Theorem climprod1

Description: The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017)

	Ref	Expression
Hypotheses	climprod1.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	climprod1.2	$$⊢\phi \to M\in \mathbb{Z}$$
Assertion	climprod1	$$⊢\phi \to seqM\left(\times ,\left(Z\times \left\{1\right\}\right)\right)⇝1$$

Proof

Step	Hyp	Ref	Expression
1		climprod1.1	$$⊢Z={\mathbb{Z}}_{\ge M}$$
2		climprod1.2	$$⊢\phi \to M\in \mathbb{Z}$$
3	1	prodfclim1	$$⊢M\in \mathbb{Z}\to seqM\left(\times ,\left(Z\times \left\{1\right\}\right)\right)⇝1$$
4	2 3	syl	$$⊢\phi \to seqM\left(\times ,\left(Z\times \left\{1\right\}\right)\right)⇝1$$