Step |
Hyp |
Ref |
Expression |
1 |
|
gcdn0cl |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑀 gcd 𝑁 ) ∈ ℕ ) |
2 |
1
|
nnne0d |
⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) → ( 𝑀 gcd 𝑁 ) ≠ 0 ) |
3 |
2
|
ex |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ¬ ( 𝑀 = 0 ∧ 𝑁 = 0 ) → ( 𝑀 gcd 𝑁 ) ≠ 0 ) ) |
4 |
3
|
necon4bd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 gcd 𝑁 ) = 0 → ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) ) |
5 |
|
oveq12 |
⊢ ( ( 𝑀 = 0 ∧ 𝑁 = 0 ) → ( 𝑀 gcd 𝑁 ) = ( 0 gcd 0 ) ) |
6 |
|
gcd0val |
⊢ ( 0 gcd 0 ) = 0 |
7 |
5 6
|
eqtrdi |
⊢ ( ( 𝑀 = 0 ∧ 𝑁 = 0 ) → ( 𝑀 gcd 𝑁 ) = 0 ) |
8 |
4 7
|
impbid1 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 gcd 𝑁 ) = 0 ↔ ( 𝑀 = 0 ∧ 𝑁 = 0 ) ) ) |