Theorem clmzss

Description: The scalar ring of a subcomplex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	clm0.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	clmsub.k	$$⊢K={\mathrm{Base}}_F$$
Assertion	clmzss	$$⊢W\in \mathrm{CMod}\to \mathbb{Z}\subseteq K$$

Proof

Step	Hyp	Ref	Expression
1		clm0.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		clmsub.k	$$⊢K={\mathrm{Base}}_F$$
3	1 2	clmsubrg	$$⊢W\in \mathrm{CMod}\to K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$
4		zsssubrg	$$⊢K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\to \mathbb{Z}\subseteq K$$
5	3 4	syl	$$⊢W\in \mathrm{CMod}\to \mathbb{Z}\subseteq K$$