P. 230 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	decpmatval 22901*	The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power, general version for arbitrary matrices. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
Theorem	decpmate 22902	An entry of the matrix consisting of the coefficients in the entries of a polynomial matrix is the corresponding coefficient in the polynomial entry of the given matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑀 decompPMat 𝐾)𝐽) = ((coe1‘(𝐼𝑀𝐽))‘𝐾))
Theorem	decpmatcl 22903	Closure of the decomposition of a polynomial matrix: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power is a matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ 𝐷)
Theorem	decpmataa0 22904*	The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power is 0 for almost all powers. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 0 = (0g‘𝐴)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝑀 decompPMat 𝑥) = 0 ))
Theorem	decpmatfsupp 22905*	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.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 0 = (0g‘𝐴)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp 0 )
Theorem	decpmatid 22906	The matrix consisting of the coefficients in the polynomial entries of the identity matrix is an identity or a zero matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐼 = (1r‘𝐶)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 1 = (1r‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))
Theorem	decpmatmullem 22907*	Lemma for decpmatmul 22908. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐾 ∈ ℕ0)) → (𝐼((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝐽) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑙 ∈ (0...𝐾) ↦ (((coe1‘(𝐼𝑈𝑡))‘𝑙)(.r‘𝑅)((coe1‘(𝑡𝑊𝐽))‘(𝐾 − 𝑙))))))))
Theorem	decpmatmul 22908*	The matrix consisting of the coefficients in the polynomial entries of the product of two polynomial matrices is a sum of products of the matrices consisting of the coefficients in the polynomial entries of the polynomial matrices for the same power. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘))))))
Theorem	decpmatmulsumfsupp 22909*	Lemma 0 for pm2mpmhm 22956. (Contributed by AV, 21-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ · = (.r‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘)))))) finSupp 0 )
Theorem	pmatcollpw1lem1 22910*	Lemma 1 for pmatcollpw1 22912. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝐼(𝑀 decompPMat 𝑛)𝐽) × (𝑛 ↑ 𝑋))) finSupp (0g‘𝑃))
Theorem	pmatcollpw1lem2 22911*	Lemma 2 for pmatcollpw1 22912: An entry of a polynomial matrix is the sum of the entries of the matrix consisting of the coefficients in the entries of the polynomial matrix multiplied with the corresponding power of the variable. (Contributed by AV, 25-Sep-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋)))))
Theorem	pmatcollpw1 22912*	Write a polynomial matrix as a matrix of sums of scaled monomials. (Contributed by AV, 29-Sep-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))))
Theorem	pmatcollpw2lem 22913*	Lemma for pmatcollpw2 22914. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶))
Theorem	pmatcollpw2 22914*	Write a polynomial matrix as a sum of matrices whose entries are products of variable powers and constant polynomials collecting like powers. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ × = ( ·𝑠 ‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋))))))
Theorem	monmatcollpw 22915	The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix having scaled monomials with the same power as entries is the matrix of the coefficients of the monomials or a zero matrix. Generalization of decpmatid 22906 (but requires 𝑅 to be commutative!). (Contributed by AV, 11-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐾 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))
Theorem	pmatcollpwlem 22916	Lemma for pmatcollpw 22917. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
Theorem	pmatcollpw 22917*	Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
Theorem	pmatcollpwfi 22918*	Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 3-Jul-2022.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))))
Theorem	pmatcollpw3lem 22919*	Lemma for pmatcollpw3 22920 and pmatcollpw3fi 22921: Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 8-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑m 𝐼)𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
Theorem	pmatcollpw3 22920*	Write a polynomial matrix (over a commutative ring) as a sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 27-Oct-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 8-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑓 ∈ (𝐷 ↑m ℕ0)𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
Theorem	pmatcollpw3fi 22921*	Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 8-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
Theorem	pmatcollpw3fi1lem1 22922*	Lemma 1 for pmatcollpw3fi1 22924. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛))))))
Theorem	pmatcollpw3fi1lem2 22923*	Lemma 2 for pmatcollpw3fi1 22924. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑓 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))))
Theorem	pmatcollpw3fi1 22924*	Write a polynomial matrix (over a commutative ring) as a finite sum of (at least two) products of variable powers and constant matrices with scalar entries. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))
Theorem	pmatcollpwscmatlem1 22925	Lemma 1 for pmatcollpwscmat 22927. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    &   ⊢ 𝑈 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝐶)    &   ⊢ 𝑀 = (𝑄 ∗ 1 )    ⇒   ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1‘𝑄)‘𝐿)), (0g‘𝑃)))
Theorem	pmatcollpwscmatlem2 22926	Lemma 2 for pmatcollpwscmat 22927. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    &   ⊢ 𝑈 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝐶)    &   ⊢ 𝑀 = (𝑄 ∗ 1 )    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 ))
Theorem	pmatcollpwscmat 22927*	Write a scalar matrix over polynomials (over a commutative ring) as a sum of the product of variable powers and constant scalar matrices with scalar entries. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (Base‘𝐴)    &   ⊢ 𝑈 = (algSc‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝐶)    &   ⊢ 𝑀 = (𝑄 ∗ 1 )    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 )))))
11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices The main result of this section is Theorem pmmpric 22959, which shows that the ring of polynomial matrices and the ring of polynomials having matrices as coefficients (called "polynomials over matrices" in the following) are isomorphic: (Poly1‘(𝑁 Mat 𝑅)) ≃ (𝑁 Mat (Poly1‘𝑅)) Or in a more common notation: (𝑁 Mat (Poly1‘𝑅)) corresponds to M(n, R[t]), the ring of n x n polynomial matrices over the ring R. (Poly1‘(𝑁 Mat 𝑅)) corresponds to M(n, R)[t], the polynomial ring over the ring of n x n matrices with entries in ring R. 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) with 𝐵 = (Base‘(𝑁 Mat (Poly1‘𝑅))) and (𝑚 decompPMat 𝑘) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1 ( i m j ) ) 𝑘))) is an isomorphism between these rings: 𝑇:𝐵–1-1-onto→𝐿 with 𝐿 = (Base‘(Poly1‘(𝑁 Mat 𝑅))) (see pm2mpf1o 22951 and pm2mprngiso 22958), and 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1 p ) 𝑘)𝑗) · (𝑘𝐸𝑌)))))) is the corresponding inverse function: (𝑇‘(𝐼‘𝑂)) = 𝑂) (see mp2pm2mp 22947). In this section, the following conventions are mostly used: 𝑅 is a (unital) ring (see df-ring 20316) 𝑃 = (Poly1‘𝑅) is the polynomial algebra over (the ring) 𝑅 (see df-ply1 22321) 𝐾 = (Base‘𝑃) is its base set (see df-base 17269) 𝑌 = (var1‘𝑅) is its variable (see df-vr1 22320) · = ( ·𝑠 ‘𝑃) is its scalar multiplication (see df-vsca 17326 or lmodvscl 20978) 𝐸 = (.g‘(mulGrp‘𝑃)) is its exponentiation (see df-mulg 19133) 𝐴 = (𝑁 Mat 𝑅) is the algebra of N x N matrices over (the ring) 𝑅 (see df-mat 22544) 𝐶 = (𝑁 Mat 𝑃) is the algebra of N x N matrices over (the polynomial ring) 𝑃. 𝐵 = (Base‘𝐶) is its base set 𝑀 ∈ 𝐵 is a concrete polynomial matrix 𝑄 = (Poly1‘𝐴) is the polynomial algebra over (the matrix ring) 𝐴. 𝐿 = (Base‘𝑄) is its base set 𝑂 ∈ 𝐿 is a concrete polynomial with matrix coefficients 𝑋 = (var1‘𝐴) is its variable ∗ = ( ·𝑠 ‘𝑄) is its scalar multiplication ↑ = (.g‘(mulGrp‘𝑄)) is its exponentiation
Syntax	cpm2mp 22928	Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices.
class pMatToMatPoly
Definition	df-pm2mp 22929*	Transformation of a polynomial matrix (over a ring) into a polynomial over matrices (over the same ring). (Contributed by AV, 5-Dec-2019.)
⊢ pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎)))))))
Theorem	pm2mpf1lem 22930*	Lemma for pm2mpf1 22935. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))
Theorem	pm2mpval 22931*	Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))))
Theorem	pm2mpfval 22932*	A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
Theorem	pm2mpcl 22933	The transformation of polynomial matrices into polynomials over matrices maps polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   ⊢ 𝐿 = (Base‘𝑄)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿)
Theorem	pm2mpf 22934	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.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   ⊢ 𝐿 = (Base‘𝑄)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐿)
Theorem	pm2mpf1 22935	The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 6-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   ⊢ 𝐿 = (Base‘𝑄)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1→𝐿)
Theorem	pm2mpcoe1 22936	A coefficient of the polynomial over matrices which is the result of the transformation of a polynomial matrix is the matrix consisting of the coefficients in the polynomial entries of the polynomial matrix. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0)) → ((coe1‘(𝑇‘𝑀))‘𝐾) = (𝑀 decompPMat 𝐾))
Theorem	idpm2idmp 22937	The transformation of the identity polynomial matrix into polynomials over matrices results in the identity of the polynomials over matrices. (Contributed by AV, 18-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄))
Theorem	mptcoe1matfsupp 22938*	The mapping extracting the entries of the coefficient matrices of a polynomial over matrices at a fixed position is finitely supported. (Contributed by AV, 6-Oct-2019.) (Proof shortened by AV, 23-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ (𝐼((coe1‘𝑂)‘𝑘)𝐽)) finSupp (0g‘𝑅))
Theorem	mply1topmatcllem 22939*	Lemma for mply1topmatcl 22941. (Contributed by AV, 6-Oct-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝐼((coe1‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0g‘𝑃))
Theorem	mply1topmatval 22940*	A polynomial over matrices transformed into a polynomial matrix. 𝐼 is the inverse function of the transformation 𝑇 of polynomial matrices into polynomials over matrices: (𝑇‘(𝐼‘𝑂)) = 𝑂) (see mp2pm2mp 22947). (Contributed by AV, 6-Oct-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Theorem	mply1topmatcl 22941*	A polynomial over matrices transformed into a polynomial matrix is a polynomial matrix. (Contributed by AV, 6-Oct-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ 𝐵)
Theorem	mp2pm2mplem1 22942*	Lemma 1 for mp2pm2mp 22947. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
Theorem	mp2pm2mplem2 22943*	Lemma 2 for mp2pm2mp 22947. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)
Theorem	mp2pm2mplem3 22944*	Lemma 3 for mp2pm2mp 22947. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   ⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))‘𝐾)))
Theorem	mp2pm2mplem4 22945*	Lemma 4 for mp2pm2mp 22947. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   ⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝐾) = ((coe1‘𝑂)‘𝐾))
Theorem	mp2pm2mplem5 22946*	Lemma 5 for mp2pm2mp 22947. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄))
Theorem	mp2pm2mp 22947*	A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)
Theorem	pm2mpghmlem2 22948*	Lemma 2 for pm2mpghm 22952. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄))
Theorem	pm2mpghmlem1 22949	Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) ∗ (𝐾 ↑ 𝑋)) ∈ 𝐿)
Theorem	pm2mpfo 22950	The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 6-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–onto→𝐿)
Theorem	pm2mpf1o 22951	The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 14-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1-onto→𝐿)
Theorem	pm2mpghm 22952	The transformation of polynomial matrices into polynomials over matrices is an additive group homomorphism. (Contributed by AV, 16-Oct-2019.) (Revised by AV, 6-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄))
Theorem	pm2mpgrpiso 22953	The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpIso 𝑄))
Theorem	pm2mpmhmlem1 22954*	Lemma 1 for pm2mpmhm 22956. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐿 = (Base‘𝑄)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋))) finSupp (0g‘𝑄))
Theorem	pm2mpmhmlem2 22955*	Lemma 2 for pm2mpmhm 22956. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐶)𝑦)) = ((𝑇‘𝑥)(.r‘𝑄)(𝑇‘𝑦)))
Theorem	pm2mpmhm 22956	The transformation of polynomial matrices into polynomials over matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))
Theorem	pm2mprhm 22957	The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄))
Theorem	pm2mprngiso 22958	The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄))
Theorem	pmmpric 22959	The ring of polynomial matrices over a ring is isomorphic to the ring of polynomials over matrices of the same dimension over the same ring. (Contributed by AV, 30-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ≃𝑟 𝑄)
Theorem	monmat2matmon 22960	The transformation of a polynomial matrix having scaled monomials with the same power as entries into a scaled monomial as a polynomial over matrices. (Contributed by AV, 11-Nov-2019.) (Revised by AV, 7-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐾 = (Base‘𝐴)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
Theorem	pm2mp 22961*	The transformation of a sum of matrices having scaled monomials with the same power as entries into a sum of scaled monomials as a polynomial over matrices. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 7-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑄))    &   ⊢ 𝑋 = (var1‘𝐴)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐾 = (Base‘𝐴)    &   ⊢ 𝑄 = (Poly1‘𝐴)    &   ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)    &   ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑌 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))))
11.7  The characteristic polynomial According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients.". Based on the definition of the characteristic polynomial of a square matrix (df-chpmat 22963) the eigenvalues and corresponding eigenvectors can be defined.
11.7.1  Definition and basic properties The characteristic polynomial of a matrix 𝐴 is the determinant of the characteristic matrix of 𝐴: (𝑡𝐼 − 𝐴).
Syntax	cchpmat 22962	Extend class notation with the characteristic polynomial.
class CharPlyMat
Definition	df-chpmat 22963*	Define the characteristic polynomial of a square matrix. According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "The characteristic polynomial of [an n x n matrix] A, denoted by pA(t), is the polynomial defined by pA ( t ) = det ( t I - A ) where I denotes the n-by-n identity matrix.". In addition, however, the underlying ring must be commutative, see definition in [Lang], p. 561: " Let k be a commutative ring ... Let M be any n x n matrix in k ... We define the characteristic polynomial PM(t) to be the determinant det ( t In - M ) where In is the unit n x n matrix." To be more precise, the matrices A and I on the right hand side are matrices with coefficients of a polynomial ring. Therefore, the original matrix A over a given commutative ring must be transformed into corresponding matrices over the polynomial ring over the given ring. (Contributed by AV, 2-Aug-2019.)
⊢ CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
Theorem	chmatcl 22964	Closure of the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.) (Proof shortened by AV, 29-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ − = (-g‘𝑌)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 1 = (1r‘𝑌)    &   ⊢ 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀))    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌))
Theorem	chmatval 22965	The entries of the characteristic matrix of a matrix. (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 10-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ − = (-g‘𝑌)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 1 = (1r‘𝑌)    &   ⊢ 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀))    &   ⊢ ∼ = (-g‘𝑃)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐻𝐽) = if(𝐼 = 𝐽, (𝑋 ∼ (𝐼(𝑇‘𝑀)𝐽)), ( 0 ∼ (𝐼(𝑇‘𝑀)𝐽))))
Theorem	chpmatfval 22966*	Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝐷 = (𝑁 maDet 𝑃)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 1 = (1r‘𝑌)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
Theorem	chpmatval 22967	The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝐷 = (𝑁 maDet 𝑃)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 1 = (1r‘𝑌)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
Theorem	chpmatply1 22968	The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸)
Theorem	chpmatval2 22969*	The characteristic polynomial of a (square) matrix (expressed with the Leibnitz formula for the determinant). (Contributed by AV, 2-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 1 = (1r‘𝑌)    &   ⊢ 𝐺 = (SymGrp‘𝑁)    &   ⊢ 𝐻 = (Base‘𝐺)    &   ⊢ 𝑍 = (ℤRHom‘𝑃)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑈 = (mulGrp‘𝑃)    &   ⊢ × = (.r‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑃 Σg (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥)))))))
Theorem	chpmat0d 22970	The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐶 = (∅ CharPlyMat 𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐶‘∅) = (1r‘(Poly1‘𝑅)))
Theorem	chpmat1dlem 22971	Lemma for chpmat1d 22972. (Contributed by AV, 7-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐺 = (𝑁 Mat 𝑃)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐼((𝑋( ·𝑠 ‘𝐺)(1r‘𝐺))(-g‘𝐺)(𝑇‘𝑀))𝐼) = (𝑋 − (𝑆‘(𝐼𝑀𝐼))))
Theorem	chpmat1d 22972	The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝑆 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑋 − (𝑆‘(𝐼𝑀𝐼))))
Theorem	chpdmatlem0 22973	Lemma 0 for chpdmat 22977. (Contributed by AV, 18-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝑄 = (𝑁 Mat 𝑃)    &   ⊢ 1 = (1r‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑄)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄))
Theorem	chpdmatlem1 22974	Lemma 1 for chpdmat 22977. (Contributed by AV, 18-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝑄 = (𝑁 Mat 𝑃)    &   ⊢ 1 = (1r‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑄)    &   ⊢ 𝑍 = (-g‘𝑄)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 )𝑍(𝑇‘𝑀)) ∈ (Base‘𝑄))
Theorem	chpdmatlem2 22975	Lemma 2 for chpdmat 22977. (Contributed by AV, 18-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝑄 = (𝑁 Mat 𝑃)    &   ⊢ 1 = (1r‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑄)    &   ⊢ 𝑍 = (-g‘𝑄)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑖 ≠ 𝑗) ∧ (𝑖𝑀𝑗) = 0 ) → (𝑖((𝑋 · 1 )𝑍(𝑇‘𝑀))𝑗) = (0g‘𝑃))
Theorem	chpdmatlem3 22976	Lemma 3 for chpdmat 22977. (Contributed by AV, 18-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝑄 = (𝑁 Mat 𝑃)    &   ⊢ 1 = (1r‘𝑄)    &   ⊢ · = ( ·𝑠 ‘𝑄)    &   ⊢ 𝑍 = (-g‘𝑄)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ 𝑁) → (𝐾((𝑋 · 1 )𝑍(𝑇‘𝑀))𝐾) = (𝑋 − (𝑆‘(𝐾𝑀𝐾))))
Theorem	chpdmat 22977*	The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ − = (-g‘𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))))
Theorem	chpscmat 22978*	The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))}    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ − = (-g‘𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))
Theorem	chpscmat0 22979*	The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))}    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ − = (-g‘𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐼𝑀𝐼))) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘(𝐼𝑀𝐼)))))
Theorem	chpscmatgsumbin 22980*	The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))}    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐹 = (.g‘𝑃)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶‘𝑀) = (𝑃 Σg (𝑙 ∈ (0...(♯‘𝑁)) ↦ (((♯‘𝑁)C𝑙)𝐹((((♯‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽))) · (𝑙 ↑ 𝑋))))))
Theorem	chpscmatgsummon 22981*	The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))}    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ − = (-g‘𝑃)    &   ⊢ 𝐹 = (.g‘𝑃)    &   ⊢ 𝐻 = (mulGrp‘𝑅)    &   ⊢ 𝐸 = (.g‘𝐻)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝑍 = (.g‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶‘𝑀) = (𝑃 Σg (𝑙 ∈ (0...(♯‘𝑁)) ↦ ((((♯‘𝑁)C𝑙)𝑍(((♯‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽)))) · (𝑙 ↑ 𝑋)))))
Theorem	chp0mat 22982	The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐴)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶‘ 0 ) = ((♯‘𝑁) ↑ 𝑋))
Theorem	chpidmat 22983	The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝐼 = (1r‘𝐴)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ − = (-g‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶‘𝐼) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘ 1 ))))
Theorem	chmaidscmat 22984	The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 5-Jul-2022.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝐾 = (Base‘𝑌)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 1 = (1r‘𝑌)    &   ⊢ 𝑆 = (𝑁 ScMat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆)
11.7.2  The characteristic factor function G In this subsection the function 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 23014. Therefore, this function is called "characteristic factor function" (in short "chfacf") in the following. It plays an important role in the proof of the Cayley-Hamilton theorem, see cayhamlem1 23002, cayhamlem3 23023 and cayhamlem4 23024.
Theorem	fvmptnn04if 22985*	The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐴)    &   ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐵)    &   ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶)    &   ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷)    ⇒   ⊢ (𝜑 → (𝐺‘𝑁) = 𝑌)
Theorem	fvmptnn04ifa 22986*	The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐴)
Theorem	fvmptnn04ifb 22987*	The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐵)
Theorem	fvmptnn04ifc 22988*	The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐶)
Theorem	fvmptnn04ifd 22989*	The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝑆 < 𝑁 ∧ ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐷)
Theorem	chfacfisf 22990*	The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
Theorem	chfacfisfcpmat 22991*	The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆)
Theorem	chfacffsupp 22992*	The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌))
Theorem	chfacfscmulcl 22993*	Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌))
Theorem	chfacfscmul0 22994*	A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) = 0 )
Theorem	chfacfscmulfsupp 22995*	A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 )
Theorem	chfacfscmulgsum 22996*	Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ + = (+g‘𝑌)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))
Theorem	chfacfpmmulcl 22997*	Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑌))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) ∈ (Base‘𝑌))
Theorem	chfacfpmmul0 22998*	The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑌))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 )
Theorem	chfacfpmmulfsupp 22999*	A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑌))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))) finSupp 0 )
Theorem	chfacfpmmulgsum 23000*	Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ × = (.r‘𝑌)    &   ⊢ − = (-g‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑌))    &   ⊢ + = (+g‘𝑌)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))