Theorem fuzxrpmcn

Description: A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	fuzxrpmcn.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	fuzxrpmcn.2	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
Assertion	fuzxrpmcn	$$⊢\phi \to F\in \left({ℝ}^{*}{\uparrow }_{𝑝𝑚}ℂ\right)$$

Proof

Step	Hyp	Ref	Expression
1		fuzxrpmcn.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
2		fuzxrpmcn.2	$$⊢\phi \to F:Z⟶{ℝ}^{*}$$
3		cnex	$$⊢ℂ\in \mathrm{V}$$
4	3	a1i	$$⊢\phi \to ℂ\in \mathrm{V}$$
5		xrex	$$⊢{ℝ}^{*}\in \mathrm{V}$$
6	5	a1i	$$⊢\phi \to {ℝ}^{*}\in \mathrm{V}$$
7	1	uzsscn2	$$⊢Z\subseteq ℂ$$
8	7	a1i	$$⊢\phi \to Z\subseteq ℂ$$
9	4 6 8 2	fpmd	$$⊢\phi \to F\in \left({ℝ}^{*}{\uparrow }_{𝑝𝑚}ℂ\right)$$