P. 211 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	smadiadetglem2 21001	Lemma 2 for smadiadetg 21002. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘𝑓 · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))
Theorem	smadiadetg 21002	The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
Theorem	smadiadetg0 21003	Lemma for smadiadetr 21004: version of smadiadetg 21002 with all hypotheses defining class variables removed, i.e. all class variables defined in the hypotheses replaced in the theorem by their definition. (Contributed by AV, 15-Feb-2019.)
⊢ 𝑅 ∈ CRing    ⇒   ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
Theorem	smadiadetr 21004	The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. Closed form of smadiadetg 21002. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝑁 Mat 𝑅))) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
11.3.5  Inverse matrix
Theorem	invrvald 21005	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 · 𝑌) = 1 )    &   ⊢ (𝜑 → (𝑌 · 𝑋) = 1 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∧ (𝐼‘𝑋) = 𝑌))
Theorem	matinv 21006	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 = (invr‘𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ (𝐷‘𝑀) ∈ 𝑉) → (𝑀 ∈ 𝑈 ∧ (𝐼‘𝑀) = ((𝐻‘(𝐷‘𝑀)) ∙ (𝐽‘𝑀))))
Theorem	matunit 21007	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 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑈 ↔ (𝐷‘𝑀) ∈ 𝑉))
11.3.6  Cramer's rule In the following, Cramer's rule cramer 21020 is proven. 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." 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), 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 (𝐴‘𝑖) be the matrix obtained by replacing the i-th column of the coefficient matrix 𝐴 by the right-hand side vector 𝐵. Additionally, let (𝑋‘𝑖) be the matrix obtained by replacing the i-th column of the identity matrix by the solution vector 𝑋, with 𝑋 = (𝑥‘𝑖). Finally, it is assumed that det 𝐴 is a unit in the underlying ring. With these definitions, it follows that 𝐴 × (𝑋‘𝑖) = (𝐴‘𝑖) (cramerimplem2 21013), using matrix multiplication (mamuval 20715) and multiplication of a vector with a matrix (mulmarep1gsum2 20903). By using the multiplicativity of the determinant (mdetmul 20952) it follows that det (𝐴‘𝑖) = det (𝐴 × (𝑋‘𝑖)) = det 𝐴 · det (𝑋‘𝑖) (cramerimplem3 21014). Furthermore, it follows that det (𝑋‘𝑖) = (𝑥‘𝑖) (cramerimplem1 21012). To show this, a special case of the Laplace expansion is used (smadiadetg 21002). From these equations and the cancellation law for division in a ring (dvrcan3 19178) it follows that (𝑥‘𝑖) = det (𝑋‘𝑖) = det (𝐴‘𝑖) / det 𝐴. This is the right to left implication (cramerimp 21015, cramerlem1 21016, cramerlem2 21017) of Cramer's rule (cramer 21020). The left to right implication is shown by cramerlem3 21018, using the fact that a solution of the system of linear equations exists (slesolex 21011). Notice that for the special case of 0-dimensional matrices/vectors only the left to right implication is valid (see cramer0 21019), because assuming the right-hand side of the implication ((𝑋 · 𝑍) = 𝑌), 𝑍 could be anything (see mavmul0g 20882).
Theorem	slesolvec 21008	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) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
Theorem	slesolinv 21009	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 = (invr‘𝐴)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = ((𝐼‘𝑋) · 𝑌))
Theorem	slesolinvbi 21010	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 = (invr‘𝐴)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑍) = 𝑌 ↔ 𝑍 = ((𝐼‘𝑋) · 𝑌)))
Theorem	slesolex 21011*	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) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌)
Theorem	cramerimplem1 21012	Lemma 1 for cramerimp 21015: 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 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ 𝑍 ∈ 𝑉) → (𝐷‘𝐸) = (𝑍‘𝐼))
Theorem	cramerimplem2 21013	Lemma 2 for cramerimp 21015: 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 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋 × 𝐸) = 𝐻)
Theorem	cramerimplem3 21014	Lemma 3 for cramerimp 21015: 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 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ ⊗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)) = (𝐷‘𝐻))
Theorem	cramerimp 21015	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 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝐼) = ((𝐷‘𝐻) / (𝐷‘𝑋)))
Theorem	cramerlem1 21016*	Lemma 1 for cramer 21020. (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⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))
Theorem	cramerlem2 21017*	Lemma 2 for cramer 21020. (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⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))))
Theorem	cramerlem3 21018*	Lemma 3 for cramer 21020. (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⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))
Theorem	cramer0 21019*	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⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))
Theorem	cramer 21020*	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 21015). 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 21007 or slesolinv 21009. 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⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ≠ ∅) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) ↔ (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 (Poly1‘𝑅)), usually represented by the class variable 𝐶 (or 𝑌, if 𝐶 is already occupied): 𝐶 = (𝑁 Mat 𝑃) with 𝑃 = (Poly1‘𝑅).
11.4.1  Basic properties
Theorem	pmatring 21021	The set of polynomial matrices over a ring is a ring. (Contributed by AV, 6-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
Theorem	pmatlmod 21022	The set of polynomial matrices over a ring is a left module. (Contributed by AV, 6-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
Theorem	pmat0op 21023*	The zero polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 ))
Theorem	pmat1op 21024*	The identity polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 1 = (1r‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))
Theorem	pmat1ovd 21025	Entries of the identity polynomial matrix over a ring, deduction form. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 1 = (1r‘𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ 𝑈 = (1r‘𝐶)    ⇒   ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))
Theorem	pmat0opsc 21026*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐴‘ 0 )))
Theorem	pmat1opsc 21027*	The identity polynomial matrix over a ring represented as operation with "lifted scalars". (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝐴‘ 1 ), (𝐴‘ 0 ))))
Theorem	pmat1ovscd 21028	Entries of the identity polynomial matrix over a ring represented with "lifted scalars", deduction form. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ 𝑈 = (1r‘𝐶)    ⇒   ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (𝐴‘ 1 ), (𝐴‘ 0 )))
Theorem	pmatcoe1fsupp 21029*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
Theorem	1pmatscmul 21030	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ 1 = (1r‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (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 19821) to a scalar 𝐴 ∈ 𝑅: ((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 21035), 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 21049) 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 21086). 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 21087. 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 21088.
Syntax	ccpmat 21031	Extend class notation with the set of all constant polynomial matrices.
class ConstPolyMat
Syntax	cmat2pmat 21032	Extend class notation with the transformation of a matrix into a matrix of polynomials.
class matToPolyMat
Syntax	ccpmat2mat 21033	Extend class notation with the transformation of a constant polynomial matrix into a matrix.
class cPolyMatToMat
Definition	df-cpmat 21034*	The set of all constant polynomial matrices, which are all matrices whose entries are constant polynomials (or "scalar polynomials", see ply1sclf 20172). (Contributed by AV, 15-Nov-2019.)
⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)})
Definition	df-mat2pmat 21035*	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)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))))
Definition	df-cpmat2mat 21036*	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 21088, 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) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Theorem	cpmat 21037*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)})
Theorem	cpmatpmat 21038	A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵)
Theorem	cpmatel 21039*	Property of a constant polynomial matrix. (Contributed by AV, 15-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅)))
Theorem	cpmatelimp 21040*	Implication of a set being a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))))
Theorem	cpmatel2 21041*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))
Theorem	cpmatelimp2 21042*	Another implication of a set being a constant polynomial matrix. (Contributed by AV, 17-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘))))
Theorem	1elcpmat 21043	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆)
Theorem	cpmatacl 21044*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆)
Theorem	cpmatinvcl 21045*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆)
Theorem	cpmatmcllem 21046*	Lemma for cpmatmcl 21047. (Contributed by AV, 18-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅))
Theorem	cpmatmcl 21047*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆)
Theorem	cpmatsubgpmat 21048	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))
Theorem	cpmatsrgpmat 21049	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶))
Theorem	0elcpmat 21050	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) ∈ 𝑆)
Theorem	mat2pmatfval 21051*	Value of the matrix transformation. (Contributed by AV, 31-Jul-2019.)
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
Theorem	mat2pmatval 21052*	The result of a matrix transformation. (Contributed by AV, 31-Jul-2019.)
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
Theorem	mat2pmatvalel 21053	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 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝑇‘𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))
Theorem	mat2pmatbas 21054	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶))
Theorem	mat2pmatbas0 21055	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐻)
Theorem	mat2pmatf 21056	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐻)
Theorem	mat2pmatf1 21057	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1→𝐻)
Theorem	mat2pmatghm 21058	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
Theorem	mat2pmatmul 21059*	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐴)𝑦)) = ((𝑇‘𝑥)(.r‘𝐶)(𝑇‘𝑦)))
Theorem	mat2pmat1 21060	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐴)) = (1r‘𝐶))
Theorem	mat2pmatmhm 21061	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)))
Theorem	mat2pmatrhm 21062	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝐶))
Theorem	mat2pmatlin 21063	The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 19542. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 19529, see lmhmsca 19537. (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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝐴)    &   ⊢ × = ( ·𝑠 ‘𝐶)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌)))
Theorem	0mat2pmat 21064	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 = (Poly1‘𝑅)    &   ⊢ 0 = (0g‘(𝑁 Mat 𝑅))    &   ⊢ 𝑍 = (0g‘(𝑁 Mat 𝑃))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘ 0 ) = 𝑍)
Theorem	idmatidpmat 21065	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 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘(𝑁 Mat 𝑅))    &   ⊢ 𝐼 = (1r‘(𝑁 Mat 𝑃))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘ 1 ) = 𝐼)
Theorem	d0mat2pmat 21066	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.)
⊢ (𝑅 ∈ 𝑉 → ((∅ matToPolyMat 𝑅)‘∅) = ∅)
Theorem	d1mat2pmat 21067	The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑆 = (algSc‘𝑃)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
Theorem	mat2pmatscmxcl 21068	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∗ (𝑇‘𝑀)) ∈ 𝐵)
Theorem	m2cpm 21069	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 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝑆)
Theorem	m2cpmf 21070	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:𝐵⟶𝑆)
Theorem	m2cpmf1 21071	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:𝐵–1-1→𝑆)
Theorem	m2cpmghm 21072	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝑈 = (𝐶 ↾s 𝑆)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈))
Theorem	m2cpmmhm 21073	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝑈 = (𝐶 ↾s 𝑆)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))
Theorem	m2cpmrhm 21074	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝑈 = (𝐶 ↾s 𝑆)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈))
Theorem	m2pmfzmap 21075	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 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑𝑚 (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑇‘(𝑏‘𝐼)) ∈ (Base‘𝑌))
Theorem	m2pmfzgsumcl 21076*	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 = (Poly1‘𝑅)    &   ⊢ 𝑌 = (𝑁 Mat 𝑃)    &   ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ · = ( ·𝑠 ‘𝑌)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌))
Theorem	cpm2mfval 21077*	Value of the inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)
⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐼 = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Theorem	cpm2mval 21078*	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 ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
Theorem	cpm2mvalel 21079	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 ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝐼‘𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0))
Theorem	cpm2mf 21080	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:𝑆⟶𝐾)
Theorem	m2cpminvid 21081	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 ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = 𝑀)
Theorem	m2cpminvid2lem 21082*	Lemma for m2cpminvid2 21083. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
Theorem	m2cpminvid2 21083	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 ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀)
Theorem	m2cpmfo 21084	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:𝐾–onto→𝑆)
Theorem	m2cpmf1o 21085	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:𝐾–1-1-onto→𝑆)
Theorem	m2cpmrngiso 21086	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝑈 = (𝐶 ↾s 𝑆)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))
Theorem	matcpmric 21087	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑈 = (𝐶 ↾s 𝑆)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ≃𝑟 𝑈)
Theorem	m2cpminv 21088	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:𝑆–1-1-onto→𝐾 ∧ ◡𝐼 = 𝑇))
Theorem	m2cpminv0 21089	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 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝐴)    &   ⊢ 𝑍 = (0g‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = 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 21091), which maps a polynomial matrix 𝑀 to a constant matrix consisting of the coefficients of the scaled monomials ((𝑐‘𝑘) ∗ (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 21109.
Syntax	cdecpmat 21090	Extend class notation to include the decomposition of polynomial matrices.
class decompPMat
Definition	df-decpmat 21091*	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)))
Theorem	decpmatval0 21092*	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.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
Theorem	decpmatval 21093*	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 21094	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 21095	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 21096*	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 21097*	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 21098	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 21099*	Lemma for decpmatmul 21100. (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 21100*	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))))))