Theorem mat2pmatbas0

Description: The result of a matrix transformation is a polynomial matrix. (Contributed by AV, 27-Oct-2019)

	Ref	Expression
Hypotheses	mat2pmatbas.t	|- T = ( N matToPolyMat R )
	mat2pmatbas.a	|- A = ( N Mat R )
	mat2pmatbas.b	|- B = ( Base ` A )
	mat2pmatbas.p	|- P = ( Poly1 ` R )
	mat2pmatbas.c	|- C = ( N Mat P )
	mat2pmatbas0.h	|- H = ( Base ` C )
Assertion	mat2pmatbas0	|- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( T ` M ) e. H )

Proof

Step	Hyp	Ref	Expression
1		mat2pmatbas.t	|- T = ( N matToPolyMat R )
2		mat2pmatbas.a	|- A = ( N Mat R )
3		mat2pmatbas.b	|- B = ( Base ` A )
4		mat2pmatbas.p	|- P = ( Poly1 ` R )
5		mat2pmatbas.c	|- C = ( N Mat P )
6		mat2pmatbas0.h	|- H = ( Base ` C )
7	1 2 3 4 5	mat2pmatbas	|- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( T ` M ) e. ( Base ` C ) )
8	7 6	eleqtrrdi	|- ( ( N e. Fin /\ R e. Ring /\ M e. B ) -> ( T ` M ) e. H )