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Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmat2pmatfval 20701* 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 20702* 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 20703 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 20704 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 20705 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 20706 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 20707 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 20708 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 20709* 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 20710 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 20711 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 20712 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 20713 The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 19208. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 19195, see lmhmsca 19203. (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 20714 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 20715 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 20716 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 20717 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 20718 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 20719 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 20720 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 20721 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 20722 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 20723 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 20724 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 20725 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 20726* 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 20727* 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 20728* 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 20729 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 20730 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 20731 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 20732* Lemma for m2cpminvid2 20733. (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 20733 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 20734 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 20735 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 20736 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 20737 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 20738 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 20739 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 20741), 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 20759.

Syntaxcdecpmat 20740 Extend class notation to include the decomposition of polynomial matrices.
class decompPMat

Definitiondf-decpmat 20741* 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 20742* 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 20743* 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 20744 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 20745 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 20746* 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 20747* 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 20748 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 20749* Lemma for decpmatmul 20750. (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 20750* 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 20751* Lemma 0 for pm2mpmhm 20798. (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 20752* Lemma 1 for pmatcollpw1 20754. (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 20753* Lemma 2 for pmatcollpw1 20754: 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 20754* 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))))))

Theorempmatcollpw2lem 20755* Lemma for pmatcollpw2 20756. (Contributed by AV, 3-Oct-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) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) finSupp (0g𝐶))

Theorempmatcollpw2 20756* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    × = ( ·𝑠𝑃)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))))

Theoremmonmatcollpw 20757 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 20748 (but requires 𝑅 to be commutative!). (Contributed by AV, 11-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &    0 = (0g𝐴)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝐶)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))

Theorempmatcollpwlem 20758 Lemma for pmatcollpw 20759. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Theorempmatcollpw 20759* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))

Theorempmatcollpwfi 20760* 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.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))

Theorempmatcollpw3lem 20761* Lemma for pmatcollpw3 20762 and pmatcollpw3fi 20763: 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷𝑚 𝐼)𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))

Theorempmatcollpw3 20762* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑓 ∈ (𝐷𝑚0)𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))

Theorempmatcollpw3fi 20763* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))

Theorempmatcollpw3fi1lem1 20764* Lemma 1 for pmatcollpw3fi1 20766. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))

Theorempmatcollpw3fi1lem2 20765* Lemma 2 for pmatcollpw3fi1 20766. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑓 ∈ (𝐷𝑚 {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))

Theorempmatcollpw3fi1 20766* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))

Theorempmatcollpwscmatlem1 20767 Lemma 1 for pmatcollpwscmat 20769. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &   𝑈 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝐶)    &   𝑀 = (𝑄 1 )       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))

Theorempmatcollpwscmatlem2 20768 Lemma 2 for pmatcollpwscmat 20769. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &   𝑈 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝐶)    &   𝑀 = (𝑄 1 )       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))

Theorempmatcollpwscmat 20769* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &   𝑈 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝐶)    &   𝑀 = (𝑄 1 )       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) ((𝑈‘((coe1𝑄)‘𝑛)) 1 )))))

11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices

The main result of this section is theorem pmmpric 20801, 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 (Poly1𝑅))

Or in a more common notation:
(𝑁 Mat (Poly1𝑅)) 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 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))

with 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅))) and (𝑚 decompPMat 𝑘) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1 ( i m j ) ) 𝑘))) is an isomorphism between these rings:

𝑇:𝐵1-1-onto𝐿 with 𝐿 = (Base‘(Poly1‘(𝑁 Mat 𝑅))) (see pm2mpf1o 20793 and pm2mprngiso 20800), and

𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1 p ) 𝑘)𝑗) · (𝑘𝐸𝑌))))))

is the corresponding inverse function:

(𝑇‘(𝐼𝑂)) = 𝑂) (see mp2pm2mp 20789).

In this section, the following conventions are mostly used:

• 𝑅 is a (unital) ring (see df-ring 18720)
• 𝑃 = (Poly1𝑅) is the polynomial algebra over (the ring) 𝑅 (see df-ply1 19725)
• 𝐾 = (Base‘𝑃) is its base set (see df-base 16036)
• 𝑌 = (var1𝑅) is its variable (see df-vr1 19724)
• · = ( ·𝑠𝑃) is its scalar multiplication (see df-vsca 16131 or lmodvscl 19053)
• 𝐸 = (.g‘(mulGrp‘𝑃)) is its exponentiation (see df-mulg 17713)
• 𝐴 = (𝑁 Mat 𝑅) is the algebra of N x N matrices over (the ring) 𝑅 (see df-mat 20387)
• 𝐶 = (𝑁 Mat 𝑃) is the algebra of N x N matrices over (the polynomial ring) 𝑃.
• 𝐵 = (Base‘𝐶) is its base set
• 𝑀𝐵 is a concrete polynomial matrix
• 𝑄 = (Poly1𝐴) is the polynomial algebra over (the matrix ring) 𝐴.
• 𝐿 = (Base‘𝑄) is its base set
• 𝑂𝐿 is a concrete polynomial with matrix coefficients
• 𝑋 = (var1𝐴) is its variable
• = ( ·𝑠𝑄) is its scalar multiplication
• = (.g‘(mulGrp‘𝑄)) is its exponentiation

Syntaxcpm2mp 20770 Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices.
class pMatToMatPoly

Definitiondf-pm2mp 20771* 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 (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))))

Theorempm2mpf1lem 20772* Lemma for pm2mpf1 20777. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈𝐵𝐾 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑈 decompPMat 𝑘) (𝑘 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))

Theorempm2mpval 20773* Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))

Theorempm2mpfval 20774* A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))))

Theorempm2mpcl 20775 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐿 = (Base‘𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝐿)

Theorempm2mpf 20776 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐿 = (Base‘𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐿)

Theorempm2mpf1 20777 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐿 = (Base‘𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐿)

Theorempm2mpcoe1 20778 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀𝐵𝐾 ∈ ℕ0)) → ((coe1‘(𝑇𝑀))‘𝐾) = (𝑀 decompPMat 𝐾))

Theoremidpm2idmp 20779 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (1r𝑄))

Theoremmptcoe1matfsupp 20780* 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 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐼𝑁𝐽𝑁) → (𝑘 ∈ ℕ0 ↦ (𝐼((coe1𝑂)‘𝑘)𝐽)) finSupp (0g𝑅))

Theoremmply1topmatcllem 20781* Lemma for mply1topmatcl 20783. (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐼𝑁𝐽𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝐼((coe1𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0g𝑃))

Theoremmply1topmatval 20782* 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‘(𝐼𝑂)) = 𝑂) (see mp2pm2mp 20789). (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))       ((𝑁𝑉𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Theoremmply1topmatcl 20783* A polynomial over matrices transformed into a polynomial matrix is a polynomial matrix. (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ 𝐵)

Theoremmp2pm2mplem1 20784* Lemma 1 for mp2pm2mp 20789. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Theoremmp2pm2mplem2 20785* Lemma 2 for mp2pm2mp 20789. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

Theoremmp2pm2mplem3 20786* Lemma 3 for mp2pm2mp 20789. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))‘𝐾)))

Theoremmp2pm2mplem4 20787* Lemma 4 for mp2pm2mp 20789. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝐾) = ((coe1𝑂)‘𝐾))

Theoremmp2pm2mplem5 20788* Lemma 5 for mp2pm2mp 20789. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑘 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑘) (𝑘 𝑋))) finSupp (0g𝑄))

Theoremmp2pm2mp 20789* 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 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)

Theorempm2mpghmlem2 20790* Lemma 2 for pm2mpghm 20794. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))) finSupp (0g𝑄))

Theorempm2mpghmlem1 20791 Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) (𝐾 𝑋)) ∈ 𝐿)

Theorempm2mpfo 20792 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵onto𝐿)

Theorempm2mpf1o 20793 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1-onto𝐿)

Theorempm2mpghm 20794 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄))

Theorempm2mpgrpiso 20795 The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpIso 𝑄))

Theorempm2mpmhmlem1 20796* Lemma 1 for pm2mpmhm 20798. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋))) finSupp (0g𝑄))

Theorempm2mpmhmlem2 20797* Lemma 2 for pm2mpmhm 20798. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))

Theorempm2mpmhm 20798 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))

Theorempm2mprhm 20799 The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄))

Theorempm2mprngiso 20800 The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄))

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