Theorem clmfgrp

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

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

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	lmodfgrp	$$⊢W\in \mathrm{LMod}\to F\in \mathrm{Grp}$$
4	2 3	syl	$$⊢W\in \mathrm{CMod}\to F\in \mathrm{Grp}$$