Step |
Hyp |
Ref |
Expression |
1 |
|
nnz |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ ) |
2 |
|
gcddvds |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) ) |
3 |
1 2
|
sylan |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ∧ ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) ) |
4 |
3
|
simprd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ) |
5 |
|
breq1 |
⊢ ( ( 𝐴 gcd 𝐵 ) = 𝐴 → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐵 ↔ 𝐴 ∥ 𝐵 ) ) |
6 |
4 5
|
syl5ibcom |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) = 𝐴 → 𝐴 ∥ 𝐵 ) ) |
7 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → 𝐴 ∈ ℤ ) |
8 |
|
iddvds |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∥ 𝐴 ) |
9 |
7 8
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → 𝐴 ∥ 𝐴 ) |
10 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → 𝐵 ∈ ℤ ) |
11 |
|
nnne0 |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ≠ 0 ) |
12 |
|
simpl |
⊢ ( ( 𝐴 = 0 ∧ 𝐵 = 0 ) → 𝐴 = 0 ) |
13 |
12
|
necon3ai |
⊢ ( 𝐴 ≠ 0 → ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) |
14 |
11 13
|
syl |
⊢ ( 𝐴 ∈ ℕ → ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) |
16 |
|
dvdslegcd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) ∧ ¬ ( 𝐴 = 0 ∧ 𝐵 = 0 ) ) → ( ( 𝐴 ∥ 𝐴 ∧ 𝐴 ∥ 𝐵 ) → 𝐴 ≤ ( 𝐴 gcd 𝐵 ) ) ) |
17 |
7 7 10 15 16
|
syl31anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 ∥ 𝐴 ∧ 𝐴 ∥ 𝐵 ) → 𝐴 ≤ ( 𝐴 gcd 𝐵 ) ) ) |
18 |
9 17
|
mpand |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ∥ 𝐵 → 𝐴 ≤ ( 𝐴 gcd 𝐵 ) ) ) |
19 |
3
|
simpld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ∥ 𝐴 ) |
20 |
|
gcdcl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ0 ) |
21 |
1 20
|
sylan |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ∈ ℕ0 ) |
22 |
21
|
nn0zd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ∈ ℤ ) |
23 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → 𝐴 ∈ ℕ ) |
24 |
|
dvdsle |
⊢ ( ( ( 𝐴 gcd 𝐵 ) ∈ ℤ ∧ 𝐴 ∈ ℕ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 → ( 𝐴 gcd 𝐵 ) ≤ 𝐴 ) ) |
25 |
22 23 24
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) ∥ 𝐴 → ( 𝐴 gcd 𝐵 ) ≤ 𝐴 ) ) |
26 |
19 25
|
mpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ≤ 𝐴 ) |
27 |
18 26
|
jctild |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ∥ 𝐵 → ( ( 𝐴 gcd 𝐵 ) ≤ 𝐴 ∧ 𝐴 ≤ ( 𝐴 gcd 𝐵 ) ) ) ) |
28 |
21
|
nn0red |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 gcd 𝐵 ) ∈ ℝ ) |
29 |
|
nnre |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ ) |
30 |
29
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → 𝐴 ∈ ℝ ) |
31 |
28 30
|
letri3d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) = 𝐴 ↔ ( ( 𝐴 gcd 𝐵 ) ≤ 𝐴 ∧ 𝐴 ≤ ( 𝐴 gcd 𝐵 ) ) ) ) |
32 |
27 31
|
sylibrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 ∥ 𝐵 → ( 𝐴 gcd 𝐵 ) = 𝐴 ) ) |
33 |
6 32
|
impbid |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℤ ) → ( ( 𝐴 gcd 𝐵 ) = 𝐴 ↔ 𝐴 ∥ 𝐵 ) ) |