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Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
11.3.5  Inverse matrix
 
Theoreminvrvald 21201 If a matrix multiplied with a given matrix (from the left as well as from the right) results in the identity matrix, this matrix is the inverse (matrix) of the given matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋 · 𝑌) = 1 )    &   (𝜑 → (𝑌 · 𝑋) = 1 )       (𝜑 → (𝑋𝑈 ∧ (𝐼𝑋) = 𝑌))
 
Theoremmatinv 21202 The inverse of a matrix is the adjunct of the matrix multiplied with the inverse of the determinant of the matrix if the determinant is a unit in the underlying ring. Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐽 = (𝑁 maAdju 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (Unit‘𝐴)    &   𝑉 = (Unit‘𝑅)    &   𝐻 = (invr𝑅)    &   𝐼 = (invr𝐴)    &    = ( ·𝑠𝐴)       ((𝑅 ∈ CRing ∧ 𝑀𝐵 ∧ (𝐷𝑀) ∈ 𝑉) → (𝑀𝑈 ∧ (𝐼𝑀) = ((𝐻‘(𝐷𝑀)) (𝐽𝑀))))
 
Theoremmatunit 21203 A matrix is a unit in the ring of matrices iff its determinant is a unit in the underlying ring. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (Unit‘𝐴)    &   𝑉 = (Unit‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑀𝑈 ↔ (𝐷𝑀) ∈ 𝑉))
 
11.3.6  Cramer's rule

In the following, Cramer's rule cramer 21216 is proven. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule 21216: "[Cramer's rule] ... expresses the [unique] solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations."

The outline of the proof for systems of linear equations with coefficients from a commutative ring, according to the proof in Wikipedia (https://en.wikipedia.org/wiki/Cramer's_rule#A_short_proof), 21216 is as follows:

The system of linear equations 𝐴 × 𝑋 = 𝐵 to be solved shall be given by the N x N coefficient matrix 𝐴 and the N-dimensional vector 𝐵. Let (𝐴𝑖) be the matrix obtained by replacing the i-th column of the coefficient matrix 𝐴 by the right-hand side vector 𝐵. Additionally, let (𝑋𝑖) be the matrix obtained by replacing the i-th column of the identity matrix by the solution vector 𝑋, with 𝑋 = (𝑥𝑖). Finally, it is assumed that det 𝐴 is a unit in the underlying ring.

With these definitions, it follows that 𝐴 × (𝑋𝑖) = (𝐴𝑖) (cramerimplem2 21209), using matrix multiplication (mamuval 20913) and multiplication of a vector with a matrix (mulmarep1gsum2 21099). By using the multiplicativity of the determinant (mdetmul 21148) it follows that det (𝐴𝑖) = det (𝐴 × (𝑋𝑖)) = det 𝐴 · det (𝑋𝑖) (cramerimplem3 21210).

Furthermore, it follows that det (𝑋𝑖) = (𝑥𝑖) (cramerimplem1 21208). To show this, a special case of the Laplace expansion is used (smadiadetg 21198).

From these equations and the cancellation law for division in a ring (dvrcan3 19364) it follows that (𝑥𝑖) = det (𝑋𝑖) = det (𝐴𝑖) / det 𝐴.

This is the right to left implication (cramerimp 21211, cramerlem1 21212, cramerlem2 21213) of Cramer's rule (cramer 21216). The left to right implication is shown by cramerlem3 21214, using the fact that a solution of the system of linear equations exists (slesolex 21207).

Notice that for the special case of 0-dimensional matrices/vectors only the left to right implication is valid (see cramer0 21215), because assuming the right-hand side of the implication ((𝑋 · 𝑍) = 𝑌), 𝑍 could be anything (see mavmul0g 21078).

 
Theoremslesolvec 21204 Every solution of a system of linear equations represented by a matrix and a vector is a vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 27-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐵𝑌𝑉)) → ((𝑋 · 𝑍) = 𝑌𝑍𝑉))
 
Theoremslesolinv 21205 The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 28-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐼 = (invr𝐴)       (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝐷𝑋) ∈ (Unit‘𝑅) ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = ((𝐼𝑋) · 𝑌))
 
Theoremslesolinvbi 21206 The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐼 = (invr𝐴)       (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑍) = 𝑌𝑍 = ((𝐼𝑋) · 𝑌)))
 
Theoremslesolex 21207* Every system of linear equations represented by a matrix with a unit as determinant has a solution. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐷 = (𝑁 maDet 𝑅)       (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → ∃𝑧𝑉 (𝑋 · 𝑧) = 𝑌)
 
Theoremcramerimplem1 21208 Lemma 1 for cramerimp 21211: The determinant of the identity matrix with the ith column replaced by a (column) vector equals the ith component of the vector. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 5-Jul-2022.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐸 = (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   𝐷 = (𝑁 maDet 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ 𝑍𝑉) → (𝐷𝐸) = (𝑍𝐼))
 
Theoremcramerimplem2 21209 Lemma 2 for cramerimp 21211: The matrix of a system of linear equations multiplied with the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐸 = (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &    × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋 × 𝐸) = 𝐻)
 
Theoremcramerimplem3 21210 Lemma 3 for cramerimp 21211: The determinant of the matrix of a system of linear equations multiplied with the determinant of the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐸 = (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐷 = (𝑁 maDet 𝑅)    &    = (.r𝑅)       (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷𝑋) (𝐷𝐸)) = (𝐷𝐻))
 
Theoremcramerimp 21211 One direction of Cramer's rule (according to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations."): The ith component of the solution vector of a system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the righthand side vector of the system of linear equations divided by the determinant of the matrix of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐸 = (((1r𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐷 = (𝑁 maDet 𝑅)    &    / = (/r𝑅)       (((𝑅 ∈ CRing ∧ 𝐼𝑁) ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷𝑋) ∈ (Unit‘𝑅))) → (𝑍𝐼) = ((𝐷𝐻) / (𝐷𝑋)))
 
Theoremcramerlem1 21212* Lemma 1 for cramer 21216. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &    / = (/r𝑅)       ((𝑅 ∈ CRing ∧ (𝑋𝐵𝑌𝑉) ∧ ((𝐷𝑋) ∈ (Unit‘𝑅) ∧ 𝑍𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))))
 
Theoremcramerlem2 21213* Lemma 2 for cramer 21216. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &    / = (/r𝑅)       ((𝑅 ∈ CRing ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → ∀𝑧𝑉 ((𝑋 · 𝑧) = 𝑌𝑧 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋)))))
 
Theoremcramerlem3 21214* Lemma 3 for cramer 21216. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &    / = (/r𝑅)       (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))
 
Theoremcramer0 21215* Special case of Cramer's rule for 0-dimensional matrices/vectors. (Contributed by AV, 28-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &    / = (/r𝑅)       (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) → (𝑋 · 𝑍) = 𝑌))
 
Theoremcramer 21216* Cramer's rule. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the [unique] solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations." If it is assumed that a (unique) solution exists, it can be obtained by Cramer's rule (see also cramerimp 21211). On the other hand, if a vector can be constructed by Cramer's rule, it is a solution of the system of linear equations, so at least one solution exists. The uniqueness is ensured by considering only systems of linear equations whose matrix has a unit (of the underlying ring) as determinant, see matunit 21203 or slesolinv 21205. For fields as underlying rings, this requirement is equivalent to the determinant not being 0. Theorem 4.4 in [Lang] p. 513. This is Metamath 100 proof #97. (Contributed by Alexander van der Vekens, 21-Feb-2019.) (Revised by Alexander van der Vekens, 1-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   𝐷 = (𝑁 maDet 𝑅)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &    / = (/r𝑅)       (((𝑅 ∈ CRing ∧ 𝑁 ≠ ∅) ∧ (𝑋𝐵𝑌𝑉) ∧ (𝐷𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷𝑋))) ↔ (𝑋 · 𝑍) = 𝑌))
 
11.4  Polynomial matrices

A polynomial matrix or matrix of polynomials is a matrix whose elements are univariate (or multivariate) polynomials. See Wikipedia "Polynomial matrix" https://en.wikipedia.org/wiki/Polynomial_matrix (18-Nov-2019). In this section, only square matrices whose elements are univariate polynomials are considered. Usually, the ring of such matrices, the ring of n x n matrices over the polynomial ring over a ring 𝑅, is denoted by M(n, R[t]). The elements of this ring are called "polynomial matrices (over the ring 𝑅)" in the following. In Metamath notation, this ring is defined by (𝑁 Mat (Poly1𝑅)), usually represented by the class variable 𝐶 (or 𝑌, if 𝐶 is already occupied): 𝐶 = (𝑁 Mat 𝑃) with 𝑃 = (Poly1𝑅).

 
11.4.1  Basic properties
 
Theorempmatring 21217 The set of polynomial matrices over a ring is a ring. (Contributed by AV, 6-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
 
Theorempmatlmod 21218 The set of polynomial matrices over a ring is a left module. (Contributed by AV, 6-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
 
Theorempmat0op 21219* The zero polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &    0 = (0g𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) = (𝑖𝑁, 𝑗𝑁0 ))
 
Theorempmat1op 21220* The identity polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &    0 = (0g𝑃)    &    1 = (1r𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))
 
Theorempmat1ovd 21221 Entries of the identity polynomial matrix over a ring, deduction form. (Contributed by AV, 16-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &    0 = (0g𝑃)    &    1 = (1r𝑃)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   𝑈 = (1r𝐶)       (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))
 
Theorempmat0opsc 21222* The zero polynomial matrix over a ring represented as operation with "lifted scalars" (i.e. elements of the ring underlying the polynomial ring embedded into the polynomial ring by the scalar injection/algebraic scalars function algSc). (Contributed by AV, 16-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ (𝐴0 )))
 
Theorempmat1opsc 21223* The identity polynomial matrix over a ring represented as operation with "lifted scalars". (Contributed by AV, 16-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (𝐴1 ), (𝐴0 ))))
 
Theorempmat1ovscd 21224 Entries of the identity polynomial matrix over a ring represented with "lifted scalars", deduction form. (Contributed by AV, 16-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (algSc‘𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   𝑈 = (1r𝐶)       (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (𝐴1 ), (𝐴0 )))
 
Theorempmatcoe1fsupp 21225* For a polynomial matrix there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
 
Theorem1pmatscmul 21226 The scalar product of the identity polynomial matrix with a polynomial is a polynomial matrix. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐸 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &    1 = (1r𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → (𝑄 1 ) ∈ 𝐵)
 
11.4.2  Constant polynomial matrices

A constant polynomial matrix is a polynomial matrix whose elements are constant polynomials, i.e. polynomials with no indeterminates. Constant polynomials are obtained by "lifting" a "scalar" (i.e. an element of the underlying ring) into the polynomial ring/algebra by a "scalar injection", i.e. applying the "algebra scalar injection function" algSc (see df-ascl 20008) to a scalar 𝐴𝑅: ((algSc‘𝑃)‘𝐴). In an analogous way, constant polynomial matrices (over the ring 𝑅) are obtained by "lifting" matrices over the ring 𝑅 by the function matToPolyMat (see df-mat2pmat 21231), called "matrix transformation" in the following.

In this section it is shown that the set 𝑆 = (𝑁 ConstPolyMat 𝑅) of constant polynomial 𝑁 x 𝑁 matrices over the ring 𝑅 is a subring of the ring of polynomial 𝑁 x 𝑁 matrices over the ring 𝑅 (cpmatsrgpmat 21245) and that 𝑇 = (𝑁 matToPolyMat 𝑅) is a ring isomorphism between the ring of matrices over a ring 𝑅 and the ring of constant polynomial matrices over the ring 𝑅 (see m2cpmrngiso 21282). By this, it is shown that the ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring, see matcpmric 21283. Finally 𝐼 = (𝑁 cPolyMatToMat 𝑅), the transformation of a constant polynomial matrix into a matrix, is the inverse function of the matrix transformation 𝑇 = (𝑁 matToPolyMat 𝑅), see m2cpminv 21284.

 
Syntaxccpmat 21227 Extend class notation with the set of all constant polynomial matrices.
class ConstPolyMat
 
Syntaxcmat2pmat 21228 Extend class notation with the transformation of a matrix into a matrix of polynomials.
class matToPolyMat
 
Syntaxccpmat2mat 21229 Extend class notation with the transformation of a constant polynomial matrix into a matrix.
class cPolyMatToMat
 
Definitiondf-cpmat 21230* The set of all constant polynomial matrices, which are all matrices whose entries are constant polynomials (or "scalar polynomials", see ply1sclf 20370). (Contributed by AV, 15-Nov-2019.)
ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)})
 
Definitiondf-mat2pmat 21231* Transformation of a matrix (over a ring) into a matrix over the corresponding polynomial ring. (Contributed by AV, 31-Jul-2019.)
matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
 
Definitiondf-cpmat2mat 21232* Transformation of a constant polynomial matrix (over a ring) into a matrix over the corresponding ring. Since this function is the inverse function of matToPolyMat, see m2cpminv 21284, it is also called "inverse matrix transformation" in the following. (Contributed by AV, 14-Dec-2019.)
cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
 
Theoremcpmat 21233* Value of the constructor of the set of all constant polynomial matrices, i.e. the set of all 𝑁 x 𝑁 matrices of polynomials over a ring 𝑅. (Contributed by AV, 15-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
 
Theoremcpmatpmat 21234 A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝐵)
 
Theoremcpmatel 21235* Property of a constant polynomial matrix. (Contributed by AV, 15-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
 
Theoremcpmatelimp 21236* Implication of a set being a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆 → (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))))
 
Theoremcpmatel2 21237* Another property of a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.) (Proof shortened by AV, 27-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘𝐾 (𝑖𝑀𝑗) = (𝐴𝑘)))
 
Theoremcpmatelimp2 21238* Another implication of a set being a constant polynomial matrix. (Contributed by AV, 17-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆 → (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘𝐾 (𝑖𝑀𝑗) = (𝐴𝑘))))
 
Theorem1elcpmat 21239 The identity of the ring of all polynomial matrices over the ring 𝑅 is a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐶) ∈ 𝑆)
 
Theoremcpmatacl 21240* The set of all constant polynomial matrices over a ring 𝑅 is closed under addition. (Contributed by AV, 17-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)𝑦) ∈ 𝑆)
 
Theoremcpmatinvcl 21241* The set of all constant polynomial matrices over a ring 𝑅 is closed under inversion. (Contributed by AV, 17-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆 ((invg𝐶)‘𝑥) ∈ 𝑆)
 
Theoremcpmatmcllem 21242* Lemma for cpmatmcl 21243. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
 
Theoremcpmatmcl 21243* The set of all constant polynomial matrices over a ring 𝑅 is closed under multiplication. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(.r𝐶)𝑦) ∈ 𝑆)
 
Theoremcpmatsubgpmat 21244 The set of all constant polynomial matrices over a ring 𝑅 is an additive subgroup of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 15-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))
 
Theoremcpmatsrgpmat 21245 The set of all constant polynomial matrices over a ring 𝑅 is a subring of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶))
 
Theorem0elcpmat 21246 The zero of the ring of all polynomial matrices over the ring 𝑅 is a constant polynomial matrix. (Contributed by AV, 27-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) ∈ 𝑆)
 
Theoremmat2pmatfval 21247* Value of the matrix transformation. (Contributed by AV, 31-Jul-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
 
Theoremmat2pmatval 21248* The result of a matrix transformation. (Contributed by AV, 31-Jul-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
 
Theoremmat2pmatvalel 21249 A (matrix) element of the result of a matrix transformation. (Contributed by AV, 31-Jul-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝑇𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))
 
Theoremmat2pmatbas 21250 The result of a matrix transformation is a polynomial matrix. (Contributed by AV, 1-Aug-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝐶))
 
Theoremmat2pmatbas0 21251 The result of a matrix transformation is a polynomial matrix. (Contributed by AV, 27-Oct-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝐻)
 
Theoremmat2pmatf 21252 The matrix transformation is a function from the matrices to the polynomial matrices. (Contributed by AV, 27-Oct-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐻)
 
Theoremmat2pmatf1 21253 The matrix transformation is a 1-1 function from the matrices to the polynomial matrices. (Contributed by AV, 28-Oct-2019.) (Proof shortened by AV, 27-Nov-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐻)
 
Theoremmat2pmatghm 21254 The transformation of matrices into polynomial matrices is an additive group homomorphism. (Contributed by AV, 28-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
 
Theoremmat2pmatmul 21255* The transformation of matrices into polynomial matrices preserves the multiplication. (Contributed by AV, 29-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐴)𝑦)) = ((𝑇𝑥)(.r𝐶)(𝑇𝑦)))
 
Theoremmat2pmat1 21256 The transformation of the identity matrix results in the identity polynomial matrix. (Contributed by AV, 29-Oct-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐴)) = (1r𝐶))
 
Theoremmat2pmatmhm 21257 The transformation of matrices into polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 29-Oct-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)))
 
Theoremmat2pmatrhm 21258 The transformation of matrices into polynomial matrices is a ring homomorphism. (Contributed by AV, 29-Oct-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝐶))
 
Theoremmat2pmatlin 21259 The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 19729. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 19716, see lmhmsca 19724. (Contributed by AV, 13-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)    &   𝐾 = (Base‘𝑅)    &   𝑆 = (algSc‘𝑃)    &    · = ( ·𝑠𝐴)    &    × = ( ·𝑠𝐶)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)))
 
Theorem0mat2pmat 21260 The transformed zero matrix is the zero polynomial matrix. (Contributed by AV, 5-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g‘(𝑁 Mat 𝑅))    &   𝑍 = (0g‘(𝑁 Mat 𝑃))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇0 ) = 𝑍)
 
Theoremidmatidpmat 21261 The transformed identity matrix is the identity polynomial matrix. (Contributed by AV, 1-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &    1 = (1r‘(𝑁 Mat 𝑅))    &   𝐼 = (1r‘(𝑁 Mat 𝑃))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇1 ) = 𝐼)
 
Theoremd0mat2pmat 21262 The transformed empty set as matrix of dimenson 0 is the empty set (i.e. the polynomial matrix of dimension 0). (Contributed by AV, 4-Aug-2019.)
(𝑅𝑉 → ((∅ matToPolyMat 𝑅)‘∅) = ∅)
 
Theoremd1mat2pmat 21263 The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐵 = (Base‘(𝑁 Mat 𝑅))    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
 
Theoremmat2pmatscmxcl 21264 A transformed matrix multiplied with a power of the variable of a polynomial is a polynomial matrix. (Contributed by AV, 6-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀𝐾𝐿 ∈ ℕ0)) → ((𝐿 𝑋) (𝑇𝑀)) ∈ 𝐵)
 
Theoremm2cpm 21265 The result of a matrix transformation is a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝑆)
 
Theoremm2cpmf 21266 The matrix transformation is a function from the matrices to the constant polynomial matrices. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝑆)
 
Theoremm2cpmf1 21267 The matrix transformation is a 1-1 function from the matrices to the constant polynomial matrices. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝑆)
 
Theoremm2cpmghm 21268 The transformation of matrices into constant polynomial matrices is an additive group homomorphism. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈))
 
Theoremm2cpmmhm 21269 The transformation of matrices into constant polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))
 
Theoremm2cpmrhm 21270 The transformation of matrices into constant polynomial matrices is a ring homomorphism. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈))
 
Theoremm2pmfzmap 21271 The transformed values of a (finite) mapping of integers to matrices. (Contributed by AV, 4-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑇‘(𝑏𝐼)) ∈ (Base‘𝑌))
 
Theoremm2pmfzgsumcl 21272* Closure of the sum of scaled transformed matrices. (Contributed by AV, 4-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))))) ∈ (Base‘𝑌))
 
Theoremcpm2mfval 21273* Value of the inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)
𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
 
Theoremcpm2mval 21274* The result of an inverse matrix transformation. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
 
Theoremcpm2mvalel 21275 A (matrix) element of the result of an inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)
𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝐼𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0))
 
Theoremcpm2mf 21276 The inverse matrix transformation is a function from the constant polynomial matrices to the matrices over the base ring of the polynomials. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝐼 = (𝑁 cPolyMatToMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆𝐾)
 
Theoremm2cpminvid 21277 The inverse transformation applied to the transformation of a matrix over a ring R results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 13-Dec-2019.)
𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾) → (𝐼‘(𝑇𝑀)) = 𝑀)
 
Theoremm2cpminvid2lem 21278* Lemma for m2cpminvid2 21279. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
 
Theoremm2cpminvid2 21279 The transformation applied to the inverse transformation of a constant polynomial matrix over the ring 𝑅 results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = 𝑀)
 
Theoremm2cpmfo 21280 The matrix transformation is a function from the matrices onto the constant polynomial matrices. (Contributed by AV, 19-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾onto𝑆)
 
Theoremm2cpmf1o 21281 The matrix transformation is a 1-1 function from the matrices onto the constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾1-1-onto𝑆)
 
Theoremm2cpmrngiso 21282 The transformation of matrices into constant polynomial matrices is a ring isomorphism. (Contributed by AV, 19-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))
 
Theoremmatcpmric 21283 The ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring. (Contributed by AV, 30-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴𝑟 𝑈)
 
Theoremm2cpminv 21284 The inverse matrix transformation is a 1-1 function from the constant polynomial matrices onto the matrices over the base ring of the polynomials. (Contributed by AV, 27-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼:𝑆1-1-onto𝐾𝐼 = 𝑇))
 
Theoremm2cpminv0 21285 The inverse matrix transformation applied to the zero polynomial matrix results in the zero of the matrices over the base ring of the polynomials. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &    0 = (0g𝐴)    &   𝑍 = (0g𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼𝑍) = 0 )
 
11.4.3  Collecting coefficients of polynomial matrices

In this section, the decomposition of polynomial matrices into (polynomial) multiples of constant (polynomial) matrices is prepared by collecting the coefficients of a polynomial matrix which belong to the same power of the polynomial variable. Such a collection is given by the functiondecompPMat (see df-decpmat 21287), which maps a polynomial matrix 𝑀 to a constant matrix consisting of the coefficients of the scaled monomials ((𝑐𝑘) (𝑘 𝑋)), i.e. the coefficients belonging to the k-th power of the polynomial variable 𝑋, of each entry in the polynomial matrix 𝑀. The resulting decomposition is provided by theorem pmatcollpw 21305.

 
Syntaxcdecpmat 21286 Extend class notation to include the decomposition of polynomial matrices.
class decompPMat
 
Definitiondf-decpmat 21287* Define the decomposition of polynomial matrices. This function collects the coefficients of a polynomial matrix 𝑚 belong to the 𝑘 th power of the polynomial variable for each entry of 𝑚. (Contributed by AV, 2-Dec-2019.)
decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
 
Theoremdecpmatval0 21288* The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power, most general version. (Contributed by AV, 2-Dec-2019.)
((𝑀𝑉𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
 
Theoremdecpmatval 21289* 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)       ((𝑀𝐵𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
 
Theoremdecpmate 21290 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       (((𝑅𝑉𝑀𝐵𝐾 ∈ ℕ0) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑀 decompPMat 𝐾)𝐽) = ((coe1‘(𝐼𝑀𝐽))‘𝐾))
 
Theoremdecpmatcl 21291 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑅𝑉𝑀𝐵𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ 𝐷)
 
Theoremdecpmataa0 21292* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝐴)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝑀 decompPMat 𝑥) = 0 ))
 
Theoremdecpmatfsupp 21293* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝐴)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp 0 )
 
Theoremdecpmatid 21294 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐼 = (1r𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝐴)    &    1 = (1r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))
 
Theoremdecpmatmullem 21295* Lemma for decpmatmul 21296. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈𝐵𝑊𝐵) ∧ (𝐼𝑁𝐽𝑁𝐾 ∈ ℕ0)) → (𝐼((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝐽) = (𝑅 Σg (𝑡𝑁 ↦ (𝑅 Σg (𝑙 ∈ (0...𝐾) ↦ (((coe1‘(𝐼𝑈𝑡))‘𝑙)(.r𝑅)((coe1‘(𝑡𝑊𝐽))‘(𝐾𝑙))))))))
 
Theoremdecpmatmul 21296* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)       ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))))
 
Theoremdecpmatmulsumfsupp 21297* Lemma 0 for pm2mpmhm 21344. (Contributed by AV, 21-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    · = (.r𝐴)    &    0 = (0g𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙𝑘)))))) finSupp 0 )
 
Theorempmatcollpw1lem1 21298* Lemma 1 for pmatcollpw1 21300. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    × = ( ·𝑠𝑃)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐽𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝐼(𝑀 decompPMat 𝑛)𝐽) × (𝑛 𝑋))) finSupp (0g𝑃))
 
Theorempmatcollpw1lem2 21299* Lemma 2 for pmatcollpw1 21300: 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    × = ( ·𝑠𝑃)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑀𝑏) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 𝑋)))))
 
Theorempmatcollpw1 21300* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    × = ( ·𝑠𝑃)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑀 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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