Step |
Hyp |
Ref |
Expression |
1 |
|
gcdmultiple |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝐾 gcd ( 𝐾 · 𝑁 ) ) = 𝐾 ) |
2 |
1
|
3adant2 |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝐾 gcd ( 𝐾 · 𝑁 ) ) = 𝐾 ) |
3 |
2
|
oveq1d |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐾 gcd ( 𝐾 · 𝑁 ) ) gcd ( 𝑀 · 𝑁 ) ) = ( 𝐾 gcd ( 𝑀 · 𝑁 ) ) ) |
4 |
|
nnz |
⊢ ( 𝐾 ∈ ℕ → 𝐾 ∈ ℤ ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → 𝐾 ∈ ℤ ) |
6 |
|
nnz |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) |
7 |
|
zmulcl |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 · 𝑁 ) ∈ ℤ ) |
8 |
4 6 7
|
syl2an |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝐾 · 𝑁 ) ∈ ℤ ) |
9 |
8
|
3adant2 |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝐾 · 𝑁 ) ∈ ℤ ) |
10 |
|
nnz |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ ) |
11 |
|
zmulcl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 · 𝑁 ) ∈ ℤ ) |
12 |
10 6 11
|
syl2an |
⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 · 𝑁 ) ∈ ℤ ) |
13 |
12
|
3adant1 |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 · 𝑁 ) ∈ ℤ ) |
14 |
|
gcdass |
⊢ ( ( 𝐾 ∈ ℤ ∧ ( 𝐾 · 𝑁 ) ∈ ℤ ∧ ( 𝑀 · 𝑁 ) ∈ ℤ ) → ( ( 𝐾 gcd ( 𝐾 · 𝑁 ) ) gcd ( 𝑀 · 𝑁 ) ) = ( 𝐾 gcd ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) ) ) |
15 |
5 9 13 14
|
syl3anc |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐾 gcd ( 𝐾 · 𝑁 ) ) gcd ( 𝑀 · 𝑁 ) ) = ( 𝐾 gcd ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) ) ) |
16 |
3 15
|
eqtr3d |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝐾 gcd ( 𝑀 · 𝑁 ) ) = ( 𝐾 gcd ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) ) ) |
17 |
16
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( 𝐾 gcd ( 𝑀 · 𝑁 ) ) = ( 𝐾 gcd ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) ) ) |
18 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
19 |
|
mulgcdr |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) = ( ( 𝐾 gcd 𝑀 ) · 𝑁 ) ) |
20 |
4 10 18 19
|
syl3an |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) = ( ( 𝐾 gcd 𝑀 ) · 𝑁 ) ) |
21 |
|
oveq1 |
⊢ ( ( 𝐾 gcd 𝑀 ) = 1 → ( ( 𝐾 gcd 𝑀 ) · 𝑁 ) = ( 1 · 𝑁 ) ) |
22 |
20 21
|
sylan9eq |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) = ( 1 · 𝑁 ) ) |
23 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
24 |
23
|
3ad2ant3 |
⊢ ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ℂ ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → 𝑁 ∈ ℂ ) |
26 |
25
|
mulid2d |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( 1 · 𝑁 ) = 𝑁 ) |
27 |
22 26
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) = 𝑁 ) |
28 |
27
|
oveq2d |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( 𝐾 gcd ( ( 𝐾 · 𝑁 ) gcd ( 𝑀 · 𝑁 ) ) ) = ( 𝐾 gcd 𝑁 ) ) |
29 |
17 28
|
eqtrd |
⊢ ( ( ( 𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) ∧ ( 𝐾 gcd 𝑀 ) = 1 ) → ( 𝐾 gcd ( 𝑀 · 𝑁 ) ) = ( 𝐾 gcd 𝑁 ) ) |