Theorem pmatlmod

Description: The set of polynomial matrices over a ring is a left module. (Contributed by AV, 6-Nov-2019)

	Ref	Expression
Hypotheses	pmatring.p	|- P = ( Poly1 ` R )
	pmatring.c	|- C = ( N Mat P )
Assertion	pmatlmod	|- ( ( N e. Fin /\ R e. Ring ) -> C e. LMod )

Proof

Step	Hyp	Ref	Expression
1		pmatring.p	|- P = ( Poly1 ` R )
2		pmatring.c	|- C = ( N Mat P )
3	1	ply1ring	|- ( R e. Ring -> P e. Ring )
4	2	matlmod	|- ( ( N e. Fin /\ P e. Ring ) -> C e. LMod )
5	3 4	sylan2	|- ( ( N e. Fin /\ R e. Ring ) -> C e. LMod )