Step |
Hyp |
Ref |
Expression |
1 |
|
gcdcl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ0 ) |
2 |
|
0re |
⊢ 0 ∈ ℝ |
3 |
|
nn0re |
⊢ ( ( 𝑀 gcd 𝑁 ) ∈ ℕ0 → ( 𝑀 gcd 𝑁 ) ∈ ℝ ) |
4 |
|
nn0ge0 |
⊢ ( ( 𝑀 gcd 𝑁 ) ∈ ℕ0 → 0 ≤ ( 𝑀 gcd 𝑁 ) ) |
5 |
|
leltne |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝑀 gcd 𝑁 ) ∈ ℝ ∧ 0 ≤ ( 𝑀 gcd 𝑁 ) ) → ( 0 < ( 𝑀 gcd 𝑁 ) ↔ ( 𝑀 gcd 𝑁 ) ≠ 0 ) ) |
6 |
2 3 4 5
|
mp3an2i |
⊢ ( ( 𝑀 gcd 𝑁 ) ∈ ℕ0 → ( 0 < ( 𝑀 gcd 𝑁 ) ↔ ( 𝑀 gcd 𝑁 ) ≠ 0 ) ) |
7 |
1 6
|
syl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 0 < ( 𝑀 gcd 𝑁 ) ↔ ( 𝑀 gcd 𝑁 ) ≠ 0 ) ) |
8 |
|
gcdeq0 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 gcd 𝑁 ) = 0 ↔ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) ) |
9 |
8
|
necon3abid |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 gcd 𝑁 ) ≠ 0 ↔ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) ) |
10 |
7 9
|
bitr2d |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ↔ 0 < ( 𝑀 gcd 𝑁 ) ) ) |