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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchpmatfval 22101* Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π· = (𝑁 maDet 𝑃)    &    βˆ’ = (-gβ€˜π‘Œ)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    1 = (1rβ€˜π‘Œ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) β†’ 𝐢 = (π‘š ∈ 𝐡 ↦ (π·β€˜((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘š)))))
 
Theoremchpmatval 22102 The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π· = (𝑁 maDet 𝑃)    &    βˆ’ = (-gβ€˜π‘Œ)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    1 = (1rβ€˜π‘Œ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐡) β†’ (πΆβ€˜π‘€) = (π·β€˜((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))))
 
Theoremchpmatply1 22103 The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   πΈ = (Baseβ€˜π‘ƒ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (πΆβ€˜π‘€) ∈ 𝐸)
 
Theoremchpmatval2 22104* The characteristic polynomial of a (square) matrix (expressed with the Leibnitz formula for the determinant). (Contributed by AV, 2-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    βˆ’ = (-gβ€˜π‘Œ)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    1 = (1rβ€˜π‘Œ)    &   πΊ = (SymGrpβ€˜π‘)    &   π» = (Baseβ€˜πΊ)    &   π‘ = (β„€RHomβ€˜π‘ƒ)    &   π‘† = (pmSgnβ€˜π‘)    &   π‘ˆ = (mulGrpβ€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘ƒ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ (πΆβ€˜π‘€) = (𝑃 Ξ£g (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)β€˜π‘) Γ— (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))π‘₯)))))))
 
Theoremchpmat0d 22105 The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
𝐢 = (βˆ… CharPlyMat 𝑅)    β‡’   (𝑅 ∈ Ring β†’ (πΆβ€˜βˆ…) = (1rβ€˜(Poly1β€˜π‘…)))
 
Theoremchpmat1dlem 22106 Lemma for chpmat1d 22107. (Contributed by AV, 7-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π‘† = (algScβ€˜π‘ƒ)    &   πΊ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    β‡’   ((𝑅 ∈ Ring ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (𝐼((𝑋( ·𝑠 β€˜πΊ)(1rβ€˜πΊ))(-gβ€˜πΊ)(π‘‡β€˜π‘€))𝐼) = (𝑋 βˆ’ (π‘†β€˜(𝐼𝑀𝐼))))
 
Theoremchpmat1d 22107 The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π‘† = (algScβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (πΆβ€˜π‘€) = (𝑋 βˆ’ (π‘†β€˜(𝐼𝑀𝐼))))
 
Theoremchpdmatlem0 22108 Lemma 0 for chpdmat 22112. (Contributed by AV, 18-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘† = (algScβ€˜π‘ƒ)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π‘„ = (𝑁 Mat 𝑃)    &    1 = (1rβ€˜π‘„)    &    Β· = ( ·𝑠 β€˜π‘„)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (𝑋 Β· 1 ) ∈ (Baseβ€˜π‘„))
 
Theoremchpdmatlem1 22109 Lemma 1 for chpdmat 22112. (Contributed by AV, 18-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘† = (algScβ€˜π‘ƒ)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π‘„ = (𝑁 Mat 𝑃)    &    1 = (1rβ€˜π‘„)    &    Β· = ( ·𝑠 β€˜π‘„)    &   π‘ = (-gβ€˜π‘„)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) β†’ ((𝑋 Β· 1 )𝑍(π‘‡β€˜π‘€)) ∈ (Baseβ€˜π‘„))
 
Theoremchpdmatlem2 22110 Lemma 2 for chpdmat 22112. (Contributed by AV, 18-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘† = (algScβ€˜π‘ƒ)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π‘„ = (𝑁 Mat 𝑃)    &    1 = (1rβ€˜π‘„)    &    Β· = ( ·𝑠 β€˜π‘„)    &   π‘ = (-gβ€˜π‘„)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    β‡’   ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑖 β‰  𝑗) ∧ (𝑖𝑀𝑗) = 0 ) β†’ (𝑖((𝑋 Β· 1 )𝑍(π‘‡β€˜π‘€))𝑗) = (0gβ€˜π‘ƒ))
 
Theoremchpdmatlem3 22111 Lemma 3 for chpdmat 22112. (Contributed by AV, 18-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘† = (algScβ€˜π‘ƒ)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π‘„ = (𝑁 Mat 𝑃)    &    1 = (1rβ€˜π‘„)    &    Β· = ( ·𝑠 β€˜π‘„)    &   π‘ = (-gβ€˜π‘„)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ 𝐾 ∈ 𝑁) β†’ (𝐾((𝑋 Β· 1 )𝑍(π‘‡β€˜π‘€))𝐾) = (𝑋 βˆ’ (π‘†β€˜(𝐾𝑀𝐾))))
 
Theoremchpdmat 22112* The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘† = (algScβ€˜π‘ƒ)    &   π΅ = (Baseβ€˜π΄)    &   π‘‹ = (var1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (𝑖 β‰  𝑗 β†’ (𝑖𝑀𝑗) = 0 )) β†’ (πΆβ€˜π‘€) = (𝐺 Ξ£g (π‘˜ ∈ 𝑁 ↦ (𝑋 βˆ’ (π‘†β€˜(π‘˜π‘€π‘˜))))))
 
Theoremchpscmat 22113* The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   π· = {π‘š ∈ (Baseβ€˜π΄) ∣ βˆƒπ‘ ∈ (Baseβ€˜π‘…)βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (π‘–π‘šπ‘—) = if(𝑖 = 𝑗, 𝑐, (0gβ€˜π‘…))}    &   π‘† = (algScβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ βˆ€π‘› ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) β†’ (πΆβ€˜π‘€) = ((β™―β€˜π‘) ↑ (𝑋 βˆ’ (π‘†β€˜πΈ))))
 
Theoremchpscmat0 22114* The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   π· = {π‘š ∈ (Baseβ€˜π΄) ∣ βˆƒπ‘ ∈ (Baseβ€˜π‘…)βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (π‘–π‘šπ‘—) = if(𝑖 = 𝑗, 𝑐, (0gβ€˜π‘…))}    &   π‘† = (algScβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ βˆ€π‘› ∈ 𝑁 (𝑛𝑀𝑛) = (𝐼𝑀𝐼))) β†’ (πΆβ€˜π‘€) = ((β™―β€˜π‘) ↑ (𝑋 βˆ’ (π‘†β€˜(𝐼𝑀𝐼)))))
 
Theoremchpscmatgsumbin 22115* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   π· = {π‘š ∈ (Baseβ€˜π΄) ∣ βˆƒπ‘ ∈ (Baseβ€˜π‘…)βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (π‘–π‘šπ‘—) = if(𝑖 = 𝑗, 𝑐, (0gβ€˜π‘…))}    &   π‘† = (algScβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   πΉ = (.gβ€˜π‘ƒ)    &   π» = (mulGrpβ€˜π‘…)    &   πΈ = (.gβ€˜π»)    &   πΌ = (invgβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ βˆ€π‘› ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) β†’ (πΆβ€˜π‘€) = (𝑃 Ξ£g (𝑙 ∈ (0...(β™―β€˜π‘)) ↦ (((β™―β€˜π‘)C𝑙)𝐹((((β™―β€˜π‘) βˆ’ 𝑙)𝐸(πΌβ€˜(𝐽𝑀𝐽))) Β· (𝑙 ↑ 𝑋))))))
 
Theoremchpscmatgsummon 22116* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   π· = {π‘š ∈ (Baseβ€˜π΄) ∣ βˆƒπ‘ ∈ (Baseβ€˜π‘…)βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (π‘–π‘šπ‘—) = if(𝑖 = 𝑗, 𝑐, (0gβ€˜π‘…))}    &   π‘† = (algScβ€˜π‘ƒ)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   πΉ = (.gβ€˜π‘ƒ)    &   π» = (mulGrpβ€˜π‘…)    &   πΈ = (.gβ€˜π»)    &   πΌ = (invgβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   π‘ = (.gβ€˜π‘…)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ βˆ€π‘› ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) β†’ (πΆβ€˜π‘€) = (𝑃 Ξ£g (𝑙 ∈ (0...(β™―β€˜π‘)) ↦ ((((β™―β€˜π‘)C𝑙)𝑍(((β™―β€˜π‘) βˆ’ 𝑙)𝐸(πΌβ€˜(𝐽𝑀𝐽)))) Β· (𝑙 ↑ 𝑋)))))
 
Theoremchp0mat 22117 The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &    0 = (0gβ€˜π΄)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) β†’ (πΆβ€˜ 0 ) = ((β™―β€˜π‘) ↑ 𝑋))
 
Theoremchpidmat 22118 The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
𝐢 = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   πΌ = (1rβ€˜π΄)    &   π‘† = (algScβ€˜π‘ƒ)    &    1 = (1rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘ƒ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) β†’ (πΆβ€˜πΌ) = ((β™―β€˜π‘) ↑ (𝑋 βˆ’ (π‘†β€˜ 1 ))))
 
Theoremchmaidscmat 22119 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 5-Jul-2022.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   πΈ = (Baseβ€˜π‘ƒ)    &   π‘Œ = (𝑁 Mat 𝑃)    &   πΎ = (Baseβ€˜π‘Œ)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘† = (𝑁 ScMat 𝑃)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ ((πΆβ€˜π‘€) Β· 1 ) ∈ 𝑆)
 
11.7.2  The characteristic factor function G

In this subsection the function 𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›)))))))) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 22149. Therefore, this function is called "characteristic factor function" (in short "chfacf") in the following. It plays an important role in the proof of the Cayley-Hamilton theorem, see cayhamlem1 22137, cayhamlem3 22158 and cayhamlem4 22159.

 
Theoremfvmptnn04if 22120* The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐢, if(𝑆 < 𝑛, 𝐷, 𝐡))))    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   ((πœ‘ ∧ 𝑁 = 0) β†’ π‘Œ = ⦋𝑁 / π‘›β¦Œπ΄)    &   ((πœ‘ ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) β†’ π‘Œ = ⦋𝑁 / π‘›β¦Œπ΅)    &   ((πœ‘ ∧ 𝑁 = 𝑆) β†’ π‘Œ = ⦋𝑁 / π‘›β¦ŒπΆ)    &   ((πœ‘ ∧ 𝑆 < 𝑁) β†’ π‘Œ = ⦋𝑁 / π‘›β¦Œπ·)    β‡’   (πœ‘ β†’ (πΊβ€˜π‘) = π‘Œ)
 
Theoremfvmptnn04ifa 22121* The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐢, if(𝑆 < 𝑛, 𝐷, 𝐡))))    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ 𝑁 = 0 ∧ ⦋𝑁 / π‘›β¦Œπ΄ ∈ 𝑉) β†’ (πΊβ€˜π‘) = ⦋𝑁 / π‘›β¦Œπ΄)
 
Theoremfvmptnn04ifb 22122* The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐢, if(𝑆 < 𝑛, 𝐷, 𝐡))))    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / π‘›β¦Œπ΅ ∈ 𝑉) β†’ (πΊβ€˜π‘) = ⦋𝑁 / π‘›β¦Œπ΅)
 
Theoremfvmptnn04ifc 22123* The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐢, if(𝑆 < 𝑛, 𝐷, 𝐡))))    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / π‘›β¦ŒπΆ ∈ 𝑉) β†’ (πΊβ€˜π‘) = ⦋𝑁 / π‘›β¦ŒπΆ)
 
Theoremfvmptnn04ifd 22124* The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐢, if(𝑆 < 𝑛, 𝐷, 𝐡))))    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ 𝑆 < 𝑁 ∧ ⦋𝑁 / π‘›β¦Œπ· ∈ 𝑉) β†’ (πΊβ€˜π‘) = ⦋𝑁 / π‘›β¦Œπ·)
 
Theoremchfacfisf 22125* The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝐺:β„•0⟢(Baseβ€˜π‘Œ))
 
Theoremchfacfisfcpmat 22126* The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘† = (𝑁 ConstPolyMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝐺:β„•0βŸΆπ‘†)
 
Theoremchfacffsupp 22127* The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ 𝐺 finSupp (0gβ€˜π‘Œ))
 
Theoremchfacfscmulcl 22128* Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) ∧ 𝐾 ∈ β„•0) β†’ ((𝐾 ↑ 𝑋) Β· (πΊβ€˜πΎ)) ∈ (Baseβ€˜π‘Œ))
 
Theoremchfacfscmul0 22129* A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) ∧ 𝐾 ∈ (β„€β‰₯β€˜(𝑠 + 2))) β†’ ((𝐾 ↑ 𝑋) Β· (πΊβ€˜πΎ)) = 0 )
 
Theoremchfacfscmulfsupp 22130* A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ 𝑋) Β· (πΊβ€˜π‘–))) finSupp 0 )
 
Theoremchfacfscmulgsum 22131* Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    + = (+gβ€˜π‘Œ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ 𝑋) Β· (πΊβ€˜π‘–)))) = ((π‘Œ Ξ£g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· ((π‘‡β€˜(π‘β€˜(𝑖 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘–))))))) + ((((𝑠 + 1) ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘ ))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))))))
 
Theoremchfacfpmmulcl 22132* Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘Œ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) ∧ 𝐾 ∈ β„•0) β†’ ((𝐾 ↑ (π‘‡β€˜π‘€)) Γ— (πΊβ€˜πΎ)) ∈ (Baseβ€˜π‘Œ))
 
Theoremchfacfpmmul0 22133* The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘Œ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) ∧ 𝐾 ∈ (β„€β‰₯β€˜(𝑠 + 2))) β†’ ((𝐾 ↑ (π‘‡β€˜π‘€)) Γ— (πΊβ€˜πΎ)) = 0 )
 
Theoremchfacfpmmulfsupp 22134* A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘Œ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ (π‘‡β€˜π‘€)) Γ— (πΊβ€˜π‘–))) finSupp 0 )
 
Theoremchfacfpmmulgsum 22135* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘Œ))    &    + = (+gβ€˜π‘Œ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ (π‘‡β€˜π‘€)) Γ— (πΊβ€˜π‘–)))) = ((π‘Œ Ξ£g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (π‘‡β€˜π‘€)) Γ— ((π‘‡β€˜(π‘β€˜(𝑖 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘–))))))) + ((((𝑠 + 1) ↑ (π‘‡β€˜π‘€)) Γ— (π‘‡β€˜(π‘β€˜π‘ ))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))))))
 
Theoremchfacfpmmulgsum2 22136* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘Œ))    &    + = (+gβ€˜π‘Œ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ (π‘‡β€˜π‘€)) Γ— (πΊβ€˜π‘–)))) = ((π‘Œ Ξ£g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (π‘‡β€˜π‘€)) Γ— (π‘‡β€˜(π‘β€˜(𝑖 βˆ’ 1)))) βˆ’ (((𝑖 + 1) ↑ (π‘‡β€˜π‘€)) Γ— (π‘‡β€˜(π‘β€˜π‘–)))))) + ((((𝑠 + 1) ↑ (π‘‡β€˜π‘€)) Γ— (π‘‡β€˜(π‘β€˜π‘ ))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))))))
 
Theoremcayhamlem1 22137* Lemma 1 for cayleyhamilton 22161. (Contributed by AV, 11-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘Œ))    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ (π‘‡β€˜π‘€)) Γ— (πΊβ€˜π‘–)))) = 0 )
 
11.7.3  The Cayley-Hamilton theorem

In this section, a direct algebraic proof for the Cayley-Hamilton theorem is provided, according to Wikipedia ("Cayley-Hamilton theorem", 09-Nov-2019, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem, section "A direct algebraic proof" (this approach is also used for proving Lemma 1.9 in [Hefferon] p. 427):

"This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries. The matrix (t * In - A) whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials [over a commutative ring] form a commutative ring, it has an adjugate

(1) B = adj(t * In - A) .

Then, according to the right-hand fundamental relation of the adjugate, one has

(2) ( t * In - A ) x B = det(t * In - A) x In = p(t) * In .

Since B is also a matrix with polynomials in t as entries, one can, for each i, collect the coefficients of t^i in each entry to form a matrix Bi of numbers, such that one has

(3) B = sumi = 0 to (n-1) t^i Bi .

(The way the entries of B are defined makes clear that no powers higher than t^(n-1) occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t^i has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity

(4) p(t) * In = ( t * In - A ) x B
= ( t * In - A ) x sumi = 0 to (n-1) t^i * Bi
= sumi = 0 to (n-1) t * In x t^i Bi - sumi = 0 to (n-1) A * t^i Bi
= sumi = 0 to (n-1) t^(i+1) * Bi - sumi = 0 to (n-1) t^i * A x Bi
= t^n Bn-1 + sumi = 1 to (n-1) t^i * ( Bi-1 - A x Bi ) - A x B0 .

Writing

(5) p(t) In = t^n * In + t^(n-1) * c(n-1) x In + ... + t * c1 In + c0 In ,

one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t^i is the same on both sides; it follows that the constant matrices with coefficient t^i in both expressions must be equal. Writing these equations then for i from n down to 0, one finds

(6) Bn-1 = In , Bi-1 - A x Bi = ci * In for 1 <= i <= n-1 , - A x B0 = c0 * In .

Finally, multiply the equation of the coefficients of t^i from the left by A^i, and sum up:

(7) A^n Bn-1 + sumi = 1 to (n-1) ( A^i x Bi-1 - A^(i+1) x Bi ) - A x B0 = A^n + cn-1 * A^(n-1) + ... + c1 * A + c0 * In .

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to p(A):

(8) 0 = p(A) .

This completes the proof."

To formalize this approach, the steps mentioned in Wikipedia must be elaborated in more detail.

The first step is to formalize the preliminaries and the objective of the theorem. In Wikipedia, the Cayley-Hamilton theorem is stated as follows: "... the Cayley-Hamilton theorem ... states that every square matrix over a commutative ring ... satisfies its own characteristic equation." Or in more detail: "If A is a given n x n matrix and In is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * In - A), where det is the determinant operation and t is a variable for a scalar element of the base ring. Since the entries of the matrix (t * In - A) are (linear or constant) polynomials in t, the determinant is also an n-th order monic polynomial in t. The Cayley-Hamilton theorem states that if one defines an analogous matrix equation, p(A), consisting of the replacement of the scalar eigenvalues t with the matrix A, then this polynomial in the matrix A results in the zero matrix,

p(A) = 0.

The powers of A, obtained by substitution from powers of t, are defined by repeated matrix multiplication; the constant term of p(t) gives a multiple of the power A^0, which is defined as the identity matrix. The theorem allows A^n to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial."

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 22098, the characteristic polynomial of an 𝑁 x 𝑁 matrix 𝑀 over a ring 𝑅 is defined as

((𝑁 CharPlyMat 𝑅)β€˜π‘€) = (π·β€˜((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))))

where 𝐷 = (𝑁 maDet 𝑃) is the function mapping an 𝑁 x 𝑁 matrix over the polynomial ring over the ring 𝑅 to its determinant, 𝑋 = (var1β€˜π‘…) is the variable of the polynomials over 𝑅, 1 is the 𝑁 x 𝑁 identity matrix as matrix over the polynomial ring over the ring 𝑅 (not the 𝑁 x 𝑁 identity matrix of the matrices over the ring 𝑅!) and (π‘‡β€˜π‘€) = ((𝑁 matToPolyMat 𝑅)β€˜π‘€) is the matrix 𝑀 over a ring 𝑅 transformed into a constant matrix over the polynomial ring over the ring 𝑅. Thus Β· is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 22102).

By this definition, it is ensured that ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€)), the matrix whose determinant is the characteristic polynomial of the matrix 𝑀, is actually a matrix over the polynomial ring over the ring 𝑅, as stated in Wikipedia ("matrix with polynomials as entries"). This matrix is called the characteristic matrix of matrix 𝑀 (see Wikipedia "Polynomial matrix", 16-Nov-2019, https://en.wikipedia.org/wiki/Polynomial_matrix 22102). Following the notation in Wikipedia, we denote the characteristic polynomial of the matrix 𝑀, formally defined by ((𝑁 CharPlyMat 𝑅)β€˜π‘€) as "p(M)" in the comments. The characteristric matrix ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€)) will be denoted by "c(M)", so that "p(M) = det( c(M) )".

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix 𝑀 must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 21589), also the characteristic polynomial can be written in this way:

p(M) = sumi ( pi * M^i )

Here, * is the scalar multiplication in the algebra of the polynomials over the ring 𝑅, and the coefficients are elements of the ring 𝑅.

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( pi * M^i)", see also cayleyhamilton1 22163. Here, * is the scalar multiplication in the algebra of the matrices over the ring 𝑅.

To prove the Cayley-Hamilton theorem, we have to show that "p(M) = 0", where 0 is the zero matrix.

In this section, the following class variables and informal identifiers (acronyms in the form "A(B)" or "AB") are used:

class variable/ informal identifier definiens explanation
𝑁 An arbitrary finite set, used as dimension for matrices
𝑅 An arbitrary (commutative) ring: 𝑅 ∈ CRing
B(R) (Baseβ€˜π‘…) Base set of (the ring) 𝑅
𝐴 (𝑁 Mat 𝑅) Algebra of 𝑁 x 𝑁 matrices over (the ring) 𝑅
𝐡 (Baseβ€˜π΄) Base set of the algebra of 𝑁 x 𝑁 matrices, i .e. the set of all 𝑁 x 𝑁 matrices
𝑀 An arbitrary 𝑁 x 𝑁 matrix
𝑃 (Poly1β€˜π‘…) The algebra of polynomials over (the ring) 𝑅
B(P) (Baseβ€˜π‘ƒ) Base set of the algebra of polynomials, i .e. the set of all polynomials
𝑋, XR (var1β€˜π‘…) The variable of polynomials over (the ring) 𝑅
π‘Œ, XA (var1β€˜π΄) The variable of polynomials over matrices of the algebra 𝐴
↑ (.gβ€˜(mulGrpβ€˜π‘ƒ)) The group exponentiation for polynomials over (the ring) 𝑅
^ Arbitrary group exponentiation
π‘ˆ (algScβ€˜π‘ƒ) The injection of scalars, i.e. elements of (the ring) 𝑅 into the base set of the algebra of polynomials over 𝑅
(π‘ˆβ€˜π‘), S(p) ((algScβ€˜π‘ƒ)β€˜π‘) The element 𝑝 of (the ring) 𝑅 represented as polynomial over 𝑅
π‘Œ (𝑁 Mat 𝑃) Algebra of 𝑁 x 𝑁 matrices over (the polynomial ring) 𝑃 over the ring 𝑅
B(Y) (Baseβ€˜π‘Œ) Base set of the algebra of polynomial 𝑁 x 𝑁 matrices, i .e. the set of all polynomial 𝑁 x 𝑁 matrices
𝑄 (Poly1β€˜π΄) Algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅
B(Q) (Baseβ€˜π‘„) Base set of the algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅, i .e. the set of all polynomials having 𝑁 x 𝑁 matrices as coefficients
+, + (+gβ€˜π‘Œ) The addition of polynomial matrices
βˆ’, - (-gβ€˜π‘Œ) The subtraction of polynomial matrices
Β·, *Y ( ·𝑠 β€˜π‘Œ) The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*A ( ·𝑠 β€˜π΄) The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*Q ( ·𝑠 β€˜π‘„) The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
Γ—, xY (.rβ€˜π‘Œ) The multiplication of polynomial matrices
xA (.rβ€˜π΄) The multiplication of matrices
xQ (.rβ€˜π‘„) The multiplication of polynomials over matrices
1, 1Y (1rβ€˜π‘Œ) The identity matrix in the algebra of polynomial matrices over 𝑅
1A (1rβ€˜π΄) The identity matrix in the algebra of matrices over 𝑅
0, 0Y (0gβ€˜π‘Œ) The zero matrix in the algebra of matrices consisting of polynomials
𝑇 (𝑁 matToPolyMat 𝑅) The transformation of an 𝑁 x 𝑁 matrix over 𝑅 into a polynomial 𝑁 x 𝑁 matrix over 𝑅
T1(M) (π‘‡β€˜π‘€) The matrix M transformed into a polynomial 𝑁 x 𝑁 matrix over 𝑅
U(M) (π‘ˆβ€˜π‘€) The (constant) polynomial 𝑁 x 𝑁 matrix M transformed into a matrix over the ring 𝑅. Inverse function of 𝑇: (π‘‡β€˜(π‘ˆβ€˜π‘€)) = 𝑀 (see m2cpminvid2 22026 )
T2(M) ((𝑁 pMatToMatPoly 𝑅)β€˜π‘€) The polynomial 𝑁 x 𝑁 matrix M transformed into a polynomial over the 𝑁 x 𝑁 matrices over 𝑅
𝐼, c(M) ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€)) The characteristic matrix of a matrix 𝑀, i.e. the matrix whose determinant is the characteristic polynomial of the matrix 𝑀
𝐢 (𝑁 CharPlyMat 𝑅) The function mapping a matrix over (a ring) 𝑅 to its characteristic polynomial
𝐾, p(M) (πΆβ€˜π‘€) The characteristic polynomial of a matrix over (a ring) 𝑅
𝐻 (𝐾 · 1 ) The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements
𝐽 (𝑁 maAdju 𝑃) The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")
π‘Š, adj(cm(M)) (π½β€˜πΌ) The adjugate of the characteristic matrix of the matrix 𝑀


Using this notation, we have:
  1. "c(M) e. B(Y)", or 𝐼 ∈ (Baseβ€˜π‘Œ), see chmatcl 22099
  2. "p(M) e. B(P)", or 𝐾 ∈ (Baseβ€˜π‘ƒ), see chpmatply1 22103
  3. "T(M) e. B(Y)", or (π‘‡β€˜π‘€) ∈ (Baseβ€˜π‘Œ), see mat2pmatbas 21997
  4. 𝐽:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘Œ), see maduf 21912
  5. "adj(cm(M)) e. B(Y)", or π‘Š ∈ (Baseβ€˜π‘Œ)


Following the proof shown in Wikipedia, the following steps are performed:
  1. Write π‘Š, the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 22059:
    adj(cm(M)) = sumi=0 to s ( XR ^i *Y T1(b(i)) )
    where b(i) are matrices over the ring 𝑅, so T1(b(i)) are constant polynomial matrices.
    This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write π‘Š as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of (𝐼 Γ— π‘Š) as infinite, but finitely supported sum, see step 3.
  2. Write (𝐼 Γ— π‘Š), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 22148. This representation is obtained by replacing π‘Š by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
    cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
    where b(i) are matrices over 𝑅, so T1(b(i)) are constant polynomial matrices:
    cm(M) *Y adj(cm(M))
    = cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 22059 (step 1.)]
    = ( ( XA *Y 1Y ) - T1(M) ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [def. of cm(M)]
    = ( XA *Y 1Y ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) - T1(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see rngsubdir 19947]
    = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0)) [see cpmadugsumlemF 22147]
    This step corresponds partially to (4) in Wikipedia.
  3. Write (𝐼 Γ— π‘Š) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 22149:
    cm(M) * adj(cm(M)) = sumi ( XR ^i *Y G(i) )
    This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 22125, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 22126). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
  4. Write 𝐻 = (𝐾 Β· 1 ), the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 22139:
    p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
    The proof of cpmidgsum 22139 is making use of pmatcollpwscmat 22062, because 𝐻 = (𝐾 Β· 1 ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since pi is an element of the ring 𝑅, S(pi) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
  5. Transform the sum representation of (𝐼 Γ— π‘Š) from step 3. into polynomials over matrices:
    T2(cm(M) * adj(cm(M))) = sumi ( U(G(i)) *Q XA ^i ) [see cpmadumatpoly 22154]
    where U(G(i)) is a matrix over the ring 𝑅.
  6. Transform the sum representation of 𝐻 from step 4. into polynomials over matrices:
    T2(p(m) *Y IY) = sumi ( pi *A IA ) *Q XA ^i ) [see cpmidpmat 22144]
  7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 22138 to obtain the equation
    sumi ( U(G(i)) *Q XA ^i ) = sumi ( pi *A IA ) *Q XA ^i ):
    sumi ( U(G(i)) *Q XA ^i )
    = T2(cm(M) * adj(cm(M))) [see step 5.]
    = T2(p(m) *Y IY) [see cpmadurid 22138]
    = sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
    Note that this step is contained in the proof of chcoeffeq 22157, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
  8. Compare the sum representations of step 7. to obtain the equations U(G(i)) = pi *A IA , see chcoeffeqlem 22156. This corresponds to (6) in Wikipedia. Since the coefficients of the transformed representations and the original representations are identical, the equations of the coefficients are also valid for the original representations of steps 3. and 4.
  9. Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 22157:
    sumi ( M^i xA U(G(i)) ) = sumi ( M^i xA ( pi *A IA) )
    This corresponds to (7) in Wikipedia.
  10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 22158:
    sumi ( pi *A M^i )
    = sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 22155]
    = sumi ( M^i xA U(G(i)) ) [see chcoeffeq 22157]
  11. Apply the theorem for telescoping sums, see telgsumfz 19696, to the sum sumi ( T1(M)^i xY G(i) ), which results in an equation to 0:
    sumi ( T1(M)^i xY G(i) ) = 0Y, see cayhamlem1 22137:
    sumi ( T1(M)^i xY G(i) )
    = sumi=1 to s ( T1(M)^i xY T1(b(i-1)) - T1(M)^(i+1) xY T1(b(i)) )
    + ( T1(M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see chfacfpmmulgsum2 22136]
    = ( T1(M) xY T1(b(0)) - T1(M)^(s+1) xY T1(b(s)) ) + ( T1 M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see telgsumfz 19696]
    = 0Y [see grpnpncan0 18777] This step corresponds partially to (8) in Wikipedia.
  12. Since 𝑇 is a ring homomorphism (see mat2pmatrhm 22005), the left hand side of the equation in step 10. can be transformed into a representation appropriate to apply the result of step 11., see cayhamlem4 22159:
    sumi ( pi *A M^i )
    = sumi ( M^i xA U(G(i)) ) [see cayhamlem3 22158 (step 10.)]
    = U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 22024]
    = U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 19646]
    = U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 20082]
    = U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 18850]
    = U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 22026 ]
  13. Finally, combine the results of steps 11. and 12., and use the fact that 𝑇 (and therefore also its inverse π‘ˆ) is an injective ring homomorphism (see mat2pmatf1 22000 and mat2pmatrhm 22005) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 22161 resp. cayleyhamilton0 22160:
    sumi ( pi *A M^i )
    = U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 22159 (step 12.)]
    = U(0Y ) [see cayhamlem1 22137 (step 11.)]
    = 0A [see m2cpminv0 22032]
The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring 𝑅 and its representation as matrix over the polynomial ring over the ring 𝑅 in general!
 
Theoremcpmadurid 22138 The right-hand fundamental relation of the adjugate (see madurid 21915) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    βˆ’ = (-gβ€˜π‘Œ)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   πΌ = ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (𝐼 Γ— (π½β€˜πΌ)) = ((πΆβ€˜π‘€) Β· 1 ))
 
Theoremcpmidgsum 22139* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ˆ = (algScβ€˜π‘ƒ)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐻 = (π‘Œ Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛 ↑ 𝑋) Β· ((π‘ˆβ€˜((coe1β€˜πΎ)β€˜π‘›)) Β· 1 )))))
 
Theoremcpmidgsumm2pm 22140* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ˆ = (algScβ€˜π‘ƒ)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    &   π‘‚ = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐻 = (π‘Œ Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛 ↑ 𝑋) Β· (π‘‡β€˜(((coe1β€˜πΎ)β€˜π‘›) βˆ— 𝑂))))))
 
Theoremcpmidpmatlem1 22141* Lemma 1 for cpmidpmat 22144. (Contributed by AV, 13-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ˆ = (algScβ€˜π‘ƒ)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    &   π‘‚ = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (π‘˜ ∈ β„•0 ↦ (((coe1β€˜πΎ)β€˜π‘˜) βˆ— 𝑂))    β‡’   (𝐿 ∈ β„•0 β†’ (πΊβ€˜πΏ) = (((coe1β€˜πΎ)β€˜πΏ) βˆ— 𝑂))
 
Theoremcpmidpmatlem2 22142* Lemma 2 for cpmidpmat 22144. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ˆ = (algScβ€˜π‘ƒ)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    &   π‘‚ = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (π‘˜ ∈ β„•0 ↦ (((coe1β€˜πΎ)β€˜π‘˜) βˆ— 𝑂))    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐺 ∈ (𝐡 ↑m β„•0))
 
Theoremcpmidpmatlem3 22143* Lemma 3 for cpmidpmat 22144. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ˆ = (algScβ€˜π‘ƒ)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    &   π‘‚ = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (π‘˜ ∈ β„•0 ↦ (((coe1β€˜πΎ)β€˜π‘˜) βˆ— 𝑂))    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ 𝐺 finSupp (0gβ€˜π΄))
 
Theoremcpmidpmat 22144* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ˆ = (algScβ€˜π‘ƒ)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    &   π‘‚ = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘Š = (Baseβ€˜π‘Œ)    &   π‘„ = (Poly1β€˜π΄)    &   π‘ = (var1β€˜π΄)    &    βˆ™ = ( ·𝑠 β€˜π‘„)    &   πΈ = (.gβ€˜(mulGrpβ€˜π‘„))    &   πΌ = (𝑁 pMatToMatPoly 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (πΌβ€˜π») = (𝑄 Ξ£g (𝑛 ∈ β„•0 ↦ ((((coe1β€˜πΎ)β€˜π‘›) βˆ— 𝑂) βˆ™ (𝑛𝐸𝑍)))))
 
TheoremcpmadugsumlemB 22145* Lemma B for cpmadugsum 22149. (Contributed by AV, 2-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„•0 ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ ((𝑋 Β· 1 ) Γ— (π‘Œ Ξ£g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘–)))))) = (π‘Œ Ξ£g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘–))))))
 
TheoremcpmadugsumlemC 22146* Lemma C for cpmadugsum 22149. (Contributed by AV, 2-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„•0 ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ ((π‘‡β€˜π‘€) Γ— (π‘Œ Ξ£g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘–)))))) = (π‘Œ Ξ£g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘–)))))))
 
TheoremcpmadugsumlemF 22147* Lemma F for cpmadugsum 22149. (Contributed by AV, 7-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &    + = (+gβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (((𝑋 Β· 1 ) Γ— (π‘Œ Ξ£g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘–)))))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘Œ Ξ£g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘–))))))) = ((π‘Œ Ξ£g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· ((π‘‡β€˜(π‘β€˜(𝑖 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘–))))))) + ((((𝑠 + 1) ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘ ))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))))))
 
Theoremcpmadugsumfi 22148* The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &    + = (+gβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &   πΌ = ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))(𝐼 Γ— (π½β€˜πΌ)) = ((π‘Œ Ξ£g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) Β· ((π‘‡β€˜(π‘β€˜(𝑖 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘–))))))) + ((((𝑠 + 1) ↑ 𝑋) Β· (π‘‡β€˜(π‘β€˜π‘ ))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0))))))
 
Theoremcpmadugsum 22149* The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &    + = (+gβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &   πΌ = ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &    0 = (0gβ€˜π‘Œ)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))(𝐼 Γ— (π½β€˜πΌ)) = (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ 𝑋) Β· (πΊβ€˜π‘–)))))
 
Theoremcpmidgsum2 22150* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &    + = (+gβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &   πΌ = ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &    0 = (0gβ€˜π‘Œ)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π» = (𝐾 Β· 1 )    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))𝐻 = (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ 𝑋) Β· (πΊβ€˜π‘–)))))
 
Theoremcpmidg2sum 22151* Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   π‘‹ = (var1β€˜π‘…)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    Γ— = (.rβ€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &    + = (+gβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &   πΌ = ((𝑋 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &    0 = (0gβ€˜π‘Œ)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   π‘ˆ = (algScβ€˜π‘ƒ)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))(π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ 𝑋) Β· ((π‘ˆβ€˜((coe1β€˜πΎ)β€˜π‘–)) Β· 1 )))) = (π‘Œ Ξ£g (𝑖 ∈ β„•0 ↦ ((𝑖 ↑ 𝑋) Β· (πΊβ€˜π‘–)))))
 
Theoremcpmadumatpolylem1 22152* Lemma 1 for cpmadumatpoly 22154. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘† = (𝑁 ConstPolyMat 𝑅)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ = (var1β€˜π‘…)    &   π· = ((𝑍 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &   π‘Š = (Baseβ€˜π‘Œ)    &   π‘„ = (Poly1β€˜π΄)    &   π‘‹ = (var1β€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π‘„)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘„))    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    β‡’   ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•) ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) β†’ (π‘ˆ ∘ 𝐺) ∈ (𝐡 ↑m β„•0))
 
Theoremcpmadumatpolylem2 22153* Lemma 2 for cpmadumatpoly 22154. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘† = (𝑁 ConstPolyMat 𝑅)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ = (var1β€˜π‘…)    &   π· = ((𝑍 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &   π‘Š = (Baseβ€˜π‘Œ)    &   π‘„ = (Poly1β€˜π΄)    &   π‘‹ = (var1β€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π‘„)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘„))    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    β‡’   ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑠 ∈ β„•) ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠))) β†’ (π‘ˆ ∘ 𝐺) finSupp (0gβ€˜π΄))
 
Theoremcpmadumatpoly 22154* The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘† = (𝑁 ConstPolyMat 𝑅)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &    1 = (1rβ€˜π‘Œ)    &   π‘ = (var1β€˜π‘…)    &   π· = ((𝑍 Β· 1 ) βˆ’ (π‘‡β€˜π‘€))    &   π½ = (𝑁 maAdju 𝑃)    &   π‘Š = (Baseβ€˜π‘Œ)    &   π‘„ = (Poly1β€˜π΄)    &   π‘‹ = (var1β€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π‘„)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘„))    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    &   πΌ = (𝑁 pMatToMatPoly 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))(πΌβ€˜(𝐷 Γ— (π½β€˜π·))) = (𝑄 Ξ£g (𝑛 ∈ β„•0 ↦ ((π‘ˆβ€˜(πΊβ€˜π‘›)) βˆ— (𝑛 ↑ 𝑋)))))
 
Theoremcayhamlem2 22155 Lemma for cayhamlem3 22158. (Contributed by AV, 24-Nov-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    &    Β· = (.rβ€˜π΄)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝐻 ∈ (𝐾 ↑m β„•0) ∧ 𝐿 ∈ β„•0)) β†’ ((π»β€˜πΏ) βˆ— (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) Β· ((π»β€˜πΏ) βˆ— 1 )))
 
Theoremchcoeffeqlem 22156* Lemma for chcoeffeq 22157. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘Š = (Baseβ€˜π‘Œ)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝑠 ∈ β„• ∧ 𝑏 ∈ (𝐡 ↑m (0...𝑠)))) β†’ (((Poly1β€˜π΄) Ξ£g (𝑛 ∈ β„•0 ↦ ((π‘ˆβ€˜(πΊβ€˜π‘›))( ·𝑠 β€˜(Poly1β€˜π΄))(𝑛(.gβ€˜(mulGrpβ€˜(Poly1β€˜π΄)))(var1β€˜π΄))))) = ((Poly1β€˜π΄) Ξ£g (𝑛 ∈ β„•0 ↦ ((((coe1β€˜πΎ)β€˜π‘›) βˆ— 1 )( ·𝑠 β€˜(Poly1β€˜π΄))(𝑛(.gβ€˜(mulGrpβ€˜(Poly1β€˜π΄)))(var1β€˜π΄))))) β†’ βˆ€π‘› ∈ β„•0 (π‘ˆβ€˜(πΊβ€˜π‘›)) = (((coe1β€˜πΎ)β€˜π‘›) βˆ— 1 )))
 
Theoremchcoeffeq 22157* The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘Š = (Baseβ€˜π‘Œ)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))βˆ€π‘› ∈ β„•0 (π‘ˆβ€˜(πΊβ€˜π‘›)) = (((coe1β€˜πΎ)β€˜π‘›) βˆ— 1 ))
 
Theoremcayhamlem3 22158* Lemma for cayhamlem4 22159. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘Š = (Baseβ€˜π‘Œ)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    &    Β· = (.rβ€˜π΄)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))(𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ (((coe1β€˜πΎ)β€˜π‘›) βˆ— (𝑛 ↑ 𝑀)))) = (𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛 ↑ 𝑀) Β· (π‘ˆβ€˜(πΊβ€˜π‘›))))))
 
Theoremcayhamlem4 22159* Lemma for cayleyhamilton 22161. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (πΆβ€˜π‘€)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( 0 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 0 , ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘Š = (Baseβ€˜π‘Œ)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    &   πΈ = (.gβ€˜(mulGrpβ€˜π‘Œ))    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„• βˆƒπ‘ ∈ (𝐡 ↑m (0...𝑠))(𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ (((coe1β€˜πΎ)β€˜π‘›) βˆ— (𝑛 ↑ 𝑀)))) = (π‘ˆβ€˜(π‘Œ Ξ£g (𝑛 ∈ β„•0 ↦ ((𝑛𝐸(π‘‡β€˜π‘€)) Γ— (πΊβ€˜π‘›))))))
 
Theoremcayleyhamilton0 22160* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22161 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 22162)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π΄)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (coe1β€˜(πΆβ€˜π‘€))    &   π‘ƒ = (Poly1β€˜π‘…)    &   π‘Œ = (𝑁 Mat 𝑃)    &    Γ— = (.rβ€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘Œ)    &   π‘ = (0gβ€˜π‘Œ)    &   π‘Š = (Baseβ€˜π‘Œ)    &   πΈ = (.gβ€˜(mulGrpβ€˜π‘Œ))    &   π‘‡ = (𝑁 matToPolyMat 𝑅)    &   πΊ = (𝑛 ∈ β„•0 ↦ if(𝑛 = 0, (𝑍 βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜0)))), if(𝑛 = (𝑠 + 1), (π‘‡β€˜(π‘β€˜π‘ )), if((𝑠 + 1) < 𝑛, 𝑍, ((π‘‡β€˜(π‘β€˜(𝑛 βˆ’ 1))) βˆ’ ((π‘‡β€˜π‘€) Γ— (π‘‡β€˜(π‘β€˜π‘›))))))))    &   π‘ˆ = (𝑁 cPolyMatToMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ ((πΎβ€˜π‘›) βˆ— (𝑛 ↑ 𝑀)))) = 0 )
 
Theoremcayleyhamilton 22161* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π΄)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (coe1β€˜(πΆβ€˜π‘€))    &    βˆ— = ( ·𝑠 β€˜π΄)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ ((πΎβ€˜π‘›) βˆ— (𝑛 ↑ 𝑀)))) = 0 )
 
TheoremcayleyhamiltonALT 22162* Alternate proof of cayleyhamilton 22161, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22160 directly, but has the same structure as the proof of cayleyhamilton0 22160. In contrast to the proof of cayleyhamilton0 22160, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π΄)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (coe1β€˜(πΆβ€˜π‘€))    &    βˆ— = ( ·𝑠 β€˜π΄)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ ((πΎβ€˜π‘›) βˆ— (𝑛 ↑ 𝑀)))) = 0 )
 
Theoremcayleyhamilton1 22163* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 22161, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (πΉβ€˜π‘›), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π΄)    &   πΆ = (𝑁 CharPlyMat 𝑅)    &   πΎ = (coe1β€˜(πΆβ€˜π‘€))    &    βˆ— = ( ·𝑠 β€˜π΄)    &    ↑ = (.gβ€˜(mulGrpβ€˜π΄))    &   πΏ = (Baseβ€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   πΈ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &   π‘ = (0gβ€˜π‘…)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ (𝐹 ∈ (𝐿 ↑m β„•0) ∧ 𝐹 finSupp 𝑍)) β†’ ((πΆβ€˜π‘€) = (𝑃 Ξ£g (𝑛 ∈ β„•0 ↦ ((πΉβ€˜π‘›) Β· (𝑛𝐸𝑋)))) β†’ (𝐴 Ξ£g (𝑛 ∈ β„•0 ↦ ((πΉβ€˜π‘›) βˆ— (𝑛 ↑ 𝑀)))) = 0 ))
 
PART 12  BASIC TOPOLOGY
 
12.1  Topology
 
12.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union, see toponuni 22185), and it may sometimes be more convenient to consider topologies without reference to the underlying set. This is why we define successively the class of topologies (df-top 22165), then the function which associates with a set the set of topologies on it (df-topon 22182), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 22204). Of course, a topology is the same thing as a topology on a set (see toprntopon 22196).

 
12.1.1.1  Topologies
 
Syntaxctop 22164 Syntax for the class of topologies.
class Top
 
Definitiondf-top 22165* Define the class of topologies. It is a proper class (see topnex 22268). See istopg 22166 and istop2g 22167 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Top = {π‘₯ ∣ (βˆ€π‘¦ ∈ 𝒫 π‘₯βˆͺ 𝑦 ∈ π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ βˆ€π‘§ ∈ π‘₯ (𝑦 ∩ 𝑧) ∈ π‘₯)}
 
Theoremistopg 22166* Express the predicate "𝐽 is a topology". See istop2g 22167 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

(𝐽 ∈ 𝐴 β†’ (𝐽 ∈ Top ↔ (βˆ€π‘₯(π‘₯ βŠ† 𝐽 β†’ βˆͺ π‘₯ ∈ 𝐽) ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ 𝐽 (π‘₯ ∩ 𝑦) ∈ 𝐽)))
 
Theoremistop2g 22167* Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 22166. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ 𝐴 β†’ (𝐽 ∈ Top ↔ (βˆ€π‘₯(π‘₯ βŠ† 𝐽 β†’ βˆͺ π‘₯ ∈ 𝐽) ∧ βˆ€π‘₯((π‘₯ βŠ† 𝐽 ∧ π‘₯ β‰  βˆ… ∧ π‘₯ ∈ Fin) β†’ ∩ π‘₯ ∈ 𝐽))))
 
Theoremuniopn 22168 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝐽) β†’ βˆͺ 𝐴 ∈ 𝐽)
 
Theoremiunopn 22169* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽) β†’ βˆͺ π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽)
 
Theoreminopn 22170 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽) β†’ (𝐴 ∩ 𝐡) ∈ 𝐽)
 
Theoremfitop 22171 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
(𝐽 ∈ Top β†’ (fiβ€˜π½) = 𝐽)
 
Theoremfiinopn 22172 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top β†’ ((𝐴 βŠ† 𝐽 ∧ 𝐴 β‰  βˆ… ∧ 𝐴 ∈ Fin) β†’ ∩ 𝐴 ∈ 𝐽))
 
Theoremiinopn 22173* The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 β‰  βˆ… ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽)) β†’ ∩ π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽)
 
Theoremunopn 22174 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐡 ∈ 𝐽) β†’ (𝐴 βˆͺ 𝐡) ∈ 𝐽)
 
Theorem0opn 22175 The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
 
Theorem0ntop 22176 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
Β¬ βˆ… ∈ Top
 
Theoremtopopn 22177 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = βˆͺ 𝐽    β‡’   (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
 
Theoremeltopss 22178 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) β†’ 𝐴 βŠ† 𝑋)
 
Theoremriinopn 22179* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ βˆ€π‘₯ ∈ 𝐴 𝐡 ∈ 𝐽) β†’ (𝑋 ∩ ∩ π‘₯ ∈ 𝐴 𝐡) ∈ 𝐽)
 
Theoremrintopn 22180 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝐽 ∧ 𝐴 ∈ Fin) β†’ (𝑋 ∩ ∩ 𝐴) ∈ 𝐽)
 
12.1.1.2  Topologies on sets
 
Syntaxctopon 22181 Syntax for the function of topologies on sets.
class TopOn
 
Definitiondf-topon 22182* Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = βˆͺ 𝑗})
 
Theoremistopon 22183 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π΅) ↔ (𝐽 ∈ Top ∧ 𝐡 = βˆͺ 𝐽))
 
Theoremtopontop 22184 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐽 ∈ Top)
 
Theoremtoponuni 22185 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 = βˆͺ 𝐽)
 
Theoremtopontopi 22186 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOnβ€˜π΅)    β‡’   π½ ∈ Top
 
Theoremtoponunii 22187 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOnβ€˜π΅)    β‡’   π΅ = βˆͺ 𝐽
 
Theoremtoptopon 22188 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = βˆͺ 𝐽    β‡’   (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
 
Theoremtoptopon2 22189 A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
 
Theoremtopontopon 22190 A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
(𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
 
Theoremfuntopon 22191 The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
Fun TopOn
 
Theoremtoponrestid 22192 Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
𝐴 ∈ (TopOnβ€˜π΅)    β‡’   π΄ = (𝐴 β†Ύt 𝐡)
 
Theoremtoponsspwpw 22193 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.)
(TopOnβ€˜π΄) βŠ† 𝒫 𝒫 𝐴
 
Theoremdmtopon 22194 The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)
dom TopOn = V
 
Theoremfntopon 22195 The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17406 for Moore collections. (Contributed by BJ, 29-Apr-2021.)
TopOn Fn V
 
Theoremtoprntopon 22196 A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)
Top = βˆͺ ran TopOn
 
Theoremtoponmax 22197 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOnβ€˜π΅) β†’ 𝐡 ∈ 𝐽)
 
Theoremtoponss 22198 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝐽) β†’ 𝐴 βŠ† 𝑋)
 
Theoremtoponcom 22199 If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))
 
Theoremtoponcomb 22200 Biconditional form of toponcom 22199. (Contributed by BJ, 5-Dec-2021.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) β†’ (𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾) ↔ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)))
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