Step |
Hyp |
Ref |
Expression |
1 |
|
gcddvds |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) ) |
2 |
1
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) ) |
3 |
2
|
simpld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ) |
4 |
|
gcd2n0cl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ ) |
5 |
|
nnz |
⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ℕ → ( 𝐴 gcd 𝐵 ) ∈ ℤ ) |
6 |
|
nnne0 |
⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ℕ → ( 𝐴 gcd 𝐵 ) ≠ 0 ) |
7 |
5 6
|
jca |
⊢ ( ( 𝐴 gcd 𝐵 ) ∈ ℕ → ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ≠ 0 ) ) |
8 |
4 7
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ≠ 0 ) ) |
9 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℤ ) |
10 |
|
df-3an |
⊢ ( ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ≠ 0 ∧ 𝐴 ∈ ℤ ) ↔ ( ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ≠ 0 ) ∧ 𝐴 ∈ ℤ ) ) |
11 |
8 9 10
|
sylanbrc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ≠ 0 ∧ 𝐴 ∈ ℤ ) ) |
12 |
|
dvdsval2 |
⊢ ( ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ ( 𝐴 gcd 𝐵 ) ≠ 0 ∧ 𝐴 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ↔ ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ∈ ℤ ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ↔ ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ∈ ℤ ) ) |
14 |
3 13
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / ( 𝐴 gcd 𝐵 ) ) ∈ ℤ ) |