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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcpmadugsumfi 20501* 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)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐼 × (𝐽𝐼)) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))

Theoremcpmadugsum 20502* 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)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐼 × (𝐽𝐼)) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))))

Theoremcpmidgsum2 20503* 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)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))𝐻 = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))))

Theoremcpmidg2sum 20504* 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)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝑈 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · ((𝑈‘((coe1𝐾)‘𝑖)) · 1 )))) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))))

Theoremcpmadumatpolylem1 20505* Lemma 1 for cpmadumatpoly 20507. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑍 = (var1𝑅)    &   𝐷 = ((𝑍 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑋 = (var1𝐴)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) ∈ (𝐵𝑚0))

Theoremcpmadumatpolylem2 20506* Lemma 2 for cpmadumatpoly 20507. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑍 = (var1𝑅)    &   𝐷 = ((𝑍 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑋 = (var1𝐴)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) finSupp (0g𝐴))

Theoremcpmadumatpoly 20507* 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)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑍 = (var1𝑅)    &   𝐷 = ((𝑍 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑋 = (var1𝐴)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐼‘(𝐷 × (𝐽𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛)) (𝑛 𝑋)))))

Theoremcayhamlem2 20508 Lemma for cayhamlem3 20511. (Contributed by AV, 24-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))    &    · = (.r𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐻 ∈ (𝐾𝑚0) ∧ 𝐿 ∈ ℕ0)) → ((𝐻𝐿) (𝐿 𝑀)) = ((𝐿 𝑀) · ((𝐻𝐿) 1 )))

Theoremchcoeffeqlem 20509* Lemma for chcoeffeq 20510. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))

Theoremchcoeffeq 20510* 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)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))

Theoremcayhamlem3 20511* Lemma for cayhamlem4 20512. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)    &    = (.g‘(mulGrp‘𝐴))    &    · = (.r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀) · (𝑈‘(𝐺𝑛))))))

Theoremcayhamlem4 20512* Lemma for cayleyhamilton 20514. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)    &    = (.g‘(mulGrp‘𝐴))    &   𝐸 = (.g‘(mulGrp‘𝑌))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))

Theoremcayleyhamilton0 20513* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 20514 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 20515)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &   𝑍 = (0g𝑌)    &   𝑊 = (Base‘𝑌)    &   𝐸 = (.g‘(mulGrp‘𝑌))    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )

Theoremcayleyhamilton 20514* 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)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )

TheoremcayleyhamiltonALT 20515* Alternate proof of cayleyhamilton 20514, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 20513 directly, but has the same structure as the proof of cayleyhamilton0 20513. In contrast to the proof of cayleyhamilton0 20513, 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)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )

Theoremcayleyhamilton1 20516* 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 20514, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹𝑛), 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)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))    &   𝐿 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑍 = (0g𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 ))

PART 12  BASIC TOPOLOGY

12.1  Topology

12.1.1  Topological spaces

Syntaxctop 20517 Extend class notation with the class of all topologies.
class Top

Syntaxctopon 20518 The class function of all topologies over a base set.
class TopOn

Syntaxctps 20519 Extend class notation with the class of all topological spaces.
class TopSp

Syntaxctb 20520 Extend class notation with the class of all topological bases.
class TopBases

Definitiondf-top 20521* Define the (proper) class of all topologies. See istop2g 20526 for an alternate way to express finite intersection.

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 = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}

Definitiondf-bases 20522* Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 20563). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)
TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}

Definitiondf-topon 20523* Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = 𝑗})

Definitiondf-topsp 20524 Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}

Theoremistopg 20525* Express the predicate "𝐽 is a topology." Note: In the literature, a topology is often represented by a script 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.)
(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))

Theoremistop2g 20526* Express the predicate "𝐽 is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)
(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽))))

Theoremuniopn 20527 The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)

Theoremiunopn 20528* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)

Theoreminopn 20529 The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theoremfitop 20530 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
(𝐽 ∈ Top → (fi‘𝐽) = 𝐽)

Theoremfiinopn 20531 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))

Theoremiinopn 20532* The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐽)) → 𝑥𝐴 𝐵𝐽)

Theoremunopn 20533 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)

Theorem0opn 20534 The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)
(𝐽 ∈ Top → ∅ ∈ 𝐽)

Theorem0ntop 20535 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
¬ ∅ ∈ Top

Theoremtopopn 20536 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
𝑋 = 𝐽       (𝐽 ∈ Top → 𝑋𝐽)

Theoremeltopss 20537 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝑋)

Theoremriinopn 20538* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵𝐽) → (𝑋 𝑥𝐴 𝐵) ∈ 𝐽)

Theoremrintopn 20539 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝐽𝐴 ∈ Fin) → (𝑋 𝐴) ∈ 𝐽)

Theoremistopon 20540 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 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))

Theoremtopontop 20541 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)

Theoremtoponuni 20542 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵 = 𝐽)

Theoremtoponmax 20543 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝐽 ∈ (TopOn‘𝐵) → 𝐵𝐽)

Theoremtoponss 20544 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝐽) → 𝐴𝑋)

Theoremtoponcom 20545 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 ∈ (TopOn‘ 𝐽)) → 𝐽 ∈ (TopOn‘ 𝐾))

Theoremtopontopi 20546 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐽 ∈ Top

Theoremtoponunii 20547 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐽 ∈ (TopOn‘𝐵)       𝐵 = 𝐽

Theoremtoptopon 20548 Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))

Theoremtopgele 20549 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
(𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))

Theoremtopsn 20550 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4366). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
(𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})

Theoremistps 20551 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Theoremistps2 20552 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))

Theoremtpsuni 20553 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐴 = 𝐽)

Theoremtpstop 20554 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
𝐽 = (TopOpen‘𝐾)       (𝐾 ∈ TopSp → 𝐽 ∈ Top)

Theoremtpspropd 20555 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Theoremtpsprop2d 20556 A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))       (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Theoremtopontopn 20557 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopSet‘𝐾)       (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾))

Theoremtsettps 20558 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐴 = (Base‘𝐾)    &   𝐽 = (TopSet‘𝐾)       (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Theoremistpsi 20559 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
(Base‘𝐾) = 𝐴    &   (TopOpen‘𝐾) = 𝐽    &   𝐴 = 𝐽    &   𝐽 ∈ Top       𝐾 ∈ TopSp

Theoremeltpsg 20560 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}       (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)

Theoremeltpsi 20561 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    &   𝐴 = 𝐽    &   𝐽 ∈ Top       𝐾 ∈ TopSp

12.1.2  TopBases for topologies

Theoremisbasisg 20562* Express the predicate "𝐵 is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))

Theoremisbasis2g 20563* Express the predicate "𝐵 is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))

Theoremisbasis3g 20564* Express the predicate "𝐵 is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
(𝐵𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥𝐵 𝑥 𝐵 ∧ ∀𝑥 𝐵𝑦𝐵 𝑥𝑦 ∧ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))))

Theorembasis1 20565 Property of a basis. (Contributed by NM, 16-Jul-2006.)
((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))

Theorembasis2 20566* Property of a basis. (Contributed by NM, 17-Jul-2006.)
(((𝐵 ∈ TopBases ∧ 𝐶𝐵) ∧ (𝐷𝐵𝐴 ∈ (𝐶𝐷))) → ∃𝑥𝐵 (𝐴𝑥𝑥 ⊆ (𝐶𝐷)))

Theoremfiinbas 20567* If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

Theorembasdif0 20568 A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)
((𝐵 ∖ {∅}) ∈ TopBases ↔ 𝐵 ∈ TopBases)

Theorembaspartn 20569* A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝑃𝑉 ∧ ∀𝑥𝑃𝑦𝑃 (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → 𝑃 ∈ TopBases)

Theoremtgval 20570* The topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥𝑥 (𝐵 ∩ 𝒫 𝑥)})

Theoremtgval2 20571* Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 20584) that (topGen‘𝐵) is indeed a topology (on 𝐵; see unitg 20582). (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 𝐵 ∧ ∀𝑦𝑥𝑧𝐵 (𝑦𝑧𝑧𝑥))})

Theoremeltg 20572 Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 (𝐵 ∩ 𝒫 𝐴)))

Theoremeltg2 20573* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴))))

Theoremeltg2b 20574* Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝑦𝐴)))

Theoremeltg4i 20575 An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))

Theoremeltg3i 20576 The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
((𝐵𝑉𝐴𝐵) → 𝐴 ∈ (topGen‘𝐵))

Theoremeltg3 20577* Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))

Theoremtgval3 20578* Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
(𝐵𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦𝐵𝑥 = 𝑦)})

Theoremtg1 20579 Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
(𝐴 ∈ (topGen‘𝐵) → 𝐴 𝐵)

Theoremtg2 20580* Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶𝐴) → ∃𝑥𝐵 (𝐶𝑥𝑥𝐴))

Theorembastg 20581 A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
(𝐵𝑉𝐵 ⊆ (topGen‘𝐵))

Theoremunitg 20582 The topology generated by a basis 𝐵 is a topology on 𝐵. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
(𝐵𝑉 (topGen‘𝐵) = 𝐵)

Theoremtgss 20583 Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
((𝐶𝑉𝐵𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))

Theoremtgcl 20584 Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Theoremtgclb 20585 The property tgcl 20584 can be reversed: if the topology generated by 𝐵 is actually a topology, then 𝐵 must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
(𝐵 ∈ TopBases ↔ (topGen‘𝐵) ∈ Top)

Theoremtgtopon 20586 A basis generates a topology on 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘ 𝐵))

Theoremtopbas 20587 A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
(𝐽 ∈ Top → 𝐽 ∈ TopBases)

Theoremtgtop 20588 A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
(𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)

Theoremeltop 20589 Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽𝐴 (𝐽 ∩ 𝒫 𝐴)))

Theoremeltop2 20590* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∀𝑥𝐴𝑦𝐽 (𝑥𝑦𝑦𝐴)))

Theoremeltop3 20591* Membership in a topology. (Contributed by NM, 19-Jul-2006.)
(𝐽 ∈ Top → (𝐴𝐽 ↔ ∃𝑥(𝑥𝐽𝐴 = 𝑥)))

Theoremfibas 20592 A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(fi‘𝐴) ∈ TopBases

Theoremtgdom 20593 A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
(𝐵𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)

Theoremtgiun 20594* The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐵𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ∈ (topGen‘𝐵))

Theoremtgidm 20595 The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
(𝐵𝑉 → (topGen‘(topGen‘𝐵)) = (topGen‘𝐵))

Theorembastop 20596 Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
(𝐵 ∈ TopBases → (𝐵 ∈ Top ↔ (topGen‘𝐵) = 𝐵))

Theoremtgtop11 20597 The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (topGen‘𝐽) = (topGen‘𝐾)) → 𝐽 = 𝐾)

Theorem0top 20598 The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
(𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))

Theoremen1top 20599 {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
(𝐽 ∈ Top → (𝐽 ≈ 1𝑜𝐽 = {∅}))

Theoremen2top 20600 If a topology has two elements, it is the indiscrete topology. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
(𝐽 ∈ (TopOn‘𝑋) → (𝐽 ≈ 2𝑜 ↔ (𝐽 = {∅, 𝑋} ∧ 𝑋 ≠ ∅)))

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