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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | smatcl 33801 | Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) & ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ∈ 𝐶) | ||
| Theorem | matmpo 33802* | Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀𝑗))) | ||
| Theorem | 1smat1 33803 | The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 22589. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
| ⊢ 1 = (1r‘((1...𝑁) Mat 𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅))) | ||
| Theorem | submat1n 33804 | One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁)) | ||
| Theorem | submatres 33805 | Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))) | ||
| Theorem | submateqlem1 33806 | Lemma for submateq 33808. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) ⇒ ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾}))) | ||
| Theorem | submateqlem2 33807 | Lemma for submateq 33808. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑀 < 𝐾) ⇒ ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) | ||
| Theorem | submateq 33808* | Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽)) | ||
| Theorem | submatminr1 33809 | If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽)) | ||
| Syntax | clmat 33810 | Extend class notation with the literal matrix conversion function. |
| class litMat | ||
| Definition | df-lmat 33811* | Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)))) | ||
| Theorem | lmatval 33812* | Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))) | ||
| Theorem | lmatfval 33813* | Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ 𝑀 = (litMat‘𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) & ⊢ (𝜑 → (♯‘𝑊) = 𝑁) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) ⇒ ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) | ||
| Theorem | lmatfvlem 33814* | Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) & ⊢ (𝜑 → (♯‘𝑊) = 𝑁) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) & ⊢ 𝐾 ∈ ℕ0 & ⊢ 𝐿 ∈ ℕ0 & ⊢ 𝐼 ≤ 𝑁 & ⊢ 𝐽 ≤ 𝑁 & ⊢ (𝐾 + 1) = 𝐼 & ⊢ (𝐿 + 1) = 𝐽 & ⊢ (𝑊‘𝐾) = 𝑋 & ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) ⇒ ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) | ||
| Theorem | lmatcl 33815* | Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) & ⊢ (𝜑 → (♯‘𝑊) = 𝑁) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝑂 = ((1...𝑁) Mat 𝑅) & ⊢ 𝑃 = (Base‘𝑂) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑃) | ||
| Theorem | lmat22lem 33816* | Lemma for lmat22e11 33817 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉‘𝑖)) = 2) | ||
| Theorem | lmat22e11 33817 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (1𝑀1) = 𝐴) | ||
| Theorem | lmat22e12 33818 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (1𝑀2) = 𝐵) | ||
| Theorem | lmat22e21 33819 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (2𝑀1) = 𝐶) | ||
| Theorem | lmat22e22 33820 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (2𝑀2) = 𝐷) | ||
| Theorem | lmat22det 33821 | The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘〈“〈“𝐴𝐵”〉〈“𝐶𝐷”〉”〉) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐽 = ((1...2) maDet 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) | ||
| Theorem | mdetpmtr1 33822* | The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐺 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗)) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸))) | ||
| Theorem | mdetpmtr2 33823* | The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐺 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸))) | ||
| Theorem | mdetpmtr12 33824* | The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐺 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀(𝑄‘𝑗))) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ∈ 𝐺) & ⊢ (𝜑 → 𝑄 ∈ 𝐺) ⇒ ⊢ (𝜑 → (𝐷‘𝑀) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝐸))) | ||
| Theorem | mdetlap1 33825* | A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐾 = (𝑁 maAdju 𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁) → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾‘𝑀)𝐼))))) | ||
| Theorem | madjusmdetlem1 33826* | Lemma for madjusmdet 33830. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁))) & ⊢ 𝑆 = (pmSgn‘(1...𝑁)) & ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) & ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) & ⊢ (𝜑 → 𝑃 ∈ 𝐺) & ⊢ (𝜑 → 𝑄 ∈ 𝐺) & ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼) & ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽) & ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁)) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
| Theorem | madjusmdetlem2 33827* | Lemma for madjusmdet 33830. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◡𝑆)‘𝑋)) | ||
| Theorem | madjusmdetlem3 33828* | Lemma for madjusmdet 33830. (Contributed by Thierry Arnoux, 27-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) & ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) & ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) & ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◡𝑆)‘𝑖)𝑈((𝑄 ∘ ◡𝑇)‘𝑗))) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁)) | ||
| Theorem | madjusmdetlem4 33829* | Lemma for madjusmdet 33830. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) & ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) & ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
| Theorem | madjusmdet 33830 | Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
| Theorem | mdetlap 33831* | Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗))))))) | ||
| Theorem | ist0cld 33832* | The predicate "is a T0 space", using closed sets. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
| ⊢ (𝜑 → 𝐵 = ∪ 𝐽) & ⊢ (𝜑 → 𝐷 = (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∀𝑑 ∈ 𝐷 (𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦)))) | ||
| Theorem | txomap 33833* | Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:𝑋⟶𝑍) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑇) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾)) & ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑦)〉) ⇒ ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀)) | ||
| Theorem | qtopt1 33834* | If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Fre) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre) | ||
| Theorem | qtophaus 33835* | If an open map's graph in the product space (𝐽 ×t 𝐽) is closed, then its quotient topology is Hausdorff. (Contributed by Thierry Arnoux, 4-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ ∼ = (◡𝐹 ∘ 𝐹) & ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ 〈(𝐹‘𝑥), (𝐹‘𝑦)〉) & ⊢ (𝜑 → 𝐽 ∈ Haus) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ (𝐽 qTop 𝐹)) & ⊢ (𝜑 → ∼ ∈ (Clsd‘(𝐽 ×t 𝐽))) ⇒ ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Haus) | ||
| Theorem | circtopn 33836* | The topology of the unit circle is generated by open intervals of the polar coordinate. (Contributed by Thierry Arnoux, 4-Jan-2020.) |
| ⊢ 𝐼 = (0[,](2 · π)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) & ⊢ 𝐶 = (◡abs “ {1}) ⇒ ⊢ (𝐽 qTop 𝐹) = (TopOpen‘(𝐹 “s ℝfld)) | ||
| Theorem | circcn 33837* | The function gluing the real line into the unit circle is continuous. (Contributed by Thierry Arnoux, 5-Jan-2020.) |
| ⊢ 𝐼 = (0[,](2 · π)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ (exp‘(i · 𝑥))) & ⊢ 𝐶 = (◡abs “ {1}) ⇒ ⊢ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)) | ||
| Theorem | reff 33838* | For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 10528. (Contributed by Thierry Arnoux, 12-Jan-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))) | ||
| Theorem | locfinreflem 33839* | A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑉 ⊆ 𝐽) & ⊢ (𝜑 → 𝑉Ref𝑈) & ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽)) ⇒ ⊢ (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓 ⊆ 𝑈 ∧ ran 𝑓 ⊆ 𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))) | ||
| Theorem | locfinref 33840* | A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑉 ⊆ 𝐽) & ⊢ (𝜑 → 𝑉Ref𝑈) & ⊢ (𝜑 → 𝑉 ∈ (LocFin‘𝐽)) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) | ||
| Syntax | ccref 33841 | The "every open cover has an 𝐴 refinement" predicate. |
| class CovHasRef𝐴 | ||
| Definition | df-cref 33842* | Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} | ||
| Theorem | iscref 33843* | The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))) | ||
| Theorem | crefeq 33844 | Equality theorem for the "every open cover has an A refinement" predicate. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐴 = 𝐵 → CovHasRef𝐴 = CovHasRef𝐵) | ||
| Theorem | creftop 33845 | A space where every open cover has an 𝐴 refinement is a topological space. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐽 ∈ CovHasRef𝐴 → 𝐽 ∈ Top) | ||
| Theorem | crefi 33846* | The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶) | ||
| Theorem | crefdf 33847* | A formulation of crefi 33846 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐵 = CovHasRef𝐴 & ⊢ (𝑧 ∈ 𝐴 → 𝜑) ⇒ ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶)) | ||
| Theorem | crefss 33848 | The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵) | ||
| Theorem | cmpcref 33849 | Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ Comp = CovHasRefFin | ||
| Theorem | cmpfiref 33850* | Every open cover of a Compact space has a finite refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Comp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ Fin ∧ 𝑣Ref𝑈)) | ||
| Syntax | cldlf 33851 | Extend class notation with the class of all Lindelöf spaces. |
| class Ldlf | ||
| Definition | df-ldlf 33852 | Definition of a Lindelöf space. A Lindelöf space is a topological space in which every open cover has a countable subcover. Definition 1 of [BourbakiTop2] p. 195. (Contributed by Thierry Arnoux, 30-Jan-2020.) |
| ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} | ||
| Theorem | ldlfcntref 33853* | Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) | ||
| Syntax | cpcmp 33854 | Extend class notation with the class of all paracompact topologies. |
| class Paracomp | ||
| Definition | df-pcmp 33855 | Definition of a paracompact topology. A topology is said to be paracompact iff every open cover has an open refinement that is locally finite. The definition 6 of [BourbakiTop1] p. I.69. also requires the topology to be Hausdorff, but this is dropped here. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} | ||
| Theorem | ispcmp 33856 | The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) | ||
| Theorem | cmppcmp 33857 | Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp) | ||
| Theorem | dispcmp 33858 | Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp) | ||
| Theorem | pcmplfin 33859* | Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement 𝑣 that is locally finite. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) | ||
| Theorem | pcmplfinf 33860* | Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) | ||
The prime ideals of a ring 𝑅 can be endowed with the Zariski topology. This is done by defining a function 𝑉 which maps ideals of 𝑅 to closed sets (see for example zarcls0 33867 for the definition of 𝑉). The closed sets of the topology are in the range of 𝑉 (see zartopon 33876). The correspondence with the open sets is made in zarcls 33873. As proved in zart0 33878, the Zariski topology is T0 , but generally not T1 . | ||
| Syntax | crspec 33861 | Extend class notation with the spectrum of a ring. |
| class Spec | ||
| Definition | df-rspec 33862 | Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟))) | ||
| Theorem | rspecval 33863 | Value of the spectrum of the ring 𝑅. Notation 1.1.1 of [EGA] p. 80. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ (𝑅 ∈ Ring → (Spec‘𝑅) = ((IDLsrg‘𝑅) ↾s (PrmIdeal‘𝑅))) | ||
| Theorem | rspecbas 33864 | The prime ideals form the base of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = (Base‘𝑆)) | ||
| Theorem | rspectset 33865* | Topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 2-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ Ring → 𝐽 = (TopSet‘𝑆)) | ||
| Theorem | rspectopn 33866* | The topology component of the spectrum of a ring. (Contributed by Thierry Arnoux, 4-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝑃 = (PrmIdeal‘𝑅) & ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝑃 ∣ ¬ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ Ring → 𝐽 = (TopOpen‘𝑆)) | ||
| Theorem | zarcls0 33867* | The closure of the identity ideal in the Zariski topology. Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) & ⊢ 𝑃 = (PrmIdeal‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑉‘{ 0 }) = 𝑃) | ||
| Theorem | zarcls1 33868* | The unit ideal 𝐵 is the only ideal whose closure in the Zariski topology is the empty set. Stronger form of the Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝐼) = ∅ ↔ 𝐼 = 𝐵)) | ||
| Theorem | zarclsun 33869* | The union of two closed sets of the Zariski topology is closed. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ ran 𝑉 ∧ 𝑌 ∈ ran 𝑉) → (𝑋 ∪ 𝑌) ∈ ran 𝑉) | ||
| Theorem | zarclsiin 33870* | In a Zariski topology, the intersection of the closures of a family of ideals is the closure of the span of their union. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑇 ⊆ (LIdeal‘𝑅) ∧ 𝑇 ≠ ∅) → ∩ 𝑙 ∈ 𝑇 (𝑉‘𝑙) = (𝑉‘(𝐾‘∪ 𝑇))) | ||
| Theorem | zarclsint 33871* | The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ ran 𝑉) | ||
| Theorem | zarclssn 33872* | The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) & ⊢ 𝐵 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅))) | ||
| Theorem | zarcls 33873* | The open sets of the Zariski topology are the complements of the closed sets. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑃 = (PrmIdeal‘𝑅) & ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}) | ||
| Theorem | zartopn 33874* | The Zariski topology is a topology, and its closed sets are images by 𝑉 of the ideals of 𝑅. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑃 = (PrmIdeal‘𝑅) & ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽))) | ||
| Theorem | zartop 33875 | The Zariski topology is a topology. Proposition 1.1.2 of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top) | ||
| Theorem | zartopon 33876 | The points of the Zariski topology are the prime ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑃 = (PrmIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃)) | ||
| Theorem | zar0ring 33877 | The Zariski Topology of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐽 = {∅}) | ||
| Theorem | zart0 33878 | The Zariski topology is T0 . Corollary 1.1.8 of [EGA] p. 81. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2) | ||
| Theorem | zarmxt1 33879 | The Zariski topology restricted to maximal ideals is T1 . (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑀 = (MaxIdeal‘𝑅) & ⊢ 𝑇 = (𝐽 ↾t 𝑀) ⇒ ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre) | ||
| Theorem | zarcmplem 33880* | Lemma for zarcmp 33881. (Contributed by Thierry Arnoux, 2-Jul-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) & ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp) | ||
| Theorem | zarcmp 33881 | The Zariski topology is compact. Proposition 1.1.10(ii) of [EGA], p. 82. (Contributed by Thierry Arnoux, 2-Jul-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑆) ⇒ ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp) | ||
| Theorem | rspectps 33882 | The spectrum of a ring 𝑅 is a topological space. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
| ⊢ 𝑆 = (Spec‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝑆 ∈ TopSp) | ||
| Theorem | rhmpreimacnlem 33883* | Lemma for rhmpreimacn 33884. (Contributed by Thierry Arnoux, 7-Jul-2024.) |
| ⊢ 𝑇 = (Spec‘𝑅) & ⊢ 𝑈 = (Spec‘𝑆) & ⊢ 𝐴 = (PrmIdeal‘𝑅) & ⊢ 𝐵 = (PrmIdeal‘𝑆) & ⊢ 𝐽 = (TopOpen‘𝑇) & ⊢ 𝐾 = (TopOpen‘𝑈) & ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆)) & ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) & ⊢ 𝑉 = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘}) & ⊢ 𝑊 = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘}) ⇒ ⊢ (𝜑 → (𝑊‘(𝐹 “ 𝐼)) = (◡𝐺 “ (𝑉‘𝐼))) | ||
| Theorem | rhmpreimacn 33884* | The function mapping a prime ideal to its preimage by a surjective ring homomorphism is continuous, when considering the Zariski topology. Corollary 1.2.3 of [EGA], p. 83. Notice that the direction of the continuous map 𝐺 is reverse: the original ring homomorphism 𝐹 goes from 𝑅 to 𝑆, but the continuous map 𝐺 goes from 𝐵 to 𝐴. This mapping is also called "induced map on prime spectra" or "pullback on primes". (Contributed by Thierry Arnoux, 8-Jul-2024.) |
| ⊢ 𝑇 = (Spec‘𝑅) & ⊢ 𝑈 = (Spec‘𝑆) & ⊢ 𝐴 = (PrmIdeal‘𝑅) & ⊢ 𝐵 = (PrmIdeal‘𝑆) & ⊢ 𝐽 = (TopOpen‘𝑇) & ⊢ 𝐾 = (TopOpen‘𝑈) & ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) | ||
| Syntax | cmetid 33885 | Extend class notation with the class of metric identifications. |
| class ~Met | ||
| Syntax | cpstm 33886 | Extend class notation with the metric induced by a pseudometric. |
| class pstoMet | ||
| Definition | df-metid 33887* | Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ~Met = (𝑑 ∈ ∪ ran PsMet ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)}) | ||
| Definition | df-pstm 33888* | Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)})) | ||
| Theorem | metidval 33889* | Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑥𝐷𝑦) = 0)}) | ||
| Theorem | metidss 33890 | As a relation, the metric identification is a subset of a Cartesian product. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) ⊆ (𝑋 × 𝑋)) | ||
| Theorem | metidv 33891 | 𝐴 and 𝐵 identify by the metric 𝐷 if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴(~Met‘𝐷)𝐵 ↔ (𝐴𝐷𝐵) = 0)) | ||
| Theorem | metideq 33892 | Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴(~Met‘𝐷)𝐵 ∧ 𝐸(~Met‘𝐷)𝐹)) → (𝐴𝐷𝐸) = (𝐵𝐷𝐹)) | ||
| Theorem | metider 33893 | The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋) | ||
| Theorem | pstmval 33894* | Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.) |
| ⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)})) | ||
| Theorem | pstmfval 33895 | Function value of the metric induced by a pseudometric 𝐷 (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ([𝐴] ∼ (pstoMet‘𝐷)[𝐵] ∼ ) = (𝐴𝐷𝐵)) | ||
| Theorem | pstmxmet 33896 | The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ ∼ = (~Met‘𝐷) ⇒ ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ ))) | ||
| Theorem | hauseqcn 33897 | In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐾 ∈ Haus) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑋) & ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
| Theorem | elunitge0 33898 | An element of the closed unit interval is positive. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 20-Dec-2016.) |
| ⊢ (𝐴 ∈ (0[,]1) → 0 ≤ 𝐴) | ||
| Theorem | unitssxrge0 33899 | The closed unit interval is a subset of the set of the extended nonnegative reals. Useful lemma for manipulating probabilities within the closed unit interval. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
| ⊢ (0[,]1) ⊆ (0[,]+∞) | ||
| Theorem | unitdivcld 33900 | Necessary conditions for a quotient to be in the closed unit interval. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.) |
| ⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1) ∧ 𝐵 ≠ 0) → (𝐴 ≤ 𝐵 ↔ (𝐴 / 𝐵) ∈ (0[,]1))) | ||
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