Theorem pmat0op

Description: The zero polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019)

	Ref	Expression
Hypotheses	pmatring.p	|- P = ( Poly1 ` R )
	pmatring.c	|- C = ( N Mat P )
	pmat0op.z	|- .0. = ( 0g ` P )
Assertion	pmat0op	|- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` C ) = ( i e. N , j e. N |-> .0. ) )

Proof

Step	Hyp	Ref	Expression
1		pmatring.p	|- P = ( Poly1 ` R )
2		pmatring.c	|- C = ( N Mat P )
3		pmat0op.z	|- .0. = ( 0g ` P )
4	1	ply1ring	|- ( R e. Ring -> P e. Ring )
5	2 3	mat0op	|- ( ( N e. Fin /\ P e. Ring ) -> ( 0g ` C ) = ( i e. N , j e. N |-> .0. ) )
6	4 5	sylan2	|- ( ( N e. Fin /\ R e. Ring ) -> ( 0g ` C ) = ( i e. N , j e. N |-> .0. ) )