Step |
Hyp |
Ref |
Expression |
1 |
|
lcmineqlem14.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
2 |
|
lcmineqlem14.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
3 |
|
lcmineqlem14.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℕ ) |
4 |
|
lcmineqlem14.4 |
⊢ ( 𝜑 → 𝐷 ∈ ℕ ) |
5 |
|
lcmineqlem14.5 |
⊢ ( 𝜑 → 𝐸 ∈ ℕ ) |
6 |
|
lcmineqlem14.6 |
⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ∥ 𝐷 ) |
7 |
|
lcmineqlem14.7 |
⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) ∥ 𝐸 ) |
8 |
|
lcmineqlem14.8 |
⊢ ( 𝜑 → 𝐷 ∥ 𝐸 ) |
9 |
|
lcmineqlem14.9 |
⊢ ( 𝜑 → ( 𝐴 gcd 𝐵 ) = 1 ) |
10 |
1
|
nnzd |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
11 |
2
|
nnzd |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
12 |
2 3 5
|
nnproddivdvdsd |
⊢ ( 𝜑 → ( ( 𝐵 · 𝐶 ) ∥ 𝐸 ↔ 𝐵 ∥ ( 𝐸 / 𝐶 ) ) ) |
13 |
7 12
|
mpbid |
⊢ ( 𝜑 → 𝐵 ∥ ( 𝐸 / 𝐶 ) ) |
14 |
|
dvdszrcl |
⊢ ( 𝐵 ∥ ( 𝐸 / 𝐶 ) → ( 𝐵 ∈ ℤ ∧ ( 𝐸 / 𝐶 ) ∈ ℤ ) ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → ( 𝐵 ∈ ℤ ∧ ( 𝐸 / 𝐶 ) ∈ ℤ ) ) |
16 |
15
|
simprd |
⊢ ( 𝜑 → ( 𝐸 / 𝐶 ) ∈ ℤ ) |
17 |
3
|
nnzd |
⊢ ( 𝜑 → 𝐶 ∈ ℤ ) |
18 |
10 17
|
zmulcld |
⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ∈ ℤ ) |
19 |
4
|
nnzd |
⊢ ( 𝜑 → 𝐷 ∈ ℤ ) |
20 |
5
|
nnzd |
⊢ ( 𝜑 → 𝐸 ∈ ℤ ) |
21 |
18 19 20 6 8
|
dvdstrd |
⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) ∥ 𝐸 ) |
22 |
1 3 5
|
nnproddivdvdsd |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) ∥ 𝐸 ↔ 𝐴 ∥ ( 𝐸 / 𝐶 ) ) ) |
23 |
21 22
|
mpbid |
⊢ ( 𝜑 → 𝐴 ∥ ( 𝐸 / 𝐶 ) ) |
24 |
10 11 16 9 23 13
|
coprmdvds2d |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∥ ( 𝐸 / 𝐶 ) ) |
25 |
1 2
|
nnmulcld |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℕ ) |
26 |
25 3 5
|
nnproddivdvdsd |
⊢ ( 𝜑 → ( ( ( 𝐴 · 𝐵 ) · 𝐶 ) ∥ 𝐸 ↔ ( 𝐴 · 𝐵 ) ∥ ( 𝐸 / 𝐶 ) ) ) |
27 |
24 26
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · 𝐶 ) ∥ 𝐸 ) |