Theorem lmodmcl

Description: Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodmcl.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	lmodmcl.k	$$⊢K={\mathrm{Base}}_F$$
	lmodmcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_F$$
Assertion	lmodmcl	$$⊢\left(W\in \mathrm{LMod}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{·}Y\in K$$

Proof

Step	Hyp	Ref	Expression
1		lmodmcl.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		lmodmcl.k	$$⊢K={\mathrm{Base}}_F$$
3		lmodmcl.t	$$⊢\underset{\dot{}}{·}={\cdot }_F$$
4	1	lmodring	$$⊢W\in \mathrm{LMod}\to F\in \mathrm{Ring}$$
5	2 3	ringcl	$$⊢\left(F\in \mathrm{Ring}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{·}Y\in K$$
6	4 5	syl3an1	$$⊢\left(W\in \mathrm{LMod}\wedge X\in K\wedge Y\in K\right)\to X\underset{\dot{}}{·}Y\in K$$