( 𝑌 / ( 𝑋 gcd 𝑌 ) ) · ( 𝑋 gcd 𝑌 ) ) ) ) = ( ( 𝐿 ‘ 𝑋 ) / ( 𝐿 ‘ 𝑌 ) ) ) |
151 |
81 144 150
|
3eqtr3d |
⊢ ( ( ( 𝑅 ∈ DivRing ∧ ( chr ‘ 𝑅 ) = 0 ) ∧ ( 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0 ) ) → ( ( 𝐿 ‘ ( numer ‘ ( 𝑋 / 𝑌 ) ) ) / ( 𝐿 ‘ ( denom ‘ ( 𝑋 / 𝑌 ) ) ) ) = ( ( 𝐿 ‘ 𝑋 ) / ( 𝐿 ‘ 𝑌 ) ) ) |