Step |
Hyp |
Ref |
Expression |
1 |
|
1z |
⊢ 1 ∈ ℤ |
2 |
|
0z |
⊢ 0 ∈ ℤ |
3 |
|
gcdaddm |
⊢ ( ( 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 0 ∈ ℤ ) → ( 𝑁 gcd 0 ) = ( 𝑁 gcd ( 0 + ( 1 · 𝑁 ) ) ) ) |
4 |
1 2 3
|
mp3an13 |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 gcd 0 ) = ( 𝑁 gcd ( 0 + ( 1 · 𝑁 ) ) ) ) |
5 |
|
gcdid0 |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 gcd 0 ) = ( abs ‘ 𝑁 ) ) |
6 |
|
zcn |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℂ ) |
7 |
|
mulid2 |
⊢ ( 𝑁 ∈ ℂ → ( 1 · 𝑁 ) = 𝑁 ) |
8 |
7
|
oveq2d |
⊢ ( 𝑁 ∈ ℂ → ( 0 + ( 1 · 𝑁 ) ) = ( 0 + 𝑁 ) ) |
9 |
|
addid2 |
⊢ ( 𝑁 ∈ ℂ → ( 0 + 𝑁 ) = 𝑁 ) |
10 |
8 9
|
eqtrd |
⊢ ( 𝑁 ∈ ℂ → ( 0 + ( 1 · 𝑁 ) ) = 𝑁 ) |
11 |
6 10
|
syl |
⊢ ( 𝑁 ∈ ℤ → ( 0 + ( 1 · 𝑁 ) ) = 𝑁 ) |
12 |
11
|
oveq2d |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 gcd ( 0 + ( 1 · 𝑁 ) ) ) = ( 𝑁 gcd 𝑁 ) ) |
13 |
4 5 12
|
3eqtr3rd |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 gcd 𝑁 ) = ( abs ‘ 𝑁 ) ) |