Theorem clmring

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

	Ref	Expression
Hypothesis	clm0.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	clmring	$$⊢W\in \mathrm{CMod}\to F\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		clm0.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		clmlmod	$$⊢W\in \mathrm{CMod}\to W\in \mathrm{LMod}$$
3	1	lmodring	$$⊢W\in \mathrm{LMod}\to F\in \mathrm{Ring}$$
4	2 3	syl	$$⊢W\in \mathrm{CMod}\to F\in \mathrm{Ring}$$