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 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ 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 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ ( Base ‘ 𝐶 ) )
8	7 6	eleqtrrdi	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑀 ) ∈ 𝐻 )