Theorem decpmatfsupp

Description: The mapping to the matrices consisting of the coefficients in the polynomial entries of a given matrix for the same power is finitely supported. (Contributed by AV, 5-Oct-2019) (Revised by AV, 3-Dec-2019)

	Ref	Expression
Hypotheses	decpmate.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	decpmate.c	⊢ 𝐶 = ( 𝑁 Mat 𝑃 )
	decpmate.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	decpmatcl.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	decpmatfsupp.0	⊢ 0 = ( 0g ‘ 𝐴 )
Assertion	decpmatfsupp	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑘 ∈ ℕ0 ↦ ( 𝑀 decompPMat 𝑘 ) ) finSupp 0 )

Proof

Step	Hyp	Ref	Expression
1		decpmate.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		decpmate.c	⊢ 𝐶 = ( 𝑁 Mat 𝑃 )
3		decpmate.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
4		decpmatcl.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
5		decpmatfsupp.0	⊢ 0 = ( 0g ‘ 𝐴 )
6	5	fvexi	⊢ 0 ∈ V
7	6	a1i	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → 0 ∈ V )
8		ovexd	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑀 decompPMat 𝑘 ) ∈ V )
9		oveq2	⊢ ( 𝑘 = 𝑥 → ( 𝑀 decompPMat 𝑘 ) = ( 𝑀 decompPMat 𝑥 ) )
10	1 2 3 4 5	decpmataa0	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ∃ 𝑠 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑠 < 𝑥 → ( 𝑀 decompPMat 𝑥 ) = 0 ) )
11	7 8 9 10	mptnn0fsuppd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ) → ( 𝑘 ∈ ℕ0 ↦ ( 𝑀 decompPMat 𝑘 ) ) finSupp 0 )