Step |
Hyp |
Ref |
Expression |
1 |
|
rabdiophlem1 |
⊢ ( ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ) |
2 |
|
eluz |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐴 ∈ ℤ ) → ( 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝐴 ) ) |
3 |
2
|
ex |
⊢ ( 𝑀 ∈ ℤ → ( 𝐴 ∈ ℤ → ( 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝐴 ) ) ) |
4 |
3
|
ralimdv |
⊢ ( 𝑀 ∈ ℤ → ( ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ → ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ( 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝐴 ) ) ) |
5 |
4
|
imp |
⊢ ( ( 𝑀 ∈ ℤ ∧ ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ) → ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ( 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝐴 ) ) |
6 |
1 5
|
sylan2 |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) → ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ( 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝐴 ) ) |
7 |
|
rabbi |
⊢ ( ∀ 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ( 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ 𝑀 ≤ 𝐴 ) ↔ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) } = { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝑀 ≤ 𝐴 } ) |
8 |
6 7
|
sylib |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) → { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) } = { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝑀 ≤ 𝐴 } ) |
9 |
8
|
3adant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) → { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) } = { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝑀 ≤ 𝐴 } ) |
10 |
|
ovex |
⊢ ( 1 ... 𝑁 ) ∈ V |
11 |
|
mzpconstmpt |
⊢ ( ( ( 1 ... 𝑁 ) ∈ V ∧ 𝑀 ∈ ℤ ) → ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝑀 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) |
12 |
10 11
|
mpan |
⊢ ( 𝑀 ∈ ℤ → ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝑀 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) |
13 |
|
lerabdioph |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝑀 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) → { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝑀 ≤ 𝐴 } ∈ ( Dioph ‘ 𝑁 ) ) |
14 |
12 13
|
syl3an2 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) → { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝑀 ≤ 𝐴 } ∈ ( Dioph ‘ 𝑁 ) ) |
15 |
9 14
|
eqeltrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) ) → { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ 𝐴 ∈ ( ℤ≥ ‘ 𝑀 ) } ∈ ( Dioph ‘ 𝑁 ) ) |