Theorem clmlmod

Description: A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Assertion	clmlmod	$$⊢W\in \mathrm{CMod}\to W\in \mathrm{LMod}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{Scalar}\left(W\right)=\mathrm{Scalar}\left(W\right)$$
2		eqid	$$⊢{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}={\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}$$
3	1 2	isclm	$$⊢W\in \mathrm{CMod}\leftrightarrow \left(W\in \mathrm{LMod}\wedge \mathrm{Scalar}\left(W\right)={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}\wedge {\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
4	3	simp1bi	$$⊢W\in \mathrm{CMod}\to W\in \mathrm{LMod}$$