Step |
Hyp |
Ref |
Expression |
1 |
|
1z |
⊢ 1 ∈ ℤ |
2 |
|
gcdaddm |
⊢ ( ( 1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) = ( 𝑀 gcd ( 𝑁 + ( 1 · 𝑀 ) ) ) ) |
3 |
1 2
|
mp3an1 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) = ( 𝑀 gcd ( 𝑁 + ( 1 · 𝑀 ) ) ) ) |
4 |
|
zcn |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ ) |
5 |
|
mulid2 |
⊢ ( 𝑀 ∈ ℂ → ( 1 · 𝑀 ) = 𝑀 ) |
6 |
5
|
oveq2d |
⊢ ( 𝑀 ∈ ℂ → ( 𝑁 + ( 1 · 𝑀 ) ) = ( 𝑁 + 𝑀 ) ) |
7 |
6
|
oveq2d |
⊢ ( 𝑀 ∈ ℂ → ( 𝑀 gcd ( 𝑁 + ( 1 · 𝑀 ) ) ) = ( 𝑀 gcd ( 𝑁 + 𝑀 ) ) ) |
8 |
4 7
|
syl |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 gcd ( 𝑁 + ( 1 · 𝑀 ) ) ) = ( 𝑀 gcd ( 𝑁 + 𝑀 ) ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd ( 𝑁 + ( 1 · 𝑀 ) ) ) = ( 𝑀 gcd ( 𝑁 + 𝑀 ) ) ) |
10 |
3 9
|
eqtrd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) = ( 𝑀 gcd ( 𝑁 + 𝑀 ) ) ) |