Theorem dmatmat

Description: An N x N diagonal matrix over (the ring) R is an N x N matrix over (the ring) R . (Contributed by AV, 18-Dec-2019)

	Ref	Expression
Hypotheses	dmatval.a	|- A = ( N Mat R )
	dmatval.b	|- B = ( Base ` A )
	dmatval.0	|- .0. = ( 0g ` R )
	dmatval.d	|- D = ( N DMat R )
Assertion	dmatmat	|- ( ( N e. Fin /\ R e. V ) -> ( M e. D -> M e. B ) )

Proof

Step	Hyp	Ref	Expression
1		dmatval.a	|- A = ( N Mat R )
2		dmatval.b	|- B = ( Base ` A )
3		dmatval.0	|- .0. = ( 0g ` R )
4		dmatval.d	|- D = ( N DMat R )
5	1 2 3 4	dmatel	|- ( ( N e. Fin /\ R e. V ) -> ( M e. D <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )
6		simpl	|- ( ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) -> M e. B )
7	5 6	biimtrdi	|- ( ( N e. Fin /\ R e. V ) -> ( M e. D -> M e. B ) )