Theorem dmatALTbasel

Description: An element of the base set of the algebra of N x N diagonal matrices over a ring R , i.e. an N x N diagonal matrix over the ring R . (Contributed by AV, 8-Dec-2019)

	Ref	Expression
Hypotheses	dmatALTval.a	|- A = ( N Mat R )
	dmatALTval.b	|- B = ( Base ` A )
	dmatALTval.0	|- .0. = ( 0g ` R )
	dmatALTval.d	|- D = ( N DMatALT R )
Assertion	dmatALTbasel	|- ( ( N e. Fin /\ R e. _V ) -> ( M e. ( Base ` D ) <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dmatALTval.a	|- A = ( N Mat R )
2		dmatALTval.b	|- B = ( Base ` A )
3		dmatALTval.0	|- .0. = ( 0g ` R )
4		dmatALTval.d	|- D = ( N DMatALT R )
5	1 2 3 4	dmatALTbas	|- ( ( N e. Fin /\ R e. _V ) -> ( Base ` 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. ( Base ` 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. ( Base ` D ) <-> ( M e. B /\ A. i e. N A. j e. N ( i =/= j -> ( i M j ) = .0. ) ) ) )