Step |
Hyp |
Ref |
Expression |
1 |
|
zex |
⊢ ℤ ∈ V |
2 |
|
nn0ssz |
⊢ ℕ0 ⊆ ℤ |
3 |
|
mapss |
⊢ ( ( ℤ ∈ V ∧ ℕ0 ⊆ ℤ ) → ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ⊆ ( ℤ ↑m ( 1 ... 𝑁 ) ) ) |
4 |
1 2 3
|
mp2an |
⊢ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ⊆ ( ℤ ↑m ( 1 ... 𝑁 ) ) |
5 |
|
mzpf |
⊢ ( ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) → ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) : ( ℤ ↑m ( 1 ... 𝑁 ) ) ⟶ ℤ ) |
6 |
|
eqid |
⊢ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) = ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) |
7 |
6
|
fmpt |
⊢ ( ∀ 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ↔ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) : ( ℤ ↑m ( 1 ... 𝑁 ) ) ⟶ ℤ ) |
8 |
5 7
|
sylibr |
⊢ ( ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ) |
9 |
|
ssralv |
⊢ ( ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ⊆ ( ℤ ↑m ( 1 ... 𝑁 ) ) → ( ∀ 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ → ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ) ) |
10 |
4 8 9
|
mpsyl |
⊢ ( ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ) |