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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | mdetpmtr2 34001* | 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 34002* | 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 34003* | 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 34004* | Lemma for madjusmdet 34008. (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 34005* | Lemma for madjusmdet 34008. (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 34006* | Lemma for madjusmdet 34008. (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 34007* | Lemma for madjusmdet 34008. (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 34008 | 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 34009* | 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 34010* | 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 34011* | 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 34012* | 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 34013* | 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 34014* | 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 34015* | 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 34016* | 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 10410. (Contributed by Thierry Arnoux, 12-Jan-2020.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣))))) | ||
| Theorem | locfinreflem 34017* | 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 34018* | 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 34019 | The "every open cover has an 𝐴 refinement" predicate. |
| class CovHasRef𝐴 | ||
| Definition | df-cref 34020* | 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 34021* | 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 34022 | 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 34023 | 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 34024* | 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 34025* | A formulation of crefi 34024 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 34026 | The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵) | ||
| Theorem | cmpcref 34027 | 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 34028* | 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 34029 | Extend class notation with the class of all Lindelöf spaces. |
| class Ldlf | ||
| Definition | df-ldlf 34030 | 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 34031* | 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 34032 | Extend class notation with the class of all paracompact topologies. |
| class Paracomp | ||
| Definition | df-pcmp 34033 | 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 34034 | The predicate "is a paracompact topology". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽)) | ||
| Theorem | cmppcmp 34035 | Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Paracomp) | ||
| Theorem | dispcmp 34036 | Every discrete space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
| ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Paracomp) | ||
| Theorem | pcmplfin 34037* | 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 34038* | 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 34045 for the definition of 𝑉). The closed sets of the topology are in the range of 𝑉 (see zartopon 34054). The correspondence with the open sets is made in zarcls 34051. As proved in zart0 34056, the Zariski topology is T0 , but generally not T1 . | ||
| Syntax | crspec 34039 | Extend class notation with the spectrum of a ring. |
| class Spec | ||
| Definition | df-rspec 34040 | Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.) |
| ⊢ Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟))) | ||
| Theorem | rspecval 34041 | 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 34042 | 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 34043* | 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 34044* | 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 34045* | 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 34046* | 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 34047* | 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 34048* | 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 34049* | 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 34050* | 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 34051* | 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 34052* | 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 34053 | 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 34054 | 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 34055 | 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 34056 | 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 34057 | 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 34058* | Lemma for zarcmp 34059. (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 34059 | 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 34060 | 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 34061* | Lemma for rhmpreimacn 34062. (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 34062* | 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 34063 | Extend class notation with the class of metric identifications. |
| class ~Met | ||
| Syntax | cpstm 34064 | Extend class notation with the metric induced by a pseudometric. |
| class pstoMet | ||
| Definition | df-metid 34065* | 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 34066* | 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 34067* | 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 34068 | 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 34069 | 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 34070 | 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 34071 | The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.) |
| ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋) | ||
| Theorem | pstmval 34072* | 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 34073 | 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 34074 | 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 34075 | 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 34076 | 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 34077 | 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 34078 | 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))) | ||
| Theorem | iistmd 34079 | The closed unit interval forms a topological monoid under multiplication. (Contributed by Thierry Arnoux, 25-Mar-2017.) |
| ⊢ 𝐼 = ((mulGrp‘ℂfld) ↾s (0[,]1)) ⇒ ⊢ 𝐼 ∈ TopMnd | ||
| Theorem | unicls 34080 | The union of the closed set is the underlying set of the topology. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 ∈ Top & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ∪ (Clsd‘𝐽) = 𝑋 | ||
| Theorem | tpr2tp 34081 | The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ (𝐽 ×t 𝐽) ∈ (TopOn‘(ℝ × ℝ)) | ||
| Theorem | tpr2uni 34082 | The usual topology on (ℝ × ℝ) is the product topology of the usual topology on ℝ. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ ∪ (𝐽 ×t 𝐽) = (ℝ × ℝ) | ||
| Theorem | xpinpreima 34083 | Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
| ⊢ (𝐴 × 𝐵) = ((◡(1st ↾ (V × V)) “ 𝐴) ∩ (◡(2nd ↾ (V × V)) “ 𝐵)) | ||
| Theorem | xpinpreima2 34084 | Rewrite the cartesian product of two sets as the intersection of their preimage by 1st and 2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
| ⊢ ((𝐴 ⊆ 𝐸 ∧ 𝐵 ⊆ 𝐹) → (𝐴 × 𝐵) = ((◡(1st ↾ (𝐸 × 𝐹)) “ 𝐴) ∩ (◡(2nd ↾ (𝐸 × 𝐹)) “ 𝐵))) | ||
| Theorem | sqsscirc1 34085 | The complex square of side 𝐷 is a subset of the complex circle of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
| ⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ ℝ ∧ 0 ≤ 𝑌)) ∧ 𝐷 ∈ ℝ+) → ((𝑋 < (𝐷 / 2) ∧ 𝑌 < (𝐷 / 2)) → (√‘((𝑋↑2) + (𝑌↑2))) < 𝐷)) | ||
| Theorem | sqsscirc2 34086 | The complex square of side 𝐷 is a subset of the complex disc of radius 𝐷. (Contributed by Thierry Arnoux, 25-Sep-2017.) |
| ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ 𝐷 ∈ ℝ+) → (((abs‘(ℜ‘(𝐵 − 𝐴))) < (𝐷 / 2) ∧ (abs‘(ℑ‘(𝐵 − 𝐴))) < (𝐷 / 2)) → (abs‘(𝐵 − 𝐴)) < 𝐷)) | ||
| Theorem | cnre2csqlem 34087* | Lemma for cnre2csqima 34088. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ (𝐺 ↾ (ℝ × ℝ)) = (𝐻 ∘ 𝐹) & ⊢ 𝐹 Fn (ℝ × ℝ) & ⊢ 𝐺 Fn V & ⊢ (𝑥 ∈ (ℝ × ℝ) → (𝐺‘𝑥) ∈ ℝ) & ⊢ ((𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹) → (𝐻‘(𝑥 − 𝑦)) = ((𝐻‘𝑥) − (𝐻‘𝑦))) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ (◡(𝐺 ↾ (ℝ × ℝ)) “ (((𝐺‘𝑋) − 𝐷)(,)((𝐺‘𝑋) + 𝐷))) → (abs‘(𝐻‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷)) | ||
| Theorem | cnre2csqima 34088* | Image of a centered square by the canonical bijection from (ℝ × ℝ) to ℂ. (Contributed by Thierry Arnoux, 27-Sep-2017.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) ⇒ ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑌 ∈ (ℝ × ℝ) ∧ 𝐷 ∈ ℝ+) → (𝑌 ∈ ((((1st ‘𝑋) − 𝐷)(,)((1st ‘𝑋) + 𝐷)) × (((2nd ‘𝑋) − 𝐷)(,)((2nd ‘𝑋) + 𝐷))) → ((abs‘(ℜ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷 ∧ (abs‘(ℑ‘((𝐹‘𝑌) − (𝐹‘𝑋)))) < 𝐷))) | ||
| Theorem | tpr2rico 34089* | For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣))) & ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ⇒ ⊢ ((𝐴 ∈ (𝐽 ×t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴)) | ||
| Theorem | cnvordtrestixx 34090* | The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.) |
| ⊢ 𝐴 ⊆ ℝ* & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) ⇒ ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))) | ||
| Theorem | prsdm 34091 | Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵) | ||
| Theorem | prsrn 34092 | Range of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → ran ≤ = 𝐵) | ||
| Theorem | prsss 34093 | Relation of a subproset. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) | ||
| Theorem | prsssdm 34094 | Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴) | ||
| Theorem | ordtprsval 34095* | Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) & ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ⇒ ⊢ (𝐾 ∈ Proset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸 ∪ 𝐹))))) | ||
| Theorem | ordtprsuni 34096* | Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ 𝐸 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑦 ≤ 𝑥}) & ⊢ 𝐹 = ran (𝑥 ∈ 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ¬ 𝑥 ≤ 𝑦}) ⇒ ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (𝐸 ∪ 𝐹))) | ||
| Theorem | ordtcnvNEW 34097 | The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ (𝐾 ∈ Proset → (ordTop‘◡ ≤ ) = (ordTop‘ ≤ )) | ||
| Theorem | ordtrestNEW 34098 | The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⇒ ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴)) | ||
| Theorem | ordtrest2NEWlem 34099* | Lemma for ordtrest2NEW 34100. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))) | ||
| Theorem | ordtrest2NEW 34100* | An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) ⇒ ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴)) | ||
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