Theorem idmatidpmat

Description: The transformed identity matrix is the identity polynomial matrix. (Contributed by AV, 1-Aug-2019) (Proof shortened by AV, 19-Nov-2019)

	Ref	Expression
Hypotheses	idmatidpmat.t	|- T = ( N matToPolyMat R )
	idmatidpmat.p	|- P = ( Poly1 ` R )
	idmatidpmat.1	|- .1. = ( 1r ` ( N Mat R ) )
	idmatidpmat.i	|- I = ( 1r ` ( N Mat P ) )
Assertion	idmatidpmat	|- ( ( R e. Ring /\ N e. Fin ) -> ( T ` .1. ) = I )

Proof

Step	Hyp	Ref	Expression
1		idmatidpmat.t	|- T = ( N matToPolyMat R )
2		idmatidpmat.p	|- P = ( Poly1 ` R )
3		idmatidpmat.1	|- .1. = ( 1r ` ( N Mat R ) )
4		idmatidpmat.i	|- I = ( 1r ` ( N Mat P ) )
5		eqid	|- ( N Mat R ) = ( N Mat R )
6		eqid	|- ( Base ` ( N Mat R ) ) = ( Base ` ( N Mat R ) )
7		eqid	|- ( N Mat P ) = ( N Mat P )
8		eqid	|- ( Base ` ( N Mat P ) ) = ( Base ` ( N Mat P ) )
9	1 5 6 2 7 8	mat2pmat1	|- ( ( N e. Fin /\ R e. Ring ) -> ( T ` ( 1r ` ( N Mat R ) ) ) = ( 1r ` ( N Mat P ) ) )
10	3	fveq2i	|- ( T ` .1. ) = ( T ` ( 1r ` ( N Mat R ) ) )
11	9 10 4	3eqtr4g	|- ( ( N e. Fin /\ R e. Ring ) -> ( T ` .1. ) = I )
12	11	ancoms	|- ( ( R e. Ring /\ N e. Fin ) -> ( T ` .1. ) = I )