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 ∈ Ring ∧ 𝑁 ∈ 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 ∈ Fin ∧ 𝑅 ∈ 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 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑇 ‘ 1 ) = 𝐼 )
12	11	ancoms	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ) → ( 𝑇 ‘ 1 ) = 𝐼 )