Theorem clmsubrg

Description: The base set of the ring of scalars of a subcomplex module is the base set of a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	isclm.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	isclm.k	$$⊢K={\mathrm{Base}}_F$$
Assertion	clmsubrg	$$⊢W\in \mathrm{CMod}\to K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		isclm.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		isclm.k	$$⊢K={\mathrm{Base}}_F$$
3	1 2	isclm	$$⊢W\in \mathrm{CMod}\leftrightarrow \left(W\in \mathrm{LMod}\wedge F={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\wedge K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
4	3	simp3bi	$$⊢W\in \mathrm{CMod}\to K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)$$