Theorem clmsscn

Description: The scalar ring of a subcomplex module is a subset of the complex numbers. (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	clmsscn	$$⊢W\in \mathrm{CMod}\to K\subseteq ℂ$$

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		cnfldbas	$$⊢ℂ={\mathrm{Base}}_{{ℂ}_{\mathrm{fld}}}$$
5	4	subrgss	$$⊢K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\to K\subseteq ℂ$$
6	3 5	syl	$$⊢W\in \mathrm{CMod}\to K\subseteq ℂ$$