Metamath Proof Explorer


Theorem rabdiophlem1

Description: Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi . (Contributed by Stefan O'Rear, 10-Oct-2014)

Ref Expression
Assertion rabdiophlem1 ( ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( ℕ0m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ )

Proof

Step Hyp Ref Expression
1 zex ℤ ∈ V
2 nn0ssz 0 ⊆ ℤ
3 mapss ( ( ℤ ∈ V ∧ ℕ0 ⊆ ℤ ) → ( ℕ0m ( 1 ... 𝑁 ) ) ⊆ ( ℤ ↑m ( 1 ... 𝑁 ) ) )
4 1 2 3 mp2an ( ℕ0m ( 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 ( ( ℕ0m ( 1 ... 𝑁 ) ) ⊆ ( ℤ ↑m ( 1 ... 𝑁 ) ) → ( ∀ 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ → ∀ 𝑡 ∈ ( ℕ0m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ ) )
10 4 8 9 mpsyl ( ( 𝑡 ∈ ( ℤ ↑m ( 1 ... 𝑁 ) ) ↦ 𝐴 ) ∈ ( mzPoly ‘ ( 1 ... 𝑁 ) ) → ∀ 𝑡 ∈ ( ℕ0m ( 1 ... 𝑁 ) ) 𝐴 ∈ ℤ )