Theorem pmatring

Description: The set of polynomial matrices over a ring is a ring. (Contributed by AV, 6-Nov-2019)

	Ref	Expression
Hypotheses	pmatring.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	pmatring.c	⊢ 𝐶 = ( 𝑁 Mat 𝑃 )
Assertion	pmatring	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		pmatring.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		pmatring.c	⊢ 𝐶 = ( 𝑁 Mat 𝑃 )
3	1	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
4	2	matring	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ) → 𝐶 ∈ Ring )
5	3 4	sylan2	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Ring )