Theorem fprodsubrecnncnv

Description: The sequence S of finite products, where every factor is subtracted an "always smaller" amount, converges to the finite product of the factors. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	fprodsubrecnncnv.1	$$⊢Ⅎk\phi$$
	fprodsubrecnncnv.2	$$⊢\phi \to X\in \mathrm{Fin}$$
	fprodsubrecnncnv.3	$$⊢\left(\phi \wedge k\in X\right)\to A\in ℂ$$
	fprodsubrecnncnv.4	$$⊢S=\left(n\in \mathbb{N}⟼\prod _{k\in X}\left(A-\left(\frac{1}{n}\right)\right)\right)$$
Assertion	fprodsubrecnncnv	$$⊢\phi \to S⇝\prod _{k\in X}A$$

Proof

Step	Hyp	Ref	Expression
1		fprodsubrecnncnv.1	$$⊢Ⅎk\phi$$
2		fprodsubrecnncnv.2	$$⊢\phi \to X\in \mathrm{Fin}$$
3		fprodsubrecnncnv.3	$$⊢\left(\phi \wedge k\in X\right)\to A\in ℂ$$
4		fprodsubrecnncnv.4	$$⊢S=\left(n\in \mathbb{N}⟼\prod _{k\in X}\left(A-\left(\frac{1}{n}\right)\right)\right)$$
5		eqid	$$⊢\left(x\in ℂ⟼\prod _{k\in X}\left(A-x\right)\right)=\left(x\in ℂ⟼\prod _{k\in X}\left(A-x\right)\right)$$
6		oveq2	$$⊢m=n\to \frac{1}{m}=\frac{1}{n}$$
7	6	cbvmptv	$$⊢\left(m\in \mathbb{N}⟼\frac{1}{m}\right)=\left(n\in \mathbb{N}⟼\frac{1}{n}\right)$$
8	1 2 3 4 5 7	fprodsubrecnncnvlem	$$⊢\phi \to S⇝\prod _{k\in X}A$$