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Type | Label | Description |
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Statement | ||
Theorem | cpmidg2sum 22901* | 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ × = (.r‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ + = (+g‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & ⊢ 𝐽 = (𝑁 maAdju 𝑃) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (𝐶‘𝑀) & ⊢ 𝑈 = (algSc‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑖)) · 1 )))) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))))) | ||
Theorem | cpmadumatpolylem1 22902* | Lemma 1 for cpmadumatpoly 22904. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝑍 = (var1‘𝑅) & ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & ⊢ 𝐽 = (𝑁 maAdju 𝑃) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) | ||
Theorem | cpmadumatpolylem2 22903* | Lemma 2 for cpmadumatpoly 22904. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝑍 = (var1‘𝑅) & ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & ⊢ 𝐽 = (𝑁 maAdju 𝑃) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0g‘𝐴)) | ||
Theorem | cpmadumatpoly 22904* | 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝑍 = (var1‘𝑅) & ⊢ 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & ⊢ 𝐽 = (𝑁 maAdju 𝑃) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋))))) | ||
Theorem | cayhamlem2 22905 | Lemma for cayhamlem3 22908. (Contributed by AV, 24-Nov-2019.) |
⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) & ⊢ · = (.r‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑m ℕ0) ∧ 𝐿 ∈ ℕ0)) → ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 ))) | ||
Theorem | chcoeffeqlem 22906* | Lemma for chcoeffeq 22907. (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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (𝐶‘𝑀) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))) | ||
Theorem | chcoeffeq 22907* | 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (𝐶‘𝑀) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )) | ||
Theorem | cayhamlem3 22908* | Lemma for cayhamlem4 22909. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (𝐶‘𝑀) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) & ⊢ · = (.r‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑀) · (𝑈‘(𝐺‘𝑛)))))) | ||
Theorem | cayhamlem4 22909* | Lemma for cayleyhamilton 22911. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (𝐶‘𝑀) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑌)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) | ||
Theorem | cayleyhamilton0 22910* | The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22911 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 22912)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 0 = (0g‘𝐴) & ⊢ 1 = (1r‘𝐴) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (coe1‘(𝐶‘𝑀)) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 𝑍 = (0g‘𝑌) & ⊢ 𝑊 = (Base‘𝑌) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑌)) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ) | ||
Theorem | cayleyhamilton 22911* | 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 = (0g‘𝐴) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (coe1‘(𝐶‘𝑀)) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ) | ||
Theorem | cayleyhamiltonALT 22912* | Alternate proof of cayleyhamilton 22911, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22910 directly, but has the same structure as the proof of cayleyhamilton0 22910. In contrast to the proof of cayleyhamilton0 22910, 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 = (0g‘𝐴) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (coe1‘(𝐶‘𝑀)) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ) | ||
Theorem | cayleyhamilton1 22913* | 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 22911, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹‘𝑛), 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 = (0g‘𝐴) & ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝐾 = (coe1‘(𝐶‘𝑀)) & ⊢ ∗ = ( ·𝑠 ‘𝐴) & ⊢ ↑ = (.g‘(mulGrp‘𝐴)) & ⊢ 𝐿 = (Base‘𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑍 = (0g‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑m ℕ0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )) | ||
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 22935), 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 22915), then the function which associates with a set the set of topologies on it (df-topon 22932), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 22954). Of course, a topology is the same thing as a topology on a set (see toprntopon 22946). | ||
Syntax | ctop 22914 | Syntax for the class of topologies. |
class Top | ||
Definition | df-top 22915* |
Define the class of topologies. It is a proper class (see topnex 23018).
See istopg 22916 and istop2g 22917 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 = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} | ||
Theorem | istopg 22916* |
Express the predicate "𝐽 is a topology". See istop2g 22917 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 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽))) | ||
Theorem | istop2g 22917* | Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 22916. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽)))) | ||
Theorem | uniopn 22918 | 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 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽) | ||
Theorem | iunopn 22919* | The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.) |
⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | ||
Theorem | inopn 22920 | The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽) | ||
Theorem | fitop 22921 | A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.) |
⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽) | ||
Theorem | fiinopn 22922 | The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
⊢ (𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽)) | ||
Theorem | iinopn 22923* | The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.) |
⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) | ||
Theorem | unopn 22924 | The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽) | ||
Theorem | 0opn 22925 | The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.) |
⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | ||
Theorem | 0ntop 22926 | The empty set is not a topology. (Contributed by FL, 1-Jun-2008.) |
⊢ ¬ ∅ ∈ Top | ||
Theorem | topopn 22927 | The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) | ||
Theorem | eltopss 22928 | A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | riinopn 22929* | A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → (𝑋 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) ∈ 𝐽) | ||
Theorem | rintopn 22930 | A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽 ∧ 𝐴 ∈ Fin) → (𝑋 ∩ ∩ 𝐴) ∈ 𝐽) | ||
Syntax | ctopon 22931 | Syntax for the function of topologies on sets. |
class TopOn | ||
Definition | df-topon 22932* | 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 = ∪ 𝑗}) | ||
Theorem | istopon 22933 | 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)) | ||
Theorem | topontop 22934 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | ||
Theorem | toponuni 22935 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | ||
Theorem | topontopi 22936 | A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐽 ∈ Top | ||
Theorem | toponunii 22937 | The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝐽 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐵 = ∪ 𝐽 | ||
Theorem | toptopon 22938 | Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) | ||
Theorem | toptopon2 22939 | 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 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | topontopon 22940 | A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽)) | ||
Theorem | funtopon 22941 | The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.) |
⊢ Fun TopOn | ||
Theorem | toponrestid 22942 | Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.) |
⊢ 𝐴 ∈ (TopOn‘𝐵) ⇒ ⊢ 𝐴 = (𝐴 ↾t 𝐵) | ||
Theorem | toponsspwpw 22943 | The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) |
⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 | ||
Theorem | dmtopon 22944 | The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.) |
⊢ dom TopOn = V | ||
Theorem | fntopon 22945 | The class TopOn is a function with domain the universal class V. Analogue for topologies of fnmre 17635 for Moore collections. (Contributed by BJ, 29-Apr-2021.) |
⊢ TopOn Fn V | ||
Theorem | toprntopon 22946 | A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.) |
⊢ Top = ∪ ran TopOn | ||
Theorem | toponmax 22947 | The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ 𝐽) | ||
Theorem | toponss 22948 | A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | ||
Theorem | toponcom 22949 | 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)) | ||
Theorem | toponcomb 22950 | Biconditional form of toponcom 22949. (Contributed by BJ, 5-Dec-2021.) |
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽))) | ||
Theorem | topgele 22951 | 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} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) | ||
Theorem | topsn 22952 | The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4904). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴}) | ||
Syntax | ctps 22953 | Syntax for the class of topological spaces. |
class TopSp | ||
Definition | df-topsp 22954 | Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.) |
⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} | ||
Theorem | istps 22955 | 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)) | ||
Theorem | istps2 22956 | 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)) | ||
Theorem | tpsuni 22957 | The base set of a topological space. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp → 𝐴 = ∪ 𝐽) | ||
Theorem | tpstop 22958 | The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.) |
⊢ 𝐽 = (TopOpen‘𝐾) ⇒ ⊢ (𝐾 ∈ TopSp → 𝐽 ∈ Top) | ||
Theorem | tpspropd 22959 | 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)) | ||
Theorem | tpsprop2d 22960 | 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)) | ||
Theorem | topontopn 22961 | 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)) | ||
Theorem | tsettps 22962 | 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) | ||
Theorem | istpsi 22963 | Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.) |
⊢ (Base‘𝐾) = 𝐴 & ⊢ (TopOpen‘𝐾) = 𝐽 & ⊢ 𝐴 = ∪ 𝐽 & ⊢ 𝐽 ∈ Top ⇒ ⊢ 𝐾 ∈ TopSp | ||
Theorem | eltpsg 22964 | Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by AV, 31-Oct-2024.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) | ||
Theorem | eltpsgOLD 22965 | Obsolete version of eltpsg 22964 as of 31-Oct-2024. Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝐽〉} ⇒ ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp) | ||
Theorem | eltpsi 22966 | 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 | ||
Syntax | ctb 22967 | Syntax for the class of topological bases. |
class TopBases | ||
Definition | df-bases 22968* | Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 22970). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.) |
⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))} | ||
Theorem | isbasisg 22969* | Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))) | ||
Theorem | isbasis2g 22970* | Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))) | ||
Theorem | isbasis3g 22971* | Express the predicate "the set 𝐵 is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ (∀𝑥 ∈ 𝐵 𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝐵∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))) | ||
Theorem | basis1 22972 | Property of a basis. (Contributed by NM, 16-Jul-2006.) |
⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))) | ||
Theorem | basis2 22973* | Property of a basis. (Contributed by NM, 17-Jul-2006.) |
⊢ (((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵) ∧ (𝐷 ∈ 𝐵 ∧ 𝐴 ∈ (𝐶 ∩ 𝐷))) → ∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (𝐶 ∩ 𝐷))) | ||
Theorem | fiinbas 22974* | If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases) | ||
Theorem | basdif0 22975 | 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) | ||
Theorem | baspartn 22976* | A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑃 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → 𝑃 ∈ TopBases) | ||
Theorem | tgval 22977* | The topology generated by a basis. See also tgval2 22978 and tgval3 22985. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) | ||
Theorem | tgval2 22978* | Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 22991) that (topGen‘𝐵) is indeed a topology (on ∪ 𝐵, see unitg 22989). See also tgval 22977 and tgval3 22985. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ (𝑥 ⊆ ∪ 𝐵 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))}) | ||
Theorem | eltg 22979 | Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))) | ||
Theorem | eltg2 22980* | Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴)))) | ||
Theorem | eltg2b 22981* | Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | ||
Theorem | eltg4i 22982 | 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)) | ||
Theorem | eltg3i 22983 | The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → ∪ 𝐴 ∈ (topGen‘𝐵)) | ||
Theorem | eltg3 22984* | Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥 ⊆ 𝐵 ∧ 𝐴 = ∪ 𝑥))) | ||
Theorem | tgval3 22985* | Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 22977 and tgval2 22978. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑦)}) | ||
Theorem | tg1 22986 | Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ 𝐵) | ||
Theorem | tg2 22987* | Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.) |
⊢ ((𝐴 ∈ (topGen‘𝐵) ∧ 𝐶 ∈ 𝐴) → ∃𝑥 ∈ 𝐵 (𝐶 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴)) | ||
Theorem | bastg 22988 | 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.) |
⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) | ||
Theorem | unitg 22989 | 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.) |
⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵) | ||
Theorem | tgss 22990 | Subset relation for generated topologies. (Contributed by NM, 7-May-2007.) |
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶)) | ||
Theorem | tgcl 22991 | Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top) | ||
Theorem | tgclb 22992 | The property tgcl 22991 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) | ||
Theorem | tgtopon 22993 | A basis generates a topology on ∪ 𝐵. (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ (TopOn‘∪ 𝐵)) | ||
Theorem | topbas 22994 | A topology is its own basis. (Contributed by NM, 17-Jul-2006.) |
⊢ (𝐽 ∈ Top → 𝐽 ∈ TopBases) | ||
Theorem | tgtop 22995 | A topology is its own basis. (Contributed by NM, 18-Jul-2006.) |
⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | ||
Theorem | eltop 22996 | Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ 𝐴 ⊆ ∪ (𝐽 ∩ 𝒫 𝐴))) | ||
Theorem | eltop2 22997* | Membership in a topology. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | ||
Theorem | eltop3 22998* | Membership in a topology. (Contributed by NM, 19-Jul-2006.) |
⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥))) | ||
Theorem | fibas 22999 | 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 | ||
Theorem | tgdom 23000 | 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.) |
⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵) |
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