Theorem dmatel

Description: A N x N diagonal matrix over (a 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	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. ) ) ) )

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	dmatval	|- ( ( N e. Fin /\ R e. V ) -> D = { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } )
6	5	eleq2d	|- ( ( N e. Fin /\ R e. V ) -> ( M e. D <-> M e. { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } ) )
7		oveq	|- ( m = M -> ( i m j ) = ( i M j ) )
8	7	eqeq1d	|- ( m = M -> ( ( i m j ) = .0. <-> ( i M j ) = .0. ) )
9	8	imbi2d	|- ( m = M -> ( ( i =/= j -> ( i m j ) = .0. ) <-> ( i =/= j -> ( i M j ) = .0. ) ) )
10	9	2ralbidv	|- ( m = M -> ( A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) <-> A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) )
11	10	elrab	|- ( M e. { m e. B | A. i e. N A. j e. N ( i =/= j -> ( i m j ) = .0. ) } <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) )
12	6 11	bitrdi	|- ( ( 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. ) ) ) )