Theorem pm2mpf

Description: The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019) (Revised by AV, 5-Dec-2019)

	Ref	Expression
Hypotheses	pm2mpval.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	pm2mpval.c	⊢ 𝐶 = ( 𝑁 Mat 𝑃 )
	pm2mpval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	pm2mpval.m	⊢ ∗ = ( ·𝑠 ‘ 𝑄 )
	pm2mpval.e	⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑄 ) )
	pm2mpval.x	⊢ 𝑋 = ( var1 ‘ 𝐴 )
	pm2mpval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	pm2mpval.q	⊢ 𝑄 = ( Poly1 ‘ 𝐴 )
	pm2mpval.t	⊢ 𝑇 = ( 𝑁 pMatToMatPoly 𝑅 )
	pm2mpcl.l	⊢ 𝐿 = ( Base ‘ 𝑄 )
Assertion	pm2mpf	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 ⟶ 𝐿 )

Proof

Step	Hyp	Ref	Expression
1		pm2mpval.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		pm2mpval.c	⊢ 𝐶 = ( 𝑁 Mat 𝑃 )
3		pm2mpval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
4		pm2mpval.m	⊢ ∗ = ( ·𝑠 ‘ 𝑄 )
5		pm2mpval.e	⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑄 ) )
6		pm2mpval.x	⊢ 𝑋 = ( var1 ‘ 𝐴 )
7		pm2mpval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
8		pm2mpval.q	⊢ 𝑄 = ( Poly1 ‘ 𝐴 )
9		pm2mpval.t	⊢ 𝑇 = ( 𝑁 pMatToMatPoly 𝑅 )
10		pm2mpcl.l	⊢ 𝐿 = ( Base ‘ 𝑄 )
11		ovexd	⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑚 ∈ 𝐵 ) → ( 𝑄 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑚 decompPMat 𝑘 ) ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ∈ V )
12	1 2 3 4 5 6 7 8 9	pm2mpval	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 = ( 𝑚 ∈ 𝐵 ↦ ( 𝑄 Σg ( 𝑘 ∈ ℕ0 ↦ ( ( 𝑚 decompPMat 𝑘 ) ∗ ( 𝑘 ↑ 𝑋 ) ) ) ) ) )
13	1 2 3 4 5 6 7 8 9 10	pm2mpcl	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑏 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑏 ) ∈ 𝐿 )
14	13	3expa	⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑏 ∈ 𝐵 ) → ( 𝑇 ‘ 𝑏 ) ∈ 𝐿 )
15	11 12 14	fmpt2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑇 : 𝐵 ⟶ 𝐿 )