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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsymgmatr01 21801* Applying a permutation that does not fix a certain element of a set to a second element to an index of a matrix a row with 0's and a 1. (Contributed by AV, 3-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 (𝑘(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑘)) = 0 ))
 
Theoremgsummatr01lem1 21802* Lemma A for gsummatr01 21806. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}       ((𝑄𝑅𝑋𝑁) → (𝑄𝑋) ∈ 𝑁)
 
Theoremgsummatr01lem2 21803* Lemma B for gsummatr01 21806. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}       ((𝑄𝑅𝑋𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ (Base‘𝐺) → (𝑋𝐴(𝑄𝑋)) ∈ (Base‘𝐺)))
 
Theoremgsummatr01lem3 21804* Lemma 1 for gsummatr01 21806. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)))) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛))))(+g𝐺)(𝐾(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝐾))))
 
Theoremgsummatr01lem4 21805* Lemma 2 for gsummatr01 21806. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       ((((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) ∧ 𝑛 ∈ (𝑁 ∖ {𝐾})) → (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)) = (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄𝑛)))
 
Theoremgsummatr01 21806* Lemma 1 for smadiadetlem4 21816. (Contributed by AV, 8-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑅 = {𝑟𝑃 ∣ (𝑟𝐾) = 𝐿}    &    0 = (0g𝐺)    &   𝑆 = (Base‘𝐺)       (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖𝑁𝑗𝑁 (𝑖𝐴𝑗) ∈ 𝑆𝐵𝑆) ∧ (𝐾𝑁𝐿𝑁𝑄𝑅)) → (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄𝑛)))) = (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄𝑛)))))
 
Theoremmarep01ma 21807* Replacing a row of a square matrix by a row with 0's and a 1 results in a square matrix of the same dimension. (Contributed by AV, 30-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑀𝐵 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵)
 
Theoremsmadiadetlem0 21808* Lemma 0 for smadiadet 21817: The products of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄𝑛)))) = 0 ))
 
Theoremsmadiadetlem1 21809* Lemma 1 for smadiadet 21817: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       (((𝑀𝐵𝐾𝑁) ∧ 𝑝𝑃) → (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))) ∈ (Base‘𝑅))
 
Theoremsmadiadetlem1a 21810* Lemma 1a for smadiadet 21817: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↦ (((𝑌𝑆)‘𝑝) · (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = 0 )
 
Theoremsmadiadetlem2 21811* Lemma 2 for smadiadet 21817: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to itself. (Contributed by AV, 31-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ↦ (((𝑌𝑆)‘𝑝) · (𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = 0 )
 
Theoremsmadiadetlem3lem0 21812* Lemma 0 for smadiadetlem3 21815. (Contributed by AV, 12-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       (((𝑀𝐵𝐾𝑁) ∧ 𝑄𝑊) → (((𝑌𝑍)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑄𝑛))))) ∈ (Base‘𝑅))
 
Theoremsmadiadetlem3lem1 21813* Lemma 1 for smadiadetlem3 21815. (Contributed by AV, 12-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛)))))):𝑊⟶(Base‘𝑅))
 
Theoremsmadiadetlem3lem2 21814* Lemma 2 for smadiadetlem3 21815. (Contributed by AV, 12-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → ran (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛)))))) ⊆ ((Cntz‘𝑅)‘ran (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))))
 
Theoremsmadiadetlem3 21815* Lemma 3 for smadiadet 21817. (Contributed by AV, 31-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} ↦ (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))) = (𝑅 Σg (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))))
 
Theoremsmadiadetlem4 21816* Lemma 4 for smadiadet 21817. (Contributed by AV, 31-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝐺 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))       ((𝑀𝐵𝐾𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾} ↦ (((𝑌𝑆)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛(𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝𝑛))))))) = (𝑅 Σg (𝑝𝑊 ↦ (((𝑌𝑍)‘𝑝)(.r𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝𝑛))))))))
 
Theoremsmadiadet 21817 The determinant of a submatrix of a square matrix obtained by removing a row and a column at the same index equals the determinant of the original matrix with the row replaced with 0's and a 1 at the diagonal position. (Contributed by AV, 31-Jan-2019.) (Proof shortened by AV, 24-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)       ((𝑀𝐵𝐾𝑁) → (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾)))
 
Theoremsmadiadetglem1 21818 Lemma 1 for smadiadetg 21820. (Contributed by AV, 13-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)       ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
 
Theoremsmadiadetglem2 21819 Lemma 2 for smadiadetg 21820. (Contributed by AV, 14-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑅 ∈ CRing    &   𝐷 = (𝑁 maDet 𝑅)    &   𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))
 
Theoremsmadiadetg 21820 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)    &    · = (.r𝑅)       ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
 
Theoremsmadiadetg0 21821 Lemma for smadiadetr 21822: version of smadiadetg 21820 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)) ∧ 𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
 
Theoremsmadiadetr 21822 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 21820. 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))) ∧ (𝐾𝑁𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
 
11.5.5  Inverse matrix
 
Theoreminvrvald 21823 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 21824 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 21825 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))
 
11.5.6  Cramer's rule

In the following, Cramer's rule cramer 21838 is proven. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule 21838: "[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), 21838 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 21831), using matrix multiplication (mamuval 21533) and multiplication of a vector with a matrix (mulmarep1gsum2 21721). By using the multiplicativity of the determinant (mdetmul 21770) it follows that det (𝐴𝑖) = det (𝐴 × (𝑋𝑖)) = det 𝐴 · det (𝑋𝑖) (cramerimplem3 21832).

Furthermore, it follows that det (𝑋𝑖) = (𝑥𝑖) (cramerimplem1 21830). To show this, a special case of the Laplace expansion is used (smadiadetg 21820).

From these equations and the cancellation law for division in a ring (dvrcan3 19932) it follows that (𝑥𝑖) = det (𝑋𝑖) = det (𝐴𝑖) / det 𝐴.

This is the right to left implication (cramerimp 21833, cramerlem1 21834, cramerlem2 21835) of Cramer's rule (cramer 21838). The left to right implication is shown by cramerlem3 21836, using the fact that a solution of the system of linear equations exists (slesolex 21829).

Notice that for the special case of 0-dimensional matrices/vectors only the left to right implication is valid (see cramer0 21837), because assuming the right-hand side of the implication ((𝑋 · 𝑍) = 𝑌), 𝑍 could be anything (see mavmul0g 21700).

 
Theoremslesolvec 21826 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 21827 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 21828 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 21829* 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 21830 Lemma 1 for cramerimp 21833: 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 21831 Lemma 2 for cramerimp 21833: 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 21832 Lemma 3 for cramerimp 21833: 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 21833 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 21834* Lemma 1 for cramer 21838. (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 21835* Lemma 2 for cramer 21838. (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 21836* Lemma 3 for cramer 21838. (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 21837* 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 21838* 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 21833). 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 21825 or slesolinv 21827. 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))
 
11.6  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).

 
11.6.1  Basic properties
 
Theorempmatring 21839 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 21840 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)
 
Theorempmatassa 21841 The set of polynomial matrices over a commutative ring is an associative algebra. (Contributed by AV, 16-Jun-2024.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ AssAlg)
 
Theorempmat0op 21842* 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 21843* 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 21844 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 21845* 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 21846* 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 21847 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 21848* 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 21849 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)
 
11.6.2  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 21060) to a scalar 𝐴𝑅: ((algSc‘𝑃)‘𝐴). Analogously, constant polynomial matrices (over the ring 𝑅) are obtained by "lifting" matrices over the ring 𝑅 by the function matToPolyMat (see df-mat2pmat 21854), 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 21868) and that 𝑇 = (𝑁 matToPolyMat 𝑅) is a ring isomorphism from the ring of matrices over a ring 𝑅 onto the ring of constant polynomial matrices over the ring 𝑅 (see m2cpmrngiso 21905). Thus, the ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring, see matcpmric 21906. 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 21907.

 
Syntaxccpmat 21850 Extend class notation with the set of all constant polynomial matrices.
class ConstPolyMat
 
Syntaxcmat2pmat 21851 Extend class notation with the transformation of a matrix into a matrix of polynomials.
class matToPolyMat
 
Syntaxccpmat2mat 21852 Extend class notation with the transformation of a constant polynomial matrix into a matrix.
class cPolyMatToMat
 
Definitiondf-cpmat 21853* The set of all constant polynomial matrices, which are all matrices whose entries are constant polynomials (or "scalar polynomials", see ply1sclf 21454). (Contributed by AV, 15-Nov-2019.)
ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)})
 
Definitiondf-mat2pmat 21854* 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 21855* 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 21907, 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 21856* 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 21857 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 21858* 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 21859* 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 21860* 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 21861* 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 21862 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 21863* 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 21864* 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 21865* Lemma for cpmatmcl 21866. (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 21866* 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 21867 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 21868 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 21869 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 21870* 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 21871* 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 21872 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 21873 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 21874 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 21875 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 21876 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 21877 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 21878* 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 21879 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 21880 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 21881 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 21882 The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 20295. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 20282, see lmhmsca 20290. (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 21883 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 21884 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 21885 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 21886 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 21887 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 21888 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 21889 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 21890 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 21891 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 21892 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 21893 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 21894 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 21895* 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 21896* 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 21897* 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 21898 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 21899 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 21900 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)
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