Theorem mzpproj

Description: A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014)

	Ref	Expression
Assertion	mzpproj	⊢ ( ( 𝑉 ∈ V ∧ 𝑋 ∈ 𝑉 ) → ( 𝑔 ∈ ( ℤ ↑m 𝑉 ) ↦ ( 𝑔 ‘ 𝑋 ) ) ∈ ( mzPoly ‘ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		mzpincl	⊢ ( 𝑉 ∈ V → ( mzPoly ‘ 𝑉 ) ∈ ( mzPolyCld ‘ 𝑉 ) )
2		mzpcl2	⊢ ( ( ( mzPoly ‘ 𝑉 ) ∈ ( mzPolyCld ‘ 𝑉 ) ∧ 𝑋 ∈ 𝑉 ) → ( 𝑔 ∈ ( ℤ ↑m 𝑉 ) ↦ ( 𝑔 ‘ 𝑋 ) ) ∈ ( mzPoly ‘ 𝑉 ) )
3	1 2	sylan	⊢ ( ( 𝑉 ∈ V ∧ 𝑋 ∈ 𝑉 ) → ( 𝑔 ∈ ( ℤ ↑m 𝑉 ) ↦ ( 𝑔 ‘ 𝑋 ) ) ∈ ( mzPoly ‘ 𝑉 ) )