This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem plybss

Description: Reverse closure of the parameter S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014)

		Ref	Expression
	Assertion	plybss	$⊢ F \in Poly (S) \to S \subseteq ℂ$

Step	Hyp	Ref	Expression
1		df-ply	$⊢ Poly = (x \in 𝒫 ℂ ⟼ \{f \| \exists n \in ℕ_{0} \exists a \in ({(x \cup \{0\})}^{ℕ_{0}}) f = (z \in ℂ ⟼ \sum_{k = 0}^{n} a (k) z^{k})\})$
2	1	mptrcl	$⊢ F \in Poly (S) \to S \in 𝒫 ℂ$
3	2	elpwid	$⊢ F \in Poly (S) \to S \subseteq ℂ$